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abstract: 'Observational arguments supporting the binary explanation of the long secondary periods (LSP) phenomenon in red giants are presented. Photometry of about 1200 semiregular variables with the LSP in the Large Magellanic Cloud are analyzed using the MACHO and OGLE photometry. For about 5% of these objects additional ellipsoidal-like or eclipsing-like modulation with the same periods as the LSP is detectable. These double-humped variations are usually shifted in phase comparing to the LSP light curves. I discuss the model of binary system with a red giant as the primary component and a low-mass object as the secondary one. The mass lost by the red giant through the wind follows the spiral pattern in the orbit around the primary star and obscures it causing the LSP variations.'
author:
- 'I. Soszy[ń]{}ski'
title: Long Secondary Periods and Binarity in Red Giant Stars
---
Introduction
============
Among numerous classes of variable stars only one type of large-amplitude stellar variability remains completely unexplained. This is the long secondary periods (LSP) observed in luminous red giant stars. The LSP variability with periods between 200 and 1500 days and with $V$-band amplitudes up to 1 mag occurs in $\sim30$% of Semiregular Variables (SRVs) and OGLE Small Amplitude Red Giants (OSARGs). This phenomenon has been known for decades [@pg54; @hou63], but an interest in the stars with the LSP has been renewed since @woo99 showed that these objects follow a period–luminosity (PL) relation (sequence D).
In recent years our knowledge about the LSP phenomenon has significantly increased, although its origin remains a mystery. @hin02 and @woo04 studied spectral features of several Galactic stars with the LSP and detected radial velocity variations with amplitudes of a few km/s which agree with photometric long-period variations. @sos04 used Optical Gravitational Lensing Experiment (OGLE) data to select and analyze close binary systems in the Large Magellanic Cloud (LMC) with a red giant as one of the components. They noticed that the sequence D in the PL diagram overlaps and is a direct continuation of the PL sequence of ellipsoidal and eclipsing red giants[^1] (sequence E), suggesting that the LSP phenomenon is related to binarity. @sos04 also showed that some evident ellipsoidal red giants exhibit simultaneously OSARG-type variability, thus, it was directly demonstrated that in some cases the binary explanation of the LSP is true. Nevertheless, there is no doubt that the bulk of the LSP variables are not typical ellipsoidal or eclipsing binaries.
@sos05 discovered that the sequence D split into two ridges in the period – Wesenheit index ($W_I=I-1.55(V-I)$) plane, which corresponds to the spectral division into oxygen-rich and carbon-rich AGB stars. The same feature was noticed for Miras and SRVs (sequences C and C$'$), and the sequence D contains relatively much smaller number of carbon-rich stars.
Recently @der06 presented a period–luminosity–amplitude analysis of variable red giants in the LMC. They examined amplitudes of ellipsoidal, LSP and Mira-like variables using MACHO red ($R_M$) and blue ($B_M$) photometry. The amplitude distribution for the LSP stars turned out to be different than for ellipsoidal and pulsating variables, but blue-to-red amplitude ratios of the LSP stars (typically 1.3) is more similar to this quantity in pulsating variables ($\sim1.4$) than to ellipsoidal/eclipsing binaries ($\sim1.1$). This last feature is used in present work for separation of the LSP and ellipsoidal/eclipsing variability in the same light curves.
Various hypotheses have been proposed to explain the origin of the LSP variability: rotation of a spotted star, episodic dust ejections, a radial and non-radial pulsation and binary companions including planets or brown dwarfs. @woo99 suggested that the sequence D stars are components of semidetached binary systems. The matter lost by the AGB star forms a dusty cloud around the companion, and regularly obscures the primary component causing the LSP variability. @hin02 and then @ow03 mentioned that the radial velocity measurements are consistent only with the binary or pulsation explanations of the LSP. @woo04 ruled out the binary hypothesis, because a short ($\sim1000$ yr) timescale on which the companion should merge with the red giant. They suggested that the most likely explanation of the LSP are low degree g$^+$ pulsation modes trapped in the outer radiative layers of the star.
The main goal of this paper is to find observational evidences for or against the binary explanation of the LSP. Since the PL sequence populated by the ellipsoidal and eclipsing variables overlaps with the LSP sequence [@sos04], it should be possible to find stars revealing simultaneously both types of variability. If periods are the same, the LSP phenomenon must be related to binarity. If not, the LSP is presumably caused by another reason.
Data Analysis
=============
Since the LSP are sometimes as long as 1500 days, presented analysis is based on observational data originated in two sources: MACHO and OGLE surveys. Merged light curves from both projects covered 15 years of observations: from 1992 to 2006. The sequence D stars selected by @sos04 in the OGLE database were cross-identified with objects collected in the MACHO archive[^2]. I found about 1200 counterparts of the previously selected variables. Then, $R_M$-band MACHO observations have been merged with the OGLE points by scaling amplitudes and shifting zero points of the photometry. The parameters of this transformation have been found by the least square fitting to the measurements obtained between 1997 and 2000, i.e., when both projects observed the LMC fields at the same time.
Searching for orbital periods different than the LSP
----------------------------------------------------
Searching for ellipsoidal or eclipsing variability with different periods than the LSP was a relatively easy procedure. For each object a third order Fourier series was fitted to the LSP light curve and subtracted from the points. Then, the period search was performed for the residual data. This procedure was repeated until four additional periods per star have been found.
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From these results I selected and visually inspected light curves with periods lying between sequences C and D, i.e., where one can expect ellipsoidal or eclipsing variables (automatic procedure gives a half of orbital periods). The best seven light curves of that type are plotted in Fig. 1. Phased LSP light curves are shown in the left column, the right column presents the light curves of these objects after subtracting the LSP variations and phased with periods two times longer than obtained automatically.
As one can see none of these secondary modulations is distinct, indisputable eclipsing or ellipsoidal light curve. The most likely explanation of these modulations are variations of phases or amplitudes of the LSP variability which produce such “artificial” variability in residual light curves. A very similar behavior occurred in eclipsing binary described in Section 3.
Searching for orbital periods the same as the LSP
-------------------------------------------------
If the LSP is related to binarity, one can expect that a number of objects shows ellipsoidal or eclipsing variability of the same period as the LSP. Unfortunately, detecting such a modulation in the LSP light curves is not an easy task, because (i) the ellipsoidal light curves have usually much smaller amplitudes than LSP, (ii) shorter semiregular variability is superimposed, (iii) the LSP light curves often change amplitudes, phases and periods. A careful investigation of the LSP light curves reveals that some of them show shallow secondary minima, what may be a sign of ellipsoidal variations superimposed on the LSP. However, since the LSP light curves appear in several variants, it is possible that such behavior is not related to binarity.
To separate LSP and possible ellipsoidal/eclipsing variations I used a feature noticed by @der06. They studied MACHO $B_M$ and $R_M$ photometry of long period variables in the LMC and found that blue-to-red amplitude ratios are different for ellipsoidal and LSP variables. In the ellipsoidal red giants, where the variability is dominated by the geometric changes, amplitudes in different filters are very similar. The median value of $A(B_M)/A(R_M)$ is equal to 1.1. For the LSP stars $A(B_M)/A(R_M)$ is more similar to pulsating variables and equal on average to 1.3. It means that if the amplitudes of the $R_M$-band light curves are scaled by a factor 1.3 and the $B_M$ observations are subtracted, the LSP variations will be canceled, but not the possible ellipsoidal/eclipsing modulation.
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The procedure was as follows. The $B_M$ and $R_M$-band amplitudes were determined by fitting the spline function to the folded light curves. Then, the $R_M$ magnitudes were converted into flux and linearly scaled to obtain the same amplitude (in magnitudes) as in the $B_M$ bandpass. The flux was again converted into magnitudes, and $B_M$ measurements were subtracted point by point from scaled $R_M$ data. The final step of the procedure was fitting a sinusoid of a period 1 year, and subtracting the function from the residual data. This way I removed a differential refraction effect distinctly visible in MACHO data.
For the vast majority of light curves the LSP and pulsation (semiregular) variability subtracted very well. No significant long-period variability was detectable in the residual points. However, for about 5% of stars I found clear periods equal to half the LSP, i.e. I noticed double-humped curves with periods the same as of the LSP. All these residual data are available from the electronic edition of the Astrophysical Journal. The most prominent curves of this type are presented in Fig. 2. In these cases the residual data seem to arrange in eclipsing-like or ellipsoidal-like light curves.
Fig. 3 shows period–$W_I$ diagram (where $W_I=I-1.55(V-I)$ is the reddening-free Wesenheit index) for stars from sequences D and E. Full circles show the position of the LSP stars with the double-humped curves. Note that these objects appear along the complete length of sequence D, although there is an overdensity for shorter-period stars, i.e. where sequences D and E overlap. Note also that double-humped objects populate both, O-rich and C-rich, sequences of the LSP stars [@sos05].
{width="12cm"}
It is worth mentioning that the described procedure does not demand any assumptions concerning periods or light curve shapes. It is a simple transformation, the same for each observing point. The only parameter of this transformation is a blue-to-red amplitude ratio of the LSP variability, but I checked that the double-humped residual light curves are visible for relatively wide range of this parameter, so it is not very important to measure amplitudes very accurately.
Discussion
==========
I have shown that about 5% of sequence D objects have ellipsoidal or eclipsing-like modulation with periods the same as the LSP. In the sample of 1200 sequence D stars I did not find any distinct ellipsoidal or eclipsing binary with period different than the LSP. Although various scenarios are still possible, it is justified to re-invoke the binary hypothesis proposed by @woo99. In this explanation the AGB star in a close binary system losses mass to the secondary component. The matter forms a dusty cloud around and behind the companion and regularly obscures the primary star.
An argument for this hypothesis is a phase lag between ellipsoidal/eclipsing and the LSP variations clearly visible in Fig. 2. Only one object of six – the star with the shortest period – does not exhibit such behavior. I checked that it might be a rule for the shortest period sequence D variables. For longer periods the minima of the LSP brightness variation occur about 0.05–0.10 of a cycle after the minima of the ellipsoidal/eclipsing light curves. Consistent results were presented by @woo01, who discovered a phase lag of $\sim$1/8 between LSP light curves and radial velocity curves. Such phase offsets between ellipsoidal/eclipsing and LSP minima agree very well with hydrodynamical simulations of a wind driven accretion flow in binary systems [@tj93; @mm98; @nag04]. These models predict that a matter lost by a red giant in a binary system follows the spiral pattern with maximum density located behind the secondary component.
An exhaustive discussion about possible origins of the LSP was done by @woo04. They argued that the merger timescale for red giants and its companion in close binary systems is of the order of $10^3$ years, while the lifetime in the AGB phase is two orders of magnitude longer. Thus, 30% of SRVs showing the LSP is highly inconsistent with the binary scenario.
However, the mass transfer in a binary system may be due to the Roche-lobe overflow [e.g. @pac71], or through the stellar wind [e.g. @abm87; @tj93]. The former process tends to circularize the orbits and to synchronize the spin of the stars with the orbital rotation, which results in shrinking the orbits and finally in merging the components. The latter phenomenon increases the eccentricity of the orbits and, if the bulk of the mass lost by the red giants escapes from the system, the distance between components may even increase. Thus, the main argument of @woo04 against the binary explanation may be not valid, if we assume that the red giants in the binary systems do not fully fill their Roche lobe, and the bulk of mass transfer is driven by the stellar wind. The non-sinusoidal radial velocity curves observed by @hin02 and @woo04, which can be explained by the eccentricity of the orbits, are in good agreement with this scenario.
{width="17cm"}
Presented hypothesis nevertheless requires that at least 30% of AGB stars exist in close binary systems. Moreover, while the studied LSP variables have velocity amplitudes of only a few km/s [@hin02; @woo04], the confirmed binary systems (sequence E stars) appear to have velocity amplitudes about ten times larger [@awc06]. The radial velocity measurements suggest that the second component in the LSP stars may be a brown dwarf. @ret05 proposed that the Jupiter-like planets may accrete the matter from its host star and increase mass into the brown dwarfs range. This hypothesis would explain such large number of the LSP cases among AGB stars, if we assume that planets with a separation of 1–5 AU are common. To test the binary hypothesis it would be interesting to measure the radial velocity changes for the LSP stars with the double-humped variations. If these stars are ellipsoidal or eclipsing variables indeed, their velocity amplitudes should be similar to these observed in sequence E stars, i.e. significantly larger than for the remaining LSP stars.
The brown dwarf scenario can also explain why only 5% of our sample show the double-humped variability. For low-mass secondary components the ellipsoidal and eclipsing variations have too small amplitudes to be detected in our procedure. One should remember that the residual data obtained by scaling $R_M$ and subtracting $B_M$ magnitudes can show variability with amplitudes of about 0.2 of the original ellipsoidal amplitudes ($R_M$ were scaled by a factor of 1.3 and $A(B_M)/A(R_M)$ is on average 1.1 for ellipsoidal variables). Moreover, presented model does not assume that the red giant fills entirely the Roche lobe, because the bulk of the mass flow is through a stellar wind, so the separation between the components can be too large to cause significant ellipsoidal variability. Note also that observed number of possible ellipsoidal and eclipsing variables among the LSP variables is in agreement with relative number of sequence E stars. The LSP modulation is observed for about 30% of the AGB stars, so 5% of these objects gives about 1-2% of the whole population. Exactly the same relative number of ellipsoidal or eclipsing variables is observed for fainter red giants [@sos04], so one should not expect to detect many more such objects among brighter stars.
Of course, it cannot be absolutely excluded that the double-humped variations have different explanation than the binarity. However, the non-binarity model of the LSP phenomenon has to explain why produced residual light curves have ellipsoidal or eclipsing-like shapes, why it appears only in a few percent of the sequence D stars, and why there is phase lag between LSP and double-humped variations. All these facts can be explained by the binary scenario.
Additional clues on the origin of the LSP can be given by an analysis of typical and peculiar cases of the light curves. A common feature of the LSP variations is the modulation of the amplitudes and phases. A few such light curves are shown in in Fig. 4. It seems that the phase shifts are correlated with the depth of minima – the deeper minimum occurs earlier than the shallower one (see the folded light curve in Fig. 4a). Sometimes the LSP variations completely disappear or appear in red giants which did not show such modulation before (Fig. 4b). The conclusion that can be drawn from these cases is that the regular (with no LSP) apparent luminosity of the red giants is the same as in the maximum apparent luminosity of the LSP variations, i.e. the LSP phenomenon decreases the total luminosity of the star. This fact must be taken into consideration by any theory of the LSP variability.
The hypothesis of a binary system with a mass-losing AGB star seems to agree with these results, because the obscuration by a cloud of matter reduces the total luminosity of the star. The amplitude and phase changes are likely connected with the variable mass loss rate. An interesting LSP light curve is presented in Fig. 4c. Starting from the beginning of the observations the depth and width of the minima were increasing. After a few cycles this process affected the maximum of the light curve, and the total apparent optical luminosity of the star dropped down. Then, the object returned, more or less, to the previous stage. This behavior can be interpreted as a sudden rise of a mass loss rate, which caused the whole system to be hidden in the cloud of matter.
In Fig. 4d I show a probable eclipsing binary system with the AGB star as one of the components. This object presents striking similarities to the LSP stars. First, it is located in the sequence D in the PL diagram. Second, the light curve changes the amplitudes and phases of variations. Third, after subtracting a function fitted to the primary (eclipsing) variability I obtained secondary long-period variations caused probably by modulation of the primary period. The same behavior I observed for the LSP stars presented in Fig. 1.
Summary and Conclusions
=======================
In this paper I show that careful analysis of available data may shed new light on the nature of the last unexplained type of stellar variability. Arguments in favor of the binary explanation of the LSP phenomenon are as follows:
1. There are no reliable examples of the LSP stars with ellipsoidal or eclipsing variations with different periods.
2. At least 5% of the sequence D stars exhibit ellipsoidal or eclipsing-like variability with the same period as the LSP.
3. Phase lag between ellipsoidal and LSP variations agrees well with models of wind accretion in binary systems.
4. The LSP light curves with variable amplitudes can be explained by a variable mass-loss rate in binary systems.
The binary scenario can explain many features of the LSP variables. The PL relation (which is a direct continuation of “binary” sequence E) is a projection of a radius–luminosity relation for red giants. Positive correlation between amplitudes of the LSP variability and mean luminosity [@sos04; @der06] may be caused by the increasing mass loss rate with the luminosity. The dimming during the LSP minima are consistent with the obscuration by a dusty cloud of matter orbiting the red giant. Finally, the radial velocity variations are in agreement with eccentric motion of the low-mass companion. The long-term project of radial velocity measurements of selected sequence D stars have been recently finished (P. Wood, private communication). I expect that these data will definitively solve the LSP problem.
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[^1]: if the real orbital periods of binary variables are considered, i.e. periods two times longer than obtained automatically.
[^2]: http://wwwmacho.mcmaster.ca/Data/MachoData.html
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abstract: 'We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.'
address:
- Masato Hoshino
- Yuzuru Inahama
- Nobuaki Naganuma
author:
- 'Masato Hoshino, Yuzuru Inahama and Nobuaki Naganuma'
title: 'Stochastic complex Ginzburg-Landau equation with space-time white noise '
---
Keywords: Stochastic partial differential equation, Complex Ginzburg-Landau equation, Regularity structure, Paracontrolled distribution, Renormalization.
MSC2010: 60H15, 82C28.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank Professor Reika Fukuizumi of Tohoku University for stimulating discussions and valuable advice. They also thank Professor Makoto Katori of Chuo University for helpful comments.
The first named author was partially supported by JSPS KAKENHI Grant Number JP16J03010. The second named author was partially supported by JSPS KAKENHI Grant Number JP15K04922. The third named author was partially supported by JSPS KAKENHI Grant Number JP17K14202.
[99]{}
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Nils Berglund and Christian Kuehn. Regularity structures and renormalisation of [F]{}itz[H]{}ugh-[N]{}agumo [SPDE]{}s in three space dimensions. , 21:Paper No. 18, 1–48, 2016.
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Masato Hoshino. Paracontrolled calculus and [F]{}unaki-[Q]{}uastel approximation for the [KPZ]{} equation. , in press.
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[ School of Fundamental Science and Engineering\
Waseda University\
Okubo, Shinjuku-ku, Tokyo, 169-8555\
Japan ]{} [hoshino@ms.u-tokyo.ac.jp]{}
[ Graduate School of Mathematics\
Kyushu University\
Motooka, Nishi-ku, Fukuoka, 819-0395\
Japan ]{} [inahama@math.kyushu-u.ac.jp]{}
[ Graduate School of Engineering Science\
Osaka University\
Machikaneyama, Toyonaka, Osaka, 560-8531\
Japan ]{} [naganuma@sigmath.es.osaka-u.ac.jp]{}
|
---
abstract: 'Associated kernels have been introduced to improve the classical continuous kernels for smoothing any functional on several kinds of supports such as bounded continuous and discrete sets. This work deals with the effects of combined associated kernels on nonparametric multiple regression functions. Using the Nadaraya-Watson estimator with optimal bandwidth matrices selected by cross-validation procedure, different behaviours of multiple regression estimations are pointed out according the type of multivariate associated kernels with correlation or not. Through simulation studies, there are no effect of correlation structures for the continuous regression functions and also for the associated continuous kernels; however, there exist really effects of the choice of multivariate associated kernels following the support of the multiple regression functions bounded continuous or discrete. Applications are made on two real datasets.'
address: 'University of Franche-Comté, Besançon, France'
author:
- 'Sobom M. Somé'
- 'Célestin C. Kokonendji'
date: 'February 5, 2014'
title: Effects of associated kernels in nonparametric multiple regressions
---
Bandwidth matrix ,continuous associated kernel ,correlation structure ,cross-validation ,discrete associated kernel ,Nadaraya-Watson estimator.: 62G05(08); 62H12
[**Short Running Title**]{}: Associated kernels in multiple regressions
Introduction
============
Considering the relation between a response variable $Y$ and a $d$-vector $(d\geq1)$ of explanatory variables $\mathbf{x}$ given by $$\label{Regfonction}
Y = m\left(\mathbf{x}\right) + \epsilon,$$where $m$ is the unknown regression function from $\mathbb{T}_{d} \subseteq \mathbb{R}^{d}$ to $\mathbb{R}$ and $\epsilon$ the disturbance term with null mean and finite variance. Let $(\bold{X}_{1}, Y_{1}), \ldots, (\bold{X}_{n}, Y_{n})$ be a sequence of independent and identically distributed (iid) random vectors on $\mathbb{T}_{d} \times \mathbb{R} (\subseteq \mathbb{R}^{d+1})$ with $m(\bold{x}) = \mathbb{E}\left(Y|\bold{X} = \bold{x}\right)$ of (\[Regfonction\]). The [@Nadaraya69] and [@Watson64] estimator $\widehat{m}_{n}$ of $m$, using continuous classical (symmetric) kernels is $$\label{RegClassic}
\widehat{m}_{n}(\mathbf{x};K,\mathbf{H}) = \displaystyle \sum_{i = 1}^{n} \dfrac{Y_iK\left\{\mathbf{H}^{-1}(\mathbf{x} - \mathbf{X}_i)\right\}}{ \sum_{i = 1}^{n}K\left\{\mathbf{H}^{-1}(\mathbf{x} - \mathbf{X}_i)\right\}}= \widehat{m}_{n}(\mathbf{x};\mathbf{H}) , ~~\forall \mathbf{x} \in \mathbb{T}_{d} := \mathbb{R}^{d},$$ where $\mathbf{H}$ is the symmetric and positive definite bandwidth matrix of dimension $d\times d$ and the function $K(\cdot)$ is the multivariate kernel assumed to be spherically symmetric probability density function. Since the choice of the kernel $K$ is not important in classical case, we use the common notation $\widehat{m}_{n}(\mathbf{x};\mathbf{H})$ for classical kernel regression. The multivariate classical kernel (e.g. Gaussian) suits only for regression functions on unbounded supports (i.e. $\mathbb{R}^d$); see also [@S92]. proposed product of kernels composed by univariate Gaussian kernels for continuous variables and [@AJG76] kernels for categorical variables; see also for some implementations and uses of these multiple kernels. Notice that the use of symmetric kernels gives weights outside variables with unbounded supports. In the univariate continuous case, is one of the pioneers who has proposed asymmetric kernels (i.e. beta and gamma) which supports coincide with those of the functions to be estimated. [@Zhang2010] and [@ZhangandKarunamuni2010] studied the performance of these beta and gamma kernel estimators at the boundaries in comparison with those of the classical kernels. Recently, [@L13] investigated several families of these univariate continuous kernels that he called univariate associated kernels; see also [@KSKZ07], [@KSK11], [@ZAK12] and [@WKK14] for univariate discrete situations. A continuous multivariate version of these associated kernels have been studied by for density estimation.
The main goal of this work is to consider multivariate associated kernels and then to investigate the importance of their choice in multiple regression. These associated kernels are appropriated for both continuous and count explanatory variables. In fact, in order to estimate the regression function $m$ in (\[Regfonction\]), we propose multiple (or product of) associated kernels composed by univariate discrete associated kernels (e.g. binomial, discrete triangular) and continuous ones (e.g. beta, Epanechnikov). We will also use a bivariate beta kernel with correlation structure. Another motivation of this work is to investigate the effect of correlation structure for explanatory variables in continuous regression estimation. These associated kernels suit for this situation of mixing axes as they fully respect the support of each explanatory variable. In other words, we will measure the effect of type of associated kernels, denoted $\boldsymbol{\kappa}$, in multiple regression by simulations and applications.
The rest of the paper is organized as follows. Section \[sec:Multiple regression\] gives a general definition of multivariate associated kernels which includes the continuous classical symmetric and the multiple composed by univariate discrete and continuous. For each definition, the corresponding kernel regression appropriated for both continuous and discrete explanatory variables are given. In Section \[sec:Simulation studies and real data analisys\], we explore the importance of the choice of appropriated associated kernels according to the support of the variables through simulations studies and real data analysis. Finally, summary and final remarks are drawn in Section \[sec:Summary and final remarks\].
Multiple regression by associated kernels {#sec:Multiple regression}
=========================================
Definition {#sec:Multivariate associated regressions}
----------
In order to include both discrete and continuous regressors, we assume $\mathbb{T}_{d}$ is any subset of $\mathbb{R}^d$. More precisely, for $j = 1, \ldots, n$, let us consider on $\mathbb{T}_{d} = \displaystyle \otimes_{j=1}^{d}\mathbb{T}^{[j]}_1$ the measure $\boldsymbol{\nu} = \nu_{1} \otimes \ldots \otimes \nu_{d}$ where $\nu_{j}$ is a Lesbesgue or count measure on the corresponding univariate support $\mathbb{T}^{[j]}_1$. Under these assumptions, the associated kernel $K_{\bold{x}, \mathbf{H}}(\cdot)$ which replaces the classical kernel $K(\cdot)$ of (\[RegClassic\]) is a probability density function (pdf) in relation to a measure $\boldsymbol{\nu}$. This kernel $K_{\bold{x}, \mathbf{H}}(\cdot)$ can be defined as follows.
\[Defkern\] Let $\mathbb{T}_{d}\left(\subseteq \mathbb{R}^{d}\right)$ be the support of the regressors, $\mathbf{x} \in \mathbb{T}_{d}$ a target vector and $\mathbf{H}$ a bandwidth matrix. A parametrized pdf $K_{\mathbf{x}, \mathbf{H}}(\cdot)$ of support $\mathbb{S}_{\mathbf{x}, \mathbf{H}} \left(\subseteq \mathbb{R}^{d}\right)$ is called “multivariate (or general) associated kernel” if the following conditions are satisfied: $$\begin{aligned}
&\mathbf{x} \in \mathbb{S}_{\mathbf{x}, \mathbf{H}}, \label{NoyAss1}\\
&\mathbb{E}\left(\mathcal{Z}_{\mathbf{x}, \mathbf{H}}\right) = \mathbf{x} + \mathbf{a}(\mathbf{x}, \mathbf{H}), \label{NoyAss2} \\
& \rm{Cov}\left(\mathcal{Z}_{\mathbf{x}, \mathbf{H}}\right) = \mathbf{B}(\mathbf{x}, \mathbf{H}), \label{NoyAss3}\end{aligned}$$ where $\mathcal{Z}_{\mathbf{x}, \mathbf{H}}$ denotes the random vector with pdf $K_{\mathbf{x}, \mathbf{H}}$ and both $\mathbf{a}(\mathbf{x}, \mathbf{H}) = \left(a_{1}(\mathbf{x}, \mathbf{H}), \ldots, a_{d}(\mathbf{x}, \mathbf{H})\right)^{\top}$ and $\mathbf{B}(\mathbf{x}, \mathbf{H}) = \left(b_{ij}(\mathbf{x}, \mathbf{H})\right)_{i,j = 1, \ldots, d}$ tend, respectively, to the null vector $\mathbf{0}$ and the null matrix $\mathbf{0}_d$ as $\mathbf{H}$ goes to $\mathbf{0}_d$.
From this definition and in comparison with (\[RegClassic\]), the Nadaraya-Watson estimator using associated kernels is
$$\label{RegFull}
\widetilde{m}_{n}(\bold{x};K_{\bold{x}, \mathbf{H}}) = \displaystyle \sum_{i = 1}^{n} \dfrac{Y_iK_{\bold{x}, \mathbf{H}}\left(\bold{X}_i\right)}{ \sum_{i = 1}^{n}K_{\bold{x}, \mathbf{H}}\left(\bold{X}_i\right)}=\widetilde{m}_{n}(\bold{x};\boldsymbol{\kappa},\mathbf{H}), ~~~ \forall \bold{x} \in \mathbb{T}_{d} \subseteq \mathbb{R}^{d},$$
where $\mathbf{H} \equiv \mathbf{H}_{n} $ is the bandwidth matrix such that $ \mathbf{H}_{n} \rightarrow \mathbf{0}$ as $n \rightarrow \infty$, and $\boldsymbol{\kappa}$ represents the type of associated kernel $K_{\bold{x}, \mathbf{H}}$, parametrized by $\bold{x}$ and $\mathbf{H}$. Without loss of generality and to point out the effect of $\boldsymbol{\kappa}$, we will in hereafter use $\widetilde{m}_{n}(\bold{x};\boldsymbol{\kappa},\mathbf{H}) = \widetilde{m}_{n}(\bold{x};\boldsymbol{\kappa})$ since the bandwidth matrix is here investigated only by cross validation.
The following two examples provide the well-known and also interesting particular cases of multivariate associated kernel estimators. The first can be seen as an interpretation of classical associated kernels through continuous symmetric kernels. The second deals on non-classical associated kernels without correlation structure.
Given a target vector $\mathbf{x}\in\mathbb{R}^{d} =:\mathbb{T}_d$ and a bandwidth matrix $\mathbf{H}$, it follows that the classical kernel in (\[RegClassic\]) with null mean vector and covariance matrix $\mathbf{\Sigma}$ induces the so-called (multivariate) classical associated kernel: $$\begin{aligned}
\label{classcical}
(i)~~K_{\mathbf{x}, \mathbf{H}}(\cdot) = \dfrac{1}{\det\mathbf{H}} K\left\{\mathbf{H}^{-1}(\mathbf{x} - \cdot)\right\}\end{aligned}$$ on $\mathbb{S}_{\mathbf{x}, \mathbf{H}} = \mathbf{x} - \mathbf{H}\mathbb{S}_{d}$ with $\mathbb{E}\left(\mathcal{Z}_{\mathbf{x}, \bf{H}}\right) = \mathbf{x}$ (i.e. $\mathbf{a}(\mathbf{x}, \mathbf{H}) = \mathbf{0}$) and $\rm{Cov}\left(\mathcal{Z}_{\mathbf{x}, \mathbf{H}}\right) = \mathbf{H}\mathbf{\Sigma}\mathbf{H}$; $$\begin{aligned}
(ii)~~K_{\mathbf{x}, \mathbf{H}}(\cdot) = \dfrac{1}{(\det\mathbf{H})^{1/2}} K\left\{\mathbf{H}^{-1/2}(\mathbf{x} - \cdot)\right\}\end{aligned}$$ on $\mathbb{S}_{\mathbf{x}, \mathbf{H}} = \mathbf{x} - \mathbf{H}^{1/2}\mathbb{S}_{d}$ with $\mathbb{E}\left(\mathcal{Z}_{\mathbf{x}, \bf{H}}\right) = \mathbf{x}$ (i.e. $\mathbf{a}(\mathbf{x}, \mathbf{H}) = \mathbf{0}$) and $\rm{Cov}\left(\mathcal{Z}_{\mathbf{x}, \mathbf{H}}\right) = \mathbf{H}^{1/2}\mathbf{\Sigma}\mathbf{H}^{1/2}$.
A second particular case of Definition \[Defkern\], appropriate for both continuous and count explanatory variables without correlation structure is presented as follows.
Let $\mathbf{x} = (x_{1}, \ldots, x_{d})^{\top} \in \times_{j=1}^{d}\mathbb{T}^{[j]}_{1} =:\mathbb{T}_d$ and $\mathbf{H} = \mathbf{Diag} (h_{11}, \ldots, h_{dd})$ with $h_{jj} > 0$. Let $K^{[j]}_{x_{j}, h_{jj}}$ be a (discrete or continuous) univariate associated kernel (see Definition \[Defkern\] for $d = 1$) with its corresponding random variable $\mathcal{Z}^{[j]}_{x_{j}, h_{jj}}$ on $\mathbb{S}_{x_{j}, h_{jj}} (\subseteq \mathbb{R})$ for all $j = 1, \ldots, d$. Then, the multiple associated kernel is also a multivariate associated kernel: $$\begin{aligned}
\label{prodkern1}
K_{\mathbf{x}, \mathbf{H}}(\cdot) = \prod_{j=1}^{d}K^{[j]}_{x_{j}, h_{jj}}(\cdot)\end{aligned}$$on $\mathbb{S}_{\mathbf{x}, \mathbf{H}} = \displaystyle \times_{j=1}^{d}\mathbb{S}_{x_{j}, h_{jj}}$ with $\mathbb{E}\left(\mathcal{Z}_{\mathbf{x}, \bf{H}}\right) = \left(x_{1} + a_{1}(x_{1}, h_{11}), \ldots, x_{d} + a_{d}(x_{d}, h_{dd})\right)^{\top}$ and $\operatorname{Cov}\left(\mathcal{Z}_{\mathbf{x}, \mathbf{H}}\right)$ = $ \mathbf{Diag}\left(b_{jj}(x_{j}, h_{jj})\right)_{j = 1, \ldots, d}$. In other words, the random variables $\mathcal{Z}^{[j]}_{x_{j}, h_{jj}}$ are independent components of the random vector $\mathcal{Z}_{\mathbf{x}, \bf{H}}$.
Here, in addition to the Nadaraya-Watson estimator using general associated kernels given in (\[RegFull\]), we proposed a slight one. In fact, for multivariate supports composed of continuous and discrete univariate support, we lack appropriate general associated kernels. Therefore, the estimator (\[RegFull\]) becomes with multiple associated kernels (\[prodkern1\]): $$\widetilde{m}_{n}(\bold{x};\boldsymbol{\kappa})= \displaystyle \sum_{i = 1}^{n} \dfrac{Y_i\prod_{j=1}^{d}K^{[j]}_{x_{j}, h_{jj}}(X_{ij})}{ \sum_{i = 1}^{n}\prod_{j=1}^{d}K^{[j]}_{x_{j}, h_{jj}}(X_{ij})},~~~\forall \bold{x} = \left(x_{1}, \ldots, x_{d}\right)^{\top} \in \mathbb{T}_{d} := \displaystyle \times_{j=1}^{d} \mathbb{T}^{[j]}_{1}. \label{Regprod}$$ In theory and in practice, one often uses (\[Regprod\]) from multiple associated kernels (\[prodkern1\]) which are more manageable than (\[RegFull\]); see, e.g., [@S92] and also [@BR10] for density estimation.
Associated kernels for illustration {#ssec:Associated kernels for illustration}
-----------------------------------
In order to point out the importance of the type of kernel $\boldsymbol{\kappa}$ in a regression study, we motivate below some kernels that will be used in simulations. These concern seven basic associated kernels for which three of them are univariate discrete, three others are univariate continuous and the last one is a bivariate beta with correlation structure.
- The binomial kernel ($\texttt{Bin}$) is defined on the support $\mathbb{S}_x = \{0, 1, \ldots, x+1\}$ with $x \in \mathbb{T}_1 :=\mathbb{N}= \{0,1,\ldots\}$ and then $h \in (0, 1]$: $$\label{c}
B_{x,h}(u)=\frac{(x+1)!}{u!(x+1-u)!}\left(\frac{x+h}{x+1}\right)^{u}\left(\frac{1-h}{x+1}\right)^{x+1-u} \mathds{1}_{\mathbb{S}_x}(u),$$ where $\mathds{1}_{A}$ denote the indicator function of any given event $A$. Note that $B_{x,h}$ is the probability mass function (pmf) of the binomial distribution $\mathcal{B}(x+1; (x+h)/(x+1))$ with its number of trials $x+1$ and its success probability in each trial $(x+h)/(x+1)$. It is appropriated for count data with small or moderate sample sizes and, also, it does not satisfy (\[NoyAss3\]); see [@KSK11] and also [@ZAK12] for a bandwidth selection by Bayesian method.
- For fixed arm $a \in \mathbb{N}$, the discrete triangular kernel ($\texttt{DTr}a$) is defined on $\mathbb{S}_{x,a} = \left\{x, x \pm 1, \ldots, x \pm a \right\}$ with $x \in \mathbb{T}_1 =\mathbb{N}$: $$\label{c}
DT_{x, h;a}(u)=\frac{(a+1)^{h} - |u - a|^{h}}{P(a, h)}\mathds{1}_{\mathbb{S}_{x\setminus \{a\}} }(u),$$where $h >0$ and $P(a, h)= (2a + 1)(a + 1) - 2 \sum_{k=0}^{a}k^{h}$ is the normalizing constant. It is symmetric around the target $x$, satisfying Definition \[Defkern\] and suitable for count variables; see [@KSKZ07] and also for an asymmetric version.
- From [@AJG76], [@KSK11] deduced the following discrete kernel that we here label DiracDU (`DirDU`) as “Dirac Discrete Uniform”. For fixed $c \in \{2,3,\ldots\}$ the number of categories, we define $\mathbb{S}_{x,c} =\{0, 1, \ldots, c-1\}$ and $$DU_{x,h;c}(u) = \left( 1-h\right) \mathds{1}_{\left\{{x}\right\}}(u)+\dfrac{h}{c-1}\mathds{1}_{\mathbb{S}_{x,c}\setminus\left\{{x}\right\}}(u),$$ where $h \in (0,1]$ and $x \in \mathbb{T}_1$. This DiracDU kernel is symmetric around the target, satisfying Definition \[Defkern\] and appropriated for categorical set $\mathbb{T}_1$. See, e.g., for some uses.
- From the well known [@E69] kernel $K^{E}(u) = \frac{3}{4}(1 - u^{2})\mathds{1}_{[ -1,1]}(u)$, we define its associated version (`Epan`) on $\mathbb{S}_{x,h}= [ x-h,x+h]$ with $x \in\mathbb{T}_1:=\mathbb{R}$ and $h>0$: $$K^E_{x,h}(u)=\frac{3}{4h}\left\{1 - \left(\frac{u-x}{h}\right)^{2}\right\}\mathds{1}_{[ x-h,x+h]}(u). \label{gam2}$$ It is obtained through (\[classcical\]) and is well adapted for continuous variables with unbounded supports.
- The gamma kernel (`Gamma`) is defined on $\mathbb{S}_{x,h}= [0,\infty)=\mathbb{T}_1$ with $x\in \mathbb{T}_1$ and $h>0$: $$GA_{x,h}(u)=\dfrac{u^{x/h}}{\Gamma\left(1+x/h\right)h^{1+x/h}}\exp{\left(-\dfrac{u}{h}\right)}
\mathds{1}_{[ 0,\infty)}(u), \label{gam2}$$ where $\Gamma(\cdot)$ is the classical gamma function. It is the pdf of the gamma distribution $\mathcal{G}a(1 + x/h,h)$ with scale parameter $1 + x/h$ and shape parameter $h$. It satisfies Definition \[Defkern\] and suits for non-negative real set $\mathbb{T}_1$; see [@Chen00a].
- The beta kernel (`Beta`) is however defined on $\mathbb{S}_{x,h}= [0,1]=\mathbb{T}_1$ with $x\in \mathbb{T}_1$ and $h>0$: $$BE_{x, h}(u) =\frac{u^{x/h}(1-u)^{(1 - x)/h}}{\mathscr{B}\left(1 + x/h, 1 + (1 - x)/h\right)} \mathds{1}_{[0, 1]}(u), \label{gam2}$$ where $\mathscr{B}(r, s) = \int_{0}^{1}t^{r-1}(1 - t)^{s - 1}dt$ is the usual beta function with $r>0$ and $s>0$. It is the pdf of the beta distribution $\mathcal{B}e(1+x/h,(1-x)/h)$ with shape parameters $1+x/h$ and $(1-x)/h$. This pdf satisfies Definition \[Defkern\] and is appropriated for rates, proportions and percentages dataset $\mathbb{T}_1$; see [@Chen99].
- We finally consider the bivariate beta kernel (`Bivariate beta`) defined by $$\begin{aligned}
\label{betakern}
BS_{\mathbf{x}, \mathbf{H}}(u_1,u_2) &=& \left(\frac{u_{1}^{x_{1}/h_{11}}(1-u_{1})^{(1 - x_{1})/h_{11}} }{\mathscr{B}(1 + x_{1}/h_{11}, 1 + (1 - x_{1})/h_{11})}\right) \left(\frac{u_{2}^{x_{2}/h_{22}}(1-u_{2})^{(1 - x_{2})/h_{22}}}{\mathscr{B}(1 + x_{2} / h_{22}, 1 + (1 - x_{2})/h_{22})} \right) \nonumber \\
&& \times \left(1 + h_{12}\times\dfrac{u_{1} - \widetilde{\mu}_{1}(x_{1}, h_{11})}{h_{11}^{1/2}\widetilde{\sigma}_{1}(x_{1}, h_{11})}\times\dfrac{u_{2} - \widetilde{\mu}_{2}(x_{2}, h_{22})}{h_{22}^{1/2}\widetilde{\sigma}_{2}(x_{2}, h_{22})}\right)\mathds{1}_{\left[0, 1\right]^2}(u_1,u_2),\end{aligned}$$ with $\mathbb{S}_{\mathbf{x}, \mathbf{H}}=\mathbb{T}_2=\left[0, 1\right]^2$, $\mathbf{x}=(x_1,x_2)^{\top} \in \mathbb{T}_2$ and $\mathbf{H} = \begin{pmatrix}h_{11} & h_{12} \\ h_{12} & h_{22}\end{pmatrix}$. For $j=1,2$, the characteristics in (\[betakern\]) are given by $h_{jj}>0$, $\widetilde{\mu}_{j}(x_{j}, h_{jj}) = (x_{j} + h_{jj})/(1 + 2h_{jj})$, $\widetilde{\sigma}_{j}^{2}(x_{j}, h_{jj}) = (x_{j} + h_{jj})(1 + h_{jj} - x_{j})(1 + 2h_{jj})^{-2}(1+3h_{jj})^{-1}h_{jj}$ , and the constraints $$h_{12} \in \left[-\beta, \beta^{\prime}\right] \cap \left(-\sqrt{h_{11}h_{22}}\:,\sqrt{h_{11}h_{22}}\right) \label{constrainteSarmanov}$$ with $
\beta = \left(\max_{v_1,v_2} \left\{\dfrac{v_{1} - \widetilde{\mu}_{1}(x_{1}, h_{11})}{h_{11}^{1/2}\widetilde{\sigma}_{1}(x_{1}, h_{11})}\times\dfrac{v_{2} - \widetilde{\mu}_{2}(x_{2}, h_{22})}{h_{22}^{1/2}\widetilde{\sigma}_{2}(x_{2}, h_{22})}\right\}\right)^{-1}
$ and\
$
\beta^{\prime} = \left\lvert\left(\min_{v_1,v_2} \left\{\dfrac{v_{1} - \widetilde{\mu}_{1}(x_{1}, h_{11})}{h_{11}^{1/2}\widetilde{\sigma}_{1}(x_{1}, h_{11})}\times\dfrac{v_{2} - \widetilde{\mu}_{2}(x_{2}, h_{22})}{h_{22}^{1/2}\widetilde{\sigma}_{2}(x_{2}, h_{22})}\right\}\right)^{-1}\right\lvert.
$ It satisfies Definition \[Defkern\] and is adapted for bivariate rates. The full bandwidth matrix $\mathbf{H}$ allows any orientation of the kernel. Therefore, it can reach any point of the space which might be inaccessible with diagonal matrix. This type of kernel is called beta-Sarmanov kernel by ; see [@S66] and also [@L96] for this construction of multivariate densities with correlation structure from independent components. Like , the miminax properties of this bivariate beta kernel are also possible and more generally for associated kernels.
Figure \[Associatedkernels\] shows some forms of the above-mentioned univariate associated kernels. The plots highlight the importance given to the target point and around it in both discrete and continuous cases. Furthermore, for a fixed bandwidth $h$, the classical associated kernel of Epanechnikov, and also the categorical DiracDU kernel, keep their respective same shapes along the support; however, they change according to the target for the others non-classical associated kernels. This explains the inappropriateness of the Epanechnikov kernel for density or regression estimation in any bounded interval (Figure \[Associatedkernels\](a)) and of the DiracDU kernel for count regression estimation (see simulations below).
Bandwidth matrix selection by cross validation {#ssec:bandwidth matrix selection}
----------------------------------------------
In the context of multivariate kernel regression, the bandwidth matrix selection is here obtained by the well-known least squares cross-validation. In fact, for a given associated kernel, the optimal bandwidth matrix is $\mathbf{\widehat{H}} = \displaystyle \mbox{arg } \underset{\mathbf{H} \in \mathcal{H} }{ \min} \mbox{ LSCV}(\mathbf{H})$ with $$\label{Hcv}
\mbox{LSCV}(\mathbf{H}) = \frac{1}{n} \displaystyle\sum_{i = 1}^{n}\left\{Y_i - \widetilde{m}_{-i}(\mathbf{X}_{i}; \boldsymbol{\kappa})\right\}^{2},$$ where $\widetilde{m}_{-i}(\mathbf{X}_{i}; \boldsymbol{\kappa})$ is computed as $\widetilde{m}_{n}$ of (\[RegFull\]) excluding $\mathbf{X}_{i}$ and, $\mathcal{H}$ is the set of bandwidth matrices $\mathbf{H}$; see, e.g., in univariate case and also [@Zhangetal2014] and [@ZAK1a] for univariate bandwidth estimation by sampling algorithm methods. For diagonal bandwidth matrices (i.e. multiple associated kernels) the LSCV method use the set of diagonal matrices $\mathcal{D}$. Concerning the beta-Sarmanov kernel (\[betakern\]) with full bandwidth matrix, this LSCV method is used under $\mathcal{H}_1$, a subset of $\mathcal{H}$ verifying the constraint (\[constrainteSarmanov\]) of the associated kernel. Their algorithms are described below and used for numerical studies in the following section.
### *Algorithms of LSCV method (\[Hcv\]) for some type of associated kernels and their correponsding bandwidth matrices* {#par:Algorithme .unnumbered}
1. Bivariate beta (\[betakern\]) with full bandwidth matrices and dimension $d=2$.
1. Choose two intervals $H_{11}$ and $H_{22}$ related to $h_{11}$ and $h_{22}$, respectively.
2. For $\delta = 1, \ldots, \ell(H_{11})$ and $\gamma = 1, \ldots, \ell(H_{22})$,
(a) Compute the interval $H_{12}[\delta,\gamma]$ related to $h_{12}$ from constraints in (\[constrainteSarmanov\]);
(b) For $\lambda = 1, \ldots, \ell(H_{12}[\delta,\gamma])$,\
Compose the full bandwidth matrix $\mathbf{H}(\delta,\gamma,\lambda):=\left(h_{ij}(\delta,\gamma,\lambda)\right)_{i,j=1,2}$ with $h_{11}(\delta,\gamma,\lambda)=H_{11}(\delta)$, $h_{22}(\delta,\gamma,\lambda)=H_{22}(\gamma)$ and $h_{12}(\delta,\gamma,\lambda)=H_{12}[\delta,\gamma](\lambda)$.
3. Apply LSCV method on the set $\mathcal{H}_1$ of all full bandwidth matrices $\mathbf{H}(\delta,\gamma,\lambda)$.
2. Multiple associated kernels (i.e. diagonal bandwidth matrices) for $d \geq 2$.
1. Choose two intervals $H_{11}$, $\ldots$, $H_{dd}$ related to $h_{11}$, $\ldots$, $h_{dd}$, respectively.
2. For $\delta_{1} = 1, \ldots, \ell(H_{11})$, $\ldots$, $\delta_{d} = 1, \ldots, \ell(H_{dd})$,\
Compose the diagonal bandwidth matrix $\mathbf{H}(\delta_{1},\ldots, \delta_{d}):=\mathbf{Diag}\left(H_{11}(\delta_1), \ldots, H_{dd}(\delta_d)\right)$.
3. Apply LSCV method on the set $\mathcal{D}$ of all diagonal bandwidth matrices $\mathbf{H}(\delta_1,\ldots, \delta_d)$.
For a given interval $I$, the notation $\ell(I)$ is the total number of subdivisions of $I$ and $I(\eta)$ denotes the real value at the subdivision $\eta$ of $I$. Also, for practical uses of (A1) and (A2), the intervals $H_{11},\ldots, H_{dd}$ are taken generally according to the chosen associated kernel.
Simulation studies and real data analysis {#sec:Simulation studies and real data analisys}
=========================================
We apply the multivariate associated kernel estimators $\widetilde{m}_{n}$ of (\[RegFull\]) and (\[Regprod\]) to some simulated target regressions functions $m$ and then to two real datasets. The multivariate and multiple associated kernels used are built from those of Section \[ssec:Associated kernels for illustration\]. The optimal bandwidth matrix is here chosen by LSCV method (\[Hcv\]) using Algorithms A1 and A2 of Section \[par:Algorithme\] and their indications. Besides the criterion of kernel support, we retain three measures to examine the effect of different associated kernels $\boldsymbol{\kappa}$ on multiple regression. In simulations, it is the average squared errors (ASE) defined as $$ASE(\boldsymbol{\kappa})=\dfrac{1}{n}\sum_{i=1}^{n}
\left\lbrace m(\mathbf{x}_i)-\widetilde{m}_{n}(\mathbf{x}_i;\boldsymbol{\kappa}) \right\rbrace^{2}.$$ For real datasets, we use the root mean squared error (RMSE) which linked to ASE through squared root and by changing the simulated value $m(x_i)$ into the observed value $y_i$: $$RMSE( \boldsymbol{\kappa}) = \sqrt{\frac{1}{n}\displaystyle\sum_{i=1}^n \left\{y_i -\widetilde{m}_{n}(\bold{x}_i;\boldsymbol{\kappa})\right\}^2 }.$$ Also, we consider the practical coefficient of determination $R^2$ which quantifies the proportion of variation of the response variable $Y_i$ explained by the non-intercept regressor $\bold{x}_i$ $$\label{R2}
R^2( \boldsymbol{\kappa}) = \frac{\sum_{i=1}^n \left\{\widetilde{m}_{n}(\bold{x}_i; \boldsymbol{\kappa}) - \overline{y}\right\}^2}{\sum_{i=1}^n (y_i - \overline{y})^2},$$ with $\overline{y} = n^{-1}(y_1 + \ldots + y_n)$. All these criteria above have their simulated or real data counterparts by replacing $y_i$ with $m(\bold{x_i})$ and vice versa. Computations have been performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comté using the R software; see [@R13].
Simulation studies {#sec:Simulation studies}
------------------
Expect as otherwise, each result is obtained with the number of replications $N_{sim}=100$.
### Bivariate cases
We consider seven target regression functions labelled A, B, C, D and E with dimension $d=2$.
- Function A is a bivariate beta without correlation $\rho(x_1,x_2) = 0$: $$\label{Betabiv}
m(x_{1}, x_{2}) = \frac{x_1^{p_{1}-1}(1 - x_1)^{q_{1}-1}x_2^{p_{2}-1}(1 - x_2)^{q_{2}-1}}{\mathscr{B}(p_{1}, q_{1})\mathscr{B}(p_{2}, q_{2})} \mbox{ } \mathds{1}_{\left[0, 1\right]}(x_{1})\mathds{1}_{\left[0, 1\right]}(x_{2}),$$ with $(p_{1}, q_{1}) = (3,2)$ and $(p_{2},q_{2}) = (5,2)$ as parameter values in univariate beta density.
- Function B is the bivariate Dirichlet density $$\label{Dirichletbiv}
m(x_{1}, x_{2}) = \frac{\Gamma(\alpha_{1} + \alpha_{2} + \alpha_{3})}{\Gamma(\alpha_{1})\Gamma(\alpha_{2})\Gamma( \alpha_{3})}x_{1}^{\alpha_{1} - 1}x_{2}^{\alpha_{2} - 1}(1 - x_{1} - x_{2})^{\alpha_{3} - 1} \mathds{1}_{\{x_{1},~ x_{2} \geq 0,~ x_{1} + x_{2} \leq 1\}}(x_{1},x_2),$$ where $\Gamma(\cdot)$ is the classical gamma function, with parameter values $\alpha_{1} = \alpha_{2} = 5$, $\alpha_{3} =6$ and, therefore, the moderate value of $\rho(x_1,x_2)=-(\alpha_{1}\alpha_{2})^{1/2}(\alpha_{1}+\alpha_{3})^{-1/2}(\alpha_{2}+\alpha_{3})^{-1/2}= -0.454$.
- Function C is a bivariate Poisson with null correlation $\rho(x_1,x_2) = 0$: $$\label{Poissonbiv}
m(x_{1}, x_{2}) = \frac{e^{-5}2^{x_1}3^{x_2}}{x_1!x_2!} \mathds{1}_{\mathbb{N}}(x_{1})\mathds{1}_{ \mathbb{N}}(x_{2}).$$
- Function D is a bivariate Poisson with correlation structure $$\label{PoissonbivCor}
m(x_{1}, x_{2}) = e^{-(\theta_1+\theta_2+\theta_{12})} \displaystyle \sum_{i=0}^{min(x_1,x_2)}\frac{\theta_{1}^{x_1+i} \theta_{2}^{x_2+i} \theta_{12}^{i}}{(x_1+i)!(x_2+i)!i!}\mathds{1}_{\mathbb{N}\times \mathbb{N}}(x_{1}, x_{2}),$$ with parameter values $\theta_1 = 2$, $\theta_2 = 3$ and $\theta_{12} = 4$ and, therefore, the moderate value of $\rho(x_1,x_2)=\theta_{12}(\theta_{1}+\theta_{12})^{-1/2}(\theta_{2}+\theta_{12})^{-1/2}= 0.617$; see, e.g., [@YS].
- Function E is a bivariate beta without correlation $\rho(x_1,x_2) = 0$: $$\label{BetaPoissonbiv}
m(x_{1}, x_{2}) = \frac{x_1^{p_{1}-1}(1 - x_1)^{q_{1}-1}3^{x_2}}{e^{3}\mathscr{B}(p_{1}, q_{1})x_2!} \mbox{ } \mathds{1}_{\left[0, 1\right]}(x_{1}) \mathds{1}_{ \mathbb{N}}(x_{2}),$$ with $(p_{1}, q_{1}) = (3,3)$.
$n$
------- ----------- ----------
$50$ $276.198$ $7.551$
$100$ $647.255$ $30.081$
: Typical Central Processing Unit (CPU) times (in seconds) for one replication of LSCV method (\[Hcv\]) by using Algorithms A1 and A2 of Section \[ssec:bandwidth matrix selection\].[]{data-label="Timehcv"}
Table \[Timehcv\] presents the execution times needed for computing the LSCV method for both bivariate beta kernels with respect to only one replication of sample sizes $n=50$ and $100$ for the target function A. The computational times of the LSCV method for the bivariate beta with correlation structure (\[betakern\]) are obviously longer than those without correlation structure. Let us note that for full bandwidth matrices, the execution times become very large when the number of observations is large; however, these CPU times can be considerably reduced by parallelism processing, in particular for the bivariate beta kernel with full LSCV method (\[Hcv\]). These constraints (\[constrainteSarmanov\]) reflect the difficulty for finding the appropriate bandwidth matrix with correlation structure by LSCV method.
$n$
-- ----- ------------------ ------------------ ------------------
$0.4368(0.3754)$ $0.4266(0.3724)$ $0.7483(0.2342)$
$0.1727(0.0664)$ $0.1952(0.0816)$ 0.6727(0.1413)
50 $1.2564(0.5875)$ $1.4267(0.4024)$ $1.6675(2.0353)$
100 $0.3041(0.1151)$ $0.3362(0.1042)$ $1.3975(1.5758)$
: Some expected values of $\overline{ASE}(\boldsymbol{\kappa})$ and their standard errors in parentheses with $N_{sim}=100$ of some multiple associated kernel regressions for simulated continuous data from functions A with $\rho(x_1,x_2) = 0$ and B with $\rho(x_1,x_2) = -0.454$.[]{data-label="ErrBetabiv"}
Table \[ErrBetabiv\] reports the average $ASE(\boldsymbol{\kappa})$ which we denote $\overline{ASE}(\boldsymbol{\kappa})$ for three continuous associated kernels $\boldsymbol{\kappa}$ with respect to functions A and B and according to sample sizes $n \in \{50, 100\}$. We can see that both beta kernels in dimension $d=2$ work better than the multiple Epanechnikov kernel for all sample sizes and all correlation structure in the regressors. This reflects the appropriateness of the beta kernels which are suitable to the support of rate regressors. Then, the explanatory variables with correlation structure give larger $\overline{ASE}(\boldsymbol{\kappa})$ than those without correlation structure. Also, both beta kernels give quite similar results. Furthermore, all $\overline{ASE}(\boldsymbol{\kappa})$ are better when the sample size increases.
Finally, Tables \[Timehcv\] and \[ErrBetabiv\] highlight that the use of bivariate beta kernels with correlation structure is not recommend in regression with rates explanatory variables. Thus, we focus on multiple associated kernels for the rest of the simulations studies.
$n$
-- ----- ---------------- ---------------- ---------------- ---------------- ------------------ -- -- -- --
1.5e-6(2.2e-6) 3.3e-6(4.1e-6) 3.6e-5(9.7e-6) 4.0e-5(3.5e-5) 1.6e-8(1.8e-8)
3.1e-7(6.9e-7) 4.7e-7(9.7e-7) 3.6e-5(7.4e-6) 3.8e-5(2.8e-5) 3.7e-9(2.3e-9)
8.6e-8(1.2e-7) 2.9e-7(3.1e-7) 3.7e-5(4.8e-6) 3.6e-5(2.3e-5) 4.1e-10(3.5e-10)
2.4e-6(2.8e-6) 4.5e-6(4.9e-6) 7.1e-6(2.6e-6) 4.2e-6(2.5e-6) 2.7e-8(2.1e-8)
2.5e-7(3.4e-7) 1.8e-7(2.5e-7) 8.1e-5(4.3e-6) 5.1e-6(1.2e-6) 4.3e-9(3.2e-9)
2.6e-8(6.2e-8) 4.8e-8(9.5e-8) 9.3e-6(8.2e-7) 7.2e-6(7.8e-7) 5.3e-10(4.6e-10)
: Some expected values of $\overline{ASE}(\boldsymbol{\kappa})$ and their standard errors in parentheses with $N_{sim}=100$ of some multiple associated kernel regressions for simulated count data from functions C with $\rho(x_1,x_2) = 0$ and D with $\rho(x_1,x_2) = 0.617$.[]{data-label="ErrPoissonbiv"}
Table \[ErrPoissonbiv\] shows the values $\overline{ASE}(\boldsymbol{\kappa})$ with respect to five associated kernels $\boldsymbol{\kappa}$ for sample size $n=20, 50$ and $100$ and count datasets generated from C and D. Globally, the discrete associated kernels in multiple case perform better than the multiple Epanechnikov kernel for all sample sizes and correlation structure in the regressors. The use of categorical DiracDU kernels gives the best result in term of $\overline{ASE}(\boldsymbol{\kappa})$ but DiracDU does not suit for these count datasets. Also, the discrete triangular kernels gives the most interesting result with an advantage to the discrete triangular with small arm $a=2$. This discrete triangular is the best since it concentrates always on the target and a few observations around it; see Figure \[Associatedkernels\](a). The results become much better when the sample size increases. The values $\overline{ASE}(\boldsymbol{\kappa})$ for regressors with or without correlation structure are comparable; and thus, we can focus on target regression functions without correlation structure for the remaining simulations.
$n$
-- ----- -------------- -------------- -------------- -------------- -------------- -- -- --
3.738(1.883) 1.966(1.382) 3.884(1.298) 6.361(2.134) 0.162(0.201)
3.978(1.404) 2.106(1.119) 3.683(0.833) 7.143(1.732) 0.138(0.171)
3.951(1.052) 1.956(0.806) 3.835(0.834) 7.277(1.574) 0.113(0.147)
: Some expected values ($\times 10^3$) of $\overline{ASE}(\boldsymbol{\kappa})$ and their standard errors in parentheses with $N_{sim}=100$ of some multiple associated kernel regressions of simulated mixed data from function E with $\rho(x_1,x_2) = 0$. []{data-label="ErrBetaPoissonbiv"}
Table \[ErrBetaPoissonbiv\] presents the values for sample sizes $n \in \{30, 50, 100\}$ and for five associated kernels $\boldsymbol{\kappa}$. The datasets are generated from E and the beta kernel is applied on the continuous rate variable of E. We observe the superiority of the multiple associated kernels using discrete kernels over those defined with the Epanechnikov kernel for all sample sizes. Then, the multiple associated kernel with the categorical DiracDU gives the best $\overline{ASE}(\boldsymbol{\kappa})$ but it is not appropriate for the count variable of E. Also, the values $\overline{ASE}(\boldsymbol{\kappa})$ are getting better when the sample size increases.
From Tables \[ErrBetabiv\], \[ErrPoissonbiv\] and \[ErrBetaPoissonbiv\], the importance of the type of associated kernel $\boldsymbol{\kappa}$ which respect the support of the explanatory variables is proven.
### Multivariate cases
Since the appropriate associated kernels perform better than the inappropriate ones, we focus in higher dimension $d>2$ on regression with only suitable associated kernels. Then, we consider two target regression functions labelled F and G for $d=3$ and 4 respectively. The formulas of the functions are given below.
- Function F is a 3-variate with null correlation: $$\label{BetaandPoissonbiv}
m(x_{1}, x_{2},x_{3}) = \frac{x_1^{p_{1}-1}(1 - x_1)^{q_{1}-1}2^{x_2}3^{x_3}}{e^{5}\mathscr{B}(p_{1}, q_{1})x_2!x_3!}\mathds{1}_{\left[0, 1\right]}(x_{1})\mathds{1}_{\mathbb{N}}(x_{2})\mathds{1}_{\mathbb{N}}(x_{3}),$$ with $(p_{1}, q_{1}) = (3,2)$.
- Function G is a 4-variate without correlation: $$\label{BetabivandPoissonbiv}
m(x_{1}, x_{2},x_{3},x_{4}) = \frac{x_1^{p_{1}-1}(1 - x_1)^{q_{1}-1}x_2^{p_{2}-1}(1 - x_2)^{q_{2}-1}2^{x_3}3^{x_4}}{e^{5}\mathscr{B}(p_{1}, q_{1})\mathscr{B}(p_{2}, q_{2})x_3!x_4!}\mathds{1}_{\left[0, 1\right]}(x_{1})\mathds{1}_{\left[0, 1\right]}(x_{2}) \mathds{1}_{\mathbb{N}}(x_{3})\mathds{1}_{\mathbb{N}}(x_{4}),$$ with $(p_{1}, q_{1}) = (3,2)$ and $(p_{2},q_{2}) = (5,2)$.
$n$
----- ---------------- ---------------- ---------------- -- -- -- -- -- --
0.2501(0.1264) 0.3038(0.1258) 0.7448(0.5481)
0.2381(0.0661) 0.2895(0.0162) 0.6055(0.2291)
0.2282(0.0649) 0.2822(0.0608) 0.5012(0.2166)
: Some expected values ($\times 10^3$) of $\overline{ASE}(\boldsymbol{\kappa})$ and their standard errors in parentheses with $N_{sim}=100$ of some multiple associated kernel regressions of simulated mixed data from 3-variate F and 4-variate G.[]{data-label="ErrBetaandPoissonbiv"}
Table \[ErrBetaandPoissonbiv\] presents the regression study for dimension $d=3$ and 4 with respect to functions F and G and for sample size $n \in \{30, 50, 100\}$. The values $\overline{ASE}(\boldsymbol{\kappa})$ show the superiority of the multiple associated kernels using the discrete triangular kernel with $a=3$ over the one with the binomial kernel. Some results with respect to function G for an associated kernel $\boldsymbol{\kappa}$ composed by two beta and two discrete triangular kernels with $a=3$ are also provided. The errors become smaller when the sample size increases.
Real data analysis
------------------
The dataset consists on a sample of 38 family economies from a US large city and is available as the [*FoodExpenditure*]{} object in the [*betareg*]{} package of . The dataset in its current form gives not available (NA) responses for associated kernel regressions especially when we use the discrete triangular or the DiracDU kernel. Then, we extend the original [*FoodExpenditure*]{} dataset with its first 20 observations which guarantees some results for the regression, and thus $n=58$. The dependent variable is [*food/income*]{}, the proportion of household *income* spent on *food*. Two explanatory variables are available: the previously mentioned household [*income*]{} ($x_1$) and the [*number of residents*]{} ($x_2$) living in the household with $\widehat{\rho}(x_1,x_2) = 0.028$. We use the Gamma or the Epanechnikov kernel for the continuous variable $income$ and the discrete (of Figure \[Associatedkernels\](a)) or the Epanechnikov for the count variable [*number of residents*]{}.
The results of the multiple associated kernels for regression are divided in two in Table \[dataset1\]. The appropriate associated kernels which strictly follow the support of each variable give comparable results in terms of both RMSE$(\boldsymbol{\kappa})$ and R$^2(\boldsymbol{\kappa})$. In fact, the associated kernels that use the discrete triangular with arm $a=2$ and 3 give some R$^2(\boldsymbol{\kappa})$ approximately equal to $64 \%$. The inappropriate kernels give various results. The multiple Epanechnikov kernel and the type of kernel with DiracDU give R$^2(\boldsymbol{\kappa})$ higher than $80 \%$ while the Gamma$\times$Epanechnikov gives R$^2(\boldsymbol{\kappa})$ less than $50\%$. Then, a little difference in terms of RMSE$(\boldsymbol{\kappa})$ can induce a high incidence on the R$^2(\boldsymbol{\kappa})$.
-- --------- --------- --------- -- -- -- --
0.01409 0.01426 0.01730
64.2681 64.2708 56.1091
0.01451 0.03266 0.01278
86.0011 47.0181 89.3462
-- --------- --------- --------- -- -- -- --
: Some expected values of RMSE$(\boldsymbol{\kappa})$ and in percentages R$^2(\boldsymbol{\kappa})$ of some multiple associated kernel regressions for the $FoodExpenditure$ dataset with $n=58$.[]{data-label="dataset1"}
------ ------ ----- ------ ------ ----- ------ ------ -----
68.1 54.6 0.8 80.3 9.1 1.5 78.6 31.0 1.6
60.8 4.4 1.3 23.1 83.1 2.3 44.9 3.2 1.0
34.4 36.2 1.2 16.9 90.4 2.0 78.2 13.9 1.8
59.4 27.5 1.3 9.4 79.2 2.8 60.2 35.2 1.1
4.7 81.0 2.9 55.8 21.9 1.3 65.6 26.1 1.6
19.9 97.4 1.2 27.5 75.0 2.2 74.4 12.6 1.6
20.6 73.6 2.4 59.1 12.9 1.4 83.5 13.3 1.8
16.4 42.9 1.1 2.7 93.9 2.4 10.9 83.5 2.6
29.9 74.4 2.0 13.9 56.9 1.4 27.0 77.1 2.2
84.8 26.6 1.6 14.0 92.9 2.1 3.1 67.0 2.2
46.1 66.9 1.2 22.9 43.9 1.1 14.8 72.9 2.5
10.2 86.3 2.5 53.8 56.2 1.0 80.6 16.5 1.6
89.4 32.5 1.6 23.7 61.5 1.5 64.1 28.6 1.5
30.9 46.3 1.1 39.6 67.2 1.4 15.6 90.5 2.0
24.3 37.8 1.2 59.5 45.1 0.9 3.9 68.6 2.5
27.4 74.6 1.9 17.3 81.2 2.6 66.9 43.7 0.9
47.7 61.7 1.1 93.7 28.5 1.5 1.5 65.8 2.3
33.1 83.8 1.5 28.7 82.7 2.0 35.6 43.7 1.0
0.3 83.3 3.0 61.3 70.9 0.6 13.9 25.0 0.8
76.9 35.4 1.2 67.1 24.0 1.7 13.2 70.8 2.2
29.5 44.6 1.3 85.8 36.5 1.2 34.5 73.7 1.8
19.6 67.7 1.9 35.5 76.9 1.8 55.6 6.9 1.3
96.2 26.1 1.7 18.8 55.9 1.3 30.7 9.1 0.9
85.9 28.0 1.5 50.4 17.7 1.4 43.5 15.1 1.0
5.6 39.1 1.1 67.2 8.7 1.5 31.5 36.7 1.2
99.9 7.15 1.3 13.1 59.4 1.7 30.0 21.5 0.8
61.0 31.1 1.4 13.7 75.8 2.5
------ ------ ----- ------ ------ ----- ------ ------ -----
: Proportions (in $\%$) of folks who like the company, those who like its strong product and turnover of a company, designed respectively by the variables $x_{1i}$, $x_{2i}$ and $y_i$, with $\widehat{\rho}(x_1,x_2) = -0.6949$ and $n=80$.[]{data-label="data2"}
Table \[data2\] of the second dataset aims to explain the turnover of a large company by two proportions explanatory variables obtained by survey. The first variable $x_1$ is the rate of people who like the company and the second one $x_2$ is the percentage of people who like the strong product of this company. The dataset is obtained in 80 branch of this company. Obviously, there is a significant correlation between these explanatory variables: $\widehat{\rho}(x_1,x_2)=-0.6949$.
Table \[dataset2\] presents the results for the nonparametric regressions with three associated kernels $\boldsymbol{\kappa}$. Both beta kernels offer the most interesting results with R$^2(\boldsymbol{\kappa})$ approximately equal to $86\%$. Note that, the multiple Epanechnikov kernel gives lower performance mainly because this continuous unbounded kernel does not suit for these bounded explanatory variables.
-- ----------- ----------- ----------- -- -- -- -- --
$0.10524$ $0.10523$ $0.18886$
$86.6875$ $86.6874$ $76.3431$
-- ----------- ----------- ----------- -- -- -- -- --
: Some expected values of RMSE$(\boldsymbol{\kappa})$ and in percentages R$^2(\boldsymbol{\kappa})$ of some bivariate associated kernel regressions for tunover dataset in Table \[data2\] with $\widehat{\rho}(x_1,x_2) = -0.6949$ and $n=80$.[]{data-label="dataset2"}
Summary and final remarks {#sec:Summary and final remarks}
=========================
We have presented associated kernels for nonparametric multiple regression and in presence of a mixture of discrete and continuous explanatory variables; see, e.g., [@ZAK14b] for a choice of the bandwidth matrix by Bayesian methods. Two particular cases including the continuous classical and the multiple (or product of) associated kernels are highlight with the bandwidth matrix selection by cross-validation. Also, six univariate associated kernels and a bivariate beta with correlation structure are presented and used for computational studies.
Simulation experiments and analysis of two real datasets provide insight into the behaviour of the type of associated kernel $\boldsymbol{\kappa}$ for small and moderate sample sizes. Tables \[Timehcv\], \[ErrBetabiv\] and \[dataset2\] on bivariate rate regressions can be conceptually summarized as follows. The use of associated kernels with correlation structure is not recommend. In fact, it is time consuming and have the same performance as the multiple beta kernel. Also, these appropriate beta kernels are better than the inappropriate multiple Epanechnikov. For count regressions, the multiple associated kernels built from the binomial and the discrete triangular with small arms are superior to those with the optimal continuous Epanechnikov. Furthermore, the categorical DiracDU kernel gives misleading results since it does not suit for count variables, see Tables \[ErrPoissonbiv\] and \[ErrBetaPoissonbiv\]. We advise beta kernels for rates variables and gamma kernels for non-negative dataset for small and moderate sample sizes, and also for all dimension $d \geq 2$; see, e.g., Tables \[ErrBetaandPoissonbiv\] and \[dataset1\]. Finally, more than the performance of the regression, it is the correct choice of the associated kernel according to the explanatory variables which is the most important. In other words, the criterion for choosing an associated kernel is the support; however, for several kernels matching the support, we use common measures such as the mean integrated squared error. It should be noted that a large coefficient of determination $R^2$ does not mean good adjustment of the data; see Tables \[dataset1\] and \[dataset2\]. Further research on associated kernels for functional regression is conceivable; see, e.g., [@ACT14] for classical kernels.
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---
abstract: 'A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function $\G$ of directed animals on the square lattice, counted according to the area and the perimeter, and the problem of solving a system of quadratic equations involving unknown matrices. We present some solid evidence that some infinite explicit matrices, the fixed points of a rewriting like system are the natural solutions to this system of equations: some strong evidence is given that the problem of finding $\G$ reduces to the problem of finding an eigenvector to an explicit infinite matrix. Similar properties are shown for other combinatorial questions concerning directed animals, and for different lattices.'
---
\[lem\][Definition]{} \[lem\][Proposition]{} \[lem\][Theorem]{} \[lem\][Corollary]{} \[lem\][Note]{}
**Directed animals, quadratic and rewriting systems\
[**Jean-François Marckert**]{}\
**
> [ The author is partially supported by the ANR-08-BLAN-0190-04 A3. ]{}
Introduction
============
We are mainly interested in the study of the area and perimeter generating function $\G^\Sq$ of directed animals on the square lattice $\Sq$, but other lattices and questions will also be addressed. The computation of $\G^\Sq$ is a central question in enumeration problems for directed animals on two dimensional lattices, since it is deeply related to the study of directed percolation on the square lattice. In this paper, even if we do not find an explicit formula for $\G^\Sq$, we show that to compute $\G^\Sq$ it suffices to solve a quadratic system of equations involving 4 unknown finite matrices. We are unable to find a solution, but we provide some infinite size matrices which appear as the natural solution to this system of equations. They appear to be a fixed point of a rewriting system, the rewriting rules involving the tensorial product of matrices. We give strong evidence that finding a right and a left eigenvector to these matrices should lead to $\G^\Sq$. We hope that this gives some insight on the algebraic structure of this problem, and that this will allow some readers to compute $\G^\Sq$.
In Section \[sec:extensions\], we show that numerous similar problems can be treated similarly.
{height="3cm"}
The set of oriented graphs with no cycle and no multiple edges which have a finite or countable number of vertices and bounded degree is denoted ${\cal G}$. For any graph $G=(V,E)$ in ${\cal G}$, $V$ is the set of vertices and $E\subset V^2$ the set of oriented edges. The orientation of the edges leads to the notion of a descendant: for $(x_1,x_2) \in E$, $x_2$ is said to be a child of $x_1$ and the set of children of $x_1$ is denoted $\Ch(x_1)$. A directed path $d$ in $G$ is a sequence of vertices $(x_1,\dots,x_k)$ such that for any $l\geq 2$, $x_l\in \Ch(x_{l-1})$. The vertex $x_1$ (resp. $x_k$) is called the origin (resp. the target) of $d$.
\[diran\] Let $G=(V,E)$ be in ${\cal G}$, and $S$ be a subset of $V$.\
$\bullet$ A directed animal (DA) $A$ with source $S$ is a subset of $V$ containing $S$, such that for every $a\in A$ there exists a directed path having target $a$ and its origin in $S$ entirely contained in $A$. The cardinality $\#A$ of $A$ is called the area of $A$.\
$\bullet$ A perimeter site $c$ of a DA $A$ with source $S$ is an element of $V\setminus A$ such that $\{c\} \cup A$ is still a DA with source $S$. The set of perimeter sites of $A$ is denoted $P(A)$.\
We denote by ${\cal A}^G_S$ the set of finite DA on $G$ with source $S$. The generating function (GF) $\G_S^G$ counts the DA with source $S$ according to the area and perimeter: $$\G_S^G(x,y):=\sum_{A\in {\cal A}^G_S}x^{\# A}y^{\#P(A)}.$$ Hence, the area generating function is $\G_S^G(x,1)$.
The search for a formula for $\G_S^G(x,y)$ may be seen as the combinatorial contribution to the study of directed percolation per site models. Indeed, on a probability space $(\Omega,{\cal A},`P)$ consider a random colouring of the vertices of $V$ by the colours 0 and 1. Formally, this is given by a family of i.i.d. Bernoulli random variables $(B^v(p),v\in V)$ indexed by the vertex set (we then have $`P(B^v(p)=1)=1-`P(B^v(p)=0)=p$). The directed percolation cluster with source $v \in V$ is the maximum DA with source $S=\{v\}$ included in the set of 1-coloured vertices, that is $\{u \in V~:B^u(p)=1\}$ (the empty case, possible here, arises with probability $1-p$): denote it ${\bf A}^v(p)$. Since for any DA A with source $v$, $`P({\bf A}^v(p)=A)=p^{\#A}(1-p)^{\#P(A)}$, the percolation cluster is finite with probability 1 if $(1-p)+\sum_{A\in{\cal A}^G_{\{v\}}} p^{\#A}(1-p)^{\#P(A)}=1$, which is equivalent to $\G_{\{v\}}^G(p,1-p)=p$. Hence a computation of $\G_{\{v\}}^G(x,y)$ would probably allows one to compute the directed percolation threshold, and/or the associated critical exponent.
Denote by $\Sq=(V_\Sq,E_\Sq)$ the directed square lattice where $V_\Sq=\mathbb{Z}^2$ and $$E_\Sq=\l\{((x_1,y_1),(x_2,y_2))\in V_\Sq^2, \textrm{ such that } (x_2,y_2)-(x_1,y_1)\in\{(0,1),(1,0)\}\r\}.$$ Surveys on the study of DA on two dimensional lattices exist: Bousquet-Mélou [@BM1] and Le Borgne & Marckert [@LB-M]. When $S$ is reduced to a singleton, the area GF $\G_S^{\Sq}(x,1)$ is well known, and numerous different approaches are possible to compute it: the gas approach (Dhar [@DH1; @DH5], but also [@BM1], [@LB-M], Albenque [@Al]), heap of pieces approach (Viennot [@VI1]), combinatorial decomposition (Corteel & al [@CDG], Bétréma & Penaud [@BEPE]). On the other side, almost nothing is known about $\G_S^{\Sq}(x,y)$ (except for Bacher [@BACH1] who computed $\sum p(A)x^{|A|}=\frac{\partial \G_S^{\Sq}(x,u)}{\partial u}(x,1)$ on the square lattice with or without periodic conditions, proving conjectures by Conway [@CC] and Le Borgne [@LB2]). $\G_S^{\Sq}(x,y)$ is not believed to be $D$-finite.
The aim of this paper is to use some algebra to search for a formula for $\G_S^{\Sq}(x,y)$. We will use the idea of Nadal & al. [@NDV] and Hakim and Nadal [@HV], also used extensively in [@BM1]. First, the work is done on a so-called cylinder $\Sq(n)$, a vertical strip of $\Sq$ with periodic conditions (see Figure \[fig:STH\]). Second, the corresponding for $\Sq$ is obtained by taking a formal limit since small DA on $\Sq(n)$ and $\Sq$ are the same. We will proceed similarly here, starting with $\G^{\Sq(n)}(x,y)$.
In order to highlight the different considerations leading to the introduction of infinite matrices, we have decided to simultaneously treat the study of $\G^{\Sq(n)}(x,y)$ and a case where infinite matrices can be avoided: the computation of $\G^{\Sq(n)}(x,1)$. This leads to a new derivation of $\G^{\Sq}(x,1)$.
Two gases
---------
Let $G=(V,E)\in{\cal G}$ be an oriented graph. Following the ideas developed in [@LB-M], we define two processes $\bX_G=(X_v,v \in V)$ and $\bY_G=(Y_v, v\in V)$ indexed by the vertex set, and taking their values in $\{0,1\}$. For this latter reason, the processes are called “gases”, the value 1 (resp. 0) representing the presence (resp. absence) of a particle.
Both processes $\bX_G$ and $\bY_G$ are defined on a probability space $(\Omega,{\cal A},`P)$, on which are defined some families of i.i.d. random variables $(\Xi^v,v\in V)$ indexed by the vertex set, where $\Xi^v=(B_1^v(p),B_2^v(q))$ is a pair of independent Bernoulli random variables whose parameters are $p$ and $q$.
#### Gas of type 1 :
For any $v\in V$, set $$\label{eq:gas1}
X_v=B^v_1(p)\prod_{c \in \Ch(v)} (1-X_c).$$ That is, if $X_c=0$ for all $c\in \Ch(v)$, then $X_v=1$ with probability $p$; otherwise $X_v=0$. Since two neighbouring sites can not be simultaneously occupied, this model is called a hard particle model in the physics literature.
#### Gas of type 2 :
For any $v\in V$, set $$\label{eq:gas2}
Y_v=B^v_1(p)\min\{Y_c~: c \in \Ch(v)\} +(1-B^v_1(p))B_2^v(q).$$ Here $Y_v$ is equal to $\min\{Y_c~: c\in \Ch(v)\}$ with probability $p$, and to $B_2^v(q)$. with probability $1-p$.
\[lem:welldefi\] Let $G\in {\cal G}$. If $p\in(0,1)$ is small enough, both processes $\bX_G$ and $\bY_G$ are almost surely well defined.
We use the argument in [@LB-M] (the argument being already present in the PhD thesis of Le Borgne [@LBPhD]). For both gases, when $B^v_1(p)=1$, the set of values $\{X_c,c\in \Ch(v)\}$ (resp. $\{Y_c,c\in \Ch(v)\}$) is needed to compute $X_v$ (resp. $Y_v$), but they are not needed when $B^v_1(p)=0$, in which case $X_v=0$ and $Y_v=B_2^v(q)$. The fact that these “recursive definitions” and indeed define some objects is not clear, but the values of both $X_v$ and $Y_v$ are certainly well defined if ${\bf A}^v(p)$ is finite, since in this case, the recursive computation of $X_v$ (and $Y_v$) using the values of the children ends since the value of $X$ and $Y$ on perimeter sites of ${\bf A}^v(p)$ – sites where $B_1(p)$ is zero – is well defined. Hence, if the family of DA $({\bf A}^v(p),v\in V)$ is a family of finite DA, both processes are defined. Now consider the standard problem of the directed percolation threshold. Let $$p_{crit}=\sup\{p~: `P(\forall v, |{\bf A}^v(p)|<+\infty)=1\}.$$ Since we assume that the maximum degree if the graph is finite, $p_{crit}\in(0,1]$. For all $0\leq p<p_{crit}$, $\bX_G$ and $\bY_G$ are then a.s. defined. $\Box$
A subset $S$ of $V$ is said to be free if for any $s_1,s_2 \in S$ with $s_1\neq s_2$, there does not exist any directed path with origin $s_1$ and target $s_2$ in $G$. The following Proposition says that the computation of the finite dimensional distribution of the gas of type 1 (resp. type 2) is equivalent to the computation of the DA GF according to the area (resp. area and perimeter), for general source. A DA is said to have over-source $S$ if is a DA with source $S'$, with $S'\subset S$. If $S'\neq S$, the set $S\setminus S'$ is taken to be a subset of the perimeter. We denote by $\bar{\G_S^G}$ the generating function of DA with over-source $S$.
\[pro:fo\] For any directed graph $G=(V,E)$ in ${\cal G}$ and any free subset $S$ of $V$, ‘P(X\_v=1,vS)&=&(-1)\^[\#S]{}\_S\^G(-p,1),\
‘P(Y\_v=1,vS)&=&|[\_S\^G]{}(p,(1-p)q), where the first equality holds for $|p|$ smaller than the radius of convergence of $\G_S^G(-p,1)$ and the second one holds if $0 < p< p_{crit}$.
The first assertion is proved in [@LB-M] (Theorem 2.7 for a single source, Proposition 2.16 for any source) on a general graph, and was already used in [@BM1] on lattices for a single source. Dhar [@DH1], who made the connection on lattices between GF of DA and the problem of finding the density of a hard particle system on an associated graph, did not use the construction of the process $X$, but different considerations of the same process. The second assertion is well-known, and it is also proved in [@LB-M] (Theorem 4.3) and valid on any graph of ${\cal G}$. The reason is simple: $Y_v=1$ if and only if $B^u_2(q)=1$ for all perimeter sites $u$ of $\A^v(p)$ (where $v$ is considered as a perimeter site of $\A^v(p)$ in the case where $\A^v(p)$ is empty). $\Box$
Although $\G(x,1)$ is a projection of $\G(x,y)$, and $\bar{\G_S^G}(x,y)$ can be computed easily thanks to $(\G_{S'}^G(x,y),S'\subset S)$, the gas of type 2 is not an extension of the gas of type 1. To compute $\G^G(p,1)$ using the gas of type 2, $q$ needs to be $1/(1-p)$, which is larger than 1; this is not possible for probabilistic considerations. Nevertheless, given the polynomial form of $\bT^{\bY}$ (see Formulas and ), $\bT^\bY$ still has a meaning when $q=1/(1-p)$; for this value, it is no longer a positive kernel, but $\sum_{\bb \in E_n} \bT^{\bY}_{\ba,\bb}=1$ for any $\ba\in E_n$. A non-negative solution $\mu$ to $\mu=\mu\bT^{\bY}$ still exists, as can be checked by rewriting the following system of equations: let, for any $C \subset\{0,\dots,n-1\}$, $$W_C=\mu(\{x \in E_n ~: i\in C \imp x_i= 1\}).$$ $\mu$ solves the system $\mu=\mu\bT^{\bY}$ if and only if $W:=(W_C,C \subset\{0,\dots,n-1\})$ is solution to $W_C =\sum_{D\subset C} ((1-p)q)^{C\setminus D} p^D W_{\Ch(D)}$, which when $(1-p)q=1$ is a rewriting of $W_C =\sum_{D\subset C} p^D W_{\Ch(D)}$. Clearly, $W$ satisfies the same system of equations as $\bar{\G_C^G}:=(\bar{\G_C^G}(p,1),C \subset\{0,\dots,n-1\})$. Since $\bar{\G_C^G}$ exists and is non negative, $\mu=\mu\bT^{\bY}$ admits some solutions. The initial conditions $W_\varnothing=\bar{\G_\varnothing^G}(p,1)=1$ allows one to identify the two set of series $W$ and $\bar{\G_C^G}$. As in Dhar [@DH1] or Bousquet-Mélou [@BM1], $W_C$ has a product form on $E_n$, meaning that ($W_\ba=\alpha_n \prod_{i=1}^n Q_{a_i,a_{i+1\mod n}}$ for some numbers $Q_{0,0},Q_{0,1},Q_{1,0},Q_{1,1}$). The behaviour of $`P(Y_v=1,v\in S)$ becomes singular at $q=1/(1-p)$: some drastic simplifications of the involved algebra appear, but only for that value of $q$. This leads to a Markovian type structure of $\mu$.
Some extensions of the gas of type 1 can also be introduced, for example $X_x=B_p\prod_{c\in\Ch(x)}(1-X_c)+(1-B_p)B_q$. These extensions are either of the same type as this gas, meaning that an easy-to-prove Markovian behaviour occurs, or will present the same kind of difficulties as for the gas of type 2 (as illustrated by the different cases discussed in Section \[sec:extensions\]). We think that any gas allowing us to compute $\G^{\Sq(n)}(x,y)$ will be (at best) as difficult to describe as the gas of type 2.
The square lattice : the cylinder approach
------------------------------------------
We study some properties of the processes of type 1 and 2 defined on $\Sq(n)$, the square lattice with periodic conditions shown in Figure \[fig:STH\]. We first need to label the sites of this lattice and its rows, in order to describe its Markov chain structure, row by row: the $l$th row is $\row^{(l)}=\l\{(x \mod n,l-x \mod n),x \in \mathbb{Z}\r\}$ and let $x^{(l)}(i)=(i \mod n,l-i\mod n)$ be the $i$th element of this row. The two children of $x^{(l)}(i)$ are $x^{(l+1)}(i+1)$ and $x^{(l+1)}(i)$. Note that $\row^{(l+1)}$ is above $\row^{(l)}$ in Figure \[fig:STH\].
It is useful to provide a row decomposition of the processes $\bX_{\Sq(n)}$ and $\bY_{\Sq(n)}$. For brievity, let $\bZ\in\{\bX_{\Sq(n)},\bY_{\Sq(n)}\}$ be one of these processes, and let $Z^{(l)}(i)$ be the value of $\bZ$ at $x^{(l)}(i)$, and $$\bZ^{(l)}=\l(Z^{(l)}(i),i=0,\dots,n-1\r)$$ the value of $\bZ$ on $\row^{(l)}$. Clearly $\bZ^{(l)}$ depends on $\bZ^{(l+1)}$ and the values of the Bernoulli random variables $(\Xi_v,v\in \row^{(l)})$ only. Then, the sequence $\bZ^{(l)}$, when $l$ goes from $+\infty$ to $-\infty$, is a Markov chain. The transition kernel $\bT_n^\bZ$ can be expressed using and : denote by $E_n$ the state space $\{0,1\}^n$, by $\ba\nn:=(a_0,\dots,a_{n-1})$, and $\bb\nn=(b_0,\dots,b_{n-1})$ some generic elements of $E_n$, and by $\oplus$ the addition in $\mathbb{Z}/n\mathbb{Z}$. We have $$\label{main:equa1}\bT^\bZ(\bb,\ba):=`P(\bZ^{(l)}=\ba ~|~\bZ^{(l+1)}=\bb)=\prod_{i=0}^{n-1} T^{\bZ}_{b_i,b_{i\oplus1},a_{i}},$$ and where $T^{\bX}_{x,y,z}$ and $T^{\bY}_{x,y,z}$ are explicit: \[eq:trans\] T\^\_[x,y,z]{}&=& ‘P(B\_1(p)(1-x)(1-y)=z)\
&=&1\_[(x,y)(0,0)]{}1\_[z=0]{}+1\_[(x,y)=0]{}ł( p 1\_[z=1]{}+(1-p)1\_[z=0]{}),\
T\^\_[x,y,z]{}&=& ‘P(B\_1(p)xy+(1-B\_1(p))B\_2(q)=z)\
&=&1\_[(x,y)=0]{}ł( (p+(1-p)(1-q))1\_[z=0]{}+(1-p)q1\_[z=1]{})\
&&+1\_[(x,y)=1]{}ł( (p+(1-p)q)1\_[z=1]{}+(1-p)(1-q)1\_[z=0]{}). If, for a given $l\in\mathbb{Z}$, $\bZ^{(l+1)}$ has distribution $\nu$, then the distribution $\mu$ of $\bZ^{(l)}$ satisfies $$\label{main:equa0}
\mu(\ba)=\sum_{\bb\in E_n} \nu(\bb)\, \bT^{\bZ}_n(\bb,\ba).$$ For clarity, we use an operator-type notation and write $\mu=\nu\,\bT^\bZ_n$; accordingly, $\mu=\nu\,(\bT^\bZ_n)^\kappa$ designates the law of $\bZ^{(-\kappa)}$ if $\bZ^{(0)}$ has distribution $\nu$.
Using the row decomposition of $\Sq(n)$, we have
\[lem:MC\] 1) Let $n\geq 1$ be fixed and $\bZ\in\{\bX_{\Sq(n)},\bY_{\Sq(n)}\}$. For $p,q\in (0,1)$, the process $(\bZ^ {(-j)},j\in \mathbb{Z})$ (indexed by decreasing $j$’s) is a Markov chain with finite state spaces $E_n$ under its stationary distribution. Since this Markov chain is irreducible and aperiodic, the distribution of $\bZ^{(j)}$ for any $j$ is the only probability measure solution $\mu^\bZ$ of $$\label{main:equa}
\mu^\bZ=\mu^\bZ\bT^\bZ_n.$$ 2) The distribution $\mu^\bZ$ is the limit distribution of any Markov chain with transition $\bT^\bZ_n$ on $\Sq(n)$, i.e. for any distribution $\nu$ on $E_n$, we have $\nu (\bT_n^\bZ)^\kappa\sous{\longrightarrow}{\kappa\to +\infty}\mu^\bZ$ .
The matrix $\bT^Z_n$, indexed by the elements of $E_n$ is called a (row) transfer matrix in statistical physics literature. By the Perron-Frobenius theorem, the equation has a unique solution up to a multiplicative constant, which can be turned into a probability distribution. Finding a $\mu^\bZ$ solution of is a problem of linear algebra, and a solution can be found by using a computer for small values of $n$. The fact that $\bT_n^\bZ$ has the product representation given in plays a secondary role in that respect. In the following, it will play a role of primary importance.
The only unclear assertion in (1) is that the row process $\bZ^{(-l)}$ follows under its stationary distribution. This comes from the infinite construction that we have, under which $\bZ\nn^{(l)}$ and $\bZ\nn^{(l+1)}$ have the same distribution. Assertion (2) is a well known property for aperiodic irreducible Markov chains on a finite state space. $\Box$
In the case of the square (or triangular) lattice the gas of type 1 is a Markov chain on the horizontal lines of the lattice [@LB-M]. In [@BM1], it is observed that it is a Markovian field on a zigzag formed with two consecutive lines of $\Sq(n)$; this Markovian field converges when the row size goes to $+\infty$ to a Markov chain on the line according to [@Al]. It turns out that the gas of type 2 is not Markovian on a line of the cylinder, on a line of the lattice, or on the zig-zag [@BM1]. In other words, there does not exist any $2\times 2$ matrix $A$ such that \[eq:prodd\] \^\_[n]{}(x\_1,…,x\_n)= c\_n \_[i=1]{}\^n A\_[x\_i,x\_[i+1]{}]{}. To show this, one can solve with a computer for $n=3,4,\dots$, to check that no factorisation of the invariant measure compatible with is possible (a similar work can be done directly on the rows of the entire lattice). It can also be checked that no solution corresponding to a Markov chain with longer memory exists, at least for small memories. When a computer is used to compute the invariant distribution on a cylinder of small size, no “regularity” of the measure has been observed: it turns out that on the cylinder of size 12, if $\mu_{12}^{\bY}(x_1,\dots,x_n)=\mu_{12}^{\bY}(y_1,\dots,y_n)$, then $\bx$ and $\by$ are equal up to a rotation and/or symmetry. This implies that what happens is drastically more complex that what appears for the gas of type 1 where $\mu^\bX_n$ has the form .
We now look at new considerations.
A new paradigm
==============
Proposition \[pro:fo\] says that $\mu_n^\bY(1,\star,\dots,\star):=\sum_{x_i\in\{0,1\},i=2,\dots,n}\mu_n^\bY(1,x_2,\dots,x_n)$, i.e. the probability that $\bY^v=1$ at some position $v$ on $\Sq(n)$ is, up to change of variables, equal to $\bar{\G^{\Sq(n)}}(x,y)$. A nice description of $\mu^{\bY}_{n}$ would thus be helpful. For this, two new ideas arise:\
The first one is to search for a representation of the same type as , but for matrices $A_{x,y}$ (taking the trace afterwards) instead of “real numbers $A_{x,y}$”. This is not just a way to add some degrees of freedom: Proposition \[pro:rep\] below says that all measures invariant under rotation on $E_n$ have such a representation, and moreover have a representation of the form $\operatorname{Trace}(\prod_{i=0}^{n-1} Q^{x_i})$ for some matrices $Q^0$ and $Q^1$,\
The second idea comes from the product form of the transition kernel $\bT_n^\bZ$ for $\bZ \in \{\bX,\bY\}$. It suggests that a “local equation” linking the matrix $Q^x$ “weighting the probability to observe $x$ somewhere on $\row(l)$” and $Q^y$, $Q^{y'}$, the matrices weighting their two children in $\row(l+1)$ could suffice. But two neighbours in $\row(l)$ share a neighbour in $\row(l+1)$, and this must be taken into account. The second idea is to share the neighbour, its suffices to “split” the matrices, and search for a solution of the form $Q^x=V^xH^x$ for some matrices $V^x$ and $H^x$ having right splitting properties. This is what is done and proved to be possible, up to taking infinite matrices $Q^x$.
Transfer matrix and measure representations {#note:o}
-------------------------------------------
We recall the definition and some properties of the Kronecker product (or tensor product) between matrices that we will use in the paper. If $A$ is an $m\times k$ matrix and $B$ is a $p\times q$, then the Kronecker product $A\otimes{B}$ is the $mp\times kq$ matrix $${A}\otimes{B} = \begin{bmatrix} A_{1,1} B & \cdots & A_{1,k}B \\ \vdots & \ddots & \vdots \\ A_{m,1} B & \cdots & A_{m,k} B \end{bmatrix}.$$ The Kronecker product is associative. For matrices $A,B,C,D$, \[eq:t1\](AB) (C D)= (AC) (BD),\
\[eq:t2\](AB)=(A)(B) where it is assumed in that $A,B,C,D$ have sizes such that $AC$ and $BD$ are well defined, and in that the matrices are square matrices. The Kronecker product extends to infinite matrices $A\otimes B$, but it is “interesting” only for finite $B$.
For $n\geq 1$, let ${\cal M}_n$ be the set of probability measures on $E_n$, invariant under rotation: if $\mu\in{\cal M}_n$, then for any $\bx=(x_0,\dots,x_{n-1})\in E_n$, $\mu(\bx)=\mu(x_1,\dots,x_{n-1},x_0)$.
\[pro:rep\] For any $\mu\in{\cal M}_n$, there exists two square matrices $Q^0$ and $Q^1$ of finite size, and four rectangular finite matrices $V^0,V^1,H^0,H^1$ such that:\
$(1)$ for any $\bx\in E_n$, $$\mu(\bx)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q^{x_i}\r),$$ $(2)$ $Q^1=V^1H^1,~~ Q^0=V^0H^0,~~ V^1H^0=\matz$ and $V^0H^1=\matz,$ where $\matz$ stands for the null matrix with the appropriate size.
First, assume that there exists some matrices $(Q^x, x\in\{0,1\})$ of size $m\times m$, such that $\mu(\bx)=\operatorname{Trace}(\prod_{i=0}^{n-1} Q^{x_i})$. Take $\tilde V^1=Q^1\otimes \begin{bmatrix}1&0 \end{bmatrix}$, $\tilde H^1=\operatorname{Id}(m) \otimes \begin{bmatrix}1\\0 \end{bmatrix}$, $\tilde V^0=Q^0\otimes \begin{bmatrix}0&1 \end{bmatrix}$ and $\tilde H^0=\operatorname{Id}(m) \otimes \begin{bmatrix}0\\1 \end{bmatrix}$, where $\operatorname{Id}(m)$ is the identity matrix of size $m\times m$. Using $\operatorname{Trace}(\prod a_i\otimes b_i)=\operatorname{Trace}(\prod a_i) \operatorname{Trace}(\prod b_i)$, we see that for any $\bx\in E_n$, $$\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q^{x_i}\r)= \operatorname{Trace}\l(\prod_{i=0}^{n-1} \tilde{Q}^{x_i}\r),$$ and clearly $\tilde{Q}^1=\tilde{V}^1\tilde{H}^1,\tilde{Q}^0=\tilde{V}^0\tilde{H}^0, \tilde{V}^0\tilde{H}^1=\tilde{V}^1\tilde{H}^0=\matz.$
Hence, only (1) remains to be proved. For this, denote by ${\cal M}_n'$ the set of probability distributions having the representation $\mu(\bx)=\operatorname{Trace}(\prod_{i=0}^{n-1} Q^{x_i})$ for some finite matrices $Q^0$ and $Q^1$. We show that ${\cal M}_n'={\cal M}_n$. For this, observe that ${\cal M}_n'$ is closed under finite mixture: if $\mu=\alpha\mu_a+(1-\alpha)\mu_b$ and $\mu_j(\bx)= \operatorname{Trace}(\prod_{i=0}^{n-1} Q_j^{x_i})$ for $j\in\{a,b\}$, then taking $$Q^x:=\begin{bmatrix}\alpha^{1/n}Q_a^x&0\\0&\beta^{1/n}Q_b^x \end{bmatrix},~~~~x\in\{0,1\},$$ we obtain $\mu(\bx)= \operatorname{Trace}(\prod_{i=0}^{n-1} Q^{x_i})$. This shows that ${\cal M}_n'$ is closed under finite mixture.
Since $\operatorname{Trace}(AB)=\operatorname{Trace}(BA)$, any probability distribution in ${\cal M}_n'$ is invariant under rotation. It remains to see that ${\cal M}_n'$ contains the uniform distribution on “simple” classes of rotation in $E_n$, since any measure in ${\cal M}_n$ is a mixture of these distributions. Take an element ${\bf \alpha}=(\alpha_0,\dots,\alpha_{n-1})$ in $E_n$. For any $i\in\cro{1,n}$, let ${\bf \alpha}(i)=(\alpha_i,\dots,\alpha_{n-1},\alpha_0,\dots,\alpha_{i-1})$ and $R:=\{{\bf \alpha}(i),i\in\{1,\dots,n\}\}$, the rotation class of ${\bf \alpha}$. We now show that the probability measure $\#R^{-1}\sum_{\beta\in R}\delta_{\bf \beta}$ belongs to ${\cal M}_n'$, which completes the proof. For this, denote by $z_i$ the number whose binary expansion is ${\bf \alpha}(i)$, and define $Q^1$ and $Q^0$ to be $2^n\times 2^n$ matrices where all entries are 0 except for $$Q^{\alpha_i}_{1+z_i,1+z_{i+1}}=\#R^{-1/n},~~ \textrm{ for any }i.$$ It is then simple to check that $\mu(\bx)=\operatorname{Trace}(\prod_{i=1}^n Q^{x_i})$ coincides with $\#R^{-1}\sum_{\beta\in R}\delta_{\bf \beta}$. $\Box$
The fundamental Lemmas {#sec:tosol }
----------------------
The following two lemmas \[lem:finite\] and \[pro:dec\] – that are among the main contributions of the present paper – serve two different purposes. The first Lemma can be used to identify the solution $\mu(\bx)=\operatorname{Trace}(\prod_i Q^{x_i})$ (and/or to prove that such a measure is a solution) of $\mu=\mu\bT_n$ when $\bT_n$ has a product form. It gives a sufficient condition on $(Q^0,Q^1)$ in the form of a “finite” system of quadratic equations. In the generic case, this system of equations does not depend on $n$, and then a right pair $(Q^0,Q^1)$ will provide a representation of $\mu$ for any $n$.
The second Lemma can be used when no such solution has been found. It permits us to describe the measure $\nu \bT^\kappa$ on the $k$th line starting from $\nu$, for some particular $\nu$ using some matrices $\bQ^x_{(\kappa)}$. Since $\nu \bT^\kappa\sous{\to}{\kappa\to +\infty}\mu$ the solution of $\mu=\mu \bT$, this provides a way to approach $\mu$. The problem is that the matrices $\bQ_{(\kappa)}^x$ appear to grow with $\kappa$. The discussion of the convergence of the sequence $(\bQ_{(\kappa)}^x,\kappa)$ is addressed later in this paper: this gives the clues we mentioned in the abstract that $\G(x,y)$ should have an expression using some eigenvectors of some explicit infinite matrices.
Lemmas \[lem:finite\] and \[pro:dec\] are stated in the case of $\Sq(n)$ for processes with values in $\{0,1\}$. They will adapted to the triangular lattice and to processes with more than 2 values in Section \[sec:tri-lat\].
A matrix $\bT_n$ indexed by $E_n$ is called a transfer matrix. If for all $\ba\in E_n$, $\sum_{\bb} \bT_n(\ba,\bb)=1$, then $\bT_n$ is a probability transfer matrix. Moreover if, for any $\ba,\bb \in E_n$, \[eq:prodform\] \_n(,)=\_[i=0]{}\^[n-1]{} T\_[a\_i,a\_[i1]{},b\_i]{} for some $T_{a,b,c}$, $a,b,c,\in\{0,1\}$, we say that the matrix transfer has a product form.
\[lem:finite\] Assume that $\bT_n$ is a probability transfer matrix with a product form. Assume that there exists square matrices $(V^x, H^x,x\in\{0,1\})$ such that, $V^x H^{y}=\begin{bmatrix}0 \end{bmatrix}$ for $x\neq y$, and such that for any $x$ \[eq:finite\] V\^x H\^x=\_[y,y’{0,1}]{} H\^y V\^[y’]{} T\_[y,y’,x]{}. Finally let $Q^x=V^{x}H^{x}$ for $x\in\{0,1\}$, and $\mu(\bx)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q^{x_i}\r)$. We have $\mu=\mu\bT_n$.
\(1) Equation is the system of quadratic equations mentioned in the introduction and in the Abstract. To find a solution to $\mu=\mu \bT_n$ it suffices to find $(V^x,H^x,x\in\{0,1\})$ satisfying .\
(2) One may weaken the conditions in Lemma \[lem:finite\], replacing by “there exists an invertible matrix $P$”, such that \[eq:finite-2\] V\^x H\^x=Pł(\_[y,y’{0,1}]{} H\^y V\^[y’]{} T\_[y,y’,x]{})P\^[-1]{}.
Before proving this Lemma we make two remarks. Firstly, we are only interested in non zero solutions! Under the hypothesis of this Lemma, the positivity of the measure $\mu$ is not guaranteed, nor the fact that it is non-zero or real. In the case where a non trivial solution exists, in the sense where $\mu(\bx)\neq0$ for some $\bx\in E_n$, then $\mu$ is a multiple of an eigenvector of $\bT_n$ with eigenvalue 1. Since $\bT_n$ is a Markov probability kernel, by the Perron-Frobenius theorem there is exactly one such eigenvector, which has non-negative entries. Note that if $(V^x,H^x,x\in \{0,1\})$ is a solution of then so is $(cV^x,H^x,x\in \{0,1\})$ for any $c\neq 0$. A $c$ (which may depend on $n$) can be taken such that $\mu$ is exactly the invariant distribution. Hence, a condition for non triviality (CNT) for a solution of is as follows: \[eref:CNT1\] CNT\_1: \_[x]{} V\^xH\^x.\
Secondly, finding a $(V^x,H^x, x\in \{0,1\})$ solution of is difficult in general. The number of variables and of equations quadratically increase with the size of the matrices $(V^x,H^x,x\in\{0,1\})$.
Write $\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q^{x_i}\r)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} \sum_{\forall i, y_i,y'_{i}\in \{0,1\}} H^{y_i} V^{y'_{i\oplus1}} T_{y_i,y'_{i\oplus1},x_i}\r)$. Since $V^{y'_{i}}H^{y_{i}}=0$ if $y'_{i}\neq y_{i}$, we impose $y'_{i}=y_{i}$. It remains to write, by some usual commutations $$\sum_{\by \in E_n}\operatorname{Trace}\l(\prod_{i=0}^{n-1} H^{y_i} V^{y_{i\oplus1}}\r)\prod_{i=0}^{n-1} T_{y_i,y_{i\oplus1},x_i}=
\sum_{\by \in E_n}\mu(\by)\bT_n(\by,\bx)=\mu \bT_n.~\Box$$
We now state the second fundamental lemma.
\[pro:dec\] Let $\bT_n$ be a transfer matrix on $E_n$. Consider two matrices $Q^x_{(0)}, x\in\{0,1\}$, of size $m\times m$ and some rectangular matrices $V^x_{(0)}, x\in\{0,1\}$ of size $m\times l$, and $H_{(0)}^x, x\in\{0,1\}$ of size $l\times m$, such that $$Q^x_{(0)}=V^x_{(0)}H^x_{(0)}, \textrm{ and, for }x\neq y,~~V^x_{(0)}H^y_{(0)}=[0].$$ Consider the measure $\mu_{(0)}$ defined on $E_n$ by $\mu_{(0)}(\bx)=\operatorname{Trace}\l(\prod_{i=1} Q_{(0)}^{x_i}\r).$ Assume that there exist 8 matrices $(h_{xy},(x,y)\in\{0,1\}^2)$ and $(v_{xy},(x,y)\in\{0,1\}^2)$ (of size $i\times j$ and $j\times i$ so that the products $h_{xy}v_{x'y'}$ and $v_{xy}h_{x'y'}$ are defined) such that for any $\by$, $\bx$ in $E_n$, $$\label{eq:f1}
\operatorname{Trace}\l(\prod_{i=0}^{n-1} h_{y_i,x_{{i\oplus1}}}v_{y_{{i\oplus1}},x_{{i\oplus1}}}\r)= \bT_n(\by,\bx).$$ Now, set for $x \in \{0,1\}$ \[eq:reecriture\] V\_[(1)]{}\^x=\_[y{0,1}]{} H\_[(0)]{}\^y h\_[y,x]{}, H\_[(1)]{}\^x=\_[y{0,1}]{} V\_[(0)]{}\^y v\_[y,x]{}, Q\_[(1)]{}\^x=V\_[(1)]{}\^xH\_[(1)]{}\^x and $\mu_{(1)}(\bx)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q_{(1)}^{x_i}\r).$ Then under these conditions,\
1) $\mu_{(1)}=\mu_{(0)}\bT_n$. If moreover $\mu_{(0)}$ is a probability measure and $\bT_n$ is a probability transfer matrix, then $\mu_{(1)}$ is also a probability measure.\
2) moreover if for any $x,x'$, $$\label{eq:cond2}
h_{xy}v_{x'y'}=\begin{bmatrix}0 \end{bmatrix} \textrm{ for }y\neq y',$$ then $V^x_{(1)}H^y_{(1)}=\begin{bmatrix}0 \end{bmatrix}$ for $x \neq y$.\
3) Assume that the probability transfer matrix $\bT_n$ has the product form and that there exist $h_{x,y}$ with one line, and $v_{x,y}$ with one column such that $$\label{eq:tra}
h_{y,x}v_{y',x}=T_{y,y',x}, \textrm{ for any }y,y',x.$$ Then condition is satisfied, and $Q_{(1)}^x$ and $Q_{(0)}^x$ have the same size. If moreover $Q_{(1)}^x=Q_{(0)}^x$ for $x\in\{0,1\}$, then $\mu_{(1)}=\mu_{(0)}$.
### Important note {#important-note .unnumbered}
$\bullet$ The size of the cylinder $n$ plays no role at all in the hypothesis. Hence, the same matrices $H^x_{(j)}$ , $V^x_{(j)}$, $Q^x_{(j)}$ will (or will not) work for all $n$.
$\bullet$ In the case where $Q_{(1)}^x=Q_{(0)}^x, x\in\{0,1\}$, we say that $Q_{(0)}$ is a fixed point for the matrix equation. That is, for any $n$, $\mu_n$ defined on $E_n$ by $$\mu_n(\bx)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q_{(0)}^{x_i}\r),$$ is a solution of $\mu_n=\mu_n\bT_n$.
Clearly, condition (2) is the condition needed to iterate the construction, that is to be able to give a matrix representation of the measure on the $\kappa$’th line, as stated in the next Corollary.
\[cor:big-construction\] Let $\mu_{(0)}$, $Q_{(0)}^x$, $V_{(0)}^x$, and $H_{(0)}^x$ satisfy the hypothesis of Lemma \[pro:dec\] (for example $V_{(0)}^1=H_{(0)}^1:=\begin{bmatrix}1 \end{bmatrix}$ and $V_{(0)}^0=H_{(0)}^0:=\begin{bmatrix}0 \end{bmatrix}$), and $h_{x,y}$ and $v_{x,y}$ satisfy , and . Then, for any $\kappa\geq 1$, $x\in\{0,1\}$, let \[eq:it-cons\] .
[l]{}V\_[()]{}\^x=\_[y{0,1}]{} H\_[(-1)]{}\^y h\_[y,x]{},\
H\_[()]{}\^x=\_[y{0,1}]{} V\_[(-1)]{}\^y h\_[y,x]{},
. and $Q^{x}_{(\kappa)}:=V_{(\kappa)}^xH_{(\kappa)}^x,~~ x\in\{0,1\}$, and $$\label{eq:mu}
\mu_{(\kappa)}(\bx)=\operatorname{Trace}\l(\prod_{i=0}^{n-1} Q^{x_i}_{(\kappa)}\r)$$ Then, for any $\kappa\geq 1$, $\mu_{(\kappa)}=\mu_{(\kappa-1)}\bT=\mu_{(0)}\bT^\kappa$.
This proof relies on and ; we also make heavy use of $\operatorname{Trace}(A_1\dots A_j)=\operatorname{Trace}(A_jA_1\dots A_{j-1})$. For $(1)$, write ł(\_[i=1]{} Q\_[(1)]{}\^[x\_i]{})&=&ł(\_[i=1]{} ł(\_y H\_[(0)]{}\^y h\_[y,x\_i]{}) ł(\_[y’]{} V\_[(0)]{}\^[y’]{} v\_[y’,x\_i]{}))\
&=&ł(\_[i=1]{} ł(\_[y’]{} V\_[(0)]{}\^[y’]{} v\_[y’,x\_i]{}) ł(\_y H\_[(0)]{}\^y h\_[y,x\_[i1]{}]{}) )\
&=&ł(\_[i=1]{} ł(\_[y’]{} V\_[(0)]{}\^[y’]{} v\_[y’,x\_i]{}) ł( H\_[(0)]{}\^[y’]{} h\_[y’,x\_[i1]{}]{}) )\
&=& \_ł(\_[i=1]{} ł(V\_[(0)]{}\^[y\_i]{} v\_[y\_i,x\_i]{}) ł(H\_[(0)]{}\^[y\_i]{} h\_[y\_i,x\_[i1]{}]{}) )\
&=&\_\_[(0)]{} () ł(\_[i=0]{}\^[n-1]{} v\_[y\_i,x\_i]{}h\_[y\_i,x\_[i1]{}]{})\
&=&\_\_[(0)]{} () ł(\_[i=0]{}\^[n-1]{} h\_[y\_i,x\_[i1]{}]{}v\_[y\_[i1]{},x\_[i1]{}]{}) $(1)$ now follows from .\
For (2), write for $x\neq y$, V\^x\_[(1)]{}H\^y\_[(1)]{}&=&ł(\_[w]{} H\_[(0)]{}\^w h\_[w,x]{})ł(\_z V\_[(0)]{}\^z v\_[z,y]{})\
&=&\_[w]{}\_z (H\_[(0)]{}\^w V\_[(0)]{}\^z) (h\_[w,x]{}v\_[z,y]{}) which is indeed $\matz$, since $h_{w,x}v_{z,y}=\matz$ for any $w,z$. $(3)$ follows immediately. $\Box$
Relation with ASEP/PASEP/TASEP
------------------------------
The system is quite close to the systems associated with the ASEP/PASEP/TASEP problems which use the “matrix ansatz” proposed by B. Derrida & al. [@DEHP]. In this seminal paper, it is shown that the invariant distribution of the ASEP can be expressed as the formal result of a computation in some algebraic structure, where some operators $D$ and $E$ satisfy some quadratic relations of the form $DE=ED + E +D$ (or more generally with some multiplicative parameters added) and some additional “border conditions”of the type $D|V\!\!>=\beta|V\!\!>$, $<\!\!W|E=\alpha^{-1}<\!\!W|$. For these problems, matrix solutions $D,E,V,W$ are explicitly found. These solutions have, depending on the values of the parameters and border conditions, a finite or an infinite size. In subsequent studies of these exclusion processes, finite/infinite matrices appear as solutions of quadratic equations of a type appearing in [@DEHP]. Each time, the questions of commutation of matrices, tensor products, and existence of limits with some growing matrices arise. We send the interested reader to [@BE] and [@HP], and references therein.
These quadratic problems/algebras are also at the core of several different problems of enumerative combinatorics, as observed by Viennot (this is discussed at length in several of his talks and courses, available on his web page, see e.g. [@VG2 section 7]).
Computation of fixed point solutions for gases of type 1 and 2
==============================================================
As explained in the previous subsection, two main cases emerge. First, there may exist a fixed point for the matrix equation involving finite matrices according to Lemma \[pro:dec\](3). If no such fixed point solution exists (or even if it does), Corollary \[cor:big-construction\] allows one to build bigger and bigger matrices to describe the distribution on the $k$th line. In some cases, taking the limit give rise to some infinite matrices. We first discuss, what happens when matrices of finite size are fixed points of the matrix equation. This is the case for the gas of type 1 on $\Sq(n)$.
Gas of type 1
-------------
This happens in the case $\bZ=\bX_{\Sq(n)}$, for which numerous different computations of the gas density exist. Let us add one way to find the solution.
### Application of Lemma \[lem:finite\] {#sec:al}
It suffices to search for finite matrices $(V^x,H^x,x\in\{0,1\})$ which solves . Take $$\label{eq:alpha}
V_{(0)}^0= \left[ \begin {array}{cc} 0&s_{{1,2}}\\ \noalign{\medskip}0&s_{{2,2}}
\end {array} \right],
V_{(0)}^1:= \left[ \begin {array}{cc} s_{{1,1}}&0\\ \noalign{\medskip}s_{{2,1}}&0
\end {array} \right],
H_{(0)}^0:=\left[ \begin {array}{cc} 0&0\\ \noalign{\medskip}t_{{1,2}}&t_{{2,2}}
\end {array} \right],
H_{(0)}^1:= \left[ \begin {array}{cc} t_{{1,1}}&t_{{1,2}}\\ \noalign{\medskip}0&0
\end {array} \right].$$ They are chosen to trivially satisfy $V_{(0)}^xH_{(0)}^y=\matz$ if $x\neq y$.
The choice of the letters $V$ and $H$ comes from these vertical and horizontal structures.
The system of equations $V^xH^x=\sum_{y,y'} H^y V^{y'}\bT^\bX_{y,y',x}$ is equivalent to: $$\Sys_1^\bX:=\left\{\begin{array}{l}
s_{1,1}t_{2,1}=0,s_{2,1}t_{1,1}=0,t_{1,1}s_{1,1}=0,s_{2,2}t_{2,2}= ( 1-p )( s_{1,2}t_{1,2}+s_{2,2}t_{2,2}),\\
t_{2,1}s_{2,1}=p ( s_{1,2}t_{1,2}+s_{2,2}t_{2,2} ) ,s_{1,2}t_{1,2}=t_{1,1}s_{1,1}+t_{2,1}s_{2,1},\\
s_{1,2}t_{2,2}=t_{1,1}s_{1,2}+t_{2,1}s_{2,2},s_{2,2}t_{1,2}=t_{1,2}s_{1,1}+t_{2,2}s_{2,1}
\end{array}\right.$$ This system has non trivial solutions. For example $s_{1,1}=0,s_{1,2}=p,s_{2,1}= 1/( 1-p )
,s_{2,2}=1,t_{1,1}=0,t_{1,2}=1,t_{2,1}= ( 1-p ) p
,t_{2,2}=1-p$, in which case $$Q^0=\left[ \begin {array}{cc}p&p(1-p)
\\ 1&1-p\end {array} \right] ,Q^1= \left[ \begin {array}
{cc} 0&0\\ \noalign{\medskip}0&p\end {array} \right].$$ From here the density $\operatorname{Trace}((Q^0+Q^1)^{n-1}Q^1)/\operatorname{Trace}((Q^0+Q^1)^n)$ of the gas of type 1 on the cylinder can be computed, and from that $\G^{\Sq(n)}(x,1)$ thanks to Proposition \[pro:fo\]. Taking the limit gives $\G^{\Sq}(x,1)$. This representation was known and is present under different forms in [@NDV; @DH1; @BM1]. What is remarkable is that it follows from a simple computation.
### Application of Lemma \[pro:dec\]
For the same result, one may use Lemma \[pro:dec\] since a solution to both and exists. The idea is to search for solutions under the following form, for which automatically holds : \[eq:les-h\]
[ccc]{} h\_[0,0]{}&=
0,a\_1,0,a\_2
, h\_[1,0]{}&=
0,b\_1,0,b\_2
\
h\_[0,1]{}&=
c\_1,0,c\_2,0
, h\_[1,1]{}&=
d\_1,0,d\_2,0
\
and $v_{x,y}={}^t h_{x,y}$ for all $x,y$, this last condition not being needed. The system can be rewritten as : \[eq:sys1\] \_2\^:={
[l]{} d\_1\^2+d\_2\^2=0, c\_1d\_1+c\_2d\_2=0, b\_1\^2+b\_2\^2-1=0,\
c\_1\^2+c\_2\^2-p=0, b\_1a\_1+b\_2a\_2-1=0, a\_1\^2+a\_2\^2-1+p=0.
. Clearly, $d_1=0, d_2=0$, and the rest of the system becomes $$\Sys^\bX:=\{b_1^2+b_2^2-1=0,~{c_{1}}^{2}+{c_{2}}^{2}-p=0,~{a_{1}}^{2}+{a_{2}}^{2}-1+p=0,~b_{1}a_{1}+b_{2}a_{2}-1=0\}.$$ Solutions to this system exist. We then take $V_{(0)}^x$, $H_{(0)}^x$ as in for example, and take $V_{(1)}^x$,$H_{(1)}^x$ as defined in . Again, $Q^x_{(1)}=Q^x_{(0)}$ is possible since $h_{x,y}$ and $v_{x,y}$ are respectively single line and single column (according to Lemma \[pro:dec\]). Of course, since $h_{y,x}v_{y',x}=T^\bX(y,y',x)$, we are back to the previous problem treated in Section \[sec:al\].
The appearance of non real numbers in the considerations ($\Sys^\bX$ has no real solution) does not harm the reasoning at all since $\bT^\bY$ is still a probability transfer matrix. The possibility to use non real matrices $V^x,H^x$ as a solution of the equations of interest enriches the space of solutions.
No finite solution for the gas of type 2
----------------------------------------
In the case of the gas of type 2, we were not able to find finite (non-trivial) matrices $(V^x,H^x,x\in\{0,1\})$ which solve . If they exist, they must have size $>5$ (this can be seen via the computation of a Gröbner basis). Also, numerical computations – which in principle does not guarantee any result – indicate that no non-trivial matrices $(V^x,H^x,x\in\{0,1\})$ with complex coefficients having size $\leq 8$ are solution. Hence we were unable to use Lemma \[lem:finite\] to go on.
\(a) There it exist finite square matrices $V^0,H^0,V^1,H^1$ which solve for $T=T^\bY$? (and such that $\sum_x V^xH^x$ has eigenvalue 1?)\
(b) If not, is it possible to find a solution to for $T=T^\bY$?
Toward an infinite size solution
================================
This section prospective explains how some infinite matrices may arise. Even if no complete solution is provided, we hope that this new point of view will allow some reader to tackle the problem of computing $\G(x,y)$.
Recall that Lemma \[pro:dec\] and Corollary \[cor:big-construction\] say that if there exist single line and single column matrices $h_{x,y}$ and $v_{x,y}$ solving $h_{x,y}v_{x',y'}=T^\bZ(x,x',y) 1_{y=y'}$, then on a cylinder of the square lattice the distribution on the $\kappa+1$’th line $\mu_{(\kappa)}:=\mu_{(\kappa-1)}\bT^\bZ$ has a representation of the form , provided that the distribution on the first line $\mu_{(0)}$ has this same form with $V_{(0)}^xH_{(0)}^y=0$ for $x\neq y$. Such matrices $h_{x,y}$ and $v_{x',y'}$ exist for $\bZ\in\{\bX,\bY\}$:
\[theo:mach\] For any $\bZ\in\{\bX,\bY\}$,\
(1) there exist single line matrices $(h_{x,y},x,y\in\{0,1\})$ and single column matrices $(v_{x,y},x,y\in\{0,1\})$ which solves and , namely $$\label{eq:master-equation}
h_{x,y}v_{x',y'}=T^\bZ(x,x',y) 1_{y=y'} \textrm{ for any } (x,y,x',y')\in\{0,1\}^4.$$ (2) For any $\kappa$, the entries of the matrices $V_{(\kappa)}^x,H_{(\kappa)}^x,Q_{(\kappa)}^x$ (as introduced in Corollary \[cor:big-construction\]) can be computed.\
(3) The solutions of and the initial matrices $V_{(0)}^x,H_{(0)}^x$ can be chosen in such a way that for any $x$, $V_{(2\kappa)}^x, H_{(2\kappa)}^x, Q_{(2\kappa)}^x$ converges simply when $\kappa\to +\infty$ (that is, each fixed entry converges).
We have already discussed the existence of solutions to $\Sys_2^\bX$, this implying the existence of solutions to in the case $\bZ=\bX$. To prove Theorem \[theo:mach\](1), let us write the following system of equations for $h_{x,y}$ and $v_{x,y}$ defined as in , in the case $\bZ=\bY$. In this case is equivalent to $$\label{eq:sys2}
\Sys_2^\bY:=\left\{
\begin{array}{l}
{c_1}^2+{c_2}^2-q+pq=0,~{c_1}^2+q{d_1}^2-q{c_1}^2=1, {d_1}^2+{d_2}^2-p+pq-q=0 \\
d_1c_1+d_2c_2-q+pq=0, ~{a_1}^2+{a_2}^2-1+q-pq=0,\\
~b_1a_1+b_2a_2-1+q-pq=0,~{b_1}^2+{b_2}^2-1+q+p-pq=0.
\end{array}\right.$$ It is not difficult to check that this system has a solution (using Maple or Mathematica, for example, or the computation of a Gröbner basis).
The proofs of Theorem \[theo:mach\] (2) and (3) are more delicate. Their respective proofs are the object of Sections \[v:const\] and \[seq:conv\] below.
**Notation. For any pair of matrices $(A^0,A^1)$, $A^\star:=A^0+A^1$. Similarly, for any doubly indexed quantity $a_{x,y}$, $a_{x,\star}=a_{x,0}+a_{x,1}$.**
Computations of the entries of $Q_{(k)}^x, V_{(k)}^x, H_{(k)}^x$ {#v:const}
----------------------------------------------------------------
Let $\bZ\in\{\bX,\bY\}$ be fixed, and let $h_{x,y}$ and $v_{x,y}$ as defined in be solution of $\Sys_2^\bZ$ (the sizes of $h$ and $v$ are $1\times m$ and $m\times 1$ respectively for some $m\geq 1$). Let us compute the entries of $\l((Q^x_{(\kappa)},V^x_{(\kappa)}, H^x_{(\kappa)}), \kappa \geq 1\r)$ starting from some matrices $V^1_{(0)}$, $H^1_{(0)}$, $V^0_{(0)}$ and $H^0_{(0)}$ such that $V_{(0)}^xH_{(0)}^{x'}=\matz$ for $x\neq x'$ of size $m_0\times m_0$ (for example, $V^1_{(0)}=H^1_{(0)}=[1], V^0_{(0)}=H^0_{(0)}=[0]$).
For $\kappa\geq 2$, by associativity of the Kronecker product, $$\label{eq:V}
V^x_{(\kappa)}=\sum_y \sum_z (V_{(\kappa-2)}^z\otimes v_{z,y}) \otimes h_{y,x} = \sum_z V_{(\kappa-2)}^z\otimes \bv_{z,x}$$ with $$\bv_{z,x}= \sum_y v_{z,y}\otimes h_{y,x},$$ which entails, iteratively that $$\label{eq:rew-W}
V_{(2\kappa)}^{z_\kappa}= \sum_{z_0,z_1,\dots,z_{\kappa-1}} V_{(0)}^{z_0} \otimes \bv_{z_0,z_1} \otimes \bv_{z_1,z_2}\dots\otimes \bv_{z_{\kappa-1},z_{\kappa}}.$$ (The matrices $\bv_{z,y}$ have size $m\times m$.) A similar formula exists for $H_{(2\kappa)}^x$, obtained by replacing $h$ by $v$ and vice versa in the previous considerations, leading to the definition of $\bh_{z,x}= \sum_y h_{z,y}\otimes v_{y,x}$.
In the same manner, $Q_{(\kappa)}^x=V_{(\kappa)}^x H_{(\kappa)}^x$ can be computed: write Q\_[()]{}\^x&=&ł(\_z V\_[(-2)]{}\^z\_[z,x]{})ł(\_[z’]{} H\_[(-2)]{}\^[z’]{}\_[z’,x]{}). Using the structure of $V_{(\kappa-2)}^z$ and $H_{(\kappa-2)}^{z'}$, we get Q\^[x]{}\_[()]{} &=& \_[w]{} Q\^w\_[(-2)]{} \_[w,x]{} where \_[w,x]{}&=&\_y \_zł(v\_[w,y]{}h\_[w,z]{}) ł(h\_[y,x]{}v\_[z,x]{}) =\_y \_z T\^\_[y,z,x]{}ł(v\_[w,y]{}h\_[w,z]{}) (we used here that $v_{a,b},h_{a,b}$ is a solution of $\Sys_2^\bZ$). Again, using the same methods above $$\label{eq:rew-Q}
Q_{(2\kappa)}^{z_\kappa}= \sum_{z_0,z_1,\dots,z_{\kappa-1}} Q_{(0)}^{z_0} \otimes \bq_{z_0,z_1} \otimes \bq_{z_1,z_2}\otimes\dots\otimes \bq_{z_{\kappa-1},z_{\kappa}}.$$ With these formulas, the entries of $V_{(2\kappa)}^x, H_{(2\kappa)}^x$ as well as that of $Q_{(2\kappa)}^x$ can be computed (and with a simple adaptation those of odd indices also). For this, recall that if $$A=B\otimes C$$ where $C$ is a $c\times c$ matrix, $B$ a $b\times b$ matrix then, for any $i,j \in \{0,\dots, bc-1\}$, \[eq:ABC-entries\] A\[[i,j]{}\]=B\[[i m, j m]{}\]C\[[i m, j m]{}\] with the convention that for any matrix $M$, $M[i,j]:=M_{i+1,j+1}$, and as usual $x \operatorname{div}m$ and $x \operatorname{mod}m$ denote the quotient and the remainder in the division of $x$ by $m$.
Assume now that some matrices $\bW_{(2\kappa)}^x,x\in\{0,1\}$, $\kappa\geq0$, satisfy \[eq:WW\] \_[(2)]{}\^[z\_]{}= \_[z\_0,z\_1,…,z\_[-1]{}]{} \_[(0)]{}\^[z\_0]{} \_[z\_0,z\_1]{} \_[z\_1,z\_2]{}…\_[z\_[-1]{},z\_]{} with $\bw_{x,y}$ having size $m\times m$, and $\bW_{(0)}^x$ having size $m_0\times m_0$. Therefore $\bW^x_{(2\kappa)}$ has size $m_0 \times m^\kappa$; any $i,j$ in $\{0,1,\dots,m_0 \times m^\kappa\}$ can be written under the following form : $$i=a_{\kappa+1}(i) m^\kappa+ \sum_{l=1}^\kappa m^{l-1}a_l(i),~~j=a_{\kappa+1}(j) m^\kappa + \sum_{l=1}^\kappa m^{l-1}a_l(j),$$ where $0\leq a_{\kappa+1}(i),a_{\kappa+1}(j) <m_0$, and $0\leq a_l(i),a_l(j) <m$ for $l \in \cro{1,\kappa}$ (apart from $a_{\kappa+1}$ which may play a special role if $m_0\neq m$, the $a_{l}(i)$’s are the digits of $i$ in base $m$). Therefore, from , \_[(2)]{}\^[z\_]{}\[i,j\]=\_[z\_0,…,z\_[-1]{}]{} \_[(0)]{}\^[z\_0]{}\[a\_[+1]{}(i),a\_[+1]{}(j)\] \_[z\_1,z\_2]{}\[a\_(i),a\_(j)\]…\_[z\_[-1]{},z\_]{}\[a\_1(i),a\_1(j)\]. There is also a way to represent this with matrices, very similar to the representation of the distribution of a Markov chain; for any $(a,b)$ in $\{0,\dots,m-1\}^2$, let $M_\bw(a,b)$ be the $2\times 2$ matrix defined by: $$M_\bw(a,b):=
\begin{bmatrix}
\bw_{0,0}[a,b]&\bw_{0,1}[a,b] \\
\bw_{1,0}[a,b]&\bw_{1,1}[a,b]
\end{bmatrix},$$ and for $a,b \in \{0,\dots,m_0-1\}$, $$\rho_{\bW}[a,b]:=
\begin{bmatrix}
\bW_{(0)}^0[a,b]&\bW_{(0)}^1[a,b]
\end{bmatrix}.$$ We have \[eq:V-kappa\] \_[(2)]{}\^z\[i,j\]=\_ M\_\[a\_(i),a\_(j)\]…M\_\[a\_[1]{}(i),a\_[1]{}(j)\]
1\_[z=0]{}\
1\_[z=1]{}
. This fact, together with and , proves Theorem \[theo:mach\](2).
Convergence of the entries of $V_{(2\kappa)}^x,H_{(2\kappa)}^x$, and $Q_{(2\kappa)}^x$ {#seq:conv}
--------------------------------------------------------------------------------------
We continue from the previous section. Notice that for a fixed $(i,j)$, $a_{l}(i)$ and all $a_l(j)$ are zero for large $l$. We then immediately have:
\[lem:conv1\] $\bW_{(2\kappa)}^z[i,j]$ converges when $\kappa\to +\infty$ in $\mathbb{C}$ for any $i,j$ in $\mathbb{N}$ if and only if $$\rho_\bW[0,0] M_\bw[0,0]^l$$ converges when $l$ goes to $+\infty$; a sufficient condition is the convergence of $M_\bw[0,0]^l$.
Again, the convergence stated in Lemma \[lem:conv1\] will be only interesting if the limit is not zero. We examine the simple convergence of $V_{(2\kappa)}^x, H_{(2\kappa)}^x$ and $Q_{(2\kappa)}^x$ when $\kappa\to+\infty$, to some infinite matrices $(V_{\infty}^x, H_{\infty}^x,Q_{\infty}^x)$, meaning that, for $x\in \{0,1\}$ and any $i,j\geq 0$, V\_[(2)]{}\^x\[i,j\]V\_\^x\[i,j\], H\_[(2)]{}\^x\[i,j\]H\_\^x\[i,j\] Q\_[(2)]{}\^x\[i,j\]Q\_\^x\[i,j\] starting with some suitable matrices $((V_{(0)}^x,H_{(0)}^x,Q_{(0)}^x),x\in\{0,1\})$. If such a convergence holds, the limiting infinite matrices $Q_{\infty}^1$ and $Q_{(\infty)}^0$ are moreover a solution of the following rewriting like system: \[eq:rew1\] x{0,1}, Q\^x\_ = \_[w=0]{}\^1 Q\^w\_ \_[w,x]{}. Rewriting rules such as rely entirely on the corners of the matrices $(Q^x_{\infty}[0,0],x\in\{0,1\})$. Writing $\rho_M[0,0]:=[M^0[0,0],M^1[0,0]]$, formula allows us to compute the only possible corners: \_[Q\_]{}&=&\_[Q\_]{}\_\
\_[V\_]{}&=&\_[V\_]{}\_\
\_[H\_]{}&=&\_[H\_]{}\_. We examine separately the two cases $\bZ=\bX$ and $\bZ=\bY$ in the next subsections.
### Case of the gas of type 1.
We work with $h$ defined in . We then find $$\begin{array}{ll}
M_\bv(0,0)=\begin{bmatrix}
0&c_1d_1 \\
0&d_1^2
\end{bmatrix}, &
M_\bq(0,0)=\begin{bmatrix}
c_1^2&0 \\
d_1^2&0
\end{bmatrix}\end{array}.$$ Therefore, the convergence of the sequence $(V_{(2\kappa)}^z[i,j],\kappa >0)$ to a non zero limit is equivalent to $d_1=1$, in which case, for any $l\geq 1$, $$\label{eq:Mcvv}M_\bv(0,0)^l= M_\bv(0,0)$$ and the convergence of $M_\bq(0,0)^l$ arises if $c_1^2=1$, in which case, for any $l\geq 1$, $$\label{eq:Mcvq}M_\bq(0,0)^l=M_\bq(0,0).$$ We may wonder if $\Sys_2^\bX$ still has some solutions if we add these conditions.\
The answer is yes for the condition $c_1=1$ (for example, $a_1^2=-p,a_2=1, b_1=0,b_2=1,c_1=1,c_2^2=p-1,d_1=d_2=0$). Then it is possible to have simple convergence for $Q_{(2\kappa)}^x$.\
The answer is no for the condition $d_1=1$. In order to find $h_{x,y},v_{x,y}$ satisfying (and $\bZ=\bX$) and such that $V_{(2\kappa)}$ simply converges, it suffices to increase the size of the matrices $h$ and $v$ defined in . Take instead \[eq:les-h2\]
[ccc]{} h\_[0,0]{}&=
0,a\_1,0,a\_2,0,a\_3
, h\_[1,0]{}&=
0,b\_1,0,b\_2,0,b\_3
\
h\_[0,1]{}&=
c\_1,0,c\_2,0,c\_3,0
, h\_[1,1]{}&=
d\_1,0,d\_2,0,d\_3,0
\
and again $v_{x,y}={}^t h_{x,y}$. The values of $M_\bq(0,0)$ and $M_\bv(0,0)$ are unchanged, but this time there are some solutions for and and where $d_1=1$. Again, this is not difficult to check with a program like Maple or Mathematica. Note also that if we want to solve $\Sys_2^\bX$ together with the two equations $c_1=1$ and $d_1=1$, there exists solutions for $h$ and $v$ having size 6. For example: $$a_1^2+p=0,a_2=0,a_3=1,b_1=0,b_2=0,b_3=1,c_1=1,c_2^2=p,c_3=i,d_1=1,d_2=0,d_3=i.$$ It remains to specify $\rho_\bq$ and $\rho_\bv$, namely the starting condition of the construction. We may take $m_0=1$ (that is, starting with $1\times 1$ matrices). Taking $Q^{(0)}=V^{(0)}=H^{(0)}=[0]$ and $Q^{(1)}=V^{(1)}=H^{(1)}=[1]$ leads to \[eq:start\] \_V=\_H=\_Q:=
0,1
. With this convention, by and , the upper-left corner of $Q_{(2\kappa)}^x$ coincides with $Q_{(2\kappa-2)}^x$, and the same thing hold for $V$ and $H$ as well. We then have $\rho_{Q_\infty}=\rho_{H_{\infty}}=\rho_{V_{\infty}}=\rho_{Q}$.
### Case of the gas of type 2.
First, for $h$ and $v$ defined in , $$M_\bq(0,0)=
\begin{bmatrix} ( -1+p) ( -1+q ) c_1^2& ( q+p-pq ) c_1^2\\
( -1+p ) ( -1+q ) {d_1}^2& ( q+p-pq ) d_1^2
\end{bmatrix},~~
M_\bv(0,0)=\begin{bmatrix} 0&d_1c_1\\ 0&{d_1}^2\end{bmatrix} .$$ The convergence of $M_\bq(0,0)^l$ to a non zero limit happens if $( p-pq+q ) {d_1}^2+ ( 1-q-p+pq ) c_1^2=1$, in which case $M_\bq(0,0)^l= M_\bq(0,0)$ for any $l\geq1$, and the convergence of $M_\bv(0,0)^l$ to a non zero limit arises if $d_1=1$, in which case $M_\bv(0,0)^l= \begin{bmatrix} 0&c_1\\ 0&1
\end{bmatrix} $ for $l\geq 1$. In this case, solutions $h_{x,y}$ of size $1\times 4$ exist : there exists solutions to $\Sys_2^\bY$ with the additional condition $( p-pq+q ) {d_1}^2+ ( 1-q-p+pq ) c_1^2=1$ or $d_1=1$. Again, if if we want to solve $\Sys_2^\bY$ with both conditions together there exist solutions for $h$ and $v$ having size 6, for example: \^2&=& (1-q)(1-p), [d\_2]{}\^2=-(1-q)(1-p),\
[a\_1]{}\^2 & =& - ,a\_2=0,a\_3=,b\_1=0,b\_2=0,c\_1=1,\
c\_2&=&-, c\_3\^2= ,d\_1=1,d\_3=0. This suffices to imply the simple convergence of $Q_{(2\kappa)}^x, V_{(2\kappa)}^x, H_{(2\kappa)}^x$. Here $\rho_{Q_{\infty}}=[(-1+p)(-1+q),p-pq+q]$, $\rho_{V_{\infty}}=[0,1]$ and $\rho_{H_{\infty}}=[0,1]$.\
This ends the proof of Theorem \[theo:mach\](3).
Trace of the limit and limit of the trace
-----------------------------------------
As said above, Theorem \[theo:mach\] gives a representation of $\mu_{(\kappa)}^\bZ$ starting from some simple $\mu_{(0)}^\bZ$ (this could be useful to make advances on enumeration issues concerning DA with height $\kappa$). The important and natural question is the following one : do we have, for any $\bx=(x_1,\dots,x_k)$, \_[(2)]{}()=ł(\_[i=0]{}\^[n-1]{}Q\_[(2)]{}\^[x\_i]{})\_()=ł(\_[i=0]{}\^[n-1]{}Q\_\^[x\_i]{}) ? Since the simple convergence of $Q_{(2\kappa)}^x$ to $Q_{\infty}^x$ does not imply the simple convergence of $Q_{(2\kappa)}^xQ_{(2\kappa)}^y$ to $Q_{\infty}^x Q_{\infty}^y$, the answer to this question is certainly not an immediate issue. For the TASEP, the choices of matrices $D,E$ satisfying the different matrix ansatz, leads or not to the convergence of the product (see discussions in [@DEHP; @BE]).
Moreover, simple convergence of a sequence of matrices $A_n$ to some matrice $A_{\infty}$ does imply the convergence of the trace, since the trace involves an infinite number of entries. Nevertheless, $\mu_{(2\kappa)}$ converges when $\kappa$ goes to +$\infty$ by Lemma \[lem:MC\]. Let $\mu^\infty(\bx)=\lim_\kappa \mu_{(2\kappa)}(\bx)$ the limit of the measure. The question is: do we have $\mu^\infty=\mu_\infty ?$ Since $\mu^\infty$ is the only non trivial probability measure fixed point of $\mu^\infty= \mu^\infty\bT^\bY$, it suffices to show that $\mu_\infty$ satisfies the same property, which would imply $\mu_\infty=\lambda\mu^\infty$, for some $\lambda$ (which must be shown to be $\neq 0$). In fact, by construction $q_{y,x}$ is associated with two-row transitions, since $\operatorname{Trace}(\prod_{i=1}^n q_{y_i,x_i})=\sum_{\bz \in E_n} \prod T^\bY_{y_i,y_{i\oplus 1},z_i}T^\bY_{z_i,z_{i\oplus1},x_i}=(\bT^\bY)^2(\bx,\by)$. Clearly $(\bT^\bY)^2$ is also a probability transfer matrix, corresponding to an aperiodic irreducible Markov chain on a finite state space. Subsequently, by uniqueness, it is easy to check that $\mu_\infty$ is the unique solution to $\mu = \mu(\bT^\bY)^2$.
Using that $Q_{\infty}$ is solution of the rewriting system , if one ignores convergence and commutation issues, then \[eq:mult-fix-point\] ł(\_[i=0]{}\^[n-1]{}Q\_\^[x\_i]{})&=&\_[E\_n]{} ł(\_[i=0]{}\^[n-1]{}Q\_\^[y\_i]{})ł(\_[i=0]{}\^[n-1]{} q\_[y\_i,x\_i]{})\
&=& \_[E\_n]{} ł(\_[i=0]{}\^[n-1]{}Q\_\^[y\_i]{})(\^)\^2(,), and one sees that we just have to justify convergence of $\prod_{i=0}^{n-1}Q_{\infty}^{x_i}$ and the validity of the rearrangements in the infinite sum.
We were unable to prove the validity of this. Besides, the entries of $Q^y_\infty$ seems no to converge to 0 (due to the choice of the value of $M_\bq(0,0)$, needed to have the convergence of $Q_{(\kappa)}^x$ to $Q_\infty^x$); also, seen as series in $p,q$, the degrees of the entries $Q^\star_{i,j}$ do not go to $+\infty$ with $i,j\to \infty$.
Is it possible to prove that $\operatorname{Trace}\l(\prod_{i=0}^{n-1}Q_{\infty}^{x_i}\r)$ is well defined (for some notion of convergence) and solution of ?
We here review some properties of the matrices $V_{(\kappa)}^x,H_{(\kappa)}^x$ and of $\mu_{(2\kappa)}$. These properties lead to some questions about the structure of $Q_\infty^x$, and its eigenvectors (if any). First, we have \_[()]{}()&=&ł(\_[i=0]{}\^[n-1]{} V\^[x\_i]{}\_[()]{}H\^[x\_i]{}\_[()]{}) =ł(\_[i=0]{}\^[n-1]{} H\^[x\_i]{}\_[()]{}V\^[x\_[i1]{}]{}\_[()]{}). This representation is very close to the standard representation of Markov chain, where here $ H^{a}_{(\kappa)}V^{b}_{(\kappa)}$ plays the role of a probability transition and $$\bP_{(\kappa)}:=\sum_{a,b} H^{a}_{(\kappa)}V^{b}_{(\kappa)}$$ plays the role of the transition matrix.
For any $\kappa$, $Q_{(\kappa)}^\star$ has eigenvalues 1 and 0. The eigenvalue 1 has multiplicity 1.
We first claim that $(Q_{(2\kappa)}^\star)^{2\kappa}(1-Q_{(2\kappa)}^\star)=(Q_{(2\kappa+1)}^\star)^{2\kappa}(1-Q_{(2\kappa+1)}^\star)=\matz$. The claim implies that the minimal polynomial of $Q_{(2\kappa+`e)}^\star$ (with $`e\in\{0,1\}$) divides $x^{2\kappa}(1-x)$, which implies that the eigenvalues are 0 and 1. Since $\operatorname{Trace}(Q^\star_{(\kappa)})=\operatorname{Trace}(Q^\star_{(\kappa-2)})$, and since $\operatorname{Trace}(Q_{(0)}^\star)=\operatorname{Trace}(Q_{(1)}^\star)=1$, the eigenvalue 1 has multiplicity 1. It remains to show the claim. For this write Q\_[()]{}\^&=&V\_[()]{}\^H\_[()]{}\^=ł(\_[y,x]{}H\_[\_[(-1)]{}]{}\^[y]{}h\_[y,x]{})ł(\_[y’,x’]{}V\_[\_[(-1)]{}]{}\^[y’]{}v\_[y’,x’]{})\
&=& \_[y,y’]{}H\_[\_[(-1)]{}]{}\^[y]{}V\_[\_[(-1)]{}]{}\^[y’]{} \_[x]{}T\_[y,y’,x]{} and since this last sum is 1, we have \[eq:QP\] Q\_[()]{}\^=\_[(-1)]{}. Also, using , \_[()]{}&=&\_y Q\^y\_[(-1)]{}\_y where $\rho_y=v_{y,\star}h_{y,\star}$. It turns out, that $h_{y,\star}v_{x,\star}=1$ for all $x$ and $y$; hence, any product of the form $v_{y_1,\star}h_{x_1,\star}v_{y_2,\star}h_{y_2,\star}\dots,v_{y_n,\star}h_{y_n,\star}$ equals $v_{y_1,\star}h_{y_n,\star}$. Henceforth, $\kappa\geq 0$, and $m\geq 0$, $$\l(Q_{(\kappa+2)}^\star\r)^{m+2}=\bP_{(\kappa+1)}^{m+2}=\sum_{x,x'} \l[Q_{\kappa}^x(Q_{\kappa}^\star)^{m}Q_{\kappa}^{x'}\r]\otimes v_{x,\star}h_{x',\star}.$$ Hence, if for some $\kappa\geq 0$, and $m\geq 0$, $(Q_{(\kappa)}^\star)^{m}=(Q_{(\kappa)}^\star)^{m-1}$ then $(Q_{(\kappa+2)}^\star)^{m+2}=(Q_{(\kappa+2)}^\star)^{m+1}.$ The initial conditions being $Q_{(0)}^\star=\begin{bmatrix} 1\end{bmatrix}$ (and $Q_{(1)}^\star=\begin{bmatrix} 1\end{bmatrix}$) we get $(Q_{(0)}^\star)^1=(Q_{(0)}^\star)^0$, and then $(Q_{(2\kappa)}^\star)^{2\kappa+1}=(Q_{(2\kappa)}^\star)^{2\kappa}$ (and $(Q_{(2\kappa+1)}^\star)^{2\kappa+1}=(Q_{(2\kappa+1)}^\star)^{2\kappa}$ as well). $\Box$
Denote by $L_{(\kappa)}$ and $R_{(\kappa)}$ the left and right eigenvectors of $Q_{(\kappa)}^\star$ associated with the eigenvalue 1. Since $(Q_{(\kappa)}^\star)^{m}$ converges to $R_{(\kappa)}L_{(\kappa)}$ when $m\to+\infty$, and since $(Q_{(\kappa)}^\star)^{m+1}=(Q_{(\kappa)}^\star)^{m}$, for $m$ large enough, \[eq:le\] (Q\_\^)\^[m]{}=R\_[()]{}L\_[()]{}. Moreover $L_{(\kappa)}$ and $R_{(\kappa)}$ can be normalised such that $L_{(\kappa)}R_{(\kappa)}=1$. Notice in the equality; in similar situations only convergence holds. For $m$ large enough the quantity of interest $$\operatorname{Trace}(Q_{(\kappa)}^1(Q_{(\kappa)}^\star)^{m})=\operatorname{Trace}(Q_{(\kappa)}^1R_{(\kappa)}L_{(\kappa)}).$$
Using $$\label{eq:122}
(Q_{(\kappa)}^\star)^m=\bP_{(\kappa-1)}^m=H_{(\kappa-1)}^\star (Q_{(\kappa-1)}^\star)^{m-1}V_{(\kappa-1)}^\star,$$ which leads to R\_[()]{}L\_[()]{}&=&V\_[()]{}\^\_[[(+1)]{}]{}[L]{}\_[[(+1)]{}]{}H\_[()]{}\^,\
[R]{}\_[()]{}[L]{}\_[()]{}&=&H\_[(-1)]{}\^R\_[(-1)]{}L\_[(-1)]{}V\_[(-1)]{}\^, and then (since all matrices have rank 1 and $L_{(\kappa)}={}^tR_{(\kappa)}$) \[eq:link\] [L]{}\_[()]{}&=&H\_[(-1)]{}\^L\_[(-1)]{},\
&=&H\_[(-1)]{}\^H\_[(-2)]{}\^…H\_[(0)]{}\^. Since $\operatorname{Trace}(Q_{(\kappa)}^1R_{(\kappa)}L_{(\kappa)})=\operatorname{Trace}(H_{(\kappa)}^1R_{(\kappa)} L_{(\kappa)}V_{(\kappa)}^1)$ we have also $$H_{(\kappa)}^1L_{(\kappa)}=H_{(\kappa)}^1H_{(\kappa-1)}^\star\dots H_{(0)}^\star$$ a triangular product whose computation seems to be quite difficult. We may also note the following
Let $d^{(n)}(1)=`P(Y^{\Sq(n)}_v=1)$, the density of the gas process of type 2 (this is $\G^{\Sq(n)}(x,y))$ up to change of variables, by Proposition \[pro:fo\]). For $n$ large enough $$d^{(n)}(1)= \frac{\sum_{i\geq 0} L_{(\kappa)}[1,2i+1]R_{(\kappa)}[{2i+1,1}]}{\sum_{j\geq 0} L_{(\kappa)}[{1,j}]R_{(\kappa)}[{j,1}]}.$$
For short, we don’t write the indices ${(\kappa)}$. For $n$ large enough (by ) we have $(Q^\star)^{n}= RL$. Hence, $$d^{(n)}(1)=\operatorname{Trace}(Q_1(Q^\star)^{n-1})/\operatorname{Trace}((Q^\star)^{n})=\operatorname{Trace}(LV^1 H^1R)/\operatorname{Trace}(LR).$$ Introduce $L(1)=\begin{bmatrix}L[1,i] \1_{i \mod 2=1} \end{bmatrix}$ and $R(1)=\begin{bmatrix}R[{i,1}] \1_{i \mod 2=1} \end{bmatrix}$, the vectors $L$ and $R$, where the even entries are sent to 0. Now, clearly $LV^1=(LV^\star)(1)$ and $H^1 R=(H^\star R)(1)$, then $(LV^\star)(1)=L(1)$ and $(H^\star R)(1)=R(1)$. $\Box$
Let us come back to the matrices $Q_{\infty}^1$ and $Q_{(\infty)}^0$ solution of the rewriting like system . Again, the value of the corner of $Q^x[0,0]$ is given by $\rho_Q[0,0]\bM_{\bq}[0,0]\begin{bmatrix} 1_{x=0}\\ 1_{x=1}\end{bmatrix}$.
The matrix $Q^\star_{\infty}$ has eigenvalue 1 with multiplicity 1. Let $L^{\infty}$ and $R^{\infty}$ be the left and right eigenvector, such that $L^{\infty}={}^{t}R^{\infty}$. We have $$\label{eq:inf}
`P(Y_x=1)=\sum_i L^{\infty}[1,2i+1]R^{\infty}[1,2i+1]/ \sum_j L^{\infty}[{1,j}]R^{\infty}[{j,1}].$$
We have seen that $Q_{(\kappa)}^x\sous{\longrightarrow}{\kappa\to +\infty} Q_{(\infty)}^x$ (simply) and $Q_{(\kappa)}^\star$ has a unique eigenvalue 1, the other ones being 0. This convergence is not sufficient to deduce that the infinite matrix $Q^\star_{\infty}$ has eigenvalue 1, and even if it is the case, the convergence of the numerator and denominator in may not converge. In the general case, for 4 given matrices $(q_{x,y}, (x,y)\in\{0,1\}^2)$, the same question arises2: can we find the eigenvectors (and eigenvalues) of the matrix $Q^\star_{\infty}$ that solves the rewriting systems .
Is it possible to find solutions to with infinite matrices $V^1, H^1, V^0, H^0$, in the case $T=T^\bY$ and such that moreover any product of the form $\prod_{i=1}^n V^{x_i}H^{x_i}$ converges ?
Other similar but different considerations
==========================================
We present some alternatives to Lemma \[pro:dec\]. Even if morally what is done has more of less the same taste as this Lemma, we were not able to reduce the following considerations to it.
Research of a solution on the zigzag {#sec:zz}
------------------------------------
We discussed above the construction of an invariant measure relying on the research of a measure of the type $\operatorname{Trace}(\prod_{i=1}^k Q^{x_i})$ on the rows of the cylinder. Two closely related constructions can be proposed. The first one is quite close to that discussed in [@BM1] around the question of “Markovian Field”.
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The idea is to search for an invariant measure on a zigzag on the cylinder having the following forms (idea used with success by Dhar): for any $(\bx,\by)\in E_n^2$, \[eq:formprod\] (,)= c\_n \_[i=1]{}\^n d\_[x\_i,y\_i]{}u\_[y\_i,x\_[i1]{}]{}, for some complex numbers $(d_{a,b},u_{a,b},a,b \in\{0,1\})$, where “$d$” is chosen for “down”, and $u$ for “up”. But such a measure has a cyclic Markovian structure on the lines since for $m_{a,b}=\sum_c d_{a,c}u_{c,b}$ and $\tilde m_{a,b}=\sum_c u_{a,c}d_{c,b}$ the induced law on the line above (up) is $\mu^{u}$ when the measure below (down) is $\mu^d$ with $\mu^u(\bx)=c_n\prod m_{x_i,x_{i \oplus 1}}$ and $\mu^d(\by)=c_n\prod \tilde{m}_{y_i,y_{i \oplus 1}}$. Since $`P(Y_i=y_i | X_i=x_i,X_{i\oplus 1}=x_{i\oplus 1})=T_{x_i,x_{i\oplus 1},y_i}$ a sufficient condition for $\mu^u=\mu^d$ and $\mu^d= \mu^u \bT$ is that for any $a,b,c,x,y$ \[eq:two-equ\] {
[l]{} [d\_[a,c]{}u\_[c,b]{}]{}/[m\_[a,b]{}]{}=T\_[a,b,c]{},\
m\_[x,y]{}= m\_[x,y]{}.
. Notice that condition can be weaken a bit since $\mu^u=\mu^d$ does not imply $\tilde m_{x,y} = m_{x,y}$, but rather $\tilde m_{a,b}=w_{a,b} {m}_{a,b}$ for $w_{a,b}$ such that, for any $\bx \in E_n$ (or sufficiently in the support of $\mu^a$), \[eq:prod1\] \_[i=1]{}\^n w\_[x\_i,x\_[i1]{}]{}=1. For measures with support $E_n$, letting $N_{a,b}(\bx)=\#\{i\in\{0,\dots,n-1\}: (x_i,x_{i\oplus1})=(a,b)\}$, rewrites $$\prod_{(a,b)\in \{0,1\}^2}w_{a,b}^{N_{a,b}(\bx)}=1.$$ Since on $E_n$, $N_{1,0}=N_{0,1}$ and $N_{1,0}+N_{1,1}=\#\{i:x_i=1\}=n-\#\{i:x_i=0\}=n-N_{0,1}-N_{0,0}$, and$N_{1,1}=n-2N_{1,0}-N_{0,0}$, this condition is fulfilled if $w_{0,0}=w_{1,1}=1$, $w_{0,1}w_{1,0}=1$, giving us one degree of freedom. If we deal with measures on $\{0,1,2,\dots, \kappa\}$, it suffices that the product on all finite cycles $w_{x_1,x_2}\dots w_{x_{l-1}x_1}$ equal 1 for $l\leq k$. This provides also some degrees of freedom. Hence, the existence of a product form as is equivalent to the existence of solutions to the following system of equations \[eq:two-equ2\] {
[l]{} =T\_[a,b,c]{},\
m\_[x,y]{}w\_[x,y]{}= m\_[x,y]{}.\
w\_[0,0]{}=w\_[1,1]{}=1, w\_[0,1]{}w\_[1,0]{}=1
. This is a finite algebraic system, and solutions can be found using the computation of a Gröbner basis. Again, in order to avoid trivial solutions, an additional equation has to be added: letting $M:=(m_{x,y})_{(x,y)\in\{0,1\}^2}$, the equation $\operatorname{Trace}(M^n)=\mu(E_n)$ let us sees that the existence of a non zero eigenvalue for $M$ is necessary and sufficient for non triviality: \[eref:CNT\] CNT\_2: M. The computation of the Gröbner basis of the set of polynomials corresponding to the equations gives all transitions $T$ for which solution exists (of course, as usual $T^\bY$ is not in this set).\
One can go further searching for a matrix type solution. It suffices to mix with the matrix considerations of the previous subsection. We then search matrices $\l(D_{a,b},U_{a,b},\textrm{ for }(a,b)\in \{0,1\}^2\r)$ of size say $k\times k$ such that for $m_{a,b}=\sum_c D_{a,c}U_{c,b}$ and $\tilde m_{a,b}=\sum_c U_{a,c}D_{c,b}$, we have \[eq:two-equ-Mat\] {
[l]{} D\_[a,c]{}U\_[c,b]{}=T\_[a,b,c]{}m\_[a,b]{},\
P m\_[x,y]{}= w\_[x,y]{}m\_[x,y]{} P
. for some $k\times k$ invertible matrix $P$, $(w_{x,y},x,y\in\{0,1\})$ solution of . Again some non-degeneracy conditions must be added: define $K:=(m_{x,y})_{(x,y)\in\{0,1\}}$ (defined by block, and having size $2k \times 2k$). The needed condition is $\operatorname{Trace}(M^n)\neq 0$, for $n$ large. By triangulation of $M$, it appears clearly that a necessary and sufficient condition is that $M$ has a non zero eigenvalue, which amounts to imposing $CNT_2$.
We were unable to show existence/non-existence of such matrices $(P,D,U)$ and weights $(w_{x,y})$ solution of + $CNT_2$ in the case $T=T^\bY$ for $m\geq 3$.
Research of a solution by projection
------------------------------------
Recall a simple fact : if $(X_i,i\geq 0)$ is a Markov chain with state spaces $S$ (with $\#S>2$), and if $\phi:S\to S'$, then in general $(\phi(Z_i),i\geq 0)$ is not a Markov chain. Here, one may search if the process $\bY$ (on $E_n$), gas of type 2, whose measure $\mu$ is solution of $\mu^\bY= \mu^\bY\bT^\bY$, can be written $Y_i=\phi(Z_i)$ for some function $\phi$, and some process $(Z_i,1\leq i\leq n)$ taking its values in a state space $S$ with $|S|>2$, and having a simple representation. We were not able to find such a solution.
Extensions {#sec:extensions}
==========
In this section we discuss some extensions of the method we developed above: first to the triangular lattice, second to processes with more than 2 states.
Triangular lattice {#sec:tri-lat}
------------------
We proceed as in Section \[sec:zz\] where a zigzag is considered. Since no new result are provided for DA on the triangular lattice, we just explain how the previous method can be adapted here. First the two probabilistic local transitions needed for the definition of the gases of type 1 and 2 on the triangular lattice are T\^\_[abc,d]{}&=& ‘P(B\_p(1-a)(1-b)(1-c)=d)\
T\^\_[abc,d]{}&=& ‘P(B\_pabc+(1-B\_p)B\_q=d). Again Proposition \[pro:fo\] says that the density of the corresponding gas provides up to a change of variables $\G(x,1)$ and $\G(x,y)$, the area, and area-perimeter GF of DA on this lattice.
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As explained in Section \[sec:zz\], one searches for a representation of the zigzag process distribution as follows: \[eq:dec-mes-zz\] \_[u\_1,d\_1,u\_2,…,u\_n,d\_n]{}=ł(D\^[u\_1d\_1]{}…U\^[d\_nu\_1]{}) for some matrices $D^{u,d}$ and $U^{d,u}$ for $u,d\in\{0,1\}^2$ (see Figure \[fig:tri\] to see the respective positions of the $u_i$’s and $d_i$’s). To find some finite matrices doing the job, it it sufficient to find matrices $D^{u,d}$ and $U^{d,u}$ for $u,d\in\{0,1\}^2$ solving the following system: \[eq:tri-fi\] {
[l]{} D\^[ab]{} U\^[|[b]{}c]{}=0,U\^[ab]{}D\^[|[b]{}c]{}=0, a,b,c {0,1},\
D\^[df]{} U\^[fd’]{}= \_u U\^[du]{}D\^[ud’]{}T\_[dud’,f]{} d,f,d’ {0,1}.
. For $T=T^\bX$ there is a solution with “matrices” $D^{x,y},U^{x,y}$ of size 2, for example $D^{x,y}=U^{x,y}$ for any $x,y$, $D^{1,1}=\matz, D^{1,0}=\begin{bmatrix} -1 & -r \\ -(rp+1+2p)/p & 1 \end{bmatrix}$, $ D^{0,1}=\begin{bmatrix} -rp & rp \\ -rp & rp \end{bmatrix}$, $D^{0,0}=\begin{bmatrix} 1 & r \\ 1 & r \end{bmatrix}$, where $r$ satisfies $p+(1-2p)r+r^2p=0$. This leads to the searched density (similar approach are present in [@Al; @BM1]).
Again no such chance arises for $T^\bY$. To adapt the construction of growing matrices as explained in Corollary \[cor:big-construction\], some single line and single column matrices $h_{a,b,c}, v_{a,b,c}$ indexed by $(a,b,c)\in\{0,1\}^3$ which solves the following system must be found: \[eq:tri-lat\] h\_[dxz]{}v\_[xd’z’]{}=T\_[dxd’,z]{} 1\_[z=z’]{}. The idea then is to grow the matrices $D_{(\kappa)},U_{(\kappa)}$ as follows: $$\l\{\begin{array}{l}
D^{dz}_{(\kappa+1)}=\sum_{x}U_{(\kappa)}^{dx}\otimes h_{dxz}\\
U^{zd'}_{(\kappa+1)}=\sum_{x'}D_{(\kappa)}^{x'd'}\otimes v_{x'd'z}
\end{array}\r.$$ Under this condition if $\mu_{(\kappa)}$ has a representation as that given in with some matrices $D_{(\kappa)},U_{(\kappa)}$ (instead of $D, U$) such that $D_{(\kappa)}^{xy}U_{(\kappa)}^{y',z}=U_{(\kappa)}^{xy}D_{(\kappa)}^{y',z}=\matz$ if $y\neq y'$, then this property is inherited for $D_{(\kappa+1)}, U_{(\kappa+1)}$. On the zigzag below (as drawn on Figure \[fig:tri\]), then measure $\mu_{(\kappa+1)}$ will also be given as in , with $D_{(\kappa+1)},U_{(\kappa+1)}$ instead of $D_{(\kappa)},U_{(\kappa)}$. Again, solutions to the system exist: this permits one to grow some matrices $D_{(\kappa)},U_{(\kappa)}$, and then to have a representation of the measure $\mu_{(\kappa)}$. Infinite matrices $U_{(\infty)}^{ab}$ and $D_{(\infty)}^{ab}$ appear again by some passage to the limit. We were not able to deduce from them $\G$.
Bond percolation
----------------
Let $A$ be a DA on $G=(V,E)$. A bond in $A$ is an edge $e\in E$ between elements of $A$; let $N(A)$ be the number of bonds in $A$. The GF $B_{C}^G(x,y)=\sum_{A} x^{|A|} y^{\#N(A)}$ of DA with source $C$ counted according to the area and number of bonds can be obtained also by computing the density of some gas process (this is explained in [@BM1], page 21, in the case $G=\Sq(n)$). Indeed, for any $G\in{\cal G}$, \[eq:bp\] B\_C\^G=x\^[|C|]{}\_[D(C)]{}B\_[D]{}\^G y\^[CD]{} where $C\to D$ is the number of bonds between $C$ and $D$. For any cell $d\in \Ch(C)$, let $S_C(d)=\{(c,d)\in E~| c\in C\}$ be the set of bonds from $C$ to $d$. Further, denote by $D_{C}(i)=\{d \in D~|~|S_C(d)|= i\}$ be the subset of $D$ of cells being extremities of $i$ edges coming from $C$. We have \[eq:bp-2\] B\_C\^G=x\^[|C|]{}\_[D(C)]{} B\_[D]{}\^G y\^[i |D\_C(i)|]{}. We will define a gas whose density will coincide up to a change of variables to $B_C$. For this, associate with the set of vertices $V$ i.i.d. Bernoulli$(p)$ random variables (denoted $(B_p^x,x\in V)$), and with the edges of $E$, i.i.d. Bernoulli$(q)$ random variables (denoted $(B_q^x,x \in E)$. Consider now the gas defined by \[eq:tran\] X\_x= B\_p\^x\_[d(x)]{} 1- ł(X\_d \_[(x,d)E]{}B\_q\^[(x,d)]{}). Taking the expectation in the previous line, leads to ‘P(X\_x=1, xC) &=&p\^[|C|]{}\_[D(C)]{} ‘P(X\_x=1,xD)(-1)\^[|D|]{} q\^[i |D\_C(i)|]{}. Set now $H_C(p,q)=(-1)^{|C|}`P(X_x=1, x\in C)$. The last equation rewrites $$H_C(p,q)=(-p)^{|C|}\sum_{D\subset \Ch(C)} H_D(p,q) q^{\sum i |D_C(i)|}$$ in other words, the family $(H_C(-p,q),C)$ satisfies the same equations as the family $(B_C(p,q),C)$, the initial conditions being $B_{\varnothing}=1$ and $H_{\varnothing}=1$. On the square lattice, set $$T^\bB_{x,y,z}=`P(X=z | X_1=x,X_2=y)=`P(z= B_p ( 1- x B_q^{(1)}) ( 1- y B_q^{(2)}).$$ Again, there exist solutions to the equations $h_{x,y}v_{x',y'}=1_{y=y'}T^\bB_{x,x',y}$, and we may also impose that $M_\bq(0,0)^l$ converges. This permits again to use Corollary \[cor:big-construction\] to represent $\mu_{(\kappa)}$.
Bicolouration {#seq:bic}
-------------
We present a new model of gas $X$ taking three values 1, 2 or 3 (we see the value as a colour), having an interest from a combinatorial point of view, and illustrating the universality of the present approach. Let $G=(V,E)$ be in ${\cal G}$, and let $(C_x,x\in V)$ be a i.i.d. random colouring of the vertices of $G$, such that $`P(C=i)=p_i$ for $i\leq 2$, and $\sum p_i=1$. We set $$\label{eq:3coul}
X_x:= C_x \prod_{c\in \Ch(x)} 1_{X_c\neq C_x}.$$ The arguments given in Lemma \[lem:welldefi\] implies that $X$ is a.s. well defined for $p_1+p_2$ small enough. According to , if $C_x=0$ then $X_x=0$ (with proba. 1), and if $C_x=i$ for $i\in\{1,2\}$, then $X_x=i$ with proba. $p_i$ if no child of $x$ has colour $i$, and $X_x=0$ in the other cases.
Clearly $`P(X_1=1)=-\G^{G}(-p_1,1)$ and $`P(X=2)=-\G^G(-p_2,1)$ since the gases $Y=1_{X=1}X$ and $Y'=1_{X=2}X/2$ have the same transitions as the gas of type 1 with parameters $p_1$ and $p_2$ (and by Lemma \[pro:fo\]).
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This gas is related to the counting of bi-coloured DA, having no bicoulor neighbouring sites (this model is interesting only when several sources are involved).
Formally consider two sets $S_1:=\{c_1,\dots,c_k\}$ and $S_2:=\{d_1,\dots,d_l\}$ such that $S_1\cap S_2=\varnothing$ and such that $S_1\cup S_2$ is a free set. We call bicoloured DA a pair $(A,l)$, where $l:A\to\{1,2\}$ (where $l(a)$ is seen as the colour of $a$). A bicoloured DA is said to be well coloured, if for any $(a,b)\in A^2\cap E$, $l(a)=l(b)$, meaning that neighbouring sites have the same colour.
Denote by ${\cal A}_{S_1,S_2}$ the set of well coloured DA $(A,l)$ with source $S_1 \cup S_2$ such that $l(S_i)=\{i\}$ (the cells of the sources $S_i$ have colour $i$). This model is hard to deal with using heap of pieces arguments. Let $\G_{S_1,S_2}$ be the corresponding GF, counted well coloured DA according to their number of cells of each colour.
We have \[eq:bi-col\] \_[S\_1,S\_2]{}(-p\_1,-p\_2)=(-1)\^[\#S\_1+\#S\_2]{}‘P(X\_x=1, xS\_1, X\_x=2,xS\_2).
Here again, the gas transition $T(a,b,c)=`P(C_x = c ~| C_{c_1}=a, C_{c_2}=b)$ can be written $T(a,b,c)=h_{a,c}v_{b,c'}1_{c=c'}$ for some monoline and monocolumn $h$ and $v$, for any $a,b,c,c'\in\{0,1,2\}$. Again, this ensures the existence of a representation of the measure on the $\kappa$th line of the lattice using some matrices $V^x_{(\kappa)},H^x_{(\kappa)},Q^x_{(\kappa)}$, for $x\in\{0,1,2\}$, starting for some measure $\mu_{(0)}(\bx)=\operatorname{Trace}(\prod_{i=0}^n V^{x_i}_{(0)}H^{x_i}_{(0)})$ and some matrices $V^x_{(0)},H^x_{(0)}$, such that $V^x_{(0)}H^y_{(0)}=\matz$ for $x\neq y$.
Here the case is particularly interesting: $`P(X_x=j)$ for $j\in\{0,1,2\}$ as well as $`P(X_x=j, x\in C)$ are easy to compute: the reason is that the projection $Y$ and $Y'$ (as defined above) are well known, and have a simple product form: they correspond in the first case to identify the states $2$ and $0$ and in the second one to identify $0$ and $1$. Hence, both $Y$ and $Y'$ are Markovian on a line of the lattice (and hard particle model on the zigzag), but $X$ is not Markovian on the lines (or on the zigzag). The combinatorial issue is not to find the density of $X$, but rather to compute a quantity as $`P(X_0=1,X_1=2)$. At this moment, I am not able to do this.
**Thanks : Grateful thanks to David Renault for numerous stimulating discussions about this work. Many thanks are due to the anonymous referees for their comments.**
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abstract: 'Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees of freedom we establish that the non-Hermitian extensions of higher-order topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian phenomena in the absence thereof.'
author:
- 'Elisabet Edvardsson, Flore K. Kunst, and Emil J. Bergholtz'
title: |
Non-Hermitian extensions of higher-order topological phases\
and their biorthogonal bulk-boundary correspondence
---
*Introduction.* Topological phases of matter are at the forefront of condensed-matter research with a recent focus on higher-order topological phases [@benalcazarbernevighughes; @langhehnpentrifuoppenbrouwer; @linhughes; @schindlercookvergnio; @parameswaranwan; @imhofbergerbayerbrehmmolenkampkiesslingschindlerleegreiterneupertthomale; @ashraf; @benalcazarbernevighughesagain; @garciaperissstrunkbilallarsenvillanuevahuber; @petersonbenalcazarhughesbahl; @songfangfang; @ezawapap; @KuMiBe2018], where a subtle interplay between topology and crystalline symmetry results in the appearance of boundary states on boundaries with a codimension higher than one, i.e., corners or hinges. Another increasingly popular direction of research revolves around studying topology in the context of non-Hermitian physics, which is a relevant approach for describing a wide range of dissipative systems [@MaAlVaVaBeFoTo2018; @HatanoNelson; @Xi2018; @KuEdBuBe2018; @KuDw2018; @lee; @yaosongwang; @yaowang; @leethomale; @schomerus; @gong; @carlstroembergholtz; @jan; @koziifu; @knots; @knots2; @yoshidapeterskawakmi; @NHarc; @wiemannkremerplotniklumernoltemakrissegevrechtsmanszameit; @NHtransition; @NHexp; @EPringExp; @NHexp2; @NHlaser; @asymhop1; @asymhop2; @yuce]. Saliently these models feature a breakdown of the conventional bulk-boundary correspondence [@Xi2018; @KuEdBuBe2018; @KuDw2018; @lee; @yaosongwang; @yaowang; @leethomale], which is intimately linked to the piling up of “bulk" states at the boundaries known as the *non-Hermitian skin effect* [@MaAlVaVaBeFoTo2018; @KuEdBuBe2018; @yaowang]. These models can be understood with open boundaries directly by defining a *biorthogonal bulk-boundary correspondence* [@KuEdBuBe2018], which combines the right and left wave functions of the boundary modes to each other to form a “biorthogonal state." By studying the behavior of this state, it is then possible to reconcile the physics of open non-Hermitian systems.
Here we show that the concept of a biorthogonal bulk-boundary correspondence can be generalized to capture non-Hermitian extensions of higher-order topological phases. Indeed, such phases have very recently been studied in a number of works resulting in the observation of variations to the skin effect and the suggestion of topological invariants [@LiZhAiGoKaUeNo2018; @LeLiGo2018; @Ez2018; @Ez20182]. Here, unlike Refs. , we focus on the biorthogonal properties of the open boundary systems, and show that this provides a comprehensive and transparent interpretation of the physical features of non-Hermitian extensions of higher-order topological phases. In particular, it unravels a subtle interplay between crystalline lattice symmetries, sample geometry, and boundary/bulk states that goes qualitatively beyond that of the Hermitian realm.
To elucidate these results we introduce several pertinent examples that admit an exact analytical treatment. First we investigate a non-Hermitian chiral hinge insulator where the conventional bulk-boundary correspondence is broken: the presence of open boundaries drastically rearranges the entire energy spectrum concomitant with a macroscopic piling up of states at the hinges. Second, we find corner modes on two geometries of the breathing kagome lattice, the rhombus and the triangle. Interestingly, in the case of the rhombus the open boundary conditions lead to the appearance of additional *biorthogonal bulk states* in a regime that is traditionally associated with edge bands. For the triangle geometry, however, no such effect is observed, but instead the corner states disappear to the bulk first via an edge transition, which has no counterpart in the Hermitian limit. In each case we show that these features can be naturally understood at a microscopic level by analyzing the biorthogonal set of exact analytical expressions for the higher-order boundary states.
{width="0.88\linewidth"}
*Non-Hermitian chiral hinge insulator.* We start by studying the lattice in Fig. \[fig:hinge\_model\](a) with open boundaries in two directions while being periodic in the third dimension parametrized by $t$. Each red-blue and green-black chain represents a one-dimensional charge pump, and the Hamiltonian of each of these chains, explicitly shown in Fig. \[fig:hinge\_model\](a), corresponds to the Rice-Mele model in the Hermitian limit [@RiMe1982], such that these chains individually realize a Chern insulator with opposite Chern number on the differently colored chains. The Hermitian limit of this model is known to have exactly solvable temporal chiral hinge states, which are protected by a mirror Chern number [@KuMiBe2018; @KuMiBe20182].
We implement non-Hermiticity by introducing a preferred hopping direction between unit cells on the individual red-blue and green-black chains. Explicitly, we change the magnitude of hopping to the right with respect to hopping to the left yielding a nonreciprocal tight-binding model [@footnote2]. The chains are then coupled to each other in a Hermitian fashion \[cf. Fig. \[fig:hinge\_model\](a)\], such that the non-Hermiticity only presides in one direction.
The absolute value of the band spectrum for this model with open boundary conditions is shown in Fig. \[fig:hinge\_model\](b) [@footnote], where the chiral hinge state is shown in red, the bulk bands in blue, and the bunched blue bands appearing in the gap are traditionally identified as surface bands. While the open spectrum is that of a chiral hinge insulator, the periodic Bloch spectrum indicated in gray is semimetallic, thus manifesting a striking breakdown of conventional bulk-boundary correspondence.
 for the cut $t=-0.5$. The left and middle columns show the localization of the right and left wave function individually, and the right column shows the biorthogonal localization.[]{data-label="fig:hinge_skin_effect"}](skin_effect_bulk.pdf){width="0.88\linewidth"}
{width="0.88\linewidth"}
To rationalize this behavior we study the distribution of bulk bands $E_n$ in the lattice at a specific cut of $t$ in Fig. \[fig:hinge\_model\](b) by computing $\left<\Pi_{m_1,m_2}\right>_{\alpha \alpha'}^n\equiv\braket{\Psi_{\alpha,n}|\Pi_{m_1,m_2}|\Psi_{\alpha',n}}$ for $\alpha, \alpha' \in \{R,L\}$, where $\Pi_{m_1,m_2} = \sum_{\beta \in \{A,B,B',B''\}} \ket{e_{\beta,m_1,m_2}}\bra{e_{\beta,m_1,m_2}}$ is a projection operator onto each site in unit cell $\{m_1,m_2\}$ and $\ket{\Psi_{R,n}}$ ($\ket{\Psi_{L,n}}$) the associated right (left) wave function. The quantities $\left<\Pi_{m_1,m_2}\right>_{RR}^n$ and $\left<\Pi_{m_1,m_2}\right>_{LL}^n$ are similar to what is known in ordinary quantum mechanics as the expectation value, and we show them in the left and middle panel of Fig. \[fig:hinge\_skin\_effect\](a), respectively. We see that the bulk state is localized to the right and left hinge, respectively, such that the breaking of bulk-boundary correspondence indeed goes hand in hand with the piling up of states, as was also observed in Refs. . Interestingly, if we now consider the *biorthogonal expectation value* $\left<\Pi_{m_1,m_2}\right>_{LR}^n$, we find the distribution displayed in the right panel of Fig. \[fig:hinge\_skin\_effect\](a), which is in accordance with expected bulk-band behavior. Therefore, we label the blue bands *biorthogonal bulk bands*. Similarly, when studying the localization of a surface band, i.e., a band that belongs to the bunched blue bands in the band gap, we find that the right wave function is localized to the top and bottom hinge \[cf. left panel of Fig. \[fig:hinge\_skin\_effect\](b)\] while the left wave function lives on the left hinge \[cf. middle panel of Fig. \[fig:hinge\_skin\_effect\](b)\]. The biorthogonal expectation value, however, reveals that the weight of the state is indeed distributed on the surfaces \[cf. right panel of Fig. \[fig:hinge\_skin\_effect\](b)\], and we call this a *biorthogonal surface state*. While we thus observe anomalous “skin" behavior when investigating the spectrum with reference to its right (or left) wave functions only, we observe an “ordinary" distribution of the bulk and surface states when approaching the problem from a biorthogonal perspective. These observations are natural considering that the relation between the eigenstates, the Hamiltonian, and the energies involves the left [*and*]{} right eigenstates as $E_n=\left<H\right>_{LR}^n$.
Next we turn to the hinge state \[red in Fig. \[fig:hinge\_model\](b)\], and generalize Refs. to write down the exact solutions $$\ket{\psi_\alpha} = \mathcal{N}_\alpha\sum_{m_1,m_2}^{}\left(r_{\alpha,1}\right)^{m_1}\left(r_{\alpha,2}\right)^{m_2}c^{\dagger}_{A,m_1,m_2}\ket{0}, \label{eq:exact_sol}$$ which have the remarkable property that they may localize on opposite hinges depending on $\alpha\,{\in}\,\{R, L\}$ which labels the right and left eigenvectors. Here $m_1$ and $m_2$ label the unit cells in the lattice with a total of $M_1M_2$ unit cells, $\mathcal{N}_\alpha$ is the normalization constant, $c^{\dagger}_{A,m_1,m_2}$ creates a particle on the $A$ sublattice \[in red in Fig. \[fig:hinge\_model\](a)\] in unit cell $\{m_1,m_2\}$, and $r_{\alpha,1}$ and $r_{\alpha,2}$ can be computed analytically and read $r_{R,1}=-\frac{-t_1+\cos(t)-\gamma/2}{-t_1-\cos(t)},
r_{L,1} =-\frac{-t_1+\cos(t)+\gamma/2}{-t_1-\cos(t)},
r_{R,2} = r_{L,2} = -1.$ These wave functions have zero amplitude on all blue, green, and black sites, and the associated eigenenergy corresponds to the eigenenergy on the $A$ sublattice, which is $E_0\,{=}\, {-}\,\textrm{sin}(t)$ in accordance with the red band in Fig. \[fig:hinge\_model\](b). Depending on the values of $|r_{R,1}|$ and $|r_{L,1}|$, the state $\ket{\psi_\alpha}$ behaves as a hinge or a bulk state [@footnote3]. In particular, the right (left) wave function is equally localized on each $A$ sublattice, in which case it behaves as a bulk state, when $|r_{\alpha,1}| \,{=}\,1$ corresponding to the orange and green dashed lines in Fig. \[fig:hinge\_model\](b). This, however, is in disagreement with the attachment of the red band to the bulk bands, where there is indeed a small gap between the red band and the blue surface bands at the orange lines.
Instead, we consider $|r_{L,1} r_{R,1}| = |r_{L,2} r_{R,2}| = 1$, and find an accurate prediction for hinge-state attachment to the bulk \[cf. the black solid lines in Fig. \[fig:hinge\_model\](b)\]. This quantity follows from considering the biorthogonal expectation value of the projection operator $\left<\Pi_{m_1,m_2}\right>_{LR}^0$ using the solutions in Eq. (\[eq:exact\_sol\]) [@KuEdBuBe2018], which is plotted in the top row of Fig. \[fig:hinge\_model\](c) for three cuts in $t$. Indeed, we see that while the biorthogonal product indicates bulk-band behavior in the middle column, the right and left wave functions shown in the middle and bottom rows of Fig. \[fig:hinge\_model\](c), respectively, suggest the band is localized to the hinge. We thus find that to accurately describe the physics of a non-Hermitian system with open boundary conditions, one has to invoke a biorthogonal bulk-boundary correspondence [@KuEdBuBe2018]. Moreover, we find that the hinge state only changes localization upon attachment to the bulk bands in full agreement with the Hermitian version of this model [@KuMiBe2018].
*Breathing kagome lattice.* Next we study the two-dimensional, breathing kagome lattice in the geometry of a rhombus \[cf. Fig. \[fig:kagome\_rhombus\](a)\] and a triangle \[cf. Fig. \[fig:kagome\_triangle\](a)\]. We implement non-Hermiticity in these lattices by changing the magnitude of the hopping terms in the down triangles such that the hopping amplitude in the clockwise direction, $t_+= t_2 + \gamma/2$, is unequal to the hopping in the anticlockwise direction, $t_- = t_2 - \gamma/2$ while keeping the hopping on up triangles, $t_1$, nonchiral. The real-space Hamiltonian for both models is explicitly shown in Figs. \[fig:kagome\_rhombus\](a) and \[fig:kagome\_triangle\](a), respectively. The Hermitian versions of these systems were previously studied in Refs. .
{width="0.88\linewidth"}
We start by focusing on the rhombus, and plot the absolute value of the energy spectrum in Fig. \[fig:kagome\_rhombus\](b) as a function of $t_1$ for fixed $t_2$ and $\gamma$ with the bulk bands in blue, and the red band corresponding to a zero-energy corner mode [@footnote]. In addition, we plot the spectrum with periodic boundary conditions in gray, and find that it is qualitatively in accordance with the spectrum in blue. This can be understood from the fact the bulk states can move around in loops in the lattice, such that they do not get trapped, and thus do not pile up. Nevertheless, when considering the behavior of the blue bands that do not overlap with the gray bands, which are thus expected to be associated with surface states, we find that some of these states are in fact *additional* biorthogonal bulk states [@footnote2].
To study the behavior of the zero-energy corner mode in more detail, we consider its associated exact wave-function solution given in Eq. (\[eq:exact\_sol\]) with $$\begin{aligned}
r_{R,1} &= r_{L,2} = - \frac{t_1}{t_2 {+} \gamma/2}, \quad r_{R,2} = r_{L,1} = - \frac{t_1}{t_2 {-} \gamma/2}. \label{eq:sol_kagome_rhombus}\end{aligned}$$ Depending on the values of $|r_{\alpha,1}|$ and $|r_{\alpha,2}|$ with $\alpha \in \{R,L\}$, the state $\ket{\Psi_\alpha}$ thus behaves as a corner, edge or bulk state. Considering $|r_{R,1}| = |r_{R,2}|=1$ and $|r_{L,1}| = |r_{L,2}|=1$, which would predict a bulk state in the framework of ordinary quantum mechanics, we find solutions corresponding to the orange and green dashed lines, respectively, in Fig. \[fig:kagome\_rhombus\](b), and clearly see that this does not predict any particular behavior in the spectrum. Instead, considering the biorthogonal delocalization criteria for $\left<\Pi_{m_1,m_2}\right>_{LR}^0$, corresponding to $|r_{L,1} r_{R,1}| = |r_{L,2} r_{R,2}| = 1$, we obtain the black solid lines in Fig. \[fig:kagome\_rhombus\](b), in complete agreement with attachment to the bulk bands.
To corroborate this picture we plot $\left<\Pi_{m_1,m_2}\right>_{\alpha \alpha'}^0$ with $\alpha, \alpha' \in \{R,L\}$ in Fig. \[fig:kagome\_rhombus\](c) for four different choices of $t_1$. This reveals several aspects. First of all, the right and left wave functions (middle and bottom row, respectively) individually suggest a transition of the corner state from one corner to the other via the *edges*, while the biorthogonal state (top row) reveals a transition via the *bulk*. Indeed, only the latter interpretation is in accordance with the Hermitian version of this model [@KuMiBe2018; @ezawapap; @xuxuewan]. Secondly, we study the blue bands to which the corner state attaches at $|r_{L,1} r_{R,1}| = |r_{L,2} r_{R,2}| = 1$, and find that they are edge bands in the context of their right and left wave functions, while admitting a bulk channel when considering their biorthogonal properties [@footnote2], i.e., they correspond to the aforementioned biorthogonal bulk states. This thus explains the migration of the biorthogonal corner state into the bulk, as opposed to it being transmitted via the edges in the case of the right and left wave functions. Third, we observe that the right (left) wave function of the corner state localizes to the top left (bottom right) corner for $t=0.55$ \[cf. third column from the left in Fig. \[fig:kagome\_rhombus\](c)\], while the corner mode is well separated from the other bands in the spectrum. This is noteworthy because in the Hermitian limit, localization of the corner state to these specific corners is not possible [@KuMiBe2018; @xuxuewan].
Next, we turn to the triangular geometry in Fig. \[fig:kagome\_triangle\](a). We plot the absolute value of the band spectrum in Fig. \[fig:kagome\_triangle\](b) with the bands for open (periodic) boundary conditions in blue (gray) [@footnote], and again find that the bulk spectrum is not qualitatively rearranged signaling the absence of the skin effect. As in the Hermitian case [@ezawapap], there is a threefold degenerate zero-energy mode for a certain parameter regime. To understand the behavior of this mode in more detail, we note that the solutions in Eq. (\[eq:exact\_sol\]) with $r_{\alpha,1}$ and $r_{\alpha,2}$ given in Eq. (\[eq:sol\_kagome\_rhombus\]) in the large system limit can be mapped onto each corner of the triangle for $|r_{\alpha,1}|,|r_{\alpha,2}| <1$. Once $|r_{\alpha,1}|$ and/or $|r_{\alpha,2}|$ become(s) equal to or larger than $1$, the states $\ket{\psi_\alpha}$ leak into the edge or the bulk such that the three corner states start to interfere with each other. This renders the predictive power of Eq. (\[eq:exact\_sol\]) invalid, while also resulting in the lifting of the zero-energy mode away from zero, which, assuming $t_1, t_2, \gamma \in \mathbb{R}$, indeed happens once $|t_1| = |t_2 - \gamma/2|$, i.e., $|r_{R,2}| = |r_{L,1}| = 1$ \[cf. the black dashed lines in Fig. \[fig:kagome\_triangle\](b)\]. Studying the localization of the lowest-lying energy mode for different choices of $t_1$ in Fig. \[fig:kagome\_triangle\](c), we see that the right, left, and biorthogonal distributions predict the same qualitative behavior [@footnote2]. Indeed, in the left column we consider $t_1=0.15$ for which the lowest-energy mode is the zero-energy corner state, and see that they are equally distributed over the three corners. When investigating the distribution of the states when they attach to the edge bands \[cf. black dashed lines in Fig. \[fig:kagome\_triangle\](b)\], we indeed see edge band behavior, while attachment to the bulk bands results in bulk-band behavior \[cf. right column in Fig. \[fig:kagome\_triangle\](c)\]. In sharp contrast to the Hermitian version of this model [@ezawapap], we find that there is a corner to edge to bulk transition, which confirms the prediction following from Eqs. (\[eq:exact\_sol\]) and (\[eq:sol\_kagome\_rhombus\]) that the corner mode first attaches to the edge states before merging with the bulk bands. This behavior can be understood from symmetry arguments—the threefold rotation dictates the same behavior for all three corner states, which due to the presence of non-Hermitian terms necessarily leak into the edge *before* they enter the bulk.
*Conclusion*. We have considered three explicit examples of non-Hermitian extensions of second-order topological systems, and shown that while conventional bulk-boundary correspondence may be strongly broken, we can exploit the biorthogonal properties of these models to fully reconcile their behavior also in the presence of a skin effect. By making use of exact solutions for the second-order boundary states \[cf. Eq. (\[eq:exact\_sol\])\], we explicitly studied the localization of these states both in the context of their right and left wave functions as well as their biorthogonal product, and related that to the spectrum with open boundary conditions. Moreover, by studying the distribution of edge/surface and bulk bands of these models we showed that additional biorthogonal bulk bands may appear when taking open boundary conditions, and that the interplay between edge, corner, and bulk can be qualitatively distinct from that of their Hermitian counterparts even in the absence of the non-Hermitian skin effect.
We thank Jan Carl Budich and Guido van Miert for related collaborations. E.E. thanks Eva Mossberg for useful discussions about MATLAB. This work was supported by the Swedish Research Council (VR) and the Wallenberg Academy Fellows program of the Knut and Alice Wallenberg Foundation.
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See Supplemental Material for technical and quantitative details.
The real and imaginary parts of the energy spectrum are included in the Supplemental Material.
Note that if $|r_{R,2}|, |r_{L,2}| \neq 1$, $r_{\alpha,1}$ and $r_{\alpha,2}$ with $\alpha \in \{R,L\}$ could be chosen such that the wave function in Eq. (\[eq:exact\_sol\]) corresponds to a surface state.
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|
---
abstract: |
A *$t$-spanner* of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner’s quality – in this work we focus on the latter.
Specifically, it is shown that for any parameters $k\ge 1$ and ${\varepsilon}>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+{\varepsilon})$-stretch spanner of weight at most $w(MST(G))\cdot O_{\varepsilon}(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.
author:
- 'Michael Elkin[^1]'
- 'Ofer Neiman[^2]'
- 'Shay Solomon[^3]'
bibliography:
- 'latex8.bib'
title: Light Spanners
---
Introduction
============
Given a weighted connected graph $G=(V,E)$ with $n$ vertices and $m$ edges, let $d_G$ be its shortest path metric. A $t$-spanner $H=(V,E')$ is a subgraph that preserves all distances up to a multiplicative factor $t$. That is, for all $u,v\in V$, $d_H(u,v)\le t\cdot d_G(u,v)$. The parameter $t$ is called the [*stretch*]{}. There are several parameters that have been studied in the literature that govern the quality of $H$, two of the most notable ones are the [*size*]{} of the spanner (the number of edges) and its total [*weight*]{} (the sum of weights of its edges).
There is a basic tradeoff between the stretch and the size of a spanner. For any graph on $n$ vertices, there exists a $(2k-1)$-spanner with $O(n^{1+1/k})$ edges [@ADDJS93]. Furthermore, there is a simple greedy algorithm for constructing such a spanner, which we shall refer to as the [*greedy spanner*]{} (see ). The bound on the number of edges is known to be asymptotically tight for certain small values of $k$, and for all $k$ assuming Erdős’ girth conjecture.
In this paper we focus on the weight of a spanner. Light weight spanners are particularly useful for efficient broadcast protocols in the message-passing model of distributed computing [@ABP90; @ABP91], where efficiency is measured with respect to both the total communication cost (corresponding to the spanner’s weight) and the speed of message delivery at all destinations (corresponding to the spanner’s stretch). Additional applications of light weight spanners in distributed systems include network synchronization and computing global functions [@ABP90; @ABP91; @Peleg00]. Light weight spanners were also found useful for various data gathering and dissemination tasks in overlay networks [@BKRCV02; @KV01], in wireless and sensor networks [@SS10], for network design [@MP98; @SCRS01], and routing [@WCT02].
While a minimum spanning tree (MST) has the lowest weight among all possible connected spanners, its stretch can be quite large. Nevertheless, when measuring the weight of a spanner, we shall compare ourselves to the weight of an MST: The [*lightness*]{} of the spanner $H$ is defined as $\frac{w(H)}{w(MST)}$ (here $w(H)$ is the total edge weight of $H$). It was shown by [@ADDJS93] that the lightness of the greedy spanner is at most $O(n/k)$, and their result was improved by [@CDNS92], who showed that for any ${\varepsilon}>0$ the greedy $(2k-1)\cdot(1+{\varepsilon})$-spanner has $O_{\varepsilon}(n^{1+1/k})$ edges and lightness $O(k\cdot n^{1/k}/{\varepsilon}^{1+1/k})$. A particularly interesting special case arises when $k \approx \log n$. Specifically, in this case the result of [@CDNS92] provides stretch and lightness both bounded by $O(\log n)$. Another notable point on the tradeoff curve of [@CDNS92] (obtained by setting ${\varepsilon}=\log n$ as well) is stretch $O(\log^2 n)$ and lightness $O(1)$.
These results of [@CDNS92] remained the state-of-the-art for more than twenty years. In particular, prior to this work it was unknown if spanners with stretch $O(\log n)$ and lightness $o(\log n)$, or vice versa, exist. In this paper we answer this question in the affirmative, and show in fact something stronger – spanners with stretch and lightness both bounded by $o(\log n)$ exist. We provide a novel analysis of the classic greedy algorithm, which improves the tradeoff of [@CDNS92] by a factor of $O(\log k)$. Specifically, we prove the following theorem.
\[thm:main\] For any weighted graph $G=(V,E)$ and parameters $k\ge 1$, ${\varepsilon}>0$, there exists a $(2k-1)\cdot(1+{\varepsilon})$-spanner $H$ with $O(n^{1+1/k})$ edges[^4] and lightness $O(n^{1/k}\cdot(1+k/({\varepsilon}^{1+1/k}\log k)))$.
By substituting $k\approx\log n$ we obtain stretch $\log n$ and lightness $O(\log n/\log\log n)$ (for fixed small ${\varepsilon}$). We also allow ${\varepsilon}$ to be some large value. In particular, setting ${\varepsilon}=\log n/\log\log n$ yields stretch $\log^2n/\log\log n$ and lightness $O(1)$. Also, by substituting $k = \log n/\log\log\log n$ we can have both stretch and lightness bounded by $O(\log n/\log\log\log n)$.
Our result shows that the potentially natural tradeoff between stretch $2k-1$ and lightness $O(k\cdot n^{1/k})$ is not the right one. This can also be seen as an indication that the right tradeoff is stretch $(2k-1)$ and lightness $O(n^{1/k})$. (Note that lightness $O(n^{1/k})$ is the weighted analogue of $O(n^{1+1/k})$ edges, and so it is asymptotically tight assuming Erdős’ girth conjecture.)
Proof Overview
--------------
The main idea in the analysis of the greedy algorithm by [@CDNS92], is to partition the edges of the greedy spanner to scales according to their weight, and bound the contribution of edges in each scale separately. For each scale they create a graph from the edges selected by the greedy algorithm to the spanner, and argue that such a graph has high girth[^5] and thus few edges. The main drawback is that when analyzing larger weight edges, this argument ignores the smaller weight edges that were already inserted into the spanner.
We show that one indeed can use information on lower weight edges when analyzing the contribution of higher scales. We create a different graph from edges added to the spanner, and argue that this graph has high girth. The new ingredient in our analysis is that we add multiple edges per spanner edge, proportionally to its weight. Specifically, these new edges form a [*matching*]{} between certain neighbors of the original edge’s endpoints. Intuitively, a high weight edge enforces strong restrictions on the length of cycles containing it, so it leaves a lot of “room” for low weight edges in its neighborhood. The structure of the matching enables us to exploits this room, while maintaining high girth.
Unfortunately, with our current techniques we can only use edges of weight at most $k$ times smaller than the weight of edges in the scale which is now under inspection. Hence this gives an improvement of $O(\log k)$ to the lightness of the greedy spanner. We hope that a refinement of our method, perhaps choosing the matching more carefully, will eventually lead to an optimal lightness of $O(n^{1/k})$.
Related Work
------------
A significant amount of research attention was devoted to constructing light and sparse spanners for Euclidean and doubling metrics. A major result is that for any constant-dimensional Euclidean metric and any ${\varepsilon}>0$, there exists a $(1+{\varepsilon})$-spanner with lightness $O(1)$ [@DHN93]. Since then there has been a flurry of work on improving the running time and other parameters. See, e.g., [@CDNS92; @ADMSS95; @DES08; @ES13-focs; @CLNS13], and the references therein. An important question still left open is whether the $O(1)$ lightness bound of [@DHN93] for constant-dimensional Euclidean metrics can be extended to doubling metrics. Such a light spanner has implications for the running time of a PTAS for the traveling salesperson problem (TSP). Recently, [@GS14] showed such a spanner exists for snowflakes[^6] of doubling metrics.
Light spanners with $(1+{\varepsilon})$ stretch have been sought for other graph families as well, with the application to TSP in mind. It has been conjectured that graphs excluding a fixed minor have such spanners. Currently, some of the known results are for planar graph [@ADMSS95], bounded-genus graphs [@G00], unit disk graphs [@KPX08], and bounded pathwidth graphs [@GH12].
A lot of research focused on constructing sparse spanners efficiently, disregarding their lightness. Cohen [@Coh93] devised a randomized algorithm for constructing $((2k-1)\cdot (1+{\varepsilon}))$-spanners with $O(k \cdot n^{1+1/k} \cdot (1/{\varepsilon}) \cdot \log n)$ edges. Her algorithm requires expected $O(m \cdot n^{1/k} \cdot k \cdot (1/{\varepsilon}) \cdot \log n))$ time. Baswana and Sen [@BS03] improved Cohen’s result, and devised an algorithm that constructs $(2k-1)$-spanners with expected $O(k \cdot n^{1+1/k})$ edges, in expected $O(k \cdot m)$ time. Roditty et al. [@RTZ05] derandomized this algorithm, while maintaining the same parameters (including running time). Roditty and Zwick [@RZ04] devised a deterministic algorithm for constructing $(2k-1)$-spanners with $O(n^{1+1/k})$ edges in $O(k \cdot n^{2+1/k})$ time.
Preliminaries
=============
Let $G=(V,E)$ be a graph on $n$ vertices with weights $w:E\to{\mathbb{R}}_+$, and let $d_G$ be the shortest path metric induced by $G$. For simplicity of the presentation we shall assume that the edge weights are positive integers. (The extension of our proof to arbitrary weights is not difficult, requiring only a few minor adjustments.) For a subgraph $H=(V',E')$ define $w(H)=w(E')=\sum_{e\in E'}w(e)$. A subgraph $H=(V,E')$ is called a *$t$-spanner* if for all $u,v\in V$, $d_H(u,v)\le t\cdot d_G(u,v)$. Define the [*lightness*]{} of $H$ as $\frac{w(H)}{w(MST(G))}$, where $MST(G)$ is a minimum spanning tree of $G$. The girth $g$ of a graph is the minimal number of edges in a cycle of $G$. The following standard Lemma is implicit in [@bolo-book].
\[lem:girth\] Let $g>1$ be an integer. A graph on $n$ vertices and girth $g$ has at most $O\left(n^{1+\frac{1}{\lfloor (g-1)/2 \rfloor}}\right)$ edges.
Greedy Algorithm
----------------
The natural greedy algorithm for constructing a spanner is described in .
$H=(V,\emptyset)$. Add the edge $\{u,v\}$ to $E(H)$.
Note that whenever an edge $e\in E$ is inserted into $E(H)$, it cannot close a cycle with $t+1$ or less edges, because the edges other than $e$ of such a cycle will form a path of length at most $t\cdot w(e)$ (all the existing edges are not longer than $w(e)$). This argument suggests that $H$ (viewed as an unweighted graph) has girth $t+2$ (when $t$ is an integer), and thus by $$\label{eq:num}
|E(H)|\le O\left(n^{1+\frac{1}{\lfloor (t+1)/2 \rfloor}}\right)~.
$$ We observe that the greedy algorithm must select all edges of an MST (because when inspected they connect different connected components in $H$). We will assume without loss of generality that the graph $G$ has a unique MST, since any ties can be broken using lexicographic rules.
\[ob:mst\] If $Z$ is the MST of $G$, then $Z\subseteq H$. Furthermore, each edge in the MST does not close a cycle in $H$ when it is inspected.
Proof of Main Result
====================
Let $H$ be the greedy spanner with parameter $t=(2k-1)\cdot(1+{\varepsilon})$. Let $Z$ be the MST of $G$, and order the vertices $v_1,v_2,\dots,v_n$ according to the order they are visited in some preorder traversal of $Z$ (with some fixed arbitrary root). Since every edge of $Z$ is visited at most twice in such a tour, $$L:=\sum_{i=2}^nd_Z(v_{i-1},v_i)\le 2w(Z)~.$$
Let $I=\lceil\log_kn\rceil$. For each $i\in [I]$, define $E_i=\{e\in E(H)\setminus E(Z)\mid w(e)\in(k^{i-1},k^i]\cdot L/n\}$. We may assume the maximum weight of an edge in $H$ is bounded by $w(Z)$ (in fact $w(Z)/t$, as heavier edges surely will not be selected for the spanner), so each edge in $H \setminus Z$ of weight greater than $L/n$ is included in some $E_i$. The main technical theorem is the following.
\[thm:tech\] For each $i\in[I]$ and any ${\varepsilon}>0$, $$w(E_i)\le O(L\cdot (n/k^{i-1})^{1/k}/{\varepsilon}^{1+1/k})~.$$
Given this, the proof of quickly follows.
Using that the stretch of the spanner is $t\ge 2k-1$, by we have $|E(H)|\le O(n^{1+1/k})$. The total weight of edges in $H$ that have weight at most $L/n$ can be bounded by $L/n\cdot |E(H)|\le L/n\cdot O(n^{1+1/k})=O(w(MST)\cdot n^{1/k})$. The contribution of the other (non-MST) edges to the weight of $H$, using , is at most $$\begin{aligned}
\sum_{i=1}^IO(L\cdot (n/k^{i-1})^{1/k}/{\varepsilon}^{1+1/k}) &\le& O(L\cdot n^{1/k}/{\varepsilon}^{1+1/k})\sum_{i=0}^\infty e^{-(i\ln k)/k}\\
&=& O(L\cdot n^{1/k}/{\varepsilon}^{1+1/k})\cdot\frac{1}{1-e^{-(\ln k)/k}}\\
&=&O(w(MST))\cdot kn^{1/k}/({\varepsilon}^{1+1/k}\ln k)~.\end{aligned}$$
Proof of
---------
#### Overview:
Fix some $i\in[I]$. We shall construct a certain graph $K$ from the edges of $E_i$, and argue that this graph has high girth, and therefore few edges. The main difference from [@CDNS92] is that our construction combines into one scale edges whose weight may differ by a factor of $k$ (in the construction of [@CDNS92] all edges in a given scale are of the same weight, up to a factor of 2). In order to compensate for heavy edges, the weight of the edge determines how many edges are added to $K$. Specifically, if the edge $\{u,v\}\in E_i$ has weight $w\cdot k^{i-1}\cdot L/n$, we shall add (at least) $\lceil w\rceil$ edges to $K$ that form a [*matching*]{} between vertices in some neighborhoods of $u$ and $v$. In this way the weight of $K$ dominates $w(E_i)$. To prove that $K$ has high girth, we shall map a cycle in $K$ to a closed tour in $H$ of proportional length. The argument uses the fact that the new edges are close to the original edge, and that a potential cycle in $K$ cannot exploit more than one such new edge, since these edges form a matching.
#### Construction of the Graph $K$:
Let $P=(p_0,\dots,p_L)$ be the unweighted path on $L+1$ vertices, created from $V$ by placing $v_1,\dots,v_n$ in this order and adding Steiner vertices so that all consecutive distances are $1$, and for all $2\le j\le n$, $d_P(v_{j-1},v_j)=d_Z(v_{j-1},v_j)$. In particular, $p_0=v_1$, $p_L=v_n$, and for every $1\le j<j'\le n$, $$d_P(v_j,v_{j'})=\sum_{h=j+1}^{j'}d_Z(v_{h-1},v_h)~.$$ Note that $d_P(v_j,v_{j'})\ge d_Z(v_j,v_{j'})\ge d_G(v_j,v_{j'})$, and all the inequalities may be strict. In order to be able to map edges of $K$ back to $H$, we shall also add corresponding Steiner points to the spanner $H$: For every Steiner point $p_h$ that lies on $P$ between $v_{j-1}$ and $v_j$, add a Steiner point on the path in the MST $Z$ that connects $v_{j-1}$ to $v_j$ at distance $d_P(v_{j-1},p_h)$ from $v_{j-1}$ (unless there is a point there already). By all MST edges are indeed in $H$, and one can simply subdivide the appropriate edge on the MST path. Note that distances in $H$ do not change, as the new Steiner points have degree $2$. Denote by $\hat{H}$ the modified spanner $H$, i.e., $H$ with the Steiner points. Let $a=k^{i-1}\cdot L/n$ be a lower bound on the weight of edges in $E_i$. Divide $P$ into $s=8L/({\varepsilon}a)$ intervals $I_1,\dots,I_s$, each of length $L/s= \frac{{\varepsilon}}{8}a$ (by appropriate scaling, we assume all these are integers). For $j\in[s]$, the interval $I_j$ contains the points $p_{(j-1)L/s},\dots,p_{jL/s}$. In each interval $I_j$ pick an arbitrary (interior) point $r_j$ as a representative, and let $R$ be the set of representatives. For each representative $r_j$ and an integer $b\ge 0$ we define its neighborhood $N_b(j)=\{r_h ~:~ |j-h|\le b\}$ to be the set of (at most) $2b+1$ representatives that are at most $b$ intervals away from $I_j$. (Note that the size of the neighborhood $N_b(j)$ can be smaller than $2b+1$ if $r_j$ is too close to one of the endpoints of the path $P$.) Define an unweighted (multi) graph $K=(R,F)$ in the following manner. Let $e=\{u,v\}\in E_i$. Assume that $u\in I_h$ and $v\in I_j$ for some $h,j\in [s]$. Let $b=\lfloor w(e)/a\rfloor$, and let $M$ be an arbitrary maximal matching between $N_b(h)$ and $N_b(j)$. Add all the edges of $M$ to $F$, see . For each of the edges $\{q,q'\}\in M$ added to $F$, we say that the edge $\{u,v\}$ is its [*source*]{} when $q\in N_b(h)$ and $q'\in N_b(j)$, and write $S(q,q')=(u,v)$. We will soon show (in below) that each edge in $K$ has a single source.
The following observation suggests that if all the edges of $K$ were given weight $a$, then its total weight is greater than or equal to the weight of the edges in $E_i$.
\[ob:K-heavy\] $|F|\cdot a\ge w(E_i)$.
Note that always $|N_b(j)|\ge b+1$, which means that we add at least $b+1\ge w(e)/a$ edges to $K$ for each edge $e\in E_i$. Summing over all edges concludes the proof.
[**Mapping from $K$ to $\hat{H}$:**]{} We shall map every edge $\{q,q'\}\in F$ to a tour $T(q,q')$ in the spanner $\hat{H}$ connecting $q$ and $q'$. If $S(q,q')=(u,v)$, then $T(q,q')$ consists of the following paths:
- A path in $Z$ connecting $q$ to $u$.
- The edge $\{u,v\}$.
- A path in $Z$ connecting $v$ to $q'$.
The following proposition asserts that the length of the tour is not longer than the weight of the source edge, up to a $1+{\varepsilon}/2$ factor.
\[proposition:map\] If an edge $\{q,q'\}\in F$ has a source $S(q,q')=(u,v)$ of weight $w$, then $T(q,q')$ is a tour in $\hat{H}$ of length at most $(1+{\varepsilon}/2)w$.
First observe that the distance in $P$ between any two points in intervals $I_j$ and $I_{j+b}$ is at most $(b+1)L/s$. Since $d_P\ge d_Z$ we also have that the distance in the MST $Z$ between two such points is bounded by $(b+1)L/s$. (By definition, this holds for Steiner points as well.) Denote the representatives of $u,v$ as $r_j,r_h$, respectively. For $b=\lfloor w/a\rfloor$, the set $N_b(j)$ contains representatives of at most $b$ intervals away from $I_j$. As $u\in I_j$ we get that $d_Z(q,u)\le(b+1)L/s$. Similarly $d_Z(q',v)\le(b+1)L/s$, thus the total length of the tour is at most $w+2(b+1)L/s = w + 2(\lfloor {w \over a} \rfloor + 1){{\varepsilon}\over 8} a \le(1+{\varepsilon}/2)w$.
Our goal is to show that $K$ is a simple graph of girth at least $2k+1$. As a warmup, let us first show that $K$ does not have parallel edges.
\[proposition:simple\] The graph $K$ does not have parallel edges.
Seeking contradiction, assume there is an edge $\{q,q'\}\in F$ with two different sources $\{u,v\},\{u',v'\}\in E_i$. Without loss of generality assume that $\{u,v\}$ is the heavier edge of the two, with weight $w$. Then $\{q,q'\}$ is mapped to two tours in $\hat{H}$ connecting $q,q'$, whose total length, using , is at most $w(2+{\varepsilon})$. Consider the tour $\hat{T}=u\to q\to u'\to v'\to q'\to v$ in $\hat{H}$ which has total length at most $w(2+{\varepsilon})-w=w(1+{\varepsilon})$. Since the Steiner points have degree 2, they can be removed from $\hat{T}$ without increasing its length, and thus there is in $H$ a simple path $T$ from $u$ to $v$ of length at most $w(1+{\varepsilon})$.
We claim that $T$ must exist at the time the edge $\{u,v\}$ is inspected by the greedy algorithm. The edge $\{u',v'\}$ exists because it is lighter. The MST edges exist since by they must connect different components when inspected, while if some of them are inserted after $\{u,v\}$, at least one of them will close the cycle $T\cup\{u,v\}$. As $w(1+{\varepsilon})\le w\cdot (2k-1)(1+{\varepsilon})$, we conclude that the edge $\{u,v\}$ should not have been added to $H$, which is a contradiction.
Showing that $K$ has large girth will follow similar lines, but is slightly more involved. The difficulty arises since we added multiple edges for each edge of $H$, thus a cycle in $K$ may be mapped to a closed tour in $H$ that uses the same edge $e\in E(H)$ more than once. In such a case, $e$ may not be a part of any simple cycle contained in the closed tour, and we will not be able to derive a contradiction from the greedy choice of $e$ to $H$. [^7] To rule out such a possibility, we use the fact that the multiple edges whose source is $e$ form a matching, and that the weights are different by a factor of at most $k$.
\[lem:gir\] The graph $K=(R,F)$ has girth $2k+1$.
It will be easier to prove a stronger statement, that for any $j\in[s]$ and any $r,r'\in N_k(j)$, every path in $K$ between $r$ and $r'$ contains at least $2k+1$ edges. Once this is proven, we may use this with $r=r'$ to conclude that $\mathit{girth}(K) \ge 2k+1$.
Seeking contradiction, assume that there is a path $Q$ in $K$ from $r$ to $r'$ that contains at most $2k$ edges, and take the shortest such $Q$ (over all possible choices of $j$ and $r,r'$). Let $\{q,z\}\in F$ be the last edge added to $Q$, with source $S(q,z)=(x,y)$ (so that $\{x,y\}\in E_i$ is the heaviest among all the sources of edges in $Q$). We claim that no other edge in $Q$ has $\{x,y\}$ as a source. To see this, consider a case in which such an edge $\{q',z'\}\in F$ is also in $Q$ with $S(q',z')=(x,y)$. We may assume w.l.o.g that $q\notin\{r,r'\}$ (since the path $Q$ contains at least 2 edges), then by definition of the graph $K$, there exists some $j'\in[s]$ with $q,q'\in N_k(j')$ (recall that the neighborhood length $b$ always satisfies $b\le k$ by definition of $E_i$). But then the sub-path of $Q$ from $q$ to $q'$ is strictly shorter than $Q$, and connects two points in the same $k$-neighborhood. Since the edges in $K$ with $\{x,y\}$ as a source form a matching, we get that $q\neq q'$, and thus this path is not of length 0. This contradicts the minimality of $Q$. Next, we will show that $\{x,y\}$ should not have been chosen for $H$, because there is a short path connecting $x$ to $y$.
By every edge $e\in Q$ whose source $S(e)=e'$ has weight $w(e')$, is mapped to a tour $T(e)$ of length at most $(1+{\varepsilon}/2)w(e')$ in $\hat{H}$. Since $w(x,y)$ is the maximum weight source of all edges in $Q$, we conclude that the total length of tours connecting $x$ to $r$ and $r'$ to $y$ is at most $(2k-1)\cdot (1+{\varepsilon}/2)w(x,y)$. Note that $r,r'$ are representatives in $N_k(j)$, which are at most $2k$ intervals apart. So their distance in the MST $Z$ is at most $2k\cdot{\varepsilon}a/8\le k\cdot{\varepsilon}w(x,y)/4$. The total length of the tour $x\to r\to r'\to y$ in $\hat{H}$ is at most $$(2k-1)\cdot (1+{\varepsilon}/2)w(x,y)+k\cdot{\varepsilon}w(x,y)/4 \le (2k-1)(1+{\varepsilon})\cdot w(x,y)~.$$ When the algorithm considers the edge $\{x,y\}$, all the edges of the above tour exist in $\hat{H}$. (This follows since they are all MST edges or lighter than $w(x,y)$, similarly to the argument used in .) We conclude that there is a path between $x$ and $y$ in $H$ of length at most $(2k-1)\cdot(1+{\varepsilon})\cdot w(x,y)$, hence $\{x,y\}$ should not have been added to $E(H)$, which yields a contradiction.
Recall that the graph $K$ has $s$ vertices. By it is a simple graph, and suggests it has girth at least $2k+1$, thus by using it has at most $O(s^{1+1/k})$ edges. Using , $$\begin{aligned}
w(E_i)&\le&|F|\cdot a\\
&\le& O(s^{1+1/k})\cdot(L/n\cdot k^{i-1})\\
&=&\left(\frac{8 \cdot n}{{\varepsilon}k^{i-1}}\right)^{1+1/k}\cdot(O(L)/n\cdot k^{i-1})\\
&\le&O(L\cdot (n/k^{i-1})^{1/k}/{\varepsilon}^{1+1/k})~.\end{aligned}$$
Weighted Girth Conjecture
=========================
The girth of a graph is defined on unweighted graphs. Here we give an extension of the definition that generalizes to weighted graphs as well, and propose a conjecture on the extremal graph attaining a weighted girth.
Let $G=(V,E)$ be a weighted graph with weights $w:E\to{\mathbb{R}}_+$, the [*weighted girth*]{} of $G$ is the minimum over all cycles $C$ of the weight of $C$ divided by its heaviest edge, that is $$\min_{C\text{ cycle in G}}~\left\{\frac{w(C)}{\max_{e\in C}w(e)}\right\}~.$$
Note that this matches the standard definition of girth for unweighted graphs. Recall that the lightness of $G$ is $\frac{w(G)}{w(MST)}$. For a given weighted girth value $g$ and cardinality $n$, we ask what is the graph on $n$ vertices with weighted girth $g$ that maximizes the lightness?
\[con:wg\] For any integer $g\ge 3$, among all graphs with $n$ vertices and weighted girth $g$, the maximal lightness is attained for an unweighted graph.
Recall that Erdős’ girth conjecture asserts that there exists an (unweighted) graph with girth $g>2k$ and $\Omega(n^{1+1/k})$ edges, that is, its lightness is $\Omega(n^{1/k})$. Observe that any graph of weighted girth larger than $2k+{\varepsilon}(2k-1)$ can be thought of as the output of with parameter $t=(2k-1)\cdot(1+{\varepsilon})$. In particular, implies that its lightness is at most $O_{\varepsilon}(kn^{1/k}/\log k)$. Thus (up to the term of ${\varepsilon}(2k-1)$ in the girth), there exists an unweighted graph which is at most $O(k/\log k) = O(g/\log g)$ lighter than the heaviest weighted graph.
The intuition behind this conjecture follows from our method of replacing high weight edges by many low weight edges. We believe that such replacement should hold when performed on all possible scales simultaneously. An immediate corollary of , is that the lightness of a greedy $(2k-1)$-spanner of a weighted graph on $n$ vertices is bounded by $O(n^{1/k})$. To see why this is true, note that the spanner’s weighted girth must be strictly larger than $2k$, and $O(n^{1/k})$ is a bound on the lightness of an [*unweighted*]{} graph on $n$ vertices with girth $2k+1$.
[^1]: Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: `elkinm@cs.bgu.ac.il`
[^2]: Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: `neimano@cs.bgu.ac.il`. Supported in part by ISF grant No. (523/12) and by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement $n^\circ$303809.
[^3]: Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. Email: `shay.solomon@weizmann.ac.il`. This work is supported by the Koshland Center for basic Research.
[^4]: In fact for large ${\varepsilon}$ a better bound can be obtained. Specifically, it is $O(n^{1+1/\lfloor\lceil(2k-1)\cdot(1+{\varepsilon})\rceil/2\rfloor})$.
[^5]: The girth of a graph is the minimal number of edges in a cycle.
[^6]: For $0\le\alpha\le 1$, an $\alpha$-snowflake of a metric is obtained by taking all distances to power $\alpha$.
[^7]: In fact, this is the only reason our method improves the lightness by a factor of $\log k$ rather than the desired $k$.
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abstract: 'A first-principles computational study of the Electron Paramagnetic Resonance (EPR) parameters of Li and Ga vacancies in LiGaO$_2$ is presented. In the EPR active charge states $V_\mathrm{Li}^0$ and $V_\mathrm{Ga}^{2-}$, the spin is localized on one of the O neighbors of the vacancy. We compare the calculated EPR parameters for spin density localized on different O neighbors. Good agreement with experiment is obtained for both the $g$-tensor values and principal axes orientations and the superfhyperfine interaction parameters supporting the prior experimental identification of which O the spin is localized on. The $g$-tensor orientations are found to be close to the bond rather than the crystalline axes. The high energy of formation of $V_\mathrm{Ga}$ compared to $V_\mathrm{Li}$ also explains why $V_\mathrm{Ga}$ were only observed after high energy irradiation while $V_\mathrm{Li}$ were found in as grown samples. On the other hand, the transition levels and Fermi level position explain why $V_\mathrm{Li}$ required ionization from the $-1$ to $0$ charge state to become active while $V_\mathrm{Ga}$ were already found in the $q=-2$ EPR active state.'
author:
- Dmitry Skachkov
- 'Walter R. L. Lambrecht'
- Klichchupong Dabsamut
- Adisak Boonchun
bibliography:
- 'dft.bib'
- 'ligao2.bib'
- 'gipaw.bib'
- 'ldau.bib'
- 'defects.bib'
title: 'Computational Study of Electron Paramagnetic Resonance Spectra for Li and Ga Vacancies in LiGaO$_2$'
---
LiGaO$_2$ is an ultra-wide-band-gap material with a wurtzite-like crystal structure[@Marezio65; @Ishii98] and experimental band gap of $\sim$5.3–5.6 eV at room temperature.[@Wolan98; @Johnson2011; @Chen14; @Ohkubo2002] It can be grown in bulk form by the Czochralsky method[@Marezio65] and has been suggested as a useful substrate for GaN but can also be grown by epitaxial method on ZnO and vice versa. Mixed ZnO-LiGaO$_2$ alloys have also been reported. [@Omata08; @Omata11] In fact, this material can be viewed as a I-III-VI$_2$ analog of the II-VI material ZnO by substituting the II-element Zn by a group I (Li) and a group III (Ga) in an ordered fashion on the wurtzite lattice. It has been considered for piezoelectric properties[@Nanamatsu72; @Gupta76; @Boonchun2010] in the past and is for the most part considered an insulator. However, Boonchun and Lambrecht [@Boonchun11] suggested it might be worthwhile considering as a semiconductor electronic material and showed in particular that it could possibly be n-type doped by Ge. In view of the recent interest in $\beta$-Ga$_2$O$_3$ as ultra-wide semiconductor for power electronics, which is also n-type by doping with Si, Sn or Ge, this makes LiGaO$_2$ worth revisiting, in particular from the point of view of defects and doping.
Recently, Electron Paramagnetic Resonance (EPR) experiments on irradiated samples of LiGaO$_2$ were reported by Lenyk [*et al. *]{}[@Lenyk18] and reported EPR signals for both the $V_\mathrm{Ga}$ and $V_\mathrm{Li}$. Here we present a computational study of the EPR parameters of these defects and in particular compare the calculated $g$-tensors and superhyperfine (SHF) interactions with both Ga and Li neighbors of the O on which the spin is localized for different possible localization sites of the spin. We will show that this confirms the experimentally deduced models for the spin-localization.
The $g$-tensor is calculated using the Gauge Including Projector Augmented Wave (GIPAW) method.[@Pickard01; @Pickard02; @Gerstmann10; @Ceresoli10] This is a Density Functional Perturbation Theory (DFPT) method to calculate the linear magnetic response of a periodic system onto an external magnetic field. It is implemented in the code QE-GIPAW,[@gipaw] which is integrated within the Quantum Espresso package.[@QE-2009] At present the QE-GIPAW code is not yet capable of dealing with orbital dependent density functionals such as DFT+U [@Anisimov91; @Anisimov93; @Liechtenstein95] or hybrid functionals.[@HSE03; @HSE06] The latter are required to ensure a strong localization of the spin-density on a single O. We thus use DFT with on-site Coulomb corrections $U$ on O-$p$ orbitals with the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to relax the structure and also calculate the SHF interactions at this GGA+U level but calculate the $g$-tensor using wavefunctions at the PBE-GGA level while keeping the structure fixed. This procedure was found to be adequate in prior work on EPR parameters in $\beta$-Ga$_2$O$_3$.[@Skachkov19; @Skachkov19-mg] The GGA+U structures were in good agreement with a previous study of the same defects[@Dabsamut] using the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional.
We focus on the EPR active states $V_\mathrm{Ga}^{2-}$ and $V_\mathrm{Li}^{0}$ which both correspond to a $S=1/2$ single unpaired electron state. We find that in the DFT+U approach with a value of $U=4$ eV on O-$p$ orbitals, the spin-density becomes well localized on a single O $p$-orbital but depending on the initial displacements given to the O, we can get it to localize on different O neighbors. Keeping this relaxed structure, the spin then stays localized on the single O even when recalculating it in GGA. We need to distinguish the following O-sites. First in the crystal structure, the O$_I$ sits on top of Li and the O$_{II}$ sits on top of Ga in the ${\bf c}$ direction. Secondly, we call an O [*apical*]{} if it sits right above the vacancy (in the [**c**]{}-direction) and [*basal*]{} if it lies in the ${\bf ab}$-plane below it. The basal plane O can still be either O$_I$ or O$_{II}$. Our results for all the cases considered are summarized in Fig. \[Fig1\], the $g$-tensors are summarized in Table \[tabeprg\] and the SHF tensors are given in Table \[tabeprA\].
{width="16cm"}
Model $g$-tensor
------------------------------ ---------- -------- ------------ --------
V$_\mathrm{Li}$ (a) O$_I$ 2.0373 2.0288 2.0078
$\theta$ 89 89 2
$\phi$ -34 56
V$_\mathrm{Li}$ (b) O$_{II}$ 2.0302 2.0119 2.0356
$\theta$ 69 71 29
$\phi$ 17 -65 64
V$_\mathrm{Li}$ (b) O$_{I}$ 2.0403 2.0125 2.0301
$\theta$ 90 73 17
$\phi$ -27 64 62
Expt.[@Lenyk18] 2.0088 2.0205 2.0366
$a$ $b$ $c$
V$_\mathrm{Ga}$ (a) O$_{II}$ 2.0220 2.0514 2.0078
$\theta$ 84 88 6
$\phi$ -35 55
V$_\mathrm{Ga}$ (b) O$_I$ 2.0081 2.0228 2.0449
$\theta$ 88 85 5
$\phi$ -16 74
Expt.[@Lenyk18] 2.0155 2.0551 2.0032
$a$ $b$ $c$
: Calculated $g$-tensor for Li and Ga vacancies. In the calculated results, the $g$-tensor is given in terms of three principal values followed by the $\theta$ (polar) and $\phi$ (azimuthal) angles in degrees measured from ${\bf c}$ and ${\bf a}$ respectively. The experimental values are along the directions indicated. \[tabeprg\]
Model Ga1 Ga2 Li1 Li2
------------------------------ ---------- -------- -------- -------- -------- -------- -------- ------- ------- ------- ------ ------- -------
V$_\mathrm{Li}$ (a) O$_I$ -19.93 -18.91 -18.67 -23.56 -22.54 -22.30 0.42 -1.75 -1.40
$\theta$ 80 61 31 80 61 31 81 90 9
$\phi$ -51 33 57 -51 33 57 62 -28 60
V$_\mathrm{Li}$ (b) O$_{II}$ -18.82 -18.64 -19.30 -23.51 -25.05 -24.00 -1.66 0.52 -1.44
$\theta$ 80 88 10 68 48 51 34 68 66
$\phi$ -34 56 -48 -85 27 -14 62 9 -70
V$_\mathrm{Li}$ (b) O$_{I}$ -16.17 -16.73 -15.75 -21.89 -21.86 -22.33 1.27 -2.00 -1.86
$\theta$ 34 88 57 33 58 83 12 89 78
$\phi$ 81 -12 77 34 49 -46 65 -31 59
Expt.[@Lenyk18] 24.30 25.00 25.30 24.30 25.00 25.30
$a$ $b$ $c$ $a$ $b$ c
V$_\mathrm{Ga}$ (a) O$_{II}$ -33.11 -33.28 -34.00 -1.89 0.60 -1.51 0.68 -1.89 -1.53
$\theta$ 67 90 23 89 80 10 80 88 10
$\phi$ 67 -23 68 90 0 -62 28
V$_\mathrm{Ga}$ (b) O$_I$ -32.41 -31.91 -32.28 -1.68 -1.47 0.63 0.64 -1.68 -1.96
$\theta$ 72 55 41 89 80 10 66 72 31
$\phi$ 56 -47 -11 88 -2 -70 11 68
Expt.[@Lenyk18] 37.50 37.40 35.90
$a$ $b$ $c$
For the $V_\mathrm{Ga}$ we examine both the apical and basal plane O$_I$ as atom for the hole to localize on. As shown in Fig. \[Fig1\] (lower-left) and in Table \[tabeprg\] we find the $g$-tensor for the apical O$_{II}$ has its smallest $g$ along the direction of the spin density, which is along ${\bf c}$. This agrees with experiment.[@Lenyk18] The largest principal axis (principal axis corresponding to largest $\Delta g$) in the ${\bf ab}$-plane is $55^\circ$ from ${\bf a}$ so closer to ${\bf b}$ which also agrees with experiment. In fact it is close to the O$_{II}$-Ga direction. The $\Delta g$ themselves are in agreement to about $\pm0.005$. For the basal plane O$_I$ location of the spin, (Fig. \[Fig1\] lower-right) on the other hand the main principal axis of the $g$ tensor is along ${\bf c}$. In both cases it is perpendicular to the spin direction of the spin density $p$-orbital which corresponds to the lowest $\Delta g$ direction. The SHF interaction (given in Table \[tabeprA\]) in both cases is with one Ga atom because obviously the O on which the spin has localized has already lost one of its Ga neighbors and each O is coordinated with two Ga and two Li. It is called a SHF interaction because the nucleus with which the electron spin is interacting is not on the atom on which the spin is localized but one of its neighbors. The hyperfine tensor $A$ is nearly isotropic with a value of about 33 G in excellent agreement with the experimental values of about 37 G. Our values are about 10% underestimated. In agreement with experiment we find a slightly larger $A$ component in the ${\bf c}$ direction for the apical O. The hyperfine with O is not observed because O is more than 99.9 % isotopically in a form which does not carry a nuclear spin. The hyperfine principal axes are indicated by the small arrows in Fig. \[Fig1\] and are seen to be close to the bond directions rather than the overall crystal axes. While the Li SHF interactions were not observed we give the calculated values for them in Table \[tabeprA\] in case future measurements would be able to measure them. The reason why they are much smaller is that the atomic wavefunction on the Li nuclear sites are much smaller than on the Ga.
The $V_\mathrm{Li}$ with spin localized on an apical O$_I$ has its main $\Delta g$-tensor component at about 30$^\circ$ from the ${\bf a}$-axis and its lowest component and spin density along ${\bf c}$ as can be seen in Fig. \[Fig1\] (upper left). This, however does not agree with the experimental data of Lenyk [*et al. *]{}[@Lenyk18] who find the $\Delta g$ tensor to be oriented with its highest value along ${\bf c}$. We have calculated two distinct configurations with spin localized on a basal plane O$_{I}$ and O$_{II}$ (See Table \[tabeprg\]). For the O$_{II}$ case we find that the lowest $\Delta g$ is coincident with the direction of the spin density and is close to the bond direction from $V_\mathrm{Li}$ to the O$_{II}$. So, it is tilted away from the ${\bf ab}$-plane by about 30$^\circ$ and close to 60$^\circ$ degrees from the ${\bf a}$-axis. Note, however, that there is an equivalent O$_{II}$ along the ${\bf a}$ axis in the ${\bf ab}$-plane projection, which is simply 120$^\circ$ rotated from the one reported in Table \[tabeprg\]. The highest $g$-component principal axis has to be perpendicular to this and indeed we find it to be tilted about 30$^\circ$ from the ${\bf c}$-axis. This model agrees closely with the one proposed by Lenyk [*et al. *]{}[@Lenyk18] with the sole difference that they consider the equivalent O$_{II}$ in the ${\bf a}$-direction. As the authors mention, the occurrence of several distinct magnetic orientations prevents them from carrying out a full study of the angular variation with magnetic field because of the overlap of different signals. As for the O$_{I}$ basal plane, neighbor, in that case the lowest $g$-component is along the corresponding V$_\mathrm{Li}-\mathrm{O}_I$ direction at about 60$^\circ$ from the ${\bf a}$ axis. However, the largest $g$ is then found at about $-30^\circ$ from ${\bf a}$ and tilted toward the basal plane. This does not agree with the center identified by Lenyk [*et al. *]{}[@Lenyk18]
The SHF splitting in $V_\mathrm{Li}$ with spin localized on O$_{II}$ is with two nonequivalent Ga. Although all Ga atoms are equivalent in the perfect crystal, the local symmetry is broken. The Ga with smaller SHF $A$ tensor lies closer to the $V_\mathrm{Li}$ than the other which lies opposite to it from the $O_{II}$ on which the spin is localized. In the experiment, also a slightly nonequivalent Ga-SHF splitting was reported but they estimated the $A$’s to differ by only 4% whereas we find them to differ by about 20%. In agreement with experiment the $A$ tensors are found to be nearly isotropic. The experimental value for the SHF splitting is closer to the larger of the two calculated $A$ and is in good agreement with experiment. For the apical O$_I$ case, one would expect the two Ga neighbors to be equivalent but in the calculation, they are still found to differ by about 20%, which may result from the symmetry breaking in the relaxation calculation.
Having identified the apical O as the location of the spin density near a $V_\mathrm{Ga}$ and the basal plane O near a $V_\mathrm{Li}$ that best agree with experiment, we may ask whether these indeed correspond to the lowest total energy. It turns out, however, that the energy differences between these different localization sites is quite small. We find that the $V_\mathrm{Ga}$ has 0.01 eV higher energy per 128 atom cell in the apical than the basal plane site within PBE0,[@PBEh] (this is a hybrid functional with 25 % exact and unscreened exchange) so opposite to the experimental identification. For the $V_\mathrm{Li}$ it is the apical oxygen that was found to have the lower energy by 0.002 eV. In the HSE functional, the apical site was found upon automatic relaxation for both cases. Clearly these energy differences are too small to trust within DFT or at least this is very challenging for any level of theory. Therefore we expect that several of these slightly different forms of the vacancy EPR centers with spin localized on different O-neighbors could be present in experiment but the overlap of these signals would make it difficult to disentangle them. The apical O$_{II}$ for $V_\mathrm{Ga}$ and basal-plane O$_{II}$ for $V_\mathrm{Li}$ agree best with the experimental observations but in the $V_\mathrm{Li}$ case, there would still be two differently oriented forms of this same defect.
Finally, we address the question under what conditions these EPR signals were observed. A hybrid functional study of the native defects in LiGaO$_2$ was recently presented by some of us.[@Dabsamut] From that study, we find that the $V_\mathrm{Li}^0$ has lower energy than the $V_\mathrm{Ga}^0$ for all chemical conditions as restricted by the formation of competing binary compounds and under Li-poor conditions can be lower than 1 eV. The $V_\mathrm{Ga}$ usually has quite high energy (10 eV for Ga-rich conditions and $\sim$4.5 eV under the most Ga-poor, Li-rich conditions allowed) and is not expected to occur in significant concentration in equilibrium. In contrast, the $V_\mathrm{Li}^{-}$ is found to be the major acceptor compensating the Ga$_\mathrm{Li}^{2+}$ antisite and is thus expected to be present in the as grown samples. The $V_\mathrm{Li}$ occurs in 0 and $-1$ charge states, the former of which contains an unpaired spin and is hence EPR active. Its $0/-$ transition level lies at 1.03 eV above the valence band maximum (VBM). The Ga-vacancy accommodates four charge states, 0, $-1$,$-2$, $-3$. The Fermi level is pinned by the compensation of Ga$_\mathrm{Li}^{2+}$ antisites with $V_\mathrm{Li}^{-}$ and to some extent by Li$_\mathrm{Ga}^{2-}$ in Li-rich conditions. In both cases, the Fermi level lies deep below the conduction band between 2.7-3.8 eV above the VBM straddling the $2-/3-$ transition level of the $V_\mathrm{Ga}$, which occurs at 3.3 eV. The $V_\mathrm{Ga}$, once it is formed, may thus be expected to be found in the EPR active $q=-2$ charge state in particular for the deeper Fermi level position, which occurs for more realistic assumptions of the O-chemical potential.
The above findings agree with the observations of Lenyk [*et al. *]{}[@Lenyk18] that high-energy particle irradiation is required to create the $V_\mathrm{Ga}$. However, the fact that they do not require to be optically activated once formed indicates a $2-$ charge state after irradiation. On the other hand the Li-vacancies were found to be present already in as-grown material. This however does not imply the material was Li-poor. Even under both Li and Ga rich conditions, the $V_\mathrm{Li}$ has an energy of formation significantly lower than that of Ga. However, the fact that its $0/-$ level lies only 1.02 eV above the VBM clearly explains why the Li must be activated optically by removing an electron from it. In the experiments by Lenyk [*et al. *]{}[@Lenyk18] this is achieved by application X-rays.
In conclusion, our first-principles calculations confirm the experimental assignment of the EPR centers of $V_\mathrm{Ga}$ and $V_\mathrm{Li}$ by Lenyk [*et al. *]{}[@Lenyk18] For the $V_\mathrm{Ga}$ the spin is localized on an apical O$_{II}$ and for the $V_\mathrm{Li}$ it is on a basal plane O$_{II}$. The orientations of the principal axes of the $g$-tensor and $A$-tensors are found to be closely related to the bond directions and in the $V_\mathrm{Li}$ case two different orientations of the defect center with respect to the crystal axis should exist with overlapping spectra. The EPR parameters for alternative localizations of the spin on different O-neighbors were also calculated and found to be different. As these different forms of localization of the spin have total energies close to each other they might possibly occur in the real systems and we hope that providing the associated parameters here could assist in disentangling these different EPR centers. For the cases observed so far, our $g$-tensor and SHF interaction parameters are in good agreement with experiment. Our calculations also explain why the $V_\mathrm{Ga}$ defects require high energy radiation to be formed but no further optical activation while the opposite is the case for the $V_\mathrm{Li}$.
[**Acknowledgements:**]{} The work at CWRU was supported by the U.S. National Science Foundation under grant No. 1755479. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) Stampede2 at the UT Austin through allocation TG-DMR180118.
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abstract: 'Modal interpretations have the ambition to construe quantum mechanics as an objective, man-independent description of physical reality. Their second leading idea is probabilism: quantum mechanics does not completely fix physical reality but yields probabilities. In working out these ideas an important motif is to stay close to the standard formalism of quantum mechanics and to refrain from introducing new structure by hand. In this paper we explain how this programme can be made concrete. In particular, we show that the Born probability rule, and sets of definite-valued observables to which the Born probabilities pertain, can be uniquely defined from the quantum state and Hilbert space structure. We discuss the status of probability in modal interpretations, and to this end we make a comparison with many-worlds alternatives. An overall point that we stress is that the modal ideas define a general framework and research programme rather than one definite and finished interpretation.'
address: |
Institute for the History and Foundations of Science\
Utrecht University, P.O.Box 80.000\
3508 TA Utrecht, The Netherlands
author:
- Dennis Dieks
date: January 2007
title: Probability in modal interpretations of quantum mechanics
---
interpretation of quantum mechanics; modal interpretation; probability; many worlds
Introduction: the modal point of view
=====================================
Modal interpretations of quantum mechanics are inspired by two main ideas. The first is to adopt a realist stance, in the specific sense of interpreting the theory’s mathematical formalism in terms of properties and relations of physical systems, independent of whether or not human observers are around. The second motivating idea is that the relation between the formalism of quantum theory and physical reality is to be taken as *probabilistic*. That is, according to modal interpretations the quantum formalism does not tell us what actually is the case in the physical world, but rather provides us with a list of possibilities and their probabilities. The modal viewpoint is therefore that quantum theory is about what *may* be the case—in philosophical jargon, quantum theory is about *modalities*.
This viewpoint is at odds with the operational viewpoint that the quantum formalism should be interpreted as a recipe for predicting measurement outcomes. Modal interpretations strive for a description of the world in terms of objective, man-independent features, both on the macroscopic and (sub)microscopic level. These features may turn out to be monadic properties of physical systems (perhaps very exotic ones from a classical point of view), or perhaps rather a structural network of relations or perspectival properties; these are things to be decided during the detailed elaboration of the interpretation.
In accordance with this modal philosophy, measurement results are nothing but a subclass of the physical things existing in our world: positions of pointers attached to measuring devices, marks on computer tapes, etc. A measurement is a physical interaction between an object system and a measuring device, and should be treated like all other interactions. Modal interpretations therefore only need the quantum formalism with unitary evolution, i.e. the standard formalism without collapses. Indeed, as soon as the idea that measurements are something special is abandoned, the motivation for associating them with an own evolution mechanism, collapses, disappears; only the unitary (Schrödinger-like) evolution remains. Modal interpretations thus fall into the class of no-collapse interpretations of quantum mechanics.
So we assume that quantum mechanical states provide a description of physical systems. It is good to be explicit here about the distinction between the state as it is defined within the mathematical formalism (a vector in Hilbert space, or a density operator) and *physical features of the represented systems*—the latter are not mathematical entities. This distinction is important because it is not an a priori obvious matter what the exact relation between the mathematical state and physical reality is; it is not self-evident in what way the mathematical formalism represents. Discussions about the interpretation of quantum mechanics sometimes neglect this point by not clearly distinguishing between the formalism and what is represented by it (e.g., by accepting as self-evident that a $+$-sign in a superposition means *joint existence*).
Our task in modal interpretations is thus to endow the standard formalism, without collapses, with physical meaning. We need interpretational rules that tell us how the mathematical formalism relates to physical reality. Such rules do not constitute an addition to the formalism of quantum mechanics: they are not part of the mathematical formalism at all but establish a relation between this formalism and the world. Any interpretation of quantum mechanics will need to specify such a correspondence with reality. Mathematical theories cannot fix their own interpretation—as pieces of pure mathematics they do not contain information about their possible applications.
Quantum mechanics has a familiar history of enormously successful physical applications that make use of certain basic interpretational rules that have proved their mettle; there is no reason to doubt these. One of these basic interpretational rules is that physical quantities are represented by hermitian operators (observables). We will accept this standard correspondence (but will change some other rules).
A natural form our interpretational question now takes is: which physical quantities—represented by hermitian operators—can be assigned a *definite value*, when it is given that the physical system is represented by a specific mathematical state. Such definite values correspond to properties possessed by the system. It might turn out in later developments that it is more appropriate to focus on relations or perspectival properties instead of the monadic properties represented by definite values of physical quantities. But let us here focus on the standard modal line, which relies on the attribution of properties in the sense of definite values of physical quantities.
There may seem to be an easy answer to the question about the relation between states and properties. Standard quantum mechanics tells us that the state of a system is given by a density operator $W$, obtained by ‘partial tracing’ from the generally *entangled* state of the system and its environment. Now consider $W$’s diagonal decomposition in terms of orthogonal projections: $$\begin{aligned}
W & = & \sum_{i} p_{i}|\psi_{i}\rangle\langle\psi_{i}|,\\
& & \langle\psi_{i}|\psi_{j}\rangle = \delta_{ij}.\end{aligned}$$ This decomposition is unique if the coefficients $p_i$ are all unequal (the case of non-uniqueness will be discussed later on). There is a well-known way of interpreting such ‘mixed states’, namely via *ignorance*: according to it the physical system possesses one of the properties corresponding to the projectors $|\psi_{i}\rangle\langle\psi_{i}|$, but we don’t know which one. In the special case of a pure state this reduces to the standardly accepted eigenstate-eigenvalue rule, according to which properties are only definite if the state is an eigenvector of the corresponding projection operator.
However, there are well-known objections to the general validity of this interpretation of $W$. The most important problem is that if it were true that the partial system possessed the property corresponding to $|\psi_{j}\rangle\langle\psi_{j}|$, then according to the eigenstate-eigenvalue link the system should be in the associated eigenstate $|\psi_{j}\rangle$. Analogously, the other partial system (the environment) must be in a pure state as well. But then a well-known theorem says that the total state must be the product state of these two pure states. This is in conflict with our initial assumptions: if the total state is $|\psi_{j}\rangle \otimes |\xi_{j}\rangle$, or a mixture of such states, there can be no entanglement whereas we assumed that in general the total state *is* entangled. The attribution of one of the properties $|\psi_{j}\rangle\langle\psi_{j}|$ therefore leads to contradictions.
To side-step this objection, modal interpretations propose to *drop the rule that a system can only possess a well-defined value of a physical magnitude if it is represented by an eigenstate of the corresponding observable*. In its stead comes a new interpretative principle according to which the mathematical state represents situations with definite physical properties even if this state is a superposition of eigenstates of the corresponding observables. The basic idea of interpreting the formalism in this vein has been put forward, with a number of variations, by several authors [@fra1; @fra2; @koc; @die1; @die2]. Bas @fra1, who seems to have been the first to think along these lines, coined the term ‘modal interpretation’; but we still have to explain what the typically *modal* aspects are. Let us mention some more details in order to do so.
Consider the quantum mechanical treatment of a composite physical system, consisting of two parts. In this case, the total Hilbert space can be decomposed: $\mathcal{H} = \mathcal{H}_1 \otimes
\mathcal{H}_2$. According to a famous theorem (Schmidt, Schrödinger) there is a corresponding *biorthogonal decomposition* of every pure state in $\mathcal{H}$: $$|\psi\rangle = \sum_{k} c_{k}|\psi_{k}\rangle\otimes
|R_{k}\rangle, \label{eq:modal}$$ with $|\psi_{k}\rangle$ in $\mathcal{H}_1, |R_{k}\rangle$ in $\mathcal{H}_2$, $\langle\psi_{i}|\psi_{j}\rangle = \delta_{ij}$ and $\langle R_{i}|R_{j}\rangle = \delta_{ij}$. This decomposition is unique if there is no degeneracy among the values of $|c_{k}|^{2}$.
One well-known version of the modal interpretation gives the following physical interpretation to this mathematical state. The system represented by vectors in $\mathcal{H}_1$ possesses exactly [*o*ne]{} of the physical properties associated with the set of projectors $\{|\psi_{k}\rangle\langle\psi_{k}|\}$, and definitely does not possess the others. That is, exactly one of the mentioned projectors is assigned the definite value $1$, the others get the definite value $0$. The interpretation thus selects, on the basis of the form of the state $|\psi\rangle$, the projectors $|\psi_{k}\rangle\langle\psi_{k}|$ as definite-valued magnitudes. All physical magnitudes represented by maximal hermitian operators with spectral resolution given by $\Sigma a_{k}
|\psi_{k}\rangle\langle\psi_{k}|$ are also definite-valued, since they are functions of the definite-valued projectors; their possible values are given by the functions in question applied to the values assumed by the projections.
In the case of degeneracy, that is $|c_{j}|^{2} = |c_{i}|^{2}$, for $i,j \in I_{l}$ (with $I_{l}$ a set of indices), the biorthogonal decomposition (\[eq:modal\]) still determines a unique set of projection operators, but these will generally be multi-dimensional. The one-dimensional projectors have in this case to be replaced by projectors $P_{l} = \sum_{i\in I_{l}}
|\psi_{i}\rangle\langle\psi_{i}|$. The physical properties now correspond to this more general set of projectors. The general class of definite physical quantities contains in this case non-maximal hermitian operators in whose spectral resolution such multi-dimensional projectors occur.
We have just stipulated that only *one* of the values that can be assumed by the definite-valued observables is actually realized. This raises the question of what the *probability* is of the $l$-$th$ alternative being actual. In accordance with the standard Born rule, this probability may be taken as $|c_{l}|^{2}$ (in the case of degeneracy this becomes $\sum_{i\in
I_{l}} |c_{i}|^2$).
These details are mentioned here to clarify the spirit of the modal ideas. We will devote a more fundamental discussion to their justification, and to possible alternatives, later on. For example, how does the notion that only *one* possibility is realized compare to the many-worlds alternative, according to which *all* terms in the superposition correspond to actualities? And is it possible to derive and justify the Born probability rule, instead of just positing it?
Taking for granted the probabilistic character of the interpretation and the Born formula for the moment, we face the consequence that, in general, the physical situation could have been different from what it actually is, given the mathematical state. Here we have the ‘modal’ aspect of the interpretation: the mathematical state does not fix what is actual, but specifies what *may* be the case. It follows from the probabilistic nature of the relation between the state and the world that the same physical situation may be realized ‘contingently’ (if the associated probability is smaller than one; things could have been different in this case) or ‘necessarily’ (if the probability is one).
The just-explained interpretational rule ascribes definite physical properties to physical systems, even if the state is a superposition of eigenstates of the corresponding observables. This has the following consequence. According to the von Neumann measurement scheme, the situation after a measurement will typically be described by a superposition of the form (\[eq:modal\]), with $|\psi_{k}\rangle$ denoting states of the object system and $|R_{k}\rangle$ states of the measuring device (‘pointer position states’). The modal interpretation of this state is that exactly one of the pointer positions is realized, with a probability given by the Born rule. This is the modal solution of the measurement problem: definite measurement outcomes are predicted even though there are no collapses of the wave function.
Definite-valued observables
===========================
Let us, after this review of the main ideas take a step back and look in a more systematic way at the interpretational possibilities that are left open by the motivating ideas behind the modal interpretation. This will make it clearer to what extent the just-discussed standard version can be justified, and will also enable us to say something about the modal interpretation’s position among its competitors.
A core desideratum in devising the modal interpretation was the wish to deny a special status to measurements: measurements should be dealt with in the same way as ordinary physical interactions. Combined with the desire to keep the usual quantum formalism intact, this leads to a specific class of interpretations, namely no-collapse interpretations, in which there is only unitary evolution in Hilbert space. The task of all these interpretations is to link the unitarily evolving states in Hilbert space to physical features of the represented systems. The simplest programme for accomplishing this consists in the attempt to define a set of definite-valued observables from the mathematical state. These are observables that can be assigned a well-defined numerical value, and that thus fix physical quantities. Now, it is a notorious feature of the Hilbert space formalism that not all observables (i.e., hermitian operators) can be assigned definite values simultaneously (if we respect the functional relations between them); this follows from the Kochen-Specker theorem. The question therefore arises: What is the maximum set of observables that are definable from the quantum state and can jointly be given definite values without getting into contradictions? The usual quantum—Born—probabilities should become expressible as probability distributions over these definite values. Indeed, as already emphasized, measurement results are special cases of system properties in this approach, and the Born probabilities therefore come to pertain to the values of physical quantities—in the same way as probabilities in classical phase space.
In our search for definite-valued observables it is possible to include interpretations like the Bohm interpretation if we allow for the possibility that there is a preferred observable $R$ that is always definite, for all quantum states (in the Bohm theory position plays this role). The situation in which no privileged observable exists then becomes a special case. It should be noted, however, that there is a tension between this assumption of a preferred observable and the desideratum that the ordinary quantum formalism should be retained as much as possible: in the usual Hilbert space formalism there is no preferred observable. After discussing the case with a preferred observable we will therefore focus on the alternative, in which there is no such *a priori* fixed physical quantity, and in which the definite-valued observables are determined by the quantum state alone.
Consider an arbitrary pure quantum state represented by a ray $\psi$ in the Hilbert space $\mathcal{H}$. Let the Boolean algebra generated by the eigenspaces of the preferred observable $R$ be denoted by $\mathcal{B}(R)$. The usual quantum mechanical probabilities of the values of $R$, calculated via the Born rule applied to $\psi$, can be represented by an ordinary Kolmogorovian measure over the 2-valued homomorphisms (consistent assignments of truth values $0$ and $1$) on $\mathcal{B}(R)$. We now ask for the maximal lattice extension $\mathcal{D}(\psi, R)$ of $\mathcal{B}(R)$, formed by adding eigenspaces of other observables, such that we can represent in the same way the Born probabilities both for values of $R$ and for the values of these other observables.
Since we do not want to accept more mathematical structure not automatically present in Hilbert space than already introduced by the presence of $R$ as a preferred observable, we require these definite-valued observables to be definable solely in terms of $\psi$ and $R$. It follows that each element of $\mathcal{D}(\psi,
R)$ should be invariant under all automorphisms of Hilbert space that preserve both the ray $\psi$ and the eigenspaces of $R$. This requirement [@dieks5] is slightly stronger than the one made in [@bub0; @bub; @bub2]; we shall comment on the difference below. This invariance, expressing definability from $\psi$ and $R$, will do most of the work in determining our definite-valued observables.
Let us consider an $n$-dimensional Hilbert space $\mathcal{H}$, and an observable $R$ with $m \leq n$ distinct eigenspaces $r_{i}$ of $\mathcal{H}$. Let $\psi_{r_{i}}$, $i = 1,2, \ldots, m$, denote the orthogonal projections of $\psi$ onto these eigenspaces $r_{i}$. Now, the set of automorphisms that leave $\psi$ and $R$ invariant includes all automorphisms that are equal to the unity operator when they operate on vectors orthogonal to $r_{i}$, and are rotations around $\psi_{r_{i}}$ or reflections with respect to the space orthogonal to $\psi_{r_{i}}$ inside $r_{i}$—since these transformations leave the eigenspaces of $R$ and the projections of $\psi$ the same, $R$ and $\psi$ themselves are invariant. Now consider a projection operator $P$ that is to correspond to a definite-valued property; so $P$ must be definable from $\psi$ and $R$. If the subspace of $\mathcal{H}$ on which $P$ projects is contained in one of the $r_{i}$, this subspace should therefore be invariant under the mentioned rotations and reflections with respect to $\psi_{r_{i}}$. This leaves four possibilities for the subspace in question: it can be the null-space, $\psi_{r_{i}}$, $\psi_{r_{i}}^{\perp} \wedge r_{i}$, or $r_{i}$.
In the general case, the subspace on which $P$ projects will not be contained in one of the $r_{i}$ spaces, but will have non-zero projections on a number of them. The requirement that the subspace remains invariant under the above-mentioned automorphisms now implies that its projections on the different spaces $r_{i}$ are each either null, $\psi_{r_{i}}$, $\psi_{r_{i}}^{\perp} \wedge
r_{i}$, or $r_{i}$. All possible subspaces on which $P$ may project are therefore found by taking one of these latter spaces for each value of $i$, and constructing their span.
The lattice of subspaces that may correspond to definite propositions is therefore generated by all sublattices $\{0,\psi_{r_{i}}, \psi_{r_{i}}^{\perp} \wedge r_{i},
r_{i}\}$[^1]. In the case that $r_{i}$ is one-dimensional, $\psi_{r_{i}}$ is equal to $r_{i}$ and $\psi_{r_{i}}^{\perp} \wedge r_{i}$ equals $0$, so that the sublattice reduces to $\{0,r_{i}\}$.
It is clear from this construction that the resulting set of definite-valued projection operators is indeed a lattice: it is closed under the lattice operations of disjunction and conjunction (corresponding to taking the span or intersection of the associated eigenspaces). Moreover, the lattice is Boolean: all projection operators in it commute with each other. Therefore, no Kochen and Specker-type paradoxes can arise, and the quantum mechanical probabilities (including joint probabilities) can be represented by means of a classical measure on the lattice.
The above construction made use of the existence of a preferred observable, namely $R$. As stated above, it is important to see what happens if the state $|\psi \rangle$ and the Hilbert space structure are the only entities used to define the definite properties.
A possible way of implementing this is to take the projection on $|\psi \rangle$ itself for $R$. If we denote the subspace orthogonal to $\psi$ by $\psi^{\perp}$, we obtain the definite lattice consisting of the subspaces $\{0, \psi, \psi^{\perp},
\mathcal{H}\}$. The same result is obtained if we take the unity operator on $\mathcal{H}$ for $R$. This therefore leads to the ‘orthodox’ property assignment: only observables of which $|\psi\rangle$ is an eigenvector qualify as definite-valued [@bub]. This traditional way of assigning properties returns us to the measurement problem, because after a measurement the combined system of measuring device and object system ends up in an entangled state that according to this assignment does not correspond to a definite pointer property. However, on second thought the situation is more complicated. The projection operator $|\psi\rangle \langle\psi|$ is an observable of the *total* system, and the just-mentioned property assignment pertains likewise to this total system. But we are really interested in the *individual* properties of device and object taken by themselves. Therefore, we need substitutes for $|\psi\rangle\langle\psi|$ that represent the states of these individual systems. In the context of standard quantum mechanics such operators are readily available, namely the density operators for the partial systems. Postponing for a moment possible doubts about the status of these operators in this new context, we are thus prompted to consider the definite lattices that result if the operators $W_{1}\otimes I$ and $I\otimes W_{2}$ are taken for $R$ (the total Hilbert space is the tensor product of Hilbert spaces belonging to the partial systems, $\mathcal{H} = \mathcal{H}_{1}
\otimes \mathcal{H}_{2}$).
Denote the eigenspaces of $W_{1}\otimes I$ by $w_l \otimes
\mathcal{H}_{2}$, $l= 1,2,\ldots$. Now write $|\psi\rangle$ as a biorthogonal decomposition $$|\psi\rangle = \sum_{k,j} c_{k,j}|\alpha_{k,j}\rangle\otimes
|\beta_{k,j}\rangle, \label{eq:modal1}$$ with $|\alpha_{k,j}\rangle$ in $\mathcal{H}_1, |\beta_{k,j}\rangle$ in $\mathcal{H}_2$, $\langle\alpha_{l,i}|\alpha_{m,j}\rangle =
\delta_{lm}.\delta_{ij} = \langle \beta_{l,i}|\beta_{m,j}\rangle$. The second index, $j$, takes possible degeneracies into account: $|c_{k,j}|^{2}$ depends only on $k$, not on $j$. The projection of $|\psi\rangle$ on $w_l \otimes \mathcal{H}_2$ is given by $|\psi_{l}\rangle= \sum_{j} c_{l,j}|\alpha_{l,j}\rangle\otimes
|\beta_{l,j}\rangle$. As we have seen, it follows that the lattice of definite properties is generated by the sublattices $\{0,
\psi_{l}, \psi_{l}^{\perp} \wedge (w_l \otimes \mathcal{H}_2),
w_{l} \otimes \mathcal{H}_2\}$. We can restrict this lattice to a lattice of definite properties of the first system alone (represented in $\mathcal{H}_1$) by looking for those definite projections in the lattice that possess the form $P \otimes I$. The projection operators $P$ can in this case be taken to represent properties of system $1$ by itself. Inspection of the lattice shows that all projections of the sought form are generated by the projectors $P_{w_{l}} \otimes I$. The restriction of the lattice of definite properties of the combined system to a lattice of definite properties of system $1$ is therefore the Boolean lattice generated by the projections $P_{w_{l}}$. These are exactly the properties assigned by modal interpretations of the type discussed in the Introduction [@die1; @die2; @vermaas2]. In measurement situations these definite properties should correspond to pointer positions.[^2]
The analysis just given is similar to the one proposed by Bub and Clifton [@bub0; @bub; @bub2]. The difference is that these authors required the *set* of definite properties *as a whole* to be definable from $|\psi\rangle$ and $R$, whereas here we have imposed the stronger demand that the *individual definite properties* be so definable. Given the idea that $|\psi\rangle$ should fix as many elements of the interpretation as possible, our stronger requirement seems the more natural one; moreover, it makes the analysis considerably simpler. As was to be expected, the lattice of definite properties that we found above on the basis of our stronger requirement is included in the lattice determined by Bub and Clifton. The latter possesses more ‘fine structure’: The Bub-Clifton lattice contains projection operators that cannot be defined individually but still belong to the set of projectors defined as a whole. However, these differences are not very significant. In the case of the measurement-like situation we have just discussed, the only difference is that in the Bub-Clifton approach all individual one-dimensional projections within the null-space of $W_{1}$ are definite, whereas in our approach it is only the total projector on this null-space that is definite-valued.
In this derivation we took $W_{1}$ for the state of system $1$. This is standard practice in quantum mechanics; however, the usual justification relies on the probabilistic interpretation of the theory and the Born rule. It would be preferable not to presuppose anything about this at the present stage. Indeed, in the next section we will make an attempt to derive the Born rule. It is therefore desirable to have a derivation of the definite-valued observables of the partial systems that does not presuppose that the density operators $W_{i}$ characterize the individual systems.
In order to achieve this we again make use of the biorthogonal way of writing $|\psi \rangle$, Eq. (\[eq:modal1\]). As before, our aim is to determine maximal sets of properties of system $1$ that can be defined from this state. We will use that $\mathcal{H}$ is the tensor product of the Hilbert spaces of the individual systems $1$ and $2$, $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$: we want the definite properties of system $1$ to be invariant under automorphisms that leave $|\psi \rangle$ the same and that respect this factorization of the total Hilbert space. These automorphisms have the form $U_{1} \otimes U_{2}$, with $U_{1}$ and $U_{2}$ defined on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively [@dieks4; @dieks5]. So we ask which automorphisms $U_{1} \otimes U_{2}$ leave $|\psi \rangle$ invariant, and which projectors in $\mathcal{H}_1$ remain the same under their operation (in other words, under the operation of the associated $U_{1}$).
To investigate this we must have a closer look at the invariance properties of (\[eq:modal1\]). This biorthogonal decomposition is unique up to certain unitary transformations *within* the subspaces spanned by the vectors $\{|\alpha_{k,j}\rangle\}_j$ (and $\{|\beta_{k,j}\rangle\}_j$), with fixed values of $k$ (these are the ‘degeneracy subspaces’, labelled by values of $k$). This can be seen in the following way. The component of $|\psi \rangle$ *within* such a degeneracy subspace can (after normalization) be written as $$|\omega\rangle = \sum_{j} N^{-1/2}
\exp{i\phi_j}|\alpha_{j}\rangle\otimes |\beta_{j}\rangle,
\label{eq:modal2}$$ with $N$ the dimension of the subspace in question. Now take an arbitrary unitary operator $U_I$ in the subspace of $\mathcal{H}_1$ spanned by the vectors $|\alpha_{j}\rangle$. Define an operator $U_{II}$ in the subspace of $\mathcal{H}_2$ spanned by the vectors $|\beta_{j}\rangle$, through its matrix elements, as follows: $$\langle \beta_k|U_{II}|\beta_l\rangle = \overline{\langle \alpha_k|U_{I}|\alpha_l\rangle}. \exp{i(\phi_k - \phi_l)},
\label{eq:u2}$$ with the bar denoting complex conjugation. It follows from this definition that $U_{II}$ is unitary (given the unitarity of $U_I$). We can now construct a product unitary operator in the tensor product of the two subspaces: $U= U_I \otimes U_{II}$. This operator leaves $| \omega\rangle$ invariant: $$\begin{aligned}
\langle \omega|U|\omega\rangle = N^{-1}\sum_{i,j}\exp{i(\phi_j - \phi_i)}\langle \alpha_i|U_{I}|\alpha_j\rangle
\langle \beta_i|U_{II}|\beta_j\rangle = \nonumber \\ = N^{-1}\sum_{i,j}|\langle \alpha_i|U_{I}|\alpha_j\rangle|^2 = 1 .\end{aligned}$$ In other words, we can operate with an arbitrary unitary operator $U_I$ in one of the degeneracy subspaces of $\mathcal{H}_1$, and undo its effect on $|\psi\rangle$ by operating with a suitably chosen unitary operator $U_{II}$, as defined in Eq.(\[eq:u2\]), in the corresponding degeneracy space of $\mathcal{H}_2$. It is of course also true that operating in a similar way with an arbitrary unitary in a degeneracy subspace of $\mathcal{H}_2$ can be undone by a corresponding unitary operation in $\mathcal{H}_1$.
In the case of a one-dimensional subspace (i.e., corresponding to a non-degenerated term in the superposition), $U_I$ can only be a multiplication by a phase factor, $\exp{i\phi}$. In this case the compensating $U_{II}$ takes the form of multiplication by the inverse factor, $\exp{-i\phi}$.
Any spaces contained within the degeneracy subspaces selected by the biorthogonal decomposition are clearly not invariant under all unitary product operations in $\mathcal{H} = \mathcal{H}_1 \otimes
\mathcal{H}_2$ that preserve $|\psi \rangle$; only the degeneracy subspaces themselves are invariant. These spaces are exactly the eigenspaces of the reduced density operator $W_{1}$.
So we arrive in a quick and simple way at the same conclusion as before: the lattice of those properties of system $1$ that can be defined on the basis of $|\psi \rangle $ alone, is generated by the projection operators $P_{w_{k}}$. Since this lattice is Boolean, definite values can be jointly assigned to all its elements without contradictions, and measures on the lattice can be represented in a classical Kolmogorovian probability space.
We have thus found a uniqueness result for the set of definite-valued observables. But any uniqueness result stands or falls with its premises. In our derivation we assumed that the definite-valued observables should be definable from $|\psi
\rangle$ and the splitting of the total Hilbert space into two factor spaces, representing the system and its environment, respectively. If it is assumed that more or other ingredients play a role in determining the properties of a system, different definite observables will result. It has been suggested, for example, that we should not just look at the system and its environment, but rather at a three-fold splitting of the total Hilbert space into factors corresponding to the system, a measuring device (or, more generally, a system that is able to make records) and the remaining environment, respectively [@zur1; @schlosshauer]. If the total state can be written as $$|\Psi_{sA\varepsilon} \rangle = \sum_k a_k |s_k \rangle |A_k \rangle
|\varepsilon_k\rangle, \label{3split}$$ with $s$, $A$ and $\varepsilon$ referring to system, device and environment, respectively, with orthogonal pointer states $\{|A_k
\rangle \}$, these orthogonal pointer states will be the only ones that make the three-fold factorization of (\[3split\]) possible. In this way a set of preferred pointer states of the device can be defined. A problematic feature of this proposal is that it does not lead to a general assignment of properties to arbitrary systems; and that states of the form (\[3split\]) are very special. Such states will only result from specific interactions [@zur1 sect. 4.2]. In the context of decoherence studies it has indeed often been suggested that it is the form of the *interaction Hamiltonian* between (macroscopic) systems and their environments that does the selecting of preferred states: that pointer states are ‘memory states’ that behave in a robust and approximately classical way. The selected pointer observable commutes with the interaction Hamiltonian (perhaps in an approximate way), so that the environment effectively performs a non-demolition measurement of the pointer observable—see also [@zur].[^3] These proposals deserve further study. Questions to be asked are, e.g., about the presence of physical properties in cases without suitably interacting environments and recording devices. Can sense be made of the suggestion that the concept of physical properties becomes only applicable in special circumstances? Another issue is the status of the approximations that are usually involved in decoherence calculations (compare section 5 below). Anyway, it has to be admitted that decoherence proposals have led to plausible candidates for preferred states in many model calculations.
It should therefore be stressed that the modal ideas constitute a research programme rather than a completely fixed interpretation. The central features remain that the quantum formalism describes the world in man-independent terms, in particular without according a special role to measurements undertaken by humans; and that the relation between formalism and physical reality is probabilistic. This leaves room for differences in detailed elaborations.
The Born measure
================
The modal interpretation is probabilistic and must therefore define a probability measure on the lattice of definite-valued observables. This raises the question: Is it possible to derive a preferred measure on the lattice of definite-valued observables, along the same lines as in the derivation of the definite-valued observables? More specifically: if we impose the requirement that the measure is to depend only on the state in Hilbert space, the tensor product structure of Hilbert space and the preferred observables induced by the state, has this enough bite to single out a definite form of the measure? An affirmative answer would fit in nicely with the modal philosophy according to which the standard quantum mechanical formalism is descriptively complete; that no elements need to be added by hand.
As we will argue, the answer is ‘yes’: the Born measure is the only one that is definable from just the relation between $|\psi
\rangle $ and its associated definite-valued observables.
Denote the measure to be assigned to the definite-valued projector $P$, if the state is $|\psi \rangle $, by $\mu(|\psi\rangle, P)$. Write $|\psi\rangle$ in its biorthogonal form again: $$|\psi\rangle = \sum_{k}
c_{k}|\alpha_{k}\rangle\otimes |\beta_{k}\rangle,$$ where we now have taken the non-degenerate case for simplicity. First note that we can take the coefficients $c_{k}$ to be real numbers: all phase factors can be absorbed into the vectors $|\alpha_{k}\rangle$ or $|\beta_{k}\rangle$, without any effect on the observables that are value-definite (the projection operators are invariant under this operation). So if $\mu$ is going to depend on the coefficients $c_{k}$, only their absolute values or, what amounts to the same thing, only $|c_{k}|^2$ can enter the expression.
An alternative road to this conclusion is to use the transformations $U_I \otimes U_{II}$ under which $|\psi \rangle$ is invariant. As we have seen in the previous section, both $U_I$ and $U_{II}$ are pure phase transformations in this non-degenerate case. One could now reason as follows, like @zur1: any physical features pertaining to system $I$ alone should be invariant under the operation of any $U_I$ on $|\psi \rangle$, for the following reason. The effect of $U_I$ can be undone by $U_{II}$ ($U_I$ is what Zurek calls an ‘envariance’ operation); and $U_{II}$ should not be expected to affect the physical properties of $I$. Consequently, any effects $U_I$ may have on the mathematical state of $I$ should not be relevant to the physical features of $I$. In particular, the phases of the coefficients $\{c_k\}$ must be irrelevant, so that only their absolute values can count.
To conclude that $\mu$ indeed only depends on $\{c_{k}\}$, we need an additional argument, however. As pointed out by @caves, it is not ‘envariance’ that is doing the work here: rather, the assumption (also made by Zurek) that the probabilities and physical properties pertaining to system $I$ do not depend on the vectors $\{|\beta_{k}\rangle \}$ in the biorthogonal decomposition is central—and once we make this assumption, the notion of envariance is no longer needed. This assumption may be seen as a no-signalling condition: its violation would make it possible to change physical features of $I$ by intervening in the state of $II$, which would make it possible to signal. It can also be regarded as a non-contextuality condition: it should not make a difference for the characteristics of $I$ what unitary operations are taking place in its environment $II$. This entails invariance of the probabilities under arbitrary $U_{II}$. In particular, $\mu$ can only depend on system $I$’s definite-valued projectors $P$ and on $\{c_k\}$, and since absorbing all phase factors into $\{|\beta_{k}\rangle\}$ does not change the probabilities only the absolute values of $\{c_k\}$ can be relevant—for further discussions of Zurek’s line of argument see [@barnum; @caves; @mohrhoff; @schlosshauer1]. In brief, the non-contextuality condition entails that all probabilities must be invariant under application of arbitrary unitaries $U_I \otimes
U_{II}$. So we find that the bases $\{|\alpha_{k}\rangle\}$ and $\{|\beta_{k}\rangle\}$ are irrelevant for the probabilities, and only the values $\{|c_k|\}$ can play a role.
However, this irrelevance of $\{|\alpha_{k}\rangle\}$ and $\{|\beta_{k}\rangle\}$ for the expression of $\mu$ can be justified in a more direct way, without invoking principles about causality and contextuality, by a definability argument like the one in the previous section. We want $\mu$ to be definable exclusively from $|\psi\rangle$ and the product Hilbert space structure. We can therefore immediately impose the requirement that unitary transformations of the form $U_I \otimes U_{II}$ should not change the values taken by the measure; that these values remain the same, but now apply to transformed projectors (like ${U_I}^{-1}PU_I$). The reason is that these unitary transformations only change the orientation of $|\psi\rangle$ in Hilbert space, but do not change anything in the relation between $|\psi\rangle$ and the definite-valued observables determined by it; all changes are equivalent to those induced by a basis transformation in the Hilbert space, and can be undone by performing an inverse basis transformation. But we want $\mu$ to be determined solely by the the state and its associated definite-valued projectors—the choice of a basis in Hilbert space in terms of which the state is written down should be immaterial. In other words, the same collection of $\mu$ values must be associated with the entire class of states that follow from $|\psi\rangle$ by application of arbitrary unitary operations $U_I \otimes U_{II}$. Since the only feature that is common to all these states are the values of $|c_i|$, $\mu$ must be a function of these values only. As pointed out above, we can therefore consider $\mu$ to be a function of $\{|c_i|^2\}$.
Now compare the situation described by $|\psi\rangle$ with the one in which we discard, forget, or are unable to observe the differences between the different $|\beta_{k}\rangle$ for $k\geq2$. The total probability of not having $|\beta_{1}\rangle$ should now be the sum of the probabilities of $|\beta_{k}\rangle$ for $k\geq 2$, since distinct alternatives have been grouped together. The vector that would correspond exactly to this new situation results from $|\psi\rangle$ by erasing the differences between $|\beta_{k}\rangle$ for $k\geq 2$, and replacing all these vectors by $|\beta_{2}\rangle$. This leads to the state $$|\chi\rangle = c_{1}|\alpha_{1}\rangle\otimes
|\beta_{1}\rangle + \sqrt{\sum_{k=2}|c_k|^2}|\alpha\rangle\otimes
|\beta_{2}\rangle,$$ where $|\alpha\rangle$ is a normalized vector. The measure assigned by this state to $|\beta_2\rangle \langle\beta_2 |$ should be the sum of the original measures of the projectors that have coalesced into $|\beta_2\rangle \langle\beta_2 |$.
Finally, because $\sum_{k=2}|c_k|^2 = 1 - |c_1|^2$ we may write $\mu(|\beta_1\rangle \langle\beta_1 |)= f(|c_1|^2)$. By parity of reasoning we may write down an analogous formula for the other projectors: $\mu(|\beta_i\rangle \langle\beta_i |)= f(|c_i|^2)$.
On the basis of our above observation about the relation between the measures induced by $|\psi\rangle$ and $|\chi\rangle$, respectively, we now find that $$f(\sum|c_k|^2)=
\sum f(|c_k|^2).$$ From this it follows that $ f(|c_k|^2) = const. |c_k|^2$, and in view of normalization $$\mu(P_k) = |c_k|^2.$$ This is the Born rule.
Probability and modality
========================
As explained in the Introduction, modal interpretations understand $\mu$ as a probability: given the state $\psi$ in Hilbert space, exactly *one* of the projectors that are singled out as definite-valued by $\psi$ possesses the value $1$, and the chance that this value is taken by $P_k$ is given by $|c_k|^2$. In general there are more than one possibilities for the actual physical situation (defined by the values of the definite-valued observables), once the state in Hilbert space has been given; the state specifies a probability distribution over them. This probability quantifies our ignorance about the actually obtaining physical situation in cases in which we know the state in Hilbert space and have no additional information. It is also reflected in the relative frequencies with which physical properties occur in repetitions of situations corresponding to the same $\psi$. In other words, the probabilities occurring in the modal interpretation have the same status as classical probabilities and have the usual classical interpretations. That $\mu$ has this physical meaning in terms of probabilities and ignorance is clearly something that is not decided by the mathematical formalism itself (see for a dissenting voice [@zur1], and for a convincing critical analysis of this argument e.g.[@mohrhoff]). It is an interpretational postulate that should be judged on the basis of comparison with alternatives—we shall have more to say about this in the next section.
According to the modal interpretation the state in Hilbert space thus is about possibilities, about what may be the case; about modalities. But there is also a second aspect to $\psi$: it is the theoretical quantity that occurs in the evolution equation, and its evolution governs deterministically how the set of definite valued quantities changes. This double role of $\psi$, on the one hand probabilistic and on the other dynamical and deterministic, is a well-known feature of the Bohm interpretation. As we have seen in section 2, the Bohm interpretation can be regarded as a specific version of the modal interpretation, namely one in which there is an *a priori* given definite-valued observable. As we see now, the double deterministic-and-probabilistic aspect of $\psi$ is typical of modal interpretations quite generally.
One versus many worlds
======================
The no-collapse scheme by itself does not imply anything about probability: it just says that the Hilbert space state evolves unitarily. An interpretation, which is external to the formalism, must be supplied before anything can be stated about what the state represents. It is sometimes suggested, however, in opposition to this, that the no-collapse formalism is capable of providing its own interpretation [@dewitt p. 168]. What seems to be meant is the claim that there exists a *simplest* interpretation that does most justice to the symmetries inherent in the Hilbert space formalism. In particular, the suggestion is that, granted the usual interpretational links between eigenstates of observables and values of physical quantities, a superposition of such eigenstates should be interpreted as representing the joint existence of the corresponding values. This is the many-worlds idea: superpositions represent collections of worlds, in each one of which exactly one value—corresponding to one term from the superposition—of an observable is realized. The claim is that this many-worlds interpretation distinguishes itself by being simple and by possessing a natural fit to the formalism, respecting its symmetries.
In a superposition all terms occur in the same way, i.e. without any markers that single out one, or some, terms as corresponding to what actually is the case. The basic thought of the many-worlds interpretation is that this symmetry signifies that all terms correspond to reality in the same way: if one term refers to something actually existing, then so must all. The identification of any particular term as representing actuality is regarded as breaking the symmetry present in the state, and therefore as objectionable.
Let us have a closer look at this argument. It may be conceded that singling out any particular term from a superposition, and identifying it as the one referring to actuality, breaks the symmetry of the state. But do probabilistic interpretations really work this way; do they single out one term over the others? Consider the analogous situation in classical probability theory: the same train of thought applied there would also lead to the conclusion that all events to which a probability distribution assigns a value should be simultaneously realized, if we do not have an underlying deterministic theory. One would be led, also in the classical case, to a many worlds ontology as the one that best fits the probability formalism. But is this interpretation really simpler or more symmetric than the usual one? Answering this question requires comparing two different mappings (reference relations), with a mathematical event space as their common domain. The probabilistic mapping is from the event space to *possibilities*; whereas the many-worlds mapping maps all elements of the event space into *realities*. Apart from this difference in status of the elements of the ranges of the two mappings (possibility and reality), which as far as the mapping itself is concerned is just a difference in labels, everything is the same. It is therefore hard to see how there could be any difference in simplicity, naturalness or symmetry.
The impression that there nevertheless is such a difference evidently derives from the notion that the probabilistic interpretation identifies one of the possibilities as the actual one, and thus violates the symmetry that is present in the many-worlds option. But this notion is incorrect. *Not* singling out such a privileged event is precisely what makes an interpretation fundamentally probabilistic. The probabilistic option treats all elements of the probability space in exactly the same way, by mapping them to possibilities that *may* be realized—it does not tell us which possibility *is* realized. Each single element of the interpretation’s range may correspond to reality. There is therefore the same symmetry as in the many-worlds option.
Still, there is a difference. In the probabilistic interpretation it is stipulated from the outset that exactly one possibility is realized. Even if there is symmetry with respect to which possibility this is, is it not true that this one-world stipulation by itself introduces surplus structure that is not present in the many-worlds interpretation? I do not think this is right. There is perfect equivalence in the sense that the many-worlds interpretation is defined by the condition that each element of the measure space corresponds to an actual states of affairs, whereas the probabilistic alternative is defined by the condition that each element may correspond to the one actual (but unspecified) state of affairs. There is consequently no difference in the symmetry properties or simplicity of the interpretations, but rather a difference in the nature of their ranges: in the one case this is a collection of many real worlds, in the other it is a collection of candidates for the one real world. So, in the end the significant difference boils down to the difference between one and many—and it surely is not a principle of metaphysics or rational theory choice that many is simpler than one. General considerations concerning symmetry and simplicity do therefore not favor a many worlds interpretation over a probabilistic, modal, interpretation.
Let us briefly discuss a further general problem with the many-worlds idea, namely the well-known question of how to accommodate the notion of probability at all in a theory according to which it is certain that all possibilities will be actually realized. The dominant opinion among many-worlds adherents seems to have become that the quantum probabilities should be seen as subjective, in the sense of quantifying a subject’s degree of belief about the future experiences of his splitting self (though not subjective in the sense of purely personal: The Born probabilities should come out as governing the objectively most rational choices). This Deutsch-Wallace line of argument [@deutsch; @wallace; @wallace1] proceeds from the assumption that a subject should be indifferent between terms in the total superposition that occur with equal ‘weights’, i.e. squared absolute values of the coefficients. This apparently *presupposes* a probabilistic conception of the quantum state—even though it is now probabilistic in the subjective sense. Indeed, it is *a priori* unclear why there should be unique rational expectations defined *at all* in a situation corresponding to a particular $\psi$ if we do not start out by assuming a probabilistic meaning of the wave function. And even if we do accept that there are measures of our credence hidden in the quantum formalism, it still is not self-evident that the symmetries of the quantum state are significant for them (the latter point is also made by @price). In our modal approach we did not face these problems, because we explicitly took an interpretational step and *postulated* a probabilistic meaning of $\psi$, and moreover required the probabilities to be definable in terms of $\psi$. It is this latter requirement that makes the symmetries in $\psi$ relevant for the probability assignment.
That the universe consists of many actual worlds, each one containing exactly one possible outcome of a process, actually does not play a role in the technical part of the Deutsch-Wallace argument. It is only the quantum state $\psi$ that enters into the reasoning, as said with the assumption that $\psi$ should govern rational expectations to start with. Any conclusions that can be drawn from such reasoning in the context of interpretations with many coexisting actual worlds, can surely also be drawn in the context of the more usual probabilistic construal (namely that only one possibility will be actually realized), or so it would seem. However, @wallace1 argues that the many worlds interpretation *is* essential here. In the course of defending the idea that a rational agent should be indifferent between outcomes that occur with equal weights in the superposition $\psi$ (he calls this principle **equivalence**), Wallace says:
> “I wish to argue that the Everett interpretation necessarily plays a central role in any such defence: in other interpretations, **equivalence** is not only unmotivated as a rationality principle but is actually absurd.
>
> Why? Observe what equivalence actually claims: that if we know that two events have the same weight, then we must regard them as equally likely regardless of any other information we may have about them. Put another way, if we wish to determine which event to bet on and we are told that they have the same weight, we will be uninterested in any other information about them.
>
> But in any interpretation which does not involve branching—that is, in any non-Everettian interpretation—there is a further piece of information that cannot but be relevant to our choice: namely, *which event is actually going to happen*? If in fact we know that E rather than F will actually occur, *of course* we will bet on E, regardless of the relative weights of the events”.
It seems to me that this argument does not work. In reasoning about rational expectations—connected to subjective probabilities—about future events one standardly distinguishes ‘admissible’ from ‘inadmissible’ information. For example, David Lewis’s ‘Principal Principle’ says that a rational agent should set his subjective probability equal to the objective chance of an event, unless there is (inadmissible) information about what is actually going to happen [@davidlewis]. In other words, it is not a sound principle that if two events have the same objectively and rationally founded subjective probability, then we have to regard them as equally likely completely regardless of any other information we may receive. In our probability judgements we do not use information about the actual outcomes; if such information were to reach us we would of course adapt our expectations in spite of the probabilities. Conversely, no principle saying that our expectations should not change whatever further information reaches us should guide our search for the values of probabilities. If it did, no probability values other than $1$ and $0$ could ever be assigned, because in principle we might always receive information (e.g., by revelation) about whether things will or will not happen. This remains the case in the many-worlds scenario. It is true that all possibilities will become actual in this scenario, so that we cannot learn which event is actually going to happen. But we may still receive information about our *actual future experiences*, which according to Wallace are subject to uncertainty in the many-worlds universe because we ourselves split and do not know who of our successors we will become (this way of accommodating uncertainty and probability in the many-worlds interpretation is itself the subject of controversy—see [@greaves; @lewis; @lewis1; @price]).
I conclude that the difference between one real world (and many possible ones) on the one hand, and many real worlds with subjective uncertainty injected into them on the other hand, is irrelevant for the justification of the form of Born rule. The Born formula can be derived from the requirement that the measure $\mu$ should be definable in terms of only $\psi$ (and should therefore not depend on anything else, like information about future events). The meaning of this $\mu$, be it in terms of modal probabilities or subjective many-worlds uncertainty, is something that cannot be derived but must be added in an interpretational step.
Let us now direct our attention to a more specific comparison of modal ideas and many worlds, relating to technical details.
One would perhaps expect the many worlds interpretation to follow the earlier explained modal way of fixing the definite observables in determining the worlds (each world corresponding to a different set of values of the definite observables), because a basic idea of the many-worlds interpretation is that the formalism is self-sufficient and that knowledge of the state is enough to obtain a full description of the universe. However, the dominant opinion among many-world adherents is different, namely that a privileged decomposition of the state is determined by the dynamical mechanism of decoherence. At first sight this decoherence recipe is almost identical to the one described in sections 2 and 3. Indeed, decoherence leads to a state of the general form $$|\psi\rangle = \sum_{k} c_{k}|\psi_{k}\rangle\otimes
|E_{k}\rangle, \label{eq:decoherence}$$ with $|\psi_{k}\rangle$ representing the part of the world undergoing decoherence, and $|E_{k}\rangle$ representing the decohering part (usually the environment of the system that is undergoing decoherence). This looks similar to Eq. \[eq:modal\]. The difference is that the states $|E_{k}\rangle$ are not exactly orthogonal, so that Eq. \[eq:decoherence\] is not a biorthogonal decomposition. It is true that typically not only $\langle E_{i}|E_{j}\rangle \rightarrow 0$ when $t\rightarrow
\infty$, but that $\langle E_{i}|E_{j}\rangle \simeq 0$, if $i\neq
j$, even very soon after the onset of the decoherence process: this process is very effective. Still, this inner product can never be assumed to really vanish at finite times. This means that the branches corresponding to the terms in Eq.\[eq:decoherence\] are not disjoint: there is interference between them. In the case of a biorthogonal decomposition the expectation value—taken in the total state—of any observable of the form $A \otimes I$ is the sum of contributions from the different branches; but this is not true in the case of the state of Eq. \[eq:decoherence\]. In the latter case there are *cross terms* in addition to the contributions from the individual branches. Although these cross terms will typically be tremendously small, they are important from a conceptual point of view. They indicate that the total situation represented by the state cannot be viewed as a juxtaposition of independent alternatives or isolated worlds.
Another way of formulating the same point is that the Born probabilities of measurement results of observables of the form $A
\otimes I$, calculated in the individual branches, when added with weights equal to the Born probabilities of those individual branches themselves, do not reproduce the probabilities of these outcomes in the total state. This is a violation of a consistency condition on the interpretation of $|c_k|^2$ as a probability both within the individual worlds *and* in the universe consisting of many worlds.
A second conceptual difficulty is that the way the state $|\psi\rangle$ has been decomposed in Eq. \[eq:decoherence\] is not unique. A change of basis from ${|\psi_{k}\rangle}$ to a slightly different set of mutually orthogonal vectors will preserve the general form of Eq. \[eq:decoherence\], together with the almost orthogonality of the decohering states. So decoherence does not lead to a well-defined set of branches. Many-worlds proponents usually see this as an innocent form of vagueness: it is sufficient that on the macroscopic level the usual quantities, defined within observational precision, become definite—this is compatible with some leeway in the quantities that are definite on a microscopic scale. As @butterfield formulates it:
> “...the ubiquity and astonishing efficiency of decoherence means that for all macrosystems ... the selected quantity will be very nearly unique—so that the vagueness will be unnoticeable by the standards of precision usual for macroscopic physics”.
This may seem plausible, but it is not a solid result backed up by calculations. That is, at present there is no guarantee that the different possible choices of ${|\psi_{k}\rangle}$ that make it possible to write the state in the ‘decoherent form’ of (\[eq:decoherence\]), with $\langle E_{i}|E_{j}\rangle$ very small, are close to each other in Hilbert space. As @baccia has shown in the context of modal interpretations, instabilities may occur in the biorthogonal decomposition, and sometimes this may lead to the selection of observables that are very different from the usual macroscopic observables one would expect to be selected. These results apply in particular to situations in which there are very many degrees of freedom, for example when macroscopic bodies are immersed in a decohering environment. It is true that these results have been rigourously derived only for the biorthogonal decomposition, but if $\langle E_{i}|E_{j}\rangle$ is very small instead of exactly zero, one should expect a similar behavior. This raises the general question of whether the decoherence scheme will be capable of always defining an adequate set of worlds, or definite outcomes of experiments. This is the same question that can be asked in the case of modal interpretations (cf. footnote 2).
In general, it seems that the technical difficulties which have been suggested to exist for modal interpretations based on the biorthogonal decomposition (in conditions with continuous or very many degrees of freedom [@baccia; @donald]), should also be taken seriously in interpretations based on the idea that decoherence singles out a preferred decomposition of the state. These difficulties have been investigated in some detail only within the modal framework, because here there are precise mathematical rules that define the definite valued observables. However, although there is more ‘slippage’ in the decoherence scheme, the problem of instability requires a solution in this context as well. One should be careful here to distinguish between two different questions. In standard treatments of decoherence one starts by writing the state as a superposition of eigenstates of an *a priori* given preferred observable, often position; and then shows that these eigenstates become correlated, through the interaction with the surroundings, with almost orthogonal environment states. The question we are facing here, however, is whether the total state, after having undergone decoherence, defines adequate physical quantities. The uncontroversial reply to the first question does not answer the second; and it is to this second question that the above questions pertain.
A final point is that the use of approximations, which is very common in decoherence approaches, runs the risk of getting into vicious circles. In particular, the motivation for neglecting ‘small’ components of $\psi$ seems to presuppose a probabilistic interpretation of the coefficients with which these components occur. But this would vitiate a *deduction* of the Born rule. Indeed, the Born rule or something equivalent to it would in this case already be used in the definition of the definite events to which the Born probabilities are later to be assigned.
In summary, both from the viewpoint of general considerations and from that of more detailed quantum mechanical arguments there seems little reason to prefer many-worlds interpretations over modal ones. Decoherence is not an obvious help for the many-worlds scheme—if the problems just mentioned can be solved, it still remains unclear why decoherence would help the many-worlds theorist and would not be available to adherents of modal ideas.
Conclusion
==========
The Hilbert space formalism of quantum mechanics restricts possible interpretations in the following sense: if we stipulate that definite-valued quantities and a probability measure over them should be definable from the quantum state and the Hilbert space tensor product structure for the system and its environment, this leads to unique expressions both for these definite quantities and the measure. *That* the state vector should be thus interpreted in terms of definite quantities and probabilities is not something that can be derived from the mathematical formalism—it is an interpretational choice. Modal interpretations implement this choice by postulating that the quantum state represents possibilities of which only one is realized in physical reality.
What the definite quantities turn out to be obviously depends on what we stipulate about the elements in the formalism on which these quantities are to depend. Here we have focused on what follows from the requirement of definability from only the state and the bipartite tensor product structure of Hilbert space. Other stipulations are possible while staying within the general framework of the modal programme, which is characterized by objectivity (description of the world through objective physical quantities) and a fundamental role for probability. These alternative options deserve further investigation.
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[^1]: The set of automorphisms that leave $\psi$ and $R$ invariant contains more elements than the ones considered in this derivation—so what has been proved is that the lattice of definite properties cannot be larger than the one constructed here. As the constructed lattice is clearly definable from $\psi$ and $R$ and satisfies the other requirements, it is the maximal lattice we were looking for.
[^2]: It is an important question whether this requirement of empirical adequacy is in fact fulfilled. In situations with a limited number of degrees of freedom this has been shown to be the case [@bacciahemmo]; but there are grounds for doubt in cases in which the number of degrees of freedom is infinite or very large [@baccia]; see [@benedie] for a possible response).
[^3]: @zur1 [sect. IIB] states, however, that all ‘measurable properties’ of a system can depend only on its own state, obtained by partial tracing from the entangled state of the system and its environment. This seems to lead us back to the standard modal property attribution.
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abstract: 'We study cosmological solutions in nonlocal teleparallel gravity or $f(T)$ theory, where $T$ is the torsion scalar in teleparallel gravity. This is a natural extenstion of the usual teleparallel gravity with nonlocal terms. In this work the phase space portrait proposed to describe the dynamics of an arbitrary flat, homogeneous cosmological background with a number of matter contents, both in early and late time epochs. The aim was to convert the system of the equations of the motion to a first order autonomous dynamical system and to find fixed points and attractors using numerical codes. For this purpose, firstly we derive effective forms of cosmological field equations describing the whole cosmic evolution history in a homogeneous and isotropic cosmological background and construct the autonomous system of the first order dynamical equations. In addition, we investigate the local stability in the dynamical systems called “the stable/unstable manifold” by introducing a specific form of the interaction between matter, dark energy, radiation and a scalar field. Furthermore, we explore the exact solutions of the cosmological equations in the case of de Sitter spacetime. In particular, we examine the role of an auxiliary function called “gauge” $\eta$ in the formation of such cosmological solutions and show whether the de Sitter solutions can exist or not. Moreover, we study the stability issue of the de Sitter solutions both in vacuum and non-vacuum spacetimes. It is demonstrated that for nonlocal $f(T)$ gravity, the stable de Sitter solutions can be produced even in vacuum spacetime.'
author:
- Kazuharu Bamba
- Davood Momeni
- Mudhahir Al Ajmi
title: Phase Space description of Nonlocal Teleparallel Gravity
---
Introduction
============
It has been supported that in addition to the inflationary stage [@Inflation] in the early universe, currently the expansion of the universe is also accelerating by various cosmological observations including Type Ia Supernovae [@SN], cosmic microwave background (CMB) radiation [@Ade:2015xua], large scale structure [@LSS], baryon acoustic oscillations (BAO) [@Eisenstein:2005su] as well as weak lensing [@Jain:2003tba]. We have two representative explanations for such a late-time cosmic acceleration. One approach is to introduce “dark energy” (DE) in the context of general relativity. The other approach is to consider the modification of gravity on the large scale (for reviews on not only DE problem but also modified gravity theories, see, for example, [@R-DE-MG]).
There is a possible candidate for a theory of gravitation alternative to general relativity, namely, teleparallel gravity, which is described by using the Weitzenböck connection [@T-G]. In teleparallel gravity, there exists torsion. This is opposite to the case of general relativity, in which the Levi-Civita connection is used. The torsion scalar $T$ represents the Lagrangian density of teleparallel gravity. It can be extended to a function of $T$, that is, $f(T)$ gravity (for a recent review, see, for instance, [@Cai:2015emx]). This idea is similar to that of $f(R)$ gravity [@F-R], where $R$ is the scalar curvature. Inflation in the early universe [@F-T-Inf] and the late-time cosmic acceleration [@F-T-LC] can be realized in $f(T)$ gravity. Various cosmological and astrophysical considerations in $f(T)$ gravity have widely been executed [@F(T)-Refs]. It is known that in $f(T)$ gravity, the local Lorentz invariance is broken [@L-L-I], and the relevant investigations on this point have been discussed [@RP-LLI].
On the other hand, in Ref. [@Deser:2007jk], there has been considered a way of modifying gravitation, the so-called nonlocal gravity, which comes from quantum effects. Furthermore, in order to unify inflation in the early universe and the late-time accelerated expansion of the universe, non-local gravity has been modified by adding an $f(R)$ term in Ref. [@Nojiri:2007uq]. In addition, a possible solution for the cosmological constant problem through the nonlocal property of gravitation [@ArkaniHamed:2002fu] has been proposed. Moreover, a physical mechanism by which a cosmological constant is screened in the framework of nonlocal gravity has been investigated [@Nojiri:2010pw; @Bamba:2012ky; @Zhang:2011uv]. It has also been indicated that in nonlocal gravity, there is the issue of ghosts [@Nojiri:2010pw]. Various aspects of nonlocal gravity have widely been explored [@NL-Ref] (for a recent review on nonlocal gravity, see, e.g. [@Maggiore:2016gpx]). It is worth noting that nonlocal terms $\Box T $ was first used in the framework of modified teleparallel gravity in Ref.[@Otalora:2016dxe] Furthermore , the nonlocal deformations of teleparallel gravity have been analyzed in Refs. [@Bahamonde:2017bps],[@Channuie:2017txg]. This theory is called nonlocal $f(T)$ gravity, which can be considered as an extension of nonlocal general relativity to the Weitzenböck spacetime. It has been discussed that there is a possibility to distinguish teleparallel gravity from general relativity by future experiments detecting nonlocal effects. In this paper, we investigate exact cosmological solutions in nonlocal $f(T)$ gravity. We analyze the autonomous system of the first order dynamical equations by deriving effective forms of cosmological field equations in a homogeneous and isotropic cosmological background, describing the whole evolution history of the universe. Moreover, we propose a specific form of the interaction between matter, dark energy, radiation and a scalar field and examine the local stability in the dynamical systems, which is called “the stable/unstable manifold”. As a result, it is demonstrated that the system has a stable attractor. Furthermore, we study exact solutions of the cosmological equations in the case of de Sitter spacetime. Particularly, we explore the role of an auxiliary function called “gauge” $\eta$ in the formation of such cosmological solutions and show whether the de Sitter solutions can exist or not in this scenario. In addition, we consider the stability problem of the de Sitter solutions both in vacuum and non-vacuum spacetimes and find that even in vacuum spacetime, the stable de Sitter solutions can be produced in the framework of nonlocal $f(T)$ gravity.
The organization of the paper is the following. In Sec. 2, we explain the framework of nonlocal $f(T)$ gravity. In Sec. 3, we explore the cosmological background and effective field equations in nonlocal $f(T)$ gravity. In Sec. 4, the interaction term and phase portrait are analyzed. In Sec. 5, the de Sitter solution is derived and its stability is examined in the following Sec. 6. Finally, conclusions are provided in Sec. 7.
Formal framework of nonlocal $f(T)$ gravity
===========================================
Let us develop the formalism of nonlocal modified gravity with torsion $T$ in a same manner as the nonlocal $f(R)$ gravity is developed [@Nojiri:2007uq]. We suppose that the possible action for gravity with matter contents is given in terms of classical gauge invariant action as follows: $$\begin{aligned}
&&S=\frac{1}{2\kappa}\int d^4x eT\Big(f(\Box^{-1}T)-1\Big)+\int d^4x e \mathcal{L}_m\label{action}\end{aligned}$$ where $\kappa=8\pi G$, $G$ is Newtonian gravitational constant, $\mathcal{L}_m$ is matter Lagrangian. To describe the geometry of spacetime in teleparallel gravity, it is commonly used the tetrads formalism where the metric can be written in an orthogonal frame $e_a^{\mu}$, in a such manner that $g^{\mu\nu}=e_{a}^{\mu}e_b^{\nu}\eta^{ab}$, where Greek alphabets run from $\mu,\nu=0...3$, the flat Minkowski metric is denoted by $\eta^{ab}$. Note that $e_a^{\mu}e^{a}_{\nu}=\delta_{\nu}^{\mu}$ and $\Box^{-1}$ is considered as an integral over the entirely spacetime manifold. The local operator $\Box =\nabla^{\mu}\nabla_{\mu}$ is called d’Alembert operator defined as $\Box=e^{-1}\partial_
{\alpha}(e\partial^{\alpha})$ here $e=det(e_{a}^{\mu})=\sqrt{-det(g_{\alpha\beta})}$ and $T$ is torsion scalar and it is defined in a same form as $f(T)$ gravity.
It is always possible to reduce nonlocal theories to scalar-tensor equivalent theories and it is easy to do that for our model given in (\[action\]) using two auxiliary (non ghost) fields $\phi=\frac{1}{\Box}T$ and $\xi=-\frac{1}{\Box}(f'(\phi)T)$. The reason that those fields are considered as non ghost is that , the norm of them defined as $||\phi||=\int_{\Sigma}|\phi|^2 ed^4x$ is always positive definite and never becomes complex as long as the metric and its torsion $T$ remains real numbers. As long as we work in Riemanninan manifolds this condition will be hold and we can safely use them as an appropriate set of auxiliary fields.
The new form for the reduced action is written as follows: $$\begin{aligned}
&&S=\frac{1}{2\kappa}\int d^4x e\Big[T\Big(f(\phi)-1\Big)-\partial_{\mu}\xi\partial^{\mu}\phi-\xi T
\Big]\\&&\nonumber
+\int d^4x e \mathcal{L}_m\label{action2}.\end{aligned}$$ Note that in (\[action2\]) the action function $f(\phi)$ is supposed to have any desired form. Formally if we take the case: $\xi =1$ and $f(\phi) = 2$ in action of theory given by Eq. (2), then the action is reduced to the one which is equivalently of Teleparallel Gravity for $T\neq 0$. Classical tests for GR prove a very good agreement with observations. As a result it is very important to know whether this nonolocal teleparallel gravity has GR limit or not. At the level of action we already demonstrate it. By an enough good choosing of the function $f(\phi)$ we can recover GR as a limiting case.
The form of equations of motion is presented in [@Bahamonde:2017bps] :
$$\begin{aligned}
&&2(1-f(\phi)+\xi)\left[ e^{-1}\partial_\mu (e S_{a}{}^{\mu\beta})-E_{a}^{\lambda}T^{\rho}{}_{\mu\lambda}S_{\rho}{}^{\beta\mu}-\frac{1}{4}E^{\beta}_{a}T\right]\\&&\nonumber
-\frac{1}{2}\Big[(\partial^{\lambda}\xi)(\partial_{\lambda}\phi)E_{a}^{\beta}-(\partial^{\beta}\xi)(\partial_{a}\phi)-(\partial_{a}\xi)(\partial^{\beta}\phi)\Big] \\&&\nonumber-2\partial_{\mu}(\xi-f(\phi))E^\rho_a S_{\rho}{}^{\mu\nu}= \kappa\Theta^\beta_a\,. \label{2}\\&&
\Box\xi+Tf'(\phi)=0,\\&&
\Box\phi-T=0.\end{aligned}$$
here the tensor $E_{\alpha}^{\beta}$ is defined through the variation of $e$ as follows $\delta e=e E^{\beta}_\alpha e_{\beta}^a$, $\Theta^\beta_a$ is the energy-momentum tensor of matter contents defined by $\Theta^\beta_a=e^{-1}\frac{\delta (e \mathcal{L}_m)}{\delta e_{\beta}^a}$ and $\Box\equiv e^{-1}\partial_{\mu}(e\partial^{\mu}) $.
In Refs. [@Bahamonde:2017bps] , the authors investigated cosmological data analysis on a suitable chosen function $f(\phi)=A \exp(n\phi)$ and later in Ref. [@Channuie:2017txg] , using Noether symmetry approach. In our paper we will fix $f(\phi)$ in another simple/adequate form in next section.
Cosmological background and effective field equations {#fieldeqs}
=====================================================
The aim of this section is to write equations of motion for a cosmological background in the presence of matter fields in an effective form. Let us suppose that the non singular, physical metric of spacetime is given in the form of a Friedman-Lemaitre-Robertson-Walker (FLRW) metric given by $ds^2=dt^2-a(t)^2(dx^bdx_b)$, where $b=1,2,3$ is spatial coordinate and $a(t) $ is scale factor and measures expansion of the whole cosmological Universe as well as its acceleration/deceleration phase. The corresponding suitable, diagonal tetrads basis is given by $e^{a}_{\mu}=diag\Big(1,a(t),a(t),a(t)\Big)$. The set of FLRW equations and the equations for the scalar fields are written as follow:
$$\begin{aligned}
&&3H^2(1+\xi-f(\phi))=\frac{1}{2}\dot{\phi}\dot{\xi}+\kappa(\rho_m+\rho_{\Lambda}+\rho_r)\label{eq1}\\
&&(2\dot{H}+3H^2)(1+\xi-f(\phi))=-\frac{1}{2}\dot{\phi}\dot{\xi}+2H(\dot{\xi}-\dot{f}(\phi))-\kappa(p_{\Lambda}+p_r)\label{eq2}\\&&
\ddot{\xi}+3H\dot{\xi}-6H^2f'(\phi)=0
\label{eq3}
\\&&\ddot{\phi}+3H\dot{\phi}+6H^2=0
\label{eq4}\end{aligned}$$
The matter energy-momentum tensor is given in terms of a diagonal tensors for matter , dark energy, radiation as follows:
$$\begin{aligned}
&&\tau_{\mu}^{\nu}=e^{a}_{\mu}e_{b}^{\nu}\tau_{a}^{b}=diag\Big(\rho
_m+\rho_{\Lambda}+
\rho_r,-p_{\Lambda}-
p_r,-p_{\Lambda}-
p_r,-p_{\Lambda}-
p_r\Big)\end{aligned}$$
where $e^{a}_{\mu}e^{\mu}_{a}=\delta^{a}_{b}$ is unit matrix. The matter budget of our model is drak matter density $\rho_m$, radiation field density $\rho_r$ and scalar field density $\rho_{\phi}$. In order to preserve the acceleration expansion and the existence of late time de Sitter cosmology we inserted a non zero cosmological constant $\Lambda$ with energy density $\rho_{\Lambda}$. As an attempt to keep simplicity we assume that all matter contents are given in barotropic forms, where we define the equation of state (EoS) parameter $w_a$ for each fluid component, namely matter, radiation and cosmological constant and as a result for any component of matter field we have a linear EoS , i.e, $p_a=w_a\rho_a$. Here the roman index $a$ refers to different matter contents. Namely we denote it by $a =\{m,\Lambda,r\}$ where $m$ is for matter, $\Lambda$ is for DE and $r$ is for radiation field. Note that neither $\phi$ nor $\xi$ are considered as the DE. The reason is that both fields play the role of auxiliary fields. We can’t make guarantee that whether the fields $\phi,\xi$ will be ghost or not. Actually the appearance of ghost scalar fields in the non local theories for gravity is an important issue and should be addressed adequately. For example in the non local extensions of the GR, when the action is corrected by nonlocal terms $\frac{1}{\Box}R$ or higher order terms, one must count the number of the degrees of freedom of the localized form of the Lagrangian. Additionally, one needs to check the equivalence between local and nonlocal representations of theory both at the action level and equations of motion levels. It is possible to make a categorization based on the first form of the auxiliary fields. Based on this classification we can find the number of algebraic constraints which they will limit our ability to write the local or nonlocal representations of theory. Although in nonlocal extensions of the GR, as long as we have a linear term we can ascertain the equivalence between frames. However, with higher order terms this equivalence is broken. That means in a general nonlocal GR when we have only curvature terms our theory may suffer from ghosts. In nonlocal extensions of the teleparallel gravity (TEGR), we can deduce the same as long as we make the theory using a nonlocal action made by linear scalar torsion $T$, and, hence, no ghost will appear. That is because it was proved that the Einstein-Hilbert action GR is dynamically equivalent to the TEGR at level of action as well as the equations of motion [@Hayashi]. In our study with the nonlocal term which we will opt in next paragraph the model will be ghost free. Although probably the scalar field $\phi$ will not be a ghost but still we do not have any strong reason to keep it as the only sector for the acceleration expansion in our model. For this reason we also keep the cosmological constant $\Lambda$ and its energy density $\rho_{\Lambda}$.
The first challenge is to choose one suitable form for $f(\phi)$. Note that $\Box\xi=-f'(\phi)\Box\phi$. It is illustrative to expand and write this equation in the following equivalent form : $$\Box(\xi+f(\phi))=f''(\phi)\Box\phi$$ Here, we suppose that $f''(\phi)=0$. Thus one suitable class of models is: $$\begin{aligned}
&&f(\phi)=A\phi+B\label{fphi}.\end{aligned}$$ Note that for $A=0$ , $B=2$ the results reduce to the GR as a limiting case. Note that now, $\Box(\xi+f(\phi))=0$, and we have a freedom to take $\xi+f(\phi)=\Psi$, where $\Psi$ is a harmonic function over $\mathcal{R}^4$. A possible option is to consider $\Psi=2A\phi$, consequently the set of eqs. (\[eq1\],\[eq2\]) are simply written in the following forms: $$\begin{aligned}
&&3H^2=\frac{\frac{1}{2}A\dot{\phi}^2+\kappa(\rho_m+\rho_{\Lambda}+\rho_r)}{1-2B}\label{eq11}\\
&&2\dot{H}+3H^2=-\frac{\frac{1}{2}A\dot{\phi}^2+\kappa(p_{\Lambda}+p_r)}{1-2B}\label{eq22}
\\&&
\ddot{\xi}+3H\dot{\xi}-6AH^2=0
\label{eq33}
\\&&\ddot{\phi}+3H\dot{\phi}+6H^2=0
\label{eq44}\end{aligned}$$ Here parameter $B$ measures the difference between TEGR and nonlocal theory respectively. Because we will study time evolution of the energy densities, it is adequate to rewrite cosmological equation , presented in Eq. (\[eq11\]) in the following forms, $$\begin{aligned}
&& 3H^2=\frac{\rho_{\phi}}{1-2B}+\frac{\rho_{m}}{1-2B}+\frac{\rho_{\Lambda}}{1-2B}+\frac{\rho_{r}}{1-2B}\label{eq111}\end{aligned}$$ in the above equation, we have defined $$\begin{aligned}
&& \rho_{\phi}\equiv A\dot{\phi}^2\end{aligned}$$ Note that the other density functions can not be written explicitly in terms of the scale factor $a$ or scalar fields, till the time which we will present continuity equations for all the matter components. In our scenario we assumed that different matter contents interact with each other through some interaction forms which will present in next section.
Hartman-Grobman linearizion theorem {#Hartman-Grobman}
-----------------------------------
To investigate the phase space analysis it is needed to reduce the system of equations to an autonomous system of first order differential equations in the form $\frac{d\vec{X}}{dN}=f(\vec{X})$, where $N$ plays the role of time and $\vec{X}$ is a vector field with density functions as components.In many works, investigation of various aspects of dynamical systems in cosmology of modified gravity is discussed [@Odintsov:2015wwp].
The Hartman-Grobman linearizion theorem provides a powerful technique to study the local stability and the portrait of the phase space, when we have a set of hyperbolic fixed points . Let $\vec X(t)\in \mathcal{R}^n$ be a non trivial solution to the following system of first order differential equations, called flow, $$\label{ds11}
\frac{\mathrm{d}\vec X}{\mathrm{d}t}=g(\vec X)\, ,$$ here $g(\vec X)$ is a locally Lipschitz, one-to one continuous map $g:\mathcal R^n\rightarrow \mathcal R^n$. Let $\vec X_*$ denotes the location of the fixed points of the dynamical system (\[ds11\]), and the corresponding Jacobian matrix, which we denote as $\mathcal{J}(g)$, is equal to, $$\label{jaconiab}
(\mathcal{J})_{ij}=\Big{[}\frac{\mathrm{\partial g_i}}{\partial
X_j}\Big{]}\, .$$ In order to have stable fixed points for system (\[ds11\]) it is enough to set all eigenvalues of the Jacobian matrix so that $\lambda_i$ satisfies $\mathrm{Re}(\lambda_i)\neq 0$. The Hartman theorem predicts the existence of a homeomorphism $\mathcal{F}:U\rightarrow \mathcal{R}^n$, where $U$ is an open neighborhood of $\vec X_*$, such that $\mathcal{F}(\vec X_*)$. The homeomorphism generates a flow $\frac{\mathrm{d}h(u)}{\mathrm{d}t}$, which is, $$\label{fklow}
\frac{\mathrm{d}h(u)}{\mathrm{d}t}=\mathcal{J}h(u)\, ,$$ It is proved that (\[fklow\]) is a topologically conjugate flow to the one system given in Eq. (\[ds11\]).
Building the cosmological autonomous system of equations
--------------------------------------------------------
Now we study a model of interacting matter contents, where the continuity equation for each energy density $\rho_a$ is given by the following form: $$\begin{aligned}
&&\dot{\rho}_a+3H(1+w_a)\rho_a=\Gamma_a\label{rhoeq}. \end{aligned}$$ where $a=\{m,\Lambda,r,\phi\}$ and $\Gamma_a$ is the interaction function given by the general form $\Gamma_a=\Gamma_a(\Omega_m,\Omega_{\Lambda},\Omega_r,\Omega_{\phi})$ and it satisfies $\Sigma_{a=1}^4\Gamma_a=0$. In $f(T)$ gravity, such interacting models are widely studied in the literatures, namely [@Jamil:2012nma]-[@Jamil:2012yz]. In Ref.[@arXiv:1012.4879] authors showed that the total gravitational energy is transferred from dark matter $\rho_m$ to dark energy $\rho_{\Lambda}$, and the cosmological coincidence problem in the Lambda-Cold Dark Matter ($\Lambda$CDM) model is slightly assuaged.
In comparison to matter, DE and radiation energy densities, let us define an auxiliary scalar energy density as $$\Omega_{\phi}=\frac{A\dot{\phi}^2}{3H^2}$$ It is important to mention here that the auxiliary field $\phi$ is not a physical field. Consequently the kinetic term could be treated as tachyonic field as well as pressureless dust matter. In this paper we consider $w_{\phi}$ as a free parameter to be adjusted using observational data.
In this case we can write the following equation for the ratio between pressure and density , called effective EoS equation, as follows:
$$\begin{aligned}
&&\frac{2\dot{H}}{3H^2}=\frac{1}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r\Big)\label{hdot}.\end{aligned}$$
where we prescribed the form of $f(\phi)$ as it is given in Eq. (\[fphi\]) and we supposed that $w_m=w_{\phi}=0,w_r=\frac{1}{3},w_{\Lambda}\in(-1,-\frac{1}{3})$.
It is easy to rewrite (\[rhoeq\]) using the definition of $$\begin{aligned}
\Omega_a=\frac{\kappa \rho_a}{3H^2}\label{omegaa}\end{aligned}$$ in the following set of first order differential equations where we used (\[hdot\]) in it,
$$\begin{aligned}
\label{sys1}
&&\frac{d\Omega_a}{dN}=\frac{\kappa\Gamma_a}{3H^3}-\Omega_a\Big((1+w_a)+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)\Big).\end{aligned}$$
Recall $a=\{m,\Lambda,r,\phi\}$ and we use slow-roll variable $N=\log(\frac{a}{a_0})=-\ln(1+z)$ (the derivatives will be taken with respect to $N$) and $z$ is redshift. This is an autonomous system and should be analyzed in the vicinity of critical points where $\frac{d\Omega_a}{dN}|_{c}=0$ using techniques developed in Sec. (\[Hartman-Grobman\]).
In terms of the variables (\[omegaa\]) the Friedmann equation (\[eq11\]) becomes the restriction : $$\begin{aligned}
&& \Omega_m+\Omega_{\Lambda}+\Omega_r+\Omega_{\phi}=1-2B\label{frw1}\end{aligned}$$
Note that due to the interaction term in the model, the density parameters $\Omega_m,\Omega_{\Lambda},\Omega_r,\Omega_{\phi}$ should be interpreted very strictly as effective density parameters. We mention here that the above constraint guaranteed the existence of possible cosmological attractors, because actually the shape of density functions remains typically the same and the full 4-dimensional configuration space constructed using density functions defines a shape invariant manifold and it defines the attractor solution in the dynamical system.
The effective EoS for system is defined
$$\begin{aligned}
&&w_{eff}=\frac{p_{tot}}{\rho_{tot}}=-1-\frac{1}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r\Big)\label{eos}.\end{aligned}$$
Interaction term and phase portrait
===================================
The general linear dependent model for interaction could be in the following form: $$\begin{aligned}
&&\Gamma_a=\frac{3H^3}{\kappa}\Sigma_{b=1}^{4}\alpha_{ab}\Omega_b\label{Omega_b}.\end{aligned}$$
There are some criticisms about interacting models of DE, however the thermal properties of this model in various gravities have been discussed in the literature [@interaction]. Furthermore, in Ref. [@He:2010im], the authors proposed a systematic scheme to construct the interaction form $\Gamma_a$ in a self consistent manner both in the perturbed form and in the background. They proved that in the perturbation formalism , there are possible ways to break the degeneracy between the interaction, DE EoS and DM abundance.
With this choice, the system of equations (\[sys1\]) are written in the following form:
$$\begin{aligned}
&&\frac{d\Omega_a}{dN}=\Sigma_{b=1}^{4}\alpha_{ab}\Omega_b-\Omega_a\Big((1+w_a)+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)
\Big)\label{model}\end{aligned}$$
Let us study a class of these models where the DE interacts with both matter $\Omega_m$ and scalar field components. .
Based on our former notation given in (\[Omega\_b\]), our interaction model is parametrized as follows: $$\begin{aligned}
&&\alpha_{m\Lambda}=-6b,\ \ \alpha_{\Lambda\Lambda}=\alpha_{r\Lambda}=\alpha_{\phi\Lambda}=2b\end{aligned}$$ The autonomous system of first order differential equations for density functions are written in the following forms,
$$\begin{aligned}
\label{sys111}
&&\frac{d\Omega_m}{dN}=-6b\Omega_{\Lambda}-\Omega_m\Big(1+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)\Big)\equiv f_1\label{f1}\\&&
\frac{d\Omega_{\Lambda}}{dN}=2b\Omega_{\Lambda}-\Omega_{\Lambda}\Big((1+w_{\Lambda})+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)\Big)\equiv f_2\label{f2}
\\&&
\frac{d\Omega_{r}}{dN}=2b\Omega_{\Lambda}-\Omega_{r}\Big(\frac{4}{3}+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)\Big)\equiv f_3\label{f3}
\\&&
\frac{d\Omega_{\phi}}{dN}=2b\Omega_{\Lambda}-\Omega_{\phi}\Big((1+w_{\phi})+\frac{3}{2B-1}\Big(\Omega_{\phi}+
\Omega_m+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r
\Big)\Big)\equiv f_4\label{f4}\end{aligned}$$
These equations are related to the dynamics and the interaction form, characterizing the main properties of our model.
The critical(fixed) points
--------------------------
We stress here that the high dimensionality of the phase space, where the system is described using dynamical systems presented in previous section, restricts us to have an effective graphical description of the phase space, and thus we will focus our investigations only on the analytical results.
To make the dynamical analysis we first need to find the critical(fixed) points of the system by setting the left hand side of equations (\[f1\])-(\[f4\]) to zero. Then we use the Hartman theorem to find the type and stability of each point [@dynamics].
The location of the fixed points $ P=
(\Omega_m,\Omega_{\Lambda},\Omega_r,\Omega_{\phi})
$ and their corresponding eigenvalues of the dynamical system are in the following table, where the stability of the fixed points is determined by evaluating the eigenvalues of the Jacobian matrix associated with the system.
P $\Omega_m^c$ $\Omega_{\Lambda}^c$ $\Omega_r^c$ $\Omega_{\phi}^c$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$
----- -------------------------------------------------------- ------------------------------ -------------- ------------------------------ ------------------------------ ------------------- --------------------------- ------------------------------
$A$ $-\frac{2}{3} w_\phi B-\frac{1}{3}+\frac{1}{3} w_\phi$ $0$ $0$ $0$ $w_{\phi}$ $w_{\phi} +1$ $w_{\phi} - \frac{1}{3}$ $2b-w_{\Lambda}+w_{\phi}$
$B$ $0$ $-\frac{2}{3} B+\frac{1}{3}$ $0$ $0$ $-\frac{1}{3}$ $1$ $-w_{\phi}$ $2b-w_{\Lambda}$
$C$ $0$ $0$ $0$ $0$ $-\frac{4}{3}$ $-1$ $-w_{\phi} - 1$ $2b-w_{\Lambda}-1$
$D$ $0$ $0$ $0$ $-\frac{2}{3} B+\frac{1}{3}$ $\frac{1}{3}$ $\frac{4}{3}$ $-w_{\phi} + \frac{1}{3}$ $\frac{1}{3}+2b-w_{\Lambda}$
$E$ $x_1/x_2$ $y_1/y_2$ $z_1/x_2$ $u_1/y_2$ $\frac{1}{3}+2b-w_{\Lambda}$ $-2b+w_{\Lambda}$ $-2b+w_{\Lambda} +1$ $-2b+w_{\Lambda}-w_{\phi}$
where\
$x_2=(-9w_{\Lambda}^4+(54b+9w_{\phi}-6)w_{\Lambda}^3
+(-108b^2+(-36w_{\phi}+30)b+6w_{\phi}+3)w_{\Lambda}^2+(72b^3+(36w_{\phi}-48)b^2-3w_{\phi})w_{\Lambda} +24b^3+(-24w_{\phi}-12)b^2-12bw_{\phi})$\
$y_2=(-3w_{\Lambda}^4+(18b+3w_{\phi}-2)w_{\Lambda}^3+(-36b^2+(-12w_{\phi}+10)b+2w_{\phi} +1)w_{\Lambda}^2+(24b^3+(12w_{\phi}-16)b^2-w_{\phi})w_{\Lambda}+8b^3+(-8w_{\phi}-4)b^2-4bw_{\phi})$\
$x_1=x(2b)(6b-3w_{\Lambda}+1)(2b-w_{\Lambda})$\
$y_1=x(-2b)(6b-3w_{\Lambda}+1)(2b-w_{\Lambda} +w_{\phi})$\
$z_1=x(2b-w_{\Lambda})(2b-w_{\Lambda}+w_{\phi})(6b-3w_{\Lambda}+1)$\
$u_1=x(2b)(2b-w_{\Lambda}))(2b-w_{\Lambda}+w_{\phi})$\
and\
$x=(2b-w_{\Lambda}-1)(2B-1)$\
The corresponding Jacobian matrix, which we denote as $\mathcal{J}(g)$, is equal to,\
$$\begin{bmatrix}
\frac{\partial f_1}{\partial \Omega_m} & \frac{\partial f_1}{\partial \Omega_{\Lambda}}& \frac{\partial f_1}{\partial \Omega_r} & \frac{\partial f_1}{\partial \Omega_{\phi}} \\
\frac{\partial f_2}{\partial \Omega_m} & \frac{\partial f_2}{\partial \Omega_{\Lambda}}& \frac{\partial f_2}{\partial \Omega_r} & \frac{\partial f_2}{\partial \Omega_{\phi}} \\
\frac{\partial f_3}{\partial \Omega_m} & \frac{\partial f_3}{\partial \Omega_{\Lambda}}& \frac{\partial f_3}{\partial \Omega_r} & \frac{\partial f_3}{\partial\Omega_{\phi}} \\
\frac{\partial f_4}{\partial\Omega_m} & \frac{\partial f_4}{\partial\Omega_{\Lambda}}& \frac{\partial f_4}{\partial\Omega_r} & \frac{\partial f_4}{\partial \Omega_{\phi}}
\end{bmatrix}$$
The Eigenvalues and their stability for each point are written as following:
- Stability for point $A$:\
The enough and sufficient condition to have $A_1$ as a stable fixed point for system is that all eigenvalues of Jacobian matrix $\lambda_i$ must satisfy $\mathrm{Re}(\lambda_i)\neq 0$, i.e., $$\begin{aligned}
&&
{w_{\Lambda} \le 2b-1,\ \ b < \frac{1}{3}w_{\Lambda}-\frac{1}{3}w_{\phi}}\\&&
{w_{\phi} < -1,\ \ 2b+\frac{1}{3} < w_{\Lambda}}\\&& {w_{\Lambda} \le 2b+\frac{1}{3},\ \ w_{\phi} < -1,\ \ 2b-1 < w_{\Lambda}}
\end{aligned}$$ From these it is found that the stability occurs at:\
$2b-1 < w_{\Lambda} \le 2b+\frac{1}{3}$\
$w_\phi < -1$. The corresponding effective EoS behaves like $w_{eff}=w_{\phi}$. Depending on the $w_{\phi}$, EoS evolves from larger than $-1$ to less than $-1$, that is, it crosses the phantom divide line of $w_{eff} = -1$.
- Stability for point $B$\
This is unstable critical point and the corresponding effective EoS, $w_{eff}=w_{\Lambda}$ is always larger than $-1$ and it crosses the phantom divide line of when $w_{\Lambda}=-1$.
- Stability for point $C$:\
Stability condition is $-1 < w_{\phi}, 2b-1 < w_{\phi}$. Consequently $C$ can be stable conditionally. The corresponding effective EoS is $w_{eff}=-1$ is located at the crosses the phantom divide line.
- Stability for point $D$:\
We obviously conclude that is unstable. The corresponding effective EoS,is given $w_{eff}=0$ and is always larger than $-1$ and it can not crosses the phantom divide line.
- Stability for point $E$:\
The point is stable conditionally only and only if $w_{\Lambda} < 2b-1, \frac{1}{2}w_{\Lambda}-\frac{1}{2}w_{\phi} < b$ .
Cosmography
===========
The following types of observational data are commonly used to study cosmography, **SNe Ia**: Type Ia supernovae (SNe Ia) or the latest “joint light curves" (JLA) sample [@sn], comprised of 740 type Ia supernovae in the redshift range $0.01 \leq z \leq 1.30$.\
**BAO**: The baryon acoustic oscillations (BAO) [@bao1],[@bao2], [@bao3], and [@bao4] (see table I of [@baotot]).\
**CC+$H_0$**: The cosmic chronometers (CC) data set in the redshift range $0 < z < 2$ [@cc]. In $f(T)$ gravity cosmography introduced and invetigated in details in Ref. [@Capozziello:2011hj].
In Fig. 1, we plot the time evolution of the density functions for $b=0.5,0.7,0.9$, $w_{\Lambda}= -1/3$, $w_{\phi}= 0$, where the horizontal axis shows $\log(1+z)$ and the vertical axis does the value of the density functions. We can observe that the density functions of matter $\Omega_m$ and the cosmological constant $\Omega_\Lambda$ increase in time, whereas the density functions of radiation $\Omega_r$ and $\Omega_\phi$ decrease in time. For low redshift values, $0 < z < 0.1$, the densities $\Omega_\phi$, $\Omega_r$ are monotonically increasing functions, but matter and cosmological constant density functions decrease. At the present redshift $z\sim 0$, $\Omega_\phi \sim \Omega_r$ are negligible in comparison to the matter and cosmological constant densities. This confirms our remarkable observation about the scalar field $\phi$ that it cannot play the role of DE. So, the density of the scalar field is almost negligible at the present time. At distinct values of the redshift shown as $z^{*}$, densities of matte, radiation and scalar field become equal, i.e. $\Omega_m \sim \Omega_r \sim \Omega_{\phi}$. This occurs at $z^{*}\approx 0.1$. Furthermore, there is an era when $z^{\dagger} \sim 0.4$ in which $\Omega_m \sim \Omega_\lambda$, shows another equilibrium among matter and cosmological constant. These behaviors of the density functions are compatible with the observations.
{width="8.0cm"}
Observation of a type of deceleration to acceleration phase transition
----------------------------------------------------------------------
The deceleration parameter $q$ is defined as $$q=-1-\frac{\dot{H}}{H^2}.$$ If the expansion of the Universe is decelerating, $q >-1$, while if the cosmic expansion is accelerating, $q < -1$. In our model using (\[hdot\]) we obtain
$$\begin{aligned}
&&
q=-1-\frac{3}{2(2B-1)}\Big(\Omega_{\phi}+
\Omega_m\\&&\nonumber+(1+w_{\Lambda})\Omega_{\Lambda}+(1+w_{r})\Omega_r\Big)\label{q}.\end{aligned}$$
We find numerical solutions by using $h = 0.7127_{-0.015}^{+0.013}$ km/s/Mpc, $\Omega_{\Lambda }=0.7018_{-0.02}^{+0.018}$, and $\Omega_{m0}=0.2981_{-0.018}^{+0.02}$, with $\chi^2_{min}=707.4$, $H_0= 73. 24 \pm 1.74$ km/s/Mpc. In Fig. 2, we depict the time evolution of the deceleration parameter $q$ for $b=0.5, 0.7, 0.9$. Here, the horizontal axis shows $\log(1+z)$ and the vertical axis shows the value of $q$. All of the curves meet. From Fig. 2, it is found that in the past for lower values of redshift, the value of $q$ evolved from larger than $-1$ to less than $-1$, namely, the expansion phase of the Universe changed from the deceleration to the acceleration. This is consistent with the observations.
![\[q\] Deceleration parameter $q$ for $b=0.5, 0.7, 0.9$. All of the curves meet. Here, the horizontal axis shows $\log(1+z)$ and the vertical axis shows the value of $q$.](DecelerationVsmN.eps){width="8.0cm"}
Effective Equation of State (EoS) of the Universe
--------------------------------------------------
Effective EoS of the Universe was defined in Eq. (\[eos\]). We note that in the DE dominated stage, the value of the EoS of DE can be regarded as the effective EoS of the Universe $w_{eff}$.
In Fig. 3, we show the time evolution of the effective EoS $w_{eff}$ for $b=0.5,0.7,0.9$ , where the horizontal axis shows $\log(1+z)$ and the vertical axis shows the value of $w_{eff}$. From Fig. 3, it is found that for low redshift values , the value of $w_{eff}$ became less than $-\frac{1}{3}$ and therefore the cosmic expansion phase of the Universe changed from the deceleration to the acceleration. It is also seen that in our model, the value of $w_{eff}$ evolves from larger than $-1$ to less than $-1$; that is, it crosses the phantom divide line of $w_{eff} = -1$. The value of the Eos parameter at the present redshift is around $w_{eff}\sim -1.6<-1$ shows an acceleration expansion beyond the phantom line.
![\[weffl\] Effective EoS $w_{eff}$ for $b=0.5,b=0.7,b=0.9$. The three curves coincides. Here, the horizontal axis shows $\log(1+z)$ and the vertical axis shows the value of $w_{eff}$.](Weff.eps){width="8.0cm"}
de Sitter solution
==================
Cosmological models usually have de Sitter (dS) solution where the Hubble parameter is constant (or almost constant in inflationary scenarios) $H=H_0$ as trivial solution. In GR such solution (used in inflationary mechanism as well as late time cosmology) becomes accessible when the dominant energy density $\rho\approx\rho_0$, which means that to have dS we need matter fields with very slowly varying energy density. It is not possible to find dS solution as an empty space solution in GR . But in modified gravity because of the geometrical terms (curvature $\mathcal{R}$ or torsion $T$) it will be possible to find dS as an (almost) exact solution for field equations. In forthcoming sections, we look for dS solution both in empty and matter contents cases in model defined by Eqs. (\[eq1\],\[eq1\]).
Let us firstly perform a little investigation on the equations of motion. As a result of continuity eq, we have two additional Klein-Gordon like dissipative eqs. (\[rhoeq\]) in non interacting case, we can find equations of motion for $\phi,\xi$. If we suppose that in (\[fphi\]), $B=0,A=-1$, they are written as following: $$\begin{aligned}
&&\ddot{\phi}+3H\dot{\phi}+6H^2=0\label{phi},\\
&&\ddot{\xi}+3H\dot{\xi}+6H^2=0\label{xi}.\end{aligned}$$ Note that always with $f(\phi)=-\phi$, we have $\nabla^{\mu}\nabla_{\mu}(\phi-\xi)=0$, and we have a “gauge freedom” to write fields $\phi,\xi$ as follows:
$$\begin{aligned}
&&\phi-\xi=\eta(t)\label{gauge}.\end{aligned}$$
An exact solution for $\eta$, in FLRW background is given by, $$\begin{aligned}
&&\eta(t)=\int\frac{\eta_0}{a(t)^3} dt\label{soleta}.\end{aligned}$$ Note that $\dot{\eta}(t)\sim \rho_m(t)$. In this case we can interpret $\dot{\eta}(t)$ as cold dark matter density. It is possible to take $\eta_0=0$ or $\eta_0\neq0$. We will study both cases in next subsections.
Empty spacetime {#vacuum}
---------------
To find an exact (almost exact) dS solution for a system given by (\[eq1\]-\[eq2\]),(\[phi\]-\[xi\]) in vacuum let us relax all matter contents, to make spacetime empty (we will never consider a quantum fluctuations in this approach). Furthermore, we set $H=H_0$ for dS case.
### Case $\eta(t)=0$
: When $\eta=0$, $\phi=\xi$. Thus the system (\[phi\],\[xi\]) and (\[hdot\]) is reduced to the following system: $$\begin{aligned}
&&\frac{H_0}{\dot{\phi}}=\frac{1}{4}\\&& \ddot{\phi}+3H_0\dot{\phi}+6H_0^2=0.\end{aligned}$$ From first equation we find $\phi(t)=4H_0t+\phi_0$. If we substitute it in second equation we obtain $H_0=0$. In $f(T)$ gravity stability for Einstein Universe is well studied in [@Wu:2011xa].
This is just Einstein static Universe and it proves that no dS solution exist.
Let us check whether this solution is stable or not. We make perturbation around the solution given by $(H,\phi)=(0,\phi_0)$. The equation is given by the following: $$\begin{aligned}
&&\delta\ddot\phi+3\dot{\phi}\delta H +3H\delta\dot\phi +12H\delta H=0.\end{aligned}$$ substituting the zeroth order solution we find $$\begin{aligned}
&&\delta\ddot\phi=0.\end{aligned}$$ Exact solution for perturbation function is $$\begin{aligned}
&&\delta\phi=at+b.\end{aligned}$$ When $t\to\infty$, we clearly observe that perturbation is growing up linearly and consequently the system becomes unstable under infinitesimal field and background perturbations.
### Case $\eta(t)\neq0$
: When $\eta= \int\frac{\eta_0}{a(t)^3} dt$, we have $\dot{\phi}-\dot{\xi}=\frac{\eta_0}{a(t)^3} $. Thus the system (\[phi\],\[xi\]) and (\[hdot\]) is reduced to the following system: $$\begin{aligned}
&&\frac{1}{2H_0}=\frac{1}{\dot{\phi}}+\frac{1}{\dot{\xi}},\\&&
\dot{\phi}-\dot{\xi}=\frac{\eta_0}{a(t)^3}\end{aligned}$$ Note that in dS phase, $a(t)=a_0 e^{H_0 t}$. An exact solution for fields pair $\phi,\xi$ is given as follows:
$$\begin{aligned}
&&\phi_0(t)=C_2+2H_0t-\frac{\eta_0}{6a_0^3H_0}e^{-3H_0 t}\\&&\nonumber
+\frac{1}{6H_0a_0^3}\Big(\Delta-4a_0^3\sqrt{H_0}\tanh^{-1}(\frac{\Delta}{4a_0^3\sqrt{H_0}})
\Big)
\label{solphi1}\\
&&\xi _0( t) =C_2+C_1+2H_0t+\frac{\eta_0}{6a_0^3H_0}e^{-3H_0 t}\\&&\nonumber
+\frac{1}{6H_0a_0^3}\Big(\Delta-4a_0^3\sqrt{H_0}\tanh^{-1}(\frac{\Delta}{4a_0^3\sqrt{H_0}})\Big)
\label{solxi1}\end{aligned}$$
Here $C_1,C_2$ are arbitrary integration constants and $\Delta=16H_0 a_0^6+a_0^3H_0^2e^{-6H_0t}$. The solutions given in (\[solphi1\],\[solxi1\]) are exact solutions for dS phase of our model under study. Let us study its stability under perturbations in time representation of fields and backgrounds. Later in Sec. (\[Stability in vacuum\]) another equivalent analysis using slow-roll coordinate N will be introduced.
Perturbation of field equations (\[phi\],\[xi\]) and Eq. (\[eq2\]) around the solution given by $(H,\phi,\xi)=(H_0,\phi_0,\xi_0)$ substituted in Eqs.(\[solphi1\],\[solxi1\]) given the following system of Eqs:
$$\begin{aligned}
&&\delta\ddot\phi+3\dot{\phi}_0\delta H +3H_0\delta\dot\phi +12H_0\delta H=0\\
&&\delta\ddot\xi+3\dot{\xi}_0\delta H +3H_0\delta\dot\xi +12H_0\delta H=0\\&&
2\delta\dot H +4\phi_0\delta\dot{H}+(\dot{\xi}_0-2H_0)\delta\dot{\phi}+
\\&&\nonumber(\dot{\phi}_0-2H_0)\delta\dot{\xi}-2(\dot{\xi}_0+\dot{\phi}_0)\delta H=0.\end{aligned}$$
It is hard to find exact solutions for perturbation functions and we do not discuss it. Note that asymptotically, $\phi\sim \xi\approx 2H_0 t$, consequently we can find the following solutions which are valid only when $t\to \infty$:
$$\begin{aligned}
&&\delta\phi\sim \delta\xi\approx (H_0t)^2,\ \ \delta H\approx H_0t.\end{aligned}$$
We conclude that dS behaves as an unstable phase in our model.
Case of matter contents {#nonvacuum}
-----------------------
Now we study exact solutions in dS phase when $\rho_a\neq0$. As a general case we consider the model given in (\[eq1\],\[eq2\]) for general densities $\rho_m=\rho_m^{0}a(t)^{-3}$.
### Case $\eta(t)=0$
When $\eta=0$, $\phi=\xi$, the system (\[phi\],\[xi\]) and (\[hdot\]) is reduced to the following system: $$\begin{aligned}
&&\dot{\phi}^2-4H_0\dot{\phi}+\kappa\rho_m^{0}a(t)^{-3}
=0.\label{hdot2}\\
&&\ddot{\phi}+3H_0\dot{\phi}+6H_0^2=0.\end{aligned}$$ Exact solutions provide that $H_0=0$ is the only possible solution. Thus similar to the empty case, still we just have Einstein static Universe.
If We perturb the system around the solutions given above, we have the following system of equations:
$$\begin{aligned}
&&\delta\ddot\phi=0\\
&&
2(1+2\phi_0)\delta\dot H -\delta \rho_m(t)=0.\end{aligned}$$
where $\phi_0=\mbox{C}$ is a constant. The system is clearly asymptotically unstable, consequently no stable static Einstein Universe exists.
### Case $\eta(t)\neq0$
When $\eta= \int\frac{\eta_0}{a(t)^3} dt$, the system (\[phi\],\[xi\]) and (\[hdot\]) is reduced to the following system: $$\begin{aligned}
&&\dot{\phi}\dot{\xi}-2H_0(\dot{\phi}+\dot{\xi})+\kappa\rho_m^{0}a(t)^{-3}
=0.\label{hdot3},\\&&
\dot{\phi}-\dot{\xi}=\frac{\eta_0}{a(t)^3}\end{aligned}$$ Exact solutions are given in terms of first integrals: $$\begin{aligned}
&&\dot{\phi}=2H_0+\frac{\eta_0}{2a(t)^3}\\&&\nonumber-\frac{1}{2}\sqrt{\eta_0a(t)^{-6}-4\kappa\rho_m^{0}a(t)^{-3}+16H_0^2
}\\&&
\dot{\xi}=2H_0-\frac{\eta_0}{2a(t)^3}\\&&\nonumber-\frac{1}{2}\sqrt{\eta_0a(t)^{-6}-4\kappa\rho_m^{0}a(t)^{-3}+16H_0^2
}.\end{aligned}$$ Therefore we have dS solution. We also conclude here that the system behaves asymptotically, $\phi\sim \xi\approx 2H_0 t$, consequently we can find the perurbated solutions when $t\to \infty$ diverge and consequently system becomes unstable.
Stability of de Sitter via Hartman-Grobman linearizion theorem
===============================================================
Stability of dS solution plays an essential role in inflationary scenario to have thermalization phase. Because nonlocal theory supposed to be an alternative theories for inflation we will study the stability in this context.
Stability in vacuum {#Stability in vacuum}
-------------------
In vacuum we observed that only when $\eta\neq0$ we have dS solutions given by (\[solphi1\],\[solxi1\]). Let us see whether this solution is stable or not. Suppose that $H=H_0,$ and (\[solphi1\],\[solxi1\]) are exact solutions for (\[eq1\],\[eq2\]) and (\[phi\],\[xi\]) with $f(\phi)=-\phi$. We can find the following auxiliary system: $$\begin{aligned}
&&\ddot{\xi}+3H\dot{\xi}+6H^2=0\label{ds1}\\&&
\ddot{\phi}+3H\dot{\phi}+6H^2=0\label{ds2}\\&&
\frac{\dot{H}}{3H^2}=2H(\frac{1}{\dot{\phi}}+\frac{1}{\dot{\xi}})-1\label{ds3}.\end{aligned}$$ Note that Eq. $\dot{\phi}-\dot{\xi}=\frac{\eta_0}{a(t)^3}$ is obtained using subtraction of two KG eqs for $\phi,\xi$. To study stability the first step is to make system given in (\[ds1\]-\[ds3\]) dimensionless, using new time parameter $N$ and new derivative $'=\frac{d}{dN}$ we have: $$\begin{aligned}
&&\varphi'=-6(2+\frac{\phi}{\alpha})\label{varphi}\\&&
\alpha'=-6(2+\frac{\alpha}{\varphi})\label{alpha}\\&&
H'=-3H(1-\frac{2}{\varphi}-\frac{2}{\alpha})\label{Hprime}.\end{aligned}$$ where $\varphi\equiv \phi',\alpha\equiv \xi'$. The critical point is located at $A=(H=H_0,\alpha=0,\varphi=0)$. We linearize the system under perturbation functions $H=H_0+\delta H, \varphi=\delta\varphi, \alpha=\delta\alpha$, The corresponding Matrix has an eigenvalue $\lambda_1=0$, and it shows that dS solution is an unstable point.
Stability in matter mixture {#Stability in matter mixture}
---------------------------
FLRW equations are given as following forms for the case with matter density and when $\phi\neq \xi$: $$\begin{aligned}
&&\frac{2\dot{H}}{3H^2}=\frac{-\dot{\phi}\dot{\xi}+2H(\dot{\phi}+\dot{\xi})-\kappa\rho_m^{0}a(t)^{-3}}{\frac{1}{2}\dot{\phi}\dot{\xi}+\kappa\rho_m^{0}a(t)^{-3}}\\&&
\ddot{\xi}+3H\dot{\xi}+6H^2=0\\&&
\ddot{\phi}+3H\dot{\phi}+6H^2=0.\end{aligned}$$
In this case because of $a(t)$ the system becomes non autonomous but still we can study the local stability in the vicinity of a critical point for $t\geq t_s$. Using the time coordinate $N$ and by redefining $\varphi\equiv \phi',\alpha\equiv \xi'$ we have the following system of differential equations:
$$\begin{aligned}
&&\frac{2}{3}\frac{H'}{H}=\frac{H^2(2(\varphi+\alpha)-\varphi\alpha)-\kappa\rho_m^{0}a^{-3}}
{\frac{1}{2}H^2\varphi\alpha+\kappa\rho_m^{0}a^{-3}}\\&&
\varphi'=\,{\frac {-3(4\,{H}^{2}{a}^{3}\alpha\,\varphi+2\,{H}^{2}{a}^{3}{\varphi}^{2}+
\kappa\,\varphi\,\rho^{0}_{{m}}+4\,\kappa\,\rho^{0}_{{m}})}{{H}^{2}{a}^{3}\alpha\,
\varphi+2\,\kappa\,\rho^{0}_{{m}}}}
\\&&
\alpha'=\,{\frac {-3(2\,{H}^{2}{a}^{3}{\alpha}^{2}+4\,{H}^{2}{a}^{3}\alpha\,
\varphi+\alpha\,\kappa\,\rho^{0}_{{m}}+4\,\kappa\,\rho^{0}_{{m}})}{{H}^{2}{a}^{3}
\alpha\,\varphi+2\,\kappa\,\rho^{0}_{{m}}}}\end{aligned}$$
here $\varphi\equiv \phi',\alpha\equiv \xi'$.
The corresponding linearized system near the unique physically accepted critical point $\{H = 0,\alpha = -4, \phi = -4\}$ called a proper node, or a star point(Actually it defines Einstein static solution) has a triplet proper node degenerated eigen value $\lambda=-\frac{3}{2}$. The critical point is asymptotically stable , and it shows that system is asymptotically stable .
Conclusions
===========
In this paper, we have considered the exact cosmological solutions in nonlocal $f(T)$ gravity, which can be regarded as an extension of nonlocal general relativity to the Weitzenböck spacetime. We have explored the autonomous system of the first order dynamical equations by deriving effective forms of cosmological field equations in a homogeneous and isotropic cosmological background to describe the whole evolution history of the universe. Furthermore, we have introduced a specific form of the interaction between matter, DE, radiation and a scalar field and analyzed the local stability in the dynamical systems, which is the so-called “the stable/unstable manifold”. It has been found that the system has a stable(unstable) attractor solutions. In addition, we have investigated the exact solutions of the cosmological equations in the case of de Sitter spacetime. We have demonstrated whether the de Sitter solutions can exist or not in this scenario by examining the role of an auxiliary function called “gauge” $\eta$ in the formation of such cosmological solutions. Moreover, we have studied the stability problem of the de Sitter solutions both in vacuum and non-vacuum spacetimes. It has been shown that for nonlocal $f(T)$ gravity, we can obtain the stable de Sitter solutions even in vacuum spacetime.
We thank the anonymous referee for intuitive comments and thorough criticism on our manuscript. This work was partially supported by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds for Fukushima University Faculty (17RI017 and 18RI009) (K.B.).
[99]{}
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---
abstract: 'We present an active control scheme of a kinetic model of swarming. It has been shown previously that the global control scheme for the model, presented in [@JK04], gives rise to spontaneous collective organization of agents into a unified coherent swarm, via a long-range attractive and short-range repulsive potential. We extend these results by presenting control laws whereby a single swarm is broken into independently functioning subswarm clusters. The transition between one coordinated swarm and multiple clustered subswarms is managed simply with a homotopy parameter. Additionally, we present as an alternate formulation, a local control law for the same model, which implements dynamic barrier avoidance behavior, and in which swarm coherence emerges spontaneously.'
address: 'US Naval Research Laboratory, Plasma Physics Division, Code 6792, Nonlinear Systems Dynamics Section, Washington, DC 20375'
author:
- 'David S. Morgan and Ira B. Schwartz'
title: Dynamic coordinated control laws in multiple agent models
---
[^1]
swarming, control, dynamics, emergent behavior
Introduction
============
Multiple agent models are comprised of a multitude of simple autonomous vehicles, which are loosely coupled via communication. It is anticipated that such systems will play a key role in future deployments, as the drive to miniaturize electronic devices results in smaller and more capable self-mobile machines with limited decision making abilities. Thus, one of the main research areas of interest is the dynamic pattern formation and control of a large number of agents [@Bonabeua99]. In particular, **given a specific dynamical system composed of a large number of individual vehicles, each with specified limited decision-making and communication abilities, a vital question is under what conditions large-scale aggregate dynamics may be controlled to form coherent structures, or patterns. An example from electronics is** a concept paper [@JK01] which shows that complex patterns can arise from a large array of micro actuators interconnected to mimic a finite difference approximation of standard reaction diffusion partial differential equations (PDE). However, it is a static theory based on quite standard pattern formation theories from reaction-diffusion which assumes pure local coupling.
In contrast, many biological examples of coherent dynamical motion (swarming) exist in nature. Populations such as bees, locusts, and wolves often move in coordinated but localized efforts toward a particular target. In addition many more examples abound of populations of individuals that move according to local rules, and whose aggregate dynamics achieve an overall large-scale complex pattern or state. Bacterial colonies, which evolve in part via chemotactic response, are such an example. The mathematical biology community has been exploring models for animal swarms, and this work pinpoints some of the difficulties (see the survey paper [@EK01]). Traditional models for biology populations involve local PDE for the population density [@Murray]. Edelstein-Keshet *et al* [@EKWG98] recently considered such a model in one space dimension for African migratory locusts. These insects have a gregarious phase in which swarms of individuals can travel for days over thousands of miles. Evidence exists that the swarms remain cohesive even in the absence of a nutrient gradient. The analysis of [@EKWG98] shows that such cohesive swarms cannot be described by traveling wave solutions of their one dimensional advection-diffusion model. More recently, Mogilner and Edelstein-Keshet consider nonlocal interactions, in which the drift velocity of the population is determined by a convolution operator with the entire population [@MEK99]. These models, resulting in integro-differential equations, do sometimes produce coherent band-like structure. Earlier work by Edelstein-Keshet and Watmough [@WEK95] on army ant swarms, considers a one dimensional model and shows the existence of traveling wave solutions for the leading edge of the pack, but they do not consider band-like solutions that would describe something like a locust swarm. These particular examples involve one-dimensional models and simulations. In summary, most studies of biological swarming involve models from continuum theory, many of which are based on some form of local communication, which are modeled by way of interactions or couplings.
The statistical physics community has recently tried to understand similar problems in situations where the number of individuals are very large. Statistical information derived for large numbers is less relevant to sensor applications involving smaller numbers of individuals. However, the connection between the discrete and the continuous is an important problem that is well-studied in this field. The particle approach involves starting with simple rules of motion, involving combinations of biased random walks, sampling of motions and positions of nearby neighbors, with some governing strategy designed to mimic core components of animal interactions. For example, Schweitzer et al [@SET01] consider a theory of canonical-dissipative systems and the energetic conditions for swarming. Daniel Grünbaum [@G99] has derived advection diffusion equations for internal state-mediated biased random walks. Mogilner and Edelstein-Keshet [@MEK96] consider both continuum and cellular automata models for populations of self-aligning objects. Stöcker [@Stocker] considers a hexagonally based cellular automata model for tuna school formation. These are just a few examples. In all cases, the local rules are precisely defined and aggregate motion can be observed in numerical simulations.
As an alternative to understanding coherent swarm structures that use finite models (non-continuum theories), a recent body of work considers general particle-based models for self-propelled organisms (see for example [@Albano; @BDG97; @CBV99; @VCFH99]). Collective motion and swarming is observed along with interesting aspects of dynamic phase transitions, including crystalline like motion, liquid, solid, and gas-like states. Toner and Tu [@TT95; @TT98; @Tu00] use renormalization group ideas to study flocking motion in a particle-based model. Some of this work parallels classical statistical theory of transport which derives hydrodynamic equations from local interaction models [@Irving:1950; @Thompson:1972; @Mazo:1967]. The approach considered by Chang, *et al.* [@ChangShaddenMarsden] considers agents in a scalar potential field and utilizes gyroscopic and braking forces.
In most cases presented, the agents are self-propelled and the nature of the coupling or communication imposes a given pattern. Here we consider similar aspects, but with the idea of controlling the communication to form patterns. In this article we consider kinetic models in which, depending on the control law used, the self-propelled agents communicate, either locally within a specified radius about each agent, or globally with every other agent in the swarm. Under appropriate choices of controlling “potentials”, coherent motion of agents is observed. In general, the models considered are based on controls which involve long range attraction and short range repulsion, similar to the ideas in [@zohdi03]. However, in [@zohdi03], the computational approach to obstacle avoidance, achieved by forming clustered groups from a single coherent swarm, is to use genetic algorithms, which contain a number of restrictive rules. This violates the assumption of creating a swarm with limited computational ability.
In the work presented here, we consider the problem of dynamically deforming a single large and coherent swarm into a collection of subswarm clusters under simple control modifications. A cluster is a subset of the original swarm which functions independently as a coherent swarm, and which, when fully formed, does not interact with agents that are not members of the cluster. A primary goal of this work is to generate simple algorithmic controls for obstacle avoidance, and we consider two methods to achieve this. We also consider multiple approaches to guiding a swarm, by dynamically steering leader agent(s), and by *a priori* fixing a target to which all agents are attracted.
We formulate the first control problem using homotopy, or continuation, theory [@Allgower80; @Rheinboldt00]. The homotopy parameter controls the communication coupling, selecting between local and global communication, and may simultaneously be used to modify other characteristics of the control law. Such a control law allows one to use a single parameter to switch from a single coherent swarm state (global coupling) to a multiple cluster state (local coupling) and back again. Swarm coherence and inter-agent collision avoidance is achieved with this control law via a long-range attractive/short-range repulsive potential, and swarm navigation is implemented via group-averaged motion and leader-following controls.
We also consider an alternate formulation using only local coupling between agents, in which a convex barrier is detected and avoided, and where the barrier location is not *a priori* known. This approach to obstacle avoidance is similar in nature to that discussed in [@ChangShaddenMarsden]. In addition, swarm navigation is achieved by introducing terms in the control law so that all agents seek a common target. Whereas clustering with the homotopy control law is due to an attractive potential to other agents, clustering appears to arise naturally with this control law, as agents interact while they seek out a common target.
The layout of the paper is as follows: In section 2 we introduce the kinetic model presented in [@JK04], and discuss some important properties of its global control law. In Section 3 we introduce a modified control law implementing local coupling. A homotopy control law is presented in Section 4. In Section 5 we present the alternative control law with an alternate approach to target seeking behavior and barrier avoidance, and we conclude with a discussion.
\[sec:UAV-model-and\]Multi-agent kinetic model and properties
=============================================================
The ideas we present apply to a large class of systems. Consider a continuous dynamical system $\frac{d\mathbf{z}}{dt}=\mathbf{F}(\mathbf{z}(t))$ arising from an autonomous vector field $\mathbf{F}$, where $t\in\Re$ and $\mathbf{z}\in\Re^{n}$, describing the equations of motion. Associated to this dynamical system is the system governing trajectories, in which all orbits have unit velocities,$$\frac{d\mathbf{r}}{dt}=\mathbf{G}(\mathbf{r}(t))\equiv\frac{\mathbf{F}(\mathbf{r})}{\left\Vert \mathbf{F}(\mathbf{r})\right\Vert }.\label{eq:traj_equation}$$ Consider a (nontrivial) trajectory $\mathbf{r}(t)$ of (\[eq:traj\_equation\]), its associated unit tangent vector $\mathbf{x}=\mathbf{G}(\mathbf{r}(t))$ defined for all $t\in\Re$, and the positively oriented unit conormal vectors $\mathbf{y}_{i}(t),\:(i=1,n-1)$. The collection of vectors $\mathcal{F}=\{\mathbf{x}(t),\mathbf{y}_{i}(t)\:(i=1,\ldots,n-1)\}$ is called the (moving) reference frame associated to $\mathbf{r}$. Thus, one may recast a continuous dynamical system as a system of trajectories $\mathbf{r}(t)$ parameterized by arclength, with the associated moving frame $\mathcal{F}$. The behavior of a system of trajectories with the associated moving frame is governed by the Frenet-Serret system of equations.
Derivation of Frenet-Serret system of equations
-----------------------------------------------
We derive the Frenet-Serret equations, restricted to the plane, following the approach of [@Jurdjevic]. Consider a differentiable trajectory $\mathbf{r}(t)$ in $\Re^{2}$, parameterized by arclength, which represents the motion of an agent over time. A positively oriented orthonormal frame $\mathbf{x}$ and $\mathbf{y}$ is associated to $\mathbf{r}(t)$, by taking $\mathbf{x}$ equal to the unit tangent vector $d\mathbf{r}/dt$, and $\mathbf{y}=\mathbf{x}^{\perp}$ to be the unit normal vector positively oriented relative to $\mathbf{x}$. There exists a function $\kappa(t)$, called the curvature of $\mathbf{r}(t)$, such that$$\frac{d\mathbf{r}}{dt}=\mathbf{x}(t),\quad\frac{d\mathbf{x}}{dt}=\kappa(t)\cdot\mathbf{y}(t).\label{eq:FS_deriv1}$$ One then obtains the equation governing the unit normal vector **$\mathbf{y}$** as follows. Using the right-hand equation of Eq. (\[eq:FS\_deriv1\]) we find that $\mathbf{x}\cdot\mathbf{y}=d\mathbf{x}/dt\cdot\mathbf{y}+\mathbf{x}\cdot d\mathbf{y}/dt=\kappa(t)+\mathbf{x}\cdot d\mathbf{y}/dt=0$, and thus $d\mathbf{y}/dt=-\kappa(t)\cdot\mathbf{x}$. The moving frame $\mathbf{x}$ and $\mathbf{y}$ associated with $\mathbf{r}(t)$ can be expressed by a rotational matrix $R(t)$, which has columns consisting of coordinates of $\mathbf{x}$ and $\mathbf{y}$ relative to a fixed orthonormal frame $e_{1}$ and $e_{2}$ in $\Re^{2}$.
This formulation leads to a natural Lie group setting, but we do not consider that aspect further in this article. We also note that it is also possible to derive the Frenet-Serret equations by considering the problem of steering unit-charge, unit-mass particles in a magnetic field. For details, consult [@JK04] and references therein.
Equations of motion for multiple agents.
----------------------------------------
We consider a set of $n$ agents, restricted to smooth motions in the plane, and moving at unit speed. The system of equations modeling each agent is$$\begin{aligned}
\dot{\mathbf{r}}_{k} & = & \mathbf{x}_{k}\label{eq:two_dim_Frenet-Serret}\\
\dot{\mathbf{x}}_{k} & = & \mathbf{y}_{k}\cdot u_{k}\nonumber \\
\dot{\mathbf{y}}_{k} & = & -\mathbf{x}_{k}\cdot u_{k},\nonumber \end{aligned}$$ for $k=1,\dots,n$. The orientation of an agent is given by the moving frame $\mathbf{x}$ and $\mathbf{y}$, its trajectory is given by $\{\mathbf{r}(t)|t\in\Re\}$, and the agents are coupled together via a scalar curvature control law $u$, which is detailed below.
The control law $u_{k}$, introduced in [@JK04], is$$u_{k}=\sum_{j\neq k}u_{jk},\label{eq:control_law}$$ with$$u_{jk}=\left[-\eta\left(\frac{\mathbf{r}_{jk}}{\left|\mathbf{r}_{jk}\right|}\cdot\mathbf{x}_{k}\right)\left(\frac{\mathbf{r}_{jk}}{\left|\mathbf{r}_{jk}\right|}\cdot\mathbf{y}_{k}\right)-f\left(\left|\mathbf{r}_{jk}\right|\right)\left(\frac{\mathbf{r}_{jk}}{\left|\mathbf{r}_{jk}\right|}\cdot\mathbf{y}_{k}\right)+\mu\mathbf{x}_{j}\cdot\mathbf{y}_{k}\right]\label{eq:control_law_atom}$$ where $\mathbf{r}_{jk}\equiv\mathbf{r}_{k}-\mathbf{r}_{j}$, $f$ is $$f\left(\left|\mathbf{r}_{jk}\right|\right)=\alpha\left[1-\left(\frac{r_{0}}{\left|\mathbf{r}_{jk}\right|}\right)^{2}\right],\label{eq:distance_func}$$ and $\eta=\eta(|\mathbf{r}|)$, $\mu=\mu(|\mathbf{r}|)$, and $\alpha=\alpha(|\mathbf{r}|)$ are specified functions. We now describe this control law in some detail. We first note that when $u_{k}<0$ $(u_{k}>0)$, the Frenet frame will rotate in a clockwise (anticlockwise) fashion, respectively. In order to simplify the following discussion, we consider the case of $n=2$, but note that the discussion holds for general $n$. Let $\mathbf{r}_{1}$, $\mathbf{x}_{1}$ and $\mathbf{y}_{1}$ be the position and corresponding Frenet frame of one of the agents. We will examine each of the terms in Eq. (\[eq:control\_law\_atom\]) in turn. The first term, $-\eta(\mathbf{r}_{jk}/|\mathbf{r}_{jk}|\cdot\mathbf{x}_{k})(\mathbf{r}_{jk}/|\mathbf{r}_{jk}|\cdot\mathbf{y}_{k})$ serves to orient the vehicles perpendicular to their common baseline, $\mathbf{r}_{jk}$. To see this, let $\theta_{\mathbf{x}}$ and $\theta_{\mathbf{y}}$ be the angles the (unit) vectors $\mathbf{x}_{1}$ and ****$\mathbf{y}_{1}$ make with $\mathbf{r}_{21}/|\mathbf{r}_{21}|=(\mathbf{r}_{1}-\mathbf{r}_{2})/|\mathbf{r}_{1}-\mathbf{r}_{2}|$, respectively. Then$$\begin{aligned}
-\eta(\mathbf{r}_{21}/|\mathbf{r}_{21}|\cdot\mathbf{x}_{1})(\mathbf{r}_{21}/|\mathbf{r}_{21}|\cdot\mathbf{y}_{1}) & = & -\eta\cos(\theta_{\mathbf{x}})\cos(\theta_{\mathbf{y}})\\
& = & -\eta\cos(\theta_{\mathbf{x}})\cos(\theta_{\mathbf{x}}-\frac{\pi}{2}).\end{aligned}$$ This expression is zero for $\theta_{\mathbf{x}}=\frac{\pi}{2},\frac{3\pi}{2}$, positive for $0\le\theta_{\mathbf{x}}<\frac{\pi}{2}$ and $\pi\leq\theta_{\mathbf{x}}<\frac{3\pi}{2}$, and is negative elsewhere. Thus, this term steers the vehicle to the nearest perpendicular with the baseline $\mathbf{r}_{21}$.
Inter-agent spacing is controlled via the short-range repulsive/long-range attractive term,$$-\alpha\left[1-\left(\frac{r_{0}}{\left|\mathbf{r}_{jk}\right|}\right)^{2}\right]\left(\frac{\mathbf{r}_{jk}}{\left|\mathbf{r}_{jk}\right|}\cdot\mathbf{y}_{k}\right).\label{eq:distance_control}$$ The first factor of (\[eq:distance\_control\]), which arises from a Leonard-Jones type of potential, is negative if the distance between two agents is less than $r_{0}$, and positive if the distance is greater than $r_{0}$. The sign of the second factor is determined by the orientation of the two agents $\mathbf{r}_{j}$ and $\mathbf{r}_{k}$, relative to the baseline between them. See figure \[cap:f\_and\_potential\], which shows both the graphs of the potential, and of $f$.
It is easy to see that the third term, $\mu\mathbf{x}_{j}\cdot\mathbf{y}_{k}$, serves to drive the vehicles to a common orientation, by rewriting the dot product in terms of cosines.
The control law (\[eq:control\_law\]) is global (see Fig. \[cap:Local-coupling-scenario.\]), meaning that every agent communicates with all other agents in the swarm. Furthermore, the final orientation (heading) of the swarm is obtained by group averaged motion, which is in turn determined by the initial positions and orientations of the agents. We define this as the globally coupled, group averaged motion law. For the case $n=2$, rigorous global convergence results have been obtained, by reducing (\[eq:two\_dim\_Frenet-Serret\]) via the symmetry group $\textrm{SE(2)}$ and demonstrating explicitly the existence of a Lyapunov function, the physical result being that agents will align to the same heading, perpendicular to their common base-line, and with the appropriate distance between them. See [@JK04] (and [@JK02]) for details. Recently, local convergence results were obtained for the general case of $n$ agents. See [@JK03] for details.
\[sec:Modified-control-laws.\]The leader following control law utilizing local coupling.
========================================================================================
The control law (\[eq:control\_law\]) is global; that is, at each time-step, an agent requires information from all other agents in the swarm. Global communication is however often not practical. It is a goal to miniaturize mobile platforms as much as possible, and so not surprisingly, space constraints limit the power and sensitivity of on-board sensors and transmitters. Environmental factors, such as weather effects and local geography can also have detrimental effects on electromagnetic signals. On the other hand, a local control law only requires an agent to communicate with some subset of agents in the swarm, such as their nearest neighbors. Indeed local coupling is observed in most natural swarms, such as schooling fish and flocking birds. See Fig. \[cap:Local-coupling-scenario.\] for a comparison of the global and local coupling we employ.
We employ local coupling of agents by limiting communication to a neighborhood of each agent, so that there may be agents that are not in communication with other agents in the swarm. However, we choose initial conditions such that each agent is in communication range of at least one other agent, and such that all agents are ‘path connected’ initially, meaning that any two agents in the swarm are coupled at least through intermediary agents. See Fig. \[cap:Local-coupling-scenario.\] and caption.
The implementation of the local coupling model is straight-forward. We simply multiply the control law (\[eq:control\_law\]) with the cutoff function$$c\left(\left|\mathbf{r}_{jk}\right|,q,w\right)=\left\{ \begin{array}{cc}
1 & \textrm{if $\left|\mathbf{r}_{jk}\right|<w$,}\\
q & \textrm{otherwise.}\end{array}\right.\label{eq:cutfn}$$ By using a nonzero value for $q$, one obtains a global ‘cutoff’ function, that is useful for imposing stronger local coupling, while maintaining weak global coupling (by setting $q\ll1$). For the present discussion we set $q=0$, so that only when the Euclidean distance between two agents is less than $w$ will they interact. We thus obtain the modified law$$\begin{aligned}
u_{k}^{L} & =c\left(\left|\mathbf{r}_{jk}\right|,0,w\right) & \sum_{j\neq k}u_{jk},\label{eq:control_law_modified}\end{aligned}$$ where $u_{jk}$ is defined by Eq. (\[eq:control\_law\_atom\]) .
We note that when the distance between all agents is less than $w$, Eq. (\[eq:control\_law\_modified\]) reduces to the original control law (\[eq:control\_law\]), while if the swarm is split into subswarm clusters greater than distance $w$ from one another, the subswarms will evolve independently of one another.
The control law (\[eq:control\_law\]) uses group averaged motion for swarm control. The asymptotic heading of the swarm is thus determined by the initial conditions. We wish to control the direction of the swarm without having to steer each agent individually. We implement a leader following control which allows one to ’steer’ the swarm by controlling a designated leader agent (or agents). This provides simple directional control of a swarm, since only the leader agents are steered, and the nonleader agents, which we define to be follower agents, pursue leader agents automatically. We note that leader following behavior can be implemented with either local or global agent coupling.
The leader following, local control law is obtained by using (\[eq:control\_law\_modified\]) for follower agents, modified so there is stronger coupling between follower and leader agents, $$u_{k}^{follower}=c\left(\left|\mathbf{r}_{l(k)k}\right|,0,w\right)\ell_{c}u_{l(k)k}+\sum_{j\neq k,l(k)}c\left(\left|\mathbf{r}_{jk}\right|,0,w\right)u_{jk},\label{eq:follower_control}$$ where $\ell_{c}$ is a coupling constant and $l(k)$ is the index of the leader swarmer closest to the $k^{\textrm{th}}$ follower swarmer, while for leader agents the control law is simply$$u_{k}^{leader}=s_{k},\label{eq:leader_control}$$ where $s_{k}$ is an explicit steering program, which can be given by a trajectory from a dynamical system.
\[sec:Homotopy-control-law.\]Homotopy control law.
==================================================
We combine the leader following, local control law given by Eqs. (\[eq:follower\_control\]) and (\[eq:leader\_control\]), introduced in the previous section, with the group averaged, global control law (\[eq:control\_law\]) of section \[sec:UAV-model-and\] to obtain a hybrid control law utilizing a homotopy parameter (defined below), which we hereafter refer to as a homotopy control law. The introduction of a homotopy parameter provides a simple mechanism to dynamically switch from one control law to another.
Let $u_{k}^{G}$ be the global control law (\[eq:control\_law\]) and let $u_{k}^{L}$ be the local control law given by Eqs. (\[eq:follower\_control\]) and (\[eq:leader\_control\]). The homotopy control law $$u_{k}=u_{k}(\lambda),\;0\leq\lambda\leq1\label{eq:homotopy_ctrl_law}$$ is defined by the properties$$u_{k}(\lambda=0)=u_{k}^{G},\quad u_{k}(\lambda=1)=u_{k}^{L},$$ along with the property that the control law $u_{k}$ varies smoothly with $\lambda$.
To implement the homotopy control law (\[eq:homotopy\_ctrl\_law\]), we designate $m$ agents to be leaders, so that there will be $n-m$ follower agents. Additionally, let $l(k)$ be the index associated to the closest leader (in the Euclidean sense) of the $k^{\textrm{th}}$ follower agent. The homotopy control law for follower agents is $$\begin{aligned}
u_{k}^{follower}(\lambda) & = & c\left(\left|\mathbf{r}_{l(k)k}\right|,1-\lambda,w\right)[(\ell_{c}-1)\lambda+1]u_{l(k)k}+\label{eq:homo_flwr_ctrl}\\
& & \sum_{j\neq k,l(k)}c\left(\left|\mathbf{r}_{jk}\right|,1-\lambda,w\right)u_{jk}\nonumber \end{aligned}$$ where $u_{jk}$ is given by (\[eq:control\_law\_atom\]) and $\ell_{c}\gg1$ is a constant which couples followers more strongly to their closest leader agent. This coupling constant was found to be necessary for the proper swarm-splitting behavior to emerge. The follower agents must react more strongly to the motion of leader agents, otherwise some follower agents were observed to escape from a local neighborhood of the swarm, and would thus no longer interact with it. The homotopy control law for the leader agent(s) is
$$u_{k}^{leader}(\lambda)=\sum_{j\neq k}\left[u_{jk}(1-\lambda)+\frac{s_{k}}{n-1}\lambda\right],\label{eq:homo_ldr_ctrl}$$
where $s_{k}$ is an explicit steering program, which may be supplied by an external dynamical system.
When $\lambda=0$, the cutoff function $c\left(\left|\mathbf{r}_{jk}\right|,1,w\right)\equiv1$, and both the follower control law (\[eq:homo\_flwr\_ctrl\]) and leader control law (\[eq:homo\_ldr\_ctrl\]) reduce to the global control law (\[eq:control\_law\]). When $\lambda=1$ on the other hand, $$c\left(\left|\mathbf{r}_{jk}\right|,0,w\right)=\left\{ \begin{array}{cc}
1 & \textrm{if $\left|\mathbf{r}_{jk}\right|<w$,}\\
0 & \textrm{otherwise,}\end{array}\right.$$ and inter-agent coupling is local. The control law for the follower and leader agents is in this case given by Eqs. (\[eq:follower\_control\]) and (\[eq:leader\_control\]), respectively.
When only one leader is present, the swarm transitions from group averaged motion to leader following motion, and thus the swarm can be directed, as $\lambda\rightarrow1$. Note that in this case, the local coupling plays no significant role in the behavior, other than perhaps making the swarm more robust to communications difficulties.
When more than one leader agent is designated, other swarming behaviors are possible. In this case, the local coupling plays a crucial role. For example, assume there are two leader agents. As $\lambda\rightarrow1$, and the two leaders are directed away from one another, the swarm will effectively split into two subswarm clusters. This is the result of the local coupling and the fact that follower agents are more strongly coupled to their *closest* leader agent. As the leader agents diverge, the follower agents following one leader leave the communications range of the followers of the other leader, so that they no longer interact. The sets of equations modeling the two subswarms in this case decouple.
Homotopy control law simulation results
---------------------------------------
We present the results of simulations of the modified model using the homotopy control law (\[eq:homotopy\_ctrl\_law\]). We additionally present the results of simulations using the homotopy control law with local coupling throughout. That is, we consider the homotopy control law (\[eq:homotopy\_ctrl\_law\]) with the modification$$u_{k}(\lambda=0)=u_{k}^{GL},$$ where $u_{k}^{GL}$ utilizes group averaged motion, but with local coupling. For each control law, we ran several simulations, each with random initial data, as described below.
We first present a prototypical simulation using the homotopy control law presented in section \[sec:Homotopy-control-law.\], using $n=6$ agents in the swarm. The parameters are $\alpha=\eta=\mu=0.02$ and $r_{0}=1.5$, while the cutoff function parameters are $q=0$ and $w=4$, and simulations commence with $\lambda=0$, so that the initial control law is the globally coupled group averaged law (\[eq:control\_law\]). The initial positions of the six agents are $\{(-1,1),(0,1),(1,1),(-1,0),(0,0),(1,0)\}$, while the initial orientations $\theta_{i}$ of each agent is randomly chosen from angles constrained to lie $\phi-\pi/4<\theta_{i}<\phi+\pi/4$, and $\phi\in[0,2\pi]$ is also randomly chosen, and the system is integrated to $t=1000$. Near $t=400,$ the homotopy parameter $\lambda$ increases to one, and the control laws (\[eq:follower\_control\]) and (\[eq:leader\_control\]) are used. See figure \[cap:UAV\_hybrid\_coupling\]. As the homotopy parameter is switched on, the two leaders use the simple ’programs’ $s_{1}=0.01$ and $s_{2}=-0.01$. This causes one leader to start a gentle clockwise loop and the other leader to begin a gentle counterclockwise loop. At this point in the simulation, the entire swarm then splits into two subswarms as the follower agents move toward their nearest leader agent. As the homotopy parameter is decreased to zero near $t=600,$ the control law reverts to (\[eq:control\_law\]), global communication between all agents is restored, and the two subswarms reform as one swarm.
We next present an extension of the above, a prototypical simulation using the homotopy control law, but with local coupling throughout, so that the simulation commences with locally coupled, group averaged motion, with all agents in communications range of one another. As in the previous subsection, the homotopy parameter is initially zero, but near $t=400$ it is increased to one, and the system transitions from group averaged motion to leader following motion, with two leaders. Once again the swarm splits into two subswarm clusters for the same reasons as outlined in the previous subsection. Near $t=600,$ the homotopy parameter is then decreased to zero, and the system returns to group averaged motion. See figure \[cap:UAV\_local\_coupling\]. However, due to the local nature of the coupling, once the system returns to group averaged motion, the two subswarms remain independent, since the subswarms are not in communications range of one another.
Target seeking and barrier avoidance control laws\[sec:Target-seeking-laws\]
============================================================================
The homotopy control law of the preceding section provided a mechanism to switch between global and local coupling, and thus created a swarm which could split into subswarms, and each of the subswarms could be independently directed. We now introduce an additional method for both swarm navigation, by coding an *a priori* target-seeking behavior into the control law, and a method of barrier avoidance, by automatically sensing a barrier and splitting the swarm around the barrier. We note that control laws presented in this section may also be integrated with the previously introduced control laws. The character of the ideas in this section is similar in spirit to those presented in [@ChangShaddenMarsden], where a force law is introduced with a global potential for target seeking, but where both gyroscopic and braking forces are used for collision avoidance.
Target seeking control law.
---------------------------
A target is considered here to be a fixed point in the plane that is specified ahead of time. In the context of UAVs, this implies that the position of a target would be preprogrammed before deployment, though an alternate possibility would be to communicate target coordinates to agents in flight. We introduce a target in the model by globally coupling the agents to an ’agent’ that does not move. Let $\bar{r}$ be the fixed location of the target. The modified control law for target seeking is$$\begin{aligned}
u_{k} & = & \sum_{j\neq k}\left[c(|r_{jk}|,0,w)(-\alpha\left(1-\left(\frac{r_{0}}{\left|r_{jk}\right|}\right)^{2}\right)(r_{jk}\cdot y_{k})\right]+\label{eq:modif_ctrl_law_2}\\
& & \gamma\alpha\left(1-\left(\frac{r_{0}}{\left|\bar{r}_{k}\right|}\right)^{2}\right)(\bar{r}_{k}\cdot y_{k}).\nonumber \end{aligned}$$ where $\bar{r}_{k}$ is the vector directed from the position of the $k^{\textrm{th}}$ agent to $\bar{r}$, and $\gamma$ is a weighting constant. Note that there is no term in this control law to align agents to a common heading. In fact, the only inter-agent term is the first term, involving the summation, which provides for collision avoidance. The cutoff function $c$ implies that this term will have no effect if agents are outside of the cutoff radius $w$. The second term, which is global, steers individual agents toward the target. Though there are no terms to explicitly group agents, if the initial conditions are chosen so that the agents start in a group, then they will tend to stay together, as they collectively steer toward the target.
Barrier avoidance
-----------------
The homotopy control law presented in section \[sec:Homotopy-control-law.\] can be used for barrier avoidance by splitting a swarm into two subswarms, which are steered independently around the barrier. The swarm splitting is achieved by explicitly changing the homotopy parameter $\lambda$. In contrast, we consider here an additional term for the control law, which applies an angular force to an agent when it is within sensing range of a barrier, where the position of the barrier is *a priori* unknown.
We restrict ourselves to the case of a convex and stationary barrier in $\Re^{2}$. For our purposes, a barrier $B$ is the convex hull defined by a set of $m$ points $b_{i}\in\Re^{2}$. The location of the barrier is not *a priori* known to an agent. Instead the barrier is detected whenever an agent is within range of any of the points defining the barrier, in which case we say that the agent is within range of the barrier.
Barrier avoidance logic is implemented in the model as follows. For each agent, at every time step of the simulation, we calculate a vector, the *average barrier direction vector*, and which may be the zero vector if the agent is not within a neighborhood of a barrier. The average barrier direction vector (directed from the $k^{\textrm{th}}$ agent) is defined as,$$v_{k}=\left\{ \begin{array}{cc}
\frac{\sum_{i=1}^{m}\left[(b_{i}-y_{k})c\left(\left|b_{i}-y_{k}\right|,0,w\right)\right]}{\left|\sum_{i=1}^{m}\left[(b_{i}-y_{k})c\left(\left|b_{i}-y_{k}\right|,0,w\right)\right]\right|}, & \qquad\textrm{if }\left|\sum_{i=1}^{m}\left[(b_{i}-y_{k})c\left(\left|b_{i}-y_{k}\right|,0,w\right)\right]\right|\neq0\\
0, & \textrm{otherwise}\end{array}\right.,\label{eq:barrier_avodiance_law}$$ where $c(\cdot)$ is the cutoff function (\[eq:cutfn\]). The control law is then modified by adding the term $\pm(v_{k}\cdot y_{k})s$, where the sign is chosen to be the sign of the expression $v_{k}\cdot y_{k}^{\perp}$. This term serves to steer the agent perpendicular to the direction of the average vector $v_{k}$, and the sign is chosen to steer the agent away from the the average direction of the barrier, relative to the current heading of the vehicle. This can result in a splitting of the swarm into two subswarms, with one subswarm going around one side of the barrier, and the other subswarm going around the other side.
Simulation results for target-seeking and barrier avoidance control law
-----------------------------------------------------------------------
We present the results of a simulation of the model using the control law \[eq:modif\_ctrl\_law\_2\] with the barrier avoidance control \[eq:barrier\_avodiance\_law\]. Figure \[cap:Target\_seeking\_2\] shows the result of a typical simulation. A target point is located at $(x,y)=(400,0)$, and there is a hexagonal barrier centered at $(x,y)=(200,0)$, and is defined by the points $\{(200,2),(201,1),(201,-1),(200,-2),(199,-1),(199,1)\}$. As the swarm approaches the barrier, it again splits into two subswarms. All agents rejoin into a single swarm after the barrier is passed and continue on to the target. This time, upon arrival, the agents swarm in an irregular fashion about the target point. The irregular swarming about the target is due to the lack of inter-agent baseline controls implemented in this version of the control law.
Discussion
==========
The presented Frenet-Serret model and associated control laws exhibits robust and spontaneous coherent motion of a collection of $n$ agents with controlled clustering for any smooth dynamical system. Such emergent behavior is important in obstacle and predator avoidance. Cluster formation from a coherent structure was done via a new type of control, which we introduced as a homotopy control. Using a simple central parameter, homotopy control provides an easily implementable method to create new emergent behavior from coherent structures. The model is robust in the sense that small perturbations of constituent agents of the swarm results in little or no change in the coherent motion of the swarm as a whole. We tested this by introducing additive noise into the simulations. At each time step, the positions and angles of the agents were perturbed with a small amount of noise. Noise was taken from a uniform distribution with mean zero. Results of simulations were qualitatively similar to those presented in section \[sec:Homotopy-control-law.\].
On the other hand, we also showed in section \[sec:Target-seeking-laws\] that even with very loose coupling, involving only simultaneous target seeking, and where the only inter-agent coupling is via a collision avoidance term, ordered behavior can emerge, even when obstacle avoidance is taken into account.
Previous studies have focused on presenting unified coherent motion of a swarm. We have extended these results by presenting a method to automatically transition to subswarm clusters, formed from an original larger swarm, and functioning independently. The spontaneous coherence implies that individual agents do not need to be manually controlled. Indeed, that is one main goal of such research; to find a set of (preferably simple) rules which will result in the desired behavior with a high degree of autonomy for the swarm, and with a minimum of external inputs.
There are some limitations to the currently considered model. When using only local coupling with the homotopy control law, there is no way to reunite subswarm clusters, and a separate mechanism would have to be introduced to do so. Additionally, the leader following model presented is asymmetric, in that there is a distinction between leader and follower agents. Thus, if a leader agent is disabled, the subswarm cluster is no longer controllable. A better approach would be to consider a symmetric control law that doesn’t distinguish between leader and follower agents, but which maintains similar behaviors. This is the subject of ongoing research.
One obvious extension of the current model is to obtain a dictionary of useful controls which can be strung together in a similar fashion to what we have done with the homotopy control law, perhaps with multiple homotopy parameters, in order to obtain multiple emergent behaviors. Additionally, a stochastic control law, in which the swarm maintains a loose cohesiveness, while incorporating stochastic motion to avoid interception by predators, is also being explored as an extension.
We thank E Justh and P.S. Krishnaprasad for useful conversations which lead to the new control scheme presented.
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[^1]: IBS is supported by the Office of Naval Research, DSM is a National Research Council postdoctoral fellow.
|
---
author:
- 'J. A. Méndez-Bermúdez'
- 'A. Alcázar-López'
- Imre Varga
title: Multifractal dimensions for critical random matrix ensembles
---
Introduction
============
It is well–known that the spatial fluctuations of the eigenstates in a disordered system at the Anderson–transition show multifractal characteristics[@MJ98; @EM08] which has been demonstrated recently in a series of experiments [@expmuf]. Therefore the modeling and analysis of multifractal states has become of central importance producing many interesting results. For this purpose random matrix models have been invoked and studied recently [@BG11; @BG11b; @ROF11].
Since the exact, analytical prediction of the multifractal dimensions of the states for the experimentally relevant Anderson–transition in $d=3$ or the integer quantum–Hall transition in $d=2$ seems to be out of reach, it is desirable to search for heuristic relations in order to understand the complexity of the states at criticality. In the present paper we propose such heuristic relations that are numerically verified using various ensembles of random matrices.
The spatial fluctuations of the eigenstates can be described by a set of multifractal dimensions $D_q$ defined by the scaling of the inverse mean eigenfunction participation numbers with the system size $N$: $$\left\langle \sum_{i=1}^N |\Psi_i|^{2q} \right\rangle \sim N^{-(q-1)D_q} \ ,
\label{Dq}$$ where $\left\langle \cdots \right\rangle$ is the average over some eigenvalue window and over random realizations of the matrix. For strongly localized eigenstates these quantities do not scale with system size, i.e. $D_q\to 0$ for all $q$, while extended states always feel the entire system, i.e. $D_q\to d$ for all $q$. Multifractal states, on the other hand, should be described by the series of the $D_q$, which are a nonlinear function of the parameter $q$.
Spectral fluctuations can be characterized in many ways. A usual, often employed quantity is the level compressibility $\chi$, which is extracted from the limiting behavior of the spectral number variance as $\Sigma^{(2)}(E) = \left\langle n(E)^2 \right\rangle - \left\langle n(E)
\right\rangle^2 \sim \chi E$, where $n(E)$ is the number of eigenstates in an interval of length $E$. The spectral fluctuations in a metallic system with extended states yield a vanishing compressibility, $\chi\to 0$, while in a strongly disordered insulating system the levels are uncorrelated, so they are easily compressible, $\chi=1$. However, for the multifractal states an intermediate statistics exists, $0<\chi <1$, furthermore the spectral and eigenstate statistics are supposed to be coupled, which has been pointed out first in Ref. [@CKL96].
One of the most important generalized dimensions often used in this context is the information dimension $D_1$. It is defined through the scaling of the mean eigenfunction entropy with the logarithm of the system size: $$\left\langle -\sum_{i=1}^N |\Psi_i|^2 \ln |\Psi_i|^2 \right\rangle
\sim D_1 \ln N \ .
\label{D1}$$ A further, well–known and widely used dimension is called the correlation dimension $D_2$, which is extracted from the inverse participation number from Eq. (\[Dq\]) using $q=2$.
In a recent work [@BG11] Bogomolny and Giraud have shown that in a $d$–dimensional critical system the information dimension $D_1$ and the level compressibility $\chi$ are simply related as $$\chi + D_1/d = 1 \ ,
\label{chiD1}$$ furthermore the generalized dimensions $D_q$ can be expressed as $$\frac{D_q}{d} = \left\{
\begin{array}{ll}
\displaystyle
\frac{\Gamma(q-1/2)}{\sqrt{\pi}\Gamma(q)}(1-\chi) \ , &
1-\chi \ll 1 \\
1-q\chi \ , & \chi \ll 1
\end{array}
\right. \ .
\label{Dqchith}$$ These expressions have been shown to be valid for various critical random matrix ensembles in Ref. [@BG11].
As for the critical, three–dimensional Anderson transition and the two–dimensional quantum–Hall transition it has been shown earlier that another relation holds between the level compressibility $\chi$ and the correlation dimension $D_2$ [@CKL96]: $$2\chi + D_2/d = 1 \ .
\label{chiD2}$$ This relation should obviously hold approximately only since $0\leq D_2/d\leq 1$ but $0\leq\chi\leq 1$, leaving the range of validity for the limit of weak–multifractality.
In the present work we show a series of relations between various generalized dimensions, $D_q$ and $D_{q'}$, and the level compressibility $\chi$ allowing for a generalization that for particular cases yields Eq. (\[chiD1\]) exactly and Eq. (\[chiD2\]) in the appropriate limit. In order to prove that, numerical simulations of various critical random matrix ensembles will be used. Further implications and more details will be presented elsewhere [@tbp].
Model and heuristic relations
=============================
In Ref. [@BG11] Eqs. (\[chiD1\]) and (\[Dqchith\]) were shown to be correct numerically for the Power-Law Banded Random Matrix (PBRM) model [@EM08; @MFDQS96; @Mirlin00] at criticality. Below we will make use of this model to derive our main results.
The PBRM model describes one–dimensional (1d) samples of length $N$ with random long-range hoppings. This model is represented by $N\times N$ real symmetric ($\beta=1$) or complex hermitian ($\beta=2$) matrices whose elements are statistically independent random variables drawn from a normal distribution with zero mean and a variance given by $\langle |H_{mm}|^2 \rangle =\beta^{-1}$ and $$\langle |H_{mn}|^2 \rangle =
\frac{1}{2} \frac{1}{1+\left[
\sin\left( \pi|m-n|/N \right)/(\pi b/N) \right]^{2\mu}} \ ,
\label{PBRMp}$$ where $b$ and $\mu$ are parameters. In Eq. (\[PBRMp\]) the PBRM model is in its periodic version; i.e. the 1d sample is in a ring geometry. Theoretical considerations [@EM08; @MFDQS96; @Mirlin00; @KT00] and detailed numerical investigations [@EM08; @EM00b; @V03] have verified that the PBRM model undergoes a transition at $\mu=1$ from localized states for $\mu >1$ to delocalized states for $\mu < 1$. This transition shows all the key features of the disorder driven Anderson metal-insulator transition[@EM08], including multifractality of eigenfunctions and non-trivial spectral statistics. Thus the PBRM model possesses a line of critical points $b\in (0,\infty)$ in the case of $\mu=1$. In the following we will focus on the PBRM model at criticality, $\mu=1$. By tuning the parameter $b$ the states cross over from the nature of weak–multifractality ($b\gg 1$) which corresponds to extended–like or metallic–like states to strong–multifractality ($b\ll 1$) showing rather localized, i.e. insulator–like states. Meanwhile at the true, Anderson transition in $d=3$ or at the integer quantum–Hall transition in $d=2$, the states belong to the weakly multifractal regime, the PBRM model allows for an investigation without such a limitation. The evolution of the generalized dimensions as a function of the parameter $b$ therefore represent this behavior, i.e. $D_q\to 1$ for $b\gg 1$ and in the other limit of $b\ll 1$ the multifractal dimensions vanish as $D_q\sim b$ [@EM08; @Mirlin00].
Previously, for the PBRM model at criticality with $\beta=1$, we have observed that both, $D_1$ and $D_2$ can be approximated simply as [@MV06] $D_1 \approx [1+(\alpha_1 b)^{-1}]^{-1}$ and $D_2 \approx [1+(\alpha_2 b)^{-1}]^{-1}$ where $\alpha_{1,2}$ are fitting constants. This continuous function is a trivial interpolation between the limiting cases of low–$b$ and large–$b$ taking the half of the harmonic mean of the two as $$\frac{1}{D_q}=1 + \frac{1}{\alpha_q b}\ ,
\label{interp}$$ valid for $q=1$ and $2$. Here we generalize and propose the following heuristic expression for a wider range of the parameter $q$ $$D_q \approx \left[ 1+(\alpha_q b)^{-1} \right]^{-1} \ ,
\label{Dqofb}$$ as a global fit for the multifractal dimensions $D_q$ of the PBRM model in both symmetries, $\beta=1$ and $\beta=2$. In Fig. \[Fig1\] we show fits of Eq. (\[Dqofb\]) to numerically obtained $D_q$ as a function of $b$ for some values of $q$ and in Fig. \[Fig2\] we plot the values of $\alpha_q$ extracted from the fittings.[^1] We observe that Eq. (\[Dqofb\]) fits reasonably well the numerical $D_q$ for $q> 1/2$. It is important to stress that Eq. (\[Dqofb\]) reproduces well the $b$-dependencies predicted analytically [@EM08] for the limits $b\ll 1$ and $b\gg 1$.
We noticed that by the use of Eq. (\[Dqofb\]), Eq. (\[chiD1\]) leads to $$\chi \approx \left( 1+\alpha_1 b \right)^{-1} \ ,
\label{chiofb}$$ which also reproduces well the $b$-dependencies predicted analytically [@BG11; @EM08] in the small- and large-$b$ limits: $$\chi = \left\{
\begin{array}{ll}
1-4b \quad & b \ll 1 \\
(2 \pi b)^{-1} & b \gg 1
\end{array}
\right. \ .
\label{chithPBRM}$$ Then, by equating $b$ in Eqs. (\[Dqofb\]) and (\[chiofb\]) we get $$\chi \approx (1-D_q) \left[ 1+(\gamma_q-1)D_q \right]^{-1} \ ,
\label{chiofgamma}$$ with $\gamma_q = \alpha_1/\alpha_q$. We observed that $\gamma_q \approx q$ in the range $0.8<q<2.5$, see Fig. \[Fig2\], so in this range of $q$ values we can write simplified relations between $\chi$ and $D_q$: $$\chi \approx \frac{1-D_q}{1+(q-1)D_q} \quad \mbox{and} \quad D_q \approx \frac{1-\chi}{1+(q-1)\chi} \ .
\label{chiofDq}
\label{Dqofchi}$$ The expression for $D_q$ in Eq. (\[Dqofchi\]) reproduces Eq. (\[Dqchith\]) exactly for $q=1$ and $q=2$ and approximately for $1<q<2.5$. Moreover, Eq. (\[Dqofchi\]) combined with Eq. (\[chiD1\]) allows us to express any $D_q$ in terms of $D_1$: $$D_q \approx D_1 \left[ q+(1-q)D_1 \right]^{-1} \ .
\label{DqofD1}$$
We also noticed that by equating $\chi$ for different $D_q$’s form Eq. (\[chiofDq\]) we could get recursive relations for them: $$\frac{q'D_{q'}}{1-D_{q'}} = \frac{qD_q}{1-D_q} \quad \mbox{and} \quad D_{q'} = \frac{qD_q}{q'+(q-q')D_q} \ ,
\label{DqpDq}
\label{DqpofDq}$$ which lead to $D_{q+1} = qD_q(1+q-D_q)^{-1}$, when $q'=q+1$. These expressions also provide a relation between the correlation dimension and the information dimension or between the correlation dimension and the compressibility of the spectrum: $$D_2 = D_1\left( 2-D_1 \right)^{-1} = \left( 1-\chi \right) \left( 1+\chi \right)^{-1} \ .
\label{D2ofD1}$$ It is relevant to add that in the weak multifractal regime, i.e. when $\chi\to 1$, Eq. (\[D2ofD1\]) reproduces the relation given in Eq. (\[chiD2\]) with $d=1$, reported in [@CKL96].
Numerical results for the PBRM model
====================================
Here we verify the expressions (\[chiofgamma\]-\[D2ofD1\]) for the PBRM model at criticality. Below we concentrate on the case $\beta=1$ but we have already validated our results for $\beta=2$.
In Fig. \[Fig3\] we plot $(1-D_q)[1+(\gamma_q-1)D_q]^{-1}$ and $(1-D_q)[1+(q-1)D_q]^{-1}$ as a function of $b$ for several values of $q$ and observe good correspondence with the analytical prediction for $\chi$; that is, we verify the validity of Eqs. (\[chiofgamma\]) and (\[chiofDq\]), respectively. In the inset of Fig. \[Fig3\](b) we plot $qD_q(1-D_q)^{-1}$ as a function of $b$, see Eq. (\[DqpDq\]), which for the PBRM model acquires the simple form $$qD_q\left( 1-D_q \right)^{-1} \approx q\alpha_q b \approx \alpha_1b \ .
\label{DqpDqPBRM}$$
Then, in Fig. \[Fig4\] we compare $D_q$ and $D_{q'}$ with $D_1[q+(1-q)D_1]^{-1}$ and $qD_q[q'+(q-q')D_q]^{-1}$, respectively, for several values of $q$; that is, we verify the validity of Eqs. (\[DqofD1\]) and (\[DqpofDq\]). Eq. (\[D2ofD1\]) is also validated in Fig. \[Fig4\](b).
Additionally, in [@KOYC10] the duality relation $$D_2(B) + D_2(B^{-1}) = 1 \ , \quad B\equiv 2^{1/4}\pi b \ ,
\label{duality}$$ was shown to be valid (with maximum deviations of $1\%$) for the PBRM model at criticality. We also want to comment that by the use of Eq. (\[Dqofb\]) we could write $D_2(B) \approx [1+(\delta B)^{-1}]^{-1}$, with $\delta\equiv \alpha_2/(2^{1/4}\pi)$, so relation (\[duality\]) gets the form $$D_2(B) + D_2(B^{-1}) \approx 1-\frac{B(\delta-1)^2}{B+\delta(B^2+\delta B+1)} \ .
\label{duality2}$$ We notice that the quantity $D_2(B) + D_2(B^{-1})$ is very sensitive to the value of $\alpha_2$. So, the error in $\alpha_2$ is magnified in the r.h.s. of Eq. (\[duality2\]). The maximal deviation from 1 (of $7.3\%$ and $2\%$ for $\beta=1$ and $\beta=2$, respectively) occurs at $B=1$ where the r.h.s. of Eq. (\[duality2\]) acquires the form $1-[(\delta-1)/(\delta+1)]^2$.
Other critical ensembles
========================
Remember that relations (\[chiofDq\]-\[D2ofD1\]) were obtained form the combination of Eqs. (\[Dqofb\]) and (\[chiofb\]). That is, relations (\[chiofDq\]-\[D2ofD1\]) are expected to work in particular for the PBRM model at criticality. However, Eqs. (\[chiofDq\]) reproduce Eqs. (\[chiD1\]) and (\[Dqchith\]), which were shown to be valid for the PBRM model but also for other critical ensembles [@BG11]. Then the question is to which extent relations (\[chiofDq\]-\[D2ofD1\]) are valid for critical ensembles different to the PBRM model. So, in the following we verify the validity of Eqs. (\[chiofDq\]-\[D2ofD1\]) for other critical ensembles.[^2]
The Ruijsenaars-Schneider Ensemble (RSE)
----------------------------------------
The RSE proposed in [@BGS09] is defined as matrices of the form $$H_{mn} = \exp(i\Phi_m) \frac{1-\exp(2\pi ia)}{N[1-\exp(2\pi i(m-n+a)/N)]} \ ,
\label{RSE}$$ where $1\le m\le n$, $\Phi_m$ are independent random phases distributed between 0 and $2\pi$, and $a$ is a free parameter independent on $N$. When $0<a<1$, the compressibility and the multifractal dimensions take the form [@BG11] $$\chi \sim (a-1)^2 \quad \mbox{and} \quad D_q = 1-q(a-1)^2 \ ;
\label{chiDqRSE}$$ while in the vicinity of an integer $k\ge 2$, when $|a-k|\ll1$, $$\chi \sim (a-k)^2/k^2 \quad \mbox{and} \quad D_q = 1-q(a-k)^2/k^2 \ .
\label{chiDqRSEk}$$ As shown in [@BG11], Eqs. (\[chiDqRSE\]) and (\[chiDqRSEk\]) satisfy relation (\[chiD1\]). Moreover, by direct substitution of Eqs. (\[chiDqRSE\]) \[or Eqs. (\[chiDqRSEk\])\] we verified that Eqs. (\[chiofDq\]-\[D2ofD1\]) are also satisfied at leading order in $(a-1)^2$ \[$(a-k)^2$\].
In Fig. \[Fig5\] we plot $D_1$ and $D_2$ as a function of $a$ for the RSE. Black and red dashed lines are the theoretical predictions for $D_1$ and $D_2$, respectively, given in Eqs. (\[chiDqRSE\]) and (\[chiDqRSEk\]). As it was earlier shown in Ref. [@BG11], the analytical form of $D_q$ given in Eqs. (\[chiDqRSE\]) and (\[chiDqRSEk\]) reproduces very well the numerically obtained $D_1$. However, we notice that Eq. (\[chiDqRSE\]) does not describe well the numerical $D_2$, mainly when $a\to 0$. Now, note that by plotting the numerically obtained $D_1/(2-D_1)$ we get good agreement with the numerical data for $D_2$, that is Eq. (\[D2ofD1\]) works well for this model. Then, if we take $D_1\approx 1-(a-1)^2$ and $D_1\approx 1-(a-k)^2/k^2$ as theoretical predictions for $D_1$ and plug them into Eq. (\[D2ofD1\]) we get $$D_2 \approx \frac{1-(a-1)^2}{1+(a-1)^2} \quad \mbox{and} \quad D_2 \approx \frac{k^2-(a-k)^2}{k^2+(a-k)^2} \ ,
\label{D2RSE}$$ for $0<a<1$ and $|a-k|\ll 1$ with $k\ge 2$, respectively; which in fact work much better than $D_2\approx 1-2(a-1)^2$ and $D_2\approx 1-2(a-k)^2/k^2$, correspondingly; see Fig. \[Fig5\].
To get expressions for $D_q$ we substituted $\chi \sim (a-1)^2$ and $\chi \sim (a-k)^2/k^2$ \[or $D_1 \approx 1-(a-1)^2$ and $D_1 \approx 1-(a-k)^2/k^2$\] into Eq. (\[Dqofchi\]) \[or Eq. (\[DqofD1\])\], to get $$D_q \approx \left[ 1-(a-1)^2 \right] \left[1+(q-1)(a-1)^2 \right]^{-1}
\label{DqRSE2}$$ and $$D_q \approx \left[ k^2-(a-k)^2 \right] \left[k^2+(q-1)(a-k)^2 \right]^{-1} \ .
\label{DqRSE2k}$$ In Fig. \[Fig6\] we plot $D_q$ as a function of $q$ for the RSE for several values of $a$. We also plot Eqs. (\[DqRSE2\]) and (\[DqRSE2k\]) and observe rather good correspondence with the numerical data mainly in the range $1<q<2$. Notice that neither Eq. (\[DqRSE2\]) nor Eq. (\[DqRSE2k\]) can be used for $a=1.5$. For that case we substituted the numerically obtained value of $D_1$ into Eq. (\[DqofD1\]) and again observe good correspondence for $0<q<2$, see the red dashed line in Fig. \[Fig6\].
Intermediate quantum maps
-------------------------
A variant of the RSE was studied in [@MGG08] with the name of intermediate quantum maps (IQM) model. In this model the parameter $a$ of the RSE equals $cN/g$ with $cN=\pm 1$ mod $g$, being $g$ the parameter of the IQM model. For the IQM model the compressibility and the multifractal dimensions take the form [@MGG08] $$\chi \approx 1/g \quad \mbox{and} \quad D_q \approx 1-q/g \ .
\label{chiDqIQM}$$ As for the RSE, here Eqs. (\[chiDqIQM\]) satisfy relation (\[chiD1\]). Again, by direct substitution of Eqs. (\[chiDqIQM\]) we verified that Eqs. (\[chiofDq\]-\[D2ofD1\]) are satisfied at leading order in $1/g$, $g\gg 1$.
We want to mention that in [@MGG08] it was shown that Eq. (\[chiDqIQM\]) reproduces well the numerically obtained $D_1$ but underestimates the numerical $D_2$, in particular for small $g$, see Fig. \[Fig7\]. Now, notice that by plotting the numerically obtained $D_1/(2-D_1)$ we nicely reproduce the numerical data for $D_2$, that is Eq. (\[D2ofD1\]) works well also for this model. Then, if we take $D_1\approx 1-1/g$ as the theoretical prediction for $D_1$ and plug it into Eq. (\[D2ofD1\]) we get $$D_2 \approx \left( 1-1/g \right) \left( 1+1/g \right)^{-1} \ ,
\label{D2IQM}$$ which in fact works much better than $D_2\approx 1-2/g$ in reproducing the numerical $D_2$, see Fig. \[Fig7\].
To get the expression for $D_q$ we substituted $\chi \approx 1/g$ or $D_1\approx 1-1/g$ into Eq. (\[Dqofchi\]) or (\[DqofD1\]), respectively, to get $$D_q \approx \left( g-1 \right) \left( g+q-1 \right)^{-1} \ .
\label{DqIQM2}$$ In Fig. \[Fig8\] we plot $D_q$ as a function of $q$ for the IQM model for some values of $g$. We also plot Eq. (\[DqIQM2\]) and observe that it falls below the numerical data mainly for small $g$. However, by substituting the numerically obtained values of $D_1$ into Eq. (\[DqofD1\]) we get much better correspondence with the numerical $D_q$, mainly for $1<q<2$. In Fig. \[Fig8\] we also include $D_q$ from Eq. (\[chiDqIQM\]). We may conclude that while Eq. (\[chiDqIQM\]) reproduces well the numerical $D_q$ for $q<1$, Eq. (\[DqIQM2\]) can serve as the analytical continuation for $q>1$.
The critical ultrametric ensemble
---------------------------------
The critical ultrametric ensemble (CUE) proposed in [@FOR09] consists of $2^K \times 2^K$ Hermitian matrices whose matrix elements are Gaussian random variables with zero mean and variance $$\left\langle |H_{mm}|^2 \right\rangle = W^2 \ , \quad
\left\langle |H_{mn}|^2 \right\rangle = 2^{2-d_{mn}}J^2 \ ,
\label{CUE}$$ where $d_{mn}$ is the ultrametric distance between $m$ and $n$ on the binary tree with $K$ levels and the root of 1. The parameter in this model is the ratio $J/W$. For the CUE, when $J/W\ll1$, the compressibility and the multifractal dimensions have the form [@BG11; @FOR09] $$\chi = 1 - \frac{J}{W} \frac{\pi}{\sqrt{2}\ln 2} \quad \mbox{and} \quad
D_q = \frac{J}{W} \frac{\sqrt{\pi}\Gamma(q-1/2)}{\sqrt{2}\ln 2\Gamma(q)} \ .
\label{chiDqCUE}$$ Eqs. (\[chiDqCUE\]) satisfy relation (\[chiD1\]) at first order in $J/W$ [@BG11]. Again, as for the previous critical ensembles, by direct substitution of Eqs. (\[chiDqCUE\]) we verified that Eqs. (\[chiofDq\]-\[D2ofD1\]) are satisfied at leading order in $J/W$, for $0.8<q<2.5$; because in this range of $q$ we have that $\Gamma(q-0.5)/\sqrt{\pi}\Gamma(q)\approx 1/q$.
In Fig. \[Fig9\] we show $D_q$ as a function of $q$ for the CUE for $J/W=0.1$ and 3. The data was taken from [@FOR09]. The blue dashed line is $D_q$ from Eq. (\[chiDqCUE\]) for $J/W=0.1$. Notice that since Eq. (\[chiDqCUE\]) is only valid when $J/W\ll 1$ and for $q\ge 3/4$ one can not use it to predict $D_q$ for $J/W=3$. However, with Eq. (\[DqofD1\]) using as input the numerically obtained $D_1$ we got good predictions for $D_q$ for small and large values of $J/W$ and even for values of $q$ smaller than 3/4. This is shown in Fig. \[Fig9\] where we plot Eq. (\[DqofD1\]) (red dashed lines) using $D_1=0.2805$ and 0.8443 for $J/W=0.1$ and 3, respectively. The values of $D_1$ were obtained by the interpolation of the $D_q$ data. We observe good correspondence between Eq. (\[DqofD1\]) and the numerical $D_q$ for $0<q<10$.
Conclusions
===========
In this paper we propose heuristic relations on one hand between the generalized multifractal dimensions, $D_q$ and $D_{q'}$, for a relatively wide range of the parameters $q$ and $q'$, and on the other hand between these dimensions and the level compressibility $\chi$. As a result we find a general framework embracing an earlier [@CKL96] and a recent one [@BG11]. Our proposed relations have been backed by numerical simulation on various random matrix ensembles whose eigenstates have multifractal properties. These results call for further theoretical as well as numerical investigations.
The authors are greatly indebted to V. Kravtsov for useful discussions. This work was partially supported by VIEP-BUAP (Grant No. MEBJ-EXC10-I), the Alexander von Humboldt Foundation, and the Hungarian Research Fund (OTKA) grants K73361 and K75529.
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[^1]: The multifractal dimensions $D_q$ were extracted from the linear fit of the logarithm of the inverse mean eigenfunction participation numbers versus the logarithm of $N$, see Eq. (\[Dq\]). $D_1$ was extracted from the linear fit of the mean eigenfunction entropy versus the logarithm of $N$, see Eq. (\[D1\]). We used $N=2^n$, $8\le n\le 13$. The average was performed over $2^{n-3}$ eigenvectors with eigenvalues around the band center with $2^{16-n}$ realizations of the random matrices.
[^2]: The multifractal dimensions $D_1$ and $D_q$ for those ensembles were extracted numerically by the use of the same matrix sizes and ensemble realizations as for the PBRM model, if not indicated otherwise.
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abstract: |
Representation stability is a theory describing a way in which a sequence of representations of different groups is related, and essentially contains a finite amount of information. Starting with Church-Ellenberg-Farb’s theory of ${{{\normalfont\operatorname{FI}}}}$-modules describing sequences of representations of the symmetric groups, we now have good theories for describing representations of other collections of groups such as finite general linear groups, classical Weyl groups, and Wreath products $S_n\wr G$ for a fixed finite group $G$. This paper attempts to uncover the mechanism that makes the various examples work, and offers an axiomatic approach that generates the essentials of such a theory: character polynomials and free modules that exhibit stabilization.
We give sufficient conditions on a category ${{\operatorname{\textbf{C}}}}$ to admit such structure via the notion of categories of ${{{\normalfont\operatorname{FI}}}}$ type. This class of categories includes the examples listed above, and extends further to new types of categories such as the categorical power ${{{\normalfont\operatorname{FI}}}}^m$, whose modules encode sequences of representations of $m$-fold products of symmetric groups. The theory is applied in [@Ga] to give homological and arithmetic stability theorems for various moduli spaces, e.g. the moduli space of degree $n$ rational maps ${\mathbb{P}}^1\rightarrow {\mathbb{P}}^m$.
author:
- Nir Gadish
title: 'Categories of FI type: a unified approach to generalizing representation stability and character polynomials'
---
Introduction
============
The purpose of this paper is to describe a categorical structure that is responsible for the existence of representation stability phenomena. Our approach is centered around free modules[^1] and character polynomials (defined below). We show that our proposed categorical structure gives rise to free modules which satisfy the fundamental properties that produce representation stability, and in particular the Noetherian property. We take an axiomatic approach that applies in a broad context, generalizing many of the known examples.
Motivation
----------
Let ${{{\normalfont\operatorname{FI}}}}$ be the category of finite sets and injections. An *${{{\normalfont\operatorname{FI}}}}$-module* is a functor from ${{{\normalfont\operatorname{FI}}}}$ to the category of modules over some fixed ring $R$. An ${{{\normalfont\operatorname{FI}}}}$-module $M_\bullet$ is a single object that packages together a sequence of representations of the symmetric groups $S_n$ for every $n\in {\mathbb{N}}$ (see e.g. [@CEF]). Objects of this form arise naturally in topology and representation theory, for example:
- Cohomology of configuration spaces $\{\operatorname{PConf}^n(X)\}_{n\in {\mathbb{N}}}$ for a manifold $X$.
- Diagonal coinvariant algebras $\{ {\mathbb{Q}}[x_1,\ldots,x_n,y_1,\ldots,y_n]/\mathcal{I}_n \}_{n\in N}$ (see [@CEF]).
A fundamental result of Church-Ellenberg-Farb [@CEF] is that an ${{{\normalfont\operatorname{FI}}}}$-module over ${\mathbb{Q}}$ is finitely-generated, i.e. there exists a finite set of elements not contained in any proper submodule, if and only if the sequence of $S_n$-representations stabilizes in a precise sense (see [@CEF] for details). This phenomenon was named *representation stability*. In particular, if one defines class functions $$X_k(\sigma) = \# \text{ of $k$-cycles in } \sigma$$ simultaneously on all $S_n$, then [@CEF] show that for every finitely-generated ${{{\normalfont\operatorname{FI}}}}$-module $M_\bullet$ then there exists a single polynomial $P\in {\mathbb{Q}}[X_1,X_2,\ldots]$ – a *character polynomial* – that describes the characters of the $S_n$-representations $M_n$ independent of $n$ for all $n\gg1$.
The uniform description of the characters in terms of a single character polynomial accounts for the most direct applications of the theory, for example:
- For every manifold $X$ and $i\geq 0$, the dimensions of $\{\operatorname{H}^i(\operatorname{PConf}^n(X);{\mathbb{Q}})\}_{n\in {\mathbb{N}}}$ are given by a single polynomial in $n$ for all $n\gg1$.[^2]
- Every polynomial statistic, regarding the irreducible decomposition of degree-$n$ polynomials over ${\mathbb{F}}_q$, tends to an asymptotic limit as $n\rightarrow\infty$.[^3]
However, the above logic could be reversed: as first suggested by Gan-Li in [@GL-coinduction], Nagpal showed in [@Na Theorem A] that if $M_\bullet$ is a finitely generated ${{{\normalfont\operatorname{FI}}}}$-module, then in some range $n\gg 1$ it admits a finite resolution by free ${{{\normalfont\operatorname{FI}}}}$-modules (see below) and these have characters given by character polynomials. It follows that for every $n \gg 1$ the character of the $M_{n}$ is itself given by a character polynomial. One can then get stabilization of the decomposition of $M_n$ into irreducible representations as a corollary of this fact! We assert that the key property of character polynomials – responsible for all representation stability phenomena and applications – is the following.
\[fact:stabilization\] The $S_n$-inner product of two character polynomials $P$ and $Q$ becomes independent of $n$ for all $n\geq \deg(P)+\deg(Q)$.
The benefit of Gan-Li’s and Nagpal’s approach is that free ${{{\normalfont\operatorname{FI}}}}$-modules and character polynomials readily generalize to a wide class of categories similar to ${{{\normalfont\operatorname{FI}}}}$, and do not require any understanding of the representation theory of the individual automorphism groups. Thus representation stability extends whenever these structures exist.
Generalization to other categories
----------------------------------
Work on generalizing representation stability to other contexts has proceeded in several partially overlapping directions. A major direction on which we will be focused is that of modules over other categories ${{\operatorname{\textbf{C}}}}$ of injections, whose automorphism groups are of interest. Let ${{\operatorname{\textbf{C}}}}$ be a category.
\[def:intro\_modules\] A *${{\operatorname{\textbf{C}}}}$-module* over a ring $R$ is a covariant functor $$M_\bullet: {{\operatorname{\textbf{C}}}}{\ensuremath{\overset{}{\longrightarrow}}}{{{\normalfont\operatorname{Mod}}}_{R}}.$$ For every object $c$, the evaluation $M_c$ is naturally a representation of the group $\operatorname{Aut}_{{\operatorname{\textbf{C}}}}(c)$ in $R$-modules, and these representation are related by the morphisms of ${{\operatorname{\textbf{C}}}}$.
One then studies this category of representations, describes the simultaneous class functions that generalize character polynomials, and proves the analog of Fact \[fact:stabilization\]. For example:
1. Putman-Sam [@PS] considered the category ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{VI}}}}_q$ of finite dimensional vector spaces over ${\mathbb{F}}_q$ and injective linear maps, whose representations encode sequences of $\operatorname{Gl}_n({\mathbb{F}}_q)$-representations.
2. Wilson [@Wi] studied ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}_{\mathcal{W}}$ whose automorphism groups are the classical Weyl groups $\mathcal{W}_n$ of type $B/C$ or $D$.
3. Sam-Snowden [@SS-FIG] and Gan-Li [@GL-coinduction] considered categories ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}_G$ for some group $G$, encoding representations of Wreath products $S_n\wr G$. Casto [@Ca] extended their treatment, and defined character polynomials in this context.
4. Barter [@Ba] considered the category ${{\operatorname{\textbf{C}}}}={{\normalfont\operatorname{T}}}$ of rooted trees with root-preserving embeddings.
This approach has been further applied to topology, arithmetic and classical representation theory (see the respective citations).
Other generalizations considered categories of dimension zero, studied by Wiltshire-Gordon and Ellenberg (see [@WG] with applications in [@WGE]); homogeneous categories, studied by Randal-Williams and Wahl (see [@RWW]); and modules over twisted commutative algebras, studied by Sam-Snowden (see [@SS-tca]). We will not discuss these ideas here.
In this paper we attempt to generalize and unify the treatments in Examples 1-4 and ask:
What structure do these categories possess that supports the existence of a representation stability theory?
Here we offer an answer by fitting Examples 1-3 and others into the context of a broader theory: representation of categories *of ${{{\normalfont\operatorname{FI}}}}$ type*, i.e. categories that have structural properties similar to those of ${{{\normalfont\operatorname{FI}}}}$ (see Definition \[def:FI-type\] below). This approach is intended to subsume the individual treatments and eliminate the need to introduce a new theory in each specific case. At the same time, it allows one to consider new types of categories, such as the next example.
As a first nontrivial example, and the original motivation behind this generalization, we consider the categorical powers ${{{\normalfont\operatorname{FI}}}}^m$. These have objects that are (essentially) $m$-tuples $(n_1,\ldots,n_m)\in {\mathbb{N}}^m$ with automorphism groups the products $S_{n_1}\times\ldots\times S_{n_m}$. Such categories are the natural indexing category for various collections of linear subspace arrangements, to which our theory is applied in a companion paper [@Ga]. To see this at work consider the following example.
Fix $m\geq 1$ and let $\operatorname{Rat}^n({\mathbb{P}}^1, {\mathbb{P}}^{m-1})$ be the space of based, degree $n$ rational maps ${{\mathbb{P}}^1{\ensuremath{\overset{}{\longrightarrow}}}{\mathbb{P}}^{m-1}}$ that send $\infty$ to $[1:\ldots:1]$. This space admits an $(S_n\times\ldots\times S_n)$-covering $\operatorname{PRat}^n({\mathbb{P}}^1, {\mathbb{P}}^{m-1})$ by picking an ordering on the zeros of the restrictions to the standard homogeneous coordinates functions on ${\mathbb{P}}^{m-1}$. The coverings fit naturally into a (contravariant) ${{{\normalfont\operatorname{FI}}}}^m$-diagram of spaces, and their cohomology is an ${{{\normalfont\operatorname{FI}}}}^m$-module.
The groups $\operatorname{H}^i(\operatorname{Rat}^n({\mathbb{P}}^1, {\mathbb{P}}^{m-1});{\mathbb{Q}})$ can then be computed from the invariant part of the $(S_n\times\ldots\times S_n)$-representation $\operatorname{H}^i(\operatorname{PRat}^n({\mathbb{P}}^1, {\mathbb{P}}^{m-1});{\mathbb{Q}})$ by transfer. Representation stability for ${{{\normalfont\operatorname{FI}}}}^m$-modules then gives the following.
\[thm:intro-homological-stability\] For every $i\geq 0$ the $i$-th Betti number of $\operatorname{Rat}^n({\mathbb{P}}^1, {\mathbb{P}}^{m-1})$ does not depend on $n$ for all $n\geq i$.
In §\[sec:rep\_theory\_of\_FI\] we discuss representation stability for ${{{\normalfont\operatorname{FI}}}}^m$, which allows one to make such claims as Theorem \[thm:intro-homological-stability\]. We remark that similar treatment could be applied to any product of categories whose representation stability is understood, but we do not pursue other examples here.
Categories of ${{{\normalfont\operatorname{FI}}}}$ type and free modules
------------------------------------------------------------------------
As outlined above, we are looking for categorical structure that gives rise to character polynomials satisfying Fact \[fact:stabilization\]. We propose the following.
\[def:FI-type\] We say that a category ${{\operatorname{\textbf{C}}}}$ is *of ${{{\normalfont\operatorname{FI}}}}$ type* if it satisfies the following axioms.
1. ${{\operatorname{\textbf{C}}}}$ is locally finite, i.e. all hom-sets are finite[^4].
2. Every morphisms is a monomorphism, and every endomorphisms is an isomorphism.
3. For every pair of objects $c$ and $d$, the group of automorphisms $\operatorname{Aut}_{{\operatorname{\textbf{C}}}}(d)$ acts transitively on the set $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$.
4. For every object $d$ there exist only finitely many isomorphism classes of objects $c$ for which $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)\neq {\varnothing}$ (we denote this by $c\leq d$).
5. ${{\operatorname{\textbf{C}}}}$ has pullbacks and pushouts[^5].
Categories that satisfy the second half of condition 2 – where every endomorphism is an isomorphism – are called ${{\normalfont\operatorname{EI}}}$ categories. The representation stability of such categories satisfying additional combinatorial conditions was studied by Gan-Li in [@GL-EI].
We will denote the automorphism group of an object $c$ by $G_c$.
In §\[subsec:char\_poly\] we define the collection of *character polynomials* for a general category ${{\operatorname{\textbf{C}}}}$ of ${{{\normalfont\operatorname{FI}}}}$ type - these are certain ${\mathbb{C}}$-valued class functions simultaneously defined on all automorphism groups $G_c$. Briefly, character polynomials are linear combinations of functions of the form $\binom{X}{\lambda}$ where $\lambda\subset G_c$ is some fixed conjugacy class. $\binom{X}{\lambda}$ evaluates on $g_d\in G_d$ to give the number of ways $g_d$ can be restricted to an element $g_c\in\lambda$, i.e. via morphisms $c{\ensuremath{\overset{f}{\longrightarrow}}}d$ for which $g_d\circ f = f\circ g_c$ with $g_c\in \lambda$.
However, it is not at all clear that these functions satisfy the analog of Fact \[fact:stabilization\], or even that they can be reasonably thought of as polynomials, i.e. closed under taking products. To demonstrate these fundamental properties we propose a categorification of character polynomials, similar to the way in which group representations categorify class functions. Our categorification takes the form of *free ${{\operatorname{\textbf{C}}}}$-modules*, introduced in Section §\[sec:free-modules\].
\[def:intro\_free\] A ${{\operatorname{\textbf{C}}}}$-module is said to be *free* if it is a direct sum of modules of the form $\operatorname{Ind}_c(V)$, where $\operatorname{Ind}_c$ is the left-adjoint functor to the restriction $M_\bullet \mapsto M_c$.
Since we are only discussing finitely-generated ${{\operatorname{\textbf{C}}}}$-modules, free modules will always be taken to be *finite* direct sums. Over the field of complex numbers these ${{\operatorname{\textbf{C}}}}$-modules are precisely the finitely-generated, projective ones.
This choice of categorification is justified by the following observation.
\[thm:intro-categorification\] If $M_\bullet$ is a free ${{\operatorname{\textbf{C}}}}$-module over ${\mathbb{C}}$, then there exists a character polynomial $P$ whose restriction to $G_c$ coincides with the character of $M_c$ for every object $c$.
Conversely, the character polynomials that arise in this way span the space of all character polynomials on ${{\operatorname{\textbf{C}}}}$, defined in §\[subsec:char\_poly\] below.
The structure of ${{{\normalfont\operatorname{FI}}}}$ type then ensures that the class of free ${{\operatorname{\textbf{C}}}}$-modules, and subsequently character polynomials, has the properties that ultimately produce representation stability.
\[thm:intro-free-modules\] If ${{\operatorname{\textbf{C}}}}$ is a category of ${{{\normalfont\operatorname{FI}}}}$ type, then the class of (finitely-generated) free ${{\operatorname{\textbf{C}}}}$-modules over ${\mathbb{C}}$ has the following properties:
1. The tensor product of two free ${{\operatorname{\textbf{C}}}}$-modules is again free.
2. There is a *degree* filtration on free ${{\operatorname{\textbf{C}}}}$-modules, taking values in the objects of ${{\operatorname{\textbf{C}}}}$. Direct sums and tensor products act on this degree in the usual way with respect to an order relation $\leq$ on ${{\operatorname{\textbf{C}}}}$ and object addition $+$ defined below.
3. Every free ${{\operatorname{\textbf{C}}}}$-module $M_\bullet$ has a dual ${{\operatorname{\textbf{C}}}}$-module $M^*_\bullet: c\mapsto \operatorname{Hom}_{{\mathbb{C}}}(M_c,{\mathbb{C}})$, which is again free of the same degree.
4. If $M_\bullet$ is a free ${{\operatorname{\textbf{C}}}}$-module of degree $\leq c$, then for every object $d\geq c$ the coinvariants $(M_d)/G_d$ are canonically isomorphic.
This statement – especially closure under tensor products – is nontrivial and depends critically on the structure of ${{{\normalfont\operatorname{FI}}}}$ type. For example, the specialization to ${{{\normalfont\operatorname{FI}}}}$-modules was proved in [@CEF] using the projectivity of ${{{\normalfont\operatorname{FI}}}}\#$-modules, and is related to the fact that products of binomial coefficients $\binom{n}{k}\binom{n}{l}$ can be expressed as linear combinations of $\binom{n}{r}$ with $r\leq k+l$.
Most of the results in Theorem \[thm:intro-free-modules\] are set-theoretic in nature and follow from combinatorial properties of ${{\operatorname{\textbf{C}}}}$-sets. They thus hold in greater generality with the base field ${\mathbb{C}}$ replaced with an arbitrary commutative ring $R$. However, when trying to decategorify and conclude character-theoretic results, the assumption of characteristic $0$ becomes necessary. To simplify our exposition, we will phrase the results only for ${{\operatorname{\textbf{C}}}}$-modules over ${\mathbb{C}}$.
Theorem \[thm:intro-free-modules\] in particular gives the categorified analog of Fact \[fact:stabilization\]. This fact captures the stabilization of the sequence of representations, as we shall see in the Application 1 below.
\[cor:inro-hom-stabilization\] If $M_\bullet$ and $N_\bullet$ are free ${{\operatorname{\textbf{C}}}}$-modules of respective degrees $\leq c_1$ and $\leq c_2$, then the spaces $$\label{eq:intro_tensor_hom}
\operatorname{Hom}_{G_d}(M_d,N_d) \cong (M^* \otimes N)_d/G_d$$ are canonically isomorphic for all $d\geq c_1+c_2$.
When the objects of ${{\operatorname{\textbf{C}}}}$ are parameterized by natural numbers, the addition $c_1+c_2$ is the usual addition operations. For the general definition of addition on objects, see Definition \[def:sums\] below. Note that the identification of the two sides in Equation \[eq:intro\_tensor\_hom\] is where characteristic $0$ assumption is used.
Decategorifying back to characters, one obtains the following.
If $P$ and $Q$ are character polynomials of respective degrees $\leq c_1$ and $\leq c_2$, then the inner products $$\langle P, Q\rangle_{G_d} = \frac{1}{|G_d|}\sum_{g\in G_d} \bar{P}(g)Q(g)$$ become independent of $d$ for all $d\geq c_1+c_2$.
These claims will be proved in §\[sec:free-modules\] and §\[sec:coinvariants\].
Application 1: Stabilization of irreducible multiplicities
----------------------------------------------------------
Let $G$ be a finite group. Recall that over ${\mathbb{C}}$ the irreducible decomposition of a $G$-representation can be detected by $G$-intertwiners. Explicitly, if $V$ is a $G$-representation and $W$ is an irreducible representation, then the multiplicity at which $W$ appears in $V$ is $\dim \operatorname{Hom}_G (W,V)$. Similarly, if $V = \oplus_{i} W_i^{r_i}$ is an irreducible decomposition then $$\dim \operatorname{Hom}_G(V,V) = \sum_{i} r_i^2.$$
Corollary \[cor:inro-hom-stabilization\] then demonstrates that these dimensions stabilize in the case of free ${{\operatorname{\textbf{C}}}}$-modules.
Let $M_\bullet$ be a free ${{\operatorname{\textbf{C}}}}$-module of degree $\leq c$. At every object $d$ let $$M_d = \oplus_i W(d)_i^{r(d)_i}.$$ be an irreducible decomposition. Then the sums $\sum_{i} r(d)_i^2$ do not depend on $d$ for $d\geq c+c$. More generally, if $N_\bullet$ is any other free ${{\operatorname{\textbf{C}}}}$-module of degree $\leq c'$ with irreducible decompositions $$N_d = \oplus_i W(d)_i^{s(d)_i}$$ then the sums $\sum_i r(d)_i\cdot s(d)_i$ do not depend on $d$ for all $d\geq c+c'$.
By choosing the test module $N_\bullet$ carefully, one can gain more information as to the individual multiplicities $r(d)_i$. In particular, it is often possible to relate the irreducible representations of the different groups $G_d$ and show that the individual multiplicities in fact stabilize for all $d\geq c+c$.
Application 2: The category of ${{\operatorname{\textbf{C}}}}$-modules is Noetherian
------------------------------------------------------------------------------------
One of the most important themes in representation stability is the Noetherian property: the subcategory of finitely-generated modules is closed under taking submodules. This allows one to apply tools from homological algebra to finitely-generated modules, with far reaching applications (see e.g. [@CEF] and [@ICM]).
Let $M$ be an orientable manifold. For every finite set $S$ the space of $S$-configurations on $M$, $\operatorname{PConf}^S(M)$, is the space of injections from $S$ to $M$. The functor $S\mapsto \operatorname{PConf}^S(M)$ is an ${{{\normalfont\operatorname{FI}}}}^{\operatorname{op}}$-space, and $S\mapsto\operatorname{H}^i(\operatorname{PConf}^S(M);{\mathbb{Q}})$ is an ${{{\normalfont\operatorname{FI}}}}$-module.
Totaro [@To] proved that there is a spectral sequence converging to $\operatorname{H}^i(\operatorname{PConf}^S(M))$ and [@CEF] showed that the every $E^{p,q}_2$-term of this sequence is a finitely-generated ${{{\normalfont\operatorname{FI}}}}$-module. [@CEF] also prove that ${{{\normalfont\operatorname{FI}}}}$-modules over ${\mathbb{Q}}$ is Noetherian, and therefore finite-generation persists to the $E_\infty$-page, and subsequently to $\operatorname{H}^i$. Therefore the sequence $\operatorname{H}^i(\operatorname{PConf}^S(M);{\mathbb{Q}})$ exhibits representation stability. One direct result is that the $i$-th Betti number of $\operatorname{PConf}^S(M)$ is eventually polynomial in $|S|$.
A corollary of the theory developed here is that the same Noetherian property holds in general.
\[thm:intro-noetherian\] If ${{\operatorname{\textbf{C}}}}$ is a category of ${{{\normalfont\operatorname{FI}}}}$ type, then the category of ${{\operatorname{\textbf{C}}}}$-modules over ${\mathbb{C}}$ is Noetherian. That is, every submodule of a finitely generated ${{\operatorname{\textbf{C}}}}$-module is itself finitely generated.
Theorem \[thm:intro-noetherian\], proved below in §\[sec:noetherian\], simultaneously generalizes the results by Church-Ellenberg-Farb [@CEF Theorem 1.3] and independently by Sam-Snowden [@SS-GL-mod Theorem 1.3.2], who proved that the category of ${{{\normalfont\operatorname{FI}}}}$-modules is Neotherian; Putman-Sam [@PS], who proved the same for the category of ${{{\normalfont\operatorname{VI}}}}$-modules; and Wilson [@Wi], who proved this for ${{{\normalfont\operatorname{FI}}}}_{\mathcal{W}}$-modules.
Gan-Li [@GL-EI] generalized all of these Noetherian results and found that they hold for every category with a skeleton whose objects are parameterized by ${\mathbb{N}}$ satisfying certain combinatorial conditions see ([@GL-EI Theorem 1.1]). However, their theory does not address categories whose objects are not parameterized by ${\mathbb{N}}$, such as ${{{\normalfont\operatorname{FI}}}}^m$ treated in §\[sec:rep\_theory\_of\_FI\] below.
One reason Noetherian results are important in our context is that they ensure that finitely-generated ${{\operatorname{\textbf{C}}}}$-modules exhibit the same stabilization phenomena as with free ${{\operatorname{\textbf{C}}}}$-module discussed in Application 1 (although without the effective bounds on stable range).
If $M_\bullet$ is a finitely-generated ${{\operatorname{\textbf{C}}}}$-module, then the sequence of coinvariants $M_c/G_c$ is eventually constant, i.e. all induced maps are $M_c/G_c{\ensuremath{\overset{}{\longrightarrow}}}M_d/G_d$ are isomorphisms for sufficiently large objects $c$.
More generally, for every free ${{\operatorname{\textbf{C}}}}$-module $F_\bullet$ the sequence of spaces $\operatorname{Hom}_{G_c}(F_c,M_c)$ is eventually constant in the same sense.
Application 3: Free modules in topology
---------------------------------------
Beyond the applications of free ${{\operatorname{\textbf{C}}}}$-modules to the representation theory of the category ${{\operatorname{\textbf{C}}}}$, they also appear explicitly in topology. In a companion paper [@Ga] we consider the cohomology of ${{\operatorname{\textbf{C}}}}$-diagrams of linear subspace arrangements, for which we show that the induced cohomology ${{\operatorname{\textbf{C}}}}$-module is free. An immediate consequence, stated here somewhat informally, is the following (see [@Ga] for the precise definitions and statements).
\[cor:intro-application-to-rational-maps\] Every (contravariant) ${{\operatorname{\textbf{C}}}}$-diagram $X^\bullet$ of linear subspace arrangements, that is generated by a finite collection of subspaces, exhibits cohomological representation stability. That is, for every $i\geq 0$ the ${{\operatorname{\textbf{C}}}}$-module $\operatorname{H}^i(X^\bullet;{\mathbb{C}})$ is free. In particular, there exists a single character polynomial $P_i$ of ${{\operatorname{\textbf{C}}}}$ that uniformly describes the $G_c$-representation $\operatorname{H}^i(X^c;{\mathbb{C}})$ for every object $c$.
Moreover, the respective quotients $X^c/G_c$ exhibit homological stability for ${\mathbb{C}}$-coefficients, and for various systems of constructible sheaves.
Acknowledgments
---------------
I wish to thank Benson Farb and Jesse Wolfson for many helpful conversations and suggestions that helped shape this work into its current form. I also thank Kevin Casto for explaining the arguments in Gan-Li’s paper and how my work fits in with theirs.
Preliminaries {#sec:prelim}
=============
Let ${{\operatorname{\textbf{C}}}}$ be a category. Objects of ${{\operatorname{\textbf{C}}}}$ will typically be denoted by $c$, $d$, and so on.
The categories with which we shall be working will have only injective morphisms. This typically precludes the possibility of having push-out objects. The following definition provides a means for salvaging some notion of a push-out diagram subject to this constraint.
\[def:weak-pushout\] A *weak push-out* diagram is a pullback diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{\tilde{f}_1}"] \arrow[d, swap, "{\tilde{f}_2}"] \& {c_1} \arrow[d, "{f_1}"] \\
{c_2} \arrow[r, swap, "{f_2}"] \& {d}
\end{tikzcd}$$]{} with the following universal property: for every other pullback diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{\tilde{f}_1}"] \arrow[d, swap, "{\tilde{f}_2}"] \& {c_1} \arrow[d, "{h_1}"] \\
{c_2} \arrow[r, swap, "{h_2}"] \& {z}
\end{tikzcd}$$]{} there exists a unique morphism $d{\ensuremath{\overset{h}{\longrightarrow}}} z$ that makes all the relevant diagrams commute. We call $d$ the *weak push-out object* and denote it by $c_1\coprod_p c_2$. The unique map $h$ induced from a pair of maps $c_i{\ensuremath{\overset{h_i}{\longrightarrow}}}z$ is denoted by $h_1\coprod_p h_2$.
This is similar to a usual push-out, but with “all" commutative squares replaced by only pullback squares. When starting from a category that has push-outs, such as ${{{\normalfont\operatorname{Set}}}}$ and ${{{\normalfont\operatorname{Vect}}}}_k$, and passing to the subcategory that includes only injective maps, we lose the push-out structure. However, weak push-outs persist, and retain most of the same function.
A standard notation that we will use throughout is the following.
We say that $c\leq d$ if there exists morphisms $c{\ensuremath{\overset{}{\longrightarrow}}}d$.
In categories of ${{{\normalfont\operatorname{FI}}}}$ type (see Definition \[def:FI-type\] above) this preorder relation between objects is essentially an order, i.e. if $c\leq d$ and $d\leq c$ then every morphism $c{\ensuremath{\overset{}{\longrightarrow}}}d$ is invertible (for an explanation see [@Ga Lamma 3.28]). However, as noted in part (5) of Definition \[def:FI-type\], push-outs typically don’t exist in categories of ${{{\normalfont\operatorname{FI}}}}$-type and we adjust the definition by demanding the following property instead.
A category ${{\operatorname{\textbf{C}}}}$ is said to be *of ${{{\normalfont\operatorname{FI}}}}$ type* is it satisfies axioms (1)-(4) from Definition \[def:FI-type\], and in addition:
1. ${{\operatorname{\textbf{C}}}}$ has pullbacks and weak push-outs, i.e. for every pair of morphisms $p{\ensuremath{\overset{f_i}{\longrightarrow}}}c_i$ there exists a weak push-out $c_1\coprod_p c_2$; and for every pair $c_i {\ensuremath{\overset{g_i}{\longrightarrow}}} d$ there exists a pullback $c_1 \times_{d} c_2$.
It seems possible that some of the theory should carry over to compact groups or even to semi-simple groups, but this direction will not be perused here.
The primary objects of study are the representation of ${{\operatorname{\textbf{C}}}}$. These are the ${{\operatorname{\textbf{C}}}}$-modules defined in Definition \[def:intro\_modules\]. Our goal is to understand the category of ${{\operatorname{\textbf{C}}}}$-modules and relate it to the categories of representations of the individual automorphism groups $G_c$.
Binomial sets and Character Polynomials {#subsec:char_poly}
---------------------------------------
The character polynomials for the symmetric groups are class functions simultaneously defined on $S_n$ for all $n$. These objects are closely linked with the phenomenon of representation stability in that the character of a representation-stable sequence is eventually given by a single character polynomial (see [@CEF]). We will now define character polynomials for a general category of ${{{\normalfont\operatorname{FI}}}}$ type.
The following notion generalizes the collection of subset of size $k$ inside a set of $n$ elements. It will be used below in the definition of character polynomials.
\[def:binom\_set\] Let $c$ and $d$ be two objects of ${{\operatorname{\textbf{C}}}}$. The group of automorphisms $G_c$ acts on $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$ on the right by precomposition. Denote the quotient $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)/G_c$ by $\binom{d}{c}$. We will call this the *binomial set, $d$ choose $c$*. If $c{\ensuremath{\overset{f}{\longrightarrow}}}d$ is a morphism, we denote its class in $\binom{d}{c}$ by $[f]$.
Since the set $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$ admits a left action by $G_d$, and this action commutes with the right action of $G_c$, the binomial set $\binom{d}{c}$ acquires a $G_d$ action naturally by $\sigma([f])=[\sigma\circ f]$.
Note that in the case of ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}$, the category of finite sets and injections, the binomial set $\binom{n}{k}$ is naturally in bijection with the collection of size $k$ subsets of $n$ (hence the terminology). Replacing ${{{\normalfont\operatorname{FI}}}}$ by ${{{\normalfont\operatorname{VI}}}}_{\mathbb{F}}$, the category of finite dimensional ${\mathbb{F}}$-vector spaces and injective linear functions, the binomial set $\binom{n}{k}$ is naturally the Grassmanian of $k$-planes in ${\mathbb{F}}^n$.
\[def:char\_poly\] Let $c$ be an object of ${{\operatorname{\textbf{C}}}}$ and $\mu\subseteq G_c$ a conjugacy class. In this case we will denote $|\mu|=c$. The *indicator character polynomial* of $\mu$ is the ${\mathbb{C}}$-valued class function $\binom{X}{\mu}$ simultaneously defined on all $G_d$ by $$\label{eq:indicator_poly}
\binom{X}{\mu}: \left(\sigma\in G_d\right) \mapsto \left|\left\{ [f]\in\binom{d}{c} \mid \exists \psi\in \mu \text{ s.t. } \sigma\circ f = f\circ \psi \right\}\right|.$$ The *degree* of $\binom{X}{\mu}$ is defined to be $\deg(\binom{X}{\mu}):=|\mu|$.
A *character polynomial* $P$ is a ${\mathbb{C}}$-linear combination of such simultaneous class functions. We say that the *degree* of $P$ is $\leq d$ for an object $d$ if for every indicator $\binom{X}{\mu}$ that appears in $P$ nontrivially we have $|\mu|\leq d$. We denote this by $\deg(P)\leq d$.
The following lemma shows that the above definition indeed gives rise to well-defined class functions.
The function $\binom{X}{\mu}$ is a class function of every group $G_d$. Furthermore, its definition in Equation \[eq:indicator\_poly\] does not depend on the choice of representative $f\in [f]$.
First to see that Equation \[eq:indicator\_poly\] does not depend on the choice of $f$, suppose $f'$ is another representative of $[f]\in \binom{d}{c}$. Then there exists some $g\in G_c$ such that $f'=f\circ g$. Then for every $\sigma\in G_d$ and $\psi\in\mu$ $$\sigma\circ f = f\circ \psi \iff \sigma\circ f' = (f\circ \psi) \circ g = f' \circ (g^{-1} \psi g)$$ and $g^{-1} \psi g$ belongs to $\mu$ as well.
Lastly, to see that we get a class function take $\sigma' = h \sigma h^{-1}$. Then $[f]\in \binom{d}{c}$ satisfies $\sigma\circ f = f\circ \psi$ if and only in $[h\circ f]$ satisfies $$\sigma'\circ (h\circ f) = h \sigma h^{-1} (h\circ f) = h \circ (f\circ \psi) = (h\circ f)\circ \psi.$$ If we denote by $U_\mu(\sigma)$ the set of classes $[f]\in \binom{d}{c}$ which is counted in Equation \[eq:indicator\_poly\] for $\sigma$, then we see that $U_\mu(h\sigma h^{-1})=h(U\mu(\sigma))$, and in particular these sets have equal cardinality.
\[ex:FI\_characters\] For ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}$ the automorphism group of the object $n=\{0,1,\ldots n-1\}$ is the symmetric group $S_n$. For any $k\in {\mathbb{N}}$ a conjugacy class in $S_k$ is described by a cycle type, $\mu = (\mu_1,\mu_2,\ldots,\mu_k)$ where $\mu_i$ is the number of $i$-cycles. For any other $n\in {\mathbb{N}}$, if we denote by $X_i$ the class function on $S_n$ $$X_i(\sigma) = \# \text{ of $i$-cycles in }\sigma$$ then we claim that $$\label{eq:FI_indicator_poly}
\binom{X}{\mu}(\sigma) = \binom{X_1(\sigma)}{\mu_1}\ldots \binom{X_k(\sigma)}{\mu_k}.$$
Indeed, the class $[f]\in \binom{n}{k}$ of an injection $k{\ensuremath{\overset{f}{\longrightarrow}}}n$ corresponds to the subset $Im(f)\subseteq n$. The condition that $\sigma\circ f = f\circ \psi$ translates to saying that $Im(f)$ is invariant under $\sigma$ and that the induced permutation on this subset has cycle type $\mu$. Then for a given $\sigma\in S_n$ the right-hand side of Equation \[eq:FI\_indicator\_poly\] counts the number of ways to assemble such an invariant subset from the cycles of $\sigma$.
[@CEF] give Equation \[eq:FI\_indicator\_poly\] as the definition of $\binom{X}{\mu}$. Thus our definition of character polynomials extends the classical notion of character polynomials for the symmetric groups to other classes of groups.
For ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{VI}}}}_{\mathbb{F}}$ the automorphism group of the object $[n] = {\mathbb{F}}^n$ is $\operatorname{GL}_n({\mathbb{F}})$. We describe the degree $1$ indicators. A conjugacy class in $\operatorname{GL}_1({\mathbb{F}})={\mathbb{F}}^\times$ is just a non-zero element $\mu\in {\mathbb{F}}$. For every $n\in {\mathbb{N}}$ the function $\binom{X}{\mu}$ on a matrix $A\in\operatorname{GL}_n({\mathbb{F}})$ is given by $$\binom{X}{\mu}(A) = \# \text{ of 1D eigenspaces of $A$ with eigenvalue } \mu.$$ These are the ${{{\normalfont\operatorname{VI}}}}$ analogs of $X_1$ on $S_n$, which counts the number of fixed points of a permutation.
Free ${{\operatorname{\textbf{C}}}}$-modules {#sec:free-modules}
============================================
This section is devoted to defining free ${{\operatorname{\textbf{C}}}}$-modules and proving that, when ${{\operatorname{\textbf{C}}}}$ is of ${{{\normalfont\operatorname{FI}}}}$ type, these modules satisfy the fundamental properties stated in Theorem \[thm:intro-free-modules\]. Note that the statements and proofs in this section are essentially set-theoretic in nature, and therefore hold in a more general setting of ${{\operatorname{\textbf{C}}}}$-modules over any ring $R$. For concreteness we will only describe here the results with $R={\mathbb{C}}$.
Free ${{\operatorname{\textbf{C}}}}$-modules are defined using a collection of left-adjoint functors. For every object $c$ there is a natural restriction functor $${{{\normalfont\operatorname{Mod}}}_{{{\operatorname{\textbf{C}}}}}} {\ensuremath{\overset{\operatorname{Res}_{c}}{\longrightarrow}}} {{{\normalfont\operatorname{Mod}}}_{{\mathbb{C}}[G_c]}}\, , \; M_\bullet\mapsto M_c$$ Following [@tD], this functor admits a left-adjoint as follows.
\[**Induction ${{\operatorname{\textbf{C}}}}$-modules**\] \[def:Ind\_functor\] Let $\operatorname{Ind}_c: {{{\normalfont\operatorname{Mod}}}_{{\mathbb{C}}[G_c]}} {\ensuremath{\overset{}{\longrightarrow}}} {{{\normalfont\operatorname{Mod}}}_{{{\operatorname{\textbf{C}}}}}}$ be the functor that sends a $G_c$-representation $V$ to the ${{\operatorname{\textbf{C}}}}$-module $$\operatorname{Ind}_c(V)_\bullet = {\mathbb{C}}[\operatorname{Hom}(d,\bullet)]\otimes_{G_d} V$$ where morphisms in ${{\operatorname{\textbf{C}}}}$ act on these spaces naturally through their action on $\operatorname{Hom}(c,\bullet)$.
We call a ${{\operatorname{\textbf{C}}}}$-module of this form an *induction module* of degree $c$, and denote $$\deg(\operatorname{Ind}_c(V))=c.$$
[@tD] shows that the functor $\operatorname{Ind}_c$ is a left adjoint to $\operatorname{Res}_c$. Recall that in Definition \[def:intro\_free\] we called direct sum of induction modules free. The following additional terminology will also be useful.
\[**Degree of a free module**\] We say that a free ${{\operatorname{\textbf{C}}}}$-module $M_\bullet$ has *degree* $\leq d$ if for every induction module $\operatorname{Ind}_c(V)$ that appears in $M_\bullet$ nontrivially we have $c\leq d$. In this case we denote $\deg(M_\bullet)\leq d$.
A *virtual* free ${{\operatorname{\textbf{C}}}}$-module is a formal ${\mathbb{C}}$-linear combination of induction modules, e.g. $$\oplus_{i=1}^n \lambda_i \operatorname{Ind}_{c_i}(V_i) \; \text{where } \lambda_i\in {\mathbb{C}}.$$ We extend the induction functors $\operatorname{Ind}_c$ linearly to virtual $G_c$-representations, i.e. $$\operatorname{Ind}_c( \oplus \lambda_i V_i ) := \oplus \lambda_i \operatorname{Ind}_c (V_i).$$
We propose that (virtual) free ${{\operatorname{\textbf{C}}}}$-modules are a categorification of character polynomials, much like the case for any finite group $G$ where (virtual) $G$-representations categorify class functions on $G$ .
If $M_\bullet$ is a ${{\operatorname{\textbf{C}}}}$-module, its *character* is the simultaneous class function $$\chi_M : \coprod_{c} G_c {\ensuremath{\overset{}{\longrightarrow}}} {\mathbb{C}}$$ that for every object $c$ sends the group $G_c$ to the character of the $G_c$-representation $M_c$.
One can express the character of induction modules in terms of indicator character polynomials, as follows.
\[lem:character\_of\_ind\] If $V$ is any $G_c$-representation whose character is $\chi_V$, then the character of $\operatorname{Ind}_c(V)$ is given by $$\label{eq:character_of_ind}
\chi_{\operatorname{Ind}_c(V)} = \sum_{\mu \in \operatorname{conj}(G_c)} \chi_V(\mu) \binom{X}{\mu}$$ where $\operatorname{conj}(G_c)$ is the set of conjugacy classes of $G_c$, and $\chi_V(\mu)$ is the value $\chi_V$ takes on any $g\in\mu$. In particular we see that the character of $\operatorname{Ind}_c(V)$ is a character polynomial of degree $c$.
Since all morphisms in ${{\operatorname{\textbf{C}}}}$ are monomorphisms, it follows for every object $d$ the equivalence class $f\circ G_c = [f]\in \binom{d}{c}$ is a right $G_c$-torsor. Thus there is an isomorphism of vector spaces $$\label{eq:ind_decomposition}
\operatorname{Ind}_c(V)_d = {\mathbb{C}}[\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)]{\mathbin{\underset{\substack{G_c}}{\otimes}}} V = {\mathbin{\underset{\substack{{[f}}{\oplus}}}\in \binom{d}{c}}] {\mathbb{C}}([f]){\mathbin{\underset{\substack{G_c}}{\otimes}}}V \cong {\mathbin{\underset{\substack{{[f}}{\oplus}}}\in \binom{d}{c}}] V$$ where the group $G_d$ permutes the summands through its action on $\binom{d}{c}$. It follows that the trace of $\sigma\in G_d$ gets a contribution from the summand ${\mathbb{C}}([f])\otimes_{G_c} V$ if and only if $\sigma([f])=[f]$. Consider such $[f]\in \operatorname{Fix}(\sigma)$, i.e. there exists some $\psi\in G_c$ such that $\sigma\circ f = f\circ \psi$. We get a commutative diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{{\mathbb{C}}([f])\otimes_{G_c}V} \arrow[r, "{\sigma}"] \arrow[d, swap, "{\cong}"] \& {{\mathbb{C}}([f])\otimes_{G_c}V} \arrow[d, "{\cong}"] \\
{V} \arrow[r, swap, "{\psi}"] \& {V}
\end{tikzcd}$$]{} so the trace of $\sigma|_{{\mathbb{C}}([f])\otimes_{G_c}V}$ is precisely $\chi_V(\psi)$. We get a formula for the character $$\chi_{\operatorname{Ind}_c(V)}(\sigma) = \sum_{\substack{[f]\in \binom{d}{c}\\ \exists\psi (\sigma\circ f = f\circ \psi)}} \chi_V(\psi).$$ Arranging this sum according to the conjugacy class of $\psi$ we get the equality claimed by Equation \[eq:character\_of\_ind\].
A corollary or Lemma \[lem:character\_of\_ind\] is that free ${{\operatorname{\textbf{C}}}}$-modules indeed categorify character polynomials.
\[thm:categorification\] Character polynomials of degree $\leq d$ are precisely the characters of virtual free ${{\operatorname{\textbf{C}}}}$-modules of degree $\leq c$.
It is sufficient to show that every $\binom{X}{\mu}$ is the character of some virtual free ${{\operatorname{\textbf{C}}}}$-module of degree $\leq |\mu|$. Denote $c=|\mu|$ and consider the indicator class function on $G_c$ $$\chi_\mu(\psi)= \begin{cases}
1 \quad & \psi\in \mu \\
0 \quad & \psi\notin \mu.
\end{cases}$$ Since the characters of $G_c$-representations form a basis for the class functions on $G_c$, there exist $G_c$-representations $V_1,\ldots,V_n$ and complex numbers $\lambda_1,\ldots,\lambda_n$ such that the virtual representation $$V_\mu = \oplus_{i=1}^n \lambda_i V_i$$ has character $\chi_\mu$. Then by Lemma \[lem:character\_of\_ind\] and linearity it follows that $$\chi_{\operatorname{Ind}_c(V_{\mu})} = \binom{X}{\mu}.$$
Tensor products
---------------
The categorification of pointwise products of character polynomials is the tensor product of free ${{\operatorname{\textbf{C}}}}$-modules. The goal of this subsection is to show that the product of two free modules is itself free.
If $M_\bullet$ and $N_\bullet$ are two ${{\operatorname{\textbf{C}}}}$-module, their tensor product $\left(M\otimes N\right)_\bullet$ is the ${{\operatorname{\textbf{C}}}}$-module $$\left(M\otimes N\right)_d = M_d \otimes N_d$$ where a morphism $c{\ensuremath{\overset{f}{\longrightarrow}}}d$ acts naturally by $M(f)\otimes N(f)$.
At the level of characters, the tensor product corresponds to pointwise multiplication: $$\chi_{M\otimes N} = \chi_M \cdot \chi_N.$$
The main result of this subsection is the parts $(1)$ and $(2)$ of Theorem \[thm:intro-free-modules\]. The following definition gives meaning to addition of objects so as to make the degree additive.
\[def:sums\] If $c_1$ and $c_2$ are two object of ${{\operatorname{\textbf{C}}}}$, then $c_1+c_2$ denotes the collection of objects $d$ that satisfy $$c_1\coprod_p c_2 \leq d$$ for every weak push-out of $c_1$ and $c_2$. If $d$ belongs to the collection $c_1+c_2$ we denote $d\geq c_1+c_2$, i.e. $$d\geq c_1+c_2 \iff d\in c_1+c_2.$$
If $M_\bullet$ is a free ${{\operatorname{\textbf{C}}}}$-module, we say that $\deg(M) \leq c_1+c_2$ if the degree is $\leq d$ for every $d\in c_1+c_2$.
If the collection $c_1+c_2$ contains an essential minimum object $d_0$, then we can identify $c_1+c_2$ with this minimum. In this case saying that $\deg(M)\leq c_1+c_2$ is equivalent to saying $\deg(M)\leq d_0$. In all the examples we currently know, the essential minimum object of $c_1+c_2$ is the weak coproduct $c_1\coprod_{\varnothing}c_2$. In particular, when ${{\operatorname{\textbf{C}}}}$ has a skeleton whose objects are parameterized naturally by ${\mathbb{N}}$ then the object “$n_1+n_2$" coincides with the standard addition $n_1+n_2$ (hence the notation).
At the level of character polynomials Theorem \[thm:intro-free-modules\](1) translates into the following result.
The collection of character polynomials forms an algebra under pointwise products, and the degree is additive with respect products. Namely, if $P$ and $Q$ are character polynomials of respective degrees $\leq c_1$ and $\leq c_2$, then their product $P\cdot Q$ is a character polynomial of degree $\leq c_1+c_2$.
It is not immediately clear that the product of two expressions $\binom{X}{\mu}$ and $\binom{X}{\nu}$ can be expanded in terms of other such expressions, but we now see they can. To demonstrate the nontriviality of this statement consider the standard binomial coefficients: for $X = \binom{X}{1}$ we have an expansion $$X\cdot \binom{X}{k} = (k+1)\binom{X}{k+1}+ k\binom{X}{k}$$
Find general formula for the expansion of $\binom{X}{k_1}\binom{X}{k_2}$ in terms of $\binom{X}{k}$’s.
The proof of Theorem \[thm:intro-free-modules\](1) will use the following definitions and lemmas. First we need an easy technical observation.
\[lem:pullback-invariance\] Suppose [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{a} \arrow[r, "{f_1}"] \arrow[d, swap, "{f_2}"] \& {b_1} \arrow[d, "{g_1}"] \\
{b_2} \arrow[r, swap, "{g_2}"] \& {c}
\end{tikzcd}$$]{} is some commutative diagram in ${{\operatorname{\textbf{C}}}}$ and $c{\ensuremath{\overset{h}{\longrightarrow}}}d$ is some monomorphism. By composing with $h$ we get another diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{a} \arrow[r, "{f_1}"] \arrow[d, swap, "{f_2}"] \& {b_1} \arrow[d, "{h\circ g_1}"] \\
{b_2} \arrow[r, swap, "{h\circ g_2}"] \& {d}
\end{tikzcd}$$]{}
If one of these diagrams is a pull-back, then so is the other.
Second we define the *push-out set* of three objects.
Let $c_1$ and $c_2$ be two objects of ${{\operatorname{\textbf{C}}}}$. For any object $d$ we define the *push-out set* ${\operatorname{PO}\binom{d}{c_1,c_2}}$ to be the set of pairs of morphisms $(c_i{\ensuremath{\overset{g_i}{\longrightarrow}}}d\mid i=1,2)$ that present $d$ as a weak push-out of $c_1$ and $c_2$. That is to say that the pullback diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{c_1\times_d c_2} \arrow[r, "{}"] \arrow[d, swap, "{}"] \& {c_1} \arrow[d, "{g_1}"] \\
{c_2} \arrow[r, swap, "{g_2}"] \& {d}
\end{tikzcd}$$]{} is a weak push-out diagram.
It is straightforward to verify that the procedure of replacing $d$ by an isomorphic object $d'$ and mapping $c_1$ and $c_2$ into $d'$ through any isomorphism ${d{\ensuremath{\overset{\sim}{\longrightarrow}}}d'}$ preserves weak push-out diagrams. Therefore any such isomorphism induces a natural bijection of sets $${\operatorname{PO}\binom{d}{c_1,c_2}} {\ensuremath{\overset{\sim}{\longrightarrow}}} {\operatorname{PO}\binom{d'}{c_1,c_2}}$$ by left-composition. In particular, the group of automorphisms $G_d$ acts on ${\operatorname{PO}\binom{d}{c_1,c_2}}$ on the left. Similarly, the group $G_{c_1}\times G_{c_2}$ acts naturally on the right by precomposition.
The general philosophy of this work is the following: statements about representation stability (of which Theorem \[thm:intro-free-modules\](1) is one) are reflected by statement about ${{\operatorname{\textbf{C}}}}$-sets. Therefore closure under tensor products should be a consequence of a set-theoretic observation. This is the content of the next lemma.
\[lem:products\_set\_version\] Let ${{\operatorname{\textbf{C}}}}$ be a category of ${{{\normalfont\operatorname{FI}}}}$ type. There is a natural isomorphism between the product functor $$\operatorname{Hom}(c_1,\bullet)\times \operatorname{Hom}(c_2,\bullet)$$ and the disjoint union functor $$\coprod_{[d]} \operatorname{Hom}\left( d, \bullet \right)\times_{G_d} {\operatorname{PO}\binom{d}{c_1,c_2}}$$ where $[d]$ ranges over the isomorphism classes of ${{\operatorname{\textbf{C}}}}$ and $d$ is some representative of $[d]$. Furthermore this natural isomorphism respects the right $(G_{c_1}\times G_{c_2})$-action on the two functors.
For any object $x$ and a representative $d$ of the isomorphism class $[d]$ we define a function $$\operatorname{Hom}\left( d, x \right)\times_{G_d} {\operatorname{PO}\binom{d}{c_1,c_2}} {\ensuremath{\overset{\Psi_x^d}{\longrightarrow}}} \operatorname{Hom}(c_1,x)\times \operatorname{Hom}(c_2,x)$$ by composition, i.e. $$\left[d{\ensuremath{\overset{f}{\longrightarrow}}}x , (c_i{\ensuremath{\overset{r_i}{\longrightarrow}}}d) \right] \mapsto \left( c_i {\ensuremath{\overset{f\circ r_i}{\longrightarrow}}} x \right)$$ By the associativity of composition, this is well-defined on the product over $G_d$. Moreover, $\Psi^d_\bullet$ is clearly natural in $x$ and respects the right action of $G_{c_1}\times G_{c_2}$ given by precomposition.
Letting $d$ range over all isomorphism classes we get a natural transformation from the union $$\coprod_{[d]}\operatorname{Hom}\left( d, \bullet \right)\times_{G_d} {\operatorname{PO}\binom{d}{c_1,c_2}} {\ensuremath{\overset{\Psi_\bullet}{\longrightarrow}}} \operatorname{Hom}(c_1,\bullet)\times \operatorname{Hom}(c_2,\bullet)$$ which respects the right $G_{c_1}\times G_{c_2}$-action.
In the other direction, let $x$ again be any object. We define a function $$\operatorname{Hom}(c_1,x)\times \operatorname{Hom}(c_2,x) {\ensuremath{\overset{\Phi^d_x}{\longrightarrow}}} \operatorname{Hom}\left( d, x \right)\times_{G_d} {\operatorname{PO}\binom{d}{c_1,c_2}}$$ as follows. Let $(c_i{\ensuremath{\overset{f_i}{\longrightarrow}}}x\mid i=1,2)$ be a pair of morphisms. Construct their pull-back [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{\alpha_1}"] \arrow[d, swap, "{\alpha_2}"] \& {c_1} \arrow[d, "{f_1}"] \\
{c_2} \arrow[r, swap, "{f_2}"] \& {x}
\end{tikzcd}$$]{} and form the weak push-out for $p{\ensuremath{\overset{\alpha_i}{\longrightarrow}}}c_i$ [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{\alpha_1}"] \arrow[d, swap, "{\alpha_2}"] \& {c_1} \arrow[d, "{r_1}"] \\
{c_2} \arrow[r, swap, "{r_2}"] \& {c_1\coprod_p c_2=:d}
\end{tikzcd}$$]{} The universal property of the weak push-out then implies that there exists a unique morphism $d{\ensuremath{\overset{f}{\longrightarrow}}}x$ such that $f\circ r_i = f_i$. We define $\Phi^d_x$ by $$(c_i {\ensuremath{\overset{f_i}{\longrightarrow}}}x )\mapsto \left[(d{\ensuremath{\overset{f}{\longrightarrow}}}x), c_i{\ensuremath{\overset{r_i}{\longrightarrow}}}d \right].$$
To see that $\Phi^d_x$ is well-defined, suppose [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p'} \arrow[r, "{\alpha_1'}"] \arrow[d, swap, "{\alpha_2'}"] \& {c_1} \arrow[d, "{r_1'}"] \\
{c_2} \arrow[r, swap, "{r_2'}"] \& {d'}
\end{tikzcd}$$]{} is another weak push-out diagram produced by the same procedure and $d'{\ensuremath{\overset{f'}{\longrightarrow}}}x$ is the corresponding induced map. First we observe that since $p$ and $p'$ are both pull-backs of the pair $(f_1,f_2)$ there exists an isomorphism $p{\ensuremath{\overset{\tau}{\longrightarrow}}}p'$ for which $\alpha_i'\circ \tau = \alpha_i$ for $i=1,2$. Second, we replace $p'$ by $p$ in the weak push-out diagram, mapping it though $\tau$, i.e. [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{\alpha_1'\circ \tau}"] \arrow[d, swap, "{\alpha_2'\circ \tau}"] \& {c_1} \arrow[d, "{r_1'}"] \\
{c_2} \arrow[r, swap, "{r_2'}"] \& {d'}
\end{tikzcd}$$]{} and this is again a weak push-out diagram. Therefore, by the universal property of the weak push-out, there exists a unique morphism $d{\ensuremath{\overset{\psi}{\longrightarrow}}}d'$ for which $\psi\circ r_i = r_i'$. The same reasoning applied in reverse shows that $\psi$ admits a unique inverse, and therefore $d\cong d'$. Since we picked $d$ to be the representative for the isomorphism class $[d]$, it follows that $d=d'$ and that $\psi\in G_d$. The induced map $f$ is characterized by the property that $f\circ r_i = f_i$, and similarly for $f'$ and $r_i'$. Therefore we find that $$f_i = f'\circ r_i' = f'\circ \psi \circ r_i$$ which by the universal property of $d$ shows that in fact $f = f'\circ \psi$. Our function $\Phi^d_x$ is defined as to send the pair $(f_1,f_2)$ to $$\left[ f, (c_i{\ensuremath{\overset{g_i}{\longrightarrow}}}d) \right] = \left[ f'\circ \psi, (c_i{\ensuremath{\overset{g_i}{\longrightarrow}}}d) \right] = \left[ f', (c_i{\ensuremath{\overset{\psi\circ g_i}{\longrightarrow}}}d) \right] = \left[ f', (c_i{\ensuremath{\overset{g_i'}{\longrightarrow}}}d) \right]$$ which we now see that is uniquely defined.
The two functions $\Psi^d_x$ and $\Phi^d_x$ are clearly inverse, and therefor they together form a natural isomorphism between the two functors. As stated above, this isomorphism respects the right $G_{c_1}\times G_{c_2}$-action.
Now we can prove that free ${{\operatorname{\textbf{C}}}}$-modules are indeed closed under tensor products.
\[Proof of Theorem \[thm:intro-free-modules\](1)\] By the distributivity of tensor products, it is enough to verify the claim for induction modules of respective degrees $\leq c_1$ and $c_2$ respectively. Moreover, by the transitivity of the order relation between objects, it will suffice if we assume that the degrees are precisely $c_1$ and $c_2$ respectively. Let $\operatorname{Ind}_{c_1}(V)$ and $\operatorname{Ind}_{c_2}(W)$ be two such ${{\operatorname{\textbf{C}}}}$-modules.
We apply an easy-to-verify equality of tensor products, $$\begin{aligned}
\operatorname{Ind}_{c_1}(V)_\bullet \otimes_k \operatorname{Ind}_{c_2}(W)_\bullet &=& \left({\mathbb{C}}[\operatorname{Hom}{c_1,\bullet}]\otimes_{G_{c_1}} V\right) \otimes_k \left({\mathbb{C}}[\operatorname{Hom}{c_2,\bullet}]\otimes_{G_{c_2}} W\right) \\
&\cong& \left({\mathbb{C}}[\operatorname{Hom}(c_1,\bullet)]\otimes_k {\mathbb{C}}[\operatorname{Hom}(c_2,\bullet)]\right) \bigotimes_{G_{c_1}\times G_{c_2}} \left( V \boxtimes W \right)\end{aligned}$$ and to this we can apply the natural isomorphism $${\mathbb{C}}[\operatorname{Hom}(c_1,\bullet)]\otimes {\mathbb{C}}[\operatorname{Hom}(c_2,\bullet)] \cong {\mathbb{C}}\left[ \operatorname{Hom}(c_1,\bullet)\times \operatorname{Hom}(c_2,\bullet) \right].$$
In Lemma \[lem:products\_set\_version\] we found a natural isomorphism between the product $$\operatorname{Hom}(c_1,\bullet)\times \operatorname{Hom}(c_2,\bullet)$$ and the union $$\coprod_{[d]} \operatorname{Hom}\left( d, \bullet \right)\times_{G_d} {\operatorname{PO}\binom{d}{c_1,c_2}}$$ which when composed with the permutation representation functor $X \mapsto {\mathbb{C}}[X]$ yields a natural isomorphism $${\mathbb{C}}\left[ \operatorname{Hom}(c_1,\bullet)\times \operatorname{Hom}(c_2,\bullet) \right] \cong \bigoplus_{[d]} {\mathbb{C}}[\operatorname{Hom}(d,\bullet)]\otimes_{G_d} {\mathbb{C}}[{\operatorname{PO}\binom{d}{c_1,c_2}}]$$
By the associativity of the tensor product, we get a natural isomorphism $$\begin{aligned}
\label{eq:tensor_of_free}
\operatorname{Ind}_{c_1}(V)_\bullet \otimes_k \operatorname{Ind}_{c_2}(W)_\bullet &\cong& \bigoplus_{[d]} {\mathbb{C}}[\operatorname{Hom}(d,\bullet)] {\mathbin{\underset{\substack{G_d}}{\otimes}}} {\mathbb{C}}[{\operatorname{PO}\binom{d}{c_1,c_2}}] \bigotimes_{G_{c_1}\times G_{c_2}} (V\boxtimes W) \\
&=& \bigoplus_{[d]} \operatorname{Ind}_d\left( {\mathbb{C}}[{\operatorname{PO}\binom{d}{c_1,c_2}}] \bigotimes_{G_{c_1}\times G_{c_2}} (V\boxtimes W) \right) {}_\bullet\end{aligned}$$ as claimed.
Note that for $d$ to have a non-zero contribution to this direct sum, the set ${\operatorname{PO}\binom{d}{c_1,c_2}}$ must be non-empty. In particular, there exists a decomposition $\displaystyle{d=c_1\coprod_p c_2}$. This proves the claim regarding the degree of terms in the sum. Since there are only finitely many isomorphism classes of objects with such a presentation, the above direct sum decomposition is finite.
Dualization
-----------
One would like to define the dual of a ${{\operatorname{\textbf{C}}}}$-module $M_\bullet$ by $(M^*)_c = (M_c)^*$. Unfortunately, this will not be a ${{\operatorname{\textbf{C}}}}$-module in general (it will be a ${{\operatorname{\textbf{C}}}}^{\operatorname{op}}$-module). In this subsection we show that when dealing with free ${{\operatorname{\textbf{C}}}}$-modules there is a good notion of dualization.
For an induction module $\operatorname{Ind}_c(V)$ we define its dual ${{\operatorname{\textbf{C}}}}$-modules by $$\operatorname{Ind}_c(V)^* = \operatorname{Ind}_c(V^*)$$ where $V^*$ is the $G_c$-representation dual to $V$. Extend this definition linearly to all (virtual) free ${{\operatorname{\textbf{C}}}}$-modules.
We claim that this indeed gives a good notion of duals.
If $M_\bullet$ is a free ${{\operatorname{\textbf{C}}}}$-module then there is a homomorphism of ${{\operatorname{\textbf{C}}}}$-modules $$M_\bullet^* \otimes M_\bullet {\ensuremath{\overset{\operatorname{ev}}{\longrightarrow}}} {\mathbb{C}}_\bullet$$ where ${\mathbb{C}}_\bullet$ is the trivial ${{\operatorname{\textbf{C}}}}$-module with ${\mathbb{C}}_d = {\mathbb{C}}$ for every object. This pairing is non-degenerate and thus defines an isomorphism of $G_d$-representations $(M^*)_d\cong (M_d)^*$ for every object $d$.
We conclude that the dual of a free ${{\operatorname{\textbf{C}}}}$-module of degree $\leq c$ is again a ${{\operatorname{\textbf{C}}}}$-module of degree $\leq c$.
Suppose $M = \oplus_i \operatorname{Ind}_{c_i}(V_i)$. Then there is a decomposition of ${{\operatorname{\textbf{C}}}}$-modules $$M^*\otimes M = {\mathbin{\underset{\substack{i,j}}{\oplus}}} \operatorname{Ind}_{c_i}(V_i^*)\otimes \operatorname{Ind}_{c_j}(V_j).$$ We define the pairing to be $0$ for all $i\neq j$. For $i=j$ consider a single induction module $\operatorname{Ind}_c(V)$ and decompose it using Equation \[eq:ind\_decomposition\] $$\operatorname{Ind}_c(V)_d = \oplus_{[f]\in\binom{d}{c}} V \implies \left(\operatorname{Ind}_c(V^*)\otimes \operatorname{Ind}_c(V^*)\right)_d = \oplus_{[f],[g]\in\binom{d}{c}} V^*\otimes V.$$ Set the pairing to be $0$ on all $[f]\neq [g]$, and for $[f]=[g]$ use the natural contraction on $V^*\otimes V$. This produces a map $$\oplus_{[f],[g]\in\binom{d}{c}} V^*\otimes V {\ensuremath{\overset{}{\longrightarrow}}} \oplus_{[f]\in \binom{d}{c}} {\mathbb{C}}{\ensuremath{\overset{+}{\longrightarrow}}} {\mathbb{C}}$$ which is the pairing we sought.
Explicitly, the pairing on $\operatorname{Ind}_c(V^*)\otimes \operatorname{Ind}_c(V)$ is given by $$\label{eq:contraction}
\langle f\otimes v^*, g\otimes v \rangle = \sum_{\substack{\psi\in G_c \\ f\circ \psi = g}} \langle v^*, \psi(v)\rangle = \begin{cases}
\langle v^*, \psi(v)\rangle & f\circ \psi = g \\
0 & [f]\neq [g]
\end{cases}$$ It is straightforward to check that the above pairing is invariant under the action of morphisms in ${{\operatorname{\textbf{C}}}}$. It is thus a morphism of ${{\operatorname{\textbf{C}}}}$-modules, as claimed. One can also check that the pairing is non-degenerate, and thus defines the claimed $G_d$-equivariant isomorphism $$(M^*)_d {\ensuremath{\overset{\sim}{\longrightarrow}}} (M_d)^*.$$
\[cor:hom\_module\] If $M_\bullet$ is a free ${{\operatorname{\textbf{C}}}}$-modules and $N_\bullet$ is any ${{\operatorname{\textbf{C}}}}$-module, then there exists a ${{\operatorname{\textbf{C}}}}$-module $\operatorname{Hom}(M,N)_\bullet$ whose value at $d$ is the $G_d$-representation $\operatorname{Hom}_{\mathbb{C}}(M_d,N_d)$.
A morphism $d{\ensuremath{\overset{f}{\longrightarrow}}}e$ induces a function $\operatorname{Hom}_{\mathbb{C}}(M_d,N_d){\ensuremath{\overset{f_*}{\longrightarrow}}}\operatorname{Hom}_{\mathbb{C}}(M_e,N_e)$ satisfying the following naturality property: if $M_d{\ensuremath{\overset{T}{\longrightarrow}}}N_d$ is any linear function, then there is a commutative diagram [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{M_d} \arrow[r, "{T}"] \arrow[d, swap, "{M(f)}"] \& {N_d} \arrow[d, "{N(f)}"] \\
{M_e} \arrow[r, swap, "{f_*(T)}"] \& {N_e}
\end{tikzcd}$$]{}
Furthermore, if $N_\bullet$ is itself free, and the degrees of $M_\bullet$ and $N_\bullet$ are $\leq c_1$ and $\leq c_2$ respectively, then $\operatorname{Hom}(M,N)_\bullet$ is also free and has degree $\leq c_1+c_2$.
The desired ${{\operatorname{\textbf{C}}}}$-module is the tensor product $M^*\otimes N$. All other claims follow for the properties of tensor products and duals proved above.
The Coinvariant quotient and Stabilization {#sec:coinvariants}
==========================================
When $G$ is a finite group, the coinvariants of a $G$-representation are the categorified analog of averaging over class function: if $\chi$ is the character of a $G$-representation $V$, then $$\label{eq:coinvariants_average}
\dim V_G = \frac{1}{|G|}\sum_{g\in G}\chi(g).$$ Such averages appear in $G$-inner products, which we want to relate for the various automorphism groups $G_c$ of our category ${{\operatorname{\textbf{C}}}}$. This section will therefore analyze the behavior of free ${{\operatorname{\textbf{C}}}}$-modules under taking their coinvariants. Recall that the coinvariant quotient of a $G$-representation $V$ is its maximal invariant quotient, namely $$V_G = V/\langle v-gv \mid v\in V, g\in G \rangle.$$ We will also denote this quotient by $V/{G}$.
In the context of a ${{\operatorname{\textbf{C}}}}$-module $M_\bullet$ we can form the $G_c$-coinvariant quotient of $M_c$ for every object $c$. If $c{\ensuremath{\overset{f}{\longrightarrow}}}d$ is any morphism and $M_c{\ensuremath{\overset{M(f)}{\longrightarrow}}}M_d$ the induced map, then it descends to a well-defined map on the coinvariants. Indeed, this follows from the assumptions that $G_d$ acts transitively on $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$: if $g\in G_c$ is any automorphism, then $f$ and $f\circ g$ are two morphisms from $c$ to $d$ and thus there exists some $\tilde{g}\in G_d$ for which $\tilde{g}\circ f = f\circ g$. This shows that for every $v\in M_c$ $$v-g(v) \overset{M(f)}{\longmapsto} f(v) - f\circ g(v) = f(v) - \tilde{g}\left(f(v)\right)$$ and indeed $v-gv$ gets mapped to zero in the coinvariant quotient of $M_d$.
We call the resulting ${{\operatorname{\textbf{C}}}}$-module of coinvariant quotients *the coinvariant ${{\operatorname{\textbf{C}}}}$-module* of $M_\bullet$ and denote it by $(M/G)_\bullet$.
Every two morphisms $c{\ensuremath{\overset{f}{\longrightarrow}}}d$ and $c{\ensuremath{\overset{f'}{\longrightarrow}}}d$ give rise to the same map between coinvariants. This is because there exists some $\tilde{g}\in G_d$ for which $\tilde{g}\circ f = f'$ and this $\tilde{g}$ acts trivially on $(M/G)_d$. Thus for every pair $c\leq d$ there is a well-defined map between the coinvariants $(M/G)_c {\ensuremath{\overset{}{\longrightarrow}}} (M/G)_d$.
The coinvariant quotient forms an endofunctor on ${{\operatorname{\textbf{C}}}}$-modules. In this subsection we study the action of this functor on free ${{\operatorname{\textbf{C}}}}$-modules and demonstrate that they exhibit stability under its operation.
\[lem:coinvariants\_of\_inductions\] Let $V$ be any $G_c$-representation and $\operatorname{Ind}_c(V)$ the corresponding induction module. The $G_d$-coinvariants of $\operatorname{Ind}_c(V)_d$ are given by $$(\operatorname{Ind}_c(V)/G)_d \cong \begin{cases}
V/{G_c} & \text{if } c\leq d \\
0 & \text{otherwise}
\end{cases}$$ with all morphisms $c\leq d{\ensuremath{\overset{f}{\longrightarrow}}}d'$ inducing the identity map.
This again reflects a statement about ${{\operatorname{\textbf{C}}}}$-sets. Namely, that the set of orbits $G_d\backslash\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$ is either a singleton if $c\leq d$ or empty otherwise.
Recall that the coinvariant quotient of a $G$-representation $W$ can be defined as the tensor product $$(W)_G \cong {\mathbb{C}}\otimes_{G} W$$ where ${\mathbb{C}}$ denotes the trivial $G$-representation.
Using the associativity of tensor products, and the presentation of $\operatorname{Ind}_c(V)$ as one, we get $$\left(\operatorname{Ind}_c(V)/G\right)_d \cong {\mathbb{C}}\otimes_{G_d} {\mathbb{C}}[\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)] \otimes_{G_c} V \cong {\mathbb{C}}[G_d\backslash\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)] \otimes_{G_c} V$$
By hypothesis the $G_d$ action on $\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)$ is transitive. Therefore if $c\leq d$ then $\operatorname{Hom}(c,d)\neq {\varnothing}$ and this set forms a single orbit. Furthermore, a morphism $d{\ensuremath{\overset{f}{\longrightarrow}}}d'$ carries this single orbit corresponding to $d$ to the one corresponding to $d'$. In the case where there are no morphisms $c{\ensuremath{\overset{}{\longrightarrow}}}d$ we have the empty set. In other words we have $${\mathbb{C}}[G_d\backslash\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)] \cong \begin{cases}
{\mathbb{C}}& \text{if } c\leq d \\
0 & \text{otherwise}
\end{cases}$$ and a morphism $c\leq d{\ensuremath{\overset{f}{\longrightarrow}}}d'$ induces the identity map on ${\mathbb{C}}$. Tensoring with $V$ over $G_c$ we get $V/{G_c}$ when $c\leq d$, zero otherwise, and morphisms as stated.
Applying this result to direct sums of induction ${{\operatorname{\textbf{C}}}}$-modules, we can formulate what happens to free ${{\operatorname{\textbf{C}}}}$-modules when we take their coinvariants.
\[thm:coinvariant\_stabilization\] When the coinvariants functor is applied to any free module of degree $\leq c$, all maps induced by ${{\operatorname{\textbf{C}}}}$-morphisms are injections, and all maps induced by morphisms between objects $\geq c$ are isomorphisms.
Explicitly, the stable isomorphism type of the coinvariant quotient of a free ${{\operatorname{\textbf{C}}}}$-module $\oplus_{i} \operatorname{Ind}_{c_i}(V_i)$ is given by $$\label{eq:coinvariants_of_free}
\lim_{\bullet\rightarrow \infty}\left(\oplus_i \operatorname{Ind}_{c_i}(V_i)/G\right)_\bullet = \oplus_{i} V_i/G_{c_i}.$$
This translates to the following result regarding character polynomials.
\[cor:expectation\_stabilization\] If $P$ is a character polynomial of degree $\leq c$, then its $G_d$-expected number $$\mathbb{E}_{G_d}[P] := \frac{1}{|G_d|}\sum_{\sigma\in G_d} P(\sigma)$$ does not depend on $d$ for $d\geq c$.
Furthermore, if $P$ is the characters of free ${{\operatorname{\textbf{C}}}}$-modules then $\mathbb{E}_{G_d}[P]$ is a non-negative integer, monotonically increasing in $d$.
Recall that for a $G$-representation $V$ the expectation $$\frac{1}{|G|}\sum_{g\in G} \operatorname{Tr}(g) = \operatorname{Tr}\left(\frac{1}{|G|}\sum_{g\in G} g\right)$$ is the trace of the projection $V\twoheadrightarrow V^G$, whose existence also demonstrates that $V^G = V/G$. The expectation is thus $\dim_{\mathbb{C}}(V/G)$. In particular it is a non-negative integer.
Suppose $P$ is the character of the free ${{\operatorname{\textbf{C}}}}$-module $M_\bullet$ of degree $\leq c$. By Theorem \[thm:coinvariant\_stabilization\] the coinvariants $(M/G)_\bullet$ is a ${{\operatorname{\textbf{C}}}}$-module, all of whose induced maps are injections, and isomorphisms for objects $\geq c$. Thus the sequence of dimensions $\dim_{\mathbb{C}}(M/G)_d$ is monotonic in $d$ and becomes constant when $d\geq c$.
The general statement follows by linearity.
We are often interested in the $G$-inner product of characters: $$\langle \chi_1, \chi_2 \rangle_G = \frac{1}{|G|}\sum_{g\in G} \chi_1(g) \bar{\chi}_2(g) = \mathbb{E}_G[ \chi_1\cdot\bar{\chi}_2 ]$$ which is central to character theory. For character polynomials the previous corollary gives the following immediate stability statement.
If $P$ and $Q$ are character polynomials of respective degrees $\leq c_1$ and $\leq c_2$, then the $G_d$-inner products $$\langle P, Q\rangle_{G_d} = \frac{1}{|G_d|}\sum_{\sigma\in G_d} P(\sigma)\bar{Q}(\sigma)$$ does not depend on $d$ for all $d\geq c_1+c_2$.
Furthermore, if $P$ and $Q$ are the characters of free ${{\operatorname{\textbf{C}}}}$-modules then $\langle P,Q\rangle_{G_d}$ is a non-negative integer, monotonic in $d$.
The claim follows directly from the presentation $$\langle P, Q\rangle_{G_d} = \mathbb{E}_{G_d}[P\bar{Q}]$$ and Corollary \[cor:expectation\_stabilization\].
If $P$ and $Q$ are the characters of $M_\bullet$ and $N_\bullet$ then $P\bar{Q}$ is the character of the free ${{\operatorname{\textbf{C}}}}$-module $M\otimes N^*$. Integrality and monotonicity follow.
Noetherian property {#sec:noetherian}
===================
In this section we apply the theory developed in the previous sections to prove that the category of ${{\operatorname{\textbf{C}}}}$-modules is Noetherian. Our proof strategy follows the argument made by Gan-Li in [@GL-EI]. The main theorem of this section is the following.
\[thm:Noetherian\] Every ${{\operatorname{\textbf{C}}}}$-submodule of a finitely generated ${{\operatorname{\textbf{C}}}}$-module is itself finitely generated.
Theorem \[thm:Noetherian\] will be proved at the end of this section. First we need some preliminary results. We start with an extension result for equivariant homomorphisms between free ${{\operatorname{\textbf{C}}}}$-modules.
\[lem:extensions\] Let $M_\bullet$ be a free ${{\operatorname{\textbf{C}}}}$-module. There is a left-exact endofunctor on ${{\operatorname{\textbf{C}}}}$-modules $$N_\bullet \mapsto \operatorname{Hom}_{G_\bullet}(M_\bullet,N_\bullet)$$ whose image is contained in trivial ${{\operatorname{\textbf{C}}}}$-modules. The value of the module $\operatorname{Hom}_{G_\bullet}(M_\bullet,N_\bullet)$ at an object $d$ is the vector space $\operatorname{Hom}_{G_d}(M_d,N_d)$. In particular, for every $d\leq e$ there is a canonical map $$\operatorname{Hom}_{G_d}(M_d,N_d){\ensuremath{\overset{\Psi_d^e}{\longrightarrow}}} \operatorname{Hom}_{G_e}(M_e,N_e)$$ that promotes a $G_d$-linear map to a $G_e$-linear one.
Furthermore, if $N_\bullet$ is itself free, and the degrees of $M_\bullet$ and $N_\bullet$ are $\leq c_1$ and $\leq c_2$ respectively, then the extension map $\Psi_d^e$ is an isomorphism whenever $d\geq c_1+c_2$. In particular, equivariant morphisms extend uniquely in this range.
To get the proposed endofunctor we use dualization, tensor products and coinvariants: $$N_\bullet \mapsto \left(M^*\otimes N\right)_\bullet \mapsto \left[\left(M^*\otimes N\right)/G\right]_\bullet$$ This gives rise to an endofunctor whose value at $d$ is $$\left(M^*_d\otimes N_d\right)/G_d.$$ The tensor product is naturally isomorphic to $\operatorname{Hom}_{\mathbb{C}}(M_d,N_d)$ and averaging over $G_d$ gives a natural lift from coinvariants to invariants. Thus the value at $d$ is naturally isomorphic to $$\operatorname{Hom}_{{\mathbb{C}}}(M_d,N_d)^{G_d} = \operatorname{Hom}_{G_d}(M_d,N_d)$$ and indeed the desired functor exists. Left exactness follows from the general fact that the functor $\operatorname{Hom}(M,\bullet)$ is left exact.
Lastly, if $M_\bullet$ and $N_\bullet$ are free of respective degrees $\leq c_1$ and $\leq c_2$ then by Theorem \[thm:intro-free-modules\](1) $M^*\otimes N$ is free of degree $\leq c_1+c_2$. We then apply Theorem \[thm:coinvariant\_stabilization\] and see that its coinvariants stabilize for all $d\geq c_1+c_2$ in the sense that all induced maps $\Psi_d^e$ are isomorphisms.
When the range $N_\bullet$ is not free we cannot guarantee that the extension maps $\Psi_d^e$ be eventually isomorphisms. But in the case where the range is contained in a free module, we can at least salvage injectivity.
\[cor:injective\_extension\] If $M_\bullet$ and $N_\bullet$ are free ${{\operatorname{\textbf{C}}}}$-modules of respective degrees $\leq c_1$ and $\leq c_2$, and $X_\bullet\subseteq N_\bullet$ is any ${{\operatorname{\textbf{C}}}}$-submodule, then the extension maps $$\operatorname{Hom}_{G_d}(M_d,X_d) {\ensuremath{\overset{\Psi_d^e}{\longrightarrow}}} \operatorname{Hom}_{G_e}(M_e,X_e)$$ are injective for all $e\geq d\geq c_1+c_2$.
For every $e\geq d$ we have a commutative square of extensions [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{\operatorname{Hom}_{G_d}(M_d,X_d)} \arrow[r, "{X_d\hookrightarrow N_d}"] \arrow[d, swap, "{\Psi_d^e}"] \& {\operatorname{Hom}_{G_d}(M_d,N_d)} \arrow[d, "{\Psi_d^e}"] \\
{\operatorname{Hom}_{G_e}(M_e,X_e)} \arrow[r, swap, "{X_e\hookrightarrow N_e}"] \& {\operatorname{Hom}_{G_e}(M_e,N_e)}
\end{tikzcd}$$]{} and since $M$ and $N$ are free of the given degrees, it follows that the rightmost extension map is an isomorphism when $d\geq c_1+c_2$. Furthermore, the two horizontal maps are injective by left-exactness. Thus we have a square in which all but the leftmost map are injections. This implies that the leftmost map is injective as well.
We are now ready to prove that ${{{\normalfont\operatorname{Mod}}}_{{{\operatorname{\textbf{C}}}}}}$ has the Noetherian property.
Suppose that $M_\bullet$ is a finitely generated ${{\operatorname{\textbf{C}}}}$-module and $X^0_\bullet\subseteq X^1_\bullet \subseteq \ldots \subseteq M_\bullet$ is an ascending chain of submodules. We need to show that $X^N = X^{N+1}=\ldots$ for some $N\in {\mathbb{N}}$. As in the standard proofs of Hilbert’s Basis Theorem, we divide the task into two parts: controlling growth in all large degrees, then handling lower degrees using Noetherian property of finite direct sums.
We assume without loss of generality that $M_\bullet$ is a free, finitely-generated ${{\operatorname{\textbf{C}}}}$-module of degree $c$, as every finitely-generated ${{\operatorname{\textbf{C}}}}$-module is a quotient of a finite sum of such. For brevity we denote the functor $X_\bullet \mapsto \operatorname{Hom}_{G_\bullet}(M_\bullet, X_\bullet)$ by $F$, i.e. $$F(X)_d := \operatorname{Hom}_{G_d}(M_d, X_d).$$ Since $M_\bullet$ is free of degree $\leq c$, it follows that all induced extension maps $$F(M)_d{\ensuremath{\overset{\Psi_d^e}{\longrightarrow}}} F(M)_e$$ are isomorphisms when $d\geq c+c$. Fix an object $d\geq c+c$. We get a collection of subspaces inside $F(M)_d$ by considering the images $$\left\{F(X^n)_e \hookrightarrow F(M)_e {\ensuremath{\overset{(\Psi_d^e)^{-1}}{\longrightarrow}}} F(M)_{d} \right\}_{n\in {\mathbb{N}},\; e\geq d}$$ Since $F(M)_{d}$ is itself Noetherian (a finite dimension vector space), this collection of subspaces has a maximal element, say the image of $F(X^{N_0})_{e_0}$.
For all $n\geq N_0$ and $e\geq e_0$ we have $X^n_e = X^{N_0}_e$.
For every $n\geq N_0$ and objects $e \geq e_0$ we have a commutative diagram $$\xymatrix{
F(X^{N_0})_{e_0} \ar@{^{(}->}[r] \ar[d]_{\Psi_{e_o}^e} &
F(X^{n})_{e_0} \ar@{^{(}->}[r] \ar[d]_{\Psi_{e_o}^e} &
F(M)_{e_0} \ar[d]_{\Psi_{e_o}^e} \ar[dr]^{(\Psi_d^{e_o})^{-1}} & \\
F(X^{N_0})_{e} \ar@{^{(}->}[r] &
F(X^{n})_{e} \ar@{^{(}->}[r] &
F(M)_e \ar[r]^{(\Psi_d^e)^{-1}} &
F(M)_{d}
}$$ which by Corollary \[cor:injective\_extension\] all vertical extension maps are injective.
But we chose $F(X^{N_0})_{e_0}$ to be the subspace whose image inside $F(M)_{d}$ is maximal. It thus follows that all arrows in the above diagram are surjective. In particular, the injection $F(X^{N_0})_e \hookrightarrow F(X^n)_e$ is an isomorphism. Recalling the definition of $F$, we found that the inclusion $$\operatorname{Hom}_{G_e}(M_e, X^{N_0}_e) \hookrightarrow \operatorname{Hom}_{G_e}(M_e, X^{n}_e)$$ is an isomorphism, where $X^{N_0}_e\subseteq N^n_e\subseteq M_e$ are $G_e$-subrepresentations. By Mashke’s theorem, this happens precisely when $X^{N_0}_e = X^{n}_e$ thus proving the claim.
It remains to show that we can find some $N_1\geq N_0$ such that for all objects $e < e_0$ the term $X^{N_1}_e$ stabilized. Indeed, since ${{\operatorname{\textbf{C}}}}$ is of ${{{\normalfont\operatorname{FI}}}}$ type, there are only finitely many isomorphism classes of objects $\leq e_0$. Pick representatives for them $e_1,\ldots,e_n$ and consider the direct sum $$\oplus_{k=1}^n M_{e_k}.$$ Since each $M_{e_k}$ is Noetherian (a finite dimensional vector space), this direct sum is Noetherian as well. We can therefore find $N_1 \geq N_0$ for which the sum $$\oplus_{k=1}^n X^{N_1}_{e_k} \subseteq \oplus_{k=1}^n M_{e_k}$$ stabilized. Now for every $n \geq N_1$ and every object $e$ we have $X^{N_1}_e =X^n_e$ thus showing that $X^{N_1}_\bullet$ is a maximal element of our chain.
Making contact with related work, we remark that in [@GL-EI Theorem 1.1] Gan-Li list a set of combinatorial condition on categories of a certain type, which are sufficient for proving the Noetherian property [@GL-EI]. Their conditions are
- **Surjectivity**: The groups $G_d$ act transitively on incoming morphisms $c{\ensuremath{\overset{}{\longrightarrow}}}d$.
- **Bijectivity**: Some sequence of double-coset spaces $H_d \backslash G_d /H_d$ stabilizes as $d$ get sufficiently large.
These conditions are related to the present context as follows. First, the Surjectivity condition is incorporated into our definition of categories of ${{{\normalfont\operatorname{FI}}}}$ type. As for Bijectivity, it was explained to me by Kevin Casto that by choosing a compatible system of morphisms $c{\ensuremath{\overset{}{\longrightarrow}}}d$ for every pair $c\leq d$ one gets a natural isomorphism $$G_d \backslash \left(\operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)\times \operatorname{Hom}_{{\operatorname{\textbf{C}}}}(c,d)\right) \cong H_d \backslash G_d / H_d$$ where $H_d \backslash G_d / H_d$ is the double-coset space the appears in the Bijectivity condition. In this sense, the objects considered in this work are a coordinate-free interpretation of those the appeared in [@GL-EI]. Arguing in this coordinate-free manner allows us to consider categories whose objects are not linearly ordered, avoid having to find a compatible system of morphisms, and show that the bijectivity condition holds for all categories of ${{{\normalfont\operatorname{FI}}}}$ type. This is a direct result of Lemma \[lem:products\_set\_version\].
The following stabilization result is a central motivation for one to be interested in the Noetherian property. It shows that finitely-generated ${{\operatorname{\textbf{C}}}}$-modules exhibit the same representation stability phenomena as free ${{\operatorname{\textbf{C}}}}$-module, only without the explicit stable range.
If $M_\bullet$ is a finitely-generated ${{\operatorname{\textbf{C}}}}$-module, then all induced maps in the associated module of coinvariants are eventually isomorphisms. That is, there exists an upward-closed and cofinal set of objects $X$ such that if $c\in X$ and $d\geq c$, then the induced map $$M_c/G_c {\ensuremath{\overset{}{\longrightarrow}}} M_d/G_d$$ is an isomorphism.
More generally, if $F_\bullet$ is any free ${{\operatorname{\textbf{C}}}}$-module, then the coinvariants of $F\otimes M$ eventually stabilizes in the above sense. In particular, the spaces $\operatorname{Hom}_{G_c}(F_c,M_c)$ stabilize as well.
By the Noetherian property, a finitely-generated ${{\operatorname{\textbf{C}}}}$-module is finitely-presented, i.e. there exist free ${{\operatorname{\textbf{C}}}}$-modules $F^i_\bullet$ for $i=0,1$ and an exact sequence $$\xymatrix{
F^1 \ar[r] &
F^0 \ar[r] &
M \ar[r] &
0
}$$ Since the functor of coinvariants is right-exact we get a similar sequence of coinvariants. But by Theorem \[thm:coinvariant\_stabilization\] the coinvariants of a free ${{\operatorname{\textbf{C}}}}$-module stabilize in the desired sense. The Five-Lemma then implies that the same stabilization occurs for $M/G$.
For the more general statement, suppose $F_\bullet$ is some free ${{\operatorname{\textbf{C}}}}$-module. By the right-exactness of the tensor product it follows that $$\xymatrix{
F\otimes F^1 \ar[r] &
F\otimes F^0 \ar[r] &
F\otimes M \ar[r] &
0
}$$ is itself exact. Theorem \[thm:intro-free-modules\](1) shows that for $i=0,1$ the product $F\otimes F^i$ is free. Thus by the same reasoning as above stabilization follows.
Lastly, replacing $F$ with its dual $F^*$ (which is again free) and using the isomorphisms $$(F_c^*\otimes M_c )/G_c \cong \operatorname{Hom}(F_c,M_c)^{G_c} = \operatorname{Hom}_{G_c}(F_c,M_c)$$ we find that the spaces on the right-hand side stabilize as well.
Example: Representation stability for ${{{\normalfont\operatorname{FI}}}}^m$ {#sec:rep_theory_of_FI}
============================================================================
This section is devoted to the category ${{{\normalfont\operatorname{FI}}}}^m$, its free modules, and representation stability in this context. We also give an explicit description of ${{{\normalfont\operatorname{FI}}}}^m$-character polynomials in terms of cycle-counting functions. The results presented below generalize to the category $({{{\normalfont\operatorname{FI}}}}_G)^m$ and its representation, where $G$ is some finite group, using the technique presented in [@SS-FIG Theorem 3.1.3].
Recall that we denote the category of finite sets and injective functions by ${{{\normalfont\operatorname{FI}}}}$. Consider the categorical power ${{{\normalfont\operatorname{FI}}}}^m$, whose objects are ordered $m$-tuples of finite sets $\bar{n} = (n^{(1)},\ldots, n^{(m)})$, and whose morphisms $\bar{n}{\ensuremath{\overset{\bar{f}}{\longrightarrow}}}\bar{n}'$ are ordered $m$-tuples of injections $\bar{f} = (f^{(1)},\ldots, f^{(m)})$ where $n^{(i)}{\ensuremath{\overset{f^{(i)}}{\longrightarrow}}}n'^{(i)}$. In everything that follows we denote the ${{{\normalfont\operatorname{FI}}}}^m$ analog of notions from ${{{\normalfont\operatorname{FI}}}}$ by an over-line. The ordering on objects in ${{{\normalfont\operatorname{FI}}}}^m$ is the following: $\bar{n}\leq \bar{n}'$ if and only if for every $1\leq i\leq m$ there is an inequality of sizes $|n^{(i)}|\leq |n'^{(i)}|$. The group of automorphisms of an object $\bar{n}$ is the product of symmetric groups $S_{n^{(1)}}\times\ldots\times S_{n^{(m)}}$, which we will denote by $S_{\bar{n}}$.
Many natural sequences of spaces and varieties are naturally parameterized by ${{{\normalfont\operatorname{FI}}}}^m$. For example, fix some space $X$ and consider the following generalization of the configurations spaces $$\operatorname{PConf}^{(n_1,\ldots,n_m)}(X) := \{ [(x_1^{(1)},\ldots,x_{n_1}^{(1)}),\ldots,(x_1^{(m)},\ldots,x_{n_m}^{(m)})] \mid \forall i\neq j (x^{(i)}_{k_i}\neq x^{(j)}_{k_j}) \}$$ inside the product $X^{n_1}\times \ldots \times X^{n_m}$. Every inclusion $\bar{n}\hookrightarrow \bar{n}'$ induces a continuous map by forgetting coordinates, so this is naturally a contravariant ${{{\normalfont\operatorname{FI}}}}^m$-diagram of spaces. Applying a cohomology functor to this diagram of spaces yields an ${{{\normalfont\operatorname{FI}}}}^m$-module. The special case of based rational maps ${\mathbb{P}}^1{\ensuremath{\overset{}{\longrightarrow}}} {\mathbb{P}}^{m-1}$ was described in the introduction, to which theory below applies and gives Corollary \[thm:intro-homological-stability\].
The category ${{{\normalfont\operatorname{FI}}}}^m$ fits in with our general framework, as the following demonstrates.
${{{\normalfont\operatorname{FI}}}}^m$ is a locally finite category of ${{{\normalfont\operatorname{FI}}}}$ type. Pullbacks and weak push-outs are given by the corresponding operations in ${{{\normalfont\operatorname{FI}}}}$ applied coordinatewise.
First we consider the case $m=1$, i.e. we need to show that ${{{\normalfont\operatorname{FI}}}}$ is indeed of ${{{\normalfont\operatorname{FI}}}}$ type. ${{{\normalfont\operatorname{FI}}}}$ is a subcategory of the category of finite sets, which has pullbacks and push-outs. The ${{{\normalfont\operatorname{Set}}}}$-pullback of two injections itself has injective structure maps, and is thus naturally a pullback in ${{{\normalfont\operatorname{FI}}}}$. Regarding weak push-outs, note that if [$$\begin{tikzcd}[sep=2.5em, ampersand replacement=\&]
{p} \arrow[r, "{f_1}"] \arrow[d, swap, "{f_2}"] \& {c_1} \arrow[d, "{g_2}"] \\
{c_2} \arrow[r, swap, "{g_2}"] \& {d}
\end{tikzcd}$$]{} is a pullback diagram in ${{{\normalfont\operatorname{Set}}}}$, then the images of $g_1$ and $g_2$ intersect precisely in the image of the composition $g_1\circ f_1 = g_2\circ f_2$. Thus if all four maps are injections, the universal function from the ${{{\normalfont\operatorname{Set}}}}$ push-out $c_1\cup_p c_2$ into $d$ is injective. We therefore see that the ${{{\normalfont\operatorname{Set}}}}$ push-out is a weak push-out in ${{{\normalfont\operatorname{FI}}}}$. The other axioms of ${{{\normalfont\operatorname{FI}}}}$ type are clear.
The case $m>1$ follows easily from the previous paragraph when pullbacks and weak push-outs are computed coordinatewise.
We turn to the decomposition into irreducible subrepresentations. First we recall some of the terminology related to the case $m=1$.
Recall that a *partition* of a natural number $n$ is a sequence $\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_k)$ such that $\sum_{i=1}^k \lambda_i = n$. In this case we write $\lambda \vdash n$ and refer to $n$ as the *degree* of $\lambda$. This degree will be denoted by $|\lambda|$.
For every other natural number $d\geq |\lambda|+\lambda_1$ we define the *padded partition* $$\lambda(d) = (d-|\lambda|\geq \lambda_1 \ldots\geq \lambda_k) \vdash d$$ By deleting the largest part of a partition, we see that every partition of $d$ is of the form $\lambda(d)$ for some partition $\lambda\vdash n < d$.
Recall that the partitions on $d$ are in one-to-one correspondence with the irreducible representations of $S_d$. Denote the corresponding irreducible representation by $V_{\lambda(d)}$.
To consider the case $m>1$ recall that the irreducible representations of a product of finite groups $G\times H$ are given exactly by the pairs $V\boxtimes W$ where $V$ and $W$ are irreducible representations of $G$ and $H$ respectively. The symbol $\boxtimes$ is the usual tensor product on the underlying vector spaces $V$ and $W$ and the action of $G\times H$ on this product is defined by $(g,h).(v\otimes w) = g(v)\otimes h(w)$.
The irreducible representations of $S_{\bar{n}}=S_{n^{(1)}}\times\ldots\times S_{n^{(m)}}$ are precisely external tensor products of the form $$V_{\bar{\lambda}(\bar{n})} := V_{\lambda^{(1)}(n_1)} \boxtimes \ldots \boxtimes V_{\lambda^{(m)}(n_m)}$$ where $|\lambda^{(i)}|+\lambda^{(i)}_1 \leq n^{(i)}$ for every $1\leq i\leq m$. Furthermore the character of such a product is given by the product of the individual characters.
Following this observation we define a $\boxtimes$ operation on ${{{\normalfont\operatorname{FI}}}}$-modules.
Let $(M^{(1)},\ldots, M^{(m)})$ be an $m$-tuple of ${{{\normalfont\operatorname{FI}}}}$-modules. We define their external tensor product to be the ${{{\normalfont\operatorname{FI}}}}^m$-module $$\bar{M} = M^{(1)}\boxtimes \ldots \boxtimes M^{(m)}$$ by composing the functor $(M^{(1)},\ldots, M^{(m)}): {{{\normalfont\operatorname{FI}}}}^m {\ensuremath{\overset{}{\longrightarrow}}} ({{{\normalfont\operatorname{Mod}}}_{R}})^m$ with the $m$-fold tensor product functor on $R$-modules.
We then see that if $\bar{n}$ is any object, then the $S_{\bar{n}}$-representation $\bar{M}_{\bar{n}}$ is precisely the external tensor product $M^{(1)}_{n^{(1)}}\boxtimes \ldots \boxtimes M^{(m)}_{n^{(m)}}$. Consequently, the character of $\bar{M}$ is the product of the ${{{\normalfont\operatorname{FI}}}}$-characters of the factors.
\[rem:tensor\_commutes\_with\_induction\] It is also important to note that the external tensor operation commutes with the $\operatorname{Ind}$ functors in the following sense: $$\operatorname{Ind}_{\bar{n}}(V^{(1)}\boxtimes \ldots \boxtimes V^{(m)}) \cong \operatorname{Ind}_{n^{(1)}}(V^{(1)})\boxtimes \ldots \boxtimes \operatorname{Ind}_{n^{(m)}}(V^{(m)}).$$ This can be verified e.g. by considering the definition of $\operatorname{Ind}$ in Definition \[def:Ind\_functor\], and using the associativity and commutativity of the tensor product.
\[thm:FIm\_to\_FI\] The following relationships hold between the representation theory of ${{{\normalfont\operatorname{FI}}}}^m$ and that of ${{{\normalfont\operatorname{FI}}}}$.
1. Every free ${{{\normalfont\operatorname{FI}}}}^m$ module of degree $\leq \bar{n}$ is the direct sum of external tensor products of free ${{{\normalfont\operatorname{FI}}}}$-modules, where the $i$-th component is of degree $\leq n^{(i)}$.
2. Every ${{{\normalfont\operatorname{FI}}}}^m$-character polynomial of degree $\leq \bar{n}$ decomposes as a sum of products of ${{{\normalfont\operatorname{FI}}}}$-character polynomials, where the $i$-th factor has degree ${\leq n^{(i)}}$.
In most related work on the representation theory of the category ${{{\normalfont\operatorname{FI}}}}$, free modules are called projective or ${{{\normalfont\operatorname{FI}}}}\#$-modules. See [@CEF] for the relevant definitions and a proof that these concepts are equivalent.
We start with the first assertion. Let $\bar{\lambda} = (\lambda^{(1)},\ldots,\lambda^{(m)})$ be an $m$-tuple of partitions and $\bar{n}$ some $m$-tuple of natural numbers satisfying $n^{(i)} \geq |\lambda^{(i)}|+\lambda^{(i)}_1$ for all $i=1,\ldots, m$. We apply the fact that $\operatorname{Ind}$ commutes with external tensor products to the irreducible $S_{\bar{n}}$-representation $$V_{\bar{\lambda}(\bar{n})} = V_{\lambda^{(1)}(n^{(1)})} \boxtimes \ldots \boxtimes V_{\lambda^{(m)}(n^{(m)})}.$$ This gives a presentation $$\operatorname{Ind}_{\bar{n}}(V_{\bar{\lambda}}(\bar{n})) \cong \operatorname{Ind}_{n^{(1)}}(V_{\lambda^{(1)}(n^{(1)})})\boxtimes \ldots \boxtimes \operatorname{Ind}_{n^{(m)}}(V_{\lambda^{(m)}(n^{(m)})})$$ which proves the first assertion of the theorem for $\operatorname{Ind}_{\bar{n}}(V)$ when $V$ is irreducible.
For a general $S_{\bar{n}}$-representation $V$, decompose $V$ into irreducible subrepresentations $V = V_1\oplus \ldots \oplus V_r$. Since $\operatorname{Ind}$ commutes with direct sums, the induction module $\operatorname{Ind}_{\bar{n}}(V)$ is a direct sum of external tensor products of induction ${{{\normalfont\operatorname{FI}}}}$-modules.
Lastly, the assertion applies to all free ${{{\normalfont\operatorname{FI}}}}^m$-modules, since they are directs sum of induction modules of the form previously considered.
As for the second assertion, a character polynomials of degree $\leq \bar{n}$ is a $k$-linear combination of the characters of free ${{{\normalfont\operatorname{FI}}}}^m$-modules of degree $\leq \bar{n}$. By the first statement such a free module is the sum of external tensor products of free ${{{\normalfont\operatorname{FI}}}}$-modules with the appropriate bounds on their degrees. But the character of an external tensor product is the product of the individual characters, which in the case of products of free ${{{\normalfont\operatorname{FI}}}}$-modules are by definition ${{{\normalfont\operatorname{FI}}}}$-character polynomials. Thus every ${{{\normalfont\operatorname{FI}}}}^m$-character polynomial is indeed a $k$-linear combination of products of ${{{\normalfont\operatorname{FI}}}}$-character polynomials with the appropriate bound on degree.
Theorem \[thm:FIm\_to\_FI\] allows us to give an explicit description of the character polynomials of ${{{\normalfont\operatorname{FI}}}}^m$ is terms of cycle counting functions.
For every natural number $k$, let $X_k: \coprod_{n} S_n {\ensuremath{\overset{}{\longrightarrow}}} {\mathbb{N}}$ be the simultaneous class function on the symmetric groups $$X_k(\sigma)= \# \text{ of $k$-cycles appearing in $\sigma$}.$$
On the products $S_{n^{(1)}}\times \ldots \times S_{n^{(m)}}$ we define a similar function $X_k^{(i)}$ by $$X_k^{(i)}(\sigma^{(1)},\ldots, \sigma^{(m)})= \# \text{ of $k$-cycles appearing in $\sigma^{(i)}$}.$$
The study of polynomials in the class functions $X_k$ dates back to Frobenius, and they are what is classically known as *character polynomials*. The following proposition shows that our definition of character polynomials generalizes this classical idea.
The filtered ${\mathbb{C}}$-algebra of character polynomials of ${{{\normalfont\operatorname{FI}}}}^m$ coincides with the polynomial ring $$R = {\mathbb{C}}[X_1^{(1)},\ldots, X_1^{(m)}, X_2^{(1)},\ldots,X_2^{(m)},\ldots].$$ where we define $\deg(X_k^{(i)})=(0,\ldots,k,\ldots,0)=k \bar{e}^{(i)}$.
We first prove this when $m=1$. For the inclusion $R\subseteq \operatorname{Char}_{{{{\normalfont\operatorname{FI}}}}}$ we show that for every $k$ the function $X_k$ is indeed a character polynomial. Recall that in Example \[ex:FI\_characters\] we showed that for every cycle type $\mu = (\mu_1,\ldots,\mu_k)$ the associated character polynomial satisfies $$\binom{X}{\mu}(\sigma) = \binom{X_1(\sigma)}{\mu_1}\ldots \binom{X_k(\sigma)}{\mu_k}.$$ Thus by taking $\mu_k=1$ and $\mu_j=0$ for all $j\neq k$ we get a character polynomial $$\binom{X}{\mu}(\sigma) = \binom{X_k(\sigma)}{1} = X_k(\sigma).$$
For the reverse inclusion, one can construct the right-hand side of $$\binom{X}{\mu} = \binom{X_1}{\mu_1}\ldots \binom{X_k}{\mu_k}$$ in the algebra generated by $X_1,X_2,\ldots$, thus realizing every generator $\binom{X}{\mu}$ of $\operatorname{Char}_{{{{\normalfont\operatorname{FI}}}}}$. This concludes the proof in the case $m=1$.
For $m>1$, Theorem \[thm:FIm\_to\_FI\] states that every ${{{\normalfont\operatorname{FI}}}}^m$-character polynomial is a linear combination of external products of ${{{\normalfont\operatorname{FI}}}}$-character polynomials. We saw that the latter class of functions is precisely the ring of polynomials in $X_1,X_2,\ldots$. The function $X_k^{(i)}$ is the external product of $X_k$ in the $i$-th coordinate with $1$’s in all other coordinates, and thus polynomials in $X_k^{(i)}$ clearly generate all linear combinations of external products of $X_1,X_2,\ldots$. This proves the claim.
Our general theory of stabilization for inner products thus applies to expressions involving the functions $X_k^{(i)}$.
\[cor:FI\_product\_stability\] The $S_{\bar{n}}$-inner product of two polynomials $P,Q\in {\mathbb{C}}[X^{(i)}_k: k\in {\mathbb{N}},\, 1\leq i\leq m]$ does not depend on $\bar{n}$ for all $\bar{n}\geq \deg(P)+\deg(Q)$, where the degree of $X^{(i)}_k$ is $k \bar{e}^{(i)}$ and addition of degrees is defined coordinatewise.
In the case ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}$ this result is proved in [@CEF-pointcounts Theorem 3.9] via a direct calculation of the $S_n$-inner products. The ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}_{{\mathbb{Z}}/2{\mathbb{Z}}}$-analog is proved in [@Wi]. When $G$ is any other finite group, a non-effective analog of Corollary \[cor:FI\_product\_stability\] for ${{\operatorname{\textbf{C}}}}={{{\normalfont\operatorname{FI}}}}_G$ is implicit in [@SS-FIG Theorem 3.2.2].
We turn to discussing representation stability for ${{{\normalfont\operatorname{FI}}}}^m$. First consider the case $m=1$: the irreducible representations of symmetric groups of different orders are naturally related in the following sense.
For every partition $\lambda \vdash |\lambda|$, there exists an ${{{\normalfont\operatorname{FI}}}}$-submodule of $\operatorname{Ind}_{|\lambda|}(V_{\lambda})$, which we will denote by $V_{\lambda(\bullet)}$, whose value at every $d\geq |\lambda|+\lambda_1$ is isomorphic to the irreducible $S_d$-representation $V_{\lambda(d)}$. Moreover, for every partition $\lambda$ there exists a character polynomial $P_\lambda$ of degree $|\lambda|$ such that the character of $V_{\lambda(\bullet)}$ coincides with $P_\lambda$ on $S_d$ for all $d \geq |\lambda|+\lambda_1$.
See [@CEF] for the existence of $V_{\lambda(\bullet)}$ and [@Mac Example I.7.14] for $P_{\lambda}$.
This fact extends to all $m>1$ via the external tensor product.
For every $m$-tuple of partitions $\bar{\lambda} = (\lambda^{(1)},\ldots,\lambda^{(m)})$ there exists an ${{{\normalfont\operatorname{FI}}}}^m$-submodule of $\operatorname{Ind}_{|\bar{\lambda}|}(V_{\bar{\lambda}})$, which we will denote by $V_{\bar{\lambda}(\bullet)}$, whose value at $\bar{n}$ is the $S_{\bar{n}}$-irreducible representation $$V_{\bar{\lambda}(\bar{n})}:=V_{\lambda^{(1)}(n^{(1)})} \boxtimes \ldots \boxtimes V_{\lambda^{(m)}(n^{(m)})}$$ for all $\bar{n}\geq |\bar{\lambda}|+\bar{\lambda}_1$. Here $|\bar{\lambda}|$ is the $m$-tuple $(|\lambda^{(1)}|,\ldots, |\lambda^{(m)}|)$, the expression $\bar{\lambda}_1$ is $(\lambda^{(1)}_1,\ldots,\lambda^{(m)}_1)$ and $+$ coincides with coordinatewise addition.
Moreover, the character of $V_{\bar{\lambda}(\bullet)}$ coincides with the character polynomial ${P_{\bar{\lambda}}:=P_{\lambda^{(1)}}\cdot \ldots \cdot P_{\lambda^{(m)}}}$ of degree $|\bar{\lambda}|$.
These sequences of irreducible representations allow us to formulate the notion of representation stability for free ${{{\normalfont\operatorname{FI}}}}^m$-modules.
\[thm:rep\_stability\] Suppose $F_\bullet$ is a free ${{{\normalfont\operatorname{FI}}}}^m$-module that is finitely-generated in degree $\leq \bar{n}$. Then there exist $m$-tuples of partitions $\bar{\lambda}_1,\ldots, \bar{\lambda}_k$, satisfying $|\bar{\lambda}_j|\leq \bar{n}$ for all $j=1,\ldots,k$, such that for all $\bar{d}\geq 2\times \bar{n} = (2n^{(1)},\ldots,2n^{(m)})$ the $S_{\bar{d}}$-module $F_{\bar{d}}$ decomposes into irreducibles as $$F_{\bar{d}} \cong (V_{\bar{\lambda}_1(\bar{d})})^{r_1}\oplus \ldots \oplus (V_{\bar{\lambda}_k(\bar{d})})^{r_k}$$ and the multiplicities $r_1,\ldots,r_k$ do not depend on $\bar{d}$.
The original definition of Representation Stability given in [@CF] includes additional injectivity and surjectivity conditions on top of the stabilization of irreducible decompositions. We will not discuss these aspects of the definition, although the reader familiar with them will readily notice that they are immediately satisfied by all free ${{\operatorname{\textbf{C}}}}$-modules.
\[Proof of Theorem \[thm:rep\_stability\]\] The case $m=1$ asserts the representation stability of free finitely-generated ${{{\normalfont\operatorname{FI}}}}$-modules. This follows directly from the Branching rule for inducing representations of the symmetric group (see [@FH]), and is proved in [@CEF Theorem 1.13].
For $m>1$ the statement follows from Theorem \[thm:FIm\_to\_FI\] using the corresponding statement in the case $m=1$. Since every free ${{{\normalfont\operatorname{FI}}}}^m$-module $M_\bullet$ is a sum of the external tensor products of free ${{{\normalfont\operatorname{FI}}}}$-modules, and each of those decomposes as a stabilizing direct sum of irreducibles, the same is true for $M_\bullet$.
At the level of character polynomials Theorem \[thm:rep\_stability\] translates to the following orthonormality statement.
\[cor:orthonormal\_basis\] The character polynomials $$\left\{ P_{\bar{\lambda}}:=P_{\lambda_1}\cdot \ldots \cdot P_{\lambda_m} \right\}_{|\bar{\lambda}|\leq \bar{n}}$$ form an orthonormal basis for all ${{{\normalfont\operatorname{FI}}}}^m$-character polynomials of degree $\leq \bar{n}$ with respect to the inner product $$\langle P, Q\rangle = \lim_{\bullet\rightarrow \infty} \langle P, Q\rangle_{\bullet} = \langle P, Q\rangle_{\deg(P)+\deg(Q)}.$$
[15]{}
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[^1]: These are commonly called \#-modules in the context of the category ${{{\normalfont\operatorname{FI}}}}$.
[^2]: See [@Church-manifolds].
[^3]: See [@CEF-pointcounts].
[^4]: Finiteness is not strictly necessary for many of the definitions and subsequent results. The author will be very interested to see how far one can push this theory with infinite automorphism groups
[^5]: Pushouts are not quite what we want here, as these typically do not exists when one insists that all morphisms be injective. We replace this notion by *weak pushouts*, defined below.
|
---
abstract: 'We compute expectation values of spatial Wilson loops in the forward light cone of high-energy collisions. We consider ensembles of gauge field configurations generated from a classical Gaussian effective action as well as solutions of high-energy renormalization group evolution with fixed and running coupling. The initial fields correspond to a color field condensate exhibiting domain-like structure over distance scales of order the saturation scale. At later times universal scaling emerges at large distances for all ensembles, with a nontrivial critical exponent. Finally, we compare the results for the Wilson loop to the two-point correlator of magnetic fields.'
author:
- 'A. Dumitru'
- 'T. Lappi'
- 'Y. Nara'
bibliography:
- 'spires.bib'
title: Structure of longitudinal chromomagnetic fields in high energy collisions
---
Introduction
============
Heavy ion collisions at high energies involve non-linear dynamics of strong QCD color fields [@Mueller:1999wm]. These soft fields correspond to gluons with light-cone momentum fractions $x\ll1$, which can be described in the “Color Glass Condensate” (CGC) framework. Because of the high gluon occupation number the gluon field can be determined from the classical Yang-Mills equations with a static current on the light cone [@McLerran:1994ni; @*McLerran:1994ka; @*McLerran:1994vd]. It consists of gluons with a transverse momentum on the order of the density of valence charges per unit transverse area, ${Q_\mathrm{s}}^2$ [@JalilianMarian:1996xn]. Parametrically, the saturation momentum scale ${Q_\mathrm{s}}$ separates the regime of non-linear color field interactions from the perturbative (linear) regime. It is commonly defined using a two-point function of electric Wilson lines, the “dipole scattering amplitude” evaluated in the field of a single hadron or nucleus [@Kovchegov:1998bi] as described below.
Before the collision the individual fields of projectile and target are two dimensional pure gauges; in light cone gauge, \[eq:alphai\] \^i\_m = V\_m \^i V\_m\^where $m=1,\, 2$ labels the projectile and target, respectively. Here $V_m$ are light-like SU(${{N_\mathrm{c}}}$) Wilson lines, which correspond to the eikonal phase of a high energy projectile passing through the classical field shockwave [@Balitsky:1995ub; @Buchmuller:1995mr].
The field in the forward light cone after the collision up to the formation of a thermalized plasma is commonly called the “glasma” [@Lappi:2006fp]. Immediately after the collision longitudinal chromo-electric and magnetic fields $E_z,~B_z\sim 1/g$ dominate [@Kharzeev:2001ev; @*Fries:2006pv; @Lappi:2006fp]. They fluctuate according to the random local color charge densities of the valence sources. The magnitude of the color charge fluctuations is related to the saturation scale ${Q_\mathrm{s}}^2$. The transverse gauge potential at proper time $\tau\equiv\sqrt{t^2-z^2}\to 0$, is given by [@Kovner:1995ja] \[eq:alpha1+alpha2\] A\^i=\_1\^i + \_2\^i . Note that while the fields of the individual projectiles $\alpha_m^i$ are pure gauges, for a non-Abelian gauge theory $A^i$ is not. Hence, spatial Wilson loops evaluated in the field $A^i$ are not equal to 1. The field at later times is then obtained from the classical Yang-Mills equations of motion, which can be solved either analytically in an expansion in the field strength [@Kovner:1995ja; @Blaizot:2008yb] or numerically on a lattice [@Krasnitz:1998ns; @Krasnitz:2001qu; @*Krasnitz:2003jw; @*Lappi:2003bi; @Lappi:2007ku]. The Wilson loop, and the magnetic field correlator, provide an explicitly gauge-invariant method to study the nonperturbative dynamics of these fields, complementary to studies of the gluon spectrum [@Lappi:2011ju].
Spatial Wilson loops at very early times $\tau$ have recently been studied numerically in Ref. [@Dumitru:2013koh], using the MV model [@McLerran:1994ni; @*McLerran:1994ka; @*McLerran:1994vd] for the colliding color charge sheets. It was observed that the loops effectively satisfy area law scaling for radii $\gsim
1/{Q_\mathrm{s}}$, up to a few times this scale. Furthermore, Ref. [@Dumitru:2013wca] found that two-point correlators of $B_z$ over distances $\lesssim 1/{Q_\mathrm{s}}$ correspond to two dimensional screened propagators with a magnetic screening mass a few times ${Q_\mathrm{s}}$. This indicates that the initial fields exhibit [*structure*]{} such that magnetic flux does not spread uniformly over the transverse plane (like in a Coulomb phase) but instead is concentrated in small domains.
The present paper extends this previous work as follows. We perform lattice measurements of spatial Wilson loops over a much broader range of radii to analyze their behavior at short ($R\ll1/{Q_\mathrm{s}}$) and long ($R\gg1/{Q_\mathrm{s}}$) distances. We also implement the so-called JIMWLK [@JalilianMarian:1996xn; @Jalilian-Marian:1997jx; @*Jalilian-Marian:1997gr; @*Jalilian-Marian:1997dw; @*JalilianMarian:1998cb; @*Iancu:2000hn; @*Iancu:2001md; @*Ferreiro:2001qy; @*Iancu:2001ad; @Weigert:2000gi; @*Mueller:2001uk] high-energy functional renormalization group evolution which resums observables to all orders in ${\alpha_{\mathrm{s}}}\log(1/x)$. High-energy evolution modifies the classical ensemble of gauge field configurations [(\[eq:V\_rho\])]{}, [(\[eq:S2\])]{} to account for nearly boost invariant quantum fluctuations at rapidities far from the sources. Finally, we also solve the Yang-Mills equations in the forward light cone to study the time evolution of magnetic flux loops.
The calculation of the initial conditions and the numerical solution of the classical boost-invariant[^1] Yang-Mills fields in the initial stages of a heavy ion collision have been documented in the references given below, so here we will only describe them very briefly in Sec. \[sec:numerics\] before moving on to show our results in Secs. \[sec:wloop\] and \[sec:bbcorr\].
Lattice implementation {#sec:numerics}
======================
We work on a two dimensional square lattice of $N_\perp^2$ points with periodic boundary conditions and consider color sources that fill the whole transverse plane. The lattice spacing is denoted as $a$, thus the area of the lattice in physical units is $L^2=N_\perp^2a^2$. The calculations are performed for ${{N_\mathrm{c}}}=3$ colors. In this work we only consider symmetric collisions, where the color charges of both colliding nuclei are taken from the same probability distribution.
In this work we compare three different initial conditions for the classical Yang-Mills equations: the classical MV model [(\[eq:S2\])]{} as well as fixed and running coupling JIMWLK evolution. We define the saturation scale ${Q_\mathrm{s}}(Y)$ at rapidity $Y$ through the expectation value of the dipole operator as $$\label{eq:defqs}
\frac{1}{{{N_\mathrm{c}}}} \left\langle {\, \mathrm{Tr} \, }V^\dag({{\mathbf{x}_T}}) V({{\mathbf{y}_T}})
\right\rangle_{Y,|{{\mathbf{x}_T}}-{{\mathbf{y}_T}}|=\sqrt{2}/{Q_\mathrm{s}}}
= e^{-1/2}.$$ Throughout this paper we shall use ${Q_\mathrm{s}}$ defined in this way from the light-like Wilson lines $V({{\mathbf{x}_T}})$ in the fundamental representation. The saturation scale is the only scale in the problem and we attempt to construct the various initial conditions in such a way that the value of ${Q_\mathrm{s}}a$ is similar, to ensure a similar dependence on discretization effects.
In the MV model the Wilson lines are obtained from a classical color charge density $\rho$ as \[eq:V\_rho\] V([[\_T]{}]{}) = { i [ ]{}x\^- g\^2 \^a([[\_T]{}]{},x\^-) }, where $\mathbb{P}$ denotes path-ordering in $x^-$. The color charge density is a random variable with a local Gaussian probability distribution \[eq:S2\] P\[\^a\] \~ {- [ ]{}\^2 [[\_T]{}]{}[ ]{}x\^- }, The total color charge $\int {\, \mathrm{d}}x^- \mu^2(x^-) \sim {Q_\mathrm{s}}^2$ is proportional to the thickness of a given nucleus.
In the numerical calculation the MV model initial conditions have been constructed as described in Ref. [@Lappi:2007ku], discretizing the longitudinal coordinate $Y$ in $N_y=100$ steps. For the calculations using the MV model directly for the initial conditions [(\[eq:alphai\])]{}, [(\[eq:alpha1+alpha2\])]{} we have performed simulations on lattices of two different sizes: $N_\perp=1024$, with the MV model color charge parameter $g^2
\mu L= 156$ which translates into ${Q_\mathrm{s}}a =0.119$; and with $N_\perp=2048$, using $g^2 \mu L= 550$, which results in ${Q_\mathrm{s}}a =
0.172$.
The MV model also provides the configurations used as the initial condition for quantum evolution in rapidity via the JIMWLK renormalization group equation, starting at $Y=\log
x_0/x=0$. Performing a step $\Delta Y$ in rapidity opens phase space for radiation of additional gluons which modify the classical action [(\[eq:V\_rho\])]{}, [(\[eq:S2\])]{}. This process can be expressed as a “random walk” in the space of light-like Wilson lines $V({{\mathbf{x}_T}})$ [@Weigert:2000gi; @*Mueller:2001uk; @Blaizot:2002xy; @Lappi:2012vw]: &&\_Y V([[\_T]{}]{}) = V([[\_T]{}]{}) [ ]{}\^2[[\_T]{}]{}\
&& - [ ]{}\^2[[\_T]{}]{} V([[\_T]{}]{}) V\^([[\_T]{}]{}) V([[\_T]{}]{}), \[eq:Lgvn\] where the Gaussian white noise $\eta^i =\eta^i_a t^a$ satisfies $\langle \eta^a_i({{\mathbf{x}_T}})\rangle =0$ and, for fixed coupling, \[eq:etaeta\] \^a\_i([[\_T]{}]{}) \^b\_j([[\_T]{}]{})= [\_]{}\^[ab]{} \_[ij]{}\^[(2)]{}([[\_T]{}]{}-[[\_T]{}]{}). Here the equation is written in the left-right symmetric form introduced in [@Kovner:2005jc; @Lappi:2012vw].
The fixed coupling JIMWLK equation is solved using the numerical method developed in [@Blaizot:2002xy; @Rummukainen:2003ns; @Lappi:2012vw]. For the smaller lattice size $N_\perp=1024$ we start with the MV model with $g^2\mu L = 31$ and without a mass regulator, which corresponds to an initial ${Q_\mathrm{s}}a = 0.0218$. After $\Delta y = 1.68/{\alpha_{\mathrm{s}}}$ units of evolution in rapidity[^2] this results in ${Q_\mathrm{s}}a = 0.145$. For a $N_\perp=2048$-lattice we again start with $g^2\mu L = 31$, corresponding to ${Q_\mathrm{s}}a = 0.0107$, and after $\Delta y = 1.8/{\alpha_{\mathrm{s}}}$ units of evolution end up with ${Q_\mathrm{s}}a = 0.141$.
For running coupling the evolution is significantly slower. We use the running coupling prescription introduced in [@Lappi:2012vw], where the scale of the coupling is taken as the momentum conjugate to the distance in the noise correlator in [Eq. ]{}[(\[eq:etaeta\])]{}. For the smaller $N_\perp=1024$ lattice we again start with $g^2\mu L =
31$, i.e. ${Q_\mathrm{s}}a = 0.0218$ and evolve for $\Delta Y = 10$ units in rapidity, arriving at ${Q_\mathrm{s}}a = 0.118$. For the larger $N_\perp=2048$ lattice we test a configuration that is farther from the IR cutoff, starting the JIMWLK evolution with $g^2\mu L = 102.4$, i.e. ${Q_\mathrm{s}}a =
0.0423$ and evolve for $\Delta Y = 10$ units in rapidity, arriving at ${Q_\mathrm{s}}a = 0.172$. In the rc-JIMWLK simulations the QCD scale is taken as ${\Lambda_{\mathrm{QCD}}}a = 0.00293$ and the coupling is frozen to a value $\alpha_0 = 0.76$ in the infrared below $2.5{\Lambda_{\mathrm{QCD}}}$.
As already mentioned above, RG evolution in rapidity resums quantum corrections to the fields $\alpha_m^\mu$ of the individual charge sheets to all orders in $\alpha_s \log 1/x$, with leading logarithmic accuracy. In other words, the effective action at $Y$ is modified from that at $Y=0$, written in [Eq. ]{}[(\[eq:S2\])]{}.
Once an ensemble of Wilson lines $V({{\mathbf{x}_T}})$ at a rapidity $Y$ is constructed, separately for both projectile and target, these configurations define $\alpha_1^i$ and $\alpha_2^i$ in light-cone gauge as written in [Eq. ]{}[(\[eq:alphai\])]{}; the initial field $A^i$ of produced soft gluons at proper time $\tau=+0$ corresponds to their sum, [Eq. ]{}[(\[eq:alpha1+alpha2\])]{}. The evolution to $\tau>0$ follows from the real-time Hamiltonian evolution described in Ref. [@Krasnitz:1998ns]. This has been used in many classical field calculations, e.g. in Refs. [@Krasnitz:2001qu; @*Krasnitz:2003jw; @*Lappi:2003bi], or more recently for the first study of the effects of JIMWLK-evolution on the gluon spectrum [@Lappi:2011ju], and in the IP-glasma model for the initial conditions for hydrodynamics [@Schenke:2012hg]. On the $N_\perp=2048$ lattices we evolve the fields up to ${Q_\mathrm{s}}\tau=5$ and on the smaller $N_\perp=1024$ ones to ${Q_\mathrm{s}}\tau=10$. In this study, the nuclei are taken to fill the whole transverse lattice, with periodic boundary conditions.
Wilson loop {#sec:wloop}
===========
![Wilson loop as a function of area for different initial conditions and times measured on $N_\perp=2048$ lattices. The thicker lines at the top correspond to time ${Q_\mathrm{s}}\tau=0$, for the classical MV model as well as for fixed and running coupling JIMWLK evolution. The results for ${Q_\mathrm{s}}\tau=1,3,5$ are shown by the thinner lines, with later times corresponding to smaller values of $\ln(-\ln
W)$. \[fig:fitchk\] ](fitchk_2048_12122013_linun_pruned_v2){width="45.00000%"}
In the continuum the spatial (magnetic) Wilson loop is defined as the trace of a path ordered exponential of the gauge field around a closed path of area $A$ in the transverse plane: $$W(A) = \frac{1}{{{N_\mathrm{c}}}} \left< {\, \mathrm{Tr} \, }\mathbb{P} \exp\left\{i g
\oint_{\partial A} {\, \mathrm{d}}{{\mathbf{x}_T}}\cdot {\mathbf{A}_T}\right\} \right>.$$ On the lattice this is easily discretized as the product of link matrices around a square of area $A$. For $N_c\ge3$ colors any particular Wilson loop is complex but the ensemble average is real.
We have measured the expectation value of the Wilson loop in the glasma field, with different initial conditions and at different times ${Q_\mathrm{s}}\tau$. The results of the calculation are shown in [Fig. ]{}\[fig:fitchk\]. As expected, the magnetic flux through a loop generically increases with its area. Focusing first on the curves corresponding to the initial time $\tau=0$ we observe that the resummation of quantum fluctuations (JIMWLK evolution) increases the flux through small loops of area $A{Q_\mathrm{s}}^2<1$. This can be understood intuitively as due to emission of additional virtual soft gluons in the pure gauge fields of the colliding charge sheets. On the other hand, the flux through large loops, $A{Q_\mathrm{s}}^2\gsim 2$, decreases. This indicates uncorrelated fluctuations of magnetic flux over such areas and is consistent with the suggestion that the flux is “bundled” in domains with a typical area $\sim1/{Q_\mathrm{s}}^2$ [@Dumitru:2013koh]. Accordingly, loops of area $\sim 1.5{Q_\mathrm{s}}^2$ are invariant under high-energy evolution.
Moving on to finite times we see that the flux through loops of fixed area decreases with $\tau$. This is, of course, a consequence of the decreasing field strength in an expanding metric. For small loops the ordering corresponding to the different initial conditions (MV, rc-JIMWLK, fc-JIMWLK) is preserved even at later times. However, for large loops one observes a striking “universality” emerging at ${Q_\mathrm{s}}\tau\sim5$ as the curves for all initial conditions fall on top of each other.
The data from [Fig. ]{}\[fig:fitchk\] shows an approximately linear dependence of $\ln(-\ln W)$ on $\ln(A{Q_\mathrm{s}}^2)$, with different slopes in the regime of small $A{Q_\mathrm{s}}^2\ll 1$ vs. large $A{Q_\mathrm{s}}^2\gg 1$. Based on this observation we fit the data to $$\label{eq:fitform}
W(A) = \exp\left\{-(\sigma A )^\gamma \right\},$$ with separate parametrizations for the IR and UV regimes: && e\^[0.5]{} < A [Q\_]{}\^2 < e\^[5]{}\
&& e\^[-3.5]{} < A [Q\_]{}\^2 < e\^[-0.5]{} . In addition to limiting the fits to the quoted ranges we also restrict them to the region where $W>0.01$ and the statistical error on $W$ is less than 0.2$W$; beyond these limits the data exhibits too large fluctuations for a meaningful fit. Figures \[fig:uvexpvstau\] and \[fig:irexpvstau\] show the time dependence of the exponents $\gamma$ in the IR and UV regions. The “string tension” $\sigma$ naturally decreases as $\sim 1/\tau$ because of the longitudinal expansion of the glasma, which leads to $B_z \sim 1/\sqrt{\tau}$. We therefore show, in [Figs. ]{}\[fig:uvsigmavstau\] and \[fig:irsigmavstau\], the time dependence of the combination $\tau\sigma/{Q_\mathrm{s}}$, where this leading effect is scaled out. The values of $\sigma/{Q_\mathrm{s}}^2$ for ${Q_\mathrm{s}}\tau = 0$ are given in the captions[^3].
![ Time dependence of the exponents $\gamma$ in the parametrization [(\[eq:fitform\])]{} fitted to the UV region. []{data-label="fig:uvexpvstau"}](uvexpcollection_031013_v2){width="45.00000%"}
![ Time dependence of the exponents $\gamma$ in the parametrization [(\[eq:fitform\])]{} fitted to the IR region. []{data-label="fig:irexpvstau"}](irexpcollection_031013_v2){width="45.00000%"}
![Time dependence of “string tension” coefficient $\sigma$ fitted to the UV region. The values of $\sigma/{Q_\mathrm{s}}^2$ at $\tau=0$ are 0.59 \[0.57\]; 0.55 \[0.53\] and 0.56 \[0.56\] for the MV, rcJIMWLK and fcJIMWLK initial conditions respectively on a $N_\perp=1024$ \[$N_\perp=2048$\] lattice. []{data-label="fig:uvsigmavstau"}](uvsigmas_scaled_091213){width="45.00000%"}
![Time dependence of “string tension” coefficient $\sigma$ fitted to the IR region, multiplied by $\tau$ to separate out the natural $\sigma \sim 1/\tau$ dependence due to the expansion of the system. The values of $\sigma/{Q_\mathrm{s}}^2$ at $\tau=0$ are 0.43 \[0.44\]; 0.37 \[0.38\] and 0.39 \[0.40\] for the MV, rcJIMWLK and fcJIMWLK initial conditions respectively on a $N_\perp=1024$ \[$N_\perp=2048$\] lattice. []{data-label="fig:irsigmavstau"}](irsigmas_scaled_091213){width="45.00000%"}
The results in the “UV”-regime probed by Wilson loops of small area are shown in [Figs. ]{}\[fig:uvexpvstau\] and \[fig:uvsigmavstau\]. They are easily understood from the differences in the initial condition. The gluon spectrum in the MV model falls steeply as a function of ${k_T}$, leading to a steep dependence of short-distance correlators on the distance. Our result for the UV exponent in [Fig. ]{}\[fig:uvexpvstau\] is close to the $A^2$-scaling obtained analytically in a weak field expansion [@Petreska:2013bna]. The difference is probably due to a combination of logarithmic corrections and lattice UV cutoff effects. For the JIMWLK ensembles the gluon spectrum is much harder [@Lappi:2011ju], especially for fixed coupling. This manifests itself in smaller values of both $\gamma$ and $\sigma$. In addition, the UV exponents are remarkably time independent at ${Q_\mathrm{s}}\tau>1$: this is consistent with the expectation that at such time the UV modes can be viewed as noninteracting gluons whose spectrum is close to the expectation from a perturbative ${k_T}$-factorized calculation [@Blaizot:2010kh].
The behavior in the IR regime ([Figs. ]{}\[fig:irexpvstau\], \[fig:irsigmavstau\]) probed by large Wilson loops points to a very different picture. At $\tau=0$ the exponents $\gamma$ and, to a lesser extent, the values of $\sigma$ depend very much on the initial conditions. As already alluded to above, the scaling exponents $\gamma_{\rm IR}<1$ obtained for the JIMWLK fields indicate that quantum emissions increase magnetic flux fluctuations at the scale $\sim1/{Q_\mathrm{s}}$, much smaller than the area of the loop. It is interesting to note that for the rather strong fixed-coupling evolution the initial scaling exponent is not too far above $\gamma_{\rm IR}=1/2$ corresponding to perimeter scaling.
At times ${Q_\mathrm{s}}\tau \gtrsim 3$, however, one observes a remarkable universality in the IR as the curves corresponding to different initial conditions collapse onto a single curve in [Fig. ]{}\[fig:fitchk\]. The string tensions in [Fig. ]{}\[fig:irsigmavstau\] are within 10% of each other at late ${Q_\mathrm{s}}\tau$, and the exponents $\gamma$ in [Fig. ]{}\[fig:irexpvstau\] are very close to each other, with values around $\gamma_{\rm IR}\approx 1.2 \dots 1.3$. The exponent gradually decreases with $\tau$, potentially approaching the area law $\gamma=1$ at late times. The initial evolution points at a rapid rearrangement of “magnetic hot spots” to some universal field configurations at later time, ${Q_\mathrm{s}}\tau \gtrsim 3$.
We stress that the *universal* behavior of large magnetic loops, characterized by a nontrivial power-law dependence on the loop area, sets in at rather early time scales of a few times $1/{Q_\mathrm{s}}$, independent of initial conditions. Actual area law scaling $\gamma=1$ is approached only later. This behavior mirrors a similar universality between MV and JIMWLK results seen in the IR part of the gluon spectrum (determined from correlators of gauge fixed fields) in Ref. [@Lappi:2011ju]. Since the structure of the fields does not seem to depend on the initial conditions, we infer that this universality in due to stong interactions in the glasma phase. This universal behavior of the Wilson loop for different initial conditions at ${Q_\mathrm{s}}\tau \gtrsim
3$ and $A{Q_\mathrm{s}}^2 \gg 1$ is the main result of this paper.
Magnetic field correlator {#sec:bbcorr}
=========================
In this section we analyze gauge-invariant two-point magnetic field correlators of the form[^4] $$C_B(r) \equiv 2 g^2 {\, \mathrm{Tr} \, }\left< B_z({{\mathbf{x}_T}}) U_{{{\mathbf{x}_T}}\to {{\mathbf{y}_T}}}
B_z({{\mathbf{y}_T}}) U^\dag_{{{\mathbf{x}_T}}\to {{\mathbf{y}_T}}} \right>~.$$ The points ${{\mathbf{x}_T}}$ and ${{\mathbf{y}_T}}$ are separated in the $x$ or $y$ direction by a distance $r=|{{\mathbf{x}_T}}-{{\mathbf{y}_T}}|$, and the Wilson line $U_{{{\mathbf{x}_T}}\to {{\mathbf{y}_T}}}$ is the ordered product of links along the straight line separating these points.
The magnetic field $B_z = t^a B_z^a$ on the lattice is defined as the traceless antihermitian part of the plaquette as $$g B_z^a ({{\mathbf{x}_T}}) = 2 \, \mathrm{Re} {\, \mathrm{Tr} \, }t^a U_{x,y}({{\mathbf{x}_T}})~,$$ where the transverse plaquette is $$U_{i,j}({{\mathbf{x}_T}}) = U_i({{\mathbf{x}_T}}) U_j({{\mathbf{x}_T}}+{\mathbf{i}_T}) U^\dag_i({{\mathbf{x}_T}}+{\mathbf{j}_T}) U^\dag_j({{\mathbf{x}_T}})~.$$ Here $U_i({{\mathbf{x}_T}})$ denotes the link matrix in the $i$-direction based at ${{\mathbf{x}_T}}$ and ${\mathbf{i}_T},{\mathbf{j}_T}$ are unit vectors.
![Magnetic field correlator on a $1024^2$-lattice at ${Q_\mathrm{s}}\tau
=0$, 2, and 10; the latter have been rescaled by factors of 20 and 100, respectively.[]{data-label="fig:Bcorr"}](BBcorr_1024_28042014){width="45.00000%"}
The resulting magnetic field correlator $r\; C_B(r)$ is plotted in [Fig. ]{}\[fig:Bcorr\]. We have multiplied by $r$ to better expose the behavior around $r \sim 1/{Q_\mathrm{s}}$. At the initial time there is a significant anticorrelation at intermediate distances. It shows the domain structure of the field such that $B_z$ is likely to flip sign[^5] (or direction) over distances of order $1/{Q_\mathrm{s}}$. This structure then changes very rapidly: already at time ${Q_\mathrm{s}}\tau \sim 2$ the fields have rearranged such that the anticorrelation has disappeared. Also, the subsequent time evolution results in damping of the fluctuations at ${Q_\mathrm{s}}r \gsim 1$ which are present in the initial field configurations. On the other hand, the strong short-distance correlations around the peak are not affected much by the time evolution beyond ${Q_\mathrm{s}}\tau \sim 2$, aside from a decrease in magnitude. In particular, no “infrared diffusion” of the peak towards larger distances is observed.
![ Magnetic field correlator $C_B(r)$ at ${Q_\mathrm{s}}\tau=10$ on a $1024^2$-lattice. The line corresponds to $\sim r^{-\alpha}$ with the exponent $\alpha= 4-2\gamma_\textnormal{IR}=1.55$ extracted in the previous section from the fit of $\gamma_\textnormal{IR}$ to the Wilson loop.[]{data-label="fig:bbscaling"}](BBcorr_al_261113){width="45.00000%"}
Given the clear scaling behavior of the Wilson loop one might expect to see a similar phenomenon for the magnetic field correlator. A very naive scaling argument would assume that if $C(r) \sim r^{-\alpha}$ then the Wilson loop should scale as $$\begin{gathered}
- \ln W \sim \int\limits_A {\, \mathrm{d}}^2 {{\mathbf{x}_T}}{\, \mathrm{d}}^2 {{\mathbf{y}_T}}\, C(|{{\mathbf{x}_T}}-{{\mathbf{y}_T}}|)
\\ \sim
R^{4-\alpha} \sim A^{\frac{4-\alpha}{2}} = A^{\gamma}~.\end{gathered}$$ The area integrals in the first line extend over $|{{\mathbf{x}_T}}|,|{{\mathbf{y}_T}}|<R$. Thus, $\gamma = 1.225$ extracted from the Wilson loop at ${Q_\mathrm{s}}\tau=10$ would give $C_B(r) \sim r^{-1.55}$. On a logarithmic scale $C_B(r)$ does indeed qualitatively resemble such behavior as shown in [Fig. ]{}\[fig:bbscaling\]. However, this kind of scaling is less conclusive than for the Wilson loop (see also appendix); this could be an indication for the presence of higher cumulants in the expansion of the spatial Wilson loop [@Dosch:1988ha; @*DiGiacomo:2000va].
![Direct measurement of the Wilson loop (points) compared to an approximation in terms of the Gaussian cumulant, [Eq. ]{}[(\[eq:wlfrombb\])]{}, which reconstructs it from the magnetic field correlator (lines). []{data-label="fig:wlscfrombb"}](wlscfrombb_280613_ed2){width="45.00000%"}
Summary
=======
In this paper we have provided some insight into the fields produced initially in a high-energy collision of dense color charge sheets. We have focused, in particular, on the structure of the longitudinal magnetic field $B_z$.
We consider both purely classical as well as JIMWLK RG evolved gauge field ensembles on which we measure expectation values of spatial Wilson loops and two-point correlation functions of $B_z$. These show that the initial fields exhibit domain-like structure over distance scales of the order of the inverse saturation scale $1/{Q_\mathrm{s}}$. Classical YM evolution to later times leads to universal scaling, for all ensembles, of the magnetic loop with area, with a nontrivial critical exponent. Also, the anti-correlation of $B_z({{\mathbf{x}_T}})$ over distances $\sim 1/{Q_\mathrm{s}}$ disappears, which we interpret as rearrangement, possibly accompanied by transverse expansion, of the magnetic field domains.
The emergence of a color field condensate in high-energy collisions of dense hadrons or nuclei is a very interesting phenomenon, and its dynamics remains to be understood in more detail. In closing we only draw attention to recent arguments that the presence of such a condensate might have important implications for the process of (pre-) thermalization in high multiplicity collisions [@Floerchinger:2013kca].
Relation between the Wilson loop and magnetic field correlator
==============================================================
In an abelian theory there is a simple relation between the Wilson loop and the magnetic field due to Stokes’ theorem: $$\oint_{\partial A} {\, \mathrm{d}}{{\mathbf{x}}}\cdot {\mathbf{A}}=
\int_A {\, \mathrm{d}}^2 {{\mathbf{x}}}\, B_z({{\mathbf{x}}}) ~.$$ If we assume that in the nonabelian case the magnetic field in each color channel $a$ is independent, and that it consists of uncorrelated domains which are much smaller than the area $A$ and distributed as Gaussian random variables, we obtain the following estimate for the Wilson loop: $$\begin{gathered}
\label{eq:wlfrombb}
\frac{1}{{{N_\mathrm{c}}}} {\, \mathrm{Tr} \, }\exp\left\{i g
\oint_{\partial A} {\, \mathrm{d}}{{\mathbf{x}}}\cdot {\mathbf{A}}\right\}
\\
\approx \exp\left\{- \frac{g^2}{2{{N_\mathrm{c}}}} \left< {\, \mathrm{Tr} \, }\left[\int_A {\, \mathrm{d}}^2{{\mathbf{x}}}\, B_z({{\mathbf{x}}})\right]^2 \right>\right\}
\\
= \exp\left\{- \frac{1}{4{{N_\mathrm{c}}}}
\int_A {\, \mathrm{d}}^2{{\mathbf{x}}}{\, \mathrm{d}}^2{{\mathbf{y}}}\, C_B ({{\mathbf{x}}}-{{\mathbf{y}}})\right\}. \end{gathered}$$ In [Fig. ]{}\[fig:wlscfrombb\] we compare the result of a numerical integration of the r.h.s. of [Eq. ]{}[(\[eq:wlfrombb\])]{} using the measured magnetic field correlator, to the direct measurement of the Wilson loop. It can be seen that the two are in a relatively good agreement. This consistency check supports the interpretation of $B_z$ as independent field domains of area $\sim 1/{Q_\mathrm{s}}^2$.
Acknowledgements {#acknowledgements .unnumbered}
================
T. L. is supported by the Academy of Finland, projects 133005, 267321 and 273464. This work was done using computing resources from CSC – IT Center for Science in Espoo, Finland. A. D. acknowledges support by the DOE Office of Nuclear Physics through Grant No. DE-FG02-09ER41620 and from The City University of New York through the PSC-CUNY Research Award Program, grant 66514-0044. The authors thank the Yukawa Institute for Theoretical Physics, Kyoto University, where part of this work was done during the YITP-T-13-05 workshop on “New Frontiers in QCD”.
[^1]: The YM equations are solved in terms of the coordinates $\tau=\sqrt{t^2-z^2}$, $\eta=\frac{1}{2}\ln \frac{t+z}{t-z}$ and ${{\mathbf{x}_T}}$; hence ${\, \mathrm{d}}s^2={\, \mathrm{d}}\tau^2-\tau^2{\, \mathrm{d}}\eta^2-{\, \mathrm{d}}{{\mathbf{x}_T}}^2 $.
[^2]: For fixed coupling the evolution variable is ${\alpha_{\mathrm{s}}}y$, so we do not need to specify a particular value of ${\alpha_{\mathrm{s}}}$ separately.
[^3]: For the MV model, at $\tau=0$ we find $\sigma/{Q_\mathrm{s}}^2=0.44$ in the IR region which is about four times larger than the value reported in ref. [@Dumitru:2013koh]. Our present results refer to $N_c=3$ colors while ref. [@Dumitru:2013koh] considered $N_c=2$; also, our current definition of ${Q_\mathrm{s}}$ via [Eq. ]{}[(\[eq:defqs\])]{} leads to smaller values for this quantity than the definition used in [@Dumitru:2013koh]. Finally, $\sigma$ is extracted from fits over a somewhat different range of areas.
[^4]: We include the factor $g^2$ for convenience, because the quantity that appears naturally in the classical lattice formulation is actually $gB$.
[^5]: Recall that $B$ transforms homogeneously. Hence, unlike the links $U_i({{\mathbf{x}_T}})$, the magnetic field $B_z({{\mathbf{x}_T}})$ can be diagonalized everywhere by a suitable gauge transformation.
|
---
abstract: |
We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval $I$. The monic integer transfinite diameter ${t_{\mathrm{M}}}(I)$ is defined as the infimum of all such supremums. We show that if $I$ has length $1$ then ${t_{\mathrm{M}}}(I) = \tfrac{1}{2}$.
We make three general conjectures relating to the value of ${t_{\mathrm{M}}}(I)$ for intervals $I$ of length less that $4$. We also conjecture a value for ${t_{\mathrm{M}}}([0, b])$ where $0<b\le 1$. We give some partial results, as well as computational evidence, to support these conjectures.
We define functions ${L_{-}}(t)$ and ${L_{+}}(t)$, which measure properties of the lengths of intervals $I$ with ${t_{\mathrm{M}}}(I)$ on either side of $t$. Upper and lower bounds are given for these functions.
We also consider the problem of determining ${t_{\mathrm{M}}}(I)$ when $I$ is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.
address:
- 'Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1'
- 'School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK.'
author:
- 'K. G. Hare'
- 'C. J. Smyth'
title: The monic integer transfinite diameter
---
Introduction and Results
========================
In this paper we continue a study, recently initiated by Borwein, Pinner and Pritsker [@BorweinPinnerPritsker03], of the [*monic integer transfinite diameter*]{} of a real interval. We write the normalized supremum on an interval $I$ as $${\|}P{\|^{{}^{\scriptstyle{*}}}}_I := \sup_{x \in I} |P(x)|^{1/\deg P}.$$ Note that this is not a norm. Then the monic integer transfinite diameter ${t_{\mathrm{M}}}(I)$ is defined as $${t_{\mathrm{M}}}(I) := \inf_P {\|}P{\|^{{}^{\scriptstyle{*}}}}_I,$$ where the infimum is taken over all non-constant monic polynomials with integer coefficients. We call ${t_{\mathrm{M}}}(I)$ the [*monic integer transfinite diameter*]{} of $I$ (also called the [*monic integer Chebyshev constant*]{} [@Borwein02; @BorweinPinnerPritsker03]). Clearly ${t_{\mathrm{M}}}(I)\ge {t_{{\mathbb{Z}}}}(I)$, where ${t_{{\mathbb{Z}}}}(I)$ denotes the [*integer transfinite diameter*]{}, defined using the same infimum, but taken over the larger set of all non-constant polynomials with integer coefficients [@BorweinErdelyi96; @Chudnovsky83; @FlammangRhinSmyth97]. Further ${t_{{\mathbb{Z}}}}(I) \ge {\mathrm{cap}}(I)$, the [*capacity*]{} or [*transfinite diameter*]{} of $I$ [@Goluzin69; @Ransford95], which can be defined again using the same infimum, but this time taken over all non-constant monic polynomials with real coefficients. It is well known that ${\mathrm{cap}}(I)=|I|/4$ for an interval $I$ of length $|I|$. Further, if $|I|\ge 4$ then ${t_{{\mathbb{Z}}}}(I)={t_{\mathrm{M}}}(I)={\mathrm{cap}}(I)$ by [@BorweinPinnerPritsker03] so that the challenge for evaluating ${t_{\mathrm{M}}}(I)$, as for ${t_{{\mathbb{Z}}}}(I)$, lies in intervals with $|I|<4$. For these intervals we know from [@BorweinPinnerPritsker03 Prop. 1.2] that ${t_{\mathrm{M}}}(I)<1$. However, in contrast to the study of ${t_{{\mathbb{Z}}}}(I)$, in the monic case it is possible to evaluate ${t_{\mathrm{M}}}(I)$ exactly over some such intervals.
Our first result is the following.
\[thm:interval 1\] All intervals $I$ of length $1$ have ${t_{\mathrm{M}}}(I)
= \tfrac{1}{2}$. In fact, slightly more is true: if $1 \leq |I| \leq 1.008848$ then ${t_{\mathrm{M}}}(I) = \tfrac{1}{2}$.
Furthermore for any $b<1$ there is an interval $I$ with $|I|=b$ and ${t_{\mathrm{M}}}(I) < \tfrac{1}{2}$, while for $b>1.064961507$ there is an interval $I$ with $|I|=b$ and ${t_{\mathrm{M}}}(I) > \tfrac{1}{2}$.
The proof, which is essentially a corollary of Theorem \[thm:[L\_[-]{}]{}(1/2)\] (a) below, is discussed in Section \[sec:interval 1\].
The numbers, 1.008848 and 1.064961507 in Theorem \[thm:interval 1\], like most numerical values given in this paper, are approximations to some exact algebraic number. These numbers are rounded in the correct direction, if necessary, to ensure an inequality still holds. The polynomial equations that they satisfy is given within the text. We have tried to do this for all numerical values.
To measure the range of lengths of intervals having a particular monic integer transfinite diameter $t$, we introduce the following two functions: $$\begin{aligned}
{L_{-}}(t)&:=&\inf_I\{|I|: {t_{\mathrm{M}}}(I)>t\};\\
{L_{+}}(t)&:=&\sup_I\{|I|: {t_{\mathrm{M}}}(I)\leq t\}.\end{aligned}$$
It follows from [@BorweinPinnerPritsker03 Prop. 1.3] that both ${L_{-}}(t)$ and ${L_{+}}(t)$ are nondecreasing functions of $t$. Also ${L_{-}}(t)\le{L_{+}}(t)$ – see Lemma \[lem:LL\](a) below. We give (Proposition \[prop:L-bounds\]) general method for finding upper and lower bounds for ${L_{-}}(t)$ and ${L_{+}}(t)$, and apply these methods to get such bounds for $\tfrac{1}{2} \leq t
\leq 1$. They are constructive, using both the LLL basis-reduction algorithm and the Simplex method. These techniques were first applied in this area by Borwein and Erdélyi [@BorweinErdelyi96], and then by Habsieger and Salvy [@HabsiegerSalvy97]. These bounds are given in Theorem \[thm:[L\_[-]{}]{}(t)\] and Proposition \[P-lowerB\] – see also Figures \[fig:[L\_[-]{}]{}(t)\] and \[fig:[L\_[+]{}]{}(t)\].
At $t=\tfrac{1}{2}$, we pushed this method further, and were able to say more.
\[thm:[L\_[-]{}]{}(1/2)\] We have
- $1.008848 \leq {L_{-}}\left(\tfrac{1}{2}\right) \leq 1.064961507$
and
- $\sqrt{2} \approx 1.41421 \leq
{L_{+}}\left(\tfrac{1}{2}\right) \leq 1.4715$.
Further properties of ${L_{+}}$ and ${L_{-}}$ are given in Lemma \[lem:LL\].
Definitions, Conjectures and Further Results
============================================
In this section, we state some old and some more new results, and (perhaps a little recklessly) make four conjectures.
The following result is simple but fundamental. It is useful for determining lower bounds for ${t_{\mathrm{M}}}(I)$.
[**Lemma BPP**]{} (Borwein, Pinner and Pritsker [@BorweinPinnerPritsker03 p.1905]).
Let $Q(x) = a_d x^d + \cdots + a_0$ be a nonmonic irreducible polynomial with integer coefficients, all of whose roots lie in the interval $I$. Then ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I \geq a_d^{-1/d}$ for every monic integer polynomial $P$, so that ${t_{\mathrm{M}}}(I)\geq a_d^{-1/d}$. Furthermore, if ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I = a_d^{-1/d}$ then ${t_{\mathrm{M}}}(I) = a_d^{-1/d}$ and $|P(\beta)|^{1/\deg P}=a_d^{-1/d}$ for every root $\beta$ of $Q$, and ${\mathrm{Res}}(P,Q)=\pm 1$.
The proof follows straight from the classical fact that, for the conjugates $\beta_i$ of $\beta$ $$\label{eqn:resultant}
{\mathrm{Res}}(P,Q)=a_d^{\deg P}\prod_{i=1}^d P(\beta_i)$$ is a nonzero integer, giving $$\label{eqn:resultant2}
{\|}P{\|^{{}^{\scriptstyle{*}}}}_I\ge \left(\prod_i|P(\beta_i)|^{1/\deg
P}\right)^{\frac{1}{d}}\ge a_d^{-1/d}|{\mathrm{Res}}(P,Q)|^{\frac{1}{d\deg
P}}\ge a_d^{-1/d}.$$ This result is a variant of a similar one in the theory of ${t_{{\mathbb{Z}}}}(I)$—see Lemma \[lem:was BE\].
We call such a value $a_d^{-1/d}$ in Lemma BPP an [*obstruction*]{} for $I$, with [*obstruction polynomial*]{} $Q(x)$. From Lemma BPP we see that ${t_{\mathrm{M}}}(I)$ is bounded below by the supremum of all such obstructions. If this supremum is attained by some value $a_d^{-1/d}$ coming from $Q(x)=a_dx^d+\cdots+a_0$, then we say $a_d^{-1/d}$ is a [*maximal obstruction*]{}, and $Q(x)$ is a [*maximal obstruction polynomial*]{}. It is not known whether such a polynomial exists for all intervals $I$ of length less than $4$ (see Conjecture \[conj:maximal\]).
We say that the monic integer polynomial $P(x)$ is an [*optimal monic integer Chebyshev polynomial for $I$*]{} if ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I={t_{\mathrm{M}}}(I)$. If $I$ has a maximal obstruction $a_d^{-1/d}$ with ${t_{\mathrm{M}}}(I)=a_d^{-1/d}$ and an optimal monic integer Chebyshev polynomial $P$ then we say that [*$P$ attains the maximal obstruction $a_d^{-1/d}$*]{}.
Throughout this paper, $P(x)$ will denote a monic integer polynomial, $Q(x)$ a nonmonic integer polynomial and $R(x)$ any integer polynomial.
One very nice property of the monic integer transfinite diameter problem, not shared by its nonmonic cousin, is that often exact values can be computed for ${t_{\mathrm{M}}}(I)$. In all cases where this has been done, including Theorem \[thm:interval 1\], it was achieved by finding a maximal obstruction, and a corresponding optimal monic integer Chebyshev polynomial. Simple examples of this are given ([@BorweinPinnerPritsker03 Theorem 1.5]) by the intervals $I=[0,1/n]$ for $n\ge 2$, where $Q(x)=nx-1$ is a maximal obstruction polynomial, and $P(x)=x$ is an optimal monic integer Chebyshev polynomial. For $n=1$, ${t_{\mathrm{M}}}([0,1])=\tfrac{1}{2}$, with $Q(x)=2x-1$ and $P(x)=x(x-1)$. This was the case too in [@BorweinPinnerPritsker03 Section 5] in the proof of the Farey Interval conjecture for small-denominator intervals.
A much less obvious example is the interval $I = [-0.3319, 0.7412]$, of length $1.0731$. Here, we have ${t_{\mathrm{M}}}(I)={\|}P{\|^{{}^{\scriptstyle{*}}}}_I= 7^{-1/3} \approx 0.522$, with maximal obstruction polynomial $7 x^3 - 7 x^2 + 1$ and where $P$ is the optimal monic integer Chebyshev polynomial $$\begin{array}{rl}
P(x)=&x^{276507}(x-1)^{29858} (x^2+x-1)^{14929}\\
&(x^5-17 x^4+24 x^3-8 x^2-2 x+1)^{28848} \\
&(x^7-117 x^6+194 x^5-70 x^4-31 x^3+18 x^2+x-1)^{7935} \\
&(x^8-4 x^7+97 x^6-172 x^5+78 x^4+20 x^3-18 x^2+1)^{9795} \\
&(x^8-34 x^7+164 x^6-208 x^5+65 x^4+33 x^3-18 x^2-x+1)^{5846} \\
&(x^8-7 x^7+2 x^6-x^5-10 x^4+28 x^3-15 x^2-2 x+2)^{1148}
\end{array}$$ of degree $670320$. (Tighter endpoints for this interval, and its length, can be computed by solving the equation $P(x) = \pm \left(7^{-1/3}\right)^{\deg P}$.) The discovery of this polynomial required the use of Lemma \[lem:resultant-value\] below.
For the nonmonic transfinite diameter ${t_{{\mathbb{Z}}}}$, Pritsker [@Pritsker* Theorem 1.7] has recently proved that no integer polynomial $R(x)$ can attain ${\|}R(x){\|^{{}^{\scriptstyle{*}}}}_I={t_{{\mathbb{Z}}}}(I)$, this value being achieved only by a normalized product of infinitely many polynomials. An immediate consequence of his result is the following.
If an interval $I$ has an optimal monic integer Chebyshev polynomial then ${t_{\mathrm{M}}}(I)>{t_{{\mathbb{Z}}}}(I)$.
A fundamental question for both the monic and nonmonic integer transfinite diameter of an interval is whether its value can be computed exactly. In [@BorweinPinnerPritsker03 Conjecture 5.1], Borwein [*et al*]{} make a conjecture for [*Farey intervals*]{} (intervals ${\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}$ where $b_1, b_2, c_1, c_2\in{\mathbb{Z}}$ and $b_2 c_1 - b_1 c_2=1$) concerning the exact value of their monic transfinite diameter.
[**Conjecture BPP**]{} (Farey Interval Conjecture [@Borwein02 p. 82], [@BorweinPinnerPritsker03 Conjecture 5.1]).
Suppose that ${\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}$ is a Farey interval, neither of whose endpoints is an integer. Then $${t_{\mathrm{M}}}\left({\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}\right)=\dfrac{1}{\min(c_1,c_2)}.$$
Borwein [*et al*]{} verify their conjecture for all Farey intervals having the denominators $c_1,c_2$ less than $22$. In Section \[sec:Farey\] we extend the verification to some infinite families of Farey intervals (Theorems \[thm:farey\] and \[thm:farey 2\]).
We next investigate what happens to ${t_{\mathrm{M}}}([0, b])$ when $b$ is close to $\frac{1}{n}$. For these intervals, some surprising things happen. Using the polynomial $P(x)=x$, we know that ${t_{\mathrm{M}}}([0,b])\le b<\frac{1}{n}$ if $b<\frac{1}{n}$. In fact it appears likely that ${t_{\mathrm{M}}}([0,b])$, clearly a non-decreasing function of $b$, has a left discontinuity at $t=1/n\quad(n>1)$. On the other hand, we show in Theorem \[thm:interval 1/n\] that ${t_{\mathrm{M}}}$ is locally constant on an interval of positive length $\delta_n$ to the right of $\frac{1}{n}$. Further, Theorem \[thm:interval 1/3\] gives much larger values for $\delta_n$ for $n = 2, 3$ and $4$, as well as an upper bound for $\delta_2$.
In fact, more may be true.
\[conj:\[0,b\]\] If $I = [0,b]$ is an interval with $b \leq 1$, then ${t_{\mathrm{M}}}(I)=1/n$, where $n=\max\left(2,\left\lceil
\frac{1}{b}\right\rceil\right)$ is the smallest integer $n \geq 2$ for which $1/n \leq b$.
What little we know about ${t_{\mathrm{M}}}([0,b])$ for $b>1$ is given in Theorem \[thm:interval 1/3\] (c), (d).
Both Conjecture BPP and Conjecture \[conj:\[0,b\]\] are a consequence of the following conjecture.
\[conj:maximal=tm\] If an interval $I$ of length less than $4$ has a maximal obstruction $m$, then ${t_{\mathrm{M}}}(I)=m$.
We were at first tempted to conjecture here that ${t_{\mathrm{M}}}(I)$, as well as equaling its maximal obstruction, is always attained by some monic integer polynomial. However, the following counterexample eliminates this possibility in general.
\[thm:unattainable\] The polynomial $7 x^3 + 4 x^2 - 2 x -1$ is a maximal obstruction polynomial for the interval $I=[-0.684, 0.517]$. However, there is no monic integer polynomial $P$ with ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I$ equal to the maximal obstruction $7^{-1/3}$ for $I$.
This result is proved in Section \[sec:unattainable\].
Our next result proves the existence of maximal obstructions for many intervals.
\[thm:maximal\] Every interval not containing an integer in its interior has a maximal obstruction.
Based on Conjecture \[conj:maximal=tm\] and Theorem \[thm:maximal\] we make the following conjecture.
\[conj:maximal\] Every interval of length less than $4$ has a maximal obstruction.
We do not have much direct evidence for this conjecture. However, our next conjecture, Conjecture \[conj:critical\], implies it. To describe this implication, we need the following notion, taken from Flammang, Rhin and Smyth [@FlammangRhinSmyth97]. An irreducible polynomial $Q(x)=a_dx^d+\cdots +a_0\in {\mathbb{Z}}[x]$ with $a_d>0$, all of whose roots lie in an interval $I$, and for which $a_d^{-1/d}$ is greater than the (nonmonic) transfinite diameter ${t_{{\mathbb{Z}}}}(I)$ is called a [*critical polynomial*]{} for $I$. Here we are interested only in nonmonic critical polynomials.
It may be that every interval of length less than $4$ has infinitely many nonmonic critical polynomials – see Proposition \[prop:crit\] below. We make the following weaker conjecture.
\[conj:critical\] Every interval of length less than $4$ has at least one nonmonic critical polynomial.
From Theorem \[thm:maximal\] below, this conjecture is true for intervals not containing an integer. For intervals $I$ of length less than $4$ that do contain an integer (say $0$), then, since ${t_{{\mathbb{Z}}}}(I)<1$, the polynomial $x$ is a critical polynomial for $I$. Thus ‘nonmonic’ is an important word in this conjecture.
In Theorem \[thm:critical=maximal\] we prove that Conjecture \[conj:critical\] implies Conjecture \[conj:maximal\]. More interestingly, we also prove in Corollary \[cor:critical=maximal\] that Conjecture \[conj:maximal=tm\] and Conjecture \[conj:maximal\] together imply Conjecture \[conj:critical\].
We observe in passing the following conditional result for the integer transfinite diameter ${t_{{\mathbb{Z}}}}$.
\[prop:crit\] Suppose that an interval $I$ has infinitely many critical polynomials $Q_i(x)=a_{d_i,i}x^{d_i}+\dots+a_{0,i}$. Then $${t_{{\mathbb{Z}}}}(I)=\inf_i a_{d_i,i}^{-\frac{1}{d_i}}.$$
This result is proved in Section \[sec:crit\]. Montgomery [@Montgomery94 p.182] conjectured this result unconditionally for the interval $I=[0,1]$.
Upper and Lower bounds for ${L_{-}}(t)$ and ${L_{+}}(t)$ for fixed $t$ {#sec:Upper/Lower Bounds}
======================================================================
The following lemma contains some simple properties, as well as alternative definitions, of ${L_{-}}$ and ${L_{+}}$.
\[lem:LL\] We have
1. ${L_{-}}(t)\leq {L_{+}}(t)$ for $t\ge 0$;
2. ${L_{-}}(t)=0$ for $0\le t \le \tfrac{1}{2}$;
3. ${L_{+}}(t)\geq 2t$ for $0\leq t \le 1/2$;
4. ${L_{-}}(t)=\sup_I\{d: {t_{\mathrm{M}}}(I)\leq t \text{ for all } I \text{ with }
|I|=d\}$ for $t\geq \tfrac{1}{2}$;
5. ${L_{+}}(t)=\inf_I\{d: {t_{\mathrm{M}}}(I)> t \text{ for all } I \text{ with }
|I|=d\}$ for $t\ge 0$;
6. ${L_{+}}(t) = {L_{-}}(t) = 4 t$ for $t \geq 1$.
First note that, by [@BorweinPinnerPritsker03 equation (1.11)], ${t_{\mathrm{M}}}(I)=\tfrac{1}{2}$ for the zero-length interval $\left[\tfrac{1}{2},\tfrac{1}{2}\right]$, from which (b) follows.
Part (c) follows from the fact that ${\|}x{\|^{{}^{\scriptstyle{*}}}}_{[-t,t]}=t$.
To prove (d), take $t\ge \tfrac{1}{2}$. Then the set $$S:=\{d: {t_{\mathrm{M}}}(I)\leq t \text{ for all } I \text{ with } |I|=d\}$$ contains $0$ (by (b)), so is nonempty. Put $s=\sup_d S$, and take $d\in S$. Since $I'\subset
I$ implies that ${t_{\mathrm{M}}}(I')\le {t_{\mathrm{M}}}(I)$ ([@BorweinPinnerPritsker03 Prop. 1.3]), any $d'$ with $0\le d' < d$ also lies in $S$, so that $S=[0,s)$ or $[0,s]$. Hence ${L_{-}}(t)\ge s$. On the other hand, for each $d>s$ there is an interval $I$ with $|I|=d$ and ${t_{\mathrm{M}}}(I)>t$. Hence ${L_{-}}(t)\le d$, giving ${L_{-}}(t)=s$.
Now (a) follows straight from (b) and (d). The proof of (e), similar to that of (d), is left as an exercise for the reader.
Finally, part (f) follows from the fact that for $|I| \geq 4$ we have ${t_{\mathrm{M}}}(I) = {t_{{\mathbb{Z}}}}(I) = \textrm{cap}(I) = \frac{|I|}{4}$ (see for instance [@BorweinPinnerPritsker03]).
Next, we give a simple lemma, needed for applying Proposition \[prop:L-bounds\] below.
\[L-simple\] Suppose that $I_i=[a_i,b_i]\quad (i=1,\dots,n)$ are intervals with $a_1< a_2<\dots< a_n=a_1+1$, and put $M:=\max_{i=1}^{n-1}(b_{i+1}-a_i)$, $m:=\min_{i=1}^{n-1}(b_{i}-a_{i+1})$. Then
1. Any interval of length at least $M$ contains an integer translate of some $I_i$.
2. Any interval of length at most $m$ is contained in an integer translate of some $I_i$.
Given an interval $I$ of length $\ell$, we can, after translation by an integer, assume that $I=[a,b]$, where $a_j\le a< a_{j+1}$, for some $j<n$.
- Suppose that $\ell\ge M$. Then $b_{j+1}\le a_j+M\le a+\ell$, so that $[a_{j+1},b_{j+1}]\subset[a,a+\ell]$.
- Suppose that $\ell\le m$. Then $b_{j}\ge a_{j+1}+m> a+\ell$, so that $[a,a+\ell]\subset[a_{j},b_{j}]$.
The following proposition will be used to obtain explicit upper and lower bounds for ${L_{-}}(t)$ and ${L_{+}}(t)$ for particular values of $t$.
\[prop:L-bounds\]
1. If $Q(x)=a_dx^d+\cdots+ a_0$, with integer coefficients and $a_d>1$, has roots spanning an interval of length $\ell$, then for any $t<a_d^{-1/d}$ we have $${L_{-}}(t)\le \ell.$$
2. Suppose that we have a finite set of polynomials $Q_i(x)=a_{d_i,i}x^{d_i}+\cdots+a_{0,i}$ with all $a_{d_i,i}^{-1/d_i}>t$ with the property that every interval of length $\ell$ contains an integer translate of the roots of at least one of the polynomials $Q_i$. Then $${L_{+}}(t)\le \ell.$$
3. Suppose that we have a finite set of intervals $I_i$ such that for each $I_i$ there is a monic integer polynomial $P_i$ with ${\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I_i} \le t$. Suppose too that every interval of length $\ell$ is contained in an integer translate of some $I_i$. Then $${L_{-}}(t)\ge \ell.$$
4. If ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=t$ for some monic integer polynomial $P$ and interval $I$ of length $\ell$, then $${L_{+}}(t)\ge \ell.$$
<!-- -->
1. Given such a $Q(x), \ell$ and interval $I$ of length $\ell$, and $t<a_d^{-1/d}$, then from Lemma BPP we have ${t_{\mathrm{M}}}(I)\ge
a_d^{-1/d}>t$ so that, from the definition of ${L_{-}}(t)$, we have ${L_{-}}(t)\le
\ell$.
2. Suppose that every interval $I$ of length $\ell$ contains some integer translate of the set of roots of some $Q_i$. Then, by Lemma BPP, ${t_{\mathrm{M}}}(I)\ge a_{d_i,i}^{-1/d_i}>t$. Hence ${t_{\mathrm{M}}}(I')>t$ for any interval of length $|I'| \ge \ell$, and so ${L_{+}}(t)\le\ell$.
3. Here, for every interval $I$ of length $\ell$ with $I+r\subset I_i$ say, (with $r \in {\mathbb{Z}}$), we have $$t>{\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I_i}\ge{\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I+r}={\|}P_i(x+r){\|^{{}^{\scriptstyle{*}}}}_I\ge{t_{\mathrm{M}}}(I),$$ so that any $I'$ with ${t_{\mathrm{M}}}(I')>t$ has $|I'|>\ell$. Hence ${L_{-}}(t)\ge
\ell$.
4. If ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=t$ and $|I|=\ell$ then ${t_{\mathrm{M}}}(I)\le t$, so that ${L_{+}}(t)\ge
\ell$.
$i$ Polynomials $Q_i$ Intervals $[a_i,b_i]$
------ ------------------------------------------ -----------------------
$1$ $7x^3+7x^2-1 $ $[-0.737, 0.328]$
$2$ $57x^6+81x^5+6x^4-32x^3-9x^2+3x+1 $ $[-0.728, 0.494]$
$3$ $7x^3+4x^2-2x-1 $ $[-0.684, 0.517]$
$4$ $59x^6+28x^5-43x^4-15x^3+11x^2+2x-1 $ $[-0.669, 0.528]$
$5$ $3x^2-1 $ $[-0.577, 0.577]$
$6$ $59x^6-28x^5-43x^4+15x^3+11x^2-2x-1 $ $[-0.528, 0.669]$
$7$ $7x^3-4x^2-2x+1 $ $[-0.517, 0.684]$
$8$ $57x^6-81x^5+6x^4+32x^3-9x^2-3x+1 $ $[-0.494, 0.728]$
$9$ $7x^3-7x^2+1 $ $[-0.328, 0.737]$
$10$ $63x^6-136x^5+72x^4+16x^3-17x^2+1 $ $[-0.310, 1.115]$
$11$ $63x^6-146x^5+91x^4+7x^3-18x^2+x+1 $ $[-0.285, 1.141]$
$12$ $58x^6-139x^5+90x^4+6x^3-18x^2+x+1 $ $[-0.285, 1.178]$
$13$ $59x^6-147x^5+105x^4-3x^3-18x^2+2x+1 $ $[-0.271, 1.184]$
$14$ $63x^6-159x^5+115x^4-4x^3-19x^2+2x+1 $ $[-0.260, 1.197]$
$15$ $15x^4-29x^3+13x^2+x-1 $ $[-0.244, 1.208]$
$16$ $57x^6-171x^5+153x^4-21x^3-21x^2+3x+1 $ $[-0.228, 1.228]$
$17$ $15x^4-31x^3+16x^2-1 $ $[-0.208, 1.244]$
$18$ $63x^6-219x^5+265x^4-126x^3+14x^2+5x-1 $ $[-0.197, 1.260]$
$19$ $59x^6-207x^5+255x^4-127x^3+18x^2+4x-1 $ $[-0.184, 1.271]$
$20$ $58x^6-209x^5+265x^4-136x^3+20x^2+4x-1 $ $[-0.178, 1.285]$
$21$ $63x^6-232x^5+306x^4-171x^3+34x^2+2x-1 $ $[-0.141, 1.285]$
$22$ $63x^6-242x^5+337x^4-204x^3+48x^2-1 $ $[-0.115, 1.310]$
: Obstruction polynomials used for Theorem \[thm:[L\_[-]{}]{}(1/2)\] to prove that ${L_{+}}(\tfrac{1}{2}) < 1.4715$.[]{data-label="tab:1.48"}
-----------------------------------------------------------------------------------------------------
$i$ Polynomials $P_i$ Intervals $I_i$
------ ------------------------------------------------------------------------ ---------------------
$1$ $x^{1600} (x^3-4 x^2+1)^{36} (x^4+4 x^3-4 x^2-x+1)^{55} $[-0.5142, 0.5613]$
\newline
(x^8+236 x^7-96 x^6-167 x^5+64 x^4+39 x^3-14 x^2-3 x+1)^{39} \newline
(x^8+372 x^7-196 x^6-249 x^5+129 x^4+55 x^3-28 x^2-4 x+2)^{20}
$
$2$ $ x^{2121} (x^3-4 x^2+1)^{77} (x^4-10 x^3+5 x^2+2 x-1)^{84} $[-0.4501, 0.5783]$
\newline (x^7-43 x^6-11 x^5+44 x^4+2 x^3-12 x^2+1)^{160}
$
$3$ $ x^{12446} (x^2+x-1)^{199} (x^4-7 x^3+5 x^2+x-1)^{909} $[-0.4388, 0.5912]$
\newline (x^6-53 x^5+46 x^4+10 x^3-14 x^2+1)^{640} $
$4$ $ x^{312924} (x^4-7 x^3+5 x^2+x-1)^{45312} \newline (x^4+8 $[-0.4267, 0.6401]$
x^3-8 x^2+1)^{217} (x^4+9 x^3-7 x^2-x+1)^{23800} $
$5$ $x^{17556} (x^5+16 x^4-22 x^3 +5 x^2+3x-1)^{2256} \newline $[-0.3797, 0.6847]$
(x^4+8 x^3-8 x^2+1)^{899} $
$6$ $ x^{49329424964} (x-1)^{6557517120} (x^2+x-1)^{70328}\newline $[-0.3241, 0.7100]$
(x^4+8 x^3-8 x^2+1)^{4916965515} \newline (x^5-17 x^4+24 x^3-8
x^2-2 x+1)^{5952478752} \newline (x^5+16 x^4-22 x^3 +5
x^2+3x-1)^{541825536} $
$7$ $x^{114080} (x-1)^{9324} (x^4+8 x^3-8 x^2+1)^{529} \newline $[-0.3064, 0.7344]$
(x^4+9 x^3-9 x^2+1)^{2852} \newline (x^8+172 x^7-440 x^6+377 x^5-82
x^4-47 x^3+21 x^2+x-1)^{8184} \newline (x^8+214 x^7-531 x^6+440
x^5-90 x^4-54 x^3+23 x^2+x-1)^{6072} $
$8$ $ x^{15200} (x-1)^{5192} (x^4+9 x^3-9 x^2+1)^{192} \newline $[-0.2943, 0.7401]$
(x^8+172 x^7-440 x^6+377 x^5-82 x^4-47 x^3+21 x^2+x-1)^{1587}$
$9$ $ x^{3136} (x-1)^{1768} (x^6+3 x^5+6 x^4-18 x^3+9 $[-0.2752, 0.7645]$
x^2+x-1)^{32}
\newline (x^8+172 x^7-440 x^6+377 x^5-82 x^4-47 x^3+21 x^2+x-1)^{91}
$
$10$ $x^{146704} (x-1)^{85868} (x^2+x-1)^{6369} \newline $[-0.2622, 1.1030]$
(x^6+3x^5+6x^4-18x^3+9x^2+x-1)^{1768}$
-----------------------------------------------------------------------------------------------------
: Optimal monic integer Chebyshev polynomials used for Theorem \[thm:[L\_[-]{}]{}(1/2)\] to prove that ${L_{-}}(\tfrac{1}{2})\ge 1.008848$.[]{data-label="tab:interval 1"}
Applying Proposition \[prop:L-bounds\](a) with $Q(x)=7 x^3-7
x^2+1$, we have $${L_{-}}(\tfrac{1}{2})\le\ell=1.064961507.$$ Here, a more precise value could be determined by calculating the span of the roots of $Q(x)$ to a higher precision.
We apply Proposition \[prop:L-bounds\](b) and Lemma \[L-simple\](a) using the polynomials $Q_i$ of Table \[tab:1.48\], with the intervals $[a_i,b_i]$ containing their roots. (Here, the endpoints listed in Table \[tab:1.48\] are approximations of the minimal and maximal root of the obstruction polynomial in question. A higher precision was used for the computation of the upper bound of ${L_{+}}\left(\tfrac{1}{2}\right) < 1.4715$.) We put $Q_{23}(x)=Q_1(x-1)$, whose roots are contained in $[a_{23},b_{23}]:=[a_1+1,b_1+1]$, and apply the Proposition to the $23$ polynomials $Q_1,\cdots,Q_{23}$. Each has $a_d^{-1/d}>\tfrac{1}{2}$. Then because $\max_{i=1}^{22}
(b_{i+1}-a_i)=b_{16}-a_{15}=1.4715$, any interval $I$ of length $|I|
>1.4715$ must, by Lemma \[L-simple\](a), contain some integer translate of some interval $[a_i,b_i]$, and so all the roots of the corresponding polynomial $Q_i$. Hence ${L_{+}}(\tfrac{1}{2})<1.4715$.
We apply Proposition \[prop:L-bounds\](c) by starting with the $10$ intervals $I_i\quad(i=1,\cdots,10)$ in Table \[tab:interval 1\], and putting $I_i=1-I_{21-i}$ and $P_i(x)=P_{21-i}(1-x)$ for $i=11,\cdots,20$, with $I_{21}=1+I_1$ and $P_{21}(x)=P_1(x-1)$. (Here again, the endpoints listed in Table \[tab:interval 1\] are approximations only. To find a more accurate values, we would solve for the roots of $P(x) = \pm \left(\frac{1}{2}\right)^{\deg P}$. Higher precision values were used to compute the lower bound ${L_{-}}\left(\tfrac{1}{2}\right) > 1.008848$.) Each polynomial $P_i$ listed has a critical point at $\tfrac{1}{2}$ (and also at $-\tfrac{1}{2}$ in the case of the last polynomial), with $P_i\left(\tfrac{1}{2}\right)=\pm
\left(\tfrac{1}{2}\right)^{\deg{P_i}}$. The value of $|P_i(x)|$ at all other critical points, as well as at the interval endpoints, is strictly less than $\left(\tfrac{1}{2}\right)^{\deg{P_i}}$. This shows in each case that ${\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I_i}= \tfrac{1}{2}$. Then all $21$ intervals $I_i$ have ${t_{\mathrm{M}}}(I_i)=\tfrac{1}{2}$ and, writing $I_i=[a_i,b_i]\quad(i=1,\cdots,21)$ we have $$\label{E:20}
\min_{i=1}^{20}\left(b_{i}-a_{i+1}\right)=b_5-a_6>1.008848.$$ From this it follows by Lemma \[L-simple\](b) that every interval $I$ of length less than $1.008848$ is a subinterval of an integer translate of some $I_i$, so that ${t_{\mathrm{M}}}(I)\le{\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I}\le {\|}P_i{\|^{{}^{\scriptstyle{*}}}}_{I_i}= \tfrac{1}{2}$. This proves part (a) of the Theorem.
Part (b) of the Theorem follows on applying Proposition \[prop:L-bounds\] (d) with $P(x)=x^2-x$. We then have, for $I=\left[\frac{1-\sqrt{2}}{2}, \frac{1+\sqrt{2}}{2}\right]$, that ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=\tfrac{1}{2}$, so that ${L_{+}}(t)\ge
|I|=\sqrt{2}$.
$i$ Polynomial $Q_i$ $t_i$ $\ell_i^-$
----- ------------------------------------------ --------------------------------------- -------------
1 $7 x^3+7 x^2-1 $ $\frac{1}{\sqrt[3]{7}} \approx 0.522$ 1.064961507
2 $3 x^2-1 $ $\frac{1}{\sqrt[2]{3}} \approx 0.577$ 1.154700538
3 $5 x^3+3 x^2-2 x-1 $ $\frac{1}{\sqrt[3]{5}} \approx 0.584$ 1.390656045
4 $2 x^2-1 $ $\frac{1}{\sqrt[2]{2}} \approx 0.707$ 1.414213562
5 $3 x^4-2 x^3-4 x^2+x+1 $ $\frac{1}{\sqrt[4]{3}} \approx 0.759$ 2.173182852
6 $2 x^3-4 x^2+1 $ $\frac{1}{\sqrt[3]{2}} \approx 0.793$ 2.306243643
7 $2 x^4-8 x^3+8 x^2-1 $ $\frac{1}{\sqrt[4]{2}} \approx 0.840$ 2.613125930
8 $2 x^5-15 x^4+39 x^3-40 x^2+12 x+1 $ $\frac{1}{\sqrt[5]{2}} \approx 0.870$ 2.982466529
9 $2 x^6-12 x^5+22 x^4-8 x^3-10 x^2+4 x+1$ $\frac{1}{\sqrt[6]{2}} \approx 0.890$ 3.131521012
: Upper bounds for ${L_{-}}(t)$. Here ${L_{-}}(t)<\ell_i^-$ for $t<t_i$, where $\ell_i^-$ is the span of the roots of the $i$th polynomial (see Theorem \[thm:[L\_[-]{}]{}(t)\]).[]{data-label="tab:L_-(t)"}
$i$ $t_i$ $\ell_i^+$ $i$ $t_i$ $\ell_i^+$
----- ----------------------------------------- ------------ ----- ----------------------------------------- ------------
1 $ \frac{1}{\sqrt[6]{63}} \approx .501 $ 1.47149 31 $ \frac{1}{\sqrt[5]{15}} \approx .582 $ 1.71707
2 $ \frac{1}{\sqrt[6]{60}} \approx .505 $ 1.47887 32 $ \frac{1}{\sqrt[3]{5}} \approx .585 $ 1.72578
3 $ \frac{1}{\sqrt[5]{30}} \approx .506 $ 1.48183 33 $ \frac{1}{\sqrt[6]{24}} \approx .589 $ 1.78511
4 $ \frac{1}{\sqrt[6]{59}} \approx .507 $ 1.48424 34 $ \frac{1}{\sqrt[5]{14}} \approx .590 $ 1.79006
5 $ \frac{1}{\sqrt[4]{15}} \approx .508 $ 1.48823 35 $ \frac{1}{\sqrt[6]{23}} \approx .593 $ 1.80103
6 $ \frac{1}{\sqrt[6]{58}} \approx .508 $ 1.49541 36 $ \frac{1}{\sqrt[4]{8}} \approx .595 $ 1.80333
7 $ \frac{1}{\sqrt[6]{57}} \approx .510 $ 1.49802 37 $ \frac{1}{\sqrt[6]{22}} \approx .597 $ 1.80514
8 $ \frac{1}{\sqrt[6]{56}} \approx .511 $ 1.50442 38 $ \frac{1}{\sqrt[6]{19}} \approx .612 $ 1.82308
9 $ \frac{1}{\sqrt[4]{14}} \approx .517 $ 1.50918 39 $ \frac{1}{\sqrt[4]{7}} \approx .615 $ 1.82808
10 $ \frac{1}{\sqrt[6]{51}} \approx .519 $ 1.51232 40 $ \frac{1}{\sqrt[6]{18}} \approx .618 $ 1.85414
11 $ \frac{1}{\sqrt[3]{7}} \approx .523 $ 1.51409 41 $ \frac{1}{\sqrt[5]{11}} \approx .619 $ 1.86446
12 $ \frac{1}{\sqrt[6]{48}} \approx .525 $ 1.54721 42 $ \frac{1}{\sqrt[6]{17}} \approx .624 $ 1.86909
13 $ \frac{1}{\sqrt[5]{25}} \approx .525 $ 1.54825 43 $ \frac{1}{\sqrt[3]{4}} \approx .630 $ 1.87806
14 $ \frac{1}{\sqrt[4]{13}} \approx .527 $ 1.55329 44 $ \frac{1}{\sqrt[6]{15}} \approx .637 $ 1.92375
15 $ \frac{1}{\sqrt[6]{46}} \approx .528 $ 1.56522 45 $ \frac{1}{\sqrt[5]{9}} \approx .644 $ 1.92862
16 $ \frac{1}{\sqrt[6]{45}} \approx .530 $ 1.57021 46 $ \frac{1}{\sqrt[4]{5}} \approx .669 $ 1.95815
17 $ \frac{1}{\sqrt[5]{23}} \approx .534 $ 1.57066 47 $ \frac{1}{\sqrt[6]{11}} \approx .671 $ 2.03528
18 $ \frac{1}{\sqrt[4]{12}} \approx .537 $ 1.57390 48 $ \frac{1}{\sqrt[3]{3}} \approx .693 $ 2.05072
19 $ \frac{1}{\sqrt[5]{21}} \approx .544 $ 1.58148 49 $ \frac{1}{\sqrt{2}} \approx .707 $ 2.07313
20 $ \frac{1}{\sqrt[4]{11}} \approx .549 $ 1.59285 50 $ \frac{1}{\sqrt[6]{7}} \approx .723 $ 2.46521
21 $ \frac{1}{\sqrt[5]{20}} \approx .549 $ 1.60583 51 $ \frac{1}{\sqrt[5]{5}} \approx .725 $ 2.49418
22 $ \frac{1}{\sqrt[6]{36}} \approx .550 $ 1.62320 52 $ \frac{1}{\sqrt[6]{6}} \approx .742 $ 2.55291
23 $ \frac{1}{\sqrt[6]{34}} \approx .556 $ 1.63662 53 $ \frac{1}{\sqrt[5]{4}} \approx .758 $ 2.58796
24 $ \frac{1}{\sqrt[6]{33}} \approx .558 $ 1.64392 54 $ \frac{1}{\sqrt[4]{3}} \approx .760 $ 2.60202
25 $ \frac{1}{\sqrt[5]{18}} \approx .561 $ 1.65596 55 $ \frac{1}{\sqrt[6]{5}} \approx .765 $ 2.61238
26 $ \frac{1}{\sqrt[6]{32}} \approx .561 $ 1.65815 56 $ \frac{1}{\sqrt[6]{4}} \approx .794 $ 2.70928
27 $ \frac{1}{\sqrt[4]{10}} \approx .562 $ 1.66032 57 $ \frac{1}{\sqrt[5]{3}} \approx .803 $ 2.89569
28 $ \frac{1}{\sqrt[6]{31}} \approx .564 $ 1.66308 58 $ \frac{1}{\sqrt[6]{3}} \approx .833 $ 2.97756
29 $ \frac{1}{\sqrt[5]{16}} \approx .574 $ 1.67218 59 $ \frac{1}{\sqrt[4]{2}} \approx .841 $ 2.98928
30 $ \frac{1}{\sqrt{3}} \approx .577 $ 1.68244 60 $ \frac{1}{\sqrt[5]{2}} \approx .871 $ 3.23520
: Upper bounds for ${L_{+}}(t)$. Here ${L_{+}}(t)<\ell_i^+$ for $t<t_i$, where $\ell_i^+$ is the span of the roots of the $i$th polynomial (see Theorem \[thm:[L\_[-]{}]{}(t)\]).[]{data-label="tab:L_+(t)"}
General bounds for ${L_{-}}(t)$ and ${L_{+}}(t)$
================================================
In this section we find upper and lower bounds for ${L_{-}}(t)$ and ${L_{+}}(t)$, valid for $t$ from $0.5$ to close to $0.9$. Our first result gives the upper bounds.
\[thm:[L\_[-]{}]{}(t)\]
- For all $t_i$ and $\ell_i^-$ in Table \[tab:[L\_[-]{}]{}(t)\] and for all $t<t_i$ we have ${L_{-}}(t) < \ell_{i}^-$.
- For all $t_i$ and $\ell_i^+$ in Table \[tab:[L\_[+]{}]{}(t)\] and for all $t<t_i$ we have ${L_{+}}(t) < \ell_{i}^+$.
The Theorem is proved by applying Proposition \[prop:L-bounds\] (a) and (b) for a range of values in $[0.5,1]$. Here again, the diameter given in Table \[tab:[L\_[-]{}]{}(t)\] can be computed more exactly by considering the difference between the maximal and minimal roots of the obstruction polynomial. For Table \[tab:[L\_[+]{}]{}(t)\], a calculation similar to that done for Table \[tab:1.48\] was done for each $t_i$. The rounding procedure was that used for Table \[tab:1.48\]. Then the monotonicity of ${L_{-}}(t)$ and ${L_{+}}(t)$ gives the result for all $t$ in this range.
For the lower bounds, we first define the normalized polynomial $P_\alpha$ $$\label{eqn:normal}
P_\alpha(x)
= (x (1-x))^{\tfrac{1-\alpha}{2}}
(x^2-x-1)^{\tfrac{\alpha}{2}},$$ of degree $1$, and let ${\alpha}^*\approx 0.4358$ be the root in $(0,1)$ of the equation $$\label{eqn:root}
4{\alpha}^{\alpha}(1-{\alpha})^{1-{\alpha}}=5^{\alpha}.$$
The following result gives the lower bounds.
\[P-lowerB\] For $0\le{\alpha}\le \frac{\ln 4}{\ln 5}$ we have
- ${L_{+}}\left(\frac{5^{\alpha/2}}{2}\right)\ge \ell_{\alpha}$, where $\ell_{\alpha}$ is the root of $P_{\alpha}\left(\tfrac{1}{2}+\ell_{\alpha}/2\right)=\frac{5^{\alpha/2}}{2}$ in $$\begin{cases} (\sqrt{5},\infty) &\text{ if } {\alpha}>{\alpha}^*;\\
(1,\sqrt{5}) &\text{ if } {\alpha}\le{\alpha}^*.
\end{cases}$$
- ${L_{-}}\left(\frac{5^{\alpha/2}}{2}\right)\ge \max(\ell_{\alpha}-1,1.008848)$.
$$\psfig{file=lk.ps,height= 300pt,width= 300pt,angle=270}$$
$$\psfig{file=Lk.ps,height= 300pt,width= 300pt,angle=270}$$
For the proof, we need the following simple observation.
\[lem:Ll\] If ${L_{+}}(t) \geq \ell+1$ then ${L_{-}}(t) \geq \ell$.
This follows straight from the fact that, given an interval $I$ of length $\ell+1$, every interval of length $\ell$ has an integer translate that is a subinterval of $I$.
It should first be pointed out that this proposition is in fact true for all $\alpha$, and not just those in the range specified. That being said, for $\alpha > \frac{\ln 4}{\ln 5}$ we would have $\frac{5^{\alpha/2}}{2} > 1$, in which case we could appeal to Lemma \[lem:LL\] (f) for the exact answer.
- We will proceed to analyze ${\|}P_\alpha(x){\|^{{}^{\scriptstyle{*}}}}_{I_\ell}$, picking $\alpha$ and $\ell$ such that, at the endpoints of the interval $I_\ell$, $|P{\alpha}(x)|$ equals the largest local maximum of $|P{\alpha}(x)|$ in the interior of $I_\ell$. (Notice that $P_{\alpha}$ is already normalized, so ${\|}P_{\alpha}{\|^{{}^{\scriptstyle{*}}}}_{I_\ell} = ||P_{\alpha}||_{I_\ell}||$.) (See Figures \[fig:alpha<alpha\*\] and \[fig:alpha>alpha\*\].)
$$\psfig{file=alphaL.ps,height= 300pt,width= 300pt,angle=270}$$
$$\psfig{file=alphaU.ps,height= 300pt,width= 300pt,angle=270}$$
Notice first that $$\left|P_\alpha\left(\tfrac{1}{2}+x/2\right)\right|
=\frac{5^{\tfrac{\alpha}{2}}}{2}\left|1-x^2\right|^{\tfrac{1-\alpha}{2}}
\left|1-\frac{x^2}{5}\right|^{\tfrac{\alpha}{2}},$$ which has a local maximum of $\frac{5^{\alpha/2}}{2}$ at $x=0$, and a local maximum of $m_\alpha=|1-\alpha|^{(-\alpha)/2}|\alpha|^{\alpha/2}$ at $x^2=5-4\alpha$ . Now the equation $m_{\alpha}=\frac{5^{\alpha/2}}{2}$ has a root ${\alpha}$ defined by (\[eqn:root\]), with $m_{\alpha}>\frac{5^{\alpha/2}}{2}$ for ${\alpha}<{\alpha}^*$ and $m_{\alpha}<\frac{5^{\alpha/2}}{2}$ for ${\alpha}>{\alpha}^*$. Hence if ${\alpha}\ge
{\alpha}^*$ then $|P_\alpha(\tfrac{1}{2}+x/2)|\le\frac{5^{\alpha/2}}{2}$ for $x\le \sqrt{5}$, so that ${\|}P_{\alpha}{\|^{{}^{\scriptstyle{*}}}}_{I_{\alpha}}=\frac{5^{\alpha/2}}{2}$, where $I_{\alpha}=\left[\tfrac{1}{2}-\ell_{\alpha}/2,\tfrac{1}{2}+\ell_{\alpha}/2\right]$ with $\ell_{\alpha}$ the root $\ell_{\alpha}>\sqrt{5}$ of $P_\alpha\left(\tfrac{1}{2}+\ell_{\alpha}/2\right) =\frac{5^{\alpha/2}}{2}$. However, if ${\alpha}<{\alpha}^*$ then we have the same result, but only for $\ell_{\alpha}$ the root in $(1,\sqrt{5})$ of $P_\alpha(\tfrac{1}{2}+\ell_{\alpha}/2)=\frac{5^{\alpha/2}}{2}$. This gives the lower bound ${L_{+}}\left(\frac{5^{\alpha/2}}{2}\right)\ge \ell_{\alpha}$, but with a left discontinuity in $\ell_{\alpha}$ (as a function of ${\alpha}$) at ${\alpha}={\alpha}^*$. A plot of this lower bound, along with the upper bounds from Theorem \[thm:[L\_[-]{}]{}(t)\] and Table \[tab:[L\_[+]{}]{}(t)\], is given in Figure \[fig:[L\_[+]{}]{}(t)\].
- We know that ${L_{-}}$ is a non-decreasing function, and that ${L_{-}}(\tfrac{1}{2}) \geq 1.008848$. Combining these facts with Lemma \[lem:Ll\] we get that ${L_{-}}\left(\frac{5^{\alpha/2}}{2}\right)
\geq \max(\ell_\alpha - 1, 1.008848)$. This is displayed numerically, along with the upper bounds from Theorem \[thm:[L\_[-]{}]{}(t)\] and Table \[tab:[L\_[-]{}]{}(t)\], in Figure \[fig:[L\_[-]{}]{}(t)\].
Intervals of length 1: Proof of Theorem \[thm:interval 1\] {#sec:interval 1}
==========================================================
From Theorem \[thm:[L\_[-]{}]{}(1/2)\] (a) we know that every interval $I$ of length $\ell \le 1.008848$ has ${t_{\mathrm{M}}}(I)\le
\frac{1}{2}$. Now since every interval of length $\ell \ge 1$ has some integer translate that contains $\tfrac{1}{2}$, we have $$\tfrac{1}{2}={t_{\mathrm{M}}}\left(\left\{\tfrac{1}{2}\right\}\right)\le {t_{\mathrm{M}}}(I)$$ for all such intervals, so that ${t_{\mathrm{M}}}(I)=\tfrac{1}{2}$ for all $I$ with $1\le |I|\le 1.008848$.
If $b>1.064961507$ then again from Theorem \[thm:[L\_[-]{}]{}(1/2)\] (a), with the polynomial $Q(x) = 7 x^3 + 7 x^2 - 1$, there is an interval $I$ of length $b$ with ${t_{\mathrm{M}}}(I)>\frac{1}{2}$.
To complete the proof, note that for $b<1$ $$\begin{aligned}
{t_{\mathrm{M}}}([-b/2,b/2])&=\sqrt{{t_{\mathrm{M}}}([0,b^2/4])},\\
\intertext{on applying \cite[Prop 1.4 with the polynomial
$x^2$]{BorweinPinnerPritsker03}, and then }
\sqrt{{t_{\mathrm{M}}}([0,b^2/4])} & \le \sqrt{b^2/4}<\tfrac{1}{2},\end{aligned}$$ using the polynomial $P(x)=x$ on $[0,b^2/4]$.
Computational methods {#sec:comp}
=====================
Finding optimal monic integer Chebyshev polynomials [*P*]{}
-----------------------------------------------------------
We now describe how the polynomials of Table \[tab:interval 1\] were found. These are optimal monic integer polynomials $P$ having ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=\tfrac{1}{2}$ on various intervals of length just greater than $1$. For these intervals, the maximal obstruction polynomial is $Q(x)=2x-1$, and the maximal obstruction is $m=\tfrac{1}{2}$. The method applies more generally, however, to any interval $I$ having a maximal obstruction polynomial $Q$, so we shall describe the method for this more general situation. We suppose that the maximal obstruction is $m=a_d^{-1/d}$, where $Q(x)=a_dx^d+\dots+a_0$, so that we seek a monic integer polynomial $P$ with ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=m$.
Firstly, potential factors of $P$ of small degree $k$ were identified using LLL [@BorweinPinnerPritsker03; @Hare02a; @LenstraLenstraLovasz82]. The basis used was $[1, x, \cdots, x^k]$, with the inner product $$\langle R_1, R_2 \rangle = \int_I R_1(x) R_2(x)\ dx + b_k c_k.$$ Here $R_1(x) = b_k x^k + \cdots + b_0$ and $R_2(x) = c_k x^k + \cdots + c_0$. The $b_k c_k$ component of the inner product was inserted to discourage nonmonic polynomials from appearing in the basis returned by LLL. Now, at least one element in the basis will contain an $x^k$ term and, because of the $b_k c_k$ penalty, such an element is almost always monic. (In fact always in the examples we computed.) So we obtained a monic polynomial of degree $d$ with small $L_2$ norm, which usually also had a small supremum norm. These monic polynomials with small $L_2$ norm are not necessarily irreducible. At this point we examined each of their irreducible factors $f_i$, again monic polynomials, and applied Lemma \[lem:resultant-value\](a) below to eliminate some of them. We then used the method of Borwein and Erd' elyi [@BorweinErdelyi96] to search for exponents ${\alpha}_i \in {\mathbb{N}}$ such that $P^{1/\deg P}:=\prod_i f_i^{{\alpha}_i/\deg f_i}$ has the desired property ${\|}P{\|^{{}^{\scriptstyle{*}}}}_I=m$. To do this, we needed to minimize $t$ subject to the constraint $$\sum_i \frac{\alpha_i}{\deg f_i} \log(|f_i(x)|) \leq t,$$ for all $x \in I$ with $\sum_i \alpha_i = 1$, $0 \leq \alpha_i$. Some additional constraints on the $\alpha_i$ that we made use of are given by Lemma \[lem:resultant-value\] (b), (c). The main difference between our application and the original one is that here the polynomials $f_i$ are all monic. By choosing a large number of points $x \in I$ to discretize the problem, we get a system of linear equations, on which the Simplex method can be used to get a good estimate of $\min(t)$ [@BorweinErdelyi96; @HabsiegerSalvy97; @Schrijver86]. In practice, with a high enough precision and a large enough number of sample points, we obtain $\min(t)=m$ exactly, and the corresponding ${\alpha}_i$ then give the required $P$. We then check that $P$ is indeed an optimal monic integer Chebyshev polynomial for $I$ by checking algebraically that $|P|^{1/\deg P}= m $ at all roots of the maximal obstruction polynomial $Q$, and furthermore that all other local maxima of $|P|$ in this interval are strictly smaller than $m$.
The following lemma, used to help construct these polynomials $P$, specifies extra properties that their factors $f_i$ and normalized exponents ${\alpha}_i$ must have.
\[lem:resultant-value\] Let $I$ be an interval that has a maximal obstruction polynomial $Q(x) = a_d x^d + \cdots +a_0$. Suppose further that $P(x)$ attains the maximal obstruction, and that $P(x)^{1/\deg P}
=\prod_if_i^{\alpha_i/\deg f_i}$, with $\sum_i\alpha_i=1$. Then
1. The resultant ${\mathrm{Res}}(f_i, Q)$ is equal to $ \pm 1$ for every factor $f_i\in{\mathbb{Z}}[x]$ of $P$.
2. For every root $\beta$ of $Q$ we have $$\sum_i\frac{{\alpha}_i}{\deg f_i} \times \frac{f'_i(\beta)}{f_i(\beta)}=0.$$
3. Fix a root $\beta\in\mathbb R$ of $Q$, and put $\hat f_i=|f_i(\beta)|^{1/\deg f_i}\in\mathbb R$. Let ${{\mathcal F}}$ be the multiplicative subgroup of $\mathbb R_{>0}$ generated by $a_d$ and the $\hat f_i$ with $b_1=a_d$ and $b_2, \cdots, b_k$ an independent generating set for ${{\mathcal F}}$, with say ${\hat f_i}^{1/\deg f_i} = \prod_j b_j^{ c_{j,i}
}$ for some integers $c_{j,i}$. Then $$\sum_i c_{j,i} \alpha_i =\begin{cases}
-1/d &\text{ if } j=1;\\
0 &\text{ if } j>1.
\end{cases}$$
We have $\prod_i P(\beta_i) = \pm 1/a_d^{\deg P}$, where the product is taken over the roots $\beta_i$ of $Q$, so that, from (\[eqn:resultant\]), ${\mathrm{Res}}(P, Q) = \pm 1$. Then (a) follows from the fact that the resultant of a product with $Q$ is the product of the resultants with $Q$.
The second part follows from the fact that all the roots $\beta$ of $Q$ must be critical points of $P(x)$. Further, since $P(x)$ attains the maximal obstruction, we have from Lemma BPP that for all such $\beta$ we have $|P(\beta)|^{1/\deg P} = a_d^{-1/d}$, giving the third part.
Note that Lemma \[lem:resultant-value\] simplifies considerably when the maximal obstruction polynomial is linear, say $a_1 x - a_0$. Then it says that $f_i\left(\frac{a_0}{a_1}\right) = \pm a_1^{-\deg f_i}$ and with $P'\left(\frac{a_0}{a_1}\right) = 0$.
The independent generating set $b_1,\cdots,b_k$ for ${{\mathcal F}}$ was found using the integer relation-finding program PSLQ, which we used to search for linear integer relations between $\log a_d$ and the $\log\hat f_i$.
As we have seen, the method for finding an optimal monic integer Chebyshev polynomial $P$ depends on first finding the (in practice there was only one) maximal obstruction polynomial for the interval. We now describe how to do this.
Finding obstruction polynomials [*Q*]{}
---------------------------------------
The obstruction polynomial $7 x^3-7 x^2+1$, as well as those listed in Table \[tab:1.48\] and \[tab:[L\_[-]{}]{}(t)\], were found using the technique of Robinson [@Robinson64] (see also [@McKeeSmyth04; @Smyth84]). In this method, the aim is to search for all degree $d$ polynomials $Q(x)=a_dx^d+\cdots +a_0$ having all their roots in an interval $I_0$, for fixed degree, and fixed lead coefficient, $a_d$, with $a_d \leq 2^d$. We describe below how $I_0$ is chosen. Robinson’s method uses the fact that for $k=1, 2, \cdots, d-1$ the span of the roots of the $k$th derivative of $Q$ is contained in the span of the roots of the $(k-1)$th derivative of $Q$. In particular, these derivatives have all their roots in $I_0$.
Starting with the $(d-1)$st derivative of $Q$, we get a range of possible valid values for $a_{d-1}$. Consider then the $(d-2)$nd derivative to find valid ranges for $a_{d-2}$. Continuing in this fashion, we obtain a list of polynomials, each one having all its roots in $I_0$. We now sieve this list, first by eliminating all polynomials that are reducible, or have integer content greater than $1$. Having obtained a list of irreducible polynomials, we can then prune it further, as follows. If $Q(x)$ and $R(x)$ are both irreducible polynomials, with the same degree and lead coefficient, and the span of the roots of $R(x)$ contain the roots of $Q(x)$, then for any interval $I$ where $R(x)$ is an obstruction polynomial, $Q(x)$ is also an obstruction polynomial, and hence $R(x)$ is not needed.
After construction of these polynomials, we can, for fixed $d, a_d$, and $t < a_d^{-1/d}$ find an upper bound for ${L_{-}}(t)$ by finding the polynomial $Q$ whose roots have the smallest span, and then appealing to Proposition \[prop:L-bounds\] (a). This was done in Table \[tab:[L\_[-]{}]{}(t)\], formalized in Theorem \[thm:[L\_[-]{}]{}(t)\] (a), and displayed in Figure \[fig:[L\_[-]{}]{}(t)\].
Similarly, given this list of polynomials, we can compute the least $\ell$ such that any interval of length $\ell$ will contain an integer translate of at least one of the polynomials in our list. Then with Proposition \[prop:L-bounds\] (b) we get an upper bound for ${L_{+}}(t)$. for given $\ell$, we must choose $I_0$ carefully. If $I_0$ is too short, we might miss an important obstruction polynomial. On the other hand, if $I_0$ is long, we will find, along with the obstruction polynomials we seek, also (possibly multiple) integer translates of these polynomials. This is inefficient, as we end up doing more calculations than we need to. So we wish to pick $I_0$ so that it is long enough to ensure that we have all important obstruction polynomials, and yet small enough that we are not doing more work than necessary. We do this by ensuring that $I_0$, the interval which contains the roots of the polynomials we have found, has the property that $|I_0|$ is just greater $\ell+1$. This ensures that there are no other useful obstruction polynomials that we might have missed, since any obstruction polynomial having a span of length $\ell$ will then have some integer translate lying in $I_0$. (We might have to re-run the calculation if $|I_0|$ is too small based on the current value of $\ell$.) We can achieve tighter upper bounds for ${L_{+}}(t)$ by considering the list of all obstruction polynomials we found such that $a_d^{-1/d} \geq t$.
This computation was done for $t = \frac{1}{2}$ (Table \[tab:1.48\] and Theorem \[thm:[L\_[-]{}]{}(1/2)\]) and also for $20$ other values of $t$ (Table \[tab:[L\_[+]{}]{}(t)\], Theorem \[thm:[L\_[-]{}]{}(t)\] (b) and Figure \[fig:[L\_[+]{}]{}(t)\]). To save space, the list of relevant polynomials for each $t$ is not given in the table. (This information is available upon request from the authors.)
critical polynomials: results and proofs {#sec:crit}
========================================
We first establish a relationship between critical polynomials and maximal obstructions. We define a [*maximal*]{} nonmonic critical polynomial of an interval $I$ to be a critical polynomial $Q(x)=a_dx^d+\cdots+a_0$ such that the value $a_d^{-1/d}$ is maximal for $Q$ within the set of nonmonic critical polynomials for $I$. Such a polynomial is well defined, as a result of the following Theorem.
\[thm:critical=maximal\] Suppose that an interval $I$ has a nonmonic critical polynomial. Then $I$ has a maximal nonmonic critical polynomial, $Q(x) = a_d x^d + \cdots + a_0$ say, and furthermore $Q$ is also a maximal obstruction polynomial, so that $a_d^{-1/d}$ is the maximal obstruction.
To prove this result, we will apply the following version of a classical lemma.
\[lem:was BE\] Let $Q(x)$ and $R(x)$ be two (not necessarily monic) integer polynomials. Further suppose that $Q(x) = a_d x^d + \cdots + a_0$ is a critical polynomial for the interval $I$, and that the integer polynomial $R(x)$ satisfies ${\|}R{\|^{{}^{\scriptstyle{*}}}}_I< a_d^{-1/d}$. Then $Q$ divides $R$.
From equations (\[eqn:resultant\]) and (\[eqn:resultant2\]), with $R(x)$ replacing $P(x)$, we must have ${\mathrm{Res}}(Q,R)=0$.
This result, essentially known to early workers on integer transfinite diameter (Gorškov, Sanov, Trigub, Aparicio Bernardo, ...), has appeared in the literature in various forms – see for instance Chudnovsky [@Chudnovsky83 Lemma 2.3], Montgomery [@Montgomery94 Chapter 10], Borwein and Erdélyi [@BorweinErdelyi96], Flammang, Rhin and Smyth [@FlammangRhinSmyth97].
We first observe that nonmonic critical polynomials are obstruction polynomials. Conversely, if an obstruction is greater than ${t_{{\mathbb{Z}}}}(I)$ then its associated polynomial is also a critical polynomial.
Assume that $I$ has a nonmonic critical polynomial, and consider the nonempty set ${{\mathcal A}}= \{a_d^{-1/d}\}$ of obstructions coming from the nonmonic critical polynomials of $I$. Any integer polynomial $R(x)$ (not necessarily monic), must, by Lemma \[lem:was BE\], contain as factors all critical polynomials $Q$ whose obstructions $a_d^{-1/d}$ are strictly greater than ${\|}R(x){\|^{{}^{\scriptstyle{*}}}}_I$. Therefore ${\|}R(x){\|^{{}^{\scriptstyle{*}}}}_I\geq \ell$ for any limit point $\ell$ of ${{\mathcal A}}$, and hence ${t_{{\mathbb{Z}}}}(I) = \ell$. So if ${{\mathcal A}}$ has a limit point, then it must be $\inf({{\mathcal A}})$. Thus $\sup({{\mathcal A}})$ is attained, and there is a maximal nonmonic critical polynomial $Q$ say. Then $Q$ is also a maximal obstruction polynomial.
\[cor:critical=maximal\] Conjecture \[conj:maximal\] and Conjecture \[conj:maximal=tm\] together imply Conjecture \[conj:critical\].
From the proof above, we see that an obstruction that is greater than ${t_{{\mathbb{Z}}}}(I)$ is associated to a critical polynomial. The existence of such an obstruction is a consequence of Conjecture \[conj:maximal\] and Conjecture \[conj:maximal=tm\].
Now ${t_{{\mathbb{Z}}}}(I)\leq \inf_i a_{d_i,i}^{-{1}/{d_i}}$, by the definition of a critical polynomial. But if this inequality were strict, then we could find an integer polynomial $R$ with ${t_{{\mathbb{Z}}}}(I)\leq {\|}R{\|^{{}^{\scriptstyle{*}}}}_I<\inf_i a_{d_i,i}^{-{1}/{d_i}}$. But then, from Lemma \[lem:was BE\], $R$ would have to be divisible by all the $Q_i$, which is impossible.
Farey intervals and the proof of Theorem \[thm:maximal\] {#sec:Farey}
========================================================
Every closed interval $I$ has a least positive integer $q$ such that some rational $p/q$ with $(p,q)=1$ lies in the interior of $I$. If $q\ge 2$ then $I$ belongs to a unique Farey interval ${\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}$ whose endpoints are consecutive fractions in the Farey sequence of order $q-1$. We define this interval to be the [*minimal Farey interval containing $I$*]{}.
Theorem \[thm:maximal\] follows directly from our next result.
\[thm:rational\] Let $I$ be an interval not containing an integer in its interior, and ${\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}$ be the minimal Farey interval containing $I$. Then $(c_1+c_2) x - (b_1+b_2)$ is a critical polynomial for $I$. Moreover, the maximal obstruction for $I$ is $$=
\begin{cases}
\frac{1}{c_1} & \text{ if } c_1\ge 2, \frac{b_1}{c_1}\in I,\frac{b_2}{c_2}\not\in I;\\
\frac{1}{c_2} & \text{ if } c_2\ge 2, \frac{b_1}{c_1}\not\in I,\frac{b_2}{c_2}\in I;\\
\frac{1}{\min(c_1,c_2)} & \text{ if } c_1\ge 2, c_2\ge 2 \text{ and } I={\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]};\\
\frac{1}{c_1+c_2} & \text{ otherwise. }
\end{cases}$$
Now the polynomial $Q(x) = (c_1 x - b_1)^{c_2} (c_2 x - b_2)^{c_1}$ has a local maximum of $\left(\frac{1}{c_1+c_2}\right)^{c_1+c_2}$ at $x= \frac{b_1+ b_2}{c_1+c_2}$ . Thus, by continuity, there exist integers $r_1$ and $r_2$ such that $R(x):=Q(x)^{r_1} ((c_1+c_2) x - (b_1+b_2))^{r_2}$ has normalized supremum less than $\frac{1}{c_1+c_2}$. Hence $(c_1+c_2) x - (b_1+b_2)$ is an obstruction polynomial. Now $\frac{b_1+ b_2}{c_1+c_2}\in I$, as otherwise $I$ would be contained in one of the Farey intervals $\left[\frac{b_1}{c_1},\frac{b_1+
b_2}{c_1+c_2}\right]$ or $\left[\frac{b_1+
b_2}{c_1+c_2},\frac{b_2}{c_2}\right]$.
Since the polynomials $(c_1+c_2) x - (b_1+b_2)$, $c_1x-b_1$ and $c_2x-b_2$ are critical only if their roots are in $I$, and are, as factors of $R$, by Lemma \[lem:was BE\] the only three possible maximal critical polynomials in this Farey interval, we get the final result.
\[thm:farey\] Let ${\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}$ with $c_1\ge 2$ be a Farey interval, and suppose that $b_1^2 \equiv \pm 1 \pmod{c_1}$ and $ b_2^2\equiv B \pmod {c_2}$ where $c_1^2 |B| < c_2^2$. Then $${t_{\mathrm{M}}}\left({\left[\tfrac{b_1}{c_1},\tfrac{b_2}{c_2}\right]}\right) = \frac{1}{c_1}.$$
From [@BorweinPinnerPritsker03 p. 1905] we have that there exists a monic quadratic integer polynomial $P(x)$ which has the property that $P\left(\frac{b_1}{c_1}\right) = \pm\frac{ 1}{c_1^2}$ and $P\left(\frac{b_2}{c_2}\right) = \frac{B}{c_2^2}$. Since its critical point is at a half integer, it is strictly monotonic on the Farey interval. Hence it attains its maximum at one of its endpoints, and $\left|P\left(\frac{b_1}{c_1}\right)\right|>
\left|P\left(\frac{b_2}{c_2}\right)\right|$.
\[thm:farey 2\] Let $P(x) = x^2 + a_1 x + a_0$ be an irreducible integer polynomial with real roots. Then there exist infinitely many Farey intervals for which $P(x)$ attains the maximal obstruction.
We know (Pell’s Equation) that the equation $x^2 + a_1 x y + a_0
y^2 = \pm 1$ has an infinite number of solutions $(x,y)=(b_i,c_i)$. These solutions have the property that $P\left(\frac{b_i}{c_i}\right)=\pm\frac{ 1}{c_i^2}$. Further, by choosing a suitable subsequence we may assume that both the $c_i$ and the $b_i/c_i$ are monotonically increasing. Thus for any interval $I :=
\left[\frac{b_i}{c_i},\frac{b_{i+1}}{c_{i+1}}\right]$ not containing a half-integer, we see that $P(x)$ attains the maximal obstruction $1/c_i$ with $Q(x)=c_ix-b_i$, so that ${t_{\mathrm{M}}}(I) =1/c_i$. This happens infinitely often as the $\frac{b_i}{c_i}$ tend to a root of $P(x)$.
We can find a $\frac{b}{c}\in\left[\frac{b_i}{c_i},\frac{b_{i+1}}{c_{i+1}}\right]$ such that $\left[\frac{b_i}{c_i},\frac{b}{c}\right]$ is a Farey interval, and hence $P(x)$ attains its maximal obstruction $1/c_i$ on this interval.
It should be noted that this method of proof will not work for polynomials of degree 3 or higher, as the resulting Thue equation $$x^n + a_{n-1} x^{n-1} y + \cdots + a_0 y^n = \pm 1$$ has only a finite number of integer solutions [@Sprindzuk93].
Study of ${t_{\mathrm{M}}}(b)$ {#sec:interval 1/3}
==============================
In this section we consider intervals $[0,b]$, with ${t_{\mathrm{M}}}(b)$ denoting ${t_{\mathrm{M}}}([0,b])$. Our first result for such intervals is a consequence of Theorem \[thm:rational\].
Let $n\ge 2$ and $\frac{1}{n} < b < \frac{1}{n-1}$. Then $\frac{1}{n}$ is the maximal obstruction of $[0, b]$.
\[thm:interval 1/n\] For all $n \in {\mathbb{N}}$ there exists $\delta_n
>{\frac{2}{n+\sqrt{n^2-4}}}-\frac{1}{n}$ such that for all $0 \leq \varepsilon \leq \delta_n$ $${t_{\mathrm{M}}}\left(\left[0, \tfrac{1}{n} + \varepsilon\right]\right) = \tfrac{1}{n}.$$
Consider the polynomial $P_n(x) = x^{n^2-2} (x^2-n x + 1)$. It has the following properties:
- $P_n\left(\frac{1}{n}\right) = \left(\frac{1}{n}\right)^{n^2}$;
- $P_n(x)$ has a local maximum (with respect to $x$) at $x=\frac{1}{n}$;
- $P_n(x)$ is strictly increasing (with respect to $x$) on $\left[0,
\frac{1}{n}\right]$;
- $P_n(x)$ has a root $\beta_n = {\frac{2}{n+\sqrt{n^2-4}}}$ strictly greater than $\frac{1}{n}$;
- $P_n(x)$ is strictly decreasing on $\left[\frac{1}{n}, \beta_n\right]$.
Let $\alpha_n$ be the minimal root, strictly greater than $\beta_n$, of the equation $|P_n(x)| = \frac{1}{n^{n^2}}$. Thus $P_n(x)$ demonstrates that ${t_{\mathrm{M}}}( \alpha_n) = \frac{1}{n}$, where $\alpha_n > \beta_n = 2/(n + \sqrt{n^2-4}) > \frac{1}{n}$.
\[thm:interval 1/3\] We have that
1. ${t_{\mathrm{M}}}(b) = \tfrac{1}{4}$ for $b \in [\tfrac{1}{4}, 0.303]$; \[it:0.3\]
2. ${t_{\mathrm{M}}}(b) = \tfrac{1}{3}$ for $b \in [\tfrac{1}{3}, 0.465]$; \[it:0.46\]
3. ${t_{\mathrm{M}}}(b) = \tfrac{1}{2}$ for $b \in [\tfrac{1}{2}, 1.26]$; \[it:1.26\]
4. ${t_{\mathrm{M}}}(1.328) > \tfrac{1}{2}$. \[it:1.32\]
Hence, in the notation of Theorem \[thm:interval 1/n\], $0.76\le\delta_2<0.828$, $\delta_3 > 0.132 $ and $ \delta_4 > 0.053$.
The optimal monic polynomials needed for Parts (a) and (b) are given in Table \[tab:interval 1/3\]. In each case they attain the maximal obstruction $\tfrac{1}{4}$ and $\tfrac{1}{3}$ respectively. As before, a slightly larger interval can be computed exactly, by solving $P(x) = \pm \left(\frac{1}{4}\right)^{\deg P}$ or $P(x) = \pm \left(\frac{1}{3}\right)^{\deg P}$ respectively. The values of 0.303 and 0.465 have been rounded down to ensure that the inequality still holds. Part (c) follows from the first part of Table \[tab:interval 1\], using the map $x \mapsto 1-x$, with the same comments to the exact values as above. Part (d) is proved using Lemma BPP using the obstruction polynomial $7x^3-14x^2+7x-1$. Here 1.328 is an approximation to its largest root, rounded up to ensure that (d) holds.
The factors used for the construction of the polynomials in Table \[tab:interval 1/3\] were found using the techniques discussed in Section \[sec:comp\], making use of the constraints given by Lemma \[lem:resultant-value\].
[|l|]{} ${t_{\mathrm{M}}}(b) = \frac{1}{4}$ for $b \in [\tfrac{1}{4}, 0.303]$ by $P(x)=$\
$ x^{640}(x^5+432 x^4-456 x^3+179 x^2-31 x+2)^{47}$\
$(x^7+8760 x^6-13342 x^5+8388 x^4-2784 x^3+514 x^2-50 x+2)^{35} $\
\
${t_{\mathrm{M}}}(b) = \frac{1}{3}$ for $b \in [\tfrac{1}{3}, 0.465]$ by $P(x)=$\
$x^{1652706720}(x^7-1233 x^6+2406 x^5-1913 x^4+791 x^3-179 x^2+21 x-1)^{118037088}$\
$(x^8+4842 x^7-10935 x^6+10355 x^5-5317 x^4+1594 x^3-278 x^2+26 x-1)
^{156479575}$\
$(x^8+14184 x^7-34944 x^6+36442 x^5-20832 x^4+7041 x^3-1405 x^2+153 x-7)
^{72166388}$\
$(x^8+7812 x^7-18072 x^6+17561 x^5-9271 x^4+2864 x^3-516 x^2+50 x-2)
^{4378185}$\
Bounds have been given on the exponents of certain factors for large integer Chebyshev polynomials used for estimating ${t_{{\mathbb{Z}}}}(I)$. For example, for the interval $I = [0,1]$, Pritsker [@Pritsker99] shows that $(x (1-x))^\gamma$, where $0.2961 \le \gamma \le 0.3634$, must appear as a factor in any polynomial $R$ (normalized to have degree $1$), for which ${\|}R{\|^{{}^{\scriptstyle{*}}}}_I$ is sufficiently close to ${t_{{\mathbb{Z}}}}(I)$.
Following [@FlammangRhinSmyth97], we now determine a lower bound for $\gamma(b)$ such that $x^{\gamma(b)}$ must divide any normalized monic integer polynomial $P$ such that ${\|}P{\|^{{}^{\scriptstyle{*}}}}_{[0,b]}$ approximates ${t_{\mathrm{M}}}(b)$ sufficiently closely.
Suppose that the function $m(b)$ is an upper bound for ${t_{\mathrm{M}}}(b)$. Then by Proposition 5.3 and Lemma 5.2 of [@FlammangRhinSmyth97] we have that $\gamma(b)$ is bounded below by the least positive root of $$\frac{(1+x)^{1+x}}{(1-x)^{1-x} (2x)^{2x} b^x} = \frac{1}{m(b)}.$$
So in particular, if ${t_{\mathrm{M}}}(b) = \frac{1}{\lceil 1/b \rceil}$ for $b\in[0,1]$ as in Conjecture \[conj:\[0,b\]\], then our lower bound for $\gamma(b)$ would have infinitely many discontinuities in this range (Figure \[fig:gamma\] – black lines). However, we know, by using the polynomial $x$, that we have a provable, albeit weaker, upper bound $m(b)=b$. This gives us a proven lower bound for $\gamma(b)$ (Figure \[fig:gamma\] – grey line).
$$\psfig{file= gamma.ps,height= 300pt,width= 300pt,angle=270}$$
\[thm:gamma(b)\] We have $\displaystyle \lim_{b\to 0} \gamma(b)
= 1$.
Define $$T(x,b) = \frac{(1+x)^{1+x}}{(1-x)^{1-x} (2x)^{2x} b^x} -
\frac{1}{b}.$$
Now $T(x,b)$ has a positive local maximum at $x = \frac{1}{\sqrt{1+4 b}} \to 1$ as $b \to 0$,while $T(1-\sqrt{b}, b) < 0$ for $0< b < 0.04$, so that $T(x,b)=0$ has a root in $[1-\sqrt{b},\frac{1}{\sqrt{1+4 b}}]$. Further, since $T(x,b)$ is increasing for $x\in[0,\frac{1}{\sqrt{1+4 b}}]$ this root is the least positive root of $T(x,b)=0$. Hence $\gamma(b)>1-\sqrt{b}$, giving the result.
Proof of Counterexample \[thm:unattainable\] {#sec:unattainable}
============================================
For the proof of Counterexample \[thm:unattainable\] we need the following $p$-adic result.
\[prop:p-adic\]Suppose that $Q(x)=a_d x^d + \cdots +a_0 \in{\mathbb{Z}}[x]$ is a maximal obstruction polynomial for the interval $I$, and that the maximal obstruction is attained by some monic integer polynomial $P(x)$. Then $\gcd(a_0,a_d)=1$ and, for every prime $p$ dividing $a_d$ we have $$\left|\frac{a_{d-i}}{a_d}\right|_p\le\left|\frac{1}{a_d}\right|_p^{i/d}
\quad (i=0,\cdots,d).$$
In particular, if $a_d$ is square-free then $\frac{1}{a_d}(Q(x)-Q(0))$ has integer coefficients.
Here $|.|_p$ is the usual $p$-adic valuation on ${\mathbb{Q}}$. For the proof, it is extended to $\overline{\mathbb{Q}}$.
Take $\beta$ to be any root of $Q(x)$, and $p$ any prime factor of $a_d$. Let $P(x)$ be of degree $m$. Then, as $P(x)$ attains the obstruction, $P(\beta)=\pm a_d^{-m/d}$, so that $|P(\beta)|_p=|1/a_d|_p^{m/d}>1$. If $|\beta|_p\le 1$ then $|P(\beta)|_p\le 1$, a contradiction, as $P(x)$ has integer coefficients. Hence $|\beta|_p>1$ and $|P(\beta)|_p=|\beta|_p^m=|1/a_d|_p^{m/d}$, giving $$\label{E-1}
|\beta|_p=|a_d|_p^{-1/d}.$$ Applying (\[E-1\]) for all roots $\beta_j$ of $Q(x)$ we get $|\prod_j\beta_j|_p=|1/a_d|_p$. But also from $a_d^{-1}Q(x)=\prod_j(x-\beta_j)$ we have that $|\prod_j\beta_j|_p=|a_0/a_d|_p$. Hence $|a_0|_p=1$. Doing this for all $p|a_d$ we obtain $(a_0,a_d)=1$. Furthermore, if $\left|\frac{a_{d-i}}{a_d}\right|_p >
\left|\frac{1}{a_d}\right|_p^{i/d}$ for any $i$ then the Newton polygon of $P$ (see for instance [@Weiss63 p. 73]) tells us that $|\beta_j|_p > |1/a_d|_p^{1/d}$ for some $j$, contradicting (\[E-1\]).
In the case of $a_d$ square-free, $\left|\frac{1}{a_d}\right|_p^{i/d}< p$ for $1\le i<d$, so that $\left|\frac{a_{d-i}}{a_d}\right|_p\le 1$, and hence, using all primes $p$ dividing $a_d$, we see that $\frac{a_{d-i}}{a_d}$ is an integer.
The fact that $7 x^3 + 4 x^2 - 2 x -1$ is a maximal obstruction polynomial for the interval $I=[-0.684, 0.517]$ can be verified by showing that it is a critical polynomial. This follows from the fact that the polynomial $$\begin{aligned}
R(x)&=& x^{28728} (5 x^3+4 x^2-x-1)^{3739} (7 x^3+4 x^2-2 x-1)^{1140} \\
&& (x^6-24 x^5-20 x^4+10 x^3+9 x^2-x-1)^{420}\\
&& (3 x^5+16 x^4+3 x^3-8 x^2-x+1)^{399}\end{aligned}$$ has ${\|}R{\|^{{}^{\scriptstyle{*}}}}_I<7^{-1/3}$, so that ${t_{{\mathbb{Z}}}}(I) < 7^{-1/3}$. As $7 x^3 + 4 x^2 - 2 x - 1$ has all its roots in $I$, it is therefore a critical polynomial. As always, the interval is an approximation only, and a tighter one can easily be computed.
We now claim that $7 x^3+4 x^2-2 x-1$ is the maximal nonmonic critical polynomial for $I$. For any critical polynomial $a_dx^d+\cdots+a_0$ for $I$ with $a_d^{-1/d}>{\|}R{\|^{{}^{\scriptstyle{*}}}}_I$ must be a factor of $R$, by Lemma \[lem:was BE\]. But among the four irreducible factors of $R$, $7 x^3+4 x^2-2 x-1$ is the only one having all its roots within $I$. As it is nonmonic, it must indeed be the maximal nonmonic critical polynomial for $I$. By Theorem \[thm:critical=maximal\], this polynomial is the maximal obstruction polynomial. However, $\frac{1}{7}(7 x^3+4 x^2-2
x)$ does not have integer coefficients so that, by Proposition \[prop:p-adic\], the interval has no optimal monic integer Chebyshev polynomial.
Some Final Comments on the Computations and Figures
===================================================
Consider Figure \[fig:[L\_[-]{}]{}(t)\]. We see that ${L_{-}}(t) = 0$ for for $t < \frac{1}{2}$, and further that ${L_{-}}(t) = 4 t$ for $t > 1$. So in fact the area of interest is for $t$ between $\frac{1}{2}$ and $1$. That being said, the upper bound is only given up to approximately $0.89$. This is because the upper bound from Proposition \[prop:L-bounds\](a) is given by high degree polynomials with small lead coefficient. In our search, we compute only up to degree 6. As $2^{-1/6} \approx 0.89$ this is the limit to our knowledge of the upper bound. If we wished to extend these calculations, we could extend the knowledge of the upper bound, but the computation time becomes excessive. For example, even if we computed up to degree 10, which is probably beyond our computational range, we would only get up to $0.933$. As it was, the computations up to degree 6 took over 3000 CPU hours, and the computation time approximately triples for each additional degree. Similar comments apply to bounding ${L_{+}}(t)$ (Figure \[fig:[L\_[+]{}]{}(t)\]) for $t$ close to $1$. In this case, it actually turned out that none of the polynomials with lead coefficient 2 and degree 6 were useful in the calculations for such $t$, and hence we only get an upper bound for $L_+(t)$ for $t$ up to $t=2^{-1/5} \approx 0.871$.
While we know from Lemma \[lem:LL\](c) that ${L_{+}}(t)\geq 2t$ for $t
\le 1/2$, we do not know ${L_{+}}(t)$ exactly in this range. In order to get an upper bound for ${L_{+}}(t)$ in at least part of this range, it would in principle be possible to extend the calculation downwards from $t=\tfrac{1}{2}$. The lower bound of $\frac{1}{2}$ for $t$ was chosen, as we computed obstruction polynomials of degree $d$, with coefficients up to $2^d$. If we were to compute up to $3^d$ instead, we would be able to extend this graph down to $t=\frac{1}{3}$. This would, however, be a massive undertaking, because we would have $3^6/2^6>11$ times as many possible lead coefficients. Furthermore, we observed that, for a given degree, the computations took longer the higher the lead coefficient was, so this factor $11$ is probably an underestimate.
It may be possible to extend these calculations though in a more sophisticated manner, somehow doing a less extensive and more intelligent search for obstruction polynomials of higher degree or larger lead coefficients. This would be a worthwhile project, and could lead to some interesting new results.
Lastly, consider Figure \[fig:gamma\]. This could very easily have been extended all the way to 0. The reason that we chose not to do this is because the hypothetical lower bound (the black lines) starts to merge into itself, and the Figure becomes unreadable. (The lower bound jumps at every $\frac{1}{n}$ which get more frequent as $\frac{1}{n} \to 0$.)
[*Acknowledgement.*]{} We thank the referee for helpful comments.
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|
---
abstract: 'In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin’s identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition. [^1]'
author:
- 'Yuu Hariya[^2]'
title: 'A pathwise interpretation of the Gorin-Shkolnikov identity'
---
v
Introduction {#;intro}
============
Let $\bes =\{ \bes _{t}\} _{0\le t\le 1}$ be a normalized Brownian excursion, that is, it is identical in law with a standard $3$-dimensional Bessel bridge, which has the duration $[0,1]$, and starts from and ends at the origin; see e.g., [@by Section ] and references therein for the definition of normalized Brownian excursion and its equivalence in law with standard $3$-dimensional Bessel bridge. We denote by $\loc =\{ \loc _{x}\} _{x\ge 0}$ the total local time process of $\bes $; namely, by the occupation time formula, two processes $\bes $ and $\loc $ are related via $$\begin{aligned}
\label{;defH}
H(x):=\int _{0}^{1}\ind _{\{ \bes _{t}\le x\} }\,dt
=\int _{0}^{x}\loc _{y}\,dy \quad \text{for all $x\ge 0$, a.s.}\end{aligned}$$ In a recent paper [@gs], Gorin and Shkolnikov have found the following remarkable identity in law as a corollary to one of their results:
\[;mt\] The random variable $X$ defined by $$\begin{aligned}
X:=\int _{0}^{1}\bes _{t}\,dt-\frac{1}{2}\int _{0}^\infty
\left( \loc _{x}\right) ^{2}dx
\end{aligned}$$ is a centered Gaussian random variable with variance $1/12$.
In [@gs], they have shown that the expected value of the trace of a random operator indexed by $T>0$, arising from random matrix theory, admits the representation $$\begin{aligned}
\sqrt{\frac{2}{\pi T^{3}}}
\ex \left[
\exp \left( -\frac{T^{3/2}}{2}X\right)
\right]$$ for any $T>0$; in comparison of this expression with the existing literature asserting that the expected value is equal to $\sqrt{2/(\pi T^{3})}\exp \left( {T^{3}/96}\right)$ for every $T>0$, they have obtained by the analytic continuation and the uniqueness of characteristic functions.
In this note, we give a proof of without relying on random matrix theory; Jeulin’s identity in law ([@jeu p.264], [@by Proposition 3.6]): $$\begin{aligned}
\label{;ji}
\{ \bes _{t}\} _{0\le t\le 1}
\stackrel{(d)}{=}\left\{
\frac{1}{2}\loc _{H^{-1}(t)}
\right\} _{0\le t\le 1}\end{aligned}$$ with $$\begin{aligned}
H^{-1}(t):=\inf \left\{
x\ge 0;\,H(x)\ge t
\right\} , \end{aligned}$$ plays a central role in the proof.
Proof of {#;prf}
=========
In this section, we give a proof of and provide some relevant results.
Recall from the representation of $\bes $ by means of a stochastic differential equation (see, e.g., [@ry Chapter XI, Exercise ]) that the process $\br =\{ \br _{t}\} _{0\le t\le 1}$ defined by $$\begin{aligned}
\label{;brm}
\br _{t}:=\bes _{t}-\int _{0}^{t}\frac{ds}{\bes _{s}}
+\int _{0}^{t}\frac{\bes _{s}}{1-s}\,ds
\end{aligned}$$ is a standard Brownian motion. We integrate both sides over $[0,1]$ and use Fubini’s theorem on the right-hand side to see that $$\begin{aligned}
\int _{0}^{1}\br _{t}\,dt
=\int _{0}^{1}\bes _{t}\,dt
-\int _{0}^{1}\frac{ds}{\bes _{s}}\int _{s}^{1}dt
+\int _{0}^{1}ds\,\frac{\bes _{s}}{1-s}\int _{s}^{1}dt,
\end{aligned}$$ which entails $$\begin{aligned}
\label{;gauss1}
\frac{1}{2}\int _{0}^{1}\br _{t}\,dt
=\int _{0}^{1}\bes _{t}\,dt
-\frac{1}{2}\int _{0}^{1}\frac{1-t}{\bes _{t}}\,dt.
\end{aligned}$$ Note that the left-hand side is a centered Gaussian random variable with variance $1/12$. By Jeulin’s identity , the right-hand side of is identical in law with $$\begin{aligned}
\frac{1}{2}\int _{0}^{1}\loc _{H^{-1}(t)}\,dt
-\int _{0}^{1}\frac{1-t}{\loc _{H^{-1}(t)}}\,dt. \label{;il}
\end{aligned}$$ We change variables with $t=H(x),\,x\ge 0$, to rewrite as $$\begin{aligned}
&\frac{1}{2}\int _{0}^{\infty }\loc _{x}H'(x)\,dx
-\int _{0}^{\infty }\frac{1-H(x)}{\loc _{x}}H'(x)\,dx
\label{;prf1}\\
&=\frac{1}{2}\int _{0}^{\infty }\left( \loc _{x}\right) ^{2}dx
-\int _{0}^{\infty }dx\int _{0}^{1}dt\,\ind _{\{ \bes _{t}>x\} } \notag \\
&=\frac{1}{2}\int _{0}^{\infty }\left( \loc _{x}\right) ^{2}dx
-\int _{0}^{1}\bes _{t}\,dt, \notag
\end{aligned}$$ where the second line follows from the definition of $H$ and the third from Fubini’s theorem. Combining this expression with yields $$\begin{aligned}
\frac{1}{2}\int _{0}^{1}\br _{t}\,dt
\stackrel{(d)}{=}\frac{1}{2}\int _{0}^{\infty }
\left( \loc _{x}\right) ^{2}dx-\int _{0}^{1}\bes _{t}\,dt
\end{aligned}$$ and concludes the proof.
We give a remark on the proof. In what follows we denote $$\begin{aligned}
M(r)=\max _{0\le t\le 1}\bes _{t}. \end{aligned}$$
We see from the above proof that the random variables $$\begin{aligned}
\int _{0}^{1}\bes _{t}\,dt, \qquad
\frac{1}{2}\int _{0}^{\infty }
\left( \loc _{x}\right) ^{2}dx, \qquad
\frac{1}{2}\int _{0}^{1}\frac{1-t}{\bes _{t}}\,dt\end{aligned}$$ have the same law; they are also identical in law with $$\begin{aligned}
\frac{1}{2}\int _{0}^{t}\frac{t}{\bes _{t}}\,dt\end{aligned}$$ by the time-reversal: $
\{ \bes _{1-t}\} _{0\le t\le 1}
\stackrel{(d)}{=}\{ \bes _{t}\} _{0\le t\le 1}
$. The Laplace transform of the law of $\int _{0}^{1}\bes _{t}\,dt$ is given in [@gro Lemma 4.2] and [@by Proposition ] in terms of a series expansion.\
We see from that $$\begin{aligned}
\int _{0}^{\infty }\loc _{y}\,dy=\int _{0}^{M(r)}\loc _{y}\,dy=1. \end{aligned}$$ Therefore, to be more specific, the second integral in should be written as $$\begin{aligned}
\int _{0}^{M(r)}\frac{1-H(x)}{\loc _{x}}H'(x)\,dx. \end{aligned}$$
Using the same reasoning as the above proof, we may obtain the following extension of :
\[;pgauss\] For every positive integer $n$, the random variable $$\begin{aligned}
2\int _{[0,1]^{n}}\min \left\{
\bes _{t_{1}},\ldots ,\bes _{t_{n}}
\right\} dt_{1}\cdots dt_{n}
-\frac{n+1}{2}\int _{0}^{\infty }
\left( 1-H(x)\right) ^{n-1}\left( \loc _{x}\right) ^{2}dx
\end{aligned}$$ has the Gaussian distribution with mean zero and variance $1/(2n+1)$.
For each fixed $n$, we multiply both sides of by $(1-t)^{n-1}$ and integrate them over $[0,1]$. Then using Fubini’s theorem, we obtain $$\begin{aligned}
\label{;gauss2}
\int _{0}^{1}(1-t)^{n-1}\br _{t}\,dt
=\frac{n+1}{n}\int _{0}^{1}(1-t)^{n-1}\bes _{t}\,dt
-\frac{1}{n}\int _{0}^{1}\frac{(1-t)^{n}}{\bes _{t}}\,dt.
\end{aligned}$$ Since the left-hand side may be expressed as $(1/n)\int _{0}^{1}(1-t)^{n}\,d\br _{t}$, we see that it is a centered Gaussian random variable with variance $$\begin{aligned}
\frac{1}{n^2}\int _{0}^{1}(1-t)^{2n}\,dt=\frac{1}{n^{2}(2n+1)}.
\end{aligned}$$ On the other hand, by Jeulin’s identity , the right-hand side of is identical in law with $$\begin{aligned}
&\frac{n+1}{2n}\int _{0}^{\infty }(1-t) ^{n-1}
\loc _{H^{-1}(t)}\,dt
-\frac{2}{n}\int _{0}^{1}\frac{(1-t)^{n}}{\loc _{H^{-1}(t)}}\,dt\\
&=\frac{n+1}{2n}\int _{0}^{\infty }
\left(
1-H(x)
\right) ^{n-1}\left( \loc _{x}\right) ^{2}dx
-\frac{2}{n}\int _{0}^{M(r)}\left( 1-H(x)\right) ^{n}dx.
\end{aligned}$$ By , we may rewrite the integral in the last term as $$\begin{aligned}
\int _{0}^{M(r)}dx\left(
\int _{0}^{1}dt\,\ind _{\{ \bes _{t}>x\} }
\right) ^{n}
&=\int _{0}^{M(r)}dx\int _{[0,1]^{n}}
dt_{1}\cdots dt_{n}\,\prod_{i=1}^{n}\ind _{\{ r_{t_{i}>x}\} }\\
&=\int _{[0,1]^{n}}\min \left\{
\bes _{t_{1}},\ldots ,\bes _{t_{n}}
\right\} dt_{1}\cdots dt_{n},
\end{aligned}$$ where we used Fubini’s theorem for the second equality. Combining these leads to the conclusion.
We end this note with a comment on a relevant fact deduced from the proof of .
It is well known (see, e.g., [@by Equation ]) that $$\begin{aligned}
M(r)\stackrel{(d)}{=}\frac{1}{2}\int _{0}^{1}\frac{dt}{\bes _{t}};
\end{aligned}$$ in fact, Jeulin’s identity entails that $$\begin{aligned}
\frac{1}{2}\int _{0}^{1}\frac{dt}{\bes _{t}}
\stackrel{(d)}{=}\int _{0}^{M(r)}\frac{1}{\loc _{x}}\times \loc _{x}\,dx
=M(r).
\end{aligned}$$ Combining this fact with a part of the proof of , one sees that the sequence of random variables $$\begin{aligned}
M(r),\quad \int _{[0,1]}\bes _{t}\,dt,\quad
\int _{[0,1]^{2}}\min \left\{ \bes _{t_{1}},\bes _{t_{2}}\right\}
dt_{1}dt_{2},\ldots
\end{aligned}$$ is identical in law with $$\begin{aligned}
\frac{1}{2}\int _{0}^{1}\frac{(1-t)^{n}}{\bes _{t}}\,dt,
\quad n=0,1,2,\ldots ,
\end{aligned}$$ as well as with $$\begin{aligned}
\frac{1}{2}\int _{0}^{1}\frac{t^{n}}{\bes _{t}}\,dt,
\quad n=0,1,2,\ldots
\end{aligned}$$ by the time-reversal.
[99]{} Ph. Biane, M. Yor, Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math. (2) [**111**]{} (1987), 23–101.
V. Gorin, M. Shkolnikov, Stochastic Airy semigroup through tridiagonal matrices, arXiv:1601.06800 (2016).
P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Related Fields [**81**]{} (1989), 79–109.
T. Jeulin, Application de la théorie du grossissement à l’étude des temps locaux browniens, in Grossissements de filtrations: exemples et applications, Séminaire de Calcul Stochastique 1982/83 Université Paris VI, 197–304, Lect. Notes in Math. [**1118**]{}, Springer, Berlin, 1985.
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, Berlin, 1999.
[^1]: E-mail: hariya@math.tohoku.ac.jp
[^2]: Mathematical Institute, Tohoku University, Aoba-ku, Sendai 980-8578, Japan.
|
---
abstract: |
For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on the torus $(\Z / n \Z )^d$ for any inverse temperature $\beta\geq 0$. This stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen–Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models.
Our algorithms are based on representing the random cluster model as a contour model using Pirogov–Sinai theory, and then computing an accurate approximation of the logarithm of the partition function by inductively truncating the resulting cluster expansion. The main innovation of our approach is an algorithmic treatment of unstable ground states; this is essential for our algorithms to apply to all inverse temperatures $\beta$. By treating unstable ground states our work gives a general template for converting probabilistic applications of Pirogov–Sinai theory to efficient algorithms.
address:
- Microsoft Research
- Microsoft Research
- University of Bristol
- University of Illinois at Chicago
- Georgia Institute of Technology
author:
- Christian Borgs
- Jennifer Chayes
- Tyler Helmuth
- Will Perkins
- Prasad Tetali
title: 'Efficient sampling and counting algorithms for the Potts model on $\Z^d$ at all temperatures'
---
Introduction {#sec:intro}
============
The Potts model is a probability distribution on assignments of $q$ colors to the vertices of a finite graph $G$. Let $$H_G(\sigma) {\coloneqq}\sum_{(i,j)\in E(G)} \delta_{\sigma_i \ne \sigma_j},
\quad \quad \sigma \in [q]^{V(G)}{\coloneqq}\{1,2,\dots, q\}^{V(G)} ,$$ be the the number of bichromatic edges of $G$ under the coloring $\sigma$. Then the *$q$-state ferromagnetic Potts model* at inverse temperature $\beta\geq 0$ is the probability distribution $\mu^{\text{Potts}}_G$ on $[q]^{V(G)}$ defined by $$\label{eq:PottsDef}
\mu^{\text{Potts}}_G(\sigma) {\coloneqq}\frac{e^{-\beta
H_G(\sigma)}}{Z^{\text{Potts}}_G(\beta)}, \qquad
Z^{\text{Potts}}_G(\beta) {\coloneqq}\sum_{\sigma \in [q]^{V(G)}} e^{-\beta H_G(\sigma)}.$$ The normalizing constant $Z^{\text{Potts}}_{G}(\beta)$ is the Potts model partition function. Since $\beta \geq 0$, monochromatic edges are preferred and the model is ferromagnetic.
From a computational point of view, $Z^{\text{Potts}}_G$ and $\mu^{\text{Potts}}_G$ define families of functions and probability measures indexed by finite graphs $G$, and there are two main computational tasks associated to these objects. The first is the *approximate counting* problem: for a partition function $Z_G$ and error tolerance $\eps>0$, compute a number $\hat Z$ so that $e^{-\eps} \hat Z \le Z_G \le e^{\eps} \hat Z$. We say that such a $\hat Z$ is an *$\eps$-relative approximation* to $Z_G$. The second is the *approximate sampling* problem: for a probability measure $\mu_G$ and error tolerance $\eps>0$, output a random configuration $\hat \sigma$ with distribution $\hat \mu$ so that $\| \hat \mu - \mu_G \|_{TV} < \eps$. We say $\hat \sigma$ is an *$\eps$-approximate sample* from $\mu_G$.
A *fully polynomial-time approximation scheme* (FPTAS) is an algorithm that given $G$ and $\eps>0$ returns an $\eps$-relative approximation to $Z_G$ and runs in time polynomial in $|V(G)|$ and $1/\eps$. If the algorithm uses randomness it is a *fully polynomial-time randomized approximation scheme* (FPRAS). A randomized algorithm that given $G$ and $\eps>0$ outputs an $\eps$-approximate sample from $\mu_G$ and runs in time polynomial in both $|V(G)|$ and $1/\eps$ is an *efficient sampling scheme*.
In this paper we give an FPRAS and an efficient sampling scheme for the $q$-state Potts model on the discrete torus $\tor = (\Z / n \Z )^d$ for *all* inverse temperatures $\beta\geq 0$, provided $q$ is large enough as a function of $d$.
\[PottsTorusCrit\] For all $d\ge2$ there exists $q_0=q_{0}(d)$ so that for $q \ge q_{0}$ and all inverse temperatures $\beta\geq 0$ there is an FPRAS and efficient sampling scheme for the $q$-state Potts model at inverse temperature $\beta$ on the torus $\tor$.
If $\epsilon$ is not too small, meaning $\eps\geq \exp(-O(n^{d-1}))$, our approximate counting algorithm is deterministic. We comment on this further below Theorem \[PottsZd\], but before stating our more general results we briefly discuss the aspects of the Potts model relevant to this paper. For a more comprehensive discussion see, e.g., [@DuminilCopinLectures].
The Potts model is known to exhibit a phase transition on $\Z^{d}$ when $d\geq 2$, and when $q$ is sufficiently large the phase diagram has been completely understood for some time [@kotecky1982first; @laanait1991interfaces]. For large $q$ there is a critical temperature $\beta_{c} = \beta_c(d,q)$ satisfying $$\label{eq:betac}
\beta_{c} = \frac{\log q}{d} + O(q^{-1/d})$$ such that for $\beta<\beta_{c}$ there is a unique infinite-volume Gibbs measure, while if $\beta>\beta_{c}$ there are $q$ extremal translation-invariant Gibbs measures. Each of these low-temperature measures favor one of the $q$ colors. At the transition point $\beta= \beta_{c}$ there are $q+1$ extremal translation-invariant Gibbs measures; $q$ of these measures favor one of the $q$ colors, and the additional measure is the ‘disordered’ measure from $\beta<\beta_{c}$. We note that the phenomenology of the model is $q$-dependent [@DuminilCopinLectures]. The preceding results require $q$ large as they use $q^{-1}$ as a small parameter in proofs.
The existence of multiple measures in the low-temperature phase is reflected in the dynamical aspects of the model. While Glauber dynamics for the Potts model mix rapidly at sufficiently high temperatures, they mix in time $\exp(\Theta(n^{d-1}))$ when $\beta\geq\beta_{c}$ [@borgs2012tight; @borgs1991finite]. Even the global-move Swensden–Wang dynamics take time $\exp(\Theta(n^{d-1}))$ to mix when $\beta=\beta_{c}$ [@borgs2012tight].
The results just discussed were primarily obtained by making use of a sophisticated form of Pirogov–Sinai theory. It was recently shown that Pirogov–Sinai theory can be used to develop efficient algorithms for approximate counting and sampling for models in which all ground states are *stable*, i.e., all ground states have the same truncated free energy [@helmuth2018contours]. Most applications of Pirogov–Sinai theory, including the results concerning the Potts model described in the previous paragraphs, involve working with both stable and unstable ground states, and the main achievement of this paper is to show how to develop efficient algorithms when unstable ground states play a significant role. We discuss our methods in more detail in Section \[sec:meth\] below.
Random cluster model {#sec:random-cluster-model}
--------------------
Our methods apply more generally than to the Potts model on the torus: they also give efficient approximation algorithms for the more general random cluster model on both the torus and on a broad class of subsets of $\Z^{d}$. To make this more precise, recall that given a finite graph $G=(V(G),E(G))$ the random cluster model is a probability distribution on edge sets of $G$ given by $$\label{eqRCdef}
\mu^{\text{RC}}_{G}(A) {\coloneqq}\frac{p^{|A|} (1-p)^{|E(G)| - |A|} q^{c(G_A)} }{Z^{\text{RC}}_G(p,q) } \,, \quad\quad A \subseteq E(G) \,,$$ where $c(G_A)$ is the number of connected components of the graph $G_A = (V(G),A)$ and $$Z^{\text{RC}}_{G}(p,q) {\coloneqq}\sum_{A \subseteq E(G)} p^{|A|} (1-p)^{|E(G)| - |A|} q^{c(G_{A})}$$ is the random cluster model partition function.
The Potts model and the random cluster model can be put onto the same probability space via the Edwards–Sokal coupling. We recall this coupling in Appendix \[sec:ES\]; the result is the relation, for $\beta \ge 0$ and integer $q \ge 2$, $$\label{eqFKpotts1}
Z^{\text{Potts}}_{G}(\beta) = e^{\beta |E(G)|} Z^{\text{RC}}_G(1-e^{-\beta},q).$$
With the parameterization $p = 1- e^{-\beta}$ the random cluster model on $\Z^d$, $d \ge 2$, also has a critical inverse temperature $\beta_c= \beta_c(q,d)$ that satisfies and that coincides with the Potts critical inverse temperature for integer $q$. For $\beta<\beta_c$ the random cluster model has a unique infinite volume measure (the *disordered* measure), while for $\beta>\beta_c$ the *ordered* measure is the unique infinite volume measure. For $\beta= \beta_c$ the two measures coexist.
Our counting and sampling algorithms extend to the random cluster model on finite subgraphs of $\Z^d$ with two different types of boundary conditions. To make this precise requires a few definitions. Let $\Lam$ be a finite set of vertices of $\Z^d$ and let $G_\Lam$ be the subgraph induced by $\Lam$. We say $G_{\Lam}$ is *simply connected* if $G_{\Lam}$ is connected and the subgraph induced by $\Lam^c =\Z^d \setminus \Lam$ is connected. The random cluster model with *free boundary conditions* on $G_{\Lam}$ is just the random cluster model on the induced subgraph $G_{\Lam}$ as defined by . The random cluster model with *wired boundary conditions* on $G_{\Lam}$ is the random cluster model on the (multi-)graph $G_{\Lam}'$ obtained from $G_{\Lam}$ by identifying all of the vertices on the boundary of $\Lam$ to be one vertex; see [@DuminilCopinLectures Section 1.2.2] for a formal definition. We refer to the Gibbs measures and partition functions with free and wired boundary conditions as $\mu^f_\Lam, \mu^w_\Lam, Z^f_{\Lam}, Z^w_{\Lam}$. Explicitly, $$\begin{aligned}
\label{eq:Zfree}
Z^f_{\Lam} &{\coloneqq}\sum_{A\subset E(G_{\Lam})}
p^{{\left|A\right|}}(1-p)^{{\left|E(G_{\Lam})\right|}-{\left|A\right|}} q^{c(G_{A})}, \qquad \text{and} \\
\label{eq:Zwired}
Z^w_{\Lam}
&{\coloneqq}\sum_{A\subset E(G_{\Lam}')}
p^{{\left|A\right|}}(1-p)^{{\left|E(G_{\Lam}')\right|}-{\left|A\right|}} q^{c(G'_{A})}, \end{aligned}$$ where $c(G_{A})$ is the number of connected components of the graph $(\Lam,A)$ and $c(G'_A)$ is the number of components of the graph $(\Lam',A)$ in which we identify all vertices on the boundary of $\Lam$.
\[PottsZd\] For $d\ge2$ there exists $q_0 = q_{0}(d)$ so that for $q \ge q_{0}$ the following is true.
For $\beta\geq \beta_{c}$ there is an FPTAS and efficient sampling scheme for the random cluster model on all finite, simply connected induced subgraphs of $\Z^d$ with wired boundary conditions.
For $\beta \leq \beta_c$ there is an FPTAS and efficient sampling scheme for the random cluster model on all finite, simply connected induced subgraphs of $\Z^d$ with free boundary conditions.
Theorem \[PottsZd\] yields an FPTAS, while Theorem \[PottsTorusCrit\] gave an FPRAS for the torus. The reason for this is that our Pirogov–Sinai based methods become more difficult to implement on the torus if the error parameter $\epsilon$ is smaller than $\exp(-O(n^{d-1}))$. The algorithm for Theorem \[PottsTorusCrit\] circumvents this by making use of the Glauber dynamics for this range of $\epsilon$. This is possible because, despite being slow mixing, the Glauber dynamics are fast enough when given time $O(\eps^{-1})$ for $\epsilon$ this small by [@borgs2012tight]. By using Glauber dynamics in a similar manner we could obtain an FPRAS for the random cluster model on $\tor$.
We note that our methods are certainly capable of handling boundary conditions other than those described above, but we leave an investigation of the full scope of their applicability for the future.
Proof overview {#sec:meth}
--------------
This section briefly outlines our arguments and highlights the new ideas introduced in the paper.
1. In Section \[secPolymer\] we briefly recall the notion of a polymer model and convergence criteria for the cluster expansion, and recall from [@helmuth2018contours] how this can be used for approximation algorithms. A key improvement upon [@helmuth2018contours] is that we work directly with the cluster expansion rather than using Barvinok’s method [@barvinok2017weighted]; the importance of this is discussed in the next subsection.
In this section we also apply the polymer model algorithm to the random cluster model at very high temperatures, meaning $\beta\leq \beta_h {\coloneqq}\frac{3\log q}{4d}$.
2. In Section \[secFKcontour\] we first recall the tools from Pirogov–Sinai theory developed in [@borgs2012tight] for the random cluster model. We then use these tools to establish the necessary ingredients for an algorithmic implementation of the method.
3. Section \[secEstimates\] contains estimates for the contour model representation derived in Section \[secFKcontour\]. We first recall the inputs that we need from [@borgs2012tight], and we then prove some consequences that are needed for our algorithms. The key additional estimates concern how unstable contours rapidly ‘flip’ to stable contours; this property is essential for our algorithms to be efficient.
This section focuses on the most interesting case of $\beta\geq\beta_{c}$. The case $\beta_h<\beta<\beta_{c}$, which is very similar to $\beta>\beta_{c}$ and again uses estimates from [@borgs2012tight], is discussed in Appendix \[sec:HT\].
4. In Section \[sec:count\] we present our approximate counting algorithms. The broad idea is to use the inductive Pirogov–Sinai method of [@helmuth2018contours], but significant refinements are needed to deal with the presence of an unstable ground state. Similar refinements are then used in Section \[sec:sample\] to develop sampling algorithms.
As is clear from this outline, this paper uses the methods and framework developed in [@borgs2012tight] and [@helmuth2018contours]. For the ease of the reader who wishes to see the proofs of results we use from [@borgs2012tight] we have largely stuck to the definitions presented in that paper, and have made careful note of the situations in which we have chosen alternative definitions that facilitate our algorithms.
Discussion {#secRelated}
----------
Before this paper, algorithmic results for the Potts model on $\Z^{d}$ for $d\geq 2$ were restricted to either $\beta<\beta_{c}$, see [@bordewich2016mixing] and references therein, or $\beta\gg\beta_{c}$ [@helmuth2018contours; @barvinok2017weighted]. In the planar case of $\Z^{2}$ more detailed results are known [@ullrich2013comparison; @gheissari2018mixing; @gheissari2016quasi; @blanca2017random]. More broadly, meaning beyond $\Z^{d}$ and beyond the Potts model, algorithms for low-temperature models have only recently been developed, and have been based primarily on cluster expansion methods [@JenssenAlgorithmsSODA; @cannon2019counting; @casel2019zeros; @PolymerMarkov; @liao2019counting; @barvinok2017weighted]. These algorithms belong to the same circle of ideas as Barvinok’s interpolation method [@barvinok2017combinatorics] and the improvements due to Patel and Regts [@patel2016deterministic].
Unlike in [@helmuth2018contours] we work directly with the cluster expansion, i.e., we avoid Barvinok’s interpolation method based on the univariate Taylor series. Recall that Barvinok’s method relies on the existence of a disk in the complex plane that is free of zeros for the partition functions that one wants to approximate. For the Potts model partition functions $Z_{\tor}^{\text{Potts}}$, there is no disk centered at $\beta_{c}$ that is zero-free uniformly in the side length $n$, precisely because there is a phase transition at $\beta_{c}$. Thus any direct application of the interpolation method cannot work at $\beta=\beta_{c}$.
It may be possible to combine results and proof techniques from [@DuminilCopinRaoufiTassion; @martinelli19942; @alexander2004mixing] to prove that the Glauber dynamics mix rapidly on the torus and sufficiently regular subsets of $\Z^d$ for all $\beta<\beta_{c}$, which would yield a much faster sampling algorithm than the one we have given here. We are not aware, however, of any existing statement in the literature which would directly imply rapid mixing in the whole range $\beta<\beta_{c}$, and leave this as an open problem. Further open problems can be found in the conclusion of this paper, Section \[sec:Conc\].
Polymer models, cluster expansions, and algorithms {#secPolymer}
==================================================
This section describes how two related tools from statistical physics, abstract polymer models and the cluster expansion, can be used to design efficient algorithms to approximate partition functions.
An *abstract polymer model* [@gruber1971general; @kotecky1986cluster] consists of a set $\cC$ of *polymers* each equipped with a complex-valued *weight* $w_{\gamma}$ and a non-negative *size* $\| \gamma \|$. The set $\cC$ also comes equipped with a symmetric compatibility relation $\sim$ such that each polymer is incompatible with itself, denoted $\gamma \nsim \gamma$. Let $\cG$ denote the collection of all sets of pairwise compatible polymers from $\cC$, including the empty set of polymers. The polymer model partition function is defined to be $$\label{eq:PolyZ}
Z( \cC,w) {\coloneqq}\sum_{\Gamma \in \cG} \prod_{\gamma \in \Gamma}
w_{\gamma}.$$ In $w$ is shorthand for the collection of polymer weights.
Let $\Gamma$ be a non-empty tuple of polymers. The *incompatibility graph $H_{\Gamma}$* of $\Gamma$ has vertex set $\Gamma$ and edges linking any two incompatible polymers, i.e., $\{\gamma,\gamma'\}$ is an edge if and only if $\gamma\nsim\gamma'$. A non-empty ordered tuple $\Gamma$ of polymers is a *cluster* if its incompatibility graph $H_\Gamma$ is connected. Let $\cG^c$ be the set of all clusters of polymers from $\cC$. The cluster expansion is the following formal power series for $\log Z(\cC, w)$ in the variables $w_\gamma$: $$\label{eq:clusterexp}
\log Z(\cC,w)
=
\sum_{\Gamma \in \cG^c} \phi(H_\Gamma) \prod_{\gamma
\in \Gamma} w_\gamma .$$ In $\phi(H)$ denotes the *Ursell function* of the graph $H=(V(H),E(H))$, i.e., $$\phi(H)
{\coloneqq}\frac{1}{|V(H)|!} \sum_{\substack{ A\subseteq E(H)
\\(V(H), A) \text{ connected} }} (-1)^{|A|}.$$
For a proof of see, e.g., [@kotecky1986cluster; @friedli2017statistical]. Define $\| \Gamma \| {\coloneqq}\sum_{\gamma \in \Gamma} \| \gamma \|$, and define the truncated cluster expansion by $$T_{m}(\cC,w)
{\coloneqq}\sum_{\substack{\Gamma \in \cG^c \\ \| \Gamma \|< m}} \phi(H_\Gamma)
\prod_{\gamma \in \Gamma} w_\gamma \,.$$
Henceforth we will restrict our attention to a special class of polymer models defined in terms of a graph $G$ with maximum degree $\Delta$ on $N$ vertices. Namely, we will assume that each polymer is a connected subgraph $\gamma=(V(\gamma),E(\gamma))$ of $G$. The compatibility relation is defined by disjointness in $G$: $\gamma \sim \gamma'$ iff $V(\gamma) \cap V(\gamma') = \emptyset$. We write ${\left|\gamma\right|}$ for ${\left|V(\gamma)\right|}$, the number of vertices in the polymer $\gamma$.
A useful criteria for convergence of the formal power series in is given by the following adaptation of a theorem of Kotecký and Preiss [@kotecky1986cluster].
\[KPthm\] Suppose that polymers are connected subgraphs of a graph $G$ of maximum degree $\Delta$ on $N$ vertices. Suppose further that for some $b>0$ and all $\gamma \in \cC$, $$\begin{aligned}
\label{eqPeierls}
\| \gamma \| &\ge b | \gamma|, \\
\label{eqPolymerKP}
|w_{\gamma}| &\le e^{- (\frac{3 + \log \Delta}{b} + 3) \| \gamma \|}.
\end{aligned}$$ Then the cluster expansion converges absolutely, and for $m\in{{\mathbb{N}}}$, $$\label{eqTruncBound}
\left| T_m (\cC,w) - \log Z(\cC,w) \right| \le N e^{- 3m } \,.$$
This lemma implies that if conditions and hold, then $\exp( T_{m}(\cC,w))$ is an $\eps$-relative approximation to $Z(\cC,w)$ for $m \ge \log (N/\eps)/3$.
We append to $\cC$ a polymer $\gamma_v$ for each $v \in V(G)$ consisting only of that vertex, with size $\| \gamma_v \|=1$ and $w_{\gamma_v}=0$. By definition, $\gamma_{v}$ is incompatible with every other polymer that contains $v$. Then $$\sum_{\gamma \nsim \gamma_v} |w_{\gamma}| e^{ |\gamma| + 3\| \gamma
\|}
\le \sum_{\gamma \nsim \gamma_v} e^{ |\gamma|} e^{ - (\frac{3 +
\log \Delta}{b})\| \gamma \|}
\le \sum_{\gamma \nsim \gamma_v} e^{ |\gamma|} e^{ -(3 + \log
\Delta) | \gamma |}
\le \sum_{k \ge 1} (e \Delta)^{k}
e^{-(2 + \log \Delta) k}$$ where the first inequality is by , the second by , and the third is by bounding the number of $k$-vertex connected subgraphs of $G$ that contain $v$ by $(e\Delta)^{k}$. This yields $$\label{eq:bd}
\sum_{\gamma \nsim \gamma_v} |w_{\gamma}| e^{ |\gamma| + 3\| \gamma
\|} \leq \sum_{k \ge 1} e^{ - k } < 1.$$
Fix a polymer $\gamma$. By summing over all $v \in \gamma$ we obtain $$\sum_{\gamma' \nsim \gamma} |w_{\gamma'}| e^{|\gamma'| +3 \| \gamma ' \|} < |\gamma| \,.$$ By applying the main theorem of [@kotecky1986cluster] with $a(\gamma) = |\gamma|$, $d(\gamma) = 3 \| \gamma \|$ we obtain that the cluster expansion converges absolutely. Moreover, we also obtain that $$\sum_{\substack{\Gamma \in \cG^c \\ \Gamma \ni v}} \left|
\phi(H_{\Gamma}) \prod_{\gamma \in \Gamma} w_{\gamma} \right|
e^{3 \| \Gamma \| } \le 1 \,,$$ where the sum is over all clusters that contain a polymer containing the vertex $v$. By using this estimate and summing over all $v \in V(G)$ one obtains $$\label{eq:tailbound}
\sum_{\substack{\Gamma \in \cG^c \\ {\|\Gamma\|}\geq m}} \left|
\phi(H_{\Gamma}) \prod_{\gamma \in \Gamma} w_{\gamma} \right|
\leq N e^{-3m}$$ which is .
Because clusters are connected objects arising from a bounded-degree graph, the truncated cluster expansion can be computed efficiently. Recall that $N={\left|V(G)\right|}$.
\[lemPolyModelCount\] Suppose the conditions of Lemma \[KPthm\] hold. Then given a list of all polymers $\gamma$ of size at most $m$ along with the weights $w_{\gamma}$ of these polymers, the truncated cluster expansion $T_m(\cC,w)$ can be computed in time $O (N\exp( O(m)))$.
This is [@helmuth2018contours Theorem 2.2].
The next lemma says that, for the purposes of approximating a polymer partition function, it is sufficient to have approximate evaluations $\tilde w_{\gamma}$ of the weights $w_{\gamma}$.
\[lemPolymerApprox\] Let $v\colon \cC\to [0,\infty)$ be a non-negative function on polymers such that $v(\gamma) \le \| \gamma \|^2$. Suppose $0<\eps < N^{-1}$, and let $m= \log
(8/\eps)/3$. Suppose the conditions of Lemma \[KPthm\] hold and that for all $\gamma \in \cC$ with $\| \gamma \| \le m$, $\tilde w_\gamma$ is an $\eps v(\gamma)$-relative approximation to $w_\gamma$. Then $\exp( T_{m}(\cC, \tilde w))$ is an $N\eps /4$-relative approximation to $Z(\cC,w)$.
Using the definition of $m$ and applying Lemma \[KPthm\], we have $$|\log Z_G(\cC,w) - T_{m}(\cC,w) | \le N\eps/8,$$ so by the triangle inequality it is enough to show that $$\label{eq:tri2}
\left | T_{m}(\cC,\tilde w) - T_{m}( \cC,w) \right | \le N\eps/8.$$ Define $r_{\gamma}$ by $\log \tilde w_\gamma = \log w_\gamma + r_\gamma$. To prove , note the identity $$T_{m}(\cC,\tilde w) - T_{m}( \cC,w)
= \sum_{\substack{\Gamma \in \cG^c(G) \\ {\|\Gamma\|}< m}}
\phi(H_\Gamma) \prod_{\gamma \in \Gamma} w_\gamma \cdot \left [
\exp\left( \sum_{\gamma \in \Gamma} r_\gamma \right) - 1
\right ].$$ Our hypotheses imply $|r_{\gamma}| \le \eps v(\gamma)$, and hence by the triangle inequality we obtain $${\left|T_{m}(\cC,\tilde w) - T_{m}( \cC,w)\right|} \leq
\sum_{\substack{\Gamma \in \cG^c(G) \\ {\|\Gamma\|}< m}}
(\exp(\sum_{\gamma\in\Gamma} \epsilon v(\gamma))-1)
{\left| \phi(H_{\Gamma}) \prod_{\gamma\in\Gamma}w_{\gamma}\right|},$$ where we have used the elementary inequality ${\left|e^{a}-1\right|}\leq e^{b}-1$ when ${\left|a\right|}\leq b$ to bound the term in square brackets. Since $v(\gamma)\leq {\|\gamma\|}^{2}$ this yields, after ordering the sum over clusters according to their size $k$, $${\left|T_{m}(\cC,\tilde w) - T_{m}( \cC,w)\right|}
\leq
\sum_{k=1}^{m-1}(\exp(\eps k^{2})-1)
\sum_{\substack{\Gamma \in \cG^c(G) \\ {\|\Gamma\|}=k}}
{\left| \phi(H_{\Gamma}) \prod_{\gamma\in\Gamma}w_{\gamma}\right|}
\leq \sum_{k=1}^{m-1}(\exp(\eps k^{2})-1) Ne^{-3k}.$$ The last inequality follows from the convergence of the cluster expansion (see in the proof of Lemma \[KPthm\]). Since $\eps<N^{-1}$ we can bound $e^{\eps k^{2}}-1$ by $2\eps k^{2}$, and follows since $\sum_{k\geq 1}k^{2}e^{-3k}<1/16$.
Putting Lemmas \[KPthm\], \[lemPolyModelCount\], and \[lemPolymerApprox\] together we see that the partition function $Z(\cC, w)$ can be approximated efficiently if
1. conditions and hold
2. polymers of size at most $m$ can be enumerated efficiently, i.e., in time polynomial in $N$ and exponential in $m$, and
3. the polymer weights $w_{\gamma}$ can be approximated efficiently, i.e., in time polynomial in the size of $\gamma$.
High temperature expansion {#secHighTemp}
--------------------------
This section explains how the polymer model algorithm of the previous section yields efficient counting and sampling algorithms for the random cluster model when $q$ is sufficiently large and $\beta \le \beta_{h}=\frac{3 \log q}{4d}$. This use of the polymer model algorithm also serves as a warm-up for the more sophisticated contour-based algorithms we will use in later sections when $\beta>\beta_{h}$.
In fact, the simpler setting of $\beta\le \beta_{h}$ allows for greater generality: we will derive an algorithm that applies to the random cluster model on *any* graph $G$ of maximum degree at most $2d$.
\[thmHighTempExpansion\] Suppose $d \ge 2$ and $q=q(d)$ is sufficiently large. Then for $\beta \le \beta_h$ there is an FPTAS and efficient sampling scheme for the Potts model and the random cluster model with $p=1- e^{-\beta}$ on all graphs of maximum degree at most $2d$.
Let $G=(V(G),E(G))$ be such a graph. We define polymers to be connected subgraphs of $G$ with at least two vertices. As per our convention, polymers are compatible if they are vertex disjoint, and $|\gamma| = |V(\gamma)|$. We set ${\|\gamma\|} =
|E(\gamma)|$, and define the weight of a polymer $\gamma$ to be $$w_\gamma
{\coloneqq}\left( \frac{p}{1-p} \right)^{{\|\gamma\|}} q^{1-|\gamma|} =
(e^\beta -1)^{{\|\gamma\|}} q^{1-|\gamma|} \,.$$
Let $\cC(G)$ be the set of all polymers on $G$, $\cG(G)$ be the collection of all sets of pairwise compatible polymers from $\cC(G)$, and let $$\Xi(G) {\coloneqq}\sum_{\Gamma \in \cG(G)} \prod_{\gamma \in \Gamma} w_{\gamma}$$ be the corresponding polymer model partition function. Then we have the identity $$\label{eq:HT-Poly}
Z^{\text{RC}}_G(p,q) = (1-p)^{|E(G)|} q^{|V(G)|}\; \Xi(G) .$$ The relation follows by extracting a common prefactor of $(1-p)^{|E(G)|} q^{|V(G)|}$ from the random cluster partition function. As a result of this the connected components consisting of a single vertex have weight one inside the sum. This is what enables the sum to be rewritten in terms of vertex-disjoint connected graphs on at least two vertices.
We will show that conditions and hold with $b=1/2$ if $\beta\le \beta_{h}$ and $q$ is sufficiently large as a function of $d$. To verify with $b=1/2$, note that a connected graph on $k$ vertices has at least $k-1$ edges, and $k-1 \geq k/2$ for $k\geq 2$.
Towards , suppose there is a $q_{0}$ such that for all $\gamma$, all $\beta\leq \beta_{h}$, and all $q\geq q_{0}$ $$\label{eq:HT-KP}
w_{\gamma}\leq C^{-{\|\gamma\|}}.$$ Then if $C=C(d)$ is small enough, holds. Since $b=1/2$, $C= \exp(-9-2\log 2d)$ suffices, and we fix $C$ to be this value hereon. We now verify in three steps, by considering polymers grouped according to the value of $k={\|\gamma\|}$.
1. For $k>5d$ we will use the fact that $|\gamma| \ge {\|\gamma\|}/d$ since every edge is incident to two vertices and every vertex is incident to at most $2d$ edges. Then we have $$w_\gamma \le q(e^{\beta} -1)^k q^{-k/d} \le q^{1 - \frac{k}{4d}} \le q^{ - \frac{k}{20d}}\,,$$ which is at most $C^{-{\|\gamma\|}}$ if $q\geq C^{20d}$.
2. For $d < k \le 5d$, we will use the fact that $|\gamma| \ge \frac{1}{2}+ \sqrt{2 {\|\gamma\|}}$ since the number of edges in a graph on $r$ vertices is at most $\binom {r}{2}$. Then we have $$w_\gamma \le q q^{\frac{3k}{4d}} q^{-\frac{1}{2}-\sqrt{2k}} \leq q^{\frac{1}{2} + \frac{3c}{4} - 2 \sqrt{c}}\,,$$ where $c = k/d$ and where we use the fact that $d \ge 2$ and $q\geq 1$. Then since $\frac{1}{2} + \frac{3c}{4} - 2 \sqrt{c} \le - \frac{1}{5}$ for $c \in [1,5]$, we have $$w_\gamma \le q^{-1/5}\,,$$ which is at most $C^{-{\|\gamma\|}}$ if $q\geq C^{25d}$.
3. For $1 \le k \le d$, since $|\gamma| \ge 2$, we have $$w_\gamma \le q^{-1} (e^{\beta}-1)^k \le q^{-1} e^{\beta k} \le q^{-1/4} \,,$$ which is at most $C^{-{\|\gamma\|}}$ provided $q\geq C^{4d}$.
Thus taking $q_{0} = \exp( 25d (9+2\log 2d))$ suffices. Lemmas \[KPthm\] and \[lemPolyModelCount\] then give an FPTAS for computing the random cluster partition function $Z^{\text{RC}}_G(1-e^{-\beta},q)$ for all graphs of maximum degree $2d$, as enumerating subgraphs of size $m$ in a bounded degree graph takes time $\exp(O(m))$, and computing the weight functions only requires counting the number of edges and vertices in each subgraph.
The efficient sampling scheme follows from [@helmuth2018contours Theorem 5.1]. Counting and sampling algorithms for the random cluster model can be converted into algorithms for the Potts model via the Edwards–Sokal coupling described in Appendix \[sec:ES\].
Theorem \[PottsTorusCrit\] follows immediately from Theorem \[thmHighTempExpansion\] since $\tor$ is $2d$-regular.
By , $\beta_{h}<\beta_{c}$ when $q$ is large enough. Thus Theorem \[PottsZd\] requires we provide approximate counting and sampling algorithms for free boundary conditions. Since induced subgraphs of $\Z^{d}$ have degree bounded by $2d$, the result follows by Theorem \[thmHighTempExpansion\].
Contour model representations {#secFKcontour}
=============================
*Contour models* refer to a class of polymer models that arise in Pirogov–Sinai theory [@pirogov1975phase]. For a given spin configuration, contours represent geometric boundaries between regions dominated by different ground states; the precise definition for the purposes of this paper will be given below. This section describes an important contour model representation for the random cluster model on the torus $\tor$ that is the basic combinatorial object in our algorithms. This contour representation was originally developed for obtaining optimal lower bounds on the mixing time for Glauber and Swensden–Wang dynamics [@borgs2012tight]. In addition to recalling the construction from [@borgs2012tight] this section also develops the additional ingredients necessary for algorithmic applications of the representation.
Continuum embedding {#sec:cont}
-------------------
The contour model representation from [@borgs2012tight] is based on the natural embedding of the discrete torus $\tor = (\Z/n\Z)^{d}$ of side-length $n\in{{\mathbb{N}}}$ into the continuum torus ${{{\boldsymbol T }}^d_n}{\coloneqq}(\R/n\R)^{d}$. This subsection recalls the basic definitions, and explains how they can be rephrased in terms of discrete graph-theoretic notions.[^1]
In what follows we abuse notation slightly and write $\tor$ for the graph $(\tor, E)$, where $E$ is the edge set of the discrete torus. We will follow the convention that bold symbols, e.g., ${{\boldsymbol V }}$, denote subsets of ${{{\boldsymbol T }}^d_n}$, while objects denoted by non-bold symbols like $V$ reside in $\tor$. Thus each vertex $v\in \tor$ is identified with a point ${{\boldsymbol v }}\in {{{\boldsymbol T }}^d_n}$, and we will identify each edge $e=\{u,v\}\in E$ with the unit line segment ${{\boldsymbol e }}\subset {{{\boldsymbol T }}^d_n}$ that joins ${{\boldsymbol u }}$ to ${{\boldsymbol v }}$. We will also drop $\tor$ from the notation when possible, e.g., $E$ for $E(\tor)$.
Recall that $\Omega = 2^{E}$ is the set of configurations of the random cluster model on $\tor$. Let ${{\boldsymbol c }}\subset {{{\boldsymbol T }}^d_n}$ denote a closed $k$-dimensional hypercube with vertices in $\tor$ for some $k=1,\dots, d$. We say a hypercube ${{\boldsymbol c }}$ is *occupied* with respect to $A\in \Omega$ if for all edges $e$ with ${{\boldsymbol e }}\subset {{\boldsymbol c }}$, $e$ is in $A$. Define $$\label{eq:fattening}
{{\boldsymbol A }} {\coloneqq}\left \{{{\boldsymbol x }}\in {{{\boldsymbol T }}^d_n}\mid \text{there exists ${{\boldsymbol c }}$ occupied
s.t.\ $d_{\infty}({{\boldsymbol x }},{{\boldsymbol c }})\leq \frac{1}{4}$} \right\},$$ where $d_{\infty}$ is the $\ell_{\infty}$-distance, and the distance from a point to a set is defined in the standard way: $d_{\infty}({{\boldsymbol x }},{{\boldsymbol c }}) =
\inf_{{{\boldsymbol y }}\in{{\boldsymbol c }}}d_{\infty}({{\boldsymbol x }},{{\boldsymbol y }})$. Thus ${{\boldsymbol A }}$ is the closed $1/4$-neighborhood of the occupied hypercubes of $A$. The connected components of the (topological) boundary $\partial {{\boldsymbol A }}$ of the set ${{\boldsymbol A }}$ are the crucial objects in what follows. Since each connected component arises from an edge configuration in $\Omega$, it is clear that the set of possible connected components is a finite set. As the connected components of $\partial {{\boldsymbol A }}$ are continuum objects, it may not be immediately apparent how to represent them in a discrete manner. We briefly describe how to do this now.
Let ${\frac{1}{2}\tor}$ denote the graph $(\frac{1}{2}\Z/n\Z)^{d}$; as a graph this is equivalent to the discrete torus $(\Z/(2n)\Z)^{d}$. The notation ${\frac{1}{2}\tor}$ is better because we will embed ${\frac{1}{2}\tor}$ in ${{{\boldsymbol T }}^d_n}$ such that (i) ${{\boldsymbol 0 }}$ coincides in $\tor$ and ${\frac{1}{2}\tor}$, and (ii) the nearest neighbors of $0$ in ${\frac{1}{2}\tor}$ are the midpoints of the edges $e$ containing $0$ in $\tor$.[^2]
An important observation is that ${{\boldsymbol A }}$ can be written as a union of collections of adjacent closed $d$-dimensional hypercubes of side-length $1/2$ centered at vertices in ${\frac{1}{2}\tor}$, where two hypercubes are called *adjacent* if they share a $(d-1)$-dimensional face. Adjacency of a set of hypercubes means the set of hypercubes is connected under the binary relation of being adjacent. By construction the connected components of ${{\boldsymbol A }}$ correspond to the connected components of the edge configuration $A$.
The boundary $\partial {{\boldsymbol A }}$ of ${{\boldsymbol A }}$ is just the sum, modulo two, of the boundaries of the hypercubes whose union gives $A$. These boundaries are $(d-1)$-dimensional hypercubes dual to edges in ${\frac{1}{2}\tor}$; here dual means that the barycenter of the $(d-1)$-dimensional hypercube is the same as barycenter of the edge in ${\frac{1}{2}\tor}$. The $(d-1)$-dimensional hypercubes that arise from this duality are the vertices in ${({\frac{1}{2}\tor})^\star}$, the graph dual to ${\frac{1}{2}\tor}$; two vertices in ${({\frac{1}{2}\tor})^\star}$ are connected by an edge if and only if the corresponding $(d-1)$-dimensional hypercubes intersect in one $(d-2)$-dimensional hypercube. The preceding discussion implies $\partial {{\boldsymbol A }}$ can be identified with a subgraph of ${({\frac{1}{2}\tor})^\star}$.
In the sequel we will discuss components of $\partial {{\boldsymbol A }}$ as continuum objects; by the preceding discussion this could be reformulated in terms of subgraphs of ${({\frac{1}{2}\tor})^\star}$. In Appendix \[app:subcomp\] we show that the computations we perform involving components of $\partial {{\boldsymbol A }}$ can be efficiently computed using their representations as subgraphs of ${({\frac{1}{2}\tor})^\star}$.
Contours and Interfaces {#sec:contours-interfaces}
-----------------------
An important aspect of the analysis in [@borgs2012tight] is that it distinguishes topologically trivial and non-trivial components of $\partial {{\boldsymbol A }}$. To make this precise, for $i=1,\dots, d$ we define the *$i$th fundamental loop* ${{\boldsymbol L }}_{i}$ to be the set $\{{{\boldsymbol y }}\in {{{\boldsymbol T }}^d_n}\mid \text{${{\boldsymbol y }}_{j}=1$ for all $j\neq i$}\}$. The *winding vector* $N({{\boldsymbol \gamma }})\in \{0,1\}^{d}$ of a connected component ${{\boldsymbol \gamma }}\in \partial{{\boldsymbol A }}$ is the vector whose $i$th component is the number of intersections (mod 2) of ${{\boldsymbol \gamma }}$ with ${{\boldsymbol L }}_{i}$.
Let $A\in\Omega$ be an edge configuration.
1. The set of *contours ${\Gamma}(A)$ associated to $A$* is the set of connected components of $\partial {{\boldsymbol A }}$ with winding vector $0$.
2. The *interface network ${\mathcal{S}}(A)$ associated to $A$* is the set of connected components of $\partial {{\boldsymbol A }}$ with non-zero winding vector. Each connected component of an interface network is an *interface*.
Without reference to any particular edge configuration, a subset ${{\boldsymbol \gamma }}\subset{{{\boldsymbol T }}^d_n}$ is a *contour* if there is an $A\in\Omega$ such that ${{\boldsymbol \gamma }}\in {\Gamma}(A)$. Interfaces and interface networks are defined analogously.
Since each fundamental loop intersects each $(d-1)$-dimensional face of a hypercube centered on ${\frac{1}{2}\tor}$ exactly zero or one times, we have the following lemma, which ensures contours can be efficiently distinguished from interfaces.
\[lem:wind-comp\] Suppose ${{\boldsymbol \gamma }}\in \partial {{\boldsymbol A }}$ is comprised of $K$ $(d-1)$-dimensional faces. Then the winding vector of ${{\boldsymbol \gamma }}$ can be computed in time $O(nK)$.
Fix $i\in \{1,2,\dots, d\}$. Each fundamental loop $L_{i}$ has length $O(n)$, and hence the set $F_{i}$ of faces that have non-trivial intersection with $L_{i}$ has cardinality ${\left|F_{i}\right|}= O(n)$. Given the list of faces in ${{\boldsymbol \gamma }}$ we can compute the $i$th component of the winding vector by (i) iterating through the list of faces of ${{\boldsymbol \gamma }}$ and adding one each time we find a face in $F_{i}$, and (ii) taking the result modulo two.
The connected components of ${{{\boldsymbol T }}^d_n}\setminus\partial{{\boldsymbol A }}$ are subsets of either ${{\boldsymbol A }}$ or ${{{\boldsymbol T }}^d_n}\setminus{{\boldsymbol A }}$. In the former case we call a component *ordered* and in the latter case *disordered*. We write ${{\boldsymbol A }}_{{\text{ord}}}$ (resp. ${{\boldsymbol A }}_{{\text{dis}}}$) for the union of the ordered (resp. disordered) components associated to $A$.
\[def:labl\] The *labelling $\ell_{A}$ associated to $A$* is the map from the connected components of ${{{\boldsymbol T }}^d_n}\setminus\partial {{\boldsymbol A }}$ to the set $\{{\text{dis}},{\text{ord}}\}$ that assigns ${\text{ord}}$ to components in ${{\boldsymbol A }}_{{\text{ord}}}$ and ${\text{dis}}$ to components in ${{\boldsymbol A }}_{{\text{dis}}}$.
Two contours ${{\boldsymbol \gamma }}_{i}$, $i=1,2$ are *compatible* if $d_{\infty}({{\boldsymbol \gamma }}_{1},{{\boldsymbol \gamma }}_{2})\geq \frac{1}{2}$. We extend this definition analogously to two interfaces, or one interface and one contour.
\[def:MCI\] A *matching collection of contours ${\Gamma}$ and interfaces ${\mathcal{S}}$* is a triple $({\Gamma},{\mathcal{S}},\ell)$ such that ${\mathcal{S}}$ is an interface network and
1. The contours and interfaces in ${\Gamma}\cup{\mathcal{S}}$ are pairwise compatible, and
2. $\ell$ is a map from the set of connected components of ${{{\boldsymbol T }}^d_n}\setminus \cup_{{{\boldsymbol \gamma }}\in{\Gamma}\cup{\mathcal{S}}}{{\boldsymbol \gamma }}$ to the set $\{{\text{dis}},{\text{ord}}\}$ such that for every ${{\boldsymbol \gamma }}\in
{\Gamma}\cup{\mathcal{S}}$, distinct components adjacent to ${{\boldsymbol \gamma }}$ are assigned different labels.
\[lem:rep\] The map from edge configurations $A\in\Omega$ to triples $({\Gamma},{\mathcal{S}},\ell)$ of matching contours and interfaces is a bijection.
See [@borgs2012tight p.15].
Contour and interface formulation of $Z$ {#sec:cont-interf-form}
----------------------------------------
By Lemma \[lem:rep\] we can rewrite the partition function in terms of matching collections of contours and interfaces by re-writing the weight $w(A)$ of a configuration $A$ in terms of its contours and interfaces. By weight $w(A)$ we mean the numerator of , i.e., $w(A) = p^{{\left|A\right|}}(1-p)^{{\left|E\setminus A\right|}}q^{c(V,A)}$. To this end, define $$\label{eq:eord-etc}
e_{{\text{ord}}} {\coloneqq}-d\log(1-e^{-\beta}),\;\;\;
e_{{\text{dis}}} {\coloneqq}d\beta - \log q, \;\;\;
\kappa {\coloneqq}\frac{1}{2}\log (e^{\beta}-1).$$ Further, define the *size ${\|{{\boldsymbol \gamma }}\|}$* of a contour ${{\boldsymbol \gamma }}$ (resp. *size* ${\|{{\boldsymbol S }}\|}$ of an interface ${{\boldsymbol S }}$) by $$\label{eq:size}
{\|{{\boldsymbol \gamma }}\|} {\coloneqq}{\left| {{\boldsymbol \gamma }}\cap \bigcup_{e\in E}{{\boldsymbol e }}\right|},
\quad {\|{{\boldsymbol S }}\|} {\coloneqq}{\left| {{\boldsymbol S }} \cap \bigcup_{e\in E}{{\boldsymbol e }}\right|}.$$ This is the number of intersections of ${{\boldsymbol \gamma }}$ (resp. ${{\boldsymbol S }}$) with $\bigcup_{e\in E}{{\boldsymbol e }}$. For a continuum set ${{\boldsymbol \Lam }}$ we write $| {{\boldsymbol \Lam }}|$ for $| {{\boldsymbol \Lam }} \cap \tor|$, that is, the number of vertices of $\tor$ in ${{\boldsymbol \Lam }}$ in the embedding of $\tor$ into ${{{\boldsymbol T }}^d_n}$. This will cause no confusion as we never need to measure the volume of a continuum set.
Using these definitions, $w(A)$ can be written as $$\label{eq:eweight}
w(A) = q^{c({{\boldsymbol A }}_{{\text{ord}}})} e^{-e_{{\text{dis}}}{\left|{{\boldsymbol A }}_{{\text{dis}}} \right|}}
e^{-e_{{\text{ord}}}{\left|{{\boldsymbol A }}_{{\text{ord}}} \right|}}
\prod_{{{\boldsymbol S }}\in{\mathcal{S}}}e^{-\kappa{\|{{\boldsymbol S }}\|}}
\prod_{{{\boldsymbol \gamma }}\in\Gamma} e^{-\kappa{\|{{\boldsymbol \gamma }}\|}},$$ where $c({{\boldsymbol A }}_{{\text{ord}}})$ is the number of connected components of ${{\boldsymbol A }}_{{\text{ord}}}$. The products run over the sets of interfaces and contours associated to the edge configuration $A$, respectively. We indicate the derivation of in Section \[sec:deriv-cont-repr\] below; see also [@borgs2012tight p.13-15]. Since $$Z = Z^{\text{RC}}_{\tor}(1-e^{-\beta},q) = \sum_{A\in\Omega}w(A) \, ,$$ it follows from and Lemma \[lem:rep\] that $$\label{eq:Z-mci}
Z = \sum_{({\mathcal{S}},{\Gamma})} q^{c({{\boldsymbol A }}_{{\text{ord}}})}
e^{-e_{{\text{dis}}}{\left|{{\boldsymbol A }}_{{\text{dis}}} \right|}}
e^{-e_{{\text{ord}}}{\left|{{\boldsymbol A }}_{{\text{ord}}} \right|}}
\prod_{{{\boldsymbol S }}\in{\mathcal{S}}}e^{-\kappa{\|{{\boldsymbol S }}\|}}
\prod_{{{\boldsymbol \gamma }}\in{\Gamma}}e^{-\kappa{\|{{\boldsymbol \gamma }}\|}},$$ where the sum runs over matching collections of contours and interfaces. This is the contour and interface network representation of the random cluster model partition function.
In what follows it will be necessary to divide the contributions to $Z$. To this end, let $$\label{eq:newsplit}
\Omega_{{\text{tunnel}}} {\coloneqq}\{A\in\Omega \mid {\mathcal{S}}(A)\neq
\emptyset\}, \quad \Omega_{{\text{rest}}} {\coloneqq}\Omega\setminus\Omega_{{\text{tunnel}}},$$ and define the corresponding partition functions $$\label{eq:newsplit-1}
Z_{{\text{tunnel}}} {\coloneqq}\sum_{A\in\Omega_{{\text{tunnel}}}}w(A), \quad Z_{{\text{rest}}}
{\coloneqq}\sum_{A\in\Omega_{{\text{rest}}}} w(A).$$ By $Z_{{\text{rest}}}$ can be expressed in terms of contours alone. We will see later that $Z_{{\text{tunnel}}}$ is very small compared to $Z_{{\text{rest}}}$, and so the task of approximating $Z$ is essentially the task of approximating $Z_{{\text{rest}}}$.
### Derivation of contour representation {#sec:deriv-cont-repr}
We briefly indicate how to obtain . Recall that $G_{A}$ denotes the graph $(V(A),A)$. Let ${\|\delta A\|} = |\delta_{1}A| + |\delta_{2}A|$, where $\delta_{k}A$ is the set of edges in $E\setminus A$ that contain $k$ vertices in $V(A)$. Observe $$\begin{aligned}
\label{eq:c}
c(V,A) &= c(G_{A}) + |V\setminus V(A)| \\
\label{eq:bdry}
2|A| &= 2d|V(A)| - {\|\delta A\|}.\end{aligned}$$ The first of these relations follows since every vertex not contained in an edge of $A$ belongs to a singleton connected component, and the second is a counting argument. Using these relations one obtains $$\label{eq:eweight2}
w(A) = q^{c(G_{A})} e^{-e_{{\text{dis}}}|V\setminus V(A)|}e^{-e_{{\text{ord}}}|V(A)|}e^{-\kappa{\|\delta A\|}}.$$
To pass from to requires just a few observations. First, $c(G_{A})$ equals the number of components of ${{\boldsymbol A }}$, which is the number of connected components of ${{\boldsymbol A }}_{{\text{ord}}}$. Second, $|V(A)| = |{{\boldsymbol A }}_{{\text{ord}}}|$, and similarly $|V\setminus V(A)| = |{{\boldsymbol A }}_{{\text{dis}}}|$. Lastly, ${\|\delta A\|}$ is precisely the sum of sizes of the contours and interfaces, as each contribution to ${\|\delta A\|}$ is given by a transverse intersection of an edge ${{\boldsymbol e }}$ with the boundary of ${{\boldsymbol A }}$.
External contour representations {#sec:Dis-Ord-Rep}
--------------------------------
Next we will take the first steps to construct a representation of $Z_{\text{rest}}$ as the sum of polymer model partition functions. We begin with some basic results and definitions. Fix an arbitrary point ${{\boldsymbol x }}_{0}\in{{{\boldsymbol T }}^d_n}$ that cannot be contained in any contour, and let $\sqcup$ denote disjoint union.
\[lem:contour-split\] For any contour ${{\boldsymbol \gamma }}$, ${{{\boldsymbol T }}^d_n}\setminus{{\boldsymbol \gamma }}$ has exactly two components.
\[def:ext\] Let ${{\boldsymbol \gamma }}$ be a contour, and suppose ${{{\boldsymbol T }}^d_n}\setminus{{\boldsymbol \gamma }} = {{\boldsymbol C }}\sqcup {{\boldsymbol D }}$. Then the *exterior* ${\text{Ext}\,}{{\boldsymbol \gamma }}$ of ${{\boldsymbol \gamma }}$ is ${{\boldsymbol C }}$ if ${\left|{{\boldsymbol C }}\right|}>{\left|{{\boldsymbol D }}\right|}$, and is ${{\boldsymbol D }}$ if the inequality is reversed. In the case of equality the exterior is the component containing ${{\boldsymbol x }}_{0}$. The *interior* ${\text{Int}\,}{{\boldsymbol \gamma }}$ of ${{\boldsymbol \gamma }}$ is the component of ${{{\boldsymbol T }}^d_n}\setminus{{\boldsymbol \gamma }}$ that is not ${\text{Ext}\,}{{\boldsymbol \gamma }}$.
Note that the notion of exterior is defined relative to ${{{\boldsymbol T }}^d_n}$, though we omit this from the notation.
This is a different definition of exterior than is used in [@borgs2012tight]; our definition is more convenient for algorithmic purposes. Most of the results of [@borgs2012tight] concerning the interiors/exteriors of contours apply verbatim with this change, and whenever we use these results we will remark on why they apply.
If two contours ${{\boldsymbol \gamma }}$ and ${{\boldsymbol \gamma }}'$ are compatible, then we write (i) ${{\boldsymbol \gamma }}<{{\boldsymbol \gamma }}'$ if ${\text{Int}\,}{{\boldsymbol \gamma }} \subset {\text{Int}\,}{{\boldsymbol \gamma }}'$ and (ii) ${{\boldsymbol \gamma }} \bot {{\boldsymbol \gamma }}'$ if ${\text{Int}\,}{{\boldsymbol \gamma }} \cap {\text{Int}\,}{{\boldsymbol \gamma }}' = \emptyset$. Given a matching collection of contours ${\Gamma}$, ${{\boldsymbol \gamma }}\in{\Gamma}$ is an *external contour* if there does not exist ${{\boldsymbol \gamma }}'\in{\Gamma}$ such that ${{\boldsymbol \gamma }}'<{{\boldsymbol \gamma }}$. The *exterior* of a matching collection of contours ${\Gamma}$ is $$\label{eq:ext-pcc}
{\text{Ext}\,}{\Gamma}{\coloneqq}\bigcap_{{{\boldsymbol \gamma }}\in{\Gamma}}{\text{Ext}\,}{{\boldsymbol \gamma }}.$$ If ${\Gamma}$ is matching, then ${\text{Ext}\,}{\Gamma}$ is a connected subset of $\tor$. This follows by noting that [@borgs2012tight Lemma 5.5] holds with Definition \[def:ext\] of the interior and exterior, and given this, the connectedness of ${\text{Ext}\,}{\Gamma}$ follows by the argument in [@borgs2012tight Lemma 5.6]. Note that since ${\text{Ext}\,}{\Gamma}$ is contained in $\tor\setminus\bigcup_{{{\boldsymbol \gamma }}\in{\Gamma}}{{\boldsymbol \gamma }}$, this implies that ${\text{Ext}\,}{\Gamma}$ is labelled either ${\text{ord}}$ or ${\text{dis}}$.
As usual in Pirogov–Sinai theory, see, e.g. [@borgs2012tight Section 6.2], it is useful to resum the matching compatible contours that contribute to according to the external contours of the configuration. To make this precise, we require several definitions. A matching collection of contours ${\Gamma}$ is *mutually external* if ${{\boldsymbol \gamma }}\bot{{\boldsymbol \gamma }}'$ for all ${{\boldsymbol \gamma }}\neq{{\boldsymbol \gamma }}'\in{\Gamma}$. For a continuum set ${{\boldsymbol \Lam }} \subseteq {{{\boldsymbol T }}^d_n}$, we say a contour ${{\boldsymbol \gamma }}$ is *a contour in ${{\boldsymbol \Lam }}$* if $d_{\infty}({{\boldsymbol \gamma }}, {{{\boldsymbol T }}^d_n}\setminus{{\boldsymbol \Lam }})\geq 1/2$. The distance to the empty set is infinite by convention.
Write $\cC({{\boldsymbol \Lam }})$ for the set of contours in ${{\boldsymbol \Lam }}$, and $\cC
= \cC({{{\boldsymbol T }}^d_n})$ for the set of all contours. For ${{\boldsymbol \Lam }}\subseteq{{{\boldsymbol T }}^d_n}$ define $\cG^{{\text{ext}}}({{\boldsymbol \Lam }})$ to be the set of matching mutually external contours in ${{\boldsymbol \Lam }}$, and then define $$\begin{aligned}
\label{eq:matchextord}
Z_{{\text{ord}}}({{\boldsymbol \Lam }})
&{\coloneqq}\sum_{{\Gamma}\in \cG^{{\text{ext}}}_{{\text{ord}}} ({{\boldsymbol \Lam }})}
e^{- e_{{\text{ord}}} | {{\boldsymbol \Lam }} \cap {\text{Ext}\,}{\Gamma}|} \prod_{{{\boldsymbol \gamma }} \in
{\Gamma}} e^{-\kappa \|{{\boldsymbol \gamma }}\|} Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }}) \\
\label{eq:matchextdis}
Z_{{\text{dis}}} ({{\boldsymbol \Lam }})
&{\coloneqq}\sum_{{\Gamma}\in \cG^{{\text{ext}}}_{{\text{dis}}}({{\boldsymbol \Lam }})}
e^{- e_{{\text{dis}}} | {{\boldsymbol \Lam }} \cap {\text{Ext}\,}{\Gamma}|}
\prod_{{{\boldsymbol \gamma }} \in {\Gamma}} e^{-\kappa \|{{\boldsymbol \gamma }}\|}
q Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }}),\end{aligned}$$ where the sums in and run over sets of matching mutually external contours in which ${\text{Ext}\,}\Gamma$ is labelled ${\text{ord}}$ and ${\text{dis}}$, respectively. This is the desired resummation. In the special case ${{\boldsymbol \Lam }}={{{\boldsymbol T }}^d_n}$ these partition functions represent the sums of $w(A)$ over $$\begin{aligned}
\label{eq:Z-split-ord}
\Omega_{{\text{ord}}}
&{\coloneqq}\{A\in\Omega\setminus\Omega_{{\text{tunnel}}} \mid
\text{${\text{Ext}\,}{\Gamma}(A)$ is labelled ${\text{ord}}$}\},
\\
\label{eq:Z-split-dis}
\Omega_{{\text{dis}}}
&{\coloneqq}\{A\in\Omega\setminus\Omega_{{\text{tunnel}}} \mid
\text{${\text{Ext}\,}{\Gamma}(A)$ is labelled ${\text{dis}}$}\}.\end{aligned}$$ That is, we get a decomposition $Z_{{\text{rest}}} = qZ_{{\text{ord}}}+Z_{{\text{dis}}}$, where $$\label{eq:Zmatch}
Z_{{\text{ord}}} = q^{-1}\sum_{A\in\Omega_{{\text{ord}}}}w(A), \quad Z_{{\text{dis}}} = \sum_{A\in\Omega_{{\text{dis}}}}w(A).$$
Subsection \[sec:RCM-form\] will give interpretations of these quantities in terms of random cluster model partition functions for many other choice of ${{\boldsymbol \Lam }}$.
Labelled contours {#sec:labell-cont-rcm}
-----------------
This subsection introduces labelled contours and establishes some basic properties of these objects. These properties will ensure that we can efficient enumerate labelled contours.
In Definition \[def:labl\] we associated a labelling to an entire collection of matching and compatible contours and interfaces. For collections of contours, since each contour splits $\tor$ into two pieces, it is more convenient to associate the labelling to individual contours. We do this by assigning a label to ${\text{Int}\,}{{\boldsymbol \gamma }}$ (resp.${\text{Ext}\,}{{\boldsymbol \gamma }}$) according to the label of the region of $\tor\setminus \cup_{{{\boldsymbol \gamma }}\in{\Gamma}}{{\boldsymbol \gamma }}$ adjacent to ${{\boldsymbol \gamma }}$ contained in ${\text{Int}\,}{{\boldsymbol \gamma }}$ (resp.${\text{Ext}\,}{{\boldsymbol \gamma }}$).
A *compatible set of labelled contours ${\Gamma}$* is a set of compatible contours ${\Gamma}$ such that the connected components of $\tor\setminus \cup_{{{\boldsymbol \gamma }}\in{\Gamma}}{{\boldsymbol \gamma }}$ are assigned the same labels by the labelled contours. More precisely, for a component ${{\boldsymbol B }}$ of $\tor\setminus \cup_{{{\boldsymbol \gamma }}\in{\Gamma}}{{\boldsymbol \gamma }}$, $\partial {{\boldsymbol B }}$ is a union of compatible contours ${{\boldsymbol \gamma }}_{0}, \dots, {{\boldsymbol \gamma }}_{k}$ for some $k\geq 0$, and (up to relabelling) either (i) ${{\boldsymbol \gamma }}_{i}<{{\boldsymbol \gamma }}_{0}$ for $i=1,\dots, k$ or (ii) ${{\boldsymbol \gamma }}_{i}\perp {{\boldsymbol \gamma }}_{j}$ for $i\neq j$. The condition of compatibility of the labels in the first case is that the interior label of ${{\boldsymbol \gamma }}_{0}$ is the same as the exterior label of ${{\boldsymbol \gamma }}_{i}$ for all $i=1,\dots k$, and in the second case is that all exterior labels agree.
By construction, the set of collections of matching and compatible contours is the same as the set of collections of compatible labelled contours. The advantage of the latter is that it enables us to define a labelled contour ${{\boldsymbol \gamma }}$ to be *ordered* if its exterior label is ${\text{ord}}$, and *disordered* if its exterior label is ${\text{dis}}$. We let $\cC_{{\text{ord}}}({{\boldsymbol \Lambda }})$ and $\cC_{{\text{dis}}}({{\boldsymbol \Lambda }})$ denote the sets of labelled contours in ${{\boldsymbol \Lambda }}$ with external labels ${\text{ord}}$ and ${\text{dis}}$, respectively, with $\cC_{\text{ord}}= \cC_{{\text{ord}}}({{{\boldsymbol T }}^d_n})$ and $\cC_{{\text{dis}}} = \cC_{{\text{dis}}}({{{\boldsymbol T }}^d_n})$. The next lemma gives a way to construct a labelled contour ${{\boldsymbol \gamma }}$ from an edge configuration.
\[lem:construct\] Let $\ell\in\{{\text{ord}},{\text{dis}}\}$, let ${{\boldsymbol \gamma }}\in \cC_{\ell}$, and ${{\boldsymbol \Lambda }}={\text{Int}\,}{{\boldsymbol \gamma }}$. Then
- If $\ell={\text{dis}}$, let $E'({{\boldsymbol \Lambda }})$ be set of edges contained in ${{\boldsymbol \Lambda }}$. Then ${{\boldsymbol \gamma }}$ is the unique component of $\partial {{\boldsymbol A }}$ where $A = E'({{\boldsymbol \Lambda }})\subset E$.
- If $\ell={\text{ord}}$, let $E'({{\boldsymbol \Lambda }})$ be the set of edges whose midpoints are contained in ${{\boldsymbol \Lambda }}$. Then ${{\boldsymbol \gamma }}$ is the unique component of $\partial {{\boldsymbol A }}$ where $A=E\setminus E'({{\boldsymbol \Lambda }})$.
These claims follows from [@borgs2012tight Lemma 5.1]; see the proof of [@borgs2012tight Lemma 5.11].[^3]
Lemma \[lem:construct\] gives a way to construct a given contour from some set of edges $A$. For our algorithms it will be important to be able to generate contours from a relatively small set of edges. We first explain how to do this for disordered contours.
Suppose ${{\boldsymbol \gamma }}\in \cC_{{\text{dis}}}$ and let $\Lambda = {\text{Int}\,}{{\boldsymbol \gamma }}\cap\tor$. Define $$\label{eq:active2s}
\cE_{{{\boldsymbol \gamma }}} {\coloneqq}\{ e=\{i,j\} \mid i,j\in\Lambda, \,\,
d_{\infty}(\text{mid}({{\boldsymbol e }}),{{\boldsymbol \gamma }})\geq 3/4 \},$$ where $\text{mid}({{\boldsymbol e }})$ denotes the midpoint of the edge ${{\boldsymbol e }}$; this is the vertex of ${\frac{1}{2}\tor}$ on the two-step path from $i$ to $j$ in ${\frac{1}{2}\tor}$.
\[lem:edge-dis\] Suppose ${{\boldsymbol \gamma }}\in \cC_{{\text{dis}}}$ and let ${{\boldsymbol \Lambda }} = {\text{Int}\,}{{\boldsymbol \gamma }}$. Suppose $F\subseteq \cE_{{{\boldsymbol \gamma }}}$ and let $A = E'\setminus F$, where $E'=E'(\Lambda)$ is defined as in Lemma \[lem:construct\]. Let ${\Gamma}$ be the set of contours in $\partial {{\boldsymbol A }}$. Then ${{\boldsymbol \gamma }}\in{\Gamma}$, and for all ${{\boldsymbol \gamma }}'\in{\Gamma}$ with ${{\boldsymbol \gamma }}'\neq{{\boldsymbol \gamma }}$ we have ${{\boldsymbol \gamma }}' <
{{\boldsymbol \gamma }}$. Moreover, all sets of matching contours consisting of $\gamma$ and contours in ${\text{Int}\,}\gamma$ arise from such $F$.
We begin by recalling an alternate construction of ${{\boldsymbol A }}$ from [@borgs2012tight]. Let $E\subset
E(\tor)$, and let $D\subset E$. Set $D^{\star}$ to be the set of $(d-1)$-dimensional unit hypercubes dual to the edges of $D$, and set $$V_{-}(D) = {\left\{x\in V(\tor) \mid \{x,y\} \in D \text{ if } \{x,y\} \in E\right\}}.$$ Set ${{\boldsymbol D }}_{{\text{dis}}}$ to be the union of the open $3/4$-neighborhood of $V_{-}(D)$ and the open $1/4$-neighborhood of $D^{\star}$. Then by [@borgs2012tight Lemma 5.1, (iv)], if $D =
E\setminus A$, ${{\boldsymbol E }}\setminus {{\boldsymbol A }} =
{{\boldsymbol D }}_{{\text{dis}}}$. I.e., ${{\boldsymbol D }}_{{\text{dis}}}$ is the disordered region associated to $A$ (relative to the region ${{\boldsymbol E }}$).
To prove the lemma, we apply this construction with $E=E'(\Lam)$ and $D=F$. The definition of $\cE_{{{\boldsymbol \gamma }}}$ ensures that both the open $3/4$-neighborhoods of the included vertices and the open $1/4$-neighborhoods of the included dual facets are at distance at least $1/2$ from ${{\boldsymbol \gamma }}$. This implies that ${{\boldsymbol \gamma }}$ is a boundary component of ${{\boldsymbol E\setminus
F }}$, and the first claim follows as all other boundary components are adjacent to ${{\boldsymbol D }}_{{\text{dis}}}$. The second claim follows from the bijection of Lemma \[lem:rep\], which restricts to a bijection in this setting.
\[lem:dis-edge-con\] Suppose ${{\boldsymbol \gamma }}\in \cC_{{\text{dis}}}$. Then there is a connected graph with edge set $A$ such that (i) ${\left|A\right|}\leq 2d{\|{{\boldsymbol \gamma }}\|}$ and (ii) ${{\boldsymbol \gamma }}$ is the outermost contour in $\partial {{\boldsymbol A }}$.
Choose $F=\cE_{{{\boldsymbol \gamma }}}$ in Lemma \[lem:edge-dis\]. Then the subgraph of $\tor$ induced by $E''=E'(\Lambda)\setminus F$ is connected: if not $\partial {{\boldsymbol {E''} }}$ would contain two compatible exterior contours as the boundaries of the thickenings of the connected components of $E''$ are compatible. This would contradict the conclusion of Lemma \[lem:edge-dis\] that there is a unique exterior contour.
The bound on the size of $A$ is crude; it can be obtained by noting that the included edges all contain a vertex from which there is an edge outgoing from $\Lambda$, and the count of these vertices is a lower bound for ${\|{{\boldsymbol \gamma }}\|}$. Each of the vertices is contained in at most $2d$ edges.
We now establish a similar way to construct an ordered contour from a small edge set. The situation is slightly different due to the differences between ordered and disordered contours in Lemma \[lem:construct\]. Define, for ${{\boldsymbol \gamma }}\in \cC_{{\text{ord}}}$, $\Lambda = {\text{Int}\,}{{\boldsymbol \gamma }} \cap \tor$, $$\label{eq:active1s}
\cE_{{{\boldsymbol \gamma }}} {\coloneqq}\{ \{i,j\} \mid i,j\in\Lambda\}.$$
\[lem:edge-ord\] Suppose ${{\boldsymbol \gamma }}\in \cC_{{\text{ord}}}$ and $F\subseteq \cE_{{{\boldsymbol \gamma }}}$. Let $A = (E\setminus E'(\Lambda))\cup F$, where $E'(\Lambda)$ is defined as in Lemma \[lem:construct\]. Let ${\Gamma}$ be the set of contours in $\partial {{\boldsymbol A }}$. Then ${{\boldsymbol \gamma }}\in{\Gamma}$, and for all ${{\boldsymbol \gamma }}'\in{\Gamma}$ with ${{\boldsymbol \gamma }}'\neq{{\boldsymbol \gamma }}$ we have ${{\boldsymbol \gamma }}'<{{\boldsymbol \gamma }}$. Moreover, all sets of matching contours consisting of $\gamma$ and contours in ${\text{Int}\,}\gamma$ arise from such $F$.
The proof is essentially the same as for Lemma \[lem:edge-dis\]. Let $A' = E\setminus E'(\Lam)$. The set ${{\boldsymbol F }}$ is disjoint from ${{\boldsymbol {A'} }}$ as every vertex $i$ interior to ${{\boldsymbol \gamma }}$ is at distance at least $3/4$ from ${{\boldsymbol \gamma }}$. This implies $\partial {{\boldsymbol A }}$ is the union of $\partial {{\boldsymbol {A'} }}$ and $\partial {{\boldsymbol F }}$, which implies the first claim. The second claim follows from the bijection of Lemma \[lem:rep\], which restricts to a bijection in this setting.
Two edges $e,f\in E$ are called *$1$-adjacent* if $d_{\infty}({{\boldsymbol e }},{{\boldsymbol f }})\leq 1$. A set of edges $A$ is *$1$-connected* if for any $e,f\in A$, there is a sequence of $1$-adjacent edges in $A$ from $e$ to $f$. In the next lemma, $\partial {{\boldsymbol {A^{c}} }}$ is the boundary of the thickening of the edge set $A^{c} = E \setminus A$.
\[lem:ord-edge-con\] Suppose ${{\boldsymbol \gamma }}\in \cC_{{\text{ord}}}$. Then there is a $1$-connected set of edges $A$ of size at most ${\|{{\boldsymbol \gamma }}\|}$ such that ${{\boldsymbol \gamma }}$ is the outermost contour in $\partial {{\boldsymbol {A^{c}} }}$.
Let $A$ be the set of all edges that intersect ${{\boldsymbol \gamma }}$. By the definition of ${\|\cdot\|}$, ${\left|A\right|}\leq {\|{{\boldsymbol \gamma }}\|}$. By Lemma \[lem:edge-ord\] ${{\boldsymbol \gamma }}$ is the outermost contour in ${{\boldsymbol {A^{c}} }}$, as $A^{c} = E'(\Lam)\cup \cE_{{\text{dis}}}(\Lam)$. The $1$-connectedness of $A$ follows from the connectedness of ${{\boldsymbol \gamma }}$ and the observation that every point of ${{\boldsymbol \gamma }}$ is at most $d_{\infty}$ distance $1/2$ from an edge in $A$.
Contour Enumeration {#sec:contour-enumeration}
-------------------
This section uses the results of the previous subsection to guarantee the existence of an efficient algorithm for enumerating contours. This requires a few additional lemmas.
\[lem:iso\] For all ${{\boldsymbol \gamma }} \in \cC$, ${\left|{\text{Int}\,}{{\boldsymbol \gamma }}\right|}\leq {\|{{\boldsymbol \gamma }}\|}^{2}$, and ${\left|{\text{Int}\,}{{\boldsymbol \gamma }}\right|}\leq (n/2) {\|{{\boldsymbol \gamma }}\|}$.
This follows by [@borgs2012tight Lemma 5.7], as the interior of a contour as defined by Definition \[def:ext\] is always smaller than the definition of the interior of a contour in [@borgs2012tight].
\[lem:findext\] There is an algorithm that determines the vertex set ${\text{Int}\,}{{\boldsymbol \gamma }}\cap \tor$ in time $O({\|{{\boldsymbol \gamma }}\|}^{3})$.
Let $m ={\|{{\boldsymbol \gamma }}\|}$. Let $G$ be the subgraph of $\tor$ that arises after removing all edges that intersect some $(d-1)$-dimensional face in ${{\boldsymbol \gamma }}$. Consider the following greedy algorithm to determine the connected components of $G$: this algorithm starts at $C_{0}=x$, where $x$ is chosen such that it is contained in an edge not present in $G$. The algorithm determines the connected component containing $x$ in $G$ by adding at step $i+1$ the first vertex (with respect to lexicographic order) in $\tor\setminus C_{i}$ that neighbors $C_{i}$; if no neighbors exist the component has been determined. The $k$th step takes time at most $(2d)k$, so performing $N$ steps of this algorithm takes time $O(N^{3})$.
Since ${\left|{\text{Int}\,}{{\boldsymbol \gamma }}\right|}\leq {\|{{\boldsymbol \gamma }}\|}^{2}$ by Lemma \[lem:iso\], we can stop the greedy procedure after $m^{2}+1$ steps. If the algorithm terminates due to this condition, the component being explored is the exterior. The interior can then be determined in at most $O(m^{3})$ additional steps by running the greedy algorithm from the neighbor of $x$ that is in the interior. Otherwise the algorithm will have already terminated and determined the interior.
\[lem:enum\] Fix an edge $e\in E$. There is an algorithm to construct all contours ${{\boldsymbol \gamma }}\in \cC_{{\text{ord}}}$ that (i) can arise from a connected edge set $A$ that contains $e$ and (ii) have ${\|{{\boldsymbol \gamma }}\|}\leq m$. The algorithm runs in time $\exp(O(m))$.
Similarly, there is an $\exp(O(m))$-time algorithm to construct all contours ${{\boldsymbol \gamma }}\in \cC_{{\text{dis}}}$ that (i) can arise from an edge set $A$ such that $A^{c}$ is $1$-connected and contains $e$ and (ii) have ${\|{{\boldsymbol \gamma }}\|}\leq m$.
We first consider disordered contours, and begin by enumerating all connected sets $A$ of edges that contain $e$ that are of size at most $2dm$. This can be done in time $\exp(O(m))$. If $2m\leq n$ then we consider the enumerated edge sets as subsets of $E(\mathbb{T}_{2m}^{d})$; otherwise we consider them as subsets of $E(\tor)$.
For each edge set $A$, construct $\partial {{\boldsymbol A }}$ and take the outermost contour (if there is not a single outermost contour, discard the result). By Lemma \[lem:dis-edge-con\] this generates all disordered contours of size at most $m$ that arise from connected edge sets containing $e$. We obtain the desired list of contours by removing any duplicates, which takes time at most $\exp(O(m))$. The remainder of the proof shows that the operations in this paragraph can be done in time polynomial in $m$.
The constructions of $\partial {{\boldsymbol A }}$ takes time at most $O(m)$ as it is a $\Z_{2}$ sum of $(d-1)$-dimensional facets, and determining these facets takes a constant amount of time (depending only on the dimension $d$) for each edge. Determining if a component of $\partial {{\boldsymbol A }}$ is a contour can be done by computing the winding number of the component; this takes time $O((2dm\wedge n)K)$ for a component with $K$ facets by Lemma \[lem:wind-comp\]. Determining the interior of a given contour takes time at most $O(m^{3})$ by Lemma \[lem:findext\], and hence we can check if ${\text{Int}\,}{{\boldsymbol \gamma }}' \subset {\text{Int}\,}{{\boldsymbol \gamma }}$ for all pairs in time $O(m^{4})$ since there are at most $m^{2}$ contours. This completes the proof for disordered contours.
For ordered contours the argument applies nearly verbatim. The changes are as follows. First, enumerate $1$-connected sets $A^{c}$ that contain $e$. Secondly, to see that we get the desired contours, appeal to Lemma \[lem:ord-edge-con\]. Lastly, computing $\partial {{\boldsymbol A }}$ takes time $O( (m\wedge n)^{d})$ which is polynomial in $m$; this is by our choice of torus in the first paragraph of the proof.
The next definition is useful for inductive arguments involving contours.
\[defLevel\] The *level* $\cL({{\boldsymbol \gamma }})$ of a contour ${{\boldsymbol \gamma }}$ is defined inductively as follows. If ${{\boldsymbol \gamma }}$ is *thin*, meaning $\cC({\text{Int}\,}{{\boldsymbol \gamma }})=\emptyset$, then $\cL({{\boldsymbol \gamma }})=0$. Otherwise, $ \cL({{\boldsymbol \gamma }})=1+\max\{\cL({{\boldsymbol \gamma }}') \mid
{{\boldsymbol \gamma }}'< {{\boldsymbol \gamma }}\}.$
Call a set ${{\boldsymbol \Lambda }} \subseteq {{{\boldsymbol T }}^d_n}$ a *region* if ${{\boldsymbol \Lambda }} = {{{\boldsymbol T }}^d_n}$ or if ${{\boldsymbol \Lam }}$ is a connected component of ${{{\boldsymbol T }}^d_n}\setminus \partial {{\boldsymbol A }}$ for some $A\subset E$. In the former case set $\partial {{\boldsymbol \Lam }}= \emptyset$, and in the latter case set $\partial {{\boldsymbol \Lam }}$ to be the union of all connected components of $\partial {{\boldsymbol A }}$ incident to ${{\boldsymbol \Lam }}$. In particular if ${{\boldsymbol \Lam }} = {\text{Int}\,}{{\boldsymbol \gamma }}$ for some contour ${{\boldsymbol \gamma }}$, then ${{\boldsymbol \Lam }}$ is a region and $\partial {{\boldsymbol \Lam }} = {{\boldsymbol \gamma }}$. Finally, for compatible contours ${{\boldsymbol \gamma }}_1, \dots, {{\boldsymbol \gamma }}_t$, define $\| {{\boldsymbol \gamma }}_1 \cup \cdots \cup {{\boldsymbol \gamma }}_t\| = \| {{\boldsymbol \gamma }}_1 \| +
\cdots + \| {{\boldsymbol \gamma }}_t\|$. We conclude this subsection by stating our main algorithmic result on efficiently computing sets of contours.
\[prop:enum\] There is an $O(({\left|{{\boldsymbol \Lambda }}\right|}+{\|\partial {{\boldsymbol \Lambda }}\|})\exp(O(m)))$-time algorithm that, for all regions ${{\boldsymbol \Lam }}$, (i) enumerates all contours in $\cC_{{\text{ord}}}({{\boldsymbol \Lambda }})\cup\cC_{{\text{dis}}}({{\boldsymbol \Lambda }})$ with size at most $m$ and (ii) sorts this list consistent with the level assignments.
We begin by proving the first item. Apply Lemma \[lem:enum\] for each edge contained in ${{\boldsymbol \Lambda }}$. This takes time $O({\left|{{\boldsymbol \Lambda }}\right|}\exp(O(m)))$ as there are at most $2d$ edges in ${{\boldsymbol \Lambda }}$ for each vertex of $\tor$ in ${{\boldsymbol \Lambda }}$. The output is a (multi-)set of contours of size at most $m$ contained in $\tor$. Trim the resulting list of contours to remove duplicates.
By Lemma \[lem:findext\] in time $\exp(O(m))$ we can determine ${\text{Int}\,}{{\boldsymbol \gamma }}$ for every ${{\boldsymbol \gamma }}$ from the list obtained in the first paragraph. We determine the list of level zero contours by iterating through the list, checking for each ${{\boldsymbol \gamma }}$ if ${{\boldsymbol \gamma }}'<{{\boldsymbol \gamma }}$ for some other ${{\boldsymbol \gamma }}'\neq {{\boldsymbol \gamma }}$ in the list. If not, assign ${{\boldsymbol \gamma }}$ level $0$. This takes time at most $\exp(O(m))$. We continue by running the same operation on the sublist of all contours of level at least one, i.e., the sublist of contours not assigned level $0$. If ${{\boldsymbol \gamma }}$ has level at least one and there is no ${{\boldsymbol \gamma }}'<{{\boldsymbol \gamma }}$, ${{\boldsymbol \gamma }}'$ also of level at least one, then ${{\boldsymbol \gamma }}$ is assigned level one. By repeating this we assign a level to every contour. The maximal level of a contour is $m^{2}$, the maximal size of the interior of a contour of size $m$, and hence the total running time is at most $m^{2}\exp(O(m)) = \exp(O(m))$.
To conclude, trim the list to retain only contours ${{\boldsymbol \gamma }}'$ contained in ${{\boldsymbol \Lambda }}$. This can be done by removing contours at distance less than $1/2$ from ${{\boldsymbol \gamma }}$. Computing this distance takes time $O({\|{{\boldsymbol \gamma }}\|} {\|{{\boldsymbol \gamma }}'\|})$, which is at most $O({\|{{\boldsymbol \gamma }}\|}m)$.
Polymer representations for $Z_{{\text{ord}}}$ and $Z_{{\text{dis}}}$ {#sec:extern-cont-repr-1}
---------------------------------------------------------------------
To obtain polymer model representations of $Z_{{\text{ord}}}$ and $Z_{{\text{dis}}}$, define ${\tilde\Omega}_{{\text{ord}}}({{\boldsymbol \Lambda }})$ and ${\tilde\Omega}_{{\text{dis}}}({{\boldsymbol \Lam }})$ to be the sets of compatible collections of contours in ${{\boldsymbol \Lam }}$ that are labelled ${\text{ord}}$ and ${\text{dis}}$, respectively. Define $$\label{eq:Kweight}
K_{{\text{ord}}}({{\boldsymbol \gamma }})
= e^{- \kappa \| {{\boldsymbol \gamma }} \|} \frac{Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }})
}{ Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }}) }, \qquad
K_{{\text{dis}}}({{\boldsymbol \gamma }})
= e^{- \kappa \| {{\boldsymbol \gamma }} \|} \frac{q Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }})
}{ Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }}) }.$$ By following a well-trodden path in Pirogov–Sinai theory (see, e.g., [@borgs2012tight p.28] or [@helmuth2018contours p.29]), these definitions give the following representations for $Z_{{\text{ord}}}$ and $Z_{{\text{dis}}}$ as partition functions of abstract polymer models: $$\begin{aligned}
\label{eqZordPoly}
Z_{{\text{ord}}}({{\boldsymbol \Lam }})
&= e^{- e_{{\text{ord}}} | {{\boldsymbol \Lam }}|} \sum_{{\Gamma}\in{\tilde\Omega}_{{\text{ord}}}({{\boldsymbol \Lambda }})} \prod_{{{\boldsymbol \gamma }} \in {\Gamma}}
K_{{\text{ord}}}({{\boldsymbol \gamma }}) \\
\label{eqZdisPoly}
Z_{{\text{dis}}}({{\boldsymbol \Lam }})
&= e^{- e_{{\text{dis}}} | {{\boldsymbol \Lam }}|} \sum_{{\Gamma}\in{\tilde\Omega}_{{\text{dis}}}({{\boldsymbol \Lambda }})} \prod_{{{\boldsymbol \gamma }} \in
{\Gamma}}
K_{{\text{dis}}}({{\boldsymbol \gamma }}) \,.\end{aligned}$$ where the sums run over collections of compatible labelled contours in ${{\boldsymbol \Lam }}$ with external label ${\text{ord}}$ and ${\text{dis}}$, respectively.
In fact, for $\ell\in \{{\text{ord}},{\text{dis}}\}$, the above formulas represent $Z_{\ell}({{\boldsymbol \Lam }})$ as the partition function of a polymer model in the form discussed in Section \[secPolymer\], i.e., where polymers are subgraphs of a fixed graph $G$ with bounded degree. In detail, recalling the discussion in Section \[sec:cont\], we consider contours as subgraphs of (a subgraph of) the bounded-degree graph ${({\frac{1}{2}\tor})^\star}$. Thus $|{{\boldsymbol \gamma }}|$ is the number of vertices in a contour when represented as a subgraph. Condition holds with $b=1$ since ${\|{{\boldsymbol \gamma }}\|}\geq |\gamma|$ by . The more substantial hypothesis will be verified in later sections for appropriate choices of the label and of $\beta$.
In the sequel we will write $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}$ for the size of set of vertices of ${({\frac{1}{2}\tor})^\star}$ that are part of some contour ${{\boldsymbol \gamma }}$ in $\cC_{\ell}({{\boldsymbol \Lam }})$ for some $\ell$. The next technical lemma shows it is enough to find algorithms that are polynomial time in $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}$.
\[lem:polygraphsize\] For ${{\boldsymbol \Lam }}$ a continuum set, $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}$ is polynomial in $|\Lam|$.
By construction, contours inside ${{\boldsymbol \Lam }}$ arise from edge configurations of edges inside ${{\boldsymbol \Lam }}$. The number of such edges is at most $2d$ times the number of vertices inside. Since contours are boundaries of unions of $(d-1)$-dimensional hypercubes centered at vertices in ${({\frac{1}{2}\tor})^\star}$ that lie on edges, this proves the claim, since there are a bounded number of such hypercubes associated to each edge.
Random cluster model formulations of contour partition functions {#sec:RCM-form}
----------------------------------------------------------------
The definitions of the partition functions $Z_{{\text{ord}}}({{\boldsymbol \Lambda }})$ and $Z_{{\text{dis}}}({{\boldsymbol \Lambda }})$ in and only involve contours. In general, these contour partition functions do not correspond to random cluster model partition functions due to the exclusion of interfaces. However, we will show that when $\Lambda = {\text{Int}\,}{{\boldsymbol \gamma }} \cap \tor$ can be embedded as a subgraph of $\Z^{d}$, there is such an interpretation.
To make this precise, recall the definitions and of $Z^f_{\Lam}$ and $Z^w_{\Lam}$ for $\Lam\subset \Z^{d}$ such that the subgraph $G_{\Lam}$ induced by $\Lam$ is simply connected. Recall that $p=1-e^{-\beta}$.
\[prop:scregion\] Suppose $\Lambda\subset\Z^{d}$ is simply connected, and let $n = 3 |\Lam|$. Then there are contours ${{\boldsymbol \gamma }}_{{\text{dis}}}\in \cC_{{\text{dis}}}(\tor)$ and ${{\boldsymbol \gamma }}_{{\text{ord}}}\in \cC_{{\text{ord}}}(\tor)$ determined by $\Lam$ such that $$Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}) =
(1-p)^{-\frac{1}{2}{\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}}Z^f_{\Lam}, \quad
Z_{\text{ord}}({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{dis}}}) =
q^{-1}p^{d{\left|{\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}\right|}-{\left|E(\Lam)\right|}} Z^w_{\Lam} .$$
Since $n=3|\Lam|$, we can embed $\Lam\subset \tor$. Moreover, the set of boundary vertices $\partial \Lam {\coloneqq}\{ i \in \Lam: \exists j \in \Z^{d}\setminus
\Lam, (i,j) \in E(\Z^{d}) \}$ can be identified with $ \{i\in \Lam: \exists j\in \Lam^{c}, (i,j)\in E\}$. Thus the graphs $G_{\Lam}$ and $G_{\Lam}'$ used in the definitions of $Z^{f}_{\Lam}$ and $Z^{w}_{\Lam}$ are the same whether defined by considering $\Lam$ as a subset of $\Z^{d}$ or $\tor$. Note that by our choice of $n$ we know that any component of $\partial {{\boldsymbol A }}$ will be a contour if $A$ is a subset of edges that are at graph distance at most two from $\Lam$. To see this in an elementary way, note that we can further consider $\Lam$ as a subset of $\tor$ such that the fundamental loops of $\tor$ are at distance at least (say) ten from ${{\boldsymbol \Lam }}$.
We first consider the case of $Z^{f}_{\Lam}$. To do this, let $A_{0}\subset E$ be the set of edges with both endpoints in $\Lam^c$. Let ${{\boldsymbol \gamma }}_{\text{ord}}$ be the unique contour in $\partial {{\boldsymbol {A_{0}} }}$; the fact that there is a unique contour follows from the fact that $\Lam$ is simply connected. By Lemma \[lem:edge-ord\], for any subset $A$ of edges in $E(G_{\Lambda})=\cE_{{{\boldsymbol \gamma }}_{{\text{ord}}}}$, the contours of $\partial {{\boldsymbol A }}$ are contained in ${\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}$. Moreover, this Lemma ensures that by carrying out the contour construction of Section \[sec:deriv-cont-repr\] for subsets of edges $A' = A_{0}\cup A$ where all edges of $A$ are from $E(G_{\Lambda})$, we obtain all contour configurations ${\Gamma}= \{\gamma_{{\text{ord}}}\}\cup {\Gamma}'$ where the contours of ${\Gamma}'$ are contained in ${\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}$.
To obtain the conclusion, note that (i) $\sum_{A'}w(A')$ is proportional to $Z^{f}_{\Lam}$, where the sum runs over these $A' = A_{0}\cup A$ described above, and (ii) $\sum_{A'}w(A')$ is proportional to $Z_{{\text{dis}}}({{\boldsymbol \gamma }}_{{\text{ord}}})$. To obtain the proportionality constant we compare the contributions of the empty edge configuration (empty contour configuration). These are, respectively, $q^{{\left|\Lam\right|}}(1-p)^{{\left|E(\Lam)\right|}}$ and $q^{{\left|\Lam\right|}}(1-p)^{d{\left|\Lam\right|}}$. The ratio of these terms is $(1-p)^{{\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}}$ since ${\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}$ is exactly the number of edges between $\Lambda$ and $\Lambda^{c}$.[^4]
We now consider the case of $Z^{w}_{\Lam}$. Let $A=E(G_{\Lam})$, and consider the ordered contour ${{\boldsymbol \gamma }}'$ that arises from the edge set $E\setminus A$. Define $$\tilde A {\coloneqq}A\cup \{e\in E \mid d_{\infty}(\text{mid}({{\boldsymbol e }}),{{\boldsymbol \gamma }}')\leq
1/2\},$$ the set of edges whose midpoints are either in the interior of ${{\boldsymbol \gamma }}'$ or within distance $1/2$ of ${{\boldsymbol \gamma }}'$. Then set ${{\boldsymbol \gamma }}_{{\text{dis}}}$ to be the single contour in $\partial {{\boldsymbol {\tilde A} }}$; there is only one contour in this set by the assumption $\Lam$ is simply connected. Note that $A$ is precisely $\cE_{{{\boldsymbol \gamma }}_{{\text{dis}}}}$ as defined above Lemma \[lem:edge-dis\], and hence there is a bijection between contour configurations in ${\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{dis}}}$ and subsets of $\tilde A$ in which each edge not in $A$ is occupied. As for the case of $Z^{f}_{\Lam}$ we can now conclude, as summing over such edge sets is proportional to both $Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{dis}}})$ (recall ) and $Z^{w}_{\Lam}$. To compute the proportionality constant, we compare the all occupied configuration to the empty contour configuration. This gives, respectively, $qp^{{\left|E(\Lam)\right|}}$ and $e^{-e_{{\text{ord}}}{\left|{\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}\right|}}$, and hence $$Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{dis}}}) =
q^{-1}p^{d{\left|{\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}\right|}-{\left|E(\Lam)\right|}} Z^{w}_{\Lam}.$$
Contour model estimates {#secEstimates}
=======================
In this section we state several estimates related to the contour representations from the previous section.
\[lem:Z-split\] There are constants $c>0$, $q_{0}=q_{0}(d)<\infty$, and $n_{0}<\infty$ such that if $q\geq q_{0}$, $n\geq n_{0}$, and $\beta\geq \beta_{c}$, $$\label{eq:tunnel-small}
\frac{Z_{{\text{tunnel}}}}{Z} \leq \exp(-c\beta n^{d-1}).$$
In what follows $c$ will always denote the constant from Lemma \[lem:Z-split\], and $q_0$ and $n_0$ will always be at least as large as the constants in the lemma. Lemma \[lem:Z-split\] ensures that $Z_{{\text{tunnel}}}$ is neglectable when approximating $Z$ up to relative errors $\epsilon \gg \exp ( -c\beta n^{d-1})$. We will also need to know that $Z_{{\text{dis}}}$ is neglectable when $\beta>\beta_{c}$. This requires two lemmas.
\[lemBCTsup\] If $q \ge q_0$, $n\geq n_{0}$, and $\beta > \beta_c$ there exist $a_{{\text{dis}}}>0$ and $f>0$ so that if $\eps_{n}{\coloneqq}2\exp(-c\beta n)$, then $$Z_{{\text{ord}}} \ge \exp(-(f +\eps_n ) n^d), \quad
Z_{{\text{dis}}} \le \exp((- f + \eps_n) n^d) \max_{\Gamma \in \cG^{{\text{ext}}}_{{\text{dis}}}} e^{- \frac{a_{{\text{dis}}}}{2} |{\text{Ext}\,}\Gamma|} \prod_{{{\boldsymbol \gamma }} \in \Gamma} e^{-\frac{c}{2} \beta \| {{\boldsymbol \gamma }} \| } ,$$
With $a_{{\text{dis}}}\geq 0$ this follows from [@borgs2012tight Lemma 6.3] provided $f=f_{{\text{ord}}}$ for $\beta\geq \beta_{c}$, and that $f=f_{{\text{ord}}}$ follows from [@borgs2012tight Lemma A.3]. What remains is to prove $a_{{\text{dis}}}>0$ when $\beta>\beta_{c}$. The results of [@laanait1991interfaces] imply that there is a unique Gibbs measure for the random cluster model when $\beta>\beta_{c}$. If $a_{{\text{dis}}}$ was $0$ for some $\beta>\beta_{c}$, then the argument establishing [@borgs2012tight Lemma 6.1 (b)] implies the existence of multiple Gibbs measures, a contradiction.
\[lemSuperEstimates2a\] If $q \ge q_0$, $n\ge n_0$, and $\beta > \beta_c$, then there exists a constant $b_{{\text{dis}}}>0$ so that $$\label{eq:SuperEstimates2a}
\frac{ Z_{{\text{dis}}}}{ Z } \le 2\exp(-b_{{\text{dis}}} n^{d-1}) \,.$$
Suppose $\Gamma \in \cG_{{\text{dis}}}^{{\text{ext}}}$. Then we claim that $$\label{eq:minsize}
|{\text{Ext}\,}\Gamma| + \sum_{{{\boldsymbol \gamma }}\in\Gamma}{\|{{\boldsymbol \gamma }}\|} \geq 2n^{d-1}.$$ To see this, note that $$\label{eq:minsize1}
|{\text{Ext}\,}\Gamma| + \sum_{{{\boldsymbol \gamma }}\in\Gamma}|{\text{Int}\,}{{\boldsymbol \gamma }}|= n^{d},$$ which combined with Lemma \[lem:iso\] implies $$|{\text{Ext}\,}\Gamma| + \frac{n}{2} \sum_{{{\boldsymbol \gamma }}\in\Gamma}{\|{{\boldsymbol \gamma }}\|} \ge n^d$$ which implies when $n\geq 2$.
By Lemma \[lemBCTsup\], if $n$ is large enough, $$\label{eq:disneg}
\frac{Z_{{\text{dis}}}}{Z_{{\text{ord}}}} \le 2 \max_{\Gamma \in \cG_{{\text{dis}}}^{{\text{ext}}}} e^{- \frac{a_{{\text{dis}}}}{2} |{\text{Ext}\,}\Gamma|} \prod_{{{\boldsymbol \gamma }} \in \Gamma} e^{-\frac{c}{2} \beta \| {{\boldsymbol \gamma }} \| } .$$ Set $b_{{\text{dis}}} {\coloneqq}\min \{ a_{{\text{dis}}}, c \beta \}>0$. By , $$e^{- \frac{a_{{\text{dis}}}}{2} |{\text{Ext}\,}\Gamma|} \prod_{{{\boldsymbol \gamma }} \in \Gamma}
e^{-\frac{c}{2} \beta \| {{\boldsymbol \gamma }} \| } \le \exp(- b_{{\text{dis}}} n^{d-1})$$ for all $\Gamma \in \cG_{{\text{dis}}}^{{\text{ext}}}$. The lemma now follows from .
The next two lemmas will allow us to verify the Kotecký–Preiss condition for the contour models defined in the previous section.
\[lemKestimates\] If $q \ge q_0$ and $\beta=\beta_c$, then $$K_{{\text{ord}}} ({{\boldsymbol \gamma }}) \le e^{-c \beta \|{{\boldsymbol \gamma }}\|}, \quad \text{and}\quad
K_{{\text{dis}}} ({{\boldsymbol \gamma }}) \le e^{-c \beta \| {{\boldsymbol \gamma }} \|} \,,$$ for all ${{\boldsymbol \gamma }}$ in $\cC_{\text{ord}}$ and $\cC_{{\text{dis}}}$, respectively.
\[lemSuperEstimates\] If $q \ge q_0$ and $\beta > \beta_c$, then $$K_{{\text{ord}}} ({{\boldsymbol \gamma }}) \le e^{-c \beta \| {{\boldsymbol \gamma }} \|}, \qquad {{\boldsymbol \gamma }} \in \cC_{{\text{ord}}}$$
In particular, since $\beta \ge \frac{3 \log q}{d}$, then for sufficiently large $q$ the contour weights $K_{{\text{ord}}}$ (for $\beta \ge \beta_c$) and $K_{{\text{dis}}}$ (for $\beta = \beta_c$) will satisfy condition .
Next we will show that when $\beta > \beta_c$ and the disordered ground state is unstable, that regions with disordered boundary conditions ‘flip’ quickly to ordered regions by way of a large contour; more precisely, the dominant contribution to $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$ from collections of contours with small external volume.
For a region ${{\boldsymbol \Lam }}$ and $M >0$ we define $${\cH}^{\text{flip}}_{\text{dis}}({{\boldsymbol \Lambda }},M) {\coloneqq}\{\Gamma\in \cG_{{\text{dis}}}^{{\text{ext}}}({{\boldsymbol \Lam }})
\mid {\left|{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}\right|} \leq M\},$$ and $$\label{eq:Zflip}
Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }},M) {\coloneqq}\sum_{\Gamma\in{\cH}^{\text{flip}}_{\text{dis}}({{\boldsymbol \Lambda }},M)}e^{-e_{{\text{dis}}}|{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}|}
\prod_{{{\boldsymbol \gamma }}\in\Gamma}e^{-\kappa {\|{{\boldsymbol \gamma }}\|}}qZ_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}).$$ Thus, c.f. , $Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }},M)$ is the contribution to $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$ from contour configurations with small exterior volume.
\[lemSuperEstimates2\] Suppose $q \ge q_0$ and $\beta > \beta_c$. Then there exists $a_{\text{dis}}>0$ so that the following holds for all $n \ge n_0$. Suppose ${{\boldsymbol \gamma }} \in \cC_{{\text{ord}}}$. For any $\eps>0$, if $$\label{eq:Mlb}
M\geq \frac{2}{a_{{\text{dis}}}}\log \frac{8q}{\eps} +
\frac{2}{a_{{\text{dis}}}}(\kappa+3){\|{{\boldsymbol \gamma }}\|}$$ then $Z_{{\text{dis}}}^{\text{flip}}({\text{Int}\,}{{\boldsymbol \gamma }} ,M)$ is an $\eps$-relative approximation to $Z_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }} )$.
Let ${{\boldsymbol \Lam }} = {\text{Int}\,}{{\boldsymbol \gamma }}$. Note that the lemma is immediate if ${\text{Int}\,}{{\boldsymbol \gamma }}$ does not contain any contours. Let $$Z_{{\text{dis}}}^{\text{err}}({{\boldsymbol \Lam }}) {\coloneqq}Z_{{\text{dis}}}(\Lam) - Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }},M) \,.$$ To prove the lemma it suffices to show that $$\label{eq:fliplem}
0 \le Z_{{\text{dis}}}^{\text{err}}({{\boldsymbol \Lam }} ) / Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }} ,M) \le
\eps/2.$$ The lower bound is immediate since $Z_{{\text{dis}}}$ is a sum of non-negative terms and $Z^{\text{flip}}_{{\text{dis}}}({{\boldsymbol \Lam }},M)$ is at least one. Thus the proof of has two parts: lower bounding $Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }} ,M)$ and upper bounding $Z_{{\text{dis}}}^{\text{err}}({{\boldsymbol \Lam }})$. The combination of these bounds will prove .
We begin with the lower bound on $Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }},M)$. Recall the definition of $\cE_{{{\boldsymbol \gamma }}}$. Let ${{\boldsymbol \gamma }}' \in \cC_{\text{dis}}({{\boldsymbol \Lam }})$ be the contour obtained by thickening $\cE_{{{\boldsymbol \gamma }}}$ and taking the boundary, i.e., $\partial {{\boldsymbol {\cE_{{{\boldsymbol \gamma }}}} }}$. Let $\Gamma =
\{{{\boldsymbol \gamma }}'\}$. Note that ${\text{Ext}\,}\Gamma$ contains no vertices, because ${{\boldsymbol \Lam }}$ is connected and all edges inside ${{\boldsymbol \Lam }}$ are in $\cE_{{{\boldsymbol \gamma }}}$.
Next observe that ${\|{{\boldsymbol \gamma }}'\|}\leq {\|{{\boldsymbol \gamma }}\|}$. This is because by construction any edge contributing to ${\|{{\boldsymbol \gamma }}'\|}$ must have one vertex outside of ${{\boldsymbol \Lam }}$, and such an edge also contributes to ${\|{{\boldsymbol \gamma }}\|}$. In particular, $\Gamma \in {\cH}^{\text{flip}}_{\text{dis}}(\Lam,M)$, and hence $$\begin{aligned}
Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lambda }},M)
&\geq
e^{-e_{{\text{dis}}}{\left|{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}\right|}} e^{-\kappa{\|{{\boldsymbol \gamma }} '\|}} q
Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}') \\
&\geq e^{-\kappa{\|{{\boldsymbol \gamma }}\|}} q Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}') \\
&\geq
e^{-(\kappa+1){\|{{\boldsymbol \gamma }}\|}} qe^{-(f + \eps_n){\left|{\text{Int}\,}{{\boldsymbol \gamma }}'\right|}} \\
&\ge \frac{1}{2}e^{-(\kappa+1){\|{{\boldsymbol \gamma }}\|}} qe^{-f {\left|{{\boldsymbol \Lam }}\right|}}\, ,
\end{aligned}$$ where $\eps_n = 2 e^{-c \beta n}$ as above and $f$ is the constant from Lemma \[lemBCTsup\]. The second inequality used that ${\text{Ext}\,}\Gamma$ contains no vertices. The second-to-last inequality follows from [@borgs2012tight Lemma 6.3 (ii)], and the last inequality follows since (i) $|{\text{Int}\,}{{\boldsymbol \gamma }} | = | {\text{Int}\,}{{\boldsymbol \gamma }}'|$ and (ii) for $n$ large enough we have $e^{\eps _n | {\text{Int}\,}{{\boldsymbol \gamma }}|} \le 2$ for all ${{\boldsymbol \gamma }} \in \cC$.
Next we prove an upper bound on $Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }},M)$. In fact, the upper bound is essentially contained in [@borgs2012tight Appendices A.2 and A.3], and we explain it here. Some further notation will be helpful. Let $a_{{\text{dis}}}>0$ be the constant from Lemma \[lemBCTsup\]. We call a contour ${{\boldsymbol \gamma }} \in \cC_{{\text{dis}}}$ ‘small’ if $\text{diam}({{\boldsymbol \gamma }}) \le \frac{c \beta}{a_{{\text{dis}}}}$ and ‘large’ otherwise. Here $\text{diam}({{\boldsymbol \gamma }})$ denotes the diameter of ${{\boldsymbol \gamma }}$, the maximum over $i=1,\dots, n$ of ${\left|I_{i}({{\boldsymbol \gamma }})\right|}$, where $I_{i}({{\boldsymbol \gamma }}) = \{ k\in \Z/n\Z \mid {{\boldsymbol S }}^{(i)}_{k}\cap {{\boldsymbol \gamma }}
\neq \emptyset\}$, where ${{\boldsymbol S }}^{i}_{k}$ is the set $\{{{\boldsymbol x }}\in {{{\boldsymbol T }}^d_n}\mid {{\boldsymbol x }}_{i}=k\}$. See [@borgs2012tight p.22].
For a region ${{\boldsymbol \Lam }}'$, let $$\begin{aligned}
\cG_{{\text{dis}}}^{{\text{ext}},\text{small}}({{\boldsymbol \Lam }}')
&{\coloneqq}\{ \Gamma \in \cG_{{\text{dis}}}^{\text{ext}}({{\boldsymbol \Lam }}') | {{\boldsymbol \gamma }}' \text { is
small } \forall {{\boldsymbol \gamma }} ' \in \Gamma \}, \\
\cG_{{\text{dis}}}^{{\text{ext}},\text{large}}({{\boldsymbol \Lam }}')
&{\coloneqq}\{ \Gamma \in \cG_{{\text{dis}}}^{\text{ext}}({{\boldsymbol \Lam }}') | {{\boldsymbol \gamma }}' \text { is
large } \forall {{\boldsymbol \gamma }} ' \in \Gamma \} ,
\end{aligned}$$ and $$\begin{aligned}
Z_{{\text{dis}}}^{\text{small}} ({{\boldsymbol \Lam }} ')
&{\coloneqq}\sum_{\Gamma \in \cG_{{\text{dis}},\text{small}}^{\text{ext}}({{\boldsymbol \Lam }}')}
e^{-e_{{\text{dis}}}|{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}'|} \prod_{{{\boldsymbol \gamma }}\in\Gamma} e^{-\kappa
{\|{{\boldsymbol \gamma }}\|}}qZ_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}) \\
&=e^{- e_{{\text{dis}}} |{{\boldsymbol \Lam }}'|} \sum_{\Gamma \in
\cG_{{\text{dis}}}^{{\text{ext}}, \text{small}}({{\boldsymbol \Lam }}')}
\prod_{{{\boldsymbol \gamma }} ' \in \Gamma} K_{{\text{dis}}} ({{\boldsymbol \gamma }}') .\end{aligned}$$
Moreover, let $$\begin{aligned}
{\cH}^{\text{err}}_{{\text{dis}}}({{\boldsymbol \Lam }})
&{\coloneqq}\{\Gamma\in \cG_{{\text{dis}}}^{{\text{ext}}}({{\boldsymbol \Lam }}) \mid {\left|{\text{Ext}\,}\Gamma \cap
{{\boldsymbol \Lam }}\right|} > M\}, \qquad \text{and} \\
{\cH}^{\text{err}, \text{large}}_{{\text{dis}}}({{\boldsymbol \Lam }})
&{\coloneqq}\{\Gamma\in \cG_{{\text{dis}}}^{{\text{ext}},\text{large}}({{\boldsymbol \Lam }}) \mid
{\left|{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}\right|} > M\} . \end{aligned}$$
Following the proof of [@borgs2012tight Lemma A.1], we have that $$\begin{aligned}
Z_{{\text{dis}}}^{\text{err}}({{\boldsymbol \Lam }},M)
&= \sum_{\Gamma \in {\cH}^{\text{err}}_{{\text{dis}}}({{\boldsymbol \Lam }})}
e^{-e_{{\text{dis}}} |{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}|} \prod_{{{\boldsymbol \gamma }}' \in \Gamma}
e^{-\kappa \|{{\boldsymbol \gamma }}'\|} q Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }} ')
\\
&\le \sum_{\Gamma \in {\cH}^{\text{err},
\text{large}}_{{\text{dis}}}({{\boldsymbol \Lam }}) } Z_{{\text{dis}}}^{\text{small}}({\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }})
\prod_{{{\boldsymbol \gamma }}' \in \Gamma} q e^{-\kappa \| {{\boldsymbol \gamma }}'\| }
Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}')
\\
&\le e^{(\eps_n-f
) |{{\boldsymbol \Lam }}| + \|{{\boldsymbol \gamma }}\|} e^{-\frac{a_{{\text{dis}}}}{2} M} \sum_{\Gamma \in
{\cH}^{\text{err}, \text{large}}_{{\text{dis}}}({{\boldsymbol \Lam }})}
e^{-\frac{a_{{\text{dis}}}}{2} |{\text{Ext}\,}\Gamma \cap {{\boldsymbol \Lam }}|} \prod_{{{\boldsymbol \gamma }} ' \in \Gamma}
e^{-(\frac{\beta}{8} -3) \|{{\boldsymbol \gamma }} ' \|}
\\
&\le 2e^{-f
|\Lam| + 2\|{{\boldsymbol \gamma }}\|} e^{-\frac{a_{{\text{dis}}}}{2} M} \, .\end{aligned}$$ The first inequality follows since for each $\Gamma \in {\cH}^{\text{err}}_{{\text{dis}}}({{\boldsymbol \Lam }})$, the set of large contours in $\Gamma$ appear in ${\cH}^{\text{err}, \text{large}}_{{\text{dis}}}({{\boldsymbol \Lam }}) $. The second inequality follows from the proof of [@borgs2012tight Lemma A.1]; as above we are using that $f=f_{{\text{ord}}}$ when $\beta>\beta_{c}$. The last inequality follows from [@borgs2012tight (A.12)] and the fact that $e^{\eps_n |{{\boldsymbol \Lam }}|} \le 2$ for large enough $n$.
We can now conclude and prove : putting the bounds together and using we get $$\frac{Z_{{\text{dis}}}^{\text{err}}({{\boldsymbol \Lam }} )}{ Z_{{\text{dis}}}^{\text{flip}}({{\boldsymbol \Lam }} ,M) }\le 4 q e^{(\kappa +3) \| {{\boldsymbol \gamma }} \| -\frac{a_{{\text{dis}}}}{2} M} \le \eps/2.\qedhere$$
We conclude this section with an enumerative lemma concerning $\cH^{\text{flip}}_{{\text{dis}}}$.
\[prop:vacant\] There is an algorithm that given ${{\boldsymbol \gamma }}\in \cC_{{\text{ord}}}$ and $M\in{{\mathbb{N}}}$ outputs $\cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$ in time ${\|{{\boldsymbol \gamma }}\|}e^{O({\|{{\boldsymbol \gamma }}\|}+M)}$.
This follows from a variation on the proof of Proposition \[prop:enum\]. To determine $\cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }})$ we will consider ${{\boldsymbol \gamma }}$ to be a contour in a torus of side-length ${\|{{\boldsymbol \gamma }}\|}\wedge n$; this torus has volume polynomial in ${\|{{\boldsymbol \gamma }}\|}$.
$\cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }})$ is the set of mutually external contour configurations $\Gamma\setminus {{\boldsymbol \gamma }}$ obtained as $F$ ranges over the possibilities listed in Lemma \[lem:edge-ord\]. As in Lemma \[lem:ord-edge-con\] we can determine $E'\cup F$ by considering it as the complement of $1$-connected set of edges $A=A'\sqcup B$, where $A'$ is the set of edges that intersect ${{\boldsymbol \gamma }}$. For any choice of such an $A$, ${\text{Ext}\,}{\Gamma}\cap\tor$ is of size at least $O({\left|B\right|})$, so to determine $\cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, M)$ it is enough to consider all possible sets $B$ of size at most $M$. The claim now follows by arguing as in the proof of Proposition \[prop:enum\].
Approximate counting algorithms {#sec:count}
===============================
This section describes our approximate counting algorithms for $\beta>\beta_{h}$. The algorithms differ depending on whether $\beta=\beta_c$, $\beta>\beta_c$, or $\beta_{h}<\beta<\beta_{c}$. Recall that $Z_{\ell}({{\boldsymbol \Lam }})$ was defined for all regions ${{\boldsymbol \Lam }}$ in –. The heart of this section is the following lemma.
\[lemOrdDiscompute\] For $d\geq 2$ and $q \ge q_0$ the following hold.
1. If $\beta = \beta_c$ there is an FPTAS to approximate $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ and $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$.
2. If $\beta> \beta_c$ there is an FPTAS to approximate $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$.
3. If $\beta_h < \beta < \beta_c$ there is an FPTAS to approximate $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$.
In each case the FPTAS applies to any region ${{\boldsymbol \Lam }}$, with running time polynomial in $| {{\boldsymbol \Lam }}|$, the number of vertices of $\tor$ in ${{\boldsymbol \Lam }}$.
Sections \[sec:lbc\] and \[sec:lbgc\] prove the first two cases of Lemma \[lemOrdDiscompute\]. The case $\beta_{h}<\beta<\beta_{c}$ is very similar to $\beta>\beta_{c}$, and we defer the details to Appendix \[sec:HT\]. In Section \[secZTogether\] we show how these results, together with a result from [@borgs2012tight], suffice to give an FPRAS for $Z$ on the torus.
Proof of Lemma \[lemOrdDiscompute\] when $\beta = \beta_c$ {#sec:lbc}
----------------------------------------------------------
We begin by defining a useful variant of the truncated cluster expansion for $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ and $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$. Let $K$ be a function from contours to positive real numbers. For $\ell\in\{{\text{ord}},{\text{dis}}\}$ define
$$T_{\ell,m}({{\boldsymbol \Lam }},K) {\coloneqq}\sum_{\substack{\Gamma \in \cG^c_{\ell}({{\boldsymbol \Lam }}) \\ \|\Gamma \| < m }} \phi(\Gamma) \prod_{{{\boldsymbol \gamma }} \in \Gamma} K({{\boldsymbol \gamma }}).$$
so that by and $Z_{\ell}(\Lam) = \exp(-e_{\ell}|\Lam|) T_{\ell,\infty}({{\boldsymbol \Lam }},K_{\ell})$ provided the cluster expansion for the polymer models converge.
Recall that the level of a contour was defined in Definition \[defLevel\], and that $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}$ was defined immediately prior to Lemma \[lem:polygraphsize\].
\[lemBcInductive\] Suppose $d\geq 2$, $q \ge q_0$ and $\beta = \beta_c$. Given ${{\boldsymbol \Lam }}$ with $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}} = N$, and an error parameter $\eps > 0$, let $m=\log(8N^2/\eps)/3$. Inductively (by level) define weights $\tilde K_{{\text{ord}}}({{\boldsymbol \gamma }})$ and $\tilde K_{{\text{dis}}}({{\boldsymbol \gamma }})$ for all contours ${{\boldsymbol \gamma }}$ in $\cC_{{\text{ord}}}({{\boldsymbol \Lam }})$ and $\cC_{{\text{dis}}}({{\boldsymbol \Lam }})$ with size ${\|{{\boldsymbol \gamma }}\|}\leq m$ by:
1. If ${{\boldsymbol \gamma }}$ is thin, then set $$\tilde K_{\text{ord}}({{\boldsymbol \gamma }}) = e^{- \kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{dis}}} - e_{{\text{ord}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|}, \quad
\tilde K_{\text{dis}}({{\boldsymbol \gamma }}) = q e^{- \kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{ord}}} - e_{{\text{dis}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|}.$$
2. If ${{\boldsymbol \gamma }}$ is not thin, then set $$\begin{aligned}
\tilde K_{{\text{ord}}} ({{\boldsymbol \gamma }}) &= e^{-\kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{dis}}} - e_{{\text{ord}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|} \exp \left[ T_{m, {\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) - T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) \right ], \\
\tilde K_{{\text{dis}}} ({{\boldsymbol \gamma }}) &=qe^{-\kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{ord}}} - e_{{\text{dis}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|} \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) - T_{m, {\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) \right ] \,.\end{aligned}$$
Then for $N$ sufficiently large $e^{-e_{\ell}|\Lam|}\exp ( T_{\ell,m}({{\boldsymbol \Lam }},\tilde K_{\ell}))$ is an $\eps$-relative approximation to $Z_{\ell}({{\boldsymbol \Lam }})$ for $\ell\in\{{\text{ord}},{\text{dis}}\}$.
Suppose $\ell \in \{ \text{dis}, \text{ord} \}$. First note that the inductive definition of the weights $\tilde K_{\ell}({{\boldsymbol \gamma }})$ makes sense: to compute $\tilde K_{\ell}({{\boldsymbol \gamma }})$ for a contour ${{\boldsymbol \gamma }}$ of level $t+1$ only requires knowing $\tilde K_{\ell}({{\boldsymbol \gamma }}')$ for contours ${{\boldsymbol \gamma }}'$ of level $t$ and smaller.
Since $\beta = \beta_c$ and $q \ge q_0$, Lemma \[lemKestimates\] tells us that $$\label{eqTaubound}
K_\ell({{\boldsymbol \gamma }}) \le e^{-c \beta \| {{\boldsymbol \gamma }} \|}$$ for $\ell \in \{ \text{dis}, \text{ord} \}$ and for all ${{\boldsymbol \gamma }} \in \cC_{\ell}(\Lam)$. If $q_0$ is large enough then implies condition holds since $\beta_c$ grows like $\log q$ by . Thus by Section \[sec:extern-cont-repr-1\] the hypotheses of Lemma \[KPthm\] are satisfied and the cluster expansion for $Z_{\ell}({{\boldsymbol \Lam }})$ converges for $\ell\in \{{\text{ord}},{\text{dis}}\}$.
Now let $\eps' = \eps/N$, so that $m = \log (8N/\eps')/3$. We will apply Lemma \[lemPolymerApprox\] with $v({{\boldsymbol \gamma }}) = | {\text{Int}\,}{{\boldsymbol \gamma }}|$. This is a valid choice of $v({{\boldsymbol \gamma }})$ by Lemma \[lem:iso\]. Lemma \[lemPolymerApprox\] says that $$e^{- e_{{\text{ord}}} |\Lam|} \exp \left( T_{{\text{ord}},m}({{\boldsymbol \Lam }}, \tilde K _{{\text{ord}}}) \right)
\quad \text{and} \quad
e^{- e_{{\text{dis}}} |\Lam|} \exp \left( T_{{\text{dis}},m}({{\boldsymbol \Lam }}, \tilde K_{{\text{dis}}}) \right)$$ are $\eps$-relative approximations to $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ and $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$ if for all ${{\boldsymbol \gamma }} \in \cC_{\ell}({{\boldsymbol \Lam }})$ of size at most $m$, $\tilde K_{\ell} ({{\boldsymbol \gamma }})$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}|$-relative approximation to $K_{\ell}({{\boldsymbol \gamma }})$. We will prove this by induction on the level of ${{\boldsymbol \gamma }}$.
For a thin contour, $\tilde K_{\ell} ({{\boldsymbol \gamma }}) = K_{\ell}({{\boldsymbol \gamma }})$. Now suppose that for all contours ${{\boldsymbol \gamma }}$ of level at most $t$ and size at most $m$, $\tilde K_{\ell} ({{\boldsymbol \gamma }})$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}|$-relative approximation of $K_{\ell}({{\boldsymbol \gamma }})$. Consider a contour ${{\boldsymbol \gamma }}$ of level $t+1$ and size at most $m$. Then all contours ${{\boldsymbol \gamma }}'$ that appear in the expansions $$\begin{aligned}
T_{m, {\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{dis}}}) \quad \text{and} \quad T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{ord}}}) \end{aligned}$$ are of level at most $t$ and size at most $m$, and so for each such ${{\boldsymbol \gamma }}'$, by the inductive hypothesis $\tilde K_\ell({{\boldsymbol \gamma }}')$ is an $\eps' |{\text{Int}\,}{{\boldsymbol \gamma }} '|$-relative approximation to $K_\ell({{\boldsymbol \gamma }}')$. Then by Lemma \[lemPolymerApprox\], we have that $$\begin{aligned}
e^{ - (e_{{\text{dis}}} - e_{{\text{ord}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|} \exp \left[ T_{m, {\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{dis}}}) - T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{ord}}}) \right ] \end{aligned}$$ is an $| {\text{Int}\,}{{\boldsymbol \gamma }}| \eps'$-relative approximation to $\frac{ Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }}) }{Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}) } $ (and likewise for ${\text{dis}}$ and ${\text{ord}}$ swapped). Multiplying by the prefactor $e^{-\kappa \|{{\boldsymbol \gamma }} \|}$ for ${\text{ord}}$ and by $q e^{-\kappa \|{{\boldsymbol \gamma }} \|}$ for ${\text{dis}}$ shows that $\tilde K_{\ell}({{\boldsymbol \gamma }})$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}|$-relative approximation to $K_{\ell} ({{\boldsymbol \gamma }})$ as desired.
With this, we can prove the $\beta = \beta_{c}$ case of Lemma \[lemOrdDiscompute\].
Let $N = |{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}$ and let $m = \log ( 8N^2/\eps)/3$. We need to show that the expansion $T_{\ell,m}({{\boldsymbol \Lam }}, \tilde K_{\ell})$ and the weights $\tilde K_{\ell}({{\boldsymbol \gamma }})$ for all ${{\boldsymbol \gamma }}$ of size at most $m$ in $\cC_{\ell}({{\boldsymbol \Lam }})$ can be computed in time polynomial in $N$ and $1/\eps$ for $\ell \in \{ {\text{dis}}, {\text{ord}}\}$. We can list the sets of contours in $\cC_{{\text{ord}}} ({{\boldsymbol \Lam }})$ and $\cC_{{\text{dis}}} ({{\boldsymbol \Lam }})$ of size at most $m$, together with their labels and levels, in time $O(N \exp(O(m))$ by Proposition \[prop:enum\]. Since $m = \log ( 8N^2/\eps)/3$, $O(N\exp(O(m))$ is polynomial in $N$ and $1/\eps$. $N$ itself is polynomial in $|\Lam|$ by Lemma \[lem:polygraphsize\].
To prove the lemma we must compute the weights $\tilde K_{\ell}({{\boldsymbol \gamma }})$ and the truncated cluster expansions $T_{m, \ell}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{\ell})$ for each contour in the list. We do this inductively by level. For level zero contours $\tilde K_{\ell}({{\boldsymbol \gamma }})=K_{\ell}({{\boldsymbol \gamma }})$ only depends on ${\|{{\boldsymbol \gamma }}\|}$ and ${\left|{\text{Int}\,}{{\boldsymbol \gamma }}\right|}$, so $\tilde K_{\ell}({{\boldsymbol \gamma }})$ can be computed in time $O({\|{{\boldsymbol \gamma }}\|}^{3})$ by computing these quantities by using Lemma \[lem:findext\]. We then continue inductively; each $\tilde K_{\ell}({{\boldsymbol \gamma }})$ can be computed efficiently since the truncated cluster expansions can be computed in time polynomial in $N$ and $1/\eps$ using Lemma \[lemPolyModelCount\].
Proof of Lemma \[lemOrdDiscompute\] when $\beta > \beta_c$ {#sec:lbgc}
----------------------------------------------------------
When $\beta > \beta_c(q,d)$ the ordered ground state is stable, but the disordered state is unstable. For a definition of stability of ground states, see, e.g., [@borgs1989unified]; the upshot for this paper is that we cannot use the cluster expansion to approximate $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$ for a region ${{\boldsymbol \Lam }}$.
To deal with this complication we will appeal to Lemma \[lemSuperEstimates2\]. In words, this lemma says that for $\beta> \beta_c$, a typical contour configuration in a region with disordered boundary conditions will have very few external vertices. We will exploit this fact to enumerate all sets of typical external contours in the region. This is possible since the number of external vertices is small. Once we have fixed a set of external contours we are back to the task of approximating partition functions with ordered boundary conditions.
We now make the preceding discussion precise. Given $K \colon \cC_{{\text{ord}}}({{\boldsymbol \Lam }}) \to [0,\infty)$, define $$\Xi^M_{{\text{dis}}}({{\boldsymbol \Lam }}, K) {\coloneqq}e^{e_{{\text{dis}}}|\Lam|}\sum_ {\Gamma \in {\cH}^{\text{flip}}_{{\text{dis}}}({{\boldsymbol \Lambda }},M)} e^{-e_{{\text{dis}}} | {\text{Ext}\,}\Gamma |} \prod_{{{\boldsymbol \gamma }} \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }} \| } q \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, K) \right].$$
\[lemSuperCritApprox\] Suppose $d\geq 2$, $q \ge q_0$ and $\beta > \beta_c$. Let ${{\boldsymbol \Lam }}$ be a region with $|{{\boldsymbol \Lam }}|_{{({\frac{1}{2}\tor})^\star}}=N$, fix $\eps > 0$, and let $m=\log(8N^2/\eps)/3$. Inductively (by level) define $\tilde K_{{\text{ord}}}({{\boldsymbol \gamma }})$ for ${{\boldsymbol \gamma }}\in\cC_{{\text{ord}}}({{\boldsymbol \Lam }})$ with size ${\|{{\boldsymbol \gamma }}\|}$ at most $m$ by
1. If ${{\boldsymbol \gamma }}$ is thin, then $$\begin{aligned}
\tilde K_{\text{ord}}({{\boldsymbol \gamma }}) &= e^{- \kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{dis}}} - e_{{\text{ord}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|} \, .\end{aligned}$$
2. If ${{\boldsymbol \gamma }}$ is not thin, define $$\begin{aligned}
\tilde K_{{\text{ord}}} ({{\boldsymbol \gamma }}) &= e^{-\kappa \| {{\boldsymbol \gamma }} \| - (e_{{\text{dis}}} - e_{{\text{ord}}}) |{\text{Int}\,}{{\boldsymbol \gamma }}|} \exp \left[ - T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) \right ] \Xi^M_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{ord}}}) \, ,\end{aligned}$$ with $M= \frac{2}{a_{{\text{dis}}}} \left( \log ( \frac{32q}{\eps'}) + (\kappa+3) m\right)$.
Then for all $N$ large enough, $e^{- e_{{\text{ord}}} |{{\boldsymbol \Lam }}|} \exp \left( T_{{\text{ord}},m}({{\boldsymbol \Lam }}, \tilde K _{{\text{ord}}}) \right)$ is an $\eps$-relative approximation to $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$.
Let $\eps ' = \eps /N$ so that $m = \log (8N /\eps')/3$.
If $q_{0}$ is large enough then we have $K_{{\text{ord}}} ({{\boldsymbol \gamma }}) \le e^{-c \beta \| {{\boldsymbol \gamma }} \|} $ by Lemma \[lemSuperEstimates\], $\beta > \beta_c$, and . This implies condition holds for ordered contours, and thus by Section \[sec:extern-cont-repr-1\] the hypotheses of Lemma \[KPthm\] are satisfied and the cluster expansion for $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ converges. Applying Lemma \[lemPolymerApprox\] with $v({{\boldsymbol \gamma }}) = |{\text{Int}\,}{{\boldsymbol \gamma }}|$ then tells us that $$e^{- e_{{\text{ord}}} |{{\boldsymbol \Lam }}|} \exp \left( T_{{\text{ord}},m}({{\boldsymbol \Lam }}, \tilde K _{{\text{ord}}}) \right)$$ is an $\eps$-relative approximation to $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ if for all ${{\boldsymbol \gamma }} \in \cC_{{\text{ord}}}(\Lam)$ of size at most $m$, $\tilde K_{{\text{ord}}}({{\boldsymbol \gamma }})$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}|$-relative approximation to $K_{{\text{ord}}}({{\boldsymbol \gamma }})$. We will prove this is the case by induction. The base case of the induction (thin contours) holds since $\tilde K_{{\text{ord}}}({{\boldsymbol \gamma }}) = K_{{\text{ord}}}({{\boldsymbol \gamma }})$. Now suppose that the statement holds for all contours of level at most $t$ and size at most $m$, and consider a contour ${{\boldsymbol \gamma }}$ of level $t+1$ and size at most $m$.
The inductive hypothesis and Lemma \[lemPolymerApprox\] imply that $$e^{- e_{{\text{ord}}} |{{\boldsymbol \Lam }}|}\exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K) \right ]$$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}|/2$-relative approximation to $Z_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }})$, and so it suffices to show that $e^{-e_{dis}|{{\boldsymbol \Lam }}|}\Xi^M_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{ord}}}) $ is an $\eps' |{\text{Int}\,}{{\boldsymbol \gamma }}|/2$-relative approximation to $Z_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }})$.
By Lemma \[lemSuperEstimates2\], $Z_{{\text{dis}}}^{\text{flip}}({\text{Int}\,}{{\boldsymbol \gamma }},M) $ is an $\eps'/4$-relative approximation to $Z_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}) $ for $M= \frac{2}{a_{{\text{dis}}}} \left( \log ( \frac{32q}{\eps'}) + (\kappa+3) m\right)$, and so it suffices to show that $e^{-e_{dis}|\Lam|}\Xi^M_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K_{{\text{ord}}}) $ is an $\eps' |{\text{Int}\,}{{\boldsymbol \gamma }}| /4$-relative approximation to $Z_{{\text{dis}}}^{\text{flip}}({\text{Int}\,}{{\boldsymbol \gamma }} ,M) $. We will accomplish this by showing, for each $\Gamma \in {\cH}^{\text{flip}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$, that $$e^{-e_{{\text{dis}}} | {\text{Ext}\,}\Gamma |} \prod_{{{\boldsymbol \gamma }}' \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }}' \| } q \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K) \right]$$ is an $\eps' |{\text{Int}\,}{{\boldsymbol \gamma }}| /4$-relative approximation to $$e^{- e_{{\text{dis}}} | {\text{Ext}\,}\Gamma |} \prod_{{{\boldsymbol \gamma }} ' \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }} ' \|} q Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }} ')$$ and then summing over ${\Gamma}$. The prefactors are identical, and so it comes down to comparing $ \prod_{{{\boldsymbol \gamma }}' \in \Gamma} \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K) \right] $ to $\prod_{{{\boldsymbol \gamma }}' \in \Gamma}Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }} ')$. Since the contours in ${\Gamma}$ are mutually external, $$\sum _{{{\boldsymbol \gamma }}' \in \Gamma} | {\text{Int}\,}{{\boldsymbol \gamma }}'| \le |{\text{Int}\,}{{\boldsymbol \gamma }}| \, ,$$ and hence it suffices to show that for each ${{\boldsymbol \gamma }}'$, $ \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K) \right]$ is an $\eps' | {\text{Int}\,}{{\boldsymbol \gamma }}'|/4$-relative approximation to $Z_{{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }} ')$. This follows from Lemma \[lemPolymerApprox\] since $m =\log (8N /\eps')/3 $ and by induction we have that $\tilde K_{{\text{ord}}} ({{\boldsymbol \gamma }}'')$ is an $\eps'|{\text{Int}\,}{{\boldsymbol \gamma }}''|$-relative approximation to $ K_{{\text{ord}}} ({{\boldsymbol \gamma }}'')$ for all contours ${{\boldsymbol \gamma }}''$ that contribute to $T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K) $.
With this, we can prove the $\beta>\beta_c$ case of Lemma \[lemOrdDiscompute\].
Given Lemma \[lemSuperCritApprox\], we need to show that we can compute $\tilde K _{{\text{ord}}}({{\boldsymbol \gamma }})$ for all ${{\boldsymbol \gamma }}$ of size at most $m = \log(8 N^2/\eps)/3 $ in time polynomial in $N$ and $1/\eps$. The proof of this is the same as the proof of the $\beta=\beta_c$ case of the lemma except that now we have to account for the computation of $\Xi^M_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}, \tilde K)$ for all ${{\boldsymbol \gamma }} \in \cC_{{\text{dis}}}({{\boldsymbol \Lam }})$ of size at most $m$, with $M= \frac{2}{a_{{\text{dis}}}} \left( \log ( \frac{32q}{\eps'}) + (\kappa+3) m\right)$.
For a given $\Gamma \in {\cH}^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$, the computation of $$e^{-e_{{\text{dis}}} | {\text{Ext}\,}\Gamma |} \prod_{{{\boldsymbol \gamma }}' \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }}' \| } q \exp \left[ T_{m, {\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K) \right ]$$ can be done in time polynomial in $N$ and $1/\eps$ since it just involves computing the truncated cluster expansions $T_{m,{\text{ord}}} ({\text{Int}\,}{{\boldsymbol \gamma }}', \tilde K)$ for at most $m^2$ contours ${{\boldsymbol \gamma }}'$, and since we compute $\tilde K_{{\text{ord}}}({{\boldsymbol \gamma }}')$ in order of the level of ${{\boldsymbol \gamma }}'$, we will have already computed all the weight functions needed in the expansion.
To conclude, note the set ${\cH}^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$ can be enumerated in polynomial time by Proposition \[prop:vacant\] since both $\|{{\boldsymbol \gamma }} \|$ and $M$ are $O(\log(N^2/\eps))$. Since $N$ is polynomial in $|{{\boldsymbol \Lam }}|$ by Lemma \[lem:polygraphsize\], the proof is complete.
Note that Lemma \[lemSuperCritApprox\] used the value of $a_{{\text{dis}}}>0$ to determine the value of $M$ in the definitions of the weights $\tilde K$. It is desirable to avoid using $a_{{\text{dis}}}$ as an input of the algorithm, and hence we close this section with a lemma that shows how to bound $M$ without knowing $a_{{\text{dis}}}$ precisely.
\[lem:adis\] Suppose $d\geq 2$, $q\geq q_{0}$, and $\beta>\beta_{c}$. There is an $O(1)$-time algorithm to determine a constant $a^{\star}_{{\text{dis}}}>0$ such that $a_{{\text{dis}}}>a^{\star}_{{\text{dis}}}$. The constants in the $O(1)$ term may depend on $q,\beta,d$.
We follow the notation from [@borgs2012tight Appendix A.1]. As discussed below [@borgs2012tight (A.7)], we have ${\left|f_{\ell}-f^{(n)}_{\ell}\right|}\leq\eps_{n}$ for $\ell\in \{{\text{ord}},{\text{dis}}\}$, where $\eps_{n}=2e^{-c\beta n}$, where $n$ is the side-length of the torus $\tor$, and $f_{\ell}=\lim_{n\to\infty}f_{\ell}^{(n)}$.
Compute $f^{(n)}_{\ell}$ for $\ell\in\{{\text{ord}},{\text{dis}}\}$ until ${\left|f^{(n)}_{{\text{ord}}}-f^{(n)}_{{\text{dis}}}\right|}$ is at least $3\eps_{n}$. Let $n_{0}$ be the first such $n$ that is found. Then by the triangle inequality, $a_{{\text{dis}}}$ is at least $a^{\star}_{{\text{dis}}}=\eps_{n_{0}}$.
Note that $n_{0}$ can be bounded above in terms of the value of $a_{{\text{dis}}}=a_{{\text{dis}}}(\beta,d,q)$ and $\eps_{n}$, so the above procedure terminates in a finite time (depending on $\beta,d,q$).
Proof of Theorem \[PottsTorusCrit\] {#secZTogether}
-----------------------------------
To prove Theorem \[PottsTorusCrit\] we will need the following result from [@borgs2012tight] about the mixing time of the Glauber dynamics.
\[thmBCTglauber\] The mixing time of the Glauber dynamics for the $q$-state ferromagnetic Potts model satisfies $$\tau_{q,\beta}(\tor) = e^{O( n^{d-1})} ,$$ where the $O(\cdot)$ in the exponent hides constants that depend on $q, \beta$.
We will use this result to give an approximation algorithm when the approximation parameter $\eps$ is extremely small. The reason we are able to combine the Glauber dynamics with our contour-based algorithm to give an FPRAS is that [@borgs2012tight] proves *optimal* slow mixing results for the Glauber and Swendsen–Wang dynamics. That is, up to a constant in the exponent, the upper bound of the mixing time of the Glauber dynamics (or Swendsen–Wang dynamics) is the inverse of the bound on $Z_{\text{tunnel}}/Z$ from Lemma \[lem:Z-split\]. Thus when $\eps$ is too small for the contour algorithms to work, the Glauber dynamics can take over.
Let $N = n^d$ be the number of vertices of $\tor$. We will use a simple fact several times below: if $\eps \in (0,1)$, $Z, Z^* > 0$, and $Z^*/Z < \eps/2$, then $(Z-Z^*)$ is an $\eps$-relative approximation to $Z$.
We first consider the case $\beta = \beta_c$. To give an FPRAS for $Z=Z_{\tor}$ we consider two subcases. Let $c$ be the constant from Lemma \[lem:Z-split\].
Suppose $\eps < 4e^{-c \beta n^{d-1}}$. Since $e^{O( n^{d-1})}$ is polynomial in $N$ and $1/\eps$, we can use Glauber dynamics to obtain an $\eps$-approximate sample in polynomial time. By using simulated annealing (e.g. [@vstefankovivc2009adaptive]) we can also approximate the partition function in time polynomial in $N$ and $1/\eps$.
If $\eps \ge 4e^{-c \beta n^{d-1}}$, then by Lemma \[lem:Z-split\], $Z_{\text{rest}}= Z_{{\text{dis}}} + Z_{{\text{ord}}}$ is an $\eps/2$-relative approximation to $Z$, so it suffices to find an $\eps/4$-relative approximation to both $Z_{{\text{dis}}} $ and $Z_{{\text{ord}}}$. This can be done in time polynomial in $N$ and $1/\eps$ by Lemma \[lemOrdDiscompute\].
Next we consider the case $\beta > \beta_c$. Again there are two subcases. Let $c$ be the constant from Lemma \[lem:Z-split\] as before, and let $b_{{\text{dis}}}$ be the constant from Lemma \[lemSuperEstimates2a\]. If $\eps < 4e^{-c \beta n^{d-1}} + 4 e^{-b_{{\text{dis}}} n^{d-1}}$, then again $e^{O( n^{d-1})}$ is polynomial in $N$ and $1/\eps$ and we can approximately count and sample by using the Glauber dynamics.
If $\eps \ge 4e^{-c \beta n^{d-1}} + 4 e^{-b_{{\text{dis}}} n^{d-1}}$, then by Lemma \[lem:Z-split\] and Lemma \[lemSuperEstimates2a\], $Z_{{\text{ord}}}$ is an $\eps/2$-relative approximation to $Z$ and so it suffices to give an $\eps/2$-relative approximation to $Z_{{\text{ord}}}$. This can be done in time polynomial in $N$ and $1/\eps$ by Lemma \[lemOrdDiscompute\].
Lastly, consider $\beta<\beta_{c}$. The case $\beta\leq \beta_{h}$ was completed in Section \[secPolymer\]. The case $\beta_{h}<\beta<\beta_{c}$ is done exactly as the case $\beta>\beta_{c}$ with the roles of ${\text{ord}}$ and ${\text{dis}}$ reversed; see Appendix \[sec:HT\] for details.
Let $\Lam \subset \Z^d$ be such that the induced subgraph $G_{\Lam}$ is finite and simply connected. By Proposition \[prop:scregion\], we can construct an ordered contour ${{\boldsymbol \gamma }}_{{\text{ord}}}$ and a disordered contour ${{\boldsymbol \gamma }}_{{\text{dis}}}$ so that $$\begin{aligned}
Z_{{\text{dis}}} ({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}) = (1-p)^{-\frac{1}{2}{\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}}Z^f_{\Lam},
\qquad
Z_{\text{ord}}({\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{dis}}}) = q^{-1}p^{d{\left|{\text{Int}\,}{{\boldsymbol \gamma }}_{{\text{ord}}}\right|}-|E(\Lam)|}Z^w_{\Lam} \,.\end{aligned}$$ The FPTAS for $Z^w_{\Lam}$ for $\beta \ge \beta_c$ then follows from Lemma \[lemOrdDiscompute\], as does the FPTAS for $Z^f_{\Lam}$ for $\beta_h < \beta \le \beta_c$. The case $\beta\leq \beta_{h}$ was covered in Section \[secPolymer\].
Sampling {#sec:sample}
========
In this section we present efficient approximate sampling algorithms for the random cluster and Potts models when $\beta>\beta_{h}$. By the Edwards–Sokal coupling, see Appendix \[sec:ES\], it suffices to obtain algorithms for the random cluster model. Describing the strategy, which is based on that of [@helmuth2018contours Sections 5 and 6], requires a few definitions.
Recall the definition of the random cluster measure $\mu^{\text{RC}}$ on $\tor$. Thus $\mu^{\text{RC}}$ is a measure on subsets of edges $A\in\Omega$. Recalling the definitions and of the sets $\Omega_{{\text{ord}}}$ and $\Omega_{{\text{dis}}}$ of ordered and disordered edge configurations, we analogously define $$\mu_{\ell}(A) {\coloneqq}\frac{w(A)}{Z_{\ell}}, \quad A\in \Omega_{\ell}\, \text{ with } \ell\in \{{\text{ord}},{\text{dis}}\}.$$
For a region ${{\boldsymbol \Lam }}$, define measures $\nu_{\ell}^{{{\boldsymbol \Lam }}}$ on the sets of external contours $\cG_{\ell}^{{\text{ext}}}({{\boldsymbol \Lam }})$ as follows. $$\begin{aligned}
\label{eq:nudis}
\nu_{\text{dis}}^{{{\boldsymbol \Lam }}}(\Gamma) &{\coloneqq}\frac{ e^{ - e_{{\text{dis}}} |\Lam \cap {\text{Ext}\,}\Gamma| } \prod_{{{\boldsymbol \gamma }} \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }} \|} qZ_{{\text{ord}}}({\text{Int}\,}{{\boldsymbol \gamma }}) }{ Z_{{\text{dis}}} ({{\boldsymbol \Lam }}) } , \qquad \Gamma \in \cG_{\text{dis}}^{\text{ext}}({{\boldsymbol \Lam }}) , \\
\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}(\Gamma) &{\coloneqq}\frac{ e^{ - e_{{\text{ord}}} |\Lam \cap {\text{Ext}\,}\Gamma| } \prod_{{{\boldsymbol \gamma }} \in \Gamma} e^{-\kappa \| {{\boldsymbol \gamma }} \|} Z_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }}) }{ Z_{{\text{ord}}} ({{\boldsymbol \Lam }}) } , \qquad \Gamma \in \cG_{\text{ord}}^{\text{ext}}({{\boldsymbol \Lam }}) , \label{eq:nuord}\end{aligned}$$ where $|\Lam\cap {\text{Ext}\,}\Gamma|$ is the number of vertices contained in the continuum set ${{\boldsymbol \Lam }} \cap {\text{Ext}\,}\Gamma$.
We now outline our strategy for approximately sampling from $\mu_{{\text{ord}}}$ and $\mu_{{\text{dis}}}$; a small modification will also apply to sampling from $\mu^{\text{RC}}$ on the torus. The key idea is that the inductive representations of the partition functions in and yield a procedure to sample from $\mu_{\text{dis}}$ and $\mu_{\text{ord}}$ if we can sample from the measures $\nu_{\ell}^{{{\boldsymbol \Lam }}}$ for $\ell\in\{{\text{ord}},{\text{dis}}\}$ and for all regions ${{\boldsymbol \Lam }}$. The procedure, which we call the *inductive contour sampling algorithm*, is as follows. Consider $\mu_{{\text{ord}}}$. To sample a set of compatible, matching contours with ordered external contours, we first sample $\Gamma$ from $\nu_{{\text{ord}}}^{{{{\boldsymbol T }}^d_n}}$, then for each ${{\boldsymbol \gamma }} \in \Gamma$ we sample from $\nu_{\text{dis}}^{{\text{Int}\,}{{\boldsymbol \gamma }}}$ and repeat inductively until there are no interiors left to sample from. The union of all contours sampled is a set of matching and compatible contours, and these contours are distributed as the restriction of to contour configurations that arise from ordered edge configurations. This set of contours can then be mapped to an edge set via the bijection of Lemma \[lem:rep\], and the distribution of this edge set is $\mu_{\text{ord}}$. The procedure for sampling from $\mu_{{\text{dis}}}$ is analogous. For a more detailed discussion of the validity of this algorithm, see [@helmuth2018contours Section 5].
By using the same procedure it is possible to efficiently approximately sample from $\mu_{\text{ord}}$ and $\mu_{\text{dis}}$ provided one can efficiently approximately sample from the external contour measures $\nu_{\text{dis}}^{{{\boldsymbol \Lam }}}$ and $\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}$. Again, we refer to [@helmuth2018contours Section 5] for further details.
The next lemma is an essential input for developing efficient approximation samplers for $\nu_{\ell}^{{{\boldsymbol \Lam }}}$ as it tells us we need only consider ‘small’ contours. For $\ell\in \{{\text{ord}},{\text{dis}}\}$ let $\nu_{\ell}^{{{\boldsymbol \Lam }},m}$ be the probability measure defined as in – , but restricted to $\Gamma$ with ${\|\Gamma\|}<m$. The normalization factor for $\nu_{\ell}^{{{\boldsymbol \Lam }},m}$ is thus the contour partition function restricted to $\Gamma$ with ${\|\Gamma\|}<m$.
\[lem:sample-trunc\] Suppose $d\geq 2$, $q\geq q_{0}$, and $\eps>0$. Then, letting $N={\left|{{\boldsymbol \Lam }}\right|}_{{({\frac{1}{2}\tor})^\star}}$, for $m\geq O(\log (N /\eps))$,
1. If $\beta\geq \beta_{c}$, then $\|\nu_{{\text{ord}}}^{{{\boldsymbol \Lam }},m} - \nu_{{\text{ord}}}^{{{\boldsymbol \Lam }}} \|_{TV} < \eps$.
2. If $\beta_{h}<\beta \le \beta_{c}$, then $\| \nu_{{\text{dis}}}^{{{\boldsymbol \Lam }},m} -\nu_{{\text{dis}}}^{{{\boldsymbol \Lam }}} \|_{TV} <\eps$.
for all regions ${{\boldsymbol \Lam }}$.[^5]
This follows from the convergence of the cluster expansion for $Z_{\ell}({{\boldsymbol \Lam }})$ for the specified choices of $\ell$ and $\beta$. For details see, e.g., [@helmuth2018contours Proof of Lemma 5.4].
\[lemOuterSample\] Suppose $d \ge 2$ and $q \ge q_0$. Then
1. For $\beta = \beta_c$, there are efficient sampling schemes for $\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}$ and $\nu_{\text{dis}}^{{{\boldsymbol \Lam }}}$.
2. For $\beta > \beta_c$ there is an efficient sampling scheme for $\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}$.
3. For $\beta_h < \beta < \beta_c$ there is an efficient sampling scheme for $\nu_{\text{dis}}^{{{\boldsymbol \Lam }}}$.
In each case these algorithms apply for all regions ${{\boldsymbol \Lam }}$.
First we consider $\beta=\beta_{c}$. By Lemma \[lemOrdDiscompute\] there are efficient algorithms to approximate $Z_{{\text{dis}}}({{\boldsymbol \Lam }})$ and $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$ for all regions ${{\boldsymbol \Lam }}$. With this, we can apply the approximate sampling algorithms given in [@helmuth2018contours Theorem 5.5 and Theorem 6.2]. We summarize the algorithm here, assuming that we want to sample a collection of ordered contours (the disordered case is identical).
By Lemma \[lem:sample-trunc\] it is enough to obtain an $\eps$-approximate sample from $\nu_{\ell}^{{{\boldsymbol \Lam }},m}$ with $m = O(\log (N / \eps))$. List all contours of size at most $m$ in $\cC_{{\text{ord}}}({{\boldsymbol \Lam }})$, and call this collection $\cC$. Order the vertices of ${{\boldsymbol \Lam }}$ arbitrarily as $v_1, \dots, v_N$. We will form a random collection $\Gamma=\Gamma_N$ of mutually external ordered contours step by step. Begin with $\Gamma_0 = \emptyset$. At step $i$, let $\cC_i$ be the subset of contours ${{\boldsymbol \gamma }}$ in $\cC$ such that (i) $v_i \in {\text{Int}\,}{{\boldsymbol \gamma }}$ (ii) ${{\boldsymbol \gamma }}$ is external to $\Gamma_{i-1}$ and (iii) ${\text{Int}\,}{{\boldsymbol \gamma }} \cap \{v_{1},\dots, v_{i-1}\}=\emptyset$. We can efficiently approximate the conditional probability of each contour in $\cC_{i}$, or of adding no contour at step $i$, by using Lemma \[lemOrdDiscompute\] to approximate the relevant polymer partition functions. The result of this procedure is the desired approximate sampling algorithm.
Sampling from $\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}$ for $\beta > \beta_c$ also follows from the algorithm described above since we have an FPTAS for computing $Z_{{\text{ord}}}({{\boldsymbol \Lam }})$, and similarly for $\nu_{{\text{dis}}}^{{{\boldsymbol \Lam }}}$ when $\beta_{h}<\beta<\beta_{c}$.
Our strategy for efficiently approximately sampling from $\mu_{{\text{ord}}}$ and $\mu_{{\text{dis}}}$ requires that we can also efficiently approximately sample from $\nu_{{\text{dis}}}^{{{\boldsymbol \Lam }}}$ for small regions ${{\boldsymbol \Lam }}$ when $\beta>\beta_{c}$ (and likewise from $\nu_{{\text{ord}}}^{{{\boldsymbol \Lam }}}$ when $\beta<\beta_c$). We cannot use the cluster expansion for this task since the disordered (resp. ordered) ground state is unstable, and so instead our approach is based on the intuition from Lemma \[lemSuperEstimates2\] that a disordered region will quickly ‘flip’ to being ordered when $\beta>\beta_{c}$.
\[lemOuterSampleU\] Suppose $d \ge 2$ and $q \ge q_0$. Then
1. For $\beta > \beta_c$ there is an $\eps$-approximate sampling algorithm for $\nu_{\text{dis}}^{{{\boldsymbol \Lam }}}$ that runs in time polynomial in $1/\eps$ and exponential in $\| \partial {{\boldsymbol \Lam }} \|$.
2. For $\beta_h < \beta < \beta_c$ there is an $\eps$-approximate sampling algorithm for $\nu_{\text{ord}}^{{{\boldsymbol \Lam }}}$ that runs in time polynomial in $1/\eps$ and exponential in $\| \partial {{\boldsymbol \Lam }} \|$.
In each case these algorithms apply for all regions ${{\boldsymbol \Lam }}$.
In our sampling algorithms we can allow exponential dependence on $\| \partial {{\boldsymbol \Lam }} \|$ since by Lemma \[lem:sample-trunc\] we need only consider contours $\gamma$ with $\| \gamma \| = O( \log (N/\eps) )$.
Consider the case $\beta>\beta_{c}$ and suppose ${{\boldsymbol \Lam }} = {\text{Int}\,}{{\boldsymbol \gamma }}$. The lemma follows from Proposition \[prop:vacant\] and Lemma \[lemSuperEstimates2\]. More precisely, set $M$ according to Lemma \[lemSuperEstimates2\], and then compute $\cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$ by Proposition \[prop:vacant\]. As in the proof of Lemma \[lemOrdDiscompute\], compute accurate approximations to the weight of each summand in $Z^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$. These approximations determine the probabilities according to which we sample $\Gamma\in \cH^{\text{flip}}_{{\text{dis}}}({\text{Int}\,}{{\boldsymbol \gamma }},M)$. By Lemma \[lemSuperEstimates2\] the result is an $\eps$-approximation to $\nu^{{\text{Int}\,}{{\boldsymbol \gamma }}}_{{\text{ord}}}$.
For $\beta_{h}<\beta<\beta_{c}$ the proof is essentially the same given the inputs discussed in Appendix \[sec:HT\].
We first consider the sampling part of Theorem \[PottsZd\], which follows similarly to the proof of the approximate counting algorithm given in the previous section. Given (i) $\Lambda\subset\Z^{d}$ such that $G_{\Lambda}$ is simply connected and (ii) a choice of wired or free boundary conditions, Proposition \[prop:scregion\] gives a contour ${{\boldsymbol \gamma }}$ such that the partition function associated to ${\text{Int}\,}{{\boldsymbol \gamma }}$ is $Z_{\Lambda}^{w}$ or $Z_{\Lam}^{f}$. Thus if $\beta=\beta_{c}$ we can use Lemma \[lemOuterSample\] to implement the inductive contour algorithm, but using $\eps'$-approximations to $\nu^{{{\boldsymbol \Lam }}}_{{\text{ord}}}$ and $\nu^{{{\boldsymbol \Lam }}}_{{\text{dis}}}$ in place of the true measures. If $\eps'=\eps^{2}/(9N^{2})$ where $N={\left|{{\boldsymbol \Lambda }}\right|}_{{({\frac{1}{2}\tor})^\star}}$, the result is an $\eps$-approximate sample by [@helmuth2018contours Lemma 5.3]. Here we are using $N$ as a crude bound for the depth of the inductive contour algorithm.
If $\beta>\beta_{c}$, then Lemma \[lem:sample-trunc\] tells us that it suffices to sample from $\nu^{{{\boldsymbol \Lam }},m}_{{\text{ord}}}$ with $m = O(\log (N/\eps))$. The consequence of this fact is that we can use the algorithm described above for $\beta=\beta_{c}$, as each call for an $\eps$-approximate sample of $\nu^{{{\boldsymbol \Lam }}}_{{\text{dis}}}$ takes time $\exp (O (\log N/\eps))$ by Lemma \[lemOuterSampleU\] since each contour is of size at most $O(\log(N/\eps))$. For $\beta_{h}<\beta<\beta_{c}$ an analogous argument applies with the roles of ${\text{ord}}$ and ${\text{dis}}$ reversed.
For Theorem \[PottsTorusCrit\] the situation is similar to what we have just discussed, except for the fact that $\mu^{\text{RC}}$ is not an ordered or a disordered measure: it includes configurations with ordered and disordered external contours and includes the configurations with interfaces. If $\beta>\beta_{c}$, however, we have $\| \mu^{\text{RC}} - \mu_{{\text{ord}}} \|_{TV} = \exp ( - \Omega(n^{d-1}))$, and hence if $\eps$ is not too small, we can sample from $\mu_{{\text{ord}}}$ as above. *Mutatis mutandis* the same argument applies for $\mu_{{\text{dis}}}$ if $\beta_{h}<\beta<\beta_{c}$. On the other hand if $\eps = \exp ( - \Omega(n^{d-1}))$, then we can use the Glauber dynamics to sample efficiently by Theorem \[thmBCTglauber\].
For $\beta=\beta_{c}$ the situation is slightly different as the probability of both the ordered and disordered configurations are both of constant order, while the probability of configurations with interfaces is still $\exp ( - \Omega(n^{d-1}))$. The solution is to use the approximate counting algorithm of Lemma \[lemOrdDiscompute\] to approximate the relative probabilities of $\Omega_{\text{ord}}$ and $\Omega_{{\text{dis}}}$ under $\mu^{\text{RC}}$ and then to sample from each using the procedure above. Again if $\eps = \exp ( - \Omega(n^{d-1}))$ we can use the Glauber dynamics.
Conclusions {#sec:Conc}
===========
In this paper we have given efficient approximate counting and sampling algorithms for the random cluster and $q$-state Potts models on $\Z^d$ at all inverse temperatures $\beta\geq 0$, provided $q\geq q_{0}(d)$ and $d\geq 2$. We believe the ideas of this paper will, however, allow for approximate counting and sampling algorithms to be developed for a much broader class of statistical mechanics models. The necessary conditions for the development of algorithms for a given model is that there are only finitely many ground states, and that there is ‘sufficient $\tau$-functionality’. These are the necessary ingredients for the implementation of Pirogov–Sinai theory, see [@borgs1989unified]. Our methods allow for the presence of unstable ground states, a significant improvement compared to the algorithms in [@helmuth2018contours].
Our results suggest that the algorithmic tasks of counting and sampling may be performed efficiently for a fairly broad class of statistical mechanics models with first-order phase transitions, but we leave a fuller investigation of this for future work. A related interesting questions is the existence of efficient algorithms for all $\beta\geq \beta_{c}$ in the presence of a second-order transition; we are not aware of any results in this direction with the exception of the Ising model, i.e., the $q=2$ state Potts model [@jerrum1993polynomial; @guo2018]. To conclude we list some further open questions related to this paper.
1. Our algorithms are restricted to $q\geq q_{0}(d)$ with $q_{0}(d)>\exp(25d\log d)$. Do efficient algorithms exist that avoid this constraint? Since the physical phenomena behind our results are believed to hold for $q\geq 3$ when $d\geq 3$, there is likely room for improvement.
2. On the torus, we obtained an FPRAS (as opposed to an FPTAS) for the partition function because of the estimate on $Z_{{\text{tunnel}}}$ from Lemma \[lem:Z-split\]: the contribution of $Z_{{\text{tunnel}}}$ cannot be ignored when $\eps\leq \exp(-\Omega(n^{d-1}))$. Fortunately, it is exactly when $\eps$ is this small that the Glauber dynamics mix in time polynomial in $1/\eps$, but of course Markov Chain Monte Carlo is a randomized algorithm. A method for systematically accounting for the interfaces that contribute to $Z_{{\text{tunnel}}}$ would likely enable the development of an FPTAS. We leave this as an open problem.
3. Our algorithms have at least two other features that could be improved. The first is the running time: while our algorithms are polynomial time, the degree of the polynomial is not small. The second is that our algorithms rely on *a priori* knowledge of whether or not $\beta=\beta_{c}$.
Both of these deficiencies have the potential to be addressed by Glauber-type dynamics as described in [@PolymerMarkov]; see also [@helmuth2018contours Section 7.2]. Proving the efficiency of these proposed algorithms would be very interesting.
4. Our deterministic algorithms for $\beta>\beta_{c}$ (and $\beta<\beta_{c}$) have diverging running times as $\beta\downarrow\beta_{c}$ ($\beta\uparrow\beta_{c}$). Are there *deterministic* algorithms that do not suffer from this dependence?
5. The algorithmic adaptation of other sophisticated contour-based methods, e.g., [@peled2018rigidity], would be also be quite interesting, particularly for applications to problems such as counting the number of proper $q$-colorings of a graph. For recent progress on approximation algorithms for $q$-colorings, see [@Liu2Delta2019; @bencs2018zero; @JenssenAlgorithmsSODA; @liao2019counting].
Acknowledgements {#acknowledgements .unnumbered}
================
Part of this work was done while WP and PT were visiting Microsoft Research New England. Part of this work was done while TH and WP were visiting the Simons Institute for the Theory of Computing. WP is supported in part by NSF Career award DMS-1847451. TH is supported by EPSRC grant EP/P003656/1. PT is supported in part by the NSF grant DMS-1811935.
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Coupling the Potts and random cluster models {#sec:ES}
============================================
Here we review the standard Edwards–Sokal coupling between the Potts and random cluster models and indicate how one can obtain counting and sampling algorithms for the Potts model from counting and sampling algorithms for the random cluster model. For more details on the couplings between the Potts model and random-cluster measures, see [@DuminilCopinLectures Section 1.2.2].
Let $G= (V,E(G))$ be a finite graph. Then the standard Edwards–Sokal coupling put the $q$-color Potts model at inverse temperature $\beta$ on the same probability space as the random cluster model with paramaters $q$ and $p = 1- e^{-\beta}$. To obtain a Potts configuration we sample a random cluster configuration $A$, then assign one of the $q$ colors uniformly at random to each of the connected components of the graph $G_A = (V,A)$; note that isolated vertices are connected components. Each vertex is then assigned the color of its connected component. This gives an efficient algorithm to sample from the Potts model given a sample from the random cluster model. Moreover, $$Z^{\text{Potts}}_{G}(\beta) = e^{\beta |E(G)|} Z^{\text{RC}}_G(1-e^{-\beta},q)\, ,$$ which gives us an FPTAS (FPRAS) for $Z^{\text{Potts}}$ given an FPTAS (FPRAS) for $Z^{\text{RC}}$.
We can also couple the Potts model with monochromatic boundary conditions to the random cluster model with wired boundary conditions. For this, let us specialize to finite induced subgraphs $(\Lam, E(\Lam))$ of $\Z^d$. Define the boundary of $\Lam$ to be $\partial \Lam {\coloneqq}\{ i \in \Lam: \exists j \in \Lam^c, (i,j) \in
E(\Z^d) \}$. Recall the definition of the random cluster model $\mu^{f}_{\Lam}$ with wired boundary conditions from page three. Given a color $r \in [q]$, the allowed colorings for the Potts model with $r$-monochromatic boundary conditions on $\Lam$ are $$\Omega_r(\Lam) = \left \{ \sigma \in [q]^\Lam : \sigma_v = r \, \forall \, v \in \partial \Lam \right \} \,.$$ The corresponding Gibbs measure and partition function are: $$\begin{aligned}
\mu_\Lam^{\text{Potts},r}( \sigma) &= \frac{ \prod_{(i,j)\in E(\Lam)} e^{-\beta \mathbf 1_{\sigma_i \ne \sigma_j}} }{Z_\Lam^{\text{Potts},r}(\beta) } \, , \quad \quad \sigma \in \Omega_r(\Lam) \\
Z_\Lam^{\text{Potts},r}(\beta) &= \sum_{ \sigma \in \Omega_r(\Lam)} e^{-\beta \mathbf 1_{\sigma_i \ne \sigma_j}} \, .\end{aligned}$$
A simple extension of the Edwards-Sokal coupling then gives the following facts. Given a sample $A$ from $\mu^{w}_\Lam$ one can obtain a sample from $\mu_\Lam^{\text{Potts},r}$ by coloring all vertices in $\partial \Lam$ or connected to $\partial \Lam$ by the edges in $A$ with color $r$, and assigning one of the $q$ colors uniformly at random to the remaining connected components of the graph $(\Lam, A)$. Moreover, we have the relation $$q Z_\Lam^{\text{Potts},r} (\beta) = e^{-\beta | E(\Lam)|} Z^w_\Lam( (1- e^{-\beta},q) \,.$$ Again this shows that efficient counting and sampling algorithms for the Potts model with monochromatic boundary conditions follow from efficient counting and sampling algorithms for the random cluster model with wired boundary conditions.
Proofs for $\beta_{h}<\beta<\beta_{c}$ {#sec:HT}
======================================
Lemma \[lemOrdDiscompute\] (iii) {#sec:lemma-refl-iii}
--------------------------------
The proof of Lemma \[lemOrdDiscompute\] in the case $\beta_h < \beta <\beta_{c}$ is the same, *mutatis mutandis*, as for $\beta>\beta_{c}$. The necessary changes are that (i) the roles of the ordered and disordered contours are exchanged, and (ii) some of the ingredients from Sections \[sec:count\] and \[sec:sample\] were stated only for $\beta>\beta_{c}$, and hence versions for $\beta_{h}<\beta<\beta_{c}$ are necessary. We outline how to obtain these versions here.
As explained in [@borgs2012tight Appendix A], [@borgs2012tight Lemma 6.3 (i) and (ii)] applies when [@borgs2012tight (A.1)] holds. In fact, the arguments apply if $$\label{eq:A1p}
\beta \geq \max \{ C_{1}\log (dC), \frac{3\log q}{4d}\}$$ where $C$ is the constant from [@borgs2012tight Lemma 5.8] and $C_{1}$ is a sufficiently large constant depending only on $d$. To verify this it is enough to check that [@borgs2012tight (A.2)] holds (up to a change in the constant $8$).[^6] Thus for $q_{0}$ sufficiently large [@borgs2012tight Lemma 6.3 (i) and (ii)] apply when $\beta_{h}<\beta<\beta_{c}$. In particular, by following the proofs from $\beta>\beta_{c}$ we obtain that when $\beta_{h}<\beta<\beta_{c}$
1. the conclusions of Lemma \[lemBCTsup\] hold with the roles of ${\text{ord}}$ and ${\text{dis}}$ reversed. The fact that $a_{{\text{ord}}}>0$ is contained in [@borgs2012tight Lemma A.3].
2. the conclusion of Lemma \[lemSuperEstimates\] holds with ${\text{ord}}$ replaced by ${\text{dis}}$.
3. the conclusion of Lemma \[lemSuperEstimates2\] holds with the roles of ${\text{ord}}$ and ${\text{dis}}$ reversed and $$M\geq \frac{2}{a_{{\text{ord}}}}\log \frac{8q}{\eps} +
\frac{2}{a_{{\text{ord}}}}(\kappa+4){\|{{\boldsymbol \gamma }}\|}.$$ The factor four (as opposed to three) in $M$ arises in the computation of the lower bound on $Z^{\text{flip}}_{{\text{ord}}}({{\boldsymbol \Lam }},M)$, as (in the notation of the proof of Lemma \[lemSuperEstimates2\]) ${\text{Ext}\,}{\Gamma}$ may be of size ${\|{{\boldsymbol \gamma }}\|}$.
Lastly, the conclusion of Proposition \[prop:vacant\] holds with ${\text{dis}}$ changed to ${\text{ord}}$. The proof is very similar to the proof of Proposition \[prop:vacant\], but using Lemmas \[lem:edge-dis\] and \[lem:dis-edge-con\] in place of Lemmas \[lem:edge-ord\] and \[lem:ord-edge-con\].
Using the ingredients above, this follows exactly as in the proof of Lemma \[lemOrdDiscompute\] (ii), i.e., for $\beta>\beta_{c}$.
Theorems \[PottsTorusCrit\] and \[PottsZd\]
-------------------------------------------
These proofs are exactly as for $\beta>\beta_{c}$ provided the conclusions of Lemma \[lemSuperEstimates2a\] hold ${\text{dis}}$ replaced by ${\text{ord}}$. This is straightforward to obtain by imitating the proof of Lemma \[lemSuperEstimates2a\], using (as discussed in the previous section) that the conclusion of Lemma \[lemBCTsup\] hold with the roles of ${\text{ord}}$ and ${\text{dis}}$ reversed.
Contour computations using subgraphs of ${({\frac{1}{2}\tor})^\star}$ {#app:subcomp}
=====================================================================
The next lemma shows that computations relating to contours ${{\boldsymbol \gamma }}$ can be implemented using only $\gamma$, the connected subgraph of ${({\frac{1}{2}\tor})^\star}$ that corresponds to ${{\boldsymbol \gamma }}$ by the construction in Section \[sec:cont\].
\[lem:subgcomp\] Let ${{\boldsymbol \gamma }}$ and ${{\boldsymbol \gamma }}'$ be contours, and let $\gamma$ and $\gamma'$ be the corresponding subgraphs of ${({\frac{1}{2}\tor})^\star}$. Then given $\gamma$, $\gamma'$,
1. $d_{\infty}({{\boldsymbol \gamma }},{{\boldsymbol \gamma }}')$ can be computed in time $O({\left|V(\gamma)\right|}{\left|V(\gamma')\right|})$,
2. The set ${\text{Int}\,}{{\boldsymbol \gamma }} \cap \tor$ can be computed in time $O({\left|V(\gamma)\right|}^{3})$,
3. ${\|{{\boldsymbol \gamma }}\|}$ can be computed in time $O({\left|V(\gamma)\right|})$.
Each vertex in ${({\frac{1}{2}\tor})^\star}$ corresponds to a $(d-1)$-dimensional hypercube in ${{{\boldsymbol T }}^d_n}$. For each pair of such hypercubes we can compute the distance between them in constant time, which implies the first claim. The third claim follows similarly, since the set of edges passing through a given $(d-1)$-dimensional hypercube can be determined in constant time.
For the second claim, we first determine the set of edges intersecting the $(d-1)$-dimensional hypercubes corresponding to ${{\boldsymbol \gamma }}$. We can then determine ${\text{Int}\,}{{\boldsymbol \gamma }}\cap \tor$ in time $O({\|{{\boldsymbol \gamma }}\|}^{3})$ by Lemma \[lem:findext\].
[^1]: This continuum construction allows for tools from algebraic topology to be used. We have chosen to follow the continuum terminology to allow the interested reader to easily consult [@borgs2012tight].
[^2]: More formally, since $\Z^{d}\subset
\frac{1}{2}\Z^{d}\subset \R^{d}$, we obtain a common embedding of ${\frac{1}{2}\tor}$ and $\tor$ in ${{{\boldsymbol T }}^d_n}$.
[^3]: These results rely only on the geometry of hypercubes and not on the definitions of interior/exterior.
[^4]: Alternatively, we can compute this directly. The proportionality constant in the first case is $p^{|E(\Lam^{c})}(1-p)^{{\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}}$, and $p^{d{\left|\Lam^{c}\right|}}e^{-\kappa {\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}}$ in the second. This agrees, since $d{\left|\Lam^{c}\right|}-{\|{{\boldsymbol \gamma }}_{{\text{ord}}}\|}/2={\left|E(\Lam^{c})\right|}$.
[^5]: The constants implicit in the $O(\cdot)$ notation depend only on the constants $c$ in the bounds on $K_{\ell}({{\boldsymbol \gamma }}) \leq \exp(-c\beta {\|{{\boldsymbol \gamma }}\|})$. These bounds are given by Lemma \[lemKestimates\] and Lemma \[lemSuperEstimates\] for $\beta\geq \beta_{c}$, and in Appendix \[sec:HT\] for $\beta_{h}<\beta<\beta_{c}$.
[^6]: Our choice of $3/4$ in is somewhat arbitrary; the same conclusion would hold for any number strictly larger than $2/3$.
|
---
abstract: 'Existing optimal estimators of nonequilibrium path-ensemble averages are shown to fall within the framework of extended bridge sampling. Using this framework, we derive a general minimal-variance estimator that can combine nonequilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations. The framework is also applied to obtaining asymptotic variance estimates, which are a useful measure of statistical uncertainty. In particular, we develop asymptotic variance estimates pertaining to Jarzynski’s equality for free energies and the Hummer-Szabo expressions for the potential of mean force, calculated from uni- or bidirectional path samples. Lastly, they are demonstrated on a model single-molecule pulling experiment. In these simulations, the asymptotic variance expression is found to accurately characterize the confidence intervals around estimators when the bias is small. Hence, it does not work well for unidirectional estimates with large bias, but for this model it largely reflects the true error in a bidirectional estimator derived by Minh and Adib.'
author:
- 'David D. L. Minh'
- 'John D. Chodera'
title: 'Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages'
---
Introduction
============
Path-ensemble averages play a central role in nonequilibrium statistical mechanics, akin to the role of configurational ensemble averages in equilibrium statistical mechanics. Expectations of various functionals over processes where a system is driven out of equilibrium by a time-dependent external potential have been shown to be related to equilibrium properties, including free energy differences [@Jarzynski1997a; @Jarzynski1997b] and thermodynamic expectations. [@Crooks2000; @Neal2001] The latter relationship, between equilibrium and nonequilibrium expectations, has been applied to several specific cases, such as: the potential of mean force (PMF) along the pulling coordinate [@Hummer2001a; @Hummer2005; @Minh2006] (or other observed coordinates [@Minh2007]) in single-molecule pulling experiments; RNA folding free energies as a function of a control parameter; [@Junier2009] the root mean square deviation from a reference structure; [@Lyman2007] the potential energy distribution [@Lyman2007] and average; [@Nummela2008] and the thermodynamic length. [@Feng2009]
Compared to equilibrium sampling, nonequilibrium processes may be advantageous for traversing energetic barriers and accessing larger regions of phase space per unit time. This is useful, for example, in reducing the effects of experimental apparatus drift or increasing the sampling of barrier-crossing events. Thus, there has been interest in calculating equilibrium properties from nonequilibrium trajectories collected in simulations or laboratory experiments. Indeed, single-molecule pulling data has been used to experimentally verify relationships between equilibrium and nonequilibrium quantities. [@Liphardt2002; @Collin2005]
While many estimators for free energy differences [@Bennett1976; @Crooks2000; @Maragakis2006; @Maragakis2008] and equilibrium ensemble averages can be constructed from nonequilibrium relationships, they will differ in the efficiency with which they utilize finite data sets, leading to varying amounts of statistical bias and uncertainty. Characterization of this bias and uncertainty is helpful for comparing the quality of different estimators [@Shirts2005] and assessing the accuracy of a particular estimate. The statistical uncertainty of an estimator is usually quantified by its variance in the asymptotic, or large sample, limit, where estimates from independent repetitions of the experiment often approach a normal distribution about the true value due to the central limit theorem. It is an important goal to find an optimal estimator which minimizes this asymptotic variance.
Although numerical estimates of the asymptotic variance may be provided by bootstrapping (e.g. Ref. [@Calderon2009]), closed-form expressions can provide computational advantages in the computation of confidence intervals, allow comparison of asymptotic efficiency, [@Tan2004; @Shirts2005] and facilitate the design of adaptive sampling strategies to target data collection in a manner that most rapidly reduces statistical error. [@Singhal2005; @Singhal2007; @Hahn2009] In the asymptotic limit, the statistical error in functions of the estimated parameters can be estimated by propagating this variance estimate via a first-order Taylor series expansion. While this procedure is relatively straightforward for simple estimators, it can be difficult for estimators that involve arbitrary functions (e.g. nonlinear or implicit equations) of nonequilibrium path-ensemble averages.
Fortunately, the extended bridge sampling (EBS) estimators, [@Vardi1985; @Gill1988; @Kong2003; @Tan2004] a class of equations for estimating the ratios of normalizing constants, are known to have both minimal-variance forms and associated asymptotic variance expressions. Recently, Shirts and Chodera [@Shirts2008] applied the EBS formalism to generalize the Bennett acceptance ratio, [@Bennett1976] producing an optimal estimator combining data from multiple equilibrium states to compute free energy differences, thermodynamic expectations, and their associated uncertainties. Here, we apply the EBS formalism to estimators utilizing nonequilibrium trajectories. We first construct a general minimal-variance path-average estimator that can use samples collected from multiple nonequilibrium path-ensembles. We then show that some existing path-average estimators using uni- and bidirectional data are special cases of this general estimator, proving their optimality. This also allows us to develop asymptotic variance expressions for estimators based on Jarzynski’s equality [@Jarzynski1997a; @Jarzynski1997b] and the Hummer-Szabo expressions for the PMF. [@Hummer2001a; @Hummer2005; @Minh2008prl] We then demonstrate them on simulation data from a simple one-dimensional system and comment on their applicability.
Extended Bridge Sampling
========================
Suppose that we sample $N_i$ paths (trajectories) from each of $K$ path-ensembles indexed by $i = 1,2,...,K$. The path-ensemble average of an arbitrary functional $\mathcal F[X]$ in path-ensemble $i$ is defined by $$\left< \mathcal F \right>_i \equiv \int dX \, \mathcal F[X] \, \rho_i[X],
\label{eq:path_average}$$ where $\rho_i[X]$ is a probability density over trajectories, $$\begin{aligned}
\rho_i[X] = c_i^{-1} q_i[X] \:\: ; \:\: c_i = \int dX \, q_i[X],
\label{eq:path_density}\end{aligned}$$ with unnormalized density $q_i[X] > 0$ and the normalization constant $c_i$ (a path partition function). The above integrals, in which $dX$ is an infinitesimal path element, are taken over all possible paths, $X$. Extended bridge sampling estimators provide a way of estimating ratios of normalization constants $c_i/c_j$, which will prove useful in estimating free energies and thermodynamic expectations.
To construct these estimators, we first note the importance sampling identity, $$\begin{aligned}
c_i \left< \alpha_{ij} \, q_j \right>_i
& = & \left[ \int dX \, q_i[X] \right] \frac{ \int dX \, \alpha_{ij}[X] \, q_i[X] \, q_j[X] }{ \int dX \, q_i[X] } \nonumber \\
& = & \left[ \int dX \, q_j[X] \right] \frac{ \int dX \, \alpha_{ij}[X] \, q_i[X] \, q_j[X] }{ \int dX \, q_j[X] } \nonumber \\
& = & c_j \left< \alpha_{ij} \, q_i \right>_j,
\label{equation:importance-sampling-identity}\end{aligned}$$ where $j$ is another path-ensemble index, $\alpha_{ij}[X]$ is an arbitrary functional of $X$, and all normalization constants are nonzero.
Summing over the index $j$ in Eq. \[equation:importance-sampling-identity\] and using the sample mean, $N_i^{-1} \sum_{n=1}^{N_i} \mathcal F[X_{in}]$, as an estimator for $\left< \mathcal F \right>_i$, we obtain a set of $K$ estimating equations, $$\begin{aligned}
\sum_{j=1}^{K} \frac{\hat{c}_i}{N_i} \sum_{n=1}^{N_i} \alpha_{ij}[X_{in}] \, q_j[X_{in}] =
\sum_{j=1}^{K} \frac{\hat{c}_j}{N_i} \sum_{n=1}^{N_j} \alpha_{ij}[X_{jn}] \, q_i[X_{jn}],
\label{eq:ext_bridge}\end{aligned}$$ whose solutions yield estimates $\hat{c}_i$ for the normalization constants $c_i$, up to an irrelevant scalar multiple. Each path, $X_{in}$, is indexed by the ensemble $i$ from which it is sampled, and the sample number $n = 1,2,...,N_i$. This coupled set of nonlinear equations defines a *family* of estimators parameterized by the choice of $\alpha_{ij}[X]$, all of which are asymptotically consistent, but whose statistical efficiencies will vary. [@Tan2004]
With the choice, $$\alpha_{ij}[X] = \frac{N_j \hat{c}_j^{-1} }{ \sum\limits_{k=1}^K N_k \, \hat{c}_k^{-1} \, q_k[X]},$$ Eq. \[eq:ext\_bridge\] simplifies to the optimal EBS estimator, $$\hat{c}_i = \sum_{j=1}^K \sum_{n=1}^{N_j}
\left[ \sum_{k=1}^K \frac{N_k}{\hat{c}_k} \frac{q_k[X_{jn}]}{q_i[X_{jn}]} \right]^{-1}. \label{eq:opt_bridge}$$ This choice for $\alpha_{ij}[X]$ is optimal in that the asymptotic variance of the ratios $\hat{c}_i / \hat{c}_j$ is minimal. [@Tan2004; @Shirts2008] These equations may be solved by any appropriate algorithm, including a number of efficient and stable methods suggested by Shirts and Chodera. [@Shirts2008]
The asymptotic covariance of Eq. \[eq:opt\_bridge\] is estimated by, $$\hat{\bm{\Theta}} = \bm{M}^{\mathrm{T}}(\bm{I}_N - \bm{MNM}^{\mathrm{T}})^+ \bm{M}
\label{eq:Theta}$$ where the elements of $\bm{\Theta}$ are the covariances of the logarithms of the estimated normalization constants, $\Theta_{ij} = \mathrm{cov}\,(\hat{\gamma}_i, \hat{\gamma}_j)$, and $\hat{\gamma}_i = \ln \hat{c}_i$.[@Kong2003] The superscript $(...)^+$ denotes an appropriate generalized inverse, such as the Moore-Penrose pseudoinverse, $\bm{I}_N$ is the $N \times N$ identity matrix (where $N=\sum_{i=1}^K N_i$ is the total number of samples), $\bm{N} = \mathrm{diag} \, (N_1,N_2,...,N_K)$ is the diagonal matrix of sample sizes, and $\bm{M}$ is the $N \times K$ weight matrix with elements, $$M_{ni} = \hat{c}_i^{-1} \frac{ q_i[X_n] }{ \sum\limits_{k=1}^K N_k \, \hat{c}_k^{-1} \, q_k[X_n] }.
\label{eq:weight_elements}$$ In this matrix, the distribution from which samples are drawn from is irrelevant, and $X$ is only indexed by $n = 1,\ldots,N$. We note that the sum over each column, $\sum_{n=1}^{N} M_{ni}$, is one.
For arbitrary functions of the logarithms of the normalization constants, $\phi(\hat{\gamma}_1,...,\hat{\gamma}_K)$ and $\psi(\hat{\gamma}_1,...,\hat{\gamma}_K)$, the asymptotic covariance $\mathrm{cov}(\hat{\phi},\hat{\psi})$ can be estimated from $\hat{\bm{\Theta}}$ according to, $$\mathrm{cov} (\hat{\phi},\hat{\psi}) \approx
\sum_{i,j=1}^K \frac{ \partial \phi }{ \partial \hat{\gamma}_i } \hat{\Theta}_{ij} \frac{ \partial \psi }{ \partial \hat{\gamma}_j },
\label{eq:cov}$$ through first-order Taylor series expansion of $\phi$ and $\psi$.
General Path-Ensemble Averages
==============================
Following previous work, [@Doss2003; @Shirts2008] we estimate nonequilibrium expectations by defining additional path-ensembles with “unnormalized densities” $$q_{\mathcal F_i}[X] = \mathcal F[X] \, q_i[X] \:\: ; \:\: c_{\mathcal F_i} = \int dX \, q_{\mathcal F_i}[X] .
\label{eq:functional_density}$$ Using Eqs. (\[eq:path\_average\]), (\[eq:path\_density\]), and (\[eq:functional\_density\]), we can express nonequilibrium expectations as a ratio of the appropriate normalization constants, $\left< \mathcal F \right>_i = c_{\mathcal F_i} / c_i.$ Notably, this can be estimated *without* actually sampling path-ensembles biased by some function of $\mathcal F[X]$ (although it is sometimes possible to do so in computer simulations [@Sun2003; @Ytreberg2004] via transition path sampling [@Pratt1986; @Dellago1998]). If no paths are drawn from the path-ensemble corresponding to $q_{\mathcal F_i}[X]$, then $N_{\mathcal F_i} = 0$ and it is no longer required that $q_{\mathcal F_i}[X] > 0$. [@Tan2004; @Shirts2008]
For each defined path-ensemble, the weight matrix $\bm{M}$ is augmented by one column with elements, $$M_{n \mathcal F_i} = \hat{c}_{\mathcal F_i}^{-1} \frac{\mathcal F[X_n] \, q_i[X_n] }{ \sum\limits_{k=1}^K \, N_k \, \hat{c}_k^{-1} \, q_k[X_n] } .
\label{eq:F_weight_elements}$$ The estimator for the path-ensemble average, $\bar{\mathcal F}_i \approx \left< \mathcal F \right>_i$, can be expressed in terms of weight matrix elements, $$\bar{\mathcal F}_i = \sum_{n=1}^N \, M_{ni} \, \mathcal F[X_n] ,$$ and its uncertainty estimated by $$\begin{aligned}
\sigma^2 ( \bar{\mathcal F}_i ) &\approx& \bar{\mathcal F}_i^2 (\hat{\Theta}_{\mathcal F_i \, \mathcal F_i} - 2 \hat{\Theta}_{\mathcal F_i \, i} + \hat{\Theta}_{i \, i} ).
\label{eq:path_average_variance}\end{aligned}$$
Experimentally Relevant Path-Ensembles
======================================
The above formalism is fully general, and may be applied to *any* situation where the ratio $q_i[X] / q_j[X]$ can be computed. For arbitrary path-ensembles, unfortunately, calculating this ratio is only possible in computer simulations unless certain assumptions are made about the dynamics. [@Nummela2007] In a few special path-ensembles, however, we can use the Crooks fluctuation theorem [@Crooks1998; @Crooks1999] to estimate this ratio, allowing us to apply the EBS estimator to laboratory experiments. We examine these here.
First, consider a *forward process*, in which a system, initially in equilibrium, is propagated under some time-dependent dynamics for a time $\tau$ which may cause it to be driven out of equilibrium. The time-dependence of the evolution law (e.g. Hamiltonian dynamics in a time-dependent potential) is the same for all paths sampled from this ensemble.
For a sample of paths only drawn from this ensemble, the optimal EBS estimator of $\left< \mathcal F \right>_f$ reduces to the sample mean estimator, which we call the *unidirectional* path-ensemble average estimator $$\begin{aligned}
\bar{\mathcal{F}}_f &=& \frac{1}{N_f} \sum_{n=1}^{N_f} \mathcal F[X_{fn}],
\label{equation:unidirectional-path-estimator}\end{aligned}$$ and the associated asymptotic variance from Eq. \[eq:cov\] reduces to the variance of the sample mean (see Appendix \[sec:uni-var\]) $$\begin{aligned}
\sigma^2 ( \bar{\mathcal{F}}_f ) &\approx&
\frac{1}{N_f} \left[ \frac{1}{N_f} \sum_{n=1}^{N_f} \left( \mathcal F[X_{fn}] - \bar{\mathcal F}_f \right)^2 \right]\end{aligned}$$
The forward process has a unique counterpart known as the *reverse process*. Here, the system moves via the opposite protocol in thermodynamic state space; after initial configurations are drawn from the final thermodynamic state of the forward path-ensemble, they are driven towards the initial state. If the dynamical law satisfies detailed balance when the control parameters are held constant at each fixed time $t$, the path probabilities in the conjugate forward and reverse path-ensembles are related according to the Crooks fluctuation theorem: [@Crooks1998; @Crooks1999] $$\frac{\rho_f[X]}{\rho_r[\tilde{X}]} = \frac{ q_f[X] }{q_r[\tilde{X}] } \frac{ c_r }{ c_f } = e^{w_\tau[X] - \Delta f_\tau} \equiv e^{\Omega[X]},
\label{eq:CFT}$$ in which $\tilde{X}$ is the time-reversal, or *conjugate twin*, [@Jarzynski2006] of $X$, $\Delta f_t = -\ln (c_t / c_0)$ is the dimensionless free energy difference between thermodynamic states at times $0$ and $t$ (with $\tau$ being the fixed total trajectory length) and $w_t[X]$ is the appropriate dimensionless work. In Hamiltonian dynamics, for example, this work is $w_t[X] = \beta \int_0^t dt' \, (\partial H/\partial t')$. For convenience, we define the total *dissipative work* as $\Omega[X] \equiv w_\tau[X] - \Delta f_\tau$.
We will refer to data sets which only include realizations from the forward path-ensemble as ‘unidirectional’, and those with paths from both path-ensembles as ‘bidirectional’. Notably, sampling paths from these conjugate ensembles and calculating the associated work $w_t[X]$ is possible in single-molecule pulling experiments as well as computer simulations (c.f. Refs. [@Collin2005; @Hummer2005]). To combine bidirectional data to estimate $\left< \mathcal F \right>_f$, we apply the Crooks fluctuation theorem [@Crooks1998; @Crooks1999] to Eq. \[eq:opt\_bridge\] and divide by $\hat{c}_f$, leading to, $$\bar{\mathcal F}_f =
\sum_{n=1}^{N_f} \frac{ \mathcal F[X_{fn}] }{ N_f + N_r \, e^{-\hat{\Omega}[X_{fn}] } } +
\sum_{n=1}^{N_r} \frac{ \mathcal F[X_{rn}] }{ N_f + N_r \, e^{-\hat{\Omega}[X_{rn}] } }
\label{equation:bidirectional-path-estimator}$$ which is bidirectional path-average estimator of Minh and Adib, [@Minh2008prl] derived here by a different route which demonstrates its optimality. (The asymptotic variance estimator for this equation is written in a closed form in Appendix \[sec:bi-var\].) In these bidirectional expressions, samples drawn from the reverse path-ensemble are time-reversed to obtain the paths $X_{rn}$. The dissipated work estimate, $\hat{\Omega}[X] \equiv w_\tau[X] - \Delta \hat{f}_\tau$, requires an estimate of $\Delta f_\tau$. A method for obtaining this estimate will be described next.
Free Energy\[sec:FE\]
=====================
Jarzynski’s equality, [@Jarzynski1997a; @Jarzynski1997b] $$e^{-\Delta f_t} = \left< e^{-w_t} \right>_f,
\label{eq:JE}$$ relates nonequilibrium work and free energy differences. To facilitate the use of EBS in Jarzynski’s equality, we define a path-ensemble by choosing $\mathcal F[X] = e^{-w_t[X]}$ in Eq. \[eq:functional\_density\], leading to $$q_{w_t}[X] = e^{-w_t[X]} \, q_f[X] \:\: ; \:\: c_{w_t} = \int dX \, e^{-w_t[X]} \, q_f[X] .$$
When only unidirectional data is available, the optimal EBS estimator for Jarzynski’s equality is $$\begin{aligned}
e^{-\Delta \hat{f}_t} = \frac{1}{N_f} \sum_{n=1}^{N_f} e^{-w_t[X_{fn}]} \label{equation:unidirectional-ft}\end{aligned}$$ and its asymptotic variance is straightforwardly given by error propagation. [@Chipot2007] Estimators [@Sun2003; @Ytreberg2004; @Minh2009b] and asymptotic variances [@Oberhofer2005; @Minh2009b] have also been developed for unidirectional *importance sampling* forms of the equality.
When bidirectional data is available, the same choice of $\mathcal F[X]$ in Eq. \[equation:bidirectional-path-estimator\] gives the estimator $$\begin{aligned}
e^{-\Delta \hat{f}_t} =
\sum_{n=1}^{N_f} \frac{ e^{-w_t[X_{fn}]} }{ N_f + N_r \, e^{-\hat{\Omega}[X_{fn}] } } +
\sum_{n=1}^{N_r} \frac{ e^{-w_t[X_{rn}]} }{ N_f + N_r \, e^{-\hat{\Omega}[X_{rn}] } } \label{equation:bidirectional-ft}\end{aligned}$$ In this equation, choosing $t=0$ or $t=\tau$ leads to an implicit function mathematically equivalent to the Bennett acceptance ratio method, [@Bennett1976; @Crooks2000] as previously explained. [@Shirts2003; @Minh2008prl] The asymptotic variance of $\Delta \hat{f}_t$ is calculated by augmenting the matrices $\bm{M}$ and $\hat{\bm{\Theta}}$ and using $\phi = \psi = \Delta f_t = - \ln (c_{w_t} / c_f)$ in Eq. \[eq:cov\], such that, $$\sigma^2 ( \Delta \hat{f}_t ) = \hat{\Theta}_{w_t \, w_t} - 2\hat{\Theta}_{w_t \, f} + \hat{\Theta}_{f \, f} .
\label{eq:vFt}$$
Potential of Mean Force
=======================
Building on Jarzynski’s equality, Hummer and Szabo developed expressions for the PMF, [@Hummer2001a; @Hummer2005] the free energy as a function of a *reaction coordinate* rather than a thermodynamic state, that may be used to interpret single-molecule pulling experiments. In these experiments, a molecule is mechanically stretched by a force-transducing apparatus, such as an laser optical trap or atomic force microscope tip (c.f. [@Hummer2005]). The Hamiltonian governing the time evolution in these experiments, $H(x;t) = H_0(x) + V(z(x);t)$, is assumed to contain both a term corresponding to the unperturbed system, $H_0(x)$, and a time-dependent (typically harmonic) external bias potential imposed by the apparatus, $V(z;t)$, which acts along a pulling coordinate, $z(x)$. As the coordinate $z_t \equiv z(x(t))$ is observed at fixed intervals $\Delta t$ over the course of the experiment, we will henceforth use $t = 0,1,...,T$ as an integer time index. We calculate the work with a discrete sum as $w_t = \sum_{n=1}^{t} \, [V_{n}(z_n) - V_{n-1}(z_n)]$, where $V_n(z) \equiv V(z; n \Delta t)$.
While the expressions in Section \[sec:FE\] provide an estimate of relative free energies of the equilibrium thermodynamic states defined by $H(x;t)$, they are not immediately useful as an estimate for the PMF along $z$. [@Hummer2001a; @Hummer2005; @Minh2008] By applying the nonequilibrium estimator for thermodynamic expectations, [@Crooks2000; @Neal2001] it was shown that the PMF in the absence of an external potential is given by [@Hummer2001a; @Hummer2005] $$\begin{aligned}
e^{-g_0(z)} &=& \left< \delta(z - z_t) \, e^{-w_t} \right>_f e^{V(z_t;t)},
\label{eq:HS_time_slice}\end{aligned}$$ where the dimensionless PMF, $g_0(z)$, is defined in relation to the normalized density as $g_0(z) = - \ln p_0(z) - \delta g$. In this equation, $\delta g$ is a time-independent constant, $e^{-\delta g} = \int dx ~ e^{-H(x;0)} / \int dx ~ e^{-H_0(x)}$.[@Hummer2005]
This theorem can be used to develop estimators for the PMF by replacing the delta function using a kernel function of finite width, such as, $$\begin{aligned}
h(z-z_t) =
\begin{cases}
\frac{1}{\Delta z}, & \text{if}~ |z-z_t|< \frac{\Delta z}{2} \\
0, & \text{else}.
\end{cases}\end{aligned}$$ The width $\Delta z$ must be small so that $e^{V(z;t)}$ does not vary substantially across it.
As this theorem is valid at all times, it is possible to obtain an asymptotically unbiased density estimate $\hat{p}_t$ from each time slice. It is far more efficient, however, to estimate the PMF using *all* recorded time slices. While any linear combination of time slices will lead to a valid estimate, certain choices will be more statistically efficient (leading to lower variance) than others. One way to combine time slices is to use the asymptotic covariance matrix in the method of control variates,[@Tan2004] leading to a generalized least-squares optimal estimate of the PMF. Unfortunately, we empirically found this approach to be numerically unstable. A more numerically stable approach, which was proposed by Hummer and Szabo, [@Hummer2001a; @Hummer2005] is based on the weighted histogram analysis method, [@Ferrenberg1989; @Kumar1992] $$\begin{aligned}
\hat{p}_0(z) = \frac{\sum_t \mu_t(z) \, \hat{p}_t(z)}{\sum_t \mu_t(z)} \:\: ; \:\: \mu_t(z) \equiv e^{-V(z;t) + \Delta \hat{f}_t } .\end{aligned}$$ While this weighting scheme is optimal, in a minimal-variance sense, for independent samples from multiple *equilibrium* distributions, these assumptions do not hold for time slices from nonequilibrium trajectories. However, Oberhofer and Dellago did not observe substantial improvement in PMF estimates when using other time-slice weighting schemes. [@Oberhofer2009]
By defining the path-ensemble, $$q_{z_t}[X] = \delta(z - z_t) \, e^{-w_t[X]} \, q_f[X] \:\: ; \:\: c_{z_t} = \int dX \, q_{z_t}[X]$$ and making use of Jarzynski’s equality (Eq. \[eq:JE\]) for $e^{-\Delta \hat{f}_t}$, we can write Hummer and Szabo’s PMF estimator as $$e^{-\hat{g}_0(z)} = \frac{ \sum_t (\hat{c}_{z_t} / \hat{c}_{w_t}) }{ \sum_t e^{-V(z;t)} \, (\hat{c}_f / \hat{c}_{w_t}) },
\label{eq:HS}$$ which can be readily analyzed in terms of EBS. While Hummer and Szabo proposed using the unidirectional path average estimator (Eq. \[equation:unidirectional-path-estimator\]) to estimate the expectations in Eq. \[eq:HS\], Minh and Adib later applied a bidirectional estimator (Eq. \[equation:bidirectional-path-estimator\]), leading to significantly improved statistical properties. [@Minh2008prl]
The asymptotic variance of these estimators can be determined by choosing $\phi = \psi = p_0(z)$ in Eq. \[eq:cov\]. For the bidirectional estimator, the matrices $\bm{M}$ and $\hat{\bm{\Theta}}$ will contain one column each for the $f$ and $r$ path-ensembles, and $T+1$ columns each for the path-ensembles associated with $\{w_t\}_{t=0}^{\mathrm{T}}$ and $\{z_t\}_{t=0}^{\mathrm{T}}$. The relevant partial derivatives are, $$\begin{aligned}
\frac{\partial p_0(z)}{\partial \gamma_{f}} & = &
-p_0(z) \\
\frac{\partial p_0(z)}{\partial \gamma_{w_t}} & = &
-\frac{1}{\mathcal D} \frac{c_{z_t}}{c_{w_t}} +
\frac{\mathcal N}{\mathcal D^2} \left( e^{-V(z;t)} \frac{c_f}{c_{w_t}} \right) \\
\frac{\partial p_0(z)}{\partial \gamma_{z_t}} & = &
\frac{1}{\mathcal D} \frac{c_{z_t}}{c_{w_t}},\end{aligned}$$ where $\gamma_i = \ln c_i$, $\mathcal N = \sum_t (c_{z_t}/c_{w_t})$ is the numerator of Eq. \[eq:HS\], and $\mathcal D = \sum_t e^{-V(z;t)} \, (c_f/c_{w_t})$ is its denominator. These lead to an estimate for $\sigma^2 (\hat{p}_0(z))$. Finally, the asymptotic variance in the PMF is given by the error propagation formula, $\sigma^2 (\hat{g}_0(z)) \approx \sigma^2 (\hat{p}_0(z)) / \hat{p}_0(z)^2$.
Illustrative Example
====================
We demonstrate these results with Brownian dynamics simulations on a one-dimensional potential with $U_0(z) = (5z^3 - 10z + 3)z$, which were run as previously described. [@Minh2008prl] A time-dependent external perturbation, $V(z;t) = k_s(z-\bar{z}(t))^2/2$, with $k_s = 15$ is applied, such that the total potential is $U(z;t) = U_0(z) + V(z;t)$. After 100 steps of equilibration at the initial $\bar{z}(t)$, $\bar{z}(t)$ is linearly moved over 750 steps from $-1.5$ to $1.5$ in forward processes and $1.5$ to $-1.5$ in the reverse. The position at each time step is calculated using the equation $z_t = z_{t-1} - \frac{ dU(x_{t-1}) }{dx} D\Delta t + (2D\Delta t)^{1/2} R_t$, where the diffusion coefficient is $D = 1$, the time step is $\Delta t = 0.001$, and $R_t \sim N(0,1)$ is a random number from the standard normal distribution.
As previously noted, [@Zuckerman2002; @Gore2003; @Zuckerman2004; @Minh2008prl] unidirectional sampling leads to significant apparent bias in estimates of $\Delta f_t$ (Fig. \[fig:Ft\]). In addition to the increased bias as the system is driven further from equilibrium, we further observe that the estimated variance also increases. Bidirectional sampling, on the other hand, leads to a significant reduction in bias and variance, [@Minh2008prl] such that free energy estimate is within error bars of the actual free energy. Because $\Delta \hat{f}_t$ represents the estimated free energy difference with respect to $t$, the estimated $\sigma^2 ( \Delta \hat{F}_t )$ increases with $t$, becoming equal to the well-known Bennett acceptance ratio asymptotic variance estimate [@Bennett1976; @Shirts2003] when $t = \tau$.
![\[fig:Ft\] Comparison of estimators for $\Delta f_t$: This figure is similar to Fig. 1 of Ref. [@Minh2008prl], except that error bars are now included and the sample size is halved. The unidirectional estimator (Eq. \[equation:unidirectional-ft\]) is applied to 250 forward (rightward triangles) or reverse (leftward triangles, time reversed) sampled paths, and the bidirectional estimator (Eq. \[equation:bidirectional-ft\]) to 125 paths in each direction (upward triangles). The exact $\Delta f_t$ is shown as a solid line. Error bars (sometimes smaller than the markers) denote one standard deviation of $\Delta \hat f_t$, estimated using the expressions presented here. The vertical dashed lines are at the times considered in Fig. (\[fig:error\_validation\]).](Ft)
Similar trends are observed with the Hummer-Szabo PMF estimates (Fig. \[fig:pmf\]). For unidirectional sampling, the finite-sampling bias and estimated variance increases when the PMF is far from the region sampled by the initial state. With bidirectional sampling, the bias is significantly reduced; the PMF estimate is largely within error bars of the actual PMF.
![\[fig:pmf\] Comparison of PMF estimators: This figure is similar to Fig 2 of Ref. [@Minh2008prl], except that error bars are now included and the sample size is halved. In the left panel, the unidirectional Hummer and Szabo estimator is applied to (a) 250 forward (rightward triangles) or 250 reverse (leftward triangles) sampled paths. In the right panel, the bidirectional estimator is applied to 125 sampled paths in each direction (upward triangles). The exact PMF is shown as a solid line in both panels. Error bars (sometimes smaller than the markers) denote one standard deviation of $\Delta \hat g_0(z)$, estimated using the expressions presented here. The vertical dashed lines are at the positions considered in Fig. (\[fig:error\_validation\]).](pmf)
To analyze these trends more quantitatively, we repeated the experiment 1000 times. For both $\Delta f_t$ and $g_0(z)$, we calculated the bias as $\bar{B}(\bar{\mathcal F}_f) = \frac{1}{S} \sum_{s=1}^{S} (\bar{\mathcal F}_{f,s} - \left< \mathcal F \right>)$ and the standard deviation as $\bar{\sigma}(\bar{\mathcal F}_f) = \sqrt{ \frac{1}{S} \sum_{s=1}^{S} (\bar{\mathcal F}_{f,s} - \left< \mathcal F \right> )^2 }$, where $S = 1000$ is the number of replicates. The results from these more extensive simulations support our described trends (Fig. \[fig:bias\_vs\_variance\]). For unidirectional sampling, the bias in both $\Delta f_t$ and $g_0(x)$ appear to significantly increase around the barrier crossing. In the bidirectional free energy estimate, however, the bias is small relative to the variance at all times. Notably, in the bidirectional PMF estimate, there is a small spike in the bias near the barrier, potentially due to reduced sampling in the region.
![\[fig:bias\_vs\_variance\] Ratio of estimator bias to standard deviation: This ratio is calculated for the (a) free energy and (b) PMF, using 1000 independent estimates. Each estimate is obtained and the type of path sample is indicated as in Figs. (\[fig:Ft\]) and (\[fig:pmf\]). The vertical dashed lines are at the times/positions considered in Fig. (\[fig:error\_validation\]). ](bias_vs_variance)
While in the large sample limit, the bias in the unidirectional estimate is expected to be small compared to the variance, [@Gore2003] our distribution of unidirectional $e^{-\Delta f_t}$ estimates is significantly skewed and does not resemble a Gaussian distribution expected by the central limit theorem (data not shown). Hence, the asymptotic limit has not been reached and the large relative bias is caused by insufficient sampling of rare events with low work values that dominate the exponential average. [@Jarzynski2006] Larger sample sizes would be necessary for the distribution of estimates to be normally distributed and for the error to be dominated by the variance (which we estimate here) rather than the bias.
The accuracy of variance estimates may be assessed by comparing predicted and observed confidence intervals. If the estimates are indeed normally distributed about the true value, about 68% of estimates from many independent replicates of the experiment should be within one standard deviation of the true value, 95% within two, and so forth. Fig. (\[fig:error\_validation\]) compares confidence intervals predicted using the described asymptotic variance estimators and the actual fraction of estimates within the interval.
![\[fig:error\_validation\] Validation of asymptotic variance estimators: Predicted vs. observed fraction of 1000 independent estimates that are within an interval of the true value for (a)-(c) $\Delta f_t$ and (d)-(f) $g_0(z)$ at the indicated times or positions. Each estimate is obtained and the type of path sample is indicated as in Figs. (\[fig:Ft\]) and (\[fig:pmf\]). Error bars on these fractions are 95% confidence intervals calculated using a Bayesian scheme described in Appendix B of Chodera et. al., [@Chodera2007] except that, for numerical reasons, the confidence interval was estimated from the variance of the Beta distribution assuming approximate normality, rather than from the inverse Beta cumulative distribution function. ](error_validation)
We observe that the accuracy of our asymptotic variance estimate in characterizing the confidence interval largely depends on the presence of bias. In the bidirectional $\Delta \hat{f}_t$ estimate, where there is little bias, the asymptotic variance estimate works very well. For the unidirectional $\Delta \hat{f}_t$ estimates, it works well near the initial state but underestimates the error as the system is driven further away from equilibrium, concurring with the bias trend. In the bidirectional PMF estimate, the asymptotic variance estimate accurately describes the confidence interval except near the barrier, where it slightly underestimates the uncertainty, probably due to the small spike in bias.
In the regime where the bias is much smaller than the variance, $\bar B \ll \bar \sigma$, the asymptotic variance estimate provides a good estimate of the actual statistical error in the estimate. This also permits us to model the posterior distribution of quantity being estimated as a multivariate normal distribution with mean $\bar{\mathcal F}$ and covariance $\hat{\bm{\Theta}}$. Doing so provides a route to combining estimates from independent datasets collected from different path ensembles — such as different pulling speeds or from equilibrium and nonequilibrium path ensembles — without knowledge of path probability ratios. This is achieved by maximizing the product of these posterior distributions in a manner similar to the Bayesian approach for estimating $\Delta f_\tau$ described in Ref. [@Maragakis2008].
Acknowledgements
================
We thank Attila Szabo and Zhiqiang Tan for helpful discussions, and Christopher Calderon for useful comments on the manuscript. D.M. thanks Artur Adib for supporting a postdoctoral fellowship. This research was supported by the Intramural Research Program of the NIH, NIDDK.
Closed-form expression for the asymptotic variance, given unidirectional data \[sec:uni-var\]
=============================================================================================
In this appendix, we show that given unidirectional data, the optimal EBS estimate is the sample mean and its variance simplifies to the variance of a sample mean. First, consider the application of the optimal EBS estimator, Eq. \[eq:opt\_bridge\], to estimating a nonequilibrium path-ensemble average from a unidirectional data set, $$\begin{aligned}
\hat{c}_{\mathcal F_f} & = &
\sum_{n=1}^{N_f} \left[ \frac{N_f}{\hat{c}_f} \frac{q_f[X_{fn}]}{q_{\mathcal F_f}[X_{fn}]} \right]^{-1} \\
& = & \sum_{n=1}^{N_f} \frac{ \mathcal F[X_{fn}] \hat{c}_f }{N_f}.\end{aligned}$$ Dividing both sides by $\hat{c}_f$, we obtain the sample mean estimator, $$\begin{aligned}
\bar{\mathcal F}_f = \frac{1}{N_f} \sum_{n=1}^{N_f} \mathcal F[X_{fn}].\end{aligned}$$
We shall now simplify the asymptotic variance estimate by closely following the procedure of Shirts and Chodera. [@Shirts2008] When $\bm{M}$ has full column rank, $\hat{\bm{\Theta}}$ can be written as (Eq. D7 of Ref. [@Shirts2008]), $$\begin{aligned}
\hat{\bm{\Theta}} = [(\bm{M}^{\mathrm{T}}\bm{M})^{-1} - \bm{N} + b \bm{1}_K \bm{1}_K^{\mathrm{T}}]^{-1},\end{aligned}$$ where $b$ is an arbitrary multiplicative factor and $\bm{1}_K$ is a $1~X~K$ matrix of ones.
The weight matrix $\bm{M}$ consists of two columns, $$\begin{aligned}
M_{nf} & = & \frac{ \hat{c}_f^{-1} q_f[X_{fn}] }{ N_f \hat{c}_f^{-1} q_f[X_{fn}] } = \frac{1}{N_f} \\
M_{n \mathcal F_f} & = &
\frac{ \hat{c}_{\mathcal F_f}^{-1} q_{\mathcal F_f}[X_{fn}] }{N_f \hat{c}_f^{-1} q_f[X_{fn}] } =
\frac{\mathcal F[X_{fn}] }{ N_f \bar{\mathcal F}_f },\end{aligned}$$ obtained by applying Eqs. \[eq:weight\_elements\] and \[eq:F\_weight\_elements\]. This leads to, $$\begin{aligned}
\bm{M}^{\mathrm{T}}\bm{M} =
\left[
\begin{array}{cc}
N_f^{-1} & N_f^{-1} \\
N_f^{-1} & \sum_{n=1}^{N_f} M_{n \mathcal F_f}^2
\end{array}
\right]
\equiv
\left[
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}
\right].\end{aligned}$$ The matrix $\bm{M}^{\mathrm{T}}\bm{M}$ has the determinant, $$\begin{aligned}
D = \frac{1}{N_f} \sum_{n=1}^{N_f} M_{n \mathcal F_f}^2 - \frac{1}{N_f^2}.\end{aligned}$$ which allows us to write the inverse covariance matrix as, $$\begin{aligned}
\hat{\bm{\Theta}}^{-1} =
\left[
\begin{array}{cc}
\frac{a_{22}}{D} - N_f + b & -\frac{a_{21}}{D} + b \\
-\frac{a_{12}}{D} + b & \frac{a_{11}}{D} + b
\end{array}
\right].\end{aligned}$$ By applying the same steps as Appendix E of Shirts and Chodera, [@Shirts2008] we then obtain the determinant $$\begin{aligned}
| \hat{\bm{\Theta}}^{-1} | = \frac{4ab}{D},\end{aligned}$$ where $a = a_{12} = a_{21}$. We then obtain the asymptotic covariance estimate, $$\begin{aligned}
\hat{\bm{\Theta}} =
\frac{D}{4ab}
\left[
\begin{array}{cc}
\frac{a_{11}}{D} + b & \frac{a}{D} - b \\
\frac{a}{D} - b & \frac{a_{22}}{D} - N_f + b
\end{array}
\right]\end{aligned}$$
To estimate the variance, we apply Eq. \[eq:path\_average\_variance\], leading to $$\begin{aligned}
\sigma^2(\bar{\mathcal F}_f^2)
& \approx & \bar{\mathcal F}_f^2 (\Theta_{\mathcal F_f \, \mathcal F_f} - 2 \Theta_{\mathcal F_f \, f} + \Theta_{f \, f}) \\
& = & \bar{\mathcal F}_f^2 \left(\sum_{n=1}^{N_f} M_{n \mathcal F_f}^2 - \frac{1}{N_f} \right) \\
& = & \sum_{n=1}^{N_f} \frac{ \mathcal F[X_{fn}]^2 }{N_f^2} - \frac{\bar{\mathcal F}_f^2}{N_f} \\
& = & \frac{1}{N_f} \left[ \frac{1}{N_f} \sum_{n=1}^{N_f} \mathcal F[X_{fn}]^2 -
\left( \frac{1}{N_f} \sum_{n=1}^{N_f} \mathcal F[X_{fn}] \right)^2 \right] \\
& = & \frac{1}{N_f} \left[ \frac{1}{N_f} \sum_{n=1}^{N_f} \left( \mathcal F[X_{fn}] - \bar{\mathcal F}_f \right)^2 \right],\end{aligned}$$ which is the variance of a sample mean estimate.
Closed-form expression for the asymptotic variance, given bidirectional data \[sec:bi-var\]
===========================================================================================
In this appendix, we obtain a closed-form expression for the asymptotic variance of the optimal EBS estimate for $\bar{\mathcal F}_f$, given bidirectional data. We will follow a similar procedure as in Appendix \[sec:uni-var\]. For the bidirectional case, the weight matrix $\bm{M}$ consists of three columns, $\bm{M} = [\bm{m}_f \, \bm{m}_r \, \bm{m}_{\mathcal F_f}]$, where $\bm{m}_i$ is a column matrix of weights from Eqs. \[eq:weight\_elements\] and \[eq:F\_weight\_elements\] corresponding to path-ensemble $i$. The elements of $\bm{M}$ are, $$\begin{aligned}
M_{nf} & = & \frac{ \hat{c}_f^{-1} q_f[X_n] }{ N_f \hat{c}_f^{-1} q_f[X_n] + N_r \hat{c}_r^{-1} q_r[\tilde{X}_n] }
= \frac{ 1 }{ N_f + N_r e^{-\hat{\Omega}[X_n]} }
= N_f^{-1} \epsilon(L_n) \\
M_{nr} & = & \frac{ \hat{c}_r^{-1} q_r[\tilde{X}_{fn}] }{ N_f \hat{c}_f^{-1} q_f[X_n] + N_r \hat{c}_r^{-1} q_r[\tilde{X}_n] }
= \frac{ 1 }{ N_f e^{\hat{\Omega}[X_n] } + N_r }
= N_r^{-1} \epsilon(-L_n) \\
M_{n \mathcal F_f} & = &
\left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) \frac{ 1}{ N_f + N_r e^{-\hat{\Omega}[X_n]} }
= \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) N_f^{-1} \epsilon(L_n),\end{aligned}$$ where $\epsilon$ is defined as the Fermi function, $\epsilon(L_n) = \frac{1}{1+e^{-L_n}}$, and we define $L_n = W[X_n] - \Delta \hat{f}_t + \ln \left( \frac{N_f}{N_r} \right)$. This allows us to write $\bm{M}^{\mathrm{T}}\bm{M}$ as, $$\begin{aligned}
\bm{M}^{\mathrm{T}}\bm{M} & = &
\sum_{n=1}^N
\left[
\begin{array}{ccc}
\frac{1}{N_f^2} \epsilon(L_n)^2 & \frac{1}{N_f N_r} \epsilon(L_n) \epsilon(-L_n)
& \frac{1}{N_f^2} \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) \epsilon(L_n)^2 \\
\frac{1}{N_f N_r} \epsilon(L_n) \epsilon(-L_n) & \frac{1}{N_r^2} \epsilon(-L_n)^2
& \frac{1}{N_f N_r} \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) \epsilon(L_n) \epsilon(-L_n) \\
\frac{1}{N_f^2} \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) \epsilon(L_n)^2
& \frac{1}{N_f N_r} \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right) \epsilon(L_n) \epsilon(-L_n)
& \frac{1}{N_f^2} \left( \frac{\mathcal F[X] }{\bar{\mathcal F}_f} \right)^2 \epsilon(L_n)^2
\end{array}
\right] \nonumber \\
& \equiv &
\left[
\begin{array}{ccc}
a_{ff} & a_{fr} & a_{f \mathcal F_f} \\
a_{fr} & a_{rr} & a_{r \mathcal F_f} \\
a_{\mathcal F_f \mathcal F} & a_{r \mathcal F_f} & a_{\mathcal F_f \mathcal F_f} \\
\end{array}
\right].\end{aligned}$$ Using the determinant, $$\begin{aligned}
D = -a_{\mathcal F_f \mathcal F_f} a_{fr}^2
+ 2 a_{f \mathcal F_f} a_{fr} a_{r \mathcal F_f}
- a_{ff} a_{r \mathcal F_f}^2
- a_{f \mathcal F_f}^2 a_{rr}
+ a_{\mathcal F_f \mathcal F_f} a_{ff} a_{rr},\end{aligned}$$ we write the inverse covariance matrix estimator as, $$\begin{aligned}
\hat{\bm{\Theta}}^{-1} =
\left[
\begin{array}{ccc}
\frac{ -a_{r \mathcal F_f}^2 + a_{\mathcal F_f \mathcal F_f} a_{rr} }{D} - N_f + b
& \frac{ -a_{\mathcal F_f \mathcal F_f} a_{fr} + a_{f \mathcal F_f} a_{r \mathcal F_f} }{D} + b
& \frac{ a_{fr} a_{r \mathcal F_f} - a_{f \mathcal F_f} a_{rr} }{D} + b \\
\frac{ -a_{\mathcal F_f \mathcal F_f} a_{fr} + a_{f \mathcal F_f} a_{r \mathcal F_f} }{D} + b
& \frac{ -a_{f \mathcal F_f}^2 + a_{\mathcal F_f \mathcal F_f} a_{ff} }{D} - N_r + b
& \frac{ a_{f \mathcal F_f} a_{fr} - a_{ff} a_{r\mathcal F_f} }{D} + b \\
\frac{ a_{fr} a_{r \mathcal F_f} - a_{f \mathcal F_f} a_{rr} }{D} + b
& \frac{ a_{f \mathcal F_f} a_{fr} - a_{ff} a_{r\mathcal F_f} }{D} + b
& \frac{ -a_{fr}^2 + a_{ff} a_{rr} }{D} + b
\end{array}
\right].\end{aligned}$$
By applying the same steps as Appendix E of Shirts and Chodera, [@Shirts2008] we obtain the determinant $$\begin{aligned}
| \hat{\bm{\Theta}}^{-1} | = \frac{9b(a_{fr}^2 N_f + a_{fr} a_{rr} N_f) }{D}.\end{aligned}$$
Applying Eq. \[eq:path\_average\_variance\] to $\hat{\bm{\Theta}}$ and simplifying, it can be shown that the variance estimate is, $$\begin{aligned}
\sigma^2(\bar{\mathcal F}_f^2)
& \approx & \bar{\mathcal F}_f^2 (\Theta_{\mathcal F_f \, \mathcal F_f} - 2 \Theta_{\mathcal F_f \, f} + \Theta_{f \, f}) \\
& = &
\frac{ a_{\mathcal F_f \mathcal F_f} a_{fr}
- a_{f \mathcal F_f} a_{fr}
- a_{f \mathcal F_f} a_{r \mathcal F_f}
+ a_{ff} a_{r \mathcal F_f}}
{a_{fr} (a_{fr} N_f + a_{rr} N_r)}\end{aligned}$$
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abstract: 'A new construction is presented of scalar-flat Kähler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that ${\mathbb{CP}}^2$ blown up at $10$ suitably chosen points, admits a scalar-flat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual $4$-manifolds.'
address:
- 'Yann Rollin, MIT, 77 Massachusetts Avenue, Cambridge MA 02139, USA'
- 'Michael Singer, School of Mathematics, King’s buildings, Edinburgh Scotland'
author:
- Yann Rollin
- Michael Singer
date: April 2004
title: 'Non-minimal scalar-flat Kähler surfaces and parabolic stability'
---
****
Introduction
============
The theory of Kähler metrics with constant scalar curvature (CSC) has seen significant progress in the last ten years: some of the obvious highlights are:
- The work of LeBrun and his co-workers [@L2; @L3; @KLP; @LS] which gives many compact examples in complex dimension $2$;
- The work of Donaldson [@D] which shows that for projective varieties with no non-trivial holomorphic vector fields there is at most one metric of constant scalar curvature in any given Kähler class; and that if such a metric [*does*]{} exist, then the underlying polarized variety must be [*stable*]{} in a suitable algebro-geometric sense;
- The work of X.X. Chen and Tian [@CT] which extends Donaldson’s uniqueness result, by different methods, to arbitrary compact Kähler manifolds (and to [*extremal*]{} Kähler metrics).
The conjecture that the existence of CSC Kähler metrics should be related to the algebro-geometric notion of stability seems to go back to Yau, but, despite the work of Donaldson, Chen and Tian, a proof of the ‘obvious conjecture’ (stability $\Rightarrow$ existence) is still lacking. In the absence of a general theorem of this kind, special constructions still have an important role to play.
In this paper we give a new construction of compact [*scalar-flat Kähler*]{} (SFK) surfaces, in other words Kähler metrics on compact complex surfaces having scalar curvature equal to zero. If $M$ is such a surface and $[\omega]$ is the Kähler class, then $c_1\cdot[\omega]=0$, for this is just the integral over $M$ of the scalar curvature. It follows (cf. [@L1; @LS]) that $c_1^2(M) {\leqslant}0$, with equality if and only $c_1(M)=0$ and the metric is Kähler–Einstein. It follows from surface classification that if $c_1^2(M){\leqslant}0$ but $c_1(M)\not=0$, then $M$ is rational or ruled, though not necessarily [*minimal*]{}[^1].
The simplest examples are therefore blow-ups of the complex projective plane ${\mathbb{CP}}^2$. Since $c_1^2({\mathbb{CP}}^2)=9$ and every blow-up reduces $c_1^2$ by $1$, we see that a 10-point blow-up of ${\mathbb{CP}}^2$ is the first such surface that could possibly admit a SFK metric. In [@KLP] it was shown that a $14$-point blow-up of ${\mathbb{CP}}^2$ does admit a SFK metric. Our first result is a sharp improvement of this result, answering a long-standing question [@L1] of LeBrun’s:
\[theop2\] The complex projective plane ${\mathbb{CP}}^2$, blown up at $10$ suitably chosen points, admits a scalar flat Kähler metric. Any further blow-up of the resulting complex surface admits a scalar-flat Kähler metric.
We also obtain new constructions of SFK metrics on ruled surfaces with base an elliptic curve:
\[theot\] Let $\TT$ be a compact Riemann surface of genus 1.
1. Let $L_1$ and $L_2$ be two non-isomorphic holomorphic line bundles of same degree over $\TT$. Then there is a $2$-point blow-up of $\PP(L_1\oplus L_2)$ which admits a scalar-flat Kähler metric.
2. There is a $4$-point blow-up of $\TT\times{\mathbb{CP}}^1$ which admits a scalar-flat Kähler metric.
Any further blow-up of the resulting complex surfaces admit a scalar-flat Kähler metric.
In addition to these specific examples, our construction provides some support for the slogan “stability $\Rightarrow$ existence”. For [*minimal*]{} ruled surfaces (i.e., no blow-ups) the relation between stability and existence was noticed in [@BB]: a ruled surface of the form $\PP(E) \to \Sigma$, where $\Sigma$ is a Riemann surface of genus ${\geqslant}2$ and $E\to
\Sigma$ is a rank-2 holomorphic vector bundle, admits a SFK metric if and only if $E$ is [*polystable*]{}. This result depends on the celebrated theorem of Narasimhan and Seshadri [@NS], which allows to construct the metric on $\PP(E)$ as a quotient of the Riemannian product metric ${\mathbb{CP}}^1\times \HH^2$, where the two factors are equipped with the standard metrics of constant curvature $+1$ and $-1$ respectively.
For non-minimal ruled surfaces, the following result was proved by LeBrun and the second author:
Let $M$ be some blow-up of a compact geometrically ruled surface $\pi:\PP(E)\to \Sigma$. Suppose that $M$ admits a non-zero, periodic holomorphic vector field. Then a Kähler class $[\omega]$ on $M$ with $c_1(M)\cdot [\omega]=0$ contains a representative of zero scalar curvature if and only if the parabolic bundle $E$ is quasi-stable.\[p1.26.11\]
In this statement, the blow-up and Kähler class are encoded by a parabolic $GL_2(\CC)$-structure on $E$ as follows. If the centres of the blow-ups are the points $Q_1,\ldots, Q_k$, then we obtain $k$ marked points $P_j = \pi(Q_j)$ in $\Sigma$ and flags $0 \subset L_j \subset
\pi^{-1}(P_j)$ in the corresponding fibres. The corresponding parabolic weight $(\beta_j,\gamma_j)$ is not uniquely defined, but is chosen to satisfy $$\gamma_j-\beta_j = \frac{\int_{S_j} \omega}{\int_{F}\omega},\;\;
\beta_j,\gamma_j \in ]0,1[$$ where $S_j$ is the exceptional divisor introduced by blowing up $P_j$.
At the end of [@LS] it was conjectured that Proposition \[p1.26.11\] should continue to hold if there is no periodic holomorphic vector field, with “quasi-stable” replaced by “stable”. The methods of this article do not prove this conjecture; instead, we use the parabolic structure to encode an iterated blow-up of $\PP(E)$ and define a “map” of the following kind:
-- --------------- --
$\rightarrow$
-- --------------- --
The reason for the quotation marks is that this “map” is only defined for rational values of the parabolic weights, and will not be smooth in any obvious sense. [*We state again that although the blow-up $\widehat M$ on the right-hand side is encoded by the parabolic structure, its construction is completely different from the one involved in Proposition \[p1.26.11\]*]{}.
By the theorem of Mehta–Seshadri [@MS], there is a correspondence between parabolically stable bundles and representations of the fundamental group of the punctured Riemann surface. We use these representations in the statement of our main theorem:
\[maintheo\] Let $\widehat \Sigma $ be a compact Riemann surface of genus $g$ with a finite set of marked points $\{P_1,P_2,\cdots, P_k\}$ and $\rho:
\pi_1(\widehat \Sigma \setminus \{ P_j\}) \rightarrow {\mathrm{SU}}(2)/\ZZ_2$ be a homomorphism. Assume in addition that
1. if $l_j$ is the homotopy class of a small loop around $P_j$, then $\rho(l_j)$ has finite order $q_j$;
2. $ \displaystyle 2 -2g - \sum_{j=1}^k (1- \frac 1 {q_j}) < 0$;
3. $\rho$ defines an irreducible representation in the sense that the induced action of $\pi_1(\widehat \Sigma \setminus \{ P_j\})$ fixes no point of ${\mathbb{CP}}^1$.
Then there is non-minimal ruled surface ${\widehat{M}}_\rho \to {\widehat{\Sigma}}$ associated canonically to $\rho$, which admits a scalar-flat Kähler metric.
The construction of ${\widehat{M}}_\rho$ will be sketched later in the Introduction and is given in detail in §§2–3.
We shall see in §2 that the blow-ups made in the construction of our SFK ruled surfaces ${\widehat{M}}$ are rather non-generic: they all involve iterated blow-ups (a sequence of blow-ups where each centre lies on the exceptional divisor introduced by the previous blow-up). Thus we get SFK metrics on a rather “thin” set in the moduli space of complex structures on ${\widehat{M}}$. The general problem of existence of a SFK metric in a Kähler class satisfying $c_1\cdot [\omega]=0$ remains mysterious.
It is not clear whether Theorem B is sharp: for example, do there exist SFK metrics on a 1-point blow-up of $\PP(L_1\oplus L_2)\to
\TT$, if $L_1$ and $L_2$ are not isomorphic? Do there exist such metrics on a 3-point blow-up of $\TT \times {\mathbb{CP}}^1$? In this direction, we remark that [@LS Prop. 3.1] shows that there do not exist SFK metrics in the first case if the centre of the blow-up lies on $L_1$ or $L_2$ and in the second case if 2 or fewer points are blown up. (The obstructions come from the non-trivial holomorphic vector field on these spaces.)
In the light of the recent work of Donaldson and Chen–Tian, the following seems a reasonable
Let $E\to \Sigma$ be a parabolic holomorphic bundle of rank $2$ (with rational weights) over a Riemann surface, such that the ruled surface $\PP(E)$ has no non-trivial holomorphic vector field. If the corresponding iterated blow-up $\widehat M$ of $\PP(E)$ (cf. Section \[secitbup\] for a precise definition) admits a scalar-flat Kähler metric, then $E$ must be parabolically stable.
It is ironic that the above conjecture runs in the “easier” direction (existence $\Rightarrow$ stability) and yet we are unable to prove it; the new numerical criterion for stability due to Ross and Thomas [@RT] should be useful here, but so far we have not succeeded in applying it.
Another possible approach might be to apply the Tian–Viaclovsky compactness theorem [@TV] which shows that under certain conditions, a sequence of SFK metrics can only degenerate to a SFK orbifold metric. (Our gluing theorem, Theorem \[theoglue\], gives explicit examples of this degeneration process.)
Outline {#outline .unnumbered}
-------
This work began from the observation that in the Burns–de Bartolomeis construction, the smooth base $\Sigma$ can be replaced by an orbifold Riemann surface ${\overline{\Sigma}}$. More precisely, with the notation of Theorem \[maintheo\], ${\overline{\Sigma}}$ is the smooth Riemann surface $\Sigma$ with a finite set of marked points $P_j$ and corresponding integer weights $q_j$. By a theorem of Troyanov [@Tr], if the orbifold Euler characteristic is negative (this is condition (ii) of Theorem \[maintheo\]), then ${\overline{\Sigma}}$ carries a (Kähler) orbifold metric $\overline{g}$ of constant curvature $-1$. In particular, $\overline{g}$ is smooth on ${\overline{\Sigma}}\setminus\{P_j\}$ and has a conical singularity at $P_j$, with cone angle $2\pi/q_j$. The riemannian product ${\overline{\Sigma}}\times
{\mathbb{CP}}^1$ is obviously scalar-flat Kähler, with non-isolated orbifold singularities around the fibres $F_j = \{P_j\}\times {\mathbb{CP}}^1$.
In order to replace these by isolated singularities, we twist by a representation $\rho$ of the orbifold fundamental group of ${\overline{\Sigma}}$, just as was done in the smooth case by Burns and de Bartolomeis. This $\rho$ must be as in condition (i) of Theorem \[maintheo\]: using it gives an orbifold SFK metric on an orbifold ruled surface, ${\overline{M}}$, say. It turns out that ${\overline{M}}$ has precisely two isolated cyclic singularities in each fibre $F_j$. Denote by ${\widehat{M}}$ the minimal resolution of singularities of ${\overline{M}}$.
In §§4–5, it will be shown that ${\widehat{M}}$ has a SFK metric, by an analytical gluing theorem. On the other hand, we shall see in §§2–3 that ${\widehat{M}}$ is also a multiple blow-up of a smooth minimal ruled surface ${\check{M}}= \PP(E)\to {\widehat{\Sigma}}$, say. In fact, ${\check{M}}$ and ${\overline{M}}$ can be viewed as two different compactifications (one smooth, the other an orbifold) of a non-compact ruled surface $M^* \to
\Sigma\setminus\{P_j\}$, each canonically associated to the representation $\rho$. Such representations are related by the Mehta–Seshadri Theorem [@MS] to parabolic stability of the underlying holomorphic vector bundle. In §2 we shall start from this point, defining a notion of parabolically stable ruled surface ${\check{M}}$ and the corresponding multiple blow-up ${\widehat{M}}$. In §3, we shall compare this with the orbifold ${\overline{M}}$.
In §2, we shall restate Theorem \[maintheo\] in the language of stable ruled surfaces and show how Theorems A and B follow. The advantage of working with parabolically stable bundles is that it is often quite easy to verify stability, whereas it can be rather difficult to find explicit representations of the fundamental group of a punctured Riemann surface.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Olivier Biquard for useful conversations and for pointing out a mistake in the proof of the gluing theorem for scalar-flat Kähler metrics in [@KS]. This problem in now fixed in section \[secgluing\] where a self-contained exposition of the result can be found. We also thank Claude LeBrun and Rafe Mazzeo for encouragement and several useful discussions. Finally, we thank the anonymous referee for several suggestions which improved the original manuscript. This work was carried out while the second author was visiting the Mathematics Department at MIT; he thanks MIT for its hospitality and financial support during this visit.
Parabolically stable ruled surfaces {#secparab}
===================================
A [*geometrically ruled surface*]{} $\check M$ is by definition a minimal complex surface obtained as $\check M= \PP(E)$, where $E\rightarrow \widehat \Sigma
$ is a holomorphic vector bundle of rank $2$ over a Riemann surface $\widehat \Sigma$. The induced map $\pi:\check M \rightarrow \widehat \Sigma$ is called the [*ruling*]{}.
A parabolic structure on $\check M$ consists of the following data:
- A finite set of distinct points $P_1,P_2,\cdots,P_n$ in $\widehat \Sigma$;
- for each $j$, a choice of point $Q_j \in F_j = \pi^{-1}(P_j)$;
- for each $j$, a choice of weight $\alpha_j \in ]0,1[\cap\QQ$.
A geometrically ruled surface with a parabolic structure will be called a [*parabolic ruled surface*]{}.
If $S\subset\check M$ is a holomorphic section of $\pi$, we define its slope $$\mu(S) = S^2 +\sum_{Q_j\not\in S}\alpha_j - \sum_{Q_j\in S}\alpha_j;$$ we say that a parabolic ruled surface is *stable* if for every holomorphic section $S$, we have $\mu(S) >0$.
If we return to the vector bundle $E$, then $Q_j$ defines a line $L_j$ in the fibre of $E$ over $P_j$. For each $j$, select $0{\leqslant}\beta_j < \gamma_j < 1$ with $\alpha_j = \gamma_j- \beta_j$. In this way $E$ is endowed with (a family of) parabolic structures. Our notion of stability of a parabolic ruled surface corresponds with the Mehta–Seshadri notion of parabolic stability for $E$. Indeed, holomorphic cross-sections $S\subset{\check{M}}$ correspond exactly to holomorphic sub-bundles $L\subset
E$. We know that $H^2(\PP(E),\ZZ)$ is generated by the class of a fibre $F$ and $H = c_1(\cO(1))$ (the fibrewise hyperplane section bundle) on each fibre (cf. for instance [@Beau]). They verify $$H^2=\deg(E),\quad F^2=0,\quad F\cdot H=1.$$ Moreover, we have $$S = H - \deg(L) F$$ hence $$S^2 = \deg(E)- 2\deg(L).$$ It follows that $\mu(S)$ is equal to twice the difference of the parabolic slopes of $E$ and $L$ in the sense of Mehta–Seshadri [@MS], so $\check M$ is stable if and only if $E$ is parabolically stable.
Iterated blow-up of a parabolic ruled surface {#secitbup}
---------------------------------------------
Let ${\check{M}}$ be a parabolic ruled surface. We shall now define a multiple blow-up $\Phi:{\widehat{M}}\to {\check{M}}$ which is canonically determined by the parabolic structure of ${\check{M}}$.
In order to simplify the notation, suppose that the parabolic structure on $\check M$ is reduced to a single point $P\in \widehat
\Sigma$; let $Q$ be the corresponding point in $F = \pi^{-1}(P)$ and let $\alpha = \frac pq$ be the weight, where $p$ and $q$ are two coprime integers, $0<p<q$. Denote the Hirzebruch–Jung continued fraction expansion of $\alpha$ by $$\label{e1.844}
\frac pq = \cfrac{1}{e_1-\cfrac{1}{e_2-\cdots\cfrac{1}{e_k}}};$$ define also $$\label{e20.844}
\frac {q-p}{q} = \cfrac{1}{e'_1-\cfrac{1}{e'_2-\cdots
\cfrac{1}{e'_l}}}.$$ These expansions are unique if, as we shall assume, the $e_j$ and $e'_j$ are all ${\geqslant}2$.
There exists a unique iterated blow-up $\Phi:{\widehat{M}}\to{\check{M}}$ with $\Phi^{-1}(F)$ equal to the following chain of curves: $$\xymatrix{
{}\ar@{-}[r]^{-e_1} & *+[o][F-]{}
\ar@{-}[r]^{ -e_{2}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-e_{k-1}} & *+[o][F-]{}
\ar@{-}[r]^{-e_k} & *+[o][F-]{}
\ar@{-}[r]^{-1} & *+[o][F-]{}
\ar@{-}[r]^{-e'_{l}} & *+[o][F-]{}
\ar@{-}[r]^{-e'_{l-1}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-e'_{2}} & *+[o][F-]{}
\ar@{-}[r]^{ -e'_{1}} &
}.$$
Here the edges represent rational curves, the number above each edge is the self-intersection of the curve, the hollow dots represent transverse intersections with intersection number $+1$, and the curve of self-intersection $-e_1$ is the proper transform of the exceptional divisor of the first blow-up. \[p1.16.4.4\]
We give an iterative construction. The first step is to blow up $Q$, to get a diagram of the form $$\label{e2.844}
\xymatrix{
{}\ar@{-}[rr]^{-1} && *+[o][F-]{}
\ar@{-}[rr]^{-1} &&{}
}$$ By blowing up the intersection point of these two curves we get the diagram $$\label{e3.844}
\xymatrix{
{}\ar@{-}[rr]^{-2} && *+[o][F-]{}
\ar@{-}[rr]^{-1} && *+[o][F]{}
\ar@{-}[rr]^{-2} &&{}
}$$ in which we see two $(-2)$-curves separated by a $-1$ curve. Suppose by induction that we have used a sequence of blow-ups so that the following chain of curves sits over $F$: $$\label{e4.844}
\xymatrix{
{}\ar@{-}[r]^{-a_1} & *+[o][F-]{}
\ar@{-}[r]^{ -a_{2}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-a_{j-1}} & *+[o][F-]{}
\ar@{-}[r]^{-a_j} & *+[o][F-]_{A}{}
\ar@{-}[r]^{-1} & *+[o][F-]_{B}{}
\ar@{-}[r]^{-b_{r}} & *+[o][F-]{}
\ar@{-}[r]^{-b_{r-1}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-b_{2}} & *+[o][F-]{}
\ar@{-}[r]^{ -b_{1}} &
}$$
for integers $a_i$ and $b_i$ ${\geqslant}2$. Then we can increase $a_j$ by one unit by blowing up the point marked $A$ and we can introduce a new curve of self-intersection $-2$ by blowing up $B$: $$\label{e5.844}
\xymatrix{
{}\ar@{-}[r]^{-a_1} & *+[o][F-]{}
\ar@{-}[r]^{ -a_{2}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-a_{j-1}} & *+[o][F-]{}
\ar@{-}[r]^{-a_j-1} & *+[o][F-]{}
\ar@{-}[r]^{-1} & *+[o][F-]{}
\ar@{-}[r]^{-2} & *+[o][F-]{}
\ar@{-}[r]^{-b_{r}} & *+[o][F-]{}
\ar@{-}[r]^{-b_{r-1}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-b_{2}} & *+[o][F-]{}
\ar@{-}[r]^{ -b_{1}} &
}$$
or
$$\label{e6.844}
\xymatrix{
{}\ar@{-}[r]^{-a_1} & *+[o][F-]{}
\ar@{-}[r]^{ -a_{2}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-a_{j-1}} & *+[o][F-]{}
\ar@{-}[r]^{-a_j} & *+[o][F-]{}
\ar@{-}[r]^{-2} & *+[o][F-]{}
\ar@{-}[r]^{-1} & *+[o][F-]{}
\ar@{-}[r]^{-b_{r}-1} & *+[o][F-]{}
\ar@{-}[r]^{-b_{r-1}} & *+[o][F-]{}
\ar@{--}[r] & *+[o][F-]{}
\ar@{-}[r]^{-b_{2}} & *+[o][F-]{}
\ar@{-}[r]^{ -b_{1}} &
}$$
It is now clear that a suitable sequence of $\sum(e_j -1)$ blow-ups, starting from the configuration yields the required configuration of curves. All that remains is the proof that the integers $e'_j$ in do satisfy . For this we use the fact if $$\label{e15.844}
\frac{\lambda}{\mu} =
\cfrac{1}{a_1-\cfrac{1}{a_2- \cdots \frac{1}{a_j}}}$$ then the reversed continued fraction $$\label{e16.844}
\cfrac{1}{a_j-\cfrac{1}{a_{j-1}- \cdots \frac{1}{a_1}}}
= \frac{\lambda'}{\mu}$$ where $\lambda \lambda' \equiv 1 \pmod \mu$, $0< \lambda' <
\mu$. Suppose by induction that in we have $$\label{e7.844}
\cfrac{1}{a_1-\cfrac{1}{a_2- \cdots \frac{1}{a_j}}} +
\cfrac{1}{b_1-\cfrac{1}{b_2- \cdots \frac{1}{b_r}}} = 1$$ Since $(\mu - \lambda')(\mu - \lambda)$ is congruent to 1 mod $\mu$, implies that $$\label{e8.844}
\cfrac{1}{a_j-\cfrac{1}{a_{j-1}- \cdots \frac{1}{a_1}}} +
\cfrac{1}{b_r-\cfrac{1}{b_{r-1}- \cdots \frac{1}{b_1}}} = 1$$ Now consider how these fractions change under the blow-up. We have that $a_j$ is replaced by $a_j+1$, and a new $b_{r+1}$ is introduced, equal to $2$. So the inductive step is completed by the trivial identity $$\frac{1}{1 + \mu/\lambda'} + \frac{1}{2- (1-\lambda'/\mu)} = 1.$$ Since the induction clearly starts (consider the diagram ), the proof is now complete.
Our main theorem may now be stated as follows:
\[maintheoparab\] Let $\check M\rightarrow \widehat \Sigma$ be a parabolically stable ruled surface. Suppose that $$\label{eqhyp}
\chi(\widehat \Sigma) - \sum_{j=1}^k(1 - \frac 1{q_j}) <0,$$ where $\alpha_j = p_j/q_j$, with $p_j$ and $q_j$ coprime. Then the iterated blow-up $\widehat M$ deduced from the parabolic structure carries a SFK metric. Furthermore, any blow-up of ${\widehat{M}}$ also carries a SFK metric.
The relationship between this statement and Theorem \[maintheo\] corresponds to the relationship between representations and parabolically stable bundles, given by the theorem of Mehta and Seshadri [@MS]. According to this theorem, given a parabolically stable ${\check{M}}$, there is a corresponding irreducible representation $$\label{e1p.16.4.4}
\rho' :\pi_1({\widehat{\Sigma}}\setminus \{P_j\}) \to U_2$$ with $$\label{e2p.16.4.4}
\rho'(l_j) \mbox{ conjugate to }\pm \mbox{diag}(e^{2 \pi i\beta_j},
e^{2\pi i\gamma_j})$$ where $l_j$ is the homotopy class of a small loop around $P_j$ and $\gamma_j-\beta_j= \alpha_j$ as before. This representation defines a parabolically stable bundle $E$ with $\PP(E) = {\check{M}}$. We note that $E$ is determined only up to tensoring with a line-bundle; we make use of this freedom (and the freedom of choice in $\beta_j$, $\gamma_j$) to ensure that the parabolic degree of $E$ is zero, as is necessary for the existence of $\rho'$.
If $A\in U_2$, write $A = z(A)A_0$, where $z(A)$ is a scalar multiple of the identity and $A_0\in SU_2/\ZZ_2$. Letting $\rho(\gamma) = \rho'(\gamma)_0$, we obtain an irreducible representation $$\label{e1.16.4.4}
\rho :\pi_1({\widehat{\Sigma}}\setminus \{P_j\}) \to SU_2/\ZZ_2$$ with $$\label{e2.16.4.4}
\rho(l_j) \mbox{ conjugate to }\pm \mbox{diag}(e^{\pi i\alpha_j},
e^{-\pi i\alpha_j})$$ as required for Theorem \[maintheo\].
Conversely, given a representation with properties and , we clearly get a ruled surface $M^* \to {\widehat{\Sigma}}\setminus\{P_j\}$. However, $M^*$ can now be compactified to give ${\check{M}}$ in a standard way: let $P$ stand for one of the punctures, and identify a neighbourhood of $P$ in ${\widehat{\Sigma}}$ with the unit disc $\Delta$. Write $L^*= \pi^{-1}(\Delta\setminus\{0\})$ and note that $$\label{e30.844}
L^* = \HH^2\times {\mathbb{CP}}^1/ \ZZ$$ where $\HH^2$ is the hyperbolic plane $\{{\mathrm{Im}}(\xi) > 0\}$ and the $\ZZ$-action is generated by $$\label{e31.8.44}
(\xi,[w_0:w_1]) \mapsto (\xi + 2\pi, [e^{\pi i \alpha}w_0,
e^{-\pi i \alpha}w_1]).$$ Let ${\check{L}}= \Delta \times {\mathbb{CP}}^1$ be covered by two standard holomorphic coordinate charts, $(x_1,y_1)\in \CC^2$, $(x_2,y_2)\in
\CC^2$ glued together by $x_2= x_1$, $y_2 = y_1^{-1}$. Then the natural embedding of $L^*$ in ${\check{L}}$ is given by the following map $$\label{e31.844}
x_1 = x_2 =e^{i\xi},\; y_1 = e^{-i\xi \alpha}w_0/w_1,\;
y_2 = e^{i\xi \alpha}w_1/w_0.$$ Notice that the pre-image of $Q$, which corresponds to $(x_1,y_1) =
(0,0)$ is given by $\xi = +i\infty$, $w_0=0$, but every other point corresponding $\xi= i\infty$ is mapped to $(x_2,y_2) = (0,0)$.
In general, we obtain ${\check{M}}$ as a smooth compactification of $M^*$ by following this recipe at each puncture.
Let us show how Theorem \[theop2\] and Theorem \[theot\] follow from Theorem \[maintheoparab\].
There is a $9$-point iterated blow-up of ${\mathbb{CP}}^1\times{\mathbb{CP}}^1$ which admits a SFK metric. As a consequence ${\mathbb{CP}}^2$ has a $10$-point iterated blow-up which admits a scalar-flat Kähler metric.
Let ${\check{M}}= {\mathbb{CP}}^1\times {\mathbb{CP}}^1$, and let $\pi:{\check{M}}\rightarrow {\mathbb{CP}}^1$ denote projection on the first factor. Pick any $4$ points $P_1,P_2,P_3,P_4$ in ${\mathbb{CP}}^1$, with weights $\alpha_1=\alpha_2= \alpha_3 = 1/2$ and $\alpha_4=1/3$, so that (\[eqhyp\]) is satisfied. Pick $Q_j \in \pi^{-1}(P_j)$. To check when this parabolic structure is stable, note first that any section $S$ is in this case the graph of a meromorphic function $f$, and $S^2=2\deg f$. Provided that no two of the $Q_j$ lie on the graph of a function of degree $0$, we have a parabolically stable ruled surface. In particular, the set of stable configurations of the $Q_j$ is a Zariski open set in the set of all such configurations.
The multiple blow-up ${\widehat{M}}$ of ${\check{M}}$ involves a total of 9 blow-ups, 2 each at $Q_1$, $Q_2$ and $Q_3$ and $3$ at $Q_4$. Hence ${\widehat{M}}$ is indeed a $9$-point blow-up of ${\mathbb{CP}}^1\times{\mathbb{CP}}^1$. However, if $n{\geqslant}1$, then any $n$-point blow-up of ${\mathbb{CP}}^1\times{\mathbb{CP}}^1$ is isomorphic to an $(n+1)$-point blow-up of ${\mathbb{CP}}^2$.
Now we turn to ruled surfaces with base an elliptic curve.
\[cortt\] Let $\TT$ be an elliptic curve (Riemann surface of genus 1) and let $L_1$ and $L_2$ be two non-isomorphic line-bundles of degree $0$ over $\TT$. Then, any double blow-up $\widehat M$ of ${\check{M}}=\PP(L_1\oplus L_2)$ at a point which is not on $\PP(L_1)$ or $\PP(L_2)$ admits a SFK metric. Any further blow-up of $\widehat
M$ also admits a SFK metric.
Endow ${\check{M}}$ with a parabolic structure by picking an arbitrary point $P\in \TT$ and a point $Q \in \pi^{-1}(P)$ so that $Q$ does not lie on $S_1=\PP(L_1 \oplus 0)$ or $S_2=\PP(0\oplus
L_2)$. Set the weight of $Q$ equal to $1/2$. Then condition (\[eqhyp\]) is satisfied and it remains to check that this parabolic structure is stable.
To see this, note first that every section $S$ of $\pi$ satisfies $S^2{\geqslant}0$; this follows because at the level of cohomology, $[S] = [S_1] + r[F]$ for some integer $r$. Then $S^2=0$ if $r=0$; otherwise $S\not= S_1$, so $r =S\cdot S_1 {\geqslant}0$ Hence $S^2
= 2r {\geqslant}0$.
Therefore, $\mu(S)>0$ unless $S^2=0$ and $Q\in S$. Suppose now that $S^2=0$, $S\neq S_1$ and $S\neq S_2$ . It follows that $S\cdot S_1= S\cdot S_2=0$, and $S$ meets neither $S_1$ nor $S_2$. Such an $S$ defines an isomorphism $L_1 \simeq L_2$ contradicting the hypothesis of the corollary.
The corresponding 2-point blow-up ${\widehat{M}}$ of ${\check{M}}$ carries a scalar-flat Kähler metric.
In the next corollary, we recover a result of Kim, LeBrun and Pontecorvo [@KLP] by a completely different method.
\[corklp\] Let $\TT$ be an elliptic Riemann surface. There is a $6$-point blow-up of $\TT\times {\mathbb{CP}}^1$ which admits a SFK metric.
Let ${\check{M}}= \TT\times {\mathbb{CP}}^1$, let $\pi$ be the projection on the first factor. Endow ${\check{M}}$ with a parabolic structure by choosing any $3$ points $P_j$ in $\TT$, points $Q_j \in
\pi^{-1}(P_j)$ each of weight $1/2$. We note that the condition is satisfied as before: it remains to analyze the stability of this parabolic structure.
Any holomorphic section $S$ of $\pi$ corresponds to the graph of a meromorphic function $s$ on $\TT$. Let $H$ be the section corresponding to the constant function $0$. Then $S\cdot F=1$, and $S\cdot H =\sum a_n$, where the $a_n{\geqslant}0$ are the multiplicities of the zeroes of $s$; hence $$S= H + (\sum a_n) F.$$
If $S^2=2 \sum a_n =0$, then $a_n=0$ and $s$ must be constant. If we choose the points $Q_j$ such that no two of them lie on the graph of a constant meromorphic function, we must have $\mu(S) {\geqslant}0 + 2/2> 0$. If $S^2>0$, we have $S^2{\geqslant}2$ for $S^2$ is even and then $\mu(S){\geqslant}2-3/2>0$ whatever the positions of the $Q_j$.
Applying Theorem \[maintheoparab\], there exists a SFK metric on ${\widehat{M}}$, which is in this case a $6$-point blow-up of ${\check{M}}$; more precisely ${\check{M}}$ is obtained by performing a double blow-up at each of $Q_1$, $Q_2$ and $Q_3$.
We now improve the last result.
Let $\TT$ be an elliptic curve. There is a $4$-point blow-up of $\TT\times {\mathbb{CP}}^1$ which admits a scalar-flat Kähler metric.
Let $P_1$ and $P_2$ be two points in $\TT$. Let $S$ be a constant section which is not the zero section $S_0$ or the section at infinity $S_\infty$, and $M'$ be the blow-up of $\TT\times{\mathbb{CP}}^1$ at $\{P_1\}\times\{0\}$ and $\{P_2\}\times\{\infty\}$. In $M'$, the proper transform of $\{P_j\}\times {\mathbb{CP}}^1$ is a $(-1)$-curve; we blow each of these curves down, getting a new minimal ruled surface ${\check{M}}$, giving the following diagram $$\xymatrix{& M'\ar[dl]_\phi \ar[dr]^{\psi} & \\
{\check{M}}\ar[dr]& & \TT\times{\mathbb{CP}}^1 \ar[dl] \\
&\TT & }$$ Here $\phi$ is a blow-up map with centres at $C_1$ and $C_2$, say, and $\phi^{-1}(C_j)$ is the $\psi$-proper transform of $\{P_j\}\times {\mathbb{CP}}^1$.
We shall show that ${\check{M}}= \PP(L_1\oplus L_2)$ satisfies the conditions of Corollary \[cortt\]. Then the double blow-up $\Phi:{\widehat{M}}\to {\check{M}}$ at some point $Q$ is scalar-flat Kähler, and so is any further blow-up of ${\widehat{M}}$. In particular, the blow-up $\widehat{\phi}:{\widehat{M}}'\to {\widehat{M}}$ of ${\widehat{M}}$ at $\Phi^{-1}(C_1)$ and $\Phi^{-1}(C_2)$ is scalar-flat Kähler. Taking $Q$ to be different from $C_1$ and $C_2$, we see that ${\widehat{M}}'$ is also the double blow-up at $\phi^{-1}(Q)$ of $M'$, as in the following diagram: $$\xymatrix{& {\widehat{M}}'\ar[dl]_{\widehat{\phi}} \ar[dr]^{\Phi'} & \\
{\widehat{M}}\ar[dr]^{\Phi}& & M' \ar[dl]_\phi \\
&{\check{M}}& }$$ In particular, ${\widehat{M}}'$ is a $4$-point blow-up of $\TT\times {\mathbb{CP}}^1$.
Let us check that ${\check{M}}$ satisfies the hypotheses of Corollary \[cortt\]. Suppose not. Let $S', S'_0, S'_\infty$ be the images by $\phi$ in ${\check{M}}$ of the proper transforms of the corresponding curves $S$, $S_1$ and $S_\infty$ in $\TT\times {\mathbb{CP}}^1$. We have $$(S_0')^2= (S'_\infty)^2= S_0'\cdot S_\infty' =0,\quad S_0'\cdot S'
= S_\infty'\cdot S'=1,\quad (S')^2 = 2.$$ This shows that $M =\PP(L_0\oplus L_\infty)$, where $L_0$ and $L_\infty$ are two line bundles of degree $0$ over $\TT$.
Suppose that $M\simeq \TT\times {\mathbb{CP}}^1$. Then $S_0'$ and $S_\infty'$ must be two distinct constant sections of $\TT\times {\mathbb{CP}}^1\to\TT$ as seen in the proof of Corollary \[corklp\]. Up to an ismorphism of ${\mathbb{CP}}^1$, we may assume that they are the $0$ section and the section at infinity of $\TT\times{\mathbb{CP}}^1$. Now $S'$ is the graph of a meromorphic function $s'$ on $\TT$. Let $z_j$ be the zeroes of $s'$ and $w_j$ its poles. The divisor of $s'$ is $$(s')= \sum a_j z_j -
\sum b_j w_j,$$ where $a_j, b_j{\geqslant}0$ are the multiplicities. Then, $S'\cdot S'_0= \sum a_j=1$ hence there is a unique pole with multiplicity $1$. Therefore $$(s')= z-w,$$ which is impossible by Abel’s theorem.
It is a general principle in algebraic geometry that “stability is an open condition”. We have seen in these examples that the set of stable parabolic structures is Zariski dense in the set of all such structures. This will be true in general: if stable parabolic structures exist on a given ruled surface, then they will form a Zariski-open subset in the set of all such structures. Furthermore, stability will be preserved under perturbation of the parabolic weights, keeping the $Q_j$ fixed. However the iterated blow-up ${\widehat{M}}$ will behave rather wildly under such perturbations: for example, $b_2({\widehat{M}})$ will not remain constant.
Scalar-flat Kähler orbifolds
============================
The goal of this section is to introduce an orbifold ${\overline{M}}$ associated to a parabolic ruled surface ${\check{M}}$. We shall see that ${\check{M}}$ and ${\overline{M}}$ are bimeromorphic to each other, and indeed both are compactifications of the punctured ruled surface $M^*$. We shall also show that ${\overline{M}}$ carries an orbifold SFK metric if ${\check{M}}$ is stable, and that ${\overline{M}}$ carries no non-trivial holomorphic vector fields.
Generalities
------------
A complex [*orbifold*]{} $X$ of complex dimension $n$ may be defined as a complex variety having only quotient singularities. More explicitly, for every point $P\in X$, one requires that there is a finite group $G= G_P$ (called the [*local isotropy group*]{}) and a [*local uniformizing chart*]{} $$\label{e1.10.12.03}
{\widetilde{U}}\to {\widetilde{U}}/G \stackrel{\phi}{\longrightarrow} U.$$ Here, ${\widetilde{U}}$ is a neighbourhood of $0$ in $\CC^n$, with a given biholomorphic action of $G$, $$\label{e1.23.12.03}
G\times {\widetilde{U}}\to {\widetilde{U}},$$ $U$ is a neighbourhood of $P$ in $X$ and $\phi$ is a homeomorphism with $\phi(0) = P$. It is a fact that one can always choose $\phi$ so as to linearize the action ; that is, one can assume $G\subset GL_n(\CC)$.
The standard notions of differential geometry extend to orbifolds by working $G$-equivariantly in a local uniformizing chart. For example a smooth orbifold Riemannian metric $g$ on $X$ is defined as usual away from the singular points, and is given by a $G_P$-invariant smooth metric on ${\widetilde{U}}$ near the singular point $P$.
Orbifold Riemann surfaces
-------------------------
A compact orbifold Riemann surface ${\overline{\Sigma}}$ can be identified with a smooth compact Riemann surface ${\widehat{\Sigma}}$ together with a finite set of marked points $P_j$, each with a given weight $q_j\in \ZZ_{{\geqslant}2}$. It is important to note that a smooth orbifold metric on ${\overline{\Sigma}}$ is not the same as a smooth metric on ${\widehat{\Sigma}}$: a smooth orbifold metric is smooth on $\Sigma = {\overline{\Sigma}}\setminus\{P_j\}$ but with respect to such a metric, the length of a small circle of radius $r$ centred at $P_j$, will be approximately $2 \pi r/q_j$.
The Euler characteristic of ${\overline{\Sigma}}$ is defined as follows $$\label{e2.10.12.03}
\chi({\overline{\Sigma}}) = \chi({\widehat{\Sigma}}) - \sum \left(1 - \frac{1}{q_j}\right),$$ where $\chi({\widehat{\Sigma}})$ is the Euler characteristic of the underlying smooth surface ${\widehat{\Sigma}}$.
Just as for smooth Riemann surfaces with negative Euler characteristic, we have the following result of Troyanov [@Tr] (see also [@McO]).
The orbifold Riemann surface ${\overline{\Sigma}}$ admits an orbifold metric of constant curvature $-1$ compatible with the given complex structure if and only if $\chi({\overline{\Sigma}}) <0$.\[theotr\]
In view of this theorem we shall refer to such orbifold Riemann surfaces as [*hyperbolic*]{}. In the previous section we saw several examples of orbifold hyperbolic Riemann surfaces.
Next we come to the fundamental group of an orbifold Riemann surface. Recall first the description of the fundamental group of the punctured Riemann surface $\Sigma$: $$\label{e3.10.12.03}
\pi_1(\Sigma)= \langle a_1,b_1,\ldots,a_g,b_g,l_1,\ldots l_k:
[a_1,b_1][a_2,b_2]\ldots[a_g,b_g]l_1\ldots l_k = 1\rangle$$ Here the $a_j$ and $b_j$ are standard generators of $\pi_1({\widehat{\Sigma}})$ and $l_j$ is (the homotopy class of) a small loop around $P_j$. The orbifold fundamental group is defined by imposing the additional conditions $$\label{e4.10.12.03}
\pi^{orb}_1({\overline{\Sigma}}) = \langle
a_1,b_1,\ldots,a_g,b_g,l_1,\ldots l_k:
[a_1,b_1][a_2,b_2]\ldots[a_g,b_g]l_1\ldots l_k = l_1^{q_1}
= \ldots = l_k^{q_k} = 1
\rangle$$ From Theorem \[theotr\] it follows that for any orbifold hyperbolic Riemann surface ${\overline{\Sigma}}$, we have ${\overline{\Sigma}}= \HH^2/\Gamma$, where $\Gamma$ is the image of the uniformizing representation of $\pi_1^{orb}({\overline{\Sigma}})$ in $SL_2(\RR) = {\rm Isom}(\HH^2)$. The only difference from the smooth case is that $\Gamma$ will not act freely on $\HH^2$.
Orbifold ruled surfaces
-----------------------
Let ${\overline{\Sigma}}$ be a hyperbolic orbifold Riemann surface as in the last section, and let $\rho: \pi_1^{orb}({\overline{\Sigma}}) \to PSL_2(\CC)$ be a representation. We suppose that $\rho$ is faithful on the loops $l_j$, so that $\rho(l_j)$ has order precisely $q_j$. We can form the quotient $$\label{e1.154}
{\overline{M}}= \HH^2\times {\mathbb{CP}}^1/\pi_1^{orb}({\overline{\Sigma}})$$ by letting $\pi_1^{orb}({\overline{\Sigma}})$ act by the uniformizing representation on the upper half-space $\HH^2$ and by $\rho$ on ${\mathbb{CP}}^1$. It is clear that ${\overline{M}}$ is a complex orbifold ruled surface equipped with a ruling $${\overline{\pi}}: {\overline{M}}\to {\overline{\Sigma}}$$ and singularities only in the fibres ${\overline{F}}_j={\overline{\pi}}^{-1}(P_j)$.
In order to analyze these singularities, choose a complex disc $\Delta$ with centre at one of the $P_j$, let ${\overline{L}}$ denote ${\overline{\pi}}^{-1}(\Delta)$ and let $L^* = {\overline{L}}\setminus {\overline{\pi}}^{-1}(0)$. Complex analytically, we can identify $L^*$ with a quotient as before (see and ). (We we do not use the cusp-metric structure on $\Delta\setminus\{0\}$ which comes from this identification.)
Then the $q$-fold cover $\tilde{L}^*$ of $L^*$ is given by $$\tilde{L}^* = \HH^2\times {\mathbb{CP}}^1/ q\ZZ.$$ Since $\alpha = p/q$, the action here is trivial on ${\mathbb{CP}}^1$, and so we can introduce coordinates $$\label{e32.844}
(u_1,v_1) = (e^{i\xi/q}, w_0/w_1),\;\;(u_2,v_2) = (e^{i\xi/q},
w_1/w_0), u_1=u_2,\; v_1 = v_2^{-1}.$$ If $\omega = e^{2\pi i/q}$, then the action of $\ZZ/q\ZZ$ on $\tilde{L}^*$ is given by the standard action of $\Gamma_{p,q}$ on the coordinates $(u_1,v_1)$ and by $\Gamma_{q-p,q}$ on $(u_2,v_2)$. Here we have written $\Gamma_{r,s}$ for the cyclic subgroup $$\label{e5.10.12.03}
\Gamma_{r,s} = \left\{\begin{pmatrix} e^{2\pi in/s} & 0\cr 0&
e^{2\pi i r n/s}\end{pmatrix}: n=0,1,\ldots,s-1\right\}.$$ of $U_2$. Thus in the orbifold ${\overline{L}}$, there are two singularities in the fibre $u_1=u_2=0$.
Denote by $\Psi: {\widehat{M}}\to {\overline{M}}$ the minimal resolution of singularities. This is obtained by replacing each $\Gamma_{p,q}$-singularity of ${\overline{M}}$ by the corresponding Hirzebruch–Jung string (cf. [@BPVV; @Fu]). In this resolution, $\Psi^{-1}(F_j)$ is exactly the configuration of curves constructed in Proposition \[p1.16.4.4\]. Indeed, as the notation anticipates, this resolution of singularities is isomorphic to the iterated blow-up constructed there:
Let ${\check{M}}$ be a parabolically ruled surface and let $M^* = {\check{M}}\backslash \cup F_j$. Let ${\overline{M}}$ by the corresponding orbifold. Then we have the diagram $$\label{e10.844}
\xymatrix{& {\widehat{M}}\ar_\Psi[dl] \ar^\Phi[dr] & \\
{\overline{M}}& & {\check{M}}\\
& M^*\ar[ul]\ar[ur]&
}$$ where the lower arrows are the natural inclusions of $M^*$. \[bimero\]
The problem is local to the base, so we use the local models and coordinates introduced above. In terms of these coordinates, the identity map on $L^*$ becomes the singular map ${\overline{L}}\to {\check{L}}$ $$\label{e33.844}
F:(u_1,v_1) \mapsto (x_1,y_1) = (u_1^q, u_1^{-p}v_1),\;
F:(u_2,v_2) \mapsto (x_2,y_2) = (u_1^q, u_1^{p}v_1),\;$$ which is smooth away from the central fibre $u_1=u_2=0$.
We claim that this map has a very simple description in the language of toric geometry, which makes clear the existence of the diagram .
Fix the standard lattice $\ZZ^2 \subset \RR^2$. The toric description[^2] of ${\check{L}}$ is in terms of the fan with 2-dimensional cones $$\sigma_1 = \{(\mu,\nu): \mu{\geqslant}0,\nu {\geqslant}0\},\;\;
\sigma_2 = \{(\mu,\nu): \mu{\geqslant}0,\nu {\leqslant}0\}.$$ The dual cones are $$\sigma_1^* = \{(m,n): m {\geqslant}0, n{\geqslant}0\},\;\;
\sigma_2^* = \{(m,n): m {\geqslant}0, n{\leqslant}0\}.$$ The corresponding coordinate rings are just $\CC[X,Y]$ and $\CC[X,Y^{-1}]$. Identifying $(X,Y)$ with $(x_1,y_1)$ in the first case and with $(x_2,y_2^{-1})$ in the second, we arrive at ${\check{L}}$, coordinatized as in . Similarly, ${\overline{L}}$ is the toric variety corresponding to the fan with two-dimensional cones $$\tau_1 = \{(\mu,\nu): \mu {\geqslant}0, p\mu + q\nu {\geqslant}0\},\;
\tau_2 = \{(\mu,\nu): \mu {\geqslant}0, p\mu + q\nu {\leqslant}0\}.$$ The dual cones are $$\tau_1^* = \{(m,n): qm-pn {\geqslant}0, n{\geqslant}0\},\;\;
\tau_2^* = \{(m,n): qm-pn {\geqslant}0, n{\leqslant}0\}.$$ Using the same indeterminates as before, we have two affine varieties with coordinate rings $$A_1 = \bigoplus_{qm {\geqslant}pn, n{\geqslant}0} \CC X^mY^n,\;\;
A_2 = \bigoplus_{qm {\geqslant}pn, n{\leqslant}0} \CC X^mY^n.$$ Following Fulton [@Fu], the first of these is identified with the coordinate ring of $\CC^2/\Gamma_{p,q}$ by introducing variables $(u_1,v_1)$ with $u_1^q = X$, $Y=u_1^{-p}v$. Then $A_1$ is precisely the $\Gamma_{p,q}$-invariant part of $\CC[u_1,v_1]$. Similarly $A_2$ is identified with the coordinate ring of $\CC^2/\Gamma_{q-p,q}$ by setting $X = u_2^q$, $Y = u_2^{-p}v_2^{-1}$. Then it is clear that the identity map of $\ZZ^2$ gives rise to our singular holomorphic map.
The map $F$ is singular because the identity map does not map the cone $\tau_1$ into either of the cones $\sigma_1$, $\sigma_2$. The Hirzebruch–Jung resolution of the two singularities in ${\overline{L}}$ corresponds to a subdivision of the fan $\{\tau_1,\tau_2\}$. Recall the continued-fraction expansions , and set $$v_0 = (0,1),v_1 = (1,0), v_j = (m_j,-n_j),$$ where $$\frac{n_j}{m_j} = \cfrac{1}{e_1 - \cfrac{1}{e_2-\cdots \frac{1}{e_{j-1}}}},$$ so that $v_{k+1} = (q,-p)$. Define similarly $$v_0' = (0,-1), v_1' = (1,-1), v'_j = (m'_j , n'_j - m'_j)$$ where $$\frac{n'_j}{m'_j} = \cfrac{1}{e'_1 - \cfrac{1}{e'_2-\cdots
\frac{1}{e'_{j-1}}}}$$ are the approximants to $(q-p)/q$. Again $v'_{l+1} = (q,-p)$. Put $$\rho_j = \RR_{{\geqslant}0} v_j \oplus \RR_{{\geqslant}0}v_{j+1},
\rho'_j = \RR_{{\geqslant}0} v'_j \oplus \RR_{{\geqslant}0}v'_{j+1}$$ for every $j$. Then the fan consisting of all the $\rho_j$ and $\rho'_j$ corresponds precisely to the minimal resolution $\Psi:\widehat L\to\overline L$ and the identity map of $\ZZ^2$ now induces the (restriction of the) holomorphic map $\Phi: \widehat L \to \check{L}$.
It is a pleasant exercise to verify from this point of view that $\widehat L \to \check L$ is a multiple blow-up. Combinatorially, a $(-1)$-curve corresponds to the occurence of adjacent cones of the form $$\RR_{{\geqslant}0} u \oplus \RR_{{\geqslant}0} (u+v),\;\;
\RR_{{\geqslant}0} (u+v) \oplus \RR_{{\geqslant}0} v$$ in a fan. The blow-down operation corresponds to the replacement of these two cones by the single cone $$\RR_{{\geqslant}0} u \oplus \RR_{{\geqslant}0} v.$$ In the above fan, one has to show that there is a sequence of such deletions that always results in a non-singular fan. In fact, at each stage, one deletes the ray generated by the vector $v_j$ or $v'_j$ with the [*largest*]{} $x$-coordinate. The details are left to the interested reader.
Scalar-flat Kähler orbifold metrics and holomorphic vector fields
-----------------------------------------------------------------
Let ${\check{M}}$, ${\widehat{M}}$ and ${\overline{M}}$ be as before, and assume that ${\check{M}}$ is a stable parabolic ruled surface. Then ${\overline{M}}$ carries a scalar-flat Kähler orbifold metric ${\overline{g}}$; and the algebra ${\mathfrak a}({\overline{M}})$ of holomorphic vector fields is $0$. \[theodefmbar\]
Recall from that ${\overline{M}}$ is a quotient of $\HH^2\times {\mathbb{CP}}^1$; if the parabolic structure is stable, then $\pi_1^{orb}({\overline{\Sigma}})$ acts by isometries of this product, so its SFK structure descends to define an orbifold SFK structure on ${\overline{M}}$.
The proof that ${\mathfrak a}({\overline{M}})=0$ is very close to the proof of [@LS Prop. 3.1]. Suppose that $\xi$ is a holomorphic vector field on ${\overline{M}}$. We claim first that $\xi$ must be vertical. To see this, consider the exact sequence $$0 \longrightarrow \cF \longrightarrow T\overline M \stackrel h\longrightarrow \cN
\longrightarrow 0,$$ on ${\overline{M}}$, where $\cF$ is the tangent space to the fibres of ${\overline{\pi}}$, while $\cN$ is the normal bundle to the fibres. If $F={\overline{\pi}}^{-1}(P)$ is a smooth fibre, then $\cN|F$ is the trivial line-bundle on $F$ and so $h(\xi)|F$ is constant on $F$ and is given by the pull-back of a vector in $T_P{\overline{\Sigma}}$. These vectors clearly patch together to give a holomorphic vector field $V$ on the punctured Riemann surface $\Sigma$. The same argument works near the singular fibres of ${\overline{\pi}}$, so $V$ extends to define a smooth (orbifold) vector field on ${\overline{\Sigma}}$. But the base is an orbifold Riemann surface with negative scalar curvature, and it follows as in the smooth case that such a surface carries no non-trivial holomorphic vector fields. So $V=0$ and $\xi$ is vertical.
The vector field $\xi$ must have zeros, for it is tangent to the smooth fibre $F$ which is a $2$-sphere. In particular, it must lie in the subalgebra ${\mathfrak a}_0({\overline{X}})$ of non-parallel holomorphic vector fields, so ${\mathfrak a}({\overline{X}})= {\mathfrak a}_0({\overline{X}})$.
Because ${\overline{M}}$ has a SFK metric ${\overline{g}}$, ${\mathfrak a}_0(X)$ is the complexification of the Lie algebra of non-parallel infinitesimal isometries of ${\overline{g}}$ (cf. [@Be Chapter 2]). So we may assume that $\xi$ is such an infinitesimal isometry. But then $\xi$ lifts to an infinitesimal isometry of $\HH^2\times {\mathbb{CP}}^1$, i.e. an element of the Lie algebra $$\label{e1.7.1.04}
\mathfrak{sl}_2(\RR) \times \mathfrak{su}_2$$ which is invariant under the induced action of $\pi_1^{orb}({\overline{\Sigma}})$. This action is given by composing the representation $$\pi_1^{orb}({\overline{\Sigma}}) \to SL_2(\RR)\times SU_2/\ZZ_2$$ with the adjoint action of $SL_2(\RR)\times SU_2/\ZZ_2$ on its Lie algebra. Now the adjoint action of $SU_2$ on its lie algebra is precisely the action of $SU_2/\ZZ_2 = SO_3$ on $\RR^3$, so a non-zero $\pi_1^{orb}({\overline{\Sigma}})$-invariant element in will determine an invariant line in $\RR^3$. An intersection of this line with the unit sphere $S^2\subset \RR^3$ defines an invariant point of ${\mathbb{CP}}^1\simeq S^2$, and we obtain an invariant complex line in $\CC^2$. This contradicts the irreducibility of the representation $\rho$.
A remark on uniqueness {#secunique}
----------------------
Before we proceed to the proof of Theorem \[maintheo\] (or Theorem \[maintheoparab\]), it is worth mentioning that the SFK metrics which are produced, are in fact unique in their Kähler class. More generally, we have the following result which is a direct consequence of [@CT Theorem 1.1].
\[theouniq\] Let $\widehat M$ be a complex surface as in Theorem \[maintheo\]. Then, there is at most one SFK metric in each Kähler class.
We give here an alternative proof of the proposition, based on the work of Donaldson. If the proposition is false, there are two distinct SFK metrics $\omega_1$, $\omega_2$ with the same Kähler class $\Omega$. The deformation theory for SFK metrics is unobstructed [@LS Theorem 2.8]. By density, we can find arbitrarily small SFK deformations $\omega'_1$ and $\omega'_2$ with Kähler class $\Omega' \in
H^2(\widehat M,\QQ)$. We may assume that $\omega'_1$ and $\omega'_2$ are distinct, which is the case for deformations small enough, and that $\Omega'$ is an integral class after multiplication by a suitable constant factor. By Kodaira’s embedding theorem $(\widehat M,\Omega')$ is a projective variety.
We remark now that $\widehat M$ cannot have any non trivial holomorphic vector field. Let $\Xi$ be a holomorphic vector fields on $\widehat M$. The complex surface $\widehat M$ is by definition the resolution $\pi:\widehat M \rightarrow \overline M$ of the orbifold $\overline
M$. The exceptional fibres $\pi$ have negative self-intersection hence $\Xi$ must be tangent to them. Therefore the vector field $\Xi$ is projectible. By Theorem [(\[theodefmbar\])]{} that $\overline
M$ does not admit any non trivial holomorphic vector field. Thus $\pi_*\Xi=0$ which forces $\Xi=0$.
According to [@D Corollary 5], there is at most one Kähler metric of constant scalar curvature on a projective manifold with no non-trivial holomorphic vector field. Thus, $\omega'_1$ and $\omega'_2$ must be equal. This is a contradiction, and the proposition holds.
Scalar-flat metrics on $\widehat{M}$ {#secgluing}
====================================
In this section we prove a gluing theorem that implies Theorem \[maintheo\] (or equivalently Theorem \[maintheoparab\]). We begin with a statement of the result and an outline of the gluing argument. We then give a rapid description of the needed perturbation theory of SFK and hermitian-ASD metrics and the necessary linear theory, before application of the implicit function theorem to prove the theorem.
Statement of gluing theorem and outline of proof
------------------------------------------------
\[theoglue\] Let $({\overline{M}}, {\overline{\omega}})$ be a compact SFK orbifold of complex dimension $2$, with all singularities isolated and cyclic. Suppose further that the algebra ${\mathfrak a}_0({\overline{M}})$ of non-parallel holomorphic vector fields on ${\overline{M}}$ is zero. Then the minimal resolution $\widehat{M}$ of ${\overline{M}}$ admits a family of SFK metrics ${\widehat{\omega}}^{\varepsilon}$ such that ${\widehat{\omega}}^{\varepsilon}\to {\overline{\omega}}$ on any compact subset $K \subset {\widehat{M}}\setminus{E}$, where $E$ is the exceptional divisor. Moreover, any blow-up of ${\widehat{M}}$ carries a similar family of SFK metrics.
We have just seen that the orbifold ${\overline{M}}$ associated to a stable parabolic ruled surface ${\check{M}}$ carries a SFK metric and satisfies ${\mathfrak a}_0({\overline{M}})=0$. Our main theorems \[maintheo\] and \[maintheoparab\] therefore follow at once from Theorem \[theoglue\].
This theorem would follow from the gluing theorems on gluing hermitian–ASD conformal structures stated in [@KS]. Unfortunately, the argument given there is not globally consistent; we therefore give a complete proof here. While many of the ingredients are the same, the new argument given here is perhaps a little more direct. We shall now outline the main points; the technical details follow in the rest of this section.
The first step in the proof of Theorem \[theoglue\] is to realize the resolution ${\widehat{M}}$ as a “generalized connected sum”. For this, denote by $X_{p,q}$ the minimal resolution of $\CC^2/\Gamma_{p,q}$. If $E\subset X_{p,q}$ is the exceptional divisor, then $X_{p,q}\setminus
E$ is canonically biholomorphic to $(\CC^2\setminus 0)/\Gamma_{p,q}$, and $E$ is the Hirzebruch–Jung string constructed from the continued fraction expansion of $p/q$ as before. The resolution of a $\Gamma_{p,q}$-singularity at a point $0\in {\overline{M}}$ can be realized explicitly as follows. Choose two small positive numbers $a$ and $b$; cut off $X_{p,q}$ at a large radius $a^{-1}$ and remove a ball $B(0,b)$ from ${\overline{M}}$. If $z$ is an asymptotic uniformizing holomorphic coordinate for $X_{p,q}$ and $u$ is a local uniformizing holomorphic system near $0\in {\overline{M}}$, then we can perform the resolution by making the identification $u = abz$.
If this model of ${\widehat{M}}$ is to carry an “approximately” SFK metric, then it is essential that $X_{p,q}$ should itself carry an asymptotically locally euclidean (ALE) SFK metric. The existence of such metrics is guaranteed by the following:
\[propcaldsingmet\] For all relatively prime positive integers $p<q$, $X_{p,q}$ carries an asymptotically locally euclidean SFK metric $g$. More precisely, there exists a compact subset $K\subset X_{p,q}$ and holomorphic coordinates $z$ on the universal cover of $X_{p,q}\backslash K$ such that $$|g - |{{\mathrm d}}z|^2| = O(|z|^{-1})$$ for all sufficiently large $|z|$. Moreover, $$|\partial^m g| = O( |z|^{-m-1})$$ for all positive integers $m$. \[p1.30.1.04\]
Such metrics were constructed in [@CS] for general $p$ and $q$. Note that $X_{q-1,q}$ is the complex manifold underlying the $A_{q-1}$ gravitational instanton of Gibbons–Hawking, Hitchin and Kronheimer [@K]. On the other hand, $X_{1,q}$ is the total space of $\cO(-q)\to {\mathbb{CP}}^1$, and the SFK metric in this case is due to LeBrun [@L4]. For general $p$ and $q$, these metrics were known to Joyce, though they do not seem to have appeared explicitly in his published work: they are implicit in [@J1] and [@J2].
The proof that the metrics of [@CS] have the correct asymptotic properties is given in §\[s1.1.30.04\].
Returning to our outline of the gluing theorem, we use cut-off functions to define a sequence of approximately SFK metrics on ${\widehat{M}}$, by gluing the orbifold metric to the ALE metric of Proposition \[propcaldsingmet\].
Finally we use the implicit function theorem to find a genuine SFK metric close to the approximate one. The successful completion of this step requires in particular that a certain linear operator be surjective in a controlled way as $a$ and $b$ go to $0$. This is why we pay a lot of attention to the linear theory in §\[linth\]. The output of the implicit function theorem is only a $C^{2,\alpha}$ metric. However, any such solution will be smooth by elliptic regularity.
For technical reasons, we shall use the above methods to find hermitian-ASD metrics on ${\widehat{M}}$. By a result of Boyer [@Boyer], any such conformal class must have a Kähler representative, and this will automatically be scalar-flat. Indeed, if $\omega_0$ is any representative $2$-form in the conformal class, there is a $1$-form $\beta$ defined by $$\label{e1.6.2.04}
{{\mathrm d}}\omega_0 + \beta \wedge \omega_0 = 0.$$ Boyer proves (using the compactness of ${\widehat{M}}$, and the fact that $b_1({\widehat{M}})$ is even) that $\beta = {{\mathrm d}}f$, for some $f$. But then ${{\mathrm d}}(e^{-f}\omega_0) =
0$ and we have found the desired SFK representative.
We start our analysis now by discussing the perturbation theory of SFK metrics and hermitian-ASD conformal structures.
Perturbation theory of SFK metrics
----------------------------------
Let $M$ be a smooth compact complex surface, with complex structure $J$. To any hermitian conformal structure $c$ on $M$ we associate the (conformal) fundamental $2$-form ${\omega}$ as follows: $$c(\xi,\eta) \leftrightarrow {\omega}(\xi,\eta) = c(\xi, J\eta)$$ Here we think of $c$ as a positive-definite section of ${\Omega}^{-1/2}S^2T^*M$, where ${\Omega}$ is the bundle of densities on $M$; accordingly, ${\omega}$ is a weightless $(1,1)$-form, in other words, a section of the bundle ${\Omega}^{-1/2}\Lambda^{1,1}$.) Fix a background conformal structure $c$ with corresponding $(1,1)$-form $\omega$. Then the set of $J$-hermitian conformal structures near $c$ is identified with a neighbourhood $U$ of $0$ in ${C^{\infty}}(X,{\Omega}^{-1/2}\Lambda_0^{1,1})$; if $A\in U$, we have the new weightless $(1,1)$-form $\omega + A$. $A$ is taken point-wise orthogonal to $\omega$ to avoid replacing $c$ by a multiple of $c$. On a complex surface, $\Lambda_0^{1,1} = \Lambda^-$, so we have parameterized the $J$-hermitian conformal structures near $c$ by a neighbourhood of $0$ in ${C^{\infty}}(X,{\Omega}^{-1/2}\Lambda^-)$.
Denote by $L_A$ the operator $\theta \mapsto (\omega +A)\wedge
\theta$, and by $\Lambda_A$ the adjoint “trace” map. It is shown by Boyer [@Boyer] that $$\Lambda_A W^+[\omega +A ] = 0 \mbox{ iff }W^+[\omega +A]=0.$$ On the LHS we have a section of $\Lambda_{\omega+A}^+\subset
\Lambda^2$. Denote by $\cP$ the projection $\Lambda^2 \to \Lambda^+$ to the [*fixed*]{} subspace of $2$-forms self-dual with respect to the fixed background $c$ and set $$\cF(A) = \cP[\Lambda_A W^+[\omega + A]].$$ (Where necessary, we shall denote the dependence of $\cF$ on the background conformal structure $c$ by a sub- or super-script.) The necessary facts about $\cF$ are summarized as follows:
Given a fixed $J$-hermitian conformal structure $c$, there is a map $$\cF : U \to {C^{\infty}}(X,\Lambda^+)$$ where $U$ is a neighbourhood of $0 \in {C^{\infty}}({\Omega}^{-1/2}\Lambda^-)$ with the property that $\cF^{-1}(0)$ is the set of $J$-hermitian ASD conformal structures near $c$ on $X$. Furthermore, there is an expansion $$\label{e5.14.4.4}
\cF(A) = \cF(0) + S[A] +
{\varepsilon}_1(A, A\otimes \nabla\nabla A)+
{\varepsilon}_2(A, \nabla A\otimes \nabla A)$$ where $$\cF(0) = \Lambda W^+[c],\;\;\;
S:{C^{\infty}}({\Omega}^{-1/2}\Lambda^-) \to {C^{\infty}}(\Lambda^+)$$ is a conformally invariant linear elliptic operator, and the nonlinear terms ${\varepsilon}_j(A,f)$ are real-analytic in the $0$-jet of $A$ and linear in the $0$-jet of $f$. \[p3.1.30.04\]
We note further that if $g$ is a SFK metric in the conformal class $c$ then, trivializing ${\Omega}^{-1/2}$, the operator $S$ gets identified with the operator $$\label{e2.6.2.04}
S: \alpha \longmapsto {{\mathrm d}}^+\delta\alpha + \langle\rho,\alpha\rangle\omega.$$ where $\rho$ is the Ricci form [@LS].
Linear theory {#linth}
-------------
From now on, in order to streamline the discussion, denote by $(X_1,g_1)$ the ALE SFK space from Proposition \[p1.30.1.04\] and denote by $(X_2,g_2)$ the non-compact space ${\overline{M}}\setminus {\overline{M}}_{sing}$. In order to save on notation, we assume that ${\overline{M}}$ has just one singular point, giving a conical singularity that matches the infinity of $X_1$. We know that there exist asymptotic (uniformizing) holomorphic coordinates $z$, defined, say, for $|z| {\geqslant}1/2$ and such that $$g_1 = |{{\mathrm d}}z|^2 + \eta_1(z)$$ where $$|\nabla^m\eta_1|_{g_1} = O(|z|^{-m-1}).$$ Similarly, on $X_2$, we have (uniformizing) holomorphic coordinates $u$, say, again defined for $|u|{\leqslant}2$ with respect to which $$g_2 = |{{\mathrm d}}u|^2 + \eta_2(u),\;\;
|\eta_2(u)| = O(|u|^2),\;
|\nabla \eta_2(u)| = O(|u|),\; |\nabla^{m+2} \eta_2(u)| = O(1)$$ for $m = 0, 1,\ldots$.
From now on we shall write, for example, $\{|z|{\leqslant}1\}$ as short-hand for $X_1\setminus\{|z| >1\}$; and similarly for subsets of $X_2$ such as $\{|u|{\geqslant}1\}$.
Denote by $S_1$ and $S_2$ the linear operators from Proposition \[p3.1.30.04\] determined by the metrics $g_1$ and $g_2$. These operators are defined over the non-compact spaces $X_1$ and $X_2$, so some care is needed in arranging for them to be Fredholm. The needed results are now standard, having been worked out in various forms by a number of different authors. The most refined results can be found in Melrose’s book [@APS]; another useful account is [@LoMcO]. This does not cover the Hölder spaces that we shall use, however: for that one can consult Mazzeo’s paper on edge operators [@Ma].
It turns out that $S_1$ and $S_2$ are Fredholm when made to operate between suitably defined weighted Hölder (or Sobolev) spaces. In order to define these Hölder spaces, put $$r_1 = |z|\mbox{ for }
|z|{\geqslant}2, r_1=1\mbox{ for }|z|{\leqslant}1/2$$ and $r_1{\geqslant}1$ everywhere. Define $r_2$ similarly by continuing $$r_2 = |u|\mbox{
for }|u|{\leqslant}1/2$$ to $1$ for $|u|{\geqslant}2$, so $r_2{\geqslant}1/2$ for $|u|{\geqslant}1/2$. Define the norms $$\|f\|_{\alpha}=\sup |f| + \sup_{z\not=z'}\left(
(r_1(z)+r_1(z'))^\alpha
\frac{|f(z) - f(z')|}{|z-z'|^\alpha}\right)$$ and $$\|f\|_{2,\alpha}
=
\sup |f| + \sup |r_1\nabla f| + \sup |r_1^2\nabla\nabla f|$$ $$\hspace{4cm}
+
\sup_{z\not=z'}\left(
(r_1(z)+r_1(z'))^{2+\alpha}
\frac{|\nabla\nabla f(z) - \nabla \nabla f(z')|}{|z-z'|^\alpha}\right)$$ The completion of ${C^{\infty}}_0(X_1)$ in $\|\cdot\|_{\alpha}$ will be denoted by $B^\alpha(X_1)$; its completion in $\|\cdot\|_{2,\alpha}$ will be denoted by $B^{2,\alpha}$. The weighted versions of these spaces are $$\label{e1.30.1.04}
r_1^\delta B^{n,\alpha} =
\{f: r_1^{-\delta}f \in B^{n,\alpha}(X_1)\},\;\;
\|f\|_{n,\alpha,\delta} = \|r_1^{-\delta}f\|_{n,\alpha},\;\; (n=0,2)$$
We define $r_2^\delta B^{n,\alpha}(X_2)$ in exactly the same way, using $u$ and $r_2(u)$ in place of $z$ and $r_1(z)$. Similar norms can be introduced in bundles by patching the local definitions.
The basic facts are then that $$\label{e3.30.1.04}
S_j : r_j^\delta B^{2,\alpha}(X_j, {\Omega}^{-1/2}\Lambda^{-})
\to r_j^{\delta -2}B^{0,\alpha}(X_j,\Lambda^+)$$ is a bounded linear operator, Fredholm for all but a discrete set of values of $\delta$. Moreover for any one of these “good” weights, $S_j$ is surjective iff the formal adjoint has no null space acting on $r_j^{-2-\delta} B^{0,\alpha}$.
Let $$\label{e2.1.30.04}
S_0 = {{\mathrm d}}^+{{\mathrm d}}^* : {C^{\infty}}(\CC^2\setminus 0,{\Omega}^{-1/2}\Lambda^-) \to
{C^{\infty}}(\CC^2\setminus 0, \Lambda^+)$$ (the linearized operator at the euclidean metric). Denote by $x$ a standard system of euclidean coordinates on $\CC^2$. The bad weights $\lambda$ correspond to homogeneous solutions $\phi$, $S_0\phi =0$, $|\phi| = |x|^\lambda$; more precisely, they correspond to the $\Gamma$-invariant such solutions, i.e. the solutions that descend to $(\CC^2\setminus 0)/\Gamma$.
The set of bad weights is contained in $\ZZ\setminus \{-1\}$. \[l1.1.30.04\]
Suppose $\phi$ satisfies $S_0\phi=0$ and is homogeneous of degree $\lambda\in \CC$. By this we mean that if $\phi = \sum
\phi_je_j$, where $e_j$ is the standard parallel orthonormal basis of ${\Omega}^{-1/2}\Lambda^-$, then each $\phi_j$ is homogeneous of degree $\lambda$. If $\lambda \in \CC \backslash \{-4,-5,-6,\cdots\}$, then by [@Horm1 Thm. 3.2.3], $\phi$ has a unique extension $\dot{\phi}$ to $\CC^2$ as a homogeneous distribution. We have $$S_0\dot{\phi} = f$$ where $f$ is a distribution supported at $0$ homogeneous of degree $\lambda - 2$, because $S_0$ is of second order. Now any distribution supported at $0$ is a finite linear combination of derivatives of the $\delta$-function $\delta_0$. Since a $k$-th order derivative of $\delta_0$ is homogeneous of degree $-4-k$, it follows that $$\mbox{If }f\not=0,\; \lambda = -2-k,\mbox{ where }k=0,1,2,\ldots.$$ On the other hand, if $f=0$, then $\dot{\phi}$ must be smooth because $S_0$ is elliptic. In particular, the degree of homogeneity of $\phi$ must be a non-negative integer.
The other essential fact about the operator $S_j$ in is that any $\phi_j$ with $S_j \phi_j =0$, must have an asymptotic expansion for $|z|\to\infty$ or $|u|\to 0$; moreover, the leading term must behave exactly like $|z|^\lambda$ or $|u|^\lambda$, where $\lambda$ is one of the “bad weights” found in Lemma \[l1.1.30.04\].
If $0< \delta < 2$ and $0<\alpha < 1$ then $$S_1: r_1^{-\delta} B^{2,\alpha}(X_1,{\Omega}^{-1/2}\Lambda^-)
\to r_1^{-\delta -2}B^{0,\alpha}(X_1,\Lambda^+)$$ has a bounded right-inverse $G_1$. If the orbifold ${\overline{M}}$ satisfies ${\mathfrak a}_0({\overline{M}})=0$, then $$S_2: r_2^{-\delta} B^{2,\alpha}(X_2,{\Omega}^{-1/2}\Lambda^-)
\to r_2^{-\delta -2}B^{0,\alpha}(X_2,\Lambda^+)$$ has a bounded right-inverse $G_2$. \[p6.30.1.04\]
For $\delta$ in the given range, there are no bad weights, so $S_j$ is Fredholm, and is surjective if and only if $$S_j^* \psi = 0, |\psi| = O(r_j^{\delta-2}) \Rightarrow \psi = 0.$$ If $j=1$, then we note that $\delta-2<0$ and so $|\psi| =
O(|z|^{-2})$. This is enough to force vanishing of $\psi$ by [@KS Theorem 8.4]. If $j=2$, we use that $\delta-2 >-2$, from which it follows that $\psi = O(1)$. By elliptic regularity, it follows that $\psi$ extends to a solution on the whole of the orbifold. To show that $\psi=0$, we follow the analysis of $S$ that was given in [@LS Thm. 2.7]—the argument goes through without change for compact orbifolds. The essential point is that the component of $\psi$ in the direction of the Kähler form $\omega$ is a function $f$ that satisfies Lichnerowicz’s equation $$\Delta^2 f + 2\langle \rho, {{\mathrm d}}{{\mathrm d}}^c f\rangle = 0.$$ Because ${\overline{M}}$ has constant scalar curvature, $\nabla^{1,0}f \in {\mathfrak a}_0({\overline{M}})$. Thus $f=0$. One shows further that $f=0$ implies that the component of $\psi$ orthogonal to $\omega$ also vanishes.
Gluing construction {#secgluingconstr}
-------------------
We continue with the notation of the previous section, now picking two small numbers $a$ and $b$. We begin by gluing $X_1$ and $X_2$ to produce the complex manifold ${\widehat{M}}= X_{a,b}$ which is the resolution of singularities of ${\overline{M}}$. This is easy: we just identify an annular region $\{a^{-1}{\leqslant}|z| {\leqslant}4a^{-1}\}$ in $X_1$ with a similar region $\{b{\leqslant}|u|{\leqslant}4b\}$ in $X_2$ by the holomorphic map $$\label{e4.30.1.04}
u = abz.$$
### Gluing metrics {#glum}
The next step is to glue the metrics. Pick a standard cut-off function $\theta_1$, $0{\leqslant}\theta_1 {\leqslant}1$, with $\theta_1 =1$ for $t{\leqslant}1$, $\theta_1 = 0$ for $t{\geqslant}2$. Set $\theta_2 = 1 - \theta_1$. Define new metrics $g_1^a$ on $X_1$ and $g_2^b$ on $X_2$ by “flattening” $g_1$ near infinity and $g_2$ near $0$; thus $g_1^a =
g_1$ for $|z|{\leqslant}a^{-1}$ and $$g_1^a = |{{\mathrm d}}z|^2 + \theta_1(a|z|)\eta_1(z)\mbox{ for }|z|{\geqslant}a^{-1}.$$ Similarly, $g_2^b = g_2$ for $|u|{\geqslant}4b$ and $$g_2^b = |{{\mathrm d}}u|^2 + \theta_2((2b)^{-1}|u|)\eta_2(u)\mbox{ for
}|u|{\leqslant}4b.$$ One computes that any curvature quantity $R$ satisfies $$\label{er1}
|R(g_1^a)| =
\left\{\begin{array}{ll}
|R(g_1)| & \mbox{ for } |z| {\leqslant}a^{-1} \\
O(a^{3}) & \mbox{ for } a^{-1} {\leqslant}|z| {\leqslant}2a^{-1} \\
0 & \mbox{ for }|z| {\geqslant}2a^{-1}.\end{array}\right.$$ Similarly, $$\label{er2}
|R(g_2^b)| =
\left\{\begin{array}{ll}
|R(g_2)| & \mbox{ for } |u| {\geqslant}4b \\
O(1) & \mbox{ for } 2b {\leqslant}|u| {\leqslant}4b \\
0 & \mbox{ for }|u| {\leqslant}2b.\end{array}\right.$$
We now define a metric on $X_{a,b}$ by matching $g_1^a$ with $a^{-2}b^{-2}g_2^b$ by along the spheres $\{|u|=2a^{-1}\}$ and $\{|z| = 2b\}$: $$g^{a,b} =
\left\{\begin{array}{ll}
g^a_1 & \mbox{ for } |z| {\leqslant}2a^{-1} \\
a^{-2}b^{-2}g_2^b & \mbox{ for } |u| {\geqslant}2b.
\end{array}\right.$$ It is clear that $g^{a,b}$ is [*hermitian*]{} with respect to the complex structure of $X_{a,b}$.
Consider now the map $\cF^{a,b}$ of Proposition \[p3.1.30.04\], associated to $g^{a,b}$ (or more accurately to the underlying conformal structure $c^{a,b}$). The main technical theorem can be stated as follows.
There exist Banach spaces $\EE$ and $\FF$ (depending upon $a$ and $b$) such that
1. $\cF:= \cF^{a,b}$ extends to a smooth map from a neighbourhood $U$ of $0\in \EE$ to $\FF$;
2. for $x\in U$, $$\cF(x) = \cF(0) + S[x] + Q(x),$$ where $S$ is a Fredholm linear operator and $Q$ satisfies $$\|Q(x) - Q(y)\| {\leqslant}C_1(\|x\| + \|y\|)\|x-y\|.$$
3. $\|\cF(0)\| \to 0$ as $a, b\to 0$.
4. If $a$ and $b$ are sufficiently small, then $S$ has a right inverse $G$, with norm uniformly bounded by $C_2$, say.
\[t4.1.30.03\]
Once one has this result, one obtains a parameterization of $\cF^{-1}(0)$ as a graph of a map $f$ from a small ball in $\EE_0$ into $\EE_1$, where $\EE_0$ is the finite-dimensional null-space of $S$ and $\EE_1$ is the range of $G$. Indeed, suppose that $$\|\cF(0)\| {\leqslant}\frac{\lambda_0}{C_1C_2^2},\;\;
k_1 =\frac{\lambda_1}{C_1C_2},\;\;
k_2 =\frac{\lambda_2}{C_1C_2^2}.$$ If we replace $x$ by $x + Gy$, where $x\in \EE_0$, then the equation $\cF(x+Gy)=0$ becomes $$y = T_x(y):= - \cF(0) - Q(x+Gy)$$ If $x$ is fixed in $\EE_0\cap\{\|x\| {\leqslant}k_1\}$ and $\|y\|{\leqslant}k_2$, then $$\|T_x(y)\| {\leqslant}\|\cF(0)\| + C_1\|x+Gy\|^2
{\leqslant}\frac{\lambda_0 + \lambda_1^2 + \lambda_2^2}{C_1C_2^2}$$ Therefore, $T_x$ maps the $k_2$-ball in $\FF$ into itself if $$\label{e5.30.1.04}
\lambda_0 + \lambda_1^2 + \lambda_2^2 {\leqslant}\lambda_2$$ Furthermore, $$\|T_x(y) - T_x(y')\| = \|Q(x+Gy) - Q(x+Gy')\| {\leqslant}2(\lambda_1+\lambda_2)\|y-y'\|$$ so that $T_x$ is a contraction mapping if $$\label{e6.30.1.04}
\lambda_1 + \lambda_2 < \frac{1}{2}.$$ It is easy to find positive numbers $\lambda_0$, $\lambda_1$ and $\lambda_2$ that simultaneously satisfy and . It follows from the contraction mapping theorem that there is a unique fixed point $y = T_x(y)$ for any given $x$ in the $k_1$-ball of $\EE_0$, and that $$\|y\| {\leqslant}\frac{1}{1 - 2(\lambda_1 +\lambda_2)}\|\cF(0)\|.$$ The norm of the corresponding zero $x + Gy$ of $\cF$ satisfies $$\|x + Gy\| {\leqslant}\|x\| + \frac{C_2}{1 - 2(\lambda_1 +\lambda_2)}\|\cF(0)\|.$$
### Gluing function spaces
The functions $r_1$ and $r_2$ agree in the gluing region $b{\leqslant}|u| {\leqslant}4b$, up to a factor of $ab$. We therefore define $$w = w^{a,b} =
\left\{\begin{array}{ll}
r_1(z) & \mbox{ for } |z| {\leqslant}2a^{-1} \\
a^{-1}b^{-1}r_2(u) & \mbox{ for }|u| {\geqslant}2b.
\end{array}\right.$$ Note that $$\label{e1.14.4.4}
1 {\leqslant}w {\leqslant}a^{-1}b^{-1} \mbox{ on }X^{a,b}.$$
Set $$\begin{aligned}
{\mbox{}^w}\|f\|_{n,\alpha} &=&
\sup |f| + \sup|w \nabla f| + \cdots + \sup|w^n\nabla^n f| + \nonumber \\
&& \sup_{P\not= Q}(w(P) + w(Q))^{n+\alpha}
\frac{|\nabla^n f(P) - \nabla^n f(Q)|}{d(P,Q)^\alpha}.\end{aligned}$$ Here all lengths are measured by the metric $g^{a,b}$. Denote the completion of ${C^{\infty}}(X)$ in this norm by $B^{n,\alpha}(X)$. We now define $$\EE = w^{-\delta}B^{2,\alpha}(X,{\Omega}^{-1/2}\Lambda^-),\;
\FF = w^{-\delta-2}B^{0,\alpha}(X,\Lambda^+).$$ The norms in $\EE$ and $\FF$ will be denoted by subscripts $\EE$ and $\FF$ or by the notation $${\mbox{}^w}\|\cdot\|_{2,\alpha,-\delta},\;\;
{\mbox{}^w}\|\cdot\|_{0,\alpha,-2-\delta}.$$ By design $\EE$ and $\FF$ are closely related to the function spaces that were introduced in Proposition \[p6.30.1.04\]. We make a couple of observations concerning $\EE$ and $\FF$. Let $A \in {C^{\infty}}(X,{\Omega}^{1/2}\Lambda^-)$, $W\in {C^{\infty}}(X,\Lambda^+)$. If $A$ and $W$ have support contained in $|z|{\leqslant}2/a$, then clearly $$\label{e2.14.4.4}
\|A\|_{\EE} = \|A\|_{2,\alpha,-\delta},\;\; \|W\|_{\FF} =
\|W\|_{0,\alpha,-2-\delta}.$$ On the other hand, if $A$ and $W$ are supported in the region $|u|{\geqslant}2b$, then $$\label{e3.14.4.4}
\|A\|_{\EE} = a^{-\delta}b^{-\delta}\|A\|_{2,\alpha,-\delta},\;\;
\|W\|_{\FF} = a^{-\delta}b^{-\delta}
\|W\|_{0,\alpha,-2-\delta}.$$
### Proof of Theorem \[t4.1.30.03\] {#proof-of-theoremt4.1.30.03 .unnumbered}
Parts (i)–(iii) will follow from the expansion in Proposition \[p3.1.30.04\]. Let us begin by computing the $\FF$-norm of $\cF(0)$. This is zero away from the gluing region $b {\leqslant}|u|{\leqslant}4b$, and can be estimated there by combining and . The result is $$\label{e2.22.4.4}
\|\cF(0)\|_{\FF} = O(a^{1-\delta} + a^{-\delta}b^2).$$ Thus to guarantee (iii), we shall need to choose $0 <\delta < 1$ and make a suitable choice of the relative sizes of $a$ and $b$, for example $a = b^2$.
In order to show that $S$ extends to a bounded linear map from $\EE$ to $\FF$, use a partition of unity to split $S(A)$ into two pieces, the relations and , and the fact that the operators $S_1$ and $S_2$ in Proposition \[p6.30.1.04\] are bounded; here it is important that the same scale factor appears in comparing the norms of $A$ and $W$ in .
Finally we turn to the non-linear terms in . These are only defined if $\sup|A|$ is sufficiently small. But $$\label{e6.14.4.4}
\sup |A| {\leqslant}\sup w^\delta |A| {\leqslant}\|A\|_{\EE}$$ by . Assuming, then, that $\|A\|_{\EE}$ is sufficiently small, we have $$\sup|w^{2+\delta}{\varepsilon}_1(A,A\otimes \nabla^2 A) {\leqslant}\sup|{\varepsilon}_1(A,w^{\delta}A\otimes w^{2+\delta}
\nabla^2 A) {\leqslant}C\|A\|^2_{\EE},$$ again using . Since ${\varepsilon}_1$ is linear in its second variable, we have $${\varepsilon}_1(A, A\otimes\nabla^2 A) - {\varepsilon}_1(B, B\otimes\nabla^2 B)
= {\varepsilon}_1(A, A\otimes \nabla^2 (A - B)) +
{\varepsilon}_1(A, (A-B)\otimes\nabla^2 B) +$$ $$\hspace{4cm} + {\varepsilon}_1(A, B\otimes\nabla^2 B)
- {\varepsilon}_1(B, B\otimes\nabla^2 B)$$ and it is straightforward to use this to prove that the Hölder quotient in the definition of $\FF$ is controlled by $\|A\|$ and also the quadratic estimate of $Q$ in (ii). The term ${\varepsilon}_2(A,\nabla
A\otimes \nabla A)$ is estimated in the same way.
It remains to prove that $S$ has a uniformly bounded right inverse. Note that to save on notation, we have not indicated explicitly that $S$ depends upon $a$ and $b$. One should not lose sight of this dependence, however, in what follows.
First note that if $a$ and $b$ are small, then the operator norms of $S_1- S_1^a$ and $S_2- S_2^b$ are small, and so there are operators $G_1^a$, $G_2^b$ such that $$S_1^a G_1^a = 1,\;\; S_2^b G_2^b = 1, \|G_1 - G_1^a\|{\leqslant}1/2,
\|G_2 - G_2^b\|{\leqslant}1/2.$$ (Here the operator norms are defined by viewing $S_j$ as operators between the Banach spaces of Proposition \[p6.30.1.04\].) We shall now follow the approach of [@DK Chapter 7] to splice these right-inverses to give first an approximate and then an exact right-inverse for $S$. Recall the partition of unity $\theta_1+\theta_2=1$ on $\RR$ that was introduced at the beginning of §\[glum\]. For any small positive number $\lambda$, which will be fixed later, define $$\label{e1.16.9.4}
\beta_1(z) = \theta_1\left((a|z|/4)^{\lambda}\right),\;
\beta_2(u) = \theta_2\left(2(|u|/2b)^{\lambda}\right).$$ Define also $$\label{e2.16.9.4}
\gamma_1(z) = \theta_1(a|z|/2),\;\gamma_2(u) = \theta_2(|u|/2b).$$ Then $\gamma_1 + \gamma_2=1$ on $X^{a,b}$, and $$\label{idemp}
\beta_1\gamma_1 = \gamma_1,\;\;\beta_2\gamma_2 = \gamma_2.$$ Moreover, we have estimates of the form $$\label{e1.22.4.4}
\sup |r_1^k\nabla^k \beta_1|=O(\lambda^k),\;\;
\sup |r_2^k\nabla^k \beta_2| = O(\lambda^k)$$ for each positive integer $k$, with the $O$’s uniform in $a$ and $b$.
Now form the operator on $X^{a,b}$, $$\label{Gdef}
G_0 = \beta_1 G_1^a \gamma_1 + \beta_2 G_2^b \gamma_2;$$ we claim first that $\|G_0\|$, regarded as an operator $\FF \to \EE$, is bounded independent of $a$ and $b$. Indeed, we have $$\|G_0 W \|_{\EE} {\leqslant}\|\beta_1 G_1^a (\gamma_1 W)\|_{\EE}
+\| \beta_2 G_2^b (\gamma_2 W)\|_{\EE}$$ $$\hspace{4cm}
= \|\beta_1 G_1^a (\gamma_1 W)\|_{2,\alpha,-\delta}
+
(ab)^{-\delta}\|\beta_2 G_2^b (\gamma_2 W)\|_{2,\alpha,-\delta}$$ using the scaling formulae and . Since the operator norms of $G_1^a$ and $G_2^b$ are uniformly bounded, we obtain $$\|G_0 W\|_{\EE} {\leqslant}C[\|\gamma_1 W\|_{\FF} +\|\gamma_2 W\|_{\FF}]
{\leqslant}C'\|W\|_{\FF}$$ since $\gamma_1 + \gamma_2 = 1$.
On the other hand, $G_0$ is an approximate right-inverse for $S$: $$SG_0 = \beta_1 S G_1^a\gamma_1
+ \beta_2 S^{a,b}G_2^a\gamma_2
+ [S^{a,b},\beta_1]G_1^a\gamma_1
+ [S^{a,b},\beta_2]G_2^a\gamma_2.$$ On the support of $\beta_1$, $S$ is close to $S^a_1$, if $b$ is small. Therefore, $$\beta_1 SG_1^a\gamma_1 =
\beta_1 S^{a}_1G_1^a\gamma_1 +o(1) = \beta_1\gamma_1 + o(1) = \gamma_1
+o(1).$$ (Here $o(1)$ indicates an operator whose norm tends to zero with $a$ and $b$.) Similarly, $$\beta_2 SG_2^b\gamma_2 = \gamma_2 + o(1).$$ We show now that the commutator terms are $O(\lambda)$. Clearly $[S,\beta_1]$ is a first-order differential operator supported where $\nabla \beta_1 \not=0$, i.e. the interval $[(4/a), (4/a)\cdot
2^{1/\lambda}]$. For fixed $\lambda$ and small $a$, $S$ is very close to $S_2^b$ over this interval, and we can estimate norms as before by passing from $\EE$ and $\FF$ to the corresponding fixed weighted Hölder spaces on $X_2$. The coefficients of $[S,\beta_1]$ are therefore bounded functions on $X_2$, multiplied by $\nabla\beta_1$ and $\nabla^2\beta_1$. Hence by the operator norm of $[S,\beta_1]$ is $O(\lambda)$, as claimed. The same is true of the other commutator term. In sum, we have found that if $\lambda$ is chosen small enough, then as $a$ and $b$ tend to zero, $$SG_0 = 1 + R,\;\; \|R\| {\leqslant}1/2$$ Hence $G = G_0(1+R)^{-1}$ is a controlled right-inverse of $S$, and the proof of Theorem \[t4.1.30.03\] is complete.
Applying the implicit function theorem as described above, we obtain solutions in $\EE$ of our equation for all sufficiently small $a$, $b$, subject to the constraints imposed by , and by elliptic regularity these solutions will be smooth.
Finally we claim that the corresponding SFK metric is close to the original one. More precisely, it is close to $g_1$ on any subset of the form $\{|z|{\leqslant}C_1\}$ of $X_1$, and it is close to $a^{-2}b^{-2}g_2$ on any subset of the form $\{|u| {\geqslant}C_2\}$ of $X_2$. From the implicit function theorem, our solution has fundamental 2-form $$\omega^{a,b} + A$$ where $\omega^{a,b}$ is the fundamental 2-form of $g^{a,b}$ and $$\|A\|_{\EE} = O(a^{1-\delta}+ a^{-\delta}b^{2}).$$ Choose $b^2 =a$, $\delta =1/2$, so that $$\sup|w^\delta A| + \sup|w^{\delta+1}\nabla A| +
\sup|w^{\delta+2}\nabla^2 A| = O(b)$$ Then the equation becomes $${{\mathrm d}}(\omega^{a,b}+A) + \beta\wedge(\omega^{a,b} + A) = 0.$$ and so $$\sup w^{\delta+1}|\beta| = O(b).$$ If we define a function $f$ with ${{\mathrm d}}f = \beta$ by integrating along curves starting from any base-point fixed in the region $|z| {\leqslant}1/2$ in $X_1$, it is not hard to see that such $f$ satisfies $$\sup |f| = O(b)$$ so that the Kähler form $\omega_{SFK}$ satisfies $$\omega_{SFK} = \omega^{a,b} + O(b)$$ as claimed.
Summary: the proof of Theorem \[maintheoparab\]
-----------------------------------------------
Let us explain first how our work applies to prove that ${\widehat{M}}$ carries a SFK metric. By Theorem \[bimero\], ${\widehat{M}}$ is the minimal resolution of singularities of the orbifold ${\overline{M}}$. By Theorem \[theodefmbar\], ${\overline{M}}$ is SFK, and ${\mathfrak a}_0({\overline{M}}) = 0$. Hence by Theorem \[theoglue\], ${\widehat{M}}$ carries SFK metrics.
Next let us note how to handle blow-ups of ${\widehat{M}}$. The whole argument of this section goes through to handle ordinary blow-ups: there is a SFK metric on blow-up of the origin of $\CC^2$; this is the “Burns metric”, but it also arises by taking $p=0$, $q=1$ in Proposition \[propcaldsingmet\]. The only thing to check is that ${\mathfrak a}_0({\widehat{M}})=0$, but this follows from the fact that ${\mathfrak a}_0({\overline{M}}) =0$.
Asymptotics of the ALE scalar-flat Kähler metrics {#s1.1.30.04}
=================================================
This section is devoted to a proof of Proposition \[p1.30.1.04\]. We shall make extensive use of the notation of [@CS].
Recall that the metric on $X_{p,q}$ is determined by a choice of real numbers $y_0> y_1 > y_2 > \cdots > y_k > y_{k+1}=0$, and the pairs $(m_j,n_j)$ coming from the continued fraction expansion of $p/q$. Here $(m_{k+1},n_{k+1}) = (q,p)$ and we define $(m_{k+2},n_{k+2}) = (0,1)$. For comparison with [@CS], we note that $p$ and $q$ have been reversed and we normalize $y_{k+1}$ to be $0$. We shall also use half-space coordinates $x>0$ and $y$, (rather than $(\rho,\eta)$) so the hyperbolic metric becomes $x^{-2}(dx^2+dy^2)$.
If $$(a_j,b_j) = (m_j - m_{j+1}, n_j - n_{j+1})\mbox{ for }j=0,1,\ldots, k+1,$$ then we can define $$\label{e1.14.11.03}
v_1 = \frac{x}{2}\sum_{j=0}^{k+1}
\frac{(a_j, b_{j})}{\sqrt{x^2 + (y - y_j)^2}}
=
\frac{x}{2\sqrt{x^2 + y^2}}(q,p-1) +
\frac{x}{2}\sum_{j=0}^k\frac{(a_j, b_{j})}{\sqrt{x^2 + (y - y_j)^2}},$$ and $$\label{e2.14.11.03}
v_2 = \frac{1}{2}\sum_{j=0}^{k+1}
\frac{(y-y_j)(a_j, b_j)}{\sqrt{x^2 + (y - y_j)^2}}
=
\frac{1}{2\sqrt{x^2 + y^2}}(q,p-1) +
\frac{1}{2}\sum_{j=0}^k\frac{(y-y_j)(a_j, b_{j})}{\sqrt{x^2 + (y - y_j)^2}},$$ giving an ALE scalar-flat Kähler metric $$\label{e3.14.11.03}
g
=\frac{ x|\langle v_1, v_2\rangle|}{x^2 + y^2}\left(
\frac{{{\mathrm d}}x^2 + {{\mathrm d}}y^2}{x^2}
+ \frac{\langle v_1, {{\mathrm d}}t \rangle^2 + \langle v_2, {{\mathrm d}}t \rangle^2}{
\langle v_1, v_2\rangle^2}\right)$$ where $t$ is an $T^2$-valued flat coordinate and $\langle\cdot,\cdot\rangle$ denotes the standard symplectic form on $\RR^2$.
The asymptotic region of this metric corresponds to the point $(x,y)=(0,0)$ in the boundary of the hyperbolic plane. We shall analyze this metric first by introducing coordinates $$\label{e4.14.11.03}
R^2e^{2i\theta} = \frac{1}{y - i x}$$ so that $$\label{e5.14.11.03}
x = R^{-2}\sin 2\theta,\; y = R^{-2}\cos 2\theta,\;\;
R>0,0{\leqslant}\theta {\leqslant}\pi/2.$$
Before attempting the computation of this metric in these coordinates, it is worth writing the standard metric on $\CC^2$ in analogous coordinates. Namely, if $({\widetilde{z}}_1,{\widetilde{z}}_2)$ are standard complex coordinates, let $$\label{e10.6.2.04}
{\widetilde{z}}_1 = Re^{i\phi}\cos\theta,\;\;{\widetilde{z}}_2 = Re^{i\psi}\sin\theta,\;
R>0, 0{\leqslant}\theta {\leqslant}\pi/2.$$ Then the standard euclidean metric becomes $$|{{\mathrm d}}{\widetilde{z}}_1|^2 + |{{\mathrm d}}{\widetilde{z}}_2|^2 =
{{\mathrm d}}R^2 + R^2{{\mathrm d}}\theta^2 + R^2(\cos^2\theta {{\mathrm d}}\phi^2
+ \sin^2\theta {{\mathrm d}}\psi^2).$$
Now return to the metric . We shall try to understand it for $R\gg 0$. Referring first to we note that each term in the sum from $0$ to $k$ is $O(R^{-2})$, so $$\label{e6.14.11.03}
v_1 = \frac{\sin 2\theta}{2}(q,p-1) + O(R^{-2}).$$ The $j$-th term in the sum in is $-(a_j,b_j )+ O(R^{-2})$, so $$\label{e3.24.11.03}
v_2 = \frac{1}{2}(1+\cos 2\theta)(q,p-1)) + (0,1) + O(R^{-2}).$$
Now define new angular variables $$\label{e7.14.11.03}
\psi = t_1/q,\; \phi = (p/q) t_1 -t_2.$$ The fact that the determinant is $q^{-1}$ means that $(\phi,\psi)$ really live on a $q$-fold cover. $$\label{e8.14.11.03}
\langle v_1,{{\mathrm d}}t\rangle = q \sin\theta\cos\theta({{\mathrm d}}\psi - {{\mathrm d}}\phi) + O(R^{-2})$$ and $$\label{e9.14.11.03}
\langle v_2,{{\mathrm d}}t\rangle = -q\sin^2\theta\,{{\mathrm d}}\psi - q
\cos^2\theta\,{{\mathrm d}}\phi
+ O(R^{-2})$$ Hence the “angular part” of the metric is given by $$\langle v_1, {{\mathrm d}}t \rangle^2 + \langle v_2, {{\mathrm d}}t \rangle^2
= q^2 \sin^2\theta {{\mathrm d}}\psi^2 + q^2 \cos^2\theta {{\mathrm d}}\phi^2 + O(R^{-2})$$
On the other hand, from , $$\label{e1.19.11.03}
y - i x = R^{-2}e^{-2i\theta},\;\; {{\mathrm d}}x^2 + {{\mathrm d}}y^2 = 4R^{-4}({{\mathrm d}}R^2
+ R^2{{\mathrm d}}\theta^2).$$ Since $$\label{e2.19.11.03}
\langle v_1, v_2\rangle = \frac{q}{2}\sin 2\theta + O(R^{-2})$$ we obtain for the metric $$\label{e3.19.11.03}
2q\left( {{\mathrm d}}R^2 + R^2{{\mathrm d}}\theta^2 + R^2[\alpha^2
\sin^2\theta + \beta^2\cos^2\theta]\right) + \mbox{ lower order terms
}.$$ Here the “lower order terms” are just terms that are of order 2 less than in the given expression for the metric (i.e. there are terms like $O(R^{-2}){{\mathrm d}}R^2$ and $O(1)\alpha^2$ etc.)
Thus the metric, pulled back to a uniformizing chart, differs from a constant multiple of the standard flat metric by tensors of order $R^{-2}$, as measured by the flat metric.
The complex structure
---------------------
The metric is Kähler with respect to the complex structure $J$, $$\label{e1.20.11.03}
J {{\mathrm d}}t =
\frac{1}{\sqrt{x^2+y^2}}\left(
(xv_1 - yv_2)\frac{{{\mathrm d}}x}{x} + (yv_1 + x v_2)\frac{{{\mathrm d}}y}{x}\right).$$ It follows that the two components of ${{\mathrm d}}t + iJ{{\mathrm d}}t$ are $(1,0)$-forms. In fact, a simple calculation shows that these components are closed, hence holomorphic.
With respect to the coordinates introduced above, we can rewrite as follows $$\label{e1.24.11.03}
J {{\mathrm d}}t = - \frac{v_1}{\sin\theta \cos\theta}{{\mathrm d}}\log R
- \frac{v_2}{\sin \theta \cos\theta} {{\mathrm d}}\theta.$$ (To check this, collect the terms in $v_1$ and $v_2$ before changing variables.) Now with $\psi$ and $\phi$ as before, one obtains from and that $$\omega_1:= {{\mathrm d}}[ i\phi + \log R + \cos\theta] + O(1/R^2) \mbox{ and }
\omega_2 := {{\mathrm d}}[ i\psi + \log R + \sin\theta] + O(1/R^2);$$ are closed holomorphic $1$-forms. Now shift to the complex coordinates $({\widetilde{z}}_1,{\widetilde{z}}_2)$, which are [*not*]{} $J$-holomorphic. We have $$\omega_j = \frac{{{\mathrm d}}{\widetilde{z}}_j}{{\widetilde{z}}_j} + F_j({\widetilde{z}}_1,{\widetilde{z}}_2)$$ where the $1$-forms $F_j$ are $O(1/R^2)$. Now set $$f_j({\widetilde{z}}_1,{\widetilde{z}}_2) = -\int_{({\widetilde{z}}_1,{\widetilde{z}}_2)}^\infty F_j$$ where the path of integral is $t\mapsto (t{\widetilde{z}}_1,t{\widetilde{z}}_2)$, $t$ from $1$ to $\infty$. Then ${{\mathrm d}}f_j = F_j$ and ${z}_j :={\widetilde{z}}_j\exp(f_j)$ are $J$-holomorphic coordinates near infinity close to the standard ones: $$|z_j - {\widetilde{z}}_j | = O(1/R)\mbox{ for }R\gg 0,$$ since $f_j= O(1/R)$. Re-expanding the metric in the coordinates $(z_1,z_2)$ will add some $1/R$-terms to the expansion to , but this is sufficient to complete the proof of Proposition \[p1.30.1.04\].
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[^1]: I.e. $M$ could contain divisors that can be blown down without introducing singularities
[^2]: Strictly, of $\CC\times {\mathbb{CP}}^1$ rather than $\Delta\times {\mathbb{CP}}^1$, but this is not important in this discussion.
|
---
abstract: 'Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $\Phi^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $\Phi^*_n(q){\leqslant}cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.'
address:
- 'Centre for Mathematics of Symmetry and Computation, University of Western Australia; also affiliated with The Department of Mathematics, University of Canberra.'
- 'Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany.'
- 'Department of Mathematics and Statistics, NUI Maynooth, Ireland. '
- 'Centre for Mathematics of Symmetry and Computation, University of Western Australia; also affiliated with King Abdulaziz University, Jeddah, Saudi Arabia.'
author:
- 'S.P. Glasby'
- Frank Lübeck
- 'Alice C. Niemeyer'
- 'Cheryl E. Praeger'
title: |
Primitive prime divisors and the\
${\large \boldsymbol{n}}$-th cyclotomic polynomial
---
*Dedicated to the memory of our esteemed colleague L.G. (Laci) Kovács*
Introduction
============
In 1974 Christoph Hering [@Hering] classified the subgroups $G$ of the general linear group ${\textup{GL}}(n,{\mathbb{F}}_q)$ which act transitively on the nonzero vectors $({\mathbb{F}}_q)^n\setminus\{0\}$. In his investigations a certain number theoretic function, $\Phi^*_n(q)$, plays an important role. It divides the $n$th cyclotomic polynomial evaluated at a prime power $q$, and hence divides $|({\mathbb{F}}_q)^n\setminus\{0\}|=q^n-1$. It is not hard to prove that ${\textup{GL}}(n,{\mathbb{F}}_q)$ contains an element of order $\Phi^*_n(q)$, and every element $g$ of ${\textup{GL}}(n,{\mathbb{F}}_q)$ whose order is not coprime to $\Phi^*_n(q)$ acts irreducibly on the natural module $({\mathbb{F}}_q)^n$, c.f. [@Hering Theorem 3.5]. A key result [@Hering p.1] shows that if $1<\gcd(|G|,\Phi^*_n(q)){\leqslant}(n+1)(2n+1)$, then the structure of $G$ is severely constrained.
Our definition below of $\Phi_n^*(q)$ differs from the one used by Hering [@Hering p.1], Lüneburg [@L Satz 2] and Camina and Whelan [@CW Theorem 3.23], who used the definition in Lemma \[Ldef\]. We show in Section \[sec:equivalentdefs\] that our definition is equivalent to theirs and that $\Phi_n^\ast(q)$ could have also been defined in several other ways.
\[def:phistar\] Suppose $n,q\in{\mathbb{Z}}$ are such that $n{\geqslant}1$ and $q{\geqslant}2$. Write $\Phi_n(X)$ for the $n$-th cyclotomic polynomial $\prod_\zeta (X-\zeta)$ where $\zeta$ ranges over the primitive complex $n$-th roots of unity. Let $\Phi_n^*(q)$ be the largest divisor of $\Phi_n(q)$ which is coprime to $\prod_{1 {\leqslant}k < n}(q^k-1)$.
Our definition of $\Phi_n^*(q)$ is motivated by the numerous applications of primitive prime divisors, see [@NieP] or [@BambergPenttila; @GPPS]. As our primary motivation is geometric, we will assume later (after Section \[M\]) that $q$ is a prime power; before this point $q{\geqslant}2$ is arbitrary unless otherwise stated. A divisor $m$ of $q^n-1$ is called a [*strong primitive divisor of $q^n-1$*]{} if $\gcd(m,q^k-1)=1$ for $1 {\leqslant}k < n$, and a [*weak primitive divisor of $q^n-1$*]{} if $m\nmid (q^k-1)$ for $1 {\leqslant}k < n$. By our definition, $\Phi_n^*(q)$ is the largest strong primitive divisor of $q^n-1$. A primitive divisor of $q^n-1$ which is prime is called a [*primitive prime divisor*]{} ([*ppd*]{}) of $q^n-1$ or a Zsigmondy prime (“strong” equals “weak” for primes). DiMuro [@DiMuro] uses weak primitive [*prime power*]{} divisors or [*pppds*]{} to extend the classification in [@GPPS] to $d/3< n{\leqslant}d$. Our application in Section \[sec:Ac\] has $d/4{\leqslant}n{\leqslant}d$.
Primitive prime divisors have been studied since Bang [@Bang] proved in 1886 that $q^n-1$ has a primitive prime divisor for all $q{\geqslant}2$ and $n>2$ except for $q=2$ and $n=6$. Given coprime integers $q>r{\geqslant}1$ and $n>2$, Zsigmondy [@Z] proved in 1892 that there exists a prime $p$ dividing $q^n-r^n$ but not $q^k-r^k$ for $1{\leqslant}k<n$ except when $q=2$, $r=1$, and $n=6$. The Bang-Zsigmondy theorem has been reproved many times as explained in [@Rib p.27] and [@DHP p.3]; modern proofs appear in [@L; @R]. Feit [@Feit] studied ‘large Zsigmondy primes’, and these play a fundamental role in the recognition algorithm in [@NieP]. Hering’s results in [@Hering] influenced subsequent work on linear groups, including the classification of linear groups containing primitive prime divisor (ppd)-elements [@GPPS], and its refinements in [@BambergPenttila; @DiMuro; @NieP].
We describe algorithms in Sections \[M\] and \[M\*\] which, given positive constants $c$ and $k$, list all pairs $(n, q)$ for which $n{\geqslant}3$ and $\Phi^*_n(q){\leqslant}c n^k$. The behaviour of $\Phi_n^*(q)$ for $n=2$ is different from that for larger $n$ (see Lemma \[Ldef\](\[LdefC\]) and Algorithm \[alg:Mstarn2\]).
\[the:one\] Let $q{\geqslant}2$ be a prime power.
1. There is an algorithm which, given constants $c,k>0$ as input, outputs all pairs $(n,q)$ with $n{\geqslant}3$ and $q{\geqslant}2$ a prime power such that $\Phi^*_n(q){\leqslant}cn^k$.
2. If $n{\geqslant}3$, then $\Phi^*_n(q){\leqslant}n^4$ if and only if $(n,q)$ is listed in *Tables \[tab:onea\], \[tab:six\]* or *\[tab:oneb\]*. Moreover, the prime powers $q$ with $q{\leqslant}5000$ and $\Phi^*_2(q){\leqslant}2^4=16$ are listed in *Table \[tab:q=2\]*.
In some group theoretic applications we need explicit information about $\Phi_n^*(q)$ when this quantity is considerably larger than $n^4$, but we have tight control over the sizes of its ppd divisors (each of which must be of the form $in+1$ by Lemma \[lemma:phiq\](\[lemma:phiq1modn\])). We give an example of this kind of result in Theorem \[the:two\], where we require that the ppd divisors are sufficiently small for our group theoretic application in Section \[sec:Ac\]. This motivated our effort to strengthen Hering’s result and we discovered two missing cases in [@Hering Theorem 3.9]; see Remark \[rem:one\]. We list in Theorem \[the:one\] all pairs $(n,q)$ with $n {\geqslant}3$ and $q{\geqslant}2$ a prime power for which $\Phi^*_n(q){\leqslant}n^4$; the implementations in [@G] can handle much larger cases like $\Phi^*_n(q){\leqslant}n^{20}$. In Theorem \[the:two\] we also require that the ppd divisors of $\Phi^*_n(q)$ be small for our group theoretic application in Section \[sec:Ac\].
\[the:two\] Suppose that $q{\geqslant}2$ is a prime power and $n{\geqslant}3$. Then all possible values of $(n, q)$ such that $\Phi^*_n(q)$ has a prime factorisation of the form $\prod_{i=1}^4 (i n+1)^{m_i},$ with $0{\leqslant}m_1{\leqslant}3$ and $0{\leqslant}m_2,m_3,m_4{\leqslant}1$ are listed in *Table \[tab:two\]*.
The proof of Theorem \[the:one\](a) rests on the correctness of Algorithms \[alg:M\] and \[alg:Mstarnbig\] which are proved in Sections \[M\] and \[M\*\]. Theorem \[the:one\](b) and \[the:two\] follow by applying these algorithms. For Theorem \[the:two\] we observe that $\Phi_n^\ast(q) {\leqslant}(n+1)^3\prod_{i=2}^4 (i n+1) {\leqslant}16n^7$ for all $n{\geqslant}4$, whereas for $n=3$ only $2n+1$ and $4n+1$ are primes and again $\Phi_n^\ast(q) {\leqslant}7\cdot13 {\leqslant}16n^7.$ Thus the entries in Table \[tab:two\] were obtained by searching the output of our algorithms to find the pairs $(n,q)$ for which $\Phi^*_n(q){\leqslant}16n^7$ and has the given factorisation. This factorisation arose from the application (Theorem \[the:Ac\]) in Section \[sec:Ac\].
\[rem:one\] The missing cases in part (d) of [@Hering Theorem 3.9] had $\Phi^*_n(q)=(n+1)^2$. We discovered the possibilities $n=2, q=17,$ and $n=2, q=71$ when comparing Hering’s result with output of the [@BCP] and [@GAP] implementations of our algorithms, see Table \[tab:q=2\].
Cyclotomic polynomials: elementary facts
========================================
The product $\prod_{1 {\leqslant}k < n}(q^k-1)$ has no factors when $n=1$. An empty product is 1, by convention, and so $\Phi_1^*(q)=\Phi_1(q)=q-1$.
The Möbius function $\mu$ satisfies $\mu(n)=(-1)^k$ if $n=p_1\cdots p_k$ is a product of distinct primes, and $\mu(n)=0$ otherwise. Our algorithm uses the following elementary facts.
\[lemma:phiq\] Let $n$ and $q$ be integers satisfying $n{\geqslant}1$ and $q{\geqslant}2$.
1. \[lemma:phiqmoebius\] The polynomial $\Phi_n(X)$ lies in ${\mathbb{Z}}[X]$ and is irreducible. Moreover, $$X^n-1 = \prod_{d|n} \Phi_d(X)\qquad\textrm{and}\qquad
\Phi_n(X) = \prod_{d|n} (X^\frac{n}{d}-1)^{\mu(d)}.$$
2. \[lemma:phidivide\] If $d\mid n$ and $d>1$, then $\Phi_n(X)$ divides $(X^n-1)/(X^{n/d}-1)=\sum_{i=0}^{d-1} (X^{n/d})^i$.
3. \[lemma:phiq1modn\] If $r$ is a prime and $r\mid \Phi_n^*(q)$, then $n$ divides $r-1$, equivalently $r \equiv 1\pmod{n}$.
4. \[lemma:phiqincreasing\] For any fixed integer $n{\geqslant}1$ the function $\Phi_n(q)$ is strictly increasing for $q>1$.
5. \[lemma:phiqeuler\] Let $\varphi$ be Euler’s totient function which satisfies $\varphi(n) = \deg(\Phi_n(X))$. Then $$\varphi(n) {\geqslant}\frac{n}{\log_2(n)+1}\qquad\textup{for $n{\geqslant}1$.}$$
6. \[lemma:phiqbound\] For all $n{\geqslant}2$ and $q{\geqslant}2$ we have $q^{\varphi(n)}/4< \Phi_n(q)<4q^{\varphi(n)}$.
(\[lemma:phiqmoebius\]) The irreducibility of $\Phi_n(X)\in{\mathbb{Z}}[X]$ and the other facts, are proved in [@DF §13.4].
(\[lemma:phidivide\]) By part $(X^n-1)/(X^{n/d}-1)$ equals $\prod_k\Phi_k(X)$ where $k\mid n$ and $k\nmid (n/d)$. Since $d>1$, it follows that $\Phi_n(X)$ is a factor in this product.
(\[lemma:phiq1modn\]) If $r\mid\Phi_n^*(q)$ then $r\mid(q^n-1)$ and $n$ is the order of $q$ modulo $r$, so $n\mid(r-1)$.
(\[lemma:phiqincreasing\]) This follows from Definition \[def:phistar\] because $\Phi_n(q)=|\Phi_n(q)|=\prod_\zeta|q-\zeta|$ and $|\zeta|=1$.
(\[lemma:phiqeuler\]) We use the formula $\varphi(n) = n \prod_{i=1}^t \frac{p_i-1}{p_i}$ where $p_1 < p_2 < \cdots < p_t$ are the prime divisors of $n$. Using the trivial estimate $p_i {\geqslant}i+1$ we get $\varphi(n) {\geqslant}n/(t+1)$. It follows from $2^t{\leqslant}p_1p_2\cdots p_t{\leqslant}n$ that $t {\leqslant}\log_2(n)$. Hence $\varphi(n) {\geqslant}n/(\log_2(n)+1)$ as claimed.
(\[lemma:phiqbound\]) Using the product formula for $\Phi_n(X)$ in (\[lemma:phiqmoebius\]) and $\mu(d) \in \{0,-1,1\}$, we see that $\Phi_n(q)$ equals $q^{\varphi(n)}$ times a product of distinct factors of the form $(1-1/q^i)^{\pm 1}$ with $1 {\leqslant}i {\leqslant}n$. Since $\prod_{i=1}^\infty (1-1/q^i) {\geqslant}\prod_{i=1}^\infty
(1-1/2^i) = 0.28878\cdots > 1/4$ we get $$\frac{q^{\varphi(n)}}{4} < \Phi_n(q) < 4 q^{\varphi(n)}.$$
Hering [@Hering Theorem 3.6] gives sharper estimates than those in Lemma \[lemma:phiq\](\[lemma:phiqbound\]). But our (easily established) estimates suffice for the efficient algorithms below.
Equivalent definitions of {#sec:equivalentdefs}
==========================
We now state equivalent ways in which to define $\Phi_n^\ast(q)$ where $q{\geqslant}2$ is an integer. Because our motivation for studying $\Phi^*_n(q)$ arose from finite geometry, we assume after the proof of Lemma \[Ldef\] that $q$ is a prime power. Observe that Lemma \[Ldef\] suggests a much faster algorithm for computing $\Phi_n^*(q)$ than does Definition \[def:phistar\].
\[Ldef\] Let $n,q$ be integers such that $n{\geqslant}2$ and $q{\geqslant}2$. The following statements could be used as alternatives to the definition of $\Phi^*_n(q)$ given in Definition \[def:phistar\].
1. \[LdefA\] $\Phi_n^*(q)$ is the largest divisor of $\Phi_n(q)$ coprime to $\prod_{k\mid n,\,k<n}\Phi_k(q)$.
2. \[LdefC\] Let $(q+1)_2$ be the largest power of $2$ dividing $q+1$, and let $r$ be the largest prime divisor of $n$. Then $$\Phi_n^*(q) = \begin{cases} (q+1)/(q+1)_2\quad\quad&\textup{if $n=2$,}\\
\Phi_n(q)&\textup{if $n>2$ and $r\nmid\Phi_n(q)$,}\\
\Phi_n(q)/r &\textup{if $n>2$ and $r\mid\Phi_n(q)$.}\end{cases}$$
3. \[LdefD\] $\Phi_n^*(q)=\Phi_n(q)/f^i$ where $f^i$ is the largest power of $f:=\gcd(\Phi_n(q),n)$ dividing $\Phi_n(q)$.
\[Rem\] For $n>2$ the last paragraph of the proof of part shows that $d:=\gcd(\Phi_n(q),\prod_{1{\leqslant}k<n}(q^i-1))$ equals $f:=\gcd(\Phi_n(q),n)$. Either $d=f=1$ and $r\nmid\Phi_n(q)$, of $d=f=r$ and $r\mid\Phi_n(q)$. Thus, part simplifies to $\Phi_n^*(q)=\Phi_n(q)/f$ when $n>2$.
We use the following notation where $m$ is a divisor of $\Phi_n(q)$: $$\begin{aligned}
P_n&=\prod_{1{\leqslant}k<n}(q^k-1), &P_n^{\,\prime}&=\prod_{k\mid n,\,k<n}\Phi_k(q),\\
d_n(m)&=\gcd(m,P_n), \quad &d^{\,\prime}_n(m)&=\gcd(m,P_n^{\,\prime}).\end{aligned}$$
Fix a divisor $m$ of $\Phi_n(q)$. We prove that $d_n(m)=1$ holds if and only if $d^{\,\prime}_n(m)=1$. Certainly $d_n(m)=1$ implies $d^{\,\prime}_n(m)=1$ as $P_n^{\,\prime}\mid P_n$. Conversely, suppose that $d_n(m)\ne1$. Then there exists a prime divisor $r$ of $m$ that divides $q^k-1$ for some $k$ with $1{\leqslant}k<n$. However, $r\mid\Phi_n(q)\mid (q^n-1)$ and $\gcd(q^n-1,q^k-1)=q^{\gcd(n,k)}-1$, so $r$ divides $q^{\gcd(n,k)}-1$. Hence $r$ divides $\Phi_\ell(q)$ for some $\ell\mid\gcd(n,k)$ by Lemma \[lemma:phiq\](\[lemma:phiqmoebius\]). In summary, $r\mid d_n(m)$ implies $r\mid d^{\,\prime}_n(m)$, so $d_n(m)\ne1$ implies $d^{\,\prime}_n(m)\ne1$.
For any divisor $m$ of $\Phi_n(q)$ we have shown that $\gcd(m,P_n)=1$ holds if and only if $\gcd(m,P_n^{\,\prime})=1$. Thus the largest divisor of $\Phi_n(q)$ coprime to $P_n^{\,\prime}$ is equal to the largest such divisor which is coprime to $P_n$, and this is $\Phi^*_n(q)$ by Definition \[def:phistar\].
First consider the case $n=2$. Now $d:=d_2(\Phi_2(q))=\gcd(q+1, q-1)$ divides 2. Indeed, $d=1$ for even $q$, and $d=2$ for odd $q$. In both cases, $(q+1)/(q+1)_2$ is the largest divisor of $q+1$ coprime to $q-1$. Thus $\Phi^*_2(q)=(q+1)/(q+1)_2$ by Definition \[def:phistar\].
Assume now that $n>2$. Let $d=\gcd(\Phi_n(q),P_n)$ where $P_n=\prod_{1{\leqslant}k<n}(q^k-1)$. If $d=1$, then $\Phi_n^*(q) = \Phi_n(q)$ by Definition \[def:phistar\]. Suppose that $d>1$ and $p$ is a prime divisor of $d$. Then the order of $q$ modulo $p$ is less than $n$, and Feit [@Feit] calls $p$ a non-Zsigmondy prime. It follows from [@R Proposition 2] or Lüneburg [@L Satz 1] that the prime $p$ divides $\Phi_n(q)$ exactly once, and $p=r$ is the largest prime divisor of $n$. Thus we see that $\gcd(\Phi_n(q)/r,P_n)=1$ and $\Phi_n^*(q) = \Phi_n(q)/r$ by Definition \[def:phistar\]. This proves .
To connect with part , we prove when $n>2$ that $d$ equals $f:=\gcd(\Phi_n(q),n)$. Indeed, we prove Remark \[Rem\] that either $d=f=1$ and $r\nmid\Phi_n(q)$, or $d=f=r$ and $r\mid\Phi_n(q)$. If $d=1$, then $\Phi_n^*(q) = \Phi_n(q)$ and a prime divisor $p$ of $\Phi_n^*(q)$ satisfies $p\equiv1\pmod n$ by Lemma \[lemma:phiq\](\[lemma:phiq1modn\]) and hence $p\nmid n$. Thus $f=1$ and $r\nmid\Phi_n(q)$ since $r\mid n$. Conversely, suppose that $d>1$. The previous paragraph shows that $d=r$ and $r^2\not\mid\Phi_n(q)$. Thus $r\mid f$. Let $p$ be a prime dividing $f=\gcd(\Phi_n(q),n)$. Since $\Phi_n(q) \mid (q^n-1)$, we have $p\mid(q^n-1)$, and hence $p\not\mid\Phi^*_n(q)$ by Lemma \[lemma:phiq\](\[lemma:phiq1modn\]). Thus $p$ divides $P_n$ by Definition \[def:phistar\], and hence $p$ divides $d=\gcd(\Phi_n(q),P_n)$. However, $d=r$ and so $p=r=f$, and in this case $r\mid\Phi_n(q)$.
By part and the last paragraph of the proof of , Definition \[def:phistar\] is equivalent to Hering’s definition [@Hering] in part .
\[REM\] When $q$ is a prime power, there is a fourth equivalent definition: $\Phi_n^*(q)$ is the order of the largest subgroup of ${\mathbb{F}}_{q^n}^\times$ (the multiplicative group of $q^n-1$ nonzero elements of ${\mathbb{F}}_{q^n}$) that intersects trivially all the subgroups ${\mathbb{F}}_{q^d}^\times$ for $d\mid n$, $d<n$.
The correspondence $H\leftrightarrow |H|$ is a bijection between the subgroups $H$ of the cyclic group ${\mathbb{F}}_{q^n}^\times$ and the divisors of $q^n-1$. Suppose $d\mid n$. Note that $H\cap {\mathbb{F}}_{q^d}^\times=\{1\}$ holds if and only if $\gcd(|H|,q^d-1)=1$ as ${\mathbb{F}}_{q^n}^\times$ is cyclic. Thus there exists a unique subgroup $H$ whose order $m$ is maximal subject to $H\cap {\mathbb{F}}_{q^d}^\times=\{1\}$ for all $d\mid n$, $d<n$. Hence $m$ is the largest divisor of $q^n-1$ satisfying $\gcd(m,q^d-1)=1$ for all $d\mid n$, $d<n$. Since $q^n-1=\prod_{d\mid n}\Phi_d(q)$ and $\Phi_d(q)\mid q^d-1$, we see that $m\mid \Phi_n(q)$. It follows from Lemma \[Ldef\] that $\Phi^*_n(q)=m$.
The polynomial bound {#M}
=====================
As we will discuss in Section \[M\*\], the number of pairs $(2,q)$ with $q$ a prime power satisfying $\Phi_2(q){\leqslant}c 2^k$ is potentially infinite. We therefore deal here with pairs $(n,q)$ for $n{\geqslant}3$. Given positive constants $c$ and $k$, we now describe an algorithm for determining all pairs in the set $$M(c,k) := \{(n,q)\in {\mathbb{Z}}\times{\mathbb{Z}}\mid n{\geqslant}3, q{\geqslant}2\textup { a prime power, and }\Phi_n(q) {\leqslant}c n^k\}.$$
\[alg:M\] 1.5mm [**Input:**]{} Positive constants $c$ and $k$.[**Output:**]{} The finite set $M(c,k)$.
\#1[\[\#1\]]{}
1. Set $s := 2+\log_2(c)$, $t := (s+k)/\ln(2)$, $u := k/\ln(2)^2$ and $b := e^{1-t/(2u)}$ and define for $x{\geqslant}3$ the function $g(x) := x-s-t\ln(x)-u\ln(x)^2$ where $\ln(x)=\log_e(x)$. Note that $g(x)$ has derivative $g'(x) := 1-t/x-2u\ln(x)/x$.
2. Set $n:=3$ and set $M(c,k)$ to be the empty set.
3. If $n > b$ and $g(n)>0$ and $g'(n)>0$ then return $M(c,k)$.
4. If $g(n) < 0$ and $2^{\varphi(n)-2} < cn^k$ then compute $\Phi_n(X)$ and find the smallest prime power $\tilde q$ such that $\Phi_n(\tilde q) > cn^k$; add $(n,q)$ to $M(c,k)$ for all prime powers $q<\tilde q$.
5. Set $n := n+1$ and go back to step \[alg:M\].3.
Algorithm \[alg:M\] starts with $n=3$ and it continues to increment $n$. We must prove that it does terminate at step \[alg:M\].3, and that it correctly returns $M(c,k)$. Note first that for fixed $n$ the values $\Phi_n(q)$ are strictly increasing with $q$ by Lemma \[lemma:phiq\](\[lemma:phiqincreasing\]). Thus it follows from Lemma \[lemma:phiq\] and that $$\Phi_n(q){\geqslant}\Phi_n(2) > \frac{2^{\varphi(n)}}{4}=2^{\varphi(n)-2}
{\geqslant}2^{n/(\log_2(n)+1) - 2}.$$ Consider the inequality $2^{n/(\log_2(n)+1) - 2} {\geqslant}c n^k$. Taking base-2 logarithms shows $$\begin{aligned}
n&{\geqslant}(k\log_2(n)+\log_2(c)+2)(\log_2(n)+1)\\
&=(\log_2(c)+2)+(k+\log_2(c)+2)\log_2(n)+k\log_2(n)^2\\
&=s+t\ln(n)+u\ln(n)^2\end{aligned}$$ where the last step uses $\log_2(n)=\ln(n)/\ln(2)$ and the definitions in step \[alg:M\].1. In summary, $2^{n/(\log_2(n)+1) - 2} {\geqslant}c n^k$ is equivalent to $g(n) {\geqslant}0$ with $g(n)$ as defined in step \[alg:M\].1.
The inequalities above show that the conditions $g(n) < 0$ and $2^{\varphi(n)-2} < cn^k$, which we test in step \[alg:M\].4, are necessary for $\Phi_n(2) {\leqslant}c n^k$. We noted above that for fixed $n$ the values of $\Phi_n(q)$ strictly increase with $q$. Thus (if executed for a particular $n$) step \[alg:M\].4 correctly adds to $M(c,k)$ all pairs $(n,q)$ for prime powers $q$ such that $\Phi_n(q) {\leqslant}cn^k$.
It remains to show (i) that the algorithm terminates, and (ii) that the returned set $M(c,k)$ contains *all* pairs $(n,q)$ such that $\Phi_n(q) {\leqslant}cn^k$. The second derivative of $g(x)$ equals $g''(x) = (t- 2u(1-\ln(x)))/x^2$. Since $u>0$ this shows that $g''(x) > 0$ if and only if $x > b = e^{1-t/(2u)}$. Thus $g'(x)$ is increasing for all $x>b$. Because $x$ grows faster than any power of $\ln(x)$ we have that $g(x) > 0$ and $g'(x) > 0$ for $x$ sufficiently large. Thus there exists a (smallest) integer $\tilde n$ fulfilling the conditions in step \[alg:M\].3, that is, $\tilde n > b$, $g(\tilde n)>0$ and $g'(\tilde n)>0$. The algorithm terminates when step \[alg:M\].3 is executed for the integer $\tilde n$. To prove that the returned set $M(c,k)$ is complete, we verify that, for all $n{\geqslant}\tilde n$, there is no prime power $q$ such that $\Phi_n(q){\leqslant}cn^k$. Now, for all $x {\geqslant}\tilde n$, we have $x >b$ so that $g'(x)$ is increasing for $x {\geqslant}\tilde n$, and so $g'(x){\geqslant}g'(\tilde n)>0$, whence $g(x)$ is increasing for $x {\geqslant}\tilde n$. In particular, $n {\geqslant}\tilde n$ implies that $g(n){\geqslant}g(\tilde n) > 0$ and so (from our displayed computation above), for all prime powers $q$, $\Phi_n(q){\geqslant}\Phi_n(2) > c n^k$. Thus there are no pairs $(n,q)\in M(c,k)$ with $n{\geqslant}\tilde n$, so the returned set $M(c,k)$ is complete.
Determining when {#M*}
=================
We describe an algorithm to determine all pairs $(n, q)$, with $n,q{\geqslant}2$ and $q$ a prime power, such that the value $\Phi_n^*(q)$ is bounded by a given polynomial in $n$, say $f(n)$. For $n{\geqslant}3$ the algorithm determines the finite list of possible $(n,q)$. For $n=2$ the output is split between a finite list which we determine, and a potentially infinite (but very restrictive) set of prime powers $q$ of the form $2^am-1$ where $m{\leqslant}f(2)$ is odd. Table \[tab:q=2\] lists the prime powers $q{\leqslant}5000$ such that $\Phi_2^*(q){\leqslant}16$; we see that some proper powers occur, though the majority of the entries are primes. For example, if $\Phi_2^*(q) =1$ then the prime powers $q$ of the form $2^a-1$, must be a prime by [@Z]. Such primes are called Mersenne primes. The set $M(c,k)$ of all pairs $(n,q)$ satisfying $\Phi_n(q){\leqslant}cn^k$ is finite by Lemma \[lemma:phiq\](\[lemma:phiqbound\]). By contrast the set of pairs $(n,q)$ satisfying $\Phi^*_n(q){\leqslant}cn^k$ may be infinite as $\Phi^*_2(q)=m$, $m$ odd, may have infinitely many (but highly restricted) solutions for $q$. Algorithm \[alg:Mstarnbig\] computes the following set (which we see below is a finite set) $$M^*_{{\geqslant}3}(c,k) = \left\{ (n,q)\in {\mathbb{Z}}\times{\mathbb{Z}}\mid n{\geqslant}3, q{\geqslant}2\textup { a prime power, and }\Phi^*_n(q) {\leqslant}c n^k\right\}.$$
\[alg:Mstarnbig\] 1.5mm [**Input:**]{} Positive constants $c$ and $k$.[**Output:**]{} The finite set $M^*_{{\geqslant}3}(c,k)$.
1. Compute $M(c,k+1)$ with Algorithm $\ref{alg:M}$.
2. Initialise $M^*_{{\geqslant}3}(c,k)$ as the empty set. For all $(n,q)\in
M(c,k+1)$ with $n {\geqslant}3$ check if $\Phi^*_n(q) {\leqslant}c n^k$. If yes, add $(n,q)$ to $M^*_{{\geqslant}3}(c,k)$.
3. Return $M^*_{{\geqslant}3}(c,k)$.
We need to show that all $M^*_{{\geqslant}3}(c,k) \subseteq M(c,k+1)$. This follows from Lemma \[Ldef\] which shows that $n \Phi^*_n(q) {\geqslant}\Phi_n(q)$ whenever $n {\geqslant}3$.
<span style="font-variant:small-caps;">Case $n=2$.</span> We treat the case $n=2$ separately as the classification has a finite part and a potentially infinite part. Suppose $q$ is odd and $\Phi_2^*(q)=\frac{q+1}{2^a}=m{\leqslant}cn^k$ where $m$ is odd by Lemma \[Ldef\]. Then solving for $q$ gives $q=2^am-1$.
If $m=1$ then $q=2^a-1$ is a (Mersenne) prime as remarked in the first paragraph of this section. Lenstra-Pomerance-Wagstaff conjectured [@LPW] that there are infinitely many Mersenne primes, and the asymptotic density of the set $\{a<x\mid 2^a-1\textrm{ prime}\}$ is ${\rm O}(\log x)$. For fixed $m$ with $m>1$, the number of prime powers of the form $2^am-1$ may also be infinite (although in this case we cannot conclude that $a$ must be prime). The set $$M_2^*(c,k)=\{(2,q)\mid \Phi^*_n(q) {\leqslant}c 2^k \textup{ and $q$ is a prime power} \}$$ is a disjoint union of three subsets: $$\begin{aligned}
R(c,k)&:=\{(2,q)\mid (2,q) \in M_2^*(c,k)\text{ and } q\not\equiv{3}\!\!\!\!\pmod4 \},\\
S(c,k)&:=\{(2,q)\mid (2,q) \in M_2^*(c,k)\text{ and }q\equiv 3\!\!\!\pmod4 \text{ and } q \text{ not prime } \},\\
T(c,k)&:=\{(2,q)\mid (2,q) \in M_2^*(c,k)\text{ and } q \equiv 3\!\!\!\pmod4\textup{
and $q$ prime}\}\\\end{aligned}$$ As the set $T(c,k)$ may be infinite Algorithm \[alg:Mstarn2\] below takes as input a constant $B>0$ and computes the finite subset $T(c,k,B) =\{ (2,q)\mid q \in T(c,k)\textup{ and } q{\leqslant}B\}$ of $M_2^\ast(c,k).$ Table \[tab:q=2\] has $n=2$ and $q{\leqslant}5000$, so we input $B=5000$.
\[alg:Mstarn2\] [**Input:**]{} Positive constants $c, k$ and $B$. [**Output:**]{} The (finite) set $R(c,k)\cup S(c,k)\cup T(c,k,B)$, see the notation above.
1. Initialise each of $R(c,k), S(c,k), T(c,k,B)$ as the empty set.
2. Add $(2,q)$ to $R(c,k)$ when $q$ is a power of $2$ with $q+1 {\leqslant}c 2^k$.
3. Add $(2,q)$ to $R(c,k)$ when $q$ is a prime power, $q \equiv 1
\pmod{4}$ and $(q+1)/2 {\leqslant}c 2^k$.
4. For all primes $p\equiv 3 \pmod{4}$ with $p {\leqslant}B$ and $(p+1)/(p+1)_2{\leqslant}c2^k$ add $(2,p)$ to $T(c,k,B)$. For all primes $p\equiv 3 \pmod{4}$ (where $p{\leqslant}c2^{k-1}$ is allowed) and all odd $\ell{\geqslant}3$ with $\sum_{i=0}^{\ell-1}(-p)^i{\leqslant}c 2^k$ add $(2,p^\ell)$ to $S(c,k)$ if $\Phi_2^*(p^\ell){\leqslant}c 2^k$.
5. Return $R(c,k)\cup S(c,k)\cup T(c,k,B)$.
By Lemma \[Ldef\], $\Phi^*_2(q) = \Phi_2(q) = q+1$ when $q$ is an even prime power and $\Phi^*_2(q) = \Phi_2(q)/2 = (q+1)/2$ if $q \equiv 1 \pmod{4}$. It is clear that steps \[alg:Mstarn2\].2 and \[alg:Mstarn2\].3 find all pairs $(2,q) \in R(c,k)$ with $q \not\equiv 3 \pmod{4}$, and there are finitely many choices for $q$.
Any prime power $q\equiv 3 \pmod{4}$ is an odd power $q=p^\ell$ of a prime $p \equiv 3 \pmod{4}$. Write $q+1 = 2^a m$ with $m$ odd and $a{\geqslant}2$, then $\Phi_2^*(q) = m$. If $q$ is a prime $(2,q) \in T(c,k,B)$ if and only if $q{\leqslant}B$ and $\Phi_2^*(q) {\leqslant}c 2^k$, so step \[alg:Mstarn2\].4 adds such pairs. This is because, when $q\equiv 3\pmod{4}$ and $q{\leqslant}B$ we have, by Lemma \[Ldef\], that $\Phi_2^*(q)=(q+1)/2 {\leqslant}B$. Suppose $q$ is not a prime, that is $\ell>1$. Then we have the factorisation $q+1 = (p+1)(\sum_{i=0}^{\ell-1}(-p)^i)$ where the second factor is odd and so divides $m$. Since $2p^{\ell-2}{\leqslant}p^{\ell-2}(p-1)<\sum_{i=0}^{\ell-1}(-p)^i{\leqslant}m$ and we require $m{\leqslant}c2^k$, we see $p^{\ell-2}{\leqslant}c2^{k-1}$. Since there are finitely many solutions to $p^{\ell-2}{\leqslant}c2^{k-1}$ with $\ell>1$ odd, $S(c,k)$ is a finitely set, and step \[alg:Mstarn2\].4 correctly computes $S(c,k)$. Finally, the disjoint union $R(c,k)\cup S(c,k)\cup T(c,k,B)$ is the desired output set.
Theorem \[the:one\](a) follows from the correctness of Algorithms \[alg:M\] and \[alg:Mstarnbig\], and Theorem \[the:one\](b) uses these algorithms with $(c,k)=(1,4)$. Similarly, Theorem \[the:two\] uses these algorithms with $(c,k)=(16,7)$. It is shown that in the penultimate paragraph of the proof of Theorem \[the:Ac\] that $\Phi^*_n(q){\leqslant}16n^7$ holds for $n{\geqslant}4$. If $n=3$ and $1{\leqslant}i{\leqslant}4$, then $in+1$ is prime for $i=2,4$, and again $\Phi_n^\ast(q) {\leqslant}7\cdot13 {\leqslant}16n^7$ holds. We then search the (rather large) output set for the pairs $(n,q)$ for which $\Phi^*_n(q)$ has the prescribed prime factorisation. [@BCP] code generating the data for Tables 1–5 mentioned in Theorems \[the:one\] and \[the:two\] is available at [@G].
The tables {#sec:tables}
==========
By Lemma \[lemma:phiq\] the prime factorisation of $\Phi_n^*(q)$ has the form $\prod_{i{\geqslant}1}(in+1)^{m_i}$ where $m_i=0$ if $in+1$ is not a prime. It is convenient to encode this prime factorisation as $\Phi_n^*(q)=\prod_{i\in I}(in+1)$ where $I$ is a multiset, and for each $i\in I$ the prime divisor $in+1$ of $\Phi_n^*(q)$ is repeated $m_i$ times in $I=I(n,q)$. For example, $\Phi^*_4(8)=65=(4+1)(3\cdot 4+1)$ so $I(4,8)={\{\kern-1.2pt\{ {1,3}\}\kern-1.2pt\}}$ and $\Phi^*_5(3)=121=(2\cdot 5+1)^2$ so $I(5,3)={\{\kern-1.2pt\{ {2,2}\}\kern-1.2pt\}}$. To save space, we omit the double braces in our tables and denote the empty multiset (corresponding to $\Phi^*_6(2)=1$) by ‘$-$’. All of our data did not conveniently fit into Table \[tab:onea\], so we created subsidiary tables \[tab:q=2\], \[tab:six\], \[tab:oneb\] for $n=2$, $n=6$ and $n{\geqslant}19$, respectively. For $n$ and $q$ such that $\Phi_n^*(q){\leqslant}n^4$ Tables \[tab:onea\] and \[tab:oneb\] record in row $n$ and column $q$ the multiset $I(n,q).$ The tables are the output from Algorithm \[alg:Mstarnbig\] with $c=1$ and $k = 4$.
Table \[tab:two\] exhibits data for two different theorems. For Theorem \[the:two\] we record the triples $(n,q,I)$ for which $n{\geqslant}3$ and $\Phi_n^*(q)$ has prime factorisation $\prod_{i \in I}(i\,n+1)$ where $I\subseteq{\{\kern-1.2pt\{ {1, 1, 1, 2, 3, 4}\}\kern-1.2pt\}}.$ For Theorem \[the:Ac\] we also list the possible degrees $c$ that can arise, namely $c_0{\leqslant}c{\leqslant}c_1$.
$n\backslash q$ $2$ $3$ $4$ $5$ $7$ $8$ $9$ $11$ $13$ $17$ $19$
----------------- ----- ------ ------- ------ ------ ------ ------ ------ ------ ------ ------
2
3 2 4 2 10 6 24 20
4 1 1 4 3 1,1 1,3 10 15 1,4 1,7 45
5 6 2, 2 2,6
6
7 18 156
8 2 5 32 39 150 2,24
9 8 84 2,8
10 1 6 4 52 1,19 1,33 118
11 2,8
12 1 6 20 50 1,15 3,9 540 1,93
13 630
14 3 39 2,8 2,32
15 10 304 10,22
16 16 1,12
18 1 1,2 2,6 287 4845
${\geqslant}19$
: Triples $(n,q,I)$ with $\Phi^*_n(q){\leqslant}n^4$ and prime factorisation $\Phi^*_n(q)=\prod_{i\in I} (in+1) .$[]{data-label="tab:onea"}
\#1[-2.5pt[\#1]{}-2.5pt]{}
$q$ 2 3 $2^2$ 5 7 $2^3$ $3^2$ 11 13 17 19 23 $5^2$ $3^3$ 29 31
----- --- --- ------- --- --- ------- ------- ---- ---- ---- ---- ---- ------- ------- ---- ----
$q$
: Prime powers $q{\leqslant}5000$ with $\Phi^*_2(q){\leqslant}2^4=16$, see Remark \[rem:one\].[]{data-label="tab:q=2"}
----- ------- ------ ------ ------ ------ ------- ------ ------ ------ ------ ------
$q$ $2$ $3$ $4$ $5$ $7$ $8$ $9$ $11$ $13$ $16$ $17$
$I$ $-$ 1 2 1 7 3 12 6 26 40 1,2
$q$ $19$ $23$ $25$ $27$ $29$ $31$ $32$ $41$ $47$ $53$ $59$
$I$ 1,1,1 2,2 100 3,6 45 1,1,3 55 91 1,17 153 1,27
----- ------- ------ ------ ------ ------ ------- ------ ------ ------ ------ ------
: Pairs $(q,I)$ with $\Phi^*_6(q){\leqslant}6^4$ and prime factorisation $\Phi^*_6(q)=\prod_{i\in I} (in+1)$ where $-$ means ${\{\kern-1.2pt\{ {\,}\}\kern-1.2pt\}}$.[]{data-label="tab:six"}
$n\backslash q$ $2$ $3$ $4$ $5$ $n\backslash q$ 2 $3$ $n\backslash q$ $2$
----------------- ------ ------- ------ ------- ----------------- ------- ------- ----------------- -------- --
20 2 59 3084 33 18166 50 5,81
21 16 34 1285 54 1615
22 31 3,30 36 1,3 14742 60 1,22
24 10 270 4,28 38 4599 66 1,316
26 105 15330 40 1542 72 6,538
27 9728 42 129 1,54 78 286755
28 1,4 1,589 44 9,48 84 17,172
30 11 1,9 2,44 2,254 46 60787 90 209300
32 2048 48 2,14
: All $(n,q,I)$ with $n{\geqslant}19$, $\Phi^*_n(q){\leqslant}n^4$, and factorisation $\Phi^*_n(q)=\prod_{i\in I} (in+1).$[]{data-label="tab:oneb"}
$\,n\,$ $\,q\,\,$ $\,\,I\,\,$ $\,c_0$ $c_1\,$ $\,n\,$ $\,q\,\,$ $\,\,I\,\,$ $\,c_0$ $c_1\,$ $\,n\,$ $\,q\,\,$ $\,\,I\,\,$ $\,c_0$ $c_1\,$
--------- ----------- ------------- --------- --------- --------- ----------- ------------- --------- --------- --------- ----------- ------------- --------- ---------
3 2 2 4 13 1,4 17 17 8 2 2 17 34
3 3 4 4 47 1,3,4 17 17 10 2 1 15 42
3 4 2 6 2 $-$ 15 26 10 4 4 41 42
3 9 2,4 6 3 1 15 25 12 2 1 15 50
3 16 2,4 6 4 2 15 26 14 2 3 43 58
4 2 1 15 18 6 5 1 15 25 18 2 1 19 74
4 3 1 15 18 6 8 3 19 26 18 3 1,2 37 73
4 4 4 17 18 6 17 1,2 15 25 20 2 2 41 82
4 5 3 15 17 6 19 1,1,1 15 25 28 2 1,4 113 114
4 7 1,1 15 17 6 31 1,1,3 19 25 36 2 1,3 109 146
4 8 1,3 15 18
: For Theorem \[the:two\] we list all $(n,q,I)$ where $n{\geqslant}3$ and $\Phi_n^*(q)$ has prime factorisation $\prod_{i\in I} (in+1)$ with $I\subseteq{\{\kern-1.2pt\{ {1,1,1,2,3,4}\}\kern-1.2pt\}}$. For Theorem \[the:Ac\] we also list the possible degrees $c$ where $c_0{\leqslant}c{\leqslant}c_1$ and in this case we must have $n{\geqslant}4$. Here $-$ denotes the empty multiset.[]{data-label="tab:two"}
An Application {#sec:Ac}
==============
Various studies of configurations in finite projective spaces have involved a subgroup $G$ of a projective group ${\textup{PGL}}(d,q)$ (or equivalently, a subgroup of ${\textup{GL}}(d,q)$) with order divisible by $\Phi^*_n(q)$ for certain $n, q$. This situation was analysed in detail by Bamberg and Penttila [@BambergPenttila] for the cases where $n > d/2$, making use of the classification in [@GPPS]. In turn, Bamberg and Penttila applied their analysis to certain geometrical questions, in particular proving a conjecture of Cameron and Liebler from 1982 about irreducible subgroups with equally many orbits on points and lines [@BambergPenttila Section 8]. In their group theoretic analysis Hering’s theorem [@Hering Theorem 3.9] was used repeatedly, notably to deal with the ‘nearly simple cases’ where $G$ has a normal subgroup $H$ containing $\mathrm{Z}(G)$ such that $H$ is absolutely irreducible, $H/\mathrm{Z}(G)$ is a nonabelian simple group, and $G/\mathrm{Z}(G){\leqslant}\mathrm{Aut}(H/\mathrm{Z}(G))$. Incidentally, the missing cases $(n,q)=(2,17)$ and $(2,71)$ mentioned in Remark \[rem:one\] do not affect the conclusions in [@BambergPenttila].
To study other related geometric questions we have needed similar results which allow the parameter $n$ to be as small as $d/4$. We give here an example of how our extension of Hering’s results might be used to deal with nearly simple groups in this more general case where no existing general classifications are applicable. For example, there are several theorems about translation planes that include restrictive hypotheses such as two-transitivity [@BJJMa; @BJJMb; @BJJMc]. In order to remove some of these restrictions, we require results similar to Theorem \[the:Ac\] for all nearly simple groups. For simplicity we now consider representations of the alternating or symmetric groups of degree $c{\geqslant}15$ with $\Phi^*_n(q)\mid c!$ and, as we see below, $c-1{\geqslant}n{\geqslant}(c-2)/4$.
\[the:Ac\] Let $G{\leqslant}{\textup{GL}}(d,q)$ where $G\cong{\mathrm{Alt}}(c),{\mathrm{Sym}}(c)$, for some $c{\geqslant}15$, and suppose that ${\mathrm{Alt}}(c)$ acts absolutely irreducibly on $({\mathbb{F}}_q)^d$ where $q$ is a power of the prime $p$. Suppose $\Phi^*_n(q)$ divides $c!$ for some $n{\geqslant}d/4$. Then $n{\geqslant}4$, $d=c-\delta(c,q)$ where $\delta(c,q)$ equals $1$ if $p\nmid c$, and $2$ if $p\mid c$, also $c_0 {\leqslant}c {\leqslant}c_1$, and $\Phi^*_n(q)$ has prime factorisation $\prod_{i\in I}(in+1)$, where all possible values for $(n,q,I, c_0, c_1)$ are listed in *Table $\ref{tab:two}$*.
The smallest and the second smallest dimensions for ${\mathrm{Alt}}(c)$ and ${\mathrm{Sym}}(c)$ modules over ${\mathbb{F}}_q$ are very roughly, $c$ and $c^2/2$ respectively. The precise statement below follows from James [@J Theorem 7], where the dimension formula $(\ast)$ on p.420 of [@J] is used for part (ii). Since $c{\geqslant}15$, these results show that either:
1. $({\mathbb{F}}_q)^d$ is the fully deleted permutation module for ${\mathrm{Alt}}(c)$ with $d=c-\delta(c,q)$, or
2. $d{\geqslant}c(c-5)/2$.
In particular, since $c{\geqslant}15$ and $n{\geqslant}d/4$, we have $n{\geqslant}4$. Since $n>2$ it follows from Theorem 3.23 of [@CW] that $\Phi_n^*(q) > 1$ except when $n=6$ and $q=2$. As the case $(n,q)=(6,2)$ is included in Table \[tab:two\], we assume henceforth that $\Phi_n^*(q) > 1$. Thus $\Phi^*_n(q)=r_1^{m_1}\cdots r_\ell^{m_\ell}$ where $\ell{\geqslant}1$, each $r_i$ is a prime, and each $m_i{\geqslant}1$. Then $r_i=a_i n+1$ for some $a_i{\geqslant}1$ by Lemma \[lemma:phiq\], and since $r_i$ divides $|\mathrm{S}_c|=c!$ we see $c{\geqslant}r_i$. Let $r$ be the largest prime divisor of $\Phi^*_n(q)$, so $c{\geqslant}r{\geqslant}n+1 > d/4$. In case (ii) this implies that $c > c(c-5)/8$ which contradicts the assumption $c{\geqslant}15$. Thus case (i) holds.
The inequalities $c-2{\leqslant}d$ and $d{\leqslant}4n$ show $a_i n+1{\leqslant}c{\leqslant}4n+2$ and hence $a_i{\leqslant}4$. The exponent $m_i$ of $r_i$ is severely constrained. If $a_i{\geqslant}2$, then $$r_i=a_i n+1{\geqslant}2n+1{\geqslant}\frac {d+3}2{\geqslant}\frac{c+1}2>\frac c2.$$ Thus the prime $r_i$ divides $c!$ exactly once, and $m_i=1$. If $a_i=1$, then a similar argument shows $r_i=n+1{\geqslant}\frac {d+4}{4}{\geqslant}\frac{c+2}{4}{\geqslant}\frac{17}{4} >4$. The inequalities $r_i>\frac{c}{4}$ and $r_i>4$ imply that $r_i$ divides $c!$ at most three times, and $m_i{\leqslant}3$. In summary, $\Phi^*_n(q)$ divides $f(n):=(n+1)^3(2n+1)(3n+1)(4n+1)$. Since $n{\geqslant}4$, we have $f(n){\leqslant}16n^7$. All possible pairs $(n,q)$ for which $\Phi^*_n(q)\mid f(n)$ can be computed using Algorithm \[alg:Mstarnbig\] with input $c=16$, $k=7$. The output is listed in Table \[tab:two\], and computed using [@G].
For given $n$ and $q$ the possible values for $c$ form an interval $c_0{\leqslant}c{\leqslant}c_1$. Since $c-\delta(c,q)=d{\leqslant}4n$ the entries $c_0, c_1$ in Table \[tab:two\] can be determined as follows: $c_0=\max(r,15)$ where $r$ is the largest prime divisor of $\Phi_n^*(q)$, and $c_1=4n+\delta(4n+2,q)$. $\hfill\ensuremath{\square}$
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank the referee for numerous helpful suggestions. The first, third and fourth authors acknowledge the support of ARC Discovery grant DP140100416.
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---
title: 'Simulating Cherenkov Telescope Array observation of RX J1713.7–3946'
---
Introduction
============
More than 100 years have passed since the discovery of cosmic rays but its origin has been long in question despite many observational and theoretical researches. The observed spectrum of cosmic rays is a power-law shape and has a break around $10^{15.5}$ eV which is so-called “knee”. Cosmic rays below the knee energies are thought to be accelerated somewhere in our Galaxy. One of the most probable candidates are supernova remnants (SNRs), where the diffusive shock acceleration may work at the shock front of SNR blast waves. Evidence for electron acceleration has been identified by the detection of synchrotron emission with spatially thin filamentary structures at shells of young SNRs (e.g., [@koyama95]). On the other hand, gamma rays with hadronic origin was detected by Large Area Telescope (LAT) onboard [*Fermi*]{} from middle-aged SNRs IC443 and W44, which are known to be interacting with molecular clouds (MCs)[@ackermann13]. The observed gamma-ray spectra are interpreted as neutral pion decay, which is characterized by a cutoff below 300 MeV, due to the interaction between accelerated cosmic-ray hadrons and MCs. However, it is also observed that that the gamma-ray spectra are suppressed above 100 GeV. We hence expect that young SNRs could be more plausible as cosmic-ray accelerators to PeV energies, i.e. PeVatron.
The Cherenkov Telescope Array (CTA) is a next-generation of Imaging Air Cherenkov Telescopes (IACT) observatory which consists of array of the large, middle, and small-sized telescopes expanding over km$^2$ area[@actis2011cta; @ctaconcept]. With higher performance in comparison to the current generation IACTs, such as better spatial resolution and sensitivity, a search for cosmic-ray PeVatron is one of the major scientific objectives of the CTA. A young SNR RX J1713.7$-$3946 is one of the brightest Very High-Energy (VHE) gamma ray sources and spatially extended emission was observed [@enomoto02; @aharonian04; @hess06; @hess07]. The VHE gamma-ray spectrum of RX J1713.7–3946 is the most precisely measured over a wide energy band from 0.3 to 100 TeV. Besides, plenty of multi-wavelength observations have been performed. [*Fermi*]{} measured the gamma-ray spectrum of RX J1713.7–3946 in the 3–300 GeV energy range, where the observed photon index of $1.5\pm0.1$ is favorable for inverse-Compton emission from accelerated electrons with a spectral index of 2.0[@fermi1713]. X-ray emission is dominated by synchrotron radiation with a good spatial correlation between VHE morphology, although the angular resolution is not good enough to be conclusive. On the other hand, radio observations of CO and H$_{\rm I}$ gas have revealed a clumpy molecular clouds (MCs) surrounding the SNR, and reported evidence for interaction between the MCs and the SNR shock (e.g. [@fukui03; @fukui12; @sano14]. The hadronic gamma-ray emission is naturally expected to reproduce the obseved spetrum (e.g. [@gab14]) . However, our idea here is that, if the hadronic gamma rays do exist, such component might be hidden by the dominant leptonic gamma-ray emission. It is of a great interest whether the improved sensitivity of the CTA could detect the possible but dim hadronic gamma rays. Hence this object is a very good target for deep observations, and also for constraining theoretical models of cosmic-ray acceleration.
Aims and methods of simulations
===============================
The major purpose of our simulation studies is to show an example of analysis strategy when we will obtain real data, and to evaluate the capabilities of CTA on finding a clue for the hadronic gamma rays.
First we perform morphological analyses in order to find out the dominant component of the VHE gamma-ray emission from RX J1713.7–3946. As for the hadronic gamma rays, the morphology should be related to spatial distribution of the accelerated protons and that of the interacting matter density which is indicated by the CO and H$_{\rm I}$ morphology obtained by the radio observations. Since we currently do not know the CR distribution, we here roughly assume that CR would be filled homogeneously inside the SNR. On the other hand, the morphology of the leptonic gamma-ray emission may be traced by that of synchrotron X-ray. We should note that the X-ray and VHE morphologies are not always completely same. The brightness of the synchrotron emission reflects not only for spatial electron distribution but also local magnetic fields. And the inverse-Compton process is also coupled to energy densities of target photons which includes infrared and/or optical photon field in addition to the cosmic microwave background. However, the overall structure could be approximated by the X-ray morphology. We hence apply the radio and X-ray images as templates for the hadronic and leptonic gamma-ray morphology, respectively. We perform maximum likelihood test to quantitatively determine which component dominates the VHE emission from the SNR. The templates also contain spectral information. Here we simply assume the same spectral shape is assumed over the SNR image. The spectrum of the leptonic component is modeled as $$\frac{ {\rm d}N_1(E)}{ {\rm d} E} = A_1 \left(\frac{E}{\rm TeV}\right)^{-\Gamma _e}\exp\left( -\frac{E}{E_{\rm c}^e}\right) ~~,$$ where $A_1$ is a normalization factor, $\Gamma _e$ is a photon index, and $E_{\rm c}^e$ is a cutoff energy. Input values for $\Gamma _e$ and $E_{\rm c}^e$ are 2.04 and 17.9 TeV, respectively, as reported by H.E.S.S. observations[@hess07]. For the hadronic emission, the spectrum is described as follows, $$\frac{ {\rm d}N_2(E)}{ {\rm d} E} = A_2 \left(\frac{E}{\rm TeV}\right)^{-\Gamma _p}\exp\left( -\frac{E}{E_{\rm c}^p}\right) ~~,$$ where $A_2$ is a normalization factor, $\Gamma _p$ is a photon index, and $E_{\rm c}^p$ is a cutoff energy. We adopt $\Gamma _p =2.0$ and $E_{\rm c}^p = 300$ TeV as fiducial parameters. Therefore $A_1$ and $A_2$, or their ratio $A_2/A_1$, are the parameter to be investigated, requiring that the sum of the integral fluxes between 1 and 10 TeV are equal to that measured by H.E.S.S.. If the hadronic gamma ray is greater, we could conclude that dominant part of the hadronic component were not accelerated to the knee energies in RX J1713.7–3946. Searching for a spectral component that extends to PeV energies, we also look for a dimmer hadronic component by spectral analysis. The maximum likelihood fit will be performed in order to unfold spectra for each component and evaluate statistical significance of the hadronic gamma-ray detection. Here the spatial templates are also considered to calculate the likelihood.
We also evaluate the capability of detecting the time variation of the spectral cutoff energy, $E_{\rm max}$, with longer time scale. The maximum energy of the CR spectrum is determined by a balance among acceleration, cooling and escape. Hence $E_{\rm max}$ variation, increase or decrease, depends on the SNR age and also on acceleration theories. In the case of RX J1713.7–3946, $\sim$10% variation in 10–20 years may be expected, where $E_{\rm max}$ could vary faster in the leptonic scenario[@ohira10]. Since the CTA will be operational for a few tens of years, such a long-term study in VHE energies will become possible. This may be a unique approach for identifying the VHE emission mechanism.
The simulation software package that we use in this study is [*ctools*]{} version 00-07-01[@jur13]. We use a preliminary instrumental response function which corresponds to the CTA southern array located at the candidate site Aar (southern Namibia). When we perform the simulations, the Galactic diffuse emission and isotropic background due to gamma-like charged cosmic rays are taken into account for background photons in the field of view.
Results
=======
$\gamma$-ray image
------------------
In order to clarify the imaging capability of CTA, we first intend to simulate different gamma ray images in the energy range of 1–100 TeV by tuning $A_2/A_1$, the ratio between the hadronic and leptonic gamma rays. Figures \[fig:images\]a and \[fig:images\]b show the simulated gamma-ray images in leptonic dominant case ($A_2/A_1$ = 0.01) and hadronic dominant case ($A_2/A_1$ = 100), respectively. Each gamma-ray image is similar to each overlaid contour, which corresponds to the non-thermal X rays and the total ISM protons including both molecular and atomic hydrogen, respectively. On the other hand, the spatial distributions of gamma rays are apparently different from each other, particularly the north and the southwest. In this extreme case, we can therefore determine the major component of the VHE emission by the morphological study with CTA. Incidentally, we found that $A_2/A_1$ = 1–10 showed the best spatial correspondence with the H.E.S.S. excess counts map [@hess07] with a correlation coefficient of $\sim$0.7–0.8. We continue to study the systematic error estimation and quantitative evaluation for the morphological difference.
![ Simulated gamma-ray images of (a) $A_2/A_1=0.01$ (leptonic dominant case) and (b) $A_2/A_1=100$ (hadronic dominant case) with $\Gamma _p =2.0$ and $E_{\rm c}^p = 300$ TeV. The green contours show (a) $XMM$-$Newton$ X-ray intensity [@acero09] and (b) total interstellar proton column density [@fukui12], which smoothed to match the PSF of CTA. The unit of color axis is counts pixel$^{-1}$ for both panels. \[fig:images\]](f1_new.eps){width="\linewidth"}
Spectrum
--------
Assuming that the leptonic component is dominant, we subsequently proceed to search for a “hidden” hard component with a hadronic origin. Using reconstructed energy band of 0.5 – 100 TeV with 50-hour observation, likelihood analyses show the significance $>10\sigma$ to observe a dimmer hard component even for a small $A_2/A_1=0.02$. Note that the result may be rather optimistic since the fitting templates are the same as the input for the simulation.
We then proceed to perform maximum likelihood fittings for the simulation data (with a ratio $A_2/A_1= 0.1$, for the safety) in 12 logarithmically spaced energy bands. Figure \[fig:bin\_by\_bin\_spec\] shows the resulting spectrum from our ‘bin-by-bin’ analysis of the same 50 hr of simulation data. It is clear that our likelihood fits reproduce the simulated spectrum for each spatial template (i.e. hadronic or leptonic morphology), which demonstrates the capability of detecting the hidden hadronic component in the best case scenario.
![ Spectral energy distribution of the gamma ray emission obtained by analyzing the CTA simulation data for RX J1713.7$-$3946 with $A_2/A_1=0.1$. The blue and red squares are the spectral points for the leptonic and hadronic spatial templates, respectively. Only statistical errors are presented. The black squares are the total fluxes of the leptonic and hadronic components. The black vertical bars are the errors for the total fluxes obtained by adding the errors for two components in quadrature. The blue, red, and black solid lines show the input spectra for the leptonic component, the hadronic component, and the total, respectively. \[fig:bin\_by\_bin\_spec\]](plotFlux_50_v2.eps){width="\linewidth"}
Time variation of cutoff energy
-------------------------------
Detecting the time variation of the cutoff energy of the gamma ray spectrum can provide a clue to the emission scenario and acceleration theories. The simulations are performed for three sets of intrinsic cutoff energies at 17.9 (nominal), 19.7 ($+10$% case) and 16.1 TeV ($-10$% case) each. As a start, we consider a variation of $\Delta E_{\rm c}/E_{\rm c} = \pm 10\%$ to show how sensitive CTA will be to such fractional changes in the spectral cutoff. Here we consider the pure leptonic scenario as an example, and used 0.2–100 TeV photons.
We define a significance, $s$, for the observed $E_{\rm c}$ variation as $
s _{\pm} (t)= \frac{|E_\pm(t) - E_0(t)|}{\sqrt{\sigma _\pm^2(t) + \sigma _0^2(t)}},
\label{eq:spar}
$ where $E_0$ is the nominal value and $E_\pm$ is the best fit $E_{\rm c}$ for the cases with a $\pm 10\%$ variation. $\sigma _0$ and $\sigma _\pm$ are the corresponding errors. We repeat all of our simulations for 100 times and take the average of the calculated $s(t)$ for each run, and show the result on Figure\[fig-ecsigma\] . Our result indicates that a decrease of $E_{\rm c}$ is slightly easier to identify than an increase. As a result, a lower cutoff energy can actually be easier to measure and precisely for a given exposure. If we observe $> 60$ hrs in the two epochs, we are able to achieve a $3\sigma$ detection for the $\Delta E_{\rm c}/E_{\rm c}=- 10\%$ case, whereas $\sim 70$ hrs are necessary for the $+10\%$ case.
![Significance of the detected variation of $E_{\rm c}$ as a function of exposure time. Dashed lines represent the best-fit curve to each dataset which is proportional to $\sqrt{t}$. Squares and circles represent results for the $\Delta E_{\rm c}/E{\rm c} = -10$% and $+10$% case, respectively. []{data-label="fig-ecsigma"}](fig3_v3.eps){width="0.8\linewidth"}
Summary
=======
In this paper, we have briefly introduced our feasibility studies for CTA observations of RX J1713.7–3946, mostly with 50 hrs observations. We showed that a 50-hr observation may be enough to identify the dominant gamma ray emission component by the morphology obtained with CTA. And in the case that the leptonic emission would be dominant, we should be able to quantify both the leptonic and hadronic components through spectral analysis if they are mixed with a ratio of $A_2/A_1 =0.1$ or less. Interestingly, we also found that CTA will be able to reveal variations of the spectral cutoff energy over 10–20 years, for the very first time. A variation of $\Delta E_{\rm c}/E_{\rm c} = \pm 10\% $ could be detected provided that an exposure time longer than 70 hr can be secured for the two epoch. However we know our present study is based on a fairly simplified input model and may contain systematic errors of which estimation is not trivial. And also this study can be extended to use more theoretically justified models. More studies are left for future works.
We gratefully acknowledge support from the agencies and organizations listed under Funding Agencies at this website: http://www.cta-observatory.org/. We thank S. Katsuda for providing the original *XMM-Newton* image.
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---
abstract: 'Reconstituted filamentous actin networks with myosin motor proteins form active gels, in which motor proteins generate forces that drive the network far from equilibrium. This motor activity can also strongly affect the network elasticity; experiments have shown a dramatic stiffening in *in vitro* networks with molecular motors. Here we study the effects of motor generated forces on the mechanics of simulated 2D networks of athermal stiff filaments. We show how heterogeneous internal motor stresses can lead to stiffening in networks that are governed by filament bending modes. The motors are modeled as force dipoles that cause muscle like contractions. These contractions “pull out” the floppy bending modes in the system, which induces a cross-over to a stiffer stretching dominated regime. Through this mechanism, motors can lead to a nonlinear network response, even when the constituent filaments are themselves purely linear. These results have implications for the mechanics of living cells and suggest new design principles for active biomemetic materials with tunable mechanical properties.'
author:
- 'C. P. Broedersz'
- 'F. C. MacKintosh'
bibliography:
- 'broedersz.bib'
title: 'Molecular motors stiffen non-affine semiflexible polymer networks'
---
The mechanics of living cells is largely governed by the cytoskeleton, a complex assembly of various filamentous proteins. Cross-linked networks of actin filaments form one of the major structural components of the cytoskleton. However, this cytoskeleton is driven far from equilibrium by the action of molecular motors that can generate stresses within the meshwork of filaments[@alberts_molecular_2002; @brangwynne_cytoplasmic_2008; @joanny_active_2009]. Such motor activity plays a key role in various cellular functions, including morphogenesis, division and locomotion. The nonequilibrium nature of motor activity has been demonstrated in simplified reconstituted filamentous actin networks with myosin motors[@mizuno_nonequilibrium_2007; @brangwynne_nonequilibrium_2008; @bendix_quantitative_2008; @koenderink_active_2009; @schaller_polar_2010]. Even in the absence of motor proteins, such *in vitro* networks of cytoskeletal filaments already constitute a rich class of soft matter systems that exhibit unusual material properties, including a highly nonlinear elastic response to external stress [@gardel_elastic_2004; @storm_nonlinear_2005; @wagner_cytoskeletal_2006; @bausch_bottom-up_2006; @tharmann_viscoelasticity_2007; @kasza_nonlinear_2009; @broedersz_measurement_2010]. This nonlinear response can be exploited using molecular motors [@mizuno_nonequilibrium_2007; @koenderink_active_2009]; the network stiffness can be varied by orders of magnitude, depending on motor activity. A quantitative understanding of such active biological matter poses a challenge for theoretical modeling [@kruse_generic_2005; @joanny_hydrodynamic_2007; @mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @liverpool_mechanical_2009; @mackintosh_active_2010; @joanny_active_2009].
The nonlinear mechanical response of reconstituted biopolymer networks in many cases reflects the nonlinear force-extension behavior of the constituting cross-links or filaments [@gardel_elastic_2004; @storm_nonlinear_2005; @wagner_cytoskeletal_2006; @kasza_nonlinear_2009; @broedersz_nonlinear_2008]. For such networks, there is both theoretical and experimental evidence that internal stress generation by molecular motors can result in network stiffening in direct analogy to an externally applied uniform stress [@koenderink_active_2009; @mizuno_nonequilibrium_2007; @mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @liverpool_mechanical_2009; @head_nonlocal_2010]. However, the mechanical response of semiflexble polymers is highly anisotropic and is typically much softer to bending than to stretching. In some cases, this renders the network deformation highly non-affine with most of the energy stored in bending modes [@head_distinct_2003; @head_deformation_2003; @wilhelm_elasticity_2003; @heussinger_floppy_2006; @das_effective_2007]. Such non-affinely deforming stiff polymer networks can also exhibit a nonlinear mechanical response, even when the network constituents have a linear force-extension behavior [@onck_alternative_2005; @lieleg_mechanics_2007; @huisman_three-dimensional_2007; @conti_cross-linked_2009]. However, the effects of internal stresses generated by molecular motors in such networks are unknown.
![\[fig:network\] Example of a portion of the diluted 2D phantom triangular network at $\mathcal{Q}=1/4$ and $\kappa=10^{-3}$. The freely hinging binary cross-links are indicated in black. Motors generate muscle-like contractions, which we model with force dipoles. The segments along which these contractile force dipoles act are indicated with red dumbbells. The inset shows an enlargement of the network.](networkwm.pdf){width="\columnwidth"}
Here we study the effects of motor generated forces on the network mechanics in 2D networks of athermal, stiff filaments using simulations. In the absence of motors, these networks can exhibit strain stiffening under an externally applied shear. This behavior has been attributed to a cross-over between two mechanical regimes; at small strains the mechanics is governed by soft bending modes and a non-affine deformation field, while at larger strains the elastic response is governed by the stiffer stretch modes and an affine deformation field[@onck_alternative_2005]. We show that motors that generate internal stresses can also stiffen the network. The motors induce force dipoles leading to muscle like contractions, which “pull out” the floppy bending modes in the system. This induces a cross-over to a stiffer stretching dominated regime. Through this mechanism, motors can lead to network stiffening in non-affine stiff polymer networks in which the constituting filaments in the network are themselves linear elements. These results have implications for the mechanics of living cells and propose new design principles for active biomemetic materials with highly tunable mechanical properties.
The model
=========
To study the basic effects of internal stress generated by molecular motors on the macroscopic mechanical properties of stiff polymer networks we employ a minimalistic model, which is illustrated in Fig. \[fig:network\]. Filamentous networks in 2D are generated by arranging filaments spanning the system size on a triangular lattice. Since physiological cross-linking proteins typically form binary cross-links, we randomly select two out of the three filaments at every vertex between which we form a binary cross-link. The remaining filament crosses this vertex as a phantom chain, without direct mechanical interactions with the other two filaments. The cross-links themselves hinge freely with no resistance. With this procedure we can generate disordered *phantom* networks, based on a triangular network, but with local 4-fold ($z=4$) connectivity corresponding to binary cross-links. The use of a triangular lattice avoids, for example, well-known mechanical pathologies of the 4-fold square lattice. To create quenched disorder in the network, we cut and remove filament segments between vertices with a probability $\mathcal{Q}$. This also has the effect of shortening the filaments.
The filaments in the network are described by an extensible wormlike chain (EWLC) model with an energy $$\label{eq:H0}
\mathcal{H}=\frac{1}{2}\kappa \int {{\rm d}}s \left(\frac{{{\rm d}}\hat{t}}{{{\rm d}}s}\right)^2+\frac{1}{2}\mu \int {{\rm d}}s \left(\frac{{{\rm d}}\ell(s)}{{{\rm d}}s}\right)^2,$$ where $\kappa$ is the bending rigidity, $\hat{\bf t}$ is the tangent vector at a position $s$ along the polymer backbone and $\frac{{{\rm d}}\ell (s)}{{{\rm d}}s} $ is the local relative change in contour length, or longitudinal strain. We can quantify the relative importance of the stretch and bend contributions by the lengthscale $\ell_b=\sqrt{\kappa/\mu}$; this length scale forms one of the key control parameters for the network mechanics. For simple cylindrical beams with a radius $r$, the stretch modulus $\mu$ is related to $\kappa$ through $\mu_{\rm mech}=4 \kappa/r^2$, and $\ell_b=r/2$. In contrast, a thermally fluctuating semiflexible polymer segment cross-linked in a network on a length-scale $\ell_c$ also has an entropic thermal stretch modulus $\mu_{\rm th}=90 \kappa^2/k_{\rm B} T\ell_c^3$ [@mackintosh_elasticity_1995], where $k_B$ is Boltzmann’s constant and $T$ is the temperature. In this case, $\ell_b=\ell_c \sqrt{\ell_c/90 \ell_p}$, where $\ell_p=\kappa/k_{\rm B} T$ is the persistence length. The most relevant values of $\ell_b/\ell_c$ for biopolymer systems range from $10^{-2} - 10^{-1}$. This range extends from relatively stiff actin filaments to the more flexible intermediate filaments. Various actin binding proteins are capable of forming tightly coupled stiff bundles of actin filaments, which further reduces $\ell_b$. The mechanical and thermal moduli add as springs in series and the total modulus is given by $\mu^{-1}=\mu_{\rm mech}^{-1}+\mu_{\rm th}^{-1}$. In the remainder of this paper all lengths are determined in units of the distance between lattice vertices $\ell_0$ and the bending rigidity $\kappa$ is measured in units of $\mu \ell_0^2$. Here, we focus on nonlinearities arising in networks of purely linear elements. Thus, we do not include intrinsic nonlinearities associated with the force-extension curve of thermal filaments. This has been examined theoretically in Refs. [@mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @liverpool_mechanical_2009]
In our numerical simulations we use a discretized version of Eq. (\[eq:H0\]) with a node at and between every lattice vertex. The mid-node allows us to capture buckling down to the single segment length-scale. To model the effect of muscle like contractions induced by molecular motors, we introduce force dipoles in the network [@mizuno_nonequilibrium_2007; @mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @head_nonlocal_2010]. These force dipoles are randomly placed at neighboring cross-links. The force dipoles $f_{ij}$ only act along existing bonds and, therefore, do not introduce additional constraints in the network. The total energy of the system includes a sum of the EWLC Hamiltonian over all filament segments and the work extracted by the force dipoles $$\label{eq:Enetwork}
E=\sum_i \mathcal{H}_{i}-\sum_{<ij>} f_{ij} r_{ij},$$ where $r_{ij}$ is the distance between cross-link $i$ and $j$. The force dipoles are numerically implemented by shortening the effective rest length of the bond along which the motors acts in the stretch term of the energy (Eq. \[eq:H0\]). The rest length is reduced by an amount $\delta r_{ij}^{(0)}$; the resulting force is given by $\mu \delta r_{ij}^{(0)}/\ell_0 \le \mu$. The effects of internal motor generated stresses modeled in this way is illustrated in Fig. \[fig:network\].
To investigate the mechanical response of the network, an external strain $\gamma$ is applied by translating one of the horizontal boundaries to which the filaments are attached. The internal degrees of freedom of the network are relaxed by minimizing the energy using a conjugate gradient algorithm [@vetterling_numerical_2002]. To reduce edge effects periodic boundary condition are employed at all boundaries. The linear shear modulus of a network of size $W^2$ is related to the energy $G=\frac{2}{W^2}\frac{E}{\gamma^2}$ for small strains. In the nonlinear regime it is common to determine the differential modulus $K=\frac{1}{W^2}\frac{{{\rm d}}^2 E}{{{\rm d}}\gamma^2}$, which reduces to $G$ for small $\gamma$. Similarly, the stress can be calculated in the nonlinear regime through $\sigma_{\rm ext}=\frac{1}{W^2}\frac{{{\rm d}}E}{{{\rm d}}\gamma}$. These measurements allow us quantify the mechanical response of the system. Here we use system sizes ranging from $W^2\simeq 3000$ to $8000$.
Results and Discussion
======================
Passive networks
----------------
We probe the 2D phantom triangular networks by determining both the linear and nonlinear elastic response of the networks in the absence of motors. The linear mechanical response of diluted networks ($\mathcal{Q}>1$) exhibits two distinct mechanical regimes. At low $\kappa$, the shear modulus $G$ scales directly with $\kappa$, as shown in the inset of Fig. \[fig:NLpassiveK\]. This demonstrates that in this regime the macroscopic mechanics is governed by filament bending deformation modes. By contrast, at large $\kappa$ the shear modulus asymptotically approaches a limit in which $G$ is independent of $\kappa$ indicative of a stretching dominated regime. These result are consistent with previous observations on 2D mikado networks [@head_distinct_2003; @head_deformation_2003; @wilhelm_elasticity_2003].
These mechanical regimes have important implications for the nonlinear elastic response. When a large external shear is imposed on a network that is initially in the bending dominated regime, the differential modulus $K=\frac{{{\rm d}}\sigma}{{{\rm d}}\gamma}$ increases strongly as a function of external stress $\sigma_{\rm ext}$, as shown in Fig. \[fig:NLpassiveK\]. Previous studies have observed similar stiffening in networks with strictly linear elements [@onck_alternative_2005; @lieleg_mechanics_2007; @huisman_three-dimensional_2007; @conti_cross-linked_2009]. This remarkable behavior has been explained in terms of a *strain*-induced cross-over from a bending to a stretching dominated regime. At low stresses the network mechanics is governed by bending modes, which for small $\kappa$ constitute the softest modes in the system. However, when the stress is increased the deformations become correspondingly large and the stretching of filaments is no longer avoidable. This picture is consistent with our simulations. When a substantial shear is imposed the stiffening curves—over a large range of bending rigidities—converge to a single curve that is consistent with the affine prediction, shown as a red dashed line in Fig. \[fig:NLpassiveK\]. This calculation also demonstates that even an affinely deforming network of strictly linear elements stiffens under shear. This stiffening behavior is purely due to geometric effects; under shear the network becomes increasingly anisotropic and the filaments reorient to line up in the shear direction [@broedersz_effective-medium_2009]. The extent of this purely geometric stiffening is, however, limited, as can be seen in the figure. Moreover, this geometrically-stiffened limit represents an upper bound on the stiffness of networks with purely linear elements. Such systems cannot stiffen indefinitely.
In addition to $\kappa$, the average length of filaments in the system $\langle L\rangle$ constitutes an important control parameter for the linear response. We can probe this by varying $\mathcal{Q}$, since the average length of filaments is given by $\langle L\rangle=1/\mathcal{Q}$ [@broedersz_future_2010]. Consistent with previous work [@head_distinct_2003; @head_deformation_2003; @wilhelm_elasticity_2003], a cross-over from a non-affine bending regime and an affine stretching regime can also be achieved by increasing $\langle L\rangle$, as shown in the inset of Fig. \[fig:NLpassiveP\]. In the high molecular weight limit, $\langle L\rangle\rightarrow\infty$, the system responds purely affinely. We estimate that in experimental biopolymer systems $\langle L\rangle$ varies a over a range of order 5-30, in units of the network mesh size. The strong dependence of the linear elastic response on $\langle L \rangle$ is also reflected in the nonlinear response (Fig. \[fig:NLpassiveP\]). Networks with shorter filaments are increasingly governed by soft bending modes and thus exhibit a greater degree of stiffening under shear.
In the absence of motors, we find that our diluted phantom triangular networks exhibit a linear and nonlinear response to external shear that is consistent with previous work on 2D off-lattice networks of stiff filaments [@head_distinct_2003; @head_deformation_2003; @wilhelm_elasticity_2003]. Our phantom triangular networks thus provide a good model system to study the effects of internal stresses generated by molecular motors in athermal networks.
Active networks
---------------
![\[fig:Mvaryden\] The shear modulus $G$ as a function of force exerted per motor $f_0$ for various motor densities $\rho_{\rm M}$ at fixed $\mathcal{Q}=1/4$ and $\kappa=10^{-3}$. The shear modulus $G$ is normalized by the shear modulus $G_0$ of the passive network. The inset shows the shear modulus $G_0$ as a function of the generated stress $\sigma_{\rm M}$. The apparent collapse of these curves supports supports the hypotheses that $\sigma_{\rm M}$ is the appropriate control variable.](varymotors.pdf){width="\columnwidth"}
To investigate the effect of motor generated stresses we introduce force dipoles in the network at various densities $\rho_{\rm M}$. The shear modulus $G$ increases strongly when the force exerted by a single motor $f_0$ is increased beyond a threshold value, as shown in Fig. \[fig:Mvaryden\]. Interestingly, the motor forces at which the system becomes nonlinear for low motor densities is close to the buckling force threshold $f_b=\pi^2 \kappa/\ell_c^2\approx2\times10^{-3}$. The buckling force threshold has been identified as an important force-scale for stiffening of these networks under external shear [@conti_cross-linked_2009; @onck_alternative_2005]. In addition, these data imply that a minimum motor density is required for motor generated stiffening, consistent with recent experiments [@koenderink_active_2009]. The characteristic motor-generated stress can be expressed as $\sigma_{\rm M}=\rho_{\rm M} \ell_0 f_0 $. Remarkably, all stiffening curves can be collapsed by expressing the shear modulus as a function of $\sigma_{\rm M}$ (upper inset Fig. \[fig:Mvaryden\]). This demonstrates that the characteristic motor generated stress $\sigma_{\rm M}$ is a useful quantity, even though the distribution of stress is likely to be highly heterogenous.
To explore the nature of the stiffening induced by motors we study the networks’ response at various values of $\kappa$. We observe that motor activity dramatically increases the network stiffness over a range of $\kappa$ values, as shown in Fig. \[fig:Mvarykappa\]. Interestingly, the degree of stiffening induced by motors stress is substantially larger for networks with lower $\kappa$, while for large $\kappa$ we observe no stiffening at all. To compare the stiffening between the active and passive networks, we determine the critical stress for the onset of stiffening. When the linear mechanics of the networks is controlled by bending modes ($G\sim\kappa$) we find that $\sigma_c$ scales linearly with $\kappa$ for both active and passive networks, as shown in the inset Fig. \[fig:Mvarykappa\]. At larger bending rigidities $\sigma_c$ saturates to a value independent of $\kappa$. Interestingly, the values of $\sigma_c$ for active floppy networks are substantially lower than for the passive networks. This indicates that internally generated motor stress is more effective in network stiffening than an external stress.
To identify the role of filament length in motor generated stiffening we vary $\mathcal{Q}$ to tune $\langle L\rangle$. Interestingly, only networks with relatively short filaments stiffen strongly (Fig. \[fig:MvaryP\]). Networks with longer filaments are governed increasingly by the stretching modes in the system. This is consistent with the numerical data in Fig. \[fig:Mvarykappa\], for which we observed that only bending dominated networks are capable of stiffening by motor activity. The critical stress for the onset of stiffening scales in the same way with $\langle L\rangle$ for the active networks as for the passive networks (inset Fig. \[fig:MvaryP\]), similar to what we observed for the scaling of $\sigma_c$ with $\kappa$ (inset Fig. \[fig:MvaryP\]). Taken together, these results provide evidence that the motor generated stiffening in the active networks derives from the same origin as the stiffening of passive networks under external shear.
![\[fig:MvaryP\] The linear shear modulus $G$ as a function of motor generated stress $\sigma_{\rm M}$ for various values of $\langle L\rangle$ at fixes bending rigidity $\kappa=10^{-3}$. The stiffening curves for $\langle L \rangle \lesssim 5$ show an approximate scaling behavior given by $K \sim \sigma$, as shown by the dashed lines that indicate a slope of $1$. The inset shows the critical stress for the onset of stiffening as a function of $\kappa$ for both the active (red squares) and the passive (black circles) systems.](MvaryP.pdf){width="\columnwidth"}
The analogy between external stress and motor generated stress can be further explored by determining the effect of motor activity on the microscopic deformation field. The stiffening in passive networks has been attributed to a shear-induced cross-over between soft bending modes and stiffer stretching modes; concomitant with this cross-over the deformation becomes increasingly affine for larger strains [@onck_alternative_2005]. Our simulations suggest that the same basic mechanism is responsible for the motor generated stiffening in non-affine networks. To further test this picture we investigate the microscopic deformation field of the these networks under a small external shear. We subtract the affine deformation $\delta {\bf r}_i^{({\rm A})}$ of a cross-link $i$ from the actual deformation $\delta {\bf r}_i$ to isolate the non-affine contribution, $$\label{eq:NA}
\delta {\bf r}_i^{({\rm NA})}=\delta {\bf r}_i-\delta {\bf r}_i^{({\rm A})}$$ Consistent with prior work[@head_distinct_2003] for a passive networks deep in the bending dominated regime, we observe large non-affine deformations, as shown in Fig. \[fig:NA\]a. In contrast, when motors are present the non-affine contribution to the deformation field is substantially reduced, as shown in Fig. \[fig:NA\]b. Note, that the motors will initially generate highly non-affine deformations and large bends. These results show, however, that the subsequent deformation of this active network under a small external shear is considerably more affine than in the passive case. This provides insight into the motor induced stiffening we observed in our simulations (Figs. \[fig:Mvarykappa\] and \[fig:MvaryP\]). Motor activity pulls out the floppy bend modes, which renders the network deformation more affine and, thereby, induces a cross-over from a response governed by bending modes to a response governed by stretching modes.
conclusion
==========
Here we have show that molecular motors—modeled as force dipoles—stiffen non-affine networks. Interestingly, we find that only networks that are strongly governed by bending modes are capable of stiffening through motor activity. The internal stresses generated by the motors pull-out the floppy bending modes in the system, leaving the stiff stretching modes. In this way, motors induce a cross-over to a stretching dominated regime, in analogy to prior results on externally-stressed networks [@onck_alternative_2005; @conti_cross-linked_2009]. The absence of motor-induced stiffening of our networks in the stretching dominated regime can be attributed to the purely linear force-extension behavior in our model. Analytical studies based on affine *stretching* dominated networks have shown that motor activity can lead to stiffening when the expected non-linear force-extension relation is taken into account [@mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @liverpool_mechanical_2009].
Nevertheless, within the model we consider, with purely linear elements, our results support the qualitative equivalence of external and internal stress in the nonlinear network response [@mizuno_nonequilibrium_2007; @koenderink_active_2009; @mackintosh_nonequilibrium_2008]. So far, this correspondence has been understood in the context of stretching-dominated networks, with nonlinear filaments [@mackintosh_nonequilibrium_2008; @levine_mechanics_2009; @liverpool_mechanical_2009]. The present work shows that this analogy is more general. Interestingly, however, there are some quantitative differences between network stiffening by external load vs internal motor stresses. Specifically, our results support the idea that motor stresses can be more effective in generating stiffening, since they act in all directions [@koenderink_active_2009]. By contrast, when a network is externally sheared most stress is focussed on a small fraction of the filaments that are oriented the direction of extension. Furthermore, there are quantitative differences in the form of the stiffening response with stress in the present model. We find that motor contractility leads to an increase in the shear modulus with motor stress $\sigma_{\rm M}$ (Figs. 4, 5) that is approximately given by $G \sim \sigma_{\rm M}^x$, where $x\approx 1$. By contrast, the stiffening by external shear exhibits a more complex dependence on the stress, with two distinct regimes, corresponding to $x\simeq1$ and $x\simeq1/2$. One important difference that sets the passive networks apart, are the geometric effects that arise at large external shears through the collective alignment of filament in the direction of maximum extension.
The results presented here provide further insight into the mechanisms available for the active cellular cytoskeleton to regulate the mechanical behavior of the cell. Furthermore, these principles can inspire the design of novel active biomemetic materials with tunable elastic properties.
This work was funded in part by FOM/NWO. The authors thank I. Barmes, E. Conti, M. Das and M. Depken for fruitful discussions.
|
---
abstract: 'Experimental observation of a new mechanism of sandpile formation is reported. As a steady stream of dry sand is poured onto a horizontal surface, a pile forms which has a thin river of sand on one side flowing from the apex of the pile to the edge of its base. The river rotates about the pile, depositing a new layer of sand with each revolution, thereby growing the pile. For small piles the river is steady and the pile formed is smooth. For larger piles, the river becomes intermittent and the surface of the pile becomes undulating. The frequency of revolution of the river is measured as the pile grows and the results are explained with a simple scaling argument. The essential features of the system that produce the phenomena are discussed.'
author:
- 'E. Altshuler$^1$, O. Ramos$^{1,2}$, A.J. Batista-Leyva$^{1,3}$, A. Rivera$^{4}$ and K.E. Bassler$^{5}$'
title: Sandpile formation by revolving rivers
---
Sandpiles have received considerable interest because of their intrinsic scientific interest both from the fundamental and applied points of view, and also because they are simple examples of complex systems whose behavior has been used in an attempt to explain a variety of physical, chemical, biological and social phenomena[@B96]. Conventional understanding of sandpile formation is that as grains of sand are poured onto a horizontal surface, a conical pile develops which grows intermittently through avalanches that “adjust” the angle of repose of the pile about some critical value, or, at least, keep it between two critical values. This mechanism of pile formation has been widely studied in the recent years[@BTW87; @JLN88; @HSK90; @RVK93; @RVK94; @M94; @BCRE95; @F96; @CCF96; @BRG98; @DD99; @DM00; @TCB00; @ARM01; @AT01; @ARG02]. Here we report experimental observation of a remarkable new mechanism of pile formation.
Pouring a steady stream of sand into the center of a cylindrical container, as shown in Fig. 1, a pile formed. Then, a continuous river of sand developed flowing from the apex of the pile to the inner boundary of the container. The river, which was narrow compared with the radius of the container, revolved around the pile depositing a helical layer of sand a few grains thick with each revolution. Thus, the pile grew as the river revolved around it. A photograph of a revolving river can be seen in Fig. 2. Within a range of experimental parameters and conditions, the formation of a revolving river was easily reproducible, and very robust. Once formed, a typical river persisted for dozens of full turns around the growing pile, and stopped only when forced to by interrupting the pouring of sand.
In the experiments, a vertical glass tube with a 20 mm inner diameter was initially filled with sand using a funnel. Then a 4 mm hole was opened in the bottom of the tube, allowing sand to fall out of the tube by its own weight. This arrangement produced a steady flow of sand out of the tube at a steady rate of 4.5 g/s for the duration of the experiment. Video cameras recorded both lateral and top views of the piles during the experiment. (Top views were obtained with the help of a $45^o$ tilted mirror). Two different versions of the experiment were performed, each corresponding to a different boundary condition of the growing pile. In the first version (described above), the pile had a [*closed boundary*]{}. The sand was dropped at the center of a cylindrical container, so that the radius of the resulting pile was constant in time. In the second version, the pile had an [*open boundary*]{}. No container was present. Instead, the sand fell onto a flat horizontal surface and the radius of the pile increased in time.
Rivers that revolved about the pile in both clockwise and counterclockwise directions were observed. The direction chosen in a particular case depended on the initial conditions. The axial symmetry of the system was therefore spontaneously broken as the river was formed. Viewed from above, the rivers were slightly bent, and always revolved around the pile in the direction of their concavity, as shown in Figs. 2a and 2c. A steady revolving river was typically observed when sand was poured into a container with a 4-6 cm radius. In this case, the surface of the pile was smooth. However, when a container with a radius larger than 6 cm was used, an instability appeared in the flow of the river. The revolving river still developed, growing the pile as before, but the flow of the river was intermittent rather than continuous. In that case, the intermittent flow produced an undulating pattern on the pile surface, visible in Figs. 2c and 2d. The undulating pattern resembles those recently observed for rapid granular flows on an inclined plane[@FP01; @AT02], but presumably is caused by a different mechanism. The observed pattern was quite regular for containers with a radius just large enough to observe the instability, but became more irregular as the size of the container grew. If a container smaller than 3 cm radius was used, stable revolving rivers were not observed.
The revolving river mechanism of pile formation has also been observed by simply pouring the sand onto a flat surface. In that case, the crossover from a continuously flowing revolving river, observed in smaller piles, to an intermittently flowing river, observed in larger piles, occurred as the radius of the pile reached about 6 cm. The crossover appeared to be correspond to the pile size needed for the length of the ballistic motion of the sand grains in a river to begin to be damped.
We have varied the drop height in the experiment. For drop heights between 1 cm and 7 cm the results closely follow the description given above. However, for drop heights less than 1 cm or larger than 7 cm, stable rivers were not observed.
The origin of the curved shape of a revolving river and the reason for it moving in the direction of its concavity can be understood by how a river forms. Based on careful observation, revolving rivers appear to form through the following scenario, illustrated in Fig. 3. Initially, sand is poured onto the top of a conical pile and it forms a river flowing straight down one side of the pile (Fig. 3a). Sand begins to build up at the bottom of the river at the edge of the pile, forming a growing inverted V shaped delta of stationary sand (Fig. 3b). The delta grows in size until the river spontaneously chooses to begin to flow down one of the sides of the delta (Fig. 3c). Once it chooses a side, it continues to flow down that side of the delta, depositing sand all along the lower, delta side of the river. As it does so, it rotates about the pile. For smaller piles, the process of rotation was stable. However, for larger piles, it was not. Instead, in that case, a new delta would intermittently begin to form at the bottom of the river (Fig. 3d). When the delta reached sufficient height, the river would “jump” forward in its rotation, and then begin forming yet another new delta.
In order to begin to quantitatively understand the revolving rivers, we measured the time evolution of the angular velocity of river rotation with both closed and open boundaries. As shown in Fig. 4, the angular velocity of river rotation was roughly constant for piles in cylindrical containers, while it decreased in time as $t^{-\alpha}$, with $\alpha = 2/3$, for open boundary conditions. These results can be explained using the following scaling argument whose geometrical hypotheses are illustrated in Fig. 5. Assume that a new layer of sand is uniformly deposited on a conical pile of radius $r$ with an angle of repose $\theta_c$, and that the volume of sand added per unit time is $F$. For a system with a closed boundary, the thickness of an added layer is proportional to $\delta h$ (see Fig. 5a). Therefore, the volume of sand deposited in each rotation of the river $$V = {\pi \; r^2 \; \delta h \over \cos \theta_c }$$ is constant in time. The angular velocity of the river $$\omega = 2\pi {F \over V}$$ is therefore also constant in time. In our experiments, we measured $\delta h =$ 2 mm, $\phi_c = 33^o$, and $F=0.35$ cm$^{3}$/s . However, for a system with an open boundary the radius of pile grows in time. In this case, the thickness of each layer is proportional to $\delta r$ (see Fig. 5b). The volume of sand deposited in a rotation of the river is $$V = \pi \;
\tan \theta_c \;
r^2 \; \delta r$$ where $r$ is a function of time, but $\delta r= \delta h/\sin \theta_c$ is constant. Thus, from this result and Eqn. 1, $\omega \sim r^{-2}$. The pile radius increases at a rate of $${dr \over dt}
=
\omega \; \delta r$$ Integrating this expression, we get $r \sim t^{1/3}$, and therefore $$\omega \sim t^{-2/3} \; .$$ Our scaling argument matches well the experimental results shown in Fig. 4a. In the case of Fig. 4b, although this argument correctly predicts the scaling of the experimental data for larger piles, it does not properly describe the behavior of smaller piles, presumably due to the fact that our geometrical assumptions are inaccurate near the tip of the pile.
The appearance of revolving rivers is quite sensitive to the type of sand used in the experiments. In the results reported here, sand from Santa Teresa, Cuba, was used. It consists in irregularly shaped grains of size $30-250 \; {\mu} m$ made of almost pure silicon oxide. It was also quite dry. Revolving rivers were still observed if the sand was meshed to remove grains smaller than $90\;{\mu} m$ and larger than $160 \; {\mu}m$. However, other sands from Cuba, USA, Norway and Tunisia were tried, including ones high in Calcium Carbonate, and ones high in Magnetite, but no revolving rivers were observed within our experimental conditions (a river occasionally formed in those sands, but it disappeared in fractions of a second). Revolving rivers also were not observed if glass beads having roughly the same size as the Santa Teresa sand were used. It is therefore suspected that the effective coefficient of friction between grains, and the mass density of grains may be important factors determining if revolving rivers appear in the formation of piles. These elements must be included in a future “first principles” model of the revolving rivers.
We acknowledge E. Martínez for cooperation in the experiments, and S. Douady, H. Herrmann, T. H. Johansen, R. Mulet,O. Pouliquen, H. Seidler, O. Sotolongo and J.E. Wesfried for useful discussions and comments. We thank material support from the University of Havana’s “Alma Mater” grants programme. KEB acknowledges support from the NSF through grant \#DMR-0074613, from the Alfred P. Sloan Foundation, and from the Texas Center for Superconductivity.
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Figure Captions {#figure-captions .unnumbered}
---------------
Fig. 1. Experimental setup
Fig. 2. Formation of a pile of sand by revolving rivers. The sand is poured vertically on the center of cylindrical containers with flat, horizontal bottoms at a deposition rate of 0.35 $cm^3/s$, from a constant height of 1.5 cm above the apex of the pile. (a) Top view of a pile growing into a 5 cm radius container, where the continuous river can be identified. (b) Lateral view of the pile shown in (a). (c) Top view of a pile growing into a 10 cm radius container where an intermittent river and the related pattern can be identified. (d) Lateral view of the pile shown in (c) (the photo shows about 3 cm of the container’s perimeter). In all cases, arrows indicate the revolving direction.
Fig. 3. Development of a revolving river. (a) A river flows straight down the side of the pile, and a delta begins to form at its bottom. (b) The delta continues to grow. (c) When the delta is sufficient size, the river begins to flow down one side and rotate around the pile. (d) If the pile is sufficiently large, a new delta forms intermittently at the bottom of the river, causing the rotation of the river to become intermittent.
Fig. 4. Time dependence of the angular speed of revolving rivers for (a) closed boundary conditions in the continuous regime(5 cm-radius container) and (b) open boundary conditions. The solid line in (b) has a slope of -2/3.
Fig. 5. Geometrical hypotheses of our scaling argumet for (a) closed boundary conditions and (b) open boundary conditions.
|
---
abstract: 'We create the unlabeled or vertex-labeled graphs with up to 10 edges and up to 10 vertices and classify them by a set of standard properties: directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how many graphs exist in these categories.'
address: 'Max-Planck Institute of Astronomy, Königstuhl 17, 69117 Heidelberg, Germany'
author:
- 'Richard J. Mathar'
bibliography:
- 'all.bib'
title: Statistics on Small Graphs
---
Classifications {#sec.tag}
===============
A finite graph on $V$ vertices with $E$ edges may be classified by some properties, which it either does have or does not:
- Each edge in a directed graph has one of two orientations. Edges in unoriented graphs do not have orientations. We reserve the tag `d` for the directed and `-d` for the undirected graphs. The adjacency matrices of undirected graphs are symmetric.
- Graphs may have at least one loop (loops are defined as edges that start and end at the same vertex), or may be loopless. The adjacency matrices of loopless graphs have zero trace. We reserve the tag `l` for the graphs with at least one loop and the tag `-l` for the loopless graphs.
- A multiedge is a collection of two ore more edges having identical endpoints [@Gross D7]. This implies that in a directed graph two edges of opposite sense do not yet establish a multiedge. We reserve the tag `m` for the graphs with at least one multiedge and `-m` for the others.
- A undirected graph is connected if one can walk from any vertex to any other vertex of the graph along edges. A directed graph is (weakly) connected if replacing each arc with an undirected edge (defining the underlying graph [@Gross D24]) reduces to a connected undirected graph. This implies that for the sake of weak connectivity it is not required that all arcs are traversed along their orientation to walk from one vertex to the other. We reserve the tag `c` for the graphs which are (weakly) connected and `-c` for the others.
A directed graph is strongly connected if one can walk from any vertex to any other vertex of the graph along edges in the directions demanded by their orientation. We reserve the tag `C` for the digraphs which are strongly connected and `-C` for the others.
- An isolated vertex is a vertex with no edge to any other vertex (so all its edges are loops). There is a loose relation with connectivity, because an isolated vertex in a graph with two or more vertices means the graph is disconnected. (There are disconnected graphs without isolated vertices…where each component contains at least two vertices.) We reserve the tag `i` for the graphs which have at least one isolated vertex and `-i` for the others. Therefore all graphs with $V=1$ are getting the `i` tag.
There are no graphs with the following combinations of tags
- `d-Cci` A directed, weakly connected graph with at least one isolated vertex has only this vertex (because with two or more vertices the graph could not be connected), and therefore the graph must also be strongly connected. So the `-C` contradicts the other tags.
- `dC-c` If the directed graph is strongly connected, it is also weakly connected, so the `-c` tag contradicts the `C` tag.
There are some non-interesting cases, which are not tabulated explicitly:
- There is the case with the tags `dCci`: A directed strongly-connected graph with isolated vertices has only one vertex, so a table with these graphs counts at most 1 graph for any number of edges (which all are loops).
- Similarly there is the case with the tags `-dci`: An undirected connected graph with isolated vertices has only one vertex, so a table with these graphs counts at most 1 graph for any number of edges (where all edges are loops).
There are many other characterizations of graphs concerning cycles, paths, diameters, transitivity and so on which are not dealt with here.
Statistics
==========
Tables \[tabU.d-C-c-i-m-l\]–\[tabL.dCc-iml\] collect the statistics of directed graphs; Tables \[tabU.-d-c-i-m-l\]–\[tabL.-d-ciml\] collect the statistics of unndirected graphs. Rows and columns are sorted along the number $E$ of edges and along the number $V$ of vertices. There are always successive tables referring to the unlabeled graphs and referring to the vertex-labeled graphs. (The latter count is obtained by weighting each unlabeled graph by the number of distinct adjancency matrices that are created by row-column permutations of the adjacency matrix. This weight is a divisor of the order of the permutation group on $V$ elements [@Harary Thm 15.2].)
Directed Graphs
---------------
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- --- ----- ----- ----- ---- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 1 0 0 0 0 0 0
3 0 0 0 1 3 1 0 0 0 0
4 0 0 0 1 7 15 3 1 0 0
5 0 0 0 0 8 43 58 15 3
6 0 0 0 0 5 82 244 257 68
7 0 0 0 0 2 103 674
8 0 0 0 0 1 102
9 0 0 0 0 0
10 0 0 0 0
: `d-C-c-i-m-l` unlabeled[]{data-label="tabU.d-C-c-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------- --------- --------- --------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 12 0 0 0 0 0 0
3 0 0 0 12 240 120 0 0 0 0
4 0 0 0 3 520 5460 5040 1680 0 0
5 0 0 0 0 500 19770 151200 191520 120960
6 0 0 0 0 270 37135 795368 5021912 7761600
7 0 0 0 0 80 46560 2359224
8 0 0 0 0 10 42450
9 0 0 0 0 0
10 0 0 0 0
: `d-C-c-i-m-l` vertex-labeled[]{data-label="tabL.d-C-c-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------ ------ --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 0 3 0 0 0 0 0 0 0
3 0 0 3 8 0 0 0 0 0 0
4 0 0 2 21 27 0 0 0 0 0
5 0 0 0 33 107 91 0 0 0
6 0 0 0 31 319 581 350 0 0
7 0 0 0 16 609 2422 3023
8 0 0 0 5 887 7529
9 0 0 0 2 912
10 0 0 0 0
: `d-Cc-i-m-l` unlabeled[]{data-label="tabU.d-Cc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- -------- --------- ---------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 2 0 0 0 0 0 0 0 0
2 0 0 12 0 0 0 0 0 0 0
3 0 0 18 128 0 0 0 0 0 0
4 0 0 6 426 2000 0 0 0 0 0
5 0 0 0 684 11080 41472 0 0 0
6 0 0 0 604 33160 337800 1075648 0 0
7 0 0 0 300 67040 1529520 11967984
8 0 0 0 78 96610 4954230
9 0 0 0 8 101580
10 0 0 0 0
: `d-Cc-i-m-l` vertex-labeled[]{data-label="tabL.d-Cc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- ---- ------ ------ ------ ------ ---- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1 1
2 0 0 1 4 5 5 5 5 5 5
3 0 0 0 4 13 16 17 17 17 17
4 0 0 0 4 27 61 76 79 80 80
5 0 0 0 1 38 154 288 346 361
6 0 0 0 1 48 379 1043 1637 1894
7 0 0 0 0 38 707 3242
8 0 0 0 0 27 1155
9 0 0 0 0 13
10 0 0 0 0
: `d-C-ci-m-l` unlabeled[]{data-label="tabU.d-C-ci-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ------ -------- ---------- ---------- ----------- --------- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1 1
1 0 0 6 12 20 30 42 56 72 90
2 0 0 3 54 190 435 861 1540 2556 4005
3 0 0 0 80 900 3940 11480 27720 59640 117480
4 0 0 0 60 2325 21945 106890 365610 1028790 2555190
5 0 0 0 24 3900 81264 699468 3628296 13870584
6 0 0 0 4 4610 218720 3374770 27446524 148477308
7 0 0 0 0 3960 453240 12650400
8 0 0 0 0 2475 748395
9 0 0 0 0 1100
10 0 0 0 0
: `d-C-ci-m-l` vertex-labeled[]{data-label="tabL.d-C-ci-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------ ------ ---- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 1 0 0 0 0 0 0
4 0 0 0 4 7 1 0 0 0 0
5 0 0 0 7 35 42 7 1 0
6 0 0 0 12 101 271 234 48 7
7 0 0 0 16 230 1057 1848
8 0 0 0 24 462 3285
9 0 0 0 30 855
10 0 0 0 41
: `d-C-c-im-l` unlabeled[]{data-label="tabU.d-C-c-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------- --------- --------- -------- -------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 24 0 0 0 0 0 0
4 0 0 0 72 720 360 0 0 0 0
5 0 0 0 132 3520 22560 20160 6720 0
6 0 0 0 210 10100 154650 806400 974400 604800
7 0 0 0 312 23120 630360 7141008
8 0 0 0 441 46970 1991325
9 0 0 0 600 88280
10 0 0 0 792
: `d-C-c-im-l` vertex-labeled[]{data-label="tabL.d-C-c-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ----- ------ ------- ------- ------ --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 1 4 0 0 0 0 0 0 0
4 0 1 16 18 0 0 0 0 0 0
5 0 1 30 109 80 0 0 0 0
6 0 1 53 391 694 367 0 0 0
7 0 1 77 1042 3574 4207 1708
8 0 1 116 2402 14093 29082
9 0 1 156 5001 46144
10 0 1 215 9737
: `d-Cc-im-l` unlabeled[]{data-label="tabU.d-Cc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ------ -------- --------- ---------- --------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 2 0 0 0 0 0 0 0 0
3 0 2 24 0 0 0 0 0 0 0
4 0 2 90 384 0 0 0 0 0 0
5 0 2 180 2472 8000 0 0 0 0
6 0 2 300 8960 75400 207360 0 0 0
7 0 2 462 24324 405160 2648880 6453888
8 0 2 672 56322 1623440 19251960
9 0 2 936 118168 5394560
10 0 2 1260 230760
: `d-Cc-im-l` vertex-labeled[]{data-label="tabL.d-Cc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------ ------- ------- ------ ------ ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 1 1 1 1 1 1 1 1
3 0 0 2 6 7 7 7 7 7 7
4 0 0 3 20 42 49 50 50 50 50
5 0 0 3 41 158 273 315 322 323
6 0 0 4 82 506 1302 1940 2174 2222
7 0 0 4 132 1330 5174 10439
8 0 0 5 222 3213 18293
9 0 0 5 335 7097
10 0 0 6 511
: `d-C-cim-l` unlabeled[]{data-label="tabU.d-C-cim-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- --- ---- ------- -------- ---------- ---------- ---------- ---------- -- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0
2 0 0 6 12 20 30 42 56 72
3 0 0 12 120 400 900 1764 3136 5184
4 0 0 15 414 3290 13155 37065 87836 186660
5 0 0 18 948 15480 113190 499926 1634976 4483296
6 0 0 21 1802 52720 667375 4685387 22082536 80250072
7 0 0 24 3120 147320 3031920 33055848
8 0 0 27 5094 362655 11463930
9 0 0 30 7948 818780
10 0 0 33 11946
: `d-C-cim-l` vertex-labeled[]{data-label="tabL.d-C-cim-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- ----- ------ ------ ---- ---- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 2 0 0 0 0 0 0
4 0 0 0 7 13 2 0 0 0 0
5 0 0 0 7 52 70 13 2 0
6 0 0 0 6 106 373 362 82 13
7 0 0 0 2 137 1092 2392
8 0 0 0 1 125 2262
9 0 0 0 0 83
10 0 0 0 0
: `d-C-c-i-ml` unlabeled[]{data-label="tabU.d-C-c-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------- --------- --------- --------- --------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 48 0 0 0 0 0 0
4 0 0 0 120 1200 720 0 0 0 0
5 0 0 0 132 5000 34560 35280 13440 0
6 0 0 0 78 10100 202920 1164240 1579200 1088640
7 0 0 0 24 12750 630360 8919176
8 0 0 0 3 10940 1314405
9 0 0 0 0 6570
10 0 0 0 0
: `d-C-c-i-ml` vertex-labeled[]{data-label="tabL.d-C-c-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ------- ------- ------ --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 2 0 0 0 0 0 0 0 0
3 0 1 7 0 0 0 0 0 0 0
4 0 0 16 26 0 0 0 0 0 0
5 0 0 16 111 107 0 0 0 0
6 0 0 7 262 702 458 0 0 0
7 0 0 2 372 2663 4251 2058
8 0 0 0 361 6936 22925
9 0 0 0 240 13442
10 0 0 0 115
: `d-Cc-i-ml` unlabeled[]{data-label="tabU.d-Cc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ------ --------- ---------- --------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 4 0 0 0 0 0 0 0 0
3 0 2 36 0 0 0 0 0 0 0
4 0 0 90 512 0 0 0 0 0 0
5 0 0 84 2472 10000 0 0 0 0
6 0 0 36 5804 75400 248832 0 0 0
7 0 0 6 8352 296600 2648880 7529536
8 0 0 0 7986 787600 15073560
9 0 0 0 5212 1542450
10 0 0 0 2304
: `d-Cc-i-ml` vertex-labeled[]{data-label="tabL.d-Cc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ------ ------- ------- ------ ------ ----- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 1 1 1 1 1 1 1
2 0 1 4 4 4 4 4 4 4 4
3 0 0 6 17 20 20 20 20 20 20
4 0 0 3 35 83 100 103 103 103 103
5 0 0 1 46 236 457 548 565 568
6 0 0 0 40 504 1659 2756 3210 3313
7 0 0 0 25 833 4986 12171
8 0 0 0 10 1064 12330
9 0 0 0 3 1084
10 0 0 0 1
: `d-C-ci-ml` unlabeled[]{data-label="tabU.d-C-ci-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- --- ---- ----- -------- --------- ---------- ---------- ----------- -- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0
1 0 2 3 4 5 6 7 8 9
2 0 1 21 54 110 195 315 476 684
3 0 0 28 292 1160 3080 6944 13944 25680
4 0 0 15 693 6605 30780 99946 268086 634950
5 0 0 3 948 22626 199926 1020936 3791256 11630052
6 0 0 0 830 52720 902265 7573790 40926732 167212668
7 0 0 0 480 89990 3031920 42473544
8 0 0 0 180 117425 7978770
9 0 0 0 40 119900
10 0 0 0 4
: `d-C-ci-ml` vertex-labeled[]{data-label="tabL.d-C-ci-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ------ ------- ------- ------ --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 6 0 0 0 0 0 0
5 0 0 0 34 46 6 0 0 0
6 0 0 0 107 347 314 52 6 0
7 0 0 0 250 1473 2869 1995
8 0 0 0 527 4731 15676
9 0 0 0 994 12883
10 0 0 0 1797
: `d-C-c-iml` unlabeled[]{data-label="tabU.d-C-c-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ------- --------- --------- --------- ------- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 144 0 0 0 0 0 0
5 0 0 0 768 4800 2880 0 0 0
6 0 0 0 2340 37000 180000 176400 67200 0
7 0 0 0 5568 159600 1750320 7514640
8 0 0 0 11634 518350 9908640
9 0 0 0 22368 1427320
10 0 0 0 40392
: `d-C-c-iml` vertex-labeled[]{data-label="tabL.d-C-c-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ------ ------- -------- ------- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 4 0 0 0 0 0 0 0 0
4 0 9 19 0 0 0 0 0 0 0
5 0 14 98 94 0 0 0 0 0
6 0 20 286 761 479 0 0 0 0
7 0 27 645 3522 5398 2480 0
8 0 35 1290 12111 34960 36619
9 0 44 2372 34847 167682
10 0 54 4110 89361
: `d-Cc-iml` unlabeled[]{data-label="tabU.d-Cc-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------- --------- ---------- ---------- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 8 0 0 0 0 0 0 0 0
4 0 18 108 0 0 0 0 0 0 0
5 0 28 576 2048 0 0 0 0 0
6 0 40 1680 17480 50000 0 0 0 0
7 0 54 3816 82144 602400 1492992 0
8 0 70 7644 284788 4009600 23767632
9 0 88 14112 824480 19528000
10 0 108 24480 2121756
: `d-Cc-iml` vertex-labeled[]{data-label="tabL.d-Cc-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ----- ------- ------- ------- ------- ------ ------ ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 1 1 1 1 1 1 1 1
3 0 2 8 8 8 8 8 8 8 8
4 0 3 27 55 63 63 63 63 63 63
5 0 3 55 224 402 460 468 468 468
6 0 4 97 671 1956 3051 3444 3508 3516
7 0 4 154 1661 7607 17024 23868
8 0 5 235 3670 25207 81289
9 0 5 342 7505 74029
10 0 6 483 14483
: `d-C-ciml` unlabeled[]{data-label="tabU.d-C-ciml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- ---- ------ -------- --------- ---------- ---------- ---------- ---------- -- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0
2 0 2 3 4 5 6 7 8 9
3 0 4 45 112 225 396 637 960 1377
4 0 5 150 1042 3815 9831 21784 43268 79101
5 0 6 315 4744 33825 142386 442715 1157920 2696625
6 0 7 553 14864 191335 1339211 6175162 21778968 64759989
7 0 8 894 37768 805875 9075456 62861757
8 0 9 1368 84739 2785270 48311766
9 0 10 2005 175140 8398505
10 0 11 2838 340402
: `d-C-ciml` vertex-labeled[]{data-label="tabL.d-C-ciml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ----- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 0 1 0 0 0 0 0 0 0
4 0 0 2 1 0 0 0 0 0 0
5 0 0 1 4 1 0 0 0 0
6 0 0 1 16 7 1 0 0 0
7 0 0 0 22 58 10 1
8 0 0 0 22 240 165
9 0 0 0 11 565
10 0 0 0 5
: `dCc-i-m-l` unlabeled[]{data-label="tabU.dCc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------- -------- ----- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 0 2 0 0 0 0 0 0 0
4 0 0 9 6 0 0 0 0 0 0
5 0 0 6 84 24 0 0 0 0
6 0 0 1 316 720 120 0 0 0
7 0 0 0 492 6440 6480 720
8 0 0 0 417 26875 107850
9 0 0 0 212 65280
10 0 0 0 66
: `dCc-i-m-l` vertex-labeled[]{data-label="tabL.dCc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ----- ------ ------ ---- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 1 0 0 0 0 0 0 0 0
4 0 2 1 0 0 0 0 0 0 0
5 0 2 8 1 0 0 0 0 0
6 0 3 25 21 1 0 0 0 0
7 0 3 51 140 40 1 0
8 0 4 101 565 525 69
9 0 4 174 1731 3719
10 0 5 290 4602
: `dCc-im-l` unlabeled[]{data-label="tabU.dCc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ------ -------- -------- ------- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 2 0 0 0 0 0 0 0 0
4 0 3 6 0 0 0 0 0 0 0
5 0 4 48 24 0 0 0 0 0
6 0 5 140 480 120 0 0 0 0
7 0 6 306 3276 4680 720 0
8 0 7 588 13230 61040 47880
9 0 8 1036 41024 437320
10 0 9 1710 109152
: `dCc-im-l` vertex-labeled[]{data-label="tabL.dCc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------ ---- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 1 0 0 0 0 0 0 0 0
4 0 1 1 0 0 0 0 0 0 0
5 0 0 6 1 0 0 0 0 0
6 0 0 9 17 1 0 0 0 0
7 0 0 6 78 34 1 0
8 0 0 2 185 346 60
9 0 0 1 259 1775
10 0 0 0 252
: `dCc-i-ml` unlabeled[]{data-label="tabU.dCc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ------ -------- ------- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 2 0 0 0 0 0 0 0 0
4 0 1 6 0 0 0 0 0 0 0
5 0 0 33 24 0 0 0 0 0
6 0 0 47 372 120 0 0 0 0
7 0 0 30 1792 3840 720 0
8 0 0 9 4206 39640 40680
9 0 0 1 5968 206095
10 0 0 0 5634
: `dCc-i-ml` vertex-labeled[]{data-label="tabL.dCc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ------ ------- ------ --- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 3 0 0 0 0 0 0 0 0
5 0 8 4 0 0 0 0 0 0
6 0 16 38 5 0 0 0 0 0
7 0 25 151 110 6 0 0
8 0 40 431 898 250 7
9 0 56 1040 4475 3665
10 0 80 2252 17039
: `dCc-iml` unlabeled[]{data-label="tabU.dCc-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------- -------- -------- ------ --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 6 0 0 0 0 0 0 0 0
5 0 16 24 0 0 0 0 0 0
6 0 30 225 120 0 0 0 0 0
7 0 50 897 2592 720 0 0
8 0 77 2562 21196 29400 5040
9 0 112 6190 106336 431360
10 0 156 13437 405552
: `dCc-iml` vertex-labeled[]{data-label="tabL.dCc-iml"}
Undirected Graphs
-----------------
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- --- --- --- ---- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 1 0 0 0 0 0 0
3 0 0 0 0 1 1 0 0 0 0
4 0 0 0 0 1 3 1 1 0 0
5 0 0 0 0 0 3 6 3 1
6 0 0 0 0 0 2 9 15 7
7 0 0 0 0 0 1 8
8 0 0 0 0 0 0
9 0 0 0 0 0
10 0 0 0 0
: `-d-c-i-m-l` unlabeled[]{data-label="tabU.-d-c-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- ---- ----- ------ ------- -------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 3 0 0 0 0 0 0
3 0 0 0 0 30 15 0 0 0 0
4 0 0 0 0 10 330 315 105 0 0
5 0 0 0 0 0 285 4410 5880 3780
6 0 0 0 0 0 100 6797 71078 116550
7 0 0 0 0 0 15 5460
8 0 0 0 0 0 0
9 0 0 0 0 0
10 0 0 0 0
: `-d-c-i-m-l` vertex-labeled[]{data-label="tabL.-d-c-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- --- ---- ---- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 0 1 0 0 0 0 0 0 0
3 0 0 1 2 0 0 0 0 0 0
4 0 0 0 2 3 0 0 0 0 0
5 0 0 0 1 5 6 0 0 0
6 0 0 0 1 5 13 11 0 0
7 0 0 0 0 4 19 33
8 0 0 0 0 2 22
9 0 0 0 0 1
10 0 0 0 0
: `-dc-i-m-l` unlabeled[]{data-label="tabU.-dc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------ ------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 0 3 0 0 0 0 0 0 0
3 0 0 1 16 0 0 0 0 0 0
4 0 0 0 15 125 0 0 0 0 0
5 0 0 0 6 222 1296 0 0 0
6 0 0 0 1 205 3660 16807 0 0
7 0 0 0 0 120 5700 68295
8 0 0 0 0 45 6165
9 0 0 0 0 10
10 0 0 0 0
: `-dc-i-m-l` vertex-labeled[]{data-label="tabL.-dc-i-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- --- --- ---- ---- ---- ---- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1 1
2 0 0 0 1 2 2 2 2 2 2
3 0 0 0 1 3 4 5 5 5 5
4 0 0 0 0 2 6 9 10 11 11
5 0 0 0 0 1 6 15 21 24
6 0 0 0 0 1 6 21 41 56
7 0 0 0 0 0 4 24
8 0 0 0 0 0 2
9 0 0 0 0 0
10 0 0 0 0
: `-d-ci-m-l` unlabeled[]{data-label="tabU.-d-ci-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ---- ------ ------- -------- --------- -------- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1 1
1 0 0 3 6 10 15 21 28 36 45
2 0 0 0 12 45 105 210 378 630 990
3 0 0 0 4 90 440 1330 3276 7140 14190
4 0 0 0 0 75 1035 5670 20370 58905 148995
5 0 0 0 0 30 1422 15939 92400 373212
6 0 0 0 0 5 1245 30660 305662 1831242
7 0 0 0 0 0 720 42525
8 0 0 0 0 0 270
9 0 0 0 0 0
10 0 0 0 0
: `-d-ci-m-l` vertex-labeled[]{data-label="tabL.-d-ci-m-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- ---- ----- ---- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 1 0 0 0 0 0 0
4 0 0 0 2 2 1 0 0 0 0
5 0 0 0 2 6 8 2 1 0
6 0 0 0 3 10 25 21 9 2
7 0 0 0 3 16 53 80
8 0 0 0 4 23 102
9 0 0 0 4 32
10 0 0 0 5
: `-d-c-im-l` unlabeled[]{data-label="tabU.-d-c-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ------ ------- -------- ------- ------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 6 0 0 0 0 0 0
4 0 0 0 9 90 45 0 0 0 0
5 0 0 0 12 220 1410 1260 420 0
6 0 0 0 15 400 4875 25200 30450 18900
7 0 0 0 18 650 11700 113232
8 0 0 0 21 980 24045
9 0 0 0 24 1400
10 0 0 0 27
: `-d-c-im-l` vertex-labeled[]{data-label="tabL.-d-c-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ----- ----- ---- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 1 1 0 0 0 0 0 0 0
4 0 1 3 3 0 0 0 0 0 0
5 0 1 4 10 6 0 0 0 0
6 0 1 6 21 29 16 0 0 0
7 0 1 7 37 81 91 37
8 0 1 9 61 191 326
9 0 1 11 95 395
10 0 1 13 141
: `-dc-im-l` unlabeled[]{data-label="tabU.-dc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ------ ------- -------- -------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 1 6 0 0 0 0 0 0 0
4 0 1 12 48 0 0 0 0 0 0
5 0 1 18 156 500 0 0 0 0
6 0 1 25 340 2360 6480 0 0 0
7 0 1 33 636 7060 41400 100842
8 0 1 42 1092 17290 162120
9 0 1 52 1764 37740
10 0 1 63 2718
: `-dc-im-l` vertex-labeled[]{data-label="tabL.-dc-im-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ----- ----- ----- ----- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 1 1 1 1 1 1 1 1
3 0 0 1 2 3 3 3 3 3 3
4 0 0 1 4 9 11 12 12 12 12
5 0 0 1 5 17 29 37 39 40
6 0 0 1 7 31 70 111 132 141
7 0 0 1 8 48 145 289
8 0 0 1 10 75 289
9 0 0 1 12 111
10 0 0 1 14
: `-d-cim-l` unlabeled[]{data-label="tabU.-d-cim-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------ -------- -------- -------- --------- ------- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 3 6 10 15 21 28 36 45
3 0 0 3 30 100 225 441 784 1296 2025
4 0 0 3 54 415 1650 4641 10990 23346 45585
5 0 0 3 78 1030 7215 31521 102676 281016
6 0 0 3 106 2035 22400 150766 700378 2529696
7 0 0 3 138 3610 56745 557676
8 0 0 3 174 5995 127170
9 0 0 3 214 9470
10 0 0 3 258
: `-d-cim-l` vertex-labeled[]{data-label="tabL.-d-cim-l"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- --- ---- ---- ---- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 1 0 0 0 0 0 0
4 0 0 0 2 3 1 0 0 0 0
5 0 0 0 1 7 10 3 1 0
6 0 0 0 1 8 28 28 11 3
7 0 0 0 0 6 42 91
8 0 0 0 0 3 48
9 0 0 0 0 1
10 0 0 0 0
: `-d-c-i-ml` unlabeled[]{data-label="tabU.-d-c-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------- -------- ------- ------- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 12 0 0 0 0 0 0
4 0 0 0 18 150 90 0 0 0 0
5 0 0 0 12 350 2205 2205 840 0
6 0 0 0 3 400 6960 37485 49980 34020
7 0 0 0 0 250 11700 151214
8 0 0 0 0 80 12330
9 0 0 0 0 10
10 0 0 0 0
: `-d-c-i-ml` vertex-labeled[]{data-label="tabL.-d-c-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ---- ----- ---- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 1 2 0 0 0 0 0 0 0
4 0 0 3 4 0 0 0 0 0 0
5 0 0 2 10 9 0 0 0 0
6 0 0 1 12 30 20 0 0 0
7 0 0 0 10 57 93 48
8 0 0 0 5 73 240
9 0 0 0 2 67
10 0 0 0 1
: `-dc-i-ml` unlabeled[]{data-label="tabU.-dc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ------ -------- -------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 2 0 0 0 0 0 0 0 0
3 0 1 9 0 0 0 0 0 0 0
4 0 0 12 64 0 0 0 0 0 0
5 0 0 6 156 625 0 0 0 0
6 0 0 1 178 2360 7776 0 0 0
7 0 0 0 116 4495 41400 117649
8 0 0 0 45 5495 115020
9 0 0 0 10 4710
10 0 0 0 1
: `-dc-i-ml` vertex-labeled[]{data-label="tabL.-dc-i-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- --- ---- ----- ----- ----- ----- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 1 1 1 1 1 1 1
2 0 1 3 3 3 3 3 3 3 3
3 0 0 3 7 9 9 9 9 9 9
4 0 0 1 9 20 25 27 27 27 27
5 0 0 0 6 30 58 74 79 81
6 0 0 0 3 32 104 183 226 243
7 0 0 0 1 27 149 381
8 0 0 0 0 16 175
9 0 0 0 0 7
10 0 0 0 0
: `-d-ci-ml` unlabeled[]{data-label="tabU.-d-ci-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ------ ------- -------- --------- --------- -------- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 2 3 4 5 6 7 8 9 10
2 0 1 12 30 60 105 168 252 360 495
3 0 0 10 88 335 875 1946 3864 7050 12045
4 0 0 3 113 1005 4530 14490 38430 90090 192060
5 0 0 0 78 1776 15141 75726 277872 844767
6 0 0 0 28 2035 34523 284991 1521072 6163248
7 0 0 0 4 1570 56745 798897
8 0 0 0 0 815 69705
9 0 0 0 0 275
10 0 0 0 0
: `-d-ci-ml` vertex-labeled[]{data-label="tabL.-d-ci-ml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ----- ----- ----- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 3 0 0 0 0 0 0
5 0 0 0 11 10 3 0 0 0
6 0 0 0 27 51 44 11 3 0
7 0 0 0 51 157 236 153
8 0 0 0 93 386 850
9 0 0 0 150 838
10 0 0 0 241
: `-d-c-iml` unlabeled[]{data-label="tabU.-d-c-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ------ ------- -------- -------- ------ --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 36 0 0 0 0 0 0
5 0 0 0 144 600 360 0 0 0
6 0 0 0 360 3250 11925 11025 4200 0
7 0 0 0 738 10650 76635 257985
8 0 0 0 1365 27650 308385
9 0 0 0 2352 62940
10 0 0 0 3834
: `-d-c-iml` vertex-labeled[]{data-label="tabL.-d-c-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ----- ------ ------ ----- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 2 0 0 0 0 0 0 0 0
4 0 5 5 0 0 0 0 0 0 0
5 0 8 19 13 0 0 0 0 0
6 0 11 45 70 35 0 0 0 0
7 0 15 87 227 245 95 0
8 0 19 153 579 1029 840
9 0 24 252 1302 3346
10 0 29 394 2681
: `-dc-iml` unlabeled[]{data-label="tabU.-dc-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ------ ------- -------- -------- --- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 4 0 0 0 0 0 0 0 0
4 0 9 27 0 0 0 0 0 0 0
5 0 14 102 256 0 0 0 0 0
6 0 20 240 1420 3125 0 0 0 0
7 0 27 471 4688 23535 46656 0
8 0 35 840 12250 102900 453096
9 0 44 1400 28080 345730
10 0 54 2214 58914
: `-dc-iml` vertex-labeled[]{data-label="tabL.-dc-iml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ----- ------ ------ ------ ------ ----- ----- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 1 1 1 1 1 1 1 1
3 0 2 6 6 6 6 6 6 6 6
4 0 3 15 24 29 29 29 29 29 29
5 0 3 26 66 107 122 127 127 127
6 0 4 40 142 318 454 520 536 541
7 0 4 57 269 800 1464 1967
8 0 5 79 474 1813 4224
9 0 5 106 793 3810
10 0 6 138 1273
: `-d-ciml` unlabeled[]{data-label="tabU.-d-ciml"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ----- ------- -------- --------- --------- --------- --------- ------- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 2 3 4 5 6 7 8 9 10
3 0 4 27 64 125 216 343 512 729 1000
4 0 5 69 358 1190 2946 6349 12356 22239 37630
5 0 6 123 1104 6275 23796 70315 177920 404109
6 0 7 193 2554 22585 130286 543837 1813568 5197044
7 0 8 285 5102 64340 538614 3165841
8 0 9 402 9363 158520 1829799
9 0 10 547 16176 354905
10 0 11 723 26626
: `-d-ciml` vertex-labeled[]{data-label="tabL.-d-ciml"}
Accumulated Marginal statistics {#sec.marg}
===============================
Adding the contents of one or more of the previous arrays defines the union of their graph sets, and regards some of the properties as irrelevant in these tables. If we look on the flags as defining a hypertable along five or six axes, these sums are the marginal sums; they create the Tables in Section \[sec.marg\].
The properties that are not taken into account while counting the graphs are either replaced by the filler `.*` or not tagged at all, using regular expressions of the usual programming languages as the tags.
`-d.*-m-l` flags graphs that are undirected, have any type of isolated vertices or connectivity, but have no multiedges or loops.
`-dc` flags graphs that are undirected and connected, but have any type of isolated vertices, multiedges or loops.
Undirected Graphs
-----------------
Tables \[tab.046742\]–\[tabL.-d-l\] summarize statistics of undirected graphs.
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- --- --- --- --- ---- ----- ----- ----- ----- ----- -- -- -- --
0 1
1 0 1
2 0 0 1
3 0 0 1 2
4 0 0 0 2 3
5 0 0 0 1 5 6
6 0 0 0 1 5 13 11
7 0 0 0 0 4 19 33 23
8 0 0 0 0 2 22 67 89 47
9 0 0 0 0 1 20 107 236 240 106
10 0 0 0 0 1 14 132 486 797 657 235
: `( -dc.*-m-l)` unlabeled. The number of connected undirected graphs without multiedges or loops [@EIS A054923,A054924,A046742][@SteinbachVol4 Vol. 1, Sec. 7, Table 1]. With the exception of the value of 1 at $E=0$, $V=1$ this is the same as Table \[tabU.-dc-i-m-l\]. []{data-label="tab.046742"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------ ------- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 0 3 0 0 0 0 0 0 0
3 0 0 1 16 0 0 0 0 0 0
4 0 0 0 15 125 0 0 0 0 0
5 0 0 0 6 222 1296 0 0 0
6 0 0 0 1 205 3660 16807 0
7 0 0 0 0 120 5700
8 0 0 0 0 45
9 0 0 0 0
: `( -dc.*-m-l)` vertex-labeled [@EIS A062734].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ----- ------ ------ ------ ------ ----- ----- ---- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1 1
1 1 2 2 2 2 2 2 2 2 2
2 1 4 6 7 7 7 7 7 7 7
3 1 6 14 20 22 23 23 23 23 23
4 1 9 28 53 69 76 78 79 79 79
5 1 12 52 125 198 245 264 271 273
6 1 16 93 287 550 782 915 973 993
7 1 20 152 606 1441 2392 3111
8 1 25 242 1226 3611 7118
9 1 30 370 2358 8608
10 1 36 546 4356
: `( -d)` unlabeled. The number of undirected graphs allowing loops and multiedges [@EIS A290428]. Sum of Table \[tabU.-dc\] and Table \[tabU.-d-c\].[]{data-label="tab.290428"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- ---- ------ ------- -------- --------- --------- --------- ---------- -- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1
1 1 3 6 10 15 21 28 36 45
2 1 6 21 55 120 231 406 666 1035
3 1 10 56 220 680 1771 4060 8436 16215
4 1 15 126 715 3060 10626 31465 82251 194580
5 1 21 252 2002 11628 53130 201376 658008 1906884
6 1 28 462 5005 38760 230230 1107568 4496388 15890700
7 1 36 792 11440 116280 888030 5379616
8 1 45 1287 24310 319770 3108105
9 1 55 2002 48620 817190
10 1 66 3003 92378
: `( -d)` vertex-labeled [@EIS A098568]. Sum of Table \[tabL.-dc\] and Table \[tabL.-d-c\].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ----- ------ ------ ------ ------ ------ ----- -- -- -- -- -- -- --
0 1
1 0 1
2 0 1 1
3 0 1 2 2
4 0 1 3 5 3
5 0 1 4 11 11 6
6 0 1 6 22 34 29 11
7 0 1 7 37 85 110 70 23
8 0 1 9 61 193 348 339 185 47
9 0 1 11 95 396 969 1318 1067 479 106
10 0 1 13 141 771 2445 4457 4940 3294 1729 235
: `-dc.*-l` unlabeled. Undirected loopless connected multigraphs with $E$ edges and $V$ vertices [@EIS A191646]. With the exception of the 1 at $E=0$, $V=1$, the sum of Table \[tabU.-dc-i-m-l\] and Table \[tabU.-dc-im-l\].[]{data-label="tab.191646"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ------ ------- -------- -------- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 1 3 0 0 0 0 0 0 0
3 0 1 7 16 0 0 0 0 0 0
4 0 1 12 63 125 0 0 0 0 0
5 0 1 18 162 722 1296 0 0 0
6 0 1 25 341 2565 10140 16807 0 0
7 0 1 33 636 7180 47100 169137
8 0 1 42 1092 17335 168285
9 0 1 52 1764 37750
10 0 1 63 2718
: `( -dc.*-l)` vertex-labeled [@EIS A290776].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ----- ------ ------ ------ ----- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 0 0
2 1 2 1 0 0 0 0 0 0 0
3 1 4 4 2 0 0 0 0 0 0
4 1 6 11 9 3 0 0 0 0 0
5 1 9 25 34 20 6 0 0 0
6 1 12 52 104 99 49 11 0 0
7 1 16 94 274 387 298 118
8 1 20 162 645 1295 1428
9 1 25 263 1399 3809
10 1 30 407 2823
: `( -dc)` unlabeled. Row sums in [@EIS A007719] []{data-label="tabU.-dc"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ------ ------- -------- -------- -------- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 0 0
2 1 3 3 0 0 0 0 0 0 0
3 1 6 16 16 0 0 0 0 0 0
4 1 10 51 127 125 0 0 0 0 0
5 1 15 126 574 1347 1296 0 0 0
6 1 21 266 1939 8050 17916 16807 0 0
7 1 28 504 5440 35210 135156 286786
8 1 36 882 13387 125730 736401
9 1 45 1452 29854 388190
10 1 55 2277 61633
: `( -dc)` vertex-labeled.[]{data-label="tabL.-dc"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12
----------------- --- --- --- --- --- ---- ----- ----- ------ ------ ------ ------ -- -- --
0 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1 1 1 1
2 0 0 1 2 2 2 2 2 2 2 2
3 0 0 1 3 4 5 5 5 5 5 5
4 0 0 0 2 6 9 10 11 11 11 11
5 0 0 0 1 6 15 21 24 25 26 26
6 0 0 0 1 6 21 41 56 63 66 67
7 0 0 0 0 4 24 65 115 148 165 172
8 0 0 0 0 2 24 97 221 345 428 467
9 0 0 0 0 1 21 131 402 771 1103 1305 1405
10 0 0 0 0 1 15 148 663 1637 2769 3664 4191
: `( -d.*-m-l)` unlabeled. Simple graphs with $E$ edges and $V$ vertices [@EIS A008406][@SteinbachVol4 vol. 4, Tables 2.2–2.2g]. Sum of tables \[tab.046742\], \[tab.sk2\]–\[tab.sk5\] and contributions by more than 5 components. []{data-label="tab.008406"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ------ ------- -------- -------- ------- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1 1
1 0 1 3 6 10 15 21 28 36 45
2 0 0 3 15 45 105 210 378 630 990
3 0 0 1 20 120 455 1330 3276 7140 14190
4 0 0 0 15 210 1365 5985 20475 58905
5 0 0 0 6 252 3003 20349 98280 376992
6 0 0 0 1 210 5005 54264 376740
7 0 0 0 0 120 6435
8 0 0 0 0 45
9 0 0 0
: `( -d.*-m-l)` vertex-labeled [@EIS A084546].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ----- ------ ------ ------ ------ ----- ----- ---- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1 1
1 0 1 2 2 2 2 2 2 2 2
2 0 2 5 7 7 7 7 7 7 7
3 0 2 10 18 22 23 23 23 23 23
4 0 3 17 44 66 76 78 79 79 79
5 0 3 27 91 178 239 264 271 273
6 0 4 41 183 451 733 904 973 993
7 0 4 58 332 1054 2094 2993
8 0 5 80 581 2316 5690
9 0 5 107 959 4799
10 0 6 139 1533
: `( -d-c)` unlabeled. See [@EIS A007717] for the limit $V\to\infty$. []{data-label="tabU.-d-c"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9
----------------- --- ---- ----- ------- -------- --------- --------- --------- ---------- -- -- -- -- -- -- -- -- --
0 0 1 1 1 1 1 1 1 1
1 0 2 6 10 15 21 28 36 45
2 0 3 18 55 120 231 406 666 1035
3 0 4 40 204 680 1771 4060 8436 16215
4 0 5 75 588 2935 10626 31465 82251 194580
5 0 6 126 1428 10281 51834 201376 658008 1906884
6 0 7 196 3066 30710 212314 1090761 4496388 15890700
7 0 8 288 6000 81070 752874 5092830
8 0 9 405 10923 194040 2371704
9 0 10 550 18766 429000
10 0 11 726 30745
: `( -d-c)` vertex-labeled.[]{data-label="tabL.-d-c"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- ----- ----- ----- ----- ----- ---- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1 1 1
2 0 1 2 3 3 3 3 3 3 3
3 0 1 3 6 7 8 8 8 8 8
4 0 1 4 11 17 21 22 23 23 23
5 0 1 5 18 35 52 60 64 65
6 0 1 7 32 76 132 173 197 206
7 0 1 8 48 149 313 471
8 0 1 10 75 291 741
9 0 1 12 111 539
10 0 1 14 160
: `( -d.*-l)` unlabeled. Undirected loopless multigraphs with $E$ edges and $V$ vertices [@EIS A192517]. Sum of tables \[tab.191646\], \[tab.mk2\]–\[tab.mk5\] and so forth. []{data-label="tab.192517"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ------ ------- -------- -------- --------- --------- -------- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1 1
1 0 1 3 6 10 15 21 28 36 45
2 0 1 6 21 55 120 231 406 666 1035
3 0 1 10 56 220 680 1771 4060 8436 16215
4 0 1 15 126 715 3060 10626 31465 82251 194580
5 0 1 21 252 2002 11628 53130 201376 658008
6 0 1 28 462 5005 38760 230230 1107568 4496388
7 0 1 36 792 11440 116280 888030
8 0 1 45 1287 24310 319770
9 0 1 55 2002 48620
10 0 1 66 3003
: `( -d.*-l)` vertex-labeled. []{data-label="tabL.-d-l"}
Directed Graphs
---------------
Tables \[tab.139621\]–\[tab.fin\] summarize statistics of oriented/directed graphs.
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------ -------- -------- ------- ------ --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 4 3 0 0 0 0 0 0 0
3 0 8 15 8 0 0 0 0 0 0
4 0 16 57 66 27 0 0 0 0 0
5 0 25 163 353 295 91 0 0 0
6 0 40 419 1504 2203 1407 350 0 0
7 0 56 932 5302 12382 13372 6790
8 0 80 1940 16549 58237 96456
9 0 105 3743 46566 237904
10 0 140 6867 121111
: `( d.*Cc-i)` unlabeled. The number of connected directed multigraphs with loops and no isolated vertex, with $E$ arcs and $V$ vertices [@EIS A139621]. []{data-label="tab.139621"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------- --------- ---------- ---------- ---------- --- --- ---- -- -- -- -- -- -- -- --
0 0 0 0 0 0 0 0 0 0 0
1 0 2 0 0 0 0 0 0 0 0
2 0 7 12 0 0 0 0 0 0 0
3 0 16 80 128 0 0 0 0 0 0
4 0 30 315 1328 2000 0 0 0 0 0
5 0 50 951 7808 29104 41472 0 0 0
6 0 77 2429 34136 234920 794112 1075648 0 0
7 0 112 5517 123272 1386880 8328192 25952128
8 0 156 11475 388223 6674205 63248832
9 0 210 22275 1101408 27706645
10 0 275 40887 2875224
: `( d.*Cc-i)` vertex-labeled
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ---- ------ ------- ------ ----- --- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0
2 1 1 0 0 0 0 0 0 0 0
3 1 2 1 0 0 0 0 0 0 0
4 1 6 4 1 0 0 0 0 0 0
5 1 10 19 6 1 0 0 0 0
6 1 19 73 59 9 1 0 0 0
7 1 28 208 350 138 12 1
8 1 44 534 1670 1361 301
9 1 60 1215 6476 9724
10 1 85 2542 21898
: `( dC)` unlabeled. The number of strongly connected directed multigraphs with loops and no vertex of degree zero, with n arcs and k vertices [@EIS A139622]. []{data-label="tab.139622"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------- -------- --------- -------- ----- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0
2 1 1 0 0 0 0 0 0 0 0
3 1 4 2 0 0 0 0 0 0 0
4 1 10 21 6 0 0 0 0 0 0
5 1 20 111 132 24 0 0 0 0
6 1 35 413 1288 960 120 0 0 0
7 1 56 1233 8152 15680 7920 720
8 1 84 3159 39049 156955 201450
9 1 120 7227 153540 1140055
10 1 165 15147 520404
: `( dC)` vertex-labeled.
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ------ ------ ------ --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0
2 0 1 3 0 0 0 0 0 0 0
3 0 0 4 8 0 0 0 0 0 0
4 0 0 4 22 27 0 0 0 0 0
5 0 0 1 37 108 91 0 0 0
6 0 0 1 47 326 582 350 0 0
7 0 0 0 38 667 2432 3024
8 0 0 0 27 1127 7694
9 0 0 0 13 1477
10 0 0 0 5
: `( d.*Cc.*-m-l)` unlabeled. The number of weakly connected directed graphs without multiedges or loops [@EIS A054733,A283753]. The undirected variants are in Table \[tab.046742\]. []{data-label="tab.054733"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- ---- ----- -------- --------- ---------- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 2 0 0 0 0 0 0 0 0
2 0 1 12 0 0 0 0 0 0 0
3 0 0 20 128 0 0 0 0 0 0
4 0 0 15 432 2000 0 0 0 0 0
5 0 0 6 768 11104 41472 0 0 0
6 0 0 1 920 33880 337920 1075648 0 0
7 0 0 0 792 73480 1536000 11968704
8 0 0 0 495 123485 5062080
9 0 0 0 220 166860
10 0 0 0 66
: `( d.*Cc.*-m-l)` vertex-labeled. [@EIS A062735]
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ---- ----- ----- --- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 0 1 0 0 0 0 0 0 0
4 0 0 2 1 0 0 0 0 0 0
5 0 0 1 4 1 0 0 0 0
6 0 0 1 16 7 1 0 0 0
7 0 0 0 22 58 10 1
8 0 0 0 22 240 165
9 0 0 0 11 565
10 0 0 0 5
: `( dCc.*-m-l)` unlabeled. The number of strongly connected directed graphs without loops or multiedges. Strongly connected variant of Table \[tab.054733\]. With the exception of the 1 at $E=0$, $V=1$ this is the same as Table \[tabU.dCc-i-m-l\].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- --- --- ----- ------- -------- ----- --- --- ---- -- -- -- -- -- -- -- --
0 1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
2 0 1 0 0 0 0 0 0 0 0
3 0 0 2 0 0 0 0 0 0 0
4 0 0 9 6 0 0 0 0 0 0
5 0 0 6 84 24 0 0 0 0
6 0 0 1 316 720 120 0 0 0
7 0 0 0 492 6440 6480 720
8 0 0 0 417 26875 107850
9 0 0 0 212 65280
10 0 0 0 66
: `( dCc.*-m-l)` vertex-labeled. With the exception of the 1 at $E=0$, $V=1$ this is the same as Table \[tabL.dCc-i-m-l\].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10
----------------- --- ----- ------ -------- -------- -------- ------- ------- ------- ----- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1 1 1
1 1 2 2 2 2 2 2 2 2 2
2 1 6 10 11 11 11 11 11 11 11
3 1 10 31 47 51 52 52 52 52 52
4 1 19 90 198 269 291 295 296 296 296
5 1 28 222 713 1270 1596 1697 1719 1723
6 1 44 520 2423 5776 8838 10425 10922 11033
7 1 60 1090 7388 24032 46384 63419
8 1 85 2180 21003 93067 230848
9 1 110 4090 55433 333948
10 1 146 7356 137944
: `( d)` unlabeled. The number of directed graphs allowing loops and multiedges [@EIS A138107].
$E\backslash V$ 1 2 3 4 5 6 7 8
----------------- --- ----- ------- --------- ---------- ----------- ----------- ----------- -- -- -- -- -- -- -- -- -- --
0 1 1 1 1 1 1 1 1
1 1 4 9 16 25 36 49 64
2 1 10 45 136 325 666 1225 2080
3 1 20 165 816 2925 8436 20825 45760
4 1 35 495 3876 20475 82251 270725 766480
5 1 56 1287 15504 118755 658008 2869685 10424128
6 1 84 3003 54264 593775 4496388 25827165 119877472
7 1 120 6435 170544 2629575 26978328 202927725
8 1 165 12870 490314 10518300 145008513
9 1 220 24310 1307504 38567100
10 1 286 43758 3268760
: `( d)` vertex-labeled. The number of labeled directed graphs allowing loops and multiedges [@EIS A214398].[]{data-label="tab.fin"}
Connected Multigraphs up to 7 vertices
======================================
Algorithm
---------
The columns of the undirected connected multigraphs in Table \[tab.191646\] have rational ordinary generating functions. To compute them, we first classify each multigraph by the number of edges and vertices of the underlying simple graph—in as many ways as counted in Table \[tab.046742\]— and then distribute the edges of the multigraph over the edges of the underlying graph using Pólya’s counting method to deal with the symmetry of the simple graphs.
The process is illustrated in Section \[sec.4conm\] for $V=4$ vertices. Explicit intermediate results are tracked in the files `G.`$V$`.`$E$`.txt` in the ancillary directory for $V=2$–$7$. Each of these files contains the contributing underlying simple graphs with $V$ vertices and $E$ edges. The file starts with $V$ and $E$ printed in the first line. Then each graph is represented by
1. a canonical adjacency matrix (binary, symmetric and traceless),
2. the label as in Section \[sec.tag\] followed by the multiplicity of the graph as if one would create all vertex-labeled graphs by permuting rows and columns (i.e. $V!$ divided by the order of the automorphism group),
3. the cycle index multinomial. This could also be derived from the table of symmetry groups in [@SteinbachVol4 Vol. 1, Sec. 7, Table 8].
The minimum number of edges for connected simple graphs is $E\ge V-1$ (sparsest, trees on $V$ vertices), and the maximum number is $E\le \binom{V}{2}$ (complete graph on $V$ vertices). Summing over all multinomials over the underlying graphs constitutes the generating function by a finite sum of rational polynomials ([@EIS A001349] terms).
4 vertices {#sec.4conm}
----------
The ordinary generating function for the number of connected multigraphs on 4 vertices is derived by adding the contributions of the 6 distinct geometries of the underlying connected simple graph.
We consider connected multigraphs with 4 vertices and $E$ edges. The multigraph thus has at least one (unoriented) edge attached to edge vertex, so all degrees are $\ge 1$. Loops are not allowed; the vertices are not labeled.
If all multiedges are replaced by a single edge, the underlying simple graph has one of 6 shapes [@GilbertCDM8]:
1. The linear chain with 3 edges.
2. A triangle with an edge to a lone vertex of degree 1 (4 edges).
3. The star graph with 3 edges.
4. The quadrangle (cycle of 4 edges).
5. A quadrangle with a single diagonal, total of 5 edges.
6. The complete graph on 4 vertices with 6 edges.
#### Linear Chain.
We wish to distribute $E\ge 3$ edges over the 3 edges of the simple graph of the linear chain. This can be done by putting any number of $k\ge 1$ edges in the middle, and distributing the remaining $n-k$ edges over the two edges connected to two endpoints.
Due to the left-right symmetry of the graph, the distribution of the $n-k$ edges can only be done in $\lfloor (n-k)/2\rfloor$ ways. The total number of graphs of this kind with multiedges is $$\sum_{k=1}^{E-2} \lfloor \frac{E-k}{2}\rfloor
=0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25,\ldots (E\ge 0)$$ with generating function [@EIS A002620] $$g_1(x) = \sum_{E\ge 0} a_1(E) x^E = \frac{x^3}{(1+x)(1-x)^3}.$$
The generating function is the product of $t_1(x)$ representing the number of ways of placing $n$ vertices at the middle edge, by the factor $x^2/[(1+x)(1-x)^2]$. The latter factor is obtained by considering the symmetry of the cyclic group $C_2$ that swaps the edges that inhabit the first and last edges of the underlying simple graph without generating a new graph. The cycle index of the group is [@FreudensteinJM3] $$Z(C_2) = (t_1^2+t_2^1)/2 ,$$ where the associated generating functions are the number of ways of placing $n$ edges without imposing symmetry on any of them: $$t_i(x) = \frac{x}{1-x} \mapsto 0,1,1,1,1,1\ldots, \quad i\ge 1.
\label{eq.tix}$$ So the latter factor can be written as [@EIS A004526] $$\frac{t_1(x)^2+t_2(x^2)}{2} = \frac{x^2}{(1+x)(1-x)^2}.
\label{eq.c2x}$$
#### Triangle.
The contribution from the triangular graph is the number of ways of placing $2\le k\le n-2$ edges on the edge to the lone vertex and the triangle edge opposite to it, and then distributing the residual $n-k$ edges to the remaining two edges under the symmetry constraint of the group $C_2$ that swaps the other two edges: $$\sum_{k=2}^{E-2} (k-1)\lfloor \frac{E-k}{2} \rfloor
= 0, 0, 0, 0, 1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161 (E\ge 0)$$ See [@EIS A002623] $$g_2(x) = \frac{x^4}{(1+x)(1-x)^4}.
\label{eq.g2x}$$ This generating function is the product of $x^2/(1-x)^2$—contribution of two cycles of length 1, fixed points under the symmetry—by $x^2/[(1+x)(1-x)^2]$, where again the latter is , the number of ways of distributing $E$ edges symmetrically over two edges of the simple graph.
#### Star Graph.
The contribution from the star graph is the number of ways of partitioning $E$ into 3 positive integers [@EIS A069905], $$\mapsto 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16 (E\ge 0),$$ $$g_3(x) = \frac{x^3}{(1+x)(1-x)^2(1-x^3)}.
\label{eq.g3x}$$ Alternatively this expression is obtained if we consider the symmetry group of order 6 of the underlying simple graph, which can be generated by (i) the group $C_3$ of the triangle combined with (ii) the mirror symmetry along a diagonal.
with(group):
g := permgroup(3, {[[1, 2, 3]], [[2, 3]]}) ;
for i in elements(g) do
print(i) ;
end do;
The cycle index obtained with this Maple code is [@Polya1983 p 57] $$Z(S_3) = \frac{t_1^3 +3t_1t_2+2t_3^1}{6}.$$ Insertion of gives .
#### Square.
The symmetry group of the square is the Dihedral Group of order 8 which essentially is generated by rotation by 90 degrees or flips along the horizontal or vertical axes or diagonals.
with(group):
g := permgroup(4, {[[1, 2, 3, 4]], [[1, 3]]}) ;
for i in elements(g) do
print(i) ;
end do;
The cycle index is [@TuckerMM47 Fig 3][@FreudensteinJM3] $$Z(D_8) = \frac{t_1^4+2t_4^1+2t_1^2t_2^1+3t_2^2}{8}$$ The enumeration theorem turns this into the generating function [@EIS A005232] $$g_4(x) = \frac{x^4(1-x+x^2)}{(1+x^2)(1+x)^2(1-x)^4}.$$ $$\mapsto 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 16, 20, 29, 35, 47 (E\ge 0).$$
#### Square with Diagonal.
In the square with a diagonal edge, the diagonal stays inert under the symmetry operations, and contributes a factor $t_1(x)$ to the generating function. The symmetry group of the four other edges allows a flip along any of the two diagonals and generates a symmetry group of order 4:
with(group):
g := permgroup(4,{[[2, 4], [1, 3]], [[1, 4], [2, 3]]}) ;
for i in elements(g) do
print(i) ;
end do;
The cycle index is $$Z(C_2\times C_2) = \frac{t_1^4+3t_2^2}{4}.$$ Insertion of into the enumeration theorem yields $x^4(1-x+x^2)/[(1+x)^2(1-x)^4]$ [@EIS A053307], and convolved with the inert factor $$g_5(x) = \frac{x^5(1-x+x^2)}{(1+x)^2(1-x)^5}.$$ This expands to $$\mapsto 0, 0, 0, 0, 0, 1, 2, 6, 11, 22, 36, 60, 90, 135, 190, 266, 357, 476 (E\ge 0)$$
#### Complete Graph.
The cycle index of the complete graph $K_4$ is [@FreudensteinJM3], $$Z(S_4)=\frac{t_1^6+9t_1^2t_2^2+8t_3^2+6t_2t_4}{24}.$$ Insertion of into the enumeration theorem yields $$g_6(x) = \frac{x^6(1-x+x^2+x^4+x^6-x^7+x^8)}{(1-x)^6(1+x)^2(1+x^2)(1+x+x^2)^2},$$ with [@EIS A003082] $$\mapsto 0, 0, 0, 0, 0, 0, 1, 1, 3, 6, 11, 18, 32, 48, 75, 111, 160, 224, 313 (E\ge 0)$$
#### Sum.
The generating function contributed by the 6 underlying simple graphs is $$\sum_{i=1}^6 g_i(x)
=
\frac{x^3(-x^{10}+x^9+2x^7-x^6+x^5-3x^4+x^2+x+2)}{(x-1)^6(1+x)^2(1+x^2)(1+x+x^2)^2}
,$$ which expands to [@EIS A290778] $$\mapsto
0, 0, 0, 2, 5, 11, 22, 37, 61, 95, 141, 203, 288, 393, 531, 704, 918, 1180, 1504
(E\ge 0, V=4)$$
Up to 7 vertices
----------------
The generating function for the number of connected loopless multigraphs on 2 vertices is $$\frac{x}{1-x}
\mapsto 0,1,1,1,1,1, (E\ge 0, V=2)
.$$
The generating function for the number of connected loopless multigraphs on 3 vertices is [@EIS A253186] $$\begin{gathered}
{\frac { \left( {x}^{3}-x-1 \right) {x}^{2}}{ \left( -1+x \right) ^{3}
\left( x+1 \right) \left( {x}^{2}+x+1 \right) }}
\\
\mapsto
0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23 (E\ge 0, V=3)
.\end{gathered}$$
On 5 vertices $$\begin{gathered}
{\frac {{x}^{4}
p_5(x)
}{ \left( -1+x \right) ^{10} \left( {x}^{2}+x+1 \right) ^{3}
\left( x+1 \right) ^{4} \left( {x}^{2}-x+1 \right) \left( {x}^{4}+{x}
^{3}+{x}^{2}+x+1 \right) ^{2} \left( {x}^{2}+1 \right) ^{2}}}
\\
\mapsto
0, 0, 0, 0, 3, 11, 34, 85, 193, 396, 771, 1411, 2490, 4221 (E\ge 0, V=5),\end{gathered}$$ where $$\begin{gathered}
p_5 \equiv
3+5\,x+12\,{x}^{2}+17\,{x}^{3}+26\,{x}^{4}+27\,{
x}^{5}+35\,{x}^{6}+28\,{x}^{7}+38\,{x}^{8}+30\,{x}^{9}+39\,{x}^{10}\\
+37 \,{x}^{11}
+34\,{x}^{12}+24\,{x}^{13}+15\,{x}^{14}+3\,{x}^{15}
-7\,{x}^{
16}-9\,{x}^{17}+4\,{x}^{20}\\
+5\,{x}^{22}+3\,{x}^{23}-8\,{x}^{18}-{x}^{19
}+6\,{x}^{21}-2\,{x}^{24}-2\,{x}^{25}-2\,{x}^{26}-{x}^{27}+{x}^{29}
.\end{gathered}$$ On 6 vertices $$\begin{gathered}
-{\frac {{x}^{5} p_6
}{ \left(
-1+x \right) ^{15} \left( x+1 \right) ^{6} \left( {x}^{2}+1 \right) ^{3
} \left( {x}^{2}+x+1 \right) ^{5} \left( {x}^{2}-x+1 \right) ^{2}
\left( {x}^{4}+{x}^{3}+{x}^{2}+x+1 \right) ^{3}}}
\\
\mapsto 0, 0, 0, 0, 0, 6, 29, 110, 348, 969, 2445, 5746, 12736, 26843, 54256, 105669 (E\ge 0, V=6),\end{gathered}$$ where $$\begin{gathered}
p_6 \equiv
6
+11\,x
+35\,{x}^{2}
+70\,{x}^{3}
+134\,{x}^{4}
+ 217\,{x}^{5}
+348\,{x}^{6}
+533\,{x}^{7}
+726\,{x}^{8}
+1038\,{x}^{9}
+1290 \,{x}^{10}
\\
+1629\,{x}^{11}
+1810\,{x}^{12}
+2040\,{x}^{13}
+1976\,{x}^{14}
+ 1984\,{x}^{15}
+1696\,{x}^{16}
+1542\,{x}^{17}
+1206\,{x}^{18}
\\
+1050\,{x}^{19}
+787\,{x}^{20}
+636\,{x}^{21}
+474\,{x}^{22 }
+273\,{x}^{23}
+169\,{x}^{ 24}
-11\,{x}^{25}
-31\,{x}^{26}
-97\,{x}^{27}
-44\,{x}^{28}
\\
-8\,{x}^{29}
+33 \,{x}^{30}
+63\,{x}^{31}
+32\,{x}^{32}
+38\,{x}^{33}
-17\,{x}^{34}
-14\,{x}^ {35}
-31\,{x}^{36}
-8\,{x}^{37}
-5\,{x}^{38}
+8\,{x}^{39}
\\
+11\,{x}^{40}
+4\,{ x}^{41}
+3\,{x}^{42}-4\,{x}^{43}-3\,{x}^{45}
+{x}^{47}
.\end{gathered}$$ On 7 vertices $$\begin{gathered}
\frac {{x}^{6}p_7(x)
}{ \left( -1+x \right) ^{21} \left( {x
}^{4}+{x}^{3}+{x}^{2}+x+1 \right) ^{4} \left( {x}^{2}+x+1 \right) ^{7}
\left( x+1 \right) ^{9} \left( {x}^{2}+1 \right) ^{4} \left( {x}^{2}-x
+1 \right) ^{3}
}
\\
\times
\frac{1}{
\left( {x}^{4}-{x}^{2}+1 \right) \left( {x}^{4}-{x}^{3
}+{x}^{2}-x+1 \right) \left( {x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x
+1 \right) ^{3}}
\\
\mapsto
0, 0, 0, 0, 0, 0, 11, 70, 339, 1318, 4457, 13572, 38201, 100622, 251078,
\ldots (E\ge 0, V=7),\end{gathered}$$ where $$\begin{gathered}
p_7(x)\equiv
-11
-48\,x
-188\,{x}^{2}
-570\,{x}^{3}
-1526\,{x}^{4 }
-3675\,{x}^{5}
-8284\,{x}^{6}
-17431\,{x}^{7}
-35005\,{x}^{8}
\\
-66742\,{x}^ {9}
-121908\,{x}^{10}
-213342\,{x}^{11}
-359515\,{x}^{12}
-583522\,{x}^{13}
-916091\,{x}^{14}
-1391716\,{x}^{15}
\\
-2051981\,{x}^{16}
-2938963\,{x}^{17}
-4097420\,{x}^{18}
-5564508\,{x}^{19}
-7373793\,{x}^{20}
-9539279\,{x}^{21 }
\\
-12063528\,{x}^{22}
- 14919997\,{x}^{23}
-18064473\,{x}^{24}
-21418776\,{x}^{25}
-24890827\,{x}^{26}
-28355984\,{x }^{27}
\\
-31688266\,{x}^{28}
-34742272\,{x}^{29}
-37387611\,{x}^{30}
- 39493274\,{x}^{31}
-40963946\,{x}^{32}
-41717383\,{x}^{33}
\\
-41723196\,{x}^ {34}
-40973187\,{x}^{35}
-39511812\,{x}^{36}
-37405689\,{x}^{37}
-34764514 \,{x}^{38}
-31705308\,{x}^{39}
\\
-28372262\,{x}^{40}
-24898844\,{x}^{41}
- 21423490\,{x}^{42}
-18060699\,{x}^{43}
-14913079\,{ x}^{44}
-12050303\,{x}^{45}
\\
-9525196\,{x}^{46}
-7357519\,{x}^{ 47}
-5550815\,{x}^{48}
-4085547\,{x}^{49}
-2932089\,{x}^{50}
-2048825\,{x}^{51}
\\
-1393454\,{x} ^{52}
-920594\,{x}^{53}
-590477\,{x}^{54}
-366935\,{x }^{55}
-220705\,{x}^{56}
-128024\,{x}^{57}
-71511\,{x}^{58 }
\\
-37993\,{x}^{59}
-18932\,{x}^{60}
-8318\,{x}^{61}
-2668\,{x}^{62}
+247\,{x}^{63}
+1501\,{x}^{64}
+1827\,{x}^{65}
+1523\,{x}^ {66}
\\
+980\,{x}^{67}
+357\,{x}^{68}
-99\,{x}^{69}
-369\,{x}^{70}
-387\,{x}^{71}
-247\,{x}^{ 72}
-23\,{x}^{73}
+152\,{x}^{74}
\\
+230\,{x}^{75}
+205\,{x}^{76}
+118\,{x}^{77 }
+15\,{x}^{78}
-61\,{x}^{79}
-88\,{x}^{80}
-74\,{x}^{81}
-33\,{x}^{82}
+3\,{x}^{83}
+26\,{x}^{84}
\\
+28\,{ x}^{85}
+19\,{x}^{86}
+5\,{x}^{87}
-4\,{x}^{88}
-7\,{x}^{89}
-5\,{x}^{90}
-{x}^{91}
+{x}^ {92}
+{x}^{93}
.\end{gathered}$$
As a cross-check on these numbers we note that using a weight $t_i(x)=x$ instead of just counts the underlying simple graphs; it computes the generating functions down columns of Table \[tab.046742\].
Multisets of Connected Graphs
=============================
If a table of connected graphs as a function of vertex count and edge count is known, the Multiset Transformation generates tables of disconnected graphs with fixed number of components. The calculation involves creating an intermediate multiset of edge-vertex pairs of the components, and looking up a product of multiset coefficients as a function of the number of connected graphs that support the pairs. The technique is demonstrated for simple (undirected, unlabeled, loopless) graphs and for undirected, unlabeled loopless graphs allowing multiedges.
The Multiset Coefficient
------------------------
The concept of the multiset is based on the concept of the set (a collection of objectes, only one object of a given type), but allows to put more than one object of a type into the collection [@SinghNSJM37].
A Multiset is a collection of objects with some individual count (object of type $i$ appearing $f_i$ times in the collection). The objects have no order in the collection.
The number of ways of assembling a multiset with $m$ objects plugged from a set of $n$ different objects is a variant of Pascal’s triangle of binomial coefficients, $$P(n,m)=\binom{n+m-1}{m}
.
\label{eq.P}$$ The equation may be illustrated for small orders $m$:
- If there is only $n=1$ type of objects, the multiset has only one choice: it contains $m$ replicates of the unique object. $P(1,m)=1$.
- If the multiset contains $m=1$ object, it contains one object of $n$ candidates. $P(n,m)=n$.
- If the multiset contains $m=2$ objects, it contains either the same type of object twice ($n$ choices), or two different objects ($\binom{n}{2}$ choices), so $P(n,2)=n+\binom{n}{2} = \binom{n+1}{2}$.
- If the multiset contains $m=3$ objects, it either contains the same type of object thrice ($n$ choices), or one type of object once and another type of object twice ($n(n-1)$ choices), or three different types of objects ($\binom{n}{3}$ choices); so $P(n,3)=n
+n(n-1)
+\binom{n}{3}
=\binom{n+2}{3}$.
- If the multiset contains $m=4$ objects, we consider all five partitions of $m$, namely ${4^1},{1^13^1},{2^2},{1^22^1},{1^4}$: it either contains the same type of object 4 times ($n$ choices), or one type of object once and another type of object thrice ($n(n-1)$ choices), or two pairs of objects ( $\binom{n}{2}$ choices), or two different objects and one pair of objects ( $\binom{n}{2}(n-2)$ choices), or four different types of objects ($\binom{n}{4}$ choices); so $P(n,4)=n +n(n-1) +\binom{n}{2}+\binom{n}{2}(n-2)+\binom{n}{4} =\binom{n+3}{4}$.
Formula can be rephrased with [@RoyAMM94 (3.8)] (setting $r=m-1$, $k=j$, $n=1$ there) or with [@Sane (1.11)]: $$\binom{n+m-1}{m}
=
\sum_{j=1}^m \binom{n}{j} \binom{m-1}{j-1}
.$$ The first factor on the right hand side indicates that in a first step one can create a set of $j$ distinct objects out of $n$ in $\binom{n}{j}$ ways. Consider that set sorted by some lexicographic order. Then the factor $\binom{m-1}{j-1}$ counts in how many ways one can insert separators in the multiset of the same lexicographic ordering to select switch-over from one type of object to the next one.
To create multisets of $m=4$ objects given $n$ distinct objects, selecting $j=1$ type of object gives the multiset $o_1o_1o_1o_1$ (no separator), $\binom{3}{0}=1$ choices; selecting $j=2$ types of objects gives $o_1o_1o_1|o_2$ or $o_1o_1|o_2o_2$ or $o_1|o_2o_2o_2$ with $\binom{3}{1}=3$ positions of the separator; selecting $j=3$ types of objects gives $o_1o_1|o_2|o_3$ or $o_1|o_2o_2|o_3$ or $o_1|o_2|o_3o_3$ with $\binom{3}{2}=3$ positions of the $2$ separators; or selecting $j=4$ types of objects gives $o_1|o_2|o_3|o_4$ with $\binom{3}{3}=1$ positions of the $3$ separators.
Multiset Transform of An Integer Sequence
-----------------------------------------
The Multiset Transform deals with the question: if the objects of type $i$ have some weight (expressed as a positive integer), in how many ways can we assemble a multiset of the objects with some prescribed total weight? The total weight is the usual arithmetic sum of the weights of the objects.
(Money Exchange Problem) In how different ways can you combine coins (weight 5 for type nickels, weight 10 for type dimes and weight 25 for type quarter…) for a wallet worth 200 (2 dollars)?
In how different ways can you fill a bag of 10 kg (weight 100) with apples of 100 g (weight 1) and oranges of 200 g (weight 2)?
The Multiset Transform computes the number $T_{n,k}$ of multisets containing $k$ objects, drawn from a set of objects of which there are $T_{n,1}$ of some additive weight $n$ [@Flajolet]. Given $T_{0,1}=1$ and an integer sequence $T_{n,1}$ for the number of objects in the weight class $n$, the $T$ of total weight $n$ are calculated recursively by $$T_{n,k}=\sum_{f_1n_1+f_2n_2+\cdots +f_kn_k=n} \prod_{i=1}^k P(T_{n_i,1},f_i),
\label{eq.mrec}$$ where the sum is over the partitions of $n$ into parts $n_i$ which occur with frequencies $f_i$.
The row sums $\sum_{k=1}^n T_{n,k}$ are obtained by the Euler Transform of the sequence $T_{n,1}$ [@Flajolet (25)].
Given two types of nickels (2 types of weight 5, tin and copper), one type of dime (weight 10), and one type of quarter (weight 25), the integer sequence $T_{n,1}$ is $(1),0,0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25$.
Given three types of apples (brown, yellow and red, each of weight 1), one type of banana (weight 2), and one type melon (weight 4), the integer sequence is $(1),3,1,0,1,0,0,0\ldots$. The Multiset Transform generates the triangular table
$n\backslash k$ 1 2 3 4 5 6
----------------- --- --- ---- ---- ---- ---- ---- ---- --
1 3
2 1 6
3 0 3 10
4 1 1 6 15
5 0 3 3 10 21
6 0 1 7 6 15 28
7 0 0 3 13 10 21 36
8 0 1 1 7 21 15 28 45
The row sums in the table are 3, 7, 13, 23, 37, 57, 83, 118…There are $T_{3,2}=3$ ways of generating a weight of 3 with two objects (a banana and any of the three types of apples). There are $T_{3,3}=10$ ways of generating a weight of 3 with three objects \[three apples (bbb), (yyy), (rrr), (byy), (brr), (bby), (yrr), (bbr), (yyr), (bry)\]. There is $T_{6,2}=1$ way to generate a weight of 6 with two objects (a banana and a melon).
If there is one type of object of each weight, $T_{n,1}=1$, the Multiset Transform generates the partition numbers [@EIS A008284], and the row sums are the partition numbers [@EIS A000041].
Graphs specified by number of components {#sec.grexa}
----------------------------------------
If the sequence $T_{n,1}$ enumerates connected graphs of type $n$ (where the weight $n$ is either the vertex count or the edge count), one fundamental way of generating a multiset is putting $k$ of them side by side and considering them a graph with $k$ components. Vertex or edge number are additive, as required.
If $T_{n,1}$ denotes connected graphs with $n$ nodes [@EIS A001349], the Multiset Transform counts graphs with $k$ components [@EIS A201922].
If $T_{n,1}$ denotes connected graphs with $n$ edges [@EIS A002905], the Multiset Transform counts graphs with $n$ edges and $k$ components [@EIS A275421].
If $T_{n,1}$ denotes trees with $n$ nodes [@EIS A000055], the Multiset Transform counts forests with $k$ trees [@EIS A095133].
If $T_{n,1}$ denotes rooted trees with $n$ nodes [@EIS A000081], the Multiset Transform counts rooted forests with $k$ trees [@EIS A033185].
If $T_{n,1}$ denotes connected regular graphs with $n$ nodes [@EIS A005177], the Multiset Transform counts regular graphs with $k$ components [@EIS A275420]. In the case of cubic graphs the transform pair is [@EIS A002851] and [@EIS A275744].
Graphs specified by number of edges, vertices and components
============================================================
Union of connected graphs
-------------------------
Let $G(E,V,k)$ be the number of graphs with $E$ edges, $V$ vertices and $k$ components. $G(E,V,1)$ is the number of connected graphs with $E$ edges and $V$ vertices. The other properties like whether the graphs are labeled, may contain loops or multiedges, are not classified here, but assumed to be fixed while composing graphs with $k$ components from connected graphs. (A multiset of labeled connected graphs is a disconnected labeled graph; a multiset of connected oriented graphs is a disconnected oriented graph; and so on.) The unified graph is the multiset union of connected graphs $\mathcal G_1$ which individually have $e_i$ edges and $v_i$ vertices: $$\mathcal G_k(E,V) = \bigcup_{i=1}^k \mathcal G_1(e_i,v_i),$$ where both the number of edges and the number of vertices are additive: $$E=\sum_i^k e_i;\quad V=\sum_i^k v_i.
\label{eq.add}$$ Summation over a set of the variables creates marginal sums: $G(.,V,k)=\sum_{E\ge 0}G(E,V,k)$ are the graphs with $V$ vertices and $k$ components. $G(E,,k)=\sum_{V\ge 1} G(E,V,k)$ is the number of graphs with $E$ edges and $k$ components. $G(E,V,.)=\sum_{k\ge 1} G(E,V,k)$ is the number of graphs with $E$ edges and $V$ vertices.
Correlated Multiset Transforms
------------------------------
The particular case we explore here is that constructing a disconnected graph from connected components means building a multiset of connected graphs, where the number of edges and *also* the number of vertices are such an additive weight.
The examples of Section \[sec.grexa\] illustrated how $G(E,,k)$ is the Multiset Transform of $G(E,,1)$ [@EIS A076864,A275421,A191970] and $G(,V,k)$ is the Multiset Transform of $G(,V,1)$ [@EIS A054924,A275420,A281446]. The aim of this paper is to demonstrate a similar technique for $G(E,V,k)$ assuming $G(E,V,1)$ is known.
Each graph which is a component contributing to $G(E,V,k)$ has a specific pair $(e_i,v_i)$ of edge and vertex count; the union of these graphs is a multiset of such pairs—which means in the multiset of graphs contributing to $G(E,V,k)$, each pair may occur more than once, and each pair may represent (in the sense of the weights above) more than one graph because there may be more than one distinct connected graph for one pair of $E$ and $V$.
The calculation starts by constructing all weak compositions of $E$ into $k$ parts $e_i$, and all weak compositions of $V$ into $k$ parts $v_i$ of pairs $(e_i,v_i)$ compatible with the requirement . This defines a two-dimensional $\binom{E+k-1}{k-1}\times \binom{V+k-1}{k}$ outer product matrix with multisets [@WiederPAM2].
\[exa.EV\] If $E=2$ and $V=3$ and $k=3$, the compositions are $2=$$2+0+0=$$0+2+0=$$0+0+2=$$1+1+0=$$1+0+1=$$0+1+1$, $3=$$3+0+0=$$0+3+0=$$0+0+3=$$2+1+0=$$1+2+0=$$2+0+1=$$2+1+0=\ldots$, and the matrix contains multisets with $k$ pairs:
$\sum e_i\backslash \sum v_i$ 3+0+0 0+3+0 2+1+0 $\ldots$ 1+1+1
------------------------------- ----------------- ----------------- --- ------- ---------- ----------------- -- -- -- -- -- -- -- -- -- -- --
2+0+0 (2,3)(0,0)(0,0) (2,0)(0,3)(0,0) … (2,1)(0,1)(0,1)
0+2+0 (0,3)(2,0)(0,0) (0,0)(2,3)(0,0) … (0,1)(2,1)(0,1)
0+0+2 …
1+1+0 …
0+1+1 (0,3)(1,0)(1,0) (0,0)(1,3)(1,0) … (0,1)(1,1)(1,1)
$\ldots$
Each element of the matrix is a multiset $(e_1,v_1)(e_2,v_2)\cdots (e_k,v_k)$ of pairs obtained by interleaving the $e_i$ and $v_i$ components of the compositions. At that point we realize that
1. if any of the $v_i$ is zero, the method of selecting such a null-graph into the disconnected graph would not fulfill the requirement of being $k$-connected. So actually only the compositions (not the weak compositions) of $V$ need to be considered as table columns.
2. because the decompositions $(e_1,v_1),(e_2,v_2)\cdots$ are candidates for multiset compositions, the order of the pairs does not matter. In the example, $(2,1)(0,1)(0,1)$ and $(0,1),(2,1)(0,1)$ are the same scheme of selecting connected graphs. So we may sort for example each $k$-set by the $e_i$ member of the pair without loss of samples, which means, we may build the table by just considering weak partitions (not all weak compositions) of $E$ into parts $e_i$ as table rows.
Continuing Example \[exa.EV\] above, the table reduces to
$\sum e_i\backslash \sum v_i$ 1+1+1
------------------------------- -----------------
2+0+0 (2,1)(0,1)(0,1)
1+1+0 (1,1)(1,1)(0,1)
The $(2,1)(0,1)(0,1)$ entry indicates to take one connected graph with 2 edges and one vertex (obviously a double-loop) and two graphs without edges and one vertex (single points). The $(1,1)(1,1)(0,1)$ entry indicates to take 2 graphs with an edge and a vertex (2 points, each with a loop) and a graph without edges and a single vertex (a single point).
The number of $k$-compositions of graphs then is the sum over all unique remaining multisets in the table, akin to Equation : $$G(E,V,k) = \sum_{(e_i,v_i)^{f_i}}
\prod_i P(G(e_i,v_i,1),f_i)$$ where $f_i$ is the frequency (number of occurrences) of the pair $(e_i,v_i)$ in the multiset.
Simple Graphs
-------------
The most important example considers undirected, unlabeled graphs without multiedges or loops. Table \[tab.046742\] shows the base information $G(E,V,1)$ which is fed into the formula to compute the tables $G(E,V,k)$ \[tab.sk2\]–\[tab.sk5\] for $k\ge 1$. (These tables differ from Steinbach’s tables [@SteinbachVol4 Vol. 4, Table 1.2a] because our components may be/have isolated vertices.)
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14
----------------- --- --- --- --- --- --- ---- ----- ------ ------ ------- ------- ------- ------- -- -- -- -- --
0 0 1
1 0 0 1
2 0 0 0 2
3 0 0 0 1 3
4 0 0 0 0 3 6
5 0 0 0 0 1 8 11
6 0 0 0 0 1 7 22 23
7 0 0 0 0 0 5 27 58 46
8 0 0 0 0 0 2 28 101 157 99
9 0 0 0 0 0 1 23 142 358 426 216
10 0 0 0 0 0 1 15 161 660 1233 1166 488
11 0 0 0 0 0 0 10 156 1010 2873 4163 3206 1121
12 0 0 0 0 0 0 5 138 1356 5705 11987 13847 8892 2644
13 0 0 0 0 0 0 2 101 1613 9985 29652 48071 45505 24743
: $G(E,V,2)$. Simple graphs with 2 components and a total of $E$ edges and $V$ vertices. See [@EIS A274934] for column sums, [@EIS A274937] for the diagonal. []{data-label="tab.sk2"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14
----------------- --- --- --- --- --- --- --- ---- ----- ------ ------- ------- ------- ------- -- -- -- -- --
0 0 0 1
1 0 0 0 1
2 0 0 0 0 2
3 0 0 0 0 1 4
4 0 0 0 0 0 3 7
5 0 0 0 0 0 1 9 14
6 0 0 0 0 0 1 7 25 29
7 0 0 0 0 0 0 5 29 68 60
8 0 0 0 0 0 0 2 29 110 186 128
9 0 0 0 0 0 0 1 23 149 397 509 284
10 0 0 0 0 0 0 1 15 164 699 1377 1399 636
11 0 0 0 0 0 0 0 10 157 1041 3070 4685 3857 1467
12 0 0 0 0 0 0 0 5 139 1375 5919 12899 15646 10706
13 0 0 0 0 0 0 0 2 101 1625 10183 30980 52024 51622
: $G(E,V,3)$. Simple graphs with 3 components and a total of $E$ edges and $V$ vertices. Column sums are in column 3 of [@EIS A201922] $\mapsto 1,1,3,9,32,154,1065,12513,276114,12021725\ldots$.
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----------------- --- --- --- --- --- --- --- --- ---- ----- ------ ------- ------- ------- ------- -- -- -- --
0 0 0 0 1
1 0 0 0 0 1
2 0 0 0 0 0 2
3 0 0 0 0 0 1 4
4 0 0 0 0 0 0 3 8
5 0 0 0 0 0 0 1 9 15
6 0 0 0 0 0 0 1 7 26 32
7 0 0 0 0 0 0 0 5 29 71 66
8 0 0 0 0 0 0 0 2 29 112 196 143
9 0 0 0 0 0 0 0 1 23 150 406 539 315
10 0 0 0 0 0 0 0 1 15 164 706 1417 1486 710
11 0 0 0 0 0 0 0 0 10 157 1044 3110 4834 4105 1631
12 0 0 0 0 0 0 0 0 5 139 1376 5951 13102 16193 11408
13 0 0 0 0 0 0 0 0 2 101 1626 10202 31198 52966 53519
: $G(E,V,4)$. Simple graphs with 4 components and a total of $E$ edges and $V$ vertices. []{data-label="tab.sk4"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----------------- --- --- --- --- --- --- --- --- --- ---- ----- ------ ------- ------- ------- -- -- -- --
0 0 0 0 0 1
1 0 0 0 0 0 1
2 0 0 0 0 0 0 2
3 0 0 0 0 0 0 1 4
4 0 0 0 0 0 0 0 3 8
5 0 0 0 0 0 0 0 1 9 16
6 0 0 0 0 0 0 0 1 7 26 33
7 0 0 0 0 0 0 0 0 5 29 72 69
8 0 0 0 0 0 0 0 0 2 29 112 199 149
9 0 0 0 0 0 0 0 0 1 23 150 408 549 330
10 0 0 0 0 0 0 0 0 1 15 164 707 1426 1516 742
11 0 0 0 0 0 0 0 0 0 10 157 1044 3117 4874 4193
12 0 0 0 0 0 0 0 0 0 5 139 1376 5954 13142 16343
13 0 0 0 0 0 0 0 0 0 2 101 1626 10203 31230 53170
: $G(E,V,5)$. Simple graphs with 5 components and a total of $E$ edges and $V$ vertices. []{data-label="tab.sk5"}
The arithmetic sum over these tables $k\ge 1$ yields Table \[tab.008406\].
Loopless connected Multigraphs
------------------------------
Another application of the algorithm is to construct [@EIS A192517] from [@EIS A191646]. The base information of $G(E,V,k=1)$ is this time in Table \[tab.191646\], and the Multiset Transformations creates Tables \[tab.mk2\]–\[tab.mk5\] and so on ($k\ge 2$). The sum over all $k\ge 1$ converges to Table \[tab.192517\].
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12
----------------- --- --- --- ---- ----- ------ ------- ------- -------- -------- -------- ------- -- -- -- -- -- -- -- --
0 0 1
1 0 0 1
2 0 0 1 2
3 0 0 1 3 3
4 0 0 1 5 8 6
5 0 0 1 6 17 20 11
6 0 0 1 9 32 58 52 23
7 0 0 1 10 53 135 185 132 46
8 0 0 1 13 84 290 548 586 344 99
9 0 0 1 15 127 565 1441 2108 1829 900 216
10 0 0 1 18 184 1055 3456 6696 7884 5680 2834 488
11 0 0 1 20 259 1859 7774 19288 29633 28718 17546 6811
12 0 0 1 24 359 3178 16578 51799 100810 126013 102743 54469
: Undirected loopless multigraphs with 2 components and a total of $E$ edges and $V$ vertices. []{data-label="tab.mk2"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12
----------------- --- --- --- --- ---- ----- ------ ------- ------- -------- -------- -------- -- -- -- -- -- -- -- --
0 0 0 1
1 0 0 0 1
2 0 0 0 1 2
3 0 0 0 1 3 4
4 0 0 0 1 5 9 7
5 0 0 0 1 6 19 23 14
6 0 0 0 1 9 35 65 62 29
7 0 0 0 1 10 57 148 214 159 60
8 0 0 0 1 13 89 313 614 681 421 128
9 0 0 0 1 15 134 601 1577 2374 2148 1104 284
10 0 0 0 1 18 192 1110 3711 7353 8938 6683 3389
11 0 0 0 1 20 269 1938 8225 20752 32692 32639 20712
12 0 0 0 1 24 371 3289 17332 54847 108802 139316 117082
: Undirected loopless multigraphs with 3 components and a total of $E$ edges and $V$ vertices.
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13
----------------- --- --- --- --- --- ---- ----- ------ ------- ------- -------- -------- -------- -- -- -- -- -- -- --
0 0 0 0 1
1 0 0 0 0 1
2 0 0 0 0 1 2
3 0 0 0 0 1 3 4
4 0 0 0 0 1 5 9 8
5 0 0 0 0 1 6 19 24 15
6 0 0 0 0 1 9 35 67 65 32
7 0 0 0 0 1 10 57 151 221 169 66
8 0 0 0 0 1 13 89 318 628 711 449 143
9 0 0 0 0 1 15 134 607 1603 2445 2248 1185 315
10 0 0 0 0 1 18 192 1119 3754 7506 9227 7025 3608
11 0 0 0 0 1 20 269 1949 8294 21049 33426 33790 21799
12 0 0 0 0 1 24 371 3304 17437 55396 110485 142723 121385
: Undirected loopless multigraphs with 4 components and a total of $E$ edges and $V$ vertices. []{data-label="tab.mk4"}
$E\backslash V$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14
----------------- --- --- --- --- --- --- ---- ----- ------ ------- ------- -------- -------- -------- -- -- -- -- -- --
0 0 0 0 0 1
1 0 0 0 0 0 1
2 0 0 0 0 0 1 2
3 0 0 0 0 0 1 3 4
4 0 0 0 0 0 1 5 9 8
5 0 0 0 0 0 1 6 19 24 16
6 0 0 0 0 0 1 9 35 67 66 33
7 0 0 0 0 0 1 10 57 151 223 172 69
8 0 0 0 0 0 1 13 89 318 631 718 459 149
9 0 0 0 0 0 1 15 134 607 1608 2459 2278 1213 330
10 0 0 0 0 0 1 18 192 1119 3761 7533 9299 7126 3690
11 0 0 0 0 0 1 20 269 1949 8304 21095 33584 34084 22146
12 0 0 0 0 0 1 24 371 3304 17450 55472 110799 143481 122562
: Undirected loopless multigraphs with 5 components and a total of $E$ edges and $V$ vertices. []{data-label="tab.mk5"}
|
---
abstract: 'The present paper is devoted to the study of space mappings, which are more general than quasiregular mappings. The questions of the behavior of differentiable mappings having the so–called $N,$ $N^{-1},$ $ACP$ and $ACP^{-1}$ – properties are studied in the work. Under some additional conditions, it is showed that the modulus of such mappings $f$ can be more than each degree of logarithmic function at every neighborhood of the isolated essential singularity of $f.$'
author:
- '[E. SEVOST’YANOV]{}\'
title: ' [**ON THE LOCAL BEHAVIOR OF THE MAPPINGS WITH NON–BOUNDED CHARACTERISTICS**]{}'
---
Introduction
============
Here are some definitions. Everywhere below, $D$ is a domain in ${\Bbb R}^n,$ $n\ge 2,$ $m$ be a measure of Lebesgue in ${\Bbb
R}^n,$ and ${\rm dist\,}(A,B)$ is the Euclidean distance between the sets $A$ and $B$ in ${\Bbb R}^n.$ A mapping $f:D\rightarrow {\Bbb
R}^n$ is said to be a [*discrete*]{} if the pre-image $f^{-1}(y)$ of any point $y\,\in\,{\Bbb R}^n$ consists of isolated points, and an [*open*]{} if the image of any open set $U\subset D$ is open in ${\Bbb R}^n.$ The notation $f:D\rightarrow {\Bbb R}^n$ assumes that $f$ is continuous on its domain. In what follows, a mapping $f$ is supposed to be orientation preserving, i.e. the topological index $µ(y, f,G)$ is greater than zero for an arbitrary domain $G\subset
D,$ $\overline{G}\subset D$ and an arbitrary $y\in f(G)\setminus
f(\partial G),$ (see, for example, §2 of the Ch. II in ). Let $f:D\rightarrow {\Bbb R}^n$ be an arbitrary mapping and suppose that there is a domain $G\subset D,$ $\overline{G}\subset D,$ for which $
f^{\,-1}\left(f(x)\right)=\left\{x\right\}.$ Then the quantity $\mu(f(x), f, G),$ which is referred to as the local topological index, does not depend on the choice of the domain $G$ and is denoted by $i(x, f).$ In what follows $(x,y)$ denotes the standard scalar multiplication of the vectors $x,y\in {\Bbb R}^n,$ ${\rm
diam\,}A$ is Euclidean diameter of the set $A\subset {\Bbb R}^n,$ $$B(x_0, r)=\left\{x\in{\Bbb R}^n: |x-x_0|< r\right\}\,,\quad {\Bbb B}^n
:= B(0, 1)\,,$$ $$S(x_0,r) = \{ x\,\in\,{\Bbb R}^n :
|x-x_0|=r\}\,,\quad{\Bbb S}^{n-1}:=S(0, 1)\,,$$ $\omega_{n-1}$ denotes the quare of the unit sphere ${\Bbb S}^{n-1}$ in ${\Bbb
R}^n,$ $\Omega_{n}$ is a volume of the unit ball ${\Bbb B}^{n}$ in ${\Bbb R}^n.$ Given a mapping $f:D\rightarrow {\Bbb R}^n,$ a set $E\subset D,$ and a point $y\in {\Bbb R}^n$ we define the multiplicity function $N(y, f, E)$ as the number of pre-images of $y$ in $E,$ that is, $$N(y, f, E) = {\rm card}\in \left\{x \in E:
f(x) = y\right\}\,.$$ Recall that a mapping $f:D\rightarrow {\Bbb R}^n$ is said to have the [*$N$ – property (of Luzin)*]{} if $m\left(f\left(S\right)\right)=0$ whenever $m(S)=0$ for all such sets $S\subset{\Bbb R}^n.$ Similarly, $f$ has the [*$N^{-1}$ – property*]{} if $m\left(f^{\,-1}(S)\right)=0$ whenever $m(S)=0.$
We write $f\in W^{1,n}_{loc}(D),$ iff all of the coordinate functions $f_j,$ $f=(f_1,\ldots,f_n),$ have the partitional derivatives which are locally integrable in the degree $n$ in $D.$
Recall that a mapping $f:D\rightarrow {\Bbb R}^n$ is said to be [*a mapping with bounded distortion*]{}, if the following conditions hold:
1\) $f\in W_{loc}^{1,n},$
2\) a Jacobian $J(x,f):={\rm det\,}f^{\,\prime}(x) $ of the mapping $f$ at the point $x\in D$ preserves the sign almost everywhere in $D,$
3\) $\Vert f^{\,\prime}(x) \Vert^n \le K \cdot |J(x,f)|$ at a.e. $x\in D$ and some constant $K<\infty,$ where $$\Vert
f^{\,\prime}(x)\Vert:=\sup\limits_{h\in {\Bbb R}^n:
|h|=1}|f^{\,\prime}(x)h|\,,$$ see., e.g., $\S\, 3$ Ch. I in , or definition 2.1 of the section 2 Ch. I in [@Ri].
Active investigations of the mappings with bounded distortion were started by Yu.G. Reshetnyak. In particular, he has proved that the mappings $f$ with bounded distortion are open and discrete, see Theorems 6.3 and 6.4, $\S\, 6,$ Ch. II in , are differentiable a.e., see Theorem 4 in , in and have $N$ – property, see Theorem 6.2 Ch. II in . From other hand, the $N^{-1}$ – property of the mappings with bounded distortion was proved by B. Bojarski and T. Iwaniec, see Theorem 8.1 in [@BI].
We recall that an isolated point $x_0$ of the boundary $\partial D$ of a domain $D$ in ${\Bbb R}^n$ is said to be a [*removable singularity*]{} if there is a finite limit $\lim\limits_{x\rightarrow
x_0}\,f(x).$ If $f(x)\rightarrow \infty$ as $x\rightarrow x_0,$ then $x_0$ is referred to as a [*pole*]{}. An isolated point $x_0$ of $\partial D$ is called an [*essential singularity*]{} of a mapping $f:D\rightarrow {\Bbb R}^n$ if the limit $\lim\limits_{x\rightarrow
x_0}\,f(x)$ does not exist.
In 1972, in the work of J. Väisälä was proved the following, see e.g. Theorem 4.2 in .
A goal of the present paper is a proof of the analogue of the statement 1 for more general classes of mappings of finite length distortion, including the classes of mappings with bounded distortion. Mappings with finite length distortion were introduced by O. Martio, V. Ryazanov, U. Srebro and E. Yakubov in 2002, see e.g. in the work , or Chapter 8 in . The considering of it is actually in the connection with the study of the so–called mappings with finite distortion, which are actively investigated at the last time, see e.g. Chapter 20 in [@AIM] or Chapter 6 in [@IM]. In this connection, see also the works [@BGMV], [@Mikl], [@Sal] and [@UV].
A curve $\gamma$ in ${\Bbb R}^n$ is a continuous mapping $\gamma
:\Delta\rightarrow{\Bbb R}^n$ where $\Delta$ is an interval in ${\Bbb R} .$ Its locus $\gamma(\Delta)$ is denoted by $|\gamma|.$ Given a family of curves $\Gamma$ in ${\Bbb R}^n ,$ a Borel function $\rho:{\Bbb R}^n \rightarrow [0,\infty]$ is called [*admissible*]{} for $\Gamma ,$ abbr. $\rho \in {\rm adm}\, \Gamma ,$ if curvilinear integral of the first type $\int\limits_{\gamma} \rho(x)|dx|$ satisfies the condition $$\int\limits_{\gamma} \rho(x)|dx| \ge 1$$ for each $\gamma\in\Gamma.$ The [*modulus*]{} $M(\Gamma )$ of $\Gamma$ is defined as $$M(\Gamma) =\inf\limits_{ \rho \in {\rm adm}\, \Gamma}
\int\limits_{{\Bbb R}^n} \rho^n(x) dm(x)$$ interpreted as $+\infty$ if ${\rm adm}\, \Gamma = \varnothing .$ The properties of the above modulus are analogous to the properties of the measure of Lebesgue $m$ in ${\Bbb R}^n.$ Namely, a modulus of the empty family equals to zero, $M(\varnothing)=0,$ a modulus has a property of monotonicity by the relation to families of curves $\Gamma_1$ and $\Gamma_2:$ $\Gamma_1\subset\Gamma_2\Rightarrow M(\Gamma_1)\le M(\Gamma_2),$ and has a property of subadditivity, $M\left(\bigcup\limits_{i=1}^{\infty}\Gamma_i\right)\le
\sum\limits_{i=1}^{\infty}M(\Gamma_i),$ see Theorem 6.2 in .
We say that a property $P$ holds for [*almost every (a.e.)*]{} curves $\gamma$ in a family $\Gamma$ if the subfamily of all curves in $\Gamma $ for which $P$ fails has modulus zero.
If $\gamma :\Delta\rightarrow{\Bbb R}^n$ is a locally rectifiable curve, then there is the unique increasing length function $l_{\gamma}$ of $\Delta$ onto a length interval $\Delta
_{\gamma}\subset{\Bbb R}$ with a prescribed normalization $l
_{\gamma}(t_0)=0\in\Delta _{\gamma},$ $t_0\in\Delta,$ such that $l
_{\gamma}(t)$ is equal to the length of the subcurve $\gamma
|_{[t_0,t]}$ of $\gamma$ if $t>t_0,$ $t\in\Delta ,$ and $l
_{\gamma}(t)$ is equal to $-l(\gamma |_{[t,t_0]})$ if $t<t_0,$ $t\in\Delta .$ Let $g: |\gamma |\rightarrow{\Bbb R}^n$ be a continuous mapping, and suppose that the curve $\widetilde{\gamma}
=g\circ\gamma$ is also locally rectifiable. Then there is a unique increasing function $L_{\gamma ,g}: \Delta
_{\gamma}\rightarrow\Delta _{\widetilde{\gamma}}$ such that $L_{\gamma ,g}\left(l_{\gamma}(t)\right) = l_{\widetilde{\gamma}}(t)
\quad\forall\quad t\in\Delta.$ A curve $\gamma$ in $D$ is called here a [*lifting*]{} of a curve $\widetilde{\gamma}$ in ${\Bbb R}^n$ under $f:D\rightarrow {\Bbb R}^n$ if $\widetilde{\gamma} =
f\circ\gamma.$ Recall that $f\in ACP$ if and only if a curve $\widetilde{\gamma}=f\circ\gamma$ is locally rectifiable for a.e. curves $\gamma$ in $D,$ and $L_{\gamma , f} $ is absolutely continuous on closed subintervals of $\Delta_{\gamma}$ for a.e. curves $\gamma$ in $D.$ We say that a discrete mapping $f$ is [*absolute continuous on curves in the inverse direction,*]{} abbr. $ACP^{-1},$ if for a.e. curves $\widetilde{\gamma}$ a lifting $\gamma$ of $\widetilde{\gamma},$ $\widetilde{\gamma}=f\circ\gamma,$ is locally rectifiable, and $L^{-1}_{\gamma , f}$ is absolutely continuous on closed subintervals of $\Delta_{\widetilde{\gamma}}$ for a.e. curves $\widetilde{\gamma}$ in $f(D)$ and for each lifting $\gamma$ of $\widetilde{\gamma}.$ A mapping $f:D\rightarrow{\Bbb
R}^n$ is said to be of [*finite length distortion*]{}, abbr. $f\in
FLD$, if $f$ is differentiable a.e. in $D,$ has $N$ – and $N^{-1}$ – properties, and $f\in ACP\cap ACP^{-1}.$
\[rem1\] The notion of the mappings with finite length distortion can be given in more general case, when $f$ does not supposed to be a discrete, see e.g. in , see also section 8.1 in . Of course, the above definition is equivalent to the correspondent general case, see, for instance, section 8.1 and corollary 8.1 in , or Corollary 3.14 in . In this connection, the word $"$discrete$"$ will be present in the text, if it is necessary.
For the classes $W_{loc}^{1,n},$ and, in particular, for the mappings with bounded distortion, the $ACP$ property is well–known as B. Fuglede’s lemma, see, for instance, Theorem 28.2 in . Besides that, the $ACP^{-1}$ property was proved by E.A. Poletskii for it, see e.g. Lemma 6 in [@Pol]. Taking into account all of the comments given above, we conclude that every mapping with bounded distortion is a mapping with finite length distortion, see also Theorem 4.7 in , or Theorem 8.2 in in this connection.
We say that a function ${\varphi}:D\rightarrow{\Bbb R}$ has a [*finite mean oscillation*]{} at the point $x_0\in D$, write $\varphi\in
FMO(x_0),$ if $${\limsup\limits_{\varepsilon\rightarrow 0}}\
\frac{1}{\Omega_n\cdot\varepsilon^n} \int\limits_{B( x_0,
\varepsilon)} |{\varphi}(x)-\overline{{\varphi}}_{\varepsilon}|\
dm(x)<\infty\,,$$ where $\overline{{\varphi}}_{\varepsilon}=
\frac{1}{\Omega_n\cdot\varepsilon^n}\int\limits_{B(
x_0,\,\varepsilon)} {\varphi}(x)\ dm(x).$ Functions of finite mean oscillation were introduced by A. Ignat’ev and V. Ryazanov in the work [@IR], see also section 11.2 in . There are the generalization and localization of the space $BMO,$ that is bounded mean oscillation functions by F. John and L. Nirenberg, see for instance [@JN].
Set $l\left(f^{\,\prime}(x)\right):=\inf\limits_{h\in {\Bbb R}^n:
|h|=1}|f^{\,\prime}(x)h|.$ Recall that [*inner dilatation*]{} of the mapping $f$ at a point $x$ is defined as $$K_I(x,f)\quad =\quad= \left\{
\begin{array}{rr}
\frac{|J(x,f)|}{{l\left(f^{\,\prime}(x)\right)}^n}, & J(x,f)\ne 0,\\
1, & f^{\,\prime}(x)=0, \\
\infty, & {\rm otherwise}
\end{array}
\right.\,.$$ [*Outher dilatation*]{} of the mapping $f$ at the point $x$ can be defined as $$K_O(x,f)\quad =\quad= \left\{
\begin{array}{rr}
\frac{\Vert f^\prime(x)\Vert^n}{|J(x,f)|}, & J(x,f)\ne 0,\\
1, & f^{\,\prime}(x)=0, \\
\infty, & {\rm otherwise}
\end{array}
\right.\,.$$ It is well–known that $K_I(x,f)\le K_O^{n-1}(x,f)$ everywhere at the points, where there are well–defined, see for instance formulae (2.7) and (2.8) of the section 2.1 of Ch. I in . In particular, for the mappings with bounded distortion we have $K_I(x, f)\le K^{n-1}$ at a.e. $x,$ that follows from it’s definition. The main result of the paper is the following.
\[rem2\] Note that the condition (\[eq2\]) is stronger than the requirement (\[eq1\]), and that from the Statement $1^{\prime}$ it follows the Statement 1.
Preliminaries. The main Lemma
=============================
The following definitions can be found in the monograph [@Ri], Ch. II, Section. 3, see also section 3.11 in . Let $f:D \rightarrow {\Bbb R}^n$ be an arbitrary mapping, $\beta:[a,\,b)\rightarrow {\Bbb R}^n$ is a path and $x\in\,f^{\,-1}\left(\beta(a)\right).$ A path $\alpha:
[a,\,c)\rightarrow D$ is called a [*maximal $f$ – lifting*]{} of $\beta$ starting at $x,$ if $(1)\quad\alpha(a)=x;$ $(2)\quad
f\circ\alpha=\beta|_{[a,\,c)};$ $(3)$if $c<c^{\prime}\,\le b,$ then there does not exist a path $\alpha^{\prime}: [a,
c^{\prime})\rightarrow D$ such that $\alpha =
\alpha^{\prime}|_{[a,\,c)}$ and $f\circ
\alpha^{\,\prime}=\beta|_{[a,\,c^{\prime})}.$
Let $x_1,\ldots,x_k$ be $k$ different points of $f^{-1}\left(\beta(a)\right)$ and let $$m = \sum\limits_{i=1}^k i(x_i,\,f).$$ We say that the sequence $\alpha_1,\dots,\alpha_m$ is a [*maximal sequence of $f$ – lifting of $\beta$ starting at points $x_1,\ldots,x_k,$*]{} if
$(a)$each $\alpha_j$ is a maximal $f$ – lifting of $\beta,$
$(b)\quad {\rm card}\,\left\{j:a_j(a)=x_i\right\}= i(x_i,\,f),\quad
1\le i\le k\,,$
$(c)\quad {\rm card}\,\left\{j:a_j(t)=x\right\}\le i(x,\,f)$ for all $x\in D$ and for all $t.$
Let $f$ be a discrete open mapping and $x_1,\ldots,x_k$ are distinct points in $f^{\,-1}\left(\beta(a)\right).$ Then $\beta$ has a maximal sequence of $f$ – liftings starting at points $x_1,\ldots,x_k,$ see Theorem 3.2 Ch. II in [@Ri]. The following statement was proved by author, see for instance Theorem 3.1 in .
\[pr1\][ *Let $f:D\rightarrow {\Bbb R}^n$ be a discrete and an open mapping with finite length distortion, $\Gamma$ a path family in $D,$ $\Gamma^{\,\prime}$ a path family in ${\Bbb R}^n$ and $m$ a positive integer such that the following is true. Suppose that for every path $\beta$ in $\Gamma^{\,\prime}$ there are paths $\alpha_1,\ldots,\alpha_m$ in $\Gamma$ such that $f\circ
\alpha_j\subset \beta$ for all $j=1,\ldots,m$ and such that for every $x\in D$ and all $t$ the equality $\alpha_j(t)=x$ holds for at most $i(x,f)$ indices $j.$ Then $$\label{eq5}
M(\Gamma^{\,\prime} )\quad\le\quad \frac{1}{m}\quad\int\limits_D
K_I\left(x,\,f\right)\cdot \rho^n (x)\ \ dm(x)
$$ for every $\rho \in {\rm \,adm}\,\Gamma.$*]{}
In particular, the Proposition \[pr1\] generalizes the corresponding result of O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, that is the above statement under $m=1,$ see for instance Theorem 6.10 in , of Theorem 8.6 in . By $\Gamma(E,F,D)$ we denote the family of all curves $\gamma:[a,b]\rightarrow{\Bbb R}^n$ connecting $E$ and $F$ in $D,$ i.e. $\gamma(a)\in E,$ $\gamma(b)\in F$ and $\gamma(t)\in D$ as $t\in (a, b).$ A compact set $G\subset {\Bbb R}^n$ is said to be [*a set of capacity zero,*]{} write ${\rm cap\,}G =0,$ if there exists $T\subset {\Bbb R}^n,$ such that $M(\Gamma(T, G, {\Bbb
R}^n))=0,$ see, for instance, Section 2 of Ch. III and Proposition 10.2 of Ch. II in [@Ri]. By definition, an arbitrary set has a zero capacity if and only if every it’s compact subset has a zero capacity. The sets of capacity zero are totally disconnected, i.e., the condition ${\rm cap\,}G =0$ implies that ${\rm Int\,}G=0,$ see e.g. Corollary 2.5 of Ch. III in [@Ri]. Open set $U\subset D,$ $\overline{U}\subset D,$ is said to be a [*normal neighborhood*]{} of the point $x\in D$ under the mapping $f:D\rightarrow {\Bbb R}^n,$ iff $U\cap f^{\,-1}\left(f(x)\right)=\left\{x\right\}$ and $\partial
f(U)=f(\partial U),$ see e.g. Section 4 of Ch. I in [@Ri].
\[pr2\][ *Let $f:D\rightarrow {\Bbb R}^n$ be an open discrete mapping, then for every $x\in D$ there exists $s_x$ such that, for every $s\in (0,
s_x),$ the $x$ – component of $f^{-1}\left(B(f(x), s)\right),$ denoted by $U(x,f,s),$ is a normal neighborhood of $x$ under $f,$ $f\left(U(x,f,s)\right)=B(f(x), s)$ and ${\rm
diam\,}U(x,f,s)\rightarrow 0$ as $s\rightarrow 0,$ see, for instance, Lemma 4.9 of Ch. I in [@Ri].*]{}
The main tool under the proof of the basic results of the present work is the following
\[lem1\][ *Let $b\in D$ and $f:D\setminus\{b\}\rightarrow {\Bbb R}^n$ be an open and a discrete mapping with finite length distortion. Suppose that there exists $\delta>0$ such that $$\label{eq3}
|f(x)|\le C \left(\log\frac{1}{|x-b|}\right)^{p}\,,$$ at every $x\in B(b, \delta)\setminus\{b\}$ and some constants $p>0$ and $C>0.$ Follow, suppose that there exist a measurable function $Q:D\rightarrow [1, \infty],$ numbers $\varepsilon_0>0,$ $\varepsilon_0<{\rm dist\,}\left(b,
\partial D\setminus\{b\}\right),$ $A>0$ and a Borel function $\psi(t):[0, \varepsilon_0]\rightarrow
(0, \infty)$ such that $K_I(x,f)\le Q(x)$ a.e., and $$\label{eq4}
\int\limits_{\varepsilon<|x-b|<\varepsilon_0}Q(x)\cdot\psi^n(|x-x_0|)
\ dm(x)\le \frac{A\cdot I^n(\varepsilon,
\varepsilon_0)}{\left(\log\log\frac{1}{\varepsilon}\right)^{n-1}}
\qquad \forall\quad\varepsilon\in(0,\varepsilon_0/2)\,,$$ где $$\label{eq11}
0<I(\varepsilon, \varepsilon_0)
=\int\limits_{\varepsilon}^{\varepsilon_0}\psi(t)dt < \infty \qquad
\forall\quad\varepsilon \in(0, \varepsilon_0)\,.$$ Then a point $b$ is a removable singularity, or a pole of the mapping $f.$*]{}
[*Proof.*]{} Suppose the contrary, i.e., a point $b$ is an essential singularity of $f.$ Without loss of generality, we can consider that $b=0$ and $C=1.$ In this case, there exists $R>0,$ such that $$\label{eq10}
f\left(S(0, \delta)\right)\subset B(0, R)\,.$$ Since $b=0$ is an essential singularity of $f,$ from the conditions (\[eq5\]), (\[eq4\]) and another author’s result, see Lemma 3.1, Lemma 5.1 and Theorem 6.5 in , we have $$N\left(y, f,
B(0, \delta)\setminus \{0\}\right)=\infty$$ for all $y\in {\Bbb
R}^n\setminus E,$ where ${\rm cap\,}E=0.$ Since $E$ of zero capacity, ${\Bbb R}^n\setminus E$ is unbounded. Thus, there exists $y_0\in {\Bbb R}^n\setminus \left(E\cup B(0, R)\right).$
Let $k_0>\frac{4Ap^{n-1}}{\omega_{n-1}},$ $k_0\in {\Bbb N}.$ Since $N\left(y_0, f, B(0, \delta)\setminus \{0\}\right)=\infty,$ there exist the points $x_1,\ldots,x_{k_0}\in f^{-1}(y_0),$ $x_1,\ldots,x_{k_0}\in B(0, \delta)\setminus\{0\}.$ By Proposition \[pr2\], there exists $r>0,$ such that every point $x_j,$ $j=1,\ldots,k_0,$ has a normal neighborhood $U_j:=U(x_j, f , r)$ with $\overline{U_l}\cap\overline{U_m}=\varnothing$ at all $l\ne m,
$ $l, m\in {\Bbb N},$ $1\le l\le k_0$ and $1\le m\le k_0.$
Set $d:=\min\left\{\varepsilon_0, {\rm dist\,}\left(0,
\overline{U_1}\cup\ldots\cup \overline{U_{k_0}}\right)\right\}.$ Let $a\in (0, d)$ and $V:=B(0, \delta)\setminus\overline{B(0, a)}.$ By (\[eq3\]), taking into account that $\partial f(V)\subset
f(\partial V)$ and $C=1,$ we have $$\label{eq6}
f(V)\subset B\left(0, \left(\log\frac{1}{a}\right)^p\right)\,.$$ Since $z_0:=y_0+re\in \overline{B(y_0, r)}=f\left(\overline{U(x_j,
f, r)}\right),$ $j=1,\ldots, k_0,$ we have $z_0\in f(V).$ Thus there exists a sequence of the points $\widetilde{x_1},\ldots,\widetilde{x_{k_0}},$ $\widetilde{x_j}\in
\overline{U_j},$ $1\le j\le k_0,$ such that $f(\widetilde{x_j})=z_0.$ Note that $k_0\le\sum\limits_{j=1}^{k_0}
i(\widetilde{x_j},\,f)=m^{\,\prime}.$ Let $H$ be a hemisphere $H=\left\{e\in {\Bbb S}^{n-1}: (e, y_0)>0\right\},$ $\Gamma^{\,\prime}$ be a curve’s family $\beta:\left[r,
\left(\log\frac{1}{a}\right)^p\right)\rightarrow {\Bbb R}^n$ of the type $\beta(t)=y_0+te,$ $e\in H,$ and $\Gamma$ be a sequence of maximal $f$ – liftings of $\beta$ under the mapping $f$ in $V,$ starting at the points $\widetilde{x_1},\ldots,\widetilde{x_{k_0}},$ $\widetilde{x_j}\in \overline{U_j},$ $1\le j\le k_0,$ consisting from $m^{\,\prime}$ curves, where $m^{\,\prime}
=\sum\limits_{j=1}^{k_0} i(\widetilde{x_j},\,f).$ Such a sequence exists by Theorem 3.2 of Ch. II in [@Ri]. By Proposition \[pr1\], $$\label{eq9} M(\Gamma^{\,\prime} )\le
\frac{1}{m^{\,\prime}}\quad\int\limits_D K_I\left(x,\,f\right)\cdot
\rho^n (x) dm(x)\le \frac{1}{k_0}\int\limits_D
K_I\left(x,\,f\right)\cdot \rho^n (x) dm(x)
$$ for every $\rho \in {\rm }\,{\rm adm}\,\Gamma.$
Given $e\in H,$ we show that, for every curve $\beta=y_0+te$ and it’s maximal lifting $\alpha(t):[r, c)\rightarrow V$ starting at the point $\widetilde{x_{j_0}},$ $\alpha\in \Gamma,$ $1\le j_0\le k_0,$ there exists a sequence $r_k\in [r, c)$ with $r_k\rightarrow c-0$ as $k\rightarrow \infty$ such that ${\rm dist\,}(\alpha(r_k),
\partial V)\rightarrow 0$ as $k\rightarrow \infty.$ Suppose the contrary, i.e., there exists $e_0\in H,$ such that $\alpha(t),$ $t\in [r, c),$ is a maximal lifting of $\beta=y_0+te_0,$ and $\alpha(t)$ lies inside of $V$ with it’s closure. Let $C(c,\,\alpha(t))$ denotes a cluster set of $\alpha$ as $t\rightarrow c-0.$ For every $x\in C(c,\,\alpha(t))$ there exists a sequence $t_k\rightarrow \infty$ such that $x=\lim\limits_{k\rightarrow \infty}\alpha(t_k).$ Since $f$ is continuous and $C(c,\,\alpha(t))\subset V$ by the assumption, we have $f(x)=f(\lim\limits_{k\rightarrow \infty}\alpha(t_k))=
\lim\limits_{k\rightarrow \infty}\beta(t_k)= \beta(c),$ from what it follows that $f$ is a constant on $C(c,\,\alpha(t)).$ Since $f$ is a discrete and a set $C(c,\,\alpha(t))$ is connected, we have $C(c,\,\alpha(t))=p_1\in V.$ Let $c\ne
b_0:=\left(\log\frac{1}{a}\right)^p.$ In this case, we can construct a lifting $\alpha^{\,\prime}$ of $\beta$ started at $p_1.$ Uniting the liftings $\alpha$ and $\alpha^{\,\prime},$ we obtain another maximal lifting $\alpha^{\,\prime\prime}$ of $\beta$ starting at the point $\widetilde{x_{j_0}},$ that contradicts to the maximal property of the first lifting $\alpha.$ Thus, $c=b_0$ and hence $\alpha$ can be extend to closed curve defined on the segment $\left[r, \left(\log\frac{1}{a}\right)^p\right]$ (we don’t change the notion of the extended curve). Then, at every $t\in \left[r,
\left(\log\frac{1}{a}\right)^p\right],$ we have $\beta(t)=f(\alpha(t))\subset f(V).$ In particular, by (\[eq6\]) $$\label{eq7}
z_1:=y_0+\left(\log\frac{1}{a}\right)^pe_0\in f(V)\subset B\left(0,
\left(\log\frac{1}{a}\right)^p\right)\,.$$ However, since $e_0\in H,$ we have $$|z_1|=\left|y_0+\left(\log\frac{1}{a}\right)^pe_0\right|=\sqrt{|y_0|^2 +
2\left(y_0,
\left(\log\frac{1}{a}\right)^pe_0\right)+\left(\log\frac{1}{a}\right)^{2p}}\ge$$ $$\label{eq8}
\ge \sqrt{|y_0|^2 + \left(\log\frac{1}{a}\right)^{2p}}\ge
\left(\log\frac{1}{a}\right)^{p}\,.$$ However, the relation (\[eq8\]) contradicts to (\[eq7\]), which disproves the assumption that $\alpha(t)$ consists in the set $V$ with it’s closure. Consequently, ${\rm dist\,}(\alpha(r_k),
\partial V)\rightarrow 0$ as $k\rightarrow c-0$ and some sequence $r_k\in [r, c)$ such that $r_k\rightarrow c-0$ as $k\rightarrow \infty.$
Note that the situation when ${\rm dist\,}(\alpha(r_k), S(0,
\delta))\rightarrow 0$ as $k\rightarrow c-0$ is excluded. In fact, suppose that there exist $p_2\in S(0, \delta)$ and a sequence $k_l,$ $l\in {\Bbb N},$ such that $\alpha(r_{k_l})\rightarrow p_2$ as $l\rightarrow \infty.$ By the continuously of $f$ we have that $\beta(r_{k_l})\rightarrow f(p_2)$ as $l\rightarrow \infty,$ that is impossible by (\[eq10\]), because for every $e\in H$ and $t\in
\left[r, \left(\log\frac{1}{a}\right)^p\right)$ we have $|\beta(t)|=|y_0+te|=\sqrt{|y_0|^2 + 2t(y_0, e)+t^2}\ge |y_0|> R$ by the choosing of $y_0.$
It follows from above that there exists a sequence $r_k\in [r, c)$ such that $r_k\rightarrow c-0$ as $k\rightarrow \infty$ and $\alpha(r_k)\rightarrow p_3\in S(0, a).$ Besides that, every such a curve $\alpha\in \Gamma$ intersects the sphere $S(0, d)$ because $\alpha$ has a start outside of $B(0, d).$ Consider the function $$\rho_{a}(x)= \left\{
\begin{array}{rr}
\psi(|x|)/I(a, d), & x\in B(0,d)\setminus B(0,a),\\
0, & x\in {\Bbb R}^n \setminus \left(B(0,d)\setminus
B(0,a)\right)
\end{array}
\right.\,,$$ where $I(a, d)$ is defined as in (\[eq11\]) and $\psi$ be a function from the condition of Lemma. Since $\psi(t)>0,$ $I(a, d)>0$ for every $0<a<d.$ Thus, a function $\rho_{a}(x)$ which is given above is well–defined. Note that a function $\rho_{a}(x)$ is Borel, moreover, since $\rho_{a}(x)$ is a radial function, by the above properties of curves of $\Gamma$ and by Theorem 5.7 in , for every curve $\alpha\in \Gamma$ we have $$\int\limits_{\alpha}\rho_a(x)|dx|\ge\frac{1}{I(a, d)}\int\limits_a^d \psi(t)dt= 1\,,$$ i.e., $\rho_{a}(x)\in {\rm adm\,}\Gamma.$ Thus, from (\[eq4\]) and (\[eq9\]) we have $$M(\Gamma^{\,\prime} )\quad\le\quad \frac{1}{k_0\cdot I^n(a,
d)}\quad\int\limits_{a<|x|<d} K_I\left(x,\,f\right)\cdot \psi^n
(|x|)dm(x)\le$$ $$\le \frac{I^n(a, \varepsilon_0)}{k_0\cdot I^n(a,
d)\cdot I^n(a, \varepsilon_0)}\quad\int\limits_{a<|x|<\varepsilon_0}
Q(x)\cdot \psi^n (|x|)dm(x)=$$$$=\left(1+\frac{I(d,
\varepsilon_0)}{I(a, d)}\right)^n\frac{1}{k_0\cdot I^n(a,
\varepsilon_0)}\quad\int\limits_{a<|x|<\varepsilon_0} Q(x)\cdot
\psi^n (|x|)dm(x)\le$$ $$\le \frac{2}{k_0\cdot I^n(a,
\varepsilon_0)}\quad\int\limits_{a<|x|<\varepsilon_0} Q(x)\cdot
\psi^n (|x|)dm(x)$$ at every $a\in (0, d_1)$ and some $d_1,$ $d_1\le
d,$ because by (\[eq4\]), $I^n(a,d)\rightarrow\infty$ as $a\rightarrow \infty.$ Now, from (\[eq4\]), we conclude that $$\label{eq12}
M(\Gamma^{\,\prime} )\le
\frac{2A}{k_0\left(\log\log\frac{1}{a}\right)^{n-1}}$$ for $a\in (0, d_1).$ From other hand, by section 7.7 in , $$\label{eq13}
M(\Gamma^{\prime})=\frac{1}{2}\frac{\omega_{n-1}}
{\left(\log\frac{\left(\log\frac{1}{a}\right)^p}{r}\right)^{n-1}}\,.$$ By (\[eq12\]) and (\[eq13\]) we have $$\frac{1}{2}\frac{\omega_{n-1}}
{\left(\log\frac{\left(\log\frac{1}{a}\right)^p}{r}\right)^{n-1}}\le
\frac{2A}{k_0\left(\log\log\frac{1}{a}\right)^{n-1}},$$ from what $$\left(\log\left(\frac{\left(\log\frac{1}{a}\right)^p}{r}\right)^{
\left(\frac{2}{\omega_{n-1}}\right)^{\frac{1}{n-1}}}\right)^{n-1}\ge
\left(\log\left(\log\frac{1}{a}\right)^{\left(\frac{k_0}{2A}\right)^{\frac{1}{n-1}}}\right)^{n-1}\,,$$ $$\left(\frac{\left(\log\frac{1}{a}\right)^p}{r}\right)^{
\left(\frac{2}{\omega_{n-1}}\right)^{\frac{1}{n-1}}}\ge
\left(\log\frac{1}{a}\right)^{\left(\frac{k_0}{2A}\right)^{\frac{1}{n-1}}}\,,$$ $$\frac{1}{r^{{
\left(\frac{2}{\omega_{n-1}}\right)^{\frac{1}{n-1}}}}}\ge
\left(\log\frac{1}{a}\right)^{{\left(\frac{k_0}{2A}\right)^{\frac{1}{n-1}}}-p{
\left(\frac{2}{\omega_{n-1}}\right)^{\frac{1}{n-1}}}}\,.$$ Since by the choosing $k_0>\frac{4Ap^{n-1}}{\omega_{n-1}},$ in the right–hand part of the above relation the logarithmic function presents in some positive degree. Letting to the limit as $a\rightarrow 0$ in the right–hand part of it we obtain that $$\frac{1}{r^{{
\left(\frac{2}{\omega_{n-1}}\right)^{\frac{1}{n-1}}}}}\ge
\infty\,,$$ that is impossible. The contradiction obtained above disproves the assumption that $b=0$ is an essential singularity of $f.$ $\Box$
The next statement follows directly from Lemma 5 in as $\psi(t)=\frac{1}{t\log\frac{1}{t}}$ and from the estimate (\[eq5\]) at $m=1.$
\[pr3\] [ *Let $b\in D$ and $f:D\rightarrow {\Bbb R}^n$ be an open and a discrete mapping with finite length distortion. Suppose that there exist a measurable function $Q:D\rightarrow [1, \infty],$ the numbers $\varepsilon_0>0,$ $\varepsilon_0<{\rm dist\,}\left(b,
\partial D\right),$ and $A>0$ such that $K_I(x,f)\le Q(x)$ a.e., such that the relations (\[eq4\]) and (\[eq11\]) hold as $\psi(t)=\frac{1}{t\log\frac{1}{t}},$ i.e., $$\label{eq14}
\int\limits_{\varepsilon<|x-b|<\varepsilon_0}\frac{Q(x)}{|x-b|^n
\log^n\frac{1}{|x-b|}}\ dm(x)\le A\cdot\log{\frac{\log{\frac{1}
{\varepsilon}}}{\log{\frac{1}{\varepsilon_0}}}} \qquad
\forall\quad\varepsilon\in(0,\varepsilon_0)\,.$$ Then for every $x\in B(b, \varepsilon_0)$ $$\label{eq15}
|f(x)-f(b)|\le
\frac{\alpha_n(1+R^2)}{\delta}\left\{\frac{\log\frac{1}{\varepsilon_0}}
{\log\frac{1}{|x-b|}}\right\}^{\beta_n}\,,$$ where $\alpha_n$ and $\beta_n=\left(\frac{\omega_{n-1}}{A}\right)^{1/(n-1)}$ depend only on $n,$ and $\delta$ depends only on $R.$* ]{}
\[cor1\][*Under the conditions of Lemma \[lem1\], suppose that the condition (\[eq14\]) take a place instead of (\[eq4\]) and (\[eq11\]), and the condition $$\label{eq16}
\lim\limits_{x\rightarrow
b}\frac{|f(x)|}{\left(\log\frac{1}{|x-b|}\right)^{\beta_n}}=0\,,$$ take a place instead of (\[eq3\]), where $\beta_n=\left(\frac{\omega_{n-1}}{A}\right)^{1/(n-1)}.$ Then a point $x=b$ is a removable singularity of $f.$* ]{}
[*Proof.*]{} Without loss of generality, we can consider that $b=0.$ By Lemma \[lem1\] a point $b$ can not to be an essential singularity of $f.$ Suppose that $b=0$ is a pole of $f.$ Consider the composition of the mappings $h=g\circ f,$ where $g(x)=\frac{x}{|x|^2}$ is inversion under the sphere ${\Bbb
S}^{n-1}.$ Note that a mapping $h$ to be a mapping with finite length distortion, $K_I(x, f)=K_I(x, h)$ and $h(0)=0.$ Moreover, $h$ is bounded in some neighborhood of zero. Thus there exist $\varepsilon_1>0$ and $R>0$ such that $|h(x)|\le R$ at every $|x|<\varepsilon_1.$ Now we apply a Proposition \[pr3\]. By (\[eq15\]), $$|h(x)|=\frac{1}{|f(x)|}\le
\frac{\alpha_n(1+R^2)}{\delta}\left\{\frac{\log\frac{1}{\varepsilon_0}}
{\log\frac{1}{|x|}}\right\}^{\beta_n}\,.$$ Consequently, $$\frac{|f(x)|}{\left\{\log\frac{1}{|x|}\right\}^{\beta_n}}\ge
\frac{\delta}{\alpha_n(1+R^2){\left\{\log\frac{1}{\varepsilon_0}\right\}}^{\beta_n}}\,.$$ However, the last relation contradicts to the (\[eq16\]). The contradiction obtained above prove that $b=0$ is a removable singularity of $f.$ $\Box$
The proof of the main results
=============================
[**Proof of the statement $1^{\,\prime}$**]{} follows from (\[eq14\]) which holds for every function $Q\in FMO(b),$ see, for instance, Corollary 2.3 in [@IR], or Lemma 6.1 of Ch. VI in , and from the Lemma \[lem1\]. $\Box$
\[cor2\]
*Let $f:D\setminus\{b\}\rightarrow {\Bbb R}^n$ be an open and a discrete mapping with finite length distortion. Suppose that there exists a measurable function such that $Q:D\rightarrow[1, \infty],$ such that $K_I(x,f)\le Q(x)$ at a.e. $x\in D$ and $Q(x)\in FMO(b).$*
There exists $p_0>0$ such that $$\label{eq18} \lim\limits_{x\rightarrow
b}\frac{|f(x)|}{\left(\log\frac{1}{|x-b|}\right)^{p_0}}=0$$ implies that $b=0$ is a removable singularity of $f.$
[**Proof**]{} follows directly from the Statement $1^{\,\prime}$ and Corollary \[cor1\].$\Box$
In what follows $q_{x_0}(r)$ denotes the integral average of $Q(x)$ under the sphere $|x-x_0|=r,$ $$\label{eq17}
q_{x_0}(r):=\frac{1}{\omega_{n-1}r^{n-1}}\int\limits_{|x-x_0|=r}Q(x)\,dS\,,$$ where $dS$ is element of the square of the surface $S.$
\[th1\]
*Let $b\in D$ and $f:D\setminus\{b\}\rightarrow {\Bbb R}^n$ be an open and a discrete mapping with finite length distortion. Suppose that there exists $\delta>0$ such that the relation (\[eq2\]) holds at every $x\in
B(b, \delta)$ and some constants $p>0$ and $C>0.$ Let there exists a function $Q:D\rightarrow[1, \infty],$ such that $K_I(x,f)\le Q(x)$ a.e. $x\in D$ and $q_{b}(r)\le C\cdot
\left(\log\frac{1}{r}\right)^{n-1}$ as $r\rightarrow 0.$ The a point $b$ is a pole or a removable singularity of $f.$*
Moreover, in addition, if the relation (\[eq18\]) holds as $p_0=\left(\frac{1}{C}\right)^{1/(n-1)},$ then a point $b=0$ is removable for $f.$
[*Proof.*]{} We can consider that $b=0.$ Let $\varepsilon_0<\min\left\{\,\,{\rm dist\,}\left(0,
\partial D\right),\quad 1\right\}.$ Set $\psi(t)=\frac{1}{t\,\log{\frac{1}{t}}}.$ Note that $$\int\limits_{\varepsilon<|x|<\varepsilon_0}
\frac{Q(x)dm(x)}{\left(|
x|\log{\frac{1}{|x|}}\right)^n}=\int\limits_{\varepsilon}^{\varepsilon_0}
\left(\int\limits_{|x|\,=\,r}\frac{Q(x)dm(x)}{\left(|
x|\log{\frac{1}{|x|}}\right)^n}\, dS\,\right)\,dr\le C\cdot\omega_
{n-1}\cdot I(\varepsilon, \varepsilon_0)\,,$$ where as above $I(\varepsilon,
\varepsilon_0):=\int\limits_{\varepsilon}^{\varepsilon_0}\psi(t) dt
= \log{\frac{\log{\frac{1}
{\varepsilon}}}{\log{\frac{1}{\varepsilon_0}}}}.$ Thus the conditions (\[eq4\]) and (\[eq11\]) of Lemma \[lem1\] hold at $\psi$ which was given above. The second statement of the Theorem \[th1\] follows from the Corollary \[cor1\]. $\Box$
Corollaries. The precision of the conditions
============================================
Recall that a point $y_0\in D$ is said to be a [*branch point*]{} of the mapping $f:D\rightarrow {\Bbb R}^n,$ if for every neighborhood $U$ of the point $y_0$ a restriction $f|_{U}$ fails to be a homeomorphism. A set of all branch sets of $f$ is denoted by $B_f.$ The following statement can be found as Theorem 1 in . Let $f:D\rightarrow {\Bbb R}^n$ be an open discrete mapping of the class $W_{loc}^{1,n}(D)$ such that either $K_O(x,f)\in L_{loc}^{n-1},$ or $K_I(x,f)\in L_{loc}^1,$ and $m(B_f)=0.$ Then $f$ to be a mapping with finite length distortion. Founded on the Statement $1^{\,\prime}$ and on Theorem \[th1\], we have the following.
\[th2\]
*Let $b\in D$ and $f:D\setminus\{b\}\rightarrow {\Bbb R}^n$ be an open discrete mapping of the class $W_{loc}^{1,n}(D),$ for which either $K_O(x,f)\in L_{loc}^{n-1},$ or $K_I(x,f)\in L_{loc}^1,$ and $m(B_f)=0.$ Suppose that there exists $\delta>0$ such that the inequality $$|f(x)|\le C \left(\log\frac{1}{|x-b|}\right)^{p}$$ take a place for all $x\in B(b, \delta)$ and some constants $p>0$ and $C>0.$ Besides that, suppose that there exists a measurable function $Q:D\rightarrow[1, \infty]$ such that $K_I(x,f)\le Q(x)$ at a.e. $x\in D$ and $Q(x)\in FMO(b).$ Then a point $b$ is a removable singularity, or a pole of $f.$*
Moreover, there exists a number $p_0>0$ such that the condition $$\lim\limits_{x\rightarrow
b}\frac{|f(x)|}{\left(\log\frac{1}{|x-b|}\right)^{p_0}}=0$$ implies that a point $b$ to be a removable singularity of $f.$
\[th3\][ *All of the conclusions of the Theorem \[th2\] take a place if the assumption $Q(x)\in FMO(b)$ to replace on the requirement $q_{b}(r)\le C\cdot \left(\log\frac{1}{r}\right)^{n-1}$ as $r\rightarrow 0.$ In this case, we can take $p_0=\left(\frac{1}{C}\right)^{1/(n-1)}.$*]{}
The following result shows that the conditions on $Q$ which are given above can not to be done more weaker, for instance, we can not replace it by the assumption $Q\in L_{loc}^q,$ $q\ge 1,$ for every sufficiently large $q.$
\[th4\][ *Given $p>0$ and $q\in [1, \infty),$ there exists a homeomorphism $f:{\Bbb B}^n\setminus\{0\}\rightarrow {\Bbb R}^n$ with finite length distortion which belongs to $f\in W_{loc}^{1,n},$ and $f^{-1}\in W_{loc}^{1,n},$ such that $K_I\in L^q_{loc}({\Bbb
B}^n),$ $$\label{eq19}|f(x)|\le 2
\left(\log\frac{1}{|x|}\right)^{p}$$ at every $x\in B(0, 1/e)\setminus\{0\},$ and a point $b=0$ to be an essential singularity of $f.$ Moreover, $f$ is a bounded mapping in this case.*]{}
[*Proof.*]{} The desired homeomorphism $f: {\Bbb
B}^n\setminus\{0\}\rightarrow {\Bbb R}^n$ can be given as $$f(x)=\frac{1+|x|^{\alpha}}{|x|}\cdot x\,,$$ where $\alpha\in \left(0, n/q(n-1)\right).$ We can consider that $\alpha<1.$ Note that $f$ maps ${\Bbb B}^n\setminus\{0\}$ onto the ring $\{1<|y|<2\}$ in ${\Bbb R}^n,$ and the cluster set $C(f,0)=\{|y|=1\}.$ In particular, it follows from here that $x_0=0$ is an essential singularity of $f.$ It is clear that $f\in C^1({\Bbb
B}^n\setminus\{0\})$ and, consequently, $f\in W_{loc}^{1,n},$ moreover, $K_I(x,f)=\left(\frac{1+|x|^{\,\alpha}}{\alpha
|x|^{\,\alpha}}\right)^{\,n-1}\le\frac{C}{|x|^{\,(n-1)\alpha}},$ see Proposition 6.3 of Ch. VI in . Thus, $K_I(x,f)\in
L^q({\Bbb B}^n)$ because $\alpha (n-1) q<n.$ Besides that, to note that $f$ is locally quasiconformal mapping and hence $f^{\,-1}\in
W_{loc}^{1,n}.$ Thus, $f$ is a mapping with finite length distortion in ${\Bbb B}^n\setminus\{0\}$ by Theorem 4.6 in , see also Theorem 8.1 of Ch. VIII in .
The mapping $f$ is bounded in ${\Bbb B}^n\setminus\{0\},$ in particular, $f$ satisfies the inequality $|f(x)|\le 2$ at $x\in
{\Bbb B}^n\setminus\{0\}.$ From other hand, a function $s(x):=\left(\log\frac{1}{|x|}\right)^{q}$ satisfies $|s(x)|\ge 1$ for all $|x|\le 1/e.$ From here we have a relation (\[eq19\]).
Thus, we construct a mapping $f$ which have an essential isolated singularity, and satisfying all of the conditions of the Theorem \[th4\]. $\Box$
The following statement shows that the condition of the openness of the mapping $f$ is essential.
\[th5\][ *There exist a discrete mapping $f:{\Bbb R}^n\setminus\{0\}\rightarrow {\Bbb R}^n$ with finite length distortion such that $K_I\equiv 1,$ satisfying to the condition $$\label{eq20}|f(x)|\le \left(\log\frac{1}{|x|}\right)^{p}$$ at every $x\in B(0, 1/e)\setminus\{0\}$ and $p>0,$ such that a point $b=0$ is an essential singularity of $f.$* ]{}
[*Proof.*]{} Consider the division of ${\Bbb R}^n$ by the cubes $$C_{k_1,\ldots,k_n}=\prod\limits_{i=1}^{n}\left[2k_i-1, 2k_i+1\right]\,,
\quad k_i\in {\Bbb Z}\,.$$ Consider a cube $C_{k_1,\ldots,k_n}$ with $k_1,\ldots,k_n \ge 0;$ the case of the different signs of $k_i$ can be considered by analogy. Let $x=(x_1,\ldots, x_n)\in C_{k_1,\ldots,k_n}.$ If $k_1=0,$ $g_{m_1}:={\rm id}.$ Let $k_1>0.$ Set $f_{1,\ldots,1,1}(x)=y_{1,\ldots,1},$ where $y_{1,\ldots,1,1}$ be a symmetric reflection of $x$ under the hyperplane $x_1=2k_1-1.$ If $2k_1-3=-1,$ the process is finished. Let $2k_1-3>-1,$ then $f_{1,\ldots,1,2}(x)=y_{1,\ldots,1,2},$ where $y_{1,\ldots,1,2}$ be a symmetric reflection of the point $y_{1,\ldots,1}$ under the hyperplane $x_1=2k_1-3.$ If $2k_1-5=-1,$ the process is finished. In other case we continue, $f_{1,\ldots,1,3}(x)=y_{1,\ldots,1,3}.$ Etc. After a finite number of the steps $m_1$ we have a mapping $g_{m_1}=f_{1,\ldots,1,m_1}\circ\cdots\circ f_{1,\ldots,1,1},$ such that $g_{m_1}(x)\in C_{0,k_2,k_3\ldots,k_n}.$
Follow, if $k_2=0,$ then $g_{m_2}:=g_{m_1}.$ As $k_2>0,$ we repeat the above transformations with the coordinate $x_2$ and the point $x_{m_1}:=g_{m_1}(x).$ Set $f_{1,\ldots,1,2, m_1}(x)$ $=y_{1,\ldots,1,2,m_1},$ where $y_{1,\ldots,1,2,m_1}$ be a symmetric reflection of the point $x_{m_1}$ under the hyperplane $x_2=2k_2-1.$ If $2k_2-3=-1,$ the process is finished. In other case we continue. Now, we have a mapping $g_{m_2}=f_{1,\ldots,m_2,m_1}\circ\cdots\circ
f_{1,\ldots,2,m_1},$ such that $g_{m_2}(x_{m_1})\in
C_{0,0,k_3\ldots,k_n}.$
Etc. After some number of the steps $m_0=m_1+m_2+\ldots+m_n$ we obtain a mapping $G_0=g_{m_n}\circ g_{m_{n-1}}\circ\cdots
g_{m_{2}}\circ g_{m_{1}},$ such that the image $x_{m_n}$ of the point $x$ under the mapping $G_0$ lies in the cube $C_{0,0,0\ldots,0}.$ The compressing $G_1(x)=\frac{\sqrt{n}}{n}\cdot
x$ maps $C_{0,0,0\ldots,0}$ into some cube $A_0,$ which lies in $\overline{{\Bbb B}^n}.$ Set $G_2:=G_1\circ G_0.$
Note that a point $z_0=\infty$ is an essential singularity of $G_2,$ moreover, $C(G_2,\infty)=A_0\subset \overline{{\Bbb B}^n}.$ Then a mapping $$\label{eq25*}
g:=G_2\circ G_3\,,$$ $G_3(x)=\frac{x}{|x|^2},$ has an essential singularity $b=0,$ and $$C(g, 0)\subset \overline{{\Bbb B}^n}\,.$$ By the construction of $G_2,$ which is given by (\[eq25\*\]), $G_2$ is a discrete mapping, preserves the lengths of curves in ${\Bbb
R}^n,$ is differentiable a.e. and has $N$ and $N^{-1}$ – properties. Thus, $g$ is a mapping with finite length distortion, moreover, it is easy to see that $K_I(x,g)=1.$ Finally, $|g(x)|\le
1$ at every $x\in {\Bbb R}^n\setminus\{0\}.$ Thus, (\[eq20\]) holds at every $x\in B(0, 1/e)\setminus\{0\}.$
The desired mapping have been constructed. $\Box$
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\
Institute of Applied Mathematics and Mechanics,\
National Academy of Sciences of Ukraine,\
74 Roze Luxemburg str., 83114 Donetsk, UKRAINE\
Phone: +38 – (062) – 3110145,\
Email: brusin2006@rambler.ru
|
---
abstract: 'We introduce two measures of weak non-compactness ${\operatorname{Ja_E}}$ and $\bd$ that quantify, via distances, the idea of boundary behind James’ compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \overline{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James’ compactness theorem is proved using ${\operatorname{Ja_E}}$ and $\bd$, and in particular it yields the following result: [*Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an element $x_0^{**}$ in $\wscl C$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\wscl C$ at which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.*]{} We indeed establish that ${\operatorname{Ja_E}}$ and $\bd$ are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.'
address:
- 'Depto de Matemáticas. Universidad de Murcia. 30.100 Espinardo. Murcia, Spain'
- |
Charles University\
Faculty of Mathematics and Physics\
Department of Mathematical Analysis\
Sokolovská 83\
186 75 Praha 8\
Czech Republic
author:
- 'Bernardo Cascales, Ondřej F.K. Kalenda and Jiří Spurný'
title: 'A quantitative version of James’ compactness theorem'
---
[^1]
Introduction
============
The celebrated James’ compactness theorem says that a closed convex subset $C$ of a Banach space $E$ is weakly compact whenever each $x^*\in E^*$ attains its supremum on $C$, see [@jam]. In particular, $E$ is reflexive whenever each $x^*\in E^*$ attains its norm at some point of the closed unit ball $B_E$ of $E$. In the present paper we prove a quantitative version of this theorem. Such a result not only fits into the recent research on quantitative versions of various famous theorems on compactness presented amongst others in [@ang-cas1; @cas-alt11; @FHMZ; @sua-alt; @sua; @sua-san], to which we relate our results here too, but also yields a strengthening of James’ theorem itself. In particular we get the following result:
\[int-j-convex\] Let $E$ be a Banach space, $C\subset E$ a closed convex bounded set which is not weakly compact. Let $0\le c<\frac12\dh(\wscl{C},C)$ be arbitrary. Then there is some $x^*\in E^*$ such that for any $x^{**}\in\wscl C$ satisfying $x^{**}(x^*)=\sup x^*(C)$ we have $\dist(x^{**},{E})>c$.
This is our notation: if $A$ and $B$ are nonempty subsets of a Banach space $E$, then $d(A,B)$ denotes the *usual $\inf$ distance* between $A$ and $B$ and the *Hausdorff non-symmetrized distance* from $A$ to $B$ is defined by $$\dh(A,B)= \sup\{d(a,B): a\in A\}.$$ Notice that $\dh(A,B)$ can be different from $\dh(B,A)$ and that $\max\{\dh(A,B),\dh(B,A)\}$ is the Hausdorff distance between $A$ and $B$. Notice further that $\dh(A,B)=0$ if and only if $A\subset\overline B$ and that $$\label{dhinf}
\dh(A,B)=\inf \{\ep>0: A\subset B+\ep B_E\}.$$ Let us remark that we consider the space $E$ canonically embedded into its bidual $E^{**}$ and that by $\wscl{C}$ we mean the weak\* closure of $C$ in the bidual $E^{**}$.
When applying Theorem \[int-j-convex\] for $c=0$ we obtain the classical James’ compactness theorem. Our results in this paper go beyond Theorem \[int-j-convex\]. We should stress that what we really do in this paper is to introduce several measures of weak non-compactness in Banach spaces related to distances to boundaries and then study their relationship with other well known measures of weak non-compactness previously studied. Our main result is Theorem \[prop-pryce\]. Combination with known or easy results gives Corollary \[cor:AllPossibleInequalities\]. Theorem \[int-j-convex\] is then an immediate consequence.
The quantities that we introduce are the following:
Given a bounded subset $H$ of a Banach space $E$ we define: $$\begin{aligned}
{\operatorname{Ja_E}}(H)=\inf \{\epsilon > 0: \text{ for every } x^*\in E^*, \text{ there is } x^{**}\in
\overline{H}^{w^*} \\ \text{ such that }x^{**}(x^*)=\sup x^*(H) \text{ and } d(x^{**},E)\leq \epsilon\}\end{aligned}$$ and $$\begin{aligned}
\bd(H)=\inf \{\epsilon > 0: \text{ for every } x^*\in E^*, \text{ there is } x^{**}\in
\overline{H}^{w^*} \\ \text{ such that }x^{**}(x^*)=\sup x^*(H) \text{ and } d(x^{**},H)\leq \epsilon\}.\end{aligned}$$
Note that the definition of $\bd(H)$ is clearly inspired by the notion of a [*boundary*]{} that is hidden in James’ theorem. Recall that if $Y$ is a Banach space and $K\subset Y^*$ is a convex weak\*-compact set, then a subset $B\subset K$ is called a [*boundary*]{} of $K$ if for each $y\in Y$ there is $b^*\in B$ such that $$b^*(y)=\sup_{k^*\in K} k^*(y).$$
James’ compactness theorem can be rephrased now in the following way: [*let $E$ be a Banach space and $C\subset E$ a bounded closed convex set; if $C$ is a boundary of $\wscl{C}$, then $C$ is weakly compact.*]{}
We will study the relationship of ${\operatorname{Ja_E}}(C)$ and $\bd(C)$ to other quantities measuring weak non-compactness of $C$. The two most obvious quantities of this kind are $\dh(\wscl{C},C)$ and $\dh(\wscl{C},E)$. We stress that these two quantities can be different (see, e.g. examples in Section \[examples\]). The first one can be called ‘measure of weak non-compactness’ of $C$, the other one can be called ‘measure of relative weak non-compactness’ of $C$.
Using the notation introduced above, Theorem \[int-j-convex\] says that the inequality ${\operatorname{Ja_E}}(C)\ge\frac12\dh(\wscl C,C)$ holds for any closed convex bounded subset $C$ of a Banach space $E$.
In the following section we introduce several other quantities measuring weak non-compactness and sum up easy inequalities among them. In Section \[S-pryce\] we formulate and prove our main result. As a corollary we obtain that all considered quantities measuring weak non-compactness are equivalent.
In Section \[S-krein\] we discuss the relationship to the quantitative version of Krein’s theorem. Section \[examples\] contains examples showing that most of the inequalities are sharp. In the final section we study some particular cases in which some of the inequalities become equalities.
Measures of weak non-compactness {#S-measures}
================================
In this section we define and relate several quantities measuring weak non-compactness of a bounded set in a Banach space. Such quantities are called [*measures of weak non-compactness*]{}. Measures of non-compactness or weak non-compactness have been successfully applied to study of compactness, in operator theory, differential equations and integral equations, see for instance [@ang-cas1; @ast-tyl; @blasi; @cas-alt11; @FHMZ; @sua-alt; @sua; @sua-san; @kry; @kr-pr-sc]. An axiomatic approach to measures of weak non-compactness may be found in [@ban-mar; @kr-pr-sc]. But many of the natural quantities do not satisfy all the axioms, so we will not adopt this approach. Anyway, there is one property which should be pointed out: A measure of weak non-compactness should have value zero if and only if the respective set is relatively weakly compact.
Let $(x_n)$ be a bounded sequence in a Banach space $E$. We define $\clust_{E^{**}}((x_n))$ to be the set of all cluster points of this sequence in $(E^{**},w^*)$, i.e. $$\clust_{E^{**}}((x_n))=\bigcap_{n\in\N}\overline{\{x_m:m>n\}}^{w^*}.$$
Given a bounded subset $H$ of a Banach space $E$ we define: $$\gamma (H)= \sup\{|\lim_n\lim_m x^*_m(x_n)- \lim_m\lim_n x^*_m(x_n)|:
(x^*_m)\subset B_{E^*}, (x_n) \subset H
\},$$ assuming the involved limits exist, $${\operatorname{ck_E}}(H)= \sup_{(x_n)\subset H}\d(\clust_{E^{**}}((x_n)),E), \ \ \ {\operatorname{ck}}(H)= \sup_{(x_n)\subset H}\d(\clust_{E^{**}}((x_n)),H).$$ Properties of $\gamma$ can be found in [@ang-cas1; @ast-tyl; @cas-alt11; @FHMZ; @kr-pr-sc] whereas ${\operatorname{ck_E}}$ can be found in [@ang-cas1] – note that ${\operatorname{ck_E}}$ is denoted as ${\operatorname{ck}}$ in that paper; do not mistake it for ${\operatorname{ck}}$ above.
So, for a bounded set $H\subset E$ we have the following quantities measuring weak non-compactness: $$\dh(\wscl H,H),\dh(\wscl H,E),{\operatorname{ck}}(H),{\operatorname{ck_E}}(H),\gamma(H),{\operatorname{Ja}}(H),{\operatorname{Ja_E}}(H).$$ Let us stress on the different nature of these quantities:
First, the quantities $\dh(\wscl H,H)$, ${\operatorname{ck}}(H)$, $\gamma(H)$ and ${\operatorname{Ja}}(H)$ do not depend directly on the space $E$. More exactly, if $F$ is a Banach space and $H\subset E\subset F$, where $E$ is a closed linear subspace of $F$ and $H$ a bounded subset of $E$, then these quantities are the same, no matter whether we consider $H$ as a subset of $E$ or as a subset of $F$. This is trivial for $\dh(\wscl H,H)$, and ${\operatorname{ck}}(H)$ and follows from the Hahn-Banach extension theorem for $\gamma(H)$ and ${\operatorname{Ja}}(H)$.
On the other hand, the quantities $\dh(\wscl H,E)$, ${\operatorname{ck_E}}(H)$ and ${\operatorname{Ja_E}}(H)$ may decrease if the space $E$ is enlarged. More exactly, if $H\subset E\subset F$ are as above, then it may happen that $\dh(\wscl H,F)<\dh(\wscl H,E)$ and similarly for the other quantities (see examples in Section \[examples\]).
Since we are interested in James’ compactness theorem, the most important case for us is the case of a closed convex bounded set $H$. Nonetheless, we define the quantities for an arbitrary bounded set and formulate results as general as possible. Anyway, such generalization do not yield really new results in view of the following proposition.
\[prop-nonconvex\] Let $E$ be a Banach space and $H\subset E$ a bounded subset.
- All the above defined quantities have the same value for $H$ and for $\overline{H}$.
- The quantities $\dh(\wscl H,E)$, ${\operatorname{Ja_E}}(H)$ and $\gamma(H)$ have the same value for $H$ and for the weak closure of $H$.
- ${\operatorname{Ja_E}}({\operatorname{co}}H)\le{\operatorname{Ja_E}}(H)$ and $\gamma({\operatorname{co}}H)=\gamma(H)$.
The assertion (i) is obvious. Let us proceed with the assertion (iii). The first inequality is trivial. The second equality is not easy at all, it is proved in [@FHMZ Theorem 13] – see [@cas-alt11 Theorem 3.3] for a different proof.
Finally, let us show (ii). The case of $\gamma(H)$ follows from (i) and (iii). The other cases are trivial.
As for the quantities not covered by this proposition it seems not to be clear whether ${\operatorname{ck_E}}(H)$ has the same value for $H$ and for the weak closure of $H$. The quantities ${\operatorname{ck_E}}(H)$ and $\dh(\wscl H,E)$ may increase when passing to ${\operatorname{co}}H$: this follows from results of [@sua] and [@sua-alt], see Example \[exa-ghm\].
We do not know whether the quantity ${\operatorname{Ja_E}}(H)$ may really decrease when passing to ${\operatorname{co}}H$. This question seems not to be easy. Indeed, in view of the obvious inequalities ${\operatorname{Ja_E}}({\operatorname{co}}H)\le{\operatorname{Ja_E}}(H)$, ${\operatorname{ck_E}}(H)\le{\operatorname{ck_E}}({\operatorname{co}}H)$ and taking into account ${\operatorname{Ja_E}}(H)\le {\operatorname{ck_E}}(H)$ (see Proposition \[int-ineq-known\] below), if we had ${\operatorname{Ja_E}}({\operatorname{co}}H)<{\operatorname{Ja_E}}(H)$ then we would conclude that ${\operatorname{Ja_E}}({\operatorname{co}}H)<{\operatorname{ck_E}}({\operatorname{co}}H)$. The only example of a convex set $C$ satisfying ${\operatorname{Ja_E}}(C)<{\operatorname{ck_E}}(C)$ known to us is given in Example \[exa-ghm\] below and it seems that it cannot be easily improved.
As for the quantities $\dh(\wscl H,H)$, ${\operatorname{ck}}(H)$ and ${\operatorname{Ja}}(H)$ – they are natural in case of a convex set $H$. If $H$ is not convex, they are not measures of weak non-compactness in the above sense since they may be strictly positive even if $H$ is relatively weakly compact. This is witnessed by Example \[exa-nonconvex\] below.
The following proposition sums up the easy inequalities.
\[int-ineq-known\] Let $E$ be a Banach space.
- Let $H\subset E$ be a bounded set. Then the following inequalities hold true: $${\operatorname{Ja_E}}(H)\le{\operatorname{ck_E}}(H)\le\dh(\wscl H,E)\le\gamma(H).$$
- Let $C\subset E$ be a convex bounded set. Then the following inequalities hold true: $$\begin{array}{ccccccccc}
&&{\operatorname{ck}}_E(C)&\le&\dh(\wscl C,E)&&&& \\
&{\rotatebox{45}{$\leq$}}&&{\rotatebox{325}{$\leq$}}&&{\rotatebox{325}{$\leq$}}&&& \\
{\operatorname{Ja_E}}(C)& \le&{\operatorname{Ja}}(C)&\le&{\operatorname{ck}}(C)&\le & \dh(\wscl C,C)&\le& \gamma(C). \end{array}$$
Let us start by the first part. The inequality ${\operatorname{ck_E}}(H)\le\dh(\wscl H,E)$ is trivial. The inequality $\dh(\wscl{H},E)\le \gamma(H)$ is proved in [@FHMZ Proposition 8(ii)], see also [@cas-alt11 Corollary 4.3]. Let us show that ${\operatorname{Ja_E}}(H)\le{\operatorname{ck_E}}(H)$.
Note first that if ${\operatorname{Ja_E}}(H)=0$ then inequality $0\leq {\operatorname{ck_E}}(H)$ trivially holds. Assume that $0< {\operatorname{Ja_E}}(H)$ and take an arbitrary $0<\epsilon<{\operatorname{Ja_E}}(H)$. By definition there is $x^*\in E^*$ such that for any $x^{**}\in \wscl H$ with $x^{**}(x^*)=\sup x^*(H)$ we have that $\epsilon<\d(x^{**},E)$. Fix a sequence $(x_n)$ in $H$ satisfying $\sup x^*(H)=\lim_n x^*(x_n)$. Then each weak\* cluster point $x^{**}$ of $(x_n)$ satisfies $x^{**}(x^*)=\sup x^*(H)$, hence $\epsilon\leq \d(\clust_{E^{**}}((x_n)),E)$ and therefore $\epsilon \leq {\operatorname{ck_E}}(H)$. This finishes the proof for ${\operatorname{Ja_E}}(H)\leq {\operatorname{ck_E}}(H)$.
Now let us proceed with the second part. All inequalities are obvious but ${\operatorname{Ja_E}}(C)\leq {\operatorname{ck}}_E(C)$, ${\operatorname{Ja}}(C)\leq {\operatorname{ck}}(C)$ and $\dh(\wscl{C},C)\le \gamma(C)$. The first one follows from the first part. The second one can be proved in the same way.
Now we prove that $\dh(\wscl{C},C)\le \gamma(C)$. Suppose that $r>0$ is such that $\dh(\wscl C ,C)>r$. Fix $x^{**}\in \wscl C$ such that $\dist(x^{**},C)>r$. By the Hahn-Banach separation theorem there is $x^{***}\in X^{***}$ with $\|x^{***}\|=1$ and $s\in\er$ such that $$\label{1}
x^{***}(x^{**})>s+r>s>\sup_{x\in C} x^{***}(x).$$ We will construct by induction two sequences $(x_n)$ in $C$ and $(x^*_n)$ in $B_{E^*}$ such that the following conditions are satisfied for each $n\in\en$:
- $x^{**}(x^*_n)>s+r$,
- $x^*_n(x_m) < s$ for $m<n$,
- $x^*_m(x_n) > s+r$ for $m\le n$.
By (\[1\]) and the Goldstine theorem we can choose $x^*_1$ satisfying (i). Now suppose that $n\in\en$ is such that $x^*_m$ for $m\le n$ and $x_m$ for $m<n$ satisfy (i)–(iii). Using that (i) holds for $x^*_1,\dots,x^*_n$ and that $x^{**}\in \wscl C$, we can choose $x_n\in C$ satisfying (iii). Further, by (\[1\]) and the Goldstine theorem we can find $x^*_{n+1}\in B_{E^*}$ satisfying (i) and (ii). This completes the construction.
By passing to subsequences we may assume that $\lim_n x^*_n(x_m)$ exists for all $m\in\en$ and that $\lim_m x^*_n(x_m)$ exists for all $n\in \en$ and (ii) and (iii) are satisfied. By taking further subsequences we may assume also that the limits $\lim_n\lim_mx^*_n(x_m)$ and $\lim_m\lim_nx^*_n(x_m)$ exist and that again and (ii) and (iii) are satisfied. By the construction we get $$\lim_n\lim_mx^*_n(x_m)\ge s+r\mbox{\quad and\quad }\lim_m\lim_nx^*_n(x_m)\le s,$$ hence $\gamma(C)\ge r$. This completes the proof.
We note that in the second part of the proposition above we only have to use the convexity of $C$ to prove the inequality $\dh(\wscl{C},C)\le \gamma(C)$; the rest of the inequalities hold for an arbitrary bounded set. But for non-convex sets only the first part is interesting. This is witnessed by the following example which shows in particular the failure of the inequality $\dh(\wscl{C},C)\le \gamma(C)$ if $C$ is not convex.
\[exa-nonconvex\] Let $E=c_0$ or $E=\ell_p$ for some $p\in(1,\infty)$. Let $H=\{e_n:n\in\en\}$, where $e_n$ is the canonical $n$-th basic vector. Then $H$ is relatively weakly compact, hence $\dh(\wscl H,E)={\operatorname{ck_E}}(H)={\operatorname{Ja_E}}(H)=\gamma(H)=0$. However, ${\operatorname{Ja}}(H)={\operatorname{ck}}(H)=\dh(\wscl H,H)=1$.
As the sequence $(e_n)$ weakly converges to $0$, $H$ is relatively weakly compact. This finishes the proof of the first part. Moreover, $\wscl H$ is in fact the weak closure of $H$ in $E$ and is equal to $H\cup\{0\}$. Thus clearly $\dh(\wscl H,H)={\operatorname{ck}}(H)=1$. Finally, to show ${\operatorname{Ja}}(H)\ge 1$, consider $x^*\in E^*$ represented by the sequence $(-\frac1{2^n})_{n=1}^\infty$ in the respective sequence space. Then $\sup x^*(H)=0$ and the only point in $\wscl H$ at which the supremum is attained is $0$. The observation that $d(0,H)=1$ completes the proof.
We remark that, for non-convex $H$, it is more natural to consider the quantity $\dh(\wscl H,{\operatorname{co}}H)$ instead of using $\dh(\wscl H,H)$ (cf. Section \[S-krein\]). Similar versions of other quantities can be studied as well.
Quantitative versions of James’ theorem {#S-pryce}
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This section is devoted to the proof of the main results of this paper. In the course of the proof we use a proof of James’ compactness theorem due to J.D. Pryce in [@pryce].
\[prop-pryce\] Let $E$ be a Banach space and $H\subset E$ a bounded subset. Then $$\frac12\gamma(H)\leq {\operatorname{Ja_E}}(H).$$
Assume that $\gamma(H)>r$ for some $r>0$. We denote by:
- $F$ the space of all [norm]{} continuous positive homogenous real-valued functions on $E$, [[i.e.]{} continuous functions $f:E\to \reals$ satisfying $f(\alpha x)=\alpha f(x)$, $\alpha \geq 0$ and $x\in E$.]{}
- $p(f)= \sup f(H)$, $f\in F$,
- $P(f)=\sup |f|(H)$, $f\in F$.
Then $p$ is a sublinear functional and $P$ is a [seminorm]{} on $F$.
Let $(f_i)\subset B_{E^*}$ and $(z_j)\subset H$ be [sequences]{} such that $$\lim_i\lim_j f_i(z_j)-\lim_j\lim_i f_i(z_j)>r$$ and all the limits involved exist. By omitting finitely many elements of $(f_i)$ we may assume that $$\label{eq:ToBeUsedLemma15.3}
\lim_j f_i(z_j)-\lim_j\lim_i f_i(z_j)>r,\quad i\in \en.$$ Hence for every $i\in \en$ there exists $j_0\in\en$ such that $$f_i(z_j)-\lim_i f_i(z_j)>r, \quad j\geq j_0.$$
Let $X$ stand for the linear span of $\{f_i:i\in\en\}$. As $X$ is separable in the seminorm $P$ and the functionals $f_i$ are equicontinuous for the norm on $E$, it follows from [@pryce Lemma 2] that we can suppose without loss of generality that $$\label{pecko}
p(f-\liminf_i f_i)=p(f-\limsup_i f_i)\quad \text{ for all }f\in X.$$
We denote $$K_n=\conv\{f_i: i\geq n\},\quad n\in\en,$$ and thus [we]{} obtain $$F\supset E^*\supset X\supset K_1\supset K_2\supset \cdots .$$
By the proof of [@pryce Lemma 3] [and bearing in mind the inequality (\[eq:ToBeUsedLemma15.3\])]{}, we obtain $$\label{kjedna}
p(f-\liminf_i f_i)>r,\quad f\in K_1.$$
Next we quote [@pryce Lemma 4].
\[cl2\] Let $Y$ be a linear space, $\rho,\beta,\beta'$ be strictly positive numbers, $p$ be a sublinear functional on $Y$, $A\subset Y$ be a convex set and $u\in Y$ satisfy $$\inf_{a\in A} p(u+\beta a)>\beta \rho +p(u).$$ Then there exists $a_0\in A$ such that $$\inf_{a\in A} p(u+\beta a_0+\beta' a)>\beta' \rho + p(u+\beta a_0).$$
This claim will be used to prove the following one which is a mild strengthening of [@pryce Lemma 5]. Let us fix $r'\in(0,r)$ arbitrary.
\[cl3\] Let $(\beta_n)$ be a sequence of strictly positive numbers. Then there exists a sequence $(g_n)$ in $F$ such that $g_n\in K_n$ for $n\in\en$ and $$\label{nerp}
p\left(\sum_{i=1}^n \beta_i(g_i-\liminf_j f_j)\right)>\beta_n r'+p\left(\sum_{i=1}^{n-1} \beta_i(g_i-\liminf_j f_j)\right), \quad n\in\en.$$
The construction proceeds by induction. Let $f_0=\liminf_j f_j$.
If $n=1$, we use Claim \[cl2\] for $u=0$, $\beta=\beta_1$, $\beta'=\beta_2$, $\rho=r'$, and $A=K_1-f_0$. By , $$\aligned
\inf_{g\in A} p(u+\beta g)&=\inf_{g\in A} \beta p(g)=\beta_1 \inf_{f\in K_1} p(f-\liminf_j f_j)\\
&>\beta_1 r'=\beta_1 r'+p(u),
\endaligned$$ and hence Claim \[cl2\] gives the existence of $g_1\in K_1$ satisfying $$\inf_{f\in K_1} p(\beta_1(g_1-f_0)+\beta_2 (f-f_0))>\beta_2 r'+
p(\beta_1(g_1-f_0)).$$ This finishes the first step of the construction.
Assume now that we have found $g_i\in K_i$, $i=1,\dots, n-1$, for some $n\in\en$, $n\ge2$, such that $$\inf_{f\in K_{n-1}-f_0} p\left(\sum_{i=1}^{n-1} \beta_i(g_i-f_0)+\beta_n f\right)>\beta_n r'+p\left(\sum_{i=1}^{n-1} \beta_i(g_i-f_0)\right).$$ We use Claim \[cl2\] with $u=\sum_{i=1}^{n-1}\beta_i(g_i-f_0)$, $\beta=\beta_n$, $\beta'=\beta_{n+1}$, $\rho=r'$, and $A=K_n-f_0$. Since $K_n\subset K_{n-1}$, inductive hypothesis gives $$\inf_{f\in A} p(u+\beta f)\geq \inf_{f\in K_{n-1}-f_0} p(u+\beta f)>\beta_n r'+p(u).$$ By Claim \[cl2\], there exists $g_n\in K_n$ such that $$\inf_{f\in A} p\left(\sum_{i=1}^n \beta_i(g_i-f_0)+\beta_{n+1} f\right)>\beta_{n+1} r'+p\left(\sum_{i=1}^{n-1} \beta_i(g_i-f_0)+\beta_n (g_n-f_0)\right).$$ This completes the inductive construction.
We have obtained elements $g_n\in K_n$, $n\in\en$, such that $$\inf_{g\in K_n} p\left(\sum_{i=1}^n \beta_i(g_i-f_0)+\beta_{n+1}(g-f_0)\right)>\beta_{n+1}r'+p\left(\sum_{i=1}^{n} \beta_i(g_i-f_0)\right).$$ Since $g_{n+1}\in K_{n+1}\subset K_n$, this yields $$p\left(\sum_{i=1}^n \beta_i(g_i-f_0)+\beta_{n+1}(g_{n+1}-f_0)\right)>\beta_{n+1}r'+p\left(\sum_{i=1}^{n} \beta_i(g_i-f_0)\right).$$ This finishes the proof.
Let $\beta_i>0$, $i\in\en$, be chosen in such a way that $\lim_{n} \frac{1}{\beta_{n}}\sum_{i=n+1}^\infty \beta_i=0$. Let $(g_n)$ be a sequence provided by Claim \[cl3\]. Since [for every $n\in\en$ we have that $g_n\in K_n\subset B_{E^*}$]{}, we can select a weak$^*$-cluster point $g_0\in B_{E^*}$ of $(g_n)$. By [@pryce Lemma 6], we have the following observation.
\[cl4\] For any $f\in X$, $p(f-g_0)=p(f-\liminf_n f_n)$.
By Claim \[cl4\], we can replace $\liminf_j f_j$ by $g_0$ in and get the following inequalities $$\label{nerg}
p\left(\sum_{i=1}^n \beta_i(g_i-g_0)\right)>\beta_n r'+p\left(\sum_{i=1}^{n-1} \beta_i(g_i-g_0)\right), \quad n\in\en.$$
We set $M=\sup\{\|x\|: x\in H\}$ and remark that $\|g_i-g_0\|\leq 2$, $i\in\en$.
We set $g=\sum_{i=1}^\infty \beta_i(g_i-g_0)$. Let $u\in \wscl{H}$ be an arbitrary point satisfying $g(u)=\sup g(H)$. Then, for any $n\in\en$, we get from $$\aligned
\sum_{i=1}^n\beta_i(g_i-g_0)(u)&=g(u)-\sum_{i=n+1}^\infty \beta_i(g_i-g_0){(u)}
\geq p(g)- 2M\sum_{i=n+1}^\infty\beta_i \\
&\geq p\left(\sum_{i=1}^n \beta_i(g_i-g_0)\right)-p\left(\sum_{i=1}^n \beta_i(g_i-g_0)-g\right)-2M\sum_{i=n+1}^\infty \beta_i\\
&\geq p\left(\sum_{i=1}^n \beta_i(g_i-g_0)\right)-4M \sum_{i=n+1}^\infty \beta_i\\
&>\beta_n r'+p\left(\sum_{i=1}^{n-1} \beta_i(g_i-g_0)\right)-4M \sum_{i=n+1}^\infty \beta_i\\
&\geq \beta_n r'+\sum_{i=1}^{n-1} \beta_i(g_i-g_0)(u)-4M \sum_{i=n+1}^\infty \beta_i.
\endaligned$$ Hence $$(g_n-g_0)(u)\geq r'-4M \frac{1}{\beta_{n}}\sum_{i=n+1}^\infty \beta_i,\quad n\in\en,$$ which gives $$\label{limi}
\liminf_n (g_n-g_0)(u)\geq r'.$$
[Let $v\in E$ be arbitrary.]{} Then $g_0(v)\geq \liminf_n g_n(v)$, which along with gives $$\aligned
r'&\leq \liminf_n g_n(u)-\liminf_n g_n(v)+g_0(v-u)\\
&\leq -\liminf_n (g_n(v)-g_n(u))+g_0(v-u)\\
&\leq 2\|v-u\|.
\endaligned$$ By the definition of ${\operatorname{Ja_E}}(H)$ it follows ${\operatorname{Ja_E}}(H)\geq \frac12 r'$. Since $r$ satisfying $\gamma(H)>r$ and $r'\in(0,r)$ are arbitrary we conclude that ${\operatorname{Ja_E}}(H)\geq \frac12 \gamma(H)$.
As a consequence of Theorem \[prop-pryce\] we obtain that all measures of non-compactness that we have considered in this paper are equivalent. In other words, [*all classical approaches used to study weak compactness in Banach spaces (Tychonoff’s theorem, Eberlein’s theorem, Grothendieck’s theorem and James’ theorem) are qualitatively and quantitatively equivalent.*]{}
\[cor:AllPossibleInequalities\] Let $E$ be a Banach space.
- Let $H\subset E$ be a bounded set. Then the following inequalities hold true: $$\frac12\gamma(H)\le{\operatorname{Ja_E}}(H)\le{\operatorname{ck_E}}(H)\le\dh(\wscl H,E)\le\gamma(H).$$
- Let $C\subset E$ be a bounded convex set. Then the following inequalities hold true: $$\begin{array}{ccccccccccc}
&&&&{\operatorname{ck}}_E(C)&\le&\dh(\wscl C,E)&&&& \\
&&&{\rotatebox{45}{$\leq$}}&&{\rotatebox{325}{$\leq$}}&&{\rotatebox{325}{$\leq$}}&&& \\
\frac12\gamma(C)&\le&{\operatorname{Ja_E}}(C)& \le&{\operatorname{Ja}}(C)&\le&{\operatorname{ck}}(C)&\le & \dh(\wscl C,C)&\le& \gamma(C). \end{array}$$
This result follows from Proposition \[int-ineq-known\] and Theorem \[prop-pryce\].
The fact that the measures of weak non-compactness $H\mapsto \dh(\wscl H,E)$, $\gamma$ and ${\operatorname{ck_E}}$ are equivalent can be found in [@FHMZ] and [@ang-cas1] with very different approaches.
In Section \[examples\] we offer several examples showing that in the corollary above any of the inequalities may become equality and that most of them may become strict.
Let $E$ be a Banach space and $C\subset E$ be a closed convex bounded subset. Then $C$ is weakly compact provided ${\operatorname{Ja_E}}(C)=0$ (*i.e.*, if for every $\epsilon >0$ and every $x^*\in X^*$ there is $x^{**}\in \overline{C}^{w^*}$ such that $x^{**}(x^*)=\sup x^*(C)$ and $d(x^{**},E)\leq \epsilon$).
Relationship to the quantitative version of Krein’s theorem {#S-krein}
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Let $E$ be a Banach space and $C\subset E$ be a bounded convex set. Then $\ext \wscl C$, the set of extreme points of $\wscl C$, is a boundary for $\wscl C$. Therefore the following inequalities are obvious:
$$\label{eq-extreme}
\begin{gathered}
\dh(\wscl{\ext \wscl C},C)\ge \dh(\ext\wscl C,C)\ge{\operatorname{Ja}}(C),\\
\dh(\wscl{\ext \wscl C},E)\ge \dh(\ext\wscl C,E)\ge{\operatorname{Ja_E}}(C).
\end{gathered}$$
These inequalities enable us to prove the following statement.
\[cor-Krein\] Let $E$ be a Banach space and $H\subset E$ be a bounded set. Then the following inequalities hold:
- $\dh (\wscl{{\operatorname{co}}H},E)\le 2 \dh(\wscl H,E)$,
- $\dh (\wscl{{\operatorname{co}}H},{{\operatorname{co}}H})\le 2 \dh(\wscl H,{{\operatorname{co}}H})$.
Set $C=\overline{{\operatorname{co}}H}$. Then $\ext \wscl C\subset \wscl H$, so the inequalities follow from (\[eq-extreme\]) and Corollary \[cor:AllPossibleInequalities\].
We remark that the assertion (i) was proved in [@FHMZ] and independently in [@sua] and [@cas-alt11]. In [@sua] and [@sua-alt] some examples are given which show that the inequality is optimal, i.e. the equality can take place if the quantities are non-zero. However, these examples do not work for the assertion (ii). Hence, the following problem seems to be natural.
Let $E$ be a Banach space and $H\subset E$ a bounded set. Is it true that $$\dh (\wscl{{\operatorname{co}}H},{{\operatorname{co}}H}) = \dh(\wscl H,{{\operatorname{co}}H}) \quad ?$$
The assertion (i) of Corollary \[cor-Krein\] is called in [@FHMZ] a quantitative version of Krein’s theorem. Krein’s theorem asserts that a closed convex hull of a weakly compact set is again weakly compact. This is the case when the quantities are $0$. In view of this also the assertion (ii) may be called a quantitative version of Krein’s theorem. An interesting phenomenon is that there are examples showing that the inequality (i) is sharp but we do not know whether the inequality (ii) is sharp. Both examples showing sharpness of (i) are of similar nature: A set $H$ is constructed in a space $E_0$ such that $\dh (\wscl{{\operatorname{co}}H},{\operatorname{co}}H) = \dh(\wscl H,{{\operatorname{co}}H})=1$. Then the space $E_0$ is enlarged in a clever way to $E$ such that $\dh(\wscl{{\operatorname{co}}H},E)$ equals $1$ but $\dh(\wscl H,E)$ decreases to $\frac12$. If the space $E$ is enlarged even more, also the quantity $\dh(\wscl{{\operatorname{co}}H},E)$ will decrease to $\frac12$ and it will be no more a counterexample. Hence, a possible counterexample showing sharpness of (ii) should be of a quite different nature.
Moreover, one can show (although it is not obvious) that the answer to the above question is positive if $H$ is norm-separable. This is another indication of a great difference between (i) and (ii) as the example from [@sua-alt] is norm-separable (see Example \[exa-ghm\] below).
Examples
========
In this section we collect examples showing the sharpness of some of the inequalities that are collected in Corollary \[cor:AllPossibleInequalities\]. We remark that unless all the quantities are zero, at least one of the inequalities must be strict. We stress again that the examples in this section show in particular that any of the inequalities may become equality and that most of them may become strict.
\[exa-c0\] Let $E=c_0$ and $C=B_E$. Then $\gamma(C)=1$ and ${\operatorname{Ja_E}}(C)=1$. Hence all other quantities are also equal to $1$.
The equality $\gamma(C)=1$ follows from [@kr-pr-sc Example 2.7 and Theorem 2.8]. To show that ${\operatorname{Ja_E}}(C)\ge1$ take $x^*\in E^*$ represented by the sequence $(\frac1{2^n})_{n=1}^\infty$ in $\ell_1$. The only element of $\wscl C=B_{\ell_\infty}$ at which $x^*$ attains its supremum on $C$ is the constant sequence $(1)_{n=1}^\infty$ whose distance from $E$ is clearly $1$. The rest now follows from Corollary \[cor:AllPossibleInequalities\].
\[exa-ell1\] Let $E=\ell_1$ and $C=B_E$. Then $\gamma(C)=2$ and $\dh(\wscl C,C)=1$. Hence all other quantities are equal to $1$.
It is clear that $\dh(\wscl C,C)\le 1$. Further, the inequality $\gamma(C)\ge 2$ is witnessed by sequences $(x_n)$ and $(x^*_n)$, where $x_n$ is the $n$-th canonical basic vector of $\ell_1$ and $x^*_n\in B_{\ell_\infty}$ is defined by $$x^*_n(m)=\begin{cases} 1 & m\le n,\\ -1 & m>n.\end{cases}$$ The rest follows from Corollary \[cor:AllPossibleInequalities\].
\[exa-c0omega\] Let $E=C([0,\omega])$ and $C=\{x\in E: 0\le x\le 1\ \&\ x(\omega)=0\}$. Then $\dh(\wscl C,E)=\frac12$ and ${\operatorname{Ja}}(C)=1$. Hence ${\operatorname{Ja}}_E(C)={\operatorname{ck}}_E(C)=\frac12$ and ${\operatorname{ck}}(C)=\dh(\wscl C,C)=\gamma(C)=1$.
Note that $E^*$ is canonically identified with $\ell_1([0,\omega])$ and $E^{**}$ with $\ell_\infty([0,\omega])$.
To show that $\dh(\wscl C,E)\le\frac12$ we observe that the constant function $\frac12$ belongs to $E$ and that $C\subset \frac12+\frac12 B_E$. Thus $\wscl C\subset \frac12+\frac12 B_{E^{**}}$.
Further, consider the element $x^*\in E^*=\ell_1([0,\omega])$ given by $x^*(n)=\frac1{2^n}$ for $n<\omega$ and $x^*(\omega)=0$. Then the only element of $\wscl C$ at which $x^*$ attains its supremum on $C$ is $\chi_{[0,\omega)}$. Its distance to $C$ of this element is clearly equal to $1$. Thus ${\operatorname{Ja}}(C)\ge 1$.
The rest follows from Corollary \[cor:AllPossibleInequalities\].
\[exa-c00omega1\] Let $E=C_0([0,\omega_1))$ and $C=\{x\in E: 0\le x\le 1\}$. Then $\dh(\wscl C,E)=1$ and ${\operatorname{ck}}(C)=\frac12$. Hence ${\operatorname{Ja}}_E(C)={\operatorname{Ja}}(C)={\operatorname{ck}}_E(C)=\frac12$ and $\dh(\wscl C,C)=\gamma(C)=1$.
First note that the dual $E^*$ can be identified with $\ell_1([0,\omega_1))$ and the second dual $E^{**}$ with $\ell_\infty([0,\omega_1))$.
To show that $\dh(\wscl C,E)\ge1$ we note that the constant function $1$ belongs to $\wscl C$ and its distance to $E$ is $1$.
Next we will show that ${\operatorname{ck}}(C)\le\frac12$. Let $(x_n)$ be any sequence in $C$. There is some $\alpha<\omega_1$ such that $x_n|_{(\alpha,\omega_1)}=0$ for each $n\in\en$. As the interval $[0,\alpha]$ is countable, there is a subsequence $(x_{n_k})$ which converges pointwise on $[0,\omega_1)$. The limit is an element of $\ell_\infty([0,\omega_1))=E^{**}$. Denote the limit by $x^{**}$. Then the sequence $(x_{n_k})$ weak\* converges to $x^{**}$. Thus in particular $x^{**}\in \clust_{E^{**}}((x_n))$. Set $x=\frac12\chi_{[0,\alpha]}$. Then $x\in C$ and $\|x^{**}-x\|\le\frac12$ (as $0\le x^{**}\le 1$ and $x^{**}|_{(\alpha,\omega_1)}=0$). The inequality ${\operatorname{ck}}(C)\le\frac12$ now follows.
The rest follows from Corollary \[cor:AllPossibleInequalities\].
Let $E=C([0,\omega_1])$ and $C=\{x\in E: 0\le x\le 1\ \&\ x(\omega_1)=0\}$. Then $\dh(\wscl C,E)={\operatorname{ck}}(C)=\frac12$ and $\dh(\wscl C,C)=1$. Hence ${\operatorname{Ja}}_E(C)={\operatorname{Ja}}(C)={\operatorname{ck}}_E(C)=\frac12$ and $\gamma(C)=1$.
We start similarly as in Example \[exa-c0omega\]: Note that $E^*$ is canonically identified with $\ell_1([0,\omega_1])$ and $E^{**}$ with $\ell_\infty([0,\omega_1])$.
To show that $\dh(\wscl C,E)\le\frac12$ notice that the constant function $\frac12$ belongs to $E$ and that $C\subset \frac12+\frac12 B_E$. Thus $\wscl C\subset \frac12+\frac12 B_{E^{**}}$.
The inequality ${\operatorname{ck}}(C)\le \frac12$ can be proved in the same way as in Example \[exa-c00omega1\]. In fact, it follows from that example, since $C_0([0,\omega_1))$ is isometric to $\{x\in E: x(\omega_1)=0\}$, and hence our set $C$ coincides with the set $C$ from Example \[exa-c00omega1\].
Finally, $\dh(\wscl C,C)\ge 1$ as $\chi_{[0,\omega_1)}\in\wscl C$ and its distance from $C$ is equal to $1$.
The rest follows from Corollary \[cor:AllPossibleInequalities\].
\[exa-ghm\] There is a Banach space $E$ and a closed convex bounded subset $C\subset E$ such that ${\operatorname{Ja_E}}(C)=\frac12$ and ${\operatorname{ck}}_E(C)={\operatorname{Ja}}(C)=1$. Hence ${\operatorname{ck}}(C)=\dh(\wscl C,E)=\dh(\wscl C,C)=\gamma(C)=1$.
We use the example from [@sua-alt]. It is constructed there a set $K_0\subset[0,1]^{\en}$ and a free ultrafilter $u$ over $\en$ such that (in particular) the following assertions are satisfied:
- $K_0$ consists of finitely supported vectors and is closed in the topology of uniform convergence on $\en$ but not in the pointwise convergence topology.
- For each $x\in \overline{K_0}$ (the closure taken in the pointwise convergence topology) we have $\lim_u x(n) = 0$.
- For each $x\in \overline{K_0}\setminus K_0$ there are infinitely many $n\in\en$ such that $x(n)=1$.
Let $E=\{x\in C(\beta\en) : x(u)=0\}$. We remark that $\beta\en$ is canonically identified with the space of ultrafilters over $\en$ and hence we have $u\in\beta\en$. Let us consider embedding $\kappa:\overline{K_0}\to E$ defined by $$\kappa(x)(p)=\lim_p x(n), \qquad p\in\beta\en,\, x\in\overline{K_0}.$$ By (b) it is a well defined mapping with values in $E$. Let $B=\kappa(K_0)$. Then $B$ is a bounded norm-closed subset of $E$. Set $C=\overline{\conv B}$.
It is proved in [@sua-alt] that $\dh(\wscl B,E)\le\frac12$. As $\wscl B$ contains extreme points of $\wscl C$, by (\[eq-extreme\]) we get ${\operatorname{Ja_E}}(C)\le\frac12$.
In [@sua-alt] it is proved that $\dh(\wscl C,E)\ge1$. We will show that even ${\operatorname{ck}}_E(C)\ge 1$. To do this it is enough to observe that $C\subset \{x\in E: x|_{\beta\en\setminus\en}=0\}$ (this follows from (a)). The latter space is isometric to $c_0$. As $c_0^*$ is separable, each element of $\wscl C$ is a weak\* limit of a sequence from $C$. It follows that ${\operatorname{ck}}_E(C)=\dh(\wscl C,E)\ge1$.
By Corollary \[cor:AllPossibleInequalities\] it remains to prove that ${\operatorname{Ja}}(C)\ge 1$. To do that let us first recall that the dual to $E$ can be canonically identified with the space of all signed Radon measures on $\beta\en\setminus\{u\}$. This space can be decomposed as $$E^*=\ell_1\oplus_1 M(\beta\en\setminus(\en\cup\{u\})).$$ The second dual is then represented as $$E^{**}=\ell_\infty\oplus_\infty M(\beta\en\setminus(\en\cup\{u\}))^*.$$ Denote by $j$ the canonical embedding of $E$ into $E^{**}$ and by $\rho$ the embedding $\rho:\ell_\infty\to E^{**}$ given by $\rho(x)=(x,0)$ using the above representation. Now, $$\rho(\ell_\infty)=\{x^{**}\in E^{**}: x^{**}(\mu)=0\mbox{ whenever $\mu\in M(\beta\en\setminus \{u\})$ is such that }\mu|_{\en}=0\}.$$ So, $\rho(\ell_\infty)$ is weak\* closed and, moreover, $\rho$ is weak\* to weak\* homeomorphism ($\ell_\infty$ being considered as the dual to $\ell_1$).
Finally, $\rho|_{K_0}=(j\circ\kappa)|_{K_0}$ and hence $\wscl B=\rho\left(\overline{K_0}\right)$. Fix some $x\in\overline{K_0}\setminus K_0$ and let $A\subset\en$ be infinite such that $x|_A=1$. Such a set $A$ exists due to (c). Enumerate $A=\{a_n:n\in\en\}$ and define an element $u\in\ell_1$ by $$u(k)=\begin{cases} \frac1{2^{n+1}}, & k = a_n,\\ 0, & k\in\en\setminus A.\end{cases}$$ Further define the element $x^{*}\in E^*$ by $x^*=(u,0)$ (using the above representation). Then $\|x^*\|=1$, so $\sup x^*(C)\le 1$. Moreover, $\rho(x)(x^*)=1$, hence $\sup x^*(C)=1$. Let $x^{**}\in\wscl C$ be such that $x^{**}(x^*)=1$. Then $x^{**}=(\rho(y),0)$ for some $y\in\ell_\infty$. As $\|y\|\le 1$, we get $y|_A=1$. But then $d(y,c_0)=1$, hence $d(x^{**},C)\ge 1$. So, ${\operatorname{Ja}}(C)\ge 1$ and the proof is completed.
The above examples show that any of the inequalities from Corollary \[cor:AllPossibleInequalities\] can be strict, with one possible exception which is described in the following problem.
Let $E$ be a Banach space and $C\subset E$ a bounded convex set. Is then ${\operatorname{Ja}}(C)={\operatorname{ck}}(C)$?
The case of weak\* angelic dual unit ball
=========================================
In this section we collect several results saying that under some additional conditions some of the inequalities from Corollary \[cor:AllPossibleInequalities\] become equalities. The basic assumption will be that the dual unit ball $B_{E^*}$ is weak\* angelic, i.e. that whenever $A\subset B_{E^*}$ and $x^*\in\wscl A$, there is a sequence in $A$ which weak\* converges to $x^*$. Inspired by [@FHMZ] we introduce the following quantity. If $E$ is a Banach space and $H\subset E$ a bounded subset, we set $$\gamma_0(H)=\sup\{|\lim_i \lim_j x^*_i(x_j)| : (x_j)\subset H, (x^*_i)\subset B_{E^*}, x^*_i \overset{w^*}{\to} 0\},$$ assuming the involved limits exist. It is clear that $\gamma_0(H)\le \gamma(H)$. In general $\gamma_0$ is not an equivalent quantity to the other ones. Indeed, if $E=\ell_\infty$ and $C=B_E$, then $\gamma_0(C)=0$ by the Grothendieck property of $E$. But in case $B_{E^*}$ is angelic, we have the following:
\[thm-angelic\] Let $E$ be a Banach space such that $B_{E^*}$ is weak\* angelic.
- Let $H\subset E$ be any bounded subset Then we have: $$\frac12\gamma(H)\le\gamma_0(H)={\operatorname{Ja_E}}(H)={\operatorname{ck_E}}(H)=\dh(\wscl H,E)\le\gamma(H).$$
- Let $C\subset E$ be any bounded convex subset. Then the following inequalities hold true: $$\begin{gathered}
\frac12\gamma(C)\le\gamma_0(C)={\operatorname{Ja_E}}(C)={\operatorname{ck}}_E(C)=\dh(\wscl C,E) \\ \le {\operatorname{Ja}}(C)\le{\operatorname{ck}}(C)\le\dh(\wscl C,C)\le\gamma(C).\end{gathered}$$
The second part follows from the first part and Corollary \[cor:AllPossibleInequalities\]. As for the first part, in view of Corollary \[cor:AllPossibleInequalities\] it is enough to prove that ${\operatorname{Ja_E}}(H)\ge \gamma_0(H)$ and $\dh(\wscl H,E)\le\gamma_0(H)$. The second inequality follows from [@FHMZ Proposition 14(ii)].
The first inequality follows from the proof of Theorem \[prop-pryce\]. In fact, the angelicity assumption is not needed here. Let us indicate the necessary changes:
Suppose that $\gamma_0(H)>r$. The space $F$ is not needed, but define the sublinear functional $p$ on $E^*$ by $p(f)=\sup f(H)$ for $f\in E^*$. Fix a sequence $(z_j)$ in $H$ and $(f_i)$ in $B_{E^*}$ such that $f_i$ weak\* converge to $0$ and $\lim_i\lim_j f_i(z_j)>r$ and all the limits involved exist. Without loss of generality suppose that:
[ *for every $i\in \en$,there is $j_0\in\en$, such that for all $j\ge j_0$ we have $f_i(z_j)>r.$*]{}
As $\limsup_i f_i=\liminf_i f_i=0$, we get the assertion (\[pecko\]) for free. We define $K_n$ for $n\in\en$ in the same way. The assertion (\[kjedna\]) then says that $p(f)>r$ for all $f\in K_1$. Fix any $r'<r$ and a sequence $(\beta_n)$ of strictly positive numbers. Claim \[cl3\] now yield a sequence $(g_n)$ with $g_n\in K_n$ such that $$p\left(\sum_{i=1}^n \beta_i g_i\right)>\beta_n r'+p\left(\sum_{i=1}^{n-1} \beta_i g_i\right).$$ As $f_n$ weak\* converge to $0$, $g_n$ weak\* converge to $0$ as well. Thus $g_0=0$. Now, if the sequence $(\beta_n)$ quickly converges to $0$ (i.e., satisfies the same condition as in the original proof), we set $g=\sum_{i=1}^\infty \beta_i g_i$. Let $u\in\wscl H$ be an arbitrary point with $g(u)=\sup g(H)$. By the final calculation we get $\liminf_n g_n(u)\ge r'$. If $v\in E$ is arbitrary, then $g_n(v)\to 0$, and thus $$r'\le \liminf g_n(u)-\lim g_n(v)=\liminf g_n(u-v)\le \|u-v\|.$$ Thus ${\operatorname{Ja_E}}(H)\ge r'$, so ${\operatorname{Ja_E}}(H)\ge\gamma_0(H)$.
Let us remark that the spaces from Examples \[exa-c0\], \[exa-ell1\] and \[exa-c0omega\] are separable and therefore they have weak\* angelic unit ball. It follows that in Theorem \[thm-angelic\] all the inequalities, with a possible exception of ${\operatorname{Ja}}(C)\le{\operatorname{ck}}(C)$, may be strict. [Note also that under the weaker assumption of the Banach space $E$ having Corson property $\cC$, it has been proved in [@ang-cas1 Proposition 2.6] that for any bounded set $H\subset E$ we have ${\operatorname{ck}}_E(H)=\dh(\wscl H,E)$.]{}
The following theorem shows that all the quantities are equal in a very special case $E=c_0(\Gamma)$.
Let $\Gamma$ be an arbitrary set and $E=c_0(\Gamma)$.
- Let $H\subset E$ be a bounded set. Then we have: $$\gamma_0(H)={\operatorname{Ja_E}}(H)={\operatorname{ck_E}}(H)=\dh(\wscl H,E)=\gamma(H).$$
- Let $C\subset E$ be a convex bounded subset. Then we have: $$\gamma_0(C)={\operatorname{Ja_E}}(C)={\operatorname{ck}}_E(C)=\dh(\wscl C,E) = {\operatorname{Ja}}(C)={\operatorname{ck}}(C)=\dh(\wscl C,C)=\gamma(C).$$
It is enough to prove $\gamma_0(H)\ge \gamma(H)$. If $\gamma(H)=0$, this inequality is trivial. So, suppose that $\gamma(H)>0$. Fix an arbitrary $r>0$ such that $\gamma(H)>0$. We find sequences $(x_i)\subset H$, $(x^*_j)\subset B_{E^*}$ and $\eta>0$ such that $$\lim_i \lim_j x_i^*(x_j)-\lim_j \lim_i x_i^* (x_j) > r(1+\eta),$$ where all the limits involved exist. As $B_{E^*}$ is weak\* sequentially compact, by passing to a subsequence we may suppose that the sequence $(x_i^*)$ weak\* converges to some $x^*\in B_{E^*}$. Then $$\lim_i \lim_j (x_i^*-x^*)(x_j) > r(1+\eta).$$ We claim that $$\limsup \|x_i^*-x^*\|\le 1.$$ Suppose not. Then, up to passing to a subsequence, we may suppose that there is $\delta>0$ such that $\|x_i^*-x^*\|\ge 1+\delta$ for each $i\in\en$. To proceed the proof we recall that $E^*$ is canonically identified with $\ell_1(\Gamma)$ an that the weak\* topology on bounded sets coincides with the pointwise convergence topology. Using this identification we can find a finite set $F\subset \Gamma$ such that $$\sum_{\gamma\in\Gamma\setminus F}|x^*(\gamma)|<\frac\delta3.$$ Further, as $x_i^*$ weak\* converges to $x^*$, there is $i_0\in\en$ such that for each $i\ge i_0$ we have $$\sum_{\gamma\in F}|x_i^*(\gamma)-x^*(\gamma)|<\frac\delta3.$$ Fix any $i\ge i_0$. Then we have: $$\begin{aligned}
\|x_i^*\|&\ge \sum_{\gamma\in\Gamma\setminus F}|x_i^*(\gamma)|
\ge \sum_{\gamma\in\Gamma\setminus F}|x_i^*(\gamma)-x^*(\gamma)|
-\sum_{\gamma\in\Gamma\setminus F}|x^*(\gamma)|
\\ & =\|x_i^*-x^*\|-\sum_{\gamma\in F}|x_i^*(\gamma)-x^*(\gamma)|
-\sum_{\gamma\in\Gamma\setminus F}|x^*(\gamma)|
\\ & >1+\delta-\frac\delta3-\frac\delta3=1+\frac\delta3.\end{aligned}$$ This is a contradiction.
So, omitting finite number of elements, we can suppose that $\|x_i^*-x^*\|<1+\eta$ for all $i\in\en$. Set $y_i^*=\frac{x_i^*-x^*}{1+\eta}$. Then $y_i^*\in B_{E^*}$, the sequence $(y_i^*)$ weak\* converges to $0$ and $$\lim_i \lim_j y_i^*(x_i)>r.$$ Thus $\gamma_0(H)\ge r$ and the proof is completed.
The equalities ${\operatorname{ck}}_E(H)=\dh(\wscl H,E) =\gamma(H)$ in case $E=c_0$ and $H\subset E$ is a bounded subset follow also easily from [@kr-pr-sc Theorem 2.8], see also [@ang-tesis Corollary 3.4.3].
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[^1]: The research of B. Cascales was supported by FEDER and MEC Project MTM2008-05396 and by Fundación Séneca (CARM), project 08848/PI/08. The research of O. Kalenda and J. Spurný is supported by the project MSM 0021620839 financed by MSMT and partly supported by the research grant GAAV IAA 100190901.
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abstract: 'This paper gives a characterization of the group $G_2(K)$ over some [algebraically closed field]{} $K$ of characteristic not 2 inside the class of simple $K^*$-groups of [finite Morley rank]{} not interpreting a bad field using the structure of centralizers of involutions. This implies a general characterization of tame $K^*$-groups of odd type whose centralizers have a certain natural structure.'
author:
- |
[**Christine Altseimer**]{}\
\
Institut für Mathematische Logik\
Albert-Ludwigs-Universität Freiburg\
Eckerstr. 1\
79104 Freiburg\
Germany
title: '**A Characterisation of $G_2(K)$**'
---
\[lemma\][Theorem]{} \[lemma\][Corollary]{} \[lemma\][Proposition]{} \[lemma\][Remark]{} \[lemma\][Definition]{} \[lemma\][Fact]{}
Introduction
============
This paper belongs to a series of publications on the classification of of tame simple groups of [finite Morley rank]{} and odd type. The result and the methods used to achieve it are intimatly related to the characterisation of $PSp(4,K)$ given [@chr1]. The motivation for this problem and all necessary definitions can be found there. The underlying conjecture is the following.
\[klitze\] Let $G$ be a simple tame $K^*$-group of odd type and Prüfer 2-rank $2$. Let $i \in G$ be any involution and $C:=C_G(i){^{\circ}}/O(C_G(i))$.
- If $C \cong \GL_2(K)$, then $G \cong \PSL_3(K)$
- If $C \cong \PSL_2(K) \times K^* $, then $G \cong
\PSp_4(K)$.
- If $C \cong \SL_2(K)*\SL_2(K)$, where the two copies of $\SL_2(K)$ intersect non-trivially, then $G \cong \PSp_4(K)$ or $G\cong \G_2(K)$.
Furthermore one of these three cases holds.
We are going to prove cases $(2)$ and $(3)$ of the conjecture using [@chr1]. In [@diss] one can furthermore find a partial result of case $(1)$, if $O(C_G(i))=1$. To prove the complete case $(1)$ seems to be extremely challenging and one will probably need new methods. Especially we prove the following theorem:
\[super\] Let $G$ be a simple $K^*$-group of [finite Morley rank]{} that does not interpret a bad field. Assume that $G$ contains an involution $i$, such that either
- $C_G(i){^{\circ}}/O(C_G(i)) \cong \SL_2(K) * \SL_2(K)$ where the two copies of $\SL(2,K)$ intersect non-trivially or
- $C_G(i){^{\circ}}/O(C_G(i)) \cong \PSL(2,K)\times K^*$
for an [algebraically closed field]{} $K$ of characteristic neither $2$ nor $3$. Then $G \cong
\PSp_4(K)$ or $G \cong \G_2(K)$.
Basic Results
=============
We are first going to show, that case $(ii)$ of Theorem \[super\] implies case $(i)$.
\[restriction\] Let $G$ be a simple $K^*$-group of odd type that does not interpret any bad field. Assume that $pr(G)=2$ and let $D$ be the four-subgroup which is contained in the connected component of a Sylow 2-subgroup $S$ of $G$. Set $O_D:=\bigcap_{l \in D^*} O(C_G(l))$. If $T_0:=C_G(D){^{\circ}}/O_D \cong K^* \times K^*$ for some [algebraically closed field]{} of characteristic not $2$, then $C_G(i){^{\circ}}/O(C_G(i))$ is isomorphic to one of the following groups for any $i \in D^*$.
- $K^* \times K^*$.
- $\GL_2(K)$
- $\PSL_2(K) \times K^* \cong \GL_2(K)/\langle -I \rangle$ where $-I = \left( \begin{smallmatrix} -1 & 0 \\ 0 & -1
\end{smallmatrix}\right)$.
- $\SL_2(K)*\SL_2(K)$ where the two copies of $\SL_2(K)$ intersect non-trivially.
If $C_G(i){^{\circ}}/O(C_G(i))$ is isomorphic to one of the groups in $(iii)$ or $(iv)$, then $O(C_G(l))=1$ for all $l \in D^*$.
Let $i \in D^*$, then $C:=C_G(i){^{\circ}}/O(C_G(i))$ is a central product of an abelian divisible group $T$ and a semisimple group $H$ all of whose components are simple algebraic groups over algebraically closed fields of characteristic different from 2 by [@nato]. Furthermore $pr(C)=2$ by [@chr2] and $pr(H)=pr(C)-pr(T)$ by Lemma [@chr2]. Thus $H$ is the central product of simple algebraic groups of Prüfer 2-rank less than or equal to 2 by [@hum75 27.5].
Set $C_k:=C_G(k)$ and $O_k:=O(C_k)$ for all $k
\in I(G)$. Let $\theta$ be the signalizer functor defined by $\theta(s):=O_s$ for all $s \in I(G)$. Then $O_i \cap C_j = O_j
\cap C_i = O_j \cap O_i = O_D$ for all $j \in D^*$, $j \neq i$. Hence $$C_i \cap C_j / O_D \cong (C_i \cap C_j)O_i/O_i \cong (C_i \cap
C_j)O_j/O_j.$$ Thus by [@mixed 2.52] $(C_i \cap C_j) / O_D \cong C_{C_i/O_i}(jO_i)
\cong C_{C_j/O_j}(iO_j)$. This implies that $T \leq C_{C}(jO_i) \cong C_G(D) / O_D = T_0$ and either $T =1$, $T
\cong K^*$ or $T \cong T_0$, since $T$ is connected and $G$ does not interpret a bad field.
Thus $(i)$ holds, if $pr(T)=2$, as in this case $H=1$ and $T \cong
T_0$. If $pr(T)=1$, then $T \cong K^*$ and $pr(H)=1$. Hence $H$ is of type $A_1(K)$, i.e. $H \cong \SL_2(K)$ or $H \cong \PSL_2(K)$. If $C=H \times T$, then $H \cong \PSL_2(K)$, as $C_{C}(jO_i) \cong
T_0$ for any involution $j \in H$. Thus $C \cong\GL_2(K)/\langle -I \rangle$ in this case. If on the other hand $C = H * T$, where $H \cap T$ is non-trivial, then $H \cong \SL_2(K)$ and $C \cong \GL_2(K)$.
If finally $pr(H)=2$, then $T$ is trivial by [@chr2] and $H$ is of type $A_1(K)
\times A_1(K)$, $A_2(K)$, $C_2(K)$ or $\G_2(K)$. However, $H$ has to contain a central involution. Thus $H$ cannot be of type $A_2(K)$ or $\G_2(K)$ and it cannot be isomorphic to $\PSp_4(K)$. As furthermore $C_{C}(jO_i) \cong T_0$ for any $j \in D\backslash \langle i
\rangle$, $H$ cannot be isomorphic to $\Sp_4(K)$ either. Hence $H$ is of type $A_1(K) \times A_1(K)$, contains the central involution $i$ and $C_{C}(jO_i) \cong T_0$ for any other involution $j$. Especially $C \cong \SL_2(K) * \SL_2(K)$, where the two copies of $\SL_2(K)$ intersect in $\langle i \rangle$.
If $C$ is isomorphic to the group in $(iii)$ or $(iv)$, then $C_G(i){^{\circ}}$ contains an elementary abelian subgroup $E$ of order $8$ which contains $D$ by [@chr1] as $C$ does. Thus $O(C_G(i))=1$ by [@chr2 17]. $\Box$
\[ratherin\] Let $G$ be a $K^*$-group of [finite Morley rank]{}. Assume that there exist two definable subgroups $N, H \leq G$ such that $N \lhd H$ and ${\overline }H:=H/N$ is a proper simple section of $G$ allowing no graph automorphisms. Then $N_G(H) =
C_{N_G(H)}({\overline }H)H$.
Let $g \in N_G(H)$ and set $R:=d(g)$. $R \leq N_G(H)$ acts on ${\overline }H$. As $G$ is a $K^*$-group and $H$ is a proper simple section, $H$ is a simple algebraic group over an [algebraically closed field]{}. Consider the semidirect product ${\overline }H {\rtimes}R/C_R({\overline }H)$. Then, viewing $R/C_R({\overline }H)$ as a subgroup of $Aut({\overline }H)$, $R/C_R({\overline }H) \leq Inn({\overline }H) \Gamma$, where $\Gamma$ are the graph automorphisms of ${\overline }H$ by [@BN 8.4]. As $\Gamma =1$, either $g \in C_{N_G(H)}({\overline }H)$ or there exists an $h \in H$ such that $x^g \in x^hN$ for all $x \in H$ and $gh^{-1} \in C_{N_G(H)}({\overline }H)$. Thus $g \in
C_{N_G(H)}({\overline }H)H$. $\Box$
\[cicacen\] Let $H$ be a group of [finite Morley rank]{}, such that $H{^{\circ}}$ is a reductive algebraic group and let ${\overline }H{^{\circ}}:=H{^{\circ}}/Z(H{^{\circ}})$. Then $C_{C_H(Z(H{^{\circ}}))}(H{^{\circ}}) =
C_{C_H(Z(H{^{\circ}}))}({\overline }H{^{\circ}})$.
Let $g \in C_H({\overline }H{^{\circ}}) \cap C_H(Z(H{^{\circ}}))$. As $H{^{\circ}}$ is reductive, ${\overline }H{^{\circ}}$ is a simple group by [@hum75 27.5] and $m:=o({\overline }g) <
|H/H{^{\circ}}|$. Set $z_x:=x^{-1}x^g \in Z(H{^{\circ}})$ for all $x \in H{^{\circ}}$. Then, as $x^g=xz_x$, $x=x^{g^m}= xz_x^m$ and $o(z_x)|m$ for all $x \in
H{^{\circ}}$. On the other hand $(x^m)^g=(xz_x)^m = x^m z_x^m$ and $g \in
C_H(x^m)$ for all $x \in H{^{\circ}}$. Especially $C_P(g)=P$, for any maximal torus $P$ of $H{^{\circ}}$, since maximal tori are divisible. As finally $H{^{\circ}}$ is a reductive algebraic group, $H {^{\circ}}= \langle P^h | \; h \in
H{^{\circ}}\rangle \leq C_H(g)$ by [@hum75 ex. 12, p. 162] and $g \in
C_H(H{^{\circ}})$. $\Box$
\[exclude\] Let $G$ be a simple $K^*$-group of [finite Morley rank]{} that does not interpret a bad field. Assume that $C_G(i){^{\circ}}/O(C_G(i)) \cong \PSL_2(K) \times K^*$ for an involution $i \in G$ where $K$ is an [algebraically closed field]{} of characteristic not $2$. Then $G$ contains an involution $j$, such that $C_G(j){^{\circ}}\cong \SL_2(K) * \SL_2(K)$.
Let $S$ be Sylow 2-subgroup of $G$ that contains $i$ and $D \leq S{^{\circ}}$ a four-subgroup. We may assume that $i \in D$ by Proposition \[restriction\]. Set $T:=C_G(D){^{\circ}}$. Hence $O(C_G(t))=1$ for all $t \in D^*$ by Proposition \[restriction\] again. Furthermore $G=\langle C_G(k){^{\circ}}| \; k \in
D_1^*\rangle$ for any four-subgroup $D_1 \leq G$ by [@chr2]. Set $T:=C_G(D){^{\circ}}$ and let $u \in (C_G(D) \cap
C_G(i){^{\circ}}) \backslash T$. Then $C_T(u)=Z(C_G(i){^{\circ}})$.
Assume that $C_G(j){^{\circ}}\cong \GL_2(K)$ for some $j \in D^*$. Then there exists an involution $w \in C_G(j){^{\circ}}\cap N_G(D)$ such that $i^w =ij$. Then $uu^w=w^uw \in C_G(D) \cap C_G(j){^{\circ}}=T$. Furthermore, as $u$ centralises $Z(C_G(i){^{\circ}})$, $u^w$ centralises $Z(C_G(i){^{\circ}})$ as well. On the other hand $C_T(u^w) = C_T(u)^w = Z(C_G(ij){^{\circ}})$. Contradiction as $Z(C_G(ij){^{\circ}})
\neq Z(C_G(i){^{\circ}})$.
Assume that $C_G(j){^{\circ}}\cong \PSL_2(K)\times K^*$ for some $j
\in D^*$, $j \neq i$. Let $v \in
C_{C_G(j){^{\circ}}}(D)\backslash T$ be an involution. As $C_T(v){^{\circ}}=Z(C_G(j){^{\circ}})$, $v \in C_G(i) \backslash C_G(i){^{\circ}}$. Let $C:=C_G(i){^{\circ}}$ and ${\overline }C:=C/Z(C)$. By Lemma \[ratherin\] $C_G(i)=C_{C_G(i)}({\overline }C)C$.
$v$ normalises $Z(C) \cong K^*$ and $G$ does not interpret a bad field. Thus $v$ either inverts $Z(C)$ or centralizes it by [@BN 10.5]. Since $C_T(v)=Z(C_G(j){^{\circ}})$, the second case cannot occur and $v$ inverts $Z(C)$. Let $x \in C$, such that $xv \in C_{C_G(i)}({\overline }C)$. Then $x
\in N_G(D)\cap C= T \langle u \rangle$ and $xv$ inverts $Z(C)$. As $u$ inverts $Z(C_G(j){^{\circ}})$ as above, $yv$ inverts $T$ for all $y \in
uT$. Hence $x \in T$, since $xv \in C_{C_G(i)}({\overline }C)$. Furthermore we may assume that $x
\in Z(C_G(j){^{\circ}})$, as $T=Z(C_G(j){^{\circ}})Z(C)$. Thus $(xv)^2=x^2 \in
C_{C}({\overline }C)=Z(C) \cap Z(C_G(j){^{\circ}}) =1$ by Lemma \[cicacen\] and $x
\in \langle j \rangle$. Then $[xv,C]
\subseteq Z(C)$ and $C=Z(C)*C_C(xv)$ by [@BN ex. 10, p. 98] where $Z(C) \cap C_C(xv)= \langle i \rangle$. Set $H:=C_C(xv){^{\circ}}$. As $C \cong \PSL_2(K)\times K^*$, $H \cong
\PSL_2(K)$ and $C=Z(C) \times H$. As $x
\in \langle j \rangle$, $xv \in I(C_G(D))$. Thus $D_1:=\langle i, xv
\rangle$ is a four-subgroup and $G:=\langle C_G(k)|\; k \in D_1^*
\rangle$. Furthermore $H \leq C_G(D_1)$. As $H$ is a normal subgroup of $C_G(i){^{\circ}}$ and $G$ is simple, $H$ cannot be a normal subgroup of both $C_G(ux)$ and $C_G(uxi)$. As $ux$ and $uxi$ are conjugate to involutions in $D$ by elementary computation in $\GL_2(K)/<-I>$, this implies that $C_G(k) \cong \SL(2,K)*\SL(2,K)$ for $k=ux$ or $k=uxi$ by Proposition \[restriction\].
As finally $G=\langle C_G(k){^{\circ}}| \; k \in D^* \rangle$, it is impossible that $C_G(k){^{\circ}}$ is abelian for all $k \in D \backslash \langle i \rangle$. This implies the claim by Proposition \[restriction\]. $\Box$
Thus Theorem \[super\] basically consists of two parts. The first one is the characterization of $\PSp_4$ as in [@chr1].
\[main\] Let $C_1$ and $C_2$ be the non-isomorphic centralizers in $\PSp_4(K)$ of two involutions where K is an [algebraically closed field]{} of characteristic not $2$. Let $G$ be a $K^*$-group of [finite Morley rank]{} that does not interpret a bad field. Assume that $G$ contains two involutions $i$ and $j$ such that $C_G(i)\cong C_1$ and $C_G(j) \cong
C_2$. Then $G \cong \PSp_4(K)$.
The corresponding result for $\G_2(K)$ is the following theorem, which will be proven in the following section.
\[maing2\] Let $C$ be the centralizer in $\G_2(K)$ of an involutions where K is an [algebraically closed field]{} of characteristic not $2$. Let $G$ be a $K^*$-group of [finite Morley rank]{} that does not interpret a bad field. Assume that $G$ contains one conjugacy class $i^G$ of involutions such that $C_G(i)\cong C$. Then $G \cong \G_2(K)$ if $char(K) \neq 3$.
A characterization of $\G_2(K)$
===============================
In this section we prove Theorem \[maing2\]. Let $G$ be as in Theorem \[maing2\], $D:=\langle i_0, i_1 \rangle$ a four-subgroup which is contained in the connected component of a Sylow 2-subgroup of $G$ and set $i_2=i_0i_1$. Let furthermore $T:=C_G(D){^{\circ}}$. $C_G(i_0)$ contains four quasiunipotent subgroups which are normalized by $T$ and isomorphic to $K^+$. Let $X$ and $Y$ be two of them that centralize each other and let $v$ be an involution in $N_{C_G(i_0)}(T) \backslash
T$ that normalizes $X$. Write ${\overline }x:=xT$ for all $x \in N_G(T)$.
\[weyl\] Let $W:=N_G(T)/T$. Then $|W|=12$ and $W$ is generated by two involutions ${\overline }w$ and ${\overline }v$, where $w \in I(C_G(i_1))$, such that $i_0^{wv}=i_1=i_2^{vw}$ and $o(wv)=6$. The center of $W$ is $\langle {\overline }z \rangle$ where $z:=(wv)^3$. Set $y:=(wv)^4=(wv)z$ and $w_{k+1}:=w^{y^k}$, $v_k:=v^{y^k}$ for $k \in
\nn$. Then ${\overline }w_k = {\overline }{w_{k+3}}$ and ${\overline }v_{k} = {\overline }{v_{k+3}}$. The elements in $W$ are hence $${\overline }1, {\overline }z = {\overline }{(wv)^3}, {\overline }{wv}={\overline }{yz},
{\overline }{(wv)^2} = {\overline }{y^{-1}}, {\overline }{(wv)^4}={\overline }y,
{\overline }{(wv)^5} = {\overline }{vw} = {\overline }{y^{-1}z}$$ and $${\overline }w_{k+1}={\overline }{w^{(vw)^{2k}}}={\overline }{y^{k}w} \mbox{ and } {\overline }v_k =
{\overline }{v^{(vw)^{2k}}} = {\overline }{y^{k}v}$$ for $k=0,1,2$.
As $G$ contains one conjugacy class of involutions, all involutions of $D$ are conjugate in $N_G(T)$ by [@BN 10.22]. $N_G(D)/C_G(D) \cong S_3$, $N_G(T)=N_G(D)$ and $|C_G(D)/C_G(T)| = 2$, which implies that $W$ is the dihedral group of order 12. $\Box$
\[grundbn3\] Let $w_0, v_0$ as in Lemma \[weyl\]. Then $w_0TXw_0 \subseteq
TXw_0X$ and $v_0TYv_0 \subseteq TYv_0Y$. Furthermore $C_G(i_0){^{\circ}}=
\langle T, X, Y ,v_0, w_0 \rangle$.
$L:=\langle X, T, X^{w_0}\rangle$ is a reductive algebraic group of rank $1$. $L$ has thus a $BN$-pair $(B_1,N_1)$, where $B_1 := TX$ and $N_1= \langle w_0, T \rangle$. Hence $w_0TXw_0 \subseteq
TX \cup TXw_0X = TXw_0X$ and $L:=\langle T, X, w_o \rangle$. The same argument for $\langle Y ,T ,Y_0 \rangle$ yields the remaining part of the lemma. $\Box$
\[xconju\] Let $X_1:=X$, $Y_1:=Y$, $X_2:=X_1^{w_0}$, $Y_2:=Y_1^{v_0}$ and $X_n^{(\lambda)}:= X_n^{y^{\lambda}}$ as well as $Y_n^{(\lambda)}:=
Y_n^{y^{\lambda}}$ for any $n=1,2$ and $\lambda \in \zz$. Then
- $X_n^{(\lambda)}=X_n^{(\lambda+3)}$ and $Y_n^{(\lambda)}=Y_n^{(\lambda+3)}$ for all $n=1,2$ and $\lambda \in \zz$.
- $v_{\lambda}$ centralizes $X_n^{(\lambda)}$ and $w_{\lambda}$ centralizes $Y_n^{(\lambda)}$ for $\lambda = 0,1,2$ and $n=1,2$.
- $ (X_1^{(\kappa)})^{w_{\lambda}} = X_2^{(2\kappa -
\lambda)}$ and $(X_1^{(\kappa)})^{v_{\lambda}} = X_1^{(2\kappa -
\lambda)}$ for $\kappa, \lambda = 0, 1, 2$.
- $ (Y_1^{(\kappa)})^{w_{\lambda}} = Y_1^{(2\kappa -
\lambda)}$ and $ (Y_1^{(\kappa)})^{v_{\lambda}} = Y_2^{(2\kappa -
\lambda)}$ for $\kappa, \lambda = 0, 1, 2$.
Since ${\overline }{y^3} = {\overline }1$, $(i)$ follows. We have furthermore chosen $w_0$ such that $w_0$ centralizes $Y_1$. Hence $v_0$ has to centralize $X_1$ which yields $(ii)$. To prove $(iii)$ note that ${\overline }{ y^{\lambda}}={\overline }{w_\lambda w_0}={\overline }{v_{\lambda}v_0}$ and hence $${\overline }{y^{\kappa}w_{\lambda}y^{\lambda - 2 \kappa}w_0} = {\overline }{y^{3
\kappa - \lambda}w_{\lambda}w_0}= {\overline }{y^{-\lambda}y^{\lambda}}= {\overline }1$$ and $${\overline }{y^{\kappa}v_{\lambda}y^{\lambda - 2 \kappa}v_0} =
{\overline }{y^{3\kappa -\lambda}v_{\lambda}v_0}= {\overline }1$$ for $\lambda, \kappa \in 0,1,2$. Since however $X_1^{w_0}=X_2$, $X_1^{v_0}=X_1$, $Y_1^{w_0}=Y_1$ and $Y_1^{v_0}=Y_2$ by $(ii)$, $(iii)$ and $(iv)$ follow. $\Box$
\[finitealg\] Let $G$ be a connected $K$-group of [finite Morley rank]{} such that the solvable radical $\sigma$ of $G$ is finite. Then $G$ is a central product of quasi-simple algebraic groups over [algebraically closed field]{}s.
Since any definable action of a definable connected group on a finite set is trivial $\sigma = Z(G)$. Furthermore $G/\sigma$ is isomorphic to the direct product of simple algebraic groups over algebraically closed fields by [@alt94]. Then $G/\sigma=(G/\sigma)'=G'\sigma/\sigma \cong G'/(G' \cap \sigma)$ and $rk(G)=rk(G')$. As $G$ is connected, $G=G'$ and $G$ is semisimple. Assume that $G$ is quasi-simple. Then $G$ is an algebraic group by [@central]. Let now $G/\sigma \cong A_1 \times \cdots \times A_m$ for some $m \in \nn$, where $A_i$ are simple algebraic groups over [algebraically closed field]{}s for $1 \leq i \leq m$. Let $G_i$ be the preimages of $A_i$ in $G$ for $1 \leq i \leq m$. Then $G=G_1{^{\circ}}....G_m{^{\circ}}$. Now $[G_i{^{\circ}},G_j {^{\circ}}]$ is a connected subgroup of $\sigma$ and hence trivial for any $1 \leq i < j \leq m$. Furthermore $[G_i{^{\circ}},G_i{^{\circ}}]=G_i{^{\circ}}$ for any $1 \leq i \leq m$ as above. Thus $G$ is a central product of $m$ quasi-simple algebraic groups by the first case. $\Box$
\[subgroups\] If $char(K) \neq 3$ then there exists an element $t$ of order 3, such that $C_G(t){^{\circ}}\cong \SL_3(K)$. Actually $C_G(t){^{\circ}}\cong \SL_3(K)$ for exactly two elements of order 3 in $T$.
Assume that $char(K)$ is not 3. Since $y^3 \in T$, $y=y't$, where we can choose $y'$ to be a 3-element by [@BN ex. 11, p. 93]. Let $P$ be a Sylow 3-subgroup of $N_G(T)$. Since $N_G(T)$ is solvable, $P$ is nilpotent-by-finite by [@BN 6.20]. Hence there exists an element $t \in T \cap Z(P)$ by [@BN ex. 12, p. 14]. It follows that $t$ is centralized by $y$. Thus $(t^{w_0}t)^{y^2} = (t^{w_0}t)^{w_2w_0} =
t^{yw_0}t =t^{w_0}t$ and $t^{w_0}t \in C_T(y)$. Hence either $t^{w_0}=t^{-1}$ or $t^{w_0}t \in C_T(y)$ is an element of order 3 that is inverted by $w_0$. We may assume that $t$ is inverted by $w_0$. Now $C_G(t){^{\circ}}= \langle C_{C_G(t)}(l){^{\circ}}|\; l \in D^* \rangle$ and $C_{C_G(i_0)}(t)=C_{C_G(i_1)}(t)^{y^2 }=C_{C_G(i_2)}(t)^{y}$. However, $C_{C_G(i_0)}(t){^{\circ}}\cong \GL_2(K)$.
We show that the solvable radical $\sigma$ of $C_G(t){^{\circ}}$ is finite. $\sigma$ is normalized by $T$ and hence $\sigma{^{\circ}}= \langle
C_{\sigma}(l){^{\circ}}| l \in D \rangle$ by [@nato 4.12]. However, $C_{\sigma}(l){^{\circ}}\leq Z(C_{C_G(l)}(t){^{\circ}})$ for all $l \in D^*$ as $C_{\sigma}(l){^{\circ}}$ is contained in the solvable radical of $C_{C_G(l)}(t){^{\circ}}$. Thus $\sigma{^{\circ}}\leq \bigcap_{l \in
D^*}Z(C_{C_G(l)}(t){^{\circ}}) \leq T$. One the other hand $Z(C_{C_G(i_0)}(t){^{\circ}}) \cong K^*$ and as $G$ does not interpret a bad field, $\sigma$ is either finite or $i_0 \in
Z(C_{C_G(i_0)}(t){^{\circ}})=Z(C_{C_G(i_1)}(t){^{\circ}})$. The second case cannot occur since $C_G(D){^{\circ}}=T$ and $\sigma$ finite.
Thus $C_G(t){^{\circ}}$ is a central product of quasi-simple algebraic groups by Lemma \[finitealg\] and $C_G(t){^{\circ}}\cong \SL_3(K)$. Furthermore $T \leq C_G(t) {^{\circ}}$ contains eight elements of order 3 and while $y$ centralizes exactly $t$ and $t^{-1}$, $yw$ operates transitively on the remaining six. $\Box$
\
The following two propositions can be proven exactly as the corresponding results in [@chr1].
\[maxuni\] There exists a subgroup $V \leq G$ such that $V$ is a maximal quasiunipotent group and is normalized by $T$.
\[NU\] Let $V$ a maximal quasiunipotent group of $G$ which is normalized by $T$. Then $N_G(V){^{\circ}}= V {\rtimes}T$.
\
Now we can construct a $BN$-pair as in [@rusbor]
\[thisis\] If $char(K) \neq 3$ then there exists a maximal quasiunipotent group $Q$ in $G$ which is normalized by $T$ such that
- $Q=V {\rtimes}Y_1$ where $V$ is normalized by $v_0$ and
- $Q=M {\rtimes}X_2^{(1)}$ where $M$ is normalized by $w_1$.
Furthermore $Q=\langle X_1, X_2^{(1)}, X_2^{(2)}, Y_1, Y_2^{(1)},
Y_1^{(2)} \rangle$.
Since $char(K)$ is not $3$, there exists an element $t \in T$ of order 3, such that $C_G(t){^{\circ}}\cong \SL_3(K)$. We can hence assume that $C_G(t){^{\circ}}$ contains the maximal quasiunipotent subgroup $U:=X_1\langle X_2^{(1)}, X_2^{(2)}\rangle$, where $X_1$ is central in $U$. Consider $C_G(X_1)$. We proceed as in [@chr1]. $U \leq C_G(X_1)$ and $L:=\langle T, Y_1, Y_2 \rangle \leq
C_G(X_1)$. Thus $C_G(X_1){^{\circ}}$ is not solvable. Furthermore $C_G(X_1){^{\circ}}$ has Prüfer 2-rank 1, since $C_G(T){^{\circ}}=T$. Thus $C_G(X_1){^{\circ}}/\sigma \cong \PSL_2(K)$ by [@alt94] and [@chr2], where $\sigma$ is the solvable radical of $C_G(X_1)$. On the other hand $L\sigma/\sigma \cong L/(\sigma \cap L)
\cong L/Z(L) \cong \PSL_2(K)$. Hence $C_G(X_1){^{\circ}}=L\sigma{^{\circ}}$ and $U\leq \sigma$. Set $V:=Q(\sigma)$. Then $V^{v_0}=V$.
Assume that $U=V$. Then $Y_1U$ is a quasiunipotent group by [@qua]. Especially $Y_1U$ is nilpotent and $W:=Y_1X_1 < Y_1U$ has infinite index in $N:=N_{Y_1U}(W){^{\circ}}$ by [@BN 6.3]. $N$ is a quasiunipotent group that is normalized by $T$ and hence $N =
\langle C_N(l){^{\circ}}|\; l \in D^*\rangle$ by [@nato 4.6]. Furthermore $C_N(i_0)=W$ and $C_N(i_k)$ is either trivial or equals $X_2^{(k)}$ for $k=1,2$ as $T$ normalises $N$ and acts transitively on $X_2\backslash \{1\}$. We may assume that $X_2^{(1)} \leq N$.
Set $P:=WX_1^{(2)} {\rtimes}T$. Then $P \leq N_G(W)$. Let $w \in W
\backslash\{1\}$ such that $C_G(w) \cap T = \langle i_0 \rangle$. As $T$ acts transitively on $W \backslash \{1\}$, for any element $x \in P$, there exists an element $t_x \in T$ such that $w^x=w^{t_x}$. Thus $P \subseteq C_P(w)T$. As $i_1$ and $i_2$ invert $W$, $C_P(w)$ is a solvable group normalized by $D$. Thus $C_P(w){^{\circ}}=\langle C_{C_P(w)}(l){^{\circ}}|\; l \in D^* \rangle$ by [@BN 4.6]. Furthermore $(C_P(w) \cap C_G(i_0)){^{\circ}}= W$, $(C_P(w)
\cap C_G(i_1)){^{\circ}}\leq X_1^{2}$ and $(C_P(w) \cap C_G(i_2)){^{\circ}}=
1$. Thus $Q:=C_P(w){^{\circ}}\leq WX_1^{(2)}$ is a quasiunipotent subgroup of $P$ containing $W$. Since there is a definable surjective map from $C_P(w) \times T$ onto $N_P(W)$ $$rk\big(WX_1^{(2)}\big)+ rk(T) = rk\big(N_P(W)\big) \leq
rk\big(C_P(w)\big)+rk(T) = rk(Q) + rk(T).$$ Thus $Q=WX_1^{(2)}$ and $N_P(W)=C_P(w){^{\circ}}{\rtimes}T$. Especially $Q \leq
C_P(w^t)$ for all $t \in T$ and hence $Q \leq C_G(W)$ as $T$ acts transitively on $W$. Thus $X_2^{(1)} \leq C_G(W)$ and $X_2^{(1)}$ centralizes $Y_1$. This implies that $Y_1^{(1)} = Y_1^{w_2} \leq
C_G(X_2^{(1)})^{w_2}=C_G(X_1)$. As $C_G(X_1){^{\circ}}=L\sigma$ this implies that $Y_1^{(1)} \leq V$. Contradiction.
Hence $U<V$ and either $\langle
Y_2^{(1)}, (Y_2^{(1)})^{v_0}\rangle \leq V$ or $\langle
Y_1^{(1)}, (Y_1^{(1)})^{v_0}\rangle \leq V$. We may assume that $V= \langle
Y_2^{(1)}, Y_1^{(2)}, U\rangle$. Set $Q:=Y_1V$. $Q$ is obviously a maximal quasiunipotent subgroup of $G$ and $V$ is normalized by $v_0$ by construction. Set $Y:=\langle Y_1,
Y_2^{(1)}, Y_1^{(2)}\rangle$. Then $Y$ is invariant under $w_1$ and $M:=Q \cap Q^{w_1} = \langle Y
, X_1 X_2^{(2)}\rangle$. Furthermore $M$ is normalized by $X_2^{(1)}$, since $N_Q(M){^{\circ}}=Q$ by [@BN 6.3] as in the previous paragraph. $\Box$
\[char3\] If $char(K) \neq 3$, $G$ is a split BN-pair of Tits rank 2, where $B:=N_G(Q){^{\circ}}$, $N:=N_G(T)$ and $\overline w_2, \overline v_0$ are the generators of the Weyl group.
This proves our result, Theorem \[maing2\], by [@qua], noting that the gap in the proof of the necessary theorem is filled by [@kramer].
$G=\langle C_G(k)|\; k \in D^*
\rangle$ by [@chr2 18] and $\langle
N,X_1Y_1\rangle \geq C_G(i_0)$ by Lemma \[grundbn3\]. Thus $G=\langle B, N \rangle$ as $i_1, i_2$ are conjugate to $i_0$ in $N$. BN1 now follows since by Proposition \[NU\] $B=QT$ and thus $B\cap N = T \lhd N$. Furthermore $\langle \overline v_0, \overline
w_1 \rangle = N_G(T)/T$ by Lemma \[weyl\] proving BN2. Let $S:=\{
\overline v_0, \overline w_1\}$ and $W:=N/T$.
To prove BN3 we will show that $vBw \subseteq BvwB \cup BvB$ for all $v,w \in N$ such that $\overline v \in W$ and $\overline w
\in S$. By Lemma \[xconju\] and Proposition \[thisis\] $$1Bv_0 \subseteq B1v_0B$$ $$v_0Bv_0 = v_0TY_1Vv_0 \subseteq TY_1v_0Y_1V \subseteq Bv_0B$$ $$v_2Bv_0 = v_2TY_1Vv_0 = TY_2^{(1)}v_2v_0V \subseteq Bv_2v_0B$$ $$w_0Bv_0 = w_0TY_1Vv_0 = TY_1w_0v_0V \subseteq Bw_0v_0B$$ $$w_1Bv_0 = w_1TY_1Vv_0 = TY_1^{(2)}w_1v_0V \subseteq Bw_1v_0B$$ $$y^2Bv_0 = y^2TY_1Vv_0 = TY_1^{(2)}y^2v_0V \subseteq By^2v_0B$$ $$v_0w_1Bv_0 = v_0w_1TY_1Vv_0 = TY_2^{(1)}v_0w_1v_0\subseteq
Bv_0w_1v_0B$$ and thus $$zBv_0 = (zv_0)v_0Bv_0 \subseteq w_0Bv_0B \subseteq Bw_0v_0B = BzB$$ $$w_1v_0Bv_0 \subseteq w_1Bv_0B \subseteq Bw_1v_0B$$ $$w_2Bv_0=(w_2v_0)v_0Bv_0 \subseteq v_0w_1Bv_0B \subseteq
Bv_0w_1v_0B
= Bw_2B.$$ Finally $$yBv_0 = yTY_1Vv_0 = TY_1^{(1)}yv_0V = TY_1^{(1)}v_1V =
Tv_1Y_1^{(2)}V \subseteq Byv_0B$$ which gives us $$v_1Bv_0 = (v_1v_0)v_0Bv_0 \subseteq yBv_0B \subseteq
Byv_0B=Bv_1B.$$
On the other hand $$1Bw_1 \subseteq B1w_1B$$ $$w_1Bw_1 = w_1TX_2^{(1)}Mw_1 \subseteq TX_2^{(1)}w_1X_2^{(1)}M
\subseteq Bw_1B$$ $$v_1Bw_1 = v_1TX_2^{(1)}Mw_1 = TX_2^{(1)}v_1w_1M \subseteq
Bv_1w_1B$$ $$w_2Bw_1 = w_2TX_2^{(1)}Mw_1 = TX_1w_2w_1M \subseteq
Bw_2w_1B$$ $$v_0Bw_1 = v_0TX_2^{(1)}Mw_1 = TX_2^{(2)}v_0w_1M \subseteq
Bv_0w_1B$$ $$yBw_1 = yTX_2^{(1)}Mw_1 = TX_2^{(2)}yw_1M \subseteq Byw_1B$$ $$w_1v_0Bw_1 = w_1v_0TX_2^{(1)}Mw_1 = TX_1w_1v_0w_1\subseteq
Bw_1v_0w_1B$$ and thus $$zBw_1 = (zw_1)w_1Bw_1 \subseteq v_1Bw_1B \subseteq Bv_1w_1B =
BzB$$ $$v_0w_1Bw_1 \subseteq v_0Bw_1B \subseteq Bv_0w_1B$$ $$w_2Bw_1=(w_2w_1)w_1Bw_1 \subseteq yBw_1B \subseteq
Byw_1B=Bw_2B.$$ Finally $$y^2Bw_1 = y^2TX_2^{(1)}Mw_1 = TX_2y^2w_1M = TX_2w_0M =
Tw_0X_1M \subseteq By^2w_1B$$ which gives us $$v_2Bw_1 = (v_2w_1)w_1Bw_1 \subseteq w_1v_0Bw_1B \subseteq
Bw_1v_0w_1B=Bv_2B.$$
and BN3 is valid. By Proposition \[thisis\] again $Q^{v_0}=VY_2 \neq VY_1 = Q$ and $Q^{w_2}=MX_1^{(1)} \neq
MX_1^{(2)}=Q$ which proves BN4 and gives us the theorem. $\Box$
Proof of Theorem \[super\]
==========================
This section is devoted to the proof that Theorems \[main\] and \[maing2\] imply Theorem \[super\].
Let $G$ be a group as in Theorem \[super\]. By Theorem \[exclude\] there exists an involution $i \in G$, such that $C_G(i){^{\circ}}/O(C_G(i)) \cong L_1 * L_2$ where $L_n \cong \SL_2(K)$ for an [algebraically closed field]{} $K$ of characteristic neither $2$ nor $3$ and $n=1,2$. We may assume that $i \in C_G(i){^{\circ}}$ by Lemma \[restriction\]. Thus $L_1 \cap L_2 = \langle {\overline }{\mbox{\it{\i}}}
\rangle$. Let $S$ be a Sylow 2-subgroup of $C_G(i)$, $D$ a four-subgroup such that $D \leq S{^{\circ}}$ and set $T:=C_G(D){^{\circ}}$. Then $D$ contains a central involution of $S$ by [@BN ex. 12, p. 14] which means that $i$ is central in $S$, since $j$ and $ij$ are conjugate in $S$ for $j\in D\backslash \langle i \rangle$.
\[corset\] $G=\langle C_G(k){^{\circ}}| \; k \in D_1^*\rangle$ for any four-subgroup $D_1 \leq G$ and $O(C_G(k))=1$ for all $k \in D^*$.
$O(C_G(t))=1$ for all $t \in D^*$ by Proposition \[restriction\] and $G=\langle C_G(k){^{\circ}}|
\; k \in D_1^*\rangle$ for any four-subgroup $D_1 \leq G$ by [@chr2]. $\Box$
\[abelian\] $C_G(k){^{\circ}}$ is nonabelian for all $k \in D^*$.
Assume that $C_G(j){^{\circ}}$ is abelian for some $j \in D^*$. Then $C_G(j){^{\circ}}\leq C_G(i){^{\circ}}$ since $i \in C_G(j){^{\circ}}$. As furthermore $j$ and $ij$ are conjugate, $G=C_G(i){^{\circ}}$ by Proposition \[corset\]. Contradiction. $\Box$
\[structure\] Let $j \in D^*$. If $i$ and $j$ are not conjugate in $G$, then $C_G(j){^{\circ}}\cong \PSL_2(K) \times K^*$.
Let $j\in D^*$ be not conjugate to $i$. Then $j$ is conjugate to $ij$ in $C_G(i){^{\circ}}$. Now $C_G(j){^{\circ}}$ is not abelian by Corollary \[abelian\]. Furthermore $C_G(D){^{\circ}}\cong K^* \times K^*$ and there are only three possibilities by Proposition \[restriction\], namely $C_G(j){^{\circ}}\cong \GL_2(K)$, $C_G(j){^{\circ}}\cong
\PSL_2(K) \times K^*$ or $C_G(j){^{\circ}}\cong \SL_2(K)*\SL_2(K)$. The first and third case cannot occur since $i$ and $ij$ are not conjugate, which proves the claim. $\Box$
We have to distinguish two different cases:
- $C_G(i)$ is connected.
- $C_G(i)$ is not connected.
We are going to show that $G \cong \G_2(K)$ in the first case and $G
\cong \PSp_4(K)$ in the second case. Assume from now on that $D=
\langle i, j \rangle$.
\[concon\] If $C_G(i)$ is connected, then $G$ contains one conjugacy class of involutions.
$S \leq C_G(i)$, since $i \in Z(S)$. As Sylow 2-subgroups of $G$ are conjugate, and as $C_G(i)$ contains only two conjugacy classes of involutions $i$ and $j^{C_G(i)}$ – all elements of order 4 are conjugate in $\SL_2(K)$ –, it is enough to show that $i$ and $j$ are conjugate.
Assume that $j$ is not conjugate to $i$ and let $w \in N_G(T)$ such that $D\langle w \rangle$ is an elementary abelian 2-subgroup of order $8$. Then $C_G(D)=T\langle w \rangle$ and $w$ inverts $T$. As $i$ and $ij$ are not conjugate in $G$, $C_G(j){^{\circ}}\cong \PSL_2(K) \times K^*$ by Corollary \[structure\]. Let $u \in (N_G(T) \cap
C_G(j){^{\circ}})\backslash T$ be an involution. Then $u \in
C_G(D)$, i.e. since $C_G(D) = T\langle w \rangle$, $u=tw$ for some $t \in
T$. Thus $u$ has to invert $T$. Contradiction as $Z(C_G(j){^{\circ}}) \leq T$ is infinite. $\Box$
\[findg\] If $C_G(i)$ is connected, then $G \cong \G_2(K)$.
If $C_G(i)$ is connected, then $G$ contains one conjugacy class of involutions by Lemma \[concon\]. Since furthermore the centralisers of involutions in $\G_2(K)$ are isomorphic to $\SL_2(K) * \SL_2(K)$, the claim follows by Theorem \[maing2\]. $\Box$
\[normandall\] $N_G(L_1) = C_G(i){^{\circ}}C_G(C_G(i){^{\circ}})$.
As $C_G(i){^{\circ}}C_G(C_G(i){^{\circ}}) \leq N_G(L_1)$, we only need to prove the reverse inclusion. Let $g \in N_G(L_1)$. As $Z(L_1)=\langle
i \rangle$, $g \in C_G(i)$ and $g \in N_G(L_2)$ as well by [@BN 7.1]. By Lemma \[ratherin\] and Lemma \[cicacen\], $g \in C_G(L_1)L_1 \cap
C_G(L_2)L_2$. Thus there exists $l_n \in L_n$ for $n=1,2$ such that $ gl_n
\in C_G(L_n)$. As $L_1$ and $L_2$ commute, $gl_1l_2 \in C_G(L_1) \cap
C_G(L_2)$ and hence $g \in C_G(C_G(i){^{\circ}})C_G(i){^{\circ}}$. $\Box$
\
Set $K_s:=C_G(C_G(s){^{\circ}})$ for all involutions $s \in G$.
\[control\] Let $s \in I(G)$. Then $K_s \cap C_G(s){^{\circ}}= Z(C_G(s){^{\circ}})$. Furthermore
- $K_i$ is a finite group,
- $I(K_i)=i$,
- $K_i \cap K_s =1$ for all $s \in D^*$ such that $i \neq s$.
As $Z(C_G(i){^{\circ}})=\langle i \rangle$, $K_i \leq C_G(i)$ and $K_i
\cap C_G(i){^{\circ}}= \langle i \rangle$. Hence $K_i{^{\circ}}\leq
C_G(i){^{\circ}}\cap K_i = \langle i \rangle$ and $K_i$ is a finite group, proving $(i)$.
To show $(ii)$ let $t \in I(K)$ be an involution. Then $O(C_G(t))=1$ by [@chr2]. Assume that $C_G(t){^{\circ}}> C_G(i){^{\circ}}$. Then $C_G(t){^{\circ}}\cong \G_2$ or $C_G(t){^{\circ}}\cong \PSp_4(K)$ by [@nato] as $pr(C_G(t){^{\circ}})=pr(G)=2$. However this would imply that $C_G(j){^{\circ}}, C_G(ij){^{\circ}}\leq C_G(t){^{\circ}}$ by Proposition \[restriction\] and $G=C_G(t){^{\circ}}$ by Proposition \[corset\]. Contradiction. Thus $C_G(t){^{\circ}}= C_G(i){^{\circ}}=
C_G(it){^{\circ}}$ and $t=i$, as otherwise $\langle i, t \rangle$ is a four-subgroup and $G=C_G(i){^{\circ}}$ by Proposition \[corset\].
Let finally $s \in D^*$ such that $s \neq
i$. As $C_G(i){^{\circ}}, C_G(s){^{\circ}}\leq C_G(K_i \cap K_s)$ and $si$ is conjugate to $s$ in $C_G(i){^{\circ}}$, [@nato 5.14] and Proposition \[corset\] imply that $C_G(K_s \cap K_t){^{\circ}}=G$. As $G$ is simple, $K_i \cap K_s=1$. $\Box$
\[precision\] $C_G(i) = (C_G(i){^{\circ}}* K_i) {\rtimes}\langle v \rangle$ for any $v \in G$ such that $L_1^v=L_2$. In this case $v^2 \in C_G(i){^{\circ}}K_i$ and we can choose $v
\in N_G(T)$ to be a 2-element.
Assume that there exists an element $v \in G$ such that $L_1^v=L_2$. As $Z(L_1)=Z(L_2)=\langle i \rangle$, $v \in
C_G(i)\backslash C_G(i){^{\circ}}$, $L_2^v=L_1$ by [@BN 7.1] and $v^2
\in C_G(i){^{\circ}}K_i$ by Lemma \[normandall\].
Let $x \in C_G(i)$. There are two possibilities: Either $x \in
N_G(L_1) = C_G(i){^{\circ}}* K_i$ by Lemma \[normandall\] and Lemma \[control\] or $L_1^x=L_2$ and $L_2^x=L_1$ by [@BN 7.1]. In the second case $xv \in N_G(L_1) =
C_G(i){^{\circ}}K_i$ by Lemma \[normandall\] again. Thus $x \in
(C_G(i){^{\circ}}* K_i) {\rtimes}\langle v \rangle$ in all cases.
Let finally $d(v)=V \times F$, where $V$ is a connected divisible group and $F
= \langle f \rangle$ a finite cylic group by [@BN ex. 10, p.93]. Then $V \leq C_G(i){^{\circ}}$ and $f^2 \in N_G(L_1)$. Thus $f$ has even order. Let $o(f)=2^km$ where $k \geq 1$ and $m \in \nn$ is odd. Then $f^m$ is a 2-element such that $L_1^{f^m}=L_2$. Furthermore $f^m$ is contained in a Sylow 2-subgroup of $C_G(i)$ and as they are all conjugate to each other, we may assume that $f^m \in N_G(T)$. $\Box$
\[wemove\] If all involutions in $D$ are conjugate, then $K_i =
\langle i \rangle$.
Assume that all involutions in $D$ are conjugate. Then there exists an element $y \in N_G(T)$, such that $i^y =j$ by [@BN 10.22]. Let $w \in N_G(T) \cap C_G(i){^{\circ}}$ such that $E:=D\langle w \rangle$ is an elementary abelian 2-subgroup of order $8$. $w$ inverts $T$ and all other involutions that invert $T$ are contained in $wC_G(T)=wTK_i$ by Corollary \[precision\]. However, $I(wTK_i)=wT$ by Lemma \[control\] and $w^y \in wT$. $K_i$ acts hence on $C_G(j){^{\circ}}$ centralising $T\langle w \rangle$. Thus $K_i \leq C_G(j){^{\circ}}K_j$ by Corollary \[precision\]. Let $k \in K_i$ and $x \in
C_G(j){^{\circ}}$, $k_1 \in K_j$ such that $k=xk_1$. Then $x=kk_1^{-1} \in
C_G(E) \cap C_G(j){^{\circ}}=E$. Thus $K_j \leq EK_i$. Furthermore $K_{ij}=K_j^{w_1} \leq EK_i$, where $w_1 \in L_1 \cap N_G(E)$ is an involution conjugating $j$ and $ij$. Let now $k:=|K_i|$. As all involutions in $D$ are conjugate, $k=|K_l|$ for all $l \in D^*$. Furthermore $|EK_i|=4k$ as $E \cap K_i = \langle i \rangle$. On the other hand $K_i, K_j, K_{ij} \leq
EK_i$ and all three subgroups normalise each other. Hence $k^3 \leq 4k$ by Lemma \[control\] and $k=2$. Thus $K=\langle i \rangle$, if all involutions in $D$ are conjugate. $\Box$
\[exclusion\] If all involutions in $D$ are conjugate, then $C_G(i)$ is connected.
Suppose that all involutions in $D$ are conjugate. Then $K_i=\langle i \rangle$ by Lemma \[wemove\]. Assume that $C_G(i)$ is not connected. Then there exists a 2-element $v \in
N_G(T)$ such that $C_G(i) = C_G(i){^{\circ}}{\rtimes}\langle v
\rangle$ where $L_1^v=L_2$ by Lemma \[precision\]. As $v^2 \in
C_G(i){^{\circ}}$ is a 2-element, it is contained in a maximal torus of $C_G(i){^{\circ}}$. As maximal tori are divisible, there exists an element $t \in C_G(i){^{\circ}}$, such that $t^2=v^2$. Assume that $t=t_1t_2^v$, where $t_1,t_2 \in
L_1$. Then $$(*) \quad 1 = v^2t^{-2} = v^2(t_2^{-2v}t_1^{-2})=vt_2^{-2}vt_1^{-2}.$$ On the other hand $v^2 \in C_G(v)$ and thus $t_1^2t_2^{2v} =
(t_1^2t_2^{2v})^v = t_2^{2v^2}t_1^{2v}$. This implies that $$t_2^{-2v^2}t_1^2=t_1^{2v}t_2^{-2v} \in L_1 \cap L_2 = \langle i
\rangle$$ Hence either $t_1^2=t_2^2$ or $t_1^2=it_2^2$. Set $s:=t_1^2$. If $t^2=ss^v$, then $(*)$ implies that $vs^{-1}$ is an involution. If $t^2=iss^v$, then $(vs^{-1})^2=i$. By replacing $v$ with a conjugate of $vs^{-1}$, we may assume that $v \in
N_G(T)$, such that $v^2 \in \langle i \rangle$.
As all involutions in $D$ are conjugate and $j$ is conjugate to $ij$ in $C_G(i)$, there exists an element $y \in N_G(T)$, such that $i^{y^2} =j^{y} =ij$ by [@BN 10.22], where $y^3 \in T$. Furthermore $v \in C_G(D)$ and $v$ does not invert $T$. This implies that $v
\in C_G(j)\backslash C_G(j){^{\circ}}$ and $v \in C_G(ij) \backslash
C_G(ij){^{\circ}}$. Thus there exist elements $x_1, x_2 \in C_G(D)$ such that $v^y=vx_1$ and $v^{y^2}=vx_2$, where $x_1 \in C_G(j){^{\circ}}$ and $x_2 \in
C_G(ij){^{\circ}}$ by Corollary \[precision\].
Let $P:=C_T(v){^{\circ}}$. Then $P=\{ll^v| l \in T \cap L_1\}$ and $C_T(v)=P\langle i \rangle$. On the other hand $x_n$ either inverts $T$ or centralizes it for some $n=1,2$. If $x_n$ would centralize $T$ for $n=1,2$, then $P=C_T(vx_n){^{\circ}}=(C_T(v)^{y^n}){^{\circ}}=
P^{y^n}$. Contradiction as $P$ contains a unique involution from $D$. Thus $x_1$ and $x_2$ invert $T$.
Let $T[v]:=\{t \in T| t^v=t^{-1}\}$. Then $T[v] \leq T$ and $T[v]{^{\circ}}=\{ll^{-v}| l \in T \cap L_1\}$. Obviously $T=PT[v]$. As $x_n$ inverts $T$ for $n=1,2$, $C_T(x^{y^n})=C_T(vx_n)=T[v]$. Thus $T[v]{^{\circ}}= (C_T(v^y){^{\circ}})^y = (T[v]{^{\circ}})^y$. Contradiction again, as $T[v]{^{\circ}}$ contains a unique involution from $D$. Hence $C_G(i)=C_G(i){^{\circ}}$ is connected. $\Box$
\[connocon\] If $C_G(i)$ is not connected, then $C_G(i)=C_G(i){^{\circ}}{\rtimes}\langle u \rangle$, where $u \in C_G(D)$ is an involution such that $L_1^u=L_2$.
As $C_G(i)$ is not connected, $i$ and $j$ are not conjugate in $G$ by Proposition \[exclusion\] and $C:=C_G(j){^{\circ}}\cong
\PSL_2(K) \times K^*$ by Corollary \[structure\]. Let $u \in
(N_G(T) \cap C)\backslash T$ be an involution. Then $u \in
C_G(D)$ but $u \notin C_G(i){^{\circ}}$, as $u$ does not invert $T$. Thus $C_G(i)=C_G(i){^{\circ}}K_i \langle u \rangle$ by Lemma \[precision\] and we need to show that $K_i = \langle i \rangle$.
As $K_i$ centralizes $Z(C)$, $K_i \leq K_jC$ by Lemma \[ratherin\] and Lemma \[cicacen\]. We show that $C_{K_i}(u) = \langle i \rangle$. $u$ acts on $K_i$. Let $k \in C_{K_i}(u)$. Then $k$ centralizes the elementary abelian subgroup $E_1:=\langle i, j, u \rangle$. As $K_i \leq K_jC$ and $C_{C}(E_1) =Z(C)\langle u \rangle$, we must have $k \in K_j \langle u
\rangle$ by Lemma \[control\]. Thus $k^2 \in K_j \cap K_i =1$ by Lemma \[control\] and $k \in \langle i
\rangle $ by Lemma \[control\] again. Especially $C_{K_i}(u) = \langle i
\rangle$.
As $|K_i|/|C_{K_i}(u)|=|S|$, where $S =\{[k,u]| \; k \in K_i\}
\subseteq K_i$ consists of elements inverted by $u$ by [@BN ex. 17, p. 7], $2|S|=|K_i|$. Let $s \in
S$. Then $u^s \in C$ and $u^s=s^uus=us^2$. Furthermore $s$ centralizes $T$ and thus $u^s \in
N_G(T)\backslash T$ which implies that there exists a $t \in T$ such that $us^2 = u^s = ut$. Hence $s^2=t \in K_i \cap T = \langle i
\rangle$. Assume that $s^2=i$. Then $u^s=ui$. As $\langle j, u \rangle$ is conjugate to $D$ in $C$, $u$ and $uj$ belong to different conjugacy classes. However, there are at most two conjugacy classes of involutions in $uC_G(i){^{\circ}}$ by elementary computation, namely $u^T$ and $(ui)^T$. Contradiction. Hence $s^2=1$ by Lemma \[control\] and $s \in \langle i
\rangle$. Then $|K_i| = 2|S| \leq 4$.
We finally prove that $|S|=1$. Assume that $|S|=2$. Then $|K_i|=4$ and there exists an element $k \in K_i$ of order $4$ by Lemma \[control\]. As $C_{K_i}(u)=\langle i \rangle$, $k^u=k^3=k^{-1}$. Especially $u^k = k^uuk = uk^2 = ui$. Contradiction as before and we are done. $\Box$
\[connocen\] If $C_G(i)$ is not connected, $C_G(j)\cong (\GL_2(K)/\langle
-I \rangle) {\rtimes}\langle w \rangle$ where $-I = \left( \begin{smallmatrix} -1
& 0 \\ 0 & -1 \end{smallmatrix}\right)$ and $w \in C_G(D) \cap
C_G(i){^{\circ}}$ is an involution that acts as an inverse-transpose automorphism on $\GL_2(K)$, i.e. $${\left(\begin{array}{cc}}a & b \\ c & d {\end{array}\right)}^{ w} = (ad -bc)^{-1}{\left(\begin{array}{cc}}d & - c \\ - b & a
{\end{array}\right)}$$ for any $\left( \begin{smallmatrix} a & b \\ c & d
\end{smallmatrix}\right) \in \GL_2(K)$.
As $C_G(i)$ is not connected, $C_G(j){^{\circ}}\cong \PSL_2(K)
\times K^* \cong \GL_2(K)/\langle -I
\rangle$ by Proposition \[exclusion\] and Corollary \[structure\]. Set $C:=C_G(j){^{\circ}}$.
Let $w \in C_G(D) \cap C_G(i){^{\circ}}$ be an involution that inverts $T$. We show that $C_G(j)=C {\rtimes}\langle w \rangle$. As $w$ inverts $T$, $w \notin C$ and $C_G(j) \geq C {\rtimes}\langle w \rangle$. To prove the other inclusion let ${\overline }C:=C/Z(C)$. By Lemma \[ratherin\] $C_G(j)=C_{C_G(j)}({\overline }C)C$. Let $h \in
C_{C_G(j)}({\overline }C)$. Then $T^h \leq TZ(C)=T$ and $h \in N_G(T)$. As $h
\in C_G(j)$ and $i$ and $ij$ are not conjugate, $h \in
C_G(D) = T \langle u, w \rangle$. Thus, as $T \langle u \rangle \leq
C$, $C_G(j)=C\langle w\rangle = C_{C_G(j)}({\overline }C)C$.
As $w$ inverts $T$, $w \notin C_{C_G(j)}({\overline }C)$. Thus there exists $v
\in C_G(D)$ such that $wv \in C_{C_G(j)}({\overline }C)$. $w$ inverts $T$ and $v \in C$, hence $wv$ inverts $Z(C)$. Furthermore $[wv,C] \subseteq Z(C)$ and $C=Z(C)*C_C(wv)$ by [@BN ex. 10, p.98] where $Z(C) \cap C_C(wv)= \langle j \rangle$. Set $H:=C_C(wv){^{\circ}}$. As $C \cong \PSL_2(K) \times K^*$, $H \cong
PSL_2(K)$ and $C=Z(C) \times H$. We may assume that $v \in H$.
Let $x \in C$. Then there exists $z \in Z(C)$ and $h \in H$ such that $x=zh$. Furthermore $x^{wv}=z^{-1}h$, and $x^w=z^{-1}h^v$. Thus $h^w=h^v$ for all $h \in H$. Set $T_1=H \cap T$. As $w$ inverts $T_1$, $v$ has to invert $T_1$ as well. Hence $v \in I(N_G(T_1)\backslash
T_1)$ and after maybe conjugating $w$ with an element from $T$, we may assume that $v$ corresponds to $\left( \begin{smallmatrix} 0 & 1 \\
-1 & 0 \end{smallmatrix}\right)$. This implies the claim. (Compare Section 3.2.) $\Box$
[*Proof of Theorem \[super\].*]{} Let $G$ be as in Theorem \[super\]. If $C_G(i)$ is connected, then $G \cong \G_2(K)$ by Corollary \[findg\]. If on the other hand $C_G(i)$ is not connected, $C_G(i)$ and $C_G(j)$ are isomorphic to centralizers of involutions in $\PSp_4(K)$ by Proposition \[connocon\] and Proposition \[connocen\]. Thus $G \cong \PSp_4(K)$ by Theorem \[main\]. $\Box$
**ACKNOWLEDGMENTS**
This paper is part of the author’s Ph.D. thesis. She would like to thank the Landesgraduiertenförderung Baden-Württemberg for their financial support. She is very grateful to Alexandre V. Borovik, her supervisor during her stay in Manchester, who gave all the help one could wish for and without whom this work would not exist.
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C. Altseimer, Strongly Embedded Subgroups of Groups of Odd Type, in preparation.
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|
---
abstract: |
A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an *interval cyclic $t$-coloring* if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is *interval cyclically colorable* if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $\mathfrak{N}_{c}$. For a graph $G\in
\mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq
\vert V(G)\vert +\Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.\
Keywords: edge-coloring, interval coloring, interval cyclic coloring, bipartite graph, complete graph.
address:
- |
Department of Informatics and Applied Mathematics,\
Yerevan State University, 0025, Armenia
- |
Institute for Informatics and Automation Problems,\
National Academy of Sciences, 0014, Armenia
author:
- 'P.A. Petrosyan[^1], S.T. Mkhitaryan[^2]'
title: 'Interval cyclic edge-colorings of graphs'
---
Introduction
============
All graphs considered in this paper are finite, undirected, and have no loops or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of $G$, respectively. For a graph $G$, the number of connected components of $G$ is denoted by $c(G)$. A graph $G$ is Eulerian if it has a closed trail containing every edge of $G$. The degree of a vertex $v\in V(G)$ is denoted by $d_{G}(v)$ (or $d(v)$), the maximum degree of $G$ by $\Delta(G)$, the diameter of $G$ by $\mathrm{diam}(G)$, and the chromatic index of $G$ by $\chi^{\prime}(G)$. The terms and concepts that we do not define can be found in [@b3; @b33].
A proper edge-coloring of a graph $G$ with colors $1,\ldots ,t$ is an *interval $t$-coloring* if all colors are used, and the colors of edges incident to each vertex of $G$ are form an interval of integers. A graph $G$ is *interval colorable* if it has an interval $t$-coloring for some positive integer $t$. The concept of interval edge-coloring of graphs was introduced by Asratian and Kamalian [@b1]. In [@b1; @b2], the authors showed that if $G$ is interval colorable, then $\chi^{\prime}(G)=\Delta(G)$. In [@b1; @b2], they also proved that if a triangle-free graph $G$ has an interval $t$-coloring, then $t\leq \left\vert V(G)\right\vert
-1$. Later, Kamalian [@b13] showed that if $G$ admits an interval $t$-coloring, then $t\leq 2\left\vert V(G)\right\vert -3$. This upper bound was improved to $2\left\vert V(G)\right\vert -4$ for graphs $G$ with at least three vertices [@b8]. For an $r$-regular graph $G$, Kamalian and Petrosyan [@b18] showed that if $G$ with at least $2r+2$ vertices admits an interval $t$-coloring, then $t\leq 2\left\vert V(G)\right\vert -5$. For a planar graph $G$, Axenovich [@b4] showed that if $G$ has an interval $t$-coloring, then $t\leq \frac{11}{6}\left\vert
V(G)\right\vert$. In [@b12; @b13; @b24; @b26], interval colorings of complete graphs, complete bipartite graphs, trees, and $n$-dimensional cubes were investigated. The $NP$-completeness of the problem of the existence of an interval coloring of an arbitrary bipartite graph was shown in [@b29]. In [@b5; @b6; @b21; @b25; @b26; @b27], interval colorings of various products of graphs were investigated. In [@b2; @b3; @b6; @b7; @b9; @b10; @b11; @b16; @b17; @b19], the problem of the existence and construction of interval colorings was considered, and some bounds for the number of colors in such colorings of graphs were given.
A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an *interval cyclic $t$-coloring* if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$. A graph $G$ is *interval cyclically colorable* if it has an interval cyclic $t$-coloring for some positive integer $t$. This type of edge-coloring under the name of $\pi$-coloringwas first considered by Kotzig in [@b20], where he proved that every cubic graph has a $\pi$-coloring with 5 colors. However, the concept of interval cyclic edge-coloring of graphs was explicitly introduced by de Werra and Solot [@b32]. In [@b32], they proved that if $G$ is an outerplanar bipartite graph, then $G$ has an interval cyclic $t$-coloring for any $t\geq \Delta(G)$. In [@b22], Kubale and Nadolski showed that the problem of determining whether a given bipartite graph is interval cyclically colorable is $NP$-complete. Later, Nadolski [@b23] showed that if $G$ is interval colorable, then $G$ has an interval cyclic $\Delta(G)$-coloring. He also proved that if $G$ is a connected graph with $\Delta(G)=3$, then $G$ has an interval cyclic coloring with at most 4 colors. In [@b14; @b15], Kamalian investigated interval cyclic colorings of simple cycles and trees. For simple cycles and trees, he determined all possible values of $t$ for which these graphs have an interval cyclic $t$-coloring.
In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if a triangle-free graph $G$ with at least two vertices has an interval cyclic $t$-coloring, then $t\leq \vert V(G)\vert +\Delta(G)-2$. For various classes of graphs, we also obtain bounds on the least and the greatest values of $t$ for which these graphs have an interval cyclic $t$-coloring. Finally, we describe some methods for constructing of interval cyclically non-colorable graphs.
Notations, definitions and auxiliary results
============================================
We use standard notations $C_{n},K_{n}$ and $Q_{n}$ for the simple cycle, complete graph on $n$ vertices and the hypercube, respectively. We also use standard notations $K_{m,n}$ and $K_{l,m,n}$ for the complete bipartite and tripartite graph, respectively, one part of which has $m$ vertices, other part has $n$ vertices and a third part has $l$ vertices.
A *partial edge-coloring* of $G$ is a coloring of some of the edges of $G$ such that no two adjacent edges receive the same color. If $\alpha $ is a partial edge-coloring of $G$ and $v\in V(G)$, then $S\left(v,\alpha \right)$ denotes the set of colors appearing on colored edges incident to $v$.
A graph $G$ is *interval colorable* if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $\mathfrak{N}$. For a graph $G\in
\mathfrak{N}$, the least and the greatest values of $t$ for which it has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively.
A graph $G$ is *interval cyclically colorable* if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $\mathfrak{N}_{c}$. For a graph $G\in \mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. The *feasible set* $F(G)$ of a graph $G$ is the set of all $t$’s such that there exists an interval cyclic $t$-coloring of $G$. The feasible set of $G$ is *gap-free* if $F(G)=\left[w_{c}(G),W_{c}(G)\right]$. Clearly, if $G\in
\mathfrak{N}$, then $G\in \mathfrak{N}_{c}$ and $\chi^{\prime}(G)\leq w_{c}(G)\leq w(G)\leq W(G)\leq W_{c}(G)\leq
\vert E(G)\vert$.
Let $\left\lfloor a\right\rfloor$ denote the largest integer less than or equal to $a$. For two positive integers $a$ and $b$ with $a\leq b$, we denote by $\left[a,b\right]$ the interval of integers $\left\{a,\ldots,b\right\}$. By $\left[a,b\right]_{even}$ ($\left[a,b\right]_{odd}$), we denote the set of all even (odd) numbers from the interval $\left[a,b\right]$.\
In [@b1; @b2], Asratian and Kamalian obtained the following two results.
\[mytheorem1\] If $G\in \mathfrak{N}$, then $\chi^{\prime}(G)=\Delta(G)$. Moreover, if $G$ is a regular graph, then $G\in \mathfrak{N}$ if and only if $\chi^{\prime}(G)=\Delta(G)$.
\[mytheorem2\] If $G$ is a connected triangle-free graph and $G\in \mathfrak{N}$, then
$W(G)\leq \vert V(G)\vert -1$.
For general graphs, Kamalian proved the following
\[mytheorem3\]([@b13]). If $G$ is a connected graph with at least two vertices and $G\in \mathfrak{N}$, then
$W(G)\leq 2\vert V(G)\vert -3$.
Note that the upper bound in Theorem \[mytheorem3\] is sharp for $K_{2}$, but if $G\neq K_{2}$, then this upper bound can be improved.
\[mytheorem4\]([@b8]). If $G$ is a connected graph with with at least three vertices and $G\in \mathfrak{N}$, then
$W(G)\leq 2\vert V(G)\vert -4$.
In [@b30], Vizing proved the following well-known result.
\[mytheorem5\] For every graph $G$,
$\Delta(G)\leq \chi^{\prime}(G)\leq \Delta(G)+1$.
\[mycorollary1\] If $G$ is a regular graph, then $G\in
\mathfrak{N}_{c}$ and $w_{c}(G)=\chi^{\prime}(G)$.
From Theorems \[mytheorem1\] and \[mytheorem5\], we get
\[mycorollary2\] $\mathfrak{N}\subset \mathfrak{N}_{c}$.
Although all regular graphs are interval cyclically colorable, there are many graphs that have no interval cyclic coloring. In Fig. \[int-cyc-non graph\], we present the smallest known interval cyclically non-colorable graph.
![The interval cyclically non-colorable graph.[]{data-label="int-cyc-non graph"}](fig1.eps "fig:"){width="10pc" height="17pc"}\
We also need the generalizations of Theorems \[mytheorem2\],\[mytheorem3\] and \[mytheorem4\] for disconnected graphs. It can be easily proved by induction on the number of connected components that the following two lemmas hold.
\[mylemma1\] If $G$ is a triangle-free graph and $G\in
\mathfrak{N}_{c}$, then
$W(G)\leq \vert V(G)\vert -c(G)$.
\[mylemma2\] If $G$ is a graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then
$W(G)\leq 2\vert V(G)\vert -3\cdot c(G)$.
Moreover, if $G$ has at least three vertices, then
$W(G)\leq 2\vert V(G)\vert -4\cdot c(G)$.
Some general results {#part1}
====================
In this section we derive some upper bounds for $W_{c}(G)$ depending on the number of vertices, degrees and diameter for connected graphs, triangle-free graphs, and, in particular, for bipartite graphs. Next we show that there are graphs $G$ for which $w_{c}(G)>\chi^{\prime}(G)$. We also investigate the feasible sets of interval cyclically colorable graphs. In particular, we show that if $G$ is interval colorable, then $\left[\Delta(G),W(G)\right]\subseteq F(G)$. On the other hand, we also show that there are interval cyclically colorable graphs for which feasible sets are not gap-free.\
Our first two theorems give upper bounds for $W_{c}(G)$ depending on the number of vertices and the maximum degree of the interval cyclically colorable graph $G$.
\[mytheorem6\] If $G$ is a connected triangle-free graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq
\vert V(G)\vert +\Delta(G)-2$.
Consider an interval cyclic $W_{c}(G)$-coloring $\alpha$ of $G$. If for each $u\in V(G)$, $S(u,\alpha)$ is an interval of integers, then $G\in \mathfrak{N}$ and $W_{c}(G)\leq W(G)\leq \vert V(G)\vert-1
\leq \vert V(G)\vert +\Delta(G)-2$, by Theorem \[mytheorem2\] and taking into account that $G$ is a connected triangle-free graph with at least two vertices.
Now suppose that there exists $v_{0}\in V(G)$ such that $S(v_{0},\alpha)$ is not an interval of integers. Since $\alpha$ is an interval cyclic $W_{c}(G)$-coloring of $G$, for each $v\in V(G)$ such that $S(v,\alpha)$ is not an interval, there are colors $k_{v}$ and $l_{v}$ such that
$S(v,\alpha)=\{1,\ldots,k_{v}\}\cup
\{W_{c}(G)-l_{v}+1,\ldots,W_{c}(G)\}$.
Let $l^{\star}=\max_{\begin{subarray}{1}
v\in V(G),\\
S(v,\alpha)~is~not~an~interval \end{subarray}} l_{v}$. Clearly, $1\leq l^{\star}\leq \Delta(G)-1$. Define an auxiliary graph $H$ as follows:
$V(H)=V(G)$ and\
$E(H)=E(G)\setminus \{e\colon\, e\in E(G)\wedge \alpha(e)\in
\{W_{c}(G)-l^{\star}+1,\ldots,W_{c}(G)\}\}$.
Clearly, $H$ is a spanning subgraph of $G$. Now let us consider the restriction of the coloring $\alpha$ on the edges of subgraph $H$ of $G$. Let $\alpha_{H}$ be this edge-coloring. It is easy to see that $\alpha_{H}$ is an interval $(W_{c}(G)-l^{\star})$-coloring of $H$. Moreover, since $H$ is a triangle-free graph and $H\in
\mathfrak{N}$, by Lemma \[mylemma1\], we have
$W_{c}(G)-l^{\star}\leq W(H)\leq \vert V(H)\vert-c(H)\leq \vert
V(H)\vert-1=\vert V(G)\vert-1$.
This implies that $W_{c}(G)\leq \vert V(G)\vert+l^{\star}-1$. From this and taking into account that $l^{\star}\leq \Delta(G)-1$, we obtain $W_{c}(G)\leq \vert V(G)\vert +\Delta(G)-2$. $\square$
\[mycorollary3\] If $G$ is a connected triangle-free graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq 2\vert V(G)\vert -3$. Moreover, if $G$ has at least three vertices, then $W_{c}(G)\leq 2\vert V(G)\vert -4$.
Note that the upper bound in Theorem \[mytheorem6\] is sharp for simple cycles, since $W_{c}(C_{n})=n$.
\[mytheorem7\] If $G$ is a connected graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq 2\vert
V(G)\vert +\Delta(G)-4$. Moreover, if $G$ has at least three vertices, then $W_{c}(G)\leq 2\vert V(G)\vert +\Delta(G)-5$.
Consider an interval cyclic $W_{c}(G)$-coloring $\alpha$ of $G$. If for each $u\in V(G)$, $S(u,\alpha)$ is an interval of integers, then $G\in \mathfrak{N}$ and $W_{c}(G)\leq W(G)\leq 2\vert V(G)\vert-3
\leq 2\vert V(G)\vert +\Delta(G)-4$, by Theorem \[mytheorem3\] and taking into account that $G$ is a connected graph with at least two vertices. Moreover, if $G$ has at least three vertices, then $W_{c}(G)\leq W(G)\leq 2\vert V(G)\vert-4 \leq 2\vert V(G)\vert
+\Delta(G)-5$, by Theorem \[mytheorem4\].
Now suppose that there exists $v_{0}\in V(G)$ such that $S(v_{0},\alpha)$ is not an interval of integers. Since $\alpha$ is an interval cyclic $W_{c}(G)$-coloring of $G$, for each $v\in V(G)$ such that $S(v,\alpha)$ is not an interval, there are colors $k_{v}$ and $l_{v}$ such that
$S(v,\alpha)=\{1,\ldots,k_{v}\}\cup
\{W_{c}(G)-l_{v}+1,\ldots,W_{c}(G)\}$.
Let $l^{\star}=\max_{\begin{subarray}{1}
v\in V(G),\\
S(v,\alpha)~is~not~an~interval \end{subarray}} l_{v}$. Clearly, $1\leq l^{\star}\leq \Delta(G)-1$. Define an auxiliary graph $H$ as follows:
$V(H)=V(G)$ and\
$E(H)=E(G)\setminus \{e\colon\, e\in E(G)\wedge \alpha(e)\in
\{W_{c}(G)-l^{\star}+1,\ldots,W_{c}(G)\}\}$.
Clearly, $H$ is a spanning subgraph of $G$. Now let us consider the restriction of the coloring $\alpha$ on the edges of the subgraph $H$ of $G$. Let $\alpha_{H}$ be this edge-coloring. It is easy to see that $\alpha_{H}$ is an interval $(W_{c}(G)-l^{\star})$-coloring of $H$. Since $H\in \mathfrak{N}$, by Lemma \[mylemma2\], we have
$W_{c}(G)-l^{\star}\leq W(H)\leq 2\vert V(H)\vert-3\cdot c(H)\leq
2\vert V(H)\vert-3=2\vert V(G)\vert-3$.
Moreover, if $G$ has at least three vertices, then
$W_{c}(G)-l^{\star}\leq W(H)\leq 2\vert V(H)\vert-4\cdot c(H)\leq
2\vert V(H)\vert-4=2\vert V(G)\vert-4$.
This implies that $W_{c}(G)\leq 2\vert V(G)\vert+l^{\star}-3$. From this and taking into account that $l^{\star}\leq \Delta(G)-1$, we obtain $W_{c}(G)\leq 2\vert V(G)\vert +\Delta(G)-4$. Moreover, if $G$ has at least three vertices, then $W_{c}(G)\leq 2\vert V(G)\vert
+\Delta(G)-5$. $\square$
\[mycorollary4\] If $G$ is a connected graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq 3\vert
V(G)\vert -5$. Moreover, if $G$ has at least three vertices, then $W_{c}(G)\leq 3\vert V(G)\vert -6$.
Note that the first upper bound in Theorem \[mytheorem7\] is sharp for $K_{2}$ and the second upper bound is sharp for $K_{3}$, but we strongly believe that these upper bounds can be improved. Next we give some upper bounds for $W_{c}(G)$ depending on degrees and diameter of the interval cyclically colorable connected graph $G$.
\[mytheorem8\] If $G$ is a connected graph and $G\in
\mathfrak{N}_{c}$, then
$W_{c}(G)\leq 1+2\cdot{\max\limits_{P\in
\mathbf{P}}}{\sum\limits_{v\in V(P)}}\left(d_{G}(v)-1\right)$,
where $\mathbf{P}$ is the set of all shortest paths in the graph $G$.
Consider an interval cyclic $W_{c}(G)$-coloring $\alpha $ of $G$. Let us show that $W_{c}(G)\leq
1+2\cdot{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)$. Suppose, to the contrary, that $W_{c}(G)> 1+2\cdot{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)$. In the coloring $\alpha$ of $G$, we consider the edges with colors $1$ and $2+{\max\limits_{P\in
\mathbf{P}}}{\sum\limits_{v\in V(P)}}\left(d_{G}(v)-1\right)$. Let $e=u_{1}u_{2}, e^{\prime}=w_{1}w_{2}$ and $\alpha(e)=1,
\alpha(e^{\prime})=2+{\max\limits_{P\in
\mathbf{P}}}{\sum\limits_{v\in V(P)}}\left(d_{G}(v)-1\right)$. Without loss of generality we may assume that a shortest path $P$ joining $e$ with $e^{\prime}$ joins $u_{1}$ with $w_{1}$, where
$P=\left(v_{0},e_{1},v_{1},\ldots ,v_{i-1},e_{i},v_{i},\ldots
,v_{k-1},e_{k},v_{k}\right)$ and $v_{0}=u_{1}$, $v_{k}=w_{1}$.
Since $\alpha$ is an interval cyclic coloring of $G$, we have
either $\alpha(e_{1})\leq d_{G}(v_{0})$ or $\alpha(e_{1})\geq
W_{c}(G)-d_{G}(v_{0})+2$,
either $\alpha(e_{2})\leq \alpha(e_{1})+d_{G}(v_{1})-1$ or $\alpha(e_{2})\geq \alpha(e_{1})-d_{G}(v_{1})+1$,
$\cdots \cdots \cdots \cdots \cdots \cdots$
either $\alpha(e_{i})\leq \alpha(e_{i-1})+d_{G}(v_{i-1})-1$ or $\alpha(e_{i})\geq \alpha(e_{i-1})-d_{G}(v_{i-1})+1$,
$\cdots \cdots \cdots \cdots \cdots \cdots$
either $\alpha(e_{k})\leq \alpha(e_{k-1})+d_{G}(v_{k-1})-1$ or $\alpha(e_{k})\geq \alpha(e_{k-1})-d_{G}(v_{k-1})+1$.
Summing up these inequalities, we obtain
either $\alpha(e_{k})\leq
1+{\sum\limits_{j=0}^{k-1}\left(d_{G}(v_{j})-1\right)}$ or $\alpha(e_{k})\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{k-1}\left(d_{G}(v_{j})-1\right)}$.
Hence, we have either $$\label{eq:1}
\alpha(e^{\prime})\leq \alpha(e_{k})+d_{G}(v_{k})-1\leq
1+{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}$$ or $$\label{eq:2}
\alpha(e^{\prime})\geq \alpha(e_{k})-d_{G}(v_{k})+1\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}.$$
On the other hand, by (\[eq:1\]), we obtain
$2+{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)=\alpha(e^{\prime})\leq
1+{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}\leq
1+{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)$, which is a contradiction.\
Similarly, by (\[eq:2\]), we obtain
$2+{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)=\alpha(e^{\prime})\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}\geq
W_{c}(G)+1-{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)$ and thus $W_{c}(G)\leq
1+2\cdot{\max\limits_{P\in \mathbf{P}}}{\sum\limits_{v\in
V(P)}}\left(d_{G}(v)-1\right)$, which is a contradiction. $\square$
\[mycorollary5\]([@b23]). If $G$ is a connected graph and $G\in \mathfrak{N}_{c}$, then
$W_{c}(G)\leq 1+2(\mathrm{diam}(G)+1)\left(\Delta(G)-1\right)$.
\[mytheorem9\] If $G$ is a connected bipartite graph and $G\in
\mathfrak{N}_{c}$, then
$W_{c}(G)\leq 1+2\cdot \mathrm{diam}(G)\left(\Delta(G)-1\right)$.
Consider an interval cyclic $W_{c}(G)$-coloring $\alpha $ of $G$. Let us show that $W_{c}(G)\leq
1+2\cdot \mathrm{diam}(G)\left(\Delta(G)-1\right)$. Suppose, to the contrary, that $W_{c}(G)>1+2\cdot
\mathrm{diam}(G)\left(\Delta(G)-1\right)$. In the coloring $\alpha$ of $G$, we consider the edges with colors $1$ and $2+\mathrm{diam}(G)\left(\Delta(G)-1\right)$. Let $e=u_{1}u_{2},
e^{\prime}=w_{1}w_{2}$ and $\alpha(e)=1,
\alpha(e^{\prime})=2+\mathrm{diam}(G)\left(\Delta(G)-1\right)$. Since for any two edges in a bipartite graph $G$, some two of their endpoints must be at a distance of at most $\mathrm{diam}(G)-1$ from each other, we may assume that there is the path $P$ joining $e$ and $e^{\prime}$ with the length is not greater than $\mathrm{diam}(G)-1$. Also, we may assume that $P$ joining $e$ with $e^{\prime}$ joins $u_{1}$ with $w_{1}$, where
$P=\left(v_{0},e_{1},v_{1},\ldots ,v_{i-1},e_{i},v_{i},\ldots
,v_{k-1},e_{k},v_{k}\right)$ and $v_{0}=u_{1}$, $v_{k}=w_{1}$.
Since $\alpha$ is an interval cyclic coloring of $G$, for $1\leq
i\leq k$, we have
either $\alpha(v_{i-1}v_{i})\leq
1+{\sum\limits_{j=0}^{i-1}\left(d_{G}(v_{j})-1\right)}$ or $\alpha(v_{i-1}v_{i})\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{i-1}\left(d_{G}(v_{j})-1\right)}$.
From this, we have either $$\label{eq:3}
\alpha(e^{\prime})\leq
1+{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}$$ or $$\label{eq:4}
\alpha(e^{\prime})\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}.$$
On the other hand, by (\[eq:3\]) and taking into account that $k\leq \mathrm{diam}(G)-1$, we obtain
$2+\mathrm{diam}(G)\left(\Delta(G)-1\right)=\alpha(e^{\prime})\leq
1+{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}\leq
1+\mathrm{diam}(G)\left(\Delta(G)-1\right)$, which is a contradiction.\
Similarly, by (\[eq:4\]) and taking into account that $k\leq
\mathrm{diam}(G)-1$, we obtain
$2+\mathrm{diam}(G)\left(\Delta(G)-1\right)=\alpha(e^{\prime})\geq
W_{c}(G)+1-{\sum\limits_{j=0}^{k}\left(d_{G}(v_{j})-1\right)}\geq
W_{c}(G)+1-\mathrm{diam}(G)\left(\Delta(G)-1\right)$ and thus $W_{c}(G)\leq 1+2\cdot \mathrm{diam}(G)\left(\Delta(G)-1\right)$, which is a contradiction. $\square$
Now we show that the coefficient 2 in the last upper bounds cannot be improved.
\[mytheorem10\] For any integers $d\geq 2$ and $n\geq 3$, there exists a connected graph $G$ with $\Delta(G)=d$ and $\mathrm{diam}(G)=\left\lfloor\frac{n}{2}\right\rfloor+2$ such that $G\in \mathfrak{N}_{c}$ and $W_{c}(G)=n(d-1)$.
For the proof, we construct a graph $G_{d,n}$ that satisfies the specified conditions. We define a graph $G_{d,n}$ as follows:
$V(G_{d,n})=\{v_{1},\ldots,v_{n}\}\cup
\left\{u^{(i)}_{j}\colon\,1\leq i\leq
n,1\leq j\leq d-2\right\}$ and\
$E(G_{d,n})=\{v_{i}v_{i+1}\colon\, 1\leq i\leq
n-1\}\cup\left\{v_{1}v_{n}\right\}\cup\left\{v_{i}u^{(i)}_{j}\colon\,1\leq
i\leq n,1\leq j\leq d-2\right\}$.
Clearly, $G_{d,n}$ is a connected graph with $\Delta(G_{d,n})=d$ and $\mathrm{diam}(G_{d,n})=\left\lfloor\frac{n}{2}\right\rfloor+2$.
Let us show that $G_{d,n}$ has an interval cyclic $n(d-1)$-coloring.
Define an edge-coloring $\alpha$ of $G_{d,n}$ as follows:
(1)
: for $1\leq i\leq n$ and $1\leq j\leq d-2$, let
$\alpha\left(v_{i}u^{(i)}_{j}\right)=(i-1)(d-1)+j$;
(2)
: for $1\leq i\leq n-1$, let
$\alpha\left(v_{i}v_{i+1}\right)=i(d-1)$ and $\alpha\left(v_{1}v_{n}\right)=n(d-1)$.
It is easy to see that $\alpha$ is an interval cyclic $n(d-1)$-coloring of $G_{d,n}$. This implies that $G_{d,n}\in
\mathfrak{N}_{c}$ and $W_{c}(G_{d,n})\geq n(d-1)$. On the other hand, clearly $W_{c}(G_{d,n})\leq \vert E(G_{d,n})\vert=n(d-1)$ and thus $W_{c}(G_{d,n})=n(d-1)$. $\square$
In the last part of the section we investigate the feasible sets of interval cyclically colorable graphs.
\[mytheorem11\] If $G\in\mathfrak{N}$, then $G\in
\mathfrak{N}_{c}$ and $\left[\Delta(G),W(G)\right]\subseteq F(G)$.
Since any interval $t$-coloring of $G$ is also an interval cyclic $t$-coloring of $G$, we obtain that $G\in
\mathfrak{N}_{c}$.
Assume that $\Delta(G)\leq t\leq W(G)$. Let $\alpha$ be an interval $W(G)$-coloring of $G$. Define an edge-coloring $\beta$ of $G$ as follows: for every $e\in E(G)$, let
$\beta(e)=\left\{
\begin{tabular}{ll}
$(e)$, & if $(e)0$,\\
$t$, & otherwise.\\
\end{tabular}\right.$
It is easy to see that $\beta$ is an interval cyclic $t$-coloring of $G$. $\square$
\[mycorollary6\]([@b23]). If $G\in\mathfrak{N}$, then $G\in
\mathfrak{N}_{c}$ and $w_{c}(G)=\Delta(G)$.
\[mytheorem12\] If $G$ is an Eulerian graph and $\vert
E(G)\vert$ is odd, then $G$ has no interval cyclic $t$-coloring for every even positive integer $t$.
Suppose, to the contrary, that $G$ has an interval cyclic $t$-coloring $\alpha$ for some even positive integer $t$. Since $G$ is an Eulerian graph, $G$ is connected and $d_{G}(v)$ is even for any $v\in V(G)$, by Euler’s Theorem. Since $\alpha$ is an interval cyclic coloring and all degrees of vertices of $G$ are even, we have that for any $v\in V(G)$, the set $S\left(v,\alpha\right)$ contains exactly $\frac{d_{G}(v)}{2}$ even colors and $\frac{d_{G}(v)}{2}$ odd colors. Now let $m_{odd}$ be the number of edges with odd colors in the coloring $\alpha$. By Handshaking lemma, we obtain $m_{odd}=\frac{1}{2}\sum\limits_{v\in
V(G)}\frac{d_{G}(v)}{2}=\frac{\vert E(G)\vert}{2}$. Thus $\vert
E(G)\vert$ is even, which is a contradiction. $\square$
\[mycorollary7\] If $G$ is an interval cyclically colorable Eulerian graph with an odd number of edges and $\chi^{\prime}(G)=\Delta(G)$, then $w_{c}(G)>\chi^{\prime}(G)$.
![Interval cyclic $5$-colorings of $K_{1,1,3}$ and $F$.[]{data-label="K_1,1,3 and Fish graph"}](fig2.eps "fig:"){width="25pc" height="10pc"}\
Fig. \[K\_[1,1,3]{} and Fish graph\] shows the complete tripartite graph $K_{1,1,3}$ and the fish graph $F$ that are smallest interval cyclically colorable Eulerian graphs with an odd number of edges and for which the chromatic index is equal to the maximum degree. Clearly, $\chi^{\prime}(K_{1,1,3})=\chi^{\prime}(F)=4$, but $w_{c}(K_{1,1,3})=w_{c}(F)=5$.
Finally let us consider the simple path $P_{m}$ and the simple cycle $C_{n}$ ($n\geq 3$). Clearly, $P_{m}\in\mathfrak{N}_{c}$ and $w_{c}(P_{m})=\Delta(P_{m})$, $W_{c}(P_{m})=m-1$. Moreover, $F(P_{m})=\left[w_{c}(P_{m}),W_{c}(P_{m})\right]$, so the feasible set of $P_{m}$ is gap-free. Now let us consider the simple cycle $C_{n}$. Clearly, $C_{n}\in\mathfrak{N}_{c}$ and $w_{c}(C_{n})=\chi^{\prime}(C_{n})$, $W_{c}(C_{n})=n$. Moreover, it is not difficult to see that any simple cycle with an odd number of vertices has an interval cyclic $t$-coloring for every odd integer $t$, $3\leq t\leq n$, so, by Theorem \[mytheorem12\], we obtain
\[mycorollary8\] For any odd integer $n\geq 3$, we have $F(C_{n})=\left[3,n\right]_{odd}$.
This corollary implies that for any odd integer $n\geq 5$, the feasible set of $C_{n}$ is not gap-free. A more general result on the feasible set of simple cycles was obtained by Kamalian in [@b15].
\[mytheorem13\]([@b15]). For any integer $n\geq 3$, we have
$F(C_{n})=\left\{
\begin{tabular}{ll}
$\_[odd]{}$, & if $n$ is odd,\\
$\_[even]{}$, & if $n=4k,k$,\\
$\_[even]{}$, & if $n=4k+2,k$.\\
\end{tabular}\right.$
Interval cyclic edge-colorings of complete graphs {#part2}
=================================================
This section is devoted to interval cyclic colorings of complete graphs. In [@b31], Vizing proved the following
\[mytheorem14\] For the complete graph $K_{n}$ ($n\geq 2$), we have
$\chi^{\prime}(K_{n})=\left\{
\begin{tabular}{ll}
$n-1$, & if $n$ is even, \\
$n$, & if $n$ is odd. \\
\end{tabular}\right.$
From Corollary \[mycorollary1\] and Theorem \[mytheorem14\], we obtain that if $n\in \mathbb{N}$, then $K_{2n},K_{2n+1}\in
\mathfrak{N}_{c}$ and $w_{c}(K_{2n})=2n-1,w_{c}(K_{2n+1})=2n+1$. Now let us consider the parameters $W_{c}(K_{2n})$ and $W_{c}(K_{2n+1})$ when $n\in \mathbb{N}$. In [@b24], Petrosyan investigated interval colorings of complete graphs and hypercubes. In particular, he proved the following
\[mytheorem15\] If $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative, then
$W(K_{2n})\geq 4n-2-p-q$.
Moreover, if $2n-1\leq t\leq 4n-2-p-q$, then $K_{2n}$ has an interval $t$-coloring.
\[mycorollary9\] If $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative, then
$W_{c}(K_{2n})\geq 4n-2-p-q$.
Moreover, $\left[2n-1,4n-2-p-q\right]\subseteq F(K_{2n})$.
![The interval cyclic $9$-coloring of $K_{6}$ and the interval cyclic $12$-coloring of $K_{8}$.[]{data-label="K_6 and K_8"}](fig3.eps "fig:"){width="35pc" height="21pc"}\
On the other hand, Corollary \[mycorollary4\] implies that $W_{c}(K_{2n})\leq 6n-6$ for $n\geq 2$. It is not difficult to see that $W(K_{4})=W_{c}(K_{4})=4$. In [@b24], it was proved that $W(K_{6})=7$ and $W(K_{8})=11$, but Fig. \[K\_[6]{} and K\_[8]{}\] shows that $W_{c}(K_{6})\geq 9$ and $W(K_{8})\geq 12$, so $W_{c}(K_{6})>W(K_{6})$ and $W_{c}(K_{8})>W(K_{8})$. In general, we strongly believe that $W_{c}(K_{2n})>W(K_{2n})$ for $n\geq 3$. Now we give a lower bound for $W_{c}(K_{2n+1})$ when $n\in \mathbb{N}$.
\[mytheorem16\] If $n\in \mathbb{N}$, then $W_{c}(K_{2n+1})\geq
3n$.
For the proof, it suffices to construct an interval cyclic $3n$-coloring of $K_{2n+1}$. Let $V(K_{2n+1})=\left\{v_{0},v_{1},\ldots,v_{2n}\right\}$.
Define an edge-coloring $\beta$ of $K_{2n+1}$. For each edge $v_{i}v_{j}\in E(K_{2n+1})$ with $i<j$, define a color $\beta\left(v_{i}v_{j}\right)$ as follows:\
$\beta\left(v_{i}v_{j}\right)=\left\{
\begin{tabular}{ll}
$1$, & if $i=0$, $j=1$;\\
$2n+1$, & if $i=0$, $j=2$;\\
$j-1$, & if $i=0$, $3jn$;\\
$n+1+j$, & if $i=0$, $n+1j2n-2$;\\
$n$, & if $i=0$, $j=2n-1$;\\
$3n$, & if $i=0$, $j=2n$;\\
$i+j-1$, & if $1i$,
$2jn$, $i+jn+1$;\\
$i+j+n-2$, & if $2in-1$, $+2jn$, $i+jn+2$;\\
$n+1+j-i$, & if $3in$, $n+1j2n-2$, $j-in-2$;\\
$j-i+1$, & if $1in$, $n+1j2n$, $j-in$;\\
$2i-1$, & if $2i1+$, $n+1jn+$, $j-i=n-1$;\\
$i+j-1$, & if $+2in$, $n+1+j2n-1$,
$j-i=n-1$;\\
$i+j-2n+1$, & if $n+1in+-1$, $n+2j2n-2$, $i+j3n-1$;\\
$i+j-n$, & if $n+1i2n-1$, $n++1j2n$, $i+j3n$.
\end{tabular}\right.$\
Let us prove that $\beta $ is an interval cyclic $3n$-coloring of $K_{2n+1}$.
Let $G$ be the subgraph of $K_{2n+1}$ induced by $\{v_{1},\ldots,v_{2n}\}$. Clearly, $G$ is isomorphic to $K_{2n}$. This edge-coloring $\beta$ of $K_{2n+1}$ is constructed and based on the interval $(3n-2)$-coloring of $G$ which is described in the proof of Theorem 4 from [@b24]. We use this interval $(3n-2)$-coloring of $G$ and then we shift all colors of the edges of $G$ by one. Let $\alpha$ be this edge-coloring of $G$. Using the property of this edge-coloring which is described in the proof of Corollary 6 from [@b24], we get
1)
: $S\left(v_{1},\alpha\right)=S\left(v_{2},\alpha\right)=[2,2n]$,
2)
: $S\left(v_{i},\alpha\right)=S\left(v_{n+i-2},\alpha\right)=[i,2n-2+i]$ for $3\leq i\leq n$,
3)
: $S\left(v_{2n-1},\alpha\right)=S\left(v_{2n},\alpha\right)=[n+1,3n-1]$.
Now, by the definition of $\beta$, we have
1)
: $S\left(v_{0},\beta\right)=[1,n]\cup [2n+1,3n]$,
2)
: $S\left(v_{1},\beta\right)=[1,2n]$ and $S\left(v_{2},\beta\right)=[2,2n+1]$,
3)
: $S\left(v_{i},\beta\right)=[i-1,2n-2+i]$ and $S\left(v_{n+i-2},\beta\right)=[i,2n-1+i]$ for $3\leq i\leq n$,
4)
: $S\left(v_{2n-1},\beta\right)=[n,3n-1]$ and $S\left(v_{2n},\beta\right)=[n+1,3n]$.
This shows that $\beta $ is an interval cyclic $3n$-coloring of $K_{2n+1}$ and hence $W_{c}(K_{2n+1})\geq 3n$. $\square$
Note that the lower bound in Theorem \[mytheorem16\] is sharp for $K_{3}$, since $W_{c}(K_{3})=3$. On the other hand, Corollary \[mycorollary4\] implies that $W_{c}(K_{2n+1})\leq 6n-3$ for $n\in\mathbb{N}$. It is worth also noting that, in general, the feasible set of $K_{2n+1}$ is not gap-free. For example, $K_{7}$ has an interval cyclic $7$-coloring and, by Theorem \[mytheorem16\], it also has an interval cyclic $9$-coloring, but since $\vert
E\left(K_{7}\right)\vert=21$, Theorem \[mytheorem12\] implies that $K_{7}$ has no interval cyclic $8$-coloring, so $F(K_{7})$ is not gap-free.
Interval cyclic edge-colorings of complete bipartite and tripartite graphs {#part3}
==========================================================================
In this section we show that all complete bipartite and tripartite graphs are interval cyclically colorable. We also obtain some bounds for parameters $W_{c}\left(K_{m,n}\right)$ and $w_{c}\left(K_{l,m,n}\right)$ when $l,m,n\in \mathbb{N}$. In [@b12], Kamalian investigated interval colorings of complete bipartite graphs and trees. In particular, he proved the following
\[mytheorem17\] For any $m,n\in \mathbb{N}$, we have
(1)
: $K_{m,n}\in \mathfrak{N}$,
(2)
: $w\left(K_{m,n}\right)=m+n-\gcd(m,n)$,
(3)
: $W\left(K_{m,n}\right)=m+n-1$,
(4)
: if $w\left(K_{m,n}\right)\leq t\leq W\left(K_{m,n}\right)$, then $K_{m,n}$ has an interval $t$-coloring.
We first prove the theorem on the feasible set of complete bipartite graphs.
\[mytheorem18\] If $\min\{m,n\}=1$, then $w_{c}(K_{m,n})=W_{c}(K_{m,n})=m+n-1$. If $\min\{m,n\}\geq 2$, then $\left[\max\{m,n\},m+n\right]\subseteq F\left(K_{m,n}\right)$.
First note that if $\min\{m,n\}=1$, then $K_{m,n}$ is a star and hence $w_{c}(K_{m,n})=W_{c}(K_{m,n})=m+n-1$.
Assume that $\min\{m,n\}\geq 2$. Let us show that if $\max\{m,n\}\leq t\leq m+n$, then $K_{m,n}$ has an interval cyclic $t$-coloring. By Theorems \[mytheorem11\] and \[mytheorem17\], we have $\left[\max\{m,n\},m+n-1\right]\subseteq
F\left(K_{m,n}\right)$. Now we prove that $K_{m,n}$ has an interval cyclic $(m+n)$-coloring.
Let $V(K_{m,n})=\{u_{1},\ldots,u_{m},v_{1},\ldots,v_{n}\}$ and $E(K_{m,n})=\left\{u_{i}v_{j}\colon\,1\leq i\leq m,1\leq j\leq
n\right\}$.
Define an edge-coloring $\alpha$ of $K_{m,n}$ as follows: for $1\leq
i\leq m$ and $1\leq j\leq n$, let
$\alpha\left(u_{i}v_{j}\right)=\left\{
\begin{tabular}{ll}
$i+j-1$, & if $(i,j)(1,n)$,\\
$m+n$, & otherwise.\\
\end{tabular}\right.$
It is not difficult to see that $\alpha$ is an interval cyclic $(m+n)$-coloring of $K_{m,n}$ and hence $\left[\max\{m,n\},m+n\right]\subseteq F\left(K_{m,n}\right)$ when $\min\{m,n\}\geq 2$. $\square$
\[mycorollary10\] If $\min\{m,n\}=1$, then $w_{c}(K_{m,n})=W_{c}(K_{m,n})=m+n-1$. If $\min\{m,n\}\geq 2$, then $w_{c}(K_{m,n})=\max\{m,n\}$ and $W_{c}(K_{m,n})\geq m+n$.
Now we show that all complete tripartite graphs are interval cyclically colorable.
\[mytheorem19\] For any $l,m,n\in \mathbb{N}$, we have $K_{l,m,n}\in \mathfrak{N}_{c}$ and $w_{c}(K_{l,m,n})\leq l+m+n$.
Without loss of generality we may assume that $l\leq m\leq
n$. Clearly, for the proof, it suffices to construct an interval cyclic $(l+m+n)$-coloring of $K_{l,m,n}$.
Let $V(K_{l,m,n})=\{u_{1},\ldots,u_{m},v_{1},\ldots,v_{n},w_{1},\ldots,w_{l}\}$ and $E(K_{l,m,n})=\left\{u_{i}v_{j}\colon\,1\leq i\leq m,1\leq j\leq
n\right\}\cup \left\{u_{i}w_{j}\colon\,1\leq i\leq m,1\leq j\leq
l\right\}\cup \left\{w_{i}v_{j}\colon\,1\leq i\leq l,1\leq j\leq
n\right\}$.
Define an edge-coloring $\alpha$ of $K_{l,m,n}$ as follows:
(1)
: for $1\leq i\leq m$ and $1\leq j\leq n$, let
$\alpha \left(u_{i}v_{j}\right)=l+i+j-1$;
(2)
: for $1\leq i\leq l$ and $1\leq j\leq n$, let
$\alpha \left(w_{i}v_{j}\right)=i+j-1$;
(3)
: for $1\leq i\leq m$, $1\leq j\leq l$ and $i+j\leq m+1$, let
$\alpha \left(u_{i}w_{j}\right)=l+n+i+j-1$;
(4)
: for $1\leq i\leq m$, $1\leq j\leq l$ and $i+j\geq m+2$, let
$\alpha \left(u_{i}w_{j}\right)=i+j-m-1$.
Let us prove that $\alpha$ is an interval cyclic $(l+m+n)$-coloring of $K_{l,m,n}$.
By the definition of $\alpha$, we have
1)
: for $1\leq i\leq m-l+1$,
$S\left(u_{i},\alpha\right)=[l+i,l+n+i-1]\cup
[l+n+i,2l+n+i-1]=[l+i,2l+n+i-1]$ due to (1) and (3),
2)
: for $m-l+2\leq i\leq m$,
$S\left(u_{i},\alpha\right)=[l+i,l+n+i-1]\cup [l+n+i,l+m+n]\cup
[1,l-m-1+i]=[1,l-m-1+i]\cup [l+i,l+m+n]$ due to (1),(3) and (4),
3)
: for $1\leq i\leq n$,
$S\left(v_{i},\alpha\right)=[l+i,l+m+i-1]\cup [i,l+i-1]=[i,l+m+i-1]$ due to (1) and (2),
4)
: for $1\leq i\leq l$,
$S\left(w_{i},\alpha\right)=[i,n+i-1]\cup [l+n+i,l+m+n]\cup
[1,i-1]=[1,n+i-1]\cup [l+n+i,l+m+n]$ due to (2),(3) and (4).
This implies that $\alpha$ is an interval cyclic $(l+m+n)$-coloring of $K_{l,m,n}$; thus $K_{l,m,n}\in \mathfrak{N}_{c}$ and $w_{c}\left(K_{l,m,n}\right)\leq l+m+n$ for $l,m,n\in \mathbb{N}$. $\square$
\[mycorollary11\] For any $l,m,n\in \mathbb{N}$, we have $K_{l,m,n}\in \mathfrak{N}_{c}$ and $W_{c}(K_{l,m,n})\geq l+m+n$.
Note that the upper bound in Theorem \[mytheorem19\] is sharp for $K_{1,m,n}$ when $m$ and $n$ are odd, since $w_{c}\left(K_{1,1,1}\right)=\chi^{\prime}\left(C_{3}\right)=3$ and for $\max\{m,n\}\geq 3$, $m+n+1\geq
w_{c}\left(K_{1,m,n}\right)>\chi^{\prime}\left(K_{1,m,n}\right)=\Delta\left(K_{1,m,n}\right)=m+n$ due to Corollary \[mycorollary7\]. It is worth also noting that although all complete tripartite graphs are interval cyclically colorable, in [@b9] Grzesik and Khachatrian proved that $K_{1,m,n}$ is interval colorable if and only if $\gcd(m+1,n+1)=1$. This implies that there are infinitely many complete tripartite graphs from the class $\mathfrak{N}_{c}\setminus \mathfrak{N}$.
Interval cyclic edge-colorings of hypercubes {#part4}
============================================
In this section we show that hypercubes $Q_{n}$ are interval cyclically colorable. We also obtain some bounds for the parameter $W_{c}\left(Q_{n}\right)$ when $n\in \mathbb{N}$. In [@b24], Petrosyan investigated interval colorings of complete graphs and hypercubes. In particular, he proved that $Q_{n}\in \mathfrak{N}$ and $w(Q_{n})=n$, $W(Q_{n})\geq\frac{n(n+1)}{2}$ for any $n\in\mathbf{N}$. In the same paper he also conjectured that $W(Q_{n})=\frac{n(n+1)}{2}$ for any $n\in\mathbf{N}$. In [@b26], the authors confirmed this conjecture. This implies that $Q_{n}\in
\mathfrak{N}_{c}$ and $w_{c}(Q_{n})=n$, $W_{c}(Q_{n})\geq\frac{n(n+1)}{2}$. Moreover, by Theorem \[mytheorem11\], we obtain $\left[n,\frac{n(n+1)}{2}\right]\subseteq F(Q_{n})$. On the other hand, since $Q_{n}$ is a connected bipartite graph and taking into account that $\mathrm{diam}\left(Q_{n}\right)=\Delta
\left(Q_{n}\right)=n$, we get $W_{c}\left(Q_{n}\right)\leq 1+2\cdot
\mathrm{diam}\left(Q_{n}\right)\left(\Delta\left(Q_{n}\right)-1\right)=2n^{2}-2n+1$, by Theorem \[mytheorem9\]. So, we have $\frac{n(n+1)}{2}\leq
W_{c}\left(Q_{n}\right)\leq 2n^{2}-2n+1$ for any $n\in\mathbf{N}$. Now we prove a new lower bound for $W_{c}\left(Q_{n}\right)$ which improves $W_{c}(Q_{n})\geq\frac{n(n+1)}{2}$ for $2\leq n\leq 5$.
![The interval cyclic $8$-coloring of $Q_{3}$.[]{data-label="Q_3"}](fig4.eps "fig:"){width="15pc" height="13pc"}\
\[mytheorem20\] For any integer $n\geq 2$, we have $W_{c}\left(Q_{n}\right)\geq 4(n-1)$.
First let us note that for any integer $n\geq 2$, $Q_{n}$ has an interval $(n+1)$-coloring such that for one half of vertices of $Q_{n}$, the set of colors appearing on edges incident to these vertices is an interval $[1,n]$ and for remaining half of vertices of $Q_{n}$, the set of colors appearing on edges incident to these remaining vertices is an interval $[2,n+1]$. It can be easily done by induction on $n$. Also, it is not difficult to see that $W_{c}\left(Q_{2}\right)=W_{c}\left(C_{4}\right)=4$ and $W_{c}\left(Q_{3}\right)\geq 8$ (See Fig. \[Q\_[3]{}\]).
Assume that $n\geq 4$.
For $(i,j)\in \{0,1\}^{2}$, let $Q_{n-2}^{(i,j)}$ be the subgraph of $Q_{n}$ induced by the vertices
$\left\{\left(i,j,\alpha _{3},\ldots ,\alpha _{n}\right)\colon\,
\left(\alpha _{3},\ldots ,\alpha _{n}\right)\in
\left\{0,1\right\}^{n-2}\right\}$.
Each $Q_{n-2}^{(i,j)}$ is isomorphic to $Q_{n-2}$. Let $\varphi$ be an interval $(n-1)$-coloring of $Q_{n-2}^{(0,0)}$ such that for one half of vertices of $Q_{n-2}^{(0,0)}$, the set of colors appearing on edges incident to these vertices be an interval $[1,n-2]$ and for remaining half of vertices of $Q_{n-2}^{(0,0)}$, the set of colors appearing on edges incident to these remaining vertices be an interval $[2,n-1]$.
Let us define an edge-coloring $\psi $ of subgraphs $Q_{n-2}^{(0,1)}$, $Q_{n-2}^{(1,1)}$ and $Q_{n-2}^{(1,0)}$ of $Q_{n}$ as follows:
(1)
: for every edge $\left(0,1,\bar
\alpha\right)\left(0,1,\bar\beta\right)\in
E\left(Q_{n-2}^{(0,1)}\right)$, let
$\psi\left(\left(0,1,\bar
\alpha\right)\left(0,1,\bar\beta\right)\right) =\varphi
\left(\left(0,0,\bar
\alpha\right)\left(0,0,\bar\beta\right)\right)+n-1$;
(2)
: for every edge $\left(1,1,\bar
\alpha\right)\left(1,1,\bar\beta\right)\in
E\left(Q_{n-2}^{(1,1)}\right)$, let
$\psi\left(\left(1,1,\bar
\alpha\right)\left(1,1,\bar\beta\right)\right) =\varphi
\left(\left(0,0,\bar
\alpha\right)\left(0,0,\bar\beta\right)\right)+2n-2$;
(3)
: for every edge $\left(1,0,\bar
\alpha\right)\left(1,0,\bar\beta\right)\in
E\left(Q_{n-2}^{(1,0)}\right)$, let
$\psi\left(\left(1,0,\bar
\alpha\right)\left(1,0,\bar\beta\right)\right) =\varphi
\left(\left(0,0,\bar
\alpha\right)\left(0,0,\bar\beta\right)\right)+3n-3$.
Now we define an edge-coloring $\lambda$ of the graph $Q_{n}$.
For every edge $\tilde{\alpha}\tilde{\beta} \in E\left(
Q_{n}\right)$, let
$\lambda \left(\tilde{\alpha}\tilde{\beta}\right)=\left\{
\begin{tabular}{ll}
$(),$ & if $,V(Q\_[n-2]{}\^[(0,0)]{})$,\\
$(),$ & if $,V(Q\_[n-2]{}\^[(0,1)]{})$ or
$,V(Q\_[n-2]{}\^[(1,1)]{})$ or
$,V(Q\_[n-2]{}\^[(1,0)]{})$,\\
$n-1,$ & if $V(Q\_[n-2]{}\^[(0,0)]{})$,
$V(Q\_[n-2]{}\^[(0,1)]{})$, $S(,)=\[1,n-2\]$,\\
$n,$ & if $V(Q\_[n-2]{}\^[(0,0)]{})$,
$V(Q\_[n-2]{}\^[(0,1)]{})$, $S(,)=\[2,n-1\]$,\\
$2n-2,$ & if $V(Q\_[n-2]{}\^[(0,1)]{})$,
$V(Q\_[n-2]{}\^[(1,1)]{})$, $S(,)=\[n,2n-3\]$,\\
$2n-1,$ & if $V(Q\_[n-2]{}\^[(0,1)]{})$,
$V(Q\_[n-2]{}\^[(1,1)]{})$, $S(,)=\[n+1,2n-2\]$,\\
$3n-3,$ & if $V(Q\_[n-2]{}\^[(1,1)]{})$,
$V(Q\_[n-2]{}\^[(1,0)]{})$, $S(,)=\[2n-1,3n-4\]$,\\
$3n-2,$ & if $V(Q\_[n-2]{}\^[(1,1)]{})$,
$V(Q\_[n-2]{}\^[(1,0)]{})$, $S(,)=\[2n,3n-3\]$,\\
$4n-4,$ & if $V(Q\_[n-2]{}\^[(1,0)]{})$,
$V(Q\_[n-2]{}\^[(0,0)]{})$, $S(,)=\[3n-2,4n-5\]$,\\
$1,$ & if $V(Q\_[n-2]{}\^[(1,0)]{})$,
$V(Q\_[n-2]{}\^[(0,0)]{})$, $S(,)=\[3n-1,4n-4\]$.\end{tabular}\right.$
It is easy to verify that $\lambda$ is an interval cyclic $(4n-4)$-coloring of $Q_{n}$; thus $W_{c}\left(Q_{n}\right)\geq
4(n-1)$ for $n\geq 2$. $\square$
This theorem implies that $W_{c}\left(Q_{2}\right)=4,
W_{c}\left(Q_{3}\right)\geq 8, W_{c}\left(Q_{4}\right)\geq 12$ and $W_{c}\left(Q_{5}\right)\geq 16$. Moreover, it is not difficult to see that $F\left(Q_{2}\right)=[2,4]$, $F\left(Q_{3}\right)=[3,8]$, $[4,12]\subseteq F\left(Q_{4}\right)$ and $[5,16]\subseteq
F\left(Q_{5}\right)$. We strongly believe that the feasible set of $Q_{n}$ is gap-free, but this is an open problem.
Graphs that have no interval cyclic edge-coloring {#part5}
=================================================
In this section we describe two methods for constructing of interval cyclically non-colorable graphs. Our first method is based on trees and it was earlier used for constructing of interval non-colorable graphs in [@b28].
Let $T$ be a tree and $V(T)=\{v_{1},\ldots,v_{n}\}$, $n\geq 2$. Let $P(v_{i},v_{j})$ be a simple path joining $v_{i}$ and $v_{j}$ in $T$, $VP(v_{i},v_{j})$ and $EP(v_{i},v_{j})$ denote the sets of vertices and edges of this path, respectively. Also, let $L(T)=\{v\colon\,v\in V(T)\wedge d_{T}(v)=1\}$. For a simple path $P(v_{i},v_{j})$, define $LP(v_{i},v_{j})$ as follows:
$LP(v_{i},v_{j})=\vert EP(v_{i},v_{j})\vert
+\vert\left\{vw\colon\,vw\in E(T), v\in VP(v_{i},v_{j}), w\notin
VP(v_{i},v_{j})\right\}\vert$.
Define: $M(T)={\max }_{1\leq i<j\leq n}LP(v_{i},v_{j})$. Let us define the graph $\widetilde{T}$ as follows:
$V(\widetilde{T})=V(T)\cup \{u\}$, $u\notin V(T)$, $E(\widetilde{T})=E(T)\cup \{uv\colon\,v\in L(T)\}$.
Clearly, $\widetilde{T}$ is a connected graph with $\Delta(\widetilde{T})=\vert L(T)\vert$. Moreover, if $T$ is a tree in which the distance between any two pendant vertices is even, then $\widetilde{T}$ is a connected bipartite graph.
In [@b14], Kamalian proved the following result.
\[mytheorem20\] If $T$ is a tree, then
(1)
: $T\in \mathfrak{N}_{c}$,
(2)
: $w_{c}(T)=\Delta(G)$,
(3)
: $W_{c}(T)=M(T)$,
(4)
: $F(T)=[w_{c}(T),W_{c}(T)]$.
\[mytheorem21\] If $T$ is a tree and $\vert L(T)\vert \geq
2(M(T)+2)$, then $\widetilde{T}\notin \mathfrak{N}_{c}$.
Suppose, to the contrary, that $\widetilde{T}$ has an interval cyclic $t$-coloring $\alpha$ for some $t\geq \vert L(T)\vert$.
Consider the vertex $u$. Without loss of generality we may assume that $S(u,\alpha)=[1,\vert L(T)\vert]$. Let $v$ and $v^{\prime}$ be two vertices adjacent to $u$ such that $\alpha(uv)=1$ and $\alpha(uv^{\prime})=M(T)+3$. Since $\widetilde{T}-u$ is a tree, there is a unique path $P(v,v^{\prime})$ in $\widetilde{T}-u$ joining $v$ with $v^{\prime}$, where
$P(v,v^{\prime})=(x_{0},e_{1},x_{1},\ldots,x_{i-1},e_{i},x_{i},\ldots,x_{k-1},e_{k},x_{k})$, $x_{0}=v$, $x_{k}=v^{\prime}$.
Since $\alpha$ is an interval cyclic coloring of $G$, for $1\leq
i\leq k$, we have
either $\alpha(x_{i-1}x_{i})\leq
2+{\sum\limits_{j=0}^{i-1}\left(d_{T}(x_{j})-1\right)}$ or $\alpha(x_{i-1}x_{i})\geq
t-{\sum\limits_{j=0}^{i-1}\left(d_{T}(x_{j})-1\right)}$.
From this, we have either $$\label{eq:5}
\alpha(x_{k-1}x_{k})=\alpha(x_{k-1}v^{\prime})\leq
2+{\sum\limits_{j=0}^{k-1}\left(d_{T}(v_{j})-1\right)}=1+LP(v,v^{\prime})\leq
1+M(T)$$ or $$\label{eq:6}
\alpha(x_{k-1}x_{k})=\alpha(x_{k-1}v^{\prime})\geq
t-{\sum\limits_{j=0}^{k-1}\left(d_{T}(v_{j})-1\right)}=t+1-LP(v,v^{\prime})\geq
t+1-M(T).$$
On the other hand, by (\[eq:5\]), we obtain $M(T)+3=\alpha(uv^{\prime})\leq 2+M(T)$, which is a contradiction. Similarly, by (\[eq:6\]), we obtain $M(T)+3=\alpha(uv^{\prime})\geq t-M(T)$ and thus $t\leq 2M(T) +3$, which is a contradiction. $\square$
\[mycorollary12\] If $T$ is a tree in which the distance between any two pendant vertices is even and $\vert L(T)\vert \geq
2(M(T)+2)$, then the bipartite graph $\widetilde{T}$ has no interval cyclic coloring.
![The tree $T$.[]{data-label="Tree"}](fig5.eps "fig:"){width="40pc" height="7pc"}\
Now let us consider the tree $T$ shown in Fig. \[Tree\]. Since $M(T)=18$ and $\vert L(T)\vert=40$, the bipartite graph $\widetilde{T}$ with $\vert V(\widetilde{T})\vert=50$ and $\Delta(\widetilde{T})=40$ has no interval cyclic coloring.
The second method which we consider is based on complete graphs and it was first described in [@b23], but here we prove a more stronger result.
Let $K_{2n+1}$ be a complete graph on $2n+1$ vertices and $V(K_{2n+1})=\left\{v_{1},\ldots,v_{2n+1}\right\}$. For any $m,n\in
\mathbb{N}$, define the graph $K_{2n+1}^{\star m}$ as follows:
$V(K_{2n+1}^{\star m})=V(K_{2n+1})\cup
\left\{u,w_{1},\ldots,w_{m}\right\}$, $E(K_{2n+1}^{\star
m})=E(K_{2n+1})\cup \left\{v_{1}u\right\}\cup
\left\{uw_{i}\colon\,1\leq i\leq m\right\}$.
Clearly, $K_{2n+1}^{\star m}$ is a connected with $\vert
V\left(K_{2n+1}^{\star m}\right)\vert=m+2n+2$ and $\Delta\left(K_{2n+1}^{\star m}\right)=\max\{m+1,2n+1\}$.
\[mytheorem22\] If $n\geq 2$ and $m\geq 6n$, then $K_{2n+1}^{\star m}\notin \mathfrak{N}_{c}$.
Suppose, to the contrary, that $K_{2n+1}^{\star m}$ has an interval cyclic $t$-coloring $\alpha$ for some $t\geq d(u)=6n+1$.
Let $H=K_{2n+1}^{\star m}-w_{1}-w_{2}-\cdots-w_{m}$. Also, let $C=\bigcup_{v\in V(H)}S(v,\alpha)$ and $\overline C=[1,t]\setminus
C$. Since $H$ is connected, it is not difficult to see that either $C$ or $\overline C$ is an interval of integers. Let $\vert C\vert =
t^{\prime}$. Clearly, $t^{\prime}\leq t$. Now let us consider the restriction of the coloring $\alpha$ on the edges of the subgraph $H$ of $K_{2n+1}^{\star m}$. Let $\alpha_{H}$ be this edge-coloring. By rotating of colors of $C$ along the cycle with colors $1,\ldots,t$, we get a new edge-coloring $\alpha_{H}^{\prime}$ of $H$ with colors $1,\ldots,t^{\prime}$. Since $\alpha$ is an interval cyclic $t$-coloring of $K_{2n+1}^{\star m}$ and taking into account that the vertex $u$ in $H$ is pendant, we obtain that $\alpha_{H}^{\prime}$ is an interval cyclic $t^{\prime}$-coloring of $H$. Moreover, by Corollary \[mycorollary4\], we have $t^{\prime}\leq 3\vert V(H)\vert -6=3(2n+2)-6=6n$. Since $t\geq
6n+1$, we get that $\alpha_{H}^{\prime}$ is also an interval $t^{\prime}$-coloring of $H$. In [@b8], it was proved that $H\notin \mathfrak{N}$, so this contradiction proves the theorem. $\square$
\[mycorollary13\] For any integer $d\geq 13$, there exists a connected graph $G$ such that $G\notin \mathfrak{N}_{c}$ and $\Delta(G)=d$.
Now we show that $K_{5}^{\star 11}\notin \mathfrak{N}_{c}$. Note that $\vert V\left(K_{5}^{\star 11}\right)\vert=17$ and $\Delta\left(K_{5}^{\star 11}\right)=12$. Suppose that, to the contrary, that $K_{5}^{\star 11}$ has an interval cyclic $t$-coloring $\alpha$ for some $t\geq 12$. Similarly as in the proof of Theorem \[mytheorem22\], we can consider the subgraph $H=K_{5}^{\star 11}-w_{1}-w_{2}-\cdots-w_{11}$ of $K_{5}^{\star
11}$. Let $t^{\prime}=\left\vert\bigcup_{v\in
V(H)}S(v,\alpha)\right\vert$ and $\alpha_{H}$ be the restriction of the coloring $\alpha$ on the edges of the subgraph $H$ of $K_{5}^{\star 11}$. Then, let $\alpha_{H}^{\prime}$ be the edge-coloring of $H$ with colors $1,\ldots,t^{\prime}$. Since $\alpha$ is an interval cyclic $t$-coloring of $K_{5}^{\star 11}$ and taking into account that the vertex $u$ in $H$ is pendant, it can be easily seen that $\alpha_{H}^{\prime}$ is an interval cyclic $t^{\prime}$-coloring of $H$, where $t^{\prime}\leq t$. Clearly, $t^{\prime}\leq \vert E(H)\vert=11$. Since $t\geq 12$, we obtain that $\alpha_{H}^{\prime}$ is also an interval $t^{\prime}$-coloring of $H$, which is a contradiction. From here, we get the following
\[mycorollary14\] For any integer $d\geq 12$, there exists a connected graph $G$ such that $G\notin \mathfrak{N}_{c}$ and $\Delta(G)=d$.
Problems and Conjectures
========================
In this section we collected different problems and conjectures that arose in previous sections. Our first conjectures concern the parameters $w_{c}(G)$ and $W_{c}(G)$ of an interval cyclically colorable graph $G$. In section \[part1\], we proved that if $G$ is a connected triangle-free graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq \vert V(G)\vert
+\Delta(G)-2$. However, we think that the maximum degree in the upper bound can be omitted; more precisely we believe that the following is true:
\[myconjecture1\] If $G$ is a connected triangle-free graph and $G\in
\mathfrak{N}_{c}$, then $W_{c}(G)\leq \vert V(G)\vert$.
Note that if Conjecture \[myconjecture1\] is true, then this upper bound cannot be improved, since $W_{c}(K_{m,n})\geq m+n$ ($\min\{m,n\}\geq 2$), by Corollary \[mycorollary10\]. We also proved that if $G$ is a connected graph with at least two vertices and $G\in \mathfrak{N}_{c}$, then $W_{c}(G)\leq 2\vert V(G)\vert
+\Delta(G)-4$. We again think that the maximum degree in this upper bound can be omitted; more precisely we believe that the following is true:
\[myconjecture2\] If $G$ is a connected graph with at least two vertices and $G\in
\mathfrak{N}_{c}$, then $W_{c}(G)\leq 2\vert V(G)\vert-3$.
It is worth noting that there exists a connection between Conjecture \[myconjecture1\] and Conjecture \[myconjecture2\]. If one can prove that Conjecture \[myconjecture1\] is true, then we able to show that $W_{c}(G)\leq 2\vert V(G)\vert-1$ for an interval cyclically colorable connected graph $G$.\
It is known that all regular graphs $G$ are interval cyclically colorable and $w_{c}(G)=\chi^{\prime}(G)$. Moreover, if $G$ is interval colorable, then $G$ is interval cyclically colorable and $w_{c}(G)=\chi^{\prime}(G)=\Delta(G)$. On the other hand, in section \[part1\], it was shown that there are many interval cyclically colorable graphs $G$ for which $w_{c}(G)>\chi^{\prime}(G)$. So, it is interesting to investigate the following
\[myproblem1\] Characterize all interval cyclically colorable graphs $G$ for which $w_{c}(G)=\chi^{\prime}(G)$.
In section \[part1\], we also investigated the feasible sets of interval cyclically colorable graphs. In particular, we proved that if $G$ is interval colorable, then $\left[\Delta(G),W(G)\right]\subseteq F(G)$. On the other hand, we gave some examples of interval cyclically graphs $G$ for which $F(G)$ is not gap-free. So, it is interesting to investigate the following
\[myproblem2\] Characterize all interval cyclically colorable graphs $G$ for which $F(G)$ is gap-free.
For example, we know that if $T$ is a tree, then $F(T)$ is gap-free [@b14], but we also strongly believe that for any $m,n\in
\mathbb{N}$, $F(K_{2n})$, $F(K_{m,n})$ and $F(Q_{n})$ are gap-free.\
In sections \[part2\] and \[part3\], we investigated interval cyclic colorings of complete, complete bipartite and tripartite graphs, but the following problems are still open:
\[myproblem3\] What is the exact value of $W_{c}\left(K_{n}\right)$ for any $n\in
\mathbb{N}$?
\[myproblem4\] What are the exact values of $w_{c}\left(K_{l,m,n}\right)$ and $W_{c}\left(K_{m,n}\right)$, $W_{c}\left(K_{l,m,n}\right)$ for any $l,m,n\in \mathbb{N}$?
In sections \[part2\] and \[part3\], we proved that all complete bipartite and tripartite graphs are interval cyclically colorable, but we think that a more general result is true:
\[myconjecture3\] All complete multipartite graphs are interval cyclically colorable.
In section \[part4\], we investigated interval cyclic colorings of hypercubes $Q_{n}$ and proved that $W_{c}\left(Q_{n}\right)=O(n^{2})$, but the following problem remains open:
\[myproblem5\] What is the exact value of $W_{c}\left(Q_{n}\right)$ for any $n\in
\mathbb{N}$?
In [@b23], Nadolski showed that if $G$ is a connected graph with $\Delta(G)=3$, then $G\in \mathfrak{N}_{c}$ and $w_{c}(G)\leq 4$. From here and taking into account that all simple paths and cycles are interval cyclically colorable, we obtain that all subcubic graphs are interval cyclically colorable. On the other hand, in section \[part5\], we proved that for any integer $d\geq 12$, there exists a connected graph $G$ such that $G\notin
\mathfrak{N}_{c}$ and $\Delta(G)=d$. So, it is naturally to consider the following
\[myproblem6\] Is there a connected graph $G$ such that $4\leq \Delta(G)\leq 11$ and $G\notin \mathfrak{N}_{c}$?
We would like to thank the organizers of 7-th Cracow conference on Graph Theory Rytro’ 14for the nice environment and working atmosphere at the conference.
[99]{}
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[^1]: email: pet\_petros@ipia.sci.am
[^2]: email: sargismk@ymail.com
|
---
abstract: 'We report on an all-sky search with the LIGO detectors for periodic gravitational waves in the frequency range $50\,$–$\,1000$ Hz and with the frequency’s time derivative in the range $-\sci{1}{-8}~\mathrm{Hz}~\mathrm{s}^{-1}$ to zero. Data from the fourth LIGO science run (S4) have been used in this search. Three different semi-coherent methods of transforming and summing strain power from Short Fourier Transforms (SFTs) of the calibrated data have been used. The first, known as “StackSlide”, averages normalized power from each SFT. A “weighted Hough” scheme is also developed and used, and which also allows for a multi-interferometer search. The third method, known as “PowerFlux”, is a variant of the StackSlide method in which the power is weighted before summing. In both the weighted Hough and PowerFlux methods, the weights are chosen according to the noise and detector antenna-pattern to maximize the signal-to-noise ratio. The respective advantages and disadvantages of these methods are discussed. Observing no evidence of periodic gravitational radiation, we report upper limits; we interpret these as limits on this radiation from isolated rotating neutron stars. The best population-based upper limit with $95\%$ confidence on the gravitational-wave strain amplitude, found for simulated sources distributed isotropically across the sky and with isotropically distributed spin-axes, is $4.28 \times 10^{-24}$ (near 140 Hz). Strict upper limits are also obtained for small patches on the sky for best-case and worst-case inclinations of the spin axes.'
title: ' All-sky search for periodic gravitational waves in LIGO S4 data '
---
Introduction {#sec:introduction}
============
We report on a search with the LIGO (Laser Interferometer Gravitational-wave Observatory) detectors [@ligo1; @ligo2] for periodic gravitational waves in the frequency range $50\,$–$\,1000$ Hz and with the frequency’s time derivative in the range $-\sci{1}{-8}~\mathrm{Hz}~\mathrm{s}^{-1}$ to zero. The search is carried out over the entire sky using data from the fourth LIGO science run (S4). Isolated rotating neutron stars in our galaxy are the prime target.
Using data from earlier science runs, the LIGO Scientific Collaboration (LSC) has previously reported on searches for periodic gravitational radiation, using a long-period coherent method to target known pulsars [@S1PulsarPaper; @S2TDPaper; @S3S4TDPaper], using a short-period coherent method to target Scorpius X-1 in selected bands and search the entire sky in the $160.0\,$–$\,728.8$ Hz band [@S2FstatPaper], and using a long-period semi-coherent method to search the entire sky in the $200\,$–$\,400$ Hz band [@S2HoughPaper]. Einstein@Home, a distributed home computing effort running under the BOINC architecture [@BOINC], has also been searching the entire sky using a coherent first stage, followed by a simple coincidence stage [@S3EatH]. In comparison, this paper: 1) examines more sensitive data; 2) searches over a larger range in frequency and its derivative; and 3) uses three alternative semi-coherent methods for summing measured strain powers to detect excess power from a continuous gravitational-wave signal.
The first purpose of this paper is to present results from our search for periodic gravitational waves in the S4 data. Over the LIGO frequency band of sensitivity, the S4 all-sky upper limits presented here are approximately an order of magnitude better than published previously from earlier science runs [@S2FstatPaper; @S2HoughPaper]. After following up on outliers in the data, we find that no candidates survive, and thus report upper limits. These are interpreted as limits on radiation from rotating neutron stars, which can be expressed as functions of the star’s ellipticity and distance, allowing for an astrophysical interpretation. The best population-based upper limit with $95\%$ confidence on the gravitational-wave strain amplitude, found for simulated sources distributed isotropically across the sky and with isotropically distributed spin-axes, is $4.28 \times 10^{-24}$ (near 140 Hz). Strict upper limits are also obtained for small patches on the sky for best-case and worst-case inclinations of the spin axes.
The second purpose of this paper, along with the previous coherent [@S2FstatPaper] and semi-coherent [@S2HoughPaper] papers, is to lay the foundation for the methods that will be used in future searches. It is well known that the search for periodic gravitational waves is computationally bound; to obtain optimal results will require a hierarchical approach that uses coherent and semi-coherent stages [@hough04; @pss01; @BC00; @cgk]. A fifth science run (S5), which started in November 2005, is generating data at initial LIGO’s design sensitivity. We plan to search this data using the best methods possible, based on what is learned from this and previous analyses.
In the three methods considered here, one searches for cumulative excess power from a hypothetical periodic gravitational wave signal by examining successive spectral estimates based on Short Fourier Transforms (SFTs) of the calibrated detector strain data channel, taking into account the Doppler modulations of detected frequency due to the Earth’s rotational and orbital motion with respect to the Solar System Barycenter (SSB), and the time derivative of the frequency intrinsic to the source. The simplest method presented, known as “StackSlide” [@BCCS; @BC00; @cgk; @StackSlideTechNote], averages normalized power from each SFT. In the Hough method reported previously [@S2HoughPaper; @hough04], referred to here as “standard Hough”, the sum is of binary zeroes or ones, where an SFT contributes unity if the power exceeds a normalized power threshold. In this paper a “weighted Hough” scheme, henceforth also referred to as “Hough”, has been developed and is similar to that described in Ref. [@Palomba2005]. This scheme also allows for a multi-interferometer search. The third method, known as “PowerFlux” [@PowerFluxTechNote], is a variant of the StackSlide method in which the power is weighted before summing. In both the weighted Hough and PowerFlux methods, the weights are chosen according to the noise and detector antenna pattern to maximize the signal-to-noise ratio.
The Hough method is computationally faster and more robust against large transient power artifacts, but is slightly less sensitive than StackSlide for stationary data [@S2HoughPaper; @StackSlideTechNote]. The PowerFlux method is found in most frequency ranges to have better detection efficiency than the StackSlide and Hough methods, the exceptions occurring in bands with large non-stationary artifacts, for which the Hough method proves more robust. However, the StackSlide and Hough methods can be made more sensitive by starting with the maximum likelihood statistic (known as the $\cal F$-statistic [@jks; @hough04; @S2FstatPaper]) rather than SFT power as the input data, though this improvement comes with increased computational cost. The trade-offs among the methods means that each could play a role in our future searches.
In brief, this paper makes several important contributions. It sets the best all-sky upper limits on periodic gravitational waves to date, and shows that these limits are becoming astrophysically interesting. It also introduces methods that are crucial to the development of our future searches.
This paper is organized as follows: Section \[sec:detectordata\] briefly describes the LIGO interferometers, focusing on improvements made for the S4 data run, and discusses the sensitivity and relevant detector artifacts. Section \[sec:waveforms\] precisely defines the waveforms we seek and the associated assumptions we have made. Section \[sec:analysismethodoverview\] gives a detailed description of the three analysis methods used and summarizes their similarities and differences, while Section \[sec:analysismethoddetails\] gives the details of their implementations and the pipelines used. Section \[sec:valandhwinj\] discusses the validation of the software and, as an end-to-end test, shows the detection of simulated pulsar signals injected into the data stream at the hardware level. Section \[sec:results\] describes the search results, and Section \[sec:comparisonresults\] compares the results from the three respective methods. Section \[sec:summary\] concludes with a summary of the results, their astrophysical implications, and future plans.
The LIGO Detector Network and the S4 Science Run {#sec:detectordata}
================================================
The LIGO detector network consists of a 4-km interferometer in Livingston Louisiana (called L1) and two interferometers in Hanford Washington, one 4-km and another 2-km (H1 and H2, respectively).
The data analyzed in this paper were produced during LIGO’s 29.5-day fourth science run (S4) [@S4detpaper]. This run started at noon Central Standard Time (CST) on February 22 and ended at midnight CST on March 23, 2005. During the run, all three LIGO detectors had displacement spectral amplitudes near $\sci{2.5}{-19}~{\rm m}~{\rm Hz}^{-1/2}$ in their most sensitive frequency band near 150 Hz. In units of gravitational-wave strain amplitude, the sensitivity of H2 is roughly a factor of two worse than that of H1 and L1 over much of the search band. The typical strain sensitivities in this run were within a factor of two of the design goals. Figure \[fig:S4sensitivity\] shows representative strain spectral noise densities for the three interferometers during the run. As discussed in Section \[sec:analysismethoddetails\] below, however, non-stationarity of the noise was significant.
Changes to the interferometers before the S4 run included the following improvements [@S4detpaper]:
- Installation of active seismic isolation of support structures at Livingston to cope with high anthropogenic ground motion in the 1-3 Hz band.
- Thermal compensation with a CO$_2$ laser of mirrors subject to thermal lensing from the primary laser beam to a greater or lesser degree than expected.
- Replacement of a synthesized radio frequency oscillator for phase modulation with a crystal oscillator before S4 began (H1) and mid-way through the S4 run (L1), reducing noise substantially above 1000 Hz and eliminating a comb of $\sim 37$ Hz lines. (The crystal oscillator replacement for H2 occurred after the S4 run.)
- Lower-noise mirror-actuation electronics (H1, H2, & L1).
- Higher-bandwidth laser frequency stabilization (H1, H2, & L1) and intensity stabilization (H1 & L1).
- Installation of radiation pressure actuation of mirrors for calibration validation (H1).
- Commissioning of complete alignment control system for the L1 interferometer (already implemented for H1 & H2 in S3 run).
- Refurbishment of lasers and installation of photodiodes and electronics to permit interferometer operation with increased laser power (H1, H2, & L1).
- Mitigation of electromagnetic interference (H1, H2, & L1) and acoustic interference (L1).
The data were acquired and digitized at a rate of 16384 Hz. Data acquisition was periodically interrupted by disturbances such as seismic transients, reducing the net running time of the interferometers. The resulting duty factors for the interferometers were 81% for H1 and H2, and 74% for L1. While the H1 and H2 duty factors were somewhat higher than those in previous science runs, the L1 duty factor was dramatically higher than the $\simeq$40% typical of the past, thanks to the increased stability from the installation of the active seismic isolation system at Livingston.
![Median amplitude strain noise spectral densities from the three LIGO interferometers during the S4 run, along with the Initial LIGO design sensitivity goal.[]{data-label="fig:S4sensitivity"}](S4sensitivity){height="6.0cm"}
Signal Waveforms {#sec:waveforms}
================
The general form of a gravitational-wave signal is described in terms of two orthogonal transverse polarizations defined as “$+$” with waveform $h_+(t)$ and “$\times$” with waveform $h_\times(t)$. The calibrated response seen by an interferometric gravitational-wave detector is then [@jks] $$h(t) = F_+(t,\alpha,\delta,\psi)h_+(t) + F_\times(t,\alpha,\delta,\psi)h_\times(t) \label{eq:detoutput},$$ where $t$ is time in the detector frame, $\alpha$ is the source right ascension, $\delta$ is the source declination, $\psi$ is the polarization angle of the wave, and $F_{+,\times}$ are the detector antenna pattern functions for the two orthogonal polarizations. For periodic (nearly pure sinusoidal) gravitational waves, which in general are elliptically polarized, the individual components $h_{+,\times}$ have the form $$\begin{aligned}
h_+(t) \quad & = & \hpluszero \cos\Phi(t) , \label{eq:sinusoidCosPhi} \\
h_\times(t) \quad & = & \hcrosszero \sin\Phi(t) , \label{eq:sinusoidSinPhi}\end{aligned}$$ where $\hpluszero$ and $\hcrosszero$ are the amplitudes of the two polarizations, and $\Phi(t)$ is the phase of the signal at the detector. (One can also define the initial phase of the signal, $\Phi_0$, but in this paper it can be taken to be an unknown and irrelevant constant).
For an isolated quadrupolar gravitational-wave emitter, characterized by a rotating triaxial ellipsoid mass distribution, the amplitudes $\hpluszero$ and $\hcrosszero$ are related to the inclination angle of the source, $\iota$, and the wave amplitude, $h_0$, by: $$\begin{aligned}
\hpluszero &=& \frac{1}{2}h_0 \left(1+ \cos^2\iota\right),\\
\hcrosszero &=& h_0 \cos\iota, \end{aligned}$$ where $\iota$ is the angle of its spin axis with respect to the line of sight between source and detector. For such a star, the gravitational-wave frequency, $f$, is twice the rotation frequency, $\nu$, and the amplitude $h_0$ is given by $$\label{eq:h0}
h_0 = \frac{16\pi^2G}{c^4}\frac{I\epsilon \nu^2}{d}\,.$$ Here $d$ is the distance to the star, $I$ is the principal moment of inertia with respect to its spin axis, and $\epsilon$ is the equatorial ellipticity of the star [@jks]. Assuming that all of the frequency’s derivative, $\dot{f}$, is due to emission of gravitational radiation and that $I$ takes the canonical value $10^{38}~{\rm kg}{\rm m}^2$, we can relate $\epsilon$ to $f$ and $\dot{f}$ and use Eq. (\[eq:h0\]) to obtain $$\label{hsd}
h_\mathrm{sd} = 4.54\times10^{-24} \left( \mbox{1 kpc} \over d \right)
\left( \mbox{250 yr} \over -f/(4\dot{f}) \right)^{\frac{1}{2}} ,$$ by eliminating $\epsilon$, or $$\label{esd}
\epsilon_\mathrm{sd} = 7.63\times10^{-5} \left( -\dot{f} \over
10^{-10}~\mathrm{Hz}~\mathrm{s}^{-1} \right)^{\frac{1}{2}} \left( \mbox{100 Hz} \over f
\right)^{\frac{5}{2}} ,$$ by eliminating $d$. These are referred to, respectively, as the *spin-down limits* on strain and ellipticity. (See Eqs. (8), (9), and (19) of [@S2FstatPaper] for more details of the derivation.)
Note that the methods used in this paper are sensitive to periodic signals from any type of isolated gravitational-wave source (e.g., freely precessing or oscillating neutron stars as well as triaxial ones), though we present upper limits in terms of $h_0$ and $\epsilon$. Because we use semi-coherent methods, only the instantaneous signal frequency in the detector reference frame, $2\pi f(t) = d\Phi(t)/dt$, needs to be calculated. In the detector reference frame this can, to a very good approximation, be related to the instantaneous SSB-frame frequency $\fhat(t)$ by [@S2HoughPaper] $$\label{eq:freqatdetector}
f(t) - \hat{f}(t) = \hat{f}(t)\frac{ {\bf v} (t)\cdot\bf{\hat{n}}}{c} ,$$ where ${\bf v}(t)$ is the detector’s velocity with respect to the SSB frame, and $\bf{\hat{n}}$ is the unit-vector corresponding to the sky-location of the source. In this analysis, we search for $\fhat(t)$ signals well described by a nominal frequency $\fhatzero$ at the start of the S4 run $t_0$ and a constant first time derivative $\fdot$, such that $$\hat{f}(t) = \hat f_0 + \fdot\left(t - t_0\right).$$ These equations ignore corrections to the time interval $t - t_0$ at the detector compared with that at the SSB and relativistic corrections. These corrections are negligible for the one month semi-coherent searches described here, though the LSC Algorithm Library (LAL) code [@LAL] used by our searches does provide routines that make all the corrections needed to provide a timing accuracy of 3 $\mu s$. (The LAL code also can calculate $f(t)$ for signals arriving from periodic sources in binary systems. Including unknown orbital parameters in the search, however, would greatly increase the computational cost or require new methods beyond the scope of this article.)
Overview of the Methods {#sec:analysismethodoverview}
=======================
Similarities and Differences {#subsec:simsanddiffs}
----------------------------
The three different analysis methods presented here have many features in common, but also have important differences, both major and minor. In this Section we give a brief overview of the methods.
### The parameter space {#subsubsec:theparamspace}
All three methods are based on summing measures of strain power from many SFTs that have been created from 30-minute intervals of calibrated strain data. Each method also corrects explicitly for sky-position dependent Doppler modulations of the apparent source frequency due to the Earth’s rotation and its orbital motion around the SSB, and the frequency’s time derivative, intrinsic to the source (see Fig. \[fig:StackingAndSlidingGraphic\]). This requires a search in a four-dimensional parameter space; a template in the space refers to a set of values: ${\mathbf \lambda} = \{ \fhatzero, \dot{f}, \alpha, \delta \}$. The third method, PowerFlux, also searches explicitly over polarization angle, so that ${\mathbf \lambda} = \{ \fhatzero, \dot{f}, \alpha, \delta, \psi \}$.
![An illustration of the discrete frequency bins of the Short Fourier Transform (SFTs) of the data are shown vertically, with the discrete start times of the SFTs shown horizontally. The dark pixels represent a signal in the data. Its frequency changes with time due to Doppler shifts and intrinsic evolution of the source. By sliding the frequency bins, the power from a source can be lined up and summed after appropriate weighting or transformation. This is, in essence, the starting point for all of the semi-coherent search methods presented here, though the actual implementations differ significantly.[]{data-label="fig:StackingAndSlidingGraphic"}](StackingAndSlidingGraphics2){height="3.0cm"}
All three methods search for initial frequency $\fhatzero$ in the range $50\,$–$\,1000$ Hz with a uniform grid spacing equal to the size of an SFT frequency bin, $$\label{eq:deltaf}
\delta f = {1 \over \Tcoh} = 5.556 \times 10^{-4} \, {\rm Hz} \,.$$ where $\Tcoh$ is the time-baseline of each SFT. The range of $\fhatzero$ is determined by the noise curves of the interferometers, likely detectable source frequencies [@palomba], and limitations due to the increasing computational cost at high frequencies.
The range of $\dot{f}$ values searched is $[-\sci{1}{-8}$, $\,0]~\mathrm{Hz}~\mathrm{s}^{-1}$ for the StackSlide and PowerFlux methods and $[-\sci{2.2}{-9}$, $\,0]~\mathrm{Hz}~\mathrm{s}^{-1}$ for the Hough method. The ranges of $\dot{f}$ are determined by the computational cost, as well as by the low probability of finding an object with $|\dot{f}|$ higher than the values searched—in other words, the ranges of $\dot{f}$ are narrow enough to complete the search in a reasonable amount of time, yet wide enough to include likely signals. All known isolated pulsars spin down more slowly than the two values of $|\dot{f}|_\max$ used here, and as seen in the results section, the ellipticity required for higher $|\dot{f}|$ is improbably high for a source losing rotational energy primarily via gravitational radiation at low frequencies. A small number of isolated pulsars in globular clusters exhibit slight spin-up, believed to arise from acceleration in the Earth’s direction; such spin-up values have magnitudes small enough to be detectable with the zero-spin-down templates used in these searches, given a strong enough signal. The parameter ranges correspond to a minimum spin-down timescale $f/|4\dot{f}|$ (the gravitational-wave spin-down age) of 40 years for a source emitting at 50 Hz and 800 years for a source at 1000 Hz. Since for known pulsars [@ATNF] this characteristic timescale is at least hundreds of years for frequencies on the low end of our range and tens of millions of years for frequencies on the high end, we see again that the ranges of $|\dot{f}|$ are wide enough to include sources from this population.
As discussed in our previous reports [@S2HoughPaper; @S2FstatPaper], the number of sky points that must be searched grows quadratically with the frequency $\fhatzero$, ranging here from about five thousand at 50 Hz to about two million at 1000 Hz. All three methods use nearly isotropic grids which cover the entire sky. The PowerFlux search also divides the sky into regions according to susceptibility to stationary instrumental line artifacts. Sky grid and spin-down spacings and other details are provided below.
### Upper limits {#subsubsec:themethodandul}
While the parameter space searched is similar for the three methods, there are important differences in the way upper limits are set. StackSlide and Hough both set population-based frequentist limits on $h_0$ by carrying out Monte Carlo simulations of a random population of pulsar sources distributed uniformly over the sky and with isotropically distributed spin-axes. PowerFlux sets strict frequentist limits on circular and linear polarization amplitudes $h_0^{\rm Circ-limit}$ and $h_0^{\rm Lin-limit}$, which correspond to limits on most and least favorable pulsar inclinations, respectively. The limits are placed separately on tiny patches of the sky, with the highest strain upper limits presented here. In this context “strict” means that, regardless of its polarization angle $\psi$ or inclination angle $\iota$, regardless of its sky location (within fiducial regions discussed below), and regardless of its frequency value and spin-down within the frequency and spin-down step sizes of the search template, an isolated pulsar of true strain amplitude $h_0 =2h_0^{\rm Lin-limit}$, would have yielded a higher measured amplitude than what we measure, in at least 95% of independent observations. The circular polarization limits $h_0^{\rm Circ-limit}$ apply only to the most favorable inclinations ($\iota\approx0$, $\pi$), regardless of sky location and regardless of frequency and spin-down, as above.
Due to these different upper limit setting methods, sharp instrumental lines are also handled differently. StackSlide and Hough carry out removal of known instrumental lines of varying widths in individual SFTs. The measured powers in those bins are replaced with random noise generated to mimic the noise observed in neighboring bins. This line cleaning technique can lead to a true signal being missed because its apparent frequency may coincide with an instrumental line for a large number of SFTs. However, population-averaged upper limits are determined self-consistently to include loss of detection efficiency due to line removal, by using Monte Carlo simulations.
Since its limits are intended to be strict, that is, valid for any source inclination and for any source location within its fiducial area, PowerFlux must handle instrumental lines differently. Single-bin lines are flagged during data preparation so that when searching for a particular source an individual SFT bin power is ignored when it coincides with the source’s apparent frequency. If more than 80% of otherwise eligible bins are excluded for this reason, no attempt is made to set a limit on strain power from that source. In practice, however, the 80% cutoff is not used because we have found that all such sources lie in certain unfavorable regions of the sky, which we call “skybands” and which we exclude when setting upper limits. These skybands depend on source frequency and its derivative, as described in Sec. \[subsubsec:skybanding\].
### Data Preparation {#subsubsec:dataprep}
Other differences among the methods concern the data windowing and filtering used in computing Fourier transforms and concern the noise estimation. StackSlide and Hough apply high pass filters to the data above $40 {\rm Hz}$, in addition to the filter used to produce the calibrated data stream, and use Tukey windowing. PowerFlux applies no additional filtering and uses Hann windowing with 50% overlap between adjacent SFT’s. StackSlide and Hough use median-based noise floor tracking [@mohanty02b; @mohanty02a; @badri]. In contrast, Powerflux uses a time-frequency decomposition. Both of these noise estimation methods are described in Sec. \[sec:analysismethoddetails\].
The raw, uncalibrated data channels containing the strain measurements from the three interferometers are converted to a calibrated “$h(t)$” data stream, following the procedure described in [@hoftpaper], using calibration reference functions described in [@S4CalibrationNote]. SFTs are generated directly from the calibrated data stream, using 30-minute intervals of data for which the interferometer is operating in what is known as science-mode. The choice of 30 minutes is a tradeoff between intrinsic sensitivity, which increases with SFT length, and robustness against frequency drift during the SFT interval due to the Earth’s motion, source spin-down, and non-stationarity of the data [@S2HoughPaper]. The requirement that each SFT contain contiguous data at nominal sensitivity introduces duty factor loss from edge effects, especially for the Livingston interferometer ($\simeq$20%) which had typically shorter contiguous-data stretches. In the end, the StackSlide and Hough searches used 1004 SFTs from H1 and 899 from L1, the two interferometers with the best broadband sensitivty. For PowerFlux, the corresponding numbers of overlapped SFTs were 1925 and 1628. The Hough search also used 1063 H2 SFTs. In each case, modest requirements were placed on data quality to avoid short periods with known electronic saturations, unmonitored calibration strengths, and the periods immediately preceding loss of optical cavity resonance.
Definitions And Notation {#subsec:basicdefsandnotation}
------------------------
Let $N$ be the number of SFTs, $\Tcoh$ the time-baseline of each SFT, and $M$ the number of uniformly spaced data points in the time domain from which the SFT is constructed. If the time series is denoted by $x_j$ ($j=0,1,2\ldots M-1$), then our convention for the discrete Fourier transform is $$\label{eq:DFT}
\tilde{x}_k = \Delta t \sum_{j=0}^{M-1}x_j e^{-2\pi {\mathrm i} jk/M} \, ,$$ where $k=0,1,2\ldots (M-1)$, and $\Delta t = \Tcoh/M$. For $0\leq k \leq M/2$, the frequency index $k$ corresponds to a physical frequency of $f_k= k/\Tcoh$.
In each method, the “power” (in units of spectral density) associated with frequency bin $k$ and SFT $i$ is taken to be $$\label{eq:defpower}
P_k^{\iSubSupInd}=\frac{2|\tilde{x}_k^{\iSubSupInd}|^2}{\Tcoh}.$$ It proves convenient to define a normalized power by $$\label{eq:normpower}
\rho_k^{\iSubSupInd} = \frac{P_k^{\iSubSupInd}}{S_k^{\iSubSupInd}} \,.$$ The quantity $S_k^{\iSubSupInd}$ is the single-sided power spectral density of the detector noise at frequency $f_k$, the estimation of which is described below. Furthermore, a threshold, $\rho_{\th}$, can be used to define a *binary count* by [@hough04]: $$\label{eq:1}
n_k^{\iSubSupInd} = \left\{
\begin{array}{ccc}
1 & \textrm{if} & \rho_k^{\iSubSupInd} \geq \rho_{\th} \\
0 & \textrm{if} & \rho_k^{\iSubSupInd} < \rho_{\th}
\end{array} \right.\,.$$
Quantity Description
--------------------------------------- -----------------------------------------------------------------------
$P^\srchTemplateInd_{\iSubSupInd}$ Power for SFT $i$ & template ${\mathbf \lambda}$
$\rho^\srchTemplateInd_{\iSubSupInd}$ Normalized power for SFT $i$ & template ${\mathbf \lambda}$
$n^\srchTemplateInd_{\iSubSupInd}$ Binary count for SFT $i$ & template ${\mathbf \lambda}$
$S^\srchTemplateInd_{\iSubSupInd}$ Power spect. noise density for SFT $i$ & template ${\mathbf \lambda}$
$F_+^{\iSubSupInd}$ $F_+$ at midpoint of SFT $i$ for template ${\mathbf \lambda}$
$F_\times^{\iSubSupInd}$ $F_\times$ at midpoint of SFT $i$ for template ${\mathbf \lambda}$
: Summary of notation used.[]{data-label="tab:BasicDefsAndNotation"}
When searching for a signal using template ${\mathbf \lambda}$ the detector antenna pattern and frequency of the signal are found at the midpoint time of the data used to generate each SFT. Frequency dependent quantities are then evaluated at a frequency index $k$ corresponding to the bin nearest this frequency. To simplify the equations in the rest of this paper we drop the frequency index $k$ and use the notation given in Table \[tab:BasicDefsAndNotation\] to define various quantities for SFT $i$ and template ${\mathbf \lambda}$.
Basic StackSlide, Hough, and PowerFlux Formalism {#subsec:methodsummaryeqns}
------------------------------------------------
We call the detection statistics used in this search the “StackSlide Power”, $P$, the “Hough Number Count”, $n$, and the “PowerFlux Signal Estimator”, $R$. The basic definitions of these quantities are given below.
Here the simple StackSlide method described in [@StackSlideTechNote] is used; the “StackSlide Power” for a given template is defined as $$\label{eq:stackslidepower}
P = {1 \over N} \sum_{i=0}^{N-1} \rho^\srchTemplateInd_{\iSubSupInd} \,,$$ This normalization results in values of $P$ with a mean value of unity and, for Gaussian noise, a standard deviation of $1/\sqrt{N}$. Details about the value and statistics of $P$ in the presence and absence of a signal are given in Appendix \[sec:stackslidepowerandstats\] and [@StackSlideTechNote].
In the Hough search, instead of summing the normalized power, the final statistic used in this paper is a weighted sum of the binary counts, giving the “Hough Number Count”: $$\label{eq:2}
n = \sum_{i=0}^{N-1} w^{\srchTemplateInd}_{\iSubSupInd} n^{\srchTemplateInd}_{\iSubSupInd}\,.$$ where the Hough weights are defined as $$\label{eq:wipropto}
w^{\srchTemplateInd}_{\iSubSupInd} \propto \frac{1}{S^{\srchTemplateInd}_{\iSubSupInd}}\left\{
\left(F_{+}^{\iSubSupInd}\right)^2 +
\left(F_{\times}^{\iSubSupInd}\right)^2\right\},$$ and the weight normalization is chosen according to $$\label{eq:3}
\sum_{i=0}^{N-1} w^{\srchTemplateInd}_{\iSubSupInd} = N\,.$$ With this choice of normalization the Hough Number Count $n$ lies within the range $[0,N]$. Thus, we take a binary count $n^{\srchTemplateInd}_{\iSubSupInd}$ to have greater weight if the SFT $i$ has a lower noise floor and if, in the time-interval corresponding to this SFT, the beam pattern functions are larger for a particular point in the sky. Note that the sensitivity of the search is governed by the ratios of the different weights, not by the choice of overall scale. In the next section we show that these weights maximize the sensitivity, averaged over the orientation of the source. This choice of $w^{\srchTemplateInd}_{\iSubSupInd}$ was originally derived in [@Palomba2005] using a different argument and is similar to that used in the PowerFlux circular polarization projection described next. More about the Hough method is given in [@S2HoughPaper; @hough04].
The PowerFlux method takes advantage of the fact that less weight should be given to times of greater noise variance or smaller detector antenna response to a signal. Noting that power estimated from the data divided by the antenna pattern increases the variance of the data at times of small detector response, the problem reduces to finding weights that minimize the variance, or in other words that maximize the signal-to-noise ratio. The resulting PowerFlux detection statistic is [@PowerFluxTechNote], $$\label{eq:PFRk}
R = {2 \over \Tcoh}
{
\sum_{i=0}^{N-1} W^{\srchTemplateInd}_{\iSubSupInd}
P^\srchTemplateInd_{\iSubSupInd} / (F_{\psi}^{\iSubSupInd})^2
\over
\sum_{i=0}^{N-1} W^{\srchTemplateInd}_{\iSubSupInd}
} ,$$ where the PowerFlux weights are defined as $$\label{eq:PFWeights}
W^\srchTemplateInd_{\iSubSupInd} = [(F_{\psi}^{\iSubSupInd})^2]^2/S^2_{\iSubSupInd} ,$$ and where $$\label{PFFpsi}
(F_{\psi}^{\iSubSupInd})^2 = \left\{
\begin{array}{cc}
(F_{+}^{\iSubSupInd})^2 & \textrm{linear polarization} \\
(F_{+}^{\iSubSupInd})^2 + (F_{\times}^{\iSubSupInd})^2 & \textrm{circular polarization}
\end{array} \right.\,.$$ As noted previously, the PowerFlux method searches using four linear polarization projections and one circular polarization projection. For the linear polarization projections, note that $(F_{+}^{\iSubSupInd})^2$ is evaluated at the angle $\psi$, which is the same as $(F_{\times}^{\iSubSupInd})^2$ evaluated at the angle $\psi - \pi/4$; for circular polarization, the value of $(F_{+}^{\iSubSupInd})^2 + (F_{\times}^{\iSubSupInd})^2$ is independent of $\psi$. Finally note that the factor of $2/\Tcoh$ in Eq. (\[eq:PFRk\]) makes $R$ dimensionless and is chosen to make it directly related to an estimate of the squared amplitude of the signal for the given polarization. Thus $R$ is also called in this paper the “PowerFlux Signal Estimator”. (See [@PowerFluxTechNote] and Appendix \[sec:polarization\] for further discussion.)
We have shown in Eqs. (\[eq:stackslidepower\])-(\[PFFpsi\]) how to compute the detection statistic (or signal estimator) for a given template. The next section gives the details of the implementation and pipelines used, where these quantities are calculated for a set of templates ${\mathbf \lambda}$ and analyzed.
Implementations and Pipelines {#sec:analysismethoddetails}
=============================
Running Median Noise Estimation {#subsec:runningmedian}
-------------------------------
The implementations of the StackSlide and Hough methods described below use a “running median” to estimate the mean power and, from this estimate, the power spectral density of the noise, for every frequency bin of every SFT. PowerFlux uses a different noise decomposition method described in its implementation section below.
Note that for Gaussian noise, the single-sided power spectral density can be estimated using $$\label{eq:Sn}
S_k^{\iSubSupInd} \cong
\frac{2 \langle |\tilde{x}_k^{\iSubSupInd}|^2\rangle}{\Tcoh}$$ where the angle brackets represent an ensemble average. The estimation of $S_k^{\iSubSupInd}$ must guard against any biases introduced by the presence of a possible signal and also against narrow spectral disturbances. For this reason the mean, $\langle|\tilde{x}^{\iSubSupInd}_k|^2\rangle$, is estimated via the median. We assume that the noise is stationary within a single SFT, but allow for non-stationarities across different SFTs. In every SFT we calculate the “running median” of $|\tilde{x}_k^{\iSubSupInd}|^2$ for every $101$ frequency bins centered on the $k^{\mathrm{th}}$ bin, and then estimate $\langle
|\tilde{x}_k^{\iSubSupInd}|^2\rangle$ [@mohanty02b; @mohanty02a; @badri] by dividing by the expected ratio of the median to the mean.
Note, however, that in the StackSlide search, after the estimated mean power is used to compute $S_k^{\iSubSupInd}$ in the denominator of Eq. (\[eq:normpower\]) these terms are summed in Eq. (\[eq:stackslidepower\]), while the Hough search applies a cutoff to obtain binary counts in Eq. (\[eq:1\]) before summing. This results in the use of a different correction to get the mean in the StackSlide search from that used in the Hough search. For a running median using 101 frequency bins, the effective ratio of the median to mean used in the StackSlide search was $0.691162$ (which was chosen to normalize the data so that the mean value of the StackSlide Power equals one) compared with the expected ratio for an exponential distribution of $0.698073$ used in the Hough search (which is explained in Appendix A of [@S2HoughPaper]). It is important to realize that the results reported here are valid independent of the factor used, since any overall constant scaling of the data does not affect the selection of outliers or the reported upper limits, which are based on Monte Carlo injections subjected to the same normalization.
The StackSlide Implementation {#subsec:stackslide}
-----------------------------
### Algorithm and parameter space {#subsubsec:stackslideimpl}
The StackSlide method uses power averaging to gain sensitivity by decreasing the variance of the noise [@BCCS; @BC00; @cgk; @StackSlideTechNote]. Brady and Creighton [@BC00] first described this approach in the context of gravitational-wave detection as a part of a hierarchical search for periodic sources. Their method consists of averaging the power from a demodulated time series, but as an approximation did not include the beam pattern response of the detector. In Ref. [@StackSlideTechNote], a simple implementation is described that averages the normalized power given in Eq. (\[eq:normpower\]). Its extension to averaging the maximum likelihood statistic (known as the $\cal F$-statistic) which does include the beam pattern response is mentioned in Ref. [@StackSlideTechNote] (see also [@jks; @hough04; @S2FstatPaper]), and further extensions of the StackSlide method are given in [@cgk].
As noted above, the simple StackSlide method given in [@StackSlideTechNote] is used here and the detection statistic, called the “StackSlide Power”, is defined by Eq. (\[eq:stackslidepower\]). The normalization is chosen so that the mean value of $P$ is equal to $1$ and its standard deviation is $1/\sqrt{N}$ for Gaussian noise alone. For simplicity, the StackSlide Power signal-to-noise ratio (in general the value of $P$ minus its mean value and then divided by the standard deviation of $P$) will be defined in this paper as $(P - 1)\sqrt{N}$, even for non-Gaussian noise.
![Flow chart for the pipeline used to find the upper limits presented in this paper using the StackSlide method.[]{data-label="fig:StackSlideFlowChart"}](StackSlideFlowChart5){height="6.5cm"}
The StackSlide code, which implements the method described above, is part of the C-based LSC Algorithms Library Applications (LALapps) stored in the lscsoft CVS repository [@LAL]. The code is run in a pipeline with options set to produce the results from a search and from Monte Carlo simulations. Parallel jobs are run on computer clusters within the LSC, in the Condor environment [@condor], and the final post processing steps are performed using Matlab [@matlab]. The specific StackSlide pipeline used to find the upper limits presented in this paper is shown in Fig. \[fig:StackSlideFlowChart\]. The first three boxes on the left side of the pipeline can also be used to output candidates for follow-up searches.
A separate search was run for each successive $0.25$ Hz band within $50-1000$ Hz. The spacing in frequency used is given by Eq. (\[eq:deltaf\]). The spacing in $\dot{f}$ was chosen as that which changes the frequency by one SFT frequency bin during the observation time $T_{\rm obs}$, i.e., so that $\dot{f} T_{\rm obs} = \delta f$. For simplicity $\Tobs = 2.778 \times 10^{6}$ seconds $\simeq 32.15$ days was chosen, which is greater than or equal to $\Tobs$ for each interferometer. Thus, the $\dot{f}$ part of the parameter space was over-covered by choosing $$\label{eq:deltafdot}
|\delta \dot{f}| = {\delta f \over \Tobs} = {1 \over \Tcoh \Tobs} = 2 \times 10^{-10} \,
\mathrm{Hz}~\mathrm{s}^{-1} \,.$$ Values of $\dot{f}$ in the range $[-1 \times 10^{-8}~\mathrm{Hz}~\mathrm{s}^{-1}, 0~\mathrm{Hz}~\mathrm{s}^{-1}]$ were searched. This range corresponds to a search over $51$ values of $\dot{f}$, which is the same as PowerFlux used in its low-frequency search (discussed in Section. \[subsec:powerflux\]).
The sky grid used is similar to that used for the all-sky search in [@S2FstatPaper], but with a spacing between sky-grid points appropriate for the StackSlide search. This grid is isotropic on the celestial sphere, with an angular spacing between points chosen for the $50$-$225$ Hz band, such that the maximum change in Doppler shift from one sky grid point to the next would shift the frequency by half a bin. This is given by $$\label{eq:deltatheta}
\delta \theta_0 = {0.5\, c\, \delta f \over \hat{f} (v \,{\rm sin}\theta)_{\rm max}}
= 9.3 \times 10^{-3} \, {\rm rad} \left ( {300 {\rm Hz} \over \hat{f}} \right ) \,,$$ where $v$ is the magnitude of the velocity $\mathbf{v}$ of the detector in the SSB frame, and $\theta$ is the angle between $\mathbf{v}$ and the unit-vector $\mathbf{\hat{n}}$ giving the sky-position of the source. Equations (\[eq:deltafdot\]) and (\[eq:deltatheta\]) are the same as Eqs. (19) and (22) in [@S2HoughPaper], which represent conservative choices that over-cover the parameter space. Thus, the parameter space used here corresponds to that in Ref. [@S2HoughPaper], adjusted to the S4 observation time, and with the exception that a stereographic projection of the sky is not used. Rather an isotropic sky grid is used like the one used in [@S2FstatPaper].
One difficulty is that the computational cost of the search increases quadratically with frequency, due to the increasing number of points on the sky grid. To reduce the computational time, the sky grid spacing given in Eq. (\[eq:deltatheta\]) was increased by a factor of $5$ above $225$ Hz. This represents a savings of a factor of $25$ in computational cost. It was shown through a series of simulations, comparing the upper limits in various frequency bands with and without the factor of 5 increase in grid spacing, that this changes the upper limits on average by less than than $0.3\%$, with a standard deviation of $2\%$. Thus, this factor of $5$ increase was used to allow the searches in the $225-1000$ Hz band to complete in a reasonable amount of time.
It is not surprising that the sky grid spacing can be increased, for at least three reasons. First, the value for $\delta \theta_0 $ given in Eq. (\[eq:deltatheta\]) applies to only a small annular region on the sky, and is smaller than the average change. Second, only the net change in Doppler shift during the observation time is important, which is less than the maximum Doppler shift due to the Earth’s orbital motion during a one month run. (If the Doppler shift were constant during the entire observation time, one would not need to search sky positions even if the Doppler shift varied across the sky. A source frequency would be shifted by a constant amount during the observation, and would be detected, albeit in a frequency bin different from that at the SSB.) Third, because of correlations on the sky, one can detect a signal with negligible loss of SNR much farther from its sky location than the spacing above suggests.
### Line cleaning {#subsubsec:stackslidelinecleaning}
![The StackSlide Power for the $145-155$ Hz band with no sliding. Harmonics of 1 Hz instrumental lines are clearly seen in H1 (top) and L1 (bottom). These lines are removed from the data by the StackSlide and Hough searches using the method described in the text, while PowerFlux search tracks these lines and avoids them when setting upper limits.[]{data-label="fig:S4H1StackSlidePowerOneHzLines145to155Hz"}](S4H1L1StackSlidePowerOneHzLines145to155Hz){height="6.5cm"}
![The L1 amplitude spectral density in a narrow frequency band estimated from 10 SFTs before and after the line cleaning used by the StackSlide pipeline. In the band shown, the $150$ Hz bin, and one bin either side of this bin have been replaced with estimates of the noise based on neighboring bins.[]{data-label="fig:S4L1StackSlideWithoutAndWithCleaning150Hz"}](S4L1StackSlideWithoutAndWithCleaning150Hz){height="6.5cm"}
----- ----------------- ---------------- ------ ----------------------- ------------------------ -------------
IFO $f_{\rm start}$ $f_{\rm step}$ Num. $\Delta f_{\rm left}$ $\Delta f_{\rm right}$ Description
Hz Hz Hz Hz
H1 46.7 — 1 0.0 0.0 Cal. Line
H1 393.1 — 1 0.0 0.0 Cal. Line
H1 973.3 — 1 0.0 0.0 Cal. Line
H1 1144.3 — 1 0.0 0.0 Cal. Line
H1 0.0 1.0 1500 0.0006 0.0006 1 Hz Comb
L1 54.7 — 1 0.0 0.0 Cal. Line
L1 396.7 — 1 0.0 0.0 Cal. Line
L1 1151.5 — 1 0.0 0.0 Cal. Line
L1 0.0 1.0 1500 0.0006 0.0006 1 Hz Comb
----- ----------------- ---------------- ------ ----------------------- ------------------------ -------------
: Instrumental lines cleaned during the StackSlide search. The frequencies cleaned are found by starting with that given in the first column, and then taking steps in frequency given in the second column, repeating this the number of times shown in the third column; the fourth and fifth columns show how many additional Hz are cleaned to the immediate left and right of each line. []{data-label="tab:StackSlideCleanedLines"}
Coherent instrumental lines exist in the data which can mimic a continuous gravitational-wave signal for parameter space points that correspond to little Doppler modulation. Very narrow instrumental lines are removed (“cleaned”) from the data. In the StackSlide search, a line is considered “narrow” if its full width is less than $5\%$ of the $0.25$ Hz band, or less than $0.0125$ Hz. The line must also have been identified [*a priori*]{} as a known instrument artifact. Known lines with less than this width were cleaned by replacing the contents of bins corresponding to lines with random values generated by using the running median to find the mean power using 101 bins from either side of the lines. This method is also used to estimate the noise, as described in Section \[subsec:runningmedian\].
It was found when characterizing the data that a comb of narrow $1$ Hz harmonics existed in the H1 and L1 data, as shown in Fig. \[fig:S4H1StackSlidePowerOneHzLines145to155Hz\]. Table \[tab:StackSlideCleanedLines\] shows the lines cleaned during the StackSlide search. As the table shows, only this comb of narrow $1$ Hz harmonics and injected lines used for calibration were removed. As an example of the cleaning process, Fig. \[fig:S4L1StackSlideWithoutAndWithCleaning150Hz\] shows the amplitude spectral density estimated from $10$ SFTs before and after line cleaning, for the band with the $1$ Hz line at $150$ Hz.
----------------------------------- -----------------------
Excluded Bands Description
Hz
$[57, 63)$ Power lines
$[n60-1,n60 + 1)$ $n = 2$ to $16$ Power line harmonics
$[340, 350)$ Violin modes
$[685, 690)$ Violin mode harmonics
$[693, 696)$ Violin mode harmonics
----------------------------------- -----------------------
: Frequency bands excluded from the StackSlide search.[]{data-label="tab:StackSlideExcludedBands"}
The cleaning of very narrow lines has a negligible effect on the efficiency to detect signals. Very broad lines, on the other hand, cannot be handled in this way. Bands with very broad lines were searched without any line cleaning. There were also a number of highly disturbed bands, dominated either by the harmonics of $60$ Hz power lines or by the violin modes of the suspended optics, that were excluded from the StackSlide results. (Violin modes refer to resonant excitations of the steel wires that support the interferometer mirrors.) These are shown in Table \[tab:StackSlideExcludedBands\]. While these bands can be covered by adjusting the parameters used to find outliers and set upper limits, we will wait for future runs to do this.
### Upper limits method {#subsubsec:stackslideulmethod}
After the lines are cleaned, the powers in the SFTs are normalized and the parameter space searched, with each template producing a value of the StackSlide Power, defined in Eq. (\[eq:stackslidepower\]). For this paper, only the “loudest” StackSlide Power is kept, resulting in a value $P_{\rm max}$ for each $0.25$ Hz band, and these are used to set upper limits on the gravitational-wave amplitude, $h_0$. (The loudest coincident outliers are also identified, but none survive as candidates after follow-up studies described in Sec. \[subsubsec:stackslidelpsandoutliers\].) The upper limits are found by a series of Monte Carlo simulations, in which signals are injected in software with a fixed value for $h_0$, but with otherwise randomly chosen parameters, and the parameter space points that surround the injection are searched. The number of times the loudest StackSlide Power found during the Monte Carlo simulations is greater than or equal to $P_{\rm max}$ is recorded, and this is repeated for a series of $h_0$ values. The $95\%$ confidence upper limit is defined to be the value of $h_0$ that results in a detected StackSlide Power greater than or equal to $P_{\rm max}$ $95\%$ of the time. As shown in Fig. \[fig:StackSlideFlowChart\], the line cleaning described above is done after each injection is added to the input data, which folds any loss of detection efficiency due to line cleaning into the upper limits self-consistently.
![Measure confidence vs. $h_0$ for an example band ($140-140.25$ Hz in H1). A best-fit straight line is used to find the value of $h_0$ corresponding to $95\%$ confidence and to estimate the uncertainties in the results (see text).[]{data-label="fig:S4H1Exampleh0vsconf"}](S4H1StackSlideh0vsconf140to140p25Hz){height="6.5cm"}
Figure \[fig:S4H1Exampleh0vsconf\] shows the measured confidence versus $h_0$ for an example frequency band. The upper limit finding process involves first making an initial guess of its value, then refining this guess using a single set of injections to find an estimate of the upper limit, and finally using this estimate to run several sets of injections to find the final value of the upper limit. These steps are now described in detail.
To start the upper limit finding process, first an initial guess, $h_0^{\rm guess}$, is used as the gravitational-wave amplitude. The initial guess need not be near the sought-after upper limit, just sufficiently large, as explained below. A single set of $n$ injections is done (specifically $n=3000$ was used) with random sky positions and isotropically distributed spin axes, but all with amplitude $h_0^{\rm guess}$. The output list of StackSlide Powers from this set of injections is sorted in ascending order and the $0.05n$’th (specifically for $n=3000$ the $150$th) smallest value of the StackSlide Power is found, which we call $P_{0.05}$, Note that the goal is to find the value of $h_0$ that makes $P_{0.05} = P_{\rm max}$, so that $95\%$ of the output powers are greater than the maximum power found during the search. This is what we call the $95\%$ confidence upper limit. Of course, in general $P_{0.05}$ will not equal $P_{\rm max}$ unless our first guess was very lucky. However, as per the discussion concerning Eq. (\[eq:eststackslidepwr\]), $P - 1$ is proportional to $h_0^2$ (i.e, removing the mean value due to noise leaves on average the power due to the presence of a signal). Thus, an estimate of the $95\%$ $h_0$ confidence upper limits is given by the following rescaling of $h_0^{\rm guess}$, $$\label{eq:StackSlideEstUL}
h_0^{\rm est} = { \sqrt{P_{\rm max} - 1} \over \sqrt{P_{0.05} - 1} } h_0^{\rm guess} \,.$$ Thus an estimated upper limit, $h_0^{\rm est}$, is found from a single set of injections with amplitude $h_0^{\rm guess}$; the only requirement is that $h_0^{\rm guess}$ is chosen loud enough to make $P_{0.05} > 1$.
It is found that using Eq. (\[eq:StackSlideEstUL\]) results in a estimate of the upper limit that is typically within $10\%$ of the final value. For example, the estimated upper limit found in this way is indicated by the circled point in Fig. \[fig:S4H1Exampleh0vsconf\]. The value of $h_0^{\rm est}$ then becomes the first value for $h_0$ in a series of Monte Carlo simulations, each with $3000$ injections, which use this value and $8$ neighboring values, measuring the confidence each time. The Matlab [@matlab] polyfit and polyval functions are then used to find the best-fit straight line to determine the value of $h_0$ corresponding to $95\%$ confidence and to estimate the uncertainties in the results. This is the final step of the pipeline shown in Fig. \[fig:StackSlideFlowChart\].
The Hough Transform Implementation {#subsec:hough}
----------------------------------
### Description of Algorithm
The Hough transform is a general method for pattern recognition, invented originally to analyze bubble chamber pictures from CERN [@hough1; @hough2]; it has found many applications in the analysis of digital images [@ik]. This method has already been used to analyze data from the second science run (S2) of the LIGO detectors [@S2HoughPaper] and a detailed description can be found in [@hough04]. Here we present only a brief description, emphasizing the differences between the previous S2 search and the S4 search described here.
The Hough search uses a weighted sum of the binary counts as its final statistic, as given by Eqs. (\[eq:1\]) and (\[eq:3\]). In the standard Hough search as presented in [@hough04; @S2HoughPaper], the weights are all set to unity. The weighted Hough transform was originally discussed in [@Palomba2005]. The software for performing the Hough transform has been adapted to use arbitrary weights without any significant loss in computational efficiency. Furthermore, the robustness of the Hough transform method in the presence of strong transient disturbances is not compromised by using weights because each SFT contributes at most $w_i$ (which is of order unity) to the final number count.
The following statements can be proven using the methods of [@hough04]. The mean number count in the absence of a signal is $\bar{n} = Np$, where $N$ is the number of SFTs and $p$ is the probability that the normalized power, of a given frequency bin and SFT defined by Eq. (\[eq:normpower\]), exceeds a threshold $\rho_\th$, i.e., $p$ is the probability that a frequency bin is selected in the absence of a signal. For unity weighting, the standard deviation is simply $\sigma = \sqrt{Np(1-p)}$. However, with more general weighting, it can be shown that $\sigma$ is given by $$\label{eq:4}
\sigma = \sqrt{||\vec{w}||^2p(1-p)}\,,$$ where $||\vec w||^2 = \sum_{i=0}^{N-1}w_i^2$. A threshold $n_{\th}$ on the number count corresponding to a false alarm rate $\alpha_\H$ is given by $$\label{eq:nth}
n_\th = Np + \sqrt{2||\vec
w||^2p(1-p)}\,\textrm{erfc}^{-1}(2\alpha_\H) \,.$$ Therefore $n_{\th}$ depends on the weights of the corresponding template $\lambda$. In this case, the natural detection statistic is not the Hough Number Count" $n$, but the *significance* of a number count, defined by $$\label{eq:5}
s = \frac{n - \bar{n}}{\sigma}\,,$$ where $\bar{n}$ and $\sigma$ are the expected mean and standard deviation for pure noise. Values of $s$ can be compared directly across different templates characterized by differing weight distributions.
The threshold $\rho_{\th}$ (c.f. Eq. \[eq:1\]) is selected to give the minimum false dismissal probability $\beta_\H$ for a given false alarm rate. In [@S2HoughPaper] it was shown that the optimal choice for $\rho_{\th}$ is $1.6$ which correspond to a peak selection probability $p = e^{-\rho_{\th}} \approx 0.2$. It can be shown that the optimal choice is unchanged by the weights and hence $\rho_{\th} = 1.6$ is used once more [@badrisintes].
Consider a population of sources located at a given point in the sky, but having uniformly distributed spin axis directions. For a template that is perfectly matched in frequency, spin-down, and sky-position, and given the optimal peak selection threshold, it can be shown [@badrisintes] that the weakest signal that can cross the threshold $n_\th$ with a false dismissal probability $\beta_\H$ has an amplitude $$\label{eq:hough_h0}
h_0 = 3.38\> \S^{1/2} \left( \frac{||\vec{w}||}{\vec{w}\cdot
\vec{X}}\right)^{1/2}\sqrt{\frac{1}{\Tcoh}} ,$$ where $$\begin{aligned}
\mathcal{S}&=&\textrm{erfc}^{-1}(2\alpha_\H)+
\textrm{erfc}^{-1}(2\beta_\H)\,, \label{eq:sdef} \\
X_i &=& \frac{1}{S^\srchTemplateInd_{\iSubSupInd}}\left\{ \left(F_{+}^{\iSubSupInd}\right)^2 +
\left(F_{\times}^{\iSubSupInd}\right)^2\right\} \,. \label{eq:optimalweights}\end{aligned}$$ As before, $F_{+}^{\iSubSupInd}$ and $F_{\times}^{\iSubSupInd}$ are the values of the beam pattern functions at the mid-point of the $i^{th}$ SFT. To derive we have assumed that the number of SFTs $N$ is sufficiently large and that the signal is weak [@hough04].
From it is clear that the scaling of the weights does not matter; $w_i\rightarrow k w_i$ leaves $h_0$ unchanged for any constant $k$. More importantly, it is also clear that the sensitivity is best, i.e. $h_0$ is minimum, when $\vec{w}\cdot\vec{X}$ is maximum: $$w_i \propto X_i \,.$$ This result is equivalent to Eq. .
In addition to improving sensitivity in single-interferometer analysis, the weighted Hough method allows automatic optimal combination of Hough counts from multiple interferometers of differing senstivities.
![The improvement in the significance as a function of the mismatch in the sky-position. A signal is injected in fake noise at $\alpha=\delta=0$ and the weights are calculated at $\alpha=\delta=\delta\theta$. The curve is the observed significance as a function of $\delta\theta$ while the horizontal line is the observed significance when no weights are used. See main text for more details. []{data-label="fig:houghmismatch"}](houghMismatch){width="7cm"}
Ideally, to obtain the maximum increase in sensitivity, we should calculate the weights for each sky-location separately. In practice, we break up the sky into smaller patches and calculate one weight for each sky-patch center. The gain from using the weights will be reduced if the sky patches are too large. From equation (\[eq:optimalweights\]), it is clear that the dependence of the weights on the sky-position is only through the beam pattern functions. Therefore, the sky patch size is determined by the typical angular scale over which $F_+$ and $F_\times$ vary; thus for a spherical detector using the beam pattern weights would not gain us any sensitivity. For the LIGO interferometers, we have investigated this issue with Monte-Carlo simulations using random Gaussian noise. Signals are injected in this noise corresponding to the H1 interferometer at a sky-location $(\alpha_0,\delta_0)$, while the weights are calculated at a mismatched sky-position $(\alpha_0+\delta\theta,
\delta_0+\delta\theta)$. The significance values are compared with the significance when no weights are used. An example of such a study is shown in Fig. \[fig:houghmismatch\]. Here, we have injected a signal at $\alpha = \delta = 0$, $\cos\iota = 0.5$, zero spin-down, $\Phi_0 = \psi = 0$, and a signal to noise ratio corresponding approximately to a $6$-$\sigma$ level without weights. The figure shows a gain of $\sim 10\%$ at $\delta\theta=0$, decreasing to zero at $\delta\theta \approx 0.3\,$rad. We get qualitatively similar results for other sky-locations, independent of frequency and other parameters. There is an additional gain due to the non-stationarity of the noise itself, which depends, however, on the quality of the data. In practice, we have chosen to break the sky up into 92 rectangular patches in which the average sky patch size is about $0.4\,$rad wide, corresponding to a maximum sky position mismatch of $\delta\theta=0.2\,$rad in Fig. \[fig:houghmismatch\].
### The Hough Pipeline
The Hough analysis pipeline for the search and for setting upper limits follows roughly the same scheme as in [@S2HoughPaper]. In this section we present a short description of the pipeline, mostly emphasizing the differences from [@S2HoughPaper] and from the StackSlide and PowerFlux searches. As discussed in the previous subsection, the key differences from the S2 analysis [@S2HoughPaper] are (i) using the beam-pattern and noise weights, and (ii) using SFTs from multiple interferometers.
The total frequency range analyzed is 50-1000 Hz, with a resolution $\delta f = 1/\Tcoh$ as in . The resolution in $\dot{f}$ is $\sci{2.2}{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$ given in , and the reference time for defining the spin-down is the start-time of the observation. However, unlike StackSlide and PowerFlux, the Hough search is carried out over only 11 values of $\dot{f}$, including zero, in the range \[$\sci{-2.2}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$, $0~\mathrm{Hz}~\mathrm{s}^{-1}$\]. This choice is driven by the technical design of the current implementation, which uses look-up-tables and partial Hough maps as in [@S2HoughPaper]. This implementation of the Hough algorithm is efficient when analyzing all resolvable points in $\dot{f}$, as given in , but this approach is incompatible with the larger $\dot f$ step sizes used in the other search methods, which permit those searches to search a larger $\dot f$ range for comparable computational cost.
The sky resolution is similar to that used by the StackSlide method for $f < 225~{\rm Hz}$ as given by . At frequencies higher than this, the StackSlide sky-resolution is 5 times coarser, thus the Hough search is analyzing about 25 more templates at a given frequency and spin-down value. In each of the 92 sky patches, by means of the stereographic projection, the sky patch is mapped to a two dimensional plane with a uniform grid of that resolution $\delta\theta_0$. Sky Patches slightly overlap to avoid gaps among them (see [@S2HoughPaper] for further details).
![Two example histograms of the normalized Hough number count compared to a Gaussian distribution for the H1 detector in the frequency band 150-151 Hz. The upper figure corresponds to a a patch located at the north pole for the case in which the weights are used. The number of templates analyzed in this 1Hz band is of $11 \times 10^6$, the number of SFTs 1004, the corresponding mean $\bar{n} = 202.7$ and $\sigma=12.94$ is obtained from the weights. The lower figure corresponds to a patch at the equator using the same data. In this case the number of templates analyzed in this 1Hz band is of $10.5 \times 10^6$, and its corresponding $\sigma = 14.96$.[]{data-label="fig:H1histo150"}](H1histo150north "fig:"){height="6.5cm"} ![Two example histograms of the normalized Hough number count compared to a Gaussian distribution for the H1 detector in the frequency band 150-151 Hz. The upper figure corresponds to a a patch located at the north pole for the case in which the weights are used. The number of templates analyzed in this 1Hz band is of $11 \times 10^6$, the number of SFTs 1004, the corresponding mean $\bar{n} = 202.7$ and $\sigma=12.94$ is obtained from the weights. The lower figure corresponds to a patch at the equator using the same data. In this case the number of templates analyzed in this 1Hz band is of $10.5 \times 10^6$, and its corresponding $\sigma = 14.96$.[]{data-label="fig:H1histo150"}](H1histo150equ "fig:"){height="6.5cm"}
Figure \[fig:H1histo150\] shows examples of histograms of the number counts in two particular sky patches for the H1 detector in the 150-151 Hz band. In all the bands free of instrumental disturbances, the Hough number count distributions follows the expected theoretical distribution, which can be approximated by a Gaussian distribution. Since the number of SFTs for H1 is 1004, the corresponding mean $\bar{n} = 202.7$ and the standard deviation is given by Eq. (\[eq:4\]). The standard deviation is computed from the weights $\vec{w}$ and varies among different sky patches because of varying antenna pattern functions.
----- ----------------- ---------------- ----- ----------------------- ------------------------ -----------------------
IFO $f_{\rm start}$ $f_{\rm step}$ $n$ $\Delta f_{\rm left}$ $\Delta f_{\rm right}$ Description
Hz Hz Hz Hz
H1 392.365 — 1 0.01 0.01 Cal. SideBand
H1 393.835 — 1 0.01 0.01 Cal. SideBand
H2 54.1 — 1 0.0 0.0 Cal. Line
H2 407.3 — 1 0.0 0.0 Cal. Line
H2 1159.7 — 1 0.0 0.0 Cal. Line
H2 110.934 36.9787 4 0.02 0.02 37 Hz Oscillator
L1 154.6328 8.1386 110 0.01 0.01 8.14 Hz Comb
L1 0.0 36.8725 50 0.02 0.02 37 Hz Oscillator (\*)
----- ----------------- ---------------- ----- ----------------------- ------------------------ -----------------------
: Instrumental lines cleaned during the Hough search that were not listed in Table \[tab:StackSlideCleanedLines\] (see text). (\*) These lines were removed only in the multi-interferometer search.[]{data-label="tab:HoughCleanedLines"}
The upper limits on $h_0$ are derived from the *loudest event*, registered over the entire sky and spin-down range in each $0.25\,$Hz band, not from the highest number count. As for the StackSlide method, we use a frequentist method, where upper limits refer to a hypothetical population of isolated spinning neutron stars which are uniformly distributed in the sky and have a spin-down rate $\dot f$ uniformly distributed in the range \[$\sci{-2.2}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$, $0~\mathrm{Hz}~\mathrm{s}^{-1}$\]. We also assume uniform distributions for the parameters $\cos\iota \in [-1,1]$, $\psi\in [0,2\pi]$, and $\Phi_0 \in
[0,2\pi]$. The strategy for calculating the 95$\%$ upper limits is roughly the same scheme as in [@S2HoughPaper], except for the treatment of narrow instrumental lines.
Known spectral disturbances are removed from the SFTs in the same way as for the StackSlide search. The known spectral lines are, of course, also consistently removed after each signal injection when performing the Monte-Carlo simulations to obtain the upper limits.
The narrow instrumental lines “cleaned” from the SFT data are the same ones cleaned during the StackSlide search shown in Table \[tab:StackSlideCleanedLines\], together with ones listed in Table \[tab:HoughCleanedLines\]. The additional lines listed in Table \[tab:HoughCleanedLines\] are cleaned to prevent large artifacts in one instrument from increasing the false alarm rate of the Hough multi-interferometer search. Note that the L1 36.8725 Hz comb was eliminated mid-way through the S4 run by replacing a synthesized radio frequency oscillator for phase modulation with a crystal oscillator, and these lines were not removed in the Hough L1 single-interferometer analysis.
No frequency bands have been excluded from the Hough search, although the upper limits reported on the bands shown in Table \[tab:StackSlideExcludedBands\], that are dominated by 60 Hz power line harmonics or violin modes of the suspended optics, did not always give satisfactory convergence to an upper limit. In a few of these very noisy bands, upper limits were set by extrapolation, instead of interpolation, of the Monte-Carlo simulations. Therefore the results reported on those bands have larger error bars. No parameter tuning was performed on these disturbed bands to improve the upper limits.
The Powerflux Implementation {#subsec:powerflux}
----------------------------
The PowerFlux method is a variant on the StackSlide method in which the contributions from each SFT are weighted by the inverse square of the average spectral power density in each band and weighted according to the antenna pattern sensitivity of the interferometer for each point searched on the sky. This weighting scheme has two advantages: 1) variance on the signal strength estimator is minimized, improving signal-to-noise ratio; and 2) the estimator is itself a direct measure of source strain power, allowing direct parameter estimation and dramatically reducing dependence on Monte Carlo simulations. Details of software usage and algorithms can be found in a technical document [@PowerFluxTechNote]. Figure \[fig:powerfluxflowchart\] shows a flow chart of the algorithm, discussed in detail below.
![Flow chart for the pipeline used to find the upper limits presented in this paper using the PowerFlux method.[]{data-label="fig:powerfluxflowchart"}](PowerFluxFlowChart){height="10.0cm"}
### Noise decomposition {#subsubsec:noisedecomposition}
Noise estimation is carried out through a time/frequency noise decomposition procedure in which the dominant variations are factorized within each nominal 0.25 Hz band as a product of a spectral variation and a time variation across the data run. Specifically, for each 0.25 Hz band, a matrix of logarithms of power measurements across the $0.56$ mHz SFT bins and across the SFT’s of the run is created. Two vectors, denoted TMedians and FMedians, are initially set to zero and then iteratively updated according to the following algorithm:
1. For each SFT (row in matrix), the median value (logarithm of power) is computed and then added to the corresponding element of TMedians while subtracted from each matrix element in that row.
2. For each frequency bin (column in matrix), the median value is computed and then added to the corresponding FMedians element, while subtracted from each matrix element in that column.
3. The procedure repeats from step 1 until all medians computed in steps 1 and 2 are zero (or negligible).
The above algorithm typically converges quickly. The size of the frequency band treated increases with central frequency, as neighboring bins are included to allow for maximum and minimum Doppler shifts to be searched in the next step.
For stationary, Gaussian noise and for noise that follows the above assumptions of underlying factorized frequency and time dependence, the expected distribution of residual matrix values can be found from simulation. Figure \[fig:sampleresidual\] shows a sample expected residual power distribution following noise decomposition for simulated stationary, Gaussian data, along with a sample residual power distribution from the S4 data (0.25-Hz band of H1 near 575 Hz, in this case) following noise decomposition. The agreement in shape between these two distributions is very good and is typical of the S4 data, despite sometimes large variations in the corresponding TMedians and FMedians vectors, and despite, in this case, the presence of a moderately strong simulated pulsar signal (Pulsar2 in Table \[tab:ParametersHWInjections\]).
The residuals are examined for outliers. If the largest residual value is found to lie above a threshold of 1.5, that corresponding 0.25 Hz band is flagged as containing a “wandering line” because a strong but drifting instrumental line can lead to such outliers. The value 1.5 is determined empirically from Gaussian simulations. An extremely strong pulsar could also be flagged in this way, and indeed the strongest injected pulsars are labelled as wandering lines. Hence in the search, the wandering lines are followed up, but no upper limits are quoted here for the affected bands.
![Typical residual logarithmic power following noise decomposition for a sample 0.25-Hz band of H1 data (crosses) near 575 Hz in a band containing an injected pulsar. The residual is defined as the difference between a measured power for a given frequency bin in a given 30-minute period and the value predicted by the FMedians and TMedians vectors. The smooth curve is for a simulation in Gaussian noise.[]{data-label="fig:sampleresidual"}](PowerFlux_sampleresidual){height="6.5cm"}
### Line flagging {#subsubsec:lineflagging}
Sharp instrumental lines can prevent accurate noise estimation for pulsars that have detected frequencies in the same $0.56$ mHz bin as the line. In addition, strong lines tend to degrade achievable sensitivity by adding excess apparent power in an affected search. In early LIGO science runs, including the S4 run, there have been sharp instrumental lines at multiples of 1 Hz or 0.25 Hz, arising from artifacts in the data acquisition electronics.
To mitigate the most severe of these effects, the PowerFlux algorithm performs a simple line detection and flagging algorithm. For each 0.25 Hz band, the detected summed powers are ranked and an estimated Gaussian sigma computed from the difference in the 50% and 94% quantiles. Any bins with power greater than 5.0 $\sigma$ are marked for ignoring in subsequent processing. Specifically, when carrying out a search for a pulsar of a nominal true frequency, its contribution to the signal estimator is ignored when the detected frequency would lie in the same $0.56$ mHz bin as a detected line. As discussed below, for certain frequencies, spin-downs and points in the sky, the fraction of time a putative pulsar has a detected frequency in a bin containing an instrumental line can be quite large, requiring care. The deliberate ignoring of contributing bins affected by sharp instrumental lines does not lead to a bias in resulting limits, but it does degrade sensitivity, from loss of data. In any 0.25 Hz band, no more than five bins may be flagged as lines. Any band with more than five line candidates is examined manually.
### Signal estimator {#subsubsec:signalestimator}
Once the noise decomposition is complete, with estimates of the spectral noise density for each SFT, the PowerFlux algorithm computes a weighted sum of the strain powers, where the weighting takes into account the underlying time and spectral variation contained in TMedians and FMedians and the antenna pattern sensitivity for an assumed sky location and incident wave polarization. Specifically, for an assumed polarization angle $\psi$ and sky location, the following quantity is defined for each bin $k$ of each SFT $i$: $$Q_{\iSubSupInd} = \frac{P_{\iSubSupInd}}{ (F_\psi^{\iSubSupInd})^2},$$ where $F_\psi^i$ is the $\psi$-dependent antenna pattern for the sky location, defined in Eq. (\[PFFpsi\]). (See also Appendix \[sec:polarization\].)
As in Sec. \[subsec:basicdefsandnotation\], to simplify the notation we define $Q^\srchTemplateInd_{\iSubSupInd} = P^\srchTemplateInd_{\iSubSupInd}
/ (F_\psi^{\iSubSupInd})^2$ as the value of $Q_{\iSubSupInd}$ for SFT $i$ and a given template ${\mathbf \lambda}$.
For each individual SFT bin power measurement $P_{\iSubSupInd}$, one expects an underlying exponential distribution, with a standard deviation equal to the mean, a statement that holds too for $Q_{\iSubSupInd}$. To minimize the variance of a signal estimator based on a sum of these powers, each contribution is weighted by the inverse of the expected variance of the contribution. Specifically, we compute the following signal estimator: $$\begin{aligned}
R & = & {2 \over \Tcoh} \left(\sum_i\frac{1}{ (\bar{Q}^\srchTemplateInd_{\iSubSupInd})^{2} }\right)^{-1}
\sum_i \frac{ Q^\srchTemplateInd_{\iSubSupInd} }{ (\bar{Q}^\srchTemplateInd_{\iSubSupInd})^2 }, \\
& = & {2 \over \Tcoh} \left(\sum_i\frac{ [(F_\psi^{\iSubSupInd})^2]^2 }{ (\bar{P}^\srchTemplateInd_{\iSubSupInd})^2 }\right)^{-1}
\sum_i \frac{(F_\psi^{\iSubSupInd})^2 P^\srchTemplateInd_{\iSubSupInd} }{ (\bar{P}^\srchTemplateInd_{\iSubSupInd})^2} ,\end{aligned}$$ where $\bar{P}_\iSubSupInd$ and $\bar{Q}_\iSubSupInd$ are the expected uncorrected and antenna-corrected powers of SFT $i$ averaged over frequency. Since the antenna factor is constant in this average, $\bar{Q}^\srchTemplateInd_{\iSubSupInd}
= \bar{P}^\srchTemplateInd_{\iSubSupInd}/(F_\psi^{\iSubSupInd})^2$. Furthermore, $\bar{P}^\srchTemplateInd_{\iSubSupInd}$ is a estimate of the power spectral density of the noise. The replacement $\bar{P}^\srchTemplateInd_{\iSubSupInd} \cong S^\srchTemplateInd_{\iSubSupInd}$ gives Eq. (\[eq:PFRk\]).
Note that for an SFT $i$ with low antenna pattern sensitivity $|F_\psi^{\iSubSupInd}|$, the signal estimator receives a small contribution. Similarly, SFT’s $i$ for which ambient noise is high receive small contributions. Because computational time in the search grows linearly with the number of SFT’s and because of large time variations in noise, it proves efficient to ignore SFT’s with sky-dependent and polarization-dependent effective noise higher than a cutoff value. The cutoff procedure saves significant computing time, with negligible effect on search performance.
Specifically, the cutoff is computed as follows. Let $\sigma_j$ be the [*ordered*]{} estimated standard deviations in noise, taken to be the ordered means of $\bar Q_{\iSubSupInd}= {1\over k_{\rm max}}\Sigma_k \bar{Q}^{\iSubSupInd}_k$, where $k_{\rm max}$ is the number of frequency bins used in the search template. Define $j_{\rm opt}$ to be the index $j_{\rm max}$ for which the quantity ${1\over j_{\rm max}}\sqrt{\Sigma_{j=1}^{j_{\rm max}}\sigma_j^2}$ is minimized. Only SFT’s for which $\sigma_j<2\sigma_{j_{\rm opt}}$ are used for signal estimation. In words, $j_{\rm opt}$ defines the last SFT that improves rather than degrades signal estimator variance in an unweighted mean. For the weighted mean used here, the effective noise contributions are allowed to be as high as twice the value found for $j_{\rm opt}$. The choice of $2\sigma_{j_{\rm opt}}$ is determined empirically.
The PowerFlux search sets strict, frequentist, all-sky 95% confidence-level upper limits on the flux of gravitational radiation bathing the Earth. To be conservative in the strict limits, numerical corrections to the signal estimator are applied: 1) a factor of $1/\cos(\pi/8)=1.082$ for maximum linear polarization mismatch, based on twice the maximum half-angle of mismatch (see Appendix \[sec:polarization\]) and 2) a factor of $1.22$ for bin-centered signal power loss due to Hann windowing (applied during SFT generation); and 3) a factor of $1.19$ for drift of detected signal frequency across the width of the $0.56$ mHz bins used in the SFT’s. Note that the use of rectangular windowing would eliminate the need for correction 2) above, but would require a larger correction of $1.57$ for 3)
Antenna pattern and noise weighting in the PowerFlux method allows weaker sources to be detected in certain regions of the sky, where run-averaged antenna patterns discriminate in declination and diurnal noise variations discriminate in right ascension. Figure \[fig:weightedskymap\] illustrates the resulting variation in effective noise across the sky for a 0.25-Hz H1 band near 575 Hz for the circular polarization projection. By separately examining SNR, one may hope to detect a signal in a sensitive region of the sky with a strain significantly lower than suggested by the strict worst-case all-sky frequentist limits presented here, as discussed below in section \[subsec:powerfluxmcvalidation\]. Searches are carried out for four linear polarizations, ranging over polarization angle from $\psi=0$ to $\psi={3\over8}\pi$ in steps of $\pi/8$ and for (unique) circular polarization.
![Sky map of run-summed PowerFlux weights for a 0.25-Hz band near 575 Hz for one choice of linear polarization in the S4 H1 data. The normalization corresponds roughly to the effective number of median-noise SFT’s contributing to the sum.[]{data-label="fig:weightedskymap"}](PowerFlux_circular_weight_skymap_575Hz){height="8.5cm"}
A useful computational savings comes from defining two different sky resolutions. A “coarse” sky gridding is used for setting the cutoff value defined above, while fine grid points are used for both frequency and amplitude demodulation. A typical ratio of number of coarse grid points to number of fine grid points used for Doppler corrections is 25.
### Sky banding {#subsubsec:skybanding}
Stationary and near-stationary instrumental spectral lines can be mistaken for a periodic source of gravitational radiation if the nominal source parameters are consistent with small variation in detected frequency during the time of observation. The variation in the frequency at the detector can be found by taking the time derivative of Eq. (\[eq:freqatdetector\]), which gives, $$\label{eq:dfdtatdetector}
{df \over dt} = \left (1 + \frac{ {\bf v} (t)\cdot\bf{\hat{n}}}{c} \right ) \dot{f}
+ \hat{f}(t)\frac{ {\bf a} (t)\cdot\bf{\hat{n}}}{c} .$$ The detector’s acceleration, ${\bf a}$ in this equation is dominated by the Earth’s orbital acceleration ${\bf a}_{\rm Earth}$, since the diurnal part of the detector’s acceleration is small and approximately averages to zero during the observation. Thus, it should be emphasized that a single instrumental line can mimic sources with a range of slightly [*different*]{} frequencies and assumed different positions in the sky that lie in an annular band. For a source $\dot{f}$ assumed to be zero, the center of the band is defined by a circle 90 degrees away from the direction of the average acceleration of the Earth during the run where $\bar{{\bf a}}_{\rm Earth} \cdot\bf{\hat{n}} = 0$, [*i.e.,*]{} toward the average direction of the Sun during the run. For source spin-downs different from zero, there can be a cancellation between assumed spin-down (or spinup) that is largely cancelled by the Earth’s average acceleration, leading to a shift of the annular region of apparent Doppler stationarity toward (away from) the Sun.
A figure of merit found to be useful for discriminating regions of “good” sky from “bad” sky (apparent detected frequency is highly stationary) is the “$S$ parameter”: $$\label{eq:sparam}
S \quad = \quad \fdot + [({\mathbf \Omega} \times {\bf v}_{\rm Earth}/c)\cdot {\bf \hat{n}})]\hat{f}_0 ,$$ where ${\bf \Omega}$ is the Earth’s angular velocity vector about the solar system barycenter. The term ${\bf \Omega} \times {\bf v}_{\rm Earth}$ is a measure of the Earth’s average acceleration during the run, where ${\bf v}_{\rm Earth}$ is taken to be the noise-weighted velocity of the H1 detector during the run. Regions of sky with small $|S|$ for a given $\hat f$ and $\fdot$ have stationary detected frequency. As discussed below in section \[subsec:powerfluxmcvalidation\], such regions are not only prone to high false-alarm rates, but the line flagging procedure described in section \[subsubsec:lineflagging\] leads to systematically underestimated signal strength and invalid upper limits. Hence limits are presented here for only sources with $|S|$ greater than a threshold value denoted $S_{\rm large}$. The minimum acceptable value chosen for $S_{large}$ is found from software signal injections to be $\sci{1.1}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ for the 1-month S4 run and can be understood to be $$\label{eq:slarge}
S_{\rm large} \quad = \quad {N_{\rm occupied\>\>bins}\over T_{obs}\cdot T_{coh}},$$ where $N_{\rm occupied\>\>bins}\sim5$ is the minimum total number of $0.56$ mHz detection bins occupied by the source during the data run for reliable detection. In practice, we use still larger values for the H1 interferometer ($S_{large}=\sci{1.85}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$) and L1 interferometer ($S_{large}=\sci{3.08}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$) during the S4 run for the limits presented here because of a pervasive and strong comb of precise 1-Hz lines in both interferometers. These lines, caused by a GPS-second synchronized electronic disturbance and worse in L1 than in H1, lead to high false-alarm rates from that data for lower values of $S_{large}$. For the frequency and spin-down ranges searched in this analysis, the average fractions of sky lost to the skyband veto are 15% for H1 and 26% for L1.
Figures \[fig:skyband100hz\]-\[fig:skyband1000hz\] illustrate the variation in the fraction of sky marked as “bad” as assumed source frequency and spin-down are varied. Generally, at low frequencies, large sky regions are affected, but only for low spin-down magnitude, while at high frequencies, small sky regions are affected, but the effects are appreciable to larger spin-down magnitude. It should be noted that the annular regions of the sky affected depend upon the start time and duration of a data run. The longer the data run, the smaller is the region of sky for which Doppler stationarity is small. Future LIGO data runs of longer duration should have only small regions near the ecliptic poles for which stationary instrumental lines prove troublesome.
![S4 sky band regions (good - light gray, bad for L1 - medium or dark gray, bad for H1 & L1 - dark gray) for a source frequency $\hat f$ = 100 Hz and three different assumed spin-down choices: a) zero; b) $-3\times10^{-9}$ Hz s$^{-1}$; and c) $-1\times10^{-8}$ Hz s$^{-1}$. The black circle indicates the average position of the Sun during the data run.[]{data-label="fig:skyband100hz"}](PowerFlux_skybands100_b){height="9.0cm"}
![S4 sky band regions (good - light gray, bad for L1 - medium or dark gray, bad for H1 & L1 - dark gray) for a source frequency $\hat f$ = 300 Hz and three different assumed spin-down choices: a) zero; b) $-3\times10^{-9}$ Hz s$^{-1}$; and c) $-1\times10^{-8}$ Hz s$^{-1}$. The black circle indicates the average position of the Sun during the data run.[]{data-label="fig:skyband300hz"}](PowerFlux_skybands300_b){height="9.0cm"}
![S4 sky band regions (good - light gray, bad for L1 - medium or dark gray, bad for H1 & L1 - dark gray) for a source frequency $\hat f$ = 1000 Hz and three different assumed spin-down choices: a) zero; b) $-3\times10^{-9}$ Hz s$^{-1}$; and c) $-1\times10^{-8}$ Hz s$^{-1}$. The black circle indicates the average position of the Sun during the data run.[]{data-label="fig:skyband1000hz"}](PowerFlux_skybands1000_b){height="9.0cm"}
### Grid-point upper limit determination {#subsubsec:powerfluxupperlimitsetting}
An intermediate step in the PowerFlux analysis is the setting of upper limits on signal strength for each sky-point for each $0.56$ mHz bin. The limits presented here for each interferometer are the highest of these intermediate limits for each 0.25-Hz band over the entire “good” sky. The intermediate limits are set under the assumption of Gaussian residuals in noise. In brief, for each $0.56$ mHz bin and sky-point, a Feldman-Cousins [@FeldmanCousins] 95% confidence-level is set for an assumed normal distribution with a standard deviation determined robustly from quantiles of the entire 0.25 Hz band. The Feldman-Cousins approach provides the virtues of a well behaved upper limit even when background noise fluctuates well below its expectation value and of smooth transition between 1-sided and 2-sided limits, but in practice the highest upper limit for any 0.25 Hz band is invariably the highest measured power plus 1.96 times the estimated standard deviation on the background power for that bin, corresponding to a conventional [*a priori*]{} 1-sided 97.5% upper CL. A Kolmogorov-Smirnov (KS) statistic is computed to check the actual power against a Gaussian distribution for each 0.25 Hz band. Those bands that fail the KS test value of 0.07 ($>$ 5$\sigma$ deviation for the S4 data sample) are flagged as “Non-Gaussian”, and no upper limits on pulsars are quoted here for those bands, although a full search is carried out. Bands subject to violin modes and harmonics of the 60 Hz power mains tend to fail the KS test because of sharp spectral slope (and sometimes because non-stationarity of sharp features leads to poor noise factorization).
Figure \[fig:powerfluxbackground\] provides an example of derived upper limits from one narrow band. The figure shows the distribution of PowerFlux strain upper limits on linear polarization amplitude $h_0^{\rm Lin}$ for a sample 0.25 Hz band of S4 H1 data near 149 Hz. The highest upper limit found is $\sci{3.35}{-24}$ (corresponding to a worst-case pulsar upper limit on $h_0$ of $\sci{6.70}{-24}$). The bimodal distribution arises from different regions of the sky with intrinsically different antenna pattern sensitivities. The peak at $\sci{2.8}{-24}$ corresponds to points near the celestial equator where the run-averaged antenna pattern sensitivity is worst.
![Histogram of Feldman Cousins 95% confidence-level upper limits in a 0.25-Hz band near 149 Hz in S4 H1 data. Each entry corresponds to the highest upper limit in the band for a single sky location.[]{data-label="fig:powerfluxbackground"}](PowerFlux_149_pulsar_ul_hist){height="6.5cm"}
Hardware Injections and Validation {#sec:valandhwinj}
==================================
Name $f_0$ (Hz) $df/dt$ ($\mathrm{Hz}~\mathrm{s}^{-1}$) $\alpha$ (radians) $\delta$ (radians) $\psi$ (radians) $\hpluszero$ $\hcrosszero$
---------- ----------------- ----------------------------------------- -------------------- -------------------- ------------------ ------------------------- -------------------------- --
Pulsar0 $265.57693318$ $-4.15\times 10^{-12}$ $1.248816734$ $-0.981180225$ $0.770087086$ $4.0250\times 10^{-25}$ $3.9212\times 10^{-25}$
Pulsar1 $849.07086108$ $-3.00\times 10^{-10}$ $0.652645832$ $-0.514042406$ $0.35603553$ $2.5762\times 10^{-24}$ $1.9667\times 10^{-24}$
Pulsar2 $575.16356732$ $-1.37\times 10^{-13}$ $3.75692884$ $0.060108958$ $-0.221788475$ $7.4832\times 10^{-24}$ $-7.4628\times 10^{-24}$
Pulsar3 $108.85715940$ $-1.46\times 10^{-17}$ $3.113188712$ $-0.583578803$ $0.444280306$ $1.6383\times 10^{-23}$ $-2.6260\times 10^{-24}$
Pulsar4 $1402.11049084$ $-2.54\times 10^{-08}$ $4.886706854$ $-0.217583646$ $-0.647939117$ $2.4564\times 10^{-22}$ $1.2652\times 10^{-22}$
Pulsar5 $52.80832436$ $-4.03\times 10^{-18}$ $5.281831296$ $-1.463269033$ $-0.363953188$ $5.8898\times 10^{-24}$ $4.4908\times 10^{-24}$
Pulsar6 $148.44006451 $ $-6.73\times 10^{-09}$ $6.261385269$ $-1.14184021$ $0.470984879$ $1.4172\times 10^{-24}$ $-4.2565\times 10^{-25}$
Pulsar7 $1220.93315655$ $-1.12\times 10^{-09}$ $3.899512716$ $-0.356930834$ $0.512322887$ $1.0372\times 10^{-23}$ $9.9818\times 10^{-24}$
Pulsar8 $193.94977254$ $-8.65\times 10^{-09}$ $6.132905166$ $-0.583263151$ $0.170470927$ $1.5963\times 10^{-23}$ $2.3466\times 10^{-24}$
Pulsar9 $763.847316499$ $-1.45\times 10^{-17}$ $3.471208243$ $1.321032538$ $-0.008560279$ $5.6235\times 10^{-24}$ $-5.0340\times 10^{-24}$
Pulsar10 $501.23896714$ $-7.03\times 10^{-16}$ $3.113188712$ $-0.583578803$ $0.444280306$ $6.5532\times 10^{-23}$ $-1.0504\times 10^{-24}$
Pulsar11 $376.070129771$ $-4.2620\times 10^{-15}$ $6.132905166$ $-0.583263151$ $0.170470927$ $2.6213\times 10^{-22}$ $-4.2016\times 10^{-23}$
All three methods discussed in this paper have undergone extensive internal testing and review. Besides individual unit tests of the software, hardware injections provided an end-to-end validation of the entire pipelines. The next subsections discuss the hardware injections, the validations of the three methods and their pipelines. The detection of the hardware injections also shows in dramatic fashion that we can detect the extremely tiny signals that the detectors were designed to find.
Hardware injections {#hwinj}
-------------------
During a 15-day period in the S4 run, ten artificial isolated pulsar signals were injected into all three LIGO interferometers at a variety of frequencies and time derivatives of the frequency, sky locations, and strengths. Two additional artificial binary pulsar signals were injected for approximately one day. These hardware injections were implemented by modulating the interferometer mirror positions via signals sent to voice actuation coils surrounding magnets glued to the mirror edges. The injections provided an end-to-end validation of the search pipelines. Table \[tab:ParametersHWInjections\] summarizes the nominal parameters used in the isolated-pulsar injections; the parameters are defined in section \[sec:waveforms\].
Imperfect calibration knowledge at the time of these injections led to slightly different actual strain amplitude injections among the three LIGO interferometers. For the H1 and L1 comparisons between expected and detected signal strengths for these injections described in section \[subsec:StackSlideValidation\], corrections must be applied for the differences from nominal amplitudes. The corrections are the ratios of the actuation function derived from final calibration to the actuation function assumed in the preliminary calibration used during the injections. For H1 this ratio was independent of the injection frequency and equal to 1.12. For L1, this ratio varied slightly with frequency, with a ratio of 1.11 for all injected pulsars except Pulsar1 (1.15) and Pulsar9 (1.18).
--------- ---------- ---------- ------------ ------------ ------------ --------- ---------- ---------- ------------ ------------ ------------
H1 $\quad$ L1
Observed Injected Observed Injected Percent $\quad$ Observed Injected Observed Injected Percent
Pulsar SNR SNR $\sqrt{P}$ $\sqrt{P}$ Difference $\quad$ SNR SNR $\sqrt{P}$ $\sqrt{P}$ Difference
Pulsar0 0.27 0.23 1.006 1.005 0.1% $\quad$ 0.15 0.13 1.003 1.003 0.1%
Pulsar1 1.62 0.80 1.035 1.017 1.7% $\quad$ 0.27 0.69 1.006 1.016 $-$1.0%
Pulsar2 8.92 8.67 1.179 1.175 0.4% $\quad$ 8.20 9.34 1.180 1.203 $-$1.9%
Pulsar3 199.78 174.72 3.124 2.943 6.2% $\quad$ 89.89 104.76 2.304 2.454 $-$6.1%
Pulsar4 2081.64 1872.24 9.607 9.116 5.4% $\quad$ 1279.12 1425.14 7.895 8.326 $-$5.2%
Pulsar5 0.05 1.30 1.001 1.028 $-$2.6% $\quad$ 1.02 0.44 1.024 1.010 1.4%
Pulsar6 0.17 2.94 1.004 1.063 $-$5.5% $\quad$ 2.90 1.36 1.067 1.032 3.4%
Pulsar7 6.25 5.50 1.129 1.114 1.3% $\quad$ 6.07 5.11 1.136 1.116 1.8%
Pulsar8 98.12 96.21 2.303 2.285 0.8% $\quad$ 92.77 103.45 2.334 2.441 $-$4.4%
Pulsar9 6.68 6.59 1.137 1.135 0.2% $\quad$ 2.61 3.69 1.061 1.085 $-$2.2%
--------- ---------- ---------- ------------ ------------ ------------ --------- ---------- ---------- ------------ ------------ ------------
StackSlide Validation {#subsec:StackSlideValidation}
---------------------
Besides individual unit tests and review of each component of the StackSlide code, we have shown that simulated signals are detected with the expected StackSlide Power, including the hardware injections listed in Table \[tab:ParametersHWInjections\]. Table \[tab:StackSlideHWInj\] shows the observed and injected SNR, and the square root of the observed and injected StackSlide Power, $\sqrt{P}$. The percent difference of the latter is given, since this compares amplitudes, which are easier to compare with calibration errors. The observed values were obtained by running the StackSlide code using a template that exactly matches the injection parameters, while the injected values were calculated using the parameters in Table \[tab:ParametersHWInjections\] and the equations in Appendix \[sec:stackslidepowerandstats\]. The SNR’s of Pulsar0, Pulsar1, Pulsar5, and Pulsar6 were too small to be detected, and Pulsar4 and Pulsar7 were out of the frequency band of the all-sky search. Pulsar2, Pulsar3, and Pulsar8 were detected as outliers with SNR $> 7$ (as discussed in Sec. \[sec:results\]) while Pulsar9 was not loud enough to pass this requirement. In all cases the observed StackSlide Power agrees well with that predicted, giving an end-to-end validation of the StackSlide code.
![Detection of hardware injected Pulsar 2 by the StackSlide code in the H1 (top) and L1 (bottom) data.[]{data-label="fig:S4H1L1StackSlideAllSkyPulsar2SNR"}](S4H1L1StackSlideAllSkyPulsar2SNR){height="6.5cm"}
As an example of an all-sky search for a band with an injection, Fig. \[fig:S4H1L1StackSlideAllSkyPulsar2SNR\] shows the detection of Pulsar2 for a search of the H1 (top) and L1 (bottom) data, and only during the times the hardware injections were running. Later, when the entire S4 data set was analyzed Pulsar2 was still detected but with lower SNR, since this data includes times when the hardware injections were absent. Also note that, as explained in section \[subsec:powerflux\], because of strong correlations on the sky, a pulsar signal will be detected at many points that lie in an annular region in the sky that surrounds the point corresponding to the average orbital acceleration vector of the Earth, or its antipode. In fact, because of the large number of templates searched, random noise usually causes the maximum detected SNR to occur in a template other than the one which is closest to having the exact parameters of the signal. For example, for the exact template and times matching the Pulsar2 hardware injection, it was detected with SNR’s of $8.92$ and $8.20$ in H1 and L1, respectively, as given in Table \[tab:StackSlideHWInj\], while the largest SNR’s shown in Fig. \[fig:S4H1L1StackSlideAllSkyPulsar2SNR\] are $13.84$ and $13.29$. During the search of the full data set (including times when Pulsar2 was off) it was detected with SNR $11.09$ and $10.71$ in H1 and L1, respectively.
Hough Validation {#subsec:houghval}
----------------
![Maximum significance as a function of frequency corresponding to the multi-interferometer search (using the data from the three detectors) and the H1 and L1 alone.[]{data-label="fig:InjectionsSignificance"}](InjectionsSignificance){height="7.4cm"}
![Maps of the Hough significance corresponding to the multi-interferometer case for Pulsar2, Pulsar3, Pulsar8 and Pulsar9. The location of the injected pulsars are the centers of the maps. For Pulsar2, Pulsar3 and Pulsar9, the maps correspond to the frequency and spin-down values closest to the real injected ones. For Pulsar8, we show the map containing the maximum significance value. The discrepancy in sky location is due to the mismatch in frequency and spin-down values between those used in the injections and those corresponding to the Hough map.[]{data-label="fig:MapSignificancePx"}](InjMap){height="7cm"}
--------- ------ ----------- ----------------- ------------ --------------
Pulsar $\,$ Detector $f_0$ range $f_0(max)$ Significance
$\,$ (Hz) (Hz)
Pulsar2 $\,$ Multi-IFO 575.15-575.18 575.1689 15.1195
$\,$ H1 575.15-575.18 575.1667 11.1730
$\,$ L1 575.15-575.18 575.1650 9.7635
Pulsar3 $\,$ Multi-IFO 108.855-108.86 108.8572 39.1000
$\,$ H1 108.855-108.86 108.8572 32.2274
$\,$ L1 108.855-108.86 108.8589 19.2267
Pulsar8 $\,$ Multi-IFO 193.932-193.945 193.9411 39.2865
$\,$ H1 193.932-193.945 193.9394 27.9008
$\,$ L1 193.932-193.945 193.9400 23.8270
Pulsar9 $\,$ Multi-IFO 763.83-763.87 763.8511 8.3159
$\,$ H1 763.83-763.87 763.8556 6.1268
$\,$ L1 - - 5.4559
--------- ------ ----------- ----------------- ------------ --------------
: Results of the Hough search for the hardware injected signals for the multi-interferometer, H1 and L1 data.[]{data-label="tab:HoughHI"}
Using the Hough search code, four hardware-injected signals have been clearly detected by analyzing the data from the interval when the injections took place. These correspond to Pulsar2, Pulsar3, Pulsar8 and Pulsar9. For each of these injected signals, a small-area search ($0.4$ rad$\times 0.4$ rad) was performed, using a step size on the spin-down parameter of $-4.2 \times 10^{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$. Given the large spin-down value of Pulsar8 ($\sci{-8.65}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$), we have used 23 values of the spin-down spanning the range \[$\sci{-9.24}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$, $0~\mathrm{Hz}~\mathrm{s}^{-1}$\] to search for this pulsar. Because of its large amplitude, Pulsar8 can be detected even with a large mismatch in the spin-down value, although at the cost of lower SNR.
Figure \[fig:InjectionsSignificance\] shows the significance maximized over different sky locations and spin-down values for the different frequencies. These four hardware injected pulsars have been clearly detected, with the exception of Pulsar9 in the L1 data. Pulsar9 is marginally visible using the H1 data alone, with a maximum significance of 6.13, but when we combine the data from the three interferometers, the significance increases up to 8.32. Details are given in Table \[tab:HoughHI\], including the frequency range of the detected signal, the frequency at which the maximum significance is obtained and its significance value.
Figure \[fig:MapSignificancePx\] shows the Hough significance maps for the multi-interferometer case. The maps displayed correspond either to the frequency and spin-down values nearest to the injected ones, or to those in which the maximum significance was observed. The location of the injected pulsars correspond to the center of each map. Note that the true spin-down value of Pulsar8, $\sci{-8.65}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$, lies between the parameter values $\sci{-8.82}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ and $\sci{-8.40}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ of the nearest templates used.
PowerFlux validation {#subsec:powerfluxmcvalidation}
--------------------
S4\_PowerFlux\_Injections\_Table.tex
Several cross checks have been performed to validate the PowerFlux search algorithm. These validations range from simple and rapid Fourier-domain “power injections” to more precise time domain software simulations, to hardware signal injections carried out during data taking.
Signal strain power injections have been carried out as part of PowerFlux algorithm development and for parameter tuning. These software injections involve superimposing calculated powers for assumed signals upon the LIGO power measurements and carrying out searches. For computational speed, when testing signal detection efficiency, only a small region of the sky around the known source direction is searched. A critical issue is whether the strict frequentist limits set by the algorithm are sufficiently conservative to avoid undercoverage of the intended frequentist confidence band. We present here a set of figures that confirm overcoverage applies. Figure \[fig:excessvsstrain\] shows the difference (“excess”) between the Feldman-Cousins 95% confidence-level upper limit (conventional 97.5% upper limit) on strain and the injected strain for a sample of elliptic-polarization time-domain injections in the H1 interferometer for the 140.50-140.75 Hz band. Injection amplitudes were distributed logarithmically, while frequencies, spin-downs, sky locations, and orientations were distributed uniformly. One sees that there is indeed no undercoverage (every excess strain value is above zero) over the range of injection amplitudes. Figure \[fig:excessvsspindown\] shows the same “excess” plotted vs the injected spin-down value, where the search assumes a spin-down value of zero, and where the sample includes injections with actual spin-down values more than a step size away from the the assumed value for the search template. As one can see, in this frequency range, a spin-down stepsize of $\sci{1.0}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ is safe (true spin-down no more $\sci{5.0}{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$ away from the assumed search value). Figure \[fig:excessvss\] shows the “excess” plotted vs the $S$ parameter that discriminates between sky regions of low and high Doppler stationarity. As shown, a value of $S_{\rm large}\sim\sci{1}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ is safe for these injections. For this search we have chosen 51 spin-down steps of $\sci{2}{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$ for 50-225 Hz and 11 steps of $\sci{1}{-9}~\mathrm{Hz}~\mathrm{s}^{-1}$ for 200-1000 Hz.
![“Excess” (upper limit minus injected) strain plotted vs injected signal strain for sample PowerFlux H1 elliptic-polarization near 140 Hz injections.[]{data-label="fig:excessvsstrain"}](PowerFlux_excessvsstrain_140){height="8cm"}
![“Excess” (upper limit minus injected) strain plotted vs injected signal spin-down for sample PowerFlux H1 elliptic-polarization near 140 Hz injections.[]{data-label="fig:excessvsspindown"}](PowerFlux_excessvsspindown_140){height="8cm"}
![“Excess” (upper limit minus injected) strain plotted vs S parameter defined in text, where values greater than $8\times10^{-24}$ have been “capped” at that ceiling value.[]{data-label="fig:excessvss"}](PowerFlux_S_dependence){height="8cm"}
More computationally intensive full time-domain signal injections were also carried out and the results found to be consistent with those from power injections, within statistical errors.
In addition, the PowerFlux method was validated with the hardware signal injections summarized in Table \[tab:ParametersHWInjections\]. The PowerFlux algorithm was run on all 10 isolated pulsars, including two outside the 50-1000 Hz search region, and results found to agree well with expectation for the strengths of the signals and the noise levels in their bands. Table \[tab:PowerFluxHWInj\] shows the results of the analysis for the six pulsars for which a detection with SNR$>$7 is obtained by PowerFlux for one or both of the 4-km interferometers. Figure \[fig:pulsar2skymap\] shows a sky map of PowerFlux $\psi=0$ polarization SNR for the 0.25 Hz band containing pulsar 2 (575.16 Hz).
![Sample sky map of Feldman Cousins upper limits on circularly polarized strain for a 0.25-Hz band containing hardware-injected Pulsar 2 at 575.16 Hz. Only the data (half the run) during which the pulsar injection was enabled has been analyzed for this plot. The injected pulsar ($h_0=\sci{8.0}{-24}$) stands out clearly above background. (Right ascension increases positively toward the left and declination toward the top of the sky map.)[]{data-label="fig:pulsar2skymap"}](PowerFlux_circular_max_upper_strain_pulsar2_S4){height="8.5cm"}
Results {#sec:results}
=======
All three methods described in Sections \[sec:analysismethodoverview\] and \[sec:analysismethoddetails\] have been applied in an all-sky search over a frequency range 50-1000 Hz. As described below, no evidence for a gravitational wave signal is observed in any of the searches, and upper limits on sources are determined. For the StackSlide and Hough methods, 95% confidence-level frequentist upper limits are placed on putative rotating neutron stars, assuming a uniform-sky and isotropic-orientation parent sample. Depending on the source location and inclination, these limits may overcover or undercover the true 95% confidence-level band. For the PowerFlux method, strict frequentist upper limits are placed on linearly and circularly polarized periodic gravitational wave sources, assuming [*worst-case*]{} sky location, avoiding undercoverage. The limits on linear polarization are also re-interpreted as limits on rotating neutron stars, assuming worst-case sky location and worst-case star inclination. The following subsections describe these results in detail.
StackSlide Results {#subsec:stackslideresults}
------------------
### Loudest powers and coincidence outliers {#subsubsec:stackslidelpsandoutliers}
![The loudest observed StackSlide Power for H1 (top) and L1 (bottom). Frequency bands with the harmonics of 60 Hz and the violin modes have been removed.[]{data-label="fig:S4H1L1StackSlideLEs"}](S4H1L1StackSlideLEs){height="6.5cm"}
The StackSlide method was applied to the S4 H1 and L1 data set, as given in Sec. \[subsec:stackslide\]. As described in that section, only the loudest StackSlide Power was returned from a search of the entire sky, the range of the frequency’s time derivative, $[-1 \times 10^{-8}, 0]$ Hz $\mathrm{s}^{-1}$, and for each $0.25$ Hz band within $50-1000$ Hz. The results are shown in Fig. \[fig:S4H1L1StackSlideLEs\].
![The loudest observed StackSlide Power for H1 (top) and L1 (bottom) with a simple veto applied: only outliers in each $0.25$ Hz band with SNR $> 7$ in both interferometers that have a fractional frequency difference $\le 2.2 \times 10^{-4}$ are kept. These are shown against the background results that have SNR $\le 7$ in both interferometers. Frequency bands with the harmonics of the 60 Hz and the violin modes have also been removed.[]{data-label="fig:S4H1L1StackSlideLEsSimpleVetoes"}](S4H1L1StackSlideLEsSimpleVetoes){height="6.5cm"}
$f_{H1}$ (Hz) $f_{L1}$ (Hz) H1 SNR L1 SNR Comment
---- --------------- --------------- -------- -------- ------------------
1 78.618889 78.618889 14.82 13.58 Inst. Lines
2 108.856111 108.856111 152.11 69.79 HW Inj. Pulsar3
3 193.947778 193.949444 121.89 125.75 HW Inj. Pulsar8
4 244.148889 244.157778 9.00 22.89 Inst. Lines
5 375.793889 375.806667 11.68 27.09 HW Inj. Pulsar11
6 376.271111 376.281667 7.47 9.46 HW Inj. Pulsar11
7 575.162778 575.153333 11.09 10.71 HW Inj. Pulsar2
8 575.250000 575.371667 7.49 7.51 Inst. Lines
9 575.250000 575.153333 7.49 10.71 Inst. & Pulsar2
10 580.682778 580.734444 7.02 7.19 Inst. Lines
11 912.307778 912.271111 7.02 7.37 Inst. Lines
12 988.919444 988.960556 9.56 9.75 Inst. Lines
13 988.919444 989.000000 9.56 8.12 Inst. Lines
14 993.356111 993.523333 7.08 7.12 Inst. Lines
: StackSlide outliers with SNR $> 7$ in both interferometers, with fraction difference in frequency less than or equal to $2.2 \times 10^{-4}$, and after removal of the bands with $60$ Hz harmonics and the violin modes.[]{data-label="tab:StackSlideOutliers"}
Many of the StackSlide results have power greater than expected due to random chance alone (for Gaussian noise). To identify the most interesting subset of these cases, a simple coincidence test was applied: only results with an SNR greater than $7$ in both H1 and L1 and with a fractional difference in frequency, measured in the SSB, less than or equal to $2.2 \times 10^{-4}$ were identified as outliers for further follow-up. The requirement on frequency agreement comes from the worst-case scenario where a signal is detected on opposite sides of the sky with opposite Doppler shifts of $1 + v/c$ and $1 - v/c$, giving a maximum fraction difference in the detected frequency at the SSB of $2 v/c \le 2.2 \times 10^{-4}$. The results after applying this simple coincidence test are shown in Fig. \[fig:S4H1L1StackSlideLEsSimpleVetoes\]. The outliers that passed the test are shown in Table \[tab:StackSlideOutliers\].
Note that the coincidence test used on the StackSlide results is very conservative in that it only covers the worst-case frequency difference, and makes no requirement on consistency in sky position or the frequency’s time derivative. However it is meant to find only the most prominent outliers. Since an automated follow-up of possible candidates is not yet in place, the follow-up is carried out manually. This dictated using a large threshold on SNR. Also, since the false dismissal rate of the coincidence test used was not determined (though it is assumed to be essentially zero) it is not used in this paper when setting upper limits. Monte Carlo studies will be needed to find appropriate thresholds on SNR and the size of coincidence windows, so that proper false alarm and false dismissal rates can be determined; such studies will be carried out when analyzing future data sets.
![The StackSlide Power vs. frequency for H1 (top), L1 (middle) and H2 (bottom) using the sky position and the $\fdot$ value of the template that gives the outlier in H1, for outlier number $2$ given in Table \[tab:StackSlideOutliers\]. Comparing with Tables \[tab:ParametersHWInjections\] and \[tab:StackSlideHWInj\] this outlier is identified as due to hardware injection Pulsar3.[]{data-label="fig:S4H1L1H2StackSlideOutliersWithHWInjs"}](figStackSlideS4H1L1H2Outlier108p856111Hz){height="6.5cm"}
![The StackSlide Power vs. frequency for H1 (top), L1 (middle) and H2 (bottom) using the sky position and the $\fdot$ value of the template that gives the outlier in H1 for outlier number $1$ given in Table \[tab:StackSlideOutliers\].[]{data-label="fig:S4H1L1H2StackSlideOutliersOtherOutliersFirstSet"}](figStackSlideS4H1L1H2Outlier78p618889Hz){height="6.5cm"}
Three types of qualitative follow-up tests were performed on each of the outliers in Table \[tab:StackSlideOutliers\]. First, using the sky position and the $\fdot$ value of the template that gives the outlier in H1, the StackSlide Power was found using the same values for these in L1 and H2 for a frequency band around that of the outlier in H1. For a fixed sky position and $\fdot$, a true gravitational-wave signal should show up in all three detectors as a narrow line at nearly the same frequency (though with an SNR corresponding to half the length displacement in H2 compared with that in H1 and L1). Second, the StackSlide Power was computed for the frequency bands containing the outliers, with sliding turned off. If an instrumental line is the underlying cause of the outlier, a stronger and narrower peak will tend to show up in this case. Third, the StackSlide Power was computed for each H1 outlier template, using half (and some other fractions) of the data. This should reduce the SNR of a true signal by roughly the square root of the fractional reduction of the data, but identify transient signals, which would fail this test by showing up in certain stretches of the data with more SNR while dissappearing in other stretches. This would be true of the hardware injections which were not always on during the run, or temporary disturbances of the instrument which appear to look like signals only for limited periods of time. (The search described here was not designed to find truely transient gravitational-wave signals.)
The follow-up tests on the outliers given in Table \[tab:StackSlideOutliers\] found that none is qualitatively consistent with a true gravitational-wave signal. The three loudest hardware injections of periodic gravitational waves from fake isolated sources were found (indicated as Pulsar3, Pulsar8, and Pulsar2), as well as interference from a fake source in a binary system (Pulsar11). All of the outliers due to the hardware injections show up in the H1 template as relatively narrow lines in all three detectors, for example as shown in Fig. \[fig:S4H1L1H2StackSlideOutliersWithHWInjs\]. These outliers, on the other hand, fail the third test when looking at times the hardware injections were turned off. In particular, this test, along with the frequencies in Table \[tab:ParametersHWInjections\], confirms the identification of outliers $5$ and $6$ as due to Pulsar11. The other hardware injections also are identified as such via their detected frequencies in Table \[tab:ParametersHWInjections\] and SNRs in Table \[tab:StackSlideHWInj\]. In comparison, none of the other outliers qualitatively passes the first test, for example as shown in Fig. \[fig:S4H1L1H2StackSlideOutliersOtherOutliersFirstSet\]. The second test was less conclusive, since some of the outliers lie at points on the sky that receive little Doppler modulation, but based on the first test we conclude that the remaining outliers are only consistent with instrumental line artifacts. These results are summarized in column six of Table \[tab:StackSlideOutliers\]. In future searches, tests of the type used here should be studied using Monte Carlo simulations, to make them more quantitative.
### StackSlide upper limits {#subsubsec:stackslideupperlimits}
{height="13cm"}
Detector Band (Hz) $h_0^{95\%}$
---------- --------------- -----------------------
H1 139.50-139.75 $4.39\times 10^{-24}$
L1 140.75-141.00 $5.36\times 10^{-24}$
: Best StackSlide all-sky $h_0$ upper limits obtained on the strength of gravitational waves from isolated neutron stars.[]{data-label="tab:StackSlideBestULs"}
The StackSlide $95\%$ confidence upper limits on $h_0$ are shown as crosses for H1 (top) and L1 (bottom) respectively in Fig. \[fig:S4H1StackSlideCharAmpvsMeasuredULs\], while the solid curves in this figure show the corresponding characteristic amplitudes given by Eq. (\[eq:stackslidecharamplitudeallsky\]) in Appendix \[sec:stackslidepowerandstats\]. The characteristic amplitudes were calculated using an estimate of the noise from a typical time during the run, but include bands with the power line and violin line harmonics which were excluded from the StackSlide search. The best upper limits over the entire search band are given in Table \[tab:StackSlideBestULs\]. The uncertainties in the upper limits and confidence due to the method used are less than or equal to $3\%$ and $5.3\%$ respectively; random and systematic errors from the calibration increase these uncertainties to about $10\%$.
Hough results {#subsec:houghresults}
-------------
### Number Counts {#subsubsec:numbercounts}
For the S4 data set, there are a total of $N=2966$ SFTs from the three interferometers, giving an expected average number count for pure noise of $\bar{n} = Np \sim 593$. The standard deviation $\sigma$ now depends on the sky-patch according to . For reference, if we had chosen unit weights, the standard deviation assuming pure Gaussian noise would have been $\sim 22$ for the multi-interferometer search. To compare number counts directly across different sky-patches, we employ the *significance* $s$ of a number count defined in Eq. (\[eq:5\]).
Since the three interferometers have different noise floors and duty factors, we would like to know their relative contributions to the total Hough number count, and whether any of the interferometers should be excluded from the search, or if all of them should be included. For this purpose, for the moment let us ignore the beam pattern functions and consider just the noise weighting: $w_i \propto
1/S^\srchTemplateInd_{\iSubSupInd}$. The relative contribution of a particular interferometer, say $I$, is given by the ratio $$\label{eq:6}
r_I = \frac{\sum_{i\in I} w_i}{\sum_{i=1}^N w_i}\,, \qquad I =
\textrm{H1, L1, H2}\,.$$ The numerator is a sum of the weights for the $I^{th}$ interferometer while the denominator is the sum of all the weights. This figure-of-merit incorporates both the noise level of data from an interferometer, and also its duty cycle as determined by the number of SFTs available for that interferometer. Figure \[fig:HoughWeights\] shows the relative contributions from H1, L1, and H2 for the duration of the S4 run. From the plot, we see that H1 clearly contributes the most. H2 contributes least at low frequencies while L1 contributes least at higher frequencies. Hence all three LIGO interferometers are included in this search. For comparison purposes and for coincidence analysis, we have also analyzed the data from H1 and L1 separately.
![Relative contributions of the three interferometers in the Hough multi-interferometer search. The noise weights are calculated in $1\,$Hz bands.[]{data-label="fig:HoughWeights"}](HoughWeights){height="6.5cm"}
![The measured loudest significance in each 0.25 Hz from the Hough search of the multi-interferometer (top), H1 (middle) and L1(bottom) data.[]{data-label="fig:HoughSignificance"}](HoughSignificance){height="9.5cm" width="9.5cm"}
Figure \[fig:HoughSignificance\] shows the result of the Hough search using data from all three LIGO interferometers, either combined in a multi-interferometer search, or just for H1 and L1 data. This figure shows the loudest significance in every $0.25\,$Hz band, maximized over all sky-positions, frequencies and spin-downs for the three searches. Line cleaning was used as described before. In the bands in which there are no spectral disturbances the significance distribution agrees very well with the theoretical expected distribution as was shown in Fig. \[fig:H1histo150\].
### Study of coincidence outliers {#subsubsec:houghout}
There are many outliers from the Hough search with significance values higher than expected for Gaussian noise, as shown in Fig. \[fig:HoughSignificance\]. Many of the large outliers correspond to well known instrumental artifacts described earlier, such as the power mains harmonics or the violin modes.
Note the relation between significance and false alarm which can be derived from equations (\[eq:nth\]) and (\[eq:5\]) for Gaussian noise: $$\label{eq:sa}
\alpha_\H= 0.5\,\textrm{erfc} (s/\sqrt{2}) \,.$$ To identify interesting candidates, we consider only those that have a significance greater than 7 in the multi-interferometer search (the most sensitive one). This is the same threshold considered by the StackSlide and PowerFlux searches. For the Hough search, this threshold corresponds to a false alarm rate of $1.3\times
10^{-12}$. With this threshold, we would expect about 6 candidates in a 100 Hz band around 1 kHz for Gaussian noise, since the number of templates analyzed in a 1 Hz band around 1 kHz is about $n=4.4\times 10^{10}$. If we would like to set a different threshold in order to select, say one event in a 1 Hz band, then we should increase the false alarm to $\alpha_\H =1/n =2.2\times 10^{-11}$.
In order to exclude spurious events due to instrumental noise in just one detector, we pass these candidates through a simple coincidence test in both the H1 and the L1 data. Since the single detector search is less sensitive than the multi-interferometer one, we consider events from H1 and L1 with a significance greater than 6.6, corresponding to a false alarm rate of $2.0\times 10^{-11}$. The numbers of templates analyzed using the H1 or L1 data are the same as for the multi-interferometer search.
--- ----------------- ----------- -------- -------- ----------------
Band (Hz) Multi-IFO H1 L1 Comment
1 78.602-78.631 12.466 12.023 10.953 Inst. Lines
2 108.850-108.875 29.006 23.528 16.090 Inj. Pulsar3
3 130.402-130.407 7.146 6.637 6.989 ?
4 193.92-193.96 27.911 17.327 20.890 Inj. Pulsar8
5 575.15-575.23 13.584 9.620 10.097 Inj. Pulsar2
6 721.45-721.50 8.560 6.821 13.647 L1 Inst. Lines
7 988.80-988.95 7.873 8.322 7.475 Inst. Lines
--- ----------------- ----------- -------- -------- ----------------
: Hough outliers that have survived the coincidence analysis in frequency, excluding those related to $60$ Hz harmonics and the violin modes.[]{data-label="tab:HoughOutliers"}
![Hough significance of the outliers that have survived the coincidence analysis without considering the bands contaminated with $60$ Hz harmonics or the violin modes. Points are plotted only for multi-interferometer templates with significance greater than $7$ and for single-interferometer templates with significance greater than $6.6$. []{data-label="fig:HoughOutliers"}](events1 "fig:"){height="6.8cm"} ![Hough significance of the outliers that have survived the coincidence analysis without considering the bands contaminated with $60$ Hz harmonics or the violin modes. Points are plotted only for multi-interferometer templates with significance greater than $7$ and for single-interferometer templates with significance greater than $6.6$. []{data-label="fig:HoughOutliers"}](events2 "fig:"){height="6.8cm"}
Detector $s$ $f_0$ (Hz) $df/dt$ ($\mathrm{Hz}~\mathrm{s}^{-1}$) $\alpha$ (rad) $\delta$ (rad)
----------- ------- ------------ ----------------------------------------- ---------------- ----------------
Multi-IFO 7.146 130.4028 $\sci{-1.745}{-9}$ 0.8798 -1.2385
H1 6.622 130.4039 $\sci{-1.334}{-9}$ 2.1889 0.7797
H1 6.637 130.4050 $\sci{-1.334}{-9}$ 2.0556 0.6115
L1 6.989 130.4067 $\sci{-1.963}{-9}$ 1.1690 -1.0104
: Parameters of the candidate events with a significance greater than 6.6 in the multi-interferometer, H1 and L1 data searches around the Hough outlier number 3. The parameters correspond to the significance, frequency and spin-down for the reference time of the beginning of S4, and sky locations.[]{data-label="tab:HoughOutliers130"}
The coincidence test applied first in frequency is similar to the one described for the StackSlide search, using a coincidence frequency window as broad as the size of the maximum Doppler shift expected at a given frequency. Of the initial 3800 0.25-Hz bands investigated, 276 yielded outliers in the multi-interferometer search with a significance higher than 7. Requiring those bands (or neighboring bands) to have outliers in H1 higher than 6.6, reduced by half the number of surviving bands. These remaining bands were studied in detail and, after eliminating power line harmonics and the violin modes, 27 candidates remained. Applying again the same coincidence test with the L1 data, we are left with only 7 coincidence outliers that are listed on Table \[tab:HoughOutliers\] and displayed in Fig. \[fig:HoughOutliers\].
Except for the third outlier, the coincidence can be attributed to instrumental lines in the detectors or to the hardware pulsar injections. Table \[tab:HoughOutliers130\] summarizes the parameters of the third coincidence candidate in the 130.40-130.41 Hz frequency band, including all the events that in any of the searches had a significance larger than 6.6. As can be seen from the Table, the events from the different data sets correspond to widely separated sky locations. Hence no detections were made in the Hough search of the S4 data.
In future searches we plan to use lower thresholds in the semi-coherent step in order to point to interesting areas in parameter space to be followed up, using a hierarchical scheme with alternating coherent and semicoherent steps. In what follows we will concentrate on setting upper limits on the amplitude $h_0$ in each of the 0.25 Hz bands.
### Upper limits {#subsubsec:houghul}
{height="13cm" width="14cm"}
![ Ratio of the upper limits measured by means of Monte-Carlo injections in the multi-interferometer Hough search to the quantity $h_0^{95\%}/C$ as defined in Equation (\[eq:8\]). The value of $\S$ in equation (\[eq:8\]) is computed using the false alarm $\alpha_\H$ corresponding to the observed loudest event, in a given frequency band, and for a false dismissal rate $\beta_\H=0.05$, in correspondence to the desired confidence level of the upper limit. The comparison is performed in each 0.25 Hz band. Analysis of the full bandwidth, and also in different 100 Hz bands, yield a scale factor $C$ to be $9.2\pm0.5$.[]{data-label="fig:HoughScaleFit"}](HoughScaleFit){height="6.5cm"}
![Ratio of the 95$\%$ confidence all-sky upper limits on $h_0$ obtained from the Hough search by means of Monte Carlo injections to those predicted by Eq. (\[eq:8\]) of the multi-interferometer (top), H1 (middle) and L1(bottom) data. The comparison is performed in 0.25 Hz bands. The scale factors $C$ used are 9.2 for the multi-interferometer search, 9.7 for H1 and 9.3 for L1. []{data-label="fig:HoughMultiULPredictionComparison"}](HoughULcomparRatio2){width="9.5cm"}
Detector $\quad$ Band (Hz) $h_0^{95\%}$
---------- --------- --------------- -----------------------
H1+H2+L1 $\quad$ 140.00-140.25 $4.28\times 10^{-24}$
H1 $\quad$ 129.00-129.25 $5.02\times 10^{-24}$
L1 $\quad$ 140.25-140.50 $5.89\times 10^{-24}$
: Best Hough all-sky upper limits obtained on the strength of gravitational waves from isolated neutron stars.[]{data-label="tab:HoughBestULs"}
As in the previous S2 Hough search [@S2HoughPaper], we set a population based frequentist upper limit using Monte Carlo signal software injections. We draw attention to two important differences from that analysis:
- In [@S2HoughPaper], known spectral disturbances were handled by simply avoiding all the frequency bins which could have been affected by Doppler broadening. Thus, the loudest event was obtained by excluding such frequency bins, and the subsequent Monte Carlo simulations also did not perform any signal injections in these bins. Here we follow the same approach as used in the StackSlide search; we use the spectral line removal procedure described in section \[subsubsec:stackslideimpl\]. For consistency, the same line removal procedure is followed in the Monte Carlo simulation after every software injection.
- Recall that the calculation of the weights depends on the sky-patch, and the search has been carried out by breaking up the sky in 92 patches. Thus, for every randomly injected signal, we calculate the weights corresponding to the center of the corresponding sky patch. The analysis of [@S2HoughPaper] did not use any weights and this extra step was not required.
The 95$\%$ confidence all-sky upper limit results on $h_0$ from the Hough search for the multi-interferometer, H1 and L1 data are shown in Fig. \[fig:HoughUL\]. These upper limits have been obtained by means of Monte-Carlo injections in each 0.25 Hz band in the same way as described in [@S2HoughPaper]. The best upper limit over the entire search band corresponds to $4.28\times
10^{-24}$ for the multi-interferometer case in the $140.00-140.25\,$Hz band. The results are summarized in Table \[tab:HoughBestULs\].
Let us now understand some features of the upper-limit results. First, it turns out that it is possible to accurately estimate the upper limits without extensive Monte Carlo simulations. From , and setting $w_i \propto X_i$, we expect that the upper limits are: $$\label{eq:7}
h_0^{95\%} \propto \left(
\frac{1}{||\vec{X}||}\right)^{1/2}\sqrt{\frac{ \S}{\Tcoh}} \,.$$ Recall that $X_i$ contains contributions both from the sky-location-dependent antenna pattern functions and from the sky-location-independent noise floor estimates. However, since we are setting upper limits for a population uniformly distributed in the sky, we might expect that the $S^\srchTemplateInd_{\iSubSupInd}$ are more important for estimating the value of $h_0^{95\%}$. From Eq. (\[eq:optimalweights\]) and averaging over the sky we get $$||\vec{X}|| \propto \sqrt{\sum_{i=0}^{N-1} \left(\frac{1}{S^\srchTemplateInd_{\iSubSupInd}}\right)^2}
\,,$$ and thus, up to a constant factor $C$, the estimated upper limits are given by $$\label{eq:8}
h_0^{95\%} = C \left( \frac{1}{
\sum_{i=0}^{N-1}(S^\srchTemplateInd_{\iSubSupInd})^{-2}}\right)^{1/4}\sqrt{\frac{
\S}{\Tcoh}} \,.$$ The value of $\S$ is calculated from Eq. (\[eq:sdef\]) using the false alarm $\alpha_\H$ corresponding to the significance of the observed loudest event in a particular frequency band. The value of the false dismissal rate $\beta_\H$ corresponds to the desired confidence level of the upper limit (in this case $95\%$). To show that such a fit is viable, Fig. \[fig:HoughScaleFit\] plots the value of the constant $C$ appearing in the above equation for every $0.25\,$Hz frequency band, using the measured upper limits. It turns out that $C=9.2\pm 0.5$. The exact value of $C$ depends on the interferometer and the search performed, but it is still found to lie within this range. This scale factor $C=9.2\pm 0.5$ is about two times worse than we would expect if we were performing a targeted (multi-interferometer with weights) search with no mismatch. This factor of two is also in very good agreement with what was reported in the S2 search [@S2HoughPaper].
The utility of this fit is that having determined the value of $C$ in a small frequency range, it can be extrapolated to cover the full bandwidth without performing any further Monte Carlo simulations. Figure \[fig:HoughMultiULPredictionComparison\] plots the ratio of the measured upper limits to the estimated values showing the accuracy of the fit. The scale factors $C$ used are 9.2 for the multi-interferometer search, 9.7 for H1 and 9.3 for L1. The scale factors have been obtained in all cases by comparing the measured upper limits by means of Monte Carlo injections to the quantity $h_0^{95\%}/C$ as defined in Equation (\[eq:8\]), using the full bandwidth of the search. These estimated upper limits have an error smaller than $5\%$ for bands free of large instrumental disturbances.
![Comparison of the upper limits obtained using 500 Monte Carlo injections with and without weights in 0.5 Hz bands for the Hough multi-interferometer search. The use of the weights improves the upper limits by a $\sim$9$\%$ factor. []{data-label="fig:HoughULweightsComparison"}](HoughULweightsComparison){height="6.5cm"}
We conclude this section by quantifying the improvement in sensitivity caused by using the weights. Figure \[fig:HoughULweightsComparison\] shows the comparison between the weighted and un-weighted results in the $800$-$900\,$Hz frequency range. The average improvement is $\sim$$9\%$ in this band. It is easy to see that the improvement as compared to the unweighted Hough search will be larger if the variation of $S^\srchTemplateInd_{\iSubSupInd}$ and the beam pattern functions is large across the SFTs. Since the variation in $S^\srchTemplateInd_{\iSubSupInd}$ is larger in a multi-interferometer search, we expect this improvement to be much more significant in a multi-interferometer search. For the case of analyzing data from a single interferometer, for example H1, the improvement in the upper limits due to the weights turns out to be only $\sim 6\%$. Also, the improvement can be increased by choosing smaller sky-patches so that the weight calculation is more optimal. In particular, if there would not be any sky mismatch in computing the weights, only due to the amplitude modulation, i.e., in the presence of Gaussian and stationary noise, we would expect an average increase of sensitivity of $\sim$$10\%$, and it could be up to $\sim$$12\%$ for optimally oriented pulsars. These results have been verified experimentally by means of a set of Monte-Carlo tests [@badrisintes].
PowerFlux results {#subsec:powerfluxresults}
-----------------
### Single-interferometer results
The PowerFlux method has been applied to the S4 data sample in the range 50-1000 Hz. Five polarization projections are sampled for each grid point: four linear polarizations with $\psi$ = 0, $\pi/8$, $\pi/4$, $3\,\pi/8$; and circular polarization. For each sky grid point in the “good sky” defined above and each of the 501 frequency bins (there is slight overlap of 0.25 Hz bands), the Feldman-Cousins [@FeldmanCousins] 95% CL upper limit is computed, as described in section \[subsubsec:powerfluxupperlimitsetting\], for each polarization projection. Worst-case upper limits on linear polarization for each grid point and frequency are taken to be the highest linear-polarization-projection strain limit divided by $\cos(\pi/8)$ to correct for worst-case polarization mismatch. The highest limit for all frequency bins in the 0.25 Hz band and over all sampled sky points is taken to be the broad-sky limit for that 0.25 Hz band. Figures \[fig:PowerFluxLinLimitsH1\]-\[fig:PowerFluxLinLimitsL1\] show the resulting broad-sky limits on linearly polarized periodic sources from H1 and L1. Bands flagged as non-Gaussian (instrumental artifacts leading to failure of the KS test) or near 60-Hz harmonics are indicated by color. The derived upper limits for these bands are considered unreliable. Diamonds indicate bands for which wandering instrumental lines (or very strong injected signals) lead to degraded upper limits. An exceedingly strong pulsar can be identified as a wandering line, and several strong hardware-injected pulsars are marked in the figures as such.
These limits on linearly polarized radiation and the corresponding limits on circularly polarized radiation can be interpreted as worst-case and best-case limits on a triaxial-ellipsoid, non-precessing neutron star, respectively, as discussed in Appendix \[sec:polarization\]. Multiplying the linear-polarization limits by a factor of two leads to the [*worst-case*]{} H1 limits on $h_0$ shown in Figs. \[fig:H1PSH\_limits\]–\[fig:L1PSH\_limits\]. The circular-polarization limits require no scale correction. Note that the StackSlide and Hough H1 limits shown on the same figure apply to a uniform-sky, uniform-orientation population of pulsars.
![PowerFlux limits on linearly polarized CW radiation amplitude for the H1 data from the S4 run. Bands flagged as non-Gaussian (instrumental artifacts) or near 60-Hz harmonics, and for which derived upper limits are unreliable, are indicated by color. Diamonds indicate bands for which wandering instrumental lines (or very strong injected signals) lead to degraded upper limits.[]{data-label="fig:PowerFluxLinLimitsH1"}](PowerFluxLinLimitsH1_corrected){height="6.5cm"}
![PowerFlux limits on linearly polarized CW radiation amplitude for the L1 data from the S4 run, with the same color coding as in the preceding figure.[]{data-label="fig:PowerFluxLinLimitsL1"}](PowerFluxLinLimitsL1_corrected){height="6.5cm"}
### Coincidence followup of loud candidates
All outliers (SNR$>$7, diamonds, and non-Gaussian bands) in the single-interferometer analysis are checked for coincidence between H1 and L1. In this followup, agreement is required in frequency to within 10 mHz, in spin-down to within $\sci{1}{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$, and in both right ascension and declination to within 0.5 radians. The only surviving candidates are associated with hardware-injected pulsars 2, 3, 4, and 8 (see Table \[tab:PowerFluxHWInj\]), 1-Hz harmonics, violin modes, and instrumental lines in both detectors near 78.6 Hz (also seen in the StackSlide and Hough searches). The source of these lines remains unknown, but followup consistency checks described in section \[subsec:stackslideresults\] rule out an astrophysical explanation.
From this coincidence analysis, we see no evidence of a strong pulsar signal in the S4 data. It should be noted, however, that the SNR threshold of 7 is relatively high. A lower threshold and a more refined algorithm for location and frequency coincidence is under development for future searches.
Comparison of the Three Methods {#sec:comparisonresults}
===============================
{height="12cm"}
{height="12cm"}
Figures \[fig:H1PSH\_limits\] and \[fig:L1PSH\_limits\] show superimposed the final upper limits on $h_0$ from the StackSlide, Hough, and PowerFlux methods when applied to the S4 single-interferometer H1 and L1 data, respectively. As one might have expected, we see that the StackSlide and Hough population-based limits lie between the best-case and worst-case $h_0$ strict limits from PowerFlux. As indicated in Figs. \[fig:H1PSH\_limits\]–\[fig:L1PSH\_limits\], the Hough search sensitivity improves with the summing of powers from two or more interferometers.
To be more precise as to expectations, we have directly compared detection efficiencies of the three methods in frequency bands with different noise characteristics. As discussed above, we expect overall improved performance of Powerflux with respect to StackSlide and Hough, except possibly for frequency bands marked by extreme non-Gaussianity or non-stationarity, where the Hough integer truncation of extreme power outliers can provide more robustness. We do not consider computational efficiency, which could play an important role in deciding which algorithm to use in computationally limited hierarchical searches.
![Comparison of StackSlide, Hough, and PowerFlux efficiencies (SNR $>$ 7) [*vs*]{} injected strain amplitude $h_0$ for the band 140.50-140.75 Hz for H1. From left to right, the curves correspond to PowerFlux, StackSlide, and Hough. This band is typical of those without large outliers. []{data-label="fig:NewEffic3MethodsBand1"}](NewEffic3MethodsH1Band1_v5){height="5.8cm"}
![Detection efficiency curves for the frequency band 357-357.25 Hz, for H1. This band has a transient spectral disturbance affecting some of the SFTs. The Hough transform method proves to be robust against such non-stationarities and is more sensitive than StackSlide or PowerFlux in this band. The SNR thresholds used to generate these curves were 6.3, 5.2, and 30, respectively, for the StackSlide, Hough, and PowerFlux methods, where the StackSlide and PowerFlux thresholds correspond to the loudest candidates in that band in the data.[]{data-label="fig:NewEffic3MethodsBand3"}](NewEffic3MethodsH1Band3_v5){height="5.8cm"}
A comparison is shown in Figs. \[fig:NewEffic3MethodsBand1\] and \[fig:NewEffic3MethodsBand3\] among the efficiencies of the three methods for two particular 0.25 Hz bands for H1: 140.5–140.75 Hz and 357–357.25 Hz. The horizontal axis in each case is the $h_0$ of Monte Carlo software injections with random sky-locations, spin-downs and orientations. The noise in the two bands have qualitatively different features. The 140.5-140.75 Hz band is a typical “clean” band with Gaussian noise and no observable spectral features. As expected, Fig. \[fig:NewEffic3MethodsBand1\] shows that the efficiency for the PowerFlux method is higher than that for StackSlide, while that of StackSlide is better than that for Hough. In other bands, where there are stationary spectral disturbances, we find that PowerFlux remains the most efficient method.
The noise in the band 357-357.25 Hz is non-Gaussian and displays a large transient spectral disturbance, in addition to stationary line noise at 357 Hz itself. The stationary 357 Hz line was removed during the StackSlide and Hough searches, avoided during the PowerFlux search, and handled self-consistently during Monte Carlo software injections. In this band, the Hough transform method proves to be robust against transient noise, and more sensitive than the StackSlide or PowerFlux implementations (see Fig. \[fig:NewEffic3MethodsBand1\]). In fact, no PowerFlux upper limit is quoted for this band because of the large non-Gaussianity detected during noise decomposition. Note that the SNR thresholds used for Stackslide, Hough and PowerFlux in Fig. \[fig:NewEffic3MethodsBand3\] are set to 6.3, 5.2 and 30, respectively, to match their loudest events in this band of the data.
Summary, Astrophysical Reach, and Outlook {#sec:summary}
=========================================
In summary, we have set upper limits on the strength of continuous-wave gravitational radiation over a range in frequencies from 50 Hz to 1000 Hz, using three different semi-coherent methods for summing of strain power from the LIGO interferometers. Upper limits have been derived using both a population-based method applicable to the entire sky and a strict method applicable to regions of the sky for which received frequencies were not stationary during the S4 data run.
The limits have been interpreted in terms of amplitudes $h_0$ for pulsars and in terms of linear and circular polarization amplitudes, corresponding to least favorable and most favorable pulsar inclinations, respectively. As a reminder, sets of known instrumental spectral lines have been cleaned from the data prior to setting the population-based StackSlide and Hough upper limits (Tables \[tab:StackSlideCleanedLines\], \[tab:StackSlideExcludedBands\], and \[tab:HoughCleanedLines\]), while regions of the sky (defined by cutoff values on the $S$ parameter (Equations \[eq:sparam\] and \[eq:slarge\]) have been excluded in the strict PowerFlux upper limits. The numerical values of the upper limits can be obtained separately[@epaps].
We have reached an important milestone on the road to astrophysically interesting all-sky results: Our best upper limits on $h_0$ are comparable to the value of a few times $10^{-24}$ at which one might optimistically expect to see the strongest signal from a previously unknown neutron star according to a generic argument originally made by Blandford (unpublished) and extended in our previous search for such objects in S2 data [@S2FstatPaper]. The value from Blandford’s argument does not depend on the distance to the star or its ellipticity, both of which are highly uncertain.
{width="6.5in"}
We find the next milestone by considering the maximum distance to which a signal could be detected and the ellipticity needed to generate a signal of the required strength at that distance. Both quantities are of interest since there are theoretical limits on the ellipticity, and both quantities are functions of the gravitational-wave frequency $f$ and its derivative $\dot{f}$. Figure \[range-hough\] is a contour plot of both quantities simultaneously, which we explain here in more detail. The Hough transform multi-interferometer upper limits on $h_0$ are used for illustration because they fall in the middle of the range of values for the different searches (see Fig. \[fig:H1PSH\_limits\]). The maximum distance $d(f,\dot{f})$ is obtained by equating the 95% confidence upper limits on $h_0$ for the multiple-interferometer plot in Fig. \[fig:HoughUL\] to the spin-down limit given in Eq. (\[hsd\]). This tacitly assumes that $\dot{f}$ is entirely due to emission of gravitational radiation, which implies the ellipticity given in Eq. (\[esd\]) regardless of the data and the distance to the source. If we relaxed this assumption, knowing that neutron stars spin down due to electromagnetic wave emission, relativistic particle winds, and other factors as well, the maximum distance and required ellipticity for a given $f$ and $\dot{f}$ would both be reduced. The degree of reduction would, however, be highly uncertain.
We can use the combined contour plot in Fig. \[range-hough\] to answer questions about the astrophysical significance of our results. Here we ask and answer several salient questions. First, what is the maximum range of the Hough transform search? The answer is obtained from looking at the top of Fig. \[range-hough\]: We could detect isolated pulsars to about 1 kpc, but only for a star radiating at a frequency near 100 Hz and then only if that star has an ellipticity somewhat more than $10^{-4}$, which is allowed only in the most extreme equations of state [@Owen:2005fn; @Xu:2003xe; @Mannarelli:2007bs]. Second, what is the maximum range of detection for a normal neutron star? Normal neutron stars are expected to have $\epsilon < 10^{-6}$ based on theoretical predictions [@Ushomirsky:2000ax]. By tracing the $\epsilon=10^{-6}$ contour, we find that the maximum range is about 50 pc at the highest frequencies (1 kHz), falling with frequency to less than 2 pc below 100 Hz. Third, what is the maximum range for a recycled millisecond pulsar? Based on the observed sample [@ATNF], recycled pulsars usually have small $|\dot{f}|$ values, corresponding to $\epsilon_\mathrm{sd}$ usually less than $10^{-8}$. Unfortunately the $\epsilon = 10^{-8}$ contour corresponds to $d<1$ pc at all frequencies in the LIGO band.
Figure \[range-hough\] then demonstrates that we have reached a second milestone not achieved in our previous all-sky searches [@S2HoughPaper; @S2FstatPaper]: The multi-interferometer Hough transform search could have detected an object at the distance of the nearest known neutron star RX J1856.5$-$3754, which is about 110–170 pc from Earth [@Walter:2002uq; @vanKerkwijk:2006nr]. We could not have detected that particular star, since the recently observed 7 s rotation period [@Tiengo:2006eb] puts the gravitational wave frequency well out of the LIGO band. But the top of Fig. \[range-hough\] shows that we could have detected a Crab-like pulsar ($f \approx 100$ Hz, $\dot{f} \approx 10^{-10}~\mathrm{Hz}~\mathrm{s}^{-1}$) at that distance if gravitational radiation dominated its spin-down.
For the ongoing S5 data run, expected to finish data collection in late 2007, several refinements of these methods are under development. The StackSlide and Hough methods can be made more sensitive than PowerFlux by starting with the maximum likelihood statistic (known as the $\cal F$-statistic [@jks; @hough04; @S2FstatPaper]) rather than SFT power. This increases the time-baseline of the coherent step in a hierarchical search, though at increased computational cost. The lower computational cost of the Hough search would be an advantage in this case. Multi-interferometer searches also increase the sensitivity, while reducing outliers (false-alarms), without having to increase greatly the size of the parameter space used, as illustrated by the Hough search in this paper. A multi-interferometer version of PowerFlux is under development, as well as hierarchical multi-interferometer searches that use the Hough and StackSlide method on the $\cal F$-statistic.
Thus, PowerFlux will be the primary tool used for semi-coherent searches using SFTs, while the Hough and StackSlide methods will be used in multi-interferometer hierarchical searches. Strong candidates from the PowerFlux search will be fed into the latter type of search as well. The parameter space searches described here do not take into account the correlations that exist between points in the four or five dimensional parameter space (including those on the sky). A map of the mismatch between a signal and the parameter-space templates can be used to generate a parameter-space metric to reduce further the number of points needed to conduct a search, a method under development for the hierarchical searches. Finally, the strain noise of the S5 data is lower by about a factor of 2, and the run will accumulate at least $1$ year of science mode data.
PowerFlux polarization projection relations {#sec:polarization}
===========================================
The PowerFlux method uses circular and four linear polarization “projections” to increase sensitivity to different source polarizations [@PowerFluxPolarizationNote]. The projections are necessarily imperfect because the interferometer itself is a polarimeter continually changing its orientation with respect to a source on the sky. There is “leakage” of one polarization into another’s projection. In this appendix we present the formulae used by PowerFlux to define these imperfect projections and discuss corrections one can make for leakage in followup studies of candidates.
As described in section \[subsubsec:signalestimator\], the signal estimator used by PowerFlux for frequency bin $k$ and projection polarization angle $\psi'$ is $$R \quad = \quad {2 \over \Tcoh} \sum_i W_{\iSubSupInd} {P_{\iSubSupInd}\over|F_{\psi'(+)}^\iSubSupInd|^2}\>/\>\sum_i W_{\iSubSupInd},$$ where $W_{\iSubSupInd} \equiv |F_{\psi'(+)}^\iSubSupInd|^4/(\bar P_{\iSubSupInd})^2$ is the weight for SFT $i$ and $F_{\psi'([+/\times])}^\iSubSupInd$ is the antenna pattern factor for a source with $[+,\times]$ polarization with respect to a major axis of polarization angle $\psi'$.
For a source of true polarization angle $\psi$ and plus / cross amplitudes $A_+$ and $A_\times$, where $h_+'(t) = A_+ \cos(\omega t+\Phi)$ and $h_\times'(t) = A_\times\sin(\omega t+\Phi)$, the strain amplitudes projected onto the $+$ and $\times$ axes for a polarization angle $\psi'$ are $$\begin{aligned}
h_+ & = & A_+ \cos(\omega t)\cos(\Delta\psi) \nonumber\\
& & -A_\times\sin(\omega t)\sin(\Delta\psi), \\
h_\times & = & A_+ \cos(\omega t)\sin(\Delta\psi) \nonumber\\
& & +A_\times\sin(\omega t)\cos(\Delta\psi), \end{aligned}$$ where $\Delta\psi \equiv 2(\psi-\psi')$, where the SFT-dependent phase constant $\Phi_0$ has been taken to be zero, for convenience, and where frequency variation of the source during each 30-minute SFT interval has been neglected. Averaging the detectable signal power $(F_{\psi'(+)}h_++F_{\psi'(\times)} h_\times)^2$ over one SFT interval $i$, one obtains approximately (neglecting antenna rotation during the half-hour interval): $$\begin{aligned}
\langle P_{\rm signal}\rangle & & = {1\over4}\bigl[(F_+^2+F_\times^2)(A_+^2+A_\times^2) \nonumber \\
& & + (F_+^2-F_\times^2)(A_+^2-A_\times^2)\cos(2\,\Delta\psi) \nonumber \\
& & + 2\,F_+F_\times(A_+^2-A_\times^2)\sin(2\,\Delta\psi)\bigr].\end{aligned}$$
Note that for a linearly polarized source with polarization angle $\psi=\psi'$ (so that $\Delta\psi = 0$) and amplitude $A_+ = h_0^{\rm Lin}$, $A_\times = 0$, one obtains $$\langle P_{\rm signal}\rangle\quad =\quad {1\over2}F_+^2(h_0^{\rm Lin})^2,$$ and that for a circularly polarized source of amplitude $A_+ = A_\times = h_0^{\rm Circ}$, $$\langle P_{\rm signal}\rangle \quad = \quad {1\over2}(F_+^2+F_\times^2)(h_0^{\rm Circ})^2,$$ as expected.
For an average of powers from many SFT’s, weighted according to detector noise and antenna pattern via $W_i$, the expectation value of the signal estimator depends on $$\begin{aligned}
\langle P^{\rm Det}\rangle \quad & = & \quad \langle P_{\rm signal}\rangle + \langle n(\psi')^2\rangle \nonumber \\
& & + 2\langle P_{\rm signal}n(\psi')\rangle,\end{aligned}$$ where $n_{\iSubSupInd}$ is the expected power from noise alone, where $\langle P_{\rm signal}n\rangle$ is assumed to vanish (signal uncorrelated with noise), and where the frequency bin index $k$ is omitted for simplicity.
For a true source with parameters $\psi$, $A_+$, and $A_\times$, this expectation value can be written: $$\begin{aligned}
\langle P^{\rm Det}\rangle\quad & = & \langle n(\psi')^2\rangle \nonumber \\
& & + {1\over4}\bigl[(1+\beta_2)(A_+^2+A_\times^2) \nonumber \\
& & +(1-\beta_2)(A_+^2-A_\times^2)\cos(2\Delta\psi) \nonumber \\
& & +2\,\beta_1\,(A_+^2-A_\times^2)
\sin(2\Delta\psi)\bigr],\end{aligned}$$ where the correction coefficients $$\begin{aligned}
\beta_1 & = & {\sum_i W_{\iSubSupInd}\>F_\times/F_+\over\sum_i W_{\iSubSupInd}}, \\
\beta_2 & = & {\sum_i W_{\iSubSupInd}\>F_\times^2/F_+^2\over\sum_i W_{\iSubSupInd}}, \end{aligned}$$ depend implicitly on $\psi'$ through $F_+$ and $F_\times$.
For a linearly polarized source with polarization angle $\psi=\psi'$, one obtains $$\langle P^{\rm Det}\rangle \quad = \quad \langle n(\psi')^2\rangle
+ {1\over2}(h_0^{\rm Lin})^2$$ and for a circularly polarized source one obtains: $$\langle P_{\rm Det}\rangle \quad = \quad \langle n(\psi')^2\rangle + {1\over2}(h_0^{\rm Circ})^2(1+\beta_2).$$
These formulae permit corrections for polarization leakage to be applied for a hypothetical source, allowing for estimation of $\psi$, $A_+$, and $A_\times$ from a sampling of polarization projection measurements. In practice, however, the calculation of the $\beta$ coefficients is computationally costly in an all-sky search and is disabled by default. Instead, upper limits on linearly polarized sources (worst-case pulsar inclination) are derived from the maximum limit over all four linear polarization projections, as described in section \[subsubsec:signalestimator\]. In followup investigations of outliers, however, these formulae permit greater discrimination of candidates, now in use for PowerFlux searches of the data from the ongoing S5 data run.
StackSlide Power And Statistics {#sec:stackslidepowerandstats}
===============================
### Approximate Form For The StackSlide Power {#sec:stackslideapproxP}
It is useful to have an analytic approximation for the StackSlide Power $P$. For a single SFT (dropping the SFT index $i$) expressing the phase in a first-order Taylor expansion about the midpoint time, $t_{1/2}$, of the interval used to generate an SFT, we can write $$\label{eq:Phioft}
\phi(t) \cong \phi_{1/2} + 2\pi f_{1/2}(t - t_{1/2}) \,,$$ where $\phi_{1/2}$ and $f_{1/2}$ are the phase and frequency at time $t_{1/2}$. Treating the values of $F_+$ and $F_\times$ as constants equal to their values at time $t_{1/2}$, the signal strain at discrete time $t_j$ is approximately, $$\begin{aligned}
h_j \cong F_{+} \hpluszero {\rm cos} (\phi_0 + 2\pi f_{1/2}t_j) \nonumber \\
\qquad + F_{\times} \hcrosszero {\rm sin} (\phi_0 + 2\pi f_{1/2}t_j) \,, \end{aligned}$$ where $j=0$ gives the start time of the SFT, and $\phi_0$ is the approximate phase at the start of the SFT (not the initial phase at the start of the observation), i.e., $$\label{eq:phi0sft}
\phi_0 \equiv \phi_{1/2} - 2 \pi f_{1/2} (\Tcoh / 2) \,.$$ Using these approximations, the Discrete Fourier Transform, given by Eq. (\[eq:DFT\]), of $h_j$ is $$\begin{aligned}
\label{eq:hk}
{\tilde{h}_k \over \Tcoh} \cong
e^{{\mathrm i}\phi_0} \Biggl [ { F_{+} \hpluszero \over 2 }
- {\mathrm i} { F_{\times} \hcrosszero \over 2 } \Biggl ]
\Biggl [ { {\rm sin} (2\pi\Delta\kappa) \over 2 \pi \Delta\kappa } \nonumber \\
+ {\mathrm i} { 1 - {\rm cos} (2\pi\Delta\kappa) \over 2 \pi \Delta\kappa } \Biggr ]\,, \qquad \qquad \qquad\end{aligned}$$ where $\Delta \kappa \equiv \kappa - k$ and $\kappa \equiv f_{1/2} T_{\rm coh} $ is usually not an integer. Equation (\[eq:hk\]) holds for $0 < \kappa < M/2$ and $|\kappa - k| << M$, which is true for all of the frequencies over which we search.
If the discrete time samples of the data from the detector consist of a signal plus noise the expected value of $P$ is approximated by $$\label{eq:eststackslidepwr}
P \cong P_0 + {1 \over 2} \langle d^2 \rangle \,,$$ where the mean value of $P_0$ is $1$ and its standard deviation is $1/\sqrt{N}$ due to the normalization used, and $$\begin{aligned}
\label{eq:aveoptimalsnr}
\langle d^2 \rangle \cong \Biggl [ \hpluszero^2 \left \langle {F_{+}^2 \over S_k} {\sin^2(\pi \Delta \kappa)
\over \pi^2 \Delta \kappa^2} \right \rangle \qquad \qquad \qquad \nonumber \\
\qquad \qquad \qquad + \hcrosszero^2 \left \langle {F_{\times}^2 \over S_k} {\sin^2(\pi \Delta \kappa)
\over \pi^2 \Delta \kappa^2} \right \rangle \Biggr ] \Tcoh \,, \qquad\end{aligned}$$ is an approximate form for the square of the optimal SNR defined in Eq. (71) in reference [@jks] averaged over SFTs (i.e., the angle brackets on $\langle d^2 \rangle$ represent an average over SFTs) and where for each SFT the index $k$ is the nearest integer value to $\kappa$. Thus, the relevant range for $\Delta \kappa$ is $0$ to $0.5$, corresponding to a frequency mismatch of $0$ to $1/2$ of an SFT frequency bin.
### StackSlide Statistics {#sec:stackslidestats}
It can be seen from Eq. (\[eq:stackslidepower\]) that, for Gaussian noise in the absence of a signal, $2NP$ is a $\chi^2$ variable with $2N$ degrees of freedom [@StackSlideTechNote]. Thus, the quantity $$\label{eq:stackslidechi2rho}
\varrho \equiv 2NP$$ follows the $\chi^2$ distribution: $$\label{eq:chi2dist2N}
{\cal P}(\varrho; N)d\varrho = {1 \over 2^N \Gamma(N)} \varrho^{N-1}e^{-\varrho/2} d\varrho \,.$$ When a signal is present, $\varrho$ follows a non-central $\chi^2$ distribution with $2N$ degrees of freedom and a non-centrality parameter $N \langle d^2 \rangle$ such that $$\begin{aligned}
\label{eq:noncentralchi2dist2N}
{\cal P}(\varrho; N \langle d^2 \rangle) d\varrho = \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\
{ I_{N-1} \biggl ( \sqrt{\varrho N \langle d^2 \rangle} \biggr )
\over (N \langle d^2 \rangle)^{{N - 1}} }
\varrho^{{N - 1 \over 2}} e^{-( \varrho + N \langle d^2 \rangle )/2} d\varrho \, ,\end{aligned}$$ where the form given here is based on that given in [@jksIII], and $I_{N-1}$ is the modified Bessel function of the first kind and order $N-1$.
The distribution described by Eqs. (\[eq:chi2dist2N\]) and (\[eq:noncentralchi2dist2N\]) can be used to find the minimum optimal signal-to-noise ratio that can be detected using the StackSlide search for fixed false alarm and false dismissal rates, for a targeted search. For a $1\%$ false alarm rate, a $10\%$ false dismissal rate, and large $N$ Eqs. (\[eq:stackslidechi2rho\]) and (\[eq:noncentralchi2dist2N\]) give $\langle d^2 \rangle = 7.385/\sqrt{N}$ (See also [@StackSlideTechNote]), while averaging Eq. (\[eq:aveoptimalsnr\]) independently over the source sky position, inclination angle, polarization angle, and mismatch in frequency gives $\langle d^2 \rangle = 0.7737 (4 / 25)(h_0^2 T_{\rm coh} / S)$ (see also Eq. 5.35 in [@hough04] ). Equating these and solving for $h_0$, the characteristic amplitude for a targeted StackSlide search with a $1\%$ false-alarm rate, $10\%$ false-dismissal rate is: $$\label{eq:stackslidecharamplitude}
\langle h_0 \rangle_{\rm targeted} = 7.7 \sqrt{S} / ( T_{\rm coh} T_{\rm obs}^*)^{1/4} \, ,$$ where $T_{\rm obs}^* = N\Tcoh$ is the actual duration of the data, which is shorter than the total observation time, $T_{\rm obs}$, because gaps exist in the data for times when the detectors were not operating in science mode. Comparing this expression with Eq. 5.35 in [@hough04] the StackSlide characteristic amplitude given in Eq. (\[eq:stackslidecharamplitude\]) is found to be about $10\%$ lower than a similar estimate for the standard Hough search. Note that in this paper an improved version of the Hough method is presented. Also, in this paper an all-sky search for the loudest StackSlide Power is carried out, covering up to $1.88 \times 10^{9}$ templates, and only the loudest StackSlide Power is returned from the search, corresponding to a false alarm rate of $5.32 \times 10^{-10}$. Furthermore, the upper limits are found by injecting a family of signals, each of which has a StackSlide Power drawn from a different noncentral chi-squared distribution. Using the results from Sec. \[sec:results\], for an all-sky StackSlide search the $95\%$ confidence all-sky upper limits are found empirically to be approximately given by: $$\label{eq:stackslidecharamplitudeallsky}
\langle h_0 \rangle_{\rm all-sky} = 23 \sqrt{S} / ( T_{\rm coh} T_{\rm obs}^*)^{1/4} \, .$$
Acknowledgments
===============
acknowledgements.tex This document has been assigned LIGO Laboratory document number LIGO-P060010-06-Z.
[99]{} S4IncoherentBiblio.tex
|
---
abstract: 'Fast and accurate eye tracking is crucial for many applications. Current camera-based eye tracking systems, however, are fundamentally limited by their bandwidth, forcing a tradeoff between image resolution and framerate, i.e. between latency and update rate. Here, we propose a hybrid frame-event-based near-eye gaze tracking system offering update rates beyond 10,000 Hz with an accuracy that matches that of high-end desktop-mounted commercial eye trackers when evaluated in the same conditions. Our system builds on emerging event cameras that simultaneously acquire regularly sampled frames and adaptively sampled events. We develop an online 2D pupil fitting method that updates a parametric model every one or few events. Moreover, we propose a polynomial regressor for estimating the gaze vector from the parametric pupil model in real time. Using the first hybrid frame-event gaze dataset, which will be made public, we demonstrate that our system achieves accuracies of 0.45$^\circ$–1.75$^\circ$ for fields of view ranging from 45$^\circ$ to 98$^\circ$.'
author:
- 'Anastasios N. Angelopoulos$^*$'
- 'Julien N.P. Martel$^*$'
- 'Amit P.S. Kohli'
- Jörg Conradt
- Gordon Wetzstein
bibliography:
- 'egbib.bib'
title: 'Event Based, Near-Eye Gaze Tracking Beyond 10,000Hz'
---
Introduction
============
\[sec:intro\]
Related Work {#sec:related}
============
System and Methods {#sec:methods}
==================
Dataset {#sec:data}
=======
Results {#sec:results}
=======
Discussion {#sec:discussion}
==========
Acknowledgements {#acknowledgements .unnumbered}
================
A.N.A. was supported by a National Science Foundation (NSF) Fellowship and a Berkeley Fellowship. J.N.P.M. was supported by a Swiss National Foundation (SNF) Fellowship (P2EZP2 181817), G.W. was supported by an NSF CAREER Award (IIS 1553333), a Sloan Fellowship, by the KAUST Office of Sponsored Research through the Visual Computing Center CCF grant, and a PECASE by the ARL. Thanks to Stephen Boyd and Mert Pilanci for helpful conversations.
|
---
abstract: 'Half-metallicity (HM) offers great potential for engineering spintronic applications, yet only few magnetic materials present metallicity in just one spin channel. In addition, most HM systems become magnetically disordered at temperatures well below ambient conditions, which further hinders the development of spin-based electronic devices. Here, we use first-principles methods based on density functional theory (DFT) to investigate the electronic, magnetic, structural, mixing, and vibrational properties of $90$ $XYZ$ half-Heusler (HH) alloys ($X =$ Li, Na, K, Rb, Cs; $Y =$ V, Nb, Ta; $Z =$ Si, Ge, Sn, S, Se, Te). We disclose a total of $28$ new HH compounds that are ferromagnetic, vibrationally stable, and HM, with semiconductor band gaps in the range of $1$–$4$ eV and HM band gaps of $0.2$–$0.8$ eV. By performing Monte Carlo simulations of a spin Heisenberg model fitted to DFT energies, we estimate the Curie temperature, $T_{\rm C}$, of each HM compound. We find that $17$ HH HM remain magnetically ordered at and above room temperature, namely, $300 \le T_{\rm C} \le 450$ K, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$. A further materials sieve based on zero-temperature mixing energies let us to conclude $5$ overall promising ferromagnetic HH HM at and above room temperature: NaVSi, RbVTe, CsVS, CsVSe, and RbNbTe. We also predict $2$ ferromagnetic materials that are semiconductor and magnetically ordered at ambient conditions: LiVSi and LiVGe.'
author:
- Muhammad Atif Sattar
- 'S. Aftab Ahmad'
- Fayyaz Hussain
- Claudio Cazorla
title: 'First-principles prediction of half-Heusler half-metals above room temperature'
---
[^1]
Introduction {#sec:intro}
============
Half-metals (HM) with full spin polarization at the Fermi level are of great potential for spintronic applications [@felser07; @zutic03]. In particular, HM are considered to be ideal for injecting spin-polarized currents into semiconductors [@bhat16; @dash09] and for manufacturing electrodes for magnetic tunnel junctions and giant magnetoresistance devices [@tanaka99; @hordequin98]. Half-Heusler (HH) alloys comprise a relatively large family of multifunctional materials with chemical formula $XYZ$ and cubic symmetry (space group $F\overline{4}3m$); the archetypal HH compound NiMnSb was the first half-metal material to be ever reported [@groot83]. The structural compatibility of HH HM with typical cubic semiconductors, the potential huge number of current HH HM compounds [@casper12], and the possibility of combining different HH to form HM layered structures [@azadani16], open great prospects in the field of spin-based electronics.
In recent years, HH alloys have been studied extensively with computational first-principles methods [@martin12; @cazorla17]. The structural simplicity, rich variety, and predictable electronic behaviour (e.g., modified Slater-Pauling rule [@damewood15]) of HH alloys convert these materials into a perfect target for automated computational searches of HM [@ma17; @legrain17; @sattar18]. The structural stability and electronic and magnetic properties of HH calculated at zero temperature are the usual materials descriptors employed to guide the theoretical searches of candidate HM. However, analysis of the corresponding magnetic properties at room temperature, which are crucial for the engineering of practical applications [@tu16; @cao17], usually are neglected due to the large computational load associated with first-principles simulation of thermal effects [@curtarolo13]. Furthermore, ferromagnetic (FM) spin ordering is widely assumed in such computational investigations [@galanakis06; @wei12] regardless of the fact that anti-ferromagnetic (AFM) spin ordering is also possible in HM materials [@damewood15; @leuken94; @luo08; @hu11]. In order to provide improved guidance to future experiments, it is convenient then to examine the magnetic properties of HH HM at $T \neq 0$ K conditions, along with their vibrational and mixing stabilities.
In this article, we investigate the electronic, magnetic, structural, mixing, and vibrational properties of $90$ $XYZ$ HH alloys ($X =$ Li, Na, K, Rb, Cs; $Y =$ V, Nb, Ta; $Z =$ Si, Ge, Sn, S, Se, Te) with first-principles methods based on density functional theory (DFT). Our selection of materials has been motivated by recent encouraging results reported by other authors for some similar systems [@damewood15; @hussain18; @wang17]. We introduce a simple HH spin Heisenberg model fitted to FM and AFM DFT energies that allows for fast and systematic monitoring of the magnetization as a function of temperature. We predict that a total of $17$ HH HM are vibrationally stable and remain magnetically ordered at and above room temperature, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$ and semiconductor (half-metal) band gaps in the range of $1$–$4$ ($0.2$–$0.8$) eV. A further materials sieve based on zero-temperature mixing energies allows us to identify $5$ HH HM that are most promising for electronic applications. Meanwhile, a total of $21$ HH alloys are found to exhibit an anti-ferromagnetic ground state but all of them turn out to be metallic. We also predict $2$ new semiconductor FM materials that possess high thermodynamic stability and Curie temperatures. General structural, thermodynamic, and functional trends are identified across the $X$, $Y$, and $Z$ series. Hence, our computational study discloses a number of electronic materials and design strategies that should be useful for spintronic applications.
{width="1.0\linewidth"}
Methods {#sec:methods}
=======
In what follows, we explain the technical details of our first-principles calculations. We also describe the simple spin Heisenberg model that we have devised to analyse the magnetic properties of HH alloys at finite temperatures, along with the technical details of the accompanying Monte Carlo simulations.
First-principles calculations {#subsec:DFT}
-----------------------------
Density functional theory (DFT) calculations have been performed with the self-consistent full potential linearized augmented plane wave (FP-LAPW) method, as implemented in the WIEN2K code [@wien2k]. The generalised gradient approximation to the exchange-correlation energy due to Perdew-Burke-Ernzerhof (PBE) [@pbe] has been employed in this study to estimate energies and determine equilibrium geometries. The value of the R$_{\rm mt} \times$K$_{\rm max}$ product, where R$_{\rm mt}$ is the value of the muffin-tin sphere radii and K$_{\rm max}$ of the plane wave cut-off energy, was fixed to $9$ in order to provide highly converged results. Likewise, a dense [**k**]{}-point mesh of $21 \times 21 \times 21$ was used for integrations within the first Brillouin zone (IBZ). The self-consistent threshold values adopted for the calculation of energies and equilibrium geometries were $10^{-5}$ eV and $10^{-4}$ eV/Å, respectively. In order to estimate accurate electronic and magnetic properties (e.g., band gaps and magnetic moments) at reasonable computational cost, we employed the Tran-Blaha modified Becke-Johnson (TB-mBJ) meta-GGA exchange-correlation functional [@tran09] over the equilibrium structures generated with PBE.
The pseudopotential plane-wave DFT code VASP [@vasp] has been also used for determining the energy difference between FM and AFM spin orderings, and for estimating vibrational lattice phonons. We used the projector-augmented wave method to represent the ionic cores [@bloch94], considering the following electrons as valence states: $X$ $s$, $p$, and $d$; $Y$ $s$, $p$, and $d$; and $Z$ $s$ and $p$. Wave functions were represented in a plane-wave basis truncated at $650$ eV and for integrations within the IBZ we employed a dense $\Gamma$-centered [**k**]{}-point mesh of $16 \times
16 \times 16$. The calculation of phonon frequencies was performed with the small displacement method [@alfe09] and the PHONOPY code [@phonopy]. The following parameters provided sufficiently well converged phonon frequencies: 190–atom supercells (i.e., $4 \times 4 \times 4$ replication of the conventional HH unit cell) and atomic displacements of $0.02$ Å.
Spin Heisenberg model and Monte Carlo simulations {#subsec:spin}
-------------------------------------------------
To analyse the effects of temperature on the magnetic properties of HH alloys, we define the following spin Heisenberg Hamiltonian: $$E_{\rm spin} = E_{0} + \frac{1}{2}\sum_{ij} J_{ij}S_{i}S_{j}~,
\label{eq:heisenberg}$$ where $E_{0}$ is a reference energy, $S_{i}$ represent the magnetic moment of ion $i$, and $J_{ij}$ the exchange interactions of ion $i$ with the rest. In our model, we only consider spin couplings between nearest magnetic ions (which add up to $12$, given that the symmetry of the magnetic sublattice is face centered cubic) and assume that all exchange interactions are equal. Based on such simplifications, the value of the model parameters can be calculated straightforwardly with DFT methods as: $$\begin{aligned}
E_{0} = \frac{1}{2}\left( E_{\rm FM} + E_{\rm AFM} \right) \nonumber \\
J_{ij} = \frac{1}{8|S|^{2}}\left( E_{\rm FM} - E_{\rm AFM} \right)~,
\label{eq:parameter}\end{aligned}$$ where $E_{\rm FM}$ and $E_{\rm AFM}$ are the energy of the crystal considering perfect ferromagnetic and anti-ferromagnetic spin arrangements, respectively. (We recall that in the AFM spin configuration each magnetic ion sees $8$ out of its $12$ nearest neighbors with opposite spin orientation [@singh17].) It is worth noting that in spite of the simplicity of the adopted spin Heisenberg Hamiltonian, similar models have been able to provide accurate results for the $T$-dependence of the magnetization of complex magnetic materials (e.g., multiferroic oxide perovskites [@cazorla13; @cazorla17b; @cazorla18]).
Classical Monte Carlo (MC) simulations of the spin Heisenberg Hamiltonian just explained were performed to estimate the magnetization of HH alloys (that is, equal to the sum of all invidual spins) as a function of temperature. We used a periodically-repeated simulation box containing $20 \times 20 \times 20$ spins. Thermal averages were computed from runs of $50,000$ MC sweeps performed after equilibration (which consisted of the same number of MC steps). A small symmetry-breaking magnetic anisotropy was introduced in the MC simulations to facilitate the accompanying numerical analysis [@escorihuela12]. By using this setup and monitoring the evolution of the magnetization as a function of $T$, we were able to estimate magnetic transition temperatures, $T_{\rm C}$, with a numerical accuracy of $25$ K.
![Volume optimization of the HH alloy NaVTe. (a) Three different atomic structures are considered with fixed ferromagnetic (FM) spin ordering. (b) Ferromagnetic and anti-ferromagnetic (AFM) spin orderings are considered in the lowest-energy T1 structure.[]{data-label="fig2"}](fig2.pdf){width="1.0\linewidth"}
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Results and Discussion {#sec:results}
======================
We start this section by providing an overview of the computational strategy that we have followed to determine the ground-state configurations of HH alloys, and the general classification that results from our calculations based on their stuctural, electronic, and magnetic properties. Then, we explain in detail the electronic and vibrational properties of the disclosed HH HM, followed by a discussion on their magnetic phase transition temperatures and zero-temperature mixing energies. Finally, we highlight the HH alloys that according to our calculations are most promising for spintronics applications.
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Overview of the $90$ $XYZ$ HH alloys {#subsec:overview}
------------------------------------
The $XYZ$ HH alloys considered in this study feature an alkali metal in the $X$ position (Li, Na, K, Rb, and Cs), a transition metal in the $Y$ position (V, Nb, and Ta), and a non-metal $sp$-element in the $Z$ position (Si, Ge, Sn, S, Se, and Te). The cubic HH unit cell consists of three interpenetrating face centered cubic (fcc) sublattices that together render a crystal structure with $F\overline{4}3m$ symmetry. There exist three possible atomic arrangements compatible with this structure, namely, T1, T2, and T3, which are generated by exchanging the Wyckoff positions of the $X$, $Y$, and $Z$ ions (see Fig. \[fig1\]). The physical properties of HH alloys may be greatly influenced by their specific atomic arrangement [@sattar18], hence configurations T1, T2, and T3 need to be all considered when performing thorough functionality searches within this family of materials.
Systematic fixed-volume geometry optimizations have been carried out for each HH compound to determine the energetically most favorable atomic arrangement, equilibrium volume, and magnetic ordering. First, the equilibrium volume of each possible structure considering FM spin ordering is obtained by fitting the series of calculated energy points to a Birch-Murnaghan equation of state [@cazorla15]; subsequently, the energy associated with AFM spin ordering is analyzed for the lowest-energy atomic arrangement obtained in the FM case. Figures \[fig2\]a-b illustrate such a computational procedure for the particular case of NaVTe, which turns out to exhibit a T1–FM ground state and an equilibrium volume of $320$ Å$^{3}$. Once the optimal atomic arrangement, equilibrium volume, and magnetic ordering have been determined at the PBE level, we accurately calculate the corresponding electronic properties (e.g., energy band gaps and magnetic moments) with the TB-mBJ functional [@tran09].
Figure \[fig3\] shows a general classification of the $90$ HH compounds analyzed in this study made on basis to their structural, magnetic, and electronic properties. A total of $56$ materials present lowest energy on the T1 arrangement, $32$ on the T3, and only $2$ on the T2. Compounds with a light-weight alkali metal in the $X$ position (e.g., Li, Na, and K) tend to be more stable in the T1 arrangement, whereas those with heavy-weight alkali metals (e.g., Rb and Cs) in the T3. On the other hand, compounds with a heavy-weight transition metal in the $Y$ position (e.g., Nb and Ta) mostly are stabilized in the T1 configuration, made the exception of the HH alloys containing Cs which always adopt the T3 arrangement. Meanwhile, the role of the non-metal element occupying the $Z$ position on choosing the most favorable structure appears to be negligible. We note that the two compounds that adopt the T2 configuration turn out to be metallic (i.e., LiTaSn and LiNbSn); thereby, the T2 arrangement will be ignored for the remainder of the article.
At zero-temperature conditions, we find that $55$ HH alloys are FM, $21$ AFM, and $14$ non-magnetic (see Fig.\[fig3\]). A total of $39$ FM compounds are found to be half-metallic, $2$ FM semiconductor (SC), and the rest metallic. In what follows, we describe with detail the electronic, vibrational, magnetic, and mixing properties of the $41$ new HM and SC compounds that have been determined in our investigation, all of which present FM spin ordering.
![Spin-projected total and partial density of states of the half-metallic HH alloy RbVTe (T3 arrangement) predicted in this study; spin up (down) density components are represented in the positive (negative) panels. The horizontal dashed line represents the Fermi level, which has been shifted to zero. RbVTe is vibrationally stable at $\Gamma$, displays FM spin ordering with a Curie temperature of $T_{\rm C} = 375 \pm 25$ K, possesses a relatively small zero-temperature mixing energy of $0.14$ eV/atom, and has a total magnetic moment of $4$ $\mu_{\rm B}$. The energy band gap of RbVTe is $3.73$ eV and the accompanying half-metallic band gap is $0.83$ eV. The electronic properties are calculated with the Tran-Blaha modified Becke-Johnson (TB-mBJ) meta-GGA exchange-correlation functional [@tran09].[]{data-label="fig6"}](fig6.pdf){width="1.0\linewidth"}
Electronic and vibrational properties {#subsec:zerotemp}
-------------------------------------
Figure \[fig4\] shows the energy and half-metallic band gaps, $E_{\rm BG}$ and $E_{\rm HM}$, estimated for the $90$ $XYZ$ HH alloys considered in this study. $E_{\rm BG}$ is the energy difference between the top of the valence band and bottom of the conduction band in the semiconductor spin channel, and $E_{\rm HM}$ the minimum energy that an electron requires to surpass the spin gap (see Fig.\[fig5\], where we represent those quantities for CsVSe). One can observe a certain correlation between these two quantities, namely, large $E_{\rm BG}$’s are accompanied by large $E_{\rm HM}$’s. For instance, RbNbS (T1 arrangement) has an energy band gap of $5.08$ eV and HM band gap of $1.46$ eV, while CsVSn (T3 arrangement) has $1.30$ eV and $0.18$ eV, respectively. Nevertheless, there are few compounds that in spite of having a relatively small energy band gap exhibit a relatively large HM band gap (e.g., RbNbSi in the T3 configuration for which we estimate $E_{\rm BG} = 1.24$ eV and $E_{\rm HM} = 0.57$ eV).
The color code employed in Fig.\[fig4\] shows that the HH alloys consisting of heavy-weight $X$-ions (K, Rb, and Cs), light-weight $Y$-ions (V and Nb), and chalcogen elements (S, Se, and Te) in the $Z$ position, generally possess the largest $E_{\rm BG}$ and $E_{\rm HM}$. By contrast, HH compounds with a light-weight (heavy-weight) alkali (transition) metal element like Li (Ta) in the $X$ ($Y$) position mostly are metallic or present medium band gaps.
Figure \[fig6\] shows the spin-projected density of electronic states calculated for the ground state of RbVTe (T3 configuration). This compound is half-metallic with a large $E_{\rm HM}$ of $0.83$ eV and large energy band gap of $3.73$ eV, hence is an interesting case for which to analyze the electronic features. In the majority band (spin up), is appreciated that the V $d$–$t_{2g}$ and Rb $d$ orbitals are highly hybridized in the region surrounding the Fermi level, which leads to the appearance of metallicity in that channel. In the minority band (spin down), however, the V and Rb $d$ orbitals are shifted to higher energies and remain unoccupied; consequently, a HM band gap appears in that channel between the occupied $d$ bonding and unoccupied $d$ antibonding states. Meanwhile, the majority of states forming the top of the valence band in the minority band (spin down) correspond to Te $p$ orbitals, which are shifted to energies well below those of the V and Rb $d$ orbitals.
We note that when the chalcogen element (S, Se, and Te) in the $Z$ position is substituted by a group–XIV element (Si, Ge, Sn), $E_{\rm BG}$ and $E_{\rm HM}$ generally undergo a significant reduction (see Fig.\[fig4\]). For instance, RbVSn has an energy band gap of $1.13$ eV and a HM band gap of $0.20$ eV. Such a band gap cloisure is consequence of a decrease in the energy separation between the $Z$ $p$ orbitals and $X$–$Y$ $d$ bonding and antibonding states (not shown here), and it occurs when the number of valence electrons in the system changes from $12$ (S, Se, Te) to $10$ (Si, Ge, Sn). Interestingly, the energy band gap changes as induced by $Z$–element substitutions correlate directly with changes in the magnetic moment of the ions; we will discuss in detail this and other magnetic effects in the next subsection.
Out of the $90$ HH alloys investigated in this study, we have found that $39$ ($2$) are ferromagnetic and half-metallic (magnetic semiconductor) at zero-temperature conditions. In order to provide useful guides for the experiments, is necessary also to assess the vibrational stability of the candidate materials proposed by theory. Here, we have calculated the lattice phonons for each HH alloy at the high-symmetry reciprocal point $\Gamma$, and concluded that $28$ FM HM along with the $2$ FM SC are vibrationally well behaved (that is, do not present imaginary phonon frequencies at the IBZ center). For some selected cases (namely, the overall most promising materials that will be discussed in Sec.\[subsec:promise\]), we have calculated also the full phonon spectrum over the entire IBZ and found that most of them are vibrationally stable (see Fig.\[fig7\], where we enclose a couple of relevant examples). We have not been able to draw any robust correlation between vibrational instability and chemical structure for the HH alloys analyzed in this study. For the remainder of the article, we will focus on describing the magnetic and mixing properties of the $28$ ($2$) FM, HM (SC), and vibrational stable compounds that we have predicted.
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![Curie temperature, $T_{\rm C}$, of the $28$ half-metallic HH alloys found in this study to be vibrationally stable at $\Gamma$ and which display FM spin ordering. Yellowish color indicates magnetic transition points above room temperature; the numerical uncertainty in our $T_{\rm C}$ results is $\pm 25$ K.[]{data-label="fig8"}](fig8.pdf){width="1.0\linewidth"}
![Zero-temperature mixing energy, $E_{\rm mix}$, of the $28$ half-metallic HH alloys found in this study to be vibrationally stable at $\Gamma$ and which display FM spin ordering. Bluish color indicates mixing energy values below $0.2$ eV per formula unit.[]{data-label="fig9"}](fig9.pdf){width="1.0\linewidth"}
-------------------------------- --------------------------------- ----------------------- ---------------------------------- ------------------- ---------------------------- ---------------------------- --------------------------- -----------------------------
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $
$ \quad {\rm Compound} \quad $ $ \quad {\rm Structure} \quad $ $ \quad a_{0} \quad $ $ \quad {\rm Electronic} \quad $ $ \quad M \quad $ $ \quad E_{\rm BG} \quad $ $ \quad E_{\rm HM} \quad $ $ \quad T_{\rm C} \quad $ $ \quad E_{\rm mix} \quad $
$ $ $ $ ${\rm (\AA)}$ $ {\rm behavior} $ $ (\mu_{\rm B}) $ $ {\rm (eV)} $ $ {\rm (eV)} $ $ {\rm (K)} $ $ ({\rm eV/f.u.}) $
$ $ $ $ $ $ $ $ $ $ $ $ $ $
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $
${\bf LiVSi} $ ${\rm T1} $ $ 5.85 $ $ {\bf SC} $ $ {\bf 2.0} $ $ {\bf 1.43} $ $ - $ $ {\bf 300} $ $ {\bf -0.11} $
${\bf LiVGe} $ ${\rm T1} $ $ 5.93 $ $ {\bf SC} $ $ {\bf 2.0} $ $ {\bf 1.06} $ $ - $ $ {\bf 375} $ $ {\bf -0.11} $
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $
${\bf NaVSi} $ ${\rm T1} $ $ 6.25 $ $ {\bf HM} $ $ {\bf 2.0} $ $ {\bf 1.11} $ $ {\bf 0.52} $ $ {\bf 300} $ $ {\bf +0.21} $
${\rm NaVTe} $ ${\rm T1} $ $ 6.84 $ $ {\rm HM} $ $ 4.0 $ $ 4.18 $ $ 0.54 $ $ 150 $ $ -0.12 $
${\rm KVSi} $ ${\rm T3} $ $ 6.55 $ $ {\rm HM} $ $ 2.0 $ $ 1.49 $ $ 0.21 $ $ 375 $ $ +0.56 $
${\bf RbVTe} $ ${\rm T3} $ $ 7.27 $ $ {\bf HM} $ $ {\bf 4.0} $ $ {\bf 3.73} $ $ {\bf 0.83} $ $ {\bf 375} $ $ {\bf +0.14} $
${\bf CsVS} $ ${\rm T3} $ $ 6.93 $ $ {\bf HM} $ $ {\bf 4.0} $ $ {\bf 3.92} $ $ {\bf 0.68} $ $ {\bf 300} $ $ {\bf +0.01} $
${\bf CsVSe} $ ${\rm T3} $ $ 7.13 $ $ {\bf HM} $ $ {\bf 4.0} $ $ {\bf 3.66} $ $ {\bf 0.66} $ $ {\bf 375} $ $ {\bf +0.11} $
$ $ $ $ $ $ $ $ $ $ $ $ $ $
${\rm RbNbSi} $ ${\rm T3} $ $ 6.77 $ $ {\rm HM} $ $ 2.0 $ $ 1.24 $ $ 0.57 $ $ 450 $ $ +0.83 $
${\rm RbNbSn} $ ${\rm T3} $ $ 7.19 $ $ {\rm HM} $ $ 2.0 $ $ 1.19 $ $ 0.33 $ $ 450 $ $ +0.83 $
${\bf RbNbTe} $ ${\rm T3} $ $ 7.41 $ $ {\bf HM} $ $ {\bf 4.0} $ $ {\bf 3.82} $ $ {\bf 0.75} $ $ {\bf 450} $ $ {\bf +0.40} $
${\rm CsNbSi} $ ${\rm T3} $ $ 6.87 $ $ {\rm HM} $ $ 2.0 $ $ 0.47 $ $ 0.11 $ $ 375 $ $ +0.91 $
${\rm CsNbGe} $ ${\rm T3} $ $ 6.96 $ $ {\rm HM} $ $ 2.0 $ $ 0.62 $ $ 0.14 $ $ 375 $ $ +0.91 $
${\rm CsNbSn} $ ${\rm T3} $ $ 7.28 $ $ {\rm HM} $ $ 2.0 $ $ 0.69 $ $ 0.16 $ $ 375 $ $ +0.94 $
${\rm CsNbTe} $ ${\rm T3} $ $ 7.55 $ $ {\rm HM} $ $ 4.0 $ $ 3.23 $ $ 0.39 $ $ 450 $ $ +0.49 $
$ $ $ $ $ $ $ $ $ $ $ $ $ $
${\rm KTaSn} $ ${\rm T1} $ $ 7.17 $ $ {\rm HM} $ $ 2.0 $ $ 1.31 $ $ 0.54 $ $ 300 $ $ +0.88 $
${\rm RbTaSi} $ ${\rm T1} $ $ 6.89 $ $ {\rm HM} $ $ 2.0 $ $ 1.36 $ $ 0.66 $ $ 300 $ $ +0.92 $
${\rm RbTaGe} $ ${\rm T1} $ $ 6.99 $ $ {\rm HM} $ $ 2.0 $ $ 1.39 $ $ 0.55 $ $ 300 $ $ +0.97 $
${\rm RbTaSn} $ ${\rm T3} $ $ 7.16 $ $ {\rm HM} $ $ 2.0 $ $ 1.05 $ $ 0.35 $ $ 375 $ $ +1.03 $
${\rm RbTaTe} $ ${\rm T3} $ $ 7.37 $ $ {\rm HM} $ $ 4.0 $ $ 3.70 $ $ 0.24 $ $ 375 $ $ +0.75 $
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $
-------------------------------- --------------------------------- ----------------------- ---------------------------------- ------------------- ---------------------------- ---------------------------- --------------------------- -----------------------------
\[tab:summary\]
Magnetism and mixing stability {#subsec:finitetemp}
------------------------------
Recently, Damewood *et al.* [@damewood15] have proposed a modified Slater-Pauling (mSP) rule for describing the magnetic moment of the half-Heusler alloys LiMn$Z$ ($Z =$ N, P, Si). Specifically, the total magnetic moment per formula unit (expressed in units of $\mu_{\rm B}$), $M$, has been parametrized as: $$M = N_{t} - 8~,
\label{eq:msp}$$ where $N_{t}$ is the total number of valence electrons in the unit cell. In the present case, $N_{t}$ is equal to $12$ for alloys displaying S, Se, or Te in the $Z$ position, and to $10$ for Si, Ge, and Sn (we recall that $N_{t} = 1$ for the $X$ elements considered in this study, $N_{t} = 5$ for the $Y$’s, and $N_{t} = 4$ or $6$ for the $Z$’s). We have found that all non-metallic and magnetic $XYZ$ alloys disclosed in this study (i.e., either HM or SC) in fact follow the mSP rule proposed by Damewood *et al.* [@damewood15] (some examples are provided in Table I). Interestingly, in those cases we also observe a clear correlation between the total magnetization and size of the energy band gap: large $M$’s and large $E_{\rm BG}$’s occur simultaneously in the same compounds. From an applied point of view, this seems to be a positive finding since wide band gap HM alloys then are likely to be also magnetically robust. Nevertheless, the size of the magnetic moments does not provide any information on the variation of the magnetization at $T \neq 0$ conditions, which is related to the exchange interactions between the magnetic moments.
We have estimated the magnetic transition temperature, $T_{\rm C}$, for the $28$ ($2$) FM, HM (SC), and vibrational stable compounds determined in this study, by using the computational methods explained in Sec. \[subsec:spin\]. Our $T_{\rm C}$ results are shown in Fig.\[fig8\]. It is appreciated that compounds with a heavy-weight alkali metal in the $X$ position (Rb and Cs) concentrate the highest magnetic transition temperatures (see also Table I). For instance, RbNbTe displays a Curie temperature of $450 \pm 25$ K and CsVSe of $375 \pm 25$ K, while NaVTe becomes paramagnetic at $150 \pm 25$ K and KTaGe at $225 \pm 25$ K. We ascertain the lack of any correlation between the size of the ionic magnetic moments, $M$, and the Curie temperature of the system; for example, RbNbSn displays $M = 2~\mu_{\rm B}$ and $T_{\rm C} = 450 \pm 25$ K while for RbNbTe we estimate $M = 4~\mu_{\rm B}$ and the same magnetic transition temperature. Interestingly, we find that a total of $17$ ($2$) FM, HM (SC), and vibrational stable compounds remain magnetically ordered at or above room temperature. All of them are listed in Table I and will be discussed in the next subsection.
Finally, we calculated the zero-temperature mixing energy, $E_{\rm mix}$, of the $28$ ($2$) FM, HM (SC), and vibrational stable compounds disclosed in this work, by using the formula: $$E_{\rm mix} = E_{XYZ}^{\rm HH} - \left( E_{X}^{\rm fcc} + E_{Y}^{\rm fcc} + E_{Z}^{\rm fcc} \right)~,
\label{eq:mix}$$ where $E_{XYZ}^{\rm HH}$ represents the ground-state energy of the HH alloy, and $E_{A}^{\rm fcc}$ the energy of the bulk $A$ crystal considering an equilibrium fcc structure. $E_{\rm mix}$ provides a quantitative estimation of how stable the $XYZ$ system is against decomposition into $X$-, $Y$-, and $Z$-rich regions. In particular, negative (positive) values of the mixing energy indicate high (low) stability of the compound against phase decomposition. We should note, however, that the configurational entropy of the HH alloy, which is totally neglected in Eq.(\[eq:mix\]), will always contribute favourably to the free energy and enhance the chemical stability of the $XYZ$ compound at finite temperatures [@page16; @rost15]. Consequently, a positive but small value of $E_{\rm mix}$ does not necessarily imply phase separation under realistic $T \neq 0$ K conditions. Here, we (somewhat arbitrarily) consider that a $E_{\rm mix}$ threshold value of $0.2$ eV per formula unit can be used to sieve materials with reasonably good mixing stability from those with tendency for phase separation [@shenoy19].
Figure \[fig9\] shows our $E_{\rm mix}$ results; only $7$ compounds out of $30$ (i.e., $28$ HM and $2$ SC) display zero-temperature mixing energies below $0.2$ eV per formula unit. We note that all of those low mixing energy compounds contain V in the $Y$ position, namely, LiVSi (SC), LiVGe (SC), NaVTe (HM), NaVSi (HM), RbVTe (HM), CsVS (HM), and CsVSe (HM) (see Table I). Specifically, only the first three alloys listed above present negative $E_{\rm mix}$ values, while for CsVS we obtain a practically null mixing energy. On the other hand, KVSi (HM), RbNbTe (HM), and CsNbTe (HM) exhibit mixing energies close to $0.50$ eV/f.u., and for the rest of compounds we estimate $E_{\rm mix}$’s that are close to $1.00$ eV per formula unit. These results indicate that most of the HH HM reported in this work, in spite of possessing relatively high Curie temperatures, are likely to present phase separation issues, which is not desirable for practical applications.
Meanwhile, we observe that when the element occupying the $Z$ position in the HH alloy is a chalcogen (S, Se, and Te) the resulting mixing energy is noticeably smaller than when the element belongs to group–XIV of the periodic table (Si, Ge, Sn). For instance, we find that $E_{\rm mix}$ amounts to $-0.12$ eV/f.u. for NaVTe and to $0.21$ eV/f.u. for NaVSi (see Table I). The same behaviour is observed also for other compounds presenting either Nb or Ta in the $Y$ position (e.g., for RbTaSn we estimate $1.03$ eV/f.u. and for RbTaTe $0.75$ eV/f.u.). Therefore, a possible strategy for improving the mixing stability of some of the new HH HM compounds reported in this study (e.g., KVSi with $E_{\rm mix} = 0.56$ eV/f.u. and RbNbSi with $E_{\rm mix} = 0.83$ eV/f.u.) may consist in doping with light-weight alkali metals (Li and Na) in the $X$ position (although this may also lead to some unwanted decrease in the energy band gap and Curie temperature of the alloy, see present and previous sections) and with chalcogen species in the $Z$ position.
Most promising magnetic HH alloys {#subsec:promise}
---------------------------------
Table I shows the $19$ vibrationally stable and ferromagnetic compounds predicted in this study that possess a Curie temperature equal or above room temperature. Two out of those $19$ alloys are semiconductor while the rest are half-metallic. We have also included NaVTe in the table, in spite of presenting a relatively low Curie temperature of $150$ K, owing to its good mixing stability properties ($E_{\rm mix} = -0.12$ eV/f.u.). From an applied perspective, an overall promising HM (or magnetic semiconductor) material should present the following qualities: (1) being vibrationally and chemically stable, (2) high Curie temperature, (3) large energy band gaps, (4) large magnetic moment, and (5) being structurally compatible with other semiconductor materials typically employed in electronic devices (e.g., silicon and GaAs with respective lattice parameters of $5.4$ and $5.6$ Å).
Except NaVTe and CsNb$Z$ with $Z =$ Si, Ge, and Sn, all the compounds reported in Table I fulfill conditions (2), (3), and (4) above. Besides, compounds LiVSi (SC), LiVGe (SC), and NaVSi (HM) also fulfill (1) and (5), which indicates that these materials are in fact very promising for spintronics applications [@tu16; @cao17]. Meanwhile, RbVTe (HM), CsVS (HM), and CsVSe (HM) fulfill (1)–(4) and only partially (5); however, the energy band gaps estimated for these compounds are so large that they also deserve to be highlighted. Finally, we mention RbNbTe as the most encouraging case of a HH HM not containing vanadium in the $Y$ position; this alloy fulfills conditions (2)–(4), as many other compounds, but the corresponding $E_{\rm BG}$, $E_{\rm HM}$, and $T_{\rm C}$ values are exceedingly large.
Overall, the most promising HH alloys for use in spintronics applications predicted by our computational research are LiVSi (SC), LiVGe (SC), NaVSi (HM), RbVTe (HM), CsVS (HM), CsVSe (HM) and RbNbTe (HM), all of which possess magnetic transition temperatures at or above room temperature. The only apparent disadvantage of some of these compounds (i.e., NaVSi and RbNbTe) are their positive and large mixing energies (see Table I), which suggests the likely existence of phase separation issues in practice. Nevertheless, as we have mentioned earlier, a likely stategy for solving this problem may consist in doping to some extent with light-weight alkali metals (Li and Na) in the $X$ position, which in turn would improve their structural compatibility with typical semiconductor materials, and/or with chalcogen atoms in the $Z$ position.
Conclusions {#sec:conclusions}
===========
We have performed a comprehensive first-principles study of the structural, electronic, structural, vibrational, and mixing properties of $90$ $XYZ$ half-Heusler alloys ($X =$ Li, Na, K, Rb, Cs; $Y =$ V, Nb, Ta; $Z =$ Si, Ge, Sn, S, Se, Te). In contrast to previous computational studies dealing with a large number of candidate materials, we have analyzed the magnetic features of most HH alloys at finite temperatures since this piece of information is crucial for guiding the experimental searches of technologically relevant materials. A total of $17$ alloys are predicted to be vibrationally stable, half-metallic, and magnetically ordered at room temperature, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$ and semiconductor band gaps in the range of $1$–$4$ eV. On the other hand, all the HH alloys that have been identified as anti-ferromagnetic, $21$ in total, turn out to be metallic. We have also found $2$ new magnetic semiconductors that exhibit high thermodynamic stability and Curie temperatures. After analyzing the mixing stability of the vibrationally well-behaved HH alloys, we have identified the following compounds as overall most promising for spintronics applications: LiVSi (SC), LiVGe (SC), NaVSi (HM), RbVTe (HM), CsVS (HM), CsVSe (HM) and RbNbTe (HM). On the other hand, we have argued that simple doping strategies may be used to improve the mixing stability of some of the discarded HH half-metals. Hence, we hope that our computational study will stimulate new experimental efforts leading to progress in the field of spin-based electronics.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported under the Australian Research Council’s Future Fellowship funding scheme (No. FT140100135). M. A. S. acknowledges financial support from the Higher Education Commission (HEC) of Pakistan under the IRSIP scholarship (PIN:IRSIP 35 PSc 11). Computational resources and technical assistance were provided by the Australian Government and the Government of Western Australia through the National Computational Infrastructure (NCI) and Magnus under the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.
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[^1]: Corresponding Author
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abstract: 'As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preserves the asymptotically conic geometry, we prove that the solution metric $g(t)$ expands at a locally uniform linear rate; moreover, the rescaled family of metrics $t^{-1}g(t)$ exhibits a transition at infinite time inasmuch as it converges locally uniformly to a complete, finite area hyperbolic metric which is the unique uniformizing metric in the conformal class of the initial metric $g_0$.'
author:
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James Isenberg [^1]\
University of Oregon
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Rafe Mazzeo [^2]\
Stanford University
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Natasa Sesum [^3]\
University of Pennsylvania
title: Ricci flow on asymptotically conical surfaces with nontrivial topology
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Introduction
============
This paper is a continuation of an ongoing general investigation of the global properties of Ricci flow on various classes of complete surfaces. Earlier work includes the paper [@JMS] by L. Ji together with the second and third authors here, which studies Ricci flow surfaces with asymptotically hyperbolic cusp ends and negative Euler characteristic, and the paper of Albin, Aldana and Rochon [@AAR] concerning the flow on surfaces with infinite area asymptotically hyperbolic ends. In the present paper we study the behaviour of this flow on surfaces with asymptotically Euclidean ends, or slightly more generally, asymptotically conic ends, and with $\chi(M) < 0$.
An open complete Riemannian surface $(M,g)$ is called asymptotically conical (or AC for short) of order $\tau > 0$ if $M$ is topologically finite (i.e., has finite genus and a finite number of ends) and if on each end $E$ of $M$, $g$ is asymptotic to a model conic metric $dr^2 + \alpha^2 r^2 d\theta^2$ of angle $2\pi \alpha$, for some constant $\alpha > 0$ with an error term which decays like $r^{-\tau}$ at infinity along with a certain number of its derivatives. We give a more precise definition in the next section.
The general problem studied here is to determine the long-time behaviour of the Ricci flow $${\partial}_t g = -2 {\mathrm{Ric}}(g(t)), \qquad g(0) = g_0,
\label{eq:rfs0}$$ where the initial metric $g_0$ is asymptotically conical and $\dim M = 2$. Prior work in this setting includes the older paper by Wu [@Wu] and the more recent work by Javaheri and the first author [@IJ], in both of which the underlying surface is diffeomorphic to ${\mathbb R}^2$. Wu’s paper allows initial metrics $g_0$ with complicated asymptotic behaviour at infinity. In either case, the conclusion is that under appropriate hypotheses, the flow converges to either flat ${\mathbb R}^2$ or else the ‘cigar soliton’. Here we allow general topologies, but impose more stringent conditions on the asymptotic geometry. Some local aspects of this flow for asymptotically Euclidean (or asymptotically locally Euclidean–ALE) metrics in higher dimensions have been obtained by Oliynyk and Woolgar [@OW] and by Dai and Ma [@DM]. The first of these papers proves some results concerning long-time existence and convergence in the rotationally symmetric case, and the latter paper establishes, amongst other things, monotonicity of the ADM mass when $\dim M \geq 3$. Another interesting stability result with weaker conditions imposed on the asymptotic behaviour of the initial metric has been proven by Schnürer, Schulze and Simon [@SSS].
For $\dim M = 2$, one expects to be able to obtain very explicit convergence results for Ricci flow. However, there are also obstructions which indicate the possibility of some interesting new phenomena. For example, in the case considered here, if $\chi(M) < 0$ and if $(M,g_0)$ is asymptotically conic, then the conformal type of $(M,g_0)$ is that of a punctured Riemann surface and the (unique) uniformizing metric is a complete hyperbolic metric with finite area (and hence cusp ends). Therefore if in this case the flow $(M, g(t))$ exists for all times and is asymptotically conic for each $t > 0$, and if it converges in certain sense then it seems that the asymptotic geometry must change drastically at $t=\infty$. This intriguing scenario motivates our main result, which we now state:
Let $(M,g_0)$ be a surface with asymptotically conical ends and $\chi(M) < 0$. If the Ricci evolution $g(t)$ is written in the form $u(t)g_0$, then there are constants $C_1, C_2 > 0$ depending only on $g_0$ such that $C_1 \le u(\cdot,t) \leq C_2(1+t)$ for all $t \geq 0$. In addition, for each compact set $K \subset \subset M$ there is a constant $C_K > 0$ such that $C_K (1+t) \leq u(t,x)$ for all $x \in K$ and $t \geq 0$.
If one defines the rescaled metric $\tilde{g}(t) = t^{-1} g(t)$, then $\tilde{g}$ converges smoothly on each compact set to a limiting metric $\tilde{g}_\infty$ on $M$ which is a complete hyperbolic metric with finite area, and hence is the unique uniformizing metric in the conformal class of $g_0$.
A similar rescaling by $1/t$ to normalize the metric and to obtain a smooth limit has been used earlier by Lott [@Lo] in an analysis of a three-dimensional flow. Recent work by Dai and Z. Zhang (in preparation) studies Ricci flow in this same two-dimensional setting; they obtain pointed Gromov-Hausdorff convergence of the unrescaled metric to a flat plane or cigar soliton.
The plan of this paper is as follows. We begin by recalling some background information about AC geometry and the Ricci flow on surfaces. We then indicate a proof of short-time existence within the class of asymptotically conic metrics. This is fairly standard and has been proved elsewhere, so we sketch this quite briefly. Long-time existence of solutions to the flow is proved using the standard device of finding a potential function; this argument is similar to, but simpler than, the corresponding argument in [@JMS]. The linear in time upper bound and uniform lower bound for $u$ follow from simple barrier estimates, but the locally uniform linear in time lower bound emerges only as a consequence of a fairly lengthy and quite geometric argument which brings in a number of other ingredients. Once we have obtained this linear lower bound, the proof of convergence for the rescaled family of metrics $\tilde{g}(t)$ is straightforward.
Preliminaries
=============
This section contains a few geometric facts about asymptotically conic surfaces and analytic facts about the Ricci flow which we use in our study.
Let us start with a more careful discussion of the space of asymptotically conical metrics on an open surface $M$. For any number $\alpha > 0$, we define the model cone with angle $2\pi \alpha$ by $$C_\alpha = {\mathbb R}^+ \times S^1, \qquad g_\alpha = dr^2 + \alpha^2 r^2 d\theta^2.$$ Now, let $M$ be an open surface with finite topology, and suppose that we have chosen a fixed identification of each end $E_j$ of $M$ with the ‘open’ end of some $C_\alpha$. In other words, suppose that we have fixed coordinates $r \geq R$ and $\theta \in S^1$ on each $E_j$. A metric $g$ on $M$ is called asymptotically conical (AC) if for each end $E_j$, $$\left. g \right|_{E_j} = g_{\alpha_j} + k_j, \qquad \mbox{where}\qquad |k_j|_{g_{\alpha_j}} \leq C r^{-\tau}$$ for some $\tau > 0$. More precisely, we suppose that $k_j$ lies in a weighted Hölder space, defined as follows. Let $\Lambda^{k,\alpha}(M)$ denote the usual Hölder space of order $k + \alpha$ defined relative to some fixed background metric $\bar{g}$ which is exactly equal to $g_{\alpha_j}$ on each $E_j$. This definition extends immediately to sections of any Hermitian bundle $F$ over $M$ which has a fixed trivialization over each $E_j$ (e.g., a tensor bundle with the induced metric). Next, for any real number $\mu$, let $r^\mu \Lambda^{k,\alpha}(M,F)$ denote the space of sections $u$ of the form $u = r^\mu \tilde{u}$ where $\tilde{u} \in \Lambda^{k,\alpha}(M,F)$. Here and below, we fix a function $r$ on $M$ which is everywhere smooth and strictly positive and which agrees with the chosen radial function $r$ on each end. With the above notation we have the following definition.
The metric $g$ on $M$ is called asymptotically conical (or AC) of order $\tau$ if on each end $E_j$, the error term $k_j$ lies in the space $r^{-\tau}\Lambda^{2,\alpha}(M, S^2 T^*M)$.
It follows immediately that if $g$ is AC of order $\tau$, then its scalar curvature $R = R_g$ lies in the space $r^{-2-\tau}\Lambda^{0,\alpha}(M)$.
Shi ([@Sh]) has proven that if $(M,g_0)$ is a complete noncompact manifold with bounded curvature (which is true for our initial metric $g_0$) then the Ricci flow (\[eq:rfs0\]) has a solution with bounded curvature on a short time interval. In [@CZ] it has been proven that a solution with a particular set of prescribed initial data is unique within the class of solutions with bounded curvatures in space. It follows that we have short time existence for a solution to (\[eq:rfs0\]) starting with the asymptotically conical metric $g_0$ on $M$. Our goal is to understand the long time behavior of our solution $g(\cdot,t)$.
The general Ricci flow equation takes a particularly simple form in two dimensions. In this setting, ${\mathrm{Ric}}(g) = \frac12 R \, g$ where $R$ is the scalar curvature (i.e. twice the Gauss curvature), so is the same as $${\partial}_t g_{ij}(t) = - R g_{ij}(t);
\label{eq:rfs1}$$ this implies that the flow preserves conformal class. Next, we recall the transformation law $$\Delta_{0} \phi - \frac12 R_{0} + \frac12 R e^{2\phi} = 0
\label{eq:trsc}$$ relating the scalar curvatures $R_0$ and $R$ of two conformally related metrics, $g = e^{2\phi}g_0$. Using this, and writing $g(t) = u(t)g_0$, we see that is equivalent to the scalar equation $${\partial}_t u = \Delta_{g_0} \log u - R_0, \qquad u(0) = u_0,
\label{eq:rfs}$$ and hence we focus on analysis on (\[eq:rfs\]).
Combining and , and noting that $\phi = \frac12 \log u$, we obtain the simple and useful relationship $$u_t = -Ru.
\label{eq:derivu}$$
Now assume that $g(t)$ is a solution of on some interval $0 \leq t < T$. Differentiating yields the evolution equation for scalar curvature, $$\label{eq-R1}
\frac{\partial}{\partial t} R = \Delta R + R^2.$$ It is also straightforward to check that $$\frac{\partial}{\partial t} dA_t = -R dA_t,
\label{eq:evolaf}$$ where $dA_t$ is the area form for $g(t)$.
The Gauss-Bonnet formula can be extended to asymptotically conical surfaces by integrating $R$ over an increasing sequence of compact subdomains in $M$ with boundary a union of circles $\{r = \mbox{const.}\}$ on each end and then taking the limit; this yields $$\int_M R\, dA_g = 4\pi\chi(M) - 4\pi \sum_{j=1}^\ell \alpha_j,
\label{eq:GB}$$ where $\ell$ is the number of ends. In particular, it is important for our later considerations to note that if $\chi(M) \leq 0$ and if $M$ is AC, so that $\alpha_j > 0$, then $\int_M R < 0$. For surfaces with finite total curvature which are not necessarily AC, one can define the ‘aperture’ of each end, and there is an analogue of due to Shiohama, see [@Wu].
The topological quantity $\chi(M)$ is obviously time independent. We can show that $\int_M R\, dV_g$ is constant along the flow as well, provided $R(\cdot,t) \in L^1(M, g(t))$ for each $t \geq 0$. Indeed, if we integrate (\[eq-R1\]) on an expanding sequence of sets as above, and if we use that ${\partial}_r R = {{\mathcal O}}(r^{-3-\tau})$, we obtain $$\label{eq-pres-R}
\frac{d}{dt}\int_M R\, dA = \lim_{r \to \infty} \left(\int_{\partial B_0(p,r)} {\partial}_\nu R\, d\sigma + \int_{B_0(p,r)} (R^2 - R^2)\, dA\right) = 0.$$ This implies that $\sum_{j=1}^\ell \alpha_j$ is independent of $t$, but leaves open the possibility that for surfaces with more than one end, the individual cone angles might vary with $t$, so long as their sum remains constant.
Existence
=========
The starting point for our discussion of is the following basic local existence result.
Let $(M,g_0)$ be a complete Riemannian surface with AC ends of order $\tau$. Then, for some $T > 0$, there is a unique solution to defined on $[0,T) \times M$, and each metric $g(t)$ in this family is AC of order $\tau$. \[pr:ste\]
The existence of the flow solution $g(t)$ follows from the results and/or techniques appearing in several different papers; see in particular Wu [@Wu], Schnürer, Schulze and Simon [@SSS] and Shi [@Sh]. Slightly less obvious is the fact that each of the metrics $g(t)$ remains asymptotically conical, and this is not discussed explicitly in any of these sources, although it does follow from an argument by Dai and Ma [@DM] which we sketch below. However, it is not difficult to give a direct proof of short-time existence for solutions of in the class of AC metrics by proving estimates for the corresponding linearized heat equation on appropriate weighted Hölder spaces and then constructing solutions of the nonlinear equation by a standard contraction mapping argument. We omit details, but we refer the reader to the analogous discussion in [@JMS].
We next turn to long-time existence for the solution. We wish to show
\[prop-LTE\] Let $(M^2,g_0)$ be AC of order $\tau$. Then the solution to exists for all $t \geq 0$. Moreover, $\sup_{x \in M}|R(x, t)|
\leq C$ for some fixed constant $C$ and for all $t \geq 0$.
We follow the strategy set forth originally by Hamilton [@Ha] which employs a potential function $f$ for a metric $g$ to obtain uniform bounds on $R$, which leads to a priori estimates and the continuation of $u$ to all $t \geq 0$. By definition, such a function $f$ is a solution to the equation $\Delta_g f = R$ with bounded gradient.
Let $(M,g)$ be an AC surface, and $R$ the scalar curvature of $g$. Then there exists a function $f$ on $M$ which satisfies $$\Delta_g f = R, \qquad \mbox{and} \qquad \sup_M |\nabla f| \le C.$$ \[lem-potential\]
For an AC metric of order $\tau$, the scalar curvature $R_g$ lies in $r^{-2-\tau}\Lambda^{0,\alpha}$.
We invoke some well-known Fredholm properties of the Laplacian on asymptotically conical spaces. Define the $2\ell$-dimensional space (where $\ell$ is the number of ends) $$\mathcal{E} = \{\sum_{j=1}^\ell \chi_j(\beta_j \log r + \gamma_j): \beta_j, \gamma_j \in {\mathbb R}\};$$ here each $\chi_j$ is a smooth function which equals one on the end $E_j$ and vanishes away from that end. Then it is known that for each $0 < \nu < 1$, if $\nu \leq \tau$, the map $$\Delta_g: \mathcal{E} \oplus r^{-\nu}\Lambda^{2,\alpha}(M) \longrightarrow r^{-\nu-2}\Lambda^{0,\alpha}(M)
\label{eq:Fred}$$ is surjective and its nullspace ${{\mathcal N}}$ is $\ell$-dimensional. (Implicit here is the easily verified fact that $\Delta_g (\log r) \in r^{-\nu-2}\Lambda^{0,\alpha}$.) This assertion is verified by combining Theorems 4.20, 4.26 and 7.14 in [@M]. This route to the proof requires a lot of machinery, however, and especially in this low-dimensional setting, one can prove the same things in a more elementary way using barriers and sequences of solutions on compact exhaustions of $M$.
In any case, using with $\nu = \tau$, we obtain a solution $f \in {{\mathcal E}}\oplus r^{-\tau}\Lambda^{2,\alpha}(M)$ to the equation $\Delta f = R$ which decomposes as $$f = \sum_{j=1}^\ell \chi_j (\beta_j \log r + \gamma_j) + \eta, \qquad \eta \in r^{-\tau}\Lambda^{2,\alpha}(M).$$ for $\beta_j, \gamma_j \in {\mathbb R}$. It is obvious from this that $\sup |\nabla f| \leq C$, so we are done.
Every Riemann surface is a Kähler manifold, so instead of writing $g(t) = u(t)g_0$ we can alternately write $g_{i\bar{\jmath}}
= (g_0)_{i\bar{\jmath}} + \partial_i\partial_{\bar{\jmath}}\phi$, where $\phi(t,\cdot)$ is called the Kähler potential of $g(t)$ relative to $g_0$. A standard calculation shows that if $g(t)$ satisfies the Ricci flow equation, then $\phi$ satisfies the Monge-Ampere equation $$\begin{aligned}
\label{eq-pot}
{\partial}_t \phi = \log\frac{\det((g_0)_{k\bar{\ell}} + \phi_{k\bar{\ell}})}{\det(g_0)_{k\bar{\ell}}} - 2f_0, \qquad \phi(x,0) = 0,\end{aligned}$$ where $f_0$ is the potential function for the metric $g_0$ in the sense of Lemma \[lem-potential\], and $\phi_{k\bar{\ell}}$ is shorthand for $\partial_k \partial_{\bar{\ell}} \phi$.
Now define $f := -\frac{1}{2}{\partial}_t \phi$. Differentiating with respect to $t$ yields that $f$ satisfies the evolution equation $${\partial}_t f = \Delta f, \qquad f(z,0) = -\frac12 \left. {\partial}_t \phi \right|_{t=0} = f_0.$$ One can then show
The functions $f(z,t)$ defined above satisfy $\Delta f = R$ for every $t \ge 0$.
To verify this claim, we carry through the following calculations.
First, we recall that $$R g_{i\bar{\jmath}} = R_{i\bar{\jmath}} = -\partial_i\partial_{\bar{\jmath}}\log \det g_{i\bar{\jmath}}.$$ Applying $\partial_i\partial_{\bar{\jmath}}$ to (\[eq-pot\]) yields $$-2\partial_i\partial_{\bar{\jmath}} f = - Rg_{i\bar{\jmath}} + R_0 (g_0)_{i\bar{\jmath}} - 2\partial_i\partial_{\bar{\jmath}} f_0;$$ we then obtain by taking the trace with respect to $g_{i\bar{\jmath}}$ that $$\Delta f = R - \frac{1}{u}(R_0 - \Delta_0 f_0) = R;$$ this uses both $\Delta_0 f_0 = R_0$ and $g = u g_0$.
To conclude that $f$ is a potential function for $g(t)$, we still need to verify the boundedness of its gradient.
If the solution $g(t)$ exists for $0 \leq t < T$, then $\sup_M |\nabla f(\cdot,t)|$ is finite for each $t < T$. \[le:nablaf\]
We compute the evolution equation for $|\nabla f|^2$: since $$\frac{\partial\, }{\partial t}g^{ij} = -g^{ip}g^{jq}\frac{\partial\, }{\partial t}g_{pq} = R g^{ij},$$ we have $$\label{eq-nabla-f}
\begin{split}
\frac{\partial\, }{\partial t}|\nabla f|^2 &= (\frac{\partial\, }{\partial t}g^{ij})\nabla_if\nabla_j f + 2g^{ij}
\nabla_i\frac{\partial f}{\partial t}\nabla_j f \\
&= R |\nabla f|^2 + 2g^{ij}\nabla_i(\Delta f)\nabla_j f \\
&= R |\nabla f|^2 +2\langle\nabla R, \nabla f\rangle \\
&\le C_1|\nabla f|^2 + C_2t^{-1/2}|\nabla f|,
\end{split}$$ where the last inequality uses Shi’s estimates $|R(x,t)| \le C(t_1)$ and $|\nabla R| \le C(t_1)t^{-1/2}$ if the flow exists on $M\times (0,t_1]$.
Next, for each $x\in M$, fix $\tau = \tau(x)$ sufficiently small, and set $D(x,\tau) = \sup_{[0,\tau]} |\nabla f(x,t)|^2$. We know that $\sup_M D(x,0) < \infty$, so integrating (\[eq-nabla-f\]) from $0$ to $\tau$ gives $$\begin{split}
D(x,\tau) & \leq C_0 + C_1 \tau D(x,\tau) + C_2 D(x,\tau) \int_0^{\tau} t^{-1/2}\, dt \\
& \leq C\left(1 + \tau D(x,\tau) + \sqrt{\tau D(x,\tau)}\right) \leq C(1+\sqrt{\tau} D(x,\tau)),
\end{split}$$ assuming that $\tau < 1$ and $D(x,\tau) > 1$. The constant $C$ is independent of $x$, hence so is $\tau = (4C^2)^{-1}$, and with this $\tau$ we obtain the uniform upper bound $$\label{eq-first}
\sup_{M \times [0,\tau]} |\nabla f(x,t)|^2 \leq C.$$
Finally, for $\tau \leq t < T$, $$\frac{\partial\, }{\partial t}|\nabla f|^2 \le C_1|\nabla f|^2 +C_2 \tau^{-1/2}|\nabla f|$$ so integrating from $\tau$ to any other value $t < t_0$ and using (\[eq-first\]) gives $$\sup_M |\nabla f(\cdot,t)|^2 \le C(t)$$ for any $t < T$, which is the desired result.
We now complete the proof of Proposition \[prop-LTE\]. Following [@Ha], we define $h := \Delta f + |\nabla f|^2$ and the symmetric $2$-tensor $Z := \nabla^2 f - \frac{1}{2}\Delta f\cdot g$. A straightforward computation shows that $$\frac{\partial h}{\partial t} = \Delta h - 2|Z|^2.
\label{eq:evh}$$ Using both Lemma \[le:nablaf\] and $\Delta f = R$, we see that $h(\cdot, t)$ is bounded for each $t$. We can thus apply the maximum principle to (\[eq:evh\]) to get that $$\sup_M h(\cdot,t ) \le \sup_M h(x,0) \le C.$$ This implies in turn that $$\sup_M R(\cdot,t) \le C,$$ for all $t\in [0,T)$, where $C$ is independent of $T$. In other words, the curvature remains uniformly bounded for as long as the flow continues to exist. Finally, since $u_t = -R u$, or equivalently, $(\log u)_t = -R$, we see that for some constants $C_1, C_2$ independent of $T$, $0 < C_1 \leq u \leq C_2$ for $0 \leq t < T$, and hence standard bootstrapping arguments show that $u$ remains bounded in ${{\mathcal C}}^\infty$ for all $t < T$. This continues to flow to $[0,T]$, and by Proposition \[pr:ste\], the flow continues to a slightly larger open interval. This proves that $g(t)$ exists for all $0 \leq t < \infty$.
A priori bounds, I
==================
We now begin to examine the long-time behaviour of $g(t)$. In this section we obtain a number of estimates concerning this behavior which we can prove using variants of the maximum principle. These are not enough to complete the proof of convergence, so in the next section we prove a number of further estimates using quite different geometric arguments. In all that follows, we often write $\Delta_t$ and $\nabla_t$ for the Laplacian and gradient with respect to $g(t)$.
The first result we obtain is that the asymptotically conical geometry is preserved. To prove this we closely follow an argument from Dai and Ma [@DM].
Suppose that $u_0(x) \in r^{-\tau}\Lambda^{2,\alpha}(M)$ and let $u(x,t)$ be the solution to with $u(x,0) = u_0(x)$. Then for all $t \geq 0$, $u(\cdot,t) \in r^{-\tau}\Lambda^{2,\alpha}(M)$, with bounds uniform in any strip $0 \leq t \leq T$. \[pr:acpersist\]
Let $r$ denote a smooth function which agrees with the radial distance function of the model conic metric on each end, and such that $r \geq 2$ on all of $M$. We may choose $r$ so that it satisfies $|\Delta_t r| + |\nabla_t r| \leq C(t)$.
We first show that $|R(t)| \leq C(t) r^{-2-\tau}$. Using and the absolute bound $|R(t)| \leq C$, we derive that $${\partial}_t (R^2) \leq \Delta_t R^2 - 2|\nabla_t R|^2 + 2C R^2,$$ from which we calculate that $w = r^{4 + 2\tau}R^2$ satisfies $${\partial}_t w \leq \Delta_t w + A \cdot \nabla_t w + B w,$$ where $A$ and $B$ are uniformly bounded. Hence, for $C_1, C_2$ sufficiently large, $${\partial}_t (w - C_1 e^{C_2 t}) \leq \Delta_t (w - C_1 e^{C_2 t}) + A \cdot \nabla_t (w - C_1 e^{C_2 t} ),$$ and moreover, $w - C_1 e^{C_2 t} \leq 0$ at $t=0$. We now invoke the maximum principle for an evolving family of metrics, proved by Ecker and Huisken and recorded as Theorem \[thm-EH\] in the appendix here. This gives $$|R| \leq C_1' e^{C_2' t} r^{-2-\tau}.$$ Integrating $ (\log u)_t = -R$ from $0$ to $t$ yields $$\log u(x,t) - \log u(x,0) = \int_0^t R(x,s)\, ds,$$ which completes the proof.
Let $(M^2,g_0)$ be AC of order $\tau$. Then $(M,g(t))$ remains AC of order $\tau$ for each $t \geq 0$.
The next two results concern upper and lower bounds for $u$ which are global in $t$.
The linear upper bound for $u(t)$ follows from a well-known argument due to Aronson and Benilan [@AB].
\[pr:ab\] Let $u$ be a solution of . Then $u_t \le \frac{u}{t}$, and hence $$u(t) \leq C(1+t)$$ for some constant $C>0$ and all $t \geq 0$.
Following [@AB], we define $u_{\lambda}(x,t) := \lambda u(x, \lambda^{-1}t)$. This satisfies $$\frac{{\partial}u_\lambda }{{\partial}t} = \Delta_{g_0} \log u_{\lambda} - R(g_0), \qquad \mbox{and} \qquad
\left. u_{\lambda}(x,t)\right|_{\lambda = 1} = u(x,t).$$ Moreover, for $\lambda > 1$, $u_{\lambda}(x,0) = \lambda u(x,0) > u(x,0)$. Setting $v_{\lambda}(x,t) := u_{\lambda}(x,t) - u(x,t)$, we calculate that $$\label{eq-dif}
\frac{{\partial}v_\lambda}{\partial t} (x,t) = \Delta_{g_0} (a(x,t)\cdot v_{\lambda}), \qquad v_{\lambda}(x,0) \geq 0,$$ where $$a(x,t) := \int_0^{1}\frac{d\theta}{\theta u_{\lambda}(x,t) + (1-\theta)u(x,t)}\, d\theta.$$
Using Proposition \[pr:acpersist\], for any $T > 0$ and all $(x,t) \in M \times [0,T]$, $C_1(T) \leq u(x,t) \leq C_2(T)$; hence $a(x,t)$ is also bounded above and below by ($T$-dependent) constants. This shows that is strictly parabolic on any finite time interval. Since $v_{\lambda}$ is uniformly bounded on $M \times [0,T)$ and since $v_{\lambda}(x,0) \ge 0$, it follows from the maximum principle that $v_\lambda \geq 0$ for $t \geq 0$. We also have $v_1(x,t) = 0$, so we find that $\lambda \mapsto v_\lambda$ is nondecreasing in some interval $[1,\lambda_0)$; that is, $\left. {\partial}_\lambda \right|_{\lambda = 1} v_\lambda(x,t)
\geq 0$, or equivalently, $u_t \le t^{-1} u$. Finally, $u > 0$, so the other estimate in the statement of this result follows by integrating $u_t/u \leq 1/t$.
It is also not hard to prove a uniform lower bound for $u$.
There exists a constant $C_1 > 0$ such that $C_1 \leq u(x,t)$ for all $(x,t) \in M \times [0,\infty)$. \[pr:clb\]
The first step is to show that the initial metric is conformal to another AC metric (with the same cone angles on each end) with $R_0 \leq 0$. Not surprisingly, this relies on the assumption that $\chi(M) < 0$.
Recall that if $\hat{g}_0 = e^{2\psi} g_0$, then $\Delta_0 \phi - \frac12 R_0 + \frac12 \hat{R}_0 e^{2\phi} = 0$ (where we denote the scalar curvatures of $g_0$ and $\hat{g}_0$ by $R_0$ and $\hat{R}_0$, respectively). Thus, given an AC metric $g_0$, to show that there exists $\psi$ such that $\hat{g}_0$ is AC and has scalar curvature $\hat{R}_0 \le 0$, it is sufficient to obtain $\psi$ satisfying $\Delta_0 \psi \geq \frac12 R_0$.
Let us choose a function $Q \in r^{-2-\tau}\Lambda^{0,\alpha}(M) \cap {{\mathcal C}}^\infty$ which satisfies $Q(x) \geq \frac12 R_0(x)$ and $\int Q = 0$. This is possible since, by , $\int R_0 < 0$. As a consequence of the surjectivity of the map $\Delta_g$ in , we can find $\psi_1 \in {{\mathcal E}}\oplus
r^{-\tau}\Lambda^{2,\alpha}$ with $\Delta_0 \psi_1 = Q$. This solution may grow logarithmically, so we must modify it further by adding on a function $w$ in the nullspace of the mapping such that the coefficient $\beta_j$ of $\log r$ in the expansion of $w$ on each end $E_j$ equals the corresponding coefficient of $\log r$ in the expansion for $\psi_1$ on that end. Proposition 6 in [@JMS] shows that this is possible. Therefore $\psi = \psi_1 - w$ is bounded and satisfies all the required properties.
Now we write the evolving metric in the form $g(t) = u(t) g_0 = u(t) e^{-2\psi}\hat{g}_0 := \hat{u}(t) \hat{g}_0$. Then $\hat{u}(0) = u(0)\cdot e^{-2\psi} \geq C_1 > 0$ and, using that $\hat{R}_0 \leq 0$, $${\partial}_t \hat{u} = \Delta_{\hat{g}(0)} \log \hat{u} - \hat{R}_0 \geq \Delta_{\hat{g}(0)} \log \hat{u}.$$ Hence by the minimum principle, $\hat{u}(t) \geq \hat{u}(0) \geq C_1 > 0$ for all $t \geq 0$. The lower bound for $u(x,t)$ now follows from that for $\hat{u}$, together with the relation $u=\hat{u} e^{2\psi}$.
Since it is also proved using the maximum principle, we include one final result, that if the initial curvature is nonpositive, then the curvature remains nonpositive for all time. Interestingly, our main convergence result is significantly easier to prove under this assumption on $R_0$.
Let $R_0 \leq 0$. Then $R \leq 0$ for all $t \geq 0$.
Recall that ${\partial}_t R = \Delta_0 R + R^2$. For any $A > 0$ define $$Q(x,t) = R(x,t) + \frac{1}{A+t}.$$ Then $${\partial}_t Q = {\partial}_t R - \frac{1}{(A+t)^2} = \Delta_0 Q + R^2 - \frac{1}{(A+t)^2} =
\Delta_0 Q + VQ,$$ where $$V = R - \frac{1}{A+t}.$$ Since $R(x,0) \leq 0$, the function $V$ is strictly negative for $t=0$. For fixed $A$, we now define $$\begin{gathered}
T = T_A = \inf \{ t > 0: R(x,t) < \frac{1}{A+\tau}\ \mbox{for}\ 0 < \tau < t \\
\mbox{and} \ R(x_0,t) = \frac{1}{A+t} \ \mbox{for some}\ x_0 \in M\}.\end{gathered}$$ Note that, since $R(x,0) \leq 0$, $T$ is strictly positive, and furthermore, if $T= \infty$, then $R(x,t) \leq 1/(t+A)$ for all $t \geq 0$.
An application of the maximum principle to $Q(x,t)$ on $M \times [0,T)$, using that $V \leq 0$ on this domain, gives $$Q(x,t) \leq \min \{ \inf_{t \in [0,T]} \frac{1}{t+A}, \frac{1}{A} \} \leq \frac{1}{A}$$ for all $(x,t)$ in this domain, or in other words, $$R(x,t) \leq \frac{1}{A} - \frac{1}{t+A} = \frac{t}{A(t+A)}.$$ Since $t/(t+A)$ is increasing in $t$, we have $$R(x,t) \leq \frac{T}{A(T+A)} = \frac{1}{T+A} \frac{T}{A}.$$ However, we know that $R(x_0,T) = 1/(T+A)$ for some $x_0$, which implies that $T_A \geq A$.
These calculations show that if $A > 0$ is arbitrary, then either $R(x,t) \leq 1/(t+A)$ for all $t \geq 0$, or else $R(x,t) \leq \frac{t}{t(t+A)}$ for $0 \leq t \leq A$. Letting $A \to \infty$ in either case implies that $R(x,t) \leq 0$ for all $t \geq 0$.
A priori bounds, II
===================
Since $g(t)$ has a uniform linear upper bound, it is natural to consider the rescaled family of metrics
$$\tilde{g}(t) := \frac{1}{t} g(t);$$ indeed, it follows from Proposition \[pr:ab\] that $\tilde{g}(t) \leq C g(0)$ for all $t \geq 1$, with $C$ independent of $t$. This family also satisfies an evolution equation: setting $\tau = \log (t)$, we calculate that $${\partial}_\tau \tilde{g}(\tau) = - (\tilde{R} + 1) \tilde{g}(\tau),
\label{eq:evgtilde}$$ where $\tilde{R}$ is the scalar curvature of $\tilde{g}$ at time $\tau$, or equivalently, $${\partial}_\tau \tilde{u} = \Delta_0 \log\tilde{u} - \tilde{u} - R_0 = - (\tilde{R} + 1)\tilde{u},
\label{eq:evutilde}$$ where $\tilde{u}(\cdot,\tau) = u(\cdot, t)/(t) = e^{-\tau} u(\cdot, e^\tau)$.
The function $\tilde{u}(\tau,x)$ is monotone nonincreasing in $\tau$ for each fixed $x$.
One form of the original evolution equation is that $u_t = -Ru$; on the other hand, by Proposition \[pr:ab\], $$R(x,t) \ge -\frac{1}{t} \Rightarrow \tilde{R}(\tau) \geq -1.$$ From these we see that the right side of (\[eq:evutilde\]) is nonpositive; hence ${\partial}_\tau \tilde{u} \leq 0$.
We now state the main result of this paper.
\[thm-part\] The metric $\tilde{g}(\tau)$ converges in ${{\mathcal C}}^\infty$ on every compact set as $\tau\to\infty$; the limiting metric $g_\infty$ is complete, hyperbolic, and has finite area.
The proof will occupy the rest of this section. It proceeds roughly as follows: since $\tilde{u}$ is monotone nonincreasing in $\tau$, it has a limit as $\tau \nearrow \infty$. If we can show that $\tilde{u}(x_0,\tau)\geq c > 0$ for some fixed $x_0$, then a gradient estimate for $\tilde{u}$ together with a Harnack inequality for $R(\tilde{g})$ imply that $\tilde{u}$ stays bounded away from zero in a fixed neighbourhood of $x_0$. This allows us to apply Hamilton’s compactness theorem to solutions of this flow and thereby conclude that $(M,\tilde{g}(\tau), x_0)$ converges to a [*complete*]{} Riemannian surface.
The first step, that $\tilde{u}(x,\tau)$ cannot tend to $0$ for every $x$, is accomplished using a further rescaling of the metric $g$, and requires the topological hypothesis that $\chi(M) < 0$.
There exists a point $x_0 \in M $ such that $\tilde{u}(x_0,\tau) \geq \delta > 0$ for some fixed $\delta$ and all $\tau \ge 0$. \[pr:lowerboundtu\]
By monotonicity of $\tilde{u}$ in $\tau$, $\lim_{\tau \to \infty} \tilde{u}(x,\tau) := \tilde{U}(x)$ exists for every $x \in M$. Thus we must prove that $\tilde{U} \not\equiv 0$.
Suppose, to the contrary, that $\tilde{U}(x) = 0$ for every $x$. Dini’s theorem states that a monotone sequence of continuous functions which converges pointwise to a continuous function must in fact converge uniformly on compact sets. We show that this leads to a contradiction.
Let $K$ be a sufficiently large compact set so that $M \setminus K$ is a union of (asymptotically conical) ends $E_j \cong [1,\infty) \times S^1$ (using the coordinates $(r,\theta)$ where the metric has the form $dr^2 + \alpha^2 r^2 d\theta^2 + {{\mathcal O}}(r^{-\nu})$). Choose any sequence $\tau_i \nearrow \infty$ and $p_i \in K$ so that $$\alpha_i := \tilde{u}(p_i,\tau_i) = \max_{x\in K} \tilde{u}(x,\tau_i).$$ By hypothesis, $\alpha_i \to 0$.
Now perform yet another rescaling: set $$\bar{g}_i(\tau) = \alpha_i^{-1} \tilde{g}(\tau_i + \alpha_i\tau),$$ and let $\bar{u}_i$ be the corresponding conformal factor. Note that $R(\bar{g}_i) = \alpha_i R(\tilde{g}_i)$, and by construction, $\bar{u}_i(p_i, 0) = 1$.
Since $R(\tilde{g}_i) \geq -1$, we have a lower bound $R(\bar{g}_i) \geq -\alpha_i \geq -C$, but we do not yet know that the curvatures of this sequence of metrics are uniformly bounded from above. We first prove our result assuming that such an upper bound is true; i.e. that $$R(\bar{g}_i) \leq C.
\label{eq:Rbar0}$$ Then at the end we justify (\[eq:Rbar0\]), allowing the constant $C$ in (\[eq:Rbar0\]) to depend on a compact set over which we are estimating the rescaled curvature; this will be enough to finish the argument.
The immediate goal is to estimate $\nabla \bar{u}_i$, which will allow us to prove that $\bar{g}_i$ converges to a complete metric.
Let $f_0$ be the potential function associated to the original scalar curvature function $R_0$, as defined in Lemma \[lem-potential\], and let $f(x,t)$ be its evolution under the linear heat flow ${\partial}_t f = \Delta_t f$. It has been proved in §3 that $f(x,t)$ is a potential function for $g(t)$. We claim that $\log u(x,t) \equiv f_0(x) - f(x,t)$, or equivalently, that $k(x,t) := \log u(x,t) - f_0(x)
+ f(x,t) \equiv 0$. To prove this, note that ${\partial}_t k = \Delta_t k$ and $k(x,0) = 0$. Furthermore, $x \mapsto k(x,t)$ is bounded for each $t$, which holds because $|{\partial}_t f| = |\Delta f| = |R|$ is bounded for each $t$. Hence $|f(x,t) - f(x,0)| \leq Ct$, and $|\log u|$ is bounded for each $t$ as well. The maximum principle then implies that $k \equiv 0$.
Continuing on, we have $$\frac{\partial}{\partial\tau}(\log \tilde{u} - f_0) = \Delta_{\tilde{g}}(\log\tilde{u} - f_0) - 1,$$ and then a computation from [@Ha] gives us $$\frac{\partial}{\partial\tau}|\nabla(\log\tilde{u} - f_0)|_{\tilde{g}}^2 \le \Delta_{\tilde{g}}|\nabla(\log\tilde{u} - f_0)|_{\tilde{g}}^2.$$ Since $|\nabla(\log\tilde{u} - f_0)|_{\tilde{g}}^2(\cdot,\tau)$ is bounded for each $\tau$, the maximum principle shows that $$\label{eq-nabla-100}
|\nabla_{\tilde{g}} \log \tilde{u}|_{\tilde{g}}(\cdot,\tau) \le C \qquad \mbox{for all} \quad \tau \geq 0,$$ or equivalently, $$|\nabla_{\bar{g}_i} \log\bar{u}_i|_{\bar{g}_i} \le C\sqrt{\alpha_i},$$ and therefore (since $\bar{u}_i(x,0) \le 1$ on $K$) $$|\nabla\bar{u}_i(\cdot, 0)|_{\bar{g}_i} \le C\sqrt{\alpha_i} \qquad \mbox{on} \quad K.$$ Combining this with the fact that $\bar{u}_i(p_i,0) = 1$, we see that there exists an $\eta > 0$ such that for each sufficiently large $i$, $$\bar{u}_i(x,0) \geq \eta \qquad \mbox{for all} \quad x\in B_{\bar{g}_i(0)}(p_i,1);$$ from this we also see that $$\label{eq-inj-bari}
{\mathrm{Inj}}_{\bar{g}_i(0)}(p_i) \ge \delta > 0 \qquad \mbox{for all} \quad i.$$
Boundedness of $R(\bar{g}_i)$ and the estimate (\[eq-inj-bari\]) are precisely the hypotheses needed to apply Hamilton’s compactness theorem. This result states that, after passing to a subsequence, $(M,\bar{g}_i(\tau),p_i)$ converges in the pointed Cheeger-Gromov sense to a limiting family of complete metrics $(M_{\infty}, g_{\infty}(\tau), p_{\infty})$ which is an eternal solution of the Ricci flow. Now, it follows from the Aronson-Bénilan inequality that any ancient solution of the Ricci flow on surfaces is nonnegatively curved. Moreover, the work of B.L. Chen [@Chen] shows that the scalar curvature of any ancient solution is nonnegative. However, since $\tilde{u} (\tau_i) \leq \alpha_i$ on $K$, we have that $$\mbox{diam}_{\tilde{g}(\tau_i)} (K) \leq C \sqrt{\alpha_i} \Longrightarrow \mbox{diam}_{\bar{g}_i}(K) \leq C',$$ where the last constant is independent of $i$. Hence $\bar{g}_i$ is ${{\mathcal C}}^2$ close to $g_\infty$ on the compact set $K$. But this leads to a contradiction since it follows from (\[eq:GB\]) that $\limsup_{i \to \infty} \int_K R(\bar{g}_i) < 0$.
It remains to verify (\[eq:Rbar0\]).
Let $\beta_i = \alpha_i \tilde{R}(p_i, \tau_i + \alpha_i)$ (this is the curvature of $R(\bar{g}_i)$ at $p_i$ at $\tau = 1$). We claim that $\beta_i$ is bounded. If this were to fail, i.e. if $\beta_i \to \infty$, at least along some subsequence, then by the Harnack inequality for $\tilde{R}$ (the form of this inequality which we use here is stated in [@Ch]; the original Harnack estimate for Ricci flow, proven by Hamilton, appears in [@Ha2]), for $\alpha_i \leq \tau \leq \alpha_i + 1$, we would have $$\tilde{R}(p_i,\tau_i+\tau) \geq C (\tilde{R}(p_i,\tau_i+\alpha_i) + 1)\cdot e^{-C(\tau - \alpha_i)} - 1 \geq C' \frac{\beta_i}{\alpha_i},$$ with $C'$ independent of $i$. Moreover, $${\partial}_\tau\tilde{u} = -(\tilde{R} + 1)\tilde{u} \leq - C' \beta_i$$ in this interval, so that $$\tilde{u}(p_i,\tau_i+\alpha_i + 1) \leq \tilde{u}(p_i,\tau_i+\alpha_i) - C' \beta_i.$$ Using monotonicity and iteration, we have $\tilde{u}(p_i,\tau_i + \alpha_i) \to -\infty$ as $i \to \infty$. This contradicts the fact that $\tilde{u} > 0$.
We need to apply Hamilton’s compactness theorem again as in the argument on the previous page; in fact, we need a generalization, proved in Appendix E of [@KL], which requires that the curvature $R(\bar{g}_i)$ be bounded uniformly in time, but only over a fixed compact set $K \subset M$. This hypothesis is verified as follows.
The Harnack estimate for $\tilde{R}$ states that for $0 \leq s \leq 1/2$, $$\tilde{R}(p,\tau_i+s\alpha_i) + 1 \le C(1+\tilde{R}(p_i,\tau_i+\alpha_i)) \, \exp \left(
C\frac{{\mathrm{dist}}^2_{\tilde{g}(\tau_i+s\alpha_i)}(p,p_i)}{(1-s)\alpha_i}\right),$$ and hence $$\tilde{R}(p,\tau_i+s\alpha_i) \leq \frac{C(\rho)}{\alpha_i},$$ for all $p$ with ${\mathrm{dist}}_{\tilde{g}(\tau_i)}(p,p_i) \le \sqrt{\alpha_i} \, \rho$. Here we are use the fact that the distance with respect to $\tilde{g}(\tau)$ between any two fixed points decreases in $\tau$; this follows from the monotonicity of $\tilde{u}$. These facts together imply the bound we are seeking, that $$|\bar{R}_i(s,x)| \le C(\rho) \qquad x \in B_{\bar{g}_i}(p_i,\rho),$$ since $s \leq 1/2$.
The other hypothesis we must verify is that $\bar{g}_i(0)$ has a bound on its injectivity radius at $p_i$ which is uniform in $i$. This is done exactly as before.
The compactness theorem allows us to pass to a limiting metric, which is necessarily an ancient solution of the Ricci flow. As we have already discussed, any ancient solution has nonnegative curvature, but considering the Gauss-Bonnet integral over $K$, we obtain the same contradiction as before.
This completes the proof that $\alpha_i$ must remain bounded away from zero.
There exists an $x_0\in K$ so that $\tilde{u}(x_0,\tau) \ge \delta > 0$ for all $\tau \ge 0$.
Since all $p_i \in K$, the set $\{p_i\}$ has an accumulation point in $K$, call it $x_0$, with the property that $$\label{eq-closeness}
\lim_{i\to\infty}{\mathrm{dist}}_{\tilde{g}(0)}(p_i,x_0) = 0.$$ Since $\alpha_i$ is bounded, there exists $\delta > 0$ so that $$\tilde{u}(p_i,\tau_i) \ge 2\delta, \,\,\, \mbox{for all} \,\,\, i.$$ Combining (\[eq-nabla-100\]) with the uniform global upper bound on $\tilde{u}$, we have $|\nabla\tilde{u}| \le C$. This implies that $$\tilde{u}(x_0,\tau_i) \ge \tilde{u}(p_i,\tau_i) - C{\mathrm{dist}}_{\tau_i}(p_i,x_0) \ge 2\delta - C{\mathrm{dist}}_{\tilde{g}(0)}(x_0,p_i) \ge \delta,$$ for $i$ sufficiently big so that ${\mathrm{dist}}_{\tilde{g}(\tau_i)}(x_0,p_i) \le {\mathrm{dist}}_{\tilde{g}(0)}(x_0,p_i) \le \frac{\delta}{C}$.
(Here we have used (\[eq-closeness\]) together with the result that the distances are nonincreasing in $\tau$.)
With this claim, we complete the proof of Proposition \[pr:lowerboundtu\].
We now know that there exists some $x_0\in M$ such that $\tilde{u}(x_0,\tau) \geq c > 0$ for all $\tau\geq 0$. The gradient estimate (\[eq-nabla-100\]) can be applied as before to show that for some $r_1 > 0$, $$\label{eq-inj}
\tilde{u}(x,\tau) \ge \delta > 0$$ for all $x\in B_{\tilde{g}(\tau)}(x_0,r_1)$, $\tau\geq 0$. Clearly $\tilde{g}(0)$ is ‘$\kappa$-noncollapsed’; i.e., there exist constants $\kappa, r_0 > 0$ ($r_0 \le r_1$) so that if $B_0(x,r)$ is the geodesic ball around $x$ of radius $r \le r_0$, with respect to $\tilde{g}(0)$, then $\mathrm{Area}_{\, 0}\,(B_0(x,r)) \geq \kappa r^2$. If $B_\tau(x,r)$, for $r \le r_0$, is the corresponding geodesic ball with respect to $\tilde{g}(\tau)$, then by monotonicity of $\tilde{u}$, $B_0(x_0,r) \subset B_{\tau}(x_0,r)$. Hence using (\[eq-inj\]), we have $$\label{volume-lower}
\begin{array}{rcl}
\mathrm{Area}_{\, \tau}(B_{\tau}(x_0,r)) &=& \int_{B_{\tilde{g}(\tau)}(x_0,r)} \frac{\tilde{u}(x,\tau)}{\tilde{u}(x,0)}\, dV_{\tilde{g}(0)} \\
&\ge& \tilde{\delta}\int_{B_{\tilde{g}(\tau)}(x_0,r)} dV_{\tilde{g}(0)} \ge
\int_{B_0(x_0,r)} dV_{\tilde{g}(0)} \\
&=& \mathrm{Area}_{\, 0} (B_0(x_0,r))\ge \kappa' r^2.
\end{array}$$ In other words, $\tilde{g}(\tau)$ is $\kappa'$-noncollapsed, where $\kappa'$ is a $\tau$-independent multiple of $\kappa$.
We shall use the same compactness theorem as before to show that $\tilde{g}(\tau)$ converges to a complete metric. In order to do so, we must show that $|R(\tilde{g}(x,\tau)| \leq C(\rho) < \infty$ for $x \in B(x_0,\rho)$, for all $\rho > 0$ and that $\mathrm{Inj}_{\tilde{g}(\tau)}(x_0) \geq c > 0$.
For the next step, we establish a local curvature bound.
\[lem-cur-comp\] For every $\rho > 0$ there is a constant $C_\rho$ so that $$\tilde{R}(x,\tau) \le C_\rho \qquad \mbox{for all} \quad (x,\tau) \ \mbox{such that}\ x \in B_{\tau}(x_0,\rho),\ \tau \ge \tau_0.$$
Using the Harnack inequality (\[eq-har-r\]) quoted in the appendix, it suffices to show that for some $C > 0$, $\tilde{R}(x_0,\tau) \le C$ for all $\tau \ge 0$. If this were to fail, then there would exist a sequence $\tau_i\nearrow \infty$ for which $Q_i := \tilde{R}(x_0,\tau_i) \to \infty$. Taking $\tau_2 = \tau_i + \tau$, $\tau_1 = \tau_i$, $x_2 = x_1 = x_0$ in (\[eq-har-r\]), and fixing $A > 0$, we see that for $0 \leq \tau \leq A$, $$\label{eq-R-bound1}
\tilde{R}(x_0,\tau_i+\tau) \geq e^{-C\tau}\cdot (\tilde{R}(x_0,\tau_i) + 1) - 1 \geq cQ_i,$$ where $c$ depends only on $A$.
Arguing as before, ${\partial}_\tau \log\tilde{u}(x_0,\tau) \le -c Q_i \to -\infty$, so integrating over the interval $[\tau_i,\tau_i+\tau]$ (with $\tau \le A$), we obtain $$\tilde{u}(x_0,\tau_i+\tau) \le e^{-c Q_i\tau}\tilde{u}(x_0,\tau_i) \to 0,$$ which contradicts that $\tilde{u}(x_0,\tau) \geq c > 0$.
This proves that $\tilde{R}(x_0,\tau) \le C$ for all $\tau \geq 0$. Finally, putting $x_1 = x$, $x_2 = x_0$, $\tau_1 = \tau$ and $\tau_2 = \tau + 1$ in (\[eq-har-r\]), we obtain the conclusion of the Lemma.
Finally, we invoke a result due to Cheeger, who has shown that for a geodesic ball $B(x_0,r_0)$ in some Riemannian manifold $(M^n,g)$, if there are lower bounds on Ricci curvature and volume, then the injectivity radius of $(M,g)$ at $x_0$ is bounded away from $0$ by some constant depending only on these bounds. We have established all of these hypotheses (in Lemma \[lem-cur-comp\] and (\[volume-lower\])); hence $$\label{eq-injradest}
{\mathrm{Inj}}_{\tau}(x_0) \geq c > 0 \qquad \forall \, \tau \geq 0.$$
We now apply the compactness theorem to conclude that for every sequence $\tau_i\to \infty$, there is a subsequence of $(M,\tilde{g}(\tau_i + \tau), x_0)$ which converges smoothly as a family of pointed spaces to a smooth complete family of metrics $(M_\infty, g_{\infty}(\tau),x_\infty)$. More specifically, for each compact interval $I\subset [0,\infty)$ and for any compact set $K\subset M_{\infty}$ containing $x_0$, there are pointed, $\tau$-independent diffeomorphisms $\phi_{K,i}:K\to K_i \subset M$ such that the appropriate subsequence of $\phi^*_{K,i}\tilde{g}(\tau_i+\tau)$ converges smoothly to $g_{\infty}(\tau)$ on $K\times I$.
It is clear from the monotonicity of $\tilde{g}(\tau)$ that this limit is unique and does not depend on $\tau$, so $g_\infty$ satisfies the stationary equation. Thus $\tilde{u}(\cdot,\tau)$ converges uniformly on compact sets of $M$ to a continuous function $\tilde{U}(x)$. This limiting function is nonnegative, but there is still a possibility that it vanishes on some nontrivial closed set, which would make $\tilde{U}(x) g_0$ degenerate. However, this is ruled out by uniqueness of the limiting metric and the fact that $g_\infty$ is complete, so $\tilde{U} > 0$ everywhere.
We may now use the equation $$\Delta\log \tilde{u} - R_0 = -\tilde{R}\tilde{u}.$$ and the fact that $|\log \tilde{u}| \leq C$ and $-1 \le \tilde{R}(x,\tau) \le C(K)$ for $\tau \ge 0$ and $x\in K$ to obtain $\tau$-independent bounds on all higher order derivatives of $\tilde{u}$ over any compact set by standard bootstrapping. It follows that the convergence of $\tilde{u}(x, \tau)$ to $\tilde{U}(x)$ is ${{\mathcal C}}^\infty$ on compact sets. We have thus proved that $$\label{eq-smooth-conv}
\tilde{g}(x,\tau) = \tilde{u}(x,\tau) g_0 \longrightarrow \tilde{U}(x) g_0 = g_\infty$$ smoothly on compact sets.
It remains to show that $g_\infty$ is hyperbolic and has finite area. For the first of these, recall that we already know that $R_{\infty}(x) \ge -1$. If we assume that $R_{\infty}(y) \ge -1 + 2\delta$ for some $\delta > 0$ and some $y\in B_{g_{\infty}}(x_0,r_0)$, where $x_0$ and $r_0$ are as above, then $\tilde{R}(y,\tau) \ge -1 + \delta$ for $\tau \ge \tau_0$ and $y\in B_{\tilde{g}(\tau)}(x_0,r_0)$ and it follows that $${\partial}_\tau \log\tilde{u}(y,\tau) = -(\tilde{R} + 1) \le -\delta \Longrightarrow
\tilde{u}(y,\tau) \le \tilde{u}(y,\tau_0) \cdot e^{-\delta(\tau - \tau_0)},$$ which contradicts (\[eq-inj\]). Therefore, $R_{\infty} \equiv -1$ on $B_{g_{\infty}}(x_0,r_0)$. Now put $\tau_1 = \tau$ and $\tau_2 = \tau + 1$ in (\[eq-har-r\]) and use (\[eq-delta\]) to get that $$\label{eq-previous}
\tilde{R}(x_1,\tau) + 1 \le \tilde{R}(x_2,\tau+1) e^{d(x_1,x_2,\tau)^2/4 + C}.$$ Taking $x_2\in B(x_0,r)$ and $x_1$ any other point in $M$, and letting $\tau\to\infty$ in (\[eq-previous\]), the smooth convergence of $\tilde{u}$ implies that $$\tilde{R}_{\infty}(x_1) + 1 \leq \tilde{R}(x_2) e^{d_{\infty}(x_1,x_2)^2/4 + C} = 0.$$ However, since $\tilde{R}_{\infty} + 1 \ge 0$, we conclude finally that $R_{\infty} \equiv -1$ on $M_\infty$.
The final step concerns the finiteness of the area. Since $\tilde{g}_\infty$ is hyperbolic and complete, if its area were not finite, then its area growth would be exponential; i.e., there would exist constants $C_1, C_2, a, b > 0$ so that $$C_1 e^{ar} \le \mathrm{Area}_\infty (B_{\infty}(p,r)) \le C_2 e^{br}$$ for $r$ large. This would imply that for $i$ large, $$\label{eq-hyper-growth}
\frac12 C_1 e^{ar} \le \mathrm{Area}_{\tau_i} B_{\tau_i}(p_i,r) \le 2C_2 e^{br}.$$ However, $\tilde{u}(\cdot, \tau_i) \leq c_2$ uniformly on $M$, for all $i$, so $(M,\tilde{g}(\tau_i))$ can have at most quadratic area growth. This contradiction finishes the proof of Theorem \[thm-part\].
Appendix: maximum principle and Harnack inequality {#sec-maximum}
==================================================
We state two versions of the maximum principle for complete manifolds, and then a version of the Harnack estimate which holds for such geometries.
\[thm-max\] Let $g(t)$, $0 \le t < T$, be a family of complete Riemannian metrics on a noncompact manifold $M$ which vary smoothly in $t$ and which satisfy $C_1 g(0) \leq g(t) \leq C_2 g(0)$ for some fixed constants $C_1, C_2$ and for all $t \in [0,T)$. Let $f(x,t)$ be a smooth bounded function on $M\times [0,T)$ which satisfies the initial condition $f(x,0) \geq 0$, and satisfies the parabolic equation $$\frac{{\partial}\,}{{\partial}t}f = \Delta_{g(t)} f + Q(f,x,t),$$ where $Q(f,x,t) \ge 0$ whenever $f \le 0$. Then $f(x,t) \ge 0$ on $M\times [0,T)$.
The proof can be found in [@Sh] and [@Sh1].
\[thm-EH\] Let $(M,g)$ be a complete noncompact Riemannian manifold which satisfies the uniform volume growth condition $${\mathrm{Vol}}_t(B_r(p)) \le e^{k(1+r^2)}$$ for some point $p\in M$ and a uniform constant $k > 0$ for all $t\in [0,T]$. Let $w \in {{\mathcal C}}^\infty(M\times (0,T]) \cap
{{\mathcal C}}^0(M\times [0,T])$ satisy the differential inequality $$\frac{\partial}{\partial t} w \le \Delta w + {\bf a}\cdot\nabla w + b w,$$ where $\sup_{M\times[0,T]}|{\bf a}| \le c_1$ and $\sup_{M\times[0,T]}|b| \le c_2$. If in addition $w(x,0) \le 0$ for all $x\in M$, $\int_0^T\int_M e^{-c_3 r_t(p,y)^2}|\nabla w|^2(y)\, d\mu_t\, dt < \infty$ for some constant $c_3 > 0$ and $\sup_{M\times[0,T]}|\frac{\partial}{\partial t} g(t)| \le c_4$, then $w \le 0$ on $M\times [0,T]$.
We now state a Harnack estimate which holds for $\tilde{R}$, allowing one to compare its value at different space-time points as the metric evolves under Ricci flow. Hamilton first proved the Harnack estimate for Ricci flow in the case that the curvature operator is nonnegative. Chow [@Ch] generalized this to the case which allows some negative curvature. Using the maximum principles stated above, and using the uniform boundedness in space of $\tilde{R}$ and its derivatives on a fixed time slice, one can adapt Chow’s arguments to the case of Ricci flow on complete manifolds. The Harnack estimate for $\tilde{R}$ states that there exist constants $\tau_0 > 0$ and $C$ such that for every $x_1, x_2 \in M$ and $\tau_2 \ge \tau_1 \ge \tau_0$, $$\label{eq-har-r}
\tilde{R}(x_2,\tau_2) + 1 \ge e^{-\Delta/4 - C(\tau_2 - \tau_1)}(\tilde{R}(x_1,\tau_1) + 1),$$ where $$\Delta = \Delta(x_1,x_2,\tau_1,\tau_2) = \inf_{\gamma}\int_{\tau_1}^{\tau_2} |\frac{d\gamma}{dt}(t)|^2\, dt,$$ and the infimum is taken over all paths $\gamma$ in $M$ whose graphs $(\gamma(t),t)$ join $(x_1,\tau_1)$ and $(x_2,\tau_2)$. Since the metric $\tilde{g}$ is shrinking it is easy to derive (see [@Ha2]) $$\label{eq-delta}
\Delta \le \frac{d(x_1,x_2,\tau_1)^2}{\tau_2-\tau_1} \le \frac{{\mathrm{dist}}(x_1,x_2,0)^2}{\tau_2-\tau_1},$$ where $d(x_1,x_2,\tau_1)$ is a distance between points $x_1$ and $x_2$ computed at time $\tau_1$.
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[^1]: Research supported in part by NSF grant PHY-0652903
[^2]: Research supported in part by NSF grant DMS- 0805529
[^3]: Research supported in part by NSF grant DMS-0905749
|
---
abstract: |
Baryon resonances with even and odd parity are collectively investigated from the viewpoint of chiral symmetry(ChS). We propose a quartet scheme where $\Delta$’s and $N^{*}$’s with even and odd parity form a chiral multiplet. This scheme gives parameter-free constraints on the baryon masses in the quartet, which are consistent with observed masses with spin ${{1 \over 2}},{{3 \over 2}},{{5 \over 2}}$. The scheme also gives selection rules in the one-pion decay: The absence of the parity non-changing decay $N(1720) \rightarrow \pi
\Delta(1232)$ is a typical example which should be confirmed experimentally to unravel the role of ChS in baryon resonances.
address:
- '$^{(1)}$ Department of Physics, Kyoto University, Kyoto, 606-8502, Japan'
- '$^{(2)}$ Faculty of Science and Technology, Ryukoku University, Seta, Otsu-city, 520-2194, Japan'
author:
- 'D. Jido$^{(1)}$, T. Hatsuda$^{(1)}$, T. Kunihiro$^{(2)}$'
title: 'Chiral Symmetry Realization for Even- and Odd-parity Baryon Resonances'
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Chiral symmetry (ChS) and its dynamical breaking in quantum chromodynamics (QCD) are the key ingredients in low energy hadron dynamics. For instance, all hadrons can be classified in principle into some representation of the chiral group $SU(N_{f})_{L} \times SU(N_{f})_{R} $, and the interactions among hadrons are strongly constrained by this symmetry.
There are two ways to realize ChS in effective low-energy Lagrangians; non-linear and linear representations. The former has been extensively studied in the pion and nucleon sector and is summarized as the celebrated chiral perturbation theory [@cp]. The non-linear chiral transformations of the pion and the nucleon are uniquely determined once we fix the parameterization of the coordinates of the coset space $SU(N_{f})_{L} \times
SU(N_{f})_{R} / SU(N_{f})_{V} $[@ccwz] and the transformation of the nucleon under $SU(N_{f})_{V}$[@wein]. On the other hand, in the linear representation, scalar mesons as chiral partners of the Nambu-Goldstone (NG) bosons are introduced. Although such heavy mesons do not allow systematic low energy expansion at zero temperature, this representation is essential for studying critical phenomena near the chiral phase transition where both the scalars and NG-bosons act as soft modes [@pw].
Then what about the baryons in the linear representation? The Gell-Mann Lévy sigma model [@GeL] is a first example where the nucleon transforms linearly both under the vector and the axial-vector transformations. DeTar and Kunihiro [@dk] generalized the model so that $N_{+}$ (the nucleon) and its odd-parity partner $N_{-}$ form a multiplet of the chiral group. (Note that, in the non-linear representation, different baryons do not form a multiplet by construction [@wein].) A unique aspect of their model is that the finite mass of the nucleon can be introduced in a chiral invariant way, which opens a possibility that even and odd parity nucleons may be degenerate with a non-vanishing mass in the chirally symmetric phase.
In DeTar-Kunihiro’s construction, $N_{\pm}$ are represented as a superposition of $N_{1}$ and $N_{2}$ which are assigned to have opposite axial charges with each other. Subsequently this was called the “mirror assignment" and distinguished from the “naive assignment" where $N_{1}$ and $N_2$ have the same axial change [@jnoh]: The two assignments are showed to have phenomenologically distinguishable predictions [@joh].
The purpose of this Letter is to develop the idea of the mirror assignment in baryon resonances with different parity ($P = \pm)$ and different isospin ($I ={{1 \over 2}}, {{3 \over 2}}$), and to explore how ChS is realized in the excited baryons. Achieving this purpose is tantamount to constructing a linear sigma model in which both $\Delta_{\pm}$’s and $N^{*}_{\pm}$’s are incorporated for a given spin sector. (Here we call $N^*$ ($\Delta$) as a resonance with $I= {{1 \over 2}}$ $({{3 \over 2}})$, and the subscripts $\pm$ denote their parity.) Thus we shall arrive at proposing a [*quartet scheme*]{} in which $N^*_{+} , N^*_{-}, \Delta_{+}$ and $\Delta_{-}$ form a chiral multiplet. It will be shown that this quartet scheme is consistent with the observed baryon spectra without fine-tuning of the model parameters. We will also show some evidence of this scheme in the decay pattern of the resonances. Throughout the present Letter, we focus on $N_{f} = 2$, and neglect the explicit breaking of ChS due to quark masses.
To make the argument explicit, let us start with $\Delta(1232)$ ($J^P={{3 \over 2}}^{+}$) and its chiral partners. First of all, we need to choose the representation of $\Delta$ under $SU(2)_{L} \times SU(2)_{R}$. The quark fields $q= q_{l} + q_{r}$ belong to $({{1 \over 2}},0) \oplus (0,{{1 \over 2}})$, where the first and second numbers in the parentheses refer to $SU(2)_{L}$ and $SU(2)_{R}$ representations, respectively. Therefore, $({{3 \over 2}},0) \oplus (0,{{3 \over 2}})$ and $(1,{{1 \over 2}}) \oplus ({{1 \over 2}},1)$ are the two candidates for $\Delta$; both of them contain isospin $I={{3 \over 2}}$ and are constructed from three quarks $[({{1 \over 2}},0) \oplus (0,{{1 \over 2}}) ]^3$ [@cj]. Here, we choose $(1,{{1 \over 2}}) \oplus ({{1 \over 2}},1)$ for $\Delta$, because $\Delta$ is known to be a strong resonance in $N$-$\pi$ system, and $N \times \pi = [({{1 \over 2}},0) \oplus
(0,{{1 \over 2}})] \times [({{1 \over 2}},{{1 \over 2}})]$ does not contain $({{3 \over 2}},0) \oplus (0,{{3 \over 2}})$. In the quark basis, this representation may be schematically written as $ (1,{{1 \over 2}}) \oplus({{1 \over 2}},1) =(q_{_L}q_{_L})_{I=1} q_{_R}
\oplus q_{_L} (q_{_R}q_{_R})_{I=1}$ where Lorentz and color indices are suppressed [@QSR]. Note that $ (1,{{1 \over 2}}) \oplus({{1 \over 2}},1)$ contains both $I={{3 \over 2}}$ and $I={{1 \over 2}}$ baryons, thus we utilize the latter to incorporate $N^*$. From now on, we do not consider the quark structure of $\Delta$ and $N^*$, and simply introduce elementary Rarita-Schwinger (RS) fields for constructing an effective Lagrangian.
To accommodate the parity partners of the baryon resonances, let us define $\psi_{1}$ and $\psi_{2}$ as two independent $J={{3 \over 2}}$ RS fields with even and odd parity, respectively. The Lorentz index $\mu =0,\ldots, 3$ for the RS fields is suppressed for brevity. We then define the chiral decomposition; $\psi_{i} = \psi_{il} + \psi_{ir} $ with $\gamma_5 \psi_{il, ir}= \mp \psi_{il, ir} $ ($i$=1,2). In the $J={{3 \over 2}}$ chiral-quartet, $\psi_{1}$ and $\psi_{2}$ are mixed to form four resonances; $\Delta_{+}(P_{33})$, $\Delta_{-}(D_{33})$, $N^{*}_{+}(P_{13})$ and $N^{*}_{-}(D_{13})$.
In the mirror assignment, $\psi_{1l}$ and $\psi_{2 r}$ belong to $ (1,{{1 \over 2}}) $, while $\psi_{1r}$ and $\psi_{2l}$ belong to $ ({{1 \over 2}},1) $ , so that $\psi_1$ and $\psi_2$ have opposite axial charge. Thus, these fields have three indices, $(\psi_{1,2})_{\alpha \beta}^{\gamma}$, with $ \alpha, \beta$ and $ \gamma $ take 1 or 2. Here $(\alpha \beta)$ is the index for $I=1$ triplet and $\gamma$ for $I={{1 \over 2}}$ doublet. Since $\psi$ is traceless for the triplet index $(\alpha \beta)$, it is convenient to introduce a component field $(\psi_{i})^{A,\gamma}$ ($A=1,2, 3$ for triplet and $\gamma=1,2$ for doublet) as $$\begin{aligned}
\label{comp}
(\psi_{1,2})_{\alpha \beta}^{\gamma} =
\sum_{A=1,2,3} (\tau^A)_{\alpha \beta} (\psi_{1,2})^{A,\gamma} \ \ ,\end{aligned}$$ where $\tau^{A}$ $(A = 1,2,3)$ is the $2 \times 2$ Pauli matrix.
The transformation rules of $\psi_i$ under $SU(2)_{L}\times SU(2)_{R}$ are then represented by $$\begin{aligned}
\label{trapsi1l}
(\tau^A)_{\alpha \beta} (\psi_{1l,2r})^{A,\gamma}
& \rightarrow & (L \tau^A L^{\dagger})_{\alpha \beta}
(R\psi_{1l,2r})^{A,\gamma}\ \ ,\\
\label{trapsi1r}
(\tau^A)_{\alpha \beta} (\psi_{2l,1r})^{A,\gamma}
& \rightarrow & (R \tau^A R^{\dagger})_{\alpha \beta}
(L\psi_{2l,1r})^{A,\gamma}\ \ ,
\end{aligned}$$ where $L$ ($R$) corresponds to the $SU(2)_{L}$ ($SU(2)_{R}$) rotation. The meson field $M \equiv \sigma + i \vec{\pi}\cdot \vec{\tau}$ belongs to $({{1 \over 2}},{{1 \over 2}})$ multiplet, and obeys the standard transformation rule, $ M \rightarrow L M R^{\dagger}$.
Now let us construct the mass term and the Yukawa coupling of $\psi_{i}$ with $M$. Here we consider only the simplest interaction which has only single $M$ without derivatives as in the case of the Gell-Mann-Lévy and DeTar-Kunihiro models. It can be shown that the chiral invariance under eq.’s. (\[trapsi1l\],\[trapsi1r\]) together with parity and time-reversal invariance allow only three terms: $$\begin{aligned}
{\cal L}_{int} &=& m_{0}\
(\bar{\psi}^{A}_{2} \gamma_{5} \psi^{A}_{1} - \bar{\psi}^{A}_{1}
\gamma_{5} \psi^{A}_{2}) \nonumber \\
&& + a\ \bar{\psi}_{1}^{A} \tau^{B} (\sigma -i \vec{\pi}\cdot
\vec{\tau}\gamma_{5}) \tau^{A} \psi_{1}^{B} \label{lag}\\
&& + b\ \bar{\psi}_{2}^{A}
\tau^{B} (\sigma + i \vec{\pi}\cdot \vec{\tau} \gamma_{5})\tau^{A}
\psi_{2}^{B} \ \ , \nonumber\end{aligned}$$ where $m_{0}$, $a$ and $b$ are free parameters not constrained by ChS. This interaction for the $(1,{{1 \over 2}}) \oplus ({{1 \over 2}},1)$ chiral-quartet is a natural generalization of that for the $({{1 \over 2}},0) \oplus (0,{{1 \over 2}})$ chiral-doublet in [@dk].
A short cut to obtain eq. (\[lag\]) is to use $L M R^{\dagger}$ together with the rotated fields in the r.h.s. of eq.(\[trapsi1l\],\[trapsi1r\]) and to look for combinations in which $L$ and $R$ do not appear in the final expression. Since $L$ and $R$ are independent transformation, the indices related to the left (right) rotation must be always contracted with the left (right) rotation. One of the chiral invariant mass terms, for example, comes from the combination, ${\rm Tr}[(R \tau^A R^{\dagger})(R\tau^B R^{\dagger})]
[(\bar{\psi}^A_{1r}L^{\dagger})(L \psi^B_{2l})]$. Also, one of the Yukawa terms is obtained from $[(\bar{\psi}^A_{1l}R^{\dagger})(R \tau^B R^{\dagger})(R M^{\dagger}
L^{\dagger})(L \tau^A L^{\dagger} )(L \psi^B_{1r})]$.
As already mentioned, $\psi^{A,\gamma}_{i}$ contains both $I={{3 \over 2}}$ field $\Delta_{i,M}$ $(M={{3 \over 2}},{{1 \over 2}},-{{1 \over 2}},-{{3 \over 2}})$ and $I={{1 \over 2}}$ field $N^{*}_{i,m}$ $(m={{1 \over 2}},-{{1 \over 2}})$ which are obtained by the following isospin decomposition: $\psi^{A,\gamma}_{i} = \sum_{M}(T^{A}_{3/2})_{\gamma M}\Delta_{i,M}$ $+ \sum_{m}(T^{A}_{1/2})_{\gamma m} N^{*}_{i,m}$, where the isospin projection matrices $T^{A}_{3/2}$ and $T^{A}_{1/2}$ are defined through the Clebsh-Gordan coefficients, $(T^{A}_{3/2})_{\gamma M} = \sum_{r,\gamma^{\prime}}
(1r{{1 \over 2}}\gamma^{\prime} |{{3 \over 2}}M)
\epsilon_{r}^{A}\chi^{\gamma}_{\gamma^{\prime}}$ and $(T^{A}_{1/2})_{\gamma m} = \sum_{r,\gamma^{\prime}}
(1r{{1 \over 2}}\gamma^{\prime} |{{1 \over 2}}m)
\epsilon_{r}^{A}\chi^{\gamma}_{\gamma^{\prime}}$. $\vec{\epsilon}_{r}$ are vectors relating the $A=(1,2,3)$ basis to $r=(+1,0,-1)$ basis, and $\vec{\chi}_{\gamma^{\prime}}$ relates the $\gamma=(1,2)$ basis to $\gamma^{\prime}=({{1 \over 2}},-{{1 \over 2}})$ basis [@bw]. Their explicit forms are $\epsilon_{1} = -1/\sqrt{2}(1,i,0)$, $\epsilon_{0}=(0,0,1)$, $\epsilon_{-1} = 1/\sqrt{2}(1,-i,0)$, $\chi_{1/2}=(1,0)$, $\chi_{-1/2}=(0,1)$.
With the invariant Lagrangian (\[lag\]), we shall next show its phenomenological consequences on the masses of $\Delta$’s and $N^{*}$’s. After the spontaneously symmetry breaking $SU(2)_{L} \times SU(2)_{R} \rightarrow SU(2)_{V}$ due to the finite $\sigma$ condensate $\langle \sigma \rangle \equiv \sigma_{0} >0 $, the mass term in eq.(\[lag\]) becomes $$\begin{aligned}
{\cal L}_{m} &=& - (\bar{\Delta}_{1}, \bar{\Delta}_{2})
\left(
\begin{array}{cc}
- 2 a \sigma_{0} & \gamma_{5} m_{0} \\
-\gamma_{5} m_{0} & - 2 b \sigma_{0}
\end{array} \right)
\left(
\begin{array}{c}
\Delta_{1} \\
\Delta_{2}
\end{array}\right) \nonumber \\ && -
(\bar{N}^{*}_{1}, \bar{N}^{*}_{2})
\left(
\begin{array}{cc}
a \sigma_{0} & \gamma_{5} m_{0} \\
-\gamma_{5} m_{0} & b \sigma_{0}
\end{array} \right)
\left(
\begin{array}{c}
N^{*}_{1} \\
N^{*}_{2}
\end{array}\right) \ .
\end{aligned}$$
The physical bases $\Delta_{\pm}$ and $N_{\pm}^*$ diagonalizing the mass matrices are given by $$\begin{aligned}
\left(
\begin{array}{c}
\Delta_{+} \\
\Delta_{-}
\end{array} \right) &=& {1 \over \sqrt{2\cosh \xi}} \left(
\begin{array}{cc}
e^{\xi / 2} & \gamma_{5} e^{-\xi / 2} \\
\gamma_{5} e^{-\xi / 2} & -e^{\xi / 2}
\end{array} \right) \left(
\begin{array}{c}
\Delta_{ 1} \\
\Delta_{2 }
\end{array} \right) \ ,
\end{aligned}$$ together with a similar formula for $ N^{*}_{\pm}$ with the replacement $\xi \rightarrow \eta$. The mixing angles $\xi$, $\eta$ are given by $\sinh \xi =
-(a+b) \sigma_{0} /m_{0}$ and $\sinh \eta = (a+b) \sigma_{0}/ (2 m_{0})$. These bases are chosen so that the masses of $\Delta$’s and $N^{*}$’s are all reduced to the chiral-invariant mass $m_0 >0$ when ChS is unbroken ($\sigma_0 =0$).
Thus we finally reach the mass formula, $$\begin{aligned}
\label{mass-f1}
m_{\Delta_\pm} & = & \sqrt{(a+b)^{2} \sigma_{0}^{\, 2}
+ m_{0}^{\,
2}} \mp \sigma_{0}(a-b) ,\\
\label{mass-f2}
m_{N^{*}_\pm} & = & \sqrt{({{a+b \over 2}})^{2} \sigma_{0}^{\, 2} +
m_{0}^{\, 2}} \pm {{\sigma_{0} \over 2}}(a-b) .\end{aligned}$$ Eq.’s (\[mass-f1\],\[mass-f2\]) shows that the spontaneous breaking of ChS lifts the degeneracy between parity partners ($\Delta_{+}$ vs $\Delta_{-}$, and $N_{+}^*$ vs $N_{-}^*$) and the degeneracy between isospin states ($\Delta$ vs $N^{*}$) simultaneously [@note2].
A remarkable consequence of our quartet scheme is the following mass relations which hold irrespectively of the choice of the parameters ($m_0, a, b$):\
1. The ordering in parity-doublet of $N^*$ is always opposite to that of $\Delta$; $$\begin{aligned}
\label{con1}
{\rm sgn} \left[ m_{\Delta_{+}} - m_{\Delta_{-}} \right]
= - \ {\rm sgn} \left[ m_{N_{+}^{*}} - m_{N_{-}^{*}} \right] \ .
\end{aligned}$$ 2. The mass difference between the two parity-doublets is fixed; $$\label{con2}
{{1 \over 2}}(m_{\Delta_{-}}-m_{\Delta_{+}} )
= m_{N^{*}_{+}} - m_{N^{*}_{-}}\ .$$ 3. The averaged mass of the $\Delta$ parity-doublet is equal or heavier than that of $N^*$; $$\label{con3}
{{1 \over 2}}(m_{\Delta_{+}}+m_{\Delta_{-}}) \ge
{{1 \over 2}}(m_{N^{*}_{+}} + m_{N^{*}_{-}}) \ .$$ So far, we have considered only the case for $J={{3 \over 2}}$. However, all the arguments and the mass relations above hold for the resonances with arbitrary spin as long as $(1,{{1 \over 2}}) \oplus ({{1 \over 2}},1)$ chiral multiplets are concerned.
For the candidate of the quartets in the real world, we adopt the lightest baryons in each spin-parity among the established resonances with three or four stars in [@PDG]. $I=J={{1 \over 2}}$ channel is, however, an exception since $N(940)$ is supposed to form a $({{1 \over 2}},0) \oplus (0,{{1 \over 2}})$ chiral doublet with its parity partner which is either $N(1535)$ or $N(1650)$, or possibly their linear combination, in the mirror assignment[@jnoh]. Therefore, we study two cases in $J = {{1 \over 2}}$ depending on whether we take $N(1535)$ (case 1) or $N(1650)$ (case 2) as a $(1,{{1 \over 2}})\oplus ({{1 \over 2}},1) $ quartet member. In Fig.\[spec\], the observed resonances taken from [@PDG] in the above criterion are shown under the label “exp" for each spin sector.
The comparison between the mass relations in the quartet scheme and the experimental data are shown in the first three rows in Table \[tab.comp\]. Parameter free constraints (\[con1\]) and (\[con2\]) are well satisfied by the observed masses. The constraint (\[con3\]) is well satisfied in $J={{1 \over 2}}$ and $J={{5 \over 2}}$ sectors, and is marginally satisfied in $J={{3 \over 2}}$.
If we have taken so called the “naive assignment" where $\psi_{1l,2l}$ belongs to $ (1,{{1 \over 2}}) $, and $\psi_{1r,2r}$ belong to $ ({{1 \over 2}},1) $, the mass formula turns out to be the same with eq.’s(\[mass-f1\],\[mass-f2\]) with $m_0=0$. This leads to a relation, $m_{\Delta_{\pm}} = 2 m_{N^{*}_{\mp}}$, which is in contradiction to the observed spectra in our criterion. This is why we have not adopted the naive assignment in this Letter.
Encouraged by the phenomenological success of the parameter free predictions of the mirror assignment, we go one step further and determine the three parameters $m_{0}$, $a$ and $b$ in each spin-sector. For this purpose, we take the four observed masses and $\sigma_{0}=f_{\pi}=93$ MeV and use the least square fit. (For $J={{3 \over 2}}$, we adopt $a=-b$ to satisfy the equality in eq.(\[con3\]).) Resultant parameters are summarized in the last two rows of Table \[tab.comp\]. The baryon masses in these parameters are also shown under the label “QS" in Fig.\[spec\]. They agree with the experimental data within $10$ percents.
$m_{0} \sim 1500$ MeV for $(1,{{1 \over 2}}) \oplus ({{1 \over 2}},1)$ in Table \[tab.comp\], which we obtained irrespective of the spin, is considerably larger than $m_{0} = 270$ MeV for $({{1 \over 2}},0) \oplus (0,{{1 \over 2}})$ [@dk]. Further investigation on the origin of $m_0$ in QCD is necessary to understand if these values as well as their difference have physical implications. Also, it is to be studied whether the baryonic excitations with finite mass $m_0$ exist in the chiral restored phase using, e.g., the lattice simulations.
Let us return to the discussion of the $J={{3 \over 2}}$ quartet and investigate the decay patterns by the single pion emission obtained from eq.(\[lag\]). The interaction Lagrangian of $\pi$ and $\psi_{\pm}$ with $a=-b=1.2$ is $$\label{1pi}
{\cal L}_{1 \pi} =
( \bar{\psi}_{+}^{A}\, , \bar{\psi}_{-}^{A})
\left(
\begin{array}{cc}
0 & -a \\
a & 0
\end{array}
\right)
\tau^{B}(i\vec{\pi}\cdot\vec{\tau})\tau^{A}
\left(
\begin{array}{c}
\psi_{+}^{B} \\
\psi_{-}^{B}
\end{array}
\right) \, ,$$ where $\psi_{+}={1 \over \sqrt{2}}(\psi_{1}+\gamma_{5}\psi_{2})$ and $\psi_{-}={1 \over \sqrt{2}}(\gamma_{5}\psi_{1}-\psi_{2})$. The mixing angles read $\xi$=$\eta$=0 due to $a+b=0$ (see Table 1). ${\cal L}_{1 \pi}$ has only the off-diagonal components in parity space: Therefore the parity non-changing couplings such as $\pi \Delta_{\pm} N^{*}_{\pm}$, $\pi \Delta_{\pm} \Delta_{\pm}$, and $\pi N^{*}_{\pm} N^{*}_{\pm}$ are forbidden in the tree level of eq.(\[1pi\]).
Observed one-pion decay patterns are qualitatively consistent with the suppression of the $\pi\Delta_{+}N^{*}_{+}$ coupling. In fact, $N_{+}(1720)
\rightarrow \pi \Delta_{+}(1232)$, although its phase space is large enough, is insignificant or has not been shown to exist in the recent analysis of $\pi N$ scattering amplitudes [@man]. (The existence has been suggested in an old analysis of $\pi N \rightarrow \pi\pi N$ though [@lon].) On the other hand, $N_{-}(1520) \rightarrow \pi \Delta_{+}(1232)$ and $\Delta_{-}(1700) \rightarrow \pi \Delta_{+}(1232)$ in the $S$-wave channel, which are not suppressed in eq.(\[1pi\]), have been seen with the partial decay rates $5\sim 12\%$ and $ 25\sim 50\%$, respectively[@PDG]. The suppression of $\pi \Delta_{\pm} \Delta_{\pm}$, and $\pi N^{*}_{\pm} N^{*}_{\pm}$ cannot be checked in the decays, but empirical studies of the $\pi N \rightarrow \pi
\pi N$ process[@arndt] seem to suggest that the $\pi
\Delta_{+}(1232)\Delta_{+}(1232)$ coupling is less than half of the quark model prediction given by $g_{\pi\Delta\Delta}=(4/5) g_{\pi NN}$ [@bw].
For $J={{1 \over 2}}, {{5 \over 2}}$ sectors, similar analysis is not possible at present, because of large uncertainties and/or the absence of experimental data for relevant decays. More experimental data on the decays among quartet shown in Fig.1 would be quite helpful for future theoretical studies.
We note here that the selection rule discussed above may in principle be modified by chiral invariant terms not considered here, such as the terms containing derivatives as well as multi $M$ fields. This is the situation similar to that for $g_{A}$ of the nucleon in the linear sigma model, where the simplest Yukawa coupling in the tree level gives $g_A =1$ while the higher dimensional derivative coupling as well as quantum corrections could shift it to 1.25 [@lee]. Therefore, detailed studies with those terms should be also done in the future.
In summary, we have investigated baryon resonances with both parities from the viewpoint of chiral symmetry. We have constructed a linear sigma model in which $\Delta_{\pm}$’s and $N^{*}_{\pm}$’s with a given spin are assigned to be a representation $(1,{{1 \over 2}}) \oplus
({{1 \over 2}},1)$ of the chiral $SU(2)_{L} \times SU(2)_{R}$ group. Adopting the “mirror assignment” for the axial charge of baryons, we have arrived at [*quartet scheme*]{} where $N^*_{+} , N^*_{-}, \Delta_{+}$ and $\Delta_{-}$ form a chiral multiplet. We have shown that the quartet scheme gives constraints not only on the baryon masses but also their couplings; it turns out that the constraints are consistent with the observed baryon spectra. We have shown that experimental confirmation of the absence of parity non-changing decay in $J={{3 \over 2}}$ sector such as $N_{+}(1720) \rightarrow \pi \Delta_{+}(1232)$ together with the measurement of the decay patterns in $J={{1 \over 2}}, {{5 \over 2}}$ sectors is important to test the quartet scheme and to explore the role of ChS in excited baryons.
D. J. is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. T. H. was partly supported by Grant-in-Aid for Scientific Research No. 10874042 of the Japanese Ministry of Education, Science and Culture.
[9]{}
S. Weinberg, . J. Gasser and H. Leutwyler, [*Annals Phys*]{}. (N.Y.) [**158**]{}, 142 (1984). S. Weinberg, .
S. Coleman, J. Wess and B. Zumino, ; C.G. Callan, S. Coleman, J. Wess and B. Zumino, .
S. Weinberg, .
T. Hatsuda and T. Kunihiro, ; . R. D. Pisarski and F. Wilczek, . K. Rajagopal and F. Wilczek, .
M. Gell-Mann and M. Lévy, [*Nuovo Cim*]{}. [**16**]{}, 705 (1960). J. Schwinger, [*Ann. Phys*]{}. (N.Y.) [**2**]{}, 407 (1958).
C. DeTar and T. Kunihiro, .
D. Jido, Y. Nemoto, M. Oka and A. Hosaka, hep-ph/9805306.
D. Jido, M. Oka and A. Hosaka, ; H. Kim, D. Jido and M. Oka, . See also, T. Hatsuda and M. Prakash, . Y. Nemoto, D. Jido, M. Oka and A. Hosaka, .
T.D. Cohen and X. Ji, .
Incidentally, the interpolating field for $\Delta$ conventionally used in the QCD sum rules and in the lattice calculation belongs to this multiplet.
G.E. Brown and W. Weise, .
For parity doublets in excited baryons in quite different contexts, see e.g., F. Iachello, ; S.B. Khokhlachev, ; M. Kirchbach, ; L. Ya. Glozman, hep-ph/9908207.
Particle Data Group, C. Caso [*et. al.*]{},
D.M. Manley and E.M. Saleski, . T.P. Vrana, S.A. Dytman and T.-S. H. Lee, nucl-th/9910012.
R.S. Longacre and J. Dolbeau, .
R.A. Arndt [*et. al.*]{}, .
B. W. Lee, [*Chiral Dynamics*]{}, (Gordon and Breach, New York, 1972).
------------------------------------------------------------ ------------------ -------------- ------------------- ------------------- --------
QS $J={{3 \over 2}}$ $J={{5 \over 2}}$
case 1 case 2
${\rm sgn}\left( { m_{N_{+}^{*}} - m_{N_{-}^{*}} \over $-$ $-$ $-$ $-$ $-$
m_{\Delta_{+}}- m_{\Delta_{-}} } \right) $
${ m_{N_{+}^{*}} - m_{N_{-}^{*}} \over $-{{1 \over 2}}$ $-0.33$ $-0.72$ $-0.43$ $-0.2$
m_{\Delta_{+}}- m_{\Delta_{-}} } $
${ m_{N_{+}^{*}} + m_{N_{-}^{*}} \over $ \le 1 $ 0.84 0.88 1.1 0.87
m_{\Delta_{+}}+ m_{\Delta_{-}} } $
1380 1460 1540 1590
$(5.2, 6.6)$ $(4.4, 6.1)$ $(1.2,-1.2)$ $(5.8, 5.7)$
------------------------------------------------------------ ------------------ -------------- ------------------- ------------------- --------
: Comparison between parameter free predictions of the quartet scheme (QS) and the observed data. Case 1 and case 2 in the $J={{1 \over 2}}$ sector stand for the cases $N^{*}_{-}$=$N(1535)$ $N^{*}_{-}$=$N(1650)$, respectively. The last two rows are the parameters $m_0,a,b$ determined from the experimental inputs.
\[tab.comp\]
=8.0cm
|
---
abstract: 'We respond to the accompanying Comment on our paper, ’Validity of certain soft photon amplitudes’. While we hope the discussion here clarifies the issues, we have found nothing which leads to a change in the original conclusions of our paper.'
address: 'TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia,Canada V6T 2A3'
author:
- 'Mark Welsh [[^1] ]{} and Harold W. Fearing [[^2]]{}'
date: 'May 12, 1999'
title: 'Reply to the comment on ’Validity of certain soft photon amplitudes’'
---
In Ref. [[@I]]{}, hereafter referred to as I, we discussed some problems which arise with generalized soft photon approximations (SPA). Such generalized SPA’s result from the infinite number of ways one can choose variables to describe the elastic amplitude. Such choices give identical elastic amplitudes when momenta satisfying $p_1+p_2=p_3+p_4$ are used. However when the variables are evaluated using the radiative momenta, $p_1+p_2=p_3+p_4+k$, as is done in the derivation of a SPA, the results differ. Thus one obtains generalized SPA’s, which however differ only by terms $O(k)$. Our paper was motivated in part by Ref. [[@Liou1]]{}, which applied the so-called TuTts and TsTts generalized SPA’s to proton-proton bremsstrahlung. However the work of I was intended to be much more general and not just a comment on Ref..
In the accompanying Comment [[@Lioucom]]{} Liou, [*et al.*]{} raised two objections to I. While we find these objections invalid, and stand by the conclusions of the original paper, we wish to discuss them, as they reemphasize some of the ambiguities which appear in these generalized SPA’s.
In I we originally discussed two kinds of problems. The first dealt with phase space and arises because for certain choices of variables the elastic amplitude, evaluated using the radiative momenta, is required outside the region which is physically measurable. This is the case for the TsTts amplitude, if evaluated for proton-proton bremsstrahlung as described in detail by Liou, [*et al.*]{} \[Ref. [[@Liou1]]{}, p. 375, 2nd column\]. We learned however in the course of discussion of this Comment that the evaluation was actually not done as described but instead by using a prescription (TETAS) which in effect moved the variables back into the physical region as necessary. This bypasses, albeit in an ad hoc fashion, the first difficulty raised in I.
The first objection raised by Liou, [*et al.*]{} deals with the second problem discussed in I, which had to do with the proper (anti)symmetrization of amplitudes involving identical particles. We will discuss here only the spin zero case for which the amplitudes must be symmetric. The generalization to spin one-half and antisymmetric amplitudes is obvious as here spin is truly an ’inessential complication.’ The original TuTts amplitude, reproduced in Eq. (26) of I, but originally from [[@Liou3]]{}, was not properly symmetrized. Symmetrizing in $p_3 \leftrightarrow p_4$ leads to Eq. (27) of I, which however cannot obviously be expressed in terms of properly symmetrized elastic amplitudes, which is necessary for a SPA expressed in terms of measurable non radiative quantities. The problem arises with the ad hoc terms $\Delta_i$ which are added to force gauge invariance. In the discussion following Eq. (30) we derived sufficient conditions on the $\Delta_i$’s to ensure that the SPA amplitude, valid through $O(k^0)$, was gauge invariant, could be expressed in terms of symmetrized elastic amplitudes, and had the proper analyticity properties, namely the $O(k^0)$ terms do not have pieces of $O(k/k)$, all as required by the general principles of SPA. We then gave four examples for the $\Delta_i$ chosen from the infinite set of possibilities. Liou, [*et al.*]{} also recognized this symmetrization problem and in a paper [@Liou2] subsequent to their original one obtained an amplitude corresponding to one of the $\Delta_i$ examples we had given.
In their Comment Liou, [*et al.*]{} work through a lot of algebra, but the essence of their claim is that their particular choice of $\Delta_i$ is the only correct one because the others have $k/k$ type structures at $O(k)$, i. e. terms of $O(k^2/k)$. Such structures do exist as they claim, but in our view they are irrelevant for a discussion of the validity of a SPA [*as a soft photon approximation.*]{} As is well known, gauge invariance provides only one condition and is sufficient only to fix the $O(k^0)$ terms. Thus SPA’s are valid only through $O(k^0)$. There will be many terms at $O(k)$, including some with structure $O(k^2/k)$ coming from higher order expansions of the external radiation graphs, which are just not determined in a SPA and thus are not relevant to SPA discussions. This principle, that what happens at $O(k)$ is immaterial, is actually acknowledged by Liou, [*et al.*]{} in their Comment, below Eq. (6), in reference to their second objection.
On the other hand if one wants to make a model dependent choice among the infinite set of SPA’s to find one that fits data, as was done in Ref., but not in I, then such external criteria, while having nothing to do with SPA as such, would be relevant and clearly in the absence of other information one would choose the $\Delta_i$ which was in some sense ’smoothest’.
The second objection of the Comment deals with the size of the error incurred in using a TuTts amplitude which is not properly symmetrized. The authors of the Comment attribute to us a much more general statement than we intended or actually worked out in I. Specifically in I we compared Eq. (27), which was the TuTts amplitude properly symmetrized, with Eq. (28) which was the prescription we understood was used in Ref. [[@Liou1]]{} to obtain numerical results using a TuTts amplitude which did not have the proper symmetry properties. All of our remarks concerning the relative size of corrections refer to this specific comparison. Subsequently Liou, [*et al.*]{} [@Liou3] derived a properly symmetrized amplitude to be used in their numerical calculations, thus making a prescription unnecessary, and our choices of $\Delta_i$ lead to similar amplitudes. Thus in our opinion this comparison has been superseded and is not particularly important. However it does illustrate yet a further ambiguity in such SPA’s, and so for that reason is perhaps worth discussing. The problem arises because of the multiple ways one can define a properly symmetrized elastic amplitude. We define one such amplitude in Eq. (29) of I for one case, which happens to be simultaneously symmetric in $p_3 \leftrightarrow p_4$ and $u \leftrightarrow t$. For the other amplitudes however there is a choice, because of the freedom to define the elastic variables in different ways. For example we could define $A^{\prime
S}(u_{23},t_{13}) = A^\prime (u_{23},t_{13})+ A^\prime (t_{24},u_{14})$ which is symmetric in $p_3 \leftrightarrow p_4$ but not in $u \leftrightarrow t$ and which might be appropriate if the elastic amplitude comes from a diagrammatic calculation easily expressed in terms of the $p_i$. Alternatively we could take $A^{\prime S}(u_{23},t_{13}) = A^\prime (u_{23},t_{13})+ A^\prime
(t_{13},u_{23})$ which is symmetric in $u \leftrightarrow t$ but not in $p_3
\leftrightarrow p_4$ and which might be more appropriate for an elastic amplitude taken from numerical data given as a function of energy and angle. For the elastic amplitude evaluated for the nonradiative momenta $p_1+p_2=p_3+p_4$ these are of course identical, but they are formally different functions of the momenta so they differ, by terms of $O(k)$, when evaluated using the radiative momenta $p_1+p_2=p_3+p_4+k$. Thus when we compare Eq. (27) and (28) using the first of these, as we did in I, the difference is $O(k/k)$ as was stated, while if we use the second, as apparently is being done by Liou, [*et al.*]{} in the context of their second objection, the difference is $O(k)$. Clearly this difference can arise only because Eq. (28) is a prescription which may not be a ’real’ SPA (cf. discussion following Eq. (29) in I) since we know that ’real’ SPA’s can differ only at $O(k)$.
In summary, in our view neither of the objections raised by Liou, [*et al.*]{} in their Comment lead to any changes in the conclusions of I. They do however serve to reemphasize the main point of I, namely that there are a lot of ambiguities related to the use of these generalized SPA’s. One is always faced with a dilemma. If $k$ is small, the $O(k)$ terms are small and all SPA’s are the same and essentially model independent. On the other hand if $k$ is larger, as is the case for all modern measurements of proton-proton bremsstrahlung, then the $O(k)$ terms make a difference and one must make a whole set of model dependent choices which basically have nothing to do with SPA’s [*per se*]{}, but which can affect the quality of fit to the data and the predictive power, or lack thereof, of the model.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.
[99]{}
Mark Welsh and Harold W. Fearing, Phys. Rev. C [**54**]{}, 2240 (1996). M. K. Liou, R. Timmermans, and B. F. Gibson, Phys. Let. B [ **345**]{},372 (1995); [**355**]{}, 606(E) (1995). M.K. Liou, R. Timmermans, B. F. Gibson, and Yi Li, Phys. Rev. C [**xx**]{}, xxxx (1999). (Accompanying Comment) M. K. Liou, Dahang Lin, and B. F. Gibson, Phys. Rev. C [ **47**]{}, 973 (1993). M. K. Liou, R. Timmermans, and B. F. Gibson, Phys. Rev. C [ **54**]{}, 1574 (1996).
[^1]: email: markw@retrologic.com
[^2]: email: fearing@triumf.ca
|
---
abstract: 'We develop a new formulation of well localized operators as well as a new proof for the necessary and sufficient conditions to characterize their boundedness between $L^2(\mathbb{R}^n,u)$ and $L^2(\mathbb{R}^n,v)$ for general Radon measures $u$ and $v$.'
address: 'Philip Benge, School of Mathematics, Washington University in St. Louis, St. Louis, MO'
author:
- Philip Benge
title: 'A Two-Weight Inequality for Essentially Well Localized Operators with General Measures'
---
[^1]
Introduction
============
We consider the boundedness of the integral operator $$Tf(x)=\int_{\mathbb{R}^n}K(x,y)f(y)dy$$ acting from $L^2(\mathbb{R}^n,u)$ to $L^2(\mathbb{R}^n,v)$, that is, we want to characterize the following inequality $$\left\Vert Tf\right\Vert_{L^2(\mathbb{R}^n,v)}\lesssim \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,u)}$$ for all $f\in L^2(\mathbb{R}^n,u)\equiv\{f:\int_{\mathbb{R}^n} \left\vert f\right\vert^2 u<\infty\}$. As is common in two-weight problems, we will consider the change of variables $d\sigma=\frac{1}{u}dx$, $F=\frac{f}{u}$ and $d\omega=vdx$, which allows us to instead characterize the boundedness of the operator $T(\sigma \cdot)$ from $L^2(\mathbb{R}^n,\sigma)$ to $L^2(\mathbb{R}^n,\omega)$, that is, we want to characterize the inequality $$\left\Vert T(\sigma f)\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\sigma)}.$$
Nazarov, Treil and Volberg in [@NTV1] found necessary and sufficient conditions for this inequality in the case when $T$ is a so called well localized operator. The primary examples of such operators are band operators, the Haar shift, Haar multipliers and dyadic paraproducts as well as perfect dyadic operators.
In this paper we develop a new characterization of well localized operators and provide a new proof showing necessary and sufficient conditions for their boundedness in terms of Sawyer type testing on the operator. As in [@NTV1], we proceed with an axiomatic approach. Rather than assume that our operator $T$ can be represented as an integral operator (which is not always possible), we instead characterize the operator based on how it behaves on an orthonormal basis of Haar-type functions. However, our behavior of interest is a simple support condition.
For example, let $\mathcal{D}$ be the standard dyadic grid in $\mathbb{R}$ and for each dyadic interval $I\in\mathcal{D}$, let $I_R$ and $I_L$ denote the right and left halves of the interval, respectively. Define the Haar function $h_I^0\equiv\frac{1}{\sqrt{\left\vert I\right\vert}}\left({\bf{1}}_{I_R}-{\bf{1}}_{I_L}\right)$ and the averaging function $h_I^1\equiv\frac{1}{\left\vert I\right\vert}{\bf{1}}_I$. Then an operator $T$ is said to be lower triangularly localized if there exists a constant $r>0$ such that for all dyadic intervals $I,J\in\mathcal{D}$ with $\left\vert I\right\vert\leq2\left\vert J\right\vert$, we have $$\left\langle T({\bf{1}}_J),h_I^0\right\rangle=0$$ if $I\not\subset J^{(r)}$ or if $\left\vert I\right\vert\leq 2^{-r}\left\vert J\right\vert$ and $I\not\subset J$. We say that $T$ is well localized if both $T$ and $T^*$ are lower triangularly localized.
Given a sequence $b=\{b_I\}_{I\in\mathcal{D}}$ and a function $f\in L^2(\mathbb{R})$ we define the martingale transform $$T_b f\equiv\sum_{I\in\mathcal{D}}b_I\left\langle f,h_I^0\right\rangle h_I^0$$ and the paraproduct $$P_b f\equiv\sum_{I\in\mathcal{D}}b_I\left\langle f,h_I^1\right\rangle h_I^0.$$ A simple computation shows that these are both well localized with respect to the constant $r=1$.
We also see that $$T_b h_I^0=b_I h_I^0\text{ and }T^*_b h_I^0=b_I h_I^0$$ as well as $$P_b h_I^0=\sum_{J\subsetneq I}b_J\left\langle h_I^0,h_J^1\right\rangle h_J^0\text{ and }P^*_b h_I^0=b_I h_I^1.$$ Thus these operators satisfy a nice support condition when applied to any Haar function, namely, if $T$ is any of the operators above, we have $\text{supp}(Th_I^0)\subset I$.
For an additional example, we let $S$ be the Haar shift operator defined by $$Sf\equiv\sum_{I\in\mathcal{D}}b_I\left\langle f,h_I^0\right\rangle \left(h_{I_R}^0-h_{I_L}^0\right).$$ Then $S$ is well localized with associated constant $r=2$, and we also have $\text{supp}(S h_I^0)\subset I$ and $\text{supp}(S^* h_I^0)\subset I^{(1)}$, where $I^{(1)}$ denotes the dyadic parent of $I$. In the following section, we formally define this support condition, and in section \[WL\] we show that this condition is in fact the same as the well localized condition up to a change in the constant $r$.
Definitions and Statement of Results
====================================
To define our orthonormal basis, let $\mathcal{D}^n$ denote the dyadic grid in $\mathbb{R}^n$, and for any $F\in\mathcal{D}^n$, define $\mathcal{D}^{n}_{k}(F)\equiv\{F'\in\mathcal{D}^n:F'\subseteq F, \ell(F')=\frac{1}{2^k}\ell(F)\}$, where $\ell(F)$ denotes the side length of the cube $F$. We further define $\mathcal{D}^n(F)=\bigcup_{k=0}^{\infty}{D_{k}^n(F)}$. From [@Wilson], we have the following lemma.
Let $F\in\mathcal{D}^n$. Then there are $2^n-1$ pairs of sets $\{(E_{F,i}^1,E_{F,i}^2)\}_{i=1}^{2^n-1}$ such that
1. for each $i$, $|E_{F,i}^1|=|E_{F,i}^2|$;
2. for each $i$, $E_{F,i}^1$ and $E_{F,i}^2$ are non-empty unions of cubes from $\mathcal{D}^{n}(F)$;
3. for every $i\neq j$, exactly one of the following must hold:
1. $E_{F,i}^1\cup E_{F,i}^2$ is entirely contained in either $E_{F,j}^1$ or $E_{F,j}^2$;
2. $E_{F,j}^1\cup E_{F,j}^2$ is entirely contained in either $E_{F,i}^1$ or $E_{F,i}^2$;
3. $(E_{F,i}^1\cup E_{F,i}^2)\cap(E_{F,j}^1\cup E_{F,j}^2)=\emptyset$.
For simplicity, we let $E_{F,i}=E_{F,i}^1\cup E_{F,i}^2$ and we will define $$\mathcal{H}^n\equiv\{E_{F,i}:{F\in\mathcal{D}^n,1\leq i\leq 2^n-1}\}$$ to be the collection of all rectangles $E_{F,i}$. We note that for all $i$, $E_{F,i}\subseteq F$, however, $E_{F,i}\not\subseteq \bigcup_{k=1}^{\infty}D_{k}^{n}(F)$. We further note that for all $k=0,1,\ldots,n$, we have that $$F=\bigcup_{i=2^{k-1}}^{2^k-1}E_{F,i}.$$ For $r\geq 0$, we define $E_{F,i}^{(r)}$ to be the rectangle of volume $2^r\left\vert E_{F,i}\right\vert$ containing $E_{F,i}$.
We now define the Haar function $h_{F,i}^{0}$ and the averaging function $h_{F,i}^1$ associated with $E_{F,i}$ by $$h_{F,i}^0=\frac{1}{\sqrt{|E_{F,i}|}}\bigg(\mathbf{1}_{E_{F,i}^2}-\mathbf{1}_{E_{F,i}^1}\bigg)$$ and $$h_{F,i}^1=\frac{1}{|E_{F,i}|}\mathbf{1}_{E_{F,i}}.$$ The functions $\{h_{F,i}^0\}_{F\in\mathcal{D}^n,1\leq i \leq 2^n-1}$ form an orthonormal basis for $L^2(\mathbb{R}^n)$.
Given a Radon measure $\sigma$, we let $$h_{F,i}^\sigma=\sqrt{\frac{\sigma(E_{F,i}^1)}{\sigma(E_{F,i})\sigma(E_{F,i}^2)}}{\bf{1}}_{E_{F,i}^2}-\sqrt{\frac{\sigma(E_{F,i}^2)}{\sigma(E_{F,i})\sigma(E_{F,i}^1)}}{\bf{1}}_{E_{F,i}^1}$$ be the weight adapted Haar function if $\sigma(E_{F,i}^1),\sigma(E_{F,i}^2)>0$ and we set $h_{F,i}^\sigma\equiv 0$ if either $\sigma(E_{F,i}^1)=0$ or $\sigma(E_{F,i}^2)=0$.
We will impose the following structure on this operator:
An operator is said to be [*essentially well localized*]{} if there exists an $r\geq 0$ such that for all $E_{F,i}$ the following properties hold: $$\label{SimpleLemma}
\text{supp}(T(\sigma h_{F,i}^\sigma))\subseteq E_{F,i}^{(r)};\text{ }\text{supp}(T^*(\omega h_{F,i}^\omega))\subseteq E_{F,i}^{(r)}.$$
We will establish the two-weight boundedness for any Radon measures $\sigma$ and $\omega$ by adapting the proof strategy found in characterizing the two-weight inequality for the Hilbert Transform (see [@H] and [@L]). We now state our main theorem.
Let $T$ be essentially well localized for some $r\geq0$. Let $\sigma$ and $\omega$ be two Radon measures on $\mathbb{R}^n$. Then $$\label{MainIneq}
\left\Vert T(\sigma f)\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert T^*(\omega f)\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\omega)}$$ if and only if for all $E_{F,i}\in\mathcal{H}^n$ and $E_{G,j}\in\mathcal{H}^n$ with $2^{-r}\left\vert E_{F,i}\right\vert\leq\left\vert E_{G,j}\right\vert\leq 2^{r}\left\vert E_{F,i}\right\vert$ and $E_{G,j}\cap E_{F,i}^{(r)}\neq\emptyset$, the following testing conditions hold: $$\begin{aligned}
\label{TestingConditions}
\left\Vert{\bf{1}}_{E_{F,i}}T(\sigma {\bf{1}}_{E_{F,i}})\right\Vert_{L^2(\mathbb{R}^n,\omega)}&\lesssim C_1\left\Vert {\bf{1}}_{E_{F,i}}\right\Vert_{L^2(\mathbb{R}^n,\sigma)};\\
\left\Vert{\bf{1}}_{E_{F,i}}T^*(\omega {\bf{1}}_{E_{F,i}})\right\Vert_{L^2(\mathbb{R}^n,\sigma)}&\lesssim C_2\left\Vert {\bf{1}}_{E_{F,i}}\right\Vert_{L^2(\mathbb{R}^n,\omega)};\\
\label{WeakBoundedness}
\left\vert\left\langle T(\sigma {\bf{1}}_{E_{F,i}}),{\bf{1}}_{E_{G,j}}\right\rangle_\omega\right\vert&\lesssim C_3\sigma(E_{F,i})^{1/2}\omega(E_{G,j})^{1/2}.\end{aligned}$$ Moreover, we have that $C\simeq C_1+C_2+C_3$.
Now because each $E_{F,i}\in\mathcal{H}^n$ is a union of cubes $Q\in\mathcal{D}^n$ and the boundedness of $T$ would imply a similar testing condition on cubes, we immediately have the following result.
\[CubeLocal\] Let $T$ be essentially well localized for some $r\geq0$. Let $\sigma$ and $\omega$ be two Radon measures on $\mathbb{R}^n$. Then $$\label{CubeMainIneq}
\left\Vert T(\sigma f)\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert T^*(\omega f)\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\omega)}$$ if and only if for all $Q\in\mathcal{D}^n$ and $R\in\mathcal{D}^n$ with $2^{-r}\left\vert Q\right\vert\leq\left\vert R\right\vert\leq 2^{r}\left\vert Q\right\vert$ and $R\cap Q^{(r)}\neq\emptyset$, the following testing conditions hold: $$\label{CubeTestingConditions}
\left\Vert{\bf{1}}_{Q}T(\sigma {\bf{1}}_{Q})\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C_1\left\Vert {\bf{1}}_{Q}\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert{\bf{1}}_{Q}T^*(\omega {\bf{1}}_{Q})\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C_2\left\Vert {\bf{1}}_{Q}\right\Vert_{L^2(\mathbb{R}^n,\omega)};$$ $$\label{CubeWeakBoundedness}
\left\vert\left\langle T(\sigma {\bf{1}}_{Q}),{\bf{1}}_{R}\right\rangle_\omega\right\vert\lesssim C_3\sigma(Q)^{1/2}\omega(R)^{1/2}.$$ Moreover, we have that $C\simeq C_1+C_2+C_3$.
We are also able to easily extract global testing conditions by noting that the boundedness of $T$ immediately implies the global testing conditions below, which then imply the local testing conditions and . With this we state another corollary.
Let $T$ be essentially well localized for some $r\geq0$. Let $\sigma$ and $\omega$ be two Radon measures on $\mathbb{R}^n$. Then $$\label{MainGlobalIneq}
\left\Vert T(\sigma f)\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert T^*(\omega f)\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\omega)}$$ if and only if for all $E_{F,i}\in\mathcal{H}^n$, the following testing conditions hold: $$\begin{aligned}
\label{GlobalTestingConditions}
\left\Vert T(\sigma {\bf{1}}_{E_{F,i}})\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C_1\left\Vert {\bf{1}}_{E_{F,i}}\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\\
\left\Vert T^*(\omega {\bf{1}}_{E_{F,i}})\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C_2\left\Vert {\bf{1}}_{E_{F,i}}\right\Vert_{L^2(\mathbb{R}^n,\omega)}.\end{aligned}$$ Moreover, we have that $C\simeq C_1+C_2$.
A similar result can be stated for cubes by .
\[CubeGlobal\] Let $T$ be essentially well localized for some $r\geq0$. Let $\sigma$ and $\omega$ be two Radon measures on $\mathbb{R}^n$. Then $$\label{CubeGlobalMainIneq}
\left\Vert T(\sigma f)\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert T^*(\omega f)\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C \left\Vert f\right\Vert_{L^2(\mathbb{R}^n,\omega)}$$ if and only if for all $Q\in\mathcal{D}^n$, the following testing conditions hold: $$\label{CubeGlobalTestingConditions}
\left\Vert T(\sigma {\bf{1}}_{Q})\right\Vert_{L^2(\mathbb{R}^n,\omega)}\lesssim C_1\left\Vert {\bf{1}}_{Q}\right\Vert_{L^2(\mathbb{R}^n,\sigma)},\text{ }\left\Vert T^*(\omega {\bf{1}}_{Q})\right\Vert_{L^2(\mathbb{R}^n,\sigma)}\lesssim C_2\left\Vert {\bf{1}}_{Q}\right\Vert_{L^2(\mathbb{R}^n,\omega)}.$$ Moreover, we have that $C\simeq C_1+C_2$.
Initial Reductions
==================
Whenever there is no ambiguity, we will simply write $L^2(\sigma)$ instead of $L^2(\mathbb{R}^n,\sigma)$. We will also write $\sum_{E_{F,i}}$ rather than $\sum_{F\in\mathcal{D}^n}\sum_{i:1\leq i\leq 2^n-1}$. By duality, we will study the pairing $\left\langle Tf,g\right\rangle_\omega$, where $$\left\langle f,g\right\rangle_{\omega}=\int_{\mathbb{R}^n} fg\omega.$$ We will also consider the martingale expansions of $f$ and $g$ with respect to $\sigma$ and $\omega$, respectively. Namely, $f=\sum_{E_{F,i}}\Delta_{E_{F,i}}^{\sigma}f$ and $g=\sum_{E_{G,j}}\Delta_{E_{G,j}}^{\omega}g$ where $\Delta_{E_{F,i}}^\sigma f=\hat{f}_{\sigma}(E_{F,i})h_{{F,i}}^{\sigma}=\left\langle f,h_{{F,i}}^{\sigma}\right\rangle_\sigma h_{{F,i}}^\sigma$.
We first make the assumption that $f$ and $g$ are finite linear combinations of indicator functions ${\bf{1}}_{E_{F,i}}$ where $2^{-d}\left\vert Q_0\right\vert\leq\left\vert E_{F,i}\right\vert\leq\left\vert Q_0\right\vert$ for some cube $Q_0$ and some $d>0$. We will obtain our estimates independent of $Q_0$ and $d$ and noting the density of simple functions in $L^2(\sigma)$ will give the result for general $f$ and $g$.
We now want to reduce to considering functions $f$ and $g$ compactly supported on a dyadic cube $Q_0\in\mathcal{D}^n$. To do this, for $1\leq j\leq 2^n$, let $Q_j\in\mathcal{D}^n$ be dyadic cubes in the $j^\text{th}$ orthant, respectively, so that $Q_0\subseteq \bigcup_{j}Q_j$. Then we can write $f=\sum_{j}f{\bf{1}}_{Q_j}$ and similarly for $g$. So $\left\Vert f\right\Vert_{L^2(\sigma)}^2=\sum_{j}\left\Vert f{\bf{1}}_{Q_j}\right\Vert_{L^2(\sigma)}^2$. We now have $$\begin{aligned}
\left\langle T (\sigma f),g\right\rangle_\omega&=\sum_{i,j}\left\langle T (\sigma f{\bf{1}}_{Q_i}),g{\bf{1}}_{Q_j}\right\rangle_\omega.\end{aligned}$$ Analyzing the terms with $i\neq j$ gives $$\left\langle T (\sigma f{\bf{1}}_{Q_i}),g{\bf{1}}_{Q_j}\right\rangle_\omega=\sum_{E_{F,i}\cap Q_i\neq\emptyset}\sum_{E_{G,j}\cap Q_j\neq\emptyset}\left\langle T(\sigma \Delta_{E_{F,i}}^\sigma f),\Delta_{E_{G,j}}^\omega g\right\rangle_\omega$$ where the cubes $E_{F,i}$ are in the $i^\text{th}$ orthant and the cubes $E_{G,j}$ are in the $j^\text{th}$ orthant. Now by property , this is zero by support considerations. Thus it suffices to show that we have $\left\vert\left\langle T (\sigma f{\bf{1}}_{Q_j}),g{\bf{1}}_{Q_j}\right\rangle_\omega\right\vert\lesssim C\left\Vert f{\bf{1}}_{Q_j}\right\Vert_{L^2(\sigma)}\left\Vert g{\bf{1}}_{Q_j}\right\Vert_{L^2(\omega)}$ because then we have $$\begin{aligned}
\left\vert\left\langle T (\sigma f),g\right\rangle_\omega\right\vert=\left\vert\sum_j\left\langle T(\sigma f{\bf{1}}_{Q_j}),g{\bf{1}}_{Q,j}\right\rangle_\omega\right\vert&\lesssim C\sum_{j}\left\Vert f{\bf{1}}_{Q_j}\right\Vert_{L^2(\sigma)}\left\Vert g{\bf{1}}_{Q_j}\right\Vert_{L^2(\omega)}\\
&\lesssim C\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}.\end{aligned}$$ So with this, we assume $Q_0\in\mathcal{D}^n$. Now we can write $$f=\sum_{E_{F,i}\subset Q_0}\Delta_{E_{F,i}}^\sigma f + \left\langle f\right\rangle_{Q_0}^\sigma {\bf{1}}_{Q_0}$$ as well as $$\left\Vert f\right\Vert^2_{L^2(\sigma)}=\sum_{E_{F,i}\subset Q_0}\left\Vert\Delta_{E_{F,i}}^\sigma f\right\Vert^2_{L^2(\sigma)} + \left\vert\left\langle f\right\rangle_{Q_0}^\sigma\right\vert^2 \sigma(Q_0)$$ where $\left\langle f\right\rangle_{E_{F,i}}^\sigma=\frac{1}{\sigma(E_{F,i})}\int_{E_{F,i}}f\sigma$ is the average of $f$ with respect to $\sigma$. With this we have $$\begin{aligned}
\left\langle T (\sigma f),g\right\rangle_\omega&=\sum_{E_{F,i},E_{G,j}\subset Q_0}\left\langle T (\sigma \Delta_{E_{F,i}}^\sigma f),\Delta_{E_{G,j}}^\omega g\right\rangle_\omega+\sum_{E_{F,i}\subset Q_0}\left\langle T(\sigma \Delta_{E_{F,i}}^\sigma f), \left\langle g\right\rangle_{Q_0}^\omega {\bf{1}}_{Q_0}\right\rangle_\omega\\
&+\sum_{E_{G,j}\subset Q_0}\left\langle \left\langle f\right\rangle_{Q_0}^\sigma T(\sigma {\bf{1}}_{Q_0}),\Delta_{E_{G,j}}^\omega g\right\rangle_\omega+\left\langle\left\langle f\right\rangle_{Q_0}^\sigma T(\sigma{\bf{1}}_{Q_0}),\left\langle g\right\rangle_{Q_0}^\omega {\bf{1}}_{Q_0}\right\rangle_\omega.\end{aligned}$$
For the last three terms, we have the following lemma.
The following estimates hold:
1. $\left\vert\sum_{E_{F,i}\subset Q_0}\left\langle T(\sigma \Delta_{E_{F,i}}^\sigma f), \left\langle g\right\rangle_{Q_0}^\omega {\bf{1}}_{Q_0}\right\rangle_\omega\right\vert\lesssim C_2\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}$;\
2. $\left\vert\sum_{E_{G,j}\subset Q_0}\left\langle \left\langle f\right\rangle_{Q_0}^\sigma T(\sigma {\bf{1}}_{Q_0}),\Delta_{E_{G,j}}^\omega g\right\rangle_\omega\right\vert\lesssim C_1\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}$;\
3. $\left\vert\left\langle\left\langle f\right\rangle_{Q_0}^\sigma T(\sigma{\bf{1}}_{Q_0}),\left\langle g\right\rangle_{Q_0}^\omega {\bf{1}}_{Q_0}\right\rangle_\omega\right\vert\lesssim C_1\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}.$
These are immediately controlled by Cauchy-Schwarz and applying the testing hypotheses. We will show the first estimate, and note that the remaining follow similarly. $$\begin{aligned}
\left\vert\sum_{E_{F,i}\subset Q_0}\left\langle T(\sigma \Delta_{E_{F,i}}^\sigma f), \left\langle g\right\rangle_{Q_0}^\omega {\bf{1}}_{Q_0}\right\rangle_\omega\right\vert&= \left\vert\left\langle g\right\rangle_{Q_0}^\omega\right\vert\left\vert\left\langle T\left(\sigma\sum_{E_{F,i}\subset Q_0}\Delta_{E_{F,i}}^\sigma f\right),{\bf{1}}_{Q_0}\right\rangle_\omega\right\vert\\
&\leq \left\vert\left\langle g\right\rangle_{Q_0}^\omega\right\vert\left\Vert\sum_{E_{F,i}\subset Q_0}\Delta_{E_{F,i}}^\sigma f\right\Vert_{L^2(\sigma)}\left\Vert {\bf{1}}_{Q_0} T^*(\omega{\bf{1}}_{Q_0})\right\Vert_{L^2(\sigma)}\\
&\lesssim C_2 \left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}.\end{aligned}$$
Thus it is enough to control only the first term, so we consider functions $f$ and $g$ that have mean zero with respect to $\sigma$ and $\omega$, respectively.
Proof of Main Theorem
=====================
Throughout the proof, we will use the notation $\Pi(f,g)=\left\langle T(\sigma f),g\right\rangle_{\omega}$. We have $$\begin{aligned}
\left\langle T(\sigma f),g\right\rangle_{\omega}&=\sum_{E_{F,i},E_{G,j}}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g)\\
&=\left(\sum_{2^{-r}\left\vert E_{F,i}\right\vert\leq\left\vert E_{G,j}\right\vert \leq 2^r\left\vert E_{F,i}\right\vert}+\sum_{\left\vert E_{G,j}\right\vert>2^r\left\vert E_{F,i}\right\vert}+\sum_{\left\vert E_{F,i}\right\vert > 2^r\left\vert E_{G,j}\right\vert}\right)\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g)\\
&=\sum_{k}\sum_{E_{F,i}}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{F,i,k}}^\omega g)+\sum_{E_{G,j}\supsetneq E_{F,i}^{(r)}}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g)\\
&\indent+\sum_{E_{F,i}\supsetneq E_{G,j}^{(r)}}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g)\\
&={\bf{A}}(f,g)+{\bf{B}}(f,g)+{\bf{C}}(f,g)\end{aligned}$$ where we have used property in the third equality to only consider rectangles with containment and where $E_{F,i,k}$ is the $k^\text{th}$ rectangle $E_{G,j}$ such that $2^{-r}\left\vert E_{F,i}\right\vert\leq\left\vert E_{G,j}\right\vert \leq 2^r\left\vert E_{F,i}\right\vert$ and $\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g)\neq0$, for some ordering of this finite set. We note that all sets $E_{F,i,k}$ will be contained in $E_{F,i}^{(r)}$ and have length at least $2^{-r}\left\vert E_{F,i}\right\vert$, which gives that there are $M=M(r,n)=\left(2^{n(2r+1)}-1\right)/\left(2^n-1\right)$ such sets. We will consider the first two sums only, as the third sum is symmetric to ${\bf{B}}(f,g)$. $$\begin{aligned}
\left\vert{\bf{A}}(f,g)\right\vert&=\left\vert\sum_{k}\sum_{E_{F,i}\in\mathcal{D}^n}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{F,i,k}}^\omega g)\right\vert\\
&=\left\vert\sum_{k}\sum_{{E_{F,i}}\in\mathcal{D}^n}\hat{f}_\sigma({E_{F,i}})\hat{g}_\omega({E_{F,i,k}})\Pi(h_{F,i}^\sigma,h_{F,i,k}^\omega)\right\vert\\
&\leq\sum_{k}\sup_{{E_{F,i}}}\left\vert \Pi(h_{F,i}^\sigma,h_{F,i,k}^\omega)\right\vert\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}\\
&\lesssim M C_3 \left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}\end{aligned}$$ where we have used in the final inequality. We now need only estimate $${\bf{B}}(f,g)=\sum_{E_{G,j}\supsetneq E_{F,i}^{(r)}}\Pi(\Delta_{E_{F,i}}^\sigma f,\Delta_{E_{G,j}}^\omega g).$$ We now define suitable stopping rectangles in $\mathcal{H}^n$. We initialize our construction with $S_0\equiv Q_0$, and we let $\mathcal{S}\equiv\{S_0\}$. In the inductive step, for a minimal stopping rectangle $S$, we let $\text{ch}_{\mathcal{S}}(S)$ be the set of all maximal $\mathcal{H}^n$ children $S'$ of S such that the following holds: $$\label{Stoppingg}
\frac{1}{\omega(S')}\int_{S'}\left\vert g\right\vert d\omega>2\frac{1}{\omega(S)}\int_{S}\left\vert g\right\vert d\omega.$$ We see immediately that $$\begin{aligned}
\sum_{S'\in \text{ch}_{\mathcal{S}}(S)}\omega(S')\leq\frac{1}{2}\omega(S),\end{aligned}$$ which gives us the Carleson condition $$\begin{aligned}
\sum_{S\in {\mathcal{S}},S\subseteq Q}\omega(S)\leq 2\omega(Q).\end{aligned}$$ We note that by the well-known dyadic Carleson embedding theorem (see [@Chung]), this condition implies that for all $g\in L^2(\omega)$ $$\sum_{S\in\mathcal{S}}\omega(S)\left\vert\left\langle g\right\rangle_S^\omega\right\vert^2\lesssim\left\Vert g\right\Vert_{L^2(\omega)}^2.$$
For every cube ${E_{F,i}}\subseteq Q_0$, we define the stopping parent $$\pi {E_{F,i}}\equiv \text{min}\{S\in\mathcal{S}:S\supseteq {E_{F,i}}\}.$$ Now define the projections $$P_S^{\omega}g=\sum_{{E_{G,j}}:\pi {E_{G,j}}=S}\Delta_{{E_{G,j}}}^\omega g,\text{ }\tilde{P}_S^{\sigma}f=\sum_{{E_{F,i}}:\pi {E_{F,i}^{(r)}}=S}\Delta_{{E_{F,i}}}^\sigma f.$$ So $f=\sum_{S\in\mathcal{S}}\tilde{P}_{S}^\sigma f$, and similarly $g=\sum_{S\in\mathcal{S}}{P}_{S}^\omega g$. With this, we have $$\begin{aligned}
{\bf{B}}(f,g)&=\sum_{S,S'\in\mathcal{S}}{\bf{B}}(\tilde{P}_S^\sigma f,P_{S'}^\omega g)\\
&=\sum_{S\in\mathcal{S}}{\bf{B}}(\tilde{P}_S^\sigma f,P_{S}^\omega g) +\sum_{S,S'\in\mathcal{S},S'\supsetneq S}{\bf{B}}(\tilde{P}_S^\sigma f,P_{S'}^\omega g)\\
&={\bf{B}}_1(f,g)+{\bf{B}}_2(f,g).\end{aligned}$$ We note that we do not get any contribution from the stoppping cubes $S\supsetneq S'$ because we are reduced to the case ${E_{F,i}}^{(r)}\subsetneq {E_{G,j}}$. We will now handle ${\bf{B}}_2(f,g)$ first: $$\begin{aligned}
{\bf{B}}_2(f,g)&=\sum_{S,S'\in\mathcal{S},S'\supsetneq S}{\bf{B}}(\tilde{P}_S^\sigma f,P_{S'}^\omega g)\\
&=\sum_{S\in\mathcal{S}}{\bf{B}}\left(\tilde{P}_S^\sigma f,\sum_{S'\in\mathcal{S},S'\supsetneq S}P_{S'}^\omega g\right)\\
&=\sum_{S\in\mathcal{S}}\sum_{{E_{G,j}}\supsetneq S}\Pi(\tilde{P}_S^\sigma f,{\bf{1}}_S\Delta_{E_{G,j}}^\omega g)\\
&=\sum_{S\in\mathcal{S}}\left\langle g\right\rangle_{S}^\omega \Pi(\tilde{P}_S^\sigma f,{\bf{1}}_{S}).\end{aligned}$$ In the third and fourth equalities, we have used $T(\sigma \tilde{P}_S^\sigma f)={\bf{1}}_S T(\sigma\tilde{P}_S^\sigma f)$ by property and further that $$\left\langle g\right\rangle_S^\omega{\bf{1}}_S={\bf{1}}_S\sum_{E_{G,j}\supsetneq S}\Delta_{E_{G,j}}^\omega g.$$ With this we have $$\begin{aligned}
\left\vert {\bf{B}}_2(f,g)\right\vert&\leq \sum_{S\in\mathcal{S}}\left\vert\left\langle g\right\rangle_S^\omega \right\vert\left\Vert \tilde{P}_S^\sigma f\right\Vert_{L^2(\sigma)}\left\Vert{\bf{1}}_S T^*(\omega{\bf{1}}_S)\right\Vert_{L^2(\sigma)}\\
&\leq C_2 \left\Vert f\right\Vert_{L^2(\sigma)}\left(\sum_{S\in\mathcal{S}}\left\vert\left\langle g\right\rangle_S^\omega\right\vert^2\omega(S)\right)^{1/2}\\
&\lesssim C_2 \left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}\end{aligned}$$ where the last inequality follows by the Carleson Embedding Theorem.
This leaves us only needing to estimate the term ${\bf{B}}_1(f,g)$. First, we will set $B_{S}(f,g)={\bf{B}}(\tilde{P}_S^\sigma f,P_S^\omega g)$. Then we have that ${\bf{B}}_{1}(f,g)=\sum_{S\in\mathcal{S}}B_S(f,g)$. We now have $$\begin{aligned}
B_S(f,g)&= \sum_{{E_{F,i}^{(r)}}\subsetneq {E_{G,j}}\subseteq S, \pi {E_{F,i}^{(r)}}=\pi {E_{G,j}}=S} \Pi(\Delta_{{E_{F,i}}}^\sigma f,\Delta_{E_{G,j}}^\omega)\\
&=\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi\left(\Delta_{E_{F,i}}^\sigma f,{\bf{1}}_{E_{F,i}^{(r)}}\sum_{{E_{G,j}}:{E_{F,i}^{(r)}}\subsetneq {E_{G,j}}\subseteq S,\pi {E_{G,j}}=S}\Delta_{E_{G,j}}^\omega g\right)\\
&=\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi\left(\Delta_{E_{F,i}}^\sigma f,\left\langle g\right\rangle_{E_{F,i}^{(r)}}^\omega{\bf{1}}_{E_{F,i}^{(r)}}-\left\langle g\right\rangle_S^\omega{\bf{1}}_S\right)\\
&=\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi\left(\Delta_{E_{F,i}}^\sigma f,\left\langle g\right\rangle_{E_{F,i}^{(r)}}^\omega{\bf{1}}_{E_{F,i}^{(r)}}\right)-\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi\left(\Delta_{E_{F,i}}^\sigma f,\left\langle g\right\rangle_S^\omega{\bf{1}}_S\right)\\
&=\bf{I-II}.\end{aligned}$$ Recalling by the stopping condition we have that $\left\vert\left\langle g\right\rangle_{E_{F,i}^{(r)}}\right\vert\leq\left\langle\left\vert g\right\vert\right\rangle_{E_{F,i}^{(r)}}\leq 2\left\langle\left\vert g\right\vert\right\rangle_S.$ With this, we have for the first term $$\begin{aligned}
\left\vert \bf{I}\right\vert&=\left\vert \sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi\left(\Delta_{E_{F,i}}^\sigma f,\left\langle g\right\rangle_{E_{F,i}^{(r)}}^\omega{\bf{1}}_{E_{F,i}^{(r)}}\right)\right\vert\\
&\lesssim \sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\left\langle \left\vert g\right\vert\right\rangle_S^\omega\left\vert\Pi\left(\Delta_{E_{F,i}}^\sigma f,{\bf{1}}_{E_{F,i}^{(r)}}\right)\right\vert\\
&\lesssim \left\langle \left\vert g\right\vert\right\rangle_S^\omega\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\left\Vert\Delta_{E_{F,i}}^\sigma f\right\Vert_{L^2(\sigma)}\left\Vert{\bf{1}}_{E_{F,i}^{(r)}} T^*(\omega {\bf{1}}_{E_{F,i}^{(r)}})\right\Vert_{L^2(\sigma)}\\
&\lesssim C_2\left\langle \left\vert g\right\vert\right\rangle_S^\omega\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\left\Vert\Delta_{E_{F,i}}^\sigma f\right\Vert_{L^2(\sigma)}\left\Vert{\bf{1}}_{E_{F,i}^{(r)}}\right\Vert_{L^2(\omega)}\\
&\lesssim C_2\left\langle \left\vert g\right\vert\right\rangle_S^\omega \left(\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\left\Vert\Delta_{E_{F,i}}^\sigma f\right\Vert^2_{L^2(\sigma)}\right)^{1/2}\left(\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\omega({E_{F,i}})\right)^{1/2}\\
&\lesssim C_2\left\langle \left\vert g\right\vert\right\rangle_S^\omega \omega(S)^{1/2}\left\Vert \tilde{P}_S f\right\Vert_{L^2(\sigma)}.\end{aligned}$$ For the second term, we have $$\begin{aligned}
\left\vert \bf{II}\right\vert&\leq\left\langle\left\vert g\right\vert\right\rangle_S^\omega\left\vert\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Pi(\Delta_{E_{F,i}}^\sigma f,{\bf{1}}_{S})\right\vert\\
&=\left\langle\left\vert g\right\vert\right\rangle_S^\omega\left\vert\Pi\left(\sum_{{E_{F,i}^{(r)}}\subseteq S, \pi {E_{F,i}^{(r)}}=S}\Delta_{E_{F,i}}^\sigma f,{\bf{1}}_{S}\right)\right\vert\\
&\leq \left\langle\left\vert g\right\vert\right\rangle_S^\omega\left\Vert {\bf{1}}_S T^*(\omega {\bf{1}}_S) \right\Vert_{L^2(\sigma)}\left\Vert \tilde{P}_S^\sigma f\right\Vert_{L^2(\sigma)}\\
&\lesssim C_2\left\langle\left\vert g\right\vert\right\rangle_S^\omega\omega(S)^{1/2}\left\Vert \tilde{P}_S^\sigma f\right\Vert_{L^2(\sigma)}.\\\end{aligned}$$ With this, we have finally that $$\begin{aligned}
\left\vert{\bf{B}_1}(f,g)\right\vert&=\left\vert\sum_{S\in\mathcal{S}}B_S(f,g)\right\vert\\
&\lesssim C_2 \sum_{S\in\mathcal{S}}\left\langle\left\vert g\right\vert\right\rangle_S^\omega\omega(S)^{1/2}\left\Vert \tilde{P}_S^\sigma f\right\Vert_{L^2(\sigma)}\\
&\lesssim C_2\left\Vert f\right\Vert_{L^2(\sigma)}\left(\sum_{S\in\mathcal{S}}\left(\left\langle\left\vert g\right\vert\right\rangle_S^\omega\right)^2\omega(S)\right)^{1/2}\\
&\lesssim C_2\left\Vert f\right\Vert_{L^2(\sigma)}\left\Vert g\right\Vert_{L^2(\omega)}\end{aligned}$$ where we again use the Carleson Embedding Theorem in the last inequality.
So we have now established $C\lesssim C_1+C_2+C_3$. To obtain the other inequality, we notice that we have $$C\gtrsim\left\Vert T\right\Vert_{L^2(\sigma)\rightarrow L^2(\omega)}\gtrsim\sup_{{E_{F,i}}}\frac{\left\Vert{\bf{1}}_{E_{F,i}} T(\sigma{\bf{1}}_{{E_{F,i}}})\right\Vert_{L^2(\omega)}}{\left\Vert{\bf{1}}_{{E_{F,i}}}\right\Vert_{L^2(\sigma)}}=C_1.$$ Similarly, we have $C\gtrsim\left\Vert T\right\Vert_{L^2(\sigma)\rightarrow L^2(\omega)}\gtrsim C_2$ and $C\gtrsim\left\Vert T\right\Vert_{L^2(\sigma)\rightarrow L^2(\omega)}\gtrsim C_3$. This indeed gives $$C\simeq C_1+C_2+C_3.$$
Well Localized Operators {#WL}
========================
We recall from [@NTV1] that well localized operators have the following definition:
$T$ is said to be lower triangularly localized if there exists a constant $r>0$ such that for all cubes $R$ and $Q$ with $\left\vert R\right\vert\leq2\left\vert Q\right\vert$ and for all $\omega$-Haar functions on $R$ $h_R^\omega$, we have $$\left\langle T(\sigma{\bf{1}}_Q),h_R^\omega\right\rangle_\omega=0$$ if $R\not\subset Q^{(r)}$ or if $\left\vert R\right\vert\leq 2^{-r}\left\vert Q\right\vert$ and $R\not\subset Q$.
We say that $T$ is well localized if both $T$ and $T^*$ are lower triangularly localized.
We will now show that well localized operators are essentially well localized. Let $T$ be a well localized operator associated with some $r>0$. Fix a cube $E_{F,i}$ and let $E_{G,j}$ be any cube with $\left\vert E_{G,j}\right\vert=\left\vert E_{F,i}\right\vert$ and $E_{F,i}^{(r)}\cap E_{G,j}=\emptyset$. So $$T (\sigma h_{F,i}^\sigma){\bf{1}}_{E_{G,j}}=\sum_{E_{H,k}\subset E_{G,j}}\Delta_{H,k}^\omega T (\sigma h_{F,i}^\sigma) +\left\langle T (\sigma h_{F,i}^\sigma)\right\rangle_{E_{G,j}}^\omega{\bf{1}}_{E_{G,j}}.$$ Now since $T$ is well localized, it is immediate that $\left\langle T (\sigma h_{F,i}^\sigma)\right\rangle_{E_{G,j}}^\omega=0$. So we have $$T (\sigma h_{F,i}^\sigma){\bf{1}}_{E_{G,j}}=\sum_{E_{H,k}\subset E_{G,j}}\Delta_{H,k}^\omega T (\sigma h_{F,i}^\sigma).$$ Now $$\Delta_{H,k}^\omega T (\sigma h_{F,i}^\sigma)=\frac{\sqrt{\sigma(E_{F,i}^1)}}{\sqrt{\sigma(E_{F,i})\sigma(E_{F,i}^2)}}\Delta_{H,k}^\omega T(\sigma {\bf{1}}_{E_{F,i}^2})-\frac{\sqrt{\sigma(E_{F,i}^2)}}{\sqrt{\sigma(E_{F,i})\sigma(E_{F,i}^1)}}\Delta_{H,k}^\omega T(\sigma {\bf{1}}_{E_{F,i}^1}).$$ For $E_{H,k}\subseteq E_{G,j}$, we clearly have $\left\vert E_{H,k}\right\vert\leq2\left\vert E_{F,i}^1\right\vert$ and $\left\vert E_{H,k}\right\vert\leq2\left\vert E_{F,i}^2\right\vert$. So applying the well localized property to each term gives $\Delta_{H,k}^\omega T (\sigma h_{F,i}^\sigma)=0$.
Now for any rectangle $Q$ with $Q \cap E_{F,i}^{(r)}=\emptyset$, we can write $Q\subseteq\bigcup Q_k$ where $Q_k\cap E_{F,i}^{(r)}=\emptyset$ and $\left\vert Q_k\right\vert =\left\vert E_{F,i}\right\vert$. So we have $T(\sigma h_{F,i}^\sigma){\bf{1}}_Q={\bf{1}}_{Q}\sum_{k}T(\sigma h_{F,i}^\sigma){\bf{1}}_{Q_k}=0$. A similar calculation shows $T^*(\omega h_{F,i}^\omega){\bf{1}}_Q=0$. So we have that $T$ is essentially well localized.
This computation also gives the following characterization for essentially well localized operators.
An operator $T$ is essentially well localized for some $r\geq 0$ if and only if for all $Q\in\mathcal{D}^n$ and $E_{F,i}\in\mathcal{H}^n$ with $\left\vert E_{F,i}\right\vert\leq2\left\vert Q\right\vert$ and $E_{F,i}\not\subset Q^{(r+1)}$, we have $$\left\langle T(\sigma{\bf{1}}_Q),h_{F,i}^\omega\right\rangle_\omega=0$$ and $$\left\langle T^*(\omega{\bf{1}}_Q),h_{F,i}^\sigma\right\rangle_\sigma=0.$$
However, if $\left\vert E_{F,i}\right\vert\leq 2^{-(r+1)}\left\vert Q\right\vert$ and $E_{F,i}\not\subset Q$, then we have that $E_{F,i}^{(r)}\cap Q=\emptyset$. With this, we immediately have the following characterization of essentially well localized operators.
An operator $T$ is essentially well localized for some $r\geq 0$ if and only if $T$ is well localized for $r+1$.
Having an alternate characterization for well localized operators allows us to easily classify some operators as the following example shows.
An operator $T$ is said to be an essentially perfect dyadic operator if for some $r\geq 0,$ $$T(\sigma f)(x)=\int_{\mathbb{R}} K(x,y)f(y)\sigma(y)dy$$ for $x\not\in\text{supp}(f)$, where $$K(x,y)\leq\frac{1}{\left\vert x-y\right\vert}$$ and $$\left\vert K(x,y)-K(x,y')\right\vert +\left\vert K(x,y)-K(x',y)\right\vert=0$$ whenever $x,x'\in I\in\mathcal{D}$, $y,y'\in J\in\mathcal{D}$ where $I^{(r)}\cap J=\emptyset$ and $I\cap J^{(r)}=\emptyset$.
If $r=0$, we recover the perfect dyadic operators first introduced in [@AHMTT].
Let $T:L^2(\sigma)\rightarrow L^2(\omega)$ be an essentially perfect dyadic operator and let $f=\sigma h_I^\sigma$ and $x\not\in I^{(r)}$. Then for all $y,y'\in I$, we have $\left\vert K(x,y)-K(x,y')\right\vert=0$. So $K(x,\cdot)$ is constant on $I$. With this we have $$\begin{aligned}
T(\sigma h_I^\sigma)(x)&=\int_{I} K(x,y)h_I^\sigma(y)\sigma(y)dy\\
&=0.\end{aligned}$$
We can write $$T^*(\omega g)(y)=\int_{\mathbb{R}} \overline{K(x,y)}g(x)\omega(x)dx.$$ Now let $g=\omega h_J^\omega$ and let $y\not\in J^{(r)}$. Then for all $x,x'\in J$, we have $\left\vert \overline{K(x,y)}-\overline{K(x',y)}\right\vert=\left\vert K(x,y)-K(x',y)\right\vert=0$. So $\overline{K(\cdot,y)}$ is constant on $J$. As above, we have $$\begin{aligned}
T^*(\omega h_J^\omega)(y)&=\int_{J} \overline{K(x,y)}h_J^\omega(x)\omega(x)dx\\
&=0.\end{aligned}$$ So we have that $T$ is essentially well localized.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author would like to thank Brett Wick for his helpful conversations and support (NSF DMS grants \#1560955 and \#1603246).
[^1]: 2010 Mathematics Subject Classification: Primary 47A30, Secondary 42B99\
Key Words and Phrases: Well-Localized Operators, Two Weight Inequalities, Carleson Embedding
|
---
abstract: '0.5cm We consider a stochastic volatility model with jumps where the underlying asset price is driven by a process sum of a 2-dimensional Brownian motion and 2-dimensional compensated Poisson process. The market is incomplete, there is an infinity of Equivalent Martingale Measures (E.M.M) and an infinity of hedging strategies. We characterize the set of E.M.M, and we hedge by minimizing the variance using Malliavin calculus.'
---
=0.5cm
Youssef El-Khatib
[ *Département de Mathématiques, Université du Maine\
Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France[^1]* ]{}
=0.5cm [**Keywords:**]{} stochastic volatility model with jumps, incomplete markets, Malliavin calculus, Clark-Ocone formula, European options, equivalent martingale measure, mean-variance hedging.\
\
[*Mathematics Subject Classification (2000):*]{} 91B24, 91B26, 91B28, 60H07.
0.7cm
Introduction
============
Stochastic volatility models were introduced in the financial literature to take in account the *smile* effect. Most of works on these models assumes -for simplification- the continuity of the asset price trajectories (driven by Brownian motion). But an asset price can jump at any moment and randomly. We are interested here by discontinuous dynamic for the asset price with discontinuous stochastic volatility.\
Formally, let the underlying asset price is given by $$\begin{aligned}
\frac{dS_t}{S_t}&=&\mu_t dt+\sigma(t,Y_t)[a^{(1)}_t dW^{(1)}_t +a^{(3)}_t dM^{(1)}_t],\
\ \ t\in[0,T],\ \ \ S_0=x>0,\\
\mbox{with}\\
dY_t&=&\mu^Y_t dt+\sum_{i=1}^2 \sigma^{(i)}_t [a^{(i)}_t dW^{(i)}_t
+a^{(i+2)}_t dM^{(i)}_t ],\ \ \ Y_0=y \in \real.\end{aligned}$$ $W=(W^{(1)},W^{(2)})$ is a 2-dimensional Brownian motion, $M=(M^{(1)},M^{(2)})$ is a 2-dimensional compensated Poisson process with independent components and deterministic intensity $(\int_0^t
\lambda_s^{(1)} ds,\int_0^t \lambda_s^{(2)} ds)$ and for $1\leq
i\leq 4, a^{(i)}: [0,T] \longrightarrow \real$ is a deterministic function.\
The most serious problem in a stochastic volatility model is the incompleteness. These models involve the existence of an infinity of equivalent martingale measures (EMM) i.e a probability equivalent to the historical one under which the actualized prices are martingales. First we seek a characterization for an EMM. We show that a probability $Q$ equivalent to the historical probability $P$ is specified by its Radon-Nikodym density w.r.t $P$ $$\begin{aligned}
\rho_T&=&\prod_{i=1}^2 \exp\left(\int_0^T \beta^{(i)}_s dW^{(i)}_s
-\frac{1}{2}\int_0^T (\beta^{(i)}_s)^2 ds\right)\exp\left(\int_0^T
\ln (1+\beta^{(i+2)}_s)
dM^{(i)}_s\right.\\
&&\left.+\int_0^T \lambda^{(i)}_s \left[\ln (1+\beta^{(i+2)}_s)
-\beta^{(i+2)}_s\right]ds\right),\end{aligned}$$ where $(\beta_{t})_{t\in [0,T]}$ is a $\real^4$-valued predictable process such that $\beta^{(3)},\beta^{(4)}>-1.$ If $Q$ is a $P-$EMM, $\beta^{(1)}$ and $\beta^{(3)}$ are related by $$\mu_t -r_t +\beta^{(1)}_t a^{(1)}_t \sigma(t,Y_t) +\lambda^{(1)}_t
\beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t)=0,$$ see Proposition \[mmeq\].\
The process $\left(-\frac{\mu_t-r_t}{a^{(1)}_t
\sigma(t,Y_t)},0,0,0\right)$ is an example of $\real^4$-valued predictable process satisfying the above equation, and it defines a $P$-E.M.M. This means that the set of $P$-EMM is not empty. Moreover, since $\beta^{(2)}$ and $\beta^{(4)}$ doesn’t appear in the last equation so they can be choosing arbitrarily and thus there exists an infinity of $P-$EMM.
Mean-variance hedging {#mean-variance-hedging .unnumbered}
---------------------
In complete market, we have a unique hedging strategy. This is not the case for incomplete market. We have an infinity of hedging strategies. We hedge using the mean-variance hedging approach initiated by Föllmer and Sondermann (1986), and we find the strategy by applying Malliavin calculus. Consider an option with payoff $f(S_T)$ where $(S_t)_{t\in [0,T]}$ is the asset price and with maturity $T$. We work with a $P$-E.M.M ${\hat Q}$. Let $(\hat{\eta}_t, \hat{\zeta}_t)_{t\in [0,T]}$ be a self-hedging strategy and $(\hat{V}_t)_{t\in [0,T]}$ be the portfolio value process. We get using the chaotic calculus, that the strategy minimizing the variance $E_{\hat{Q}}\left[(f(S_T)-
\hat{V}_T )^2 \right],$ is given by $${\hat{\eta}}_t =\frac{a^{(1)}_t E[D^{\hat{W}^{(1)}}_t f(S_T)\mid
{\mathcal F}_t]+\lambda^{(1)}_t(1+{\hat{\beta}}^{(3)}_t)a^{(3)}_t
E[D^{N^{(1)}}_t f(S_T)\mid {\mathcal F}_t]} {((a^{(1)}_t)^2
+\lambda^{(1)}_t(1+{\hat{\beta}}^{(3)}_t)(a^{(3)}_t)^2)e^{\int_t^T
r_sds}\sigma(t,Y_t) S_t},$$ where ${\hat{W}^{(1)}}_t={W}^{(1)}_t
-\int_0^t \hat{\beta}^{(1)}ds$, and the operators $D^{\hat{W}^{(1)}}$ and $D^{N^{(1)}}$ are respectively the Malliavin derivative in the direction of the one dimensional Brownian motion ${\hat{W}}^{(1)}$ and the Malliavin operator in the direction of the Poisson process $N^{(1)}$.\
The paper is organized as follows : In Section 2, we present some necessary formulas. In the third section we introduce the model. The fourth one is devoted to the hedging by minimizing the variance via Malliavin calculus. In the last section, we characterize the E.M.M minimizing the entropy, this allows us to establish explicit formulae for the strategy.
Preliminary
===========
Let $W=(W^{(1)},W^{(2)})$ be a 2-dimensional Brownian motion and $N=(N^{(1)},N^{(2)})$ denotes a 2-dimensional Poisson process with independent components and deterministic intensity $(\int_0^t \lambda_s^{(1)} ds,\int_0^t \lambda_s^{(2)} ds)$. We work in a filtered probability space $(\Omega, {\cal{F}},({\cal{F}}_t)_{t\in
[0,T]}, P)$, where $({\cal{F}}_t)_{t\in [0,T]}$ is the naturel filtration generated by $W$ and $N$. We denote by $M=(M^{(1)},M^{(2)})$ the associated compensated Poisson process i.e for $i=1,2$ and $t\in [0,T]$ we have $dM^{(i)}_t=dN^{(i)}_t
-\lambda^{(i)}_t dt$. Both $({\cal F}_t)_{t\in [0,T]}$-martingales $W$ and $M$ are independent.
Let $\Gamma$ be the set of all ${\cal F}_t$-predictable processes $(\gamma_t)_{t\in [0,T]}$ with values in $\real^4$, such that $$\sum_{i=1}^2 E_P \left[\int_0^t (\gamma^{(i)}_s)^2 ds\right] +
\sum_{i=1}^2 E_P \left[\int_0^t (\gamma^{(i+2)}_s)^2 \lambda^{i}_s
ds\right]<\infty, \ \ \ t\in [0,T].$$
We denote by ${\cal E}(X)_t$, for a semi-martingale $X$ with $X_0=0$, the unique solution of the stochastic differential equation $$Z_t=1+\int_0^t Z_{s^-} dX_s.$$ ${\cal E}(X)_t$ is called the Doléans-Dade exponential. We have (Theorem 36 of Protter (1990))$$\label{expdade} {\cal E}(X)_t=\exp\left(X_t
-\frac{1}{2}[X_t,X_t]^c\right)\prod_{s\leq t}(1+\Delta
X_s)\exp(-\Delta X_s).$$
Notice that for $\gamma \in \Gamma$ such that $\gamma^{(3)},\gamma^{(4)}>-1$ we have for $i=1,2$ $$\begin{aligned}
{\cal E}(\gamma^{(i)} W^{(i)})_t&=&\exp\left(\int_0^t \gamma^{(i)}_s
dW^{(i)}_s
-\frac{1}{2}\int_0^t (\gamma^{(i)}_s)^2 ds\right),\\
{\cal E}(\gamma^{(i+2)} M^{(i)})_t&=&\exp\left(\int_0^t \ln
(1+\gamma^{(i+2)}_s)
dM^{(i)}_s\right.\\
&&\left.+\int_0^t \lambda^{(i)}_s \left[\ln (1+\gamma^{(i+2)}_s)
-\gamma^{(i+2)}_s\right]ds\right).\end{aligned}$$
The next lemma is the martingale representation theorem (Jacod (1979)).
\[repres\] Let $Z=(Z_t)_{t\in [0,T]}$ be a ${\cal
F}_t$-martingale. There exists a predictable process $\gamma \in
\Gamma$ such that $$dZ_t = \sum_{i=1}^2 \gamma^{(i)}_t dW^{(i)}_t +\sum_{i=1}^2
\gamma^{(i+2)}_t dM^{(i)}_t,\ \ \ t \in [0,T].$$
The Itô formula is given by the following lemma (see Protter (1990)).
\[gito\] Let $R=(R^1,\ldots,R^n)$ be a $n$-dimensional adapted process, and $\gamma=(\gamma^{1},\ldots,\gamma^{n})$ such that $$\gamma^k=(\gamma^{(k,1)},\gamma^{(k,2)},\gamma^{(k,3)},\gamma^{(k,4)})
\in \Gamma,\ \ \ 1\leq k\leq n.$$ We consider the process $X=(X^1,\ldots,X^n)$ where for $k \in \{1,\ldots,n\}$, $X^k$ is given by $$dX^k_t=R^k_t dt +\sum_{i=1}^2 \gamma^{(k,i)}_t dW^{(i)}_t
+\sum_{i=1}^2\gamma^{(k,i+2)}_t dM^{(i)}_t, \ \ \ X^k_0\in \real.$$ For any function $f\in {\cal C}^{1,2}([0,T]\times \real),$ we have $$\begin{aligned}
f(t,X_t)&=&f(0,X_0)+\int_0^t\left[\frac{\partial f}{\partial s}
(s,X_{s^-})+\langle R_s , \nabla f(s,X_s)\rangle \right.\\
&&\left.+\sum_{k,l=1}^n\frac{1}{2}\sum_{i=1}^2 \frac{\partial^2 f}
{\partial x^k x^l}(s,X_{s^-})\gamma^{(k,i)}_s \gamma^{(l,i)}_s \right.\\
&&\left.+\sum_{k=1}^n \sum_{i=1}^2 \lambda^{(i)}_s
\left(f(s,(X^1_{s^-},\ldots,X^k_{s^-}+\gamma^{(k,i+2)}_s,\ldots,X^n_{s^-}))-f(s,X_{s^-})
\right.\right.\\
&&\left.\left.-\gamma^{(k,i+2)}_s \frac{\partial f}{\partial
x^k}(s,X_{s^-})\right)\right]ds
+\sum_{k=1}\sum_{i=1}^2 \int_0^t \gamma^{(k,i)}_s \frac{\partial f}{\partial x^k}(s,X_s)dW^{(i)}_s \\
&&+\sum_{k=1}^n \sum_{i=1}^2 \int_0^t
(f(s,(X^1_{s^-},\ldots,X^k_{s^-}+\gamma^{(k,i+2)}_s,\ldots,X^n_{s^-}))-f(s,X_{s^-}))
dM^{(i)}_s. \\\end{aligned}$$
The model {#the model}
=========
Let us consider a market with two assets: a risky asset to which is related a European call option and a riskless one. The maturity is $T$ and the strike is $K$. The price of the riskless asset is given by $$dA_t=r_t A_t dt, \ \ \ t\in [0,T],\ \ \ A_0=1,$$ where $r_t$ is deterministic and denotes the interest rate. The price of the risky asset has a stochastic volatility and is given by $$\begin{aligned}
\label{stochv}
\frac{dS_t}{S_t}&=&\mu_t dt+\sigma(t,Y_t)[a^{(1)}_t dW^{(1)}_t +a^{(3)}_t dM^{(1)}_t],\
\ \ t\in[0,T],\ \ \ S_0=x>0,\\
\nonumber dY_t&=&\mu^Y_t dt+\sum_{i=1}^2 \sigma^{(i)}_t [a^{(i)}_t
dW^{(i)}_t +a^{(i+2)}_t dM^{(i)}_t ],\ \ \ t\in[0,T],\ \ \ Y_0=y \in
\real,\end{aligned}$$ where for $1\leq i\leq 4, a^{(i)}: [0,T] \longrightarrow \real$ is a deterministic function. We assume that $$\sigma(t,.)\neq 0,\ \ \ \mbox{and}\ \ \ 1+\sigma(t,.)a^{(3)}_t >0, \ \ \ t\in [0,T].$$ We have $$\begin{aligned}
S_{t} &=& x\exp \left( \int_0^t a^{(1)}_s \sigma(s,Y_s)dW^{(1)}_s
+ \int_0^t (\mu_{s}-
a^{(3)}_s \lambda^{(1)}_s \sigma(s,Y_s) -
\frac{1}{2} (a^{(1)}_s)^2 \sigma^2(s,Y_s)) ds \right)\\
&&\times \prod_{k=1}^{k=N_t}
(1+a^{(3)}_{T^{(1)}_k} \sigma(T^{(1)}_k,Y_{T^{(1)}_k})),\end{aligned}$$ $0\leq t\leq T$, where $(T^{(1)}_k)_{k\geq 1}$ denotes the jump times of $(N^{(1)}_t)_{t\in
[0,T]}$.
Change of probability
---------------------
Let $Q$ be a $P$-equivalent probability; by the Radon-Nikodym theorem there exists a ${\cal{F}}_T$-measurable random variable, $\rho_T :=\frac{dQ}{dP}$, such that $Q(A)=E_P[\rho_T 1_A]$, $A \in
{\cal P}(\Omega)$. Notice that $\rho_T$ is strictly positive $P$-a.s, since $Q$ is equivalent to $P$, and $E_P[\rho_T]=E_P[\rho_T
1_\Omega]=1$. Consider now the $P$-martingale $\rho=(\rho_t)_{t\in
[0,T]}$ defined by $$\rho_t :=E_P[\rho_T \mid {\cal F}_t]=E_P\left[\frac{dQ}{dP} \mid
{\cal F}_t\right].$$
Let ${\cal H}$ be the set of all $P-$EMM i.e $Q \in {\cal H}$ if and only if $Q \simeq P$ and the actualized prices are $Q$-martingales.
The next proposition gives the Radon-Nikodym density w.r.t $P$ of a $P$-EMM.
\[mmeq\]Let $Q \in {\cal H}$. There exists a predictable process $(\beta_t)_{t\in [0,T]}$ taking values in $\real^4$ such that $\beta^{(3)},\beta^{(4)}>-1$ and the Radon-Nikodym density of $Q$ w.r.t $P$ is given by $$\begin{aligned}
\nonumber
\rho_T&=&\prod_{i=1}^2 {\cal E}(\beta^{(i)}W^{(i)})_T
{\cal E}(\beta^{(i+2)}M^{(i)})_T\\
\nonumber &=& \prod_{i=1}^2 \exp\left(\int_0^T \beta^{(i)}_s
dW^{(i)}_s -\frac{1}{2}\int_0^T (\beta^{(i)}_s)^2
ds\right)\exp\left(\int_0^T \ln (1+\beta^{(i+2)}_s)
dM^{(i)}_s\right.\\
\label{rhob} &&\left.+\int_0^T \lambda^{(i)}_s \left[\ln
(1+\beta^{(i+2)}_s) -\beta^{(i+2)}_s\right]ds\right).\end{aligned}$$ Moreover $\beta^{(1)}$ and $\beta^{(3)}$ are related by $$\label{rel} \mu_t -r_t +\beta^{(1)}_t a^{(1)}_t \sigma(t,Y_t)
+\lambda^{(1)}_t \beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t)=0.$$
We follow Bellamy (1999) for the case of discontinuous market with deterministic volatility. By the martingale representation theorem (Lemma \[repres\]) there exists a predictable process $(\gamma_t)_{t \in [0,T]} \in \Gamma$ such that $$d\rho_t= \sum_{i=1}^2 \gamma^{(i)}_t dW^{(i)}_t
+\sum_{i=1}^2 \gamma^{(i+2)}_t dM^{(i)}_t ,\ \ \ t \in [0,T].$$ We have $P(\rho_t >0, t\in [0,T])=1$; assuming $\beta:=\frac{\gamma}{\rho}$, we obtain $$\begin{aligned}
\frac{d\rho_t}{\rho_t}&=&\sum_{i=1}^2 \beta^{(i)}_t dW^{(i)}_t
+\sum_{i=1}^2\beta^{(i+2)}_t dM^{(i)}_t=dX_t,\ \ \ t \in [0,T].\end{aligned}$$ (\[rhob\]) follows from (\[expdade\]). In addition $(e^{-\int_0^t r_s ds}S_t)_{t\in [0,T]}$ is a $Q$-martingale, which is equivalent to say that $(e^{-\int_0^t r_s ds}S_t \rho_t)_{t\in
[0,T]}$ is a $P$-martingale. The integration by parts formula (Protter (1990)) gives $$d(e^{-\int_0^t r_s ds}S_t \rho_t)=\rho_t d(e^{-\int_0^t r_s ds}S_t)
+e^{-\int_0^t r_s ds}S_t d\rho_t +d[e^{-\int_0^t r_s
ds}S_t,\rho_t],$$ with $$\begin{aligned}
d[e^{-\int_0^t r_s ds}S_t,\rho_t]&=&\beta^{(1)}_t a^{(1)}_t
\sigma(t,Y_t)dt +
\beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t)dN^{(1)}_t,\\
&=&\left(\beta^{(1)}_t a^{(1)}_t \sigma(t,Y_t)+\lambda^{(1)}_t
\beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t)\right)dt+ \beta^{(3)}_t
a^{(3)}_t \sigma(t,Y_t)dM^{(1)}_t.\end{aligned}$$ Therefore, we have $$\begin{aligned}
d(e^{-\int_0^t r_s ds}S_t \rho_t)&=&\rho_t S_t e^{-\int_0^t r_s
ds}\left[(\mu_t -r_t +\beta^{(1)}_t a^{(1)}_t
\sigma(t,Y_t) +\lambda^{(1)}_t \beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t))dt\right.\\
&&+\left.(\beta^{(1)}_t+\sigma(t,Y_t)a^{(1)}_t)dW^{(1)}_t
+\beta^{(2)}_t dW^{(2)}_t \right.\\
&&+\left.\left(\sigma(t,Y_t)a^{(3)}_t + \beta^{(3)}_t(1+
\sigma(t,Y_t)a^{(3)}_t)\right)dM^{(1)}_t+\beta^{(4)}_t
dM^{(2)}_t\right].\end{aligned}$$ Thus $Q$ is a $P$-EMM if $$\mu_t -r_t +\beta^{(1)}_t a^{(1)}_t \sigma(t,Y_t) +\lambda^{(1)}_t \beta^{(3)}_t a^{(3)}_t \sigma(t,Y_t)=0.$$
One can notice that there is no restriction on $\beta^{(2)}$ and $\beta^{(4)}$, which means that if ${\cal H} \neq \emptyset$ thus ${\cal H}$ contains an infinity of $P$-EMM.
Girsanov theorem
----------------
Let $\Gamma^{\cal H}$ be the set of processes $\beta \in \Gamma$ satisfying (\[rel\]). The Radon-Nikodym derivative $\rho_T$ associated to $\beta$ and given by (\[rhob\]) define a $P$-EMM. From now on, a $P$-EMM $Q$ in ${\cal
H}$ will be denoted by $Q^{\beta}$ where $\beta \in \Gamma^{\cal
H}$.\
Let $\beta \in \Gamma^{\cal H}$ and consider the two processes $\tilde{W}=(\tilde{W}^{(1)},\tilde{W}^{(2)})$ and $\tilde{M}=(\tilde{M}^{(1)},\tilde{M}^{(2)})$ where for $i=1,2$ $${\tilde{W}^{(i)}}_t={W}^{(i)}_t -\int_0^t \beta^{(i)}ds,\ \ \ t\in
[0,T],\ \ \ \mbox{and} \ \ \ {\tilde{M}^{(i)}}_t={M}^{(i)}_t
-\int_0^t \lambda^{(i)}_s \beta^{(i+2)}_s ds,\ \ \ t\in [0,T].$$ By Girsanov theorem (Jacod (1979)) $\tilde{W}$ is a $Q^{\beta}$-Brownian motion and $\tilde{M}$ is a $Q^{\beta}$-compensated Poisson process. The dynamic of $(S_t)_{t\in[0,T]}$ under $Q^{\beta}$ is $$\frac{dS_t}{S_t}=r_t dt+\sigma(t,Y_t)[a^{(1)}_t d{\tilde{W}^{(1)}}_t +a^{(3)}_t d{\tilde{M}^{(1)}}_t],\
\ \ t\in[0,T],\ \ \ S_0=x>0,$$ and $(Y_t)_{t\in [0,T]}$ is given by $$\begin{aligned}
dY_t&=&\left(\mu^Y_t +\sum_{i=1}^2 \sigma^{(i)}_t [a^{(i)}_t \beta^{(i)}_t +\lambda^{(i)}_t \beta^{(i+2)}_t a^{(i+2)}_t ]\right)dt\\
&&+\sum_{i=1}^2 \sigma^{(i)}_t [a^{(i)}_t
d{\tilde{W}^{(i)}}_t +a^{(i+2)}_t d{\tilde{M}^{(i)}}_t ],\ \ \ t\in
[0,T]\ \ \ Y_0=y \in \real.\end{aligned}$$
Hedging by minimizing the variance {#hed}
==================================
In this section we are interested by finding an optimal strategy for our model described in Section \[the model\]. We compute the strategy by minimizing the variance. This is on applying Malliavin calculus.\
From now on, we work with the $P$-EMM minimizing the entropy $\hat{Q}$. The price of a European option with payoff $f(S_T)$ is $
E_{\hat{Q}}\left[
e^{-\int_t^T r_s ds}f(S_T)\mid {\cal F}_{t}\right],\ \
\ t\in [0,T]$. Our aim is to determine the ${\cal F}_t$-adapted strategy $(\hat{\zeta}_t, \hat{\eta}_t)_{t\in [0,T]}$ that minimizes $$\label{vari} E_{\hat{Q}}\left[(f(S_T)- \hat{V}_T)^2\right],$$ where $\hat{\zeta}_t$, $\hat{\eta}_t$ and $\hat{V}_t$ denote respectively the number of units invested in riskless and risky asset and the value of the portfolio. We have for $t \in [0,T]$ $\hat{V}_t=\hat{\zeta}_t A_t + \hat{\eta}_t S_t$. Since the strategy is assumed to be self-financing, so $dV_t = \hat{\zeta}_t dA_t +
\hat{\eta}_t dS_t$ and $$dV_t = r_t V_t dt + \sigma(t,Y_t)\hat{\eta}_t S_t[a^{(1)}_t
d{\hat{W}^{(1)}}_t +a^{(3)}_t d{\hat{M}^{(1)}}_t], \quad t\in
[0,T].$$ Therefore $$\begin{aligned}
\label{proval} \hat{V}_T&=& \hat{V}_0 e^{\int_0^T r_sds}+\int_0^T
e^{\int_t^T r_sds}\sigma(t,Y_t)\hat{\eta}_t S_t[a^{(1)}_t
d{\hat{W}^{(1)}}_t +a^{(3)}_t d{\hat{M}^{(1)}}_t].\end{aligned}$$
Chaotic calculus
----------------
The chaotic calculus allows to obtain the strategy that minimizes the variance, that is by using the Clark-Ocone formula.\
Let us denote by $\hat{X}$, the $4$-dimensional martingale coming from the $2$-dimensional Brownian motion and compensated Poisson process introduced in section $2$, i.e $$({\hat{X}}^{(1)}_t,{\hat{X}}^{(2)}_t,{\hat{X}}^{(3)}_t,{\hat{X}}^{(4)}_t)=
({\hat{W}}^{(1)}_t,{\hat{W}}^{(2)}_t,{\hat{M}}^{(1)}_t,{\hat{M}}^{(2)}_t),
\ \ \ t \in [0,T].$$ This martingale has the Chaotic Representation Property (CRP). The CRP for $\hat{X}$ states that any square-integrable functional ${\cal{F}}_T$-measurable, can be expanded into a series of multiple stochastic integrals -w.r.t $\hat{X}_t$- of deterministic functions. Using this expansion, we define the Malliavin operator, by acting on the multiple stochastic integrals. The Clark-Ocone formula is then deduced by technical ways, allowing to make appear the Malliavin operator in the expansion, and to write this last one as a simple stochastic integral w.r.t $\hat{X}$.\
We define the multiple stochastic integral and introduce the Malliavin operator and the Clark-Ocone formula in the multidimensional Brownian-Poisson case (more precisely in the 4-dimensional case, the following definitions and formulas can be extended for the $d-$dimensional case, $d>4$). For more details we refer to L[ø]{}kka (1999), Nualart (1995), Nualart and Vives(1990), [Ø]{}ksendal (1996) and Privault (1997 a,b). Let $(e_{1},e_{2},e_{3},e_{4})$ be the canonical base of $\real^{4}$. For $g_n \in L^2([0,T]^n)$ we define the $n$-th iterated stochastic integral of the function $f_n e_{i_1}\otimes
\ldots \otimes e_{i_n}$, with $1\leq i_1,\ldots,i_n\leq 4$, by $$\begin{aligned}
I_n(g_n e_{i_1}\otimes \ldots \otimes e_{i_n}): &=& n!\int_0^T
\int_0^{t_n}\ldots
\int_0^{t_2}g_n(t_1,\ldots,t_n)d{\hat{X}}^{(i_1)}_{t_1}\ldots
d{\hat{X}}^{(i_n)}_{t_n}.
\end{aligned}$$ The iterated stochastic integral of a symmetric function $f_n=(f_n^{(i_1,\ldots,i_n)})_{1\leq i_1,\ldots,i_n \leq 4}\in$ $L^2([0,T],\real^{4})^{\otimes n}$, where $f_n^{(i_1,\ldots,i_n)} \in L^2([0,T]^n)$, is $$\begin{aligned}
I_n(f_n):&=&\sum_{i_1,\ldots,i_n=1}^{4} I_n(f_n^{(i_1,\ldots,i_n)} e_{i_1}\otimes \ldots \otimes e_{i_n})\\
&=&n! \sum_{i_1,\ldots,i_n=1}^{4} \int_0^T \int_0^{t_n}
\ldots \int_0^{t_2} f_n^{(i_1,\ldots,i_n)}(t_1,\ldots,t_n)d{\hat{X}}^{(i_1)}_{t_1}\ldots
d{\hat{X}}^{(i_n)}_{t_n}.\end{aligned}$$ For $F\in L^2(\Omega)$, there exists a unique sequence $(f_n)_{n\in
\inte}$ of deterministic symmetric functions $f_n=(f_n^{(i_1,\ldots,i_n)})_{i_1,\ldots,i_n \in \{1,\ldots,4\}}\in
L^2([0,T],\real^{4})^{\circ n}$ such that $$\label{prc} F=\sum_{n=0}^{\infty} I_n(f_n).$$
Let $l\in \{1,\ldots,4\}$, we define the operator $D^{(l)}:
\Dom(D^{(l)})\subset L^2(\Omega)\rightarrow L^2(\Omega,[0,T])$ does correspond for $F \in \Dom(D^{(l)})$ ($F$ having the representation (\[prc\])), the process $(D^{(l)}_t F)_{t\in [0,T]}$ given by $$\begin{aligned}
\nonumber \lefteqn{D^{(l)}_t F:=\sum_{n=1}^{\infty} \sum_{h=1}^n
\sum_{i_1,\ldots,i_n=1}^{4}
1_{\{i_h=l\}}}\\
&&I_{n-1}(f_n^{(i_1,\ldots,i_n)}(t_1,\dots,t_{l-1},t,t_{l+1}\ldots,t_n)e_{i_1}\otimes
\ldots \otimes e_{i_{h-1}}
\otimes e_{i_{h+1}}\ldots \otimes e_{i_n})\\
&=&\sum_{n=1}^{\infty}nI_{n-1}(f_n^l(*,t)),\ \ \
d\hat{Q}\times dt-a.e.\end{aligned}$$ with $f_n^l=(f_n^{(i_1,\ldots,i_{n-1},l)}e_{i_1}\otimes
\ldots\otimes e_{i_{n-1}})_{1\leq i_1,\ldots,i_{n-1}\leq {4}}$.
The domain of $D^{(l)}$ is $$\begin{aligned}
\Dom(D^{(l)})&=&\left\{F=\sum_{n=0}^{\infty}\sum_{i_1,\ldots,i_n=1}^{4}
I_n(f_n^{(i_1,\ldots,i_n)}e_{i_1}\otimes \ldots \otimes e_{i_n}) \in
L^2(\Omega):\right.\\
&&\left.\sum_{i_1,\ldots,i_n=1}^{4} \sum_{n=0}^{\infty}n
n!\|f_n^{(i_1,\ldots,i_n)}\|^2_{L^2([0,T]^n)}< \infty\right\}.\end{aligned}$$ We will now give the probabilistic interpretations of $D^{(l)}$ in the Brownian motion and Poisson process cases. These interpretations will allow us to compute the requested strategy.
The Brownian operator
: For $1\leq l\leq 2$, the operator $D^{(l)}$ is, in fact, the Malliavin derivative in the direction of the one dimensional Brownian motion ${\hat{W}}^{(l)}$. So, we have for $1\leq l\leq 2$ and $F=f\left({\hat{W}}_{t_1}, \ldots,{\hat{W}}_{t_n}\right)\in
L^2(\Omega)$ where $(t_1,\ldots,t_n)\in [0,T]^n$ and $f(x^{11},x^{21},\ldots,x^{1n},x^{2n})\in {\cal
C}_b^{\infty}(\real^{2n})$ $$\begin{aligned}
D^{(l)}_t F &=&\sum_{k=1}^{k=n}\frac{\partial f}{\partial x^{lk}}
\left({\hat{W}}_{t_1}, \ldots,{\hat{W}}_{t_n}\right)1_{[0,t_k]}(t).\end{aligned}$$ To calculate the Mallaivin derivative for Itô integral, we will use the following proposition (see corollary 5.13 of [Ø]{}ksendal (1996).
\[derivint\] Let $(u_t)_{t\in [0,T]}$ be a ${\cal{F}}_t-$adapted process, such that $u_t \in \Dom(D^{(l)})$, we have $$D^{(l)}_t \int_0^T u_s d{\hat{W}^{(l)}}_s=\int_t^T (D^{(l)}_t u_s)d{\hat{W}^{(l)}}_s+ u_t,$$
The Poisson operator
: For $3\leq l\leq 4$, $D^{(l)}$ is the Malliavin operator[^2] in the direction of the Poisson process $N^{(l-2)}$. For $F\in
\Dom(D^{(l)})$ $$D^{(l)}_t F(\omega^{(1)},\ldots,\omega^{4})=\left\{
\begin{array}{ll}
F(\omega^{(1)},\omega^{(2)},\omega^{(3)}+1_{[t,\infty[},\omega^{(4)}
)-F(\omega^{(1)},\ldots,\omega^{(4)}), & l=3, \\
F(\omega^{(1)},\omega^{(2)},\omega^{(3)},\omega^{(4)}+1_{[t,\infty[}
)-F(\omega^{(1)},\ldots,\omega^{(4)}), & l=4.
\end{array}
\right.$$
The Clark-Ocone formula is given by the next proposition.
Consider a square-integrable functional $F$, ${\cal{F}}_T$-measurable, such that $F \in \bigcap_{l=1}^{4}
\Dom(D^{(l)})$. $F$ has the following predictable representation $$\begin{aligned}
F&=&E[F] + \sum_{l=1}^2 \int_0^T E[D^{(l)}_t F\mid {\mathcal
F}_t]d{\hat{W}}^{(l)}_t +\sum_{l=1}^2 \int_0^T E[D^{N^{(l)}}_t F\mid
{\mathcal F}_t]d{\hat{M}}^{(l)}_t.\end{aligned}$$
Now we apply the Clark-Ocone formula to determine the strategy minimizing the variance for our model considered in the Section \[the model\].
\[malliavinstra\] The strategy minimizing (\[vari\]) of the model of Section \[the model\] is given by $$\label{couvmall} {\hat{\eta}}_t =\frac{a^{(1)}_t
E[D^{\hat{W}^{(1)}}_t f(S_T)\mid {\mathcal
F}_t]+\lambda^{(1)}_t(1+{\hat{\beta}}^{(3)}_t)a^{(3)}_t
E[D^{N^{(1)}}_t f(S_T)\mid {\mathcal F}_t]} {((a^{(1)}_t)^2
+\lambda^{(1)}_t(1+{\hat{\beta}}^{(3)}_t)(a^{(3)}_t)^2)e^{\int_t^T
r_sds}\sigma(t,Y_t) S_t}.$$
First we approach the function $x\mapsto
f(x)(=(x-K)^+\mbox{or}=(K-x)^+)$ by polynomials on compact intervals and proceed as in [Ø]{}ksendal (1996)pp. 5-13. By dominated convergence, $(f(S_T)\in \bigcap_{l=1}^{4}
\Dom(D^{(l)})$. Applying the Clark-Ocone formula to $f(S_T)$ and using (\[proval\]) we obtain $$\begin{aligned}
\lefteqn{ E_{\hat{Q}}\left[f(S_T) - \hat{V}_T )^2 \right] =}
\\
&&E_{\hat{Q}}\left[\left(\int_0^T \left(E[D^{\hat{W}^{(1)}}_t
f(S_T)\mid {\mathcal F}_t]-e^{\int_t^T r_sds}\sigma(t,Y_t)
\hat{\eta}_t S_t a^{(1)}_t \right)d{\hat{W}^{(1)}}_t\right)^2\right.\\
&&+\left.\left(\int_0^T E_{\hat{Q}}[D^{\hat{W}^{(2)}}_t f(S_T)\mid
{\mathcal F}_t]d{\hat{W}^{(2)}}_t\right)^2 +\left(\int_0^T
E_{\hat{Q}}[D^{{N}^{(2)}}_t f(S_T)\mid
{\mathcal F}_t]d{\hat{M}^{(2)}}_t\right)^2 \right.\\
&&+\left.\left( \int_0^T \left(E_{\hat{Q}}[D^{N^{(2)}}_t f(S_T)\mid
{\mathcal F}_t]-e^{\int_t^T r_sds}\sigma(t,Y_t)
\hat{\eta}_t S_t a^{(3)}_t \right)d{\hat M}^1(t)
\right )^2 \right]\\
&=&E_{\hat{Q}}\left[\int_0^T h_2({\hat{\eta}}_t)dt\right],
\end{aligned}$$ where $$\begin{aligned}
h_2(x)&=&(E_{\hat{Q}}[D^{\hat{W}^{(2)}}_t f(S_T)\mid {\mathcal
F}_t])^2 +\lambda^{(2)}_t(1+{\hat{\beta}}^{(4)}_t)(E[D^{N^{(2)}}_t
f(S_T)\mid {\mathcal F}_t])^2 \\
&&+\left(E_{\hat{Q}}[D^{\hat{W}^{(1)}}_t f(S_T)\mid {\mathcal
F}_t]-e^{\int_t^T r_sds}\sigma(t,Y_t)
xS_t a^{(1)}_t \right)^2\\
&&+\lambda^{(1)}_t(1+{\hat{\beta}}^{(3)}_t)\left(E_{\hat{Q}}[D^{N^{(1)}}_t f(S_T)\mid
{\mathcal F}_t]-e^{\int_t^T r_sds}\sigma(t,Y_t)
x S_t a^{(3)}_t \right)^2 .
\end{aligned}$$ It is easily verified that $h_2$ is convex, hence the minimum is the solution of $h_2^{'}(x)=0.$ Therefore the strategy minimizing the variance is given by (\[couvmall\]).
Explicit formulae, Equivalent Martingale Measure minimizing the entropy
=======================================================================
The process $\left(\frac{\mu_t-r_t}{a^{(1)}_t \sigma(t,Y_t)},0,0,0
\right)$ belongs to $\Gamma^{\cal H}$ and it defines a $P$-EMM, so ${\cal H}\neq \emptyset$. Thus ${\cal H}$ contains an infinity of $P$-EMM. We choose the one that minimizes the relative entropy. Let $Q^{\beta} \in \cal{H}$. Denoting by $I(Q^{\beta},P)$ the relative entropy of $Q^{\beta}$ w.r.t $P$, we have $$I(Q^{\beta},P)=E_P\left[\frac{dQ^{\beta}}{dP}\ln\frac{dQ^{\beta}}{dP}\right].$$ Our aim is to minimize $I(P,Q^{\beta})$ under $\cal{H}$. We have $$I(P,Q^{\beta})=E_{Q^{\beta}}\left[\frac{dP}{dQ^{\beta}}\ln\frac{dP}{dQ^{\beta}}\right]$$ Therefore the problem is to find a $\hat{\beta}$ which satisfy $$\label{min} I(P,Q^{\hat{\beta}}) = \min_{\beta \in \Gamma^{\cal
H}}-E_P\left[\ln\frac{dQ^{\beta}}{dP}\right].$$
\[lem\] The minimization problem (\[min\]) is equivalent to the minimization of $$\begin{aligned}
&&(\mu_t -r_t+\lambda^{(1)}_t a^{(3)}_t \beta^{(3)}_t \sigma
(t,Y_t))^2
-2\sigma^2 (t,Y_t) (a^{(1)}_t)^2 \lambda^{(1)}_t
\left[\ln(1+\beta^{(3)}_t)-\beta^{(3)}_t \right]\\
&&-2\sigma^2 (t,Y_t) (a^{(1)}_t)^2 \lambda^{(2)}_t
\left[\ln(1+\beta^{(4)}_t)-\beta^{(4)}_t\right],\end{aligned}$$ under all $\beta=\left(\frac{\mu_t -r_t+\lambda^{(1)}_t a^{(3)}_t
\sigma(t,Y_t)\beta^{(3)}_t }{\sigma(t,Y_t)a^{(1)}_t},0,\beta^{(3)},
\beta^{(4)}\right) \in \Gamma^{\cal H}.$
Let $Q^{\beta} \in \cal{H}$. By (\[rhob\]) $$\begin{aligned}
I(P,Q^{\beta})&=&-E_P\left[\ln\frac{dQ^{\beta}}{dP}\right]\\
&=&E_P\left[\int_0^T \sum_{i=1}^2 \frac{1}{2}(\beta^{(i)}_t)^2
-\lambda^{(i)}_t \left[\ln(1+\beta^{(i+2)}_t)-\beta^{(i+2)}_t\right]dt\right]\\
&=&E_P\left[\int_0^T \frac{G(\beta_t)}{2\sigma^2
(t,Y_t)(a^{(1)}_t)^2}dt\right],\end{aligned}$$ where $\beta \in \Gamma^{\cal H}$, and $G$ is the function defined by $$\begin{aligned}
G(\beta_t)&=& 2\sigma^2
(t,Y_t)(a^{(1)}_t)^2\left(\frac{1}{2}(\beta^{(1)}_t)^2
+\frac{1}{2}(\beta^{(2)}_t)^2 -\lambda^{(1)}_t
\left[\ln(1+\beta^{(3)}_t)-\beta^{(3)}_t
\right]\right.\\
&&\left.-\lambda^{(2)}_t
\left[\ln(1+\beta^{(4)}_t)-\beta^{(4)}_t\right]\right),\ \ \ t\in
[0,T].\end{aligned}$$ For a fixed $t$, we have by (\[rel\]), $$\begin{aligned}
G(\beta_t)&=&(\mu_t -r_t+\lambda^{(1)}_t a^{(3)}_t
\sigma(t,Y_t)\beta_t^{(3)} )^2
+\sigma^2 (t,Y_t) (a^{(1)}_t)^2\left((\beta^{(2)}_t)^2 \right.\\
&&\left.-2 \lambda^{(1)}_t \left[\ln(1+\beta^{(3)}_t)-\beta^{(3)}_t
\right]-2 \lambda^{(2)}_t
\left[\ln(1+\beta^{(4)}_t)-\beta^{(4)}_t\right]\right),\ \ \ t\in
[0,T].\end{aligned}$$ Now the lemma is deduced from the fact that $\beta^{(2)}_t$ appears only in the term $\sigma^2 (t,Y_t) (a^{(1)}_t)^2(\beta^{(2)}_t)^2$ which is always positive : so $\beta^{(2)}_t$ must be equal to zero.
The following proposition gives the solution to the minimization \[min\].
\[mmeqmva\] Consider $({\hat{\beta}}^{(1)}_t,{\hat{\beta}}^{(2)}_t,
{\hat{\beta}}^{(3)}_t, {\hat{\beta}}^{(4)}_t)_{t \in[0,T]} \in
\Gamma^{\cal H}$, with $${\hat{\beta}}^{(2)}_t={\hat{\beta}}^{(4)}_t=0,\ \ \
{\hat{\beta}}^{(1)}_t=
\begin{cases}
\frac{r_t -\mu_t -\lambda^{(1)}_t a^{(3)}_t \sigma(t,Y_t)
{\hat{\beta}}^{(3)}_t}{\sigma(t,Y_t) a^{(1)}_t} &\ \ \ \mbox{if} \ \ \ a^{(1)}\neq 0,\\
0 &\ \ \ \mbox{if} \ \ \ a^{(1)}=0,
\end{cases}$$ and let ${\hat{\beta}}^{(3)}_t$ be the unique solution of the equation $$\label{entropie}\lambda^{(1)}_t \sigma(t,Y_t)(a^{(3)}_t)^2
x+(a^{(1)}_t)^2 \sigma(t,Y_t)\left(\frac{x}{1+x}\right)-a^{(3)}_t
(r_t-\mu_t)=0.$$Then, the $P$-EMM $\hat{Q}$ defined by its Radon-Nikodym density $$\prod_{i=1}^2 {\cal E}({\hat{\beta}}^{(i)}W^{(i)})_T {\cal
E}({\hat{\beta}}^{(i+2)}M^{(i)})_T,$$ is the $P$-EMM minimizing $I(P,Q^{\beta})$.
By Lemma \[lem\], we have to minimize the function $F :
]-1,\infty[\times]-1,\infty[\longrightarrow \real$ defined by $$\begin{aligned}
F(x,y)&=&(\mu_t
-r_t+\lambda^{(1)}_t a^{(3)}_t \sigma(t,Y_t)x)^2-2\sigma^2 (t,Y_t)
(a^{(1)}_t)^2 \left(\lambda^{(1)}_t \left[\ln(1+x)-x\right]
+\lambda^{(2)}_t \left[\ln(1+y)-y\right]\right),\end{aligned}$$ for fixed $t$ in $[0,T]$. Let $\hat{x}$ be the solution of (\[entropie\]), it is unique since the function $$x \longrightarrow 2(\lambda^{(1)}_t)^2 \sigma^2(t,Y_t)(a^{(3)}_t)^2
x+2(a^{(1)}_t)^2 \lambda^{(1)}_t
\sigma^2(t,Y_t)\frac{x}{1+x}+2\lambda^{(1)}_t \sigma(t,Y_t)
a^{(3)}_t (\mu_t -r_t),$$ is strictly increasing from $]-1,\infty[$ to $\real$. Let $F^{'}_{x}$ and $F^{'}_{y}$ denote the first order partial derivatives of $F$. One can check that $(\hat{x},0)$ is the only point which satisfy $F^{'}_{x} (x,y) = F^{'}_{y} (x,y) = 0.$ Moreover we have $$(F^{''}_{xy}(\hat{x},0))^2
-F^{''}_{x^2}(\hat{x},0) F^{''}_{y^2}(\hat{x},0)<0\ \ \ \mbox{and}\
\ \ F^{''}_{x^2}(\hat{x},0)>0.$$ Therefore $F$ has a strict local minimum at $(\hat{x},0)$. This minimum is global since the limits of $F$ on borders go to infinity.
To obtain explicit formulas for the strategy computed in Proposition \[malliavinstra\], we will separate the two cases : Brownian motion and Poisson process. In the two following subsections we compute explicitly the strategy for a European call option in the Brownian motion and the Poisson process cases respectively. The payoff of the option is then given by $f(S_T)=(S_T-K)^+$, where $K$ denotes the Strike.
Brownian case
-------------
Assume that $a^{(1)}_t=a^{(2)}_t=1$ and $a^{(3)}_t=a^{(4)}_t=0$, so $(S_t)_{0\leq t \leq T}$ depends on Brownian information only. Under ${\hat Q}$, $(S_t)_{0\leq t \leq
T}$ is given by $$S_t=x\exp\left(\int_0^t \left(r_s - \frac{\sigma^2
(s,Y_s)}{2}\right) ds + \int_0^t \sigma (s,Y_s)d{\hat{W}^{(1)}}_s
\right),$$ with $$Y_t=y+\int_0^t \left(\mu^Y_s +\sigma^{(1)}_s
\frac{r_s-\mu_s}{\sigma(s,Y_s)}\right)ds+
\int_0^t \sigma^{(1)}_s d{\hat{W}^{(1)}}_t +\int_0^t \sigma^{(2)}_s
dW^{(2)}_s.$$ In the following proposition We compute the Malliavin derivative of the payoff $(S_T-K)^+$. We can replace the result in the formula (\[couvmall\]), and obtain an explicit formula for the strategy.
We have $$\begin{aligned}
\nonumber D^{\hat{W}^{(1)}}_t(S_T-K)^+&=&1_{\{S_T > K \}}S_T
\left(\sigma(t,Y_t)+\int_t^T \frac{\partial \sigma}{\partial
y}(s,Y_s)D^{\hat{W}^{(1)}}_s Y_t d\hat{W}^{(1)}_s \right.\\
\label{derivs}
&&-\left. \int_t^T
\sigma(s,Y_s)\frac{\partial \sigma}{\partial y}(s,Y_s)D^{\hat{W}^{(1)}}_t
Y_s ds\right)\end{aligned}$$ where $$\label{derivy}
D^{\hat{W}^{(1)}}_t Y_s=\sigma^{(1)}_t
\exp\left(-\int_t^s \sigma^{(1)}_u
\frac{r_u-\mu_u}{\sigma^2(u,Y_u)}du\right)\ \ \ s\in[t,T].$$
By the chain role of $D^{\hat{W}^{(1)}}_t$ and thanks to Proposition \[derivint\] we obtain $$\begin{aligned}
\lefteqn{ D^{\hat{W}^{(1)}}_t(S_T-K)^+ =}\\
&&1_{\{S_T > K \}}S_T \left(D^{\hat{W}^{(1)}}_t \int_0^T \left(r_s -
\frac{\sigma^2 (s,Y_s)}{2}\right) ds +D^{\hat{W}^{(1)}}_t
\int_0^T \sigma(s,Y_s)d{\hat{W}^{(1)}}_s\right)\\
&=&1_{\{S_T > K \}}S_T \left(-\int_t^T D^{\hat{W}^{(1)}}_t
\frac{\sigma^2 (s,Y_s)}{2}ds +
\int_t^T D^{\hat{W}^{(1)}}_t \sigma(s,Y_s)d{\hat{W}^{(1)}}_s
+\sigma(t,Y_t)\right),\end{aligned}$$ which gives (\[derivs\]). Concerning the other derivative, we have for $0\leq t\leq s\leq T$ $$\begin{aligned}
D^{\hat{W}^{(1)}}_t Y_s &=& \int_t^s D^{\hat{W}^{(1)}}_t
\left(\mu^Y_u +\sigma^{(1)}_u
\frac{r_u-\mu_u}{\sigma(u,Y_u)}\right)du+
\sigma^{(1)}_t\\
&=&\sigma^{(1)}_t -\int_t^s \sigma^{(1)}_u
\frac{r_u-\mu_u}{\sigma^2(u,Y_u)}D^{\hat{W}^{(1)}}_t Y_u du,\end{aligned}$$ So for $t$ fixed in $[0,T]$, the Malliavin derivative of $Y_s$ for $s\in [t,T]$ : $(D^{\hat{W}^{(1)}}_t Y_s)_{s\in [t,T]}$, satisfies a stochastic differential equation, its solution is precisely (\[derivy\]).
The Poisson case
----------------
Similarly, like in the Brownian case, we aim to compute the quantity $D^{\hat{M}^{(1)}}_t (S_T-K)^+$ and replace the result in the expression of the strategy, to obtain an explicit formula for the Poisson case. Let us suppose that we work in the Poisson space with a $2$-dimensional Poisson process. The underlying asset price $(S_t)_{0\leq t \leq T}$ depends on Poisson process only. Hence we assume that $a^{(3)}_t=a^{(4)}_t=1$ and $a^{(1)}_t=a^{(2)}_t=0$. Under ${\hat Q}$, the dynamic of $(S_t)_{0\leq t \leq T}$ is given by $$S_{t} = x\exp \left(\int_0^t
\left(\mu_{s}+
\frac{r_s -\mu_s}{\sigma(s,Y_s)}\ln(1+\sigma(s,Y_s))\right)ds
+\int_0^t \ln(1+\sigma(s,Y_s))d\hat{M}^{(1)}_s \right),$$ for $t\in[0,T]$. The process $(Y_t)_{t\in [0,T]}$ under ${\hat Q}$, have the representation $$Y_t=y+\int_0^t \left(\mu^Y_s +\sigma^{(1)}_s
\frac{r_s-\mu_s}{\sigma(s,Y_s)}\right)ds+
\int_0^t \sigma^{(1)}_s d{\hat{M}^{(1)}}_t +\int_0^t \sigma^{(2)}_s
dM^{(2)}_s.$$
$$\begin{aligned}
\nonumber \lefteqn{D^{\hat{M}^{(1)}}_t (S_T-K)^+ =}\\
\nonumber &&-(S_T-K)^+ + \left(\exp \left\{\int_t^T \left[\mu_{s}+
\frac{r_s -\mu_s}{\sigma(s,Y_s
+\sigma^{(1)}_t )}\ln(1+\sigma(s,Y_s +\sigma^{(1)}_t))\right]ds\right.\right.\\
&&+\left.\left.\int_t^T \ln(1+\sigma(s,Y_s
+\sigma^{(1)}_t))d\hat{M}^{(1)}_s \right\}\times S_t (1+\sigma(t,Y_t
+\sigma^{(1)}_t))-K\right)^+\end{aligned}$$
Using the probabilistic interpretation of $D^{\hat{M}^{(1)}}_t$ given below, we obtain $$D^{\hat{M}^{(1)}}_t (S_T-K)^+=(S_T(\omega +1_{[t,T]})-K)^{+}
-(S_T(\omega)-K)^+ .$$ But $$\begin{aligned}
\lefteqn{S_T(\omega +1_{[t,T]})=x\exp \left(\int_0^t \left[\mu_{s}+
\frac{r_s -\mu_s}{\sigma(s,Y_s(\omega
+1_{[t,T]}))}\ln(1+\sigma(s,Y_s(\omega+1_{[t,T]})))\right]ds\right.}\\
&&\left.+\int_0^t
\ln(1+\sigma(s,Y_s(\omega+1_{[t,T]})))d\hat{M}^{(1)}_s\right)\\
&&\times\exp \left(\int_t^T \left[\mu_{s}+ \frac{r_s
-\mu_s}{\sigma(s,Y_s(\omega
+1_{[t,T]}))}\ln(1+\sigma(s,Y_s(\omega+1_{[t,T]})))\right]ds\right.\\
&&\left.+\int_t^T
\ln(1+\sigma(s,Y_s(\omega+1_{[t,T]})))d\hat{M}^{(1)}_s
\right)\times(1+\sigma(t,Y_t(\omega+1_{[t,T]}))),\end{aligned}$$ and $$\begin{aligned}
Y_t(\omega+1_{[t,T]})&=&Y_t +\sigma^{(1)}_t,\ \ \ t\in [0,T],\\
Y_s(\omega+1_{[t,T]})&=&\begin{cases}
Y_s&\ \ \ \mbox{if}\ \ \ s\in [0,t[,\\
Y_s +\sigma^{(1)}_t&\ \ \ \mbox{if}\ \ \ s\in [t,T].
\end{cases}\end{aligned}$$ The proof is established.
[stoch]{}
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Jacod, J. (1979): . volume [714]{} of [*Lecture Notes in Mathematics*]{}. Springer Verlag.
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Nualart, D. and Vives, J. (1990): Anticipative calculus for the [P]{}oisson process based on the [F]{}ock space. In J. Az[é]{}ma, P.A. Meyer, and M. Yor, editors, [*S[é]{}minaire de [P]{}robabilit[é]{}s XXIV*]{}, volume [1426]{} of [*Lecture Notes in Mathematics*]{}, 154–165. Springer Verlag.
ksendal, B. (1996): Working paper no. 3, Institute of Finance and Management Science, Norwegian School of Economics and Business Administration.
Privault, N. (1997a): An extension of stochastic calculus to certain non-Markovian processes. Prépublication de l’université d’Evry, 49.
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Protter, Ph. (1990): . Springer-Verlag, Berlin.
[cc]{}
[cc]{}
[^1]: ykhatib@univ-lemans.fr
[^2]: Notice that, unlike the Brownian case, the Malliavin operator in the Poisson space does not a derivative.
|
---
abstract: 'We construct an irreducible holomorphic connection with ${\rm SL}(2,{\mathbb{R}})$–monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in [@CDHL].'
address:
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India'
- 'Université Côte d’Azur, CNRS, LJAD, France'
- 'Institute of Differential Geometry, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover'
author:
- Indranil Biswas
- Sorin Dumitrescu
- Sebastian Heller
title: |
Irreducible flat ${\rm
SL}(2,{\mathbb{R}})$-connections on the trivial holomorphic bundle
---
Introduction {#sec:intro}
============
Take a compact connected oriented topological surface $S$ of genus $g$, with $g \geq 2$. There is an equivalence between the flat $\rm{SL}(2, {\mathbb{C}})$–connections over $S$ and the conjugacy classes of group homomorphisms from the fundamental group of $S$ into $\rm{SL}(2, {\mathbb{C}})$ (two such homomorphisms are conjugate if they differ by an inner automorphism of $\rm{SL}(2, {\mathbb{C}})$). This equivalence sends a flat connection to its monodromy representation. When $S$ is equipped with a complex structure, a flat $\rm{SL}(2, {\mathbb{C}})$–connection on $S$ produces a holomorphic vector bundle of rank two and trivial determinant on the Riemann surface defined by the complex structure on $S$; this is because constant transition functions for a bundle are holomorphic. In fact, since a holomorphic connection on a compact Riemann surface $\Sigma$ is automatically flat, there is a natural bijection between the following two:
1. pairs of the form $(E,\, D)$, where $E$ is a holomorphic vector bundle of rank two on $\Sigma$ with $\bigwedge^2 E$ holomorphically trivial, and $D$ is a holomorphic connection on $E$ that induces the trivial connection on $\bigwedge^2 E$;
2. flat $\rm{SL}(2, {\mathbb{C}})$–connections on $\Sigma$.
This bijection is a special case of the Riemann–Hilbert correspondence (see, for instance, [@De; @Ka]).
Consider the flat $\rm{SL}(2, {\mathbb{C}})$–connections on a compact Riemann surface $\Sigma$ satisfying the condition that the corresponding holomorphic vector bundle of rank two on $\Sigma$ is holomorphically trivial; they are known as differential ${\mathfrak
s}{\mathfrak l}(2, {\mathbb{C}})$–systems on $\Sigma$ (see [@CDHL]), where ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$ is the Lie algebra of $\rm{SL}(2, {\mathbb{C}})$. In view of the above Riemann–Hilbert correspondence, differential ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$–systems on $\Sigma$ are parametrized by the vector space ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})\otimes H^0(\Sigma ,\, K_{\Sigma})$, where $K_{\Sigma}$ is the holomorphic cotangent bundle of $\Sigma$. The zero element of the vector space ${\mathfrak s}{\mathfrak
l}(2, {\mathbb{C}})\otimes H^0(\Sigma ,\, K_{\Sigma})$ corresponds to the trivial $\rm{SL}(2,
{\mathbb{C}})$–connection on $\Sigma$. A differential ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$–system is called irreducible if the corresponding monodromy representation of the fundamental group of $\Sigma$ is irreducible. We shall now describe a context where irreducible differential ${\mathfrak s}{\mathfrak
l}(2, {\mathbb{C}})$–systems appear.
For any cocompact lattice $\Gamma\, \subset\, {\rm SL}(2,{\mathbb C})$, the compact complex threefold ${\rm
SL}(2,{\mathbb C}) / \Gamma$ does not admit any compact complex hypersurface [@HM p. 239, Theorem 2], in particular, there is no nonconstant meromorphic function on ${\rm SL}(2,{\mathbb C}) / \Gamma$. It is easy to see that ${\rm SL}(2,{\mathbb C}) / \Gamma$ does not contain a ${\mathbb C}{\mathbb P}^1$. It is known that some elliptic curves do exist in those manifolds. A question of Margulis asks whether ${\rm SL}(2,{\mathbb
C})/\Gamma$ can contain a compact Riemann surface of genus bigger than one. Ghys has the following reformulation of Margulis’ question: Is there a pair $(\Sigma,\, D)$, where $D$ is a differential ${\mathfrak
s}{\mathfrak l}(2, {\mathbb{C}})$–system on a compact Riemann surface $\Sigma$ of genus at least two, such that the image of the monodromy homomorphism for $D$ $$\pi_1(\Sigma)\, \longrightarrow\, \rm{SL}(2, {\mathbb{C}})$$ is a conjugate of $\Gamma$ ? Existence of such a pair $(\Sigma,\, D)$ is equivalent to the existence of an embedding of $\Sigma$ in ${\rm SL}(2,{\mathbb C}) / \Gamma$.
Inspired by Ghys’ strategy, the authors of [@CDHL] study the Riemann–Hilbert mapping for the irreducible differential ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$–systems (see also [@BD]). Although some (local) results were obtained in [@CDHL] and [@BD], the question of Ghys is still open. In this direction, it was asked in [@CDHL] (p. 161) whether discrete or real subgroups of $\rm{SL}(2, {\mathbb{C}})$ can be realized as the monodromy of some irreducible differential ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$–system on some compact Riemann surface. Note that if the flat connection on a compact Riemann surface $\Sigma$ corresponding to a homomorphism $\pi_1(\Sigma)\, \longrightarrow\,
{\rm SL}(2,{\mathbb C})$ with finite image is irreducible, then the underlying holomorphic vector bundle is stable [@NSe], in particular, it is not holomorphically trivial.
Our main result (Theorem \[Main\]) is the construction of a pair $(\Sigma, \, D)$, where $\Sigma$ is a compact Riemann surface of genus bigger than one and $D$ is an irreducible differential ${\mathfrak s}{\mathfrak l}(2, {\mathbb{C}})$–system on $\Sigma$, such that the image of the monodromy representation for $D$ is contained in $\operatorname{SL}(2,{\mathbb{R}})$.
Let us mention that the related question of characterizing rank two holomorphic vector bundles $\mathcal L$ over a compact Riemann surface such that for some holomorphic connection on $\mathcal L$ the associated monodromy is real was raised in [@Ka p. 556] attributing it to Bers.
The Betti moduli space of a 1-punctured torus {#ADHS1t}
=============================================
For $\tau\, \in\, \mathbb C$ with ${\rm Im}\, \tau\, >\, 0$, let $\Gamma\,=\, {\mathbb Z}+\tau{\mathbb Z}\,\subset\,
\mathbb C$ be the corresponding lattice. Set $T^2\,:=\,{\mathbb{C}}/\Gamma$, and fix the point $o\,=\,[0]\,\in\, T^2.$ We shall always consider $T^2$ as a Riemann surface, and for simplicity we restrict to the case of $$\tau\,=\,\sqrt{-1}\, .$$
For a fixed $\rho\,\in \, [0,\, \tfrac{1}{2}[$, we are interested in the Betti moduli space $\mathcal M^\rho_{1,1}$ parametrizing flat $\operatorname{SL}(2,{\mathbb{C}})$–connections on the complement $T^2\setminus\{o\}$ whose local monodromy around $o$ lies in the conjugacy class of $$\label{locmon}{{\left(\begin{matrix} e^{2\pi \sqrt{-1} \rho} &0 \\ 0& e^{-2\pi\sqrt{-1} \rho}\end{matrix}\right)}}\,\in\,
{\rm SL}(2,{\mathbb C})\, .$$ This Betti moduli space $\mathcal M^\rho_{1,1}$ does not depend on the complex structure of $T^2$. When $\rho\,=\,0$, it is the moduli space of flat $\operatorname{SL}(2,{\mathbb{C}})$–connections on $T^2$; in that case $\mathcal M^\rho_{1,1}$ is a singular affine variety. However, for every $0\,<\,\rho\,<\,\tfrac{1}{2}$, the space $\mathcal M^\rho_{1,1}$ is a nonsingular affine variety. We shall recall an explicit description of this affine variety.
Let $x,\, y,\, z$ be the algebraic functions on $\mathcal M^\rho_{1,1}$ defined as follows: for any homomorphism $$h\,\colon\, \pi_1(T^2\setminus\{o\},\, q)\,\longrightarrow\, \operatorname{SL}(2,{\mathbb{C}})$$ representing $[h]\,\in \, {\mathcal M}^\rho_{1,1}$, $$x([h]) \,=\, \operatorname{tr}(h(\alpha)),\ y([h]) \,=\, \operatorname{tr}(h(\beta)),\ z([h])\,=\, \operatorname{tr}(h(\beta\alpha)),\,$$ where $\alpha,\,\beta$ are the standard generators of $\pi_1(T^2\setminus\{o\},\,q)$ (see Figure \[figure1\]).
Then the variety ${\mathcal M}^\rho_{1,1}$ is defined by the equation $$\label{M11eq}
{\mathcal M}^\rho_{1,1}\,=\,\{(x,y,z)\,\in\,{\mathbb{C}}^3\,\mid \, x^2+y^2+z^2-xyz-2-2\cos(2\pi \rho)\}\, ;$$ the details can be found in [@Gol], [@Magn].
\[irr\] Take any $\rho\,\in\,]0,\,\tfrac{1}{2}[$, and consider a representation $$h\,\colon\, \pi_1(T^2\setminus\{o\},\, q)\,\longrightarrow\, \operatorname{SL}(2,{\mathbb{C}})\, ,$$ with $[h]\,\in\, {\mathcal M}^\rho_{1,1}$. Then, the representation of the free group $F(s,t)$, with generators $s$ and $t$, defined by $$s\,\longmapsto\, X\,:=\,h(\alpha)h(\alpha) \ \ \text{ and } \ \ t\,\longmapsto\,
Y\,:=\, h(\beta)h(\beta)$$ is reducible if and only if $$x([h])y([h])\, =\, 0\, ,$$ where $x,\,y$ are the functions in .
It is known (see [@Gol]) that, up to conjugation, $$\label{repxy}
h(\alpha)\,=\,\begin{pmatrix} x([h])&1\\-1&0\end{pmatrix}, \ \ h(\beta)
\,=\, \begin{pmatrix} 0&-\zeta\\ \zeta^{-1}& y([h])\end{pmatrix}\, ,$$ where $$\label{zeta1}
\zeta+\zeta^{-1}\,=\,z([h])\, .$$ Note that the two solutions of $\zeta$ satisfying actually give conjugate (= equivalent) representations. A representation generated by two $\operatorname{SL}(2,{\mathbb{C}})$ matrices $A,\,B$ is reducible if and only if $$AB-BA$$ has a non-trivial kernel. Note that $$\text{Det}(XY-YX)\,=$$ $$-x([h])^2y([h])^2\frac{1+\zeta^4-\zeta x([h]) y([h])-
\zeta^3 x([h])y([h])+ \zeta^2(-2+x([h])^2+y([h])^2)}{\zeta^2}\, .$$ On the other hand, we have $$2 \cos(2\pi\rho)\,=\,\text{tr}(h(\beta)^{-1}h(\alpha)^{-1}h(\beta)h(\alpha))$$ $$=\,\zeta^{-2}+\zeta^2+x([h])^2-x([h])y([h])\zeta^{-1}-x([h])y([h])\zeta+y([h])^2\, .$$ Therefore, it follows that $$\text{Det}(XY-YX)\,=\,2x([h])^2y([h])^2(1-\cos[2\pi \rho])\, ,$$ and the proof of the lemma is complete.
Parabolic bundles and holomorphic connections
=============================================
Parabolic bundle {#sec2.1}
----------------
We briefly recall the notion of a parabolic structure, mainly for the purpose of fixing the notation. We are only concerned with the $\operatorname{SL}(2,{\mathbb{C}})$–case, so our notation differs from the standard references, e.g., [@MSe; @Biq; @B]. Instead, we follow the notation of [@Pir] (be aware that Pirola uses a scaling factor 2 of the parabolic weights); see also [@HeHe] for this notation.
Let $V\,\longrightarrow\,\Sigma$ be a holomorphic vector bundle of rank two with trivial determinant bundle over a compact Riemann surface $\Sigma$. Let $p_1,\, \cdots ,\, p_n\,\in\,\Sigma$ be pairwise distinct points, and set the divisor $$D\,=\,p_1+\ldots +p_n\, .$$ For every $k\,\in\,\{1,\,\cdots ,\, n\}$, let $$L_k\,\subset\, V_{p_k}$$ be a line in the fiber of $V$ at $p_k,$ and also take $$\rho_k\,\in\, ]0,\, \tfrac{1}{2}[\, .$$
\[def:par\] A [*parabolic structure*]{} on $V$ is given by the data $${\mathcal P}\,:=\, (D,\, \{L_1,\,\cdots ,\, L_n\},\, \{\rho_1,\,\cdots ,\, \rho_k\})\, ;$$ we call $\{L_k\}_{k=1}^n$ the quasiparabolic structure, and $\rho_k$ the parabolic weights.
A parabolic bundle over $\Sigma$ is given by a rank two holomorphic vector bundle $V$, with $\bigwedge^2 V\,=\, {\mathcal O}_\Sigma$, equipped with a parabolic structure $\mathcal P$.
It should be emphasized that Definition \[def:par\] is very specific to the case of $\operatorname{SL}(2,{\mathbb{C}})$–bundles. The parabolic degree of a holomorphic line subbundle $$F\,\subset\, V$$ is defined to be $$\text{par-deg}(F)\,=\, {\rm degree}(F)+\sum_{k=1}^n \rho^F_k\, ,$$ where $\rho^F_k\,= \, \rho_k$ if $F_{p_k}\,=\,L_k$ and $$\rho^F_k\,=\,
-\rho_k$$ if $F_{p_k}\,\neq\, L_k$.
A parabolic bundle $(V,\,\mathcal P)$ is called [*stable*]{} if and only $$\text{par-deg}(F)\, <\, 0$$ for every holomorphic line subbundle $F\,\subset \, V$.
As before, ${\mathcal P}\,=\,(D\,=\, p_1+\ldots +p_n,\, \{L_1,\,\cdots ,\,L_n\},
\,\{\rho_1,\,\cdots ,\, \rho_k\})$ is a parabolic structure on a rank two bundle $V$ of trivial determinant.
A strongly parabolic Higgs field on $(V,\,{\mathcal P})$ is a holomorphic section $$\Theta\, \in\, H^0(\Sigma,\, \text{End}(V)\otimes K_\Sigma\otimes {\mathcal O}_\Sigma(D))$$ such that
- $\text{trace}(\Theta) \,=\, 0$,
- $L_k\,\subset\,\text{kernel}(\Theta(p_k))$ for all $1\, \leq\, k\, \leq\, n$.
This implies that all the residues of a strongly parabolic Higgs field are nilpotent.
Deligne extension {#Delext}
-----------------
Using the complex structure of $T^2\,=\, {\mathbb C}/\Gamma$, an open subset of the moduli space ${\mathcal M}^\rho_{1,1}$ can be realized as a fibration over a moduli space of parabolic bundles. This map, which will be described in Section \[sec3.e\], is constructed using the Deligne extension (introduced in [@De]).
Any flat $\operatorname{SL}(2,{\mathbb{C}})$–connection $\nabla$ on a holomorphic vector bundle $E_0$ over $T^2\setminus\{o\}$, corresponding to a point in ${\mathcal M}^\rho_{1,1}$, locally, around $o\,\in\, T^2$, is holomorphic $\operatorname{SL}(2,{\mathbb{C}})$–gauge equivalent to the connection $$\label{local-normal-form-connection}
d+{{\left(\begin{matrix}\rho&0\\0&-\rho\end{matrix}\right)}}\frac{dw}{w}$$ on the trivial holomorphic bundle of rank two, where $w$ is a holomorphic coordinate function on $T^2$ defined around $o$ with $w(o)\,=\, 0$. Take such a neighborhood $U_o$ of $o$, and consider the trivial holomorphic bundle $U_o\times {\mathbb C}^2\, \longrightarrow\, U_o$ equipped with the connection in . Now glue the two holomorphic vector bundles, namely $U_o\times {\mathbb C}^2$ and $E_0$, over $U_o\setminus\{o\}$ such that the connection $\nabla\vert_{U_o\setminus\{o\}}$ is taken to the restriction of the connection in to $U_o\setminus\{o\}$. This gluing is holomorphic because it takes one holomorphic connection to another holomorphic connection. Consequently, this gluing produces a holomorphic vector bundle $$\label{dV}
V\,\longrightarrow\, T^2$$ of rank $2$ and degree $0$. Furthermore, the connection $\nabla$ on $E_0\, \longrightarrow\, T^2\setminus\{0\}$ extends to a logarithmic connection on $V$ over $T^2$; this logarithmic connection on $V$ will also be denoted by $\nabla$. (See [@De] for details.) It can be shown that
1. $\bigwedge^2 V\, =\, {\mathcal O}_{T^2}$, where $V$ is the vector bundle in , and
2. the logarithmic connection on $\bigwedge^2 V$ induced by the logarithmic connection $\nabla$ on $V$ coincides with the holomorphic connection on ${\mathcal O}_{T^2}$ induced by the de Rham differential $d$.
Indeed, the logarithmic connection on $U_o\times \bigwedge^2{\mathbb C}^2\,=\, U_o\times {\mathbb C}$ induced by the connection in coincides with the trivial connection on $U_o\times\mathbb C$ given by the de Rham differential $d$. On the other hand, the connection on $\bigwedge^2 E_0\,=\, {\mathcal O}_{T^2\setminus\{o\}}$ induced by the connection $\nabla$ on $E_0$ coincides with the trivial connection on ${\mathcal O}_{T^2\setminus\{o\}}$ given by the de Rham differential $d$. The above two statements follow from these.
From Atiyah’s classification of holomorphic vector bundles over any elliptic curve, [@At], we know the possible types of the vector bundle $V$ in .
\[cort\] The vector bundle $V$ in is one of the following three types:
1. $V\,=\, L\oplus L^{^*}$ with ${\rm degree}(L)\,=\,0$;
2. there is a spin bundle $S$ on $T^2$ (meaning a holomorphic line bundle of order two), such that $V$ is a nontrivial extension $$0\, \longrightarrow\, S\, \longrightarrow\, V\, \longrightarrow\, S \, \longrightarrow\,0$$ of $S$ by itself; and
3. $V\,=\, L\oplus L^{^*}$ with ${\rm degree}(L)\,>\,0$.
\[lem3.3\] Consider the vector bundle $V$ in for $\tfrac{1}{2}\,>\,\rho\,>\,0$. Then the last one of the three cases in Corollary \[cort\], as well as the special situation of the first case where $L\,=\, S$ is a spin bundle, cannot occur.
Assume that the third case occurs. Then consider the composition of homomorphisms $$L\, \hookrightarrow\, L\oplus L^{^*} \, \stackrel{\nabla}{\longrightarrow}\, (L\oplus L^{^*})\otimes
K_{T^2}\otimes {\mathcal O}_{T^2}(o) \, \longrightarrow\, L^{^*}\otimes
K_{T^2}\otimes {\mathcal O}_{T^2}(o)\,=\, L^{^*}\otimes {\mathcal O}_{T^2}(o)\, ,$$ where $K_{T^2}\,=\, {\mathcal O}_{T^2}$ is the holomorphic cotangent bundle of $T^2$ and the homomorphism $$(L\oplus L^{^*})\otimes
K_{T^2}\otimes {\mathcal O}_{T^2}(o) \, \longrightarrow\, L^{^*}\otimes
K_{T^2}\otimes {\mathcal O}_{T^2}(o)$$ is given by the projection $L\oplus L^{^*}\, \longrightarrow\, L^{^*}$. This composition of homomorphisms vanishes identically, because $$\text{degree}(L)\,>\, \text{degree}(L^*\otimes {\mathcal O}_{T^2}(o))
\,=\, 1- \text{degree}(L)$$ (recall that $\text{degree}(L)\, >\, 0$). Consequently, the logarithmic connection $\nabla$ on $V$ preserves the line subbundle $L$. For a holomorphic line bundle $\xi$ with a logarithmic connection singular over $o$, we have $$\label{rd}
{\rm degree}(\xi)+\text{Residue}_{\xi}(o)\,=\, 0$$ [@Oh p. 16, Theorem 3]. Now, the logarithmic connection on $L$ induced by $\nabla$ contradicts , because ${\rm degree}(L)+\text{Residue}_L(o)\,> \, 0$; note that $\text{Residue}_L(o)\,\in \,\{\rho,\, -\rho\}$. Therefore, we conclude that the third case can’t occur.
If $V\,=\, S\oplus S\,=\, S\otimes {\mathcal O}_{T^2}$, where $S$ is a holomorphic line bundle on $T^2$ of order two, then for a suitable direct summand $S$ of $V$, the residue of the logarithmic connection on it, constructed using the above composition, is $\rho$. This again contradicts .
Parabolic structure from a logarithmic connection {#sec3.e}
-------------------------------------------------
Consider a logarithmic connection $\nabla$ on a holomorphic bundle $V$ of rank two and with trivial determinant over a compact Riemann surface $\Sigma$. We assume that $\nabla$ is a $\operatorname{SL}(2,{\mathbb{C}})$–connection, i.e., the logarithmic connection on $\bigwedge^2 V\,=\, {\mathcal O}_\Sigma$ induced by $\nabla$ is the trivial connection. Let $p_1,\,\cdots ,\,p_n\,\in\,\Sigma$ be the singular points of $\nabla$. We also assume that the residue $$\text{res}_{p_k}(\nabla)\in\text{End}_0(V_{p_k})$$ of the connection $\nabla$ at every point $p_k$ has two real eigenvalues $\pm\rho_k$ with $\rho_k\,\in\,]0,\, \tfrac{1}{2}[.$ For every $1\, \leq\, k\, \leq\, n$, let $$L_k\,:=\, \text{Eig}(\text{Res}_{p_k}(\nabla),\, \rho_k)\, \subset\, V_{p_k}$$ be the eigenline of the residue of $\nabla$ at $p_k$ for the eigenvalue $\rho_k$.
The logarithmic connection $\nabla$ gives rise to the parabolic structure $${\mathcal P}\,=\,(D=p_1+\ldots +p_n,\, \{L_1,\, \cdots,\, L_n\},\,
\{\rho_1,\,\cdots ,\, \rho_n\})\, .$$ It is straightforward to check that another such logarithmic connections $\nabla^1$ on $V$ induces the same parabolic structure $P$ if and only if $\nabla -\nabla^1$ is a strongly parabolic Higgs field on $(V,\, {\mathcal P})$.
It should be mentioned that in [@MSe], the local form $$d+{{\left(\begin{matrix}\rho&0\\0& 1-\rho\end{matrix}\right)}}\frac{dw}{w}$$ of the connection is used (instead of the local form in ). In that case the Deligne extension gives a rank two holomorphic vector bundle $W$ (instead of $V$) with $\bigwedge^2 W\,=\, {\mathcal O}_{\Sigma}(-D)$ (instead of $\bigwedge^2 V\,=\, {\mathcal O}_{\Sigma}$), while the parabolic weights at $p_k$ become $\rho_k,\, 1-\rho_k$ (instead of $\rho_k,\, -\rho_k$).
A theorem of Mehta and Seshadri [@MSe p. 226, Theorem 4.1(2)], and Biquard [@Biq p. 246, Théorème 2.5] says that the above construction of a parabolic bundle $(V,\, {\mathcal P})$ from a logarithmic connection $\nabla$ produces a bijection between the stable parabolic bundles (in the sense of Section \[sec2.1\]) on $(\Sigma,\, D)$ and the space of isomorphism classes of irreducible flat ${\rm
SU}(2)$–connections on the complement $\Sigma\setminus D$. See, for example, [@Pir Theorem 3.2.2] for our specific situation. As a consequence of the above theorem of [@MSe] and [@Biq], for every logarithmic connection $\nabla$ on $V$ which produces a stable parabolic structure $\mathcal P$, there exists a unique strongly parabolic Higgs field $\Theta$ on $(V,\, {\mathcal P})$ such that the holonomy of the flat connection $\nabla+\Theta$ is contained in ${\rm SU}(2)$. Moreover, this flat ${\rm SU}(2)$–connection $\nabla+\Theta$ is irreducible.
Abelianization
==============
In [@He3], the connection $\nabla$ (or more correctly representatives for each gauge class in $\mathcal M_{1,1}^\rho$) is computed for the special case where $\rho\,=\,\tfrac{1}{6},$ $\tau\,=\,\sqrt{-1}$ and $L\,\in \,{\rm Jac}(T^2)\setminus\{S \,\mid\, S^{\otimes 2}\,=\, K_{T^2}\}$. We shall show (see Proposition \[explicit\_coeff\]) that for general $\rho$, but $\tau\,=\,\sqrt{-1}$ and $L\,\in \,{\rm Jac}(T^2)\setminus\{S \,\mid\, S^{\otimes 2}\,=\, K_{T^2}\}$, the corresponding connection $\nabla$ is of the form $$\label{abel-connection}
\nabla\,=\,\nabla^{a,\chi,\rho}\,=\,{{\left(\begin{matrix}\nabla^L &\gamma^+_\chi\\ \gamma^-_\chi &
\nabla^{L^*} \end{matrix}\right)}}\, ,$$ where $a,\, \chi\,\in\,{\mathbb{C}}$, $$\nabla^L\,=\,d+a\cdot dw+\chi\cdot d\overline{w}$$ is a holomorphic connection on $L$ and $\nabla^{L^*}$ is its dual connection on $L^{^*}$; here $w$ a complex affine coordinate on $T^2\,=\, {\mathbb C}/\Gamma$. The off–diagonal terms in can be described explicitly in terms of the theta functions as explained below.
Before doing so, we briefly describe both the Jacobian and the rank one de Rham moduli space for $T^2$ in terms of some useful coordinates. Let $$d\,=\,\partial+\overline\partial$$ be the decomposition of the de Rham differential $d$ on $T^2$ into its $(1,0)$–part $\partial$ and $(0,1)$–part $\overline\partial$. It is well–known that every holomorphic line bundle of degree zero on $T^2$ is given by a holomorphic structure $$\overline{\partial}^\chi\,=\,\overline{\partial} +\chi\cdot d\overline{w}$$ on the $C^\infty$ trivial line bundle $T^2\times {\mathbb C}\,\longrightarrow\,
T^2$ for some $\chi\,\in\,{\mathbb{C}}$, where $w$ is an affine coordinate function on ${\mathbb{C}}/({\mathbb{Z}}+\sqrt{-1}{\mathbb{Z}})\,=\,T^2$ (note that $d\overline{w}$ does not depend on the choice of the affine function $w$). Clearly, two such differential operators $$\overline{\partial}^{\chi_1}\ \ \text{ and } \ \ \overline{\partial}^{\chi_2}$$ determine isomorphic holomorphic line bundles if and only if $\overline{\partial}^{\chi_1}$ and $\overline{\partial}^{\chi_2}$ are gauge equivalent. Now, they are gauge equivalent if and only if $$\chi_2-\chi_1\,\in\, \Gamma^*$$ where $$\Gamma^*\,=\,\pi{\mathbb{Z}}+\pi \sqrt{-1}{\mathbb{Z}}$$ (recall that $\tau\,=\, \sqrt{-1}$).
\[rs\] The holomorphic line bundle $L(\overline{\partial}^{\chi})\, :=\,
[\overline{\partial}^{\chi}]$, given by the Dolbeault operator $\overline{\partial}^\chi$, is a spin bundle if and only if $2\chi\,\in\,\Gamma^*.$
Similarly, flat line bundles on $T^2$ are given by the connection operator $$d^{a,\chi}\,=\,d+a\cdot dw+\chi\cdot d\overline{w}$$ on the line bundle $T^2\times {\mathbb C}\,\longrightarrow\,
T^2$, for some $a,\, \chi\,\in\,{\mathbb{C}}$. Moreover two connections $d^{a_1,\chi_1}$ and $d^{a_2,\chi_2}$ are isomorphic if and only if $$(a_2-a_1) + (\chi_2-\chi_1) \,\in\, 2\pi \sqrt{-1} {\mathbb{Z}}\ \ \text{ and }\ \
(a_2-a_1) - (\chi_2-\chi_1)\,\in\, 2\pi \sqrt{-1} {\mathbb{Z}}\, .$$
The (shifted) theta function for ${\mathbb{C}}/ \Gamma$, where as before $\Gamma\,=\,{\mathbb{Z}}+{\mathbb{Z}}\sqrt{-1}$, will be denoted by $\vartheta$. In other words, $\vartheta$ is the unique (up to a multiplicative constant) entire function satisfying $\vartheta(0) \,=\, 0$ and $$\vartheta(w+ 1) \,=\, \vartheta (w),\,\, \vartheta(w+\sqrt{-1}) \,=\, - \vartheta (w)e^{-2\pi
\sqrt{-1}w}\, .$$ Then the function $$t_{x}(w) \,:=\, \frac{\vartheta(w-x)}{\vartheta(w)}e^{-\pi x(w-\overline{w})}$$ is doubly periodic on ${\mathbb{C}}\setminus\Gamma$ with respect to $\Gamma$ and satisfies the equation $$(\operatorname{\overline\partial}-\pi xd\overline{w})t_{x}\,=\,0\, .$$ Thus $t_x$ is a meromorphic section of the holomorphic bundle $L(\overline{\partial}^{-\pi x})\, :=\,[\overline{\partial}^{-\pi x}]$ (it is the holomorphic line bundle given by the Dolbeault operator $\operatorname{\overline\partial}-\pi xd\overline{w}$). Notice that for $x\,\notin\,\Gamma$, the section $t_x$ has a simple zero at $w\,=\,x$ and a first order pole at $w \,=\, 0$. Moreover, up to scaling by a complex number, this $t_x$ is the unique meromorphic section of $L(\overline{\partial}^{-\pi x})\, :=\, [\overline{\partial}^{-\pi x}]$ with a simple zero at $o$.
\[TdualJ\] Once a base point $o\,\in\, T^2$ has been chosen, we get the well–known isomorphism $$T^2\,\longrightarrow\, {\rm Jac}(T^2)\, ,\ \ [x]\,\longmapsto\, L(\overline{\partial}^{-\pi x})\, :=\,[\overline{\partial}^{-\pi x}]$$ that associating to $[x]$ the divisor of the meromorphic section $t_x$: $$(t_x)\,=\, [x]-o\, .$$
For $\frac{1}{2}\, >\, \rho\, >\, 0$, if $V$ in is of the form $V\,=\, L\oplus L^{^*}$, then from Corollary \[cort\] and Lemma \[lem3.3\] it follows that $\text{degree}(L)\,=\, 0$ and $L$ is not a spin bundle. In other words, $$L\,=\, L(\overline{\partial}+\chi\cdot d\overline{w})$$ for some $\chi\,\in\,\mathbb C$, and $$\chi\,\notin\,\tfrac{1}{2}\Gamma^*\, ;$$ see Remark \[rs\].
\[explicit\_coeff\] For any $\rho\,\in\, [0,\, \tfrac{1}{2}[$, take $[\nabla]\,\in\, {\mathcal M}_{1,1}^\rho$ such that its Deligne extension is given by the holomorphic vector bundle $$V\,=\, L\oplus L^*$$ (see ), where $L\,=\, L(\overline{\partial}+\chi d\overline{w})$ is a holomorphic line bundle on $T^2$ of degree zero which is not a spin bundle. Set $x\,=\,-\frac{1}{\pi }\chi$, so $x\,\notin\,\tfrac{1}{2}\Gamma.$ Then, there exists $$a\,\in\,{\mathbb{C}}$$ such that one representative of $[\nabla]$ is given by $$\nabla^{a,\chi,\rho}$$ as in , where the second fundamental forms $\gamma^+_\chi$ and $\gamma^-_\chi$ in are given by the meromorphic $1$–forms $$\label{gammamp}
\gamma^+_\chi([w])\,=\,\rho \tfrac{\vartheta'(0)}{\vartheta(-2x)}t_{2x}(w)dw\ \
\text{ and }\ \
\gamma^-_\chi([w])\,=\,\rho \tfrac{\vartheta'(0)}{\vartheta(2x)}t_{-2x}(w)dw$$ with values in the holomorphic line bundles of degree zero $L([2x]-[0])\,=\,L(\operatorname{\overline\partial}+ 2\chi d\overline{w})$ and $L([-2x]-[0])\,=\,L(\operatorname{\overline\partial}-2\chi d\overline{w})$ respectively.
Using Section \[Delext\] we know that there exists a representative $\nabla$ of $[\nabla]$ such that its $(0,1)$–part ${\overline\partial}^\nabla$ is given by $${\overline\partial}^\nabla\,=\,{\overline\partial}+\begin{pmatrix} \chi d\overline{w}&0
\\ 0& -\chi d\overline{w}\end{pmatrix}\, .$$ The $(1,0)$–part $\partial^\nabla$ is given by $$\partial^\nabla\,=\,\partial+\begin{pmatrix} A & B\\ C& -A \end{pmatrix}\, ,$$ where $$\Psi\,=\,\begin{pmatrix} A & B\\ C& -A \end{pmatrix}$$ is a $\text{End}(V)$–valued meromorphic $1$–form on $T^2$, with respect to the holomorphic structure ${\overline\partial}^\nabla$, such that $\Psi$ a simple pole at $o$ and $\Psi$ is holomorphic elsewhere. In particular, $A$ is a meromorphic $1$–form on $T^2$ with simple pole at $o$, and hence by the residue theorem it is in fact holomorphic, i.e., $$A\,=\, adw$$ for some $a\,\in\,{\mathbb{C}}$. Furthermore, $B$ and $C$ are meromorphic $1$–forms with values in the holomorphic bundles $L(\operatorname{\overline\partial}+2\chi d\overline{w})$ and $L(\operatorname{\overline\partial}-2\chi d\overline{w})$, respectively. Note that for $x\,\in\,\tfrac{1}{2}\Gamma$, $L(\operatorname{\overline\partial}+2\chi d\overline{w})$ would be the trivial holomorphic line bundle and $B$ and $C$ could not have non-trivial residues at $o$ by the residue theorem. The determinant of the residue of $\Psi$ at $o$ is $-\rho^2$ by . Therefore, from the holomorphicity of $A$ we conclude that the quadratic residue of the meromorphic quadratic differential $BC$ is $$\text{qres}_o(BC)\,=\,\rho^2\, .$$ From the discussion prior to Remark \[TdualJ\] there is a unique meromorphic section of $L(\operatorname{\overline\partial}\pm2\chi d\overline{w})$ with a simple pole at $o$. Thus, after a possible constant diagonal gauge transformation, from the uniqueness, up to scaling, of the meromorphic section of $L(\operatorname{\overline\partial}\pm2\chi d\overline{w})$ with simple pole at $o$, it follows that $$B\,=\,\gamma^+_\chi \ \ \text{ and } \ \ C\,=\,\gamma^-_\chi\, ,$$ where $\gamma^+_\chi$ and $\gamma^-_\chi$ are the second fundamental forms ; here the assumption that $L$ is not a spin bundle is used. This completes the proof.
\[rem:strongparaHiggs\] The off–diagonal parts $\gamma^+_\chi$ and $\gamma^-_\chi$ depend only on $\chi$. Note that $\chi$ also uniquely determines the parabolic structure unless $L(\operatorname{\overline\partial}+\chi d\overline{w})$ is a spin bundle, or equivalently, $2\chi\,\in\, \Gamma^*$. Also note that $L(\operatorname{\overline\partial}-\chi d
\overline{w})$ is the dual of $L(\operatorname{\overline\partial}+\chi d\overline{w})$.
We also see from Proposition \[explicit\_coeff\] that every strongly parabolic Higgs field on the parabolic bundle corresponding to the connection $\nabla$ in Proposition \[explicit\_coeff\] is of the form $$c \begin{pmatrix} dw &0\\0&-dw\end{pmatrix}$$ for some constant $c\,\in\,{\mathbb{C}}$.
\[Pro-stab\] Assume that $\rho\,\in\, ]0,\, \tfrac{1}{2}[$. Take $[\nabla]\, \in\, {\mathcal M}^\rho_{1,1}$ such that the corresponding bundle $V$ in is of the form $L\oplus L^{^*}$ (so $L$ is not a spin bundle but its degree is zero by Corollary \[cort\] and Lemma \[lem3.3\]). Then, the rank two parabolic bundle corresponding to $[\nabla]$ (see Section \[sec3.e\]) is parabolic stable.
The two holomorphic line bundles $L$ and $L^{^*}$ are not isomorphic, because $L$ is not a spin bundle. From this it can be shown that any holomorphic subbundle of degree zero $$\xi\, \subset\, V\,=\, L\oplus L^{^*}$$ is either $L$ or $L^{^*}$. Indeed, this follows by considering the two compositions of homomorphisms: $$\xi\, \hookrightarrow\, L\oplus L^{^*}\, \longrightarrow\, L
\ \text{ and }\ \xi\, \hookrightarrow\, L\oplus L^{^*}\, \longrightarrow\, L^{^*}\, ;$$ one of them has to be the zero homomorphism and the other an isomorphism.
As the residue in is off–diagonal (with respect to the holomorphic decomposition $V\,=\,L\oplus L^*$), the above observation implies that every holomorphic line subbundle $\xi\, \subset\, V$ of degree zero has parabolic degree $-\rho$. On the other hand, the parabolic degree of a holomorphic line subbundle of negative degree is less than or equal to $$-1+\rho\,<\,0\,.$$ Consequently, the parabolic bundle is stable.
Outlook: Exceptional bundles
----------------------------
The exceptional cases of non-trivial extensions of a spin bundle $S$ by itself the second case in Corollary \[cort\]) can be described as follows. After a normalization, the holomorphic structure of the vector bundle is given by the Dolbeault operator on the $C^\infty$ trivial bundle $T^2\times {\mathbb C}^2\, \longrightarrow\, T^2$ $$\operatorname{\overline\partial}\,=\,{{\left(\begin{matrix}\operatorname{\overline\partial}^S & d\overline{w}\\ 0 & \operatorname{\overline\partial}^S\end{matrix}\right)}}\, ,$$ where $w$ is the global coordinate on the universal covering ${\mathbb{C}}\,
\longrightarrow\, {\mathbb{C}}/\Gamma\,=\,T^2$. The $(1,0)$–type component $\operatorname{\partial}$ of the connection is than given by $$\operatorname{\partial}={{\left(\begin{matrix} \operatorname{\partial}^S+a dw& bdw \\ c dw & \operatorname{\partial}^S-a dw\end{matrix}\right)}}\, ,$$ where $a,\,b,\,c\,\colon \,T^2\setminus\{o\}\,\longrightarrow\, {\mathbb{C}}$ are smooth functions with first order pole like singularity at $o\,\in\, T^2.$ The connection $\nabla\,=\,\operatorname{\partial}+\operatorname{\overline\partial}$ is flat if and only if $$\label{exceptional_flatness}
{{\left(\begin{matrix}\operatorname{\overline\partial}a+c d\overline{w}&\operatorname{\overline\partial}b-2 a d\overline{w}\\ \operatorname{\overline\partial}c& -\operatorname{\overline\partial}a-c d\overline{c}\end{matrix}\right)}}\,=\,0\, .$$ Since $c$ has at most a first order pole at $o\,\in\, T^2$, and satisfies the equation $\operatorname{\overline\partial}c\,=\,0$, it must be a constant. This constant turns out to be related to the weight $\rho$ in the following way.
If $a$ has a first order pole like singularity at $o$ of the form $$a(w)\,\sim \,\frac{a_1}{w}+a_0+\ldots\, ,$$ then integration by parts yields $$2\pi\sqrt{-1} a_1\,=\,\int_{T^2} \operatorname{\overline\partial}a\wedge dw\,=\,\int_{T^2}c d\overline{w}\wedge dw\, .$$ The connection $\nabla$ is locally gauge equivalent, by a holomorphic gauge that extends smoothly to $o\,\in\, T^2$, to the connection in ; using this it follows that $$a_1\,=\,\pm\rho\, ,$$ and therefore $$\label{exceptional-c-constant}
c\,=\,\pm\frac{2\pi\sqrt{-1} \rho}{\int_{T^2}d\overline{w}\wedge dw}
\,=\,\pm \pi\rho$$ (recall that $\tau\,=\, \sqrt{-1}$). The sign in tells us whether the induced parabolic structure is stable or not. More precisely, if $0\,<\,\rho\,<\,\tfrac{1}{2},$ then we have for the plus “$+$” sign an unstable parabolic structure, as the parabolic degree of the unique holomorphic line subbundle $L\,=\,S\oplus\{0\}$ of degree $0$ is $$\text{par-deg}(L)\,=\,{\rm degree}(L)+\rho\,>\,0\, .$$ Analogously, the parabolic structure is stable for the minus “$-$” sign in .
We have not yet shown that there is actually a flat connection $\nabla$ for each case of $\pm\rho.$ The complex number $c$ is determined by $\rho$ using , and there is a unique solution of $a$, up to an additive constant, for the equation in . Then, for each solution of $a$, there is again a unique solution for $b$, with first order pole like singularity at $o\,\in\,
T^2$, of the equation $$\operatorname{\overline\partial}b-2 a d\overline{w}\,=\,0\, ;$$ indeed, this can easily be deduced from Serre duality. Hence, up to two additive constants, the flat connection is unique. But due to the option of the constant gauge transformations $$G\,=\, {{\left(\begin{matrix} 1& h\\0&1\end{matrix}\right)}}$$ of the $C^\infty$ trivial bundle $T^2\times {\mathbb C}^2\,
\longrightarrow\, T^2$, where $h\,\in\,{\mathbb{C}}$ is any constant, the isomorphism class of the flat connection does not depend on the choice of the additive constant in the solution $a$. Note that in the unstable case, the gauge transformation $G$ does not alter the parabolic structure, but in the case of the stable parabolic structure we obtain different, but nevertheless gauge equivalent, parabolic structures.
Flat connections on the 4-punctured torus
=========================================
Consider $$\widehat{T}^2\,=\,{\mathbb{C}}/(2{\mathbb{Z}}+2\sqrt{-1}{\mathbb{Z}})$$ and the 4–fold covering $$\label{mPi}
\Pi\,\colon\, \widehat{T}^2\,\longrightarrow\, T^2\,=\,{\mathbb{C}}/({\mathbb{Z}}+\sqrt{-1}{\mathbb{Z}})$$ produced by the identity map of $\mathbb C$. Let $$\{p_1,\,p_2,\, p_3,\, p_4\}\,:=\, \Pi^{-1}(o) \, \subset\, \widehat{T}$$ be the preimage of $o\, \in\, T^2$.
Fix $$\rho\,=\, 0\, .$$ We use $\Pi$ in to pull back the connection in to $\widehat{T}^2$. The traces $$T_1(\chi,a)\,=\, \text{tr}(h(\widehat{\alpha}))\ \ \text{ and }\ \
T_2(\chi,a)\,=\, \text{tr}(h(\widehat{\beta}))\, ,$$ of the monodromy representation $h$ for $\Pi^*\nabla^{a,\chi,\rho=0}$ along $$\label{tab}
\widehat{\alpha}\,=\, 2\,\in\, 2{\mathbb{Z}}+2 \sqrt{-1}{\mathbb{Z}}\, \subset\, \pi_1(\widehat{T}^2\setminus\{p_1,\cdots ,p_4\}\, ,q)\ \ \text{ and}$$ $$\widehat{\beta}\,=\,2\sqrt{-1}\,\in\, 2{\mathbb{Z}}+2 \sqrt{-1}{\mathbb{Z}}\, \subset\,
\pi_1(\widehat{T}^2\setminus\{p_1,\cdots ,p_4\},\, q)$$ (see Figure \[figure2\]), are given by $$T_1(\chi,a)\,=\,e^{-2(a+\chi)}+e^{2(a+\chi)}\ \ \text{ and}$$ $$T_2(\chi,a)\,=\,e^{2\sqrt{-1}(-a+\chi)}+e^{2\sqrt{-1}(a -\chi)}$$ respectively, while the local monodromy of $\Pi^*\nabla^{a,\chi,\rho=0}$ around each of $p_1,\,\cdots ,\,p_4$ is trivial, because $\rho\,=\,0$.
In the following, fix $$\label{chifix}
\chi\,=\,\frac{\pi}{4}(1-\sqrt{-1})\, ,$$ and consider $$a_k\,=\,-\frac{\pi}{4}(1+\sqrt{-1})+k\pi(1+\sqrt{-1})$$ for all $k\,\in\,{\mathbb{Z}}.$ Then we have $$\label{T1eq}T_1(\chi,a_k)\,=\,-(e^{-2k\pi}+e^{2k\pi})\,\in\, {\mathbb{R}}$$ $$\label{T2eq}T_2(\chi,a_k)\,=\,-(e^{-2k\pi}+e^{2k\pi})\,\in\, {\mathbb{R}}\, ;$$ as before, $T_1(\chi,a_k)$ and $T_2(\chi,a_k)$ are the traces of holonomies of $\Pi^*\nabla^{a_k,\chi,0}$ along $\widehat\alpha$ and $\widehat\beta$ respectively (see ). Moreover, $$\label{der1}
\begin{split}
\frac{\partial}{\partial s}T_1(\chi,a_k+s+\sqrt{-1}t)&\,=\,-2e^{-2k\pi}(-1+e^{4k\pi})\,\in\, {\mathbb{R}}\\
\frac{\partial}{\partial t}T_1(\chi,a_k+s+\sqrt{-1} t)&\,=\,-2\sqrt{-1}e^{-2k\pi}(-1+e^{4k\pi})
\,\in\, \sqrt{-1}{\mathbb{R}}\setminus\{0\}
\end{split}$$ and $$\label{der2}
\begin{split}
\frac{\partial}{\partial s}T_2(\chi,a_k+s+\sqrt{-1} t)&\,=\,2\sqrt{-1}e^{-2k\pi}(-1+e^{4k\pi})
\,\in\, \sqrt{-1}{\mathbb{R}}\setminus\{0\}\\
\frac{\partial}{\partial t}T_2(\chi,a_k+s+\sqrt{-1} t)&\,=\,-2e^{-2k\pi}(-1+e^{4k\pi})\,\in\, {\mathbb{R}}\, .
\end{split}$$
\[real-mon-hatT2\] Let $k\,\in\,{\mathbb{Z}}\setminus\{0\}$, $\chi\,=\,\frac{\pi}{4}(1-\sqrt{-1})$ and $a_k
\,=\,-\frac{\pi}{4}(1+\sqrt{-1})+k\pi(1+\sqrt{-1})$. Then there exists $\epsilon\,>\,0$ such that for each $\rho\,\in\,]0,\,\epsilon[$, there is a unique number $a\,\in\,{\mathbb{C}}$ near $a_k$ satisfying the condition that the monodromy of the flat connection $$\Pi^*\nabla^{a,\chi,\rho}$$ on $\widehat{T}^2\setminus\{p_1,\cdots ,p_4\}$ is irreducible and the image of the monodromy homomorphism is conjugate to a subgroup of $\operatorname{SL}(2,{\mathbb{R}})$.
Using and , and applying the implicit function theorem to the imaginary parts of the traces $T_1$ and $T_2$, there exists for each sufficiently small $\rho$ a unique complex number $a$ such that the traces $T_1$ and $T_2$, of holonomies of $\nabla^{a,\chi,\rho}$ along $\widehat\alpha$ and $\widehat\beta$ respectively, are real. Because $k\,\neq\, 0$, and $\rho$ is small, we obtain from and that these traces satisfy $$T_1\,<\,-2\ \ \text{ and }\ \ T_2\,<\,-2\, .$$
Recall the general formula $$\label{tXY}\text{tr}(X)\text{tr}(Y)\,=\,\text{tr}(XY)+\text{tr}(XY^{-1})$$ for $X,\,Y\,\in\, \text{SL}(2,{\mathbb{C}})$. Let $$x\,=\,\text{tr}(h(\alpha))\ \ \text{ and } \ \ y\,=\,\text{tr}(h(\beta))$$ be the traces of the monodromy homomorphism $h$ of the connection $\nabla^{a,\chi,\rho}$ on $T^2\setminus\{0\}$ along $\alpha$ and $\beta$ (recall the notation of Section \[ADHS1t\]).
Applying to $$X\,=\, h(\alpha)\,=\, Y \ \ \text{( respectively, }\ \ X\,=\, h(\beta)\,=\,Y)$$ we obtain that $x$ (respectively, $y$) must be purely imaginary. Then it can be checked directly that the trace along any closed curve in the 4–punctured torus is real: In fact, that $$z\,=\,\text{tr}(h(\alpha\circ\beta))$$ is real is a direct consequence of and the above observation that $x,\,y\,\in\,\sqrt{-1}{\mathbb{R}}$. Using repeatedly (compare with [@Gol]) it is deduced that the trace of the monodromy along any closed curve on $\widehat{T}^2$ is real.
For $\rho\,\neq\,0$ sufficiently small, the connection $\Pi^*\nabla^{a,\chi,\rho}$ on $\widehat{T}^2$ is irreducible as a consequence of Lemma \[irr\] — note that the condition $xy\,\neq\, 0$ follows directly from the fact that $\rho\,\neq\, 0$ — applied to $h(\widehat{\alpha})$ and $h(\widehat{\beta})$ (see ).
We will prove that the image of the monodromy homomorphism $h$ is conjugate to a subgroup of $\text{SL}(2,{\mathbb{R}})$.
To prove this, since the monodromy is irreducible and has all traces real, the homomorphism $h$ is conjugated to its complex conjugate representation $\overline h$, meaning there exists $C\,\in\,
\operatorname{SL}(2,{\mathbb{C}})$ such that $$C^{-1}\overline{h} C\,=\, h\, .$$ Applying this equation twice we get that $$\overline{C}C\,=\, \pm \text{Id}$$ because $h$ is irreducible.
Assume that $\overline{C}C\,=\, -\text{Id}$. Then a straightforward computation shows that there exists $D\,\in\, \operatorname{SL}(2,{\mathbb{C}})$ such that $$C\,=\,\pm \overline{D}^{-1} \delta D\, ,$$ with $$\delta\,=\, \begin{pmatrix}0&1\\
-1&0\end{pmatrix}\, .$$
Therefore, the conjugated representation $$H\, :=\, DhD^{-1}$$ is unitary as $$(\overline{H}^t)^{-1}\,=\, \delta^{-1}\overline{H}\delta\,=\,
(\pm1)^2\delta^{-1}\overline{D} \overline{h} \overline{D}^{-1}\delta\,=\, H\, .$$ Now, since the traces of some elements in the image of the monodromy are not contained in $[-2,\, 2]$, we get a contradiction.
Thus, $$\overline{C}C\,=\, \text{Id}\, ,$$ and a direct computation implies then that $$C\,=\,\overline{D}^{-1} D$$ for some $D\,\in\, \operatorname{SL}(2,{\mathbb{C}})$. Consequently, we have $$DhD^{-1}\,=\, \overline{D}\overline{h} \overline{D}^{-1}\, .$$ Hence the image of the monodromy homomorphism $h$ is conjugate to a subgroup of $\text{SL}(2,{\mathbb{R}})$.
Once we know that $x$ and $y$ are purely imaginary and $z$ is real with $|z|\,>\, 2$ ($|z|\,>\,2$ follows from $k\neq0$), here is an alternative argument showing that the monodromy representation in Theorem \[real-mon-hatT2\] is conjugated to an $\operatorname{SL}(2,{\mathbb{R}})$-representation. First observe that both solutions of $$\zeta+\zeta^{-1}\,=\,z$$ are real. A direct calculation shows that for $h(\alpha)$ and $h(\beta)$ as in , all the matrices for a set of generators for the fundamental group of the 4–punctured torus, for example $$\begin{split}&h(\alpha)^2,\\& h(\beta)^2,\\
& h(\beta)^{-1}h(\alpha)^{-1}h(\beta)h(\alpha),\\
h(\alpha)^{-1}&h(\beta)^{-1}h(\alpha)^{-1}h(\beta)h(\alpha)h(\alpha),\\
h(\beta)^{-1}&h(\beta)^{-1}h(\alpha)^{-1}h(\beta)h(\alpha)h(\beta),\\
h(\alpha)^{-1}h(\beta)^{-1}&h(\beta)^{-1}h(\alpha)^{-1}h(\beta)h(\alpha)h(\beta)h(\alpha),
\end{split}$$ have the property that the off–diagonal entries are purely imaginary and the diagonal entries are real. Conjugating by $$\begin{pmatrix} e^{\tfrac{\pi \sqrt{-1}}{4}}&0\\0& e^{-\tfrac{\pi \sqrt{-1}}{4}}\end{pmatrix}$$ directly gives a representation into $\operatorname{SL}(2,{\mathbb{R}}).$
We shall use the following theorem.
\[thetrivcon\] Let $\chi\,=\,\frac{\pi}{4}(1-\sqrt{-1}).$ For every $\rho\,\in\, [0,\, \tfrac{1}{2}[$, there exists $a^u\,\in\, {\mathbb{C}}$ such that $$\Pi^*\nabla^{a^u,\chi,\rho}$$ is a reducible unitary connection satisfying the following condition: the monodromies of $\Pi^*\nabla^{a^u,\chi,\rho}$ along $$\widehat\alpha\,=\,2 \,\in\, \pi_1(\widehat{T}^2\setminus\{p_1,\cdots ,p_4\},\, q)
\ \ and\ \ \widehat\beta\,=\,2\sqrt{-1} \,\in\,
\pi_1(\widehat{T}^2\setminus\{p_1,\cdots ,p_4\},\, q)$$ (see ) are both $-{\rm Id}$.
First, the parabolic bundle on $T^2$ determined by $\chi\,=\,\frac{\pi}{4}(1-\sqrt{-1})$ is stable; this stable parabolic bundle on $T^2$ will be denoted by $W_*$. Note that all the strongly parabolic Higgs fields on this parabolic bundle are given by constant multiples of $$\begin{pmatrix} dw & 0\\ 0&-dw\end{pmatrix}\, .$$ In view of the theorem of Mehta–Seshadri and Biquard ([@MSe], [@Biq]) mentioned in Section \[sec3.e\], there exists $a^u\,\in\,{\mathbb{C}}$ such that $$\nabla^{a^u,\chi,\rho}$$ has unitary monodromy on $T^2.$ Then, the flat connection $\Pi^*\nabla^{ a^u,\chi,\rho}$ on $\widehat{T}^2$ has unitary monodromy as well, where $\Pi$ is the map in . On the other hand, the pulled back parabolic bundle $\Pi^*W_*$ on $\widehat{T}^2$ is strictly semi-stable, because $\chi\,=\,\frac{\pi}{4}(1-\sqrt{-1})$ and $\widehat{T}^2\,=\,{\mathbb{C}}/(2\Gamma)$ for the specific lattice $2\Gamma\,=\,2{\mathbb{Z}}+2\sqrt{-1}{\mathbb{Z}}$ (it can be proved by a direct computation, but it also follows from [@HeHe]), so that the unitary connection $\Pi^*\nabla^{ a^u,\chi,\rho}$ is automatically reducible.
We give an alternative explanation for the semi-stability of the parabolic bundle $\Pi^*W_*$. Take $x \,=\, y \,= \,0$, and the unique positive solution of $z$ in . Note, that if $\rho\,=\,0,$ then $z\,=\,2$ and $a^u\,=\,-\overline\chi$, with $\chi$ given by . Then, using we see that the representation $h$ of the fundamental group of the $1$–punctured torus given by $x(h)\,=\, 0\, =\, y(h)$ and $z(h)\,=\,
z$ induces a unitary reducible representation of the fundamental group of the 4–punctured torus for any real $\rho$. The corresponding monodromies along $\widehat\alpha$ and $\widehat\beta$ are given by $h(\alpha) h(\alpha)$ and $h(\beta) h(\beta)$, and both are equal to $-\text{Id}$ by . It is easy to see that, for $\rho\,<\,\tfrac{1}{4}$ (this case suffices for our proof), the parabolic structure on the holomorphic bundle $$L\oplus
L^*\,\longrightarrow\, \widehat{T}^2$$ cannot be strictly semi-stable if $L^2$ is not trivial; this is because the lines giving the quasiparabolic structure are not contained in $L$ or $L^*$ by , and these two, namely $L$ and $L^{^*}$, are the only holomorphic subbundles of degree zero by the assumption that $L^2\,\neq\, {\mathcal
O}_{\widehat{T}^2}$. By continuity of the monodromy representation of $\Pi^*\nabla^{a^u,\chi,\rho}$ with respect to the parameters $(a^u,\,\chi,\,\rho)$, the representation of $\Pi^*\nabla^{ a^u,\chi,\rho}$ must be the unitary reducible representation $h$ with $x(h)\,=\,0\,=\,y(h)$ and positive $z(h)\,=\,z$. As we already know that the monodromies of $h$ along $\widehat\alpha$ and $\widehat\beta$ are both $-\text{Id}$, this finishes the proof.
Flat irreducible $\operatorname{SL}(2,{\mathbb{R}})$–connections on compact surfaces
====================================================================================
We assume that $$\rho\,=\,\frac{1}{2p}\, ,$$ for some $p\,\in\,{\mathbb{N}}$ odd, with $\rho$ being small enough so that Theorem \[real-mon-hatT2\] is applicable.
The torus $\widehat{T}^2$ in is of square conformal type, and it is given by the algebraic equation $$y^2\,=\, \frac{z^2-1}{z^2+1}\, .$$ Without loss of any generality, we can assume that the four points $$\{p_1,\,\cdots ,\,p_4\}
\,=\, \Pi^{-1}(\{o\})\, ,$$ where $\Pi$ is the map in , are the branch points of $z$, i.e., the $(y,\, z)$ coordinates of $p_1,\,\cdots ,\, p_4$ are $$p_1\,=\,(0,\,1), \ \ p_2\,=\,(\infty,\,\sqrt{-1}),\ \ p_3\,=\,(0,\,-1), \ \
p_4\,=\,(\infty,\,-\sqrt{-1})\, .$$
Define the compact Riemann surface $\Sigma$ by the algebraic equation $$\label{sigmayz}
x^{2p}\,=\,\frac{z^2-1}{z^2+1}\, .$$ Consider the $p$–fold covering $$\Phi_p\,\colon\, \Sigma\,\longrightarrow\, \widehat{T}^2\, ,\ \ (x,\, z)
\, \longmapsto \, (x^p,\, z)\, ,$$ which is totally branched over $p_1,\, \cdots,\, p_4$. Denote the inverse image $\Phi^{-1}_p(p_i)$, $1\, \leq\, i\, \leq\, 4$, by $P_i$ (see Figure \[figure3\]).
For a connection $\nabla^A$ (respectively, $\nabla^B$) on a vector bundle $A$ (respectively, $B$), the induced connection $(\nabla^A\otimes\text{Id}_B)\oplus
(\text{Id}_A\otimes\nabla^B)$ on $A\otimes B$ will be denoted by $\nabla^A\otimes\nabla^B$ for notational convenience.
There are holomorphic line bundles $$S\,\longrightarrow\, \Sigma$$ of degree $-2$ such that $$S\otimes S\,=\,{\mathcal O}_\Sigma(-P_1-P_2-P_3-P_4)\, .$$ For every such $S$, there is a unique meromorphic connection $\nabla^S$ on $S$ with the property that $$\nabla^S\otimes\nabla^S s_{-P_1-P_2-P_3-P_4}\,=\,0\, ,$$ where $s_{-P_1-P_2-P_3-P_4}$ is the meromorphic section of ${\mathcal O}_\Sigma(-P_1-P_2-P_3-P_4)$ given by the constant function $1$ on $\Sigma$ (this section has simple poles at $P_1,\,\cdots,\,P_4$). Observe that the monodromy representation of $\nabla^S$ takes values in ${\mathbb{Z}}/2{\mathbb{Z}}.$ Also, note that $(S,\,\nabla^S)$ is unique up to tensoring with an order two holomorphic line bundle $\xi$ equipped with the (unique) canonical connection that induces the trivial connection on $\xi\otimes\xi$.
\[trivialmon\] For given $\rho\,=\,\tfrac{1}{2p}$ and $\Sigma$, consider $a^u$ and $\chi$ as in Theorem \[thetrivcon\]. There exists a unique pair $(S,\,\nabla^S)$ such that the monodromy of the connection $$\nabla^S\otimes (\Pi\circ\Phi_p)^*\nabla^{a^u,\chi,\rho}$$ is trivial.
Since $p$ is odd, $\rho\,=\, \tfrac{1}{2p}$, and $\Phi_p$ is a totally branched covering, the local monodromies of $$(\Pi\circ\Phi_p)^*\nabla^{a^u,\chi,\rho}$$ around the points of $P_i$, $1\,\leq\, i\, \leq\, 4$, are all $-\text{Id}.$ Moreover, from Theorem \[thetrivcon\] it follows easily that the monodromy along any closed curve is $$\pm\text{Id}.$$ The lemma follows from these.
The connection $$\nabla^S\otimes (\Pi\circ\Phi_p)^*\nabla^{a^u,\chi,\rho}$$ is defined on the vector bundle $$S\otimes (L\oplus L^*)\, \longrightarrow\, \Sigma\, ,$$ where $L$ is the pull-back, by $\Pi\circ\Phi_p$, of the $C^\infty$ trivial line bundle $T^2\times{\mathbb C}\, \longrightarrow \,T^2$ equipped with holomorphic structure $$\overline{\partial}+\chi d\overline{w}\, .$$ For each $1\, \leq\,i\, \leq\, 4$, the residues of the connection $\nabla^S\otimes (\Pi\circ\Phi_p)^*\nabla^{a^u,\chi,\rho}$ at the points of $P_i\,=\, \Phi^{-1}_p(p_i)$ are $$\label{rescon}
\tfrac{1}{2}\begin{pmatrix} 1&-1\\-1&1\end{pmatrix}$$ with respect to any frame at points of $P_i$ compatible with the decomposition $S\otimes (L\oplus L^*)\, =\, (S\otimes L)\oplus (S\otimes L^*)$.
As in [@He3 § 3], there exists a holomorphic rank two bundle $V$ on $\Sigma$ with trivial determinant, equipped a holomorphic connection $D$, together with a holomorphic bundle map $$\label{df}
F\,\colon\, S\otimes (L\oplus L^*)\, \longrightarrow\, V\, ,$$ which is an isomorphism away from $P_1,\,\cdots ,\, P_4$, such that $$\nabla^S\otimes (\Pi\circ\Phi_p)^*\nabla^{a^u,\chi,\rho}\,=\,F^{-1}\circ D\circ F\, .$$ From Lemma \[trivialmon\] we know that $(V,\, D)$ is trivial.
\[Lemired\] Assume $p\,\geq\, 3.$ Consider the strongly parabolic Higgs field $$\Psi\,=\, \begin{pmatrix} dw&0\\0&-dw\end{pmatrix}$$ with respect to the parabolic structure induced by $\nabla^{a^u,\chi,\rho}$. Then, $$\Theta\,= \, F\circ(\Pi\circ\Phi_p)^*\Psi\circ F^{-1}$$ is a holomorphic Higgs field on the trivial holomorphic bundle $(V,\, D^{0,1})
\,=\, (V,\, D'')$ (the Dolbeault operator for the trivial holomorphic structure is denoted by $D''$).
Consider the holomorphic Higgs field $$(\Pi\circ\Phi_p)^*\Psi\,\colon\, S\otimes (L\oplus L^*)\,\longrightarrow\,
K_\Sigma\otimes S\otimes (L\oplus L^*)$$ on the rank two holomorphic bundle $S\otimes (L\oplus L^*).$ It vanishes of order $p-1\,\geq\,2$ at the singular points $P_1,\,\cdots ,\,P_4.$ Performing the local analysis (as in [@He3 § 3.2]), near $P_k$, of the normal form of the homomorphism $F$ in , we directly see that $\Theta\,=\, F\circ(\Pi\circ\Phi_p)^*\Psi\circ F^{-1}$ has no singularities, i.e., it is a holomorphic Higgs field on the trivial holomorphic bundle $(V,\, D'')$. Indeed, the homomorphism $F$ in has the local form $$\begin{pmatrix} 1&-\tfrac{z}{2}\\1&\tfrac{z}{2}\end{pmatrix}$$ with respect to the frame corresponding to and with respect to a holomorphic coordinate $z$ centered at $P_k$; so by conjugating with $F^{-1}$, the entries of $\Psi$ (with respect to a holomorphic frame) gets multiplied, at worst, with $$\frac{1}{z}\, ,$$ consequently, $\Theta$ does not have poles.
\[Main\] There exists a compact Riemann surface $\Sigma$ of genus $g\,>\,1$ with a irreducible holomorphic connection $\nabla$ on the trivial holomorphic rank two vector bundle ${\mathcal O}^{\oplus 2}_\Sigma$ such that the image of the monodromy homomorphism for $\nabla$ is contained in $\operatorname{SL}(2,{\mathbb{R}})$.
For $\rho\,=\,\tfrac{1}{2p}$ with $p$ being an odd integer, consider the connection $\nabla^{a,\chi,\rho}$, over bundle on $T^2$, given by Theorem \[real-mon-hatT2\]. Since the image of the monodromy homomorphism for $\Pi^*\nabla^{a,\chi,\rho}$ is conjugate to a subgroup of $\operatorname{SL}(2,{\mathbb{R}})$, and $\nabla^S$ has ${\mathbb{Z}}/2{\mathbb{Z}}$–monodromy, the image of the monodromy homomorphism for the connection $$D\, :=\, \nabla^S\otimes(\Pi\circ\Phi_p)^*\nabla^{a,\chi,\rho}$$ can be conjugated into $\operatorname{SL}(2,{\mathbb{R}})$ as well. The same holds for the connection $$\nabla\,:=\,F\circ(\nabla^S\otimes(\Pi\circ\Phi_p)^*\nabla^{a,\chi,\rho}) \circ F^{-1}$$ because $F$ is a (singular) gauge transformation. By Lemma \[Lemired\], $$\nabla-D$$ is a holomorphic Higgs field on the trivial holomorphic vector bundle $(V,\,D'').$
It remains to show that the monodromy homomorphism for $\nabla$ is an irreducible representation of the fundamental group. Since $\rho\,\neq\,0$ is small, this follows again from Lemma \[irr\]. Indeed, observe that the monodromies along the curves $$\widetilde{\alpha},\, \widetilde{\beta}\,\in\,\pi_1(\Sigma,q)$$ (see Figure \[figure3\]) are given by $$h(\alpha)h(\alpha) \ \ \text{ and }\ \ h(\beta)h(\beta)$$ up to a possible sign. Because $xy\,\neq\, 0$, in view of and and continuity in $\rho$, the monodromy representation must be irreducible by Lemma \[irr\].
Figures
=======
![The 1-punctured torus.[]{data-label="figure1"}](1torus.pdf){width="63.50000%"}
![The 4–punctured torus.[]{data-label="figure2"}](4torus.pdf "fig:"){width="37.50000%"} ![The 4–punctured torus.[]{data-label="figure2"}](immersion.pdf "fig:"){width="50.00000%"}
![The Riemann surface $\Sigma$ for $q\,=\,3$, shown with vertical and horizontal trajectories of $(\Pi\circ\pi_3)^*(dw)^2$. Picture by Nick Schmitt.[]{data-label="figure3"}](lawson5.pdf){width="50.00000%"}
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abstract: 'Certain hyperbolic monopoles and all hyperbolic vortices can be constructed from $\text{SO}(2)$ and $\text{SO}(3)$ invariant Euclidean instantons, respectively. This observation allows us to describe a large class of hyperbolic monopoles as hyperbolic vortices embedded into $\mathbb{H}^3$ and yields a remarkably simple relation between the two Higgs fields. This correspondence between vortices and monopoles gives new insight into the geometry of the spectral curve and the moduli space of hyperbolic monopoles. It also allows an explicit construction of the fields of a hyperbolic monopole invariant under a $\mathbb{Z}$ action, which we compare to periodic monopoles in Euclidean space.'
author:
- |
Rafael Maldonado[^1]\
*Department of Applied Mathematics and Theoretical Physics,*\
*Wilberforce Road, Cambridge CB3 0WA, U.K.*
title: Hyperbolic monopoles from hyperbolic vortices
---
=1
DAMTP-2015-47
Introduction
============
It has been known for some time that both the BPS monopole and the Abelian-Higgs vortex equations are more tractable in hyperbolic space (of a prescribed curvature) than in Euclidean space, with solutions expressible as rational functions. The reason for this simplification is that both cases are reductions of the (conformally invariant) self-duality equations in $\mathbb{R}^4$, by an $\text{SO}(2)$ and an $\text{SO}(3)$ action respectively.
In this paper we explore the relation between monopoles and vortices in hyperbolic space. In the remainder of this section we review the construction of hyperbolic monopoles and vortices. Section \[symminst\] discusses how the hyperbolic monopole and vortex equations come about from instanton reductions and shows how hyperbolic vortices can be used to construct hyperbolic monopoles. A description of this procedure in terms of JNR data is given in section \[JNRsection\]. In section \[JNRSC\] we look at the spectral curve of the resulting hyperbolic monopoles and compare to the spectral data of Euclidean monopoles. The metric on the $2$-hyperbolic-monopole moduli space (defined via the connection at infinity) is compared to the physical metric on the underlying hyperbolic vortex moduli space in section \[secmod\]. Finally, in section \[chainssection\] we use the procedure of section \[symminst\] to construct a periodic hyperbolic monopole, for which a direct construction in terms of JNR or ADHM data is not currently known.
Hyperbolic vortices {#hyperbolicvortices}
-------------------
Abelian Higgs vortices consist of a complex Higgs field $\phi$ and a two-component gauge potential $a$. At critical coupling there is a topological energy bound, and this fixes the number of zeros of $\phi$. Away from its zeros, $|\phi|^2$ obeys the Taubes equation: $$\Delta\log|\phi|^2+2(1-|\phi|^2)\,=\,0,\label{Taubeseq}$$ where $\Delta$ is the Laplace-Beltrami operator, which for a conformally flat background is given by $\Omega^{-1}\nabla^2$, where $\Omega$ is the conformal factor and $\nabla^2$ is the Euclidean Laplacian. On the hyperbolic plane of Gauss curvature $-1$ the Taubes equation can be reduced to the Liouville equation, which is integrable. Working in the Poincaré disk model, solutions to the Taubes equation are given in terms of a holomorphic function $f(w)$ satisfying $|f(w)|\leq1$, with equality on the boundary of the disk $|w|=1$. Explicitly, $$\phi\,=\,\frac{1-|w|^2}{1-|f(w)|^2}\frac{df}{dw},\qquad\qquad a_{\bar{w}}\,=\,-\text{i}\partial_{\bar{w}}\log(\phi),\label{phifformula}$$ which are defined up to a $\text{U}(1)$ gauge transformation. For prescribed vortex locations it is possible in principle to construct the required function $f(w)$ as a Blaschke product. An equivalent construction in terms of JNR data with poles on the boundary circle will be discussed in section \[JNRsection\].
Hyperbolic monopoles {#intromonopoles}
--------------------
$\text{SU}(2)$ monopoles consist of an adjoint-valued scalar $\Phi$ and a three-component gauge potential $A$. In hyperbolic $3$-space $\mathbb{H}^3$ the Bogomolny monopole equations are $$F_{ij}\,=\,\sqrt{\Omega}\,\epsilon_{ijk}D_k\Phi.\label{Bogomolnyeqs}$$ Solutions are rational if the boundary condition $\|\Phi\|^2\coloneqq-\tfrac{1}{2}\text{tr}(\Phi^2)\to v^2$ has half-integer $v$. The simplest case has $v=\tfrac{1}{2}$, when a large family of monopoles can be constructed from JNR data with poles on the boundary of $\mathbb{H}^3$. More generally, all solutions for $v=\tfrac{1}{2}$ arise from circle-invariant ADHM data, while for all half-integer $v$ one obtains a discrete version of the Nahm equations, known as the Braam-Austin equations [@BA90].
Examples of $v=\tfrac{1}{2}$ hyperbolic monopoles which have been studied include those with axial [@Coc14] and Platonic [@MS14] symmetry. More recently, monopoles of large charge have been modelled as magnetic bags [@BHS].
Symmetric instantons {#symminst}
====================
The goal of this paper is to explore the relation between hyperbolic monopoles and vortices by means of the underlying $\text{SO}(3)$-invariant instanton. We firstly lift the general vortex solution to an instanton using Witten’s approach [@Wit77], which is suited to the upper half space model of hyperbolic space. The instanton is then reduced to a monopole by imposing a circle invariance. This leads to a simple expression relating the monopole and vortex Higgs fields. We then confirm that for this class of monopoles, the Bogomolny equations imply the Taubes equation on the vortex fields.
Conformal rescalings
--------------------
Before we proceed, let us fix our conventions. The metric on $\mathbb{E}^4$ is $$ds^2_{\mathbb{E}^4}\,=\,(dx^4)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2\,=\,(dx^4)^2+(dx^1)^2+d\rho^2+\rho^2d\xi^2,$$ where $x^2=\rho\cos(\xi)$ and $x^3=\rho\sin(\xi)$. Imposing independence from the coordinate $\xi$, this metric is conformally equivalent to hyperbolic $3$-space with the upper half space metric $$ds^2_{\mathbb{H}^3}\,=\,\frac{1}{\rho^2}\left((dx^4)^2+(dx^1)^2+d\rho^2\right).\label{H3metric}$$ Now introduce the coordinates $r\geq0$ and $\theta\in[0,\pi)$ via $x^1=r\cos(\theta)$, $\rho=r\sin(\theta)$. Then $r$, $\theta$ and $\xi$ are standard spherical polar coordinates with respect to which $$ds^2_{\mathbb{E}^4}\,=\,(dx^4)^2+dr^2+r^2\left(d\theta^2+\sin^2(\theta)d\xi^2\right),$$ where $r^2=(x^1)^2+\rho^2=(x^1)^2+(x^2)^2+(x^3)^2$. Quotienting by the angular dependence now gives a metric on the hyperbolic plane $\mathbb{H}^2$, $$ds^2_{\mathbb{H}^2}\,=\,\frac{1}{r^2}\left((dx^4)^2+dr^2\right).\label{H2metric}$$ The relation between the metrics and is interesting. Restricting to $\theta=\pi/2$, reads $$\left.ds_{\mathbb{H}^2}^2\right|_{\theta=\frac{\pi}{2}}\,=\,\frac{1}{\rho^2}\left((dx^4)^2+d\rho^2\right),$$ which by comparison with is a slice of $\mathbb{H}^3$ (an equatorial slice of the unit ball model of $\mathbb{H}^3$ is a unit disc carrying a hyperbolic metric). There is a more subtle reduction if we restrict to $\theta=0$. Then becomes $$\left.ds_{\mathbb{H}^2}^2\right|_{\theta=0}\,=\,\frac{1}{(x^1)^2}\left((dx^4)^2+(dx^1)^2\right).\label{boundarymetric}$$ This is the boundary of $\mathbb{H}^3$ equipped with a hyperbolic metric, and is known as the ‘hemisphere model’ of $\mathbb{H}^2$. Since $x^1$ can take either sign, this is two copies of the hyperbolic plane glued along the $x^4$-axis. By extension there is a family of such metrics, according to a choice of the angle $\theta$.
We will frequently use the ball model of $\mathbb{H}^3$, where cyclic symmetry is more apparent than in the upper half space model. The ball model coordinates are given in terms of the upper half space coordinates by $$X^1+\text{i}X^2\,=\,\frac{2(x^4+\text{i}x^1)}{(x^1)^2+(x^4)^2+(\rho+1)^2},\qquad X^3\,=\,\frac{(x^1)^2+(x^4)^2+(\rho^2-1)}{(x^1)^2+(x^4)^2+(\rho+1)^2},$$ $$R^2\,=(X^1)^2+(X^2)^2+(X^3)^2\,=\,\frac{(x^1)^2+(x^4)^2+(\rho-1)^2}{(x^1)^2+(x^4)^2+(\rho+1)^2},\label{ballR}$$ and the ball metric is $$ds^2\,=\,\frac{4}{(1-R^2)^2}\left((dX^1)^2+(dX^2)^2+(dX^3)^2\right).$$ For completeness, we invert these expressions to give the upper half space coordinates in terms of the ball model coordinates: $$x^4+\text{i}x^1\,=\,\frac{2(X^1+\text{i}X^2)}{1+R^2-2X^3},\qquad\rho\,=\,\frac{1-R^2}{1+R^2-2X^3}.$$
Dimensional reductions {#dimreds}
----------------------
Monopoles and vortices in hyperbolic space are constructed by dimensional reductions of instantons on $\mathbb{E}^4$. The self-duality (instanton) equations are conformally invariant, so solutions are unchanged under the conformal rescalings of the background metric described above. Instantons invariant under a circle symmetry can then be dimensionally reduced to monopoles on $\mathbb{H}^3$, while $\text{SO}(3)$-invariant instantons give rise to hyperbolic vortices.
The reduction of circle-invariant instantons to hyperbolic monopoles was first considered by Atiyah [@Ati88] and carried out by Chakrabarti [@Cha86] and Nash [@Nas86]. Given an instanton gauge potential $A_i(x^4,x^1,x^2,x^3)$ which is independent of $\xi=\tan^{-1}(x^3/x^2)$, one must perform a gauge transformation $G$ such that $A^G_i$ is explicitly independent of $\xi$. In this gauge, the monopole Higgs field $\Phi$ is identified with $A^G_\xi$, and the monopole gauge potential has components $A^G_4$, $A^G_1$, $A^G_\rho$.
The relation between instantons and hyperbolic vortices first arose in Witten’s search for cylindrically symmetric instantons [@Wit77]. In the upper half plane model of $\mathbb{H}^2$, , a vortex consists of a Higgs field $\phi=\phi_1+\text{i}\,\phi_2$ and a gauge potential $a=a_4\,dx^4+a_r\,dr$, which we assume is in Coulomb gauge, $\partial_ia_i=0$. From these one constructs an $\text{SO}(3)$-invariant instanton: $$A_i\,=\,\frac{\text{i}}{2}\left(\frac{\phi_2+1}{r^2}\epsilon_{ijk}x^k\tau_j+\frac{\phi_1}{r^3}[r^2\tau_i-x^ix^j\tau_j]+\frac{a_rx^i}{r^2}\,x^j\tau_j\right),\qquad A_4\,=\,\frac{\text{i}a_4}{2r}\,x^j\tau_j,\label{instanton}$$ where $i$ runs from $1$ to $3$, $r^2=(x^1)^2+(x^2)^2+(x^3)^2$ and all the $x^4$ dependence is encoded in the vortex fields.
We would like to explore the class of hyperbolic monopoles obtained by lifting hyperbolic vortices to instantons and then reducing by a circle action. To do this, we first of all combine the $A_2$ and $A_3$ components of the instanton gauge potential into $A_\rho=(x^2A_2+x^3A_3)/\rho$ and $A_\xi=-x^3A_2+x^2A_3$,
[rcl]{} A\_&=&(x\^1\[-\_2+\_3\]+\[-\_1+x\^1\_2+x\^1\_3\]+x\^j\_j)\
A\_&=&(+\[-\_2+\_3\]),
where $\text{c}=\cos(\xi)$ and $\text{s}=\sin(\xi)$. Now the $A_i$ are rendered explicitly independent of $\xi$ by application of the gauge transformation $$A_i\,\mapsto\,A_i^G\,=\,G^{-1}A_iG+G^{-1}\partial_iG,$$ with $G=\text{exp}\left(-i\xi\tau_1/2\right)$. The monopole fields are then simply the transformed gauge potential:
[rcl]{} A\_1\^G&=&(-\_3+\[\_1-x\^1\_2\]+\[x\^1\_1+\_2\])\[A1\]\
A\_\^G&=&(x\^1\_3-\[\_1-x\^1\_2\]+\[x\^1\_1+\_2\])\[Arho\]\
=A\_\^G&=&(+\_3)-\_1\[Phi\]\
A\_4\^G&=&\[x\^1\_1+\_2\].\[A4\]
The instanton fields will match the standard JNR gauge introduced in section \[JNRsection\] if $$a_4\,=\,\frac{\phi_2+1}{r}\qquad\text{and}\qquad a_r\,=\,\frac{\phi_1}{r},\label{Mantongaugea}$$ and with this choice the monopole gauge potential is automatically in Coulomb gauge, $d_iA_i^G=0$.
From we obtain the key formula relating the norms of the vortex and monopole Higgs fields: $$\|\Phi\|^2\,=\,\frac{\rho^2|\phi|^2+(x^1)^2}{4\,r^2},\label{modphisquared}$$ where $r^2=(x^1)^2+\rho^2$ and $\phi$ is a function of $x^4$ and $r$. Let us analyse this formula in more detail. Recall that we are working on the upper half space whose boundary is the $(x^4,x^1)$ plane. $\|\Phi\|^2$ has the correct boundary behaviour for a monopole with $v=\tfrac{1}{2}$ ($\|\Phi\|^2\to\tfrac{1}{4}$ as we approach the boundary $\rho\to0$), and its zeros occur where $x^1=0$ and $\phi=0$. In the equatorial plane $x^1=0$, the monopole Higgs field $\|\Phi\|^2$ is proportional to the vortex Higgs field $|\phi|^2$, providing an obvious interpretation of the monopole as an embedded vortex.
Now take $(x^4_0,r_0)$ to be the position of a vortex zero. Setting $r=r_0$ defines a geodesic in the upper half space: a semicircle which meets the boundary at $(x^4,x^1)=(x^4_0,\pm r_0)$, as shown in figure \[fig1\]. As a function of the hyperbolic distance $d_{\text{H}}$ from the monopole zero, measured along this geodesic, the Higgs field is $$\|\Phi\|^2_{\phi=0}\,=\,\frac{(x^1)^2}{4r^2}\,=\,\frac{1}{4}\tanh^2(d_{\text{H}}).\label{Phiphi0}$$ This is precisely the radial profile function of a single hyperbolic monopole, and the result is independent of the multiplicity of the associated monopole zero and of its position relative to any other monopoles in the configuration.[^2] In section \[JNRSC\] we will see that these distinguished geodesics are always spectral lines.
![Monopoles are located on the black dots in the plane $x^1=0$. The vortex Higgs field $\phi$ vanishes along the solid lines, representing geodesics in the upper half space.[]{data-label="fig1"}](fig1-eps-converted-to.pdf){width="0.8\linewidth"}
A similar analysis to that given in this section allowed Cockburn [@Coc14] to relate axially symmetric monopoles to charge one monopoles of half-integer mass $v>\tfrac{1}{2}$.
Field equations {#fieldequations}
---------------
Now let us check that the Bogomolny equations imply the vortex equations. Note that $\partial_1\tilde{\phi}=x^1(\partial_r\tilde{\phi})/r$, $\partial_\rho\tilde{\phi}=\rho(\partial_r\tilde{\phi})/r$, where $\tilde{\phi}$ represents any of the vortex fields, which are independent of $\theta$. Then, using the fields - but suppressing the superscript ${}^G$ for clarity,
[rcl]{} F\_[41]{}&=&\_4A\_1-\_1A\_4+\[A\_4,A\_1\]\
&=&((\_4\_1+a\_4\_2)\_a+x\^1(\_4a\_r-\_ra\_4)\_b-(\_4\_2-a\_4\_1)\_3)\[F41\]
[rcl]{} D\_&=&\_+\[A\_,\]\
&=&((\_r\_2-a\_r\_1)\_a+(1-\_1\^2-\_2\^2)\_b+(\_r\_1+a\_r\_2)\_3),
where $\tau_a=[\rho\tau_1-x^1\tau_2]$ and $\tau_b=[x^1\tau_1+\rho\tau_2]$. The Bogomolny equations $$F_{ij}\,=\,\frac{1}{\rho}\,\epsilon_{ijk}D_k\Phi$$ with $\epsilon_{41\rho}=1$ imply the complex vortex equation $(D^{(\text{v})}_4+\text{i}D^{(\text{v})}_r)\phi=0$, where the superscript ${}^{(\text{v})}$ is used to distinguish the covariant deriviative for the vortex fields, $D_i^{(\text{v})}=\partial_i-\text{i}a_i$, from its counterpart for monopole fields. One similarly obtains the real vortex equation, $r^2B=(1-\phi_1^2-\phi_2^2)$. It would be interesting to study whether a similar embedding of the vortex equations into $\mathfrak{su}(2)$ is be possible for $v\neq\tfrac{1}{2}$.
Monopole number
---------------
We should check that the fields - have the correct topology to be monopoles. The fields are reflection-symmetric: the replacement $x^1\to-x^1$ reverses the orientation and changes the fields by a gauge: $$A_1'\,=\,-\tau_2A_1\tau_2,\qquad A_\rho'\,=\,\tau_2A_\rho\tau_2,\qquad\Phi'\,=\,-\tau_2\Phi\tau_2,\qquad A_4'\,=\,\tau_2A_4\tau_2.\label{orientation}$$ Now compute the Chern number by performing the integral $$c_1\,=\,-\int\frac{\text{tr}(F\Phi)}{4\pi\|\Phi\|}\label{chern}$$ over the boundary of $\mathbb{H}^3$. In upper half space coordinates this is the $x^4-x^1$ plane. Setting $\rho=0$, we have from and that $$\Phi^0\,=\,-\frac{\text{i}}{2}\,\tau_1,\qquad\qquad F_{41}^0\,=\,\frac{\text{i}}{2}\left(\partial_4a_r-\partial_ra_4\right)\tau_1\,=\,\frac{\text{i}}{2}\,B\,\tau_1.$$ Evaluating , we get $$c_1\,=\,-\frac{1}{2\pi}\int_{\partial\mathbb{H}^3}\frac{1}{2}\,B\,dx^4\wedge dx^1\label{integral}$$ Now $r^2=(x^1)^2$ (since $\rho=0$), but unlike the coordinate $r$, $x^1$ spans the entire real line and the integral is performed over two copies of the upper half plane. This gives $$c_1\,=\,-\frac{1}{2\pi}\int_{\mathbb{H}^2}\frac{1}{2}\,\frac{B}{\Omega}\,\left(2\,\Omega\,dx^4\wedge dr\right)\,=\,-N,$$ so a charge $N$ vortex lifts to a charge $N$ monopole.
Energy density {#energysection}
--------------
The energy density of a monopole is obtained by applying the Laplace-Beltrami operator to $\|\Phi\|^2$. In the upper-half-space model we are using, this is $$\mathcal{E}\,=\,\rho^2\left(\partial_1^2+\partial_4^2+\partial_\rho^2-\frac{1}{\rho}\,\partial_\rho\right)\|\Phi\|^2.$$ Written in terms of derivatives of the vortex Higgs field $|\phi|^2$ gives $$\mathcal{E}\,=\,\frac{\rho^4}{r^4}\,\left(\frac{1}{4}\,\Delta|\phi|^2+\frac{1}{2}\,(1-|\phi|^2)\right),\qquad\text{where}\qquad\Delta\,=\,r^2\left(\partial_4^2+\partial_r^2\right)\label{energy}$$ is the Laplace-Beltrami operator acting on the vortex Higgs field in the upper half plane model of $\mathbb{H}^2$. We recognise the bracketed term in as the energy density of the vortex defined in [@MN99]. Integrating over the upper half space we find $$E\,=\,\int\mathcal{E}\,\frac{1}{\rho^3}\,d\rho\,dx^1\,dx^4\,=\,2\pi N.
$$
Example: a single monopole
--------------------------
Let us illustrate our discussion with a simple example. A single vortex in the Poincaré disk has $$|\phi|\,=\,\frac{2|w|}{1+|w|^2}.\label{onevortex}$$ Now convert to upper half plane coordinates $z=x^4+\text{i}r$ using $$w\,=\,\frac{\text{i}-z}{\text{i}z-1},$$ This gives $$|\phi|^2\,=\,\frac{((x^4)^2+(r-1)^2)((x^4)^2+(r+1)^2)}{((x^4)^2+r^2+1)^2},\label{oneuhpvortex}$$ then from and we find, after some manipulation, $$\|\Phi\|^2\,=\,\frac{1}{4}-\frac{\rho^2}{\left((x^1)^2+(x^4)^2+\rho^2+1\right)^2}\,=\,\frac{R^2}{(1+R^2)^2}.\label{onemonopole}$$ Applying the energy density formula to the vortex gives $$\mathcal{E}_1\,=\,\frac{3}{2}\left(\frac{1-R^2}{1+R^2}\right)^4$$ as expected for the charge one monopole . In section \[chainssection\] we will use this method to obtain a new explicit hyperbolic monopole solution.
JNR construction {#JNRsection}
================
The JNR Ansatz [@JNR77] gives a large class of instantons. An $N$-instanton is generated by the function (harmonic in $\mathbb{R}^4$) $$\psi\,=\,\sum_{j=0}^N\frac{\lambda_j^2}{|x-\gamma_j|^2}$$ which gives the instanton gauge potentials $$A_i\,=\,\frac{\text{i}}{2}\left[\epsilon_{ijk}\,\partial_j\log(\psi)\tau_k+\partial_4\log(\psi)\tau_i\right],\qquad A_4\,=\,-\frac{\text{i}}{2}\,\partial_i\log(\psi)\tau_i$$ where $\tau_i$ are the Pauli matrices. Only for $N=1$ does the JNR construction give all possible instantons.
The dimensional reductions of the preceding section can be made at the level of JNR data. Circle-invariant JNR data gives a subset of hyperbolic monopoles. The poles of $\psi$ must lie on a plane (the fixed set of a circle action) in $\mathbb{E}^4$, which becomes the boundary of $\mathbb{H}^3$. Counting parameters suggests that all hyperbolic monopoles for $N\leq3$ can be generated in this way [@BCS15]. To reduce the monopoles to vortices, we have the additional constraint that the poles must be on the fixed set of an $\text{SO}(3)$ action, i.e. on a line in $\mathbb{E}^4$. It was shown by Manton [@Man78] that the JNR Ansatz generates all hyperbolic vortices, i.e. that it is gauge-equivalent to the formulation of section \[hyperbolicvortices\].
A suitable definition of ‘centered’ hyperbolic monopoles is given in [@MNS03]. The centered moduli space has dimension $4(N-1)$, while the moduli space of centered hyperbolic vortices has dimension $2(N-1)$. There is an $S^2$ worth of freedom in our choice of embedding of the hyperbolic vortices into $\mathbb{H}^3$, so the construction presented in this paper gives a $2N$ dimensional family of centered hyperbolic monopoles. In particular, we obtain all centered $2$-monopoles, whose moduli space is explored in section \[secmod\].
Using the same upper half space coordinates as before, a monopole Higgs field is constructed using the JNR function $$\psi\,=\,\sum_{j=0}^N\frac{\lambda_j^2}{|x^4+\text{i}x^1-\gamma_j|^2+\rho^2}\label{JNRpsi}$$ in $$\|\Phi\|^2\,=\,\frac{\rho^2}{4\psi^2}\left(\left(\frac{\partial\psi}{\partial x^4}\right)^2+\left(\frac{\partial\psi}{\partial x^1}\right)^2+\left(\frac{\psi}{\rho}+\frac{\partial\psi}{\partial\rho}\right)^2\right).\label{JNRPhi}$$ Placing all the poles of on the real $x^4$-axis gives $$\psi\,=\,\sum_{j=0}^N\frac{\lambda_j^2}{(x^4-\gamma_j)^2+r^2},\label{JNRpsi2}$$ and the vortex Higgs field is given by [@Man78] $$|\phi|^2\,=\,\frac{r^2}{\psi^2}\left(\left(\frac{\partial\psi}{\partial x^4}\right)^2+\left(\frac{\psi}{r}+\frac{\partial\psi}{\partial r}\right)^2\right)\,=\,-r^2\left(\partial_4^2+\partial_r^2\right)\log(r\psi).\label{vortexJNRformula}$$ Fixing the phase of $\phi$ by specialising to Coulomb gauge and using the relations gives the components of the gauge potential as $$a_4\,=\,-\partial_r\log\psi,\qquad\qquad a_r\,=\,\partial_4\log\psi.\label{Mantongaugeb}$$
Using in and changing variables again gives the relation . Of course, there are certain vortex configurations for which the JNR function $\psi$ is not known. The more general argument of section \[symminst\] ensures that is still valid, and it is for these configurations that the construction of monopoles as an embedding of vortices provides truly novel monopole solutions.
The remarkable similarity between and invites us to consider a further dimensional reduction. The resulting one-dimensional field theory describes the $\mathrm{SO}(4)$-invariant instanton. Using the radial coordinate $\varrho^2=r^2+(x^4)^2$ we define[^3] $$\varphi^2\,=\,\frac{\varrho^2}{\psi^2}\left(\frac{\psi}{\varrho}+\frac{d\psi}{d\varrho}\right)^2,\label{kinkphi}$$ where $\psi$ is a function of $\varrho$ only. Combining with , the corresponding vortex Higgs field is $$|\phi|^2\,=\,\frac{r^2\varphi^2+(x^4)^2}{r^2+(x^4)^2}.\label{vortexkink}$$ Mimicking what we did in section \[fieldequations\], we substitute into the Taubes equation , to yield $$\frac{d\varphi}{d\,\log(\varrho)}\,=\,1-\varphi^2,$$ which is the Bogomolny equation for a $\varphi^4$ kink. In other words, we can obtain the charge $1$ hyperbolic vortex by embedding the (essentially unique) $\varphi^4$ kink into $\mathbb{H}^2$. Lifting to $\mathbb{H}^3$, the hyperbolic tangent function describing the $\varphi^4$ kink shows up when the Higgs field of a single monopole is expressed as a function of hyperbolic distance from the Higgs zero, . By a change of coordinates we regain the BPST instanton [@BPST75]. The $\varrho$ coordinate of the kink is precisely the scale size $\lambda$ of the instanton.
Spectral data {#JNRSC}
=============
The spectral curve of a hyperbolic monopole is defined by scattering data, as the set of geodesics along which $$(D_s-\text{i}\Phi)w\,=\,0\label{specdef}$$ has normalisable solutions, where $s$ is the arc length along the curve. The spectral curve can be given explicitly in terms of the positions and weights of JNR poles [@BCS15]: $$\mathcal{S}:\qquad\sum_{j=0}^N\lambda_j^2\prod_{k\neq j}(\zeta-\gamma_k)(1+\eta\bar{\gamma}_k)\,=\,0.\label{scjnr}$$ Geodesics in $\mathbb{H}^3$ are parametrised in terms of their endpoints $\zeta$ and $-\bar{\eta}^{-1}$ on the boundary $\mathbb{R}^2\cong\mathbb{C}$. We are interested in embedded vortices, where all JNR poles lie on the real axis, so $\gamma_k=\bar{\gamma}_k$. Any $2$-monopole can be cast in this form by an appropriate choice of centre and orientation.
In the following sections we study three distinguished classes of spectral lines.
Spectral lines through the monopole zeros {#speclinesthroughzeros}
-----------------------------------------
Consider a vortex configuration embedded in $\mathbb{H}^3$ as described in section \[dimreds\], where it was observed that geodesics through monopole zeros orthogonal to the plane $x^1=0$ have $\phi=0$. The monopole Higgs field $\Phi$ along this line is the radial field of a unit charge hyperbolic monopole. It then follows from the definition that such geodesics are spectral lines, by virtue of the fact that all spectral lines of a charge $1$ monopole pass through the zero. We see this by expressing $\phi$ in terms of JNR data, such that $$\phi(z_0)\,=\,0\qquad\Rightarrow\qquad\left.\left((\bar{z}-z)\partial_z\log(\psi)\right)\right|_{z=z_0}\,=\,1\qquad\Rightarrow\qquad\sum_{j=0}^N\lambda_j^2\prod_{k\neq j}(z_0-\gamma_k)^2\,=\,0,$$ where $z=x^4+\text{i}r$. Solutions for $z=z_0$ define geodesics in $\mathbb{H}^3$ which meet the boundary of the upper half space at $\zeta=z_0$ and $\zeta=\bar{z}_0$. By comparison with , we see that these geodesics are in fact the unique spectral lines with $\eta=-\zeta^{-1}$, i.e. which intersect the plane $x^1=0$ at right angles. This observation should be contrasted with the case of Euclidean monopoles, when the spectral lines of a generic charge $2$ monopole only approximately pinpoint the zero.
Spectral lines in the plane of the vortices
-------------------------------------------
We now analyse some of the spectral lines described by . Firstly, note that geodesics between any pair of JNR poles are spectral lines. It is also clear that there are precisely $N$ spectral lines for each choice of $\zeta$ on the boundary, and that any geodesic with $\zeta\in\mathbb{R}$ also has $\eta\in\mathbb{R}$. Specialising to $N=2$ with $\zeta\in\mathbb{R}$ leads to an interesting geometric picture in terms of Poncelet’s theorem, which has already given insight into the geometry of instantons [@Har78] and indeed hyperbolic monopoles [@Hit95]. We will work through the details explicitly in our case, making use of various theorems of Daepp-Gorkin-Mortini [@DGM02] and D. Singer [@Sin06].
We work with the ball model of $\mathbb{H}^3$, where the equatorial slice defined by $\zeta\in\mathbb{R}$ is a Poincaré disk with complex coordinate $w=X^1+\text{i}X^3$. The boundary $w=\text{i}\,\text{e}^{-\text{i}\theta}$ is related to the coordinate $\zeta$ by stereographic projection: $\zeta=\cot(\theta/2)$. For notational convenience we will consider a centered $2$-monopole aligned with the $X^3$-axis, although the discussion follows through for any value of the (vortex) moduli. The spectral curve can be parametrised as $$\gamma^2(\zeta^2-\gamma^2)(1-\eta^2\gamma^2)-(1-\gamma^4)\eta(\zeta-\eta\gamma^2)\,=\,0,\label{speccharge2}$$ with $\tfrac{1}{3}\!\leq\!\gamma^2<1$, and the relation between $\gamma$ and the monopole separation will be clarified in section \[secmod\].
Recall from section \[hyperbolicvortices\] that a centered charge $2$ hyperbolic vortex can be constructed from the $C_2$ symmetric Blaschke product $$f(w)\,=\,w\,\frac{w^2+a^2}{1+a^2 w^2},\label{blas}$$ where vortex zeros are located at the critical points of $f(w)$ and $a^2$ is related to $\gamma^2$ by $(\gamma^2+1)(a^2+3)=4$. Restricting to the action of $f$ on the boundary, it is established in [@DGM02] that $f$ is a surjection and that a point $w=w_0$ has exactly $3$ preimages $\{w_1,w_2,w_3\}=f^{-1}(\{w_0\})$, defining an ideal triangle. The edges of this triangle are spectral lines, a fact that is readily checked by direct computation in simple cases, or numerically for more generic values of the parameters. The prescribed Blaschke product then generates all of the spectral lines (with $\zeta\in\mathbb{R}$) and hence a family of ideal triangles corresponding to the gauge freedom in the JNR data. It was shown in [@Sin06] that the envelope of this family of triangles is a hyperbolic ellipse (the locus of points for which the sum of the geodesic distances from the foci is constant) whose foci are at the critical points of $f$, i.e. at the vortex zeros.[^4] Figure \[fig2\] shows the hyperbolic ellipse for the monopole with $\gamma^2=\tfrac{1}{4}$.
![Some spectral lines for a charge $2$ hyperbolic monopole with $\gamma^2=\tfrac{1}{4}$, restricted to the equatorial plane $X^2=0$ in which the vortices are embedded. The monopole zeros are located at the foci of the inscribed hyperbolic ellipse.[]{data-label="fig2"}](fig2-eps-converted-to.pdf){width="0.4\linewidth"}
Principal axes and spectral radii {#principalaxes}
---------------------------------
Atiyah and Hitchin [@AH88] observed that there are two spectral lines through the centre of a charge $2$ monopole. This fact is used to define the principal axes of the monopole, which in turn define the Euler angles, as natural coordinates on the moduli space. A similar definition is possible in the hyperbolic case. Spectral lines through the origin of the hyperbolic ball have $\eta=\zeta$. Taking a configuration of the form with $\gamma^2\leq\tfrac{1}{3}$, these spectral lines are always contained in the plane $X^3=0$, and coalesce along the $X^2$ axis when $\gamma^2=\tfrac{1}{3}$. The axis $e_1$ is defined as the bisector of the angle between these spectral lines. The second bisector defines the axis $e_2$, which lies in the plane of the JNR poles. The third principal axis, $e_3$, is parallel to the line of separation of the monopole zeros.
The three spectral radii of a Euclidean $2$-monopole are defined as half the separation between the unique two spectral lines parallel to each of the three principal axes, [@AH88]. In the hyperbolic setting we will define the spectral radii as the minimal geodesic separation between each pair of spectral lines orthogonal to one of the principal axes. This gives two of the spectral radii as the semi-major and semi-minor axes of the hyperbolic ellipse discussed above: $$d_{\pm}\,=\,\cosh^{-1}\left(\frac{2}{\sqrt{3\mp2a^2-a^4}}\right).$$ In section \[speclinesthroughzeros\] we showed that the only pair of spectral lines which meet the equatorial plane at right angles are those through the monopole zeros. This gives the third spectral radius as half the hyperbolic distance between the zeros, $$d_3\,=\,\cosh^{-1}\left(\sqrt{\frac{3+2a^2-a^4}{3-2a^2-a^4}}\right).$$ Atiyah & Hitchin’s observation [@AH88] that the three spectral radii define a right-angled triangle also holds in the hyperbolic case, i.e. $\cosh(d_-)\cosh(d_3)=\cosh(d_+)$. From our description, we see that this fact follows immediately from the definition of an ellipse. The area of this triangle is minimal when $a=2^{2/3}-1$. Curiously, this corresponds precisely to the critical radius at which there is a closed geodesic in Hitchin’s metric [@Hit95].
Moduli space {#secmod}
============
Low energy scattering of solitons has successfully been modelled by geodesic motion on the moduli space. The metric on the moduli space is given by the $L^2$ norm of perturbations to the fields, subject to the gauge-fixing constraint that gauge orbits are orthogonal to such perturbations.
It is well known that the requisite integral diverges for hyperbolic monopoles, although various alternative metrics have been proposed. Examples are Hitchin’s metric on the space of spectral curves [@Hit95], and the $L^2$ metric on the space of circle-invariant instantons [@FS] (both of which have positive scalar curvature). We will focus on the metric defined via the connection on the boundary of $\mathbb{H}^3$, [@BA90; @MNS03; @BCS15], and compare this metric to the $L^2$ metric on the moduli space of the underlying hyperbolic vortices. In the charge $1$ case these metrics are both proportional to the underlying hyperbolic metric. We thus focus on vortices and monopoles of charge $2$ and fixed centre of mass.
The centered $2$-vortex metric was computed by Strachan [@Str92]. For vortices located at $z=\pm\alpha\text{e}^{\text{i}\theta}$ in the Poincaré disk, the gauge condition is $$2\,\partial_i(\delta a_i)+\text{i}\left(\bar{\phi}\delta\phi-\phi\delta\bar{\phi}\right)\,=\,0\label{Strachangauge}$$ and the metric takes the form
[rcl]{} ds\^2&=&(|+g\^[ij]{}a\_ia\_j)dxdr\
&=&(1+).\[Strachanmetric\]
Note that the gauge condition does not allow the variations in the fields to be computed by varying the JNR function $\psi$ in the gauge defined through (\[Mantongaugea\], \[Mantongaugeb\]).
Boundary fields
---------------
In order to define a metric on the hyperbolic monopole moduli space, we consider the fields on the boundary of the hyperbolic ball. In this section, we use coordinates $z=x+\text{i}r=x^4+\text{i}x^1$ with metric , $$ds^2\,=\,\frac{1}{r^2}(dx^2+dr^2).\label{bdymetric}$$ The boundary fields are obtained by taking the limit $\rho=0$ and $r=x^1$ in -: $$A_4^0\,=\,\frac{\text{i}}{2}\,a_4\tau_1,\qquad\qquad A_1^0\,=\,\frac{\text{i}}{2}\,a_r\tau_1,\label{Abdy}$$ $$A_\rho^0\,=\,\frac{\text{i}}{2}\left(\frac{\phi_1}{r}\tau_2+\frac{\phi_2+1}{r}\tau_3\right),\qquad\qquad\Phi^0\,=\,-\frac{\text{i}}{2}\,\tau_1.$$ As the Higgs field tends to a constant, the relevant gauge fixing condition is simply the Coulomb gauge $\partial_i(\delta a_i)=0$, which holds identically for fields of the form , $$a_x\,=\,-\partial_r\log\psi,\qquad\qquad a_r\,=\,\partial_x\log\psi,\label{axar}$$ allowing us to obtain the metric by varying $\psi$. The metric is then defined by $$ds^2\,=\,\int g^{ij}\,\delta a_i\,\delta a_j\,\sqrt{g}\,dx\,dr\,=\,\int\delta^{ij}\,\delta a_i\,\delta a_j\,dx\,dr,\label{metricformula}$$ where $g_{ij}$ is the hyperbolic metric on the boundary. The gauge potentials are simply those of a vortex in the hemisphere model of $\mathbb{H}^2$. However, the lack of a Higgs field contribution and the different gauge condition will give a metric different from .
The moduli space metric is invariant both under gauge transformations and conformal rescalings of the boundary metric . The Coulomb gauge condition leaves a residual gauge freedom to multiply $\psi$ by the modulus-squared of a holomorphic function, and we use this to remove the poles in $\psi$. The resulting JNR function can equivalently be obtained from the spectral curve polynomial by setting $(\zeta,\eta)=(z,-\bar{z}^{-1})$ and multiplying by $\bar{z}^N$. We denote the resulting function $h$.
Monopole metric: radial component
---------------------------------
We wish to compare to the radial component of the metric of two hyperbolic monopoles obtained from lifting a charge $2$ hyperbolic vortex to $\mathbb{H}^3$. To compute the metric for two hyperbolic monopoles whose zeros are in the plane $x^1=0$, we take the ’t Hooft function $$\psi\,=\,1+\frac{\lambda^2}{(x^4-\gamma)^2+r^2}+\frac{\lambda^2}{(x^4+\gamma)^2+r^2},\label{charge2JNR}$$ where $r^2=\rho^2+(x^1)^2$ and the poles are fixed to lie on the $x^4$ axis. A geodesic one-parameter family is obtained by imposing dihedral symmetry $D_2$, which requires that $2\lambda^2=\gamma^{-2}-\gamma^2$, and this is centered by the definition of [@MNS03]. To relate $\gamma$ to the positions of the Higgs zeros we must locate, from , the zeros of $\nabla^2\log(r\psi)$. There are two regimes: for $\gamma^2\in[0,\tfrac{1}{3}]$, the zeros are found at $x^4=x^1=0$ and $$\rho_0^{\pm2}\,=\,\frac{1}{2\gamma^2}\left(\left(1-3\gamma^4\right)+\sqrt{\left(1-3\gamma^4\right)^2-4\gamma^4}\right),\label{rhogamma}$$ while for $\gamma^2\in[\tfrac{1}{3},1]$ they are at $x^1=0$ and $$x^4_0\,=\,\pm\,\frac{1}{2\gamma}\sqrt{(1+\gamma^2)(3\gamma^2-1)},\qquad\qquad\rho_0\,=\,\frac{1}{2\gamma}\sqrt{(1-\gamma^2)(3\gamma^2+1)}.$$ Converting back to the ball model of $\mathbb{H}^3$, the monopoles are located at $$(X^1,X^2,X^3)\,=\,\left(0,\pm\frac{x^4_0}{\rho_0+1},0\right),\qquad\qquad\frac{1}{3}\leq\gamma^2\leq1$$ or $$(X^1,X^2,X^3)\,=\,\left(0,0,\pm\frac{\rho_0-1}{\rho_0+1}\right),\qquad\qquad0\leq\gamma^2\leq\frac{1}{3},\label{gammaalpha}$$ from which we define $$\alpha\,=\,\frac{\rho_0-1}{\rho_0+1}.$$ For ease of numerical computation we recast the JNR function into the form $$h\,=\,|z|^4-A(\gamma)\,(z^2+\bar{z}^2)+B(\gamma)\,|z|^2+1,\label{hfunction}$$ with $A=\gamma^2$ and $B=\gamma^{-2}-\gamma^2$. We now obtain the radial component of the moduli space metric from , using the relations and to change to the coordinate $\alpha$: $$g_{\alpha\alpha}\,d\alpha^2\,=\,\frac{4\,d\alpha^2}{(1-\alpha^2)^2}\,\rho_0^2\left(\frac{d\gamma^2}{d\rho_0}\right)^2\int\frac{\partial a_i}{\partial(\gamma^2)}\frac{\partial a_i}{\partial(\gamma^2)}\,dx\,dr\,\equiv\,f^2(\alpha)d\alpha^2.\label{monmet}$$ The integral in is evaluated numerically and the profile function is compared with the (rescaled) metric of the corresponding vortex, , in figure \[fig3\]. Note that in both cases the asymptotic metric approaches that of the underlying $\mathbb{H}^3$.
![Radial component of the metric as a function of $\alpha$, the distance of each Higgs zero from the origin. Solid line: analytic result for the vortex metric (rescaled by $32/9$). Dashed line: monopole metric . In both cases we have divided by the factor $4(1-\alpha^2)^{-2}$.[]{data-label="fig3"}](fig3-eps-converted-to.pdf){width="0.5\linewidth"}
Monopole metric: angular components
-----------------------------------
$\mathrm{SO}(3)$ and dihedral symmetry imply that the moduli space metric of two hyperbolic monopoles is diagonal when expressed in terms of the $\mathrm{SO}(3)$-invariant one-forms $\sigma_i$, [@AH88]: $$g\,=\,f^2(\alpha)\,d\alpha^2+a^2(\alpha)\,\sigma_1^2+b^2(\alpha)\,\sigma_2^2+c^2(\alpha)\,\sigma_3^2.$$ The function $f(\alpha)$ was defined in the previous section. To compute $a$, $b$, $c$ we rotate the poles of the standard JNR function so as to align each of the principal axes $e_1$, $e_2$, $e_3$ (identified in section \[principalaxes\]) with the $X^3$ coordinate axis in turn, as shown in figure \[fig4\].
![Orientation of the JNR poles used to compute the functions $a^2$, $b^2$ and $c^2$. The body-fixed $e_1$, $e_2$ and $e_3$ axes are aligned with the spatial $X^3$ axis in turn.[]{data-label="fig4"}](fig4a-eps-converted-to.pdf){width="\linewidth"}
![Orientation of the JNR poles used to compute the functions $a^2$, $b^2$ and $c^2$. The body-fixed $e_1$, $e_2$ and $e_3$ axes are aligned with the spatial $X^3$ axis in turn.[]{data-label="fig4"}](fig4b-eps-converted-to.pdf){width="\linewidth"}
![Orientation of the JNR poles used to compute the functions $a^2$, $b^2$ and $c^2$. The body-fixed $e_1$, $e_2$ and $e_3$ axes are aligned with the spatial $X^3$ axis in turn.[]{data-label="fig4"}](fig4c-eps-converted-to.pdf){width="\linewidth"}
The gauge potential is still determined by functions of the form , with
[rclcl]{} a\^2&&A=&&B=\
b\^2&&A=&&B=\
c\^2&&A=\^2&&B=.
Deformations are now parametrised by a rotation by an angle $\omega$ in the $z$ plane (which is a stereographic projection of the boundary of the unit ball). This choice of parametrisation fixes the gauge freedom in the JNR data, and $\omega$ represents a rotation about one of the principal axes. For each choice of $A$ and $B$ the relevant component of the metric is given by the integral $$\int\left[\left(\partial_\omega a_x\right)^2+\left(\partial_\omega a_r\right)^2\right]\,dx\,dr\,=\,64\,A^2\int\left[\left(\partial_x\left(\frac{rx}{h}\right)\right)^2+\left(\partial_r\left(\frac{rx}{h}\right)\right)^2\right]\,dx\,dr.$$ Plots of these functions are given in figure \[fig5\].
![Angular component of the metric as a function of $\alpha$. Solid line: $a^2(\alpha)$, dashed line: $b^2(\alpha)$, dotted line: $c^2(\alpha)$.[]{data-label="fig5"}](fig5-eps-converted-to.pdf){width="0.5\linewidth"}
Expanding near $\alpha=0$ gives the metric coefficients
[lcl]{} f\^2=c\_1\^2+(\^6),&&b\^2=c\_2+c\_3\^2+(\^4),\
a\^2=\^2f\^2+(\^6),&&c\^2=c\_2-c\_3\^2+(\^4).
where $$c_1\,=\,\frac{32\pi}{9}\,\left(81-10\sqrt{3}\,\pi\right),\qquad c_2\,=\,\frac{8\pi}{27}\,\left(2\sqrt{3}\,\pi-9\right),\qquad c_3\,=\,\frac{4\pi}{9}\,\left(8\sqrt{3}\,\pi-27\right).$$ A numerical computation of the coefficients for $\alpha\to1$ is well fitted by the expansions
[rcl]{} f\^2&=&(1-4(1-)\^4+…),\
a\^2=b\^2&=&(1-(1-)\^2+…),c\^2(1-)\^8,
which, when $\alpha$ is converted to a hyperbolic distance, is exponentially close to the background hyperbolic metric. It would be interesting to find a physical interpretation of this metric in terms of the forces between well separated monopoles, akin to Manton’s results in the Euclidean case, [@Man85]. Note in particular that the factor $16\pi/3$ is twice the value obtained for a single monopole and plays the role of a mass. It may also be possible to describe our asymptotic metric by LeBrun’s hyperbolic analogue [@LeB91] of the Gibbons-Hawking metric.
Periodic monopoles {#chainssection}
==================
The original motivation for this work was to obtain new examples of hyperbolic monopoles. The method presented in section \[symminst\] is particularly useful for periodic arrays of monopoles, for which the JNR and ADHM constructions are not currently known. However, periodic and large charge vortex configurations have been studied [@MR10; @Sut12; @MM15] and they are easily lifted to $\mathbb{H}^3$.
Periodic monopoles in Euclidean space have previously been studied in some depth via the Nahm transform and spectral curve [@CK01]. These tools demonstrated the splitting of the monopole into constituents [@War05] and allowed a study of the moduli space dynamics [@MW13].
The periodic monopole we will construct in this section is the one lifted from a vortex on the hyperbolic cylinder [@MR10], in which the Higgs zeros are strung along a geodesic in $\mathbb{H}^2$. The JNR data for this periodic vortex is not known, so the formula provides a novel example of a hyperbolic monopole. The vortex Higgs field is given in terms of elliptic functions, where the elliptic modulus $k$ determines the periodicity. Explicitly, we use the formula with $$f(w)\,=\,\frac{\text{cd}_k(2\kappa\tan^{-1}(w))-1}{\text{cd}_k(2\kappa\tan^{-1}(w))+1}\,,\label{Jacobif}$$ where $\pi\kappa=2{\text{\bf{K}}}_k$. Using the coordinate $\text{i}u=\log(x^4+\text{i}r)$ in the upper half space model of $\mathbb{H}^2$ gives the Higgs field $$|\phi|^2\,=\,\kappa^2\left|\text{cn}_k(\kappa u)\text{dn}_k(\kappa u)\right|^2\left(\frac{\text{sin}({\text{Re}}(u))}{{\text{Re}}(\text{sn}_k(\kappa u))}\right)^2.\label{MRchain}$$ The monopole constructed from this vortex has zeros at $x^4_0=x^1_0=0$ and $\rho_0=\text{e}^{n\lambda/2}$, with $n\in\mathbb{Z}$ and $$\lambda\,=\,\frac{\pi{\text{\bf{K}}}'_k}{{\text{\bf{K}}}_k},$$ where $\tfrac{1}{2}\lambda$ is the hyperbolic distance between neighbouring zeros of the Higgs field. The energy density is computed using and plotted in figure \[fig6\] for various values of $k$.
![Slices through a periodic hyperbolic monopole constructed from . Contours show energy density in intervals of $0.1$, with regions of $\mathcal{E}>0.5$ shaded. [**Left:**]{} $X^2=0$ plane with $k=0.7$. The monopoles are localised at the zeros of $\Phi$. [**Centre:**]{} $X^2=0$ plane with $k=0.9$. Each monopole has split into two constituents, and there is a saddle point in energy density at the monopole zeros. [**Right:**]{} $X^1=0$ plane with $k=0.9$. Constituents do not develop in the $X^2$ direction.[]{data-label="fig6"}](fig6a-eps-converted-to.pdf){width="0.85\linewidth"}
![Slices through a periodic hyperbolic monopole constructed from . Contours show energy density in intervals of $0.1$, with regions of $\mathcal{E}>0.5$ shaded. [**Left:**]{} $X^2=0$ plane with $k=0.7$. The monopoles are localised at the zeros of $\Phi$. [**Centre:**]{} $X^2=0$ plane with $k=0.9$. Each monopole has split into two constituents, and there is a saddle point in energy density at the monopole zeros. [**Right:**]{} $X^1=0$ plane with $k=0.9$. Constituents do not develop in the $X^2$ direction.[]{data-label="fig6"}](fig6b-eps-converted-to.pdf){width="0.85\linewidth"}
![Slices through a periodic hyperbolic monopole constructed from . Contours show energy density in intervals of $0.1$, with regions of $\mathcal{E}>0.5$ shaded. [**Left:**]{} $X^2=0$ plane with $k=0.7$. The monopoles are localised at the zeros of $\Phi$. [**Centre:**]{} $X^2=0$ plane with $k=0.9$. Each monopole has split into two constituents, and there is a saddle point in energy density at the monopole zeros. [**Right:**]{} $X^1=0$ plane with $k=0.9$. Constituents do not develop in the $X^2$ direction.[]{data-label="fig6"}](fig6c-eps-converted-to.pdf){width="0.85\linewidth"}
For small $k$ the monopoles are well separated and the energy density is peaked at the Higgs zeros. As $k$ is increased the monopoles get closer together and widen in the $X^1$ direction. Then at some critical value the energy peaks break apart and move away from the $X^3$ axis, leaving the positions of the Higgs zeros as saddle points of energy density.
Expansions of the Higgs field at the half period points of the periodic vortex (in the Poincaré disk model) were given in [@MM15]. Applying the formula to these expansions yields explicit expressions for the maximal and minimal values taken by the Higgs field along the $X^3$ axis: $$\mathcal{E}_{\text{min}}\,=\,\frac{1}{2}\left(1-\left(\frac{2{\text{\bf{K}}}_k}{\pi}(1-k)\right)^2\right)^2,\qquad\qquad\mathcal{E}_{\text{saddle}}\,=\,\frac{1}{2}+\left(\frac{2{\text{\bf{K}}}_k}{\pi}\sqrt{1-k^2}\right)^4.\label{emaxsad}$$ The critical value of $k$ at which the maximum energy on the $X^3$ axis becomes a saddle point is found by expanding $|\phi|^2$ to higher order. Performing this expansion and converting back to the upper half-plane model, the energy density at $x^4+\text{i}r=\text{i}+|\delta|\text{e}^{\text{i}\theta}$ restricted to $x^1=0$ is $$\left.\mathcal{E}\right|_{x^1=0}\,=\,\left(\frac{1}{2}+a^2\right)+3\,a|\delta|^2\left(b\cos(2\theta)-a\right)+\mathcal{O}(|\delta|^3),\label{Eexpansion}$$ where $a$ and $b$ are the coefficients at order $w^2$ and $w^4$ in an expansion of , and are given by $$a\,=\,-\frac{4{\text{\bf{K}}}_k^2}{\pi^2}(1-k^2),\qquad\qquad b\,=\,-\frac{8{\text{\bf{K}}}_k^2}{3\pi^2}(1-k^2)\left(\frac{4{\text{\bf{K}}}_k^2}{\pi^2}(1+k^2)-1\right).$$ When $\theta=0$, the order $|\delta|^2$ term in the expansion changes sign when $a=b$, i.e. $$8{\text{\bf{K}}}_{k_0}^2(1+k_0^2)\,=\,5\pi^2\qquad\Rightarrow\qquad k_0\,\approx\,0.780.$$ Considering the next term in the expansion shows that the hyperbolic distance of the energy peaks from the $X^3$ axis grows like $d\propto(k-k_0)^{1/2}$. Differentiating with respect to $x^1$ shows that the plane $x^1=0$ is everywhere a stationary point of the energy density, with a local maximum at the vortex zero. A similar splitting can be observed for chains of finite length. However, such a splitting has not been observed in the periodic monopole obtained from the axially symmetric Harrington-Shepard periodic instanton [@HS78], which gives Higgs zeros equally spaced on a horocycle.
Acknowledgements {#acknowledgements .unnumbered}
================
Many thanks to Derek Harland, Nick Manton and Paul Sutcliffe for useful comments. This work was supported by the UK Science and Technology Facilities Council, grant number ST/J000434/1.
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[^1]: [R.Maldonado@damtp.cam.ac.uk]{}
[^2]: The energy density, which depends on derivatives of $\|\Phi\|^2$, is proportional to the radial energy density profile of a single hyperbolic monopole. The constant of proportionality depends on the leading behaviour of $\|\Phi\|^2$ near its zeros.
[^3]: There is, of course, only one $\text{SO}(4)$ symmetric ’t Hooft function, namely $\psi=1+\lambda^2/\varrho^2$, but we will stick to using $\psi$ in order to highlight the analogy with the previous reduction from $3$ to $2$ dimensions.
[^4]: On the other hand, joining the triples of points $w_i$ by Euclidean triangles would yield a Euclidean ellipse with foci at $w=\pm\text{i}a$, the ‘non-zero zeros’ of $f$, [@DGM02].
|
---
abstract: 'For any classical Lie algebra $\mathfrak{g}$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of the $(m,n)$-chain, and for $m=n$, there is also a family of symmetrically reduced Toda systems, the $(m,m)_{\mathrm{Sym}}$-Toda systems, which are also integrable. In the quantum case, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all $(m,n)$-Toda systems survive after quantization.'
author:
- |
Liu Zhao[^1], Wangyun Liu[^2] and Zhanying Yang[^3]\
Institute of Modern Physics, Northwest University,\
Xi’an, 710069, P. R. China
bibliography:
- 'BCD-Toda.bib'
title: Generalized Toda mechanics associated with classical Lie algebras and their reductions
---
Introduction
============
Integrable many body systems have attracted intensive attention of theoretical physicists as well as mathematicians for over twenty years because they are related to diverse problems ranging, e.g. from physical problems such as long-range correlation [@Calogero], nonlinear wave propagation [@Toda1; @Toda2], Hall effect [@Fring1] and brane and gravitational instanton solutions [@Ivashchuk; @Ketov] to mathematical problems like inverse scattering method [@Kharchev; @Flaschka1; @Flaschka2], nonlinear Lie symmetries [@Nirov], quantum groups and algebro-geometrical properties [@Vanhaecke; @Getzler] of certain Riemanian surfaces. Among the known classes of integrable many body systems, the Toda, Calogero-Moser [@Calogero; @Calogero2; @Calogero3; @Moser1; @Moser2] and Ruijsenaars-Schneider systems [@Ruijsenaars1; @Ruijsenaars2] are the most interesting and extensively studies ones. That the above mentioned many body systems received particular attention is partly due to the recent progress on the studies of nonperturbative properties of certain SUSY gauge theories. In particular, the spectral curve for the periodic Toda chain is found to be related to the Seiberg-Witten construction of prepotential for the $N=2$ SUSY Yang-Mills theory in $4$-spacetime dimensions [@Donagi-Witten; @Marshakov1; @Marshakov:book], while the values of the integrals of motion for the Toda chain in the stationary configuration is related to the chiral effective superpotential of $N=1$ SUSY Yang-Mills theory [@Dorey1].
There are quite intensive literatures on the study of Toda type integrable systems, among which many are concentrated on their integrable generalizations. However, many of the papers on generalizations of Toda theories considered only some special cases, e.g. generalizations into higher dimensions, coupling with extra matter of certain type, or nonabelian generalizations. In a recent paper [@Zhao-Wang1], we studied the integrable generalizations of Toda type systems based on the Lie algebras $\mathfrak{gl}_{r+1}$ and $\mathfrak{sl}_{r+1}$ in full detail. Though we restricted our generalizations within the scope of $(0+1)$-dimensional mechanical systems, our constructions turn out to be extremely generic, with the equations of motion for all possible orders of generalizations given explicitly. We also discussed different variants of the generalized Toda systems, including both abelian and nonabelian versions of infinite, finite and periodic chains of different order (characterized by an ordered pair of integers $(m_{+},m_{-})$).
In this paper, we shall continue our work on the integrable generalizations of Toda mechanics to the case of arbitrary classical Lie algebras, with emphasis on the case of $B_{r},C_{r}$ and $D_{r}$. The case $B_{r}$ is considered in great detail for illustration purpose. We will show that for any classical Lie algebra $\mathfrak{g}$, there is a family of integrable generalizations of Toda mechanics characterized also by an ordered pair of integers $(m,n)$, for which we present both the universal form of the equation of motion in abstract notations and concrete examples for $m,n\leq3$. The Hamiltonian structures for the generalizations are also presented, and various possible reductions are also studied. It turns our that, upon quantization, all nontrivial generalizations (i.e. at least one of $m,n$ is bigger than $1$) will involve coordinate noncommutativity, however the integrability in the quantum case is not affected in in most cases.
Notations and conventions
=========================
In this section we shall review the necessary Lie algebra knowledge for notational convenience. For any classical Lie algebra $\mathfrak{g}$, there is a root space decomposition$$\mathfrak{g=h\oplus}{\textstyle\bigcup\limits_{\alpha\in\Delta}}
\mathfrak{g}_{\alpha},$$ where $\mathfrak{h}$ is the Cartan subalgebra and $\Delta$ is the root system of $\mathfrak{g}$. Denoting $\mathfrak{g}^{(0)}=\mathfrak{h}$ and $\mathfrak{g}^{(k)}={\textstyle\bigcup\limits_{\mathrm{ht}(\alpha)=k}}
\mathfrak{g}_{\alpha}$ where $\mathrm{ht}(\alpha)$ refers to the height of the root $\alpha$, the Lie algebra $\mathfrak{g}$ is endowed with a natural integer gradation, i.e. the $\mathbb{Z}$-gradation by the heights of the roots, $$\mathfrak{g}=\bigoplus_{n}\mathfrak{g}^{(n)}.$$ Accordingly, the dual $\mathfrak{g}^{\ast}$ of $\mathfrak{g}$ is also $\mathbb{Z}$-graded, and any element $f$ of $\mathfrak{g}^{\ast}$ (regarded as a linear function over $\mathfrak{g}$) can be decomposed into the form $$f=\sum_{n}f^{(n)},$$ where the domain of $f^{(n)}$ is $\mathfrak{g}^{(n)}$. In the next section, while writing the Lax matrices for the generalized Toda systems, we shall conform to this convention, taking the Lax matrices as linear functions over $\mathfrak{g}$.
The Lie algebra basis we shall use is the Chevalley basis which is a collection of $3r$ elements $\{h_{i},e_{i},f_{i}\}$, spanning only the subspaces $\mathfrak{g}^{(0)},$ $\mathfrak{g}^{(\pm1)}$. They satisfy the following generating relations together with the so-called Serre relations which we omit here$$\begin{aligned}
\lbrack h_{i},h_{j}] & =0,\\
\lbrack h_{i},e_{j}] & =K_{ij}e_{j},\\
\lbrack h_{i},f_{j}] & =-K_{ij}f_{j},\\
\lbrack e_{i},f_{j}] &
=\delta_{ij}h_{i}.\end{aligned}$$ One of the merits for using Chevalley basis is that the structure constants are all integers and they are actually matrix elements of the Cartan matrix $K$ of the Lie algebra $\mathfrak{g}$. The Cartan matrix elements are related to the simple roots via$$K_{ij}=\frac{2(\alpha_{i},\alpha_{j})}{(\alpha_{i},\alpha_{i})},$$ where we use $(,)$ to denote the inner product on the root lattice. We label the simple roots of the classical Lie algebras $B_{r},C_{r}$ and $D_{r}$ as in Figure \[fig1\], so that their Cartan matrices can be read out directly.
![Dynkin diagrams for the Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$[]{data-label="fig1"}](DynkinBCD)
Though the Chevalley generators do not span the subspaces $\mathfrak{g}^{(\pm k)}$ with $k>1$, they can generate the basis for $\mathfrak{g}^{(\pm k)}$ by use of iterated Lie products. For instance, if $$\pm(\alpha_{i_{k}}+\cdots+\alpha_{i_{2}}+\alpha_{i_{1}})\in\Delta$$ are roots of heights $\pm k$, the corresponding root vectors can be obtained as$$e_{(i_{k},\,\cdots,\,i_{2},\,i_{1})}=[e_{i_{k}},\cdots,[e_{i_{2}},e_{i_{1}}]],\quad f_{(i_{k},\,\cdots,\,i_{2},\,i_{1})}=(-1)^{k}[f_{i_{k}},\cdots,[f_{i_{2}},f_{i_{1}}]], \label{nonsimple}$$ where the choice of sign in the definition of $f_{(i_{k},\,\cdots
,\,i_{2},\,i_{1})}$ is such that it is the image of $e_{(i_{k},\,\cdots
,\,i_{2},\,i_{1})}$ under *Cartan involution*,$$f_{(i_{k},\,\cdots,\,i_{2},\,i_{1})}=\mathrm{Inv\,}e_{(i_{k},\,\cdots
,\,i_{2},\,i_{1})}.$$ Then, a typical basis for $\mathfrak{g}^{(k)}$ can be taken to be the set $\{e_{(i_{k},\,...,\,i_{2},\,i_{1})}\}$ for $k>1$, or $\{f_{(i_{k},\,...,\,i_{2},\,i_{1})}\}$ for $k<-1$. A crucial point in the notations in (\[nonsimple\]) is that the order of the successive Lie products must ensure that each intermediate step corresponds to a root in the same root chain. The Lie products between the non-simple root vector (\[nonsimple\]) and other generators of the same Lie algebra can be evaluated by use of the Chevalley generating relations and the Jacobi identity $$\lbrack\lbrack a,b],c]+[[b,c],a]+[[c,a],b]=0. \label{01}$$ The value of Killing form evaluated on one of these root vectors together with another Lie algebra generator can be obtained by use of the invariant property $$\langle a,[b,c]\rangle=\langle\lbrack a,b],c\rangle\label{invariance}$$ as well as the Jacobi identity.
For later use, we also list all roots of heights $\pm1,\pm2$ and $\pm3$ for the Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$ in Appendix A. Also in Appendix A we give the representation matrices for the Chevalley generators of the Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$, which are to be used while calculating the Hamiltonians of the generalized Toda models.
General construction$\label{secgen}$
====================================
In this paper, we shall construct generalized Toda chains with Lax matrix of the form $$\begin{aligned}
& L^{(m,n)}\equiv L_{+}^{(m)}+L_{-}^{(n)},\nonumber\\
& L_{+}^{(m)}=\sum_{i=1}^{m}L^{(i)},\quad L_{-}^{(n)}=\sum_{i=-n}^{-1}L^{(i)},\quad m,n>0. \label{LaxLmn}$$ The claim is that any Lax matrix of the form (\[LaxLmn\]) together with the $M$ matrix $$M^{(m,n)}=L_{+}^{(m)}-L_{-}^{(n)} \label{LaxMmn}$$ defines an integrable generalization of the Toda chain via the Lax equation $$\dot{L}^{(m,n)}=[M^{(m,n)},L^{(m,n)}], \label{Laxeq}$$ which we call the $(m,n)$-extension of the Toda chain, or simply the $(m,n)$-chain. The standard Toda chains correspond to the simplest case $m=n=1$. In the following, we shall always assume $m\geq n$ for generic values of $m$ and $n$ without loss of generality, because the cases $n\geq m$ can be easily obtained by use of a simple Cartan involution over the Lie algebra $\mathfrak{g}$.
Suppose we are given a pair of Lax matrices $L^{(m,n)}$ and $M^{(m,n)}$ as described in (\[LaxLmn\]) and (\[LaxMmn\]). Then straightforward calculation yields $$\lbrack{M}^{(m,n)},{L}^{(m,n)}]=[{L}_{+}^{(m)}-{L}_{-}^{(n)},{L}^{(0)}]+2[{L}_{+}^{(m)},{L}_{-}^{(n)}]. \label{Commu1}$$ The right hand side of the last equation falls completely within the subspace $\bigoplus_{i=-n}^{m}\mathfrak{g}^{(i)}$, which is also the domain of $\dot
{L}^{(m,n)}$. Therefore, the form of the Lax matrices we have chosen is indeed consistent with the $\mathbb{Z}$-gradation according to the heights of the roots. A more careful look at (\[Commu1\]) yields, by use of the Lax equation (\[Laxeq\]), the following abstract form of the equations of motion for the $(m,n)$-extended Toda chain,$$\begin{aligned}
\dot{L}^{(k)} & =\mathrm{sign}(k)[L^{(k)},L^{(0)}]+2\sum_{i=\max
(1,\,k+1)}^{\min(m,\,k+n)}[L^{(i)},L^{(k-i)}],\quad k=-n+1,\cdots
,m-1,\label{gene-eq1}\\
\dot{L}^{(m)} & =[L^{(m)},L^{(0)}],\qquad\dot{L}^{(-n)}=-[L^{(-n)},L^{(0)}].
\label{gene-eq2}$$ The form of the above equations is Lie algebra independent but too abstract to get any detailed information from. Therefore, we need to parameterize the $L^{(k)}$ concretely for each underlying Lie algebra to learn the actual mechanical behavior of the generalized Toda systems. This will be the task of the next section. At present, we would like to point out some universal properties of the above system of equations.
The first universal feature we shall point out is that the equations in (\[gene-eq2\]) can be integrated explicitly, so that the actual mechanical variables for the $(m,n)$-Toda chains are consisted only of $L^{(k)}$ with $k=-n+1,\cdots,m-1$. To see this, we now give the explicit integration of the equations in (\[gene-eq2\]). They are$$\begin{aligned}
L^{(m)} & =\exp\left( -\mathrm{ad}\mathcal{L}^{(0)}\right) \mathcal{L}^{(m)}=\exp\left( -\mathrm{ad}\mathcal{L}^{(0)}\right) c^{(m)},\nonumber\\
L^{(-n)} & =\exp\left( \mathrm{ad}\mathcal{L}^{(0)}\right) \mathcal{L}^{(-n)}=\exp\left( \mathrm{ad}\mathcal{L}^{(0)}\right) c^{(-n)},
\label{submn}$$ where $c^{(m)}$ and $c^{(-n)}$ are some constant elements in $\mathfrak{g}^{(m)}$ and $\mathfrak{g}^{(-n)}$ respectively, and $\mathcal{L}^{(0)}$ is related to $L^{(0)}$ via$$L^{(0)}=\dot{\mathcal{L}}^{(0)}=\frac{d}{dt}\mathcal{L}^{(0)}.$$ We can also reparametrize $L^{(k)}$ as$$L^{(k)}=\exp\left[ -\mathrm{sign}(k)\mathrm{ad}\mathcal{L}^{(0)}\right]
\mathcal{L}^{(k)},\quad k\neq0. \label{LcalL}$$ After this reparametrization, the equations in (\[gene-eq1\]) will be simplified into$$\mathcal{\ddot{L}}^{(0)}=2\sum_{i=1}^{n}[\exp\left( -\mathrm{ad}\mathcal{L}^{(0)}\right) \mathcal{L}^{(i)},\exp\left( \mathrm{ad}\mathcal{L}^{(0)}\right) \mathcal{L}^{(-i)}] \label{CALEQ:1}$$ and$$\begin{aligned}
\mathcal{\dot{L}}^{(k)} & =2\exp\left[ \mathrm{sign}(k)\mathrm{ad}\mathcal{L}^{(0)}\right] \sum_{i=\max(1,\,k+1)}^{\min(m,\,k+n)}[\exp\left[
-\mathrm{ad}\mathcal{L}^{(0)}\right] \mathcal{L}^{(i)},\exp\left[
\mathrm{ad}\mathcal{L}^{(0)}\right] \mathcal{L}^{(k-i)}],\quad\nonumber\\
k & =-n+1,\cdots,m-1,\quad k\neq0. \label{CAlEQ:2}$$ Equation (\[CAlEQ:2\]) can also be rewritten as$$\begin{aligned}
\mathcal{\dot{L}}^{(k)} & =2\sum_{i=\,\max(-n,\,k-m)}^{-1}[\mathcal{L}^{(k-i)},\exp\left[ 2\mathrm{ad}\mathcal{L}^{(0)}\right] \mathcal{L}^{(i)}],\quad k=1,\cdots,m-1,\label{CALEQ:3}\\
\mathcal{\dot{L}}^{(k)} & =2\sum_{i=1}^{\,\min(m,\,k+n)}[\exp\left[
-2\mathrm{ad}\mathcal{L}^{(0)}\right] \mathcal{L}^{(i)},\mathcal{L}^{(k-i)}],\quad k=-n+1,\cdots,-1. \label{CALEQ:4}$$ In the next section, we shall present explicit examples for the equations of motion (\[CALEQ:1\]), (\[CALEQ:3\]) and (\[CALEQ:4\]) in component form for the case of Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$ in the special case $m=n=3$.
Another universal feature of the system of equations (\[gene-eq1\]), (\[gene-eq2\]) is that they admit systematic reductions. We shall discuss two different types of possible reductions. One is the reduction $L^{(m,\,n)}\rightarrow L^{(m-1,\,n)}$ by setting $c^{(m)}=0,$ $\mathcal{L}^{(m-1)}=c^{(m-1)}$, or $L^{(m,\,n)}\rightarrow L^{(m,\,n-1)}$ by setting $c^{(-n)}=0,$ $\mathcal{L}^{(-n+1)}=c^{(-n+1)}$. It is easy to see that the equations of motion (\[CALEQ:1\]), (\[CALEQ:3\]) and (\[CALEQ:4\]) following from the evolution of $L^{(m,n)}$ will be consistently reduced to the ones which follow from $L^{(m-1,n)}$ or $L^{(m,n-1)}$ via the above reduction. This observation is quite remarkable, since it implies that the dynamics of the $(m,n)$-Toda chain is automatically a special case of the $(m^{\prime},n^{\prime})$-chain with $m^{\prime}\geq m,$ $n^{\prime}\geq
n$. If the Hamiltonian structure for both chains are defined, then the phase space of the $(m,n)$-Toda chain is a proper subspace of that of the $(m^{\prime},n^{\prime})$-chain. The other type of reduction can take place only in the special case of $m=n$. Then we can see that setting $L^{(m,m)}$ to be symmetric, i.e. letting $\mathcal{L}^{(k)}=\mathrm{Inv}(\mathcal{L}^{(-k)})$ will not spoil the correctness of the equations of motion (\[CALEQ:1\]), (\[CALEQ:3\]) and (\[CALEQ:4\]). This latter type of reduction is called symmetric reduction and was studied in some detail in [@Zhao-Wang1] for the case of $\mathfrak{g=gl}_{r+1}$. For later references, we may call the symmetrically reduced $(m,m)$-Toda chain a $(m,m)_{\mathrm{Sym}}$-chain.
The last universal feature we shall mention is, just like for all Lax integrable systems based on Lie algebras, that the integrals of motion for the $(m,n)$-Toda chains can be obtained by taking the trace of the $k$-th power of $L^{(m,\,n)}$,$$H_{k}=\frac{1}{k}\mathrm{tr}\left[ \left( L^{(m,\,n)}\right) ^{k}\right]
.$$ In particular, the second integral of motion, which is to be interpreted as the Hamiltonian of the system, is given as$$H_{2}=\mathrm{tr}\left[ \left( \mathcal{\dot{L}}^{(0)}\right) ^{2}\right]
+\sum_{i=1}^{n}\mathrm{tr}\left[ \mathcal{L}^{(i)}\exp\left( 2\mathrm{ad}\mathcal{L}^{(0)}\right) \mathcal{L}^{(-i)}\right] .$$ While writing the last equations, we assumed that there is a finite dimensional matrix representation for $\mathfrak{g}$, as is true for all classical finite dimensional Lie algebras, on which the trace is taken. Let us further assume that the dimension of the above matrix representation is $\ell
$. Then we can also write down the Poisson brackets for the system of equations in abstract form. They read$$\begin{aligned}
\{\mathcal{\dot{L}}_{1}^{(0)},\mathrm{ad}\mathcal{L}_{2}^{(0)}\}(f) &
=\frac{1}{\ell}P_{12}\mathrm{ad}f_{2},\label{genpois:1}\\
\{\mathcal{L}_{1}^{(\pm i)},\mathcal{L}_{2}^{(\pm j)}\}(f) & =\pm\frac
{2}{\ell}P_{12}\mathrm{ad}\mathcal{L}_{1}^{(\pm i\pm j)}(f), \label{genpois:2}$$ where $\mathcal{L}^{(\pm k)}\in\mathfrak{g}^{\ast},f\in\mathfrak{g,}$ $$\mathcal{L}_{1}^{(\pm k)}\equiv\mathcal{L}^{(\pm k)}\otimes I,\quad
\mathcal{L}^{(\pm k)}\equiv I\otimes\mathcal{L}^{(\pm k)},$$ $I$ is the identity matrix of dimension $\ell$, and $P_{12}$ is the permutation matrix, i.e. $$P_{12}\left( A\otimes B\right) =B\otimes A.$$ Notice that, on the right hand side of (\[genpois:2\]), we have assumed $\mathcal{L}^{(k)}=0$ for $k>m$ or $k<-n$.
Examples: the $(3,3)$-Toda chains associated with Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$
===========================================================================================
Having described the general construction of the $(m,n)$-Toda chains associated with arbitrary classical Lie algebra $\mathfrak{g}$, we now study some special examples in order to give the readers some more intuitive idea about the generalized Toda systems. For $\mathfrak{g}=\mathfrak{gl}_{r+1}$ and $A_{r}=\mathfrak{sl}_{r+1}$, the explicit equations of motion in component form together with their Liouville integrability and Hamiltonian structures for the case of arbitrary $m,n$ have already been studied in detail in [@Zhao-Wang1]. Therefore, we shall be concentrating on the cases $\mathfrak{g}=B_{r}$, $C_{r}$ and $D_{r}$. However, since the root systems for the Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$ are more complicated than that of $A_{r}$, writing down the explicit equation of motion for general $m,n$ will be an extremely cumbersome task and hence we shall be restricting ourselves to the special cases of $m,n\leq3$, when we are able to calculate everything explicitly. Moreover, according to the last section, the chains for which $m,n\leq3$ can all be obtained as proper reductions of the $(3,3)$-chain. We thus will make the construction explicitly only for $(m,n)=(3,3)$. Since the structures for the root systems for $B_{r}$, $C_{r}$ and $D_{r}$ are quite different, we shall treat the cases for each families of these Lie algebras separately, with the main emphasis focused on $B_{r}$. We shall also establish the Hamiltonian structure for each concrete examples on the fly.
The $(3,3)$-chain for $B_{r}$
-----------------------------
According to the list for roots of heights $h=\pm1,\pm2,\pm3$ given in Appendix A, we can write the most general form of the Lax matrix components $\mathcal{L}^{(0)},\mathcal{L}^{(\pm1)},\mathcal{L}^{(\pm2)}$ and $c^{(\pm3)}$ as follows,$$\begin{aligned}
\mathcal{L}^{(0)} & =\sum_{i=1}^{r}q_{i}h_{i},\nonumber\\
\mathcal{L}^{(1)} & =\mu^{(1)}\sum_{i=1}^{r}\psi_{i}^{(+1)}e_{i},\quad
\quad\mathcal{L}^{(-1)}=\mu^{(-1)}\sum_{i=1}^{r}\psi_{i}^{(-1)}f_{i},\nonumber\end{aligned}$$$$\mathcal{L}^{(2)}=\mu^{(2)}\sum_{i=1}^{r-1}\psi_{i}^{(+2)}e_{(i,\,i+1)},\quad\mathcal{L}^{(-2)}=\mu^{(-2)}\sum_{i=1}^{r-1}\psi_{i}^{(-2)}f_{(i,\,i+1)}, \label{Laxcomp}$$$$\begin{aligned}
c^{(3)} & =\mu^{(3)}\left( \sum_{i=1}^{r-2}e_{(i,\,i+1,\,i+2)}+e_{(r,\,r-1,\,r)}\right) ,\nonumber\\
c^{(-3)} & =\mu^{(-3)}\left( \sum_{i=1}^{r-2}f_{(i,\,i+1,\,i+2)}+f_{(r,\,r-1,\,r)}\right) ,\nonumber\end{aligned}$$ where $q_{i},\psi_{i}^{(\pm k)}$ $(k=1,2)$ are time dependent mechanical variables and $\mu^{(\pm k)}$ $(k=1,2,3)$ are coupling constants. Inserting these expressions into (\[CALEQ:1\]), (\[CALEQ:3\]) and (\[CALEQ:4\]), we get the following equations of motion in component form, which are divided into three groups (all suffices $i$ are in the range $1\leq i\leq r-2$):
- equations for $q_{i}$:$$\begin{aligned}
\ddot{q}_{i} & =2\omega_{i}\left[ \tau^{(1)}\psi_{i}^{(+1)}\psi_{i}^{(-1)}+\tau^{(2)}\left( \omega_{i-1}\psi_{i-1}^{(+2)}\psi_{i-1}^{(-2)}+\omega_{i+1}\psi_{i}^{(+2)}\psi_{i}^{(-2)}\right) \right. \nonumber\\
& +\left. \tau^{(3)}\left( \left( \delta_{i,\,r-2}+1\right) \omega
_{i+1}\omega_{i+2}+\omega_{i-1}\omega_{i+1}+\omega_{i-2}\omega_{i-1}\right)
\right] ,\label{33eq:1}\\
\ddot{q}_{r-1} & =2\omega_{r-1}\left[ \tau^{(1)}\psi_{r-1}^{(+1)}\psi
_{r-1}^{(-1)}+\tau^{(2)}\left( 2\omega_{r}\psi_{r-1}^{(+2)}\psi_{r-1}^{(-2)}+\omega_{r-2}\psi_{r-2}^{(+2)}\psi_{r-2}^{(-2)}\right) \right.
\nonumber\\
& +\left. \tau^{(3)}\left( 2\omega_{r-2}\omega_{r}+\omega_{r-3}\omega
_{r-2}+4\omega_{r}^{2}\right) \right] ,\\
\ddot{q}_{r} & =2\omega_{r}\left[ \tau^{(1)}\psi_{r}^{(+1)}\psi_{r}^{(-1)}+\tau^{(2)}\omega_{r-1}\psi_{r-1}^{(+2)}\psi_{r-1}^{(-2)}+\tau
^{(3)}\left( \omega_{r-2}\omega_{r-1}+4\omega_{r-1}\omega_{r}\right)
\right] ;\end{aligned}$$
- equations for $\psi_{i}^{(\pm1)}$:$$\begin{aligned}
\dot{\psi}_{i}^{(+1)} & =2\left[ \tau^{(2,1)}\left( \omega_{i+1}\psi
_{i}^{(+2)}\psi_{i+1}^{(-1)}-\omega_{i-1}\psi_{i-1}^{(+2)}\psi_{i-1}^{(-1)}\right) \right. \nonumber\\
& +\left. \tau^{(3,2)}\left( \left( \delta_{i,\,r-2}+1\right)
\omega_{i+1}\omega_{i+2}\psi_{i+1}^{(-2)}-\omega_{i-1}\omega_{i-2}\psi
_{i-2}^{(-2)}\right) \right] ,\\
\dot{\psi}_{r-1}^{(+1)} & =2\left[ \tau^{(2,1)}\left( 2\omega_{r}\psi_{r-1}^{(+2)}\psi_{r}^{(-1)}-\omega_{r-2}\psi_{r-2}^{(+2)}\psi
_{r-2}^{(-1)}\right) -\tau^{(3,2)}\omega_{r-3}\omega_{r-2}\psi_{r-3}^{(-2)}\right] ,\\
\dot{\psi}_{r}^{(+1)} & =2\left[ \tau^{(3,2)}\left( 2\omega_{r-1}\omega_{r}\psi_{r-1}^{(-2)}-\omega_{r-2}\omega_{r-1}\psi_{r-2}^{(-2)}\right)
-\tau^{(2,1)}\omega_{r-1}\psi_{r-1}^{(+2)}\psi_{r-1}^{(-1)}\right] ,\end{aligned}$$$$\begin{aligned}
\dot{\psi}_{i}^{(-1)} & =2\left[ \tau^{(1,2)}\left( \omega_{i+1}\psi
_{i+1}^{(+1)}\psi_{i}^{(-2)}-\omega_{i-1}\psi_{i-1}^{(+1)}\psi_{i-1}^{(-2)}\right) \right. \nonumber\\
& +\left. \tau^{(2,3)}\left( \left( \delta_{i,\,r-2}+1\right)
\omega_{i+1}\omega_{i+2}\psi_{i+1}^{(+2)}-\omega_{i-1}\omega_{i-2}\psi
_{i-2}^{(+2)}\right) \right] ,\\
\dot{\psi}_{r-1}^{(-1)} & =2\left[ \tau^{(1,2)}\left( 2\omega_{r}\psi
_{r}^{(+1)}\psi_{r-1}^{(-2)}-\omega_{r-2}\psi_{r-2}^{(+1)}\psi_{r-2}^{(-2)}\right) -\tau^{(2,3)}\omega_{r-3}\omega_{r-2}\psi_{r-3}^{(+2)}\right]
,\\
\dot{\psi}_{r}^{(-1)} & =2\left[ \tau^{(2,3)}\left( 2\omega_{r-1}\omega_{r}\psi_{r-1}^{(+2)}-\omega_{r-2}\omega_{r-1}\psi_{r-2}^{(+2)}\right)
-\tau^{(1,2)}\omega_{r-1}\psi_{r-1}^{(+1)}\psi_{r-1}^{(-2)}\right] ;\end{aligned}$$
- equations for $\psi_{i}^{(\pm2)}$:$$\begin{aligned}
\dot{\psi}_{i}^{(+2)} & =2\tau^{(3,1)}\left( \left( \delta_{i,\,r-2}+1\right) \omega_{i+2}\psi_{i+2}^{(-1)}-\omega_{i-1}\psi_{i-1}^{(-1)}\right)
,\\
\dot{\psi}_{r-1}^{(+2)} & =-2\tau^{(3,1)}\left( \omega_{r-2}\psi
_{r-2}^{(-1)}+2\omega_{r}\psi_{r}^{(-1)}\right) ,\\
\dot{\psi}_{i}^{(-2)} & =2\tau^{(1,3)}\left( \left( \delta_{i,\,r-2}+1\right) \omega_{i+2}\psi_{i+2}^{(+1)}-\omega_{i-1}\psi_{i-1}^{(+1)}\right)
,\\
\dot{\psi}_{r-1}^{(-2)} & =-2\tau^{(1,3)}\left( \omega_{r-2}\psi
_{r-2}^{(+1)}+2\omega_{r}\psi_{r}^{(+1)}\right) ; \label{33eq:last}$$
where, throughout this paper, we use the abbreviations$$\omega_{i}=\exp\left( -2\sum_{j=1}^{r}q_{j}K_{j,\,i}\right)
,\quad i=1,\cdots,r$$ and$$\tau^{(k)}=\mu^{(k)}\mu^{(-k)},\quad\tau^{(i,j)}=\mu^{(i)}\mu^{(-j)}.$$ We have also set $$\omega_{i}=0,\quad\psi_{i}^{(\pm k)}=0,\quad k=1,2,\quad i<1.$$ We can see that the variables $q_{i}$ behave like the standard Toda variables (i.e. they couple among themselves via exponential interactions) but now involve quasi-long range interactions. The interaction range in the above concrete case is $5$, since the term $\omega_{i-1}\omega_{i}\omega_{i+1}$ with longest interaction range contains $q_{i-2}$ through $q_{i+2}$.
The calculations for getting the equations (\[33eq:1\])-(\[33eq:last\]) are very cumbersome, and, while doing the complicated Lie brackets calculations like $[e_{(i,\,i+1,\,i+2)},f_{(j,\,j+1,\,j+2)}]$ and $[e_{(i,\,i+1)},f_{(j,\,j+1,\,j+2)}]$, we used the *Mathematica* package `Operator Linear Algebra` written by one of the authors [@Zhao:OLA].
The Hamiltonian for the $(3,3)$-chain given above can be defined as half the trace of $L^{(3,3)}$. Inserting (\[Laxcomp\]) into (\[LaxLmn\]) via (\[LcalL\]), then by use of the invariant property (\[invariance\]) and the *Mathematica* package [@Zhao:OLA], we obtain the following explicit form of the Hamiltonian$$\begin{aligned}
H_{B_{r}}^{(3,\,3)} & =\frac{1}{2}\mathrm{tr}\left[ \left( L_{B_{r}}^{(3,\,3)}\right) ^{2}\right] \nonumber\\
& =\sum_{i,j=1}^{r}S_{ij}(B_{r})\dot{q}_{i}\dot{q}_{j}+2\tau^{(1)}\left(
\sum_{i=1}^{r-1}\omega_{i}\psi_{i}^{(+1)}\psi_{i}^{(-1)}+2\omega_{r}\psi
_{r}^{(+1)}\psi_{r}^{(-1)}\right) \nonumber\\
& \,+2\tau^{(2)}\left( \sum_{i=1}^{r-2}\omega_{i}\omega_{i+1}\psi_{i}^{(+2)}\psi_{i}^{(-2)}+2\omega_{r-1}\omega_{r}\psi_{r-1}^{(+2)}\psi
_{r-1}^{(-2)}\right) \nonumber\\
& \,+2\tau^{(3)}\left( \sum_{i=1}^{r-2}\left( \delta_{i,\,r-2}+1\right)
\omega_{i}\omega_{i+1}\omega_{i+2}+4\omega_{r}^{2}\omega_{r-1}\right) ,
\label{hamb33}$$ where $S_{ij}$ is defined as$$S_{ij}(B_{r})=\frac{1}{2}\mathrm{tr}\left[ h_{i}h_{j}\right] ,$$ which is related to the Cartan matrix elements $K_{ij}$ via $$S_{ij}(B_{r})=K_{ij}^{(B_{r})}\frac{2}{(\alpha_{j},\alpha_{j})}=\left\{
\begin{array}
[c]{cc}K_{ij}^{(B_{r})} & (j\neq r)\\
2K_{ij}^{(B_{r})} & (j=r)
\end{array}
\right. .$$ The canonical Poisson brackets which are consistent with the Hamiltonian and the equations of motion are$$\{q_{i},\dot{q}_{j}\}=\frac{1}{2}\left( S^{-1}\right) _{ij}(B_{r}),\quad
i,j=1,\cdots,r; \label{PoiB33:1}$$$$\begin{aligned}
\{\psi_{i}^{(+1)},\psi_{j}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] (\delta_{j,\,i+1}\psi_{i}^{(+2)}-\delta_{j,\,i-1}\psi
_{i-1}^{(+2)}),\\
\{\psi_{i}^{(+1)},\psi_{r}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \delta_{i,\,r-1}\psi_{r-1}^{(+2)},\\
\{\psi_{i}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] (\delta_{k,\,i+1}-\delta_{k,\,i-2}),\\
\{\psi_{r}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] (\delta_{k,\,r-1}-\delta_{k,\,r-2}),\end{aligned}$$$$\begin{aligned}
\{\psi_{i}^{(-1)},\psi_{j}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] (\delta_{j,\,i+1}\psi_{i}^{(-2)}-\delta_{j,\,i-1}\psi
_{i-1}^{(-2)}),\\
\{\psi_{i}^{(-1)},\psi_{r}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] \delta_{i,\,r-1}\psi_{r-1}^{(-2)},\\
\{\psi_{i}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] (\delta_{k,\,i+1}-\delta_{k,\,i-2}),\\
\{\psi_{r}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] (\delta_{k,\,r-1}-\delta_{k,\,r-2}), \label{Poib33:last}$$ where the suffices $i,j,k$ of $\psi^{(\pm1,2)}$ take values in the range $i,j,k=1,\cdots,r-1$.
The $(3,3)$-chain for $C_{r}$
-----------------------------
The lax matrix components $\mathcal{L}^{(0)},\mathcal{L}^{(\pm1)}$ and $\mathcal{L}^{(\pm2)}$ for the $(3,3)$-chain associated with $C_{r}$ are the same as those for the $(3,3)$-chain for $B_{r}$, and only the $c^{(\pm3)}$ are different from those given in (\[Laxcomp\]), according to the list of roots of heights $\pm3$ for the Lie algebra $C_{r}$ given in Appendix A. The constant Lie algebra elements $c^{(\pm3)}$ for $C_{r}$ should be $$\begin{aligned}
c^{(3)} & =\mu^{(3)}\left( \sum_{i=1}^{r-2}e_{(i,\,i+1,\,i+2)}+e_{(r-1,\,r-1,\,r)}\right) ,\\
c^{(-3)} & =\mu^{(-3)}\left( \sum_{i=1}^{r-2}f_{(i,\,i+1,\,i+2)}+f_{(r-1,\,r-1,\,r)}\right) ,\end{aligned}$$ which, together with $\mathcal{L}^{(0)},\mathcal{L}^{(\pm1)}$ and $\mathcal{L}^{(\pm2)}$ given in (\[Laxcomp\]), yield the equations of motion for the $(3,3)$-chain associated with $C_{r}$ after being inserted into (\[CALEQ:1\]), (\[CALEQ:3\]) and (\[CALEQ:4\]). However, since these equations are quite complicated (more complicated than those for the $(3,3)$-chain associated with $B_{r}$) and we shall not use them in the sequel, we prefer to omit them here and present only the corresponding Hamiltonian and Poisson brackets.
The Hamiltonian reads $$\begin{aligned}
H_{C_{r}}^{(3,\,3)} & =\sum_{i,j=1}^{r}S_{ij}(C_{r})\dot{q}_{i}\dot{q}_{j}+2\tau^{(1)}\left( \sum_{i=1}^{r-1}\omega_{i}\psi_{i}^{(+1)}\psi
_{i}^{(-1)}+\omega_{r}\psi_{r}^{(+1)}\psi_{r}^{(-1)}\right) \nonumber\\
& +2\tau^{(2)}\sum_{i=1}^{r-1}\omega_{i}\omega_{i+1}\psi_{i}^{(+2)}\psi
_{i}^{(-2)}+2\tau^{(3)}\left( \sum_{i=1}^{r-2}\omega_{i}\omega_{i+1}\omega_{i+2}+4\omega_{r}\omega_{r-1}^{2}\right) ,\end{aligned}$$ and the Poisson brackets are given as follows,$$\{q_{i},\dot{q}_{j}\}=\frac{1}{2}\left( S^{-1}\right) _{ij}(C_{r}),\quad
i,j=1,\cdots,r;$$$$\begin{aligned}
\{\psi_{i}^{(+1)},\psi_{j}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \left( \delta_{j,\,i+1}\psi_{i}^{(+2)}-\delta_{j,\,i-1}\psi_{i-1}^{(+2)}\right) ,\\
\{\psi_{i}^{(+1)},\psi_{r}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] 2\delta_{i,\,r-1}\psi_{r-1}^{(+2)},\\
\{\psi_{i}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] \left( \delta_{k,\,i+1}-\delta_{k,\,i-2}\right) ,\\
\{\psi_{r-1}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] \left( \delta_{k,\,r-3}+2\delta_{k,\,r-1}\right) ,\end{aligned}$$$$\begin{aligned}
\{\psi_{i}^{(-1)},\psi_{j}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] \left( \delta_{j,\,i+1}\psi_{i}^{(-2)}-\delta_{j,\,i-1}\psi_{i-1}^{(-2)}\right) ,\\
\{\psi_{i}^{(-1)},\psi_{r}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] 2\delta_{i,\,r-1}\psi_{r-1}^{(-2)},\\
\{\psi_{i}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] \left( \delta_{k,\,i+1}-\delta_{k,\,i-2}\right) ,\\
\{\psi_{r-1}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] \left( \delta_{k,\,r-3}+2\delta_{k,\,r-1}\right) ,\end{aligned}$$ where the suffices $i,j$ of $\psi^{(\pm1)}$ take the values $i,j=1,\cdots
,r-2,r,$ while the suffices $k$ of $\psi^{(\pm2)}$ take the values $k=1,\cdots,r-1$. The symmetric matrix $S_{ij}(C_{r})$ is related to the Cartan matrix $K_{ij}^{(C_{r})}$ via $$S_{ij}(C_{r})=K_{ij}^{(C_{r})}\frac{2}{(\alpha_{j},\alpha_{j})}=\left\{
\begin{array}
[c]{cc}2K_{ij}^{(C_{r})} & (j\neq r)\\
K_{ij}^{(C_{r})} & (j=r)
\end{array}
\right. .$$ Though the equations of motion are omitted, they can be easily obtained from the above Hamiltonian and Poisson brackets following from the standard Hamiltonian equations.
The $(3,3)$-chain for $D_{r}$
-----------------------------
The Lax matrix components for the $(3,3)$-chain associated with $D_{r}$ are much more different from the $B_{r}$ case than those of the $C_{r}$ case are. According to the root system structure for $D_{r}$, we may parameterize the Lax matrix components $\mathcal{L}^{(0)}$, $\mathcal{L}^{(\pm1)}$, $\mathcal{L}^{(\pm2)}$ and $\mathcal{L}^{(\pm3)}$ as$$\begin{aligned}
\mathcal{L}^{(0)} & =\sum_{i=1}^{r}q_{i}h_{i},\\
\mathcal{L}^{(1)} & =\mu^{(1)}\sum_{i=1}^{r}\psi_{i}^{(+1)}e_{i},\quad
\quad\mathcal{L}^{(-1)}=\mu^{(-1)}\sum_{i=1}^{r}\psi_{i}^{(-1)}f_{i},\end{aligned}$$ $$\begin{aligned}
\mathcal{L}^{(2)} & =\mu^{(2)}\left( \sum_{i=1}^{r-2}\psi_{i}^{(+2)}e_{(i,\,i+1)}+\psi^{(+2)}e_{(r-2,\,r)}\right) ,\quad\\
\mathcal{L}^{(-2)} & =\mu^{(-2)}\sum_{i=1}^{r-2}\psi_{i}^{(-2)}f_{(i,\,i+1)}+\psi^{(-2)}f_{(r-2,\,r)},\end{aligned}$$$$\begin{aligned}
c^{(3)} & =\mu^{(3)}\left( \sum_{i=1}^{r-3}e_{(i,\,i+1,\,i+2)}+e_{(r-3,\,r-2,\,r)}+e_{(r-1,\,r-2,\,r)}\right) ,\\
c^{(-3)} & =\mu^{(-3)}\left( \sum_{i=1}^{r-3}f_{(i,\,i+1,\,i+2)}+f_{(r-3,\,r-2,\,r)}+f_{(r-1,\,r-2,\,r)}\right) .\end{aligned}$$ As in the $C_{r}$ case, we prefer not to write out the complicated set of equations of motion, but present the Hamiltonian and Poisson brackets instead.
The Hamiltonian and canonical Poisson brackets of this system read$$\begin{aligned}
H_{D_{r}}^{(3,\,3)} & =2\sum_{i,j=1}^{r}K_{ij}^{(D_{r})}\dot{q}_{i}\dot
{q}_{j}+2\tau^{(1)}\left( \sum_{i=1}^{r-1}\omega_{i}\psi_{i}^{(+1)}\psi
_{i}^{(-1)}+\omega_{r}\psi_{r}^{(+1)}\psi_{r}^{(-1)}\right) \nonumber\\
& +2\tau^{(2)}\left( \sum_{i=1}^{r-2}\omega_{i}\omega_{i+1}\psi_{i}^{(+2)}\psi_{i}^{(-2)}+\omega_{r-2}\omega_{r}\psi^{(+2)}\psi^{(-2)}\right)
\nonumber\\
& +2\tau^{(3)}\left( \sum_{i=1}^{r-3}\omega_{i}\omega_{i+1}\omega
_{i+2}+\omega_{r-3}\omega_{r-2}\omega_{r}+\omega_{r-1}\omega_{r-2}\omega
_{r}\right) ,\end{aligned}$$ and $$\begin{aligned}
\{q_{i},\dot{q}_{j}\} & =\frac{1}{4}\left( K^{-1}\right) _{ij}^{(D_{r})},\quad i,j=1,\cdots,r;\\
\{\psi_{i}^{(+1)},\psi_{j}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \left( \delta_{j,\,i+1}\psi_{i}^{(+2)}-\delta_{j,\,i-1}\psi_{i-1}^{(+2)}\right) ,\\
\{\psi_{i}^{(+1)},\psi_{r}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \delta_{i,\,r-2}\psi^{(+2)},\\
\{\psi_{i}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] \left( \delta_{k,\,i+1}-\delta_{k,\,i-2}\right) ,\\
\{\psi_{r}^{(+1)},\psi_{k}^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau
^{(2)}\right] \left( \delta_{k,\,r-2}-\delta_{k,\,r-3}\right) ,\\
\{\psi_{i}^{(+1)},\psi^{(+2)}\} & =\left[ \tau^{(3,\,2)}/\tau^{(2)}\right]
\left( \delta_{i,\,r-3}+\delta_{i,\,r-1}\right) ,\\
\{\psi_{i}^{(-1)},\psi_{j}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] \left( \delta_{j,\,i+1}\psi_{i}^{(-2)}-\delta_{j,\,i-1}\psi_{i-1}^{(-2)}\right) ,\\
\{\psi_{i}^{(-1)},\psi_{r}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] \delta_{i,\,r-2}\psi^{(-2)}\\
\{\psi_{i}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] \left( \delta_{k,\,i+1}-\delta_{k,\,i-2}\right) ,\\
\{\psi_{r}^{(-1)},\psi_{k}^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau
^{(2)}\right] \left( \delta_{k,\,r-2}-\delta_{k,\,r-3}\right) ,\\
\{\psi_{i}^{(-1)},\psi^{(-2)}\} & =\left[ \tau^{(2,\,3)}/\tau^{(2)}\right]
\left( \delta_{i,\,r-3}+\delta_{i,\,r-1}\right) ,\end{aligned}$$ where the suffices $i,j$ of $\psi^{(\pm1)}$ take the values $i,j=1,\cdots
,r-1$, while the $k$ of $\psi^{(\pm2)}$ take the values $k=1,\cdots,r-2$.
So far, we have obtained in explicit form the Hamiltonians and Poisson brackets for the $(3,3)$-Toda chains associated with the classical Lie algebras $B_{r}$, $C_{r}$ and $D_{r}$, and also the equations of motions for the $(3,3)$-chain associated with $B_{r}$. These systems shares some common features, though with distinct details in the concrete couplings between mechanical variables. Some of the common features are 1) quasi-long range interactions among the variables $q_{i}$; 2) the couplings involve at most two $\psi$’s in each term, irrespective of their upper and lower indices; 3) some of the Poisson brackets among the $\psi$’s (i.e. $\psi_{i}^{(\pm1)}$) contain mechanical variables (i.e. $\psi_{k}^{(\pm2)}$), which makes the Poisson brackets for the $(3,3)$-chains into some nontrivial Lie algebras, rather than Heisenberg algebras as in the case of most mechanical as well as field theoretic systems. Actually, the last feature begins to show up only for $m,n\geq3$, which is part of the reason why we chose to present the explicit examples at $(m,n)=(3,3)$. Later, we shall make some discussion on the quantum consequences of the above mentioned Poisson brackets. But we shall first turn to another reason that we choose to begin our study at $(m,n)=(3,3)$, that is, the $(3,3)$-chains admit a good number of ways to make the reductions we mentioned in the end of the general construction.
Reductions of the $(3,3)$-chain associated with $B_{r}$
=======================================================
In this section, we shall illustrate the reductions mentioned in the end of Section \[secgen\] in explicit examples. The aim of this section is to show that both two types of reductions mentioned before can be combined and/or nested, and most of them are Hamiltonian, i.e. preserving the Poisson structure. The explicit examples we shall take are all reductions of the $(3,3)$-chain associated with $B_{r}$.
Type 1 reductions: combined and nested
--------------------------------------
By Type 1 reductions we mean the reductions $L^{(m,\,n)}\rightarrow$ $L^{(m-1,\,n)}$ and $L^{(m,\,n)}\rightarrow L^{(m,\,n-1)}$. It is easy to see that the reduction procedures $L^{(m,\,n)}\rightarrow$ $L^{(m-1,\,n)}$ and $L^{(m,\,n)}\rightarrow L^{(m,\,n-1)}$ mutually commute and they can be combined into a single step $L^{(m,\,n)}\rightarrow$ $L^{(m-1,\,n-1)}$. For $(m,n)=(3,3)$, this amounts to $L^{(3,\,3)}\rightarrow$ $L^{(2,\,2)}$ which we show below for the case of $B_{r}$. The reduction conditions read$$\begin{aligned}
\psi_{k}^{(\pm2)} & =1,\quad k=1,\cdots,r-1;\nonumber\\
\mu^{(\pm3)} & =0. \label{redcond}$$ Substituting these conditions into the equations of motion for the $(3,3)$-chain associated with $B_{r}$, we get the following reduced system of equations, which can be easily seen to be exactly the equations of motion for the $(2,2)$-chain associated with $B_{r}$. The equations of motion for the reduced system read
- equations for $q_{i}$:$$\begin{aligned}
\ddot{q}_{i} & =2\omega_{i}\left[ \tau^{(1)}\psi_{i}^{(+1)}\psi_{i}^{(-1)}+\tau^{(2)}\left( \omega_{i-1}+\omega_{i+1}\right) \right]
,\label{b22:1}\\
\ddot{q}_{r-1} & =2\omega_{r-1}\left[ \tau^{(1)}\psi_{r-1}^{(+1)}\psi
_{r-1}^{(-1)}+\tau^{(2)}\left( 2\omega_{r}+\omega_{r-2}\right) \right] ,\\
\ddot{q}_{r} & =2\omega_{r}\left[ \tau^{(1)}\psi_{r}^{(+1)}\psi_{r}^{(-1)}+\tau^{(2)}\omega_{r-1}\right] ;\end{aligned}$$
- equations for $\psi_{i}^{(\pm1)}$:$$\begin{aligned}
\dot{\psi}_{i}^{(+1)} & =2\tau^{(2,1)}\left( \omega_{i+1}\psi_{i+1}^{(-1)}-\omega_{i-1}\psi_{i-1}^{(-1)}\right) ,\label{psi22:1}\\
\dot{\psi}_{r-1}^{(+1)} & =2\tau^{(2,1)}\left( 2\omega_{r}\psi_{r}^{(-1)}-\omega_{r-2}\psi_{r-2}^{(-1)}\right) ,\\
\dot{\psi}_{r}^{(+1)} & =-2\tau^{(2,1)}\omega_{r-1}\psi_{r-1}^{(-1)},\end{aligned}$$$$\begin{aligned}
\dot{\psi}_{i}^{(-1)} & =2\tau^{(1,2)}\left( \omega_{i+1}\psi_{i+1}^{(+1)}-\omega_{i-1}\psi_{i-1}^{(+1)}\right) ,\\
\dot{\psi}_{r-1}^{(-1)} & =2\tau^{(1,2)}\left( 2\omega_{r}\psi_{r}^{(+1)}-\omega_{r-2}\psi_{r-2}^{(+1)}\right) ,\\
\dot{\psi}_{r}^{(-1)} & =-2\tau^{(1,2)}\omega_{r-1}\psi_{r-1}^{(+1)}.
\label{b22:last}$$
It is important to note that the reduction conditions (\[redcond\]) can be substituted directly into the Hamiltonian (\[hamb33\]) and Poisson brackets (\[PoiB33:1\])-(\[Poib33:last\]) without introducing any inconsistency. This shows that the above reduction is a kind of Hamiltonian reduction, with the reduced Hamiltonian $$\begin{aligned}
H_{B_{r}}^{(2,\,2)} & =\sum_{i,j=1}^{r}S_{ij}(B_{r})\dot{q}_{i}\dot{q}_{j}+2\tau^{(1)}\left( \sum_{i=1}^{r-1}\omega_{i}\psi_{i}^{(+1)}\psi
_{i}^{(-1)}+2\omega_{r}\psi_{r}^{(+1)}\psi_{r}^{(-1)}\right) \nonumber\\
& \,+2\tau^{(2)}\left( \sum_{i=1}^{r-2}\omega_{i}\omega_{i+1}+2\omega
_{r-1}\omega_{r}\right)\end{aligned}$$ and the reduced Poisson brackets$$\{q_{i},\dot{q}_{j}\}=\frac{1}{2}\left( S^{-1}\right) _{ij}(B_{r}),\quad
i,j=1,\cdots,r;$$$$\begin{aligned}
\{\psi_{i}^{(+1)},\psi_{j}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] (\delta_{j,\,i+1}-\delta_{j,\,i-1}),\\
\{\psi_{i}^{(+1)},\psi_{r}^{(+1)}\} & =\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \delta_{i,\,r-1},\\
\{\psi_{i}^{(-1)},\psi_{j}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] (\delta_{j,\,i+1}-\delta_{j,\,i-1}),\\
\{\psi_{i}^{(-1)},\psi_{r}^{(-1)}\} & =\left[ \tau^{(1,\,2)}/\tau
^{(1)}\right] \delta_{i,\,r-1}.\end{aligned}$$ The Poisson brackets involving $\psi_{k}^{(\pm2)}$ on the left hand side become trivial identities after the reduction.
One can of course make a further reduction $L^{(2,\,2)}\rightarrow
L^{(2,\,1)}$ by setting$$\begin{aligned}
\psi_{i}^{(-1)} & =1,\quad i=1,\cdots,r,\\
\mu^{(-2)} & =0.\end{aligned}$$ The resulting system of equations read$$\begin{aligned}
\ddot{q}_{i} & =2\tau^{(1)}\omega_{i}\psi_{i}^{(+1)},\quad\\
\dot{\psi}_{i}^{(+1)} & =2\tau^{(2,1)}\left( \omega_{i+1}-\omega
_{i-1}\right) ,\\
i & =1,\cdots,r,\end{aligned}$$ where $\psi_{r+1}^{(+1)}=0$. From the $(3,3)$-chain point of view, the $(2,1)$-chain is the result of the nested reduction $L^{(3,\,3)}\rightarrow$ $L^{(2,\,2)}\rightarrow L^{(2,\,1)}$. We should mention that the $(2,2)$-chain and $(2,1)$-chain could be considered also as dimensional reductions of the $B_{r}$ generalizations of the so-called bosonic superconformal Toda model [@Hou-Zhao] and the heterotic Toda model [@Zhao-Hou] respectively, which are both integrable field theoretic models in $(1+1)$-spacetime dimensions.
Type 2 reduction (symmetric reduction)
--------------------------------------
As mentioned in the end of Section \[secgen\], for the case of $m=n$, there is another type of reductions, i.e. the symmetric one. We shall only illustrate this type of reduction in the special case of $(2,2)$-chain associated with $B_{r}$. Starting from the equations of motion (\[b22:1\])-(\[b22:last\]), we may apply the symmetric reduction condition$$\begin{aligned}
\psi_{i}^{(+1)} & =\psi_{i}^{(-1)}=\psi_{i},\label{sym22}\\
\mu^{(+k)} & =\mu^{(-k)}.\nonumber\end{aligned}$$ The equations of motion for the reduced system (the $(2,2)_{\mathrm{Sym}}$-chain) turn out to be
- equations for $q_{i}$:$$\begin{aligned}
\ddot{q}_{i} & =2\omega_{i}\left[ \tau^{(1)}\psi_{i}^{2}+\tau^{(2)}\left(
\omega_{i-1}+\omega_{i+1}\right) \right] ,\\
\ddot{q}_{r-1} & =2\omega_{r-1}\left[ \tau^{(1)}\psi_{r-1}^{2}+\tau
^{(2)}\left( 2\omega_{r}+\omega_{r-2}\right) \right] ,\\
\ddot{q}_{r} & =2\omega_{r}\left[ \tau^{(1)}\psi_{r}^{2}+\tau^{(2)}\omega_{r-1}\right] ;\end{aligned}$$
- equations for $\psi_{i}$:$$\begin{aligned}
\dot{\psi}_{i} & =2\tau^{(2,1)}\left( \omega_{i+1}\psi_{i+1}-\omega
_{i-1}\psi_{i-1}\right) ,\\
\dot{\psi}_{r-1} & =2\tau^{(2,1)}\left( 2\omega_{r}\psi_{r}-\omega
_{r-2}\psi_{r-2}\right) ,\\
\dot{\psi}_{r} & =-2\tau^{(2,1)}\omega_{r-1}\psi_{r-1},\end{aligned}$$
with the corresponding Hamiltonian$$\begin{aligned}
H_{\mathrm{Sym}}^{(2,\,2)} & =\sum_{i,j=1}^{r}S_{ij}(B_{r})\dot{q}_{i}\dot{q}_{j}+2\tau^{(1)}\left( \sum_{i=1}^{r-1}\omega_{i}\psi_{i}^{2}+2\omega_{r}\psi_{r}^{2}\right) \nonumber\\
& \,+2\tau^{(2)}\left( \sum_{i=1}^{r-2}\omega_{i}\omega_{i+1}+2\omega
_{r-1}\omega_{r}\right)\end{aligned}$$ and canonical Poisson brackets$$\{q_{i},\dot{q}_{j}\}=\frac{1}{2}\left( S^{-1}\right) _{ij}(B_{r}),\quad
i,j=1,\cdots,r;$$$$\begin{aligned}
\{\psi_{i},\psi_{j}\} & =\frac{1}{2}\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] (\delta_{j,\,i+1}-\delta_{j,\,i-1}),\label{syb22Pois:1}\\
\{\psi_{i},\psi_{r}\} & =\frac{1}{2}\left[ \tau^{(2,\,1)}/\tau
^{(1)}\right] \delta_{i,\,r-1}. \label{syb22Pois:2}$$ One should notice that the Poisson brackets (\[syb22Pois:1\])-(\[syb22Pois:2\]) are not obtained from the corresponding Poisson brackets (\[psi22:1\])-(\[b22:last\]) by a direct substitution of the reduction condition (\[sym22\]). In fact, the reduction condition (\[sym22\]) mixes the originally Poisson commuting objects $\psi_{i}^{(+1)}$ and $\psi
_{i}^{(-1)}$ and hence has some more complicated behavior from the Hamiltonian dynamics point of view. However, for $r$ even, one can easily show by use of the Dirac method that the reduction from $(2,2)$-Toda chain to $(2,2)_{\mathrm{Sym}}$-Toda chain is still a Hamiltonian reduction. For $r$ odd, however, since the $(2,2)$-Toda system itself is already not free of constraints (since the matrix $G_{ij}\equiv\{\psi_{i}^{(+1)},\psi_{j}^{(+1)}\}$ is not invertible), the proof for the Hamiltonian nature for the symmetric reduction will be more involved, and we shall not deal with this problem any more.
Discussions on the quantum aspect
=================================
So far, we have been considering the generalized Toda chains as classical integrable mechanical systems. These systems are certainly of interests, especially from the modern SUSY gauge theory point of view, given the known relationship between standard Toda chains and the ($N=1$ and $2$) SUSY Yang-Mills theories in 4 spacetime dimensions.
Moreover, we can also consider the generalized Toda systems as quantum mechanical systems in the Heisenberg picture. Doing so we only need to replace all the Poisson brackets by commutation relations via $\{\,,\,\}\rightarrow
-i[\,,\,]$. One can immediately see that, when considered as quantum mechanical systems, all the $(m,n)$-Toda systems become systems of standard Toda variables ($q_{i}$’s) coupled to noncommutative coordinate variables ($\psi_{i}^{(\pm k)}$’s). That the variables $\psi_{i}^{(\pm k)}$ become noncommutative in the quantum case does not affect the correctness of the Lax equation (and hence the integrability), because all $\psi_{i}^{(+k)}$ remain commuting with $\psi_{j}^{(-l)}$. Also, there should be no problem in the ordering of variables in the Hamiltonians due to the same reason. However, the Lax integrability for the $(m,m)_{\mathrm{Sym}}$-chains will encounter some problem, because the variables $\psi^{(\pm k)}_{i}$ are reduced into a single set of new variables $\psi_{i}$ and there is noncommutativity among these variables.
The above discussions show that, in addition to learn the classical exact solutions for the equations of motion, it is also interesting to calculate the exact quantum correlations between different variables in the $(m,n)$-Toda systems (except the $(m,m)_{\mathrm{Sym}}$-chains). It is also hopeful that the generalized Toda systems presented here not only provide concrete models of exactly solvable quantum mechanical systems, but also describe certain actual physical system with enough complexity. In this respect, we should remind the readers that in some actual physical systems, quantum mechanics involving noncommutative coordinate variables have already found important applications [@Gamboa; @Nair:1].
Appendix A {#appendix-a .unnumbered}
==========
In this appendix, we list all the roots of heights $\pm1,\pm2$ and $\pm3$ for the Lie algebras $B_{r},C_{r}$ and $D_{r}$. These roots determine the form of the Lax matrices for the $(m,n)$-Toda chains with $m,n\leq3$. We also present the matrix representation for the Chevalley generators of the above Lie algebras, which are useful while calculating the Hamiltonians of the $(m,n)$-Toda chains.
Roos of different heights can be read out directly from the Dynkin diagrams. For instance, a root of height $\pm1$ corresponds to a single node in the Dynkin diagram; a root of height $\pm2$ corresponds to two connected nodes in the Dynkin diagram; while a root of height $\pm3$ corresponds to three simply connected nodes in the Dynkin diagram or two nodes with double links in between. In the case with double links, the node corresponding to the short simple root should be counted twice. The explicit form of all the roots of heights $\pm1,\pm2$ and $\pm3$ for the Lie algebras $B_{r},C_{r}$ and $D_{r}$ and the corresponding root vectors are listed below:
- $B_{r}$: $$\begin{array}
[l]{|l|l|l|l|}\hline\hline
h & \mathrm{roots} & \mathrm{Root\,vectors} & \\\hline
\pm1 & \pm\alpha_{i} & e_{i},f_{i} & i=1,...,r\\
\pm2 & \pm\left( \alpha_{i}+\alpha_{i+1}\right) & e_{(i,\,i+1)},f_{(i,\,i+1)} & i=1,...,r-1\\
\pm3 & \pm\left( \alpha_{i}+\alpha_{i+1}+\alpha_{i+2}\right) , &
e_{(i,\,i+1,\,i+2)},f_{(i,\,i+1,\,i+2)}, & i=1,...,r-2\\
& \pm(\alpha_{r-1}+2\alpha_{r}) & e_{(r,\,r-1,\,r)},f_{(r,\,r-1,\,r)} &
\\\hline\hline
\end{array}$$
- $C_{r}$:$$\begin{array}
[l]{|l|l|l|l|}\hline\hline
h & \mathrm{roots} & \mathrm{Root\,vectors} & \\\hline
\pm1 & \pm\alpha_{i}\qquad & e_{i},f_{i} & i=1,...,r\\
\pm2 & \pm\left( \alpha_{i}+\alpha_{i+1}\right) \qquad & e_{(i,\,i+1)},f_{(i,\,i+1)} & i=1,...,r-1\\
\pm3 & \pm\left( \alpha_{i}+\alpha_{i+1}+\alpha_{i+2}\right) , &
e_{(i,\,i+1,\,i+2)},f_{(i,\,i+1,\,i+2)}, & i=1,...,r-2\\
& \pm\left( 2\alpha_{r-1}+\alpha_{r}\right) & e_{(r-1,\,r-1,\,r)},f_{(r-1,\,r-1,\,r)} & \\\hline\hline
\end{array}$$
- $D_{r}$:$$\begin{array}
[c]{|l|l|l|l|}\hline\hline
h & \mathrm{roots} & \mathrm{Root\,vectors} & \\\hline
\pm1 & \pm\alpha_{i} & e_{i},f_{i} & i=1,...,r\\
\pm2 & \pm\left( \alpha_{i}+\alpha_{i+1}\right) , & e_{(i,\,i+1)},f_{(i,\,i+1)}, & i=1,...,r-2\\
& \pm\left( \alpha_{r-2}+\alpha_{r}\right) & e_{(r-2,\,r)},f_{(r-2,\,r)} &
\\
\pm3 & \pm\left( \alpha_{i}+\alpha_{i+1}+\alpha_{i+2}\right) , &
e_{(i,\,i+1,\,i+2)},f_{(i,\,i+1,\,i+2)}, & i=1,...,r-3\\
& \pm\left( \alpha_{r-1}+\alpha_{r-2}+\alpha_{r}\right) , &
e_{(r-1,\,r-2,\,r)},f_{(r-1,\,r-2,\,r)}, & \\
& \pm\left( \alpha_{r-3}+\alpha_{r-2}+\alpha_{r}\right) &
e_{(r-3,\,r-2,\,r)},f_{(r-3,\,r-2,\,r)} & \\\hline\hline
\end{array}$$
For the Chevalley generators, we have the following representation matrices in the defining representation. For the Lie algebra $B_{r}$, $$\begin{aligned}
h_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i}-e_{r+i,\,r+i}-e_{i+1,\,i+1}+e_{r+i+1,\,r+i+1}, & i=1,2,\cdots,r-1,\\
2(e_{r,\,r}-e_{2r,\,2r}),~~~ & i=r.
\end{array}
\right. \nonumber\\
e_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i+1}-e_{r+i+1,\,r+i},~~ & i=1,2,\cdots,r-1,\\
\sqrt{2}(e_{r,\,2r+1}-e_{2r+1,\,2r}), & i=r.
\end{array}
\right. \nonumber\\
f_{i} & =\left\{
\begin{array}
[c]{ll}e_{i+1,\,i}-e_{r+i,\,r+i+1},~~~ & i=1,2,\cdots,r-1,\\
\sqrt{2}(-e_{2r,\,2r+1}+e_{2r+1,\,r}), & i=r.
\end{array}
\right. \label{05}$$ For $C_{r}$,$$\begin{aligned}
h_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i}-e_{r+i,\,r+i}-e_{i+1,\,i+1}+e_{r+i+1,\,r+i+1}, & i=1,2,\cdots,r-1,\\
e_{r,\,r}-e_{2r,\,2r},~~~ & i=r.
\end{array}
\right. \nonumber\\
e_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i+1}-e_{r+i+1,\,r+i}, & i=1,2,\cdots,r-1,\\
e_{r,\,2r},~~~ & i=r.
\end{array}
\right. \nonumber\\
f_{i} & =\left\{
\begin{array}
[c]{ll}e_{i+1,\,i}-e_{r+i,\,r+i+1}, & i=1,2,\cdots,r-1,\\
e_{2r,\,r},~~ & i=r,
\end{array}
\right. \label{06}$$ and lastly, for $D_{r}$, we have$$\begin{aligned}
h_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i}-e_{r+i,\,r+i}-e_{i+1,\,i+1}+e_{r+i+1,\,r+i+1}, & i=1,2,\cdots,r-1,\\
e_{r-1,\,r-1}-e_{2r-1,\,2r-1}+e_{r,\,r}-e_{2r,\,2r}, & i=r.
\end{array}
\right. \nonumber\\
e_{i} & =\left\{
\begin{array}
[c]{ll}e_{i,\,i+1}-e_{r+i+1,\,r+i},~ & i=1,2,\cdots,r-1,\\
e_{r-1,\,2r}-e_{r,\,2r-1},~ & i=r.
\end{array}
\right. \nonumber\\
f_{i} & =\left\{
\begin{array}
[c]{ll}e_{i+1,\,i}-e_{r+i,\,r+i+1}, & i=1,2,\cdots,r-1,\\
-(e_{2r-1,\,r}+e_{2r,\,r-1}), & i=r,
\end{array}
\right. \label{07}$$ where $e_{i,\,j}$ is the usual matrix units which should not be confused with the root vectors $e_{(i,\,j)}$.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported in part by the National Natural Science Foundation of China.
[^1]: Email:lzhao@nwu.edu.cn
[^2]: Email:wyl@phy.nwu.edu.cn
[^3]: Email:yzy@phy.nwu.edu.cn
|
---
abstract: 'Conditional quantum oscillations are investigated for quantum gate operations in superconducting flux qubits. We present an effective Hamiltonian which describes a conditional quantum oscillation in two-qubit systems. Rabi-type quantum oscillations are discussed in implementing conditional quantum oscillations to quantum gate operations. Two conditional quantum oscillations depending on the states of control qubit can be synchronized to perform controlled-gate operations by varying system parameters. It is shown that the conditional quantum oscillations with their frequency synchronization make it possible to operate the controlled-NOT and -$U$ gates with a very accurate gate performance rate in interacting qubit systems. Further, this scheme can be applicable to realize a controlled multi-qubit operation in various solid-state qubit systems.'
author:
- Ai Min
- Sam Young
title: Controllable conditional quantum oscillations and synchronization in superconducting flux qubits
---
[^1]
Introduction
============
Quantum gates lie at the heart of realization of quantum computing [@Nielsen]. Two-qubit gates as well as single-qubit gates have been demonstrated in various types of quantum systems such as cavity QED [@cavityQED], ion traps [@iontraps], NMR [@NMR], quantum dots [@QD], and superconducting charge [@charge] and flux [@Plantenberg] qubits . To perform qubit operations, normally, an electromagnetic field is applied such as microwave fields, laser pulses, and oscillating voltages, which can induce quantum oscillations between qubit states. Especially, quantum Rabi oscillations can be applicable to achieve a controlled-gate operation because it implies a complete transition between two quantum states at the half-period time of the oscillation. Of particular interests are to manipulate such quantum oscillations between qubit states by applying an external field in association with controlled-gate operations.
A gate operation depending on a control qubit state can be performed, which is called [*conditional gate operation*]{}, where the target states can be flipped for a control-NOT (CNOT) gate operation. It has been experimentally demonstrated for a pair of superconducting qubits in Refs. [@charge] and [@Plantenberg]. In the experiments, an individual conditional operation has been applied to observe a CNOT gate operation in the superconducting charge and flux qubits. Also, in Ref. [@Gagnebin], a similar scheme has been theoretically discussed via time evolution of a two qubit system for an applied pulsed-bias duration. In our study, a [*conditional quantum oscillation*]{} rather than a conditional gate operation is introduced by considering an effective interacting two-qubit Hamiltonian which can be adjusted within some system parameter ranges. We investigate how conditional quantum oscillations can be simultaneously manipulated to perform a controlled-gate operation in a controllable and accurate manner.
To clearly discuss an implementation of conditional quantum oscillations to quantum gate operations, in this paper, we restrict ourselves to a rotating wave approximation (RWA) in the presence of applied time-dependent fields, which allows us to capture an essential physics for controlled quantum gate operations based on conditional quantum oscillations. At resonant frequencies, conditional Rabi and non-Rabi oscillations are shown to characterize the time-dependent dynamics of the two qubit system. We discuss a frequency matching condition for achievable controlled-gate operations. By synchronizing the two oscillation frequencies on the matching condition, the CNOT gate operation performance and operation time are shown to be controllable to obtain a very accurate gate operation. It shows that conditional quantum oscillations and their frequency synchronization are applicable to various quantum gate operations in solid-state multi-qubit systems such as Toffoli and Fredkin gates.
Model
=====
Any two-state system can play a role of qubits. In solid-state systems, qubit parameters are controllable. In terms of the pseudo-spin-1/2 language, a single qubit in the two states of the basis $\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\}$ can be described by the Hamiltonian $H_{i}=\left[(\varepsilon_i+V_i(t))\mbox{\boldmath $\sigma$}^z_i
- \Delta_i \mbox{\boldmath $\sigma$}^x_i\right]/2$, where $\mbox{\boldmath $\sigma$}$’s are the Pauli matrices with the identity matrix $\mbox{\boldmath $\sigma$}^0$. $\varepsilon_i$ and $\Delta_i$ correspond to the energy difference and transition amplitude between the two states of the qubit $i$, respectively. $V_i(t)$ is responsible for an interplay between the states of the qubit $i$ and a time-dependent applied field. In the basis $\{\left|\uparrow\uparrow\right\rangle,\left|\uparrow\downarrow\right\rangle,
\left|\downarrow\uparrow\right\rangle,\left|\downarrow\downarrow\right\rangle\}$, let us consider the Hamiltonian [@Majer; @Plantenberg] of interacting two qubits $(i\in \{A,B\})$ written by $$H=H_A\otimes \mbox{\boldmath $\sigma$}\!_B^0+ \mbox{\boldmath $\sigma$}_A^0 \otimes
H_B+J\mbox{\boldmath $\sigma$}\!_{A}^{z}\otimes\mbox{\boldmath $\sigma$}\!_{B}^{z},$$ where $J$ is the interaction strength characterizing the interaction between the two qubits. If the transition amplitudes $\Delta_A$ or $\Delta_B$ become negligible by means of controlling system parameters, the Hamiltonian has a form of block-diagonal matrix. Each block of the matrix Hamiltonian corresponds to a conditional Hamiltonian. For instance, the negligible $\Delta_A$ leads to, for the states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ of qubit $A$, the effective qubit $B$ Hamiltonians respectively written by
$$\begin{aligned}
H_{B:\left|\uparrow\right\rangle}\!\!\!
&=&\!\!\!\frac{1}{2}\!
\left[(\varepsilon_B\! +\! 2J +\! V_B(t))\mbox{\boldmath $\sigma$}\!_B^z
\!+\!(\varepsilon_A \!+\! V_A(t))\mbox{\boldmath $\sigma$}\!_B^0
\!-\! \Delta_B \mbox{\boldmath $\sigma$}_B^x\right],
\label{Conditional:H1}
\\
H_{B:\left|\downarrow\right\rangle}\!\!\!
&=&\!\!\!\frac{1}{2}\!
\left[ (\varepsilon_B\! -\! 2J +\! V_B(t))\mbox{\boldmath $\sigma$}\!_B^z
\!-\!(\varepsilon_A\! +\! V_A(t))\mbox{\boldmath $\sigma$}\!_B^0
\!-\! \Delta_B \mbox{\boldmath $\sigma$}\!_B^x\right].
\label{Conditional:H2}
\end{aligned}$$
Equations (\[Conditional:H1\]) and (\[Conditional:H2\]) show that the time-dependent dynamics of the system can be understood in association with a combination of [*two conditional quantum oscillations of the qubit $B$*]{}. Away from the qubit degeneracy point, where the qubit energy $\varepsilon_A$ ( transition amplitude $\Delta_A$) becomes relatively bigger (smaller) than other parameters, such conditional quantum oscillations can be achievable [@Plantenberg].
Conditional quantum oscillations
=================================
In the absence of time-dependent applied fields $V_i(t)=0$ ($i \in \{A,B\})$, the time-independent conditional Hamiltonians generate the time evolution of the states of qubit $B$ through the Schrödinger equation $i \partial_t \left| \psi_{B:s}(t) \right\rangle
=H^{(0)}_{B:s}\left|\psi_{B:s}(t)\right\rangle$. For the two conditional time evolutions of the qubit states, at time $t$, the states are written by $ \left|\psi_{B:s}(t)\right\rangle = G_{B:s}(t) \left|\psi_{B:s}(0)\right\rangle$, where $G_{B:s}(t)=U^{-1}(\eta^{(0)}_{B:s}) \exp\left[-i {\tilde
H}^{(0)}_{B:s}t\right]U(\eta^{(0)}_{B:s})$ with ${\tilde H}^{(0)}_{B:s}=U(\eta^{(0)}_{B:s})
H^{(0)}_{B:s} U^{-1}(\eta^{(0)}_{B:s})$. Here, $U(\eta)$ is a unitary transformation matrix which makes the conditional Hamiltonians $H^{(0)}_{B:s}$ diagonal respectively with $\eta^{(0)}_{B:\left|\uparrow\right\rangle/\left|\downarrow\right\rangle}
= \tan^{-1}\Delta_B/\left(\varepsilon_B \pm 2 J\right)$. The unitary transformation matrix is written by $ U(\eta)
= \left( \begin{array}{cc}
\cos\frac{\eta}{2} & -\sin\frac{\eta}{2} \\
\sin\frac{\eta}{2} & \cos\frac{\eta}{2} \end{array}
\right)$. These unitary transformations give the eigenvalues of the conditional Hamiltonians, i.e., the eigenvalues of the system, without time-dependent external fields: $\varepsilon^{(0)}_{\pm:\left|\uparrow\right\rangle}
=
\frac{1}{2}\left( \varepsilon_A \pm
\left[(\varepsilon_B+2 J)^2+\Delta_B^2\right]^{1/2}\right)$ and $ \varepsilon^{(0)}_{\pm:\left|\downarrow\right\rangle}
=
-\frac{1}{2}\left( \varepsilon_A \mp
\left[(\varepsilon_B-2 J)^2+\Delta_B^2\right]^{1/2}\right)$.
In fact, the two conditional time evolutions of the one qubit can characterize the dynamics of two qubit system. If the initial state of qubit $B$ is chosen as an arbitrary state $\left|\psi_{B:s}(0)\right\rangle
=a\left|\uparrow\right\rangle+b\left|\downarrow\right\rangle$ with $a^2+b^2=1$, the occupation probabilities of the states $\{\left|\uparrow\uparrow\right\rangle,\left|\uparrow\downarrow\right\rangle,
\left|\downarrow\uparrow\right\rangle,\left|\downarrow\downarrow\right\rangle\}$ at time $t$ are obtained as $P_{s \uparrow}(t)=1-P_{s \downarrow}(t)
=a^2+[b^2-(a\sin\eta^{(0)}_{B:s}
+b\cos\eta^{(0)}_{B:s})^2
]\sin^2[\Omega^{(0)}_{B:s} t/2]
$ for the $\left|s\right\rangle$ states of qubit $A$, where the conditional oscillation frequencies are respectively denoted by the qubit Larmor frequencies $\Omega^{(0)}_{B:\left|\uparrow\right\rangle/\left|\downarrow\right\rangle}
=\left[(\varepsilon_B\pm 2
J)^2+\Delta_B^2\right]^{1/2}$ corresponding to the resultant energy level spacings. (i) For $a=1$ and $b=0$, i.e., $\left|\psi_{B:s}(0)\right\rangle=\left|\uparrow\right\rangle$ or (ii) for $a=0$ and $b=1$, i.e., $\left|\psi_{B:s}(0)\right\rangle=\left|\downarrow\right\rangle$, if $\varepsilon_B=-2J$ $(\varepsilon_B=2J)$ then the conditional Rabi oscillation between the states $\left|\uparrow\uparrow\right\rangle$ $(\left|\downarrow\uparrow\right\rangle)$ and $\left|\uparrow\downarrow\right\rangle$ $(\left|\downarrow\downarrow\right\rangle)$ occurs with the characteristic frequency $\Omega^{(0)}_{R} = \Delta_B$. These conditional Rabi oscillations show that an initial state of the qubit $B$ can be in the other flipped state of the qubit $B$ at the periodic time $t=2\pi(m-1/2)/\Delta_B$ with a positive integer $m$.
For $\varepsilon_B=2 J$, as an example, the conditional Rabi oscillation between the states $\left|\downarrow\uparrow\right\rangle$ and $\left|\downarrow\downarrow\right\rangle$ occurs with its frequency $\Omega^{(0)}_{R} = \Delta_B$, while a non-Rabi oscillation between the states $\left|\uparrow\uparrow\right\rangle$ and $\left|\uparrow\downarrow\right\rangle$ takes place with the frequency $\Omega^{(0)}_{n\mbox{-}R}
= \left[16 J^2 +\Delta^2_B\right]^{1/2}$. At the time $t=\pi/\Delta_B$, if the states $\left|\uparrow\uparrow\right\rangle$ and $\left|\uparrow\downarrow\right\rangle$ are in their original states, i.e., for instance, the non-Rabi frequency becomes $ \Omega^{(0)}_{n\mbox{-}R}= 2\Omega^{(0)}_{R}$ $
(J=\frac{\sqrt{3}}{4}\Delta_B=\frac{1}{2}\varepsilon_B)$ [@Gagnebin], the two qubit system can perform a CNOT gate operation. Then one can expect a CNOT gate operation at the half-period times of the Rabi oscillation because the states of target qubit can be in their flipped states for the one state of control qubit (Rabi oscillation) while they can stay at the original states for the other state of control qubit (non-Rabi oscillation). Therefore, adjusting the conditional time evolutions of non-Rabi and Rabi oscillations enables to perform a controlled-gate operation. We will discuss the details of possible controlled-gate operations in the presence of time-dependent applied fields below.
Controlled-gate operations with controllable conditional quantum oscillations
=============================================================================
To implement the conditional quantum oscillations to a quantum gate operation, suppose that the qubit $A$ is the control qubit and the qubit $B$ is the target qubit. Let us consider a time-dependent applied field $V_{i}(t)= V_i \cos\omega t$, where $V_i$ and $\omega$ are its amplitude and frequency, respectively. The time-dependent applied field can give rise to an external field-driven conditional Rabi oscillation. Recall a unitary transformation $U(\eta^{(0)}_{B})$, where $\eta^{(0)}_{B}=\tan^{-1} \Delta_B/\varepsilon_B$. Actually, any other unitary transformation can be used for the qubit $B$ such as $\eta^{(0)}_{B:s}$. The conditional Hamiltonians are then transformed as ${\tilde H}_{B:s}=U(\eta^{(0)}_{B}) H_{B:s} U^{-1}(\eta^{(0)}_{B})$. The basis $\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\}$ of qubit $B$ are transformed into the basis $\{\left|0\right\rangle,\left|1\right\rangle\}$ by $\left|\tilde \psi_{B:s}\right\rangle=U(\eta^{(0)}_{B})\left|
\psi_{B:s}\right\rangle$.
In order to show clearly a possible controlled-gate operation, we will employ a rotating wave approximation (RWA) which implies that an applied time-dependent field can be a static field in a rotating frame. The system parameters can also be adjusted to satisfy the regime that $\varepsilon_B V_B/\Omega_{B}^{(0)} \ll
\Omega_{B}^{(0)}$ [@Saito], $V_A \ll \varepsilon_A$, and $|2 J| \ll V_B$, where $\Omega_{B}^{(0)}=\left[\varepsilon_B^2+ \Delta_B^2\right]^{1/2}$. Within the approximations, then, the conditional Hamiltonians in the rotating frame are obtained through $\tilde H_{B:s}^{\rm eff}=-i U(t) \partial_t U^{-1}(t)
+ U(t) \tilde H_{B:s} U^{-1}(t)$ with $U(t)=\exp\left[i\omega t\, \mbox{\boldmath $\sigma$}\!_B^z/2\right]$ as
$$\begin{aligned}
\tilde H_{B:\left|\uparrow\right\rangle}^{\rm eff}
\!\!\!\!&\simeq&\!\!\!\frac{1}{2}\!
\left[\left(\Omega^{(0)}_{B}\!+\!\frac{2 J\varepsilon_B}{\Omega^{(0)}_{B}}
\!-\!\omega\right)\!
\mbox{\boldmath $\sigma$}\!_B^z + \varepsilon_A \mbox{\boldmath
$\sigma$}\!_B^0 \!+\!
\frac{V_B}{2}\!\left(\frac{\Delta_B}{\Omega^{(0)}_{B}}\right)\!
\mbox{\boldmath $\sigma$}_B^x \right]\!,
\label{RWA:effH1}
\\
\tilde H_{B:\left|\downarrow\right\rangle}^{\rm eff}
\!\!\!\!&\simeq&\!\!\!
\frac{1}{2}\!
\left[\left(\Omega^{(0)}_{B}\!-\!\frac{2 J\varepsilon_B}{\Omega^{(0)}_{B}}
\!-\!\omega\right)\!
\mbox{\boldmath $\sigma$}\!_B^z - \varepsilon_A \mbox{\boldmath
$\sigma$}\!_B^0
\!+\!
\frac{V_B}{2}\!\left(\frac{\Delta_B}{\Omega^{(0)}_{B}}\right)\!
\mbox{\boldmath $\sigma$}_B^x \right]\!.
\label{RWA:effH2}
\end{aligned}$$
To see a quantum gate operation, one can employ a probability amplitude table ${\cal U}(t)$ at time $t$. For the conditional quantum oscillations in two qubit systems, then, a two-qubit gate operation can be seen in the probability amplitude table ${\cal U}(t)$ expressed as $${\cal U}(t) = % e^{i\frac{\varepsilon_A}{2}t}
\left( \begin{array}{cccc}
%%%%%%%%%%%%%%%%%111111111111111111%%%%%%%%%%%%%%%%%%%%%%%%%%%
P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow
0\right\rangle}(t)
& P_{\left|\uparrow 1\right\rangle \leftarrow
\left|\uparrow 0\right\rangle}(t) & 0 & 0 \\
%%%%%%%%%%%%%%%%%22222222222222222222%%%%%%%%%%%%%%%%%%%%%%%%%%%
P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow
1\right\rangle}(t)
&
P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow
1\right\rangle}(t)
& 0 & 0 \\
%%%%%%%%%%%%%%%%%33333333333333333%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0& P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow
0\right\rangle} (t)
& P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow 0\right\rangle}(t) \\
%%%%%%%%%%%%%%%%%444444444444444444%%%%%%%%%%%%%%%%%%%%%%%%%%%
0 & 0 &
P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow
1\right\rangle}(t)
& P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow
1\right\rangle}(t)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{truth:1}
\end{array}
\right),$$ where $P_{\beta \leftarrow \alpha}$ denotes the probability that an $\alpha$ input state becomes a $\beta$ output state with $\alpha$, $\beta$ $\in \{ \left|\uparrow 0\right\rangle,
\left|\uparrow 1\right\rangle, \left|\downarrow 0\right\rangle,
\left|\downarrow 1\right\rangle \}$. Within the approximations, by solving the Schrödinger equations of the effective Hamiltonians $i \partial_t \left| \tilde \psi_{B:s}(t) \right\rangle
=\tilde H^{\rm eff}_{B:s}\left| \tilde \psi_{B:s}(t)\right\rangle$ in Eqs. (\[RWA:effH1\]) and (\[RWA:effH2\]), we obtain the probabilities as $$P_{\left|s 1\right\rangle \leftarrow \left|s 0\right\rangle}(t) =
\sin^2 \eta_{B:s}\sin^2\frac{\Omega_{B:s}}{2}t$$ with the relations $ P_{\left|s 0\right\rangle \leftarrow \left|s
0\right\rangle}
\! =\!
P_{\left|s 1\right\rangle \leftarrow \left|s
1\right\rangle}
\!=\!1\!-\!P_{\left|s 1\right\rangle \leftarrow \left|s
0\right\rangle}
\!=\!1\!-\!P_{\left|s 0\right\rangle \leftarrow \left|s
1\right\rangle},
$ where the conditional oscillation frequencies are $
\Omega_{B:s}
\!\! =\!\!\left[\left(
\omega-\Omega^{(0)}_{B}-2 s J\, \varepsilon_B/{\Omega^{(0)}_{B}} \right)^2
\!+\! \left(\Delta_B V_B/ \Omega^{(0)}_{B}\right)^2\!\! /4\right]^{1/2}
$ for $\left|\uparrow\right\rangle/\left|\downarrow\right\rangle =
\pm $. The transformation angles are denoted by $\eta_{B:s}
=\tan^{-1}\left[\frac{ \Delta_B V_B}{2\,
\left(\omega-\Omega^{(0)}_{B}-2 s J\, \varepsilon_B/{\Omega^{(0)}_{B}}
\right) \Omega^{(0)}_{B}}\right]$.
There are two resonant frequencies (i) $\omega =\Omega^{(0)}_{B}
+2J\,\varepsilon_B/\Omega^{(0)}_{B}$ and (ii) $\omega = \Omega^{(0)}_{B}
-2J\,\varepsilon_B/\Omega^{(0)}_{B}$, each of which can induce a conditional Rabi oscillation. (i) For $\omega =\Omega^{(0)}_{B}
+2J\,\varepsilon_B/\Omega^{(0)}_{B}$, $ P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow
0\right\rangle}(t)
= \sin^2\frac{\Omega_{R}}{2}t
$ undergoes a Rabi oscillation while $
P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow
0\right\rangle}(t)
= \sin^2 \eta_{n\mbox{-}R}
\sin^2\frac{\Omega_{n\mbox{-}R}}{2}t
$ can do a non-Rabi oscillation, where the conditional Rabi and non-Rabi oscillation frequencies are respectively given by
$$\begin{aligned}
\Omega_{R} &=& \frac{V_B}{2}\left(\frac{\Delta_B}{\Omega^{(0)}_{B}}\right),
\label{Rabi}
\\
\Omega_{n\mbox{-}R}&=&
\left[16 J^2 \left(\frac{\varepsilon_B}{\Omega^{(0)}_{B}}\right)^2
+ \frac{V^2_B}{4}
\left(\frac{\Delta_B}{\Omega^{(0)}_{B}}\right)^2\right]^{\frac{1}{2}}.
\label{nonRabi}
\end{aligned}$$
The amplitude of the non-Rabi oscillation is determined by $\eta_{n\mbox{-}R}=\tan^{-1}\left[ \Delta_B V_B/8 J\varepsilon_B \right]$. The Rabi oscillation has a longer period than the non-Rabi oscillation because the non-Rabi frequency is larger than the Rabi oscillation frequency, $\Omega_{R} < \Omega_{n\mbox{-}R}$. It should be noted that, in the chosen basis, the Rabi frequency does not depend on the interaction strength $J$ within our approximations while the non-Rabi frequency depends on the interaction. This shows that if no interaction exists between the qubits, in fact, the conditional quantum oscillations are not realizable in the basis. If one choose other basis, however, a Rabi frequency can be dependent of the interaction strength $J$. (ii) For the other resonant frequency $\omega = \Omega^{(0)}_{B}
-2J\,\varepsilon_B/\Omega^{(0)}_{B}$, the two conditional quantum oscillations exchange their roles each other, i.e., $ P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow
0\right\rangle}(t)
= \sin^2\frac{\Omega_{R}}{2}t
$ undergoes a Rabi oscillation while $
P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow
0\right\rangle}(t)
= \sin^2 \eta_{n\mbox{-}R}
\sin^2\frac{\Omega_{n\mbox{-}R}}{2}t
$ can do a non-Rabi oscillation.
To see clearly a role of two conditional quantum oscillations for CNOT gate operations, let us introduce the fidelity $F$ of the probability amplitude table for the truth table of CNOT gate as $F (t)= \frac{1}{4}\mathrm{Tr}\left[ {\cal U}(t)
{\cal U}_\mathrm{CNOT}\right]$. The fidelity $F$ and its error $\delta F(t)=1-F(t)$ give the estimations of CNOT gate performance and its reliability. In terms of the transition probability amplitudes, with ${\cal
U}_\mathrm{CNOT}=\mathrm{diag([0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1])}$, the fidelity is generally given by $$F = \frac{1}{4}
\Big( P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow 1\right\rangle}
+
P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow 0\right\rangle}
+
P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow 0\right\rangle}
+
P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow 1\right\rangle}
\Big).
\label{F}$$ For the conditional quantum oscillations, from Eq. (\[truth:1\]), the fidelity becomes $F(t) = \frac{1}{2}\Big(
P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow 1\right\rangle} (t)
+
P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow
0\right\rangle} (t)
\Big)$ because $P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow 1\right\rangle} (t)
=P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow 0\right\rangle}
(t)$ and $P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow
0\right\rangle} (t)
=P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow
1\right\rangle} (t)$. This shows that two conditional quantum oscillations with their characteristic frequencies $\Omega_{R}$ and $\Omega_{n\mbox{-}R}$ determine a CNOT operation performance and its reliability. For the resonant frequency $\omega
=\Omega^{(0)}_{B}+2J\varepsilon_B/\Omega^{(0)}_B$, the Rabi oscillation $P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow
1\right\rangle} (t)$ shows that the initial state is in its flipped state at a periodic time $t=(m-1/2)\, \tau_R$ with a positive integer $m$, where $\tau_R=2\pi/\Omega_{R}$ is the period of Rabi oscillation. The non-Rabi oscillation $P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow 0\right\rangle}
(t)$ shows that the initial state is in its original state at a periodic time $t=m\, \tau_{n\mbox{-}R}$, where $\tau_{n\mbox{-}R}=2\pi/\Omega_{n\mbox{-}R}$ is the period of non-Rabi oscillation. Performing a CNOT gate operation $(F=1)$ is, in fact, to synchronize the periods $\tau_R$ and $\tau_{n\mbox{-}R}$ of the Rabi and non-Rabi oscillations by varying the system parameters because, according to the control qubit states, the target state should be in a flipped state (Rabi oscillation: $P_{\left|\uparrow 0\right\rangle \leftarrow \left|\uparrow
1\right\rangle}=1$) or the original state (non-Rabi oscillation: $P_{\left|\downarrow 0\right\rangle \leftarrow \left|\downarrow
1\right\rangle}=0$) at a certain operation time $t_{OP}$.
Synchronization of two conditional oscillations for controlled gate operation
=============================================================================
To synchronize the two conditional quantum oscillations for a CNOT gate operation, we discuss a frequency matching condition between the quantum oscillations. If the operation time of CNOT gate is $t^{(1)}_{OP} =\tau_R/2$ for the first flipped state in the Rabi oscillation, a CNOT gate operation can be performed with a positive [*multiple-integer period*]{} of non-Rabi oscillation $ n \tau_{n\mbox{-}R} =\tau_R/2$, where $n$ is a positive integer. Once the frequencies are matched with the condition, the CNOT gate is operated periodically at the periodic operation time $t^{(m)}_{OP}=(m-1/2)\,
\tau_R$ with the $m$-th flipped state of the Rabi oscillation. More generally, if a CNOT gate operation is performed at the $l$-th flipped states in the Rabi oscillation, i.e., the operation time becomes $t_{OP} =(l-1/2) \tau_R$ with a positive integer $l$, then the matching frequencies are given by the relation $ (n+l-1) \tau_{n\mbox{-}R} = (l-1/2)\tau_R$ because of $ \tau_{n\mbox{-}R} < \tau_R $ ($\Omega_R < \Omega_{n\mbox{-}R}$). In other words, a $(n+l-1)$ multiple-integer period of non-Rabi oscillation matches with the period $(l-1/2)\tau_R$ of Rabi oscillation for the CNOT gate operation. For $l=1$, these matching frequencies are reduced to the relation $ n \tau_{n\mbox{-}R} =\tau_R/2$ at the first flipped state. Consequently, the matching frequencies between the conditional non-Rabi and Rabi oscillations for a CNOT gate operation are given by the relation $$\Omega_{n\mbox{-}R} (V_B)
=2 \left(\frac{n + l -1}{2 l -1} \right)
\Omega_{R} (V_B).
\label{RabiFrequency}$$ Both the Rabi and non-Rabi oscillations can be tuned by varying the amplitude of applied fields $V_B$ for a CNOT gate operation. By using the conditional quantum oscillations, therefore, a CNOT gate operation can be achievable by synchronizing the periods of conditional Rabi and non-Rabi oscillations. Note that, for the other resonant frequency $\omega =\Omega^{(0)}_{B}
-2J\varepsilon_B/\Omega^{(0)}_{B}$, the matching frequencies in Eq. (\[RabiFrequency\]) leads to another CNOT gate operation with the ideal truth table ${\cal U}_\mathrm{CNOT}=\mathrm{diag([1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0])}$ because $P_{\left|\uparrow 1\right\rangle \leftarrow \left|\uparrow
0\right\rangle}=0$ and $P_{\left|\downarrow 1\right\rangle \leftarrow \left|\downarrow
0\right\rangle}=1$ at the operation time $t_{OP} =(l-1/2) \tau_R$.
When the CNOT gate operation has carried out, from Eq. (\[RabiFrequency\]), the amplitudes of the applied time-dependent field $V_B$ are given by $$V^\mathrm{CNOT}_B (n,l) = \frac{2 l -1}{\sqrt{(2n-1)(2n+4 l-3)}}\left(
\frac{8J\,\varepsilon_B}{\Delta_B } \right).
\label{VB}$$ Then as $V_B$ varies a CNOT gate operation is executed for $V_B=V^{\rm CNOT}_B(n,l)$ consecutively. This implies that the synchronization of the operation time $t_{OP}=(l-1/2) \tau_R$ can be achievable by tuning the applied time-dependent field $V_B$. Also, it is shown that from Eqs. (\[Rabi\]), (\[nonRabi\]), and (\[VB\]) the CNOT gate operation can be performed with [*possible Rabi and non-Rabi frequencies*]{} given as
$$\begin{aligned}
\Omega^\mathrm{CNOT}_{R} (n,l)
\!\!&=&\!\!
\frac{2 l -1}{\sqrt{(2n-1)(2n+4 l-3)}}\left(\!
\frac{4J\,\varepsilon_B}{\sqrt{\varepsilon^2_B+\Delta^2_B} }
\right),
\\
\Omega^\mathrm{CNOT}_{n\mbox{-}R} (n,l)\!\!
&=&\!\!
\frac{2 (n + l -1)}{\sqrt{(2n-1)(2n+4 l-3)}}\left(\!
\frac{4J\,\varepsilon_B}{\sqrt{\varepsilon^2_B+\Delta^2_B} }
\right).
\end{aligned}$$
As a result, synchronizing well conditional quantum oscillations by varying system parameters makes it possible to achieve a CNOT gate operation with a very accurate performance rate, which can be applied to various types of qubit systems. In addition, the operation time can be controlled by means of the matching frequencies.
A CNOT gate is a special case of the controlled-$U$ gate. If the conditional non-Rabi oscillation is suppressed to make the states of target qubit staying in their original states during the conditional Rabi oscillation, the two qubit system can be a controlled-$U$ gate. Actually, if $\eta_{n\mbox{-}R}\ll 1$, i.e., $\Delta_B V_B \ll 8 J\,\varepsilon_B$, the amplitude of the non-Rabi oscillation becomes negligible $\sin^2\eta_{n\mbox{-}R}\approx 0$. Without the matching frequencies, a controlled-$U$ gate operation can then be obtained. Another possible way for a controlled-$U$ gate is also to be the matching frequencies. From Eq. (\[RabiFrequency\]), the amplitude of the non-Rabi oscillation is given by $$\sin^2\eta^{\rm CNOT}_{n\mbox{-}R}
% =\left(\frac{\Omega^{\rm CNOT}_R}{\Omega^{\rm
% CNOT}_{n\mbox{-}R}}\right)^2
= \frac{1}{4}\left(\frac{2l-1}{n+l-1}\right)^2.$$ As $n$ increases, i.e., the amplitude of time-dependent field decreases in Eq. (\[VB\]), the amplitude of the non-Rabi oscillation can be significantly suppressed, $\sin^2\eta_{n\mbox{-}R}\approx 0$. This results in a realization of a controlled-$U$ gate rather than a controlled-CNOT gate. In the case of $\omega =
\Omega^{(0)}_{B} \pm 2 J \varepsilon_B/\Omega^{(0)}_B$, then, the probability amplitude tables become a truth table of controlled-$U$ gate respectively given as
$$\begin{aligned}
{\cal U}^{(+)}_{CU}(t) &\simeq&
\left(\begin{array}{cccc}
\cos^2\frac{\Omega_{R}}{2}t
& \sin^2\frac{\Omega_{R}}{2}t & 0 & 0 \\
\sin^2\frac{\Omega_{R}}{2}t
& \cos^2\frac{\Omega_{R}}{2}t
& 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \right),
\\
{\cal U}^{(-)}_{CU}(t) &\simeq&
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \cos^2\frac{\Omega_{R}}{2}t
& \sin^2\frac{\Omega_{R}}{2}t \\
0 & 0 & \sin^2\frac{\Omega_{R}}{2}t
& \cos^2\frac{\Omega_{R}}{2}t
\end{array} \right).
\end{aligned}$$
As a consequence, if one of conditional quantum oscillations is suppressed by controllable system parameters, the two qubit system can provide a controlled-$U$ gate.
Multi-qubit system and conditional quantum oscillation
=======================================================
For multi-qubit systems, conditional quantum oscillation can be realizable if a similar adjustment is made in the system parameters. Once conditional quantum oscillations are achieved in multi-quibt systems, one may synchronize their characteristic frequencies to perform a controlled multi-qubit gate operation such as Toffoli and Fredkin gates which are an extension of two-qubit gates to multi-qubit gates.
Conclusion
==========
We investigated conditional quantum oscillations in interacting solid-state qubit systems. It was shown that a conditional quantum oscillation can be achievable in a way of tuning a range of system parameters. Synchronizing conditional quantum oscillations by varying applied time-dependent fields as well as system parameters enables to perform quantum gate operations such as controlled-NOT and -$U$ gate operations with a very accurate performance rate and adjustable operation time. Controlled multiple-qubit gate operations such as Toffoli and Fredkin gates can be implemented with conditional quantum oscillations and their synchronization.
We thank Mun Dae Kim for helpful discussions. This work was supported by the NSFC under Grant No.10874252 and Natural Science Foundation Project of CQ CSTC.
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[^1]: Fax: +86-23-65111531
|
---
abstract: 'Generic wave dislocations (phase singularities, optical vortices) in three dimensions have anisotropic local structure, which is analysed, with emphasis on the twist of surfaces of equal phase along the singular line, and the rotation of the local anisotropy ellipse (twirl). Various measures of twist and twirl are compared in specific examples, and a theorem is found relating the (quantised) topological twist and twirl for a closed dislocation loop with the anisotropy C line index threading the loop.'
address: 'H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK'
author:
- M R Dennis
title: 'Local phase structure of wave dislocation lines: twist and twirl'
---
Introduction {#sec:int}
============
Phase singularity lines in 3-dimensional scalar waves, that is, nodal lines or wave dislocations [@nb:34; @nye:natural], are often called (optical) vortices since they are vortices of current, or energy, flow. Being places where the phase is singular, all phases are found in the vortex neighbourhood, and the phase change is arbitrarily fast there. The surfaces on which phase is constant all meet on the singularity line, and are often twisted around the singularity as helicoids, as with the familiar screw dislocations [@nb:34]; it is this twisting that distinguishes screw dislocations from edge dislocations, whose phase structure does not rotate along the dislocation line.
In two dimensions, phase singularities are generically points where the intensity of the wave vanishes and the phase is singular. Knowledge of the phase contour lines near the point singularities can be sufficient to fill in the phase structure of the rest of the field. In 3-dimensional fields, topological singularities form a network of lines in space, often described as a ‘skeleton’ of the spatial pattern [@nye:natural; @hll:topology] - in complex scalar fields, the phase structure local to the phase singularity lines show how the global phase field is constructed upon this skeleton. The simplest 3-dimensional property of a phase singularity is its sense, or [*topological current*]{}: the direction endowed on the singularity line by the right-handed increase of phase. The topological current direction is preserved along the singularity line. The sense of the phase helicoids twisting around the singularity is independent of this; for instance (as demonstrated below), if a dislocation is embedded in a plane wave whose propagation direction is parallel to the topological current direction, the phase surfaces ending on the singularity form left-handed helicoids; if the propagation is antiparallel to the dislocation sense, they are right-handed. The twist is rather more complicated in more general situations, such as: when the singularities evolve off-axis in gaussian beams, with a noncanonical transverse shape [@ss:parameterization; @mwt:noncanonical]; when they are knotted, linked, or braided [@bd:332; @bd:333; @dennis:braided]; in isotropic random plane waves, when the singularities are tangled in a nontrivial way [@bd:321]; when they form the characteristic ‘antelope horns’ of the elliptic umbilic diffraction catastrophe [@bnw:79]. The zero contour surfaces of the real or imaginary parts of the field are often used to locate phase singularities and describe their geometry [@bd:321; @freund:trajectories]; these are, of course, special cases of the general phase contours.
My aim here is to describe this 3-dimensional phase twist structure in the general case, illustrated by simple examples of dislocated waves. As these examples show, it is very easy to find dislocations in waves that do not have a uniform phase screw structure, and this situation is, in fact, what one would expect generically. The main complication to the description comes from the fact that different phase helicoids twist at different rates, and this must somehow be averaged to give an overall sense of dislocation twist at a point on the singularity line. This is related to the fact that the local phase structure transverse to a dislocation line is generically anisotropic, squeezed into an ellipse [@ss:parameterization; @mwt:noncanonical; @bd:321; @dennis:thesis; @berry:330; @dennis:local], complicating the averaging around the dislocation. An example of two phase helicoids near a dislocation are shown in figure \[fig:surf\]; the first has a uniform helicoidal structure, the other does not.
![Two surfaces of constant phase (mod $\pi$) in the vicinity of a twisted wave dislocation. (a) Uniform twist. This is a surface of constant phase (mod $\pi$) of the simple screw dislocation (\[eq:edgescrew\]) with $\bi{k} = (0,0,k),$ and is a uniform helicoid. (b) Nonuniform twist. This constant phase surface is an example of the generic situation, calculated from (\[eq:twex\]) with $\beta = 2\alpha$ and $c = -1/2.$[]{data-label="fig:surf"}](twsurfs.eps){width="10cm"}
The notation of [@bd:321; @dennis:thesis] will be followed. In particular, $\psi = \psi(\bi{r})$ denotes a complex scalar wave, and in terms of real and imaginary parts, or amplitude and phase, $$\psi = \xi + \rmi \eta = \rho \exp(\rmi \chi).
\label{eq:psi}$$ Time dependence will not be considered here, and terms such as twist and twirl refer to variation with distance, not time. Local cartesian coordinates $\bi{r} = (x,y,z)$ or cylindrical coordinates $R,\phi,z$ will be used, with the dislocation along $x=y=r=0,$ and phase $\chi$ increasing with $\phi$ (that is, the topological current is parallel to the $+z$-direction). Important formulae will be given in coordinate-free form, with the dislocation tangent denoted by $\bi{T}$ (which is $(0,0,1)$ in local coordinates), and the directional derivative along the dislocation by $\bullet '$ (which is $\partial_z \bullet$ in local coordinates). Several of the results described here were also discussed in [@dennis:thesis], particularly in section 2.8, pages 54-58.
The epitome of a twisted wave dislocation is in the example [@nb:34] $$\psi_0 = (x + \rmi y) \exp(\rmi \bi{k}\cdot\bi{r}).
\label{eq:edgescrew}$$ This wave is a local solution of the Helmholtz equation $\nabla^2 \psi + k^2 \psi = 0.$ It is a [*screw dislocation*]{} when $\bi{k} = (0,0,k),$ and each of the surfaces of constant phase is a regular helicoid with pitch $2\pi/k,$ reminiscent of the atomic planes in a crystal lattice near a crystal screw dislocation [@read:dislocations]. It may be compared with an [*edge dislocation*]{}, where in (\[eq:edgescrew\]), $\bi{k} = (0,k,0).$ In this case, the phase structure does not change with $z,$ and the phase lines have a structure similar to the arrangement of the atomic planes near a crystal edge dislocation [@read:dislocations]. A general $\bi{k}$ in (\[eq:edgescrew\]) gives a [*mixed edge-screw dislocation.*]{} Far from the dislocation, the phase of (\[eq:edgescrew\]) has a plane wave character with wavevector $\bi{k}.$
This example, and its crystal analogy, motivated Nye [@nye:motion] to define the [*Burgers vector*]{} of a wave dislocation $$\bi{b} = \left(\lim_{x\to 0} \left(\chi_x\right)_{y=z=0}, \lim_{y\to 0}
\left(\chi_y\right)_{x=z=0}, \lim_{z\to 0} \left(\chi_z\right)_{x=y=0}\right),
\label{eq:burgers}$$ where subscripts after brackets denote quantities held constant, other subscripts denote partial derivatives, and coordinates are local. The dislocation (\[eq:edgescrew\]) has Burgers vector $\bi{b} = \bi{k},$ which is the wavevector of the asymptotic plane wave. This gives the expected results of the vector being parallel and perpendicular to the screw and edge dislocation lines respectively. However, there is a danger of taking the analogy between wave dislocations and crystal dislocations too far, and there are problems with this definition of a Burgers vector for a wave dislocation in other rather general situations.
In the derivation of (\[eq:burgers\]) in [@nye:motion], it is assumed that the dislocation is curved and moving (i.e. the wave is narrow-band); for monochromatic waves, the dislocation is stationary, and many dislocation lines occurring in optics are straight lines (often parallel to the beam direction). If the only available information about the field is local to the dislocation, for instance, a finite Taylor expansion of (\[eq:edgescrew\]) about $x = y = 0,$ $b_z$ is not defined. In more general cases of wave interference, such as isotropic random plane wave superpositions [@bd:321], there is no overall propagation direction, and only local properties of the dislocation, determined using derivatives of $\psi$ on the dislocation, are relevant.
It is desirable to have a measure of screwness, determining the twist of the local phase structure along the dislocation, in terms only of local derivatives. The following is an exploration of this twist geometry, and its realisation in simple beam superpositions.
The geometry of twist and twirl {#sec:geometry}
===============================
The [*twist*]{} $Tw$ is defined to be the rate of rotation of phase along a dislocation, according to several different measures to be described. The simplest geometric twist is the rotation of a surface of constant phase along the dislocation, which is also the rate of rotation of the normal to the phase surface. For convenience, a surface of constant phase in the vicinity of the phase singularity will be called a [*phase ribbon*]{} [@ws:filaments1; @ws:filaments2]; for most of the discussion, only the ribbon geometry is important. Each plot (a) and (b) of figure \[fig:surf\] shows two phase ribbons, with phases differing by $\pi.$
Fixing a particular phase $\chi_0,$ the normal to the $\chi_0$-ribbon is $$\begin{aligned}
\bi{U}_{\chi_0} & = \rm{Re} \{\nabla \psi \exp(-\rmi \chi_0)\} \nonumber \\
& = \nabla \xi \cos \chi_0 + \nabla \eta \sin\chi_0.
\label{eq:udef}\end{aligned}$$ Using $\phi$ as a local azimuthal coordinate, the rate of twist $Tw(\chi_0)$ of the $\chi_0$-ribbon is $$\begin{aligned}
Tw(\chi_0) & = \left(\phi_z\right)_{\chi=\chi_0} \nonumber \\
& = \partial_z \arctan \frac{U_{\chi_0,y}}{U_{\chi_0,x}} \nonumber \\
& = \frac{\bi{T}\cdot\bi{U}_{\chi_0} \times \bi{U}_{\chi_0}'}{U_{\chi_0}^2}.
\label{eq:twch}\end{aligned}$$ Applying this formula to the wave (\[eq:edgescrew\]) gives $-k_z.$ Apart from the sign, this is the $z$-component of the Burgers vector. The $-$ sign originates in the fact that the helicoid is left-handed, a general property of dislocations whose topological current is parallel to the propagation direction [@dennis:thesis; @dennis:braided].
The phase structure transverse to the dislocation in (\[eq:edgescrew\]) is isotropic, and all the phase ribbons twist at the same rate as each other, and along the dislocation (one such is shown in figure \[fig:surf\] (a)). General dislocations are anisotropic, however, and the local contours of intensity $\rho^2$ are elliptical, with a corresponding squeezing of phase gradient $\nabla\chi.$ It can be shown [@bd:321; @dennis:thesis; @dennis:local] that $(\chi_{\phi})_{z=0} = R^2 \omega /\rho^2,$ where $\omega$ is the vorticity on the dislocation, defined as $\omega \equiv |\nabla \xi \times \nabla \eta | = |\nabla \psi^{\ast} \times \nabla \psi|/2$ (the direction of this vector gives the topological current). The plots in figure \[fig:twirltab\] indicate this aspect of elliptic phase squeezing. In fact, both the ellipse and the phase squeezing are accounted for by the gradient vector $\nabla \psi$ on the dislocation. As with all complex vectors, it is associated with an ellipse, traced out by $\bi{U}_{\chi_0}$ as $\chi_0$ changes; this ellipse is the same shape as that described by $\rho^2$ and $\nabla \chi,$ but with axes exchanged.
It is more natural to quantify the twist of the entire singularity core, rather than simply for a fixed phase. Therefore, it is necessary to average $Tw(\chi_0)$ over all phases, although it is not clear whether averaging with respect to the phase $\chi$ or azimuth $\phi$ is appropriate. It is possible to calculate both averages: the phase average (integrating $Tw(\chi_0)$ with respect to $\chi_0$) has a complicated form involving the axes of the phase ellipse, and is in [@dennis:thesis] equation (2.8.6); the azimuth average (integrating $Tw(\chi_0)$ with respect to $\phi$) has a simpler form, and is ([@dennis:thesis], equation (2.8.7)) $$Tw_{\phi} = \frac{1}{2\pi} \int_0^{2\pi} \rmd \phi \,
Tw(\chi) = \frac{\rm{Re}\{\bi{T}\cdot \nabla \psi^{\ast} \times \nabla
\psi'\}}{2\omega}.
\label{eq:twph}$$ This depends only on derivatives up to the second of $\psi$ on the dislocation line. As expected, it gives $-k_z$ for (\[eq:edgescrew\]).
Berry [@berry:330] chose to average phase instead by examining the rate of change of phase at a fixed azimuth $(\chi_z)_{\phi = \phi_0}.$ The result of a particular averaging, he defined the [*screwness*]{} $\sigma$ to be $$\sigma = \frac{-\rm{Im} \{\nabla \psi^{\ast} \cdot \nabla
\psi'\}}{|\nabla \psi|^2}.
\label{eq:screwness}$$ ($\sigma$ defined here is the negative of that defined in [@berry:330].) The screwness for (\[eq:edgescrew\]) is $-k_z.$
The difficulty in defining the total twist arises because the different phase ribbons twist at different rates, due to the phase anisotropy ellipse associated with $\nabla \psi.$ Along the dislocation line, the anisotropy ellipse itself may rotate, as well as change its size and eccentricity. Because the rotation of the ellipse is independent of the phase twist, it will be referred to as the [*twirl*]{} $tw$ of the dislocation line, and may be found as follows.
The complex vector field $\nabla \psi$ shares geometric features associated with vector polarization fields [@dennis:thesis; @bd:324] [^1]. In particular, in local coordinates, $\nabla \psi$ is confined to the $xy$-plane, and therefore parameters, describing all the geometric properties of the ellipse, may be defined: $$\begin{aligned}
S_0 = |\nabla \psi|^2, \qquad & S_1 = |\psi_x|^2 -|\psi_y|^2, \nonumber \\
S_2 = \psi_x^{\ast} \psi_y + \psi_x \psi_y^{\ast}, \qquad &
S_3 = -\rmi (\psi_x^{\ast} \psi_y + \psi_x \psi_y^{\ast}) =
2\omega.
\label{eq:stokes}\end{aligned}$$ These parameters describing the anisotropy are analogous to the Stokes parameters in polarization; the anisotropy ellipse is related to $\nabla \psi$ and these parameters as the polarization ellipse is related to the electric vector and the Stokes parameters. The parameters (\[eq:stokes\]) do not, themselves, have anything to do with polarization. The anisotropy ellipse, depending only on $\nabla\psi,$ may be defined at any point of the scalar field, not only on a dislocation; the ellipse is circular (or linear) generically along lines in space [@bd:324; @nh:wavestructure; @dennis:thesis].
The azimuthal angle of orientation of the major ellipse axis is $\arg(S_1 + \rmi S_2)/2.$ The twirl $tw_{\phi}$ may therefore be defined as the rate of change of this angle along the dislocation line: $$tw_{\phi} = \frac{1}{2} \frac{S_1 S_2' - S_2 S_1'}{S_1^2 +
S_2^2}.
\label{eq:twphi}$$ The denominator is equal to $S_0^2 - S_3^2 = |\nabla \psi|^4 - 4 \omega^2,$ which is zero when the ellipse is circular and twirl is not defined.
The natural measure of phase around the ellipse associated with a complex vector is the [*rectifying phase*]{} $\chi_{\rm{r}}$ [@dennis:thesis; @dennis:polarization] (for polarization ellipses, it also called phase of the vibration [@nye:natural]). $\chi_{\rm{r}}$ is defined such that the complex vector $\exp(-\rmi \chi_{\mathrm{r}}) \nabla \psi$ has orthogonal real and imaginary parts (the real part along the ellipse major axis, the imaginary along the minor), and can be shown [@dennis:polarization] to be equal to $\arg (\nabla \psi \cdot \nabla \psi)/2.$ The [*phase twirl*]{} $tw_{\chi}$ may be defined as the rate of change of this phase along the dislocation: $$tw_{\chi} = \frac{1}{2} \left(\arctan \frac{2 \nabla \xi \cdot
\nabla \eta}{|\nabla \xi|^2 - |\nabla \eta|^2}\right)'.
\label{eq:twchi}$$ This gives a natural measure of the rate of change of phase with respect to the ellipse axes. Its form is not particularly simple when the derivative in (\[eq:twchi\]) is taken, although it is easily seen that the denominator is $|\nabla \psi|^4 - 4 \omega^2.$ This implies that $tw_{\chi}$ is not defined when the ellipse is circular (isotropic).
The two twirls here defined in (\[eq:twphi\]), (\[eq:twchi\]) may be combined to give a new measure of the phase twist. Since $tw_{\chi}$ measures the rate of change of phase with respect to the ellipse axes, its negative gives a sense for the helicoid phase twist with respect to the ellipse. Therefore, the difference $tw_{\phi} - tw_{\chi}$ gives an [*ellipse-defined twist*]{} $Tw_{\rm{ell}},$ which can be shown to be $$Tw_{\rm{ell}} = tw_{\phi} - tw_{\chi} = \frac{\rm{Re}\{ \bi{T} \cdot(\nabla \psi^{\ast} \wedge \nabla
\psi') + \rmi \nabla \psi^{\ast}\cdot \nabla \psi'\}}{|\nabla \psi|^2
+2\omega}.
\label{eq:twell}$$ Although neither type of twirl is defined when the ellipse is circular, $Tw_{\rm{ell}}$ is, and for the example (\[eq:edgescrew\]), it is $-k_z,$ as desired. It is also interesting to note that $Tw_{\rm{ell}}$ is the sum of the numerators of $Tw_{\phi}$ and $\sigma,$ divided by the sum of the denominators.
![The sequence of transverse phase lines (separated by an equal phase difference $\pi/6$) and anisotropy ellipse of a twisting, twirling dislocation of the form (\[eq:twex\]), with $\beta = 2\alpha, c = -1/2.$ the $z$-spacing between each frame is $\pi/2\alpha.$ The twist and twirl have opposite senses. The surface in figure (b) is swept out by a pair of opposite phase lines in this figure; the parameters in the two figures are the same.[]{data-label="fig:twirltab"}](twirltab.eps){width="8cm"}
A simple example of a twisting, twirling dislocation is found in the sum of two screw dislocated waves, with dislocations in opposite directions, and with different pitches $2\pi/\alpha, 2\pi/\beta:$ $$\psi_1 = (x + \rmi y) \exp(\rmi \alpha z) + c (x - \rmi y)
\exp(\rmi \beta z)
\label{eq:twex}$$ with $c$ in general complex, with $1 > |c|^2$ ensuring that the dislocation along $x = y = 0$ has topological current in the $+z$-direction.
(\[eq:twex\]) could represent, for example, the sum of two copropagating but counterrotating order one Laguerre-Gauss or Bessel beams with different $k_z$ components, in the vicinity of the $z$-axis. A possible realisation, in terms of Bessel beams, is $\exp(\rmi \phi+ \rmi \alpha z) J_1(\sqrt{1-\alpha^2} R) + c \exp(-\rmi \phi+ \rmi \beta z) J_1(\sqrt{1-\beta^2} R) \sqrt{(1-\alpha^2)/(1-\beta^2)}$. The rotation of a tranverse interference pattern along a beam, achieved though superposing beams different axial phase dependencies, was also studied by Courtial [@courtial:selfimaging], where the Gouy phases of two superposed singular beams were different; here, the $k_z$ components of the two dislocated fields are different.
The phase change around the dislocation in (\[eq:twex\]) is anisotropic for $c \neq 0;$ the Stokes parameters $S_0, S_3$ are constant, indicating the anisotropy ellipse has a constant area $\pi(1-|c|^2)$ and eccentricity $2|c|^{1/2}/(1+|c|).$ (\[eq:twex\]) is therefore a local normal form for dislocations whose twist and twirl are much greater than their curvature or rate of change of anisotropy.
The two types of twirl are computed to be $$tw_{\phi} = -(\alpha - \beta)/2, \quad tw_{\chi} = (\alpha
+\beta)/2 \; \rm{for} \, \psi_1.
\label{ex1twirl}$$ The various rates of twist are $$\left. \begin{array}{ll} Tw_{\rm{ell}} & = -\alpha \\
Tw_{\phi} & = -(\alpha - |c|^2 \beta)/(1-|c|^2) \\
\sigma & = -(\alpha + |c|^2 \beta)/(1+|c|^2) \end{array}
\right\} \, \rm{for} \, \, \psi_1.
\label{eq:ex1twist}$$
For this example of a dislocation with a uniform twirl, and constant anisotropy, the different measures of twist are, in general, different. They are equal if $c\to 0,$ (yielding the screw dislocation in (\[eq:edgescrew\])), or if the twirl is zero, i.e. $\alpha = \beta$ (this was the case discussed in [@dennis:thesis]). Of the different measures of twist, it appears that $Tw_{\rm{ell}}$ agrees numerically the most with intuition: if $\beta$ is a positive integer multiple $m \alpha$ of $\alpha,$ the ellipse will undergo $m-1$ rotations as the phase undergoes one, and the entire pattern is periodic (as in figure \[fig:twirltab\]), with period $2\pi/\alpha.$ $Tw_{\rm{ell}}$ is the only measure of twist to reflect this. Equivalently, it can be argued that since the first term on the right hand side of (\[eq:twex\]) defines the dislocation direction, it also defines the twist, and the effect of the second term is merely to modulate the pattern to produce the twirl.
Closed dislocation topology {#sec:closed}
===========================
When phase singularity lines form closed loops, certain topological identities must be satisfied. In relation to twist, continuity of the wavefunction requires that the total number of twists of each phase ribbon must be a (positive or negative) integer, which is the same for each ribbon. This [*screw number*]{} is therefore a property of the dislocation loop, and is positive if the topological twisting is right handed with respect to the dislocation direction, negative if left handed. For obvious geometric reasons, a dislocation loop with nonzero screw number will be called a [*closed screw dislocation*]{}.
The importance of the screw number is that it gives the dislocation strength threading the loop, by the [*twisted loop theorem*]{}: the screw number $m$ of a strength 1 dislocation loop is equal to minus the dislocation strength threading the loop (in a right handed sense). This result is discussed and proved in [@dennis:thesis; @ws:filaments2; @ws:filaments3; @ws:filaments4; @bd:332]. If the dislocation loop is planar, then the integral of the ribbon twist around the loop divided by $2\pi$ gives the screw number. If the loop is nonplanar, then the Cǎlugǎreanu-White-Fuller theorem [@ws:filaments3] implies that the writhe of the curve must be added to the twist integral. Only planar curves will be considered here.
A simple wave containing a closed screw dislocation can be made from a combination of polynomial waves in cylindrical coordinates [@bd:333]: $$\psi_{\rm{closed}} = R^{|m|} \exp(\rmi m \phi)\exp(\rmi k z) (R^2 -
R_0^2 + 2 \rmi (|m|+1) z/k).
\label{eq:clscrew}$$ This wave has a closed screw dislocation in the $z=0$ plane at $R=R_0,$ with screw number $m,$ its topological current directed opposite to the increase of $\phi.$ It is threaded by a strength $m$ dislocation up the $z$-axis (its sense in $+z$). In [@bd:332; @bd:333], high-strength loops with similar geometry were found. If $m=0,$ the loop is the familiar closed edge dislocation loop. $Tw_{\phi}, Tw_{\rm{ell}}$ and $\sigma$ are $-m/R_0$ on the closed loop; the screw number is $m.$ The twirl $tw_{\phi}$ is zero for this loop, and the anisotropy ellipse axes are oriented in the $R,z$ directions.
The nature of the twisted phase ribbons near the dislocation have consequences for the global topology of the total phase surface, that is, the wavefront. Figure \[fig:noncompact\] shows a surface of constant phase (modulo $\pi$) of the wave $\psi_{\rm{closed}}$ with $m = 1.$ This surface is a ‘noncompact torus’ (just as the plane is a ‘noncompact sphere’), extended to infinity because of the infinite straight dislocation on the axis. Unlike a compact torus, there is no way of distinguishing the two sides of the surface. The discussions in [@dennis:thesis; @ws:filaments2] show that such complex wavefronts are inevitable with closed screw dislocations.
![A phase surface (mod $\pi$) of the wave $\psi_{\rm{closed}}$ with $m = 1.$ A straight vertical dislocation line threads a closed screw dislocation, and the local phase ribbons join up in the form of a noncompact torus.[]{data-label="fig:noncompact"}](nct07.eps){width="6cm"}
The closed dislocation of (\[eq:clscrew\]) has zero twirl. Waves with closed twirling dislocations loops can be made by superposing waves of the form $\psi_{\rm{closed}}$ with dislocations of opposite sense and different topological twists: $$\begin{aligned}
\psi_2 & = R^{|m_1|} \exp(\rmi m_1 \phi)\exp(\rmi k z) (R^2 -
R_0^2 + 2 \rmi (|m_1|+1) z/k) \nonumber \\
& \quad + c R^{|m_2|} \exp(\rmi m_2 \phi)\exp(-\rmi k z) (R^2 -
R_0^2 - 2 \rmi (|m_2|+1) z/k).
\label{eq:cltwirl}\end{aligned}$$ When $m_1, m_2$ are different, we may expect the twirl to be nonzero, as with (\[eq:twex\]). Not every combination of $m_1, m_2$ yields the expected twirling wave however, because the complicated threading interference structure is affected in a nonlinear way. For simplicity, $k$ and $R_0$ are taken as 1.
Choosing $m_1 = 1, m_2 = 0, c = 2/3$ gives a wave with a twirling dislocation, and the measures of twist are $$\fl \left. \begin{array}{ll} Tw_{\rm{ell}} & = -1 \\
Tw_{\phi} & = -3(6 + \cos\phi)/(2(7 + 3 \cos\phi)) \\
\sigma & = -3(15 - 2 \cos \phi)/(53 - 12\cos\phi) \end{array}
\right\} \mathrm{for} \; \psi_2 \; \mathrm{with} \; m_1 = 1, m_2 = 0, c = 2/3.
\label{ex2twist}$$ The screw number of this wave is $-1,$ since the phase structure of the summand with larger coefficient dominates. Only $Tw_{\rm{ell}}$ integrates to this integer; the others give irrational numbers, and this is the case for other choices of $m_1, m_2, c.$ The integrability of $Tw_{\rm{ell}}$ may be explained by the fact that it is the derivative of the difference of angles (\[eq:twell\]), the others are averages of an angle, and taking the average does not commute with integrating around the dislocation loop. Thus, of all the various twists considered, only $Tw_{\rm{ell}}$ can be used as a topological twist.
The twirl is $$\fl tw_{\phi} = -(8 - 6\cos\phi)/(25 - 24 \cos\phi)
\qquad \mathrm{for} \; \psi_2\; \mathrm{with} \; m_1 = 1, m_2 = 0, c = 2/3,
\label{ex2twirl}$$ and, by (\[ex2twist\]) and (\[eq:twell\]), $tw_{\chi} = 1+tw_{\phi}.$ The topological twirl around a closed loop is also quantised, equal to the number of rotations of the anisotropy ellipse around the loop, although it only needs to undergo a half turn to return to itself smoothly [@nye:natural; @dennis:polarization]. Integrating $tw_{\phi}$ around the loop gives a topological twirl of $-1/2:$ with respect to the anisotropy ellipse, the loop is a M[ö]{}bius band. The sense of topological twirl and topological twist are independent.
A natural question to ask is whether the topology of twirl gives an analogue of the twisted loop theorem. In fact, it does: the ellipse rotation around a loop is related to the [*anisotropy C line index*]{} enclosed by the loop. Anisotropy C lines, where the anisotropy ellipse is circular, occur when $\nabla \psi \cdot \nabla \psi$ is zero; they correspond to phase singularities of this complex scalar, whose phase is twice the rectifying phase $ \chi_{\rm{r}}.$ The anisotropy C line index therefore is half this phase singularity strength, and is equal to the integral of $tw_{\chi}$ around the loop, divided by $2\pi.$ This fact, together with the twisted loop theorem and the definition (\[eq:twell\]) of $Tw_{\rm{ell}},$ leads to the [*twirling loop theorem*]{}, which may be stated as follows.
The anisotropy C line index threading a closed dislocation loop (in units of $1/2$) is equal to the topological twirl minus the topological twist of the loop, i.e. the number of rotations of the anisotropy ellipse around the dislocation, minus the number of rotations of the phase structure, in a right handed sense with respect to the dislocation strength.
This gives a topological role to anisotropy C lines. Anisotropy L lines, where $\omega = 0$ (that is, where all the phase surfaces share a common normal), govern the reconnection of dislocation lines [@bd:333; @nye:airy; @nye:local].
Twist and twirl in isotropic random waves {#sec:random}
=========================================
As a final example, twist and twirl are considered in 3-dimensional isotropic random waves, that is, superpositions of plane waves with isotropic random directions and phases, whose ellipse anisotropy statistics have been calculated [@bd:321; @dennis:thesis]. These random waves might occur in monochromatic waves in a large, chaotic cavity, or (a scalar caricature of) black-body radiation; the wave dislocations form a complicated tangle. Nevertheless, this random wave model has many statistical symmetries, the averages being spatially and temporally invariant.
The details of the calculations are omitted, but the calculations are reasonably straightforward, following the methods of [@bd:321; @dennis:thesis], in which statistics of many geometrical properties of dislocations were calculated. The calculations proceed by taking advantage of the fact that $\psi$ and its derivatives have gaussian distributions, the details depending on the power spectrum of the waves considered. For all of the probability density functions, the twists and twirls are in units of the [*characteristic twist*]{} $Tw_{\rm{c}} = (k_4/5k_2)^{1/2},$ where $k_n$ is the $n$th moment of the power spectrum; for monochromatic waves of wavelength $\lambda,$ this is $2\pi/\sqrt{5} \lambda.$
![The probability density functions of the different twist and twirl measures. In ascending order of $y$-axis interception, they are: twirl $tw$ (thick black line), helicoid twist $Tw(\chi_0)$ (thin black line), azimuth-averaged twist $Tw_{\phi}$ (dotted line), screwness $\sigma$ (dashed line), and ellipse-averaged twist $Tw_{\rm{ell}}$ (thick grey line). The distributions are all in units of characteristic twist $Tw_{\rm{c}}.$[]{data-label="fig:dist"}](twistdist.eps){width="8cm"}
All measures of twist and twirl discussed here ($Tw(\chi_0)$ in (\[eq:twch\]), $Tw_{\phi}$ in (\[eq:twph\]), $\sigma$ in (\[eq:screwness\]), $tw_{\phi}$ in (\[eq:twphi\]), $tw_{\chi}$ in (\[eq:twchi\]), and $Tw_{\mathrm{ell}}$ in (\[eq:twell\])) involve some combination of first derivatives of the dislocation with their dislocation directional derivative, divided by another function involving first derivatives only. In the calculation, the gaussian-distributed second derivatives are integrated first, giving a function involving $\omega$ and $G \equiv |\nabla \psi|^2,$ which is then integrated using the probability distributions described in [@bd:321; @dennis:thesis].
The probability distribution functions of the various twists and twirls are found to be $$\begin{aligned}
P_{Tw(\chi_0)}(t) & = & \frac{3}{4} \frac{1}{(1 + t^2)^{5/2}}, \nonumber \\
P_{\sigma}(t) & = & \frac{35}{32(1+t^2)^{9/2}},
\nonumber \\
P_{Tw_{\phi}}(t) & = & \frac{3}{32 t^4}\left(2-
\frac{(2+7t^2)}{(1+t^2)^{7/2}}\right),
\nonumber \\
P_{Tw_{\rm{ell}}}(t) & = & \frac{1}{8} \left(\frac{\sqrt{2}(11+ 8 t^2 +
32 t^4)}{(1+2t^2)^{7/2}} -
\frac{4}{(1+t^2)^{3/2}}\right),
\nonumber \\
P_{tw}(t) & = & \frac{1}{32t^4 (1+t^2)^{5/2}} \left( (2+t^2)(3 + 3t^2 +
8t^4) E \left(\frac{t^2}{1+t^2}\right)\right.
\nonumber \\
& & \qquad \qquad \qquad \left.-2(3+3t^2 + 2t^4)
K\left(\frac{t^2}{1+t^2}\right) \right),
\label{eq:twpdfs}\end{aligned}$$ where $E,K$ represent the complete elliptic integrals of first and second kinds [@as:handbook]. These are plotted in \[fig:dist\]; they are all symmetric, since right-handed and left-handed screw dislocations are equally weighted in the ensemble. All of the twist distributions have power law tails. The fluctuations of the $P_{tw}$ are the largest (the second moment does not converge). The fluctuations of $Tw_{\rm{ell}}$ are the smallest (second moment is $\log \sqrt{2} - 1/4$), providing further support for the preference of $Tw_{\rm{ell}}.$
Discussion
==========
Twist and twirl for anisotropic 3-dimensional wave dislocations are not important for the simplest optical vortices because they are isotropic and therefore not subject to the subtleties described here. However, it is important to note that the screw-edge distinction only applies in special cases such as (\[eq:edgescrew\]), and in more chaotic fields, only local phase geometry may be appealed to. Various measures of twist and twirl were introduced in \[sec:geometry\], their topology was examined in section \[sec:closed\], and statistics in \[sec:random\]. The conclusion from each of these deliberations was that the ellipse-defined twist $Tw_{\rm{ell}}$ (\[eq:twell\]) has the most desirable properties of the various measures.
Experimental verification of the twirling loop dislocation (\[eq:twex\]) should be straightforward; with measurements of the phase, the twist and twirl structures of more complicated, curved dislocations may be found, although the full experimental analysis of the twist and twirl of an arbitrary dislocation line is likely to be rather difficult.
It is natural to ask whether there is any curvature structure (i.e. second spatial derivative of $\psi$) transverse to a dislocation, providing a transverse analogue to twist. This would provide the geometric counterpart to the transverse component of the Burgers vector, as twist is related to the longitudinal part. For the standard edge dislocation (\[eq:edgescrew\]), the Burgers vector is normal to the phase surface (mod $\pi$) whose local transverse curvature is zero (a phase saddle in the transverse plane also occurs on this surface); it is tempting to guess this may be the desired structure. After some analysis, however, it can be shown that there is no unique phase contour whose local transverse curvature vanishes: generically, there are either one or three phase contours with this property. The number of phase contours in question is given by the number of real roots of a certain cubic in $\tan \phi,$ reminiscent of the number of straight lined terminating on a line field singularity (lemon versus star and monstar) [@bh:60; @dennis:polarization]. An example of a solution to the wave equation with three phase lines of vanishing transverse curvature is $$\psi = \left( x + \rmi y + \rmi k (x^2-y^2/2)/2\right) \exp(\rmi k z).
\label{eq:cubic}$$ Phase patterns for dislocations in two dimensions with this property are plotted in [@nhh:tides], figure 6 (f). It is not certain, therefore, whether local dislocation geometry can give a transverse direction to a Burgers vector.
The philosophy of studying phase singularities is that they give information about the global phase structure: they are a 3-dimensional ‘skeleton’ for the entire field. Twist is an important, intrinsic property of phase singularities, and shows how this information may be gleaned from the local dislocation morphology; closed screw dislocation loops indicate nontrivial wavefront topology, as in figure \[fig:noncompact\]. Phase anisotropy gives rise to further structures, such as twirl, and anisotropy C and L lines. A possibility of a further extension is analysing the gradient of the anisotropy scalar $\nabla \psi \cdot \nabla \psi$ itself: its phase singularities (the anisotropy C lines) themselves can be twisted, twirling, and therefore related to even higher singular morphologies. The twirling loop theorem shows how the first members of this hierarchy of anisotropy structures are coupled.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Michael Berry, John Hannay and John Nye for many interesting and stimulating discussions. This work was supported by the Leverhulme Trust.
References {#references .unnumbered}
==========
[10]{}
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Dennis M R 2001 Local properties and statistics of phase singularities in generic wavefields, in M S Soskin and M V Vasnetsov, eds, [*Singular Optics (Optical Vortices): Fundamentals and Applications*]{}, SPIE [**4403**]{} 13–23
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Nye J F 2004 Local solutions for the interaction of wave dislocations in press
Abramowitz M and Stegun I A, eds 1965 Dover, New York
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[^1]: The two fields do not have identical structures, since free field vector solutions of Maxwell’s equations are divergence free, whereas $\nabla \psi,$ being a gradient field, is curl free.
|
---
abstract: 'Charge transfer along the base-pair stack in DNA is modeled in terms of thermally-assisted tunneling between adjacent base pairs. Central to our approach is the notion that tunneling between fluctuating pairs is rate limited by the requirement of their optimal alignment. We focus on this aspect of the process by modeling two adjacent base pairs in terms of a classical damped oscillator subject to thermal fluctuations as described by a Fokker-Planck equation. We find that the process is characterized by two time scales, a result that is in accord with experimental findings.'
author:
- 'Maria R. D’Orsogna'
- Joseph Rudnick
title: 'Two-level system with a thermally fluctuating transfer matrix element: Application to the problem of DNA charge transfer'
---
\[sec:introduction\]Introduction
================================
In spite of the fact that a decade has passed since the first definitive observation of charge transfer along the DNA base-pair stack [@murphy], the detailed properties of this process have not been definitively elucidated. This is partly due to the inherent complexity of the molecular structure of DNA, and to the large number of external and intrinsic factors that exert an influence on DNA structure and behavior. The current unsettled situation also reflects the absence of an overall agreement on the precise mechanism by which this charge transport takes place. One of the key issues that awaits full illumination is the role of disorder—both static and dynamic—on the propagation of charge along the base-pair stack. A related, and quite fundamental, question is whether charge transport is a coherent quantum mechanical process, like conduction of electronic charge against a static, or deformable, background, or whether it it takes place as fundamentally incoherent transport, as a variation of the random walk. The answers to these and other questions will have a significant impact on both our understanding of the biological impact of charge transport in DNA and the development of applications based on this phenomenon.
Despite the often contradictory results of experimental investigations [@fink; @porath; @storm; @pablo; @giese], a few conclusions seem inescapable. The first is that long-range charge transport along the base-pair stack depends quite strongly on the sequence of the base pairs [@nunez]. In addition, base-pair mismatches can have a significant deleterious effect on charge transport [@kelley; @jackson] (see, however [@schuster]). Furthermore, strands of DNA display considerable disorder, both static [@calladinebook] and dynamic [@brauns; @swaminathan; @cheat; @troisi]. Finally, several sets of experiments on ensembles of short DNA strands have uncovered an unusual two-step charge transfer process [@barton99; @zewail]. These studies focus on fluorescent charge donors intercalated in DNA oligostrands. As the charge migrates towards the acceptor, the fluorescence is quenched and the rate of migration is determined by the decaying fluorescence profile. The data reveals that this decay process occurs according to two characteristic time-scales which are separated by more than an order of magnitude [@barton99; @zewail]. Any model that purports to explain charge transport must take all this into account.
In this paper, we discuss a model for short range charge transport along a base pair stack that undergoes substantial structural fluctuations. The process occurs via thermally-assisted quantum mechanical tunneling of charge carriers from one base pair to the next, under the assumption that this tunneling is properly characterized as occurring in the presence of a dissipative environment. A key conjecture is that charge transfer takes place only when the neighboring pairs are in a state of optimal “alignment”, and that this alignment is statistically unlikely in thermodynamic equilibrium. As we will see, this conjecture leads in a natural way to a model exhibiting the dual-time-scale feature described above. Additionally, the model generates predictions that can be readily tested. We shall relate the problem at hand to the dynamics of a simple two level system (TLS), realized by a donor and an acceptor state.
In Section \[sec:tunneling\], we briefly recapitulate what is known about the tunneling process in the presence of friction for a TLS system. We also quantify our notion of a coordinate $\theta$ associated with the “alignment” of adjacent base pairs and of the influence of the dynamics of this new coordinate on charge transfer. Section [\[sec:model\]]{} specifies the model for describing a generic collection of two-level systems (TLS), initially in the donor state and characterized by a fluctuating alignment variable $\theta$. The probability distribution of donor states, $W(\theta, \dot \theta, t)$ obeys a Kramers equation with a sink term due to charge transfer to the acceptor. The rate of charge transfer will be expressed by the fluctuating rate $\Gamma(\theta)$. This Kramers expression is recast into the form of a Volterra equation with the use of a Lie-Algebra approach defined on the Hilbert space of the eigenstates of the Kramers equation for $\Gamma(\theta)=0$. We will discuss limiting cases of the solution to obtain physical insight and to reveal the two-time-scale decay of the probability distribution due to the sink term. We conclude in Section \[sec:conclusions\] with a discussion of the possible application of our results to charge transfer in strands of DNA consisting of several base pairs.
The key result of our calculations lies in the determination of $P(\theta^*,t)$, the probability distribution of donor states evaluated at the optimal configuration $\theta^*$ and with the $\dot \theta$ variable integrated out. Indeed, under the assumption that the tunneling process is most effective at $\theta \sim \theta^*$, this quantity is directly related to the fluorescence intensity $I(t)$ of the base pair complexes, as probed by J. Barton and A. Zewail [@barton99; @zewail], through the following:
$$\label{fluorescence}
I(t) = I_0 \left[1-\Gamma \int_{0}^{t} P(\theta^*,t') dt' \right].$$
The quantity $I_0$ of the above relationship is a proportionality constant and $\Gamma$ is the integrated rate of transfer to the acceptor. We shall determine the double exponential character of $P(\theta^*,t)$, and hence of $I(t)$, in qualitative agreement with the experimental findings. The conjectures made on the existence of an optimal and unlikely configuration $\theta^*$ will be crucial in obtaining the two stage decay process, a result that justifies the assumptions made.
The model we shall construct is obviously not restricted in application to DNA oligostrands. Using our results, we may conclude that in an ensemble of generic systems the migration of a particle from donor to acceptor proceeds statistically as a two-time scale process, provided the transfering process is of rare occurrence.
\[sec:tunneling\]The tunneling process
======================================
The process of charge transfer from a donor site to an acceptor site —a two-level system—is ubiquitous in biochemical and physical phenomena [@datta]. It occurs under a broad variety of spatio-temporal conditions. Chemical bond formation or destruction, ATP production in photosynthetic reactions, or the operation of semiconducting devices, all involve the transfering of charges to and from specific sites, via thermal activation or quantum-mechanical tunneling through an energy barrier. Because of its intrinsic nature, charge transfer via quantum-mechanical tunneling takes place on a length scale of up to tens of angstroms [@muller]; larger distances are possible if other transport mechanisms are involved. These include thermal hopping among sites, which are typical in disordered systems, the creation of conduction bands in metals, or of lattice distortions of polaronic type in specific systems.
Quantum-mechanical tunneling from a donor site to an acceptor site is quite simply represented by a two-level system (TLS) [@leggett]. In this description, the tunneling particle is limited to being in the donor or in the acceptor state, while the other degrees of freedom of the system, nuclear for instance, describe the charge potential energy.
The energetic profile of the system is thus characterized by a multidimensional surface of which the acceptor and the donor states constitute relative minima, separated by a barrier. Of the many existing degrees of freedom, it is often possible to identify a “reaction coordinate” $y$ such that the energy barrier between donor and acceptor is minimized along this specific direction. The progress of the reaction is then dominated by the evolution along this coordinate and the potential energy surface can be reduced to an effective one-dimensional curve.
In certain systems the physical interpretation of the reaction coordinate is immediate: it may be the relative bond length in two diatomic molecules, or solvent polarization around the donors and acceptors [@barbara]. It is not an easy task to give a physical interpretation of the reaction coordinate in the case of DNA base pairs because of the many possibilities involved - intra-base distance, mobile counter-ion concentration, solvent concentration, or a combination of all the above. A possibility is offered by Ref. [@basko] where it is suggested that the most relevant quantity is the interaction of the charge with the polar water molecules of the solvent. In this paper we shall refer to the reaction coordinate $y$ in most general terms.
A common representation of tunneling with dissipation is through the spin-boson formalism [@leggett]. The donor and acceptor states are represented by means of a pseudo spin, which points up when the charge is in the donor state and down otherwise. The Hamiltonian of the system is given by:
$$H_{ET}= \tau \sigma_x + \frac{P^2_y} {2M} + V(y, \sigma_z) + H_{\rm
bath},
\label{spinboson}$$
where $$V(y, \sigma_z) = \frac 1 2 M \omega ^2 (y+y_0 \sigma_z) ^2 +
\frac 1 2 \epsilon \sigma_z,$$ and $\sigma_{x,z}$ are the Pauli matrices. The charge in the donor (up) state corresponds to the potential $V(y, +)$ whose equilibrium reaction coordinate is $- y_0$, and the converse state corresponds to $V(y,-)$, whose stable minimum is at $y_0$. The $H_{\rm bath}$ term represents contributions to the Hamiltonian of a dissipative environment coupled to the reaction coordinate. Figure \[fig:flucdis\] illustrates the meaning of the potential $V(
y, \sigma_{z})$ in the effective Hamiltonian of Eq.(\[spinboson\]). The curve marked A corresponds to the potential term in the donor state, while the curve marked B represents the potential function in the acceptor state.
This model has been thoroughly analyzed in the work by Garg [*e*t al.]{} [@garg] based on earlier work by Leggett [@leggett2]. A similar analysis, but within a more chemical framework, is presented by Marcus [*e*t al.]{} [@marcus-sutin]. Energy conservation requires that charge transfer takes place only when the reaction coordinate is close to the degeneracy point $y=y^*$ for which $V(y^*,+) = V(y^*,-)$; once the degeneracy point is reached, charge transfer is possible only because of the non zero off-diagonal tunneling matrix elements $\tau$.
The tunneling rate $\Gamma$ from donor to acceptor, is calculated in the above references. For moderate dissipation of the reaction coordinate, it is given by:
$$\label{gamma}
\Gamma = \frac{\tau^2}{\hbar}
\left( \frac{\pi}{E_r k_B T_{\rm eff}}\right)^{\frac 1 2} ~
\! \left( e^ {- E_f
/ k_B T_{\rm eff}} + e^{- E_b/ k_B T_{\rm eff}}\right)$$
where the reorganization energy $E_r$ and the energy barriers $E_f$ and $E_b$ depend on the details of the potential described by the reaction coordinate. In the limit of high temperatures $T_{\rm eff}$ reduces to the usual temperature $T$, whereas in the opposite limit the quantity is temperature independent.
The novelty explored in this paper is the introduction and investigation of the effect of a second reaction coordinate, $\theta$, governing the charge transfer process and coupled not to the energy, but to the off-diagonal tunneling element $\tau$, hitherto been treated as a constant, and which we now write $\tau(\theta)$.
This new coordinate reflects the conjecture that in the case of DNA the tunneling matrix element is highly sensitive to the donor-acceptor relative configuration. Charge transport along DNA in fact occurs along the stacked base pairs by means of overlapping $\pi$ orbitals, and at room temperature, these base pairs strongly fluctuate with respect to each other through variations of the twist, tilt and roll parameters [@calladine]. The existence of base pair fluctuations for DNA in solution is very well established, and is corroborated by experimental [@brauns] and molecular dynamics studies [@swaminathan; @cheat; @troisi]. For such a highly asymmetric system such as DNA, fluctuations in the relative orientation of donors and acceptors affect the magnitude of the orbital overlap between pairs, and the new collective coordinate $\theta$ embodies the effects of these fluctuations.
We will also assume that the $\theta$ variable is slowly varying compared to the motion of the reaction coordinate $y$, so as to define the lowest energy scale of the system. We may then separate the motion of the two reaction coordinates in a Born-Oppenheimer spirit. Charge transfer will be assumed to be instantaneous once the optimal $\theta = \theta^*$ value is reached, and a purely classical framework will be utilized for the $\theta$-dynamics. The new reaction coordinate $\theta$ need not necessarily be pictured as a geometrical one, although this is the framework we will be utilizing in this paper. Just as in the case of the $y$ reaction coordinate, $\theta$ may be associated to the particular chemical environment of the molecule or to any other quantity influencing the strength of the tunneling element $\tau$ between the donor and the acceptor sites.
The Hamiltonian describing the system thus, is a modified version of the spin-boson Hamiltonian introduced in Eq. (\[spinboson\]) with a $\tau(\theta) \sigma_x$ off-diagonal term, as also described in earlier work [@us]. In order for charge transfer to take place, we will assume that the reaction coordinate coupled to the energy must be close to the degeneracy point $y=y^*$, and, also, that the $\theta$ coordinate must be in the neighborhood of an optimal value $\theta^*$, which maximizes the tunneling amplitude. The physical picture to associate to this requirement is that the relative “alignment”, $\theta$, does not favor charge transfer unless an optimal configuration is reached: $\tau(\theta) \simeq 0$ unless $\theta
\simeq \theta^*$. This conjecture will prove to be crucial in yielding the two time-scale charge transfer of references [@barton99; @zewail].
In analogy to the experimental work cited above, we consider a collection of such two-level systems, with the charge initially located on the donor site. Each one of these systems is associated to a particular $\tau(\theta)$ and through Eq. (\[gamma\]) to a particular $\Gamma (\theta)$ rate. Our objective is to determine the mechanisms of charge transfer taking into account the $\theta$ time evolution and the $\Gamma(\theta)$ rates accordingly distributed. We shall assume the $\theta$ dynamics to be governed by small, Langevin type random fluctuations. At $t=0$, when the external charge is injected on the donor site, the distribution of $\theta$ values is the usual Boltzmann distribution. If the occurrence of the optimal $\theta^*$ configuration is relatively unlikely, we will indeed be able to show that the transfer process is characterized by a two time scale migration of the initial donor population.
The emergence of two time scales in the transfer process can be physically explained as follows. The existence of an initial non-zero population of TLS presenting the optimal value $\theta^*$, ensures that rapid tunneling to the acceptor. The $\theta$ distribution is thus depleted of population at the special value and other transitions are forbidden to take place. The other TLS will tunnel to the acceptor only after the system has re-equilibrated and re-populated the optimal configuration, a process which is slow, because of the assumption that the optimal configuration is a relatively unlikely one. Hence, the existence of a fast, initial decay followed by a slower decay process.
The TLS and $\theta$ fluctuations {#sec:model}
=================================
The model
---------
Consider a collection of TLS which at the initial time $t=0$ are all in the up-donor configuration, and characterized by the angular parameter $\theta$. Let us denote by $W(\theta, \dot \theta, t )$ the TLS population remaining in the up-donor state at time $t$ and for which the collective angular variable and its velocity are specified.
The physical requirement that $\theta$ be randomly, classically, fluctuating in time, translates into the fact that $W(\theta, \dot
\theta, t )$ must evolve according to a Fokker-Planck type equation as dictated by standard Langevin theory. To this probability evolution equation we must add an additional depleting term, that which represents tunneling to the donor site as given by the $\Gamma(\theta)$ term discussed above.
Different scenarios are possible for the $\theta$ dependence of $\tau$ and hence of $\Gamma$. As discussed in the above section we shall focus on the particular situation in which tunneling is possible only for a very specific subset of energetically unfavorable $\theta$ values. In this picture, tunneling is allowed only if donors and acceptors reach an optimal—but unlikely—orientation one with respect to the other. By including the tunneling term in the time evolution equation for $W(\theta, \dot \theta, t)$ we obtain a modified Fokker-Planck equation that may be used to approach any physical system in which the presence of a depleting term competes with the usual Langevin fluctuations. The most natural choice for the $\theta$ motion, the one we shall discuss in the remainder of this paper, is that of a damped harmonic oscillator. We shall see that starting from an initially equilibrated system in which the $\theta$ distribution is the the Boltzmann one, the insertion of the tunneling term will result in the emergence of the two time scales discussed above. We will refer to the time derivative of the $\theta$ coordinate as $u$. The rotational moment of inertia associated to $\theta$ is denoted by $I$ and its rotational frequency by $\Omega$.
The goal of the next subsections will be to determine $W(\theta^*, u, t)$, and in particular its integration with respect to the $u$ variable. As described in the introduction in fact, it is this quantity that is directly related to the experiments we wish to model by means of Eq.(\[fluorescence\]).
Kramers equation with a sink term {#sec:FP}
---------------------------------
The generic damped harmonic oscillator subject to random noise responds to the following Langevin-type equations: $$\begin{aligned}
\label{langevin1}
\dot \theta = u; \hspace{1cm} \dot u = -\gamma u - \Omega^2 \theta +
\eta(t),\end{aligned}$$ where the stochastic force $\eta(t)$ is assumed to be a zero-mean gaussian and whose correlation function is dictated by the fluctuation-dissipation theorem for classical variables: $$\label{flu-diss}
\left< \eta(t)\eta(t') \right> = \frac{2 \gamma k_B T}{I} \delta(t-t')
= 2q \delta(t-t').$$ The corresponding Fokker-Planck equation may be written by identifying [@risken] the proper coefficients in the Kramers-Moyal expansion from Eq. (\[langevin1\]) and is generally referred to as the Kramers equation. This equation governs the time evolution of the distribution, $W(\theta, u, t)$, of an ensemble of systems obeying the equations of motion (\[langevin1\]). It takes the form: $$\label{kramers}
\frac{\partial W}{\partial t} = -u \frac{\partial W}{\partial \theta} +
\frac{\partial}{\partial u}[(\gamma u +\Omega^2 \theta) W] + q
\frac{\partial^2 W}{\partial u ^2}.$$ The above equation is thoroughly analyzed in [@Chandrasekhar], where assuming an initial probability distribution $W(\theta, u, 0) = \delta(\theta-\theta')
\delta(u-u')$, the probability $W(\theta,u,t)$, as well as other relevant statistical quantities, are obtained. At equilibrium Kramers equation is solved by the time independent Boltzmann distribution, $W(\theta,u,t)= \psi_{0,0}(\theta,u)$ with: $$\label{ground1}
\psi_{0,0}(\theta,u) = \frac{\gamma \Omega} {2 \pi q} \exp
\left[-\frac{\gamma}{2q}\left(u^{2} + \Omega^{2}\theta^{2}\right)
\right]. \vspace{0.3cm}$$ Under the assumptions discussed earlier, the probability distribution function $W(\theta,u, t)$ for a particle localized on the donor site and describing an effective angle $\theta$ with its neighbor, will be described by the time evolution equation for a collection of damped oscillators subject to a decay term $\Gamma$, representing tunneling to the acceptor. The latter term is appreciable only for a specific value of the $\theta$ coordinate $\theta^*$: $$\label{mine}
\frac{d W}{dt} = HW - \Gamma(\theta,u,t)~ W.$$ The $H$ term is the differential operator that stems from the right hand side of Eq. (\[kramers\]). We shall assume the decay term to be introduced at time $t=0$, prior to which the system had attained its equilibration state. In other words, we choose the initial distribution $W(\theta,u,0)$ to be Boltzmann-like, as expressed in Eq. (\[ground1\]). For simplicity, we choose $\Gamma(\theta)$ to be independent of $u$ and of $t$ and to be a gaussian centered on $\theta^*$ and with width $\sigma$: $$\begin{aligned}
\Gamma(\theta)=\frac{\kappa}{\sqrt{2\pi\sigma}}~ \exp
\left[-\frac{(\theta-\theta^*)^2} {2\sigma}\right]. \vspace{0.3cm}\end{aligned}$$
The coefficient $\kappa$ contains the physical parameters of temperature and energy as expressed in Eq. (\[gamma\]). We also impose the constraint that at $t=0$ the optimal value $\theta^*$ carries a small Boltzmann weight. This is equivalent to the physical assumption that the occurrence of particle tunneling is a rather unlikely event, and that the system tends to relax to $\theta$ values that are far from the tunneling point. We also impose the width of the decay gaussian $\sqrt{\sigma}$, to be small compared to $\theta^*$, so that $\Gamma(\theta)$ is highly peaked around the optimal configuration value $\theta^*$: $\sqrt{\sigma} \ll \sqrt{q
/\gamma \Omega^2} \ll \theta^*$.
In the following subsections we will solve Eq. (\[mine\]) for the early and long time regimes. The general solution for arbitrary times is contained in the appendix. The coupling of the system to the orientational degree of freedom along the lines discussed above, manifests itself very clearly in the unusual time dependence of the probability distribution. Two different decay rates in fact arise, with a rapid initial decay of the donor population $W(\theta,u,t)$ followed by a slower transfer process. The ratio of these two time scales, and the main result of this analysis is succinctly expressed by Eq. (\[rates\]) in terms of all the physical parameters of this system.
Short time regime {#sec:shorttimes}
-----------------
In order to determine the asymptotic behavior of $W(\theta, u, t)$ in the early time regime, we consider Eq. (\[mine\]) with the gaussian choice of $\Gamma(\theta)$ and we perform a multiple time scale analysis [@bender]. This is carried out by introducing a new ad-hoc variable $\xi = \Gamma(\theta) t$, into the probability distribution, and by seeking solutions in the form $W(\theta,u,t) = W_0(\theta,u,t,\xi) + \Gamma(\theta) ~
W_1(\theta,u,t,\xi) + \dots$. The Fokker-Planck equation is thus expanded in powers of $\Gamma(\theta)$ and, for the zeroth and first order terms, it yields:
$$\begin{aligned}
\label{homo2}
\frac{\partial W_0}{\partial t} - H W_0 &=&0, \\
\label{nonhomo2}
\frac{\partial W_1}{\partial t} - H W_1 &=& - \left[\frac{\partial
W_0}{\partial \xi} + W_0 \right] + u \Gamma^{-1} \frac{\partial
\Gamma}{\partial \theta} ~ W_1.\end{aligned}$$
Note that the partial derivative with respect to $t$ in the above equations treats $\xi$ as an independent variable. The solution to the first equation is expanded in terms of the complete set of functions $\Psi_{m,n}(\theta,u,t)$ that solve Eq. (\[homo2\]) - obtained in Eq. (\[general\]) and Eq. (\[solutions\]) of the appendix - with coefficients $A_{m,n}$ that depend on $\xi$, i.e:
$$W_0(\theta,u,t) = \sum_{m,n} ~ A_{m,n}(\xi) ~ \psi_{m,n}(\theta,u) ~
e^{-\lambda_{m,n} t}.$$
Substituting this solution for $W_{0}$ into Eq. (\[nonhomo2\]), the inhomogeneous term in square brackets becomes:
$$-\sum_{m,n} ~ [\frac {\partial A_{m,n}}{\partial \xi} + A_{m,n}] ~
\psi_{m,n}(\theta,u) ~ e^{-\lambda_{m,n} t}.$$
If this were the only term present on the right hand side of Eq. (\[nonhomo2\]), then $W_1(\theta,u,t,\xi)$ would contain a secular term in its solution of the type:
$$\begin{aligned}
\hspace{-0.2cm}
\label{solution2}
&& W_1(\theta, u, t) \sim \hspace{6cm} \\
\nonumber
&& \hspace{0.5cm}
-t \sum_{m,n} ~ [\frac {\partial A_{m,n}}{\partial \xi} + A_{m,n}] ~
\psi_{m,n}(\theta,u) ~ e^{-\lambda_{m,n}t}.\end{aligned}$$
Such a solution will eventually exceed the “leading order” one. We determine the coefficients $A_{m,n}$ by requiring that there be no secular term in the solution to the equation. It is precisely this constraint that constitutes the underlying idea of multiple scale analysis. The above condition translates into requiring that the non-homogeneous term within parenthesis in Eq. ($\ref{nonhomo2}$) or equivalently in Eq. (\[solution2\]) vanish:
$$\frac{\partial A_{m,n}(\xi)}{\partial \xi} = - A_{m,n}(\xi).$$
We now solve for $A_{m,n}$. Imposing the initial condition $W(\theta,u,0) =
\psi_{0,0}(\theta,u)$ and reinserting $\xi = \Gamma(\theta) t$ the solution reads:
$$\label{slow}
W_0(\theta,u,t) =\psi_{0,0}(\theta,u) ~ \exp ~ [-\Gamma(\theta) t].$$
The above is a zero-th order approximation to the full problem presented in (\[nonhomo2\]) to the extent that the effect of $H$ acting on $t \Gamma (\theta)$ can be neglected with respect to $\Gamma (\theta)$ itself. In other words, Eq. (\[slow\]) is an approximate solution as long as:
$$\label{limit}
t \ll \frac{\Gamma(\theta)}{|u \Gamma_\theta(\theta)|} =
\frac{\sigma}{|u (\theta-\theta^*)|}.$$
This equation is valid only under the conditions expressed in $(\ref{limit})$ and up to $t \simeq \Gamma^{-1}(\theta)$. For this time limitation to be meaningful, it is necessary that the width of the decay term $\sqrt{\sigma}$ be finite. In the limit that the width vanishes the above analysis fails, since the expansion parameter diverges. At time $t \sim 0$ we cannot approximate $\Gamma(\theta)$ by a strict delta function. Note that for $\theta \sim \theta^*$, the tunneling point, and for finite $u$ the condition arising from the multiple scale analysis ($t
\simeq \sqrt{2 \pi \sigma}/ \kappa$) is the most stringent one, and the probability distribution is approximated by:
$$\label{short2}
W(\theta^*,u,t) = \psi_{0,0}(\theta^*,u) ~ \exp ~ [-\frac{\kappa
t}{\sqrt{2 \pi \sigma}}].$$
We now perform an integration over the $u$ variable on both sides of Eq. (\[slow\]) and obtain an approximation for the distribution probability function $P(\theta,t) =
\int_{-\infty}^{\infty} W(\theta,u,t) ~ du$:
$$\label{rate1}
P(\theta,t) \simeq \psi_{0}(\theta) ~ \exp ~ [- \Gamma(\theta)~ t].$$
where $\psi_{0}(\theta)$ is the Boltzmann distribution associated to the $\theta$ variable $\psi_{0}(\theta) =
\int_{-\infty}^{\infty} \psi_{0,0}(\theta,u) ~ du$. For small times, $P(\theta,t)$ retains its initial gaussian shape, with its amplitude decreasing exponentially.
Long time regime {#sec:longtimes}
----------------
In this subsection we determine the long time asymptotic behavior of $W(\theta, u, t)$, utilizing some of the results obtained in the appendix for arbitrary times. In particular, we adapt the kernel expansion of Eq. (\[kernel0\]) and Eq. (\[kernel\]) to the long time regime. Differentiating Eq. (\[kernel0\]) with respect to $t$ and with the gaussian choice for $\Gamma(\theta)$ we obtain: $$\begin{aligned}
\label{long}
\nonumber \frac {\partial W}{\partial t}= -\int \! \! \int
d \theta' du' \left.
\psi^{-1}_{0,0}(\theta',u') \frac{}{} \Gamma(\theta') \right. \\
\nonumber \left[ \frac{}{} K(\theta,\theta',u,u',0) ~ W(\theta',u',t)
\right. +\\
\left. \int^{t}_{0} dt'~ \frac{\partial K}{\partial
t'}(\theta,\theta',u,u',t') ~ W(\theta',u',t-t') \right],
\\
\nonumber\end{aligned}$$ where the integrals in $\theta'$ and in $u$ range from $-\infty$ to $+\infty$. The time-derivative of the kernel in the last integral can be obtained with the use of the expression obtained in Eq. (\[kernel\]) but with the summation restricted to non-zero values of the integers $m$ and $n$. The contribution to the kernel of the term associated with $m=n=0$ is time-independent, and it has the form $\psi_{0,0} (\theta,u) ~ \psi_{0,0} (\theta',u')$. We then replace $\partial_{t'} K$ with $\partial_{t'} K'$ where $K'$ is defined as the kernel without the first ($m,n=0$) summand.
The function $K'$ and its time derivative contain exponentially vanishing terms in $t$. The time integrand in Eq. (\[long\]) will therefore be appreciable only for $t' \leq \Omega_c^{-1}$ where $\Omega_c$ is a cutoff frequency of the order of $|\lambda_{1,0}| =
\Omega$. For $t \gg \Omega_c^{-1}$ we can approximate $W(\theta',u',t-t') \simeq W(\theta',u',t)$ and restrict the time interval from the origin to $\Omega_c^{-1}$. Integrating by parts, and using the above approximation for $W(\theta',u',t)$, the time integral yields: $$\begin{aligned}
\label{longmiddle0}
\frac {\partial W}{\partial t} &=& -\int \! \! \int
d \theta' du' ~
\psi^{-1}_{0,0}~ (\theta',u') ~ \Gamma(\theta') \\
&&
\nonumber
\left\{ \frac{}{} W(\theta', u', t)
\left[ \frac{}{}
K(\theta,\theta',u,u',0) + \right. \right. \\
&&
\nonumber
\left. \left. \frac{}{}
K'(\theta,\theta'u,u',\Omega_c^{-1})
-K'(\theta,\theta',u,u',0) \frac{}{} \right]\right\}.\end{aligned}$$ This equality is simplified by $K'(\theta,\theta'u,u',\Omega_c^{-1})$ being negligible. We can now rewrite the right hand side of Eq. (\[longmiddle0\]) as: $$\begin{aligned}
\label{long2}
\frac {\partial W}{\partial t} = -\int \! \! \int d\theta' ~
du' \left[ \psi^{-1}_{0,0}(\theta',u')
\frac{}{} \Gamma(\theta') \right.\\
\nonumber \left. \psi_{0,0}(\theta,u) ~ \psi_{0,0}(\theta',u')~
W(\theta',u',t) \frac{}{}\right].\end{aligned}$$ Since we are dealing with non-zero times, the $\theta'$ integration can be performed under the assumption that $\Gamma(\theta')$ is highly peaked around $\theta^*$ and $\Gamma(\theta) \simeq \kappa ~ \delta(\theta-\theta^*)$: $$\begin{aligned}
\frac{\partial{W}}{\partial{t}} =- \kappa ~ \psi_{0,0}(\theta,u)
\int_{-\infty}^{\infty} du' ~ W(\theta^*,u',t).\end{aligned}$$ A last integration in the $u$ variable, performed on both sides of the equation, yields the probability distribution function for the $\theta$ variable: $$\frac{\partial P(\theta,t)}{\partial t} = -\kappa ~ \psi_{0}(\theta) ~
P(\theta^*,t).$$ For $\theta=\theta^*$ the above relationship yields a decay rate of $-\kappa ~ \psi_{0}(\theta^*)$, and for arbitrary $\theta$ values we obtain the its behavior in the late time regime: $$\label{rate2}
P(\theta,t) = P_0 ~ \psi_{0}(\theta) ~ \exp \left[- \kappa ~
\psi_{0}(\theta^*) ~ t \right].$$
The two time scales
-------------------
As anticipated, two different scenarios for $P(\theta^*, t)$ emerge from the analysis carried out in the previous subsections. From Eq. (\[rate1\]), at early times, the decay to the acceptor state is rapid, occurring at a rate $r_1=\kappa /\sqrt{2 \pi
\sigma}$, whereas at latter times the rate is as given above: $r_2= \kappa \psi_{0}(\theta^*)$. The ratio between the two is $$\label{rates}
\frac{r_1}{r_2} = \sqrt{\frac{k_B T }{\sigma I \Omega^2 }} ~ \exp ~
\left[{\frac{I \Omega^2 }{2 k_B T } (\theta^*)^2} \right] \gg 1,$$ as follows from the assumptions made on the gaussian $\Gamma(\theta)$. The initial decay is much faster than that at later times.
Numerical results {#sec:numbers}
-----------------
Based on the general solution of Eq. ($\ref{kernel0}$), we present a numerical analysis of the distribution function $W(\theta,u,t)$ for different choices of its arguments. In this equation the probability distribution $W(\theta,u,t)$ is cast in a Volterra-type formulation, for which solutions can be constructed iteratively in time. The probability distribution $W(\theta,u,t)$ as expressed in Eq. (\[kernel0\]) in fact, depends only on its previous history and on the known propagator function.
For a numerical approach, it is necessary to discretize the $\theta,u,t$ variables and keep track of the value of $W(\theta,u,t)$ for every position and velocity at every temporal iteration. While feasible, this approach is rather cumbersome, since for every time step $t_k = k \Delta t $ we must create a new $O(N^2)$ matrix $W(\theta_i,u_j,t_k), ~ 1 \leq i,j \leq N$, where $N$ is the number of spacings for the position and velocity meshes. On the other hand, the evaluation at of $W(\theta^*,u_j,t_k)$ where $\theta^*$ represents the $\theta_i$ interval centered on the optimal value $\theta^*$ is greatly simplified if the corresponding mesh is chosen so that $\Gamma(\theta)$ may be replaced for all purposes by a delta function at non-zero times. The recursive equations now involve only the $O(N)$ element vector $W(\theta^*, u_j, t_k), 1 \leq j \leq N$.
At $t=0$, when the propagator itself is a point source, the gaussian shape for $\Gamma(\theta)$ must be retained for finiteness, but the iteration at a time that is far from zero does not involve values of the position that are significantly different from $\theta^*$. The $u$ mesh is chosen with $\Delta u = 0.05$ and the time interval spacing is $\Delta t=0.01$.
In order to insure consistency with the constraint $\sqrt{\sigma} \ll
\sqrt{q/\gamma \Omega^2} \ll \theta^*$ we choose the following parameters: $\sigma=10^{-4}$, $\gamma \Omega^2= 2q$, $\theta^*=1.5$. The $\alpha$ parameter for the underdamped case is chosen as $\alpha =
0.02$, whereas $\kappa$ is fixed at $\kappa = 0.4$. The resulting probability distribution $W(\theta^*,u,t)$ is plotted in Figure \[fig:dists\] as a function of $u$ for various time intervals.
Two features of the evolving distribution are noteworthy. The first is the depression around $u=0$. The second is a clear asymmetry in the velocity distribution, in that the distribution for negative values of the velocity, $u$, is lower than for positive $u$ values. The reason for the first feature is the fact that when the velocity is low, a pair will remain in a nearly optimal configuration longer, and hence a tunneling event, leading to depletion of the distribution, is more likely. The asymmetry can be ascribed to the fact that the optimal orientation is at positive values of the parameter $\theta$. The time evolution equation encapsulates two mechanisms, one pushing the distribution towards its Boltzmann limit, the other being the tunneling process that leads to depletion of the distribution at values of $\theta$ close to $\theta^{*}$. In light of the trajectory of the underdamped oscillation, a member of the ensemble with negative velocity, $u$, is likely to be within a half an oscillation period of having passed with a small velocity through $\theta^{*}$, which is positive, while a representative with positive $u$ is more likely to have spent more then half an oscillation period away from the optimal tunneling configuration. This latter, positive $u$ configuration will have had more time to experience the “restorative” effects of the mechanism that acts to generate the Boltzmann distribution.
It is also possible to perform a $u$-variable integration and obtain the time dependence of $P(\theta^*,t)$. The parameters are chosen as above, and the two time scale decay of $P(\theta^*,t)$ can be clearly seen to occur with rates $r_1$ and $r_2$ as described in Eq. $(\ref{rates})$. Also note that both at large and short times $P(\theta^*,t)$ is proportional to $W(\theta^*,0, t)$. The above results, and the expressions for $r_1$ and $r_2$ are not affected by changes in the damping variable $\alpha$. As anticipated, Figure $\ref{fig:pout2}$ clearly shows the double exponential decay of $P(\theta^*, t)$, in agreement with the experimental results of [@barton99; @zewail].
Discussion {#sec:conclusions}
==========
The model we have presented is expected to be of significant relevance to charge transfer in DNA. Thermal fluctuations strongly affect the structure of molecule, and an accurate description requires this motion to be taken into account.
Not only has the existence of fluctuations been experimentally documented [@brauns], but it has also been suggested [@troisi] that the motion that most affects the electronic coupling between base pairs - what we have referred to as $\tau(\theta)$ - is their sliding one with respect to the other. It must be pointed out that both these studies focus on DNA in solution, not on dry strands of DNA.
On the other hand, charge transport with more than one rate has been reported in the literature [@barton99]. For an oligomer with the ethedium molecule acting as the donor, charge transfer is found to occur along the same patterns as described by our model, with two time scales of 5 and 75 picoseconds. Two-time-scale decays are also observed in a series of measurements [@zewail] performed on shorter strands of donor and acceptor complexes (Ap-G). In these experiments the Ap donor can be treated, for all practical purposes, as an intrinsic purine base, and the ambiguity related to the choice of an extraneous donor (the ethedium of the previous reference) is removed.
In both these experiments, an increase in the length results in a competition between the fast and slow exponential decays in favor of the slower time component. Increasing the length of the system diminishes the possibility that multiple base pairs simultaneously arrange in the configuration that facilitates rapid charge transfer. When the process of optimal alignment does occur (a relatively likely event only for a few base pairs), the tunneling might not even require localization of the charge on each base pair, and super-exchange can take place.
For long strands of DNA, thus, we expect the two intrinisic rates associated to a single charge transfer to be averaged out in favor of the slower component. Traces of this unusual two time scale migration mechanism however, may be found in the fact that DNA conductivity is enhanced upon increasing the temperature [@gruner2], presumably allowing for greater base pair motion. Charge transfer is also hindered by disruptions to the stacking, which alter the base pair’s ability to find optimal transfer configurations, such as the insertion of bulges along the helix or of strong mismatches within the base pair stacking [@dandy; @barton99_sci] which are poorly compatible with the intrinsic conformation of the aromatic pairs. Lastly, it is noted that charge transfer effectiveness seems to be inversely proportional to the measured hypochromicity [@barton97], a quantity that determines the ordering of base pairs along a certain direction and defined as the reduction of absorption intensity due to interactions between neighboring electric dipoles. From this data it is apparent that the higher the disorder of the system, the more efficient charge transfer is. It would be interesting to see how different solvent environments affect conduction along the molecule in relation to their effect on structural fluctuations. More temperature-dependent experimental measures are desirable as well.
Conclusions {#sec:concludo}
===========
We have presented a model for a spin boson TLS whose tunneling matrix element depends on the structural conformation of the donor with respect to the acceptor. In the limit that the relative geometry between the two fluctuates in time defining the lowest energy scale, we are led to a classical problem, that of a collection of damped harmonic oscillators obeying a modified Fokker-Planck equation. If charge transfer proceeds only for specific orientations of the donor with respect to the acceptors, the resulting rate for charge transfer is divided into a fast component at short times and a subsequent slower one. These results agree with the experimental findings of two-time-scale charge transfer in the donor intercalated DNA complexes of J.Barton and coworkers [@barton99; @zewail]. It must be noted that an implicit assumption of this work is that for long range DNA conduction mediated by thermal fluctuations once the charge has undergone a transfer between base pairs it does not return to the pair at which it is originally localized. However, it is reasonable to assume that the transfer process will continue after this event has occurred and that subsequent events will, with some probability, deposit the charge at its point of origin at a later time. We have performed calculations on a two-time-scale hopping model based on the results obtained here [@future]. In these calculations, the single set of two base pairs is replaced by a linear array. We have determined the probability that the charge carrier is at its point of origin as a function of time, $t$, after its having been placed there. We find that this probability exhibits two-time-scale behavior, with an initial, brief, rapid, exponential decay followed by a much slower, power-law, decay at later times times. The long-time asymptotics of this process are those of a random walk.
We acknowledge many useful conversations with Prof. R. Bruinsma and Prof. T. Chou.
General solution of the Kramers equation {#sec:sol}
========================================
We shall adopt a Lie-Algebra approach [@note; @to; @Risken] to identify a complete set of orthonormal functions that solve the homogeneous problem in the general case of Eq. (\[kramers\]), and through them the general solution for the decay equation (\[mine\]) will be found.
Let us look for solutions of the following type, where $m$ and $n$ represent non negative integers: $$\label{general}
\Psi_{m,n}(\theta,u,t)=\psi_{m,n}(\theta,u) e^{-\lambda_{m,n} t}.$$ Upon insertion of the above expression in Eq. (\[kramers\]) a time independent Schrödinger-like equation can be written: $$\label{homo}
-(\lambda_{m,n} + \gamma) \psi _{m,n} = H' \psi_{m,n},$$ where: $$H'(\theta, u) = q p_u^2 + \gamma u p_u + \Omega^2 \theta p_u - up_\theta,$$ and the subscripts represent derivatives, $p_u = \partial
/\partial_u$. As expected, the time independent Boltzmann distribution satisfies the homogeneous equation, as can be verified by direct substitution with $\lambda_{0,0}=0$. The physical requirement that solutions must be well behaved as $t \rightarrow \infty$, i.e. that the $\lambda_{m,n}$’s be non negative, suggest that this is the ground state: $$\label{ground2}
\Psi_{\rm ground}(\theta,u,t) = \psi_{0,0}(\theta,u).$$ The other solutions are found by constructing the ladder operators. For the underdamped case, we introduce the $\alpha$ variable such that $ \cos \alpha = \gamma/(2 \omega)$ and impose that $[H', O]= l ~ O$ with $l$ and $O$ respectively complex variable and operator to be determined. In practice, the operator $O$ corresponds to either a raising or a lowering operator. Two sets of solutions exist for the following ‘quanta’ $l_{1,2}$: $$l_1 = \Omega e^{-i \alpha}, ~ ~ l_2=\Omega e^{i \alpha},$$ for which the associated raising and lowering operators $R_{1,2}$ and $L_{1,2}$ are: $$\begin{aligned}
\label{lo}
&& R_{1,2} = -p_\theta+ l_{1,2} ~ p_u; \\
&& L_{1,2} = \Omega^2 \theta +\frac{q}{\gamma} p_\theta +l_{1,2}
\left(\frac{q}{\gamma} p_u +u \right).\end{aligned}$$ The commutation rules for the above operators can be easily derived as: $$\begin{aligned}
\nonumber && [R_i, R_j] =0, ~ [L_i, L_j]=0, ~ [R_1, L_2]=0, \\
&& {[ R_1, L_1]}= \Omega^2(e^{-2 i \alpha} -1), \\
\nonumber && {[ R_2, L_2]}= \Omega^2(e^{+2 i \alpha} -1).\end{aligned}$$ The raising operators applied to the ground state yield the set of solutions $\psi_{m,n}$ for Eq. (\[homo\]) with the associated eigenvalues $\lambda_{m,n}$ as follows: $$\begin{aligned}
\label{solutions}
\psi_{m,n} (\theta,u) = R_2^n ~ R_1^m ~ \psi_{0,0} (\theta,u), \\
\lambda_{m,n}= m \Omega e^{-i \alpha} + n \Omega e^{i \alpha}.\end{aligned}$$ It is worth noting that the Hamiltonian $H'$ can also be reformulated as $ H'= (2 \Omega i \sin \alpha)^{-1} [(L_2 R_2)- (L_1
R_1)]$. In order to construct solutions to the non-homogeneous problem within the Hilbert space spanned by the set of solutions $\{\psi_{m,n}(\theta, u)\}$, it is necessary to determine the orthonormality of those solutions. To this purpose, let us consider the following $\{\phi'_{m,n}(\theta, u)\} = \{P_2^n P_1^m
\psi_{0,0}(\theta,u) \}$ where $P_{1,2}$ are operators defined as: $$\label{pi}
P_{1,2} =-p_\theta-l_{1,2} ~ p_u.$$ We can now prove an orthogonal relation between the two sets, using the commutation rules and and introducing $\psi_{0,0}^{-1}(\theta,u)$ as a weighting function:
$$\begin{aligned}
\nonumber
\int \int du ~ d\theta ~
\phi'_{m',n'}(\theta,u) ~ \psi_{0,0}^{-1}( \theta,u) ~
\psi_{m,n}(\theta,u) &=& \\
C_{m,n} ~ \delta_{m,m'} ~ \delta_{n,n'}.\end{aligned}$$
The integration limits are over the entire real axis, both for $\theta$ and $u$. The orthonormal set of eigenfunctions is thus expressed as $\{ C_{m,n}^{-1} ~ \phi'_{m,n}(\theta,u) \}$, to which we refer as $\{\phi_{m,n}(\theta,u) \}$. The constant of proportionality $C_{m,n}$ is: $$\begin{aligned}
C_{m,n} = m! ~ n! \left( \frac{\gamma \Omega^2}{q}\right) ^{m+n}
\left(1-e^{-2i\alpha}\right)^m
\left(1-e^{2i\alpha}\right)^n.\end{aligned}$$ Let us now look for the full solution $W(\theta, u, t)$ to Eq. (\[mine\]), posing it in the following form: $$\label{nonhomo}
W(\theta,u,t)= \sum_{m,n} h_{m,n}(t) ~ \psi_{m,n} (\theta,u) ~
e^{-\lambda_{m,n} t }.$$ The $h_{m,n}(t)$ functions are to be determined, in analogy to the scattering problem of particles in quantum mechanics. Let us assume that the decay term is introduced at time $t=0$, and that the initial distribution is the equilibrium solution to the homogeneous problem, i.e. the ground state as expressed in Eq. (\[ground1\]). Inserting Eq. (\[nonhomo\]) in Eq. (\[mine\]) and using the orthonormality relations, it is possible to find time evolution equations for $h_{m,n}(t)$ and to write a recursion formula for the full solution: $$\begin{aligned}
\nonumber W(\theta,u,t) &=& W(\theta,u,0) - \int_{0}^{t} dt'
\int_{-\infty}^{\infty} d\theta' \int_{-\infty}^{\infty} du' \\
\label{kernel0}
&&
\nonumber
\left [\frac{}{}K(\theta,\theta',u,u', t-t') ~
\psi_{0,0}^{-1}(\theta',u') \right. \\
&& \left.\Gamma(\theta',u',t') ~W(\theta',u',t') \frac{}{}
\right].\end{aligned}$$ Here, we have kept $\Gamma$ a generic function of all variables and the $K$ function is the response kernel of the system: $$\begin{aligned}
\nonumber
\hspace{-1.5cm}
K(\theta,\theta',u,u', t) &=& \\
\label{kernel}
&& \hspace{-2cm}\sum_{m,n} \psi_{m,n}(\theta,u) ~ \phi_{m,n}(\theta',u')
~ e^{-\lambda_{m,n} t }.\end{aligned}$$ The product $W'(\theta,u,t)= K(\theta,\theta',u,u',t) ~
\psi_{0,0}^{-1}(\theta',u')$, is the distribution function for the homogeneous system, under the initial conditions $W'(\theta,u,0) =
\delta(\theta-\theta') \delta(u-u')$. Its asymptotic behavior reduces to the Boltzmann distribution, and apart from $t=0$, it is an analytical function in all its variables. The explicit representation of the kernel may be written by inserting the expressions for [$\psi_{m,n}(\theta,u)$]{} and [$\phi_{m,n}(\theta,u)$]{} in Eq. (\[kernel\]):
$$\begin{aligned}
\label{response1}
&& K\left(\theta,u, \theta^{\prime} u^{\prime},t \right) = \\
\nonumber \\
\nonumber && \exp \left[ \frac{q \left( \partial_\theta - \Omega e^{+i
\alpha}\partial_u \right) \left( \partial_{\theta^{\prime}} + \Omega
e^{+i \alpha}\partial_{u^{\prime}} \right)} {\gamma \Omega^{2} \left(
1 - e^{+2i \alpha}\right)} e^{- \Omega e^{+i \alpha}t} \right] \\
\nonumber \\
\nonumber && \exp \left[ \frac{q \left( \partial_\theta - \Omega e^{-i
\alpha}\partial_u \right) \left( \partial_{\theta^{\prime}} + \Omega
e^{-i \alpha}\partial_{u^{\prime}} \right)}{\gamma \Omega^{2} \left( 1
- e^{-2i \alpha}\right)} e^{- \Omega e^{-i\alpha}t} \right] \\
\nonumber \\
\nonumber && \hspace{3cm} \psi_{0,0}(\theta,u) ~
\psi_{0,0}(\theta'u'),\end{aligned}$$
where the exponential terms are intended as operators acting on the ground state wave functions. As it is written, the above kernel is still expressed symbolically. In order to obtain its explicit form it will suffice to perform a Fourier transform of Eq. (\[response1\]), and then return to real space, a straightforward but tedious calculation we omit. The complete solution for the kernel is given by [@note; @to; @Chandrasekhar]: $$\begin{aligned}
&& K(\theta,\theta',u,u',t)= \left(\frac{\gamma \Omega}{2
\pi q}\right)^2 \frac{1}{\sqrt {T G}} \\
\nonumber \\
&& \nonumber \exp \left[-\frac{\gamma}{4 q T}
\left(\frac{}{}\Omega^2 (1-n)
(\theta+\theta')^2 + (1+l)(u-u')^2
\right. \right. \\
\nonumber &&\left. \left. \frac{}{} \hspace{3.5cm}
+2 m \Omega
(\theta+\theta')(u-u') \right) \right] \\
\nonumber &&\exp \left[-\frac{\gamma}{4 q G}
\left(\frac{}{}\Omega^2 (1+n) (\theta-\theta')^2 \right. \right.
\nonumber +(1-l) (u+u')^2 \\
&& \left. \left. \frac{}{}
\nonumber
\label{kernel3}
\hspace{3.5cm}+2m \Omega (\theta'-\theta)(u+u') \right) \right].\end{aligned}$$
In order to keep a lighter notation, we have suppressed the time dependence of the $T(t)$, $G(t)$, $l(t)$, $m(t)$, $n(t)$ functions. They are defined as: $$\begin{aligned}
l(t) \sin \alpha & = & e^{- \Omega t \cos \alpha}
~ \sin(\alpha + \Omega t
\sin \alpha), \\
m(t) \sin \alpha & = & e^{-\Omega t \cos \alpha}
~ \sin ( \Omega t \sin
\alpha), \\
n(t) \sin \alpha & = & e^{-\Omega t \cos \alpha}
~ \sin (\alpha - \Omega t
\sin \alpha).\end{aligned}$$ The functions $T(t)$ and $G(t)$ are combinations of the above: $$\begin{aligned}
T(t) & = & 1+l(t)-n(t)-n(t)l(t) -m^2(t), \\
G(t) & = & 1+n(t)-l(t)-n(t)l(t)-m^2(t).\end{aligned}$$
In order to ensure integrability for Eq. (\[kernel0\]), some limitations are posed on the form of the $\Gamma(\theta',u',t')$ function. For instance, the seemingly most natural choice, a delta function centered around $\theta^*$, yields a non integrable expression for $W(\theta,u,t)$ at small times, when the kernel is a product of delta functions itself. Instead, the gaussian choice introduced earlier, with its finite $\sigma$, ensures integrability at all time regimes.
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A slightly different method is also outlined in [@risken], chapter 9.
This solution, apart from the multiplicative $\psi_{0,0}^{-1}(\theta',u')$ term, is the same as discussed in [@Chandrasekhar]. Integrating the $u$ coefficient, the expression is the same as Eq.(30) in Ref. [@Chandrasekhar].
|
---
abstract: 'Adiabatic cyclic modulation of a one-dimensional periodic potential will result in quantized charge transport, which is termed the Thouless pump. In contrast to the original Thouless pump restricted by the topology of the energy band, here we experimentally observe a generalized Thouless pump that can be extensively and continuously controlled. The extraordinary features of the new pump originate from interband coherence in nonequilibrium initial states, and this fact indicates that a quantum superposition of different eigenstates individually undergoing quantum adiabatic following can also be an important ingredient unavailable in classical physics. The quantum simulation of this generalized Thouless pump in a two-band insulator is achieved by applying delicate control fields to a single spin in diamond. The experimental results demonstrate all principal characteristics of the generalized Thouless pump. Because the pumping in our system is most pronounced around a band-touching point, this work also suggests an alternative means to detect quantum or topological phase transitions.'
author:
- Wenchao Ma
- Longwen Zhou
- Qi Zhang
- Min Li
- Chunyang Cheng
- Jianpei Geng
- Xing Rong
- Fazhan Shi
- Jiangbin Gong
- Jiangfeng Du
title: Experimental Observation of a Generalized Thouless Pump with a Single Spin
---
In 1983, Thouless discovered that the charge transport across a one-dimensional lattice over an adiabatic cyclic variation of the lattice potential is quantized, equaling to the first Chern number defined over a Brillouin zone formed by quasimomentum and time [@ThoulessPump1]. This phenomenon, known as the Thouless pump, shares the same topological origin as the quantization of Hall conductivity [@TKNN1; @TKNN2; @TKNN3] and may thus be regarded as a dynamical version of the integer quantum Hall effect [@DQHE1]. In the ensuing years, the Thouless pump was investigated extensively [@TKNN3]. Up to now, several single-particle pumping experiments have been implemented in nanoscale devices [@PumpNanoExp1; @PumpNanoExp2; @PumpNanoExp3; @PumpNanoExp4]. Most recently, the Thouless pump was observed in cold atom systems [@PumpCdAtExp1; @PumpCdAtExp2]. On the application side, the Thouless pump has the potential for realizing novel current standards [@PumpApp1; @PumpApp2], characterizing many-body systems [@PumpInt1; @PumpInt2; @PumpInt3; @PumpInt4; @PumpPhoton], and exploring higher dimensional physics [@PumpHighD1].
In Thouless’ original proposal and almost all the follow-up studies, the initial-state quantum coherence between different energy bands, namely, the interband coherence in the initial states, is not taken into account. As a fundamental feature of quantum systems [@Schroedinger; @Quantify], quantum coherence is at the root of a number of fascinating phenomena in chemical physics [@CC1; @CC2; @CC3], quantum optics [@QO1; @QO2; @QO3; @QO4; @QO5], quantum information [@Chuangbook], quantum metrology [@Metrology1; @Metrology2; @Metrology3], solid-state physics [@Solid1; @Solid2], thermodynamics [@Therm1; @Therm2; @Therm4; @Therm5; @Therm6], magnetic resonance [@MR1; @MR2; @MR3], and even biology [@QB1; @QB2; @QB3]. Therefore, a question naturally arises as to how the pump will behave if the interband coherence resides in the initial state. A theoretical analysis of this issue is outlined in Fig. \[theory\], where $Q_a$ and $Q_b$ represent pumping contributed by individual bands, and $Q_{\rm IBC}$ is fueled by interband coherence [@PumpWang; @PumpZhou]. In contrast to the conventional Thouless pump where the pumping components $Q_a$ and $Q_b$ are determined by the Berry curvature of each filled band, the generalized Thouless pump is featured by the component $Q_{\rm IBC}$ that can be continuously and extensively controlled. The main aspects of $Q_{\rm IBC}$ are experimentally investigated in this work.
![Illustration about how interband coherence in the initial state leads to the generalized Thouless pump of duration $T$. Band dispersion relations are plotted via energy and quasimomentum variables $E$ vs $k$, and $\tau$ represents the time scaled by $T$. (a) With an initial state being an incoherent mixture of states from different bands, the nonadiabatic correction to the band populations is of the order of $1/T^2$. The resultant pumping $Q_a + Q_b$ is a weighted sum of the contribution from each band. (b) With an initial state having interband coherence, the pumping operation induces a population correction of the order of $1/T$, whose effect accumulated over $T$ yields a pumping term $Q_{\text{IBC}}$ that is $T$ independent and continuously tunable. []{data-label="theory"}](fig1){width="1\columnwidth"}
Consider a one-dimensional two-band insulator subject to time-dependent modulations. Its Hamiltonian in the quasimomentum space is $$\begin{aligned}
H(k,\tau)&=&\frac{\omega\sin k}{2}\big[\cos\phi(\tau)~\sigma_{x}+\sin\phi(\tau)~\sigma_{y}\big] \nonumber \\
&& + \frac{\delta_1\cos k+\delta_2}{2}~\sigma_{z}.
\label{Hamiltonian}\end{aligned}$$ Throughout, $\tau=t/T$ is the scaled time with $t$ being the real time and $T$ the duration of one pumping cycle, $k\in(-\pi,\pi]$ is the quasimomentum, $\sigma_{x,y,z}$ are the Pauli matrices, and $\hbar$ is set to $1$. The instantaneous spectrum of $H(k,\tau)$ is gapless at $k=0~(k=\pi)$ when and only when $\delta_1=-\delta_2$ ($\delta_1=\delta_2$). One pumping cycle can be realized by slowly varying $\phi$ from $0$ to $2\pi$. In a lattice representation, the parameters $\delta_1$ and $\delta_2$ represent the respective bias in the nearest-neighbor hopping strength and energy between two internal states.
For a general initial state with equal populations on quasimomenta $k$ and $-k$ on each band, the pumped amount of charge $Q$ over $N$ adiabatic cycles can be found from the first-order adiabatic perturbation theory (APT) [@Messiah; @AdPt3; @PumpWang; @PumpZhou], where $Q=N(Q_{\rm TP}+Q_{\rm IBC})+Q_{\rm NG}$, with $$\begin{aligned}
{1}
{Q_{{\rm{TP}}}}=& \frac{1}{{2\pi }}\int_{ - \pi }^\pi {dk\sum\limits_n {{{\left. {{\rho _{nn}}} \right|}_{\tau = 0}}} \int_0^1 {d\tau }\ {\rm{ }}} {\Omega_{\tau k}^{(n)}}, \\
{Q_{{\rm{IBC}}}}=& \frac{1}{{2\pi }}\int_{ - \pi }^\pi dk \sum\limits_{m,n(m < n)}\nonumber \\
&{\left. {\frac{{2\ {\mathop{\rm Im}\nolimits} \left( {{\rho _{mn}}\left\langle n \right|{\partial _\tau}\left| m \right\rangle } \right)}}{{{E_m} - {E_n}}}} \right|} _{\tau = 0}\int_0^1 {d\tau}~({v_{mm}} - {v_{nn}}),\label{QIBC}\end{aligned}$$ and $Q_{\rm NG}$ being a nongeneric term that does not build up with the number of pumping cycles (hence not of interest here) [@SM]. That is, only $Q_{\rm TP}$ and $Q_{\rm IBC}$ represent contributions from pumping over each adiabatic cycle. In above $m$ and $n$ are band indices, $|m(k,\tau)\rangle$ represent an instantaneous eigenstate of $H(k, \tau)$ with the eigenvalue $E_m(k,\tau)$, $\rho_{mn}(k,\tau)$ and $v_{nm}(k,\tau)$ refer to matrix elements of the density operator and the velocity operator $v(k,\tau)\equiv\partial_k H(k,\tau)$ in representation of $|m(k,\tau)\rangle$, and ${\Omega _{\tau k}^{(n)}}$ is the Berry curvature of the $n$th instantaneous energy band of $H(k,\tau)$. The component $Q_{\rm TP}$ ($=Q_a+Q_b$ in the case of Fig. \[theory\]), representing a weighted integral of the Berry curvature, was found previously by Thouless [@ThoulessPump1]. The component $Q_{\rm IBC}$, namely, the charge pumping induced by interband coherence in the initial state, is responsible for the generalized Thouless pump and will be our focus here. Analogous to the conventional Thouless pumping, $Q_{\rm IBC}$ arises from an accumulation of small nonadiabatic effects over one pumping cycle. As sketched in Fig. \[theory\], the initial-state interband coherence plays a crucial role in generating the underlying nonadiabatic effects. The term $Q_{\rm IBC}$ is found to be nontopological and can change continuously. As indicated by Eq. (\[QIBC\]), $Q_{\rm IBC}$ depends on $\langle n|\partial_{\tau}|m\rangle|_{\tau=0}$ and the band gap. For a pumping parameter $\phi(\tau)$ as in our two-band model depicted in Eq. (\[Hamiltonian\]), one has $\langle n|\partial_{\tau}|m\rangle|_{\tau=0} \sim \frac{d\phi(\tau)}{d\tau}|_{\tau=0}$, which can be controlled by the switching-on rate of a pumping protocol. The band gap in our model can be altered via tuning $\delta_1/\delta_2$ and $Q_{\rm IBC}$ can even diverge logarithmically as the band gap is tuned to approach zero [@SM].
The generalized Thouless pump can be experimentally realized on a qubit system because the insulator’s Hamiltonian in Eq. (\[Hamiltonian\]) is also the Hamiltonian of a qubit in a rotating field parametrized by $k$ and $\phi$. That is, by mapping the two-band insulator’s Hamiltonian to that for a qubit in a rotating field, we can experimentally demonstrate the generalized Thouless pump using a single spin [@SpHfCherNumExp]. To highlight the contribution from $Q_{{\rm IBC}}$, in our experiment the initial state is properly designed such that $Q_{\rm TP}=Q_{\rm NG}=0$; i.e., the traditional Thouless pumping and the non-generic term $Q_{\rm NG}$ have no contribution. To demonstrate the sensitivity of $Q_{{\rm IBC}}$ to the switching-on rate of the pumping protocol, namely, $\frac{d\phi(\tau)}{d\tau}|_{\tau=0}$, we consider a linear ramp $\phi(\tau)=2\pi \tau$ and a quadratic ramp $\phi(\tau)=2\pi\tau^{2}$. One directly sees that the latter choice with zero switching-on rate will make $Q_{{\rm IBC}}=0$ within the first-order APT. To demonstrate the dependence of $Q_{{\rm IBC}}$ on the band gap, we implement the Hamiltonian in Eq. (\[Hamiltonian\]) with a varying band gap.
A negatively charged nitrogen-vacancy (NV) center in diamond is used in the experiment. As shown in Fig. \[structure\](a), the NV center is composed of one substitutional nitrogen atom and an adjacent vacancy [@Doherty; @Schirhagl; @Prawer; @Wrachtrup]. In our experiment, an external static magnetic field around $510$ G is parallel to the NV symmetry axis. Such magnetic field enables both the NV electron spin and the host $^{14}$N nuclear spin to be polarized by optical excitation [@Jacques; @Sar]. As illustrated in Fig. \[structure\](b), microwaves generated by an arbitrary waveform generator drives the transition between the electronic levels $\left| {{m_s} = 0} \right\rangle$ and $\left| {{m_s} = - 1} \right\rangle$ which compose a qubit, and the level $\left| {{m_s} = 1} \right\rangle$ remains idle due to large detuning [@NV]. The Hamiltonian of the qubit in the laboratory frame is $H^{\rm{lab}} = \omega_0\sigma_{z}/2 + f(t)\sigma_{x}$, where the term $f(t)\sigma_{x}$ delineates the effect of the microwave field. The expectation value of the observable $\sigma_z$ can be read out via fluorescence detection during optical excitation. All the optical procedure are performed on a home-built confocal microscope, and a solid immersion lens is etched on the diamond above the NV center to enhance the fluorescence collection [@Robledo; @Rong].
![Experimental system and method. (a) NV center in diamond. (b) Electronic ground state of a negatively charged NV center. The energy splitting depends on the magnetic field which is parallel to the NV axis in this experiment [@NV]. The two levels $\left| {{m_s} = 0} \right\rangle$ and $\left| {{m_s} = - 1} \right\rangle$ are encoded as a qubit which is manipulated by microwaves (MW). (c) Pulse sequence for qubit control and measurement. The ellipsoid surface represents the parameter space and the sphere represents the Bloch sphere. []{data-label="structure"}](fig2){width="1\columnwidth"}
In our work, the experiment for different $k$ is performed separately in different runs of experiment. The pulse sequence for each $k$ is sketched in Fig. \[structure\](c). At first, the qubit is polarized to the state $\rho_0=(\mathbbm{1}+\sigma_z)/2$ by a laser pulse \[see the green bar in the preparation section in Fig. \[structure\](c)\], and then the initial state $\rho(k,0)$ needs to be prepared. To optimize $Q_{\text{IBC}}$ in the experiment, the initial density matrix $\rho(k,0)$ at individual values of $k$ is designed such that its associated Bloch vector is perpendicular to the direction of the field that yields $H(k,0)$. This choice is again illustrated in Fig. \[structure\](c). Specifically, $\rho(k,0)$ is chosen as $[\mathbbm{1}+ {\bm{n}(k)} \cdot {\bm{\sigma }}]/2$ with the unit vector $\bm{n}(k)$ along the direction $\left( { - {\delta _1}\cos k - {\delta _2},0,\omega \sin k } \right)$ except that $\bm{n}=(0,0,1)$ at a band touching point. This choice also makes $Q_{\rm{TP}}$ and $Q_{\rm NG }$ vanish [@SM]. To prepare such an initial state, we apply a resonance microwave pulse with the temporal dependence $f(t) = \omega_1 \cos ( {\omega _0} t +\varphi_{\rm{ini}})$, where the time $t$ starts from zero, $\omega_1$ is the Rabi frequency, and the initial phase $\varphi_{\rm{ini}}$ is set as $-\pi/2~~(\pi/2)$ if ${\delta _1}\cos k + {\delta _2}\ge0~~(<0)$. The duration of the pulse is $t_{\rm{ini}}=\alpha/\omega_1$, where $\alpha$ is the inclination angle of $\bm{n}(k)$. The orange bar in the preparation section in Fig. \[structure\](c) represents this pulse. Upon initial-state preparation, the qubit is left to evolve under $H(k,\tau)$, namely, in the presence of a field whose transverse and longitudinal magnitudes are given by $\omega\sin k$ and ${\delta _1}\cos k + {\delta _2}$, respectively. The field is then rotated around the $z$ axis according to $\phi(\tau)$, with $\phi(\tau)$ understood as the azimuthal angle. This rotating field is implemented by applying a microwave pulse with $f(t)=\omega \sin k \cos \left[ {({\omega _0} - {\delta _1}\cos k - {\delta _2})t + \phi (\tau)} + \varphi_{\rm {I}}\right]$, where $t$ starts from zero and the initial phase $\varphi_{\rm {I}}=\omega _0 t_{\rm{ini}}$ is used to match the phase of the first pulse. In Fig. \[structure\](c) this pulse is depicted by the blue bar in the evolution section. In a frame with the initial azimuthal angle $\varphi_{\rm {I}}$ and rotating around the $z$ axis with the angular frequency ${\omega}_0 - {\delta}_1 \cos k - {\delta}_2$ relative to the laboratory frame, the bare Hamiltonian $H^{\rm{lab}}$ is transformed, via the rotation transformation operator ${e^{ - i[({\omega _0} - {\delta _1}\cos k - {\delta _2})t + \varphi_{\rm {I}}]{\sigma _z}/2}}$, to our target Hamiltonian $H(k,\tau)$ under the rotating wave approximation. The parameters adopted in our experiment are $\omega = 2\pi \times 20$ MHz, $\delta_1 = 2\pi \times 10$ MHz, and $\delta_2$ between $0$ to $2\pi \times 20$ MHz. The evolution governed by $H(k,\tau)$ lasts for some duration $t_{\rm e} \in [0,T]$ (with the corresponding scaled time $\tau_{\rm e} \in [0,1]$), and then the velocity $v(k,\tau_{\rm e})$ needs to be measured. As shown in Fig. \[structure\](c), the measurement procedure begins with a microwave pulse (the magenta bar). This pulse is described by $f(t)=\omega_1 \cos ( {\omega _0}t + \varphi_{\rm {I}} + \varphi_{\rm{II}}+ \varphi_{\rm{fin}})$, with $t$ starting from zero, $\varphi_{\rm{II}}=({\omega _0} - {\delta _1}\cos k - {\delta _2})t_{\rm e}+\phi (\tau_{\rm{e}})$, and $\varphi_{\rm{fin}} = -\pi/2~~(\pi/2)$ when $\cos k \ge 0~~(<0)$. The duration of the pulse is $t_{\rm{fin}}=\beta/\omega_1$, where $\beta$ is the inclination angle of the direction of $v$. This resonant microwave pulse, which steers the direction of $v(k,\tau_{\rm e})$ to the $+z$ direction, is followed by a laser pulse \[the right green bar in Fig. \[structure\](c)\] together with fluorescence detection. The fluorescence is collected via two counting windows represented by the two red bars in Fig. \[structure\](c). The former window records the signal while the latter records the reference [@SM]. The fluorescence collection amounts to the measurement of $\sigma_z$, and the effect combined with the microwave pulse is equivalent to the observation of $v(k,\tau_{\rm e})/ \left\|v\right\|$, where $\left\|v\right\|$ is the spectral norm of $v$.
The above sequence is performed for a series of $\tau_{\rm e}\in [0,1]$, and is iterated at least a hundred thousand times to obtain the expectation value. One can then get $\langle v(k,\tau) \rangle / \left\|v\right\|$ as a function of $\tau$. Numerical integration over $\tau$ based on these experimental data, multiplied by $\left\|v\right\|$, yields the experimental value of $q(k)=T\int_{0}^{1}\langle v(k,\tau) \rangle d\tau$, the pumped charge contributed from a certain $k$. This procedure is repeated for different values of $k \in [0,\pi]$. Some experimental data with $T=1$ $\rm\mu$s and $\phi(\tau)=2 \pi \tau$ are instantiated in Fig. \[intermediate\]. The pattern of the normalized velocity $\langle v(k,\tau) \rangle / \left\|v\right\|$ depends strongly on $\delta_2/\delta_1$, and so does the shape of $q(k)$. In particular, there is a significant charge transport for $\delta_2/\delta_1\approx 1$ and $k\approx \pi$, i.e., near the band touching point.
![Normalized velocity expectation values and pumped charge per each $k$. (a),(b) Normalized value $\langle v \rangle / \left\|v\right\|$ vs $k$ and $\tau$ for $\delta_2/\delta_1=0$ and $1$, respectively. Experimental data (calculations based on the Schrödinger equation) are on the right (left). The red curves in the experimental figures are guides to the eye to clarify the patterns in the color map. These guidelines are the crest lines in the patterns of the calculated $\langle v \rangle / \left\|v\right\|$. (c) Pumped charge $q(k)$ per each synthetic quasimomentum $k$ for several values of $\delta_2/\delta_1$. Symbols (curves) represent the experimental data (the calculation). []{data-label="intermediate"}](fig3){width="0.95\columnwidth"}
![Transported charge $Q$ and band gap vs $\delta_2/\delta_1$. (a) The symbols and curves represent experimental data and theoretical results, respectively. The orange symbols and curve are for the linear ramp $\phi(\tau)=2 \pi \tau$ with $T=1$ $\rm\mu$s. The green symbols and curve are for $\phi(\tau)=2 \pi \tau$ with $T=0.5$ $\rm\mu$s. The grey theoretical curve corresponds to $\phi(\tau)=2 \pi \tau$ with $T\to\infty$. The blue symbols and curve correspond to the parabolic ramp $\phi(\tau)=2 \pi \tau^2$ with $T=1$ $\rm\mu$s. Error bars represent $\pm1$ s.d. (b) In the two-band model, the band gap is $|\delta_2-\delta_1|$. []{data-label="final"}](fig4){width="1\columnwidth"}
Because of symmetry considerations, it suffices to let our measurements cover half of the first Brillouin zone to extract the pumped charge $Q=\int_{-\pi}^{\pi}q(k)dk/(2\pi)=\int_{0}^{\pi}q(k)dk/\pi$ [@SM]. As illustrated by the orange curve and data points in Fig. \[final\](a), the pumped charge $Q$ first rises and then declines as the parameter $\delta_2/\delta_1$ sweeps from $0$ to $2$. The parameter $\delta_2/\delta_1$ also determines the band gap as sketched in Fig. \[final\](b). Though the ramp time $T=1~\rm\mu$s is still not in the true adiabatic limit $T\to\infty$, the pumped charge $Q$ for $T=1~\rm\mu$s as a function of $\delta_2/\delta_1$ bears strong resemblance with the theoretical curve for $T\rightarrow \infty$ obtained using the first-order APT, with their differences well accounted for. In particular, the theoretical logarithmic divergence of $Q$ as $\delta_2/\delta_1\rightarrow 1$ \[see the gray curve in Fig. \[final\](a)\] implicitly requires $T\rightarrow \infty$ as the condition to apply the first-order APT. The actual observed pumping for a finite $T=1$ $\rm\mu$s is thus not expected to shoot to infinity. In addition, the peak of $Q$ is not precisely at $\delta_2/\delta_1=1$, but has a rightward shift. In this linear ramp case, a non-perturbative theory can be developed [@SM]. The theoretical shift of the peak of $Q$ as a function of $\delta_2/\delta_1$ is found to be $2\pi/(\delta_1T)$, in good agreement with our observation. This clearly indicates that the observed peak shift is merely a finite-$T$ effect. For a shorter ramp time $T=0.5$ $\rm\mu$s as depicted by the green curve and data points in Fig. \[final\](a), the pumping peak slightly goes lower again and shifts further away from the exact phase transition point $\delta_2/\delta_1=1$. Overall, the two pumping curves with $T=1$ $\rm\mu$s and $T=0.5$ $\rm\mu$s have a remarkable overlap with each other, thus supporting that to the zeroth order of $1/T$, the outcome of the generalized Thouless pump is independent of $T$. We next investigate another pumping protocol $\phi(\tau)=2 \pi \tau^2$ with $T=1$ $\rm\mu$s. The initial switching-on rate of this pumping protocol now vanishes. In this case, we observe negligible pumping, as evidenced by the blue curve and data points in Fig. \[final\](a). The results for the two different protocols confirm that the generalized Thouless pump can be extensively tuned by varying the switching-on rate of a pumping protocol. Finally, one may note the differences between experimental results and the simulation results \[orange, green, and blue solid curves in Fig. \[final\](a)\] based solely on time-dependent Schrödinger equations. The experimental errors are mainly due to the imperfection of the microwave pulses. Nevertheless, in the presence of the experimental errors, our experimental results have demonstrated all principal features of the generalized Thouless pump. In conclusion, by incorporating interband coherence into the initial state as a powerful quantum resource, we are able to go beyond the traditional Thouless pump. Using a single spin in diamond, we have experimentally demonstrated a novel type of quantum adiabatic pump, which is extensively and continuously tunable by varying the switching-on rate of a pumping protocol. The tunability of our generalized Thouless pump is reminiscent of the famous Archimedes screw, where water is pumped via rotating a screw-shaped blade in a cylinder and the amount of pumped water can also be changed continuously [@Archimedes1; @Archimedes2; @Archimedes3]. Furthermore, because the coherence-based pumping in our system is most pronounced around a band-touching point, it may provide an alternative means for the detection of band touching and hence quantum or topological phase transition points. Our work thus enriches the physics of adiabatic pump and coherence-based quantum control.
The authors at University of Science and Technology of China are supported by the National Natural Science Foundation of China (Grants No. 81788101, No. 11227901, No. 31470835, No. 91636217, and No. 11722544), the CAS (Grants No. GJJSTD20170001, No. QYZDY-SSW-SLH004, No. QYZDB-SSW-SLH005, and No. YIPA2015370), the 973 Program (Grants No. 2013CB921800, No. 2016YFA0502400, and No. 2016YFB0501603), the CEBioM, and the Fundamental Research Funds for the Central Universities (WK2340000064). The authors at National University of Singapore are supported by the Singapore NRF Grant No. NRF-NRFI2017-04 (WBS No. R-144-000-378-281) and by the Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112).
W. M., L. Z., and Q. Z. contributed equally to this work.
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**Supplementary Material**
1. Theory
=========
1.1 Adiabatic charge pumping with nonequilibrium initial states
---------------------------------------------------------------
In this supplementary note, we derive Eqs. (2) and (3) in the main text, which describe the particle pumping over an adiabatic cycle in the generalized Thouless pump for initial states with interband coherence. Throughout this note, we take $\hbar=1$.
We start with the time-dependent Schrödinger equation $$i\frac{d}{d\tau}|\Psi(k,\tau)\rangle=TH(k,\tau)|\Psi(k,\tau)\rangle,\label{eq:Seq}$$ where $\tau=t/T\in[0,1]$ is the scaled time with $t$ being the real time and $T$ the duration of the evolution, $k$ represents some other time-independent parameters of the system, $|\Psi(k,\tau)\rangle$ represents the state of the system at the scaled time $\tau$, and $H(k,\tau)$ is the system’s Hamiltonian which depends on time through some parameter such as $\phi(\tau)$. In this study, we consider the class of quantum systems whose Hamiltonian $H(k,\tau)$ admits a discrete instantaneous energy spectrum $\{E_{n}(k,\tau)\}$ with eigenstates $\{|n(k,\tau)\rangle\}$, such that $$H(k,\tau)|n(k,\tau)\rangle=E_{n}(k,\tau)|n(k,\tau)\rangle,$$ with $n$ being the energy level index. At the start of the evolution ($\tau=0$), the initial state of the system $|\Psi(k,0)\rangle$ can be written in the basis $\{|n(k,0)\rangle\}$ as $$|\Psi(k,0)\rangle=\sum_{n}c_{n}(k,0)|n(k,0)\rangle,
\label{eq:Ini-State}$$ with the amplitude $c_{n}(k,0)=\langle n(k,0)|\Psi(k,0)\rangle$. At a later time $\tau$, the state of the system can be written in the basis $\{|n(k,\tau)\rangle\}$ as $$|\Psi(k,\tau)\rangle=\sum_{n}e^{-i\Theta_{n}(k,\tau)}c_{n}(k,\tau)|n(k,\tau)\rangle,\label{eq:State}$$ where $\Theta_{n}(k,\tau)=T\int_{0}^{\tau}E_{n}(k,\tau')d\tau'$ is the dynamical phase. The Schrödinger equation in Eq. (\[eq:Seq\]) is solved if all $\{c_{n}(k,\tau)\}$ are found at each $\tau$.
In quasiadiabatic evolutions, $\phi$ varies slowly in time, so that $\frac{d\phi}{dt}=\frac{1}{T}\frac{d\phi(\tau)}{d\tau}$ is much smaller than any energy gap of the instantaneous Hamiltonian $H(k,\tau)$. In this case, $c_{n}(k,\tau)$ can be expressed as a series of $1/T$ through adiabatic perturbation theory [@APT1]. Keeping terms up to ${\cal O}(1/T)$, we get $$c_{n}(k,\tau)=c_{n}(k,0)+\frac{1}{T}\sum_{m\neq n}c_{m}(k,0)\left.\left[\frac{i\langle n(k,\tau')|\partial_{\tau}m(k,\tau')\rangle}{E_{n}(k,\tau')-E_{m}(k,\tau')}e^{i\Theta_{nm}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau},\label{eq:Amp}$$ where $\Theta_{nm}(k,\tau)\equiv\Theta_{n}(k,\tau)-\Theta_{m}(k,\tau)$ is the dynamical phase difference [@APT2]. The above equation can also be expressed by the element of the density matrix, namely, $$\label{element}
\begin{aligned}
\rho_{mn}(k,\tau)=\rho_{mn}(k,0)+&\frac{1}{T}\sum_{\ell\neq n}\rho_{m\ell}(k,0)\left.\left[\frac{i\langle \ell(k,\tau')|\partial_{\tau}n(k,\tau')\rangle}{E_{n}(k,\tau')-E_{\ell}(k,\tau')}e^{i\Theta_{\ell n}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau}\\
+&\frac{1}{T}\sum_{\ell\neq m}\rho_{\ell n}(k,0)\left.\left[\frac{i\langle m(k,\tau')|\partial_{\tau}\ell(k,\tau')\rangle}{E_{m}(k,\tau')-E_{\ell}(k,\tau')}e^{i\Theta_{m\ell}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau},
\end{aligned}$$ Note that in writing down the expressions in Eqs. (\[eq:Amp\]) and (\[element\]), we have taken the parallel transport gauge convention. This means to choose the phase for the basis state $|n(k,\tau)\rangle$, in order to make $\langle n(k,\tau)|\partial_{\tau}|n(k,\tau)\rangle=0$ for all $n$ at any $\tau\in(0,1)$.
In Thouless’ setup of adiabatic charge transport [@ThouPump1], $H(k,\tau)$ describes noninteracting electrons in a one-dimensional lattice modulated by a slowly varying time-dependent potential, which is periodic in both space and time. The particle transported across the system over an adiabatic driving cycle (i.e., $\tau:0\rightarrow1$ here) is given by $$Q=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\int_{0}^{1}d\tau T\langle v(k,\tau) \rangle,\label{eq:Q}
$$ where $k\in(-\pi,\pi]$ is the quasimomentum (with lattice constant $a=1$), $v(k,\tau)\equiv\partial_{k}H(k,\tau)$ represents the group velocity operator, and $\langle v(k,\tau) \rangle\equiv{\rm{tr}}[\rho(k,\tau) v(k,\tau)]$ represents the expectation value of $v(k,\tau)$.
In the following, we give the detailed derivation of the charge pumping $Q$ discussed in the main text. Let’s first introduce a set of compact notations as $$\begin{aligned}
D_{mn}(k,\tau) & \equiv E_{m}(k,\tau)-E_{n}(k,\tau),\\
v_{mn}(k,\tau) & \equiv\langle m(k,\tau)|v(k,\tau)|n(k,\tau)\rangle,\\
M_{mn}(k,\tau) & \equiv i\langle m(k,\tau)|\partial_{\tau}|n(k,\tau)\rangle.\end{aligned}$$ In terms of these notations, we can organize the density matrix components in Eq. (\[element\]) as $$\begin{aligned}
{1}
\varrho_{{\rm I}} & \equiv\rho_{mn}(k,0),\label{eq:rhoI}\\
\varrho_{{\rm II}} & \equiv\frac{1}{T}\sum_{\ell\neq n}\rho_{m\ell}(k,0)\left.\left[\frac{M_{\ell n}(k,\tau')}{D_{n\ell}(k,\tau')}e^{i\Theta_{\ell n}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau},\label{eq:rhoII}\\
\varrho_{{\rm III}} & \equiv\frac{1}{T}\sum_{\ell\neq m}\rho_{\ell n}(k,0)\left.\left[\frac{M_{m\ell}(k,\tau')}{D_{m\ell}(k,\tau')}e^{i\Theta_{m\ell}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau}.\label{eq:rhoIII}\end{aligned}$$ Correspondingly, we will also decompose the charge pumping $Q$ into three parts as $$Q = Q_{{\rm NG}}+Q_{{\rm II}}+Q_{{\rm III}}.$$ Explicit expressions for these components will be derived in the following subsections.
### 1.1.1 Derivation of $Q_{{\rm NG}}$
The contribution of $\varrho_{{\rm I}}$ to $Q$ is denoted by $Q_{{\rm NG}}$. From Eqs. (\[eq:Q\]) and (\[eq:rhoI\]), we find $$Q_{{\rm NG}} = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n}\rho_{nm}(k,0)\int_{0}^{1} T d\tau v_{mn}(k,\tau)e^{i\Theta_{mn}(k,\tau)}.\label{eq:QTS1}$$ In the case of $m=n$, the integral becomes $\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{n}\rho_{nn}(k,0)\int_{0}^{1} T d\tau v_{nn}(k,\tau)$. If there is symmetry breaking in $k$-space (e.g., a population imbalance with respect to $k$), this term could make a contribution to the transport of order $T$, which may become very large in the adiabatic limit ($T\rightarrow\infty$). But this contribution is not due to pumping. To remove this irrelevant term, we will assume $$\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{n}\rho_{nn}(k,0)\int_{0}^{1} T d\tau v_{nn}(k,\tau)=0.\label{eq:Assumption1}$$ Practically this can be achieved if, e.g., $v_{nn}(k,\tau)$ and $\rho_{nn}(k,0)$ have opposite parities as functions of $k$. In our experiment, we studied a two-band system and choose the initial state to equally populate the two bands, i.e., $\rho_{11}(k,0)=\rho_{22}(k,0)$ for all $k$. Since the group velocities of the two bands satisfy $v_{11}(k,\tau)=-v_{22}(k,\tau)$, we will always have $\sum_{n=1}^{2}\rho_{nn}(k,0)v_{nn}(k,\tau)=0$ in our experimental situation, and therefore the assumption (\[eq:Assumption1\]) is always justified.
Under the assumption (\[eq:Assumption1\]), Eq. (\[eq:QTS1\]) simplifies to $$Q_{{\rm NG}} = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nm}(k,0)\int_{0}^{1} T d\tau v_{mn}(k,\tau)e^{i\Theta_{mn}(k,\tau)}.$$ Performing an integration by parts over the dynamical phase exponent $e^{i\Theta_{mn}(k,\tau)}$, we find $$\begin{aligned}
{1}
Q_{{\rm NG}} & = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nm}(k,0)\int_{0}^{1}\frac{v_{mn}(k,\tau)}{i D_{mn}(k,\tau)}de^{i\Theta_{mn}(k,\tau)}\nonumber \\
& =\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nm}(k,0)\left.\left[\frac{v_{mn}(k,\tau)}{iD_{mn}(k,\tau)}e^{i\Theta_{mn}(k,\tau)}\right]\right|_{\tau=0}^{\tau=1}+{\cal O}\left(\frac{1}{T}\right).\label{eq:QTS2}\end{aligned}$$ Since in the adiabatic limit ($T\rightarrow\infty$), the phase factor $\Theta_{mn}(k,\tau)$ is oscillating fast with respect to $k$, its average over $k$ will tend to vanish. This may also be seen by performing another integration by parts over $e^{i\Theta_{mn}(k,\tau)}$, which will generate a term $1/\partial_{k}\Theta_{mn}(k,\tau)\propto1/T$. So in the adiabatic limit ($T\rightarrow\infty$), Eq. (\[eq:QTS2\]) further reduces to $$Q_{{\rm NG}} = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m<n}\frac{2{\rm Im}\left[\rho_{mn}(k,0)v_{nm}(k,0)\right]}{E_{m}(k,0)-E_{n}(k,0)}.\label{eq:Qts}$$ This term is highly non-generic. It contains the memory of the state to the initial condition and is time-independent (since it is only evaluated at $\tau=0$). Furthermore, $Q_{{\rm NG}}$ will not accumulate with the increasing of the number of pumping cycles, and thus of secondary importance in long time dynamics. In our experiment, the initial states we prepared satisfy ${\rm Im}\left[\rho_{mn}(k,0)v_{nm}(k,0)\right]=0$, and therefore make $Q_{{\rm NG}}$ vanish as mentioned in the main text.
### 1.1.2 Derivation of $Q_{{\rm TP}}$ and $Q_{{\rm IBC}}$
Plugging Eq. (\[eq:rhoII\]) into Eq. (\[eq:Q\]), we find $$Q_{{\rm II}} = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n}\sum_{\ell\neq m}\rho_{n\ell}(k,0)\int_{0}^{1}d\tau v_{mn}(k,\tau)\left.\left[\frac{M_{\ell m}(k,\tau')}{D_{m\ell}(k,\tau')}e^{i\Theta_{\ell m}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau}e^{i\Theta_{mn}(k,\tau)}.\label{eq:QII1}$$ When $m=n$, the factor $\frac{M_{\ell m}(k,\tau')}{D_{m\ell}(k,\tau')}e^{i\Theta_{\ell m}(k,\tau')}$ has contribution to $Q_{{\rm II}}$ in the adiabatic limit only at $\tau'=0$, where $e^{i\Theta_{\ell m}(k,\tau')}=1$. When $m\neq n$, the factor $\frac{M_{\ell m}(k,\tau')}{D_{m\ell}(k,\tau')}e^{i\Theta_{\ell m}(k,\tau')}$ has contribution to $Q_{{\rm II}}$ in the adiabatic limit only at $\tau'=\tau$ with $\ell=n$. These can be derived by performing integration by parts over the dynamical phase exponent $e^{i\Theta_{\ell m}(k,\tau)}$, and arguments parallel to what we have in used the last subsection to obtain Eq. (\[eq:QTS2\]). Taking the adiabatic limit ($T\rightarrow\infty$) and collecting all non-vanishing terms of Eq. (\[eq:QII1\]), we obtain $$\begin{aligned}
{1}
Q_{{\rm II}} & =\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nm}(k,0)\left.\left[\frac{M_{mn}(k,\tau)}{D_{mn}(k,\tau)}\right]\right|_{\tau=0}\int_{0}^{1}d\tau v_{nn}(k,\tau)\nonumber \\
& +\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nn}(k,0)\int_{0}^{1}dsv_{mn}(k,\tau)\frac{M_{nm}(k,\tau)}{D_{mn}(k,\tau)}.\label{eq:QII2}\end{aligned}$$ Similarly, plugging Eq. (\[eq:rhoIII\]) into Eq. (\[eq:Q\]) yields $$Q_{{\rm III}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\int_{0}^{1}d\tau\sum_{m,n}\sum_{\ell\neq n}v_{mn}(k,\tau)\rho_{\ell m}(k,0)\left.\left[\frac{M_{n\ell}(k,\tau')}{D_{n\ell}(k,\tau')}e^{i\Theta_{n\ell}(k,\tau')}\right]\right|_{\tau'=0}^{\tau'=\tau}e^{i\Theta_{mn}(k,\tau)}.\label{eq:QIII1}$$ Parallel to the previous analysis, this expression reduces in the adiabatic limit to $$\begin{aligned}
{1}
Q_{{\rm III}} & =\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{mn}(k,0)\left.\left[\frac{M_{nm}(k,\tau)}{D_{mn}(k,\tau)}\right]\right|_{\tau=0}\int_{0}^{1}d\tau v_{nn}(k,\tau)\nonumber \\
& +\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m\neq n}\rho_{nn}(k,0)\int_{0}^{1}d\tau v_{nm}(k,\tau)\frac{M_{mn}(k,\tau)}{D_{mn}(k,\tau)}.\label{eq:QIII2}\end{aligned}$$ It is not hard to see that $Q_{{\rm III}}=Q_{{\rm II}}^{*}$. Thus we can collect them together and recombine relevant terms to obtain $Q_{{\rm II}}+Q_{{\rm III}}=Q_{{\rm TP}}+Q_{{\rm IBC}}$.
To summarize, the total charge pumping $Q$ can be expressed as a summation of three components as discussed in the main text: $$Q = Q_{{\rm NG}}+Q_{{\rm II}}+Q_{{\rm III}}
= Q_{{\rm NG}}+Q_{{\rm TP}}+Q_{{\rm IBC}}.\label{eq:PumpQ}$$ On the right hand side of Eq. (\[eq:PumpQ\]), the term $Q_{{\rm NG}}$ has been discussed in the last section. The $Q_{{\rm TP}}$ has the following expression: $$Q_{{\rm TP}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{n}\rho_{nn}(k,0)\int_{0}^{1}d\tau\Omega_{\tau k}^{(n)},\label{eq:Qeq}$$ where $\left.\rho_{nn}(k,\tau)\right|_{\tau=0}=|c_{n}(k,0)|^{2}$ is the initial population at the quasimomentum $k$ on the Bloch band $n$, and $\Omega_{\tau k}^{(n)}=i\langle\partial_{\tau}n(k,\tau)|\partial_{k}n(k,\tau)\rangle+{\rm c.c.}$ represents the Berry curvature. Therefore, $Q_{{\rm TP}}$ is given by an integral of the Berry curvature weighted by initial Bloch band populations. It has been found in Thouless’ original work [@ThouPump1], but has nothing to do with interband coherence in the initial state. The third term, $Q_{{\rm IBC}}$, is the focus of our experimental study. It is given by $$Q_{{\rm IBC}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\sum_{m,n,m<n}\left.\frac{2{\rm Im}\left[\rho_{mn}(k,\tau)\langle n(k,\tau)|\partial_{\tau}|m(k,\tau)\rangle\right]}{E_{m}(k,\tau)-E_{n}(k,\tau)}\right|_{\tau=0}\int_{0}^{1}d\tau[v_{mm}(k,\tau)-v_{nn}(k,\tau)].\label{eq:Qibc}$$ Through its dependence on $\rho_{mn}(k,\tau)$ at $\tau=0$ for $m\neq n$, we see that $Q_{{\rm IBC}}$ is originated from interband coherence in the initial state. As can be seen from Eq. (\[eq:Amp\]), such an initial-state coherence could induce a correction to interband population transfer of the order of $1/T$. The accumulation of this nonadiabatic effect over a long time duration $T$ finally makes $Q_{{\rm IBC}}$ an important component of the total charge pumping. Moreover, $Q_{{\rm IBC}}$ depends on the term $\langle n|\partial_{\tau}|m\rangle=\frac{d\phi}{d\tau}\langle n|\partial_{\phi}|m\rangle$ evaluated at $\tau=0$, and is therefore sensitive to the switching-on behavior of a pumping protocol. In our experiment, we considered two different adiabatic protocols. The first protocol, $\phi(\tau)=2\pi\tau$, is linear in $\tau$ with a constant rate $\frac{d\phi}{d\tau}=2\pi$. The second protocol, $\phi(\tau)=2\pi\tau^{2}$, is quadratic in $\tau$. It has a rate $\frac{d\phi}{d\tau}=4\pi\tau$, vanishing at $\tau=0$. So for the second protocol, one has $Q_{{\rm IBC}}=0$ within the first-order adiabatic perturbation theory.
### 1.1.3 Pumping over $N$ adiabatic cycles
If the pump is operated over $N$ adiabatic cycles, the non-generic part of charge pumping is still given by Eq. (\[eq:Qts\]) since its right-hand side is independent of $\tau$. Due to the periodicity of $\Omega^{(n)}_{\tau k}$ in $\tau$, i.e., $\Omega^{(n)}_{\tau+1 k}=\Omega^{(n)}_{\tau k}$, we have $\int_{0}^{N}d\tau\Omega_{\tau k}^{(n)}=N\int_{0}^{1}d\tau\Omega_{\tau k}^{(n)}$ and therefore the relevant pumping component over $N$ adiabatic cycle is $N Q_{{\rm TP}}$, where $Q_{{\rm TP}}$ is the component over one adiabatic cycle. Similarly, the velocity operator $v(k,\tau)$ is also a periodic function of $\tau$ with period $1$, thus $\int_{0}^{N}d\tau[v_{mm}(k,\tau)-v_{nn}(k,\tau)]=N\int_{0}^{1}d\tau[v_{mm}(k,\tau)-v_{nn}(k,\tau)]$ and therefore the relevant pumping component over $N$ adiabatic cycle is $N Q_{{\rm IBC}}$, where $Q_{{\rm IBC}}$ is the component over one adiabatic cycle. Collecting all these together, we conclude that the charge pumping over $N$ adiabatic cycles is $N(Q_{{\rm TP}}+Q_{{\rm IBC}})+Q_{{\rm NG}}$ as discussed in the main text. Here $N$ is a positive integer.
1.2 Model for the experiment
----------------------------
In our experiment, we map a one-dimensional two-band insulator model onto a single qubit subject to a time-dependent driving field. Explicitly, the qubit Hamiltonian is given by $$H(k,\tau)=\frac{\omega\sin (k)}{2}\big\{\cos[\phi(\tau)]~\sigma_{x}+\sin[\phi(\tau)]~\sigma_{y}\big\} + \frac{\delta_1\cos(k)+\delta_2}{2}~\sigma_{z},
\label{eq:Hqubit}$$ Its corresponding lattice Hamiltonian is $$H(\tau ) = \sum\limits_k {H(k,\tau )\left| k \right\rangle \left\langle k \right|},$$ where $k$ represents the quasimomentum. In the position representation, the Hamiltonian is given by $$H(\tau)=\frac{1}{2}\sum_{j}\left\{\left| j \right\rangle\frac{\delta_{1}\sigma_{z}-i\omega\cos[\phi(\tau)]\sigma_{x}-i\omega\sin[\phi(\tau)]\sigma_{y}}{2}\left\langle j+1 \right|+{\rm h.c.}\right\}+\frac{1}{2}\sum_{j}\left| j \right\rangle\left(\delta_{2}\sigma_{z}\right)\left\langle j \right|,$$ where $\left| j \right\rangle$ represents the lattice site basis, $\delta_{1}$ represents an energy bias in the hopping of spin up and down particles, and $\delta_{2}$ represents an energy bias between spin up and down particles in the same unit cell. The driving field modulates the spin orientation of the particle on $xy$ plane during its hopping between nearest neighbor sites.
In the experiment, we conduct the charge pumping along the synthetic dimension $k$ on a qubit with the Hamiltonian $H(k,\tau)$. To single out the contribution of $Q_{{\rm IBC}}$ from the total charge pumping $Q$ in an adiabatic cycle, the initial state (at $\tau=0$) is chosen to have equal populations on the two levels of $H(k,0)$ at each $k$. From Eq. (\[eq:Qeq\]), we observe that $Q_{{\rm TP}}=0$ for such an initial state, since the Berry curvature satisfies $\sum_{n}\Omega_{\tau k}^{(n)}=0$ at each point $(k,\tau)$ in the parameter space. Actually, in this specific model, one has $Q_{{\rm TP}}=0$ even if the initial populations on the two bands are not equal because the first Chern numbers vanish, i.e., the integral of Berry curvature $\frac{1}{2\pi}\int_{-\pi}^{\pi}dk\int_{0}^{1}d\tau\Omega_{\tau k}^{(n)}$ is zero. Despite this particularity of our two-band model, preparing the initial state with equal populations on all bands at each $k$ is an effective method to eliminate $Q_{{\rm TP}}$. It should also be noted that $Q_{{\rm IBC}}$ given by Eq. (\[eq:Qibc\]) is a natural consequence of the initial-state interband coherence and is not restricted to the model adopted in this work. The eigenenergies and eigenstates of $H(k,\tau)$ are $$E_{\pm}(k)=\pm\frac{1}{2}\sqrt{\nu^{2}+\delta^{2}},\qquad
|+(k,\tau)\rangle=\frac{1}{\sqrt{2}}\begin{bmatrix}\sqrt{1+\frac{\delta}{\Delta}}\\
{\rm sgn}(\frac{\nu}{\Delta})e^{i\phi}\sqrt{1-\frac{\delta}{\Delta}}
\end{bmatrix},\qquad|-(k,\tau)\rangle=\frac{1}{\sqrt{2}}\begin{bmatrix}{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}\\
-e^{i\phi}\sqrt{1+\frac{\delta}{\Delta}}
\end{bmatrix},
\label{eigen1}$$ with $$\Delta\equiv E_{+}(k)-E_{-}(k)=\sqrt{\nu^{2}+\delta^{2}},\qquad\nu\equiv\omega\sin(k),\qquad\delta\equiv\delta_{1}\cos(k)+\delta_{2}.\label{eq:E-nu-del}$$ Here $\Delta$ represents the level spacing, $\nu$ is the magnitude of the transverse field, and $\delta$ is the magnitude of the longitudinal field. The ${\rm sgn}$ function in Eq. (\[eigen1\]) reflects a gauge choice. Note also that in our model $E_{\pm}(k)$ and $\Delta$ are independent of time. Consider next the following initial state for each $k$, $$|\Psi(k,0)\rangle=\frac{1}{\sqrt{2}}[|+(k,0)\rangle+|-(k,0)\rangle]=\frac{1}{2}\begin{bmatrix}\sqrt{1+\frac{\delta}{\Delta}}+{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}\\
{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}-\sqrt{1+\frac{\delta}{\Delta}}
\end{bmatrix},\label{eq:StateIni}$$ and the corresponding density matrix is $$\label{MatrixIni}
\rho(k,0) = \frac{\mathbbm{1}+ {\bm{n}} \cdot {\bm{\sigma }}}{2},~~~~~~\bm{n} = \frac{{\left( { - {\delta _1}\cos k - {\delta _2},0,\omega \sin k } \right)}}{{\sqrt {{{(\omega \sin k )}^2} + {{({\delta _1}\cos k + {\delta _2})}^2}} }}.$$ In the singular case where $\delta_1=\delta_2$ and $k=\pi$, the eigenstates in Eq. (\[eigen1\]) is redefined as $$\begin{aligned}
&{\left. {|+(k,\tau)\rangle} \right|_{{\delta _1} = {\delta _2}, k = \pi}}
= \mathop {\lim }\limits_{k \to \pi^- } \Bigg\{ \mathop {\lim }\limits_{\delta _1 \to \delta _2 }\frac{1}{\sqrt{2}}\begin{bmatrix}\sqrt{1+\frac{\delta}{\Delta}}\\
{\rm sgn}(\frac{\nu}{\Delta})e^{i\phi}\sqrt{1-\frac{\delta}{\Delta}}
\end{bmatrix}\Bigg\} = \frac{1}{\sqrt{2}}\left( {\begin{array}{*{20}{c}}
1 \\
{{e^{i\phi }}} \\
\end{array}} \right),\\
&{\left. {|-(k,\tau)\rangle} \right|_{{\delta _1} = {\delta _2}, k = \pi}}
=\mathop {\lim }\limits_{k \to \pi^- } \Bigg\{ \mathop {\lim }\limits_{\delta _1 \to \delta _2 }\frac{1}{\sqrt{2}}\begin{bmatrix}{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}\\
-e^{i\phi}\sqrt{1+\frac{\delta}{\Delta}}
\end{bmatrix}\Bigg\} = \frac{1}{\sqrt{2}}\left( {\begin{array}{*{20}{c}}
1 \\
{{-e^{i\phi }}} \\
\end{array}} \right).
\end{aligned}
\label{eigensing}$$ Likewise, the initial state in Eq. (\[eq:StateIni\]) is redefined as $${\left. {|\Psi(k,0)\rangle} \right|_{{\delta _1} = {\delta _2}, k = \pi}}
=\mathop {\lim }\limits_{k \to \pi^- } \Bigg\{ \mathop {\lim }\limits_{\delta _1 \to \delta _2 }\frac{1}{2}\begin{bmatrix}\sqrt{1+\frac{\delta}{\Delta}}+{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}\\
{\rm sgn}(\frac{\nu}{\Delta})\sqrt{1-\frac{\delta}{\Delta}}-\sqrt{1+\frac{\delta}{\Delta}}
\end{bmatrix}\Bigg\} = \left( {\begin{array}{*{20}{c}}
1 \\
0 \\
\end{array}} \right),
\label{eq:StateIniSing}$$ and the corresponding Bloch vector in Eq. (\[MatrixIni\]) is redefined as $$\label{SingularVector}
{\left. {\bm{n}} \right|_{{\delta _1} = {\delta _2}, k = \pi}} = \mathop {\lim }\limits_{k \to \pi^- } \Bigg[ \mathop {\lim }\limits_{\delta _1 \to \delta _2 } {\frac{{\left( { - {\delta _1}\cos k - {\delta _2},0,\omega \sin k} \right)}}{{\sqrt {{{(\omega \sin k)}^2} + {{({\delta _1}\cos k + {\delta _2})}^2}} }}} \Bigg] = (0,0,1).$$ Plugging the initial state in Eq. (\[MatrixIni\]) into Eq. (\[eq:Qibc\]), with the help of Eqs. (\[eq:Hqubit\]) and (\[eq:E-nu-del\]), we find after some algebra that $$Q_{{\rm IBC}}=-\frac{1}{2\pi}\int_{0}^{\pi}dk\frac{(\omega^{2}-\delta_{1}^{2})\cos(k)-\delta_{1}\delta_{2}}{\left\{ \omega^{2}\sin^{2}(k)+\left[\delta_{1}\cos(k)+\delta_{2}\right]^{2}\right\} ^{3/2}}\omega\sin^{2}(k)\dot{\phi}|_{\tau=0}.
\label{eq:Qibc-2-level-generalramp}$$ This expression for $Q_{{\rm IBC}}$, now involving only the $k$ integral from $0$ to $\pi$, motivates us to restrict $k$ to the regime of $[0,\pi]$ in our actual experiment.
Around the band touching point ($\delta_1=\delta_2, k=\pi$), the numerator of the integrand in Eq. (\[eq:Qibc-2-level-generalramp\]) approaches zero as $|k-\pi|^2$, whereas the denominator of the same integrand approaches zero as $|k-\pi|^3$. Therefore, the integrand itself approaches zero as $|k-\pi|^{-1}$ and its integration over $k$ hence yields a logarithmic divergence around $k=\pi$. This explains the origin of the divergence in the theoretical pumping curve shown in Fig. \[final\](a) of the main text.
For the specific linear ramp case, $\phi=2\pi\tau$, the above equation becomes $$Q_{{\rm IBC}}=-\int_{0}^{\pi}dk\frac{(\omega^{2}-\delta_{1}^{2})\cos(k)-\delta_{1}\delta_{2}}{\left\{ \omega^{2}\sin^{2}(k)+\left[\delta_{1}\cos(k)+\delta_{2}\right]^{2}\right\} ^{3/2}}\omega\sin^{2}(k).
\label{eq:Qibc-2-level}$$ In Fig. \[final\](a) of the main text, the gray theoretical curve (for $T\rightarrow\infty$) is obtained by evaluating this term with $\omega=2\delta_{1}$ and $\delta_{2}\in[0,2\delta_{1}]$.
Additionally, since both $\rho_{-+}(k,\tau)$ and $v_{+-}(k,\tau)$ are real at $\tau=0$, the term $Q_{\rm{NG}}$ vanishes according to Eq. (\[eq:Qts\]). To summarize, for our initial state $|\Psi(k,0)\rangle$, the charge pumping over an adiabatic cycle is solely given by $Q_{{\rm IBC}}$, the contribution due to interband coherence in the initial state.
1.3 Reflection symmetry of the velocity expectation value
---------------------------------------------------------
For the specific initial state in Eq. (\[MatrixIni\]), the velocity expectation value has reflection symmetry in $k$, i.e., $\langle v(k,\tau) \rangle = \langle v(-k,\tau) \rangle$. The proof is as follows.
We turn to the rotating frame that rotates around the $z$ axis according to $\phi(\tau)$, or in other words, we apply the rotating transformation characterized by the rotation operator $R=e^{-i\phi(\tau)\sigma_z/2}$. The Hamiltonian, the evolution operator, the density operator, and the velocity operator in this rotating frame are $$\label{rotate}
\begin{aligned}
&\widetilde{H}(k,\tau) = {R^\dag }H(k,\tau)R - \frac{i}{T}{R^\dag }\frac{d}{{d\tau}}R = \nu \frac{{{\sigma _x}}}{2} + \left( {\delta - \frac{1}{T}\frac{{d\phi }}{{d\tau}}} \right)\frac{{{\sigma _z}}}{2},\\
&\widetilde{U}(k,\tau)=\mathcal{T} {{\exp \left[ - i\int_0^\tau\widetilde{H}(k,\tau)d\tau'\right]}}, \\
&\widetilde{\rho}(k,\tau) = {R^\dag }\rho(k,\tau) R = \widetilde{U}(k,\tau)\widetilde\rho(k,0)\widetilde{U}(k,\tau)^\dag,\\
&\widetilde{v}(k,\tau) = {R^\dag }v(k,\tau)R = \omega \cos k \frac{{{\sigma _x}}}{2} - {\delta _1}\sin k \frac{{{\sigma _z}}}{2}, \\
\end{aligned}$$ where $\nu\equiv\omega\sin k$ and $\delta\equiv\delta_{1}\cos k +\delta_{2}$ are the same as Eq. (\[eq:E-nu-del\]). One can observe that $$\begin{aligned}
&{\sigma _z}\widetilde{H}(k,\tau){\sigma _z} = \widetilde{H}(-k,\tau),~~~~~~{\sigma _z}\widetilde{U}(k,\tau){\sigma _z} = \widetilde{U}(-k,\tau),\\
&{\sigma _x}\widetilde{v}(k,\tau){\sigma _x} = \widetilde{v}(-k,\tau),~~~~~~{\sigma_x}\widetilde{\rho}(k,0){\sigma _x} = \widetilde{\rho}(-k,0),\\
&{\sigma _y}\widetilde{\rho}(k,0){\sigma _y} = - \widetilde{\rho}(k,0),~~~~~~{\sigma _y}\widetilde{v}(k,\tau){\sigma _y} = - \widetilde{v}(k,\tau).\\
\end{aligned}$$ The above relations entail $$\begin{aligned}
\left\langle \widetilde{v}(-k,\tau) \right\rangle &= {\rm{tr}}[\widetilde{\rho}(-k,\tau)\widetilde{v}(-k,\tau)] = {\rm{tr}}\left[ \widetilde{U}(-k,\tau)\widetilde{\rho}(-k,0)\widetilde{U}(-k,\tau)\widetilde{v}(-k,\tau) \right] \\
&= {\rm{tr}}\left[ {{\sigma _z}\widetilde{U}(k,\tau){\sigma _z}{\sigma _x}\widetilde{\rho}(k,0){\sigma _x}{\sigma _z}\widetilde{U}(k,\tau)^\dag{\sigma _z}{\sigma _x}\widetilde{v}(k,\tau){\sigma _x}} \right] \\
&= {\rm{tr}}\left[ \widetilde{U}(k,\tau){\sigma _y}\widetilde{\rho}(k,0){\sigma _y}\widetilde{U}(k,\tau)^\dag{\sigma _y}\widetilde{v}(k,\tau){\sigma _y} \right]\\
&= {\rm{tr}}\left[ \widetilde{U}(k,\tau)\widetilde{\rho}(k,0)\widetilde{U}(k,\tau)^\dag \widetilde{v}(k,\tau) \right] = {\rm{tr}}[\widetilde{\rho}(k,\tau)\widetilde{v}(k,\tau)] = \left\langle \widetilde{v}(k,\tau) \right\rangle.\\
\end{aligned}$$ By taking ${\rm{tr}}[\widetilde{\rho}(k,\tau)\widetilde{v}(k,\tau)]={\rm{tr}}[\rho(k,\tau)v(k,\tau)]$ into account, one finally obtains $\langle v(k,\tau) \rangle = \langle v(-k,\tau) \rangle$. Therefore, for the specific initial state in Eq. (\[MatrixIni\]), only half of the first Brillouin zone is enough for evaluating the pumped charge $Q$ in Eq. (\[eq:Q\]).
1.4 Detailed solution for the linear ramp case
----------------------------------------------
For the linear driving protocol $\phi=2\pi\tau$, the Hamiltonian in the rotating frame is $$\label{linearrotate}
\widetilde{H}(k) = \nu \frac{{{\sigma _x}}}{2} + \left( {\delta - \frac{2\pi}{T}} \right)\frac{{{\sigma _z}}}{2},$$ according to Eq. (\[rotate\]). Note that this Hamiltonian is time-independent. The eigenenergies and eigenstates of $\widetilde{H}(k)$ are $$\widetilde{E}_{\pm}(k)=\pm\frac{1}{2}\sqrt{\nu^{2}+\widetilde{\delta}^{2}},\qquad
|\widetilde{+}(k)\rangle=\frac{1}{\sqrt{2}}\begin{bmatrix}\sqrt{1+\frac{\widetilde{\delta}}{\widetilde{\Delta}}}\\
{\rm sgn}(\nu)\sqrt{1-\frac{\widetilde{\delta}}{\widetilde{\Delta}}}
\end{bmatrix},\qquad|\widetilde{-}(k)\rangle=\frac{1}{\sqrt{2}}\begin{bmatrix}{\rm sgn}(\nu)\sqrt{1-\frac{\widetilde{\delta}}{\widetilde{\Delta}}}\\
-\sqrt{1+\frac{\widetilde{\delta}}{\widetilde{\Delta}}}
\end{bmatrix},$$ with $$\widetilde\delta\equiv\delta-\frac{2\pi}{T},\qquad\widetilde\Delta\equiv \widetilde{E}_{+}(k)-\widetilde{E}_{-}(k)=\sqrt{\nu^{2}+\widetilde\delta^{2}}.\label{eq:rotdel}$$
The shift of peaks in the experimental observation of charge pumping may be investigated from the spectrum of $\widetilde{H}(k)$, which is gapless at $k=\pi$ if the two frequencies $\delta_{1}$ and $\delta_{2}$ satisfy the relation $\delta_{2}=\delta_{1}+2\pi/T$. So for a finite $T$, the position of peak will shift from $\delta_{2}/\delta_{1}=1$ to $\delta_{2}/\delta_{1}=1+\frac{2\pi}{\delta_{1}T}>1$. With the increasing of $T$, the peak will shift gradually to left, until being coincide with the adiabatic result at $\delta_{2}=\delta_{1}$ when $T\rightarrow\infty$. In the following we give more detailed calculations to support this argument.
Thanks to the time-independence of the Hamiltonian in Eq. (\[linearrotate\]), the evolution of a state in this rotating frame can be solved analytically. Due to ${\rm{tr}}[\widetilde{\rho}(k,\tau)\widetilde{v}(k,\tau)]={\rm{tr}}[\rho(k,\tau)v(k,\tau)]$, we can evaluate Eq. (\[eq:Q\]) in the rotating frame. After some algebra we find $$\begin{aligned}
{1}
Q & =Q_{\rm{st}}+Q_{\rm{os}},\\
Q_{\rm{st}} & \equiv\frac{1}{4\pi}\int_{-\pi}^{\pi}dkT\frac{\partial\tilde{\Delta}}{\partial k}\left[|\langle\widetilde{+}|\Psi(k,0)\rangle|^{2}-|\langle\widetilde{-}|\Psi(k,0)\rangle|^{2}\right]\label{eq:Qbardef},\\
Q_{\rm{os}} & \equiv\frac{1}{\pi}\int_{-\pi}^{\pi}dk\frac{\langle\Psi(k,0)|\widetilde{+}\rangle\langle\widetilde{+}|\frac{\partial\tilde{H}}{\partial k}|\widetilde{-}\rangle\langle\widetilde{-}|\Psi(k,0)\rangle}{\tilde{\Delta}}\sin(\tilde{\Delta}T),\label{eq:Qtiddef}\end{aligned}$$ for the initial states given by Eq. (\[eq:StateIni\]). As we will show in the following, $Q_{\rm{st}}$ is the stationary part of $Q$, and it will converge to $Q_{{\rm IBC}}$ given by Eq. (\[eq:Qibc-2-level\]) in the adiabatic limit. On the contrary, the integrand of $Q_{\rm{os}}$ is an oscillating function of $k$. When $T$ is large, the integrand of $Q_{\rm{os}}$ will oscillate fast with respect to $k$, and its contribution to $Q$ after integrating over $k$ is at least of the order of $1/T$, which will finally vanish in the adiabatic limit.
Straightforward calculations yield $$\begin{aligned}
{1}
Q_{\rm{st}} & =-\int_{0}^{\pi}dk\frac{\nu(\nu\partial_{k}\nu+\delta\partial_{k}\delta)}{\Delta\tilde{\Delta}^{2}}\nonumber \\
& =-\int_{0}^{\pi}dk\frac{\left[\left(\omega^{2}-\delta_1^2\right)\cos(k)-\delta_1\delta_{2}\right]\omega\sin^{2}(k)}{\sqrt{\omega^{2}\sin^{2}(k)+\left[\delta_1\cos(k)+\delta_{2}\right]^{2}}\left\{ \omega^{2}\sin^{2}(k)+\left[\delta_1\cos(k)+\delta_{2}-\frac{2\pi}{T}\right]^{2}\right\} }.\label{eq:Qbar}\end{aligned}$$ Comparing this with Eq. (\[eq:Qibc-2-level\]) for the charge pumping due to interband coherence, we find that in the adiabatic limit ($T\rightarrow\infty$), $Q_{\rm{st}}$ will converge to $Q_{{\rm IBC}}$, i.e., $$\lim_{T\rightarrow\infty}Q_{\rm{st}}=Q_{{\rm IBC}}.$$ In the large $T$ regime, $Q_{\rm{st}}$ will approach to $Q_{{\rm IBC}}$ algebraically, with leading correction of order $1/T$. Also we note that the integrand of $Q_{\rm{st}}$ does not diverge at $k=\pi$ for either $\delta_{2}=\delta_{1}$ or $\delta_{2}=\delta_{1}+2\pi/T$. This implies that in practice the peak is smooth and has a finite height.
Performing similar calculations, the oscillatory part $Q_{\rm{os}}$ in Eq. (\[eq:Qtiddef\]) is found to be $$Q_{\rm{os}}=-\frac{1}{2\pi}\int_{0}^{\pi}dk\frac{\omega\left[\delta_{1}+\left(\delta_{2}-\frac{2\pi}{T}\right)\cos(k)\right]\left(\tilde{\delta}\delta+\nu^{2}\right)}{\Delta\tilde{\Delta}^{3}}\sin(\tilde{\Delta}T)$$ The integrand of this integral also does not diverge at $k=\pi$ for either $\delta_{2}=\delta_{1}$ or $\delta_{2}=\delta_{1}+2\pi/T$. Furthermore, when $T$ is large, $\sin(\tilde{\Delta}T)$ is a fast oscillating function with respect to $k$. Its integral over $k$ is therefore at least of the order of $1/T$. So in the adiabatic limit we will have $\lim_{T\rightarrow\infty}Q_{\rm{os}}=0$. In the large $T$ regime, $Q_{\rm{os}}$ will decrease with the increase of $T$. Its leading contribution to $Q$ is also of the order of $1/T$.
In Fig. \[final\](a) of the main text, the orange and green curves related to the linear ramp $\phi=2\pi\tau$ are obtained by evaluating Eq. (\[eq:Q\]) based on the exact solutions of the Schrödinger equation with the initial states given by Eq. (\[eq:StateIni\]). For the blue curve related to the protocol $\phi=2\pi\tau^{2}$, the corresponding rotating-frame Hamiltonian is also time dependent. So this curve is found by first solving the Schrödinger equation numerically with the initial state (\[eq:StateIni\]) and then computing the charge pumping with Eq. (\[eq:Q\]).
2. Experiment
=============
2.1 Experimental setup
----------------------
The experiment is performed on an NV center in a {100}-face bulk diamond synthesized by chemical vapor deposition (CVD). The nitrogen impurity is less than 5 ppb and the abundance of $^{13}$C is at the natural level of about 1.1%. The dephasing time of the NV electron spin is 1.7 $\mu$s. The NV center is optically addressed by a home-built confocal microscope. Green laser is used for optical excitation. The laser beam is released and cut off by an acousto-optic modulator (power leakage ratio $\sim1/1000$). To reduce the laser leakage further, the beam passes twice through the acousto-optic modulator. The laser is focused into the diamond by an oil objective (60\*O, NA 1.42). The phonon sideband fluorescence with the wavelength between 650 and 800 nm is collected by the same oil objective and finally detected by an avalanche photodiode with a counter card. A solid immersion lens etched on the diamond by focused ion beam enhances the fluorescence counting rate up to 400 thousand counts per second. The microwaves generated by an arbitrary waveform generator (AWG) pass a 6 dB attenuator and then strengthened by a power amplifier. Finally, the microwaves are radiated to the NV center from a coplanar waveguide. The magnetic field is supplied by a permanent magnet mounted on a manual translation stage.
2.2 Calibration
---------------
The magnitude of the transverse field is $\omega\sin k$ during the evolution period. In order to feed the microwaves with proper amplitude to the NV center, we calibrate the Rabi frequency as a function of the AWG’s output amplitude. The calibration is done by performing conventional Rabi oscillation experiments with various output amplitudes. We fit the experimental data of Rabi oscillation associated with each output amplitude $V$ to extract the corresponding Rabi frequency $\omega_{\rm R}$, and then fit the Rabi frequency using $\omega_{\rm R}=a V^b$ with $a$ and $b$ being the coefficients to be determined. The relation between the Rabi frequency and the AWG’s output amplitude is thus obtained. Such calibration is carried out hourly to guard against the drift of experimental conditions.
2.3 Pulse sequence
------------------
After the qubit is polarized to the state $\left| {{\psi _0}} \right\rangle={(1,0)^{\rm{T}}}$ by a green laser pulse, a resonant microwave pulse is applied to prepare the initial state. In the laboratory frame, the Hamiltonian of the qubit irradiated by the pulse is $$\label{PrepLab}
H_{{\rm{ini}}}^{{\rm{lab}}} = \frac{\omega _0}{2}\sigma _z + \omega_1 \cos ( {\omega _0} t +\varphi_{\rm{ini}})\sigma _x.$$ where the first term on the right-hand side is the static component of the Hamiltonian with $\omega _0$ being the resonant frequency, and the second term accounts for the effect of the microwaves with $\omega_1$, $\varphi_{\rm{ini}}$, and $t$ being the Rabi frequency, the initial phase, and the time starting from zero, respectively. The value of $\varphi_{\rm{ini}}$ depends on ${\delta _1}\cos k + {\delta _2}$ as $\varphi_{\rm{ini}}=-{\pi}/{2}$ for ${\delta _1}\cos k + {\delta _2}\ge0$ and $\varphi_{\rm{ini}}={\pi}/{2}$ for ${\delta _1}\cos k + {\delta _2}<0$. In the rotating frame that rotates around the $z$ axis with the angular frequency $\omega_0$, or to put it another way, under the rotating transformation characterized by the rotation operator $R_{\rm{ini}} = {e^{ - i{\omega _0} t{\sigma _z}/2}}$, the Hamiltonian in Eq. (\[PrepLab\]) is transformed to $$\label{PrepRot}
H_{{\rm{ini}}}^{{\rm{rot}}} = {R_{\rm{ini}}^\dag }H_{{\rm{ini}}}^{{\rm{lab}}}R_{\rm{ini}} - i{R_{\rm{ini}}^\dag }\frac{d}{{dt}}R_{\rm{ini}} = \frac{\omega _1}{2} ( \sigma _x\cos \varphi_{\rm{ini}} +\sigma _y\sin \varphi_{\rm{ini}}),$$ where the second equality is based on the rotating wave approximation. The pulse lasts for $t_{\rm{ini}}=\alpha/\omega_1$, where $\alpha$ is the inclination angle of the initial state. From Eq. (\[MatrixIni\]) one can see that, in usual cases, $$\label{IniPolarAngle}
\alpha=\arccos\frac{ \omega \sin k }{{\sqrt {{{(\omega \sin k )}^2} + {{({\delta _1}\cos k + {\delta _2})}^2}} }}.$$ From Eq. (\[SingularVector\]) one can see that, in the singular case where $\delta_1=\delta_2$ and $k=\pi$, the angle $\alpha$ equals zero. Therefore, after the pulse, the state of the qubit in the rotating frame is $$\label{IniStateRot}
\left| {\psi _{{\rm{ini}}}^{{\rm{rot}}}} \right\rangle = e^{-iH_{{\rm{ini}}}^{{\rm{rot}}}t_{\rm{ini}}} \left| {{\psi _0}} \right\rangle = \left| {\Psi {\rm{(}}k,0{\rm{)}}} \right\rangle,$$ which is our desired initial state. In the laboratory frame, this state immediately after the pulse is expressed as $$\label{IniStateLab}
\left| {\psi _{{\rm{ini}}}^{{\rm{lab}}}} \right\rangle = {e^{ - i{\omega _0} t_{\rm{ini}} {\sigma _z}/2}} \left| {\psi _{{\rm{ini}}}^{{\rm{rot}}}} \right\rangle.$$
Next, a microwave pulse is applied to build the model Hamiltonian in Eq. (\[eq:Hqubit\]). In most cases, this pulse is off-resonant. In the laboratory frame, the Hamiltonian of the qubit irradiated by the pulse is $$\label{PumpLab}
H_{{\rm{pump}}}^{{\rm{lab}}} = \frac{\omega _0}{2}\sigma _z + \omega \sin k \cos \left[ {({\omega _0} - {\delta _1}\cos k - {\delta _2})t + \phi \left(\frac{t}{T}\right)} + \omega _0 t_{\rm{ini}}\right]\sigma _x,$$ where $t$ is the time starting from zero. In the rotating frame with the rotation operator $R_{\rm{pump}} = {e^{ - i[({\omega _0} - {\delta _1}\cos k - {\delta _2})t + \omega _0 t_{\rm{ini}}]{\sigma _z}/2}}$, the Hamiltonian in Eq. (\[PumpLab\]) is transformed to our target Hamiltonian, namely, $$\label{PumpRot}
H_{{\rm{pump}}}^{{\rm{rot}}} = {R_{\rm{pump}}^\dag }H_{{\rm{pump}}}^{{\rm{lab}}}R_{\rm{pump}} - i{R_{\rm{pump}}^\dag }\frac{d}{{dt}}R_{\rm{pump}} = \frac{\omega \sin k}{2} \left[ {\cos \phi \left(\frac{t}{T}\right){{\sigma _x}} + \sin \phi \left(\frac{t}{T}\right){{\sigma _y}}} \right]+\frac{{\delta _1}\cos k+{\delta _2}}{2}\sigma _z,$$ where the second equality is based on the rotating wave approximation. In this rotating frame, the state in Eq. (\[IniStateLab\]) is rewritten as $$\label{IniStateRot}
R_{\rm{pump}}^\dag(t=0)\left| {\psi _{{\rm{ini}}}^{{\rm{lab}}}} \right\rangle=\left| {\psi _{{\rm{ini}}}^{{\rm{rot}}}} \right\rangle = \left| {\Psi {\rm{(}}k,0{\rm{)}}} \right\rangle,$$ which has the same form as in the rotating frame defined by $R_{\rm{ini}}$. The pulse lasts for some duration duration $t_{\rm e}\in [0,T]$, which is a sampling point in time. Assume that the state of the qubit immediately after the pulse is $\left| {\psi _{{\rm{pump}}}^{{\rm{rot}}}} \right\rangle$ in this rotating frame. In the laboratory frame, the state is expressed as $$\label{PumpStateLab}
\left| {\psi _{{\rm{pump}}}^{{\rm{lab}}}} \right\rangle={e^{ - i[({\omega _0} - {\delta _1}\cos k - {\delta _2})t_{\rm e} + \omega _0 t_{\rm{ini}}]{\sigma _z}/2}}\left| {\psi _{{\rm{pump}}}^{{\rm{rot}}}} \right\rangle.$$
Finally, a resonant microwave pulse is applied to assist measurement. In the laboratory frame, the Hamiltonian of the qubit irradiated by the pulse is $$\label{FinLab}
H_{{\rm{fin}}}^{{\rm{lab}}} = \frac{\omega _0}{2}\sigma _z + \omega_1 \cos \left[ {{\omega _0}t + ({\omega _0} - {\delta _1}\cos k - {\delta _2})t_{\rm e} + \phi \left(\frac{t_{\rm e}}{T}\right)} + \omega _0 t_{\rm{ini}} + \varphi_{\rm{fin}}\right]\sigma _x,$$ where $\varphi_{\rm{fin}}=-\pi/2$ for $\cos k \ge 0$ and $\varphi_{\rm{fin}}=\pi/2$ for $\cos k<0$, and $t$ is also the time starting from zero. In the rotating frame with the rotation operator $R_{\rm{fin}} = {e^{ - i[{\omega _0} t+({\omega _0} - {\delta _1}\cos k - {\delta _2})t_{\rm e}+\omega _0 t_{\rm{ini}}]{\sigma _z}/2}}$, the Hamiltonian in Eq. (\[FinLab\]) is transformed to $$\label{ReadRot}
H_{{\rm{fin}}}^{{\rm{rot}}} = {R_{\rm{fin}}^\dag }H_{{\rm{fin}}}^{{\rm{lab}}}R_{\rm{fin}} - i{R_{\rm{fin}}^\dag }\frac{d}{{dt}}R_{\rm{fin}}
= \frac{\omega _1}{2} \left\{ \sigma _x\cos \left[\phi \left(\frac{t_{\rm e}}{T}\right)+\varphi_{\rm{fin}}\right] +\sigma _y\sin\left[ \phi \left(\frac{t_{\rm e}}{T}\right)+\varphi_{\rm{fin}}\right]\right\},$$ where the second equality is based on the rotating wave approximation. In this rotating frame, the state in Eq. (\[PumpStateLab\]) is rewritten as $$\label{PumpStateRot}
R_{\rm{fin}}^\dagger(t=0)\left| {\psi _{{\rm{pump}}}^{{\rm{lab}}}} \right\rangle = \left| {\psi _{{\rm{pump}}}^{{\rm{rot}}}} \right\rangle,$$ which has the same form as in the rotating frame defined by $R_{\rm{pump}}$. The pulse lasts for $t_{\rm{ini}}=\beta/\omega_1$, where $$\label{VelPolarAngle}
\beta=\arccos\frac{-\delta_1 \sin k }{{\sqrt {{{(\omega \cos k )}^2} + {{({\delta_1}\sin k})}^2}} }.$$ is the inclination angle of the direction of the velocity operator $v=\partial_k H=\omega \cos k (\cos\phi~\sigma_x+\sin\phi~\sigma_y)/2 - \delta_1\sin k~\sigma_z/2$. After the pulse, laser illumination is carried out to realize the measurement of $\sigma_z$. The combined effect of the final microwave pulse and its subsequent laser illumination amounts to the measurement of $$\label{VelMeas}
e^{iH_{{\rm{fin}}}^{{\rm{rot}}}t_{\rm{fin}}} \sigma_z e^{-iH_{{\rm{fin}}}^{{\rm{rot}}}t_{\rm{fin}}}
= \frac{v}{\left\|v\right\|}.$$
2.4 Experimental data analysis
------------------------------
As shown in Fig. \[structure\](c) of the main text, the spin state is read out during the latter laser pulse and there are two counting windows. Such sequence is iterated at least a hundred thousand times. The total photon count recorded by the first (second) window during these iterations is regarded as signal (reference) and denoted by $s$ ($r$). The raw experimental data is $x=s/r$. To normalize the data, a conventional Rabi oscillation is performed alongside. We fit the raw data of the Rabi oscillation using the function $x=x_0+a\cos(\omega_{\rm R} t+\varphi)$, and then normalize the experimental data as $x_{\rm{n}}=(x-x_0)/a$. The data thus normalized represent the expectation value of $\sigma_z$. In the experiment, the sampling interval in time $t=T\tau$ is 10 ns. For $\delta_2/\delta_1=$0, 0.5, 1.5, and 2, the sampling interval in $k$ is $\pi/18$. For other values of $\delta_2/\delta_1$ around the phase transition point $\delta_2/\delta_1=1$, the sampling interval in $k$ is $\pi/18$ when $0\le k\le8\pi/9$ and is $\pi/90$ when $8\pi/9\le k\le\pi$. The numerical integration is based on Simpson’s rule.
2.5 Experimental data
---------------------
The complete data that support the final results in Fig. \[final\](a) of the main text are as follows. \
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![Normalized velocity expectation values and pumped charge per each $k$ for $\phi(\tau)=2 \pi \tau$ with $T=1$ $\rm\mu$s. (a-i) Normalized velocity expectation values $\langle v \rangle / \left\|v\right\|$ as a function of the synthetic quasimomentum $k$ and the scaled time $\tau$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.05, 1.1, 1.15, 1.2, 1.5, and 2, respectively. The calculations based on the Schrödinger equation are on the left and the experimental data are on the right. The red curves in experimental contour maps are guides to the eye to clarify the patterns in the color map. These guidelines are the crest lines in the patterns of the calculated $\langle v \rangle / \left\|v\right\|$. The transparency of the guidelines is related to the values of $\langle v \rangle / \left\|v\right\|$ and reflects the amplitude of oscillation. (j) Pumped charge $q(k)$ contributed from each quasimomentum $k$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.05, 1.1, 1.15, 1.2, 1.5, and 2. Symbols represent the experimental data and curves represent the calculation. []{data-label="linearone"}](figs1){width="1\columnwidth"}
![Normalized velocity expectation values and pumped charge per each $k$ for $\phi(\tau)=2 \pi \tau$ with $T=0.5$ $\rm\mu$s. (a-i) Normalized velocity expectation values $\langle v \rangle / \left\|v\right\|$ as a function of the synthetic quasimomentum $k$ and the scaled time $\tau$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.1, 1.2, 1.25, 1.3, 1.5, and 2, respectively. The calculations based on the Schrödinger equation are on the left and the experimental data are on the right. The red curves in experimental contour maps are guides to the eye to clarify the patterns in the color map. These guidelines are the crest lines in the patterns of the calculated $\langle v \rangle / \left\|v\right\|$. The transparency of the guidelines is related to the values of $\langle v \rangle / \left\|v\right\|$ and reflects the amplitude of oscillation. (j) Pumped charge $q(k)$ contributed from each quasimomentum $k$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.1, 1.2, 1.25, 1.3, 1.5, and 2. Symbols represent the experimental data and curves represent the calculation. []{data-label="linearhalf"}](figs2){width="1\columnwidth"}
![Normalized velocity expectation values and pumped charge per each $k$ for $\phi(\tau)=2 \pi \tau^2$ with $T=1$ $\rm\mu$s. (a-i) Normalized velocity expectation values $\langle v \rangle / \left\|v\right\|$ as a function of the synthetic quasimomentum $k$ and the scaled time $\tau$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.05, 1.1, 1.15, 1.2, 1.5, and 2, respectively. The calculations based on the Schrödinger equation are on the left and the experimental data are on the right. The red curves in experimental contour maps are guides to the eye to clarify the patterns in the color map. These guidelines are the crest lines in the patterns of the calculated $\langle v \rangle / \left\|v\right\|$. The transparency of the guidelines is related to the values of $\langle v \rangle / \left\|v\right\|$ and reflects the amplitude of oscillation. (j) Pumped charge $q(k)$ contributed from each quasimomentum $k$ for $\delta_2/\delta_1=$0, 0.5, 1, 1.05, 1.1, 1.15, 1.2, 1.5, and 2. Symbols represent the experimental data and curves represent the calculation. []{data-label="quadratic"}](figs3){width="1\columnwidth"}
[1]{} A. Messiah, *Quantum Mechanics* (North-Holland, Amsterdam, 1962), Vol. II, p. 752.
G. Rigolin, G. Ortiz, and V. H. Ponce, Beyond the quantum adiabatic approximation: Adiabatic perturbation theory, Phys. Rev. A **78**, 052508 (2008).
D. J. Thouless, Quantization of particle transport, Phys. Rev. B **27**, 6083 (1983).
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Self-diffusion in granular gases: An impact of particles’ roughness\
Anna Bodrova, Nikolai Brilliantov\
*Department of Physics, Moscow State University, 119991 Moscow, Russia*\
*Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK*\
An impact of particles’ roughness on the self-diffusion coefficient $D$ in granular gases is investigated. For a simplified collision model where the normal, $\varepsilon$, and tangential, $\beta$, restitution coefficients are assumed to be constant we develop an analytical theory for the diffusion coefficient, which takes into account non-Maxwellain form of the velocity-angular velocity distribution function. We perform molecular dynamics simulations for a gas in a homogeneous cooling state and study the dependence of the self-diffusion coefficient on $\varepsilon$ and $\beta$. Our theoretical results are in a good agreement with the simulation data.
Introduction {#intro}
============
Among numerous contributions by Isaac Goldhirsch to the theory of granular fluids [@Goldhirsch:2003] are his pioneering works on the hydrodynamic of gas of particles with the rotational degrees of freedom [@Goldhirsch_PRL:2005]. We wish to dedicate this article, addressed to granular gases of rough particles, to the memory of Isaac Goldhirsch.
Granular fluids are systems composed of a large number of macroscopic particles which suffer dissipative collisions. In many respects they behave like ordinary molecular fluids and may be described by the standard tools of kinetic theory and hydrodynamics [@Goldhirsch:2003; @book; @PoeschelBrilliantov:2003]. Among numerous phenomena, common to molecular and granular fluids, are Brownian motion, diffusion and self-diffusion [@Brey_JSP:1999; @SD; @Garzo:2002; @Puglisi:2002; @SDGreenCubo; @physa; @Puglisi_JSP:2010]; in the latter case tracers are identical to surrounding particles. The self-diffusion coefficient $D$ has a microscopic and macroscopic (thermodynamic) meaning, e.g. [@book; @SDGreenCubo]: Microscopically, it characterizes the dependence on time of the mean-square displacement of a grain $\langle\vec{R}^{2}(t)\rangle$: $$\langle\vec{R}^{2}(t)\rangle=6\int\limits_{0}^{t}D(t^{\prime})dt^{\prime}\, , \label{R2int}$$ while macroscopically, it relates the macroscopic flux of tracers $\vec{J}_s \left(\vec{r}, t
\,\right)$ (the index [*s*]{} refers to tracers) to its concentration gradient $ \vec{\nabla} n_s \left(\vec{r}, t \,\right)$: $$\label{eq:Js_nablans} \vec{J}_s \left(\vec{r},t\,\right)= - D(t) \vec{\nabla} n_s
\left(\vec{r},t\,\right) \,.$$ With the continuity equation $\frac{\partial n_s \left(\vec{r},t\,\right) }{\partial t} + \vec{\nabla} \vec{J}_s \left(\vec{r},t
\,\right)= 0$ the local tracers density $n_s\left(\vec{r},t\,\right)$ obeys the diffusion equation: $$\label{difcanon} \frac{\partial n_s \left(\vec{r}\,\right) }{\partial t} = D \vec{\nabla}^2 n_s
\left(\vec{r}\,\right)\,.$$ If the state of a granular gas is not stationary, like, e.g. a homogeneous cooling state, the diffusion coefficient generally depends on time.
In the previous studies the self-diffusion coefficient has been calculated only for smooth particles, that is, for particles which do not exchange upon collisions the energy of their rotational motion [@Brey_JSP:1999; @SD; @Garzo:2002; @SDGreenCubo; @physa; @Puglisi_JSP:2010]. The macroscopic nature of granular particles, however, implies friction between their surfaces and hence the rotational degrees of freedom are unavoidably involved in grains dynamics.
In the present paper we analyze the impact of particles roughness on the self-diffusion in granular gases – rarified systems, where the volume of the solid fraction is much smaller than the total volume. We calculate the self-diffusion coefficient as a function of normal and tangential restitution coefficients, which we assume to be constant. We develop an analytical theory for this kinetic coefficient and perform molecular dynamics (MD) simulations. The rest of the article is organized as follows. In Sect. \[sec:2\] we formulate the model, introduce the necessary notations and give the detailed description of the theoretical approach. In Sect. \[sec:3\] we present the results of MD simulations and compare the numerical data with the predictions of our theory. In the last Sect. \[sec:4\] we summarize our findings.
Calculation of the self-diffusion coefficient {#sec:2}
=============================================
Collision rules
---------------
The dissipative collisions of grains are characterized by two quantities, $\varepsilon$ and $\beta$ – the normal and tangential restitution coefficients. These coefficients relate respectively the normal and tangential components of the relative velocity between surfaces of colliding grains, $\vec{g}=\vec{v}_{12}+\frac{\sigma}{2}\left(\vec{e}
\times\left(\vec{\omega}_1+\vec{\omega}_2\right)\right)$, before (unprimed quantities) and after (primed quantities) a collision [@book; @Zippelius:2006]: $$\label{eq:2}
\left(\vec{g}^{\, n}\right)^{ \prime} =-\varepsilon \vec{g}^{\, n} \qquad \quad \left(\vec{g}^{\,
t}\right)^{ \prime} =-\beta\vec{g}^{\, t} \, .
$$Here $\vec{v}_{12} = \vec{v}_{1}-\vec{v}_{2}$ is the relative translational velocity of particles, $\vec{\omega}_1$ and $\vec{\omega}_2$ are their rotational velocities, the unit vector $\vec{e}$ is directed along the inter-center vector $\vec{r}_{12}$ at the collision instant and $\sigma$ is the diameter of particles. The normal and tangential components of $\vec{g}$ read respectively, $\vec{g}^{\,
n} = \left( \vec{g} \cdot \vec{e} \,\right) \vec{e}$ and $\vec{g}^{\, t} = \vec{g} - \vec{g}^{\, n}$. For macroscopic particles the normal restitution coefficient $\varepsilon$ can vary from $0$ (completely inelastic impact) to $1$ (completely elastic collision)[^1], while the tangential coefficient $\beta$ ranges from $-1$ (absolutely smooth particles) to $+1$ (absolutely rough particles). From the conservation laws and Eqs. (\[eq:2\]) follow the after-collision velocities and angular velocities of particles in terms of the pre-collision ones, e.g. [@book; @Zippelius:2006; @Temp]: $$\begin{aligned}
\label{vprime}
\vec{v}_{1,2}^{\, \prime} = \vec{v}_{1,2}\mp\frac{1+\varepsilon}{2}\vec{g}^n \mp\eta\vec{g}^t \nonumber
\\
\label{omegaprime}
\vec{\omega}_{1,2}^{\, \prime} = \vec{\omega}_{1,2}+ \frac{2\eta}{q\sigma}\left[\vec{e}\times\vec{g}^t\right] \, .
$$ Here $\eta \equiv \frac{q(1+\beta)}{2(1+q)}$ characterizes friction, $q=4I/m\sigma^2$, with $I$ and $m$ being respectively the moment of inertia and mass of particles. Although the normal and tangential restitution coefficients generally depend on the translational and rotational velocities of colliding particles and an impact geometry, e.g. [@book; @Walton:1993; @Zippelius:2006; @Poesc_tan_PRE:2008; @SBHB:2010], here we assume for simplicity that the restitution coefficients are constants.
Boltzmann equation
------------------
To compute the self-diffusion coefficient in granular gases two main approaches may be exploited – Green-Kubo approach, based on the time correlation functions of a dynamical variable (a particle velocity for the case of $D$) and Chapman-Enskog method, where the evolution of velocity distribution function is analyzed. It may be shown that these two are equivalent for the case of constant normal restitution coefficient [@book; @SDGreenCubo]; here we adopt the latter approach. First we consider the force-free granular gas in a homogeneous cooling state. We start from the Boltzmann equation for the distribution function of tracers $f_s(\vec{v}, \vec{\omega}, \vec{r},t)$, $$\label{eq:collin_trac} \left( \frac{\partial}{\partial t} + \vec{v}_1 \cdot \vec{\nabla} \right)
f_s\left(\vec{v}_1,\vec{\omega}_1, \vec{r}, t\right) = g_2\left(\sigma\right)I(f,f_s) \, ,$$ where $f(\vec{v},\vec{\omega}, \vec{r}, t) $ is the velocity distribution function of the gas particles, $g_2\left(\sigma\right)$ is the contact value of the pair correlation function, which accounts for the increasing collision frequency due to the excluded volume of grains and $I(f,f_s)$ abbreviates the collision integral [@book; @SDGreenCubo; @Santos_FF:2011]: $$\begin{aligned}
\nonumber && I(f,f_s) \!=\! \sigma^2 \!\int \! d \vec{v}_2 \! \int \! d \vec{\omega}_2 \! \int \! d \vec{e}
\Theta \left(-\vec{v}_{12} \cdot \vec{e} \, \right) \left| \vec{v}_{12} \cdot \vec{e} \,\right| \times \\
&& \Big[ \frac{1}{\varepsilon^2 \beta^2 }\! f_s\left(\vec{v}^{\,\prime\prime}_1,
\vec{\omega}^{\,\prime\prime}_1\right) f\left(\vec{v}^{\,\prime\prime}_2,
\vec{\omega}^{\,\prime\prime}_2\right) -
f_s\left(\vec{v}_1,\vec{\omega}_1\right)f\left(\vec{v}_2,\vec{\omega}_2\right)
\Big] \nonumber\\
\label{eq:collin}\end{aligned}$$ Here $\vec{v}^{\,\prime\prime}_{1,2}$ and $\vec{\omega}^{\,\prime\prime}_{1,2}$ denote the pre-collision velocities and angular velocities of particles for the [*inverse*]{} collision, i.e., for the collision which ends up with $\vec{v}_{1,2}$ and $\vec{\omega}_{1,2}$, the factor $\Theta\left(-\vec{v}_{12} \cdot \vec{e} \, \right)$ selects only approaching particles, and $\left|\vec{v}_{12} \cdot \vec{e} \,\right|$ gives the length of the collision cylinder. We assume that the concentration of tracers $$\label{eq:n_s}
n_s(\vec{r},t) \equiv \int d \vec{v}_1 \int d \vec{\omega}_1 f_s(\vec{v}_1, \vec{\omega}_1, \vec{r},t)$$ is much smaller than the number density of the gas particles $n(\vec{r},t)=n={\rm const}$, so that tracers do not affect the velocity distribution function $f$ for the gas, which is the solution of the Boltzmann equation $$\label{eq:Boltz_gas} \frac{\partial}{\partial t} f\left(\vec{v},\vec{\omega}, t\right) =
g_2\left(\sigma\right)I(f,f)$$ for the homogeneous cooling state. The collision integral in the last equation is given by Eq. [[(\[eq:collin\])]{}]{} with the distribution function of tracers $f_s$ substituted by that for the gas particles $f$.
Solving the Boltzmann equation [[(\[eq:collin\_trac\])]{}]{} for $f_s \left( \vec{v},\vec{\omega},\vec{r},t
\right)$, which is also known as Boltzmann-Lorentz equation, one can write the diffusion flux $$\label{eq:J_s_def} \vec{J}_s \left( \vec{r}, t \right) = \int d\vec{v} \int d\vec{\omega}\, \vec{v}
f_s(\vec{v}, \vec{\omega}, \vec{r},t) \, ,$$ and then, using the macroscopic equation [[(\[eq:Js\_nablans\])]{}]{}, find the diffusion coefficient as the coefficient at the concentration gradient $\vec{\nabla} n_s \left(\vec{r},t\,\right)$.
In a homogeneous cooling state a density of the system remains uniform, while the translational $T(t)$ and rotational $R(t)$ granular temperatures, defined through the average energies of translational and rotational motion, respectively, $$\label{eq:T_tr} \frac32 n T = \int \! d\vec{v} d\vec{\omega}\, \frac{mv^2}{2} f, \qquad \frac32 n R=
\int d \!\vec{v} d\vec{\omega}\, \frac{I \omega^2}{2} f$$decay with time as $\partial T /\partial t = -\zeta T$ and $\partial R/\partial t = -\zeta_{R} T$, where $\zeta$ and $\zeta_R$ are the translational and rotational cooling rates, e.g. [@book; @Zippelius:2006; @Corr; @Santos_FF:2011].
The distribution function $f\left(\vec{v},\vec{\omega},t\right)$ is close to the Maxwell distribution and the deviations from the Max-wellian are twofold. Firstly, the translational and angular velocity distributions slightly differ from the Gaussian distribution, which is accounted by the expansion of $f$ in Sonine polynomials series with respect to $\vec{v}^{\, 2}$ and $\vec{\omega}^{\, 2}$. Secondly, a linear velocity and spin of particles are correlated; this is quantified using an expansion in Legendre polynomials series with respect to $\theta
= \widehat{\vec{v}\, \vec{\omega}}$ – the angle between the velocity $\vec{v}$ and angular velocity $\vec{\omega}$ of a particle. Using the reduced quantities $\vec{c} = \vec{v}/v_T$ and $\vec{w} = \vec{\omega }/\omega_T$ with $v_T \equiv \sqrt{2T/m}$ and $\omega_T \equiv \sqrt{2R/I}$, the distribution function reads $$\begin{aligned}
\nonumber
f \left(\vec{v},\vec{\omega}, t\right) = \frac{n}{\pi^2v_T^3 \, \omega_T^3} e^{-c^2 -w^2} \Big[ 1+a_{20}
S_{1/2}^{(2)}(c^2)+
\\
\nonumber
a_{02} S_{1/2}^{(2)}(w^2) + a_{11} S_{1/2}^{(1)}(c^2) S_{1/2}^{(1)}(w^2) \!+ \\ \bar{b}c^2w^2 P_2(\cos \theta) \Big] ,
\label{eq:disfunc}\end{aligned}$$ where only leading terms in the expansion are kept. The Sonine and Legendre polynomials in Eq. [[(\[eq:disfunc\])]{}]{} are $S_{1/2}^{(1)}(x) = \frac32 -x$, $S_{1/2}^{(2)}(x) = \frac18(15-20x -4x^2)$ and $P_2(x) = \frac32 \left(x^2 -\frac13 \right)$. The distribution function attains the scaling form [[(\[eq:disfunc\])]{}]{} after some relaxation time. At this stage, which will be addressed below, the coefficients $a_{20}$, $a_{11}$, $a_{02}$ and $\bar{b}$ are constants. These coefficients have been analyzed in detail in Refs. [@Santos_FF:2011; @Corr]. They are complicated nonlinear functions of $\varepsilon$, $\beta$ and $r=R/T$, which are too cumbersome to be given here.
The temperature ratio $r$ also attains a steady state value in the scaling regime and the cooling rates become equal, $\zeta=\zeta_R$ [@Santos_FF:2011; @Corr]. The steady-state value of $r=R/T$ may be obtained as an appropriate root of the eighth-order algebraic equation [@Santos_FF:2011]. The cooling rate $\zeta$ may be represented in the following way: $$\begin{aligned}
\nonumber
&&\zeta = -\frac{2}{3n T} \int d\vec{v}_1 \int d \vec{\omega}_1 \frac{mv_1^2}{2} I
(f,f) = 2\tau_{E}^{-1} \zeta^* \, , \\
&&\zeta^*= \left( \frac14(1-\varepsilon^2) +\eta (1-\eta) \right) \left( 1
+\frac{3}{16}a_{20} \right) - \nonumber \\
&&- \left( 1-\frac{a_{20}}{16} +\frac{a_{11}}{4} -
\frac{\bar{b}}{8} \right) \frac{\eta^2}{q}r
\label{eq:psi_tr}\end{aligned}$$ and $\zeta_R$ may be obtained similarly [@Santos_FF:2011]. Here $\tau_{ E}$ is the Enskog relaxation time $$\label{eq:tau_E} \tau_{E}^{-1}(t) = \frac83 g_2(\sigma) \sigma^2 n \sqrt{\frac{\pi T (t) }{m}}\, ,$$
Chapman-Enskog scheme
---------------------
To find $f_s$ we need to solve the Boltzmann-Lorentz equation [[(\[eq:collin\_trac\])]{}]{}. It may be done approximately, using the Chapman-Enskog approach based on two simplifying assumptions: (i) $f_s
\left(\vec{v}, \vec{\omega},\vec{r},t \right)$ depends on space and time only trough the macroscopic hydrodynamic fields and (ii) $f_s \left( \vec{v},\vec{\omega},\vec{r},t \right)$ can be expanded in terms of the field gradients. In the gradient expansion $$\label{eq:ChEns_fs} f_s= f_s^{(0)} + \lambda f_s^{(1)} + \lambda^2 f_s^{(2)} \ldots \,$$ a formal parameter $\lambda$ is introduced, which indicates the power of the field gradient; at the end of the computations it is set to unity, $\lambda=1$.
In a homogeneous cooling state with a lack of macroscopic fluxes only two hydrodynamic fields are relevant – the number density of tracers, $n_s(\vec{r},t)$, and the total temperature $T_{\rm tot} =
T(t)+R(t)$. The former corresponds to conservation of particles, and the latter to conservation of energy in the elastic limit. $T_{\rm tot}$ is equal for the gas and tagged particles due to their mechanical identity. In the scaling regime, when the Chapman-Enskog approach is applicable, the translational, rotational and total temperature are linearly related. Hence one can use any of these fields in the Chapman-Enskog scheme and we choose $T(t)$ here without the loss of generality. Thus, the two relevant hydrodynamic fields satisfy in the homogeneous cooling state the following equations: $$\begin{aligned}
\label{eq:dndt_dTdt_lamb}
\frac{\partial n_s}{\partial t} = \left( \frac{\partial^{(0)}}{\partial t} + \lambda \frac{\partial^{(1)}}{\partial t} + \lambda^2 \frac{\partial^{(2)}}{\partial t} + \ldots \right) n_s
=\lambda^2 D \nabla^2 n_s \nonumber \\ \\
\frac{\partial T }{\partial t} = \left( \frac{\partial^{(0)}}{\partial t} + \lambda \frac{\partial^{(1)}}{\partial t} + \lambda^2 \frac{\partial^{(2)}}{\partial t} + \ldots \right) T
= - \zeta T \, ,\nonumber
$$ where $ \partial^{(k)} \psi / \partial t $ indicates that only terms of $k$-th order with respect to the field gradients are accounted.
Substituting Eq. [[(\[eq:ChEns\_fs\])]{}]{} into the Boltzmann-Lorentz equation [[(\[eq:collin\_trac\])]{}]{} and collecting terms of the same order of $\lambda$ (that is of the same order in gradients) we obtain successive equations for $f_s^{(0)}(\vec{v},\vec{\omega},\vec{r},t)$, $ f_s^{(1)}
(\vec{v},\vec{\omega},\vec{r},t)$, etc. The zeroth-order equation in the gradients yields $f_s^{(0)}(\vec{v},\vec{\omega},\vec{r},t)$ for the homogeneous cooling state. Since the tracers are mechanically identical to the rest of the particles, $f_s^{(0)}(\vec{v},\vec{\omega},\vec{r},t)$ is simply proportional to the distribution function of the embedding gas: $$\label{eq:fs_homog} f_s^{(0)}(\vec{v},\vec{\omega},\vec{r},t) = \frac{n_s \left(\vec{r}, t \right) }{n}
f(\vec{v},\vec{\omega},t) \,.$$ The first-order equation then reads $$\label{eq:Boltz1_tagg} \frac{\partial^{(0)} f_s^{(1)}}{\partial t}\! +\!\frac{\partial^{(1)}
f_s^{(0)}}{\partial t} + \vec{v}_1 \vec{\nabla} f_s^{(0)}\! =\! g_2(\sigma) I \left( f, f_s^{(1)}
\right) \!.$$ Using Eqs. (\[eq:dndt\_dTdt\_lamb\]), one can show, that the first term in the left-hand side of Eq. [[(\[eq:Boltz1\_tagg\])]{}]{} is equal to $$\label{eq:dt0f_s1}
\frac{\partial^{(0)} f_s^{(1)}}{\partial t}\! = \! \frac{\partial^{(0)} n_s}{\partial t}
\frac{\partial f_s^{(1)} }{\partial n_s} \!+ \! \frac{\partial^{(0)} T}{\partial t}
\frac{\partial f_s^{(1)} }{\partial T }\! = \! -\zeta T \frac{\partial f_s^{(1)}}{\partial T},$$ since $\partial^{(0)} n_s/ \partial t $. Similarly, we obtain that $\partial^{(1)} f_s^{(0)}/\partial t
= 0$. Substituting Eq. [[(\[eq:fs\_homog\])]{}]{} with the space independent $n$ and $f$ into the last term in the left-hand side of Eq. [[(\[eq:Boltz1\_tagg\])]{}]{}, we arrive finally at the equation for $f_s^{(1)}$: $$\label{eq:forf1s} \zeta T \frac{\partial f_s^{(1)}}{\partial T} + g_2(\sigma) I \left( f, f_s^{(1)}
\right) = \frac{f}{n} \left( \vec{v}_1 \cdot \vec{\nabla} n_s \right) \, .$$ We search for the solution of Eq. [[(\[eq:forf1s\])]{}]{} in the form $$\label{eq:form_of_f1s}
f_s^{(1)} = \vec{G} \left(\vec{v}_1, \vec{\omega}_1, t \right) \cdot
\vec{\nabla} n_s \left( \vec{r}, t \right) \,,$$ which implies with Eq. [[(\[eq:J\_s\_def\])]{}]{} the diffusion flux $$\label{eq:D_via_Gv} \vec{J}_s \!=\!\int d \vec{v}_1 \int d \vec{\omega }_1 \, \vec{v}_1 \left( \vec{G}
\cdot \vec{\nabla} n_s \right)
\!=\!-D \vec{\nabla} n_s \, ,$$ where we take into account the isotropy of zero-order function $f_s^{(0)}=f_s^{(0)}(|\vec{v}\,|,|\vec{\omega}\,|)$. From Eq. [[(\[eq:D\_via\_Gv\])]{}]{} we obtain (cf. [@book]): $$\label{eq:D_viaGvfinal} D= - \frac13 \int d \vec{v}_1 \int d \vec{\omega }_1 \, \vec{v}_1 \cdot \vec{G}
\,.$$ We substitute $f_s^{(1)}$ from Eq. [[(\[eq:form\_of\_f1s\])]{}]{} into Eq. [[(\[eq:forf1s\])]{}]{} discarding the factor $\vec{\nabla} n_s$. Then we multiply it by $\frac13 \vec{v}_1$ and integrate over $\vec{v}_1$ and $\vec{\omega}_1$ to obtain $$\begin{aligned}
\label{eq:Gv_int_overv}
\zeta T \frac{\partial}{\partial T} \frac13 \!\!\int \!d \vec{v}_1 \!\int
\!\!d \vec{\omega}_1 \vec{v}_1 \cdot \vec{G} + \frac{g_2(\sigma)}{3} \int \! \! d \vec{v}_1 \! \int
\!\!d \vec{\omega}_1 \times \nonumber \\ \times \vec{v}_1 \cdot I \left( f, \vec{G} \right) =
\frac{2}{3nm} \int \!d \vec{v}_1 \! \int \! d \vec{\omega}_1 \frac{mv_1^2}{2} f .\end{aligned}$$ The first term in the left-hand side equals $ - \zeta T \partial D / \partial T$, while the right-hand side equals $(T/m)$, according to Eq. [[(\[eq:T\_tr\])]{}]{}. The structure of Eq. [[(\[eq:forf1s\])]{}]{} dictates the Ansatz[^2] $$\label{eq:G_via_vfb0} \vec{G} \left(\vec{v}, \vec{\omega},t \right) \propto \vec{v} f\left(\vec{v},
\vec{\omega},t \right) = B_0 \vec{v} f \, ,$$ where we assume that the mean value of the rotational velocity $\vec{\omega}$ is equal to zero. The unknown constant $B_0$ is to be determined from the above equation. With this Ansatz the second term in the left-hand side of Eq. [[(\[eq:Gv\_int\_overv\])]{}]{} reads $$\begin{aligned}
\label{eq:int_vI_Gf1}
&&\frac{g_2(\sigma)}{3} \int \!d\vec{\omega}_1 d\vec{v}_1 \vec{v}_1 I\left(f, \vec{G} \right)
= \frac{g_2(\sigma)\sigma^2 }{3} \frac{B_0}{2} \! \int \!d\vec{v}_1 d\vec{v}_2 \nonumber \\
&&~~~~~~~~\times \int \! d\vec{\omega}_1 \!
d\vec{\omega}_2 \!\int \!d\vec{e} \, \Theta \left(-\vec{v}_{12} \cdot \vec{e} \, \right)
\left|\vec{v}_{12} \cdot \vec{e} \, \right|
\\
&&~~~~~~~~ \times f\left(\vec{v}_1,\vec{\omega}_1 \right) \! f\left(\vec{v}_2, \vec{\omega}_2\right)
\left( \vec{v}_1 - \vec{v}_2 \right) \cdot \left( \vec{v}^{\,\prime}_1 - \vec{v}_1 \right) .\nonumber\end{aligned}$$ To evaluate the expression in Eq. [[(\[eq:int\_vI\_Gf1\])]{}]{} we use the collision rule [[(\[vprime\])]{}]{} for the factor $\left(
\vec{v}^{\,\prime}_1 - \vec{v}_1 \right)$ and Eq. [[(\[eq:disfunc\])]{}]{} for the distribution function $f$. After a straightforward algebra we arrive at $$\label{eq:vIG} \frac{g_2(\sigma)}{3} \int \!d\vec{\omega}_1 \!\int \!d\vec{v}_1 \vec{v}_1 I\left(f,
\vec{G} \right)
= -\frac{B_0 Tn}{m} \tau_{v,\,{\rm ad}}^{-1},$$ where $$\label{eq:tau_ad} \tau_{v,\,{\rm ad}}^{-1} = \left( \frac{1+\varepsilon}{2} + \eta \right) \left(1
+\frac{3 }{16} a_{20} \right) \tau_{E}^{-1}\, .$$ Note that $\tau_{v,\,{\rm ad}}$ has a physical meaning of the relaxation time of the velocity correlation function of granular particles, e.g. [@book; @SDGreenCubo]. Eq. [[(\[eq:tau\_ad\])]{}]{} gives the generalization of this quantity for the case of rough grains. If we express the unknown constant $B_0$ using the relation $$ D= -\frac{B_0}{3} \int d \vec{v}_1 \int d \vec{\omega}_1\, \vec{v}_1 \cdot \vec{v}_1 f(\vec{v}_1,
\vec{\omega}_1,t)
= - \frac{B_0Tn}{m}\, , \nonumber
$$ according to Eqs. (\[eq:D\_viaGvfinal\], \[eq:T\_tr\]), we recast [[(\[eq:Gv\_int\_overv\])]{}]{} into the form $$\label{eq:forD_final} -\zeta T \frac{\partial D}{\partial T} + D \tau_{v,\,{\rm ad}}^{-1} = \frac{T}{m}
\,.$$ As it follows from Eqs. [[(\[eq:psi\_tr\])]{}]{}, [[(\[eq:tau\_E\])]{}]{} and [[(\[eq:tau\_ad\])]{}]{}, $\tau_{v,\,{\rm
ad}}^{-1} \propto \tau_{{E}}^{-1} \propto \sqrt{T}$ and $\zeta T \frac{\partial}{\partial T} \propto
\zeta \propto \tau_{{E}}^{-1} \propto \sqrt{T}$. At the same time, the right-hand side of Eq. [[(\[eq:forD\_final\])]{}]{} scales as $\propto T$, which implies that $D \propto \sqrt{T}$. Therefore $T
\frac{\partial D}{\partial T} = \frac{D}{2}$ and we obtain the solution of Eq. [[(\[eq:forD\_final\])]{}]{}, $$D(t) = \frac{T}{m} \left[ \tau_{v, \,{\rm ad}}^{-1}(t) - \frac12 \zeta (t) \right]^{-1} \, .
\label{eq:D_byond_ad1}$$ Substituting $\tau_{v,\,{\rm ad}}$ and $\zeta$ given by Eqs. [[(\[eq:psi\_tr\])]{}]{} and [[(\[eq:tau\_ad\])]{}]{} into Eq. [[(\[eq:D\_byond\_ad1\])]{}]{} we finally find: $$\begin{aligned}
\label{eq:D_rough}
&&D = \\
&&\frac{D_{E}}{\left(1\!+\!\frac{3}{16}a_{20} \right)
\left(\frac{(1+\varepsilon)^2}{4} \!+\! \eta^2 \right) \!+\!\left(1 \!-\!\frac{a_{20}}{16}
\!+\!\frac{a_{11}}{4} -\frac{\bar{b}}{8}\right)\frac{\eta^2}{q}r } \nonumber\end{aligned}$$ where $D_{E}(t) \equiv \left(T(t)/m \right) \tau_{E}(t)$ is the Enskog value for the self-diffusion coefficient for given temperature $T(t)$. Note that for $\eta=0$, i.e., for smooth particles, Eq. [[(\[eq:D\_rough\])]{}]{} reproduces the previously known result, e.g. [@book; @SDGreenCubo].
All the expansion coefficients $a_{20}$, $a_{11}$, $a_{02}$ and $\bar{b}$ are small and with a reasonable accuracy one can use the Maxwell approximation for $r$ [@Temp; @Santos_FF:2011]: $$\begin{aligned}
\label{eq:r_steady} r &=& \sqrt{1+C^2}+C \\
C& \equiv & \frac{1+q}{2q(1+\beta)} \left[ \frac{1- \varepsilon^2}{1+\beta} (1+q) -(1-q)(1-\beta)
\right] \nonumber\end{aligned}$$ and for $D$, which simplifies in this case to $$\label{eq:D_roughMax}
D = D_{E} \left( \frac{(1+\varepsilon)^2}{4} +\eta^2 \left( 1+ \frac{r}{q} \right) \right)^{-1} \,.$$
It is rather straightforward to perform similar calculations for a driven granular gas in the white-noise thermostat. The Boltzmann equation reads in this case: $$\frac{\partial}{\partial t} f\left(\vec{v},\vec{\omega}, t\right)-\frac{\chi^2}{2}\left(\frac{\partial}{\partial\vec{v}_1}\right)^2 f =
g_2\left(\sigma\right)I(f,f)\,,$$ where $\chi$ characterizes the strength of the stochastic force. The gas temperature and the coefficients $a_{20}$, $a_{11}$, $a_{22}$ are then determined by the intensity of the noise [@Santos_FF:2011]. The temperature rapidly relaxes to a constant value, therefore only one hydrodynamic field $n_s(r, t)$ is relevant in the Chapman-Enskog scheme. Performing then all steps as in the case of HCS, we finally arrive at the self-diffusion coefficient for a driven granular gas: $$\begin{aligned}
\label{eq:D_termo}
D_{\rm w.n.th.} = \frac{D_{E}}{\left(1 \!+\!\frac{3}{16}a_{20} \right)
\left(\frac{1+\varepsilon}{2} + \eta \right) }\end{aligned}$$
Molecular dynamics simulations {#sec:3}
==============================
To check the predictions of our theory we perform molecular dynamics (MD) simulations [@CompBook] for a force-free system, using 8000 spherical particles of radius $\sigma/2=1$ and mass $m=1$ in a box of length $L_{\rm box}=132$ with periodic boundary conditions. We confirmed that the system remained in the HCS during its evolution. The initial translational and rotational temperatures were equal $T(0)=R(0)=1$; the ratio between translational and rotational temperatures $r$ rapidly relaxed to a constant value, indicating that the application of our theory, based on the scaling form of the distribution function, is valid. To analyze the diffusion coefficient we used the re-scaled time $\tau$ $$d\tau = dt/ \tau_c(t), \quad \tau_c^{-1}(t)= 4 \sqrt{\pi} g_2(\sigma) \sigma^2 n \sqrt{T(t)/m}.
\nonumber$$In the re-scaled time the self-diffusion coefficient attains at $\tau \gg 1$ a constant value \[see Eq. [[(\[R2int\])]{}]{}\], $ D = \langle R^2\left(\tau \right) \rangle/\left(6\tau\right) $.
The dependence of the self-diffusion coefficients on the normal and tangential restitution coefficients is presented in Fig. \[Gdv\], where the molecular dynamics data are compared with the theoretical predictions.
![ The dependence of the self-diffusion coefficient in a HCS on the normal $\varepsilon$ (upper panel) and tangential $\beta$ (lower panel) restitution coefficients. The MD simulation data (symbols) are compared with the predictions of the theory for a gas in a HCS, Eq. [[(\[eq:D\_rough\])]{}]{} (solid line) and Eq. [[(\[eq:D\_roughMax\])]{}]{} (dashed line, Maxwellian approximation). The fixed restitution coefficients are: $\beta=0.9$ for the upper and $\varepsilon =0.9$ for the lower panel. []{data-label="Gdv"}](Dvepssantos.jpg "fig:"){width="0.98\columnwidth"} ![ The dependence of the self-diffusion coefficient in a HCS on the normal $\varepsilon$ (upper panel) and tangential $\beta$ (lower panel) restitution coefficients. The MD simulation data (symbols) are compared with the predictions of the theory for a gas in a HCS, Eq. [[(\[eq:D\_rough\])]{}]{} (solid line) and Eq. [[(\[eq:D\_roughMax\])]{}]{} (dashed line, Maxwellian approximation). The fixed restitution coefficients are: $\beta=0.9$ for the upper and $\varepsilon =0.9$ for the lower panel. []{data-label="Gdv"}](Dbsantos.jpg "fig:"){width="0.98\columnwidth"}
As it may be seen from the figure, the relative diffusion coefficient $D(t)/D_{E}(t)$ increases with decreasing $\varepsilon$ – the dependence, which has been already observed for smooth particles [@Brey_JSP:1999; @SD; @Garzo:2002; @book; @SDGreenCubo; @physa]. The physical nature of the effect is very simple: With the increasing inelasticity, which suppresses the normal component of the after-collisional relative velocity of particles, their trajectories become more stretched. This leads to the increasing correlation time and and hence, to a larger $D$. At the same time self-diffusion coefficient decreases for large roughness ($\beta > -0.25$) with increasing tangential restitution coefficient, see Fig. \[Gdv\]. It is not difficult to explain the observed behavior of $D$. Indeed, when the tangential restitution coefficient $\beta$ increases from $-1$ (smooth particles) to $+1$ (absolutely rough particles), the translational and rotational motion become more and more engaged \[see Eqs. (\[vprime\])\] and the trajectories of particles more and more chaotic, that is, less stretched. This eventually causes the decrease of the diffusion coefficient with roughness.
As it is follows from the Fig. \[Gdv\], the theoretical predictions for the self-diffusion coefficient are in a reasonably good agreement with the numerical data. The diffusion coefficient, calculated in the framework of the Maxwellian approximation, practically does not differ from the full solution, Eq. [[(\[eq:D\_roughMax\])]{}]{}; the difference becomes apparent only for very high inelasticity ($\varepsilon<0.6$). A slight overestimate by the theory of the value of $D$, obtained in molecular dynamics, may be possibly attributed to the simple Ansatz [[(\[eq:G\_via\_vfb0\])]{}]{} for the first-order function $f_s^{(1)}$, where only zero-order term of the expansion in the Sonine polynomials $
S_{3/2}^{(k)}(v^2/v_T^2) $ was used (see the footnote before Eq. [[(\[eq:G\_via\_vfb0\])]{}]{}). One probably needs to extend the expansion and include the next order terms; this would be a subject of a future study.
Conclusion {#sec:4}
==========
We have analyzed an impact of particles’ roughness on the self-diffusion coefficient in granular gases. We use a simplified collision model with constant normal $\varepsilon$ and tangential $\beta$ restitution coefficients. The former characterizes the collisional dissipation of the normal relative motion of particles at a collision, the latter – of the tangential one. We develop an analytical theory for the self-diffusion coefficient, taking into account the deviation of the velocity-angular velocity distribution function from the Maxwellian; we use the leading-order terms in the expansion of this deviation in Sonine and Legendre polynomials series. We notice that the impact of the non-Maxwellian distribution on the self-diffusion coefficient is small. To check the predictions of our theory we perform the molecular dynamics simulations for a granular gas in a homogeneous cooling state for different values of the normal and tangential restitution coefficients. We find that the theoretical results are in a reasonably good agreement with the simulation data. Both the theory and molecular dynamics demonstrate that the relative diffusion coefficient $D(t)/D_{E}(t)$ ($D_{E}$ is the Enskog value of diffusion coefficient for smooth elastic particles) increases with decreasing normal restitution coefficient $\varepsilon$, similarly, as for a gas of smooth particles and decreases with increasing tangential restitution coefficient $\beta$ for large roughness ($\beta> -0.25$) .
[10]{} \[1\][[\#1]{}]{} urlstyle \[1\][DOI \#1]{}
Becker, V., Schwager, T., Poschel, T.: Coefficient of tangential restitution for the linear dashpot model. Phys. Rev. E **77**, 011304 (2008); Schwager, T., Becker, V., Poschel, T.: Coefficient of tangential restitution for viscoelastic spheres. Eur. Phys. J. E **27**, 107–114 (2008)
Bodrova, A.S., Brilliantov, N.V.: Granular gas of viscoelastic particles in a homogeneous cooling state. Physica A **388**, 3315–3324 (2009)
Brey, J.J., Dufty, J.W., Santos, A.: Kinetic models for granular flow. J. Stat. Phys. **97**, 281 (1999); Brey, J.J., Ruiz-Montero, M.J., Cubero, D., Garcia-Rojo, R.: Self-diffusion in freely evolving granular gases. Phys. Fluids **12**, 876 (2000); Brey, J.J., Ruiz-Montero, M.J., Cubero, D., Garcia-Rojo, R.: Self-diffusion in freely evolving granular gases. Physics of Fluids **12**(4), 876–883 (2000); Brey, J.J., Ruiz-Montero, M.J., Garcia-Rojo, R., Dufty, J.W.: Brownian motion in a granular gas. Phys. Rev. E **60**, 7174 (1999)
Brilliantov, N.V., Pöschel, T.: Self-diffusion in granular gases. Phys. Rev. E **61**(2), 1716–1721 (2000)
Brilliantov, N.V., Pöschel, T.: Kinetic theory of Granular Gases. Oxford University Press, Oxford (2004)
Brilliantov, N.V., Pöschel, T.: Self-diffusion in granular gases: Green-Kubo versus Chapman-Enskog. Chaos **15**, 026108 (2005)
Brilliantov, N.V., Pöschel, T., Kranz, W.T., Zippelius, A.: Translations and rotations are correlated in granular gases. Phys. Rev. Lett. **98**, 128001 (2007); Kranz, W.T., Brilliantov, N.V., Pöschel, T., Zippelius, A.: Correlation of spin and velocity in the homogenous cooling state of a granluar gas of rough particles. Eur. Phys. J. Special Topics **179**, 91 – 111 (2009)
Chapman, S., Cowling, T.G.: The mathematical theory of Nonuniform gases. Cambridge University Press, Londom (1970)
Garz´o, V.: Tracer diffusion in granular shear flows. Phys. Rev. E **66**, 021308 (2002); Garz´o, V., Montanero, J.M.: Diffusion of impurities in a granular gas. Phys. Rev. E **69**, 021301 (2004)
Goldhirsch, I.: Rapid granular flows. Annu. Rev. Fluid Mech. **35**, 267 (2003)
Goldhirsch, I., Noskowicz, S.H., Bar-Lev, O.: Nearly smooth granular gases. Phys. Rev. Lett. **95**, 068002 (2005); Goldhirsch, I., Noskowicz, S.H., Bar-Lev, O.: Hydrodynamics of nearly smooth granular gases. J. Phys. Chem. **109**, 21449–21470 (2005)
Luding, S., Huthmann, M., McNamara, S., Zippelius, A.: Homogeneous cooling of rough, dissipative particles: Theory and simulations. Phys. Rev. E **58**, 3416–3425 (1998)
van Noije, T.P.C., Ernst, M.H.: Velocity distributions in homogeneous granular fluids: the free and the heated case. Granular Matter **1**, 57–64 (1998)
Pöschel, T., Brilliantov, N.V.: Granular Gas Dynamics, *Lecture Notes in Physics*, vol. 624. Springer, Berlin (2003); Pöschel, T., Luding, S.: Granular Gases, *Lecture Notes in Physics*, vol. 564. Springer, Berlin (2001)
Pöschel, T., Schwager, T.: Computational Granular Dynamics. Springer, Berlin (2005)
Puglisi, A., Baldassarri, A., Loreto, V.: Fluctuation-dissipation relations in driven granular gases. Phys. Rev. E **66**, 061305 (2002)
Saitoh, K., Bodrova, A., Hayakawa, H., Brilliantov, N.: Negative normal restitution coefficient found in simulation of nanocluster collisions. Phys. Rev. Lett. **105**, 238001 (2010)
Santos, A., Kremer, G.M., dos Santos, M.: Sonine approximation for collisional moments of granular gases of inelastic rough spheres. Phys. Fluids **23**, 030604 (2011)
Sarracino, A., Villamaina, D., Costantini, G., Puglisi, A.: Granular brownian motion. J. Stat. Mech.: Theory and Experiment P04013 (2010)
Walton, O.R.: Numerical simulation of inelastic frictional particle-particle interactions. In: M.C. Roco (ed.) Particle Two-Phase Flow, pp. 884–907. Butterworth, London (1993)
Zippelius, A.: Granular gases. Physica A **369**, 143–158 (2006)
[^1]: For nano-particles the normal restitution coefficient can attain negative values [@SBHB:2010].
[^2]: Only the first term of the expansion $B=B_0+B_1 S_{3/2}^{(1)}(v^2/v_T^2)+B_2
S_{3/2}^{(2)}(v^2/v_T^2) + \ldots$ is used in Eq. [[(\[eq:G\_via\_vfb0\])]{}]{}, see e.g. [@ChapmanCowling]
|
---
abstract: 'There is a significant literature on methods for incorporating knowledge into multiple testing procedures so as to improve their power and precision. Some common forms of prior knowledge include (a) beliefs about which hypotheses are null, modeled by non-uniform prior weights; (b) differing importances of hypotheses, modeled by differing penalties for false discoveries; (c) multiple arbitrary partitions of the hypotheses into (possibly overlapping) groups; and (d) knowledge of independence, positive or arbitrary dependence between hypotheses or groups, suggesting the use of more aggressive or conservative procedures. We present a unified algorithmic framework called ${\textnormal{\texttt{p-filter}}}$ for global null testing and false discovery rate (FDR) control that allows the scientist to incorporate all four types of prior knowledge (a)–(d) simultaneously, recovering a variety of known algorithms as special cases.'
address:
- |
Aaditya K. Ramdas\
Department of Statistics and Data Science\
Carnegie Mellon University\
- |
Rina F. Barber\
Department of Statistics\
University of Chicago\
- |
Martin J. Wainwright\
Departments of Statistics and EECS\
University of California, Berkeley\
- |
Michael I. Jordan\
Departments of Statistics and EECS\
University of California, Berkeley\
author:
-
-
-
-
bibliography:
- 'FDR.bib'
title: 'A Unified Treatment of Multiple Testing with Prior Knowledge using the p-filter'
---
,
,\
,
Introduction {#sec:intro}
============
Multiple hypothesis testing is both a classical and highly active research area, dating back (at least) to an initially unpublished 1953 manuscript by Tukey entitled “The Problem of Multiple Comparisons” [@tukey1953problem; @tukey1994]. Given a large set of null hypotheses, the goal of multiple testing is to decide which subset to reject, while guaranteeing some notion of control on the number of false rejections. It is of practical importance to incorporate different forms of prior knowledge into existing multiple testing procedures; such prior knowledge can yield improvements in power and precision, and can also provide more interpretable results. Accordingly, we study methods that control the False Discovery Rate (FDR) or test the global null (GN) hypothesis while incorporating any number of the following strategies for incorporating prior knowledge: (a) the use of prior weights, (b) the use of penalty weights, (c) the partitioning of the hypotheses into groups, (d) the incorporation of knowledge of the dependence structure within the data, including options such as estimating and adapting to the unknown number of nulls under independence, or reshaping rejection thresholds to preserve error-control guarantees in the presence of arbitrary dependence. It is a challenge to incorporate all of these forms of structure while maintaining internal consistency (coherence and consonance) among the pattern of rejections and acceptances, and most existing work has managed to allow only one or two of the four strategies (a), (b), (c), (d) to be employed simultaneously. We present a general unified framework, called `p-filter`, for integrating these four strategies while performing a GN test or controlling the FDR. The framework is accompanied by an efficient algorithm, with code publicly available at [https://www.stat.uchicago.edu/$\sim$rina/pfilter.html](https://www.stat.uchicago.edu/~rina/pfilter.html). This framework allows scientists to mix and match techniques, and use multiple different forms of prior knowledge simultaneously. As a by-product, our proofs often simplify and unify the analysis of existing procedures, and generalize the conditions under which they are known to work.
**Organization**. The rest of this paper is organized as follows. In [Section \[sec:contributions\]]{}, we begin with an example to provide intuition, and we discuss the contributions of this paper. In [Section \[sec:pf+\]]{}, we describe the general [[`p-filter`]{.nodecor}]{} framework, along with its associated theoretical guarantees; this section lays out the central contribution of our work. In [Section \[sec:lemmas\]]{}, we present three lemmas that provide valuable intuition and are central to the proof of [Theorem \[thm:pf+\]]{}; see [Section \[sec:proof\_main\]]{} for the proof itself. We prove the three aforementioned lemmas in [Section \[sec:lemmas-proofs\]]{}, and prove some related propositions in [Section \[sec:prop-proofs\]]{}. While directly related work is discussed immediately when referenced, we end by overviewing other related work in [Section \[sec:disc\]]{}.
An example, and our contributions {#sec:contributions}
=================================
The various kinds of prior information considered in this paper have been studied in earlier works and repeatedly motivated in applied settings, and our focus is accordingly on the conceptual and mathematical aspects of multiple decision-making with prior knowledge. Before beginning our formal presentation, we consider a simple example, illustrated in Figure \[fig:partitions\], in order to provide intuition.
![Consider a set of $n = 16$ hypotheses arrayed in a $4
\times 4$ grid, with four different partitions into groups: elementary, rows, columns and blocks. On the top row is the underlying truth, with the leftmost panel showing the hypothesis-level non-nulls, and the other three panels showing which groups in each partition are hence identified as non-null. On the bottom row is an example of a set of discoveries, with the leftmost panel showing the hypothesis-level rejections, and the other three panels showing which groups are correspondingly rejected (light-grey for correct rejections, black for false rejections). The false discovery proportions (FDP) in each partition are 0.2, 0, 0.33, 0.5 respectively. The true discovery proportions (empirical power) in each partition are 0.8, 0.66, 1, 0.5 respectively.[]{data-label="fig:partitions"}](graph.jpg){width="95.00000%"}
Consider a set of sixteen hypotheses arranged in a $4\times4$ grid, as displayed in the first panel of Figure \[fig:partitions\]. One may imagine that one coordinate refers to spatial locations, and the other to temporal locations, so that each square represents an elementary null hypothesis $H_{s,t}$, stating that there is nothing of interest occurring at spatial location $s$ at time $t$. As displayed in the leftmost panel of the first row, the non-nulls may be expected to have some spatio-temporal structure. In order to exploit this structure, the scientist may choose to group the hypotheses a priori in three ways: by spatial location, by temporal location, and by spatio-temporal blocks, as displayed by the other three panels in the first row. Each such group can be associated with a group null hypothesis, which states that there is nothing of interest occurring within that group.
As displayed in the first row, the group-level non-nulls are simply implied by the elementary non-nulls. The second row of the figure displays the results of a hypothetical procedure that makes some elementary rejections (first panel), and hence some corresponding group level rejections (other three panels). The scientist may wish to not report too many false elementary rejections, but also not report discoveries at spurious locations, times or space-time blocks. One way of enabling this wish is to enforce group FDR constraints, in addition to overall FDR control. This would correspond to controlling the “spatial” FDR, “temporal” FDR, and “spatio-temporal” FDR, in addition to the overall FDR. Requiring the rejected hypotheses to satisfy additional constraints may reduce power, but it may also increase interpretability, and result in higher precision (achieved FDP).
For example, @barber2016p consider an example from neuroscience where each null hypothesis $H_{v,t}$ states that at $t$ seconds after the presentation of the stimulus, a chosen feature of the stimulus is unrelated to (a specific measure of) the brain activity at voxel $v$ in the brain. One may spatially group these hypotheses according to pre-defined regions of interest (ROIs) such as the visual cortices V1 to V5 and the left/right temporal cortices. This spatial partition allows us to capture the idea that either an ROI is unrelated to the stimulus, or many of its voxels $v$ will be related. Similarly, for a fixed voxel, one may temporally group the hypotheses, to capture the idea that either a voxel $v$ will remain unrelated to the stimulus at various delays, or it will be related at several consecutive delays (usually $t=4,6,8,10$ seconds after stimulus onset).
The earlier [[`p-filter`]{.nodecor}]{} algorithm [@barber2016p] simultaneously guarantees FDR control for multiple arbitrary and possibly non-hierarchical partitions, while ensuring that the elementary and group rejections are “internally consistent.” It is a multi-dimensional step-up procedure, one which reduces to the BH step-up procedure or Simes’ GN test in special cases. The reader may refer to the original paper for numerical simulations and more details on the neuroscience example.
Our contributions {#our-contributions .unnumbered}
-----------------
Consider a collection $\{H_1, \ldots, H_n \}$ of $n$ unordered hypotheses, along with associated $p$-values $\{P_1,\dots,P_n\}$. It is convenient to introduce the shortand notation $[n] {\ensuremath{:\, =}}\{1,\dots,n\}$. Further, consider $M$ arbitrary unordered partitions of these hypotheses into groups, where the $m$-th partition (“layer”) contains an unordered set of $G^{(m)}$ groups: $$\begin{aligned}
\small A^{(m)}_1,\dots,A^{(m)}_{G^{(m)}}\subseteq [n] \text{ for
$m=1,\dots,M$.}\end{aligned}$$ It may help the reader to imagine the first partition as being the elementary or finest partition, meaning that it contains $n$ groups of one hypothesis each, but it is important to note that this partition is entirely optional, and can be dropped if there is no desire of controlling the overall FDR. If ${\mathcal{H}_0}\subseteq n$ are the true nulls, then we call a group $g$ null if $A^{(m)}_g \subseteq
{\mathcal{H}_0}$. Let the set of null groups in partition $m$ be denoted by ${\mathcal{H}_0^{(m)}}$.
Although we continue to use the name [[`p-filter`]{.nodecor}]{} for the algorithm that we discuss in this paper, the algorithm goes significantly beyond the original algorithm; in particular, in our general setting of $M$ arbitrary partitions there are seven ways in which the new procedure to be developed here goes beyond the original framework.
1. **Overlapping groups.** We allow the groups in any partition to overlap. An elementary hypothesis need not be part of just a single group and we let $g^{(m)}(i)$ denote the set of groups in the $m$-th partition to which $P_i$ belongs—viz. $$\begin{aligned}
g^{(m)}(i) = \{g \in [G^{(m)}] : P_i \in A^{(m)}_g \}.\end{aligned}$$ For example, in the neuroscience example introduced earlier, if the scientist is unsure about the accuracy of the ROI borders, they may place boundary hypotheses into two or more ROIs to reflect this uncertainty.
2. **Incomplete partitions.** We allow partitions to be incomplete—let the $m$-th partition’s leftover set $\L^{(m)} \subset [n]$ represent all elementary hypotheses that do not belong to any group in the $m$-th partition: $$\begin{aligned}
\small \L^{(m)} = [n] \backslash \bigcup_g A^{(m)}_g
$$ This gives additional flexibility to the user who may not want to assign some hypotheses to any groups. Note that $\L^{(m)}$ is not just another group; this set is not counted when calculating the group-level FDR in layer $m$, meaning that elementary discoveries within $\L^{(m)}$ do not alter the group FDR at layer $m$. Hence, hypotheses in this leftover set have no internal consistency constraints imposed by layer $m$. For instance, in the neuroscience example, if it is determined (for example due to brain damage or surgery) that some voxels may not naturally fit into any ROI, then they can be left out of that partition.
3. **Internal consistency (IC)**. In order to maintain interpretability when dealing with overlapping groups, it is convenient to introduce two natural notions of internal consistency of the group rejections and elementary rejections:
- **Weak IC**. We reject $H_i$ if and only if in every partition, either there is at least one rejected group containing $i$, or $i\in L^{(m)}$.
- **Strong IC**. We reject $H_i$ if and only if in every partition, either every group that contains $i$ is rejected, or $i\in L^{(m)}$.
These definitions[^1] of IC are extensions to the multilayer setting of the notions of *coherence* and *consonance* as defined by @gabriel1969simultaneous, and explored in the FWER literature by Sonnemann and Finner [@sonnemann1982allgemeine; @sonnemann2008general; @sonnemann1988vollstandigkeitssatze], and @romano2011consonance. In the aforementioned neuroscience example, weak internal consistency may be more appropriate.
4. **Weights.** The $m$-th partition can be associated with two sets of positive weights, one pair for each group $g$ in that partition: $$\begin{aligned}
\text{Penalties } \{u^{(m)}_g\} \text{ and priors } \{w^{(m)}_g\},
\text{ such that } \sum_{g=1}^{G^{(m)}} u^{(m)}_g w^{(m)}_g = G^{(m)}.\end{aligned}$$ This generalizes work on doubly-weighted procedures by @blanchard2008two, who considered a single partition. Their work in turn generalizes earlier work using prior weights [@genovese2006false] and penalty weights @BH97 separately. Large prior weights indicate beliefs that the hypotheses are more likely to be non-null, and large penalties reflect which hypotheses are more scientifically important. For example, in the neuroscience example, weights can also be used to take differing ROI sizes into account, or prior knowledge of when and where effects are expected to be found.
5. **Reshaping.** Reshaping functions $\beta$ are used to guard against possible dependence among the $p$-values by *undercounting* the size (or weight) of rejected sets. Reshaping makes it possible to handle arbitrary dependence; on the other hand, this favorable robustness property is accompanied by a loss of power. Following Blanchard and Roquain [@blanchard2008two], for any probability measure $\nu$ on $[0,\infty)$, we define the reshaping function $$\begin{aligned}
\label{eqn:reshaping}
\beta(k) {\ensuremath{:\, =}}\int_{0}^k x~ \mathsf{d}\nu(x) ~\leq~ k.\end{aligned}$$ If the $p$-values within or across layers are arbitarily dependent, we may use reshaping functions $\beta^{(m)}$ to reshape thresholds in layer $m$. In the special case that Simes $p$-values are used to form the group-level $p$-values $P^{(m)}_g$, we may use reshaping functions ${\widetilde\beta}^{(m)}_g$ to protect the Simes procedure from arbitrary dependence within the group. The original procedure of @BY01 corresponds to choosing the reshaping function $\beta_{BY}(k) =
\frac{k}{\sum_{i=1}^n \frac1i}$. Many other examples and their connections to other formulations of multiple testing methods can be found in the literature [@blanchard2008two; @sarkar2008methods; @sarkar2008two]. In contrast to the discrete distributions which have been the focus of past work, in the current paper it is necessary to consider continuous measures since the penalty weight of rejected hypotheses, unlike their count, can be fractional.
6. **Adaptivity.** For any partition whose group $p$-values are known to be independent (i.e., independence *between* groups, but not necessarily *within* each group), we can incorporate “null-proportion adaptivity” for that partition [@hochberg1990more; @benjamini2000adaptive]. For partition $m$, we fix a user-defined constant $\lambda^{(m)} \in (0,1)$, and define a weighted null-proportion estimator:
$$\begin{aligned}
\label{eqn:pihatm}
{\widehat{\pi}}^{(m)} := \frac{|u^{(m)} \cdot w^{(m)}|_\infty + \sum_{g}
u^{(m)}_g w^{(m)}_g {{\bf{1}}\left\{{P^{(m)}_g >
\lambda^{(m)}}\right\}}}{G^{(m)} (1-\lambda^{(m)})} \;.
$$
The use of null-proportion adaptivity in any *one layer* may improve the power in *all layers*, since more groups being discovered in one layer leads to more elementary discoveries, and hence more discovered groups in other layers. For a single group with no weights, our approach reduces to the original suggestion of Storey et al. [@Storey02; @Storey04].
7. **Arbitrary group $p$-values.** Our new ${\textnormal{\texttt{p-filter}}}$ algorithm is no longer tied to the use of Simes $p$-values at the group layers, unlike the original algorithm. In other words, each group-level $p$-value at each layer can be formed by combining the elementary $p$-values within that group [@vovk2012combining; @heard2018choosing]. When the $p$-values are independent, some options include Fisher’s $-2\sum_i \ln P_i$ and Rosenthal’s $\sum_i \Phi^{-1}(P_i)$, where $\Phi$ is the Gaussian CDF (originally proposed by @stouffer1949american). When there are very few non-nulls, the Bonferroni correction is known to be more powerful, and it also works under arbitrary dependence, as does Rüschendorf’s proposal of $2 \sum_i P_i / n$, and Rüger’s proposal of $P_{(k)} \cdot n/k$ for a fixed $k$. Alternately the group $p$-values can be constructed directly from raw data. Accordingly, we can appropriately use adaptivity or reshaping, as needed, depending on the induced dependence.
Suppose a procedure rejects a subset ${\widehat{\mathcal{S}}}\subseteq[n]$ of hypotheses and a subset ${{\widehat{\mathcal{S}}}^{(m)}}\subseteq[G^{(m)}]$ of groups in partition $m$. Then, we may define the penalty-weighted FDR as $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u = {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}{\sum_{g \in [G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}{\sum_{g \in [G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}{\sum_{g \in [G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}{\sum_{g \in [G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]} \;.\end{aligned}$$ The rejections made by [[`p-filter`]{.nodecor}]{} will be internally consistent, and satisfy $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \alpha^{(m)} \text{ simultaneously for all } m = 1,\dots,M.\end{aligned}$$ In order to handle the possibility of a ratio “$\frac{0}{0}$” in this definition, and in our later work, we adopt the “dotfraction” notation $$\begin{aligned}
\label{eqn:define_dotfrac}
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} & {\ensuremath{:\, =}}\begin{cases}
0,&\text{ if } a=0,\\
\frac{a}{b},&\text{ if }a \neq 0,b\neq 0,\\
\text{undefined } &\text{ if }a\neq 0, b=0.
\end{cases}\end{aligned}$$ Dotfractions behave like fractions whenever the denominator is nonzero. The use of dotfractions simplifies the presentation: note that ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} \neq \frac{a}{\max(b,1)}$ since $b$ may be fractional due to the use of penalty weights. We formally derive properties of dotfractions in the supplement ([Appendix \[app:dotfrac\]]{}).
When there is only one partition, and the weights equal one, the quantity ${\textnormal{FDR}}^{(m)}_u$ reduces to the usual FDR defined by @BH95, and ${\textnormal{\texttt{p-filter}}}$ reduces to the BH procedure[^2]. Many other procedures are recovered as special cases of ${\textnormal{\texttt{p-filter}}}$, as detailed after [Theorem \[thm:pf+\]]{}.
Lg control and internal consistency for multiple partitions {#sec:pf+}
===========================================================
Even in the case of just two partitions—one partition of groups and the elementary partition of individual hypotheses—it is non-trivial to provide a guarantee of internal consistency while controlling both group-level and individual-level FDR. For example, a sequential procedure, of first rejecting groups at a target FDR level $\alpha_1$ and then rejecting individual hypotheses within rejected groups at level $\alpha_2$, may not control the elementary FDR (due to not accounting for selection bias), and may not be internally consistent (because there might be a group rejected in the first round, with none of the elementary hypotheses in this group rejected in the second round). Further, such a method is not easily generalized to non-hierarchical partitions. Similarly, a parallel procedure that independently runs FDR procedures on the groups and on the individuals, may also fail to be internally consistent. Naively intersecting these rejections—that is, rejecting those hypotheses whose groups are rejected at every layer—may also fail to control the FDR (see @barber2016p for explicit examples).
The [[`p-filter`]{.nodecor}]{} algorithm is a multivariate extension of classical step-up procedures that is roughly based on the following sequence of steps:
- Select all hypotheses in each layer whose $p$-values are smaller than some initial layer-specific threshold.
- Reject an elementary hypothesis if it is contained in a selected group in every layer.
- In each layer, reject a group hypothesis if it contains a rejected elementary hypothesis. Then, estimate the group-FDP in each layer.
- Lower the initial thresholds at each layer, and repeat the steps above, until the group-FDP is below the desired level for all partitions.
Next, we discuss the necessary dependence assumptions and then derive the [[`p-filter`]{.nodecor}]{} algorithm that implements the above scheme.
Marginal and joint distributional assumptions on $p$-values
-----------------------------------------------------------
We assume that the marginal distribution of each null $p$-value is stochastically larger than the uniform distribution, referred to as *super-uniform* for brevity. Formally, for any index $i \in {\mathcal{H}_0}$, we assume that $$\begin{aligned}
\label{EqnSuperUniform}
{\textnormal{Pr}\!\left\{{P_i \leq t}\right\}} \leq t \quad\mbox{for all $t \in [0,1]$.}\end{aligned}$$ Of course, uniformly-distributed $p$-values trivially satisfy this condition. We use the phrase *under uniformity* to describe the situation in which the null $p$-values are marginally exactly uniform. If this phrase is not employed, it is understood that the null $p$-values are marginally super-uniform.
Regarding assumptions on the joint distribution of $p$-values, three possible kinds of dependence will be considered in this paper: independence, positive dependence or arbitrary dependence. In the independent setting, null $p$-values are assumed to be mutually independent, and independent of non-nulls. In the arbitrary dependence setting, no joint dependence assumptions are made on the $p$-values. The last case is that of positive dependence, as formalized by the *Positive Regression Dependence on a Subset* (PRDS) condition [@lehmann1966some; @sarkar1969some; @BY01]. In order to understand its definition, it is helpful to introduce some basic notation. For a pair of vectors $x, y \in [0,1]^n$, we use the notation $x \preceq y$ to mean that $x \leq y$ in the orthant ordering, i.e., $x_i \leq y_i$ for all $i \in \{1, \dots, n\}$. A set ${\ensuremath{\mathcal{D}}}\subseteq [0,1]^n$ is said to be *nondecreasing* if $x \in
{\ensuremath{\mathcal{D}}}$ implies $y \in {\ensuremath{\mathcal{D}}}$ for all $y \succeq x$. We say that a function $f: [0,1]^n \mapsto {\ensuremath{[0,\infty)}}$ is *nonincreasing*, if $x
\preceq y$ implies $f(x) \geq f(y)$.
\[ass:PRDS\] We say that the vector $P$ satisfies PRDS if for any null index $i \in
{\mathcal{H}_0}$ and nondecreasing set $ {\ensuremath{\mathcal{D}}}\subseteq [0,1]^n$, the function $t~\mapsto~{\textnormal{Pr}\!\left\{{P\in {\ensuremath{\mathcal{D}}}}\ \middle| \ {P_i \leq t}\right\}}$ is nondecreasing over $t\in(0,1]$.
The original positive regression dependence assumption was introduced by @lehmann1966some in the bivariate setting and by @sarkar1969some in the multivariate setting, and extended to the PRDS assumption first made by @BY01. These previous papers used the equality $P_i = t$ instead of the inequality $P_i \leq t$ in the definitions, but one can prove that both conditions are essentially equivalent.
The PRDS condition holds trivially if the $p$-values are independent, but also allows for some amount of positive dependence. For intuition, suppose that is a multivariate Gaussian vector with covariance matrix $\Sigma$; the null components correspond to Gaussian variables with zero mean. Letting $\Phi$ be the CDF of a standard Gaussian, the vector of $p$-values is PRDS on $P_i$ for every index $i$ if and only if all entries of the covariance matrix $\Sigma$ are non-negative. See @BY01 for additional examples of this type.
Specifying the Lg algorithm
---------------------------
In order to run the ${\textnormal{\texttt{p-filter}}}$ algorithm, we need to search for rejection thresholds for each layer. These thresholds will be parametrized by *weighted* discovery counts $k^{(m)}\in[0,G^{(m)}]$ for each layer $m=1,\dots,M$. The reader is cautioned that each $k^{(m)}$ need not be an integer but instead should be viewed as a real number corresponding to the total rejected penalty weight. If the weights $u^{(m)}_g$ are all set equal to one, then $k^{(m)}$ corresponds to the number of groups in layer $m$ that are rejected. Given some prototypical vector $\vec{k} {\ensuremath{:\, =}}(k^{(1)},\dots,k^{(M)})$, we first perform an initial screening on each layer separately: $$\begin{aligned}
\label{eqn:Shm}
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k}) = \left\{g\in[G^{(m)}]: P^{(m)}_g
\leq \min \Big \{ \tfrac{ w^{(m)}_{g} \alpha^{(m)}
\beta^{(m)}(k^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)} \Big \}
\right\}.\end{aligned}$$ If the groups in partition $m$ are independent, we replace $\beta^{(m)}(k^{(m)})$ by just $k^{(m)}$, and set ${\widehat{\pi}}^{(m)}$ using ; on the other hand, if they are arbitrarily dependent, we set ${\widehat{\pi}}^{(m)}=1$ and $\lambda^{(m)}=1$. This convention allows the same expressions to be used in all settings.
For weak internal consistency, we define the elementary rejections as
$$\begin{aligned}
\label{eqn:hatS-defn}
{\widehat{\mathcal{S}}}(\vec{k}) & = {\widehat{\mathcal{S}}}_{\textnormal{weak}}(\vec{k})= \bigcap_{m=1}^M
\Biggr( \Big[\bigcup_{g\in {{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k})}
A^{(m)}_g \Big] \cup L^{(m)} \Biggr) \nonumber \\
&= \{ P_i : \text{$\forall m$, either } P_i \in L^{(m)}, \text{ or
$\exists \ g \in g^{(m)}(i)$, } A^{(m)}_g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k}) \}.\end{aligned}$$
Alternately, for strong internal consistency, we may instead define $$\begin{aligned}
\label{eqn:hatS-defn-strong}
{\widehat{\mathcal{S}}}(\vec{k}) &= {\widehat{\mathcal{S}}}_{\textnormal{strong}}(\vec{k})= \bigcap_{m=1}^M
\left([n]\backslash \bigcup_{g\in[G^{(m)}]\backslash
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k})} A^{(m)}_g\right) \nonumber \\
& = \{ P_i : \text{$\forall m$, either } P_i \in L^{(m)}, \text{ or
$\forall \ g \in g^{(m)}(i)$, } A^{(m)}_g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k}) \}.\end{aligned}$$
Finally, using either ${\widehat{\mathcal{S}}}(\vec{k})={\widehat{\mathcal{S}}}_{\textnormal{weak}}(\vec{k})$ or ${\widehat{\mathcal{S}}}(\vec{k})={\widehat{\mathcal{S}}}_{\textnormal{strong}}(\vec{k})$, we redefine the set of groups in layer $m$ which are rejected as: $$\begin{aligned}
\label{eqn:hatSm-defn}
{{\widehat{\mathcal{S}}}^{(m)}}(\vec{k}) = \left\{ g \in [G^{(m)}]: A^{(m)}_g\cap {\widehat{\mathcal{S}}}(\vec{k})
\neq \emptyset \text{ and } g \in {{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k})
\right \}.\end{aligned}$$ Examining these definitions, it may be verified that (weak or strong) internal consistency is satisfied by the rejections ${\widehat{\mathcal{S}}}(\vec{k}),
{\widehat{\mathcal{S}}}_m(\vec{k})$.
Of course, these definitions depend on the initial choice of the vector $\vec{k}$. Since we would like to make a large number of discoveries, we would like to use a $\vec{k}$ that is as large as possible (coordinatewise), while at the same time controlling the layer-specific FDRs, which are the expectations of $$\begin{aligned}
\small {{\textnormal{FDP}}^{(m)}}_u(\vec{k}) {\ensuremath{:\, =}}{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}} }{\sum_{g \in [G^{(m)}]}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}} }{\sum_{g \in [G^{(m)}]}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}} }{\sum_{g \in [G^{(m)}]}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}} }{\sum_{g \in [G^{(m)}]}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})}\right\}}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} \;.\end{aligned}$$
Now, define the data-dependent set of feasible vectors $\vec{k}$ as $$\begin{aligned}
\label{eqn:feasible-thresholds}
{\widehat{\mathcal{K}}}= \left \{ \vec{k} \in [0,G_1] \times \dots \times [0,G_M] :
\sum_{g \in {{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})} u^{(m)}_g \geq k^{(m)} \textnormal{ for all
} m \right \},\end{aligned}$$ where we suppress the implicit dependence of ${\widehat{\mathcal{K}}}$ on input parameters such as $\alpha^{(m)}, \lambda^{(m)},
\{w^{(m)}_g\},\{u^{(m)}_g\}$. In particular, if the penalty weights are all equal to one, then the consistency condition defining the “feasible” $\vec{k}$’s is equivalent to requiring that $|{{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})| \geq k^{(m)}$ for all $m=1,\dots,M$; i.e., the numbers of rejections in each layer at the vector $\vec{k}$ are elementwise $\geq \vec{k}$. This condition can be viewed as a generalization, to the multi-partition setting, of the “self-consistency” condition described by @blanchard2008two.
It is also worth noting that the ${\textnormal{\texttt{p-filter}}}$ algorithm in @barber2016p was derived in terms of thresholds $\vec{t}$ instead of number of rejections $\vec{k}$, and there the corresponding feasiblity condition was that ${\widehat{{\textnormal{FDP}}}}_m(\vec{t}) \leq \alpha^{(m)}$, where ${\widehat{{\textnormal{FDP}}}}_m(\vec{t})$ is an empirical-Bayes type estimate of the FDP. Indeed, if we avoid ${\widehat{\pi}}^{(m)},\beta^{(m)},w^{(m)},u^{(m)}$ for simplicity, then associating ${\widehat{t}}^{(m)}$ to $\alpha^{(m)} {\widehat{k}}^{(m)} /
G^{(m)}$ and comparing our derivation to that of @barber2016p, we can see that the “self-consistency” viewpoint and the “empirical-Bayes” viewpoint are equivalent and lead to the same algorithm. However, when dealing with reshaping under arbitrary dependence, the proofs are simpler in terms of $\vec{k}$ than in terms of $\vec{t}$, explaining our switch in notation.
As with the BH and BY procedures, we then choose the largest feasible thresholds $k^{(m)}$, given by: $$\begin{aligned}
\label{eqn:max-threshold-m}
{\widehat{k}}^{(m)} = \max\left\{k^{(m)} : \exists k^{(1)},\dots,k^{(m-1)},k^{(m+1)},\dots,k^{(M)}\textnormal{ s.t. } \vec{k} \in{\widehat{\mathcal{K}}}\right\} \;. \end{aligned}$$ This choice defines our algorithm: the ${\textnormal{\texttt{p-filter}}}$ algorithm rejects the hypotheses ${\widehat{\mathcal{S}}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$, as defined in or , with rejections at layer $m$ given by ${{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ as defined in . Next, we present the theoretical guarantees associated with ${\textnormal{\texttt{p-filter}}}$.
Theoretical guarantees
----------------------
The following proposition states that the set of feasible vectors ${\widehat{\mathcal{K}}}$ actually has a well-defined “maximum” corner.
\[prop:max\] Let the set of feasible vectors ${\widehat{\mathcal{K}}}$ be defined as in equation , and let the partition-specific maximum feasible vector ${\widehat{k}}^{(m)}$ be defined as in equation . Then we have $$\begin{aligned}
\label{eqn:feasible-thresholds2}
({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})\in{\widehat{\mathcal{K}}}\;.\end{aligned}$$
The proof is provided in [Section \[sec:proof\_max\]]{}; it is a generalization of the corresponding result for the original ${\textnormal{\texttt{p-filter}}}$ algorithm [@barber2016p].
The vector $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ is not just feasible from the perspective of self-consistency as captured by ${\widehat{\mathcal{K}}}$, but it is also feasible from the perspective of FDR control. Specifically, the next theorem establishes that—assuming for now that we can find $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$—selecting the set ${\widehat{\mathcal{S}}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ guarantees simultaneous control of ${\textnormal{FDR}}_u^{(m)}$ for all $M$ partitions. In this theorem, the notation ${\textnormal{Simes}}_w(P_{A^{(m)}_g})$ refers to the weighted Simes’ $p$-value (see [Appendix \[app:simes\]]{} in the supplement for details).
\[thm:pf+\] Any procedure that computes the vector $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ according to definition satisfies the following properties, for all partitions $m = 1,\dots,M$ simultaneously:
1. If the base $p$-values are independent, and all group $p$-values are given by $P^{(m)}_g = {\textnormal{Simes}}_w(P_{A^{(m)}_g})$, then employing adaptivity by defining ${\widehat{\pi}}^{(m)}$ as in guarantees that ${\textnormal{FDR}}_u^{(m)}\leq\alpha^{(m)}$.
2. If base $p$-values are positively dependent (PRDS) and group $p$-values are given by $P^{(m)}_g = {\textnormal{Simes}}_w(P_{A^{(m)}_g})$, then without adaptivity or reshaping, we have that ${\textnormal{FDR}}_u^{(m)}\leq
\alpha^{(m)} \frac{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g
w^{(m)}_g}{G^{(m)}} \leq \alpha^{(m)}$.
3. When all $p$-values are arbitrarily dependent, and are constructed arbitrarily (under the assumption that $P^{(m)}_g$ is super-uniform for any null group $g\in{\mathcal{H}_0^{(m)}}$, meaning it is a valid $p$-value), then using reshaping as in guarantees that ${\textnormal{FDR}}_u^{(m)}\leq \alpha^{(m)}\frac{\sum_{g \in
{\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g}{G^{(m)}} \leq \alpha^{(m)}$.
4. In the setting of part (c), if additionally the groups at layer $m$ are independent (that is, $P_{A^{(m)}_g}$ is independent from $P_{-A^{(m)}_g}$, for each $g\in[G^{(m)}]$), then using reshaping as in and adaptivity for layer $m$ as in , guarantees that ${\textnormal{FDR}}_u^{(m)}\leq\alpha^{(m)}$.
The proof, given in [Section \[sec:proof\_main\]]{}, uses three interpretable lemmas that we first discuss in [Section \[sec:lemmas\]]{}. It also introduces several new ideas to handle overlapping groups with dependent $p$-values. To remark on the difference between parts (c) and (d), what these two results guarantee is that if we use adaptivity for some set $\mathcal{M}_{\textnormal{adapt}}\subset[M]$ of layers, and do not use adaptivity (i.e. set ${\widehat{\pi}}^{(m)}=1$) for the remaining layers, then FDR control is maintained across [*all*]{} layers as long as, for each $m\in\mathcal{M}_{\textnormal{adapt}}$, the layer-specific independence statement holds—$P_{A^{(m)}_g}$ is independent from $P_{-A^{(m)}_g}$, for each $g\in[G^{(m)}]$. If this condition fails for some $m\in\mathcal{M}_{\textnormal{adapt}}$, the FDR control in other layers will in fact not be affected. One application of statement (d) is when the base $p$-values are independent, there are no overlapping groups, and group $p$-values are formed using a Fisher, Rosenthal, or other combinations of the base $p$-values. Recently, @katsevich17mkf proved that in case (d), the FDR is controlled even without using reshaping, albeit at a constant factor larger than the target level.
In practice, if we have accurate side information about group structures that the rejected hypotheses likely respect, then we may significantly improve our *precision*, achieving a lower FDR than the theoretical bound, without affecting our power much. However, inaccurate side information may significantly lower our power, since each $p$-value would have additional misguided constraints to meet. These issues were explored in simulations by @barber2016p.
#### Special cases
The setting with a single partition ($M=1$) recovers a wide variety of known algorithms. Considering only the finest partition with $n$ groups containing one hypothesis each, the [[`p-filter`]{.nodecor}]{} algorithm and associated [Theorem \[thm:pf+\]]{} together recover known results about (a) the BH procedure of @BH95 when weights, reshaping and adaptivity are not used, (b) the BY procedure of @BY01 when reshaping is used, (c) the prior-weighted BH procedure of @genovese2006false when only prior weights are used, (d) the penalty-weighted BH procedure of @BH97 when only penalty weights are used, (e) the doubly-weighted BH procedure of @blanchard2008two when both sets of weights and reshaping are used, and (f) the Storey-BH procedure of Storey et al. [@Storey02; @Storey04] when only adaptivity is used.
When we instantiate ${\textnormal{\texttt{p-filter}}}$ with the coarsest partitions with a single group containing all $n$ hypotheses, we recover exactly (g) the ${\textnormal{Simes}}$ test [@simes1986improved] without weights, and (h) a variant by @HL94 if prior weights are used. We recover the results of (i) the p-filter by @barber2016p under positive dependence, when we do not use weights, adaptivty, reshaping, overlapping groups, leftover sets, and restrict ourselves to Simes’ $p$-values. We also recover a host of new procedures: for example, while the past literature has not yet shown how to use either prior or penalty weights together with adaptivity, [[`p-filter`]{.nodecor}]{} reduces to (j) a doubly-weighted adaptive procedure for the finest partition under independence. Also, while the aforementioned procedures were each proved under one form of dependence or the other, we recover results for all three forms of dependence at one go, with a single unified proof technique.
An efficient implementation
---------------------------
Although one can employ a brute-force grid search to find $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$, the `p-filter` algorithm presented in Algorithm \[alg:multi-layer\_fdr\] is able to find this vector efficiently using a coordinate-descent style procedure, and is a strict generalization of the algorithm by the same name in @barber2016p.
**Input:** $M$ possibly incomplete partitions of possibly overlapping groups of indices $[n]$;\
A vector of base $p$-values $P\in[0,1]^n$;\
Group $p$-values $P^{(m)}_g$ for each group $g=1,\dots,G^{(m)}$ in layers $m=1,\dots,M$;\
$M$ target FDR levels $\{\alpha^{(m)}\}$;\
$M$ sets of prior weights and/or penalty weights $\{w^{(m)}_g, u^{(m)}_g\}$;\
$M$ thresholds for adaptive null proportion estimation $\{\lambda^{(m)}\}$;\
$M$ reshaping functions $\{\beta^{(m)}\}$, if desired. **Initialize:** Set $k^{(m)}=G^{(m)}$, and ${\widehat{\pi}}^{(m)}$ as in definition . Update the $m$th vector: defining ${{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})$ as in equation (using weak or strong internal consistency, as desired), let $$\begin{aligned}
k^{(m)} \leftarrow \max\left\{k'^{(m)} \in [0,G^{(m)}]: \sum\limits_{g
\in {{\widehat{\mathcal{S}}}^{(m)}}(k^{(1)},\dots,k^{(m-1)},k'^{(m)},k^{(m+1)},\dots,k^{(M)})}
u^{(m)}_g \geq k'^{(m)} \right\}\end{aligned}$$ **Output:** Vector ${\widehat{k}}=(k^{(1)},\dots,k^{(m)})$, rejected hypotheses ${\widehat{\mathcal{S}}}({\widehat{k}})$, and rejected groups ${{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}})$ in each partition.
The following proposition provides a correctness guarantee for Algorithm \[alg:multi-layer\_fdr\]:
\[prop:alg\] The output of Algorithm \[alg:multi-layer\_fdr\] is the maximum feasible corner $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(m)})$ defined in equations and .
This result was proved by @barber2016p in the setting of the original ${\textnormal{\texttt{p-filter}}}$ algorithm, where the $k^{(m)}$’s take only integer values; here, the algorithm is slightly more subtle, with real-valued $k^{(m)}$’s due to the presence of penalty weights $u^{(m)}_g$. The proof of the proposition for this more general setting is given in [Section \[sec:proof\_alg\]]{}.
Three lemmas {#sec:lemmas}
============
In this section, we present three lemmas that lie at the heart of the succinct proofs of the theorems in this paper. Our motivation for presenting these lemmas here is that they are interpretable, and provide valuable intuition for the proofs that follow.
A super-uniformity lemma for FDR control
----------------------------------------
In order to develop some intuition for the lemma that follows, we note that our super-uniformity assumption on the null $p$-values can be reformulated as: $$\begin{aligned}
\label{eqn:PRDS-fixed}
\text{For any $i \in {\mathcal{H}_0}$, ~}~ {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{{\bf{1}}\left\{{P_i\leq t}\right\}}}}{t}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{{\bf{1}}\left\{{P_i\leq t}\right\}}}}{t}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{{\bf{1}}\left\{{P_i\leq t}\right\}}}}{t}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{{\bf{1}}\left\{{P_i\leq t}\right\}}}}{t}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}
& \leq 1 \text{ for any fixed $t \in [0,1]$.}\end{aligned}$$ Of course, if $P_i$ is uniform then the above inequality holds with equality.
The following lemma guarantees that property continues to hold for certain random thresholds $f(P)$. Recall that the term “nonincreasing” is interpreted coordinatewise, with respect to the orthant ordering.
\[lem:power\] Let $i \in {\mathcal{H}_0}$ be a null hypothesis with $p$-value $P_i$, and let $P^{-i}$ denote the other $n-1$ $p$-values.
1. For any nonincreasing function $f:[0,1]^n\rightarrow[0,\infty)$, if $P_i$ is independent of $P^{-i}$, then we have $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\ \middle| \ { P^{-i}}\right]} \leq 1.\end{aligned}$$ Furthermore, if we additionally assume that $f$ has range $[0,1]$ and satisfies the LOOP condition (supplement, [Appendix \[app:LOOP\]]{}), and that $P_i$ is uniformly distributed, then the inequality is replaced with equality: $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\ \middle| \ { P^{-i}}\right]} = 1.\end{aligned}$$
2. For any nonincreasing function $f:[0,1]^n\rightarrow[0,\infty)$, if $P$ is PRDS with respect to $P_i$, then $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]} \leq 1.\end{aligned}$$
3. For any constant $c\geq 0$, any function $f:[0,1]^n\rightarrow[0,\infty)$, and any reshaping function $\beta$, under arbitrary dependence of the $p$-values, $$\begin{aligned}
\small {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq c\cdot \beta(f(P)) }\right\}}}{c\cdot f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq c\cdot \beta(f(P)) }\right\}}}{c\cdot f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq c\cdot \beta(f(P)) }\right\}}}{c\cdot f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq c\cdot \beta(f(P)) }\right\}}}{c\cdot f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]}
\leq 1.\end{aligned}$$
4. For any constant $c\geq 0$, any functions $f_1,\dots,f_m:[0,1]^n\rightarrow[0,\infty)$, and any reshaping functions $\beta_1,\dots,\beta^{(m)}$, under arbitrary dependence of the $p$-values, $$\begin{aligned}
\small {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq c\cdot
\prod_{\ell=1}^m\beta_\ell(f_\ell(P)) }\right\}}}{c\cdot \prod_{\ell=1}^m
f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq c\cdot
\prod_{\ell=1}^m\beta_\ell(f_\ell(P)) }\right\}}}{c\cdot \prod_{\ell=1}^m
f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq c\cdot
\prod_{\ell=1}^m\beta_\ell(f_\ell(P)) }\right\}}}{c\cdot \prod_{\ell=1}^m
f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq c\cdot
\prod_{\ell=1}^m\beta_\ell(f_\ell(P)) }\right\}}}{c\cdot \prod_{\ell=1}^m
f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]} \leq 1 .\end{aligned}$$
The proofs of statement (a) with equality, and of statement (d), are given in [Section \[sec:power-proof\]]{}. Statement (a) with inequality is recovered as a special case of statement (b), which was proved by @blanchard2008two, who also proved (c). The more general statement (d), with more than one reshaping function present in the bound, will be required in the proof of the following novel group super-uniformity [Lemma \[lem:power3\]]{}.
A group-level super-uniformity lemma
------------------------------------
In analogy to the super-uniformity [Lemma \[lem:power\]]{}, we present the following lemma, which contains analogous bounds under the settings of independent or positively dependent base $p$-values (in which case the group $p$-value is constructed with a Simes $p$-value), and in the setting of arbitrarily dependent base $p$-values (in which case the group $p$-value can be constructed by any method—reshaped Simes, Fisher, or others—as long as it is a valid $p$-value.)
\[lem:power3\] Let $g \in {\mathcal{H}_0^{{\textnormal{grp}}}}$ be a null group, that is, $A_g\subseteq
{\mathcal{H}_0}$. Let $P_{A_g}$ denote the $p$-values in this group, $P_{A_g}=(P_j)_{j\in A_g}$, and let $P_{-A_g}$ denote the remaining $p$-values, $P_{-A_g}=(P_j)_{j\not \in A_g}$.
1. If $f:[0,1]^n\rightarrow[0,\infty)$ is a nonincreasing function, and the base $p$-values $P_1,\dots,P_n$ are independent, then $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\ \middle| \ {P_{-A_g}}\right]} \leq 1.\end{aligned}$$
2. If $f:[0,1]^n\rightarrow[0,\infty)$ is a nonincreasing function, and the base $p$-values $P_1,\dots,P_n$ are positively dependent (PRDS), then $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{{\textnormal{Simes}}_w(P_{A_g})\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]} \leq 1.\end{aligned}$$
3. If the base $p$-values $P_1,\dots,P_n$ are arbitrarily dependent, then for any constant $c>0$, any reshaping function $\beta$, and any function $f:[0,1]^n\rightarrow[0,\infty)$, we have $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{T(P_{A_g})\leq c \beta(f(P)) }\right\}}}{cf(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{T(P_{A_g})\leq c \beta(f(P)) }\right\}}}{cf(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{T(P_{A_g})\leq c \beta(f(P)) }\right\}}}{cf(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{T(P_{A_g})\leq c \beta(f(P)) }\right\}}}{cf(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]} \leq 1,\end{aligned}$$ where $T:[0,1]^{|A_g|}\rightarrow[0,1]$ is any valid group $p$-value; i.e., any function with the property that $T(P_{A_g})$ is super-uniform whenever $g$ is null.
4. Let $g_1,\dots,g_k$ be a set of $k$ possibly overlapping null groups, meaning $A_{g_1},\dots,A_{g_k} \subseteq {\mathcal{H}_0}$, and $S_1,\dots,S_k$ represent the corresponding Simes’ $p$-values. If $f:[0,1]^n\rightarrow[0,\infty)$ is a nonincreasing function, and the base $p$-values $P_1,\dots,P_n$ are positively dependent (PRDS), then $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{{\textnormal{Simes}}(S_1,\dots,S_k) \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{{\textnormal{Simes}}(S_1,\dots,S_k) \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{{\textnormal{Simes}}(S_1,\dots,S_k) \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{{\textnormal{Simes}}(S_1,\dots,S_k) \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\right]} \leq 1.\end{aligned}$$
The proof of this lemma relies on [Lemma \[lem:power\]]{}, and can be found in [Section \[sec:power2-proof\]]{}. We remark that statement (d) is different from statement (b) applied to the null group $g =
\bigcup_{i=1}^k g_i$; indeed, in statement (d), the arguments to the Simes’ procedure are themselves Simes’ $p$-values, and not the original base $p$-values. If desired, statement (d) can be further bootstrapped to apply to the root of an entire tree of null groups, where each internal node stores the Simes’ $p$-value calculated on its children.
As an aside, one may wonder whether the Simes’ $p$-values are themselves positively dependent (PRDS), given that they satisfy a super-uniformity lemma much like the PRDS $p$-values. We have neither been able to prove nor disprove such a claim, and it may be of independent interest to do so.
An inverse binomial lemma for adaptivity with weights
-----------------------------------------------------
The following lemma is required for the proof of adaptivity with weights; more specifically, we use it to bound the expected inverse of the doubly-weighted null-proportion estimate.
\[lem:power2\] Given a vector $a \in [0,1]^{d}$, constant $b \in [0,1]$, and Bernoulli variables $Z_i ~\stackrel{\textnormal{i.i.d.}}{\sim}~
\text{Bernoulli}(b)$, the weighted sum $Z {\ensuremath{:\, =}}1 + \sum_{i=1}^{d} a_i
Z_i$ satisfies $$\begin{aligned}
\frac{1}{1 + b \sum_{i=1}^d a_i } \leq {\mathbb{E}\left[{\frac{1}{Z}}\right]} & \leq
\frac{1}{b (1 + \sum_{i=1}^d a_i)}.\end{aligned}$$
Since ${\mathbb{E}\left[{Z}\right]} = 1 + b \sum_{i=1}^d a_i $, the lower bound on ${\mathbb{E}\left[{1/Z}\right]}$ follows by Jensen’s inequality. We include this bound to provide context for the upper bound on ${\mathbb{E}\left[{1/Z}\right]}$, whose proof can be found in [Section \[sec:power3-proof\]]{}. When $a_i=1$ for all $i$ and $b=1$, the claim follows by a standard property of binomial distributions, as described in @benjamini2006adaptive.\
With these three lemmas in place, we now turn to the proof of the main theorem in the next section.
Proof of Theorem {#sec:proof_main}
=================
In order to be able to handle all four cases of the theorem, we define a function $\gamma^{(m)}$ to be the identity if we are not using reshaping (theorem statements (a,b)), or $\gamma^{(m)}=\beta^{(m)}$ if we are using reshaping (theorem statements (c,d)). We also let ${\widehat{\pi}}^{(m)}=1$ and $\lambda^{(m)}=1$ if we are not using adaptivity (theorem statements (b,c)), or let ${\widehat{\pi}}^{(m)}$ be defined as in equation where adaptivity is used (theorem statements (a,d)).
Fix any partition $m$. Since ${\textnormal{Pr}\!\left\{{P_i=0}\right\}}=0$ for any $i\in{\mathcal{H}_0}$ by assumption, we assume that $P_i\neq 0$ for any $i\in{\mathcal{H}_0}$ without further mention; this assumption then implies that if $
g\in{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ for some null group $
g\in{\mathcal{H}_0^{(m)}}$, we must have ${\widehat{k}}^{(m)}>0$. We can then calculate $$\begin{aligned}
{{\textnormal{FDP}}^{(m)}}_u({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)}) &= {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ \sum_{g \in
[G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ \sum_{g \in
[G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ \sum_{g \in
[G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ \sum_{g \in
[G^{(m)}]} u^{(m)}_g {{\bf{1}}\left\{{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}
\\[1.5em] & \leq {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}, \\[1em] & \leq
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)}
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)}
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)}
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{g \in
{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})}\right\}}}{ {\widehat{k}}^{(m)}
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}, \\[1em] & = {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{P^{(m)}_g
\leq \min\{w^{(m)}_g \frac{\alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)} \} }\right\}}
}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{P^{(m)}_g
\leq \min\{w^{(m)}_g \frac{\alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)} \} }\right\}}
}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{P^{(m)}_g
\leq \min\{w^{(m)}_g \frac{\alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)} \} }\right\}}
}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g {{\bf{1}}\left\{{P^{(m)}_g
\leq \min\{w^{(m)}_g \frac{\alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)} \} }\right\}}
}{ {\widehat{k}}^{(m)} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} ,\end{aligned}$$ where the first inequality follows by definition of the feasible set ${\widehat{\mathcal{K}}}$, the second follows since ${{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})\subseteq{{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k})$ for any $\vec{k}$ by definition, and the last step uses the definition of ${{\widehat{\mathcal{S}}}^{(m)}}_{\textnormal{init}}(\vec{k})$ in (without reshaping, for theorem statements (a,b), or with reshaping for theorem statements (c,d)).
Multiplying the numerator and denominator of each term by $\frac{\alpha^{(m)} w^{(m)}_g}{G^{(m)}}$, and taking expectations on both sides, it follows that
$$\begin{aligned}
\label{eqn:maintheorem-midproof-new}
\textstyle {\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in
{\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)}
\} }\right\}}}{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{ G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)}
\} }\right\}}}{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{ G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)}
\} }\right\}}}{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{ G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)}
\gamma^{(m)}({\widehat{k}}^{(m)})}{{\widehat{\pi}}^{(m)} G^{(m)}}, \lambda^{(m)}
\} }\right\}}}{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{ G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}.\end{aligned}$$
With these calculations in place, we now prove the four statements of the theorem. Given the suggestive form of the above expression, it is natural to anticipate the use of the two super-uniformity lemmas.
#### Theorem statement (a)
Define the function $f^{(m)}_{g}$ that maps the vector $P$ to $\frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)} }{{\widehat{\pi}}^{(m)} G^{(m)}}$. Note that $f^{(m)}_{g}$ is a nonincreasing function of $P$, since ${\widehat{k}}^{(m)}$ is a nonincreasing function of $P$ by definition of our procedure, while ${\widehat{\pi}}^{(m)}$ is a nondecreasing function of $P$. We also define the quantity $$\begin{aligned}
\label{eqn:pihatm_g}
{\widehat{\pi}}^{(m)}_{-g} ~{\ensuremath{:\, =}}~ \frac{|u^{(m)} w^{(m)}|_\infty + \sum_{h \neq
g} u^{(m)}_h w^{(m)}_h {{\bf{1}}\left\{{ P^{(m)}_h >\lambda^{(m)}}\right\}}}{G^{(m)}
(1-\lambda^{(m)})}.\end{aligned}$$ Returning to expression , we may then deduce that $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \notag & \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in
{\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)}
G^{(m)}}, \lambda^{(m)} \} }\right\}}}{ {\widehat{\pi}}^{(m)} \frac{w^{(m)}_g
\alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)} G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)}
G^{(m)}}, \lambda^{(m)} \} }\right\}}}{ {\widehat{\pi}}^{(m)} \frac{w^{(m)}_g
\alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)} G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)}
G^{(m)}}, \lambda^{(m)} \} }\right\}}}{ {\widehat{\pi}}^{(m)} \frac{w^{(m)}_g
\alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)} G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
\min\{ \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)}
G^{(m)}}, \lambda^{(m)} \} }\right\}}}{ {\widehat{\pi}}^{(m)} \frac{w^{(m)}_g
\alpha^{(m)} {\widehat{k}}^{(m)}}{{\widehat{\pi}}^{(m)} G^{(m)}} }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\\ \notag&=
\frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g
{\mathbb{E}\left[{{{\bf{1}}\left\{{P^{(m)}_g \leq\lambda^{(m)}}\right\}}\cdot {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g
\leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g
\leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g
\leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g
\leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]} \\
\label{eqn:maintheorem-midproof-new-a}
& \stackrel{(i)}{\leq} \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in
{\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]},\end{aligned}$$ where inequality (i) holds because the event $P^{(m)}_g\leq
\lambda^{(m)}$ implies ${\widehat{\pi}}^{(m)}={\widehat{\pi}}^{(m)}_{-g}$. Conditioning on $P_{-A^{(m)}_g}$ for each group $g$ in expression , we get:
$$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u & \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\\ &= \frac{\alpha^{(m)}}{G^{(m)}}
\sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g} f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\ \middle| \ {P_{-A^{(m)}_g}}\right]}}\right]} \\
& \stackrel{(ii)}{=} \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{
{{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\ \middle| \ {P_{-A^{(m)}_g}}\right]}}\right]} \\
& \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g
w^{(m)}_g {\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}}\right]},\end{aligned}$$
where equality (ii) holds because ${\widehat{\pi}}^{(m)}_{-g}$ is a function of only the $p$-values outside of group $g$, i.e., of $P_{-A^{(m)}_g}$, while the last inequality holds by [Lemma \[lem:power3\]]{}(a).
Finally, observe that independence between the different groups of partition $m$ implies that the indicator variables ${{\bf{1}}\left\{{P^{(m)}_h
>\lambda^{(m)}}\right\}}$ are independent Bernoullis with probabilities $\geq
1-\lambda^{(m)}$ of success. Thus, as a consequence of [Lemma \[lem:power2\]]{}, we can prove that $$\begin{aligned}
\label{eqn:storey-wbinomial-maintheorem}
{\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}}\right]} ~\leq~ \frac{G^{(m)}}{\sum\limits_{h
\in {\mathcal{H}_0^{(m)}}} u^{(m)}_h w^{(m)}_h}.\end{aligned}$$
To establish property , let $b {\ensuremath{:\, =}}(1-\lambda^{(m)}),~ d {\ensuremath{:\, =}}|{\mathcal{H}_0^{(m)}}|-1$, and define $$\begin{aligned}
Z {\ensuremath{:\, =}}1+ \sum_{h \in {\mathcal{H}_0^{(m)}}, h \neq g} a_h {{\bf{1}}\left\{{P^{(m)}_h >
\lambda^{(m)}}\right\}} \text{ where } a_h &= \frac{u^{(m)}_h
w^{(m)}_h}{|u^{(m)}\cdot w^{(m)}|_\infty}.\end{aligned}$$ Since $Z \leq \frac{G^{(m)} (1-\lambda^{(m)})}{|u^{(m)} \cdot
w^{(m)}|_\infty} {\widehat{\pi}}^{(m)}_{-g}$ as the right-hand side expression sums over more indices than the left, applying [Lemma \[lem:power2\]]{} guarantees that $$\begin{aligned}
{\mathbb{E}\left[{\frac{|u^{(m)} \cdot w^{(m)}|_\infty}{G^{(m)} (1-\lambda^{(m)})
{\widehat{\pi}}^{(m)}_{-g}}}\right]} \leq {\mathbb{E}\left[{\frac1{Z}}\right]} \leq \frac{|u^{(m)}\cdot
w^{(m)}|_\infty}{(1-\lambda^{(m)})(|u^{(m)}\cdot
w^{(m)}|_\infty+\sum_{h\in {\mathcal{H}_0^{(m)}}, h \neq g}u^{(m)}_h
w^{(m)}_h)}.\end{aligned}$$ Some simple algebra then leads to property .
Plugging back into our bounds on FDR, we finally obtain $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u & \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}}\right]}\\ &\leq
\frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g
w^{(m)}_g\frac{G^{(m)}}{\sum\limits_{h \in {\mathcal{H}_0^{(m)}}} u^{(m)}_h
w^{(m)}_h} \\ &\leq \alpha^{(m)}.\end{aligned}$$
#### Theorem statement (b)
The proof of statement (b) follows the same steps as for (a), but without the need to condition on $P_{-A^{(m)}_g}$, since we do not use adaptivity. Define the function $f^{(m)}_{g}(P) = \frac{w^{(m)}_g \alpha^{(m)} {\widehat{k}}^{(m)} }{
G^{(m)}}$. Then $f^{(m)}_{g}$ is a nonincreasing function of $P$, since ${\widehat{k}}^{(m)}$ is a nonincreasing function of $P$.
Returning to , as in the proof of statement (a), we calculate $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq f^{(m)}_g(P)}\right\}}}{
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}.\end{aligned}$$ By [Lemma \[lem:power3\]]{}(b), we know that ${\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
f^{(m)}_g(P)}\right\}}}{ f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\leq 1$, and therefore $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g,\end{aligned}$$ as claimed.
#### Theorem statement (c)
We now turn to proving the method under reshaping. Define $f^{(m)}_g(P) = {\widehat{k}}^{(m)}$, and define constant $c^{(m)}_g = \frac{w^{(m)}_g \alpha^{(m)}}{G^{(m)}}$. Returning to , as before, we calculate $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g \cdot
\beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g \cdot
\beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g \cdot
\beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g \cdot
\beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}.\end{aligned}$$ By [Lemma \[lem:power3\]]{}(c), we know that ${\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g \cdot \beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g
\cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g \cdot \beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g
\cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g \cdot \beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g
\cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g \cdot \beta^{(m)}\big(f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g
\cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\leq 1$ since $P^{(m)}_g$ is assumed to be super-uniform for any null group $g\in{\mathcal{H}_0^{(m)}}$. Therefore, $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g.\end{aligned}$$
#### Theorem statement (d)
The proof of part (d) combines the calculations of part (a) (where adaptivity is used) with part (c) (where reshaping is used). Define $f^{(m)}_g = {\widehat{k}}^{(m)}$ and $c^{(m)}_g = \frac{w^{(m)}_g \alpha^{(m)}}{{\widehat{\pi}}^{(m)}_{-g}
G^{(m)}}$, where ${\widehat{\pi}}^{(m)}_{-g}$ is defined as in equation from part (a). Note that $c^{(m)}_g$ is no longer a constant, but nonetheless, proceeding as in part (a), we can calculate $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u &\leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g}\cdot
c^{(m)}_g \cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g}\cdot
c^{(m)}_g \cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g}\cdot
c^{(m)}_g \cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ {\widehat{\pi}}^{(m)}_{-g}\cdot
c^{(m)}_g \cdot f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}.\end{aligned}$$ Next we condition on the $p$-values outside the group $A^{(m)}_g$: $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u &\leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g\cdot \beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g}\cdot c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g\cdot \beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g}\cdot c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g\cdot \beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g}\cdot c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq
c^{(m)}_g\cdot \beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{
{\widehat{\pi}}^{(m)}_{-g}\cdot c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\ \middle| \ {P_{-A^{(m)}_g}}\right]}}\right]} \\
& = \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}} u^{(m)}_g
w^{(m)}_g {\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}\cdot {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{
{{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot \beta^{(m)}\big(
f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{
{{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot \beta^{(m)}\big(
f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{
{{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot \beta^{(m)}\big(
f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{
{{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot \beta^{(m)}\big(
f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot f^{(m)}_g(P)
}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\ \middle| \ {P_{-A^{(m)}_g}}\right]}}\right]},\end{aligned}$$ where the last step holds since ${\widehat{\pi}}^{(m)}_{-g}$ is a function of $P_{-A^{(m)}_g}$.
Finally, we apply [Lemma \[lem:power3\]]{}(c) to show that each of these conditional expected values is $\leq 1$. Of course, the subtlety here is that we must condition on $P_{-A^{(m)}_g}$. To do so, note that, after fixing $P_{-A^{(m)}_g}$, the function $f^{(m)}_g(P)$ can be regarded as a function of only the remaining unknowns (i.e., of $P_{A^{(m)}_g}$), and is still nonwincreasing, the value $c^{(m)}_g$ is now a constant; and $P^{(m)}_g = T^m_g(P_{A^{(m)}_g})$ is indeed super-uniform since, due to the independence of $P_{A^{(m)}_g}$ from $P_{-A^{(m)}_g}$, its distribution has not changed. Therefore, we can apply [Lemma \[lem:power3\]]{}(c) (with the random vector $P_{A^{(m)}_g}$ in place of $P$, while $P_{-A^{(m)}_g}$ is treated as constant), to see that ${\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P^{(m)}_g \leq c^{(m)}_g\cdot
\beta^{(m)}\big( f^{(m)}_g(P)\big)}\right\}}}{ c^{(m)}_g \cdot
f^{(m)}_g(P) }$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\ \middle| \ {P_{-A^{(m)}_g}}\right]}~\leq~1$, and therefore, $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g {\mathbb{E}\left[{\frac{1}{{\widehat{\pi}}^{(m)}_{-g}}}\right]}.\end{aligned}$$ Finally, we need to bound ${\widehat{\pi}}^{(m)}_{-g}$. As in the proof of part (a), we see that the indicator variables ${{\bf{1}}\left\{{P^{(m)}_h
>\lambda^{(m)}}\right\}}$ are independent, since $P^{(m)}_h=T^{(m)}_h(P_{A^m_h})$, and the sets of $p$-values $P_{A^m_h}$ are assumed to be independent from each other. Furthermore, since $T^{(m)}_h(P_{A^m_h})$ is assumed to be a valid $p$-value, i.e., super-uniform for any $h\in{\mathcal{H}_0^{(m)}}$, this means that the variable ${{\bf{1}}\left\{{P^{(m)}_h >\lambda^{(m)}}\right\}}$ is Bernoulli with chance $\geq
1-\lambda^{(m)}$ of success. Therefore, the bound calculated in the proof of part (a) holds here as well, and so $$\begin{aligned}
{\textnormal{FDR}}^{(m)}_u \leq \frac{\alpha^{(m)}}{G^{(m)}} \sum_{g \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_g w^{(m)}_g \frac{G^{(m)}}{\sum\limits_{h \in {\mathcal{H}_0^{(m)}}}
u^{(m)}_h w^{(m)}_h} = \alpha^{(m)}.\end{aligned}$$
This concludes the proof of all four parts of [Theorem \[thm:pf+\]]{}.
Proofs of supporting lemmas {#sec:lemmas-proofs}
===========================
In this section, we collect the proofs of some supporting lemmas.
Proof of super-uniformity Lemma {#sec:power-proof}
--------------------------------
[Lemma \[lem:power\]]{} follows directly from earlier work [@blanchard2008two; @barber2016p]. Statement (a) with inequality (but not with equality) follows as a special case of (b), since independence is a special case of positive dependence, and the distribution of a null $P_i$ does not change on conditioning on an independent set of $p$-values. Statement (c) was proved also by @blanchard2008two. We now prove the statements (a), (d).
#### Statement (a)
We prove the first part of [Lemma \[lem:power\]]{}, under the assumptions that the function $P \mapsto f(P)$ satisfies the leave-one-out property with respect to index $i$, and that $P_i$ is uniformly distributed and is independent of the remaining $p$-values. Since ${\textnormal{Pr}\!\left\{{P_i = 0}\right\}}=0$, we ignore this possibility in the following calculations. Since $f$ satisfies the LOOP condition, we have $$\begin{aligned}
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} = \frac{{{\bf{1}}\left\{{P_i\leq
f({{P}^{i \to 0}})}\right\}}}{f({{P}^{i \to 0}})}.\end{aligned}$$ This can be seen by separately considering what happens when the numerator on the left-hand side is zero or one.
Since $P^{-i}$ determines $f({{P}^{i \to 0}})$, it immediately follows that $$\begin{aligned}
{\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i\leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} }\ \middle| \ {P^{-i}}\right]} &= {\mathbb{E}\left[{
\frac{{{\bf{1}}\left\{{P_i\leq
f({{P}^{i \to 0}})}\right\}}}{f({{P}^{i \to 0}})}}\ \middle| \ {P^{-i}}\right]}\\ &=
\frac{{\textnormal{Pr}\!\left\{{P_i\leq
f({{P}^{i \to 0}})}\ \middle| \ {f({{P}^{i \to 0}})}\right\}}}{f({{P}^{i \to 0}})} \\ &= 1,
\end{aligned}$$ where the last step follows since $f$ has range $[0,1]$, and $P_i$ is uniformly distributed and is independent of ${{P}^{i \to 0}}$; therefore, we may deduce that ${\textnormal{Pr}\!\left\{{P_i\leq f({{P}^{i \to 0}})}\ \middle| \ {f({{P}^{i \to 0}})}\right\}} =
f({{P}^{i \to 0}})$. This concludes the proof of the super-uniformity lemma under independence and uniformity.
#### Statement (d)
For each $\ell=1,\dots,m$, let $\nu_{\ell}$ be a probability measure on $[0,\infty)$ chosen such that $\beta_\ell(k) = \beta_{\nu_\ell}(k) = \int_{x=0}^k x \;\mathsf{d}\nu_\ell(x)$, as in the definition of a reshaping function. Let $X_\ell\sim \nu_\ell$ be drawn independently for each $\ell=1,\dots,m$, and let $\nu$ be the probability measure on $[0,\infty)$ corresponding to the distribution of $Z = \prod_{\ell=1}^m X_\ell$. Then $$\begin{aligned}
c \cdot \prod_{\ell=1}^m\beta_\ell(f_\ell(P))
&=c \cdot \prod_{\ell=1}^m \left( \int_{x_\ell=0}^{f_\ell(P)} x_\ell\;\mathsf{d}\nu_\ell(x_\ell)\right)\\
&=c \cdot \int_{x_1=0}^{\infty} \dots \int_{x_m=0}^{\infty} \left( \prod_{\ell=1}^m x_{\ell} \cdot {{\bf{1}}\left\{{x_\ell\leq f_\ell(P)}\right\}}\right)\; \mathsf{d}\nu_m(x_m)\dots \mathsf{d}\nu_1(x_1)\\
&=c \cdot{\mathbb{E}\left[{ \prod_{\ell=1}^m \left(X_\ell\cdot {{\bf{1}}\left\{{X_\ell\leq f_\ell(P)}\right\}}\right)}\right]}\\
&=c \cdot{\mathbb{E}\left[{Z \cdot {{\bf{1}}\left\{{X_1\leq f_1(P),\dots,X_m\leq f_m(P)}\right\}}}\right]}\\
&\leq c \cdot{\mathbb{E}\left[{Z \cdot {{\bf{1}}\left\{{Z \leq \prod_{\ell=1}^m f_\ell(P)}\right\}}}\right]}\\
&=c \cdot\int_{z=0}^{ \prod_{\ell=1}^m f_\ell(P)} z\;\mathsf{d}\nu(z)
=c \cdot\beta_\nu\left( \prod_{\ell=1}^m f_\ell(P)\right).\end{aligned}$$ Therefore, $${\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i \leq c\cdot \prod_{\ell=1}^m\beta_\ell(f_\ell(P))}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i \leq c\cdot \prod_{\ell=1}^m\beta_\ell(f_\ell(P))}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i \leq c\cdot \prod_{\ell=1}^m\beta_\ell(f_\ell(P))}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i \leq c\cdot \prod_{\ell=1}^m\beta_\ell(f_\ell(P))}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\\
\leq {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i \leq c \cdot\beta_\nu\left( \prod_{\ell=1}^m f_\ell(P)\right)}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i \leq c \cdot\beta_\nu\left( \prod_{\ell=1}^m f_\ell(P)\right)}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i \leq c \cdot\beta_\nu\left( \prod_{\ell=1}^m f_\ell(P)\right)}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i \leq c \cdot\beta_\nu\left( \prod_{\ell=1}^m f_\ell(P)\right)}\right\}}}{c\cdot \prod_{\ell=1}^m f_\ell(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}\leq 1,$$ where the last step holds by Lemma \[lem:power\](c).
Proof of group super-uniformity Lemma {#sec:power2-proof}
--------------------------------------
First, note that the proof of [Lemma \[lem:power3\]]{}(c) is straightforward, by applying [Lemma \[lem:power\]]{}(c). More precisely, define an augmented vector $P' = (P_1,\dots,P_n,T(P_{A_g}))\in[0,1]^{n+1}$, and define a function $f'(P') {\ensuremath{:\, =}}f(P_1,\dots,P_n) = f(P)$. Since $T(P_{A_g})$ is assumed to be super-uniform (since $g\in{\mathcal{H}_0^{{\textnormal{grp}}}}$ is a null group), this means that $P'_{n+1} = T(P_{A_g})$ is super-uniform, i.e., index $n+1$ is a null $p$-value, in the augmented vector of $p$-values $P'$. Then applying [Lemma \[lem:power\]]{}(c), with $P'$ and $f'$ in place of $P$ and $f$, and with index $i=n+1$, yields the desired bound.
[Lemma \[lem:power3\]]{}(a) is simply a special case of [Lemma \[lem:power3\]]{}(b) since independence is a special case of positive dependence, and conditioning on an independent set of $p$-values $P_{-A_g}$ doesn’t change the distribution of $P_{A_g}$.
For [Lemma \[lem:power3\]]{}(b), our proof strategy is to reduce this statement into a form where [Lemma \[lem:power\]]{}(b) becomes applicable. (Note that we cannot simply take the approach of our proof of [Lemma \[lem:power3\]]{}(c), because if we define an augmented vector of $p$-values $P'=\big(P_1,\dots,P_n,{\textnormal{Simes}}_w(P_{A_g})\big)$, we do not know if this vector is positively dependent—specifically, whether $P'$ is PRDS on entry $P'_{n+1}={\textnormal{Simes}}_w(P_{A_g})$.)
With this aim in mind, let ${\widehat{k}}_g \in \{0,\dots,n_g\}$ be the number of discoveries made by the ${\textnormal{BH}}_w$ procedure when run on the $p$-values within group $g$ at level $f(P)$. Then, using the connection between the Simes test and the BH procedure, we may write $$\begin{aligned}
{{\bf{1}}\left\{{P_g \leq f(P)}\right\}} = {{\bf{1}}\left\{{{\widehat{k}}_g > 0}\right\}} = {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{\widehat{k}}_g}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{\widehat{k}}_g}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{\widehat{k}}_g}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{\widehat{k}}_g}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} =
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{i \in A_g} {{\bf{1}}\left\{{P_i \leq \frac{w_i {\widehat{k}}_g f(P)}{n_g}
}\right\}}}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{i \in A_g} {{\bf{1}}\left\{{P_i \leq \frac{w_i {\widehat{k}}_g f(P)}{n_g}
}\right\}}}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{i \in A_g} {{\bf{1}}\left\{{P_i \leq \frac{w_i {\widehat{k}}_g f(P)}{n_g}
}\right\}}}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{i \in A_g} {{\bf{1}}\left\{{P_i \leq \frac{w_i {\widehat{k}}_g f(P)}{n_g}
}\right\}}}{{\widehat{k}}_g}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}\end{aligned}$$ since for the ${\textnormal{BH}}_w$ procedure at level $f(P)$, the $i$th $p$-value $P_i$ will be rejected if and only if $P_i\leq \frac{w_i {\widehat{k}}_g
f(P)}{n_g}$. Hence, we may conclude that $$\begin{aligned}
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_g \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_g \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_g \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_g \leq f(P)}\right\}}}{f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} = {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{i \in A_g}
{{\bf{1}}\left\{{P_i \leq \frac{w_i{\widehat{k}}_g f(P)}{n_g} }\right\}}}{{\widehat{k}}_g f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{i \in A_g}
{{\bf{1}}\left\{{P_i \leq \frac{w_i{\widehat{k}}_g f(P)}{n_g} }\right\}}}{{\widehat{k}}_g f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{i \in A_g}
{{\bf{1}}\left\{{P_i \leq \frac{w_i{\widehat{k}}_g f(P)}{n_g} }\right\}}}{{\widehat{k}}_g f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{i \in A_g}
{{\bf{1}}\left\{{P_i \leq \frac{w_i{\widehat{k}}_g f(P)}{n_g} }\right\}}}{{\widehat{k}}_g f(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} = \frac1{n_g}
\sum_{i \in A_g} w_i {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{{{\bf{1}}\left\{{P_i \leq {\widetilde{f}}_g(P)}\right\}}}{{\widetilde{f}}_g(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{{{\bf{1}}\left\{{P_i \leq {\widetilde{f}}_g(P)}\right\}}}{{\widetilde{f}}_g(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{{{\bf{1}}\left\{{P_i \leq {\widetilde{f}}_g(P)}\right\}}}{{\widetilde{f}}_g(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{{{\bf{1}}\left\{{P_i \leq {\widetilde{f}}_g(P)}\right\}}}{{\widetilde{f}}_g(P)}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
},\end{aligned}$$ where we have defined ${\widetilde{f}}_g(P) {\ensuremath{:\, =}}\frac{w_i{\widehat{k}}_g f(P)}{n_g}$.
Taking expectations on both sides and applying [Lemma \[lem:power\]]{}(b) immediately proves [Lemma \[lem:power3\]]{}(b). (Specifically, we know that $P
\mapsto {\widehat{k}}_g$ is a nonincreasing function of $P$, and $P\mapsto f(P)$ is also assumed to be nonincreasing; therefore, ${\widetilde{f}}_g$ is also nonincreasing in $P$.)
Given that [Lemma \[lem:power3\]]{}(b) is proved, the proof of [Lemma \[lem:power3\]]{}(d) follows exactly the same argument as above, except that in the very last equation, $P_i$ is replaced by $S_i$, and [Lemma \[lem:power3\]]{}(b) is invoked in place of [Lemma \[lem:power\]]{}(b).
Proof of inverse-binomial Lemma {#sec:power3-proof}
--------------------------------
The lower bound follows immediately from Jensen’s inequality, since ${\mathbb{E}\left[{Z}\right]} = 1 + b \sum_{i=1}^d a_i$. We split the argument for the upper bound into three cases.
#### Case 1: integer weights
First, suppose that all the weights $a_i$ are integers, that is, $a_i\in\{0,1\}$ for all $i$. In this case, we have $Z\sim 1 + \textnormal{Binomial}(k,b)$, where $k$ is the number of weights $a_i$ that are equal to $1$. A simple calculation shows that $$\begin{aligned}
{\mathbb{E}\left[{\frac{1}{1+\textnormal{Binomial}(k,b)}}\right]} &= \sum_{z=0}^k\frac1{1+z}
\binom{k}{z} b^{z} (1-b)^{k-z} \\
&= \frac1{b(1+k)} \sum_{z=0}^k
\binom{k+1}{z+1} b^{z+1} (1-b)^{(k+1)-(z+1)}\\
& = \frac1{b(1+k)} \cdot{\textnormal{Pr}\!\left\{{\textnormal{Binomial}(k+1,b)\leq k}\right\}}\\
&\leq \frac{1}{b(1+k)} =
\frac{1}{b(1+\sum_i a_i)}.\end{aligned}$$
#### Case 2: one non-integer weight
Suppose that exactly one of the weights $a_i$ is a non-integer. Without loss of generality we can take $a_1=\dots=a_k=1$, $a_{k+1}=c$, $a_{k+2}=\dots=a_n=0$, for some $k\in\{0,\dots,n-1\}$ and some $c\in(0,1)$. Let $A=Z_1 + \dots +
Z_{k+1}\sim \textnormal{Binomial}(k+1,b)$, and $Y=Z_{k+1}\sim\textnormal{Bernoulli}(b)$. Note that ${\textnormal{Pr}\!\left\{{Y=1}\ \middle| \ {A}\right\}}
= \frac{A}{1+k}$. Then $$\begin{aligned}
{\mathbb{E}\left[{\frac{1}{Z}}\right]} &= {\mathbb{E}\left[{\frac{1}{1+A - (1-c)Y}}\right]}\\
&=
{\mathbb{E}\left[{{\mathbb{E}\left[{\frac{1}{1+A - (1-c)Y}}\ \middle| \ {A}\right]}}\right]}\\
&= {\mathbb{E}\left[{\frac{1}{1+A} \cdot {\textnormal{Pr}\!\left\{{Y=0}\ \middle| \ {A}\right\}} +
\frac{1}{c+A} \cdot{\textnormal{Pr}\!\left\{{Y=1}\ \middle| \ {A}\right\}}}\right]}\\
&= {\mathbb{E}\left[{\frac{1}{1+A} +
\left(\frac{1}{c+A} - \frac{1}{1+A}\right)\cdot{\textnormal{Pr}\!\left\{{Y=1}\ \middle| \ {A}\right\}}}\right]}\\
&=
{\mathbb{E}\left[{\frac{1}{1+A} + \frac{1-c}{(c+A)(1+A)}\cdot\frac{A}{1+k}}\right]}\\
&\leq
{\mathbb{E}\left[{\frac{1}{1+A} + \frac{1-c}{(c+1+k)(1+A)}\cdot\frac{1+k}{1+k}}\right]},\end{aligned}$$ where the inequality holds since $\frac{A}{c+A}\leq \frac{1+k}{1+k+c}$ because $0\leq A\leq k+1$. Simplifying, we get
$$\begin{aligned}
{\mathbb{E}\left[{\frac{1}{Z}}\right]} \leq {\mathbb{E}\left[{\frac{1}{1+A}}\right]}\cdot \frac{2+k}{1+k+c} \leq \frac{1}{b(2+k)}\cdot\frac{2+k}{1+k+c} = \frac{1}{b(1+k+c)} = \frac{1}{b(1+\sum_i a_i)},\end{aligned}$$
where the inequality uses the fact that ${\mathbb{E}\left[{\frac{1}{1+\textnormal{Binomial}(k+1,b)}}\right]}\leq \frac{1}{b(2+k)}$ as calculated in Case 1.
#### Case 3: general case
Now suppose that there are at least two non-integer weights, $0<a_i\leq a_j<1$. Let $C=\sum_{\ell\neq i,j}a_{\ell}Z_{\ell}$, then $Z=1+C+a_iZ_i+a_jZ_j$. Let $\alpha = \min\{a_i,1-a_j\}>0$. Then
$$\begin{gathered}
{\mathbb{E}\left[{\frac{1}{Z}}\ \middle| \ {C}\right]} = b^{2}\cdot\frac{1}{1+C+a_i+a_j} +
b(1-b)\cdot\frac{1}{1+C+a_i} + b(1-b)\cdot\frac{1}{1+C+a_j} +
(1-b)^{2}\cdot \frac{1}{C}\\ \leq b^{2}\cdot\frac{1}{1+C+a_i+a_j} +
b(1-b)\cdot\frac{1}{1+C+(a_i-\alpha)} +
b(1-b)\cdot\frac{1}{1+C+(a_j+\alpha)} + (1-b)^{2}\cdot \frac{1}{C},\end{gathered}$$
where the inequality follows from a simple calculation using the assumption that $\alpha \leq a_i\leq a_j \leq 1- \alpha$. Now, define a new vector of weights $\tilde{a}$ where $\tilde{a}_i = a_i - \alpha,
\tilde{a}_j = a_j + \alpha$ and $\tilde{a}_\ell = a_\ell$ if $\ell
\notin \{i,j\}$. Defining $\widetilde{Z} = 1+\sum_\ell
\tilde{a}_\ell Z_\ell$, the above calculation proves that ${\mathbb{E}\left[{\frac{1}{Z}}\right]}\leq {\mathbb{E}\left[{\frac{1}{\widetilde{Z}}}\right]}$ (by marginalizing over $C$).
Note that $\sum_i a_i = \sum_i\tilde{a}_i$, but $\tilde{a}_i$ has (at least) one fewer non-integer weight. Repeating this process inductively, we see that we can reduce to the case where there is at most one non-integer weight (i.e., Case 1 or Case 2). This proves the lemma.
Proof of propositions about [[`p-filter`]{.nodecor}]{} {#sec:prop-proofs}
======================================================
Proof of “maximum-corner” Proposition {#sec:proof_max}
--------------------------------------
For each $m$, by definition of ${\widehat{k}}^{(m)}$, there is some $k^{(m)}_1,\dots,k^{(m)}_{m-1},k^{(m)}_{m+1},\dots,k^{(m)}_M$ such that $$\begin{aligned}
\label{eqn:in_T}
(k^{(m)}_1,\dots,k^{(m)}_{m-1},{\widehat{k}}^{(m)},k^{(m)}_{m+1},\dots,k^{(m)}_M)\in{\widehat{\mathcal{K}}}\;.\end{aligned}$$ Thus, for each $m'\neq m$, ${\widehat{k}}^{(m')}\geq k^{(m)}_{m'}$ by definition of ${\widehat{k}}^{(m')}$. Then $$\begin{aligned}
{\widehat{\mathcal{S}}}(k^{(m)}_1,\dots,k^{(m)}_{m-1},{\widehat{k}}^{(m)},k^{(m)}_{m+1},\dots,k^{(m)}_M)
\subseteq {\widehat{\mathcal{S}}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}_{m-1},{\widehat{k}}^{(m)},{\widehat{k}}_{m+1},\dots,{\widehat{k}}^{(M)})\;,\end{aligned}$$ because ${\widehat{\mathcal{S}}}(k^{(1)},\dots,k^{(M)})$ is a nondecreasing function of $(k^{(1)},\dots,k^{(M)})$, and this immediately implies $${{\widehat{\mathcal{S}}}^{(m)}}(k^{(m)}_1,\dots,k^{(m)}_{m-1},{\widehat{k}}^{(m)},k^{(m)}_{m+1},\dots,k^{(m)}_M)
\subseteq {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}_{m-1},{\widehat{k}}^{(m)},{\widehat{k}}_{m+1},\dots,{\widehat{k}}^{(M)}).$$ Therefore, for each layer $m$, $$\begin{aligned}
\sum_{g \in {{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(m)})} u^{(m)}_g \geq \sum_{g \in
{{\widehat{\mathcal{S}}}^{(m)}}(k^{(m)}_1,\dots,k^{(m)}_{m-1},{\widehat{k}}^{(m)},k^{(m)}_{m+1},\dots,k^{(m)}_M)}
u^{(m)}_g \geq {\widehat{k}}^{(m)},\end{aligned}$$ where the second inequality holds by observation , and by definition of ${\widehat{\mathcal{K}}}$ as the set of feasible vectors. Since this holds for all $m$, this proves that $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ is itself a feasible vector, and hence $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)}) \in {\widehat{\mathcal{K}}}$.
Proof of “halting” Proposition {#sec:proof_alg}
-------------------------------
First we introduce some notation: let $(k^{(1)}_{(s)},\dots,k^{(M)}_{(s)})$ be the vector after the $s$th pass through the algorithm. We prove that $k^{(m)}_{(s)}\geq {\widehat{k}}^{(m)}$ for all $m,s$, by induction. At initialization, $k^{(m)}_{(0)} = G^{(m)} \geq {\widehat{k}}^{(m)}$ for all $m$. Now suppose that $k^{(m)}_{(s-1)}\geq {\widehat{k}}^{(m)}$ for all $m$; we now show that $k^{(m)}_{(s)}\geq
{\widehat{k}}^{(m)}$ for all $m$.
To do this, consider the $m$-th layer of the $s$-th pass through the algorithm. Before this stage, we have vectors $k^{(1)}_{(s)},\dots,k^{(m-1)}_{(s)},k^{(m)}_{(s-1)},k^{(m+1)}_{(s-1)},\dots,k^M_{(s-1)}$, and we now update $k^{(m)}_{(s)}$. Applying induction also to this inner loop, and assuming that $k^{(m')}_{(s)}\geq {\widehat{k}}^{(m')}$ for all $m'=1,\dots,m-1$, we can now prove that $k^{(m)}_{(s)}\geq{\widehat{k}}^{(m)}$. By definition of the algorithm,
$$\begin{aligned}
\label{eqn:kms}
k^{(m)}_{(s)} = \max_{k'^{(m)} \in \{0,1,\dots,G^{(m)}\}}\left\{ k'^{(m)} : \sum\limits_{g \in
{{\widehat{\mathcal{S}}}^{(m)}}(k^{(1)}_{(s)},\dots,k^{(m-1)}_{(s)},k'^{(m)},k^{(m+1)}_{(s-1)},\dots,k^{(M)}_{(s-1)})}
u^{(m)}_g \geq k'^{(m)} \right\}.\end{aligned}$$
Since $k^{(m')}_{(s)}\geq {\widehat{k}}^{(m')}$ for all $m'=1,\dots,m-1$, and $k^{(m')}_{(s-1)}\geq {\widehat{k}}^{(m')}$ for all $m'=m+1,\dots,M$, we have $$\begin{aligned}
\sum \limits_{g \in
{{\widehat{\mathcal{S}}}^{(m)}}(k^{(1)}_{(s)},\dots,k^{(m-1)}_{(s)},{\widehat{k}}^{(m)},k^{(m+1)}_{(s-1)},\dots,k^{(M)}_{(s-1)})}
u^{(m)}_g ~\geq~ \sum\limits_{g \in
{{\widehat{\mathcal{S}}}^{(m)}}({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(m-1)},{\widehat{k}}^{(m)},{\widehat{k}}^{(m+1)},\dots,{\widehat{k}}^{(M)})} u_g^{(m)},\end{aligned}$$ since ${{\widehat{\mathcal{S}}}^{(m)}}(\vec{k})$ is a nondecreasing function of $\vec{k}$ by definition. The right-hand side of this expression is in turn $\geq
{\widehat{k}}^{(m)}$ by definition of $({\widehat{k}}^{(1)},\dots,{\widehat{k}}^{(M)})$ being feasible. Therefore, ${\widehat{k}}^{(m)}$ is in the feasible set for Eq. , and so we must have $k^{(m)}_{(s)}\geq {\widehat{k}}^{(m)}$. By induction, this is then true for all $s,m$, as desired.
Now suppose that the algorithm stabilizes at $(k^{(s)}_1,\dots,k^{(s)}_M)$, after $s$ full passes. After completing the $m$th layer of the last pass through the algorithm, we had vectors $k^{(1)}_{(s)},\dots,k^{(m)}_{(s)},k^{(m+1)}_{(s-1)},\dots,k^{(M)}_{(s-1)}$; however, since the algorithm stops after the $s$th pass, this means that $k^{(m')}_{(s-1)}=k^{(m')}_{(s)}$ for all $m'$. Using this observation in the definition of $k^{(m)}_{(s)}$, we see that $$\begin{aligned}
\sum \limits_{g \in
{{\widehat{\mathcal{S}}}^{(m)}}(k^{(1)}_{(s)},\dots,k^{m-1}_{(s)},k^{(m)}_{(s)},k^{(m+1)}_{(s)},\dots,k^{(M)}_{(s)})}
u^{(m)}_g \geq k^{(m)}_{(s)}.\end{aligned}$$ This means that $(k_{(s)}^{(1)},\dots,k_{(s)}^{(M)})\in{\widehat{\mathcal{K}}}$, and so $k_{(s)}^{(m)}\leq {\widehat{k}}^{(m)}$ for all $m$ by the definition of ${\widehat{k}}^{(1)},\ldots,{\widehat{k}}^{(m)}$ and [Proposition \[prop:max\]]{}. But by the induction above, we also know that $k_{(s)}^{(m)}\geq {\widehat{k}}^{(m)}$ for all $m,s$, thus completing the proof.
Discussion and extensions {#sec:disc}
=========================
The procedures that we have analyzed and generalized do not fully cover the huge literature on FDR-controlling procedures. For example, `p-filter` is a generalized multi-dimensional step-up procedure, but much work has also been done on alternative styles of procedures, such as step-down, step-up-down and multi-step methods. For example, @benjamini1999step propose step-down procedures that control FDR under independence. Also, procedures by @benjamini1999distribution and @romano2006stepdown provably control FDR under arbitrary dependence, with @gavrilov2009adaptive extending them to adaptive control under independence. Two-step adaptive procedures have been analyzed in @benjamini2006adaptive under independence, and by @blanchard2009adaptive under dependence. Different methods of incorporating weights into such procedures have also been studied, cf. a different notion of the weighted Simes $p$-value proposed by @BH97.
The super-uniformity lemmas (Lemma \[lem:power\] and, in the grouped setting, Lemma \[lem:power3\]), can be used to quickly prove FDR control under dependence for many of the above procedures, and may be a useful tool for designing new multiple testing procedures in broader settings[^3]. For example, it has been used to derive a decentralized procedure for FDR control on sensor networks [@ramdas2017qute] and a sequential algorithm for FDR control on directed acyclic graphs [@ramdas2018dagger]. This lemma was also used to derive a “post-selection BH procedure” [@brzyski2017controlling]: if a set $S \subseteq [n]$ of hypotheses was selected by the user in an arbitrary monotone data-dependent manner (see footnote 1), one way to find a subset $T \subseteq S$ that controls the FDR is to run BH on $S$ at level $\widetilde \alpha:= \alpha |S| / n$. Indeed, $$\small
{\textnormal{FDR}}= {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{i \in S \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \widetilde \alpha\frac{ | T|}{| S|} }\right\}}}{| T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{i \in S \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \widetilde \alpha\frac{ | T|}{| S|} }\right\}}}{| T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{i \in S \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \widetilde \alpha\frac{ | T|}{| S|} }\right\}}}{| T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{i \in S \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \widetilde \alpha\frac{ | T|}{| S|} }\right\}}}{| T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}
\leq \sum_{i \in {\mathcal{H}_0}} \frac{\alpha}{n} \cdot {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P_i \leq \alpha\frac{ | T|}{n}}\right\}} }{\alpha\frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P_i \leq \alpha\frac{ | T|}{n}}\right\}} }{\alpha\frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P_i \leq \alpha\frac{ | T|}{n}}\right\}} }{\alpha\frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P_i \leq \alpha\frac{ | T|}{n}}\right\}} }{\alpha\frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]} \leq \alpha \frac{|{\mathcal{H}_0}|}{n}.$$ Notice that the post-selection BH procedure reduces to BH in the absence of selection, that is when $S = [n]$. As another particularly simple but striking example, consider the following novel “structured BH procedure”[^4]. Suppose we wish to insist that only certain subsets of $[n]$ are allowed to be rejected; let $\mathcal{K} \subseteq 2^{[n]}$ be the set of such allowed rejection sets (these could be determined by known logical constraints or structural requirements). Then, if the $p$-values are positively dependent, we may reject the largest set $T \in \mathcal{K}$ such that all its $p$-values are less than $\alpha|T|/n$. Completely equivalently, one can define ${\widehat{{\textnormal{FDP}}}}(S) = \frac{n \cdot \max_{i \in S} P_i}{|S|}$ and reject the largest set $T \in \mathcal{K}$ such that ${\widehat{{\textnormal{FDP}}}}(T) \leq~\alpha$. This procedure controls FDR under positive dependence due to a trivial one-line proof using Lemma \[lem:power\]: $$\small
{\textnormal{FDR}}= {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{\sum_{i \in T \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \alpha \frac{ |T|}{n} }\right\}}}{|T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{\sum_{i \in T \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \alpha \frac{ |T|}{n} }\right\}}}{|T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{\sum_{i \in T \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \alpha \frac{ |T|}{n} }\right\}}}{|T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{\sum_{i \in T \cap {\mathcal{H}_0}} {{\bf{1}}\left\{{P_i \leq \alpha \frac{ |T|}{n} }\right\}}}{|T|}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]}
\leq \sum_{i \in {\mathcal{H}_0}} \frac{\alpha}{n} \cdot {\mathbb{E}\left[{{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ {{\bf{1}}\left\{{P_i \leq \alpha \frac{|T|}{n}}\right\}} }{\alpha \frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ {{\bf{1}}\left\{{P_i \leq \alpha \frac{|T|}{n}}\right\}} }{\alpha \frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ {{\bf{1}}\left\{{P_i \leq \alpha \frac{|T|}{n}}\right\}} }{\alpha \frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ {{\bf{1}}\left\{{P_i \leq \alpha \frac{|T|}{n}}\right\}} }{\alpha \frac{|T|}{n}}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}}\right]} \leq \alpha \frac{|{\mathcal{H}_0}|}{n}.$$
Again, notice that the structured BH procedure reduces to BH in the absence of structural constraints, that is when $\mathcal{K}=2^{[n]}$. Of course, except for special structured settings, the *largest* set $T \in \mathcal{K}$ may not be efficiently computable in general. When it is infeasible, *any* set $T\in \mathcal{K}$ such that ${\widehat{{\textnormal{FDP}}}}(T) \leq \alpha$ may be chosen, and FDR control will be maintained, and heuristics can be used to find large sets. (In both the above examples, one may instead use reshaping to control for arbitrary dependence.)
While there exist works that can incorporate a single layer of groups [@hu2010false], these often provide guarantees only for the finest partition. Alternative error metrics have been discussed by @benjamini2014selective, who devise a way to take a single partition of groups into account and control a *selective* FDR. This idea has been extended by @peterson2016many and @bogomolov2017testing to partitions that form a hierarchy (i.e., a tree). However, none of these aforementioned papers have been extended to handle arbitrary non-hierarchical partitions, leftover or overlapping groups, both sets of weights, adaptivity or reshaping. Recently, @katsevich17mkf derived a knockoff [[`p-filter`]{.nodecor}]{} that extended the work of @barber2016p in two ways: it allows the group $p$-values to be formed by procedures other than Simes’ (like this paper), and it can use knockoff statistics instead of $p$-values. In both settings it provides FDR control at a constant (between 1 and 2) times the target FDR. While their work can handle arbitrary non-hierarchical partitions (since it uses the same [[`p-filter`]{.nodecor}]{} framework) along with knockoff statistics, it also does not handle null-proportion adaptivity, both sets of weights, reshaping, leftover or overlapping groups, and so on. We believe that many of the algorithmic ideas and proof techniques (especially the lemmas) introduced here may generalize to these related works, and could be an avenue for future work.
Finally, as a last very general extension, it was recently noted by @katsevich2018towards that the ${\textnormal{\texttt{p-filter}}}$ framework can arbitrarily “stack” together different layers, where each layer uses a different type of FDR-controlling algorithm (ordered testing, knockoffs, online algorithms, interactive algorithms, and so on), and the ${\textnormal{\texttt{p-filter}}}$ framework can be simply used as a wrapper to ensure internal consistency.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Wenyu Chen for helping implement the new ${\textnormal{\texttt{p-filter}}}$ algorithm. We thank Eugene Katsevich, Etienne Roquain, Aditya Guntuboyina and Fanny Yang for relevant discussions. The authors are also thankful to audience members at the Statistics departments of Stanford, Wharton, UC Davis, the St. Louis Workshop on Higher Order Asymptotics and Post-Selection Inference, and the NIPS Workshop on Adaptive Data Analysis, whose questions partly shaped this work. This work was supported in part by the Office of Naval Research under grant number W911NF-16-1-0368, the Air Force Office of Scientific Resesarch under grant number AFOSR-FA9550-14-1-0016, by NSF award DMS-1654076, and by an Alfred P. Sloan fellowship.
Generalized Lg global null tests {#app:simes}
================================
@simes1986improved proposed an improvement to the Bonferroni procedure for global null testing at level $\alpha$. We first calculate the Simes p-value using a reshaping function ${\widetilde\beta}$ if required:[^5] $$\begin{aligned}
{\textnormal{Simes}}(P) = \min_{1\leq k\leq n}\frac{P_{(k)}\cdot n}{{\widetilde\beta}(k)},\end{aligned}$$ and we reject $H_{GN}$ if ${\textnormal{Simes}}(P)\leq \alpha$. The connection to the ${\textnormal{BY}}$ procedure [@BY01] is quite transparent: note that ${\textnormal{Simes}}(P) \leq \alpha$ if and only if the ${\textnormal{BY}}$ procedure makes at least one rejection at level $\alpha$.
It is well known that the Simes p-value ${\textnormal{Simes}}(P)$ is a bonafide p-value, a result to be recovered as a special case of [Proposition \[prop:simesw\]]{}.
The prior-weighted Lg test for the global null
----------------------------------------------
The Simes test [@simes1986improved] was extended by @HL94 to incorporate prior weights under independence. As before, we define weighted p-values $Q_i {\ensuremath{:\, =}}P_i/w^{(1)}_i$ for each hypothesis, and then calculate the generalized ${\textnormal{Simes}}_w$ p-value for the group as $$\begin{aligned}
\label{eqn:wSimes}
{\textnormal{Simes}}_w(P) {\ensuremath{:\, =}}\min_{1\leq k\leq n}\frac{Q_{(k)}\cdot
n}{k}.\end{aligned}$$ The global null hypothesis for the group $A_g$, i.e., the hypothesis that $A_g\subseteq{\mathcal{H}_0}$ consists entirely of nulls, is then rejected at the level $\alpha$ if ${\textnormal{Simes}}_w(P)\leq
\alpha$.
In a more general setting where the individual p-values $P_i$ within the group $A_g$ may be arbitrarily dependent, we can instead consider the reshaped weighted Simes p-value, given by $$\begin{aligned}
\label{eqn:rwSimes}
{\textnormal{rSimes}}_w(P) {\ensuremath{:\, =}}\min_{1\leq k\leq n}\frac{Q_{(k)}\cdot
n}{{\widetilde\beta}(k)},\end{aligned}$$ for a reshaping function ${\widetilde\beta}$ (recall definition ).
The following result states that the (weighted and/or reshaped) Simes p-value really is a bonafide p-value.
\[prop:simesw\] Under the global null hypothesis, the weighted Simes p-value has the following properties:
1. Under independence and uniformity, if $\max_i w_i \leq
1/\alpha$, ${\textnormal{Simes}}_w(P)$ is exactly uniformly distributed.
2. Under positive dependence (PRDS), ${\textnormal{Simes}}_w(P)$ is super-uniformly distributed.
3. Under arbitrary dependence, the reshaped Simes p-value ${\textnormal{rSimes}}_w(P)$ is super-uniformly distributed.
While statement (a) was first proven by @HL94, and statement (c) under unit weights by @hommel1983tests, all the above statements are straightforward consequences of the properties of the weighted BH and BY procedures.
Leave-one-out property (LOOP) {#app:LOOP}
=============================
For a vector $x \in \R^n$, we use ${{x}^{i \to 0}} ~:=~
(x_1,\dots,x_{i-1},0,x_{i+1},\dots,x_n) ~\in~ \R^n$ to denote a vector with the $i$-th coordinate set to zero.
\[def:LOOP\] A function $f:[0,1]^n\rightarrow[0,\infty)$ is said to satisfy the *leave-one-out property* (LOOP) if for any null index $i \in
{\mathcal{H}_0}$ and any $P \in[0,1]^n$, we have $f({{P}^{i \to 0}})>0$ and $$\begin{aligned}
\label{eqn:LOOP}
\small
\begin{cases}
\text{ if } P_i \leq f(P),\text{ then } P_i \leq f({{P}^{i \to 0}}) =
f(P), \\ \text{ if } P_i > f(P),\text{ then } P_i > f({{P}^{i \to 0}}).
\end{cases}\end{aligned}$$
When $f$ satisfies LOOP, even though threshold $f({{P}^{i \to 0}})$ may differ significantly from $f(P)$, the p-value $P_i$ will either lie below both thresholds, or above both thresholds—in other words, from the perspective of $P_i$, the threshold might as well have been $f({{P}^{i \to 0}})$ instead of $f(P)$.
Properties of dotfractions {#app:dotfrac}
==========================
In this section, we verify that “dotfractions” satisfy many of the same properties as ordinary fractions, and thus the notation ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ can be safely used throughout the proofs of our main results. In all of the following, the property will be shown to hold assuming that all dotfractions appearing in its equation or inequality are well defined. Hence, throughout, we assume that the various properties are only used if all of the dotfractions in the expression are defined—that is, we may use these properties only if we never have ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ with $a\neq 0$ and $b=0$. As a side note, observe that in the paper, we always use ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ when $a,b
\geq 0$ only.
1. Comparing two fractions: $$\begin{aligned}
\label{eqn:dotfrac_compare}
\text{If $a \geq b\geq 0$ and $c\geq 0$, then ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}\geq {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$, and ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}\leq{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$.}\end{aligned}$$ In order to prove the first bound, if $c>0$ then this reduces to $\frac{a}{c}\geq \frac{b}{c}$, while if $c=0$ then we must have $a=b=0$ (since, otherwise, ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ and ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ would be undefined) and so ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}={
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}=0$. To prove the second bound, if $b>0$, then this reduces to $\frac{c}{a}\leq
\frac{c}{b}$, while if $b=0$ then we must have $c=0$ (since, otherwise, ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ would be undefined), in which case ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{a}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}={
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}=0$.
2. Comparing against a scalar: $$\begin{aligned}
\label{eqn:dotfrac_compare_scalar}
\text{If $c\geq 0$ and $a\geq {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ then $ac\geq b$.}\end{aligned}$$ To prove this, if $c\neq 0$ then we have $a\geq \frac{b}{c}$, while if $c=0$ then we must have $b=0$ (so that ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ is not undefined), and so $ac\geq b$ is trivially true as both sides equal zero.
3. Adding numerators: $$\begin{aligned}
\label{eqn:dotfrac_add}
\text{For any $a,b,c$, \quad ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} + {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} =
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a+b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a+b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a+b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a+b}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$.}\end{aligned}$$ In order to prove this claim, we note thatif $c\neq 0$ then this reduces to $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$, while if $c=0$ then we must have $a=b=0$ (otherwise the dotfractions are undefined), and so the left-hand and right-hand sides are both equal to zero.
4. Multiplying fractions: $$\begin{aligned}
\label{eqn:dotfrac_mult}
\text{For any $a,b,c,d$, \quad ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}\cdot {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} =
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ac}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ac}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ac}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ac}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$.}\end{aligned}$$ In order to prove this claim, if $b,d\neq 0$ then this reduces to $\frac{a}{b}\cdot \frac{c}{d} = \frac{ac}{bd}$, while if $b=0$ or $d=0$, then either $a=0$ or $c=0$ (otherwise ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ or ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ would be undefined), and so the left-hand and right-hand sides are again both equal to zero.
5. Cancelling nonzero factors : $$\begin{aligned}
\label{eqn:dotfrac_mult_one}
\text{If $c\neq 0$ then for any $a,b$, \quad ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} =
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$.}\end{aligned}$$ To see why, we simply apply with $d=c$ (noting that, with the assumption $c\neq 0$, we have ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{c}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}=1$).
6. Multiplying by a scalar: $$\begin{aligned}
\label{eqn:dotfrac_mult_scalar}
\text{For any $a,b,c$, \quad $c\cdot {
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} =
{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ac}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ac}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ac}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ac}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$.}\end{aligned}$$ To see why, if $b\neq 0$ then this reduces to $c\cdot
\frac{a}{b}=\frac{ac}{b}$, while if $b=0$ then we must have $a=0$ so that ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ is not undefined, and so the left- and right-hand sides are both zero.
While the above properties all carry over from fractions to dotfractions, there are some settings where familiar manipulations with fractions may no longer be correct. For example, ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}\neq{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ac}{bc}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ when $a,b\neq 0$ while $c=0$. Relatedly, we cannot add fractions in the usual way, i.e. ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
} +{
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ may not be equal to ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ad+bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ad+bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ad+bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ad+bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$; this fails because implicitly we would be assuming that ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{a}{b}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}={
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{ad}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{ad}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{ad}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{ad}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ and ${
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{c}{d}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}={
\mathchoice
{\ooalign{$\genfrac{}{}{0pt}{0}{bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{1}{bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{2}{bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
{\ooalign{$\genfrac{}{}{0pt}{3}{bc}{bd}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}}
}$ in order to make the two denominators the same, which may fail if $d=0$ or if $b=0$.
[^1]: We remark these are not the only two notions of internal consistency that can fit into our framework: any *monotone* notion of IC can be handled, where *monotone* means that decreasing the $p$-values can only possibly increase the number of rejections at all layers.
[^2]: For a review of its history that involves Eklund and Seeger in the 1960s, and Simes, Hommel, Soric, Benjamini and Hochberg in the 1980s and 1990s, see [@seeger1968note; @benjamini2000adaptive].
[^3]: An analog of the super-uniformity lemma has also been discovered in the online FDR setting and has proved useful for designing new algorithms [@javanmard2018online; @RYWJ17; @ramdas2018saffron].
[^4]: This procedure was independently discovered recently by @katsevich2018controlling, along with several other substantial extensions.
[^5]: Here and henceforth, the tilde in ${\widetilde\beta}$ is used to signified a reshaping function for calculating a Simes p-value within a single group, and we will continue the use of notation $\beta$, without the tilde, when comparing these p-values across multiple groups.
|
---
author:
- |
Charalampos E. Tsourakakis\
Brown University\
`charalampos_tsourakakis@brown.edu`
bibliography:
- 'ref.bib'
title: Streaming Graph Partitioning in the Planted Partition Model
---
Introduction {#sec:intro}
============
Related Work {#sec:related}
============
Proposed Algorithm {#sec:algorithm}
==================
Experimental Results {#sec:experiments}
====================
Conclusions {#sec:concl}
===========
Acknowledgements
================
The author would like to thank Alan Frieze and Moez Draief for their feedback on the manuscript.
|
---
abstract: 'This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a *sketch*. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters *a priori* to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.'
author:
- 'Joel A. Tropp[^1]'
- 'Alp Yurtsever[^2]'
- 'Madeleine Udell[^3]'
- 'Volkan Cevher[^4]'
title: |
Practical Sketching Algorithms\
for Low-Rank Matrix Approximation[^5]
---
Dimension reduction; matrix approximation; numerical linear algebra; randomized algorithm; single-pass algorithm; sketching; streaming algorithm; subspace embedding.
Primary, 65F30; Secondary, 68W20.
Motivation
==========
This paper presents a framework for computing structured low-rank approximations of a matrix from a *sketch*, which is a random low-dimensional linear image of the matrix. Our goal is to develop simple, practical algorithms that can serve as reliable modules in other applications. The methods apply for the real field (${\mathbb{F}}= {\mathbb{R}}$) and for the complex field (${\mathbb{F}}= {\mathbb{C}}$).
Low-Rank Matrix Approximation
-----------------------------
Suppose that ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ is an arbitrary matrix. Let $r$ be a target rank parameter where $r \ll \min\{m ,n\}$. The computational problem is to produce a low-rank approximation $\hat{{\bm{A}}}$ of ${\bm{A}}$ whose error is comparable to a best rank-$r$ approximation: $$\label{eqn:intro-low-rank}
{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}} \approx \min_{{\operatorname{rank}}({\bm{B}}) \leq r} {{\Vert {\bm{A}} - {\bm{B}} \Vert}_{\mathrm{F}}}.$$ The notation ${{\Vert \cdot \Vert}_{\mathrm{F}}}$ refers to the Frobenius norm. We explicitly allow the rank of $\hat{{\bm{A}}}$ to exceed $r$ because we can obtain more accurate approximations of this form, and the precise rank of $\hat{{\bm{A}}}$ is unimportant in many applications. There has been extensive research on randomized algorithms for \[eqn:intro-low-rank\]; see Halko et al. [@HMT11:Finding-Structure].
Sketching
---------
Here is the twist. Imagine that our interactions with the matrix ${\bm{A}}$ are severely constrained in the following way. We construct a linear map $\mathcal{L} : {\mathbb{F}}^{m \times n} \to {\mathbb{F}}^d$ that does not depend on the matrix ${\bm{A}}$. Our only mechanism for collecting data $\mathsf{S}$ about ${\bm{A}}$ is to apply the linear map $\mathcal{L}$: $$\label{eqn:lin-sketch}
\mathsf{S} := \mathcal{L}({\bm{A}}) \in {\mathbb{F}}^d.$$ We refer to $\mathsf{S}$ as a *sketch* of the matrix, and $\mathcal{L}$ is called a *sketching map*. The number $d$ is called the *dimension* or *size* of the sketch.
The challenge is to make the sketch as small as possible while collecting enough information to approximate the matrix accurately. In particular, we want the sketch dimension $d$ to be much smaller than the total dimension $mn$ of the matrix ${\bm{A}}$. As a consequence, the sketching map $\mathcal{L}$ has a substantial null space. Therefore, it is natural to draw the sketching map *at random* so that we are likely to extract useful information from any fixed input matrix.
Why Sketch?
-----------
There are a number of situations where the sketching model \[eqn:lin-sketch\] is a natural mechanism for acquiring data about an input matrix.
First, imagine that ${\bm{A}}$ is a huge matrix that can only be stored outside of core memory. The cost of data transfer may be substantial enough that we can only afford to read the matrix into core memory once [@HMT11:Finding-Structure Sec. 5.5]. We can build a sketch as we scan through the matrix. Other types of algorithms for this problem appear in [@FSS12:Turning-Big; @FRV16:Dimensionality-Reduction].
Second, there are applications where the columns of the matrix ${\bm{A}}$ are revealed one at a time, and we must be able to compute an approximation at any instant. One approach is to maintain a sketch that is updated when a new column arrives. Other types of algorithms for this problem appear in [@BGKL15:Online-Principal; @JJK+16:Streaming-PCA].
Third, we may encounter a setting where the matrix ${\bm{A}}$ is presented as a sum of ordered updates: $$\label{eqn:additive-update}
{\bm{A}} = {\bm{H}}_1 + {\bm{H}}_2 + {\bm{H}}_3 + {\bm{H}}_4 + \cdots.$$ We must discard each innovation ${\bm{H}}_i$ after it is processed [@CW09:Numerical-Linear; @Woo14:Sketching-Tool]. In this case, the random linear sketch \[eqn:lin-sketch\] is more or less the only way to maintain a representation of ${\bm{A}}$ through an arbitrary sequence of updates [@LNW14:Turnstile-Streaming]. Our research was motivated by a variant [@YUTC16:Sketchy-Decisions] of the model ; see \[sec:updates\].
Overview of Algorithms
----------------------
Let us summarize our basic approach to sketching and low-rank approximation of a matrix. Fix a target rank $r$ and an input matrix ${\bm{A}} \in {\mathbb{F}}^{m \times n}$. Select sketch size parameters $k$ and $\ell$. Draw and fix independent standard normal matrices ${\bm{\Omega}} \in {\mathbb{F}}^{n \times k}$ and ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times m}$; see \[def:std-normal-mtx\]. We realize the randomized linear sketch \[eqn:lin-sketch\] via left and right matrix multiplication: $$\label{eqn:sketch-intro}
{\bm{Y}} := {\bm{A\Omega}}
\quad\text{and}\quad
{\bm{W}} := {\bm{\Psi A}}$$ We can store the random matrices and the sketch using $(k+\ell)(m+n)$ scalars. The arithmetic cost of forming the sketch is $\Theta((k+\ell)mn)$ floating-point operations (flops) for a general matrix ${\bm{A}}$.
Given the random matrices $({\bm{\Omega}}, {\bm{\Psi}})$ and the sketch $({\bm{Y}}, {\bm{W}})$, we compute an approximation $\hat{{\bm{A}}}$ in three steps:
1. Form an orthogonal–triangular factorization ${\bm{Y}} =: {\bm{QR}}$ where ${\bm{Q}} \in {\mathbb{F}}^{m \times k}$.
2. Solve a least-squares problem to obtain ${\bm{X}} := ({\bm{\Psi}} {\bm{Q}})^\dagger {\bm{W}} \in {\mathbb{F}}^{k \times n}$.
3. Construct the rank-$k$ approximation $\hat{{\bm{A}}} := {\bm{QX}}$.
The total cost of this computation is $\Theta(kl (m + n))$ flops. See \[sec:intuition\] for the intuition behind this approach.
Now, suppose that we set the sketch size parameters $k = 2r+1$ and $\ell = 4r+2$. For this choice, \[thm:err-frob\] yields the error bound $${\operatorname{\mathbb{E}}}{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}} \leq 2 \cdot \min_{{\operatorname{rank}}({\bm{B}}) \leq r} {{\Vert {\bm{A}} - {\bm{B}} \Vert}_{\mathrm{F}}}.$$ In other words, we typically obtain an approximation with rank $\approx 2r$ whose error lies within twice the optimal rank-$r$ error! Moreover, the total storage cost is about $6r(m+n)$, which is comparable with the number of degrees of freedom in an $m \times n$ matrix with rank $r$, so the sketch size cannot be reduced substantially.
Our Contributions {#sec:contributions}
-----------------
This paper presents a systematic treatment of sketching algorithms for low-rank approximation of a matrix. All of the methods rely on the simple sketch \[eqn:sketch-intro\] of the input matrix (\[sec:sketch\]). The main algorithm uses this sketch to compute a high-quality low-rank approximation $\hat{{\bm{A}}}$ of the input matrix (\[alg:detailed-low-rank-recon\]). We prove that this method automatically takes advantage of spectral decay in the input matrix (\[thm:err-frob\]); this result is new.
We also explain how to compute approximations with additional structure—such as symmetry, positive semidefiniteness, or fixed rank—by projecting the initial low-rank approximation onto the family of structured matrices (\[sec:structured,sec:fixed-rank\]). This approach ensures that the structured approximations also exploit spectral decay (\[fact:convex-structure,prop:fixed-rank-err\]). In the sketching context, this idea is new.
Each algorithm is accompanied by an informative error bound that provides a good description of its actual behavior. As a consequence, we can offer the first concrete guidance on algorithm parameters for various types of input matrices (\[sec:best-parameters\]), and we can implement the methods with confidence. We also include pseudocode and an accounting of computational costs.
The paper includes a collection of numerical experiments (\[sec:experiments\]). This work demonstrates that the recommended algorithms can significantly outperform alternative methods, especially when the input matrix has spectral decay. The empirical work also confirms our guidance on parameter choices.
Our technical report [@TYUC17:Randomized-Single-View-TR] contains some more error bounds for the reconstruction algorithms. It also documents additional numerical experiments.
Limitations {#sec:limitations}
-----------
The algorithms in this paper are not designed for all low-rank matrix approximation problems. They are specifically intended for environments where we can only make a single pass over the input matrix or where the data matrix is presented as a stream of linear updates. When it is possible to make multiple passes over the input matrix, we recommend the low-rank approximation algorithms documented in [@HMT11:Finding-Structure]. Multi-pass methods are significantly more accurate because they drive the error of the low-rank approximation down to the optimal low-rank approximation error exponentially fast in the number of passes.
Overview of Related Work {#sec:related}
------------------------
Randomized algorithms for matrix approximation date back to research [@PRTV00:Latent-Semantic; @FKV04:Fast-Monte-Carlo] in theoretical computer science (TCS) in the late 1990s. Starting around 2004, this work inspired numerical analysts to develop practical algorithms for matrix approximation and related problems [@MRT11:Randomized-Algorithm]. See the paper [@HMT11:Finding-Structure Sec. 2] for a comprehensive historical discussion. The surveys [@Mah11:Randomized-Algorithms; @Woo14:Sketching-Tool] provide more details about the development of these ideas within the TCS literature.
### Sketching Algorithms for Matrix Approximation
To the best of our knowledge, the first sketching algorithm for low-rank matrix approximation appears in Woolfe et al. [@WLRT08:Fast-Randomized Sec. 5.2]. Their primary motivation was to compute a low-rank matrix approximation faster than any classical algorithm, rather than to work under the constraints of a sketching model. A variant of their approach is outlined in [@HMT11:Finding-Structure Sec. 5.5].
Clarkson & Woodruff [@CW09:Numerical-Linear] explicitly frame the question of how to perform numerical linear algebra tasks under the sketching model \[eqn:lin-sketch\]. Among other things, they develop algorithms and lower bounds for low-rank matrix approximation. Some of the methods that we recommend are algebraically—but not numerically—equivalent to formulas [@CW09:Numerical-Linear Thm. 4.7, 4.8] that they propose. Their work focuses on obtaining *a priori* error bounds. In contrast, we also aim to help users implement the methods, choose parameters, and obtain good empirical performance in practice. Additional details appear throughout our presentation.
There are many subsequent theoretical papers on sketching algorithms for low-rank matrix approximation, including [@Woo14:Sketching-Tool; @CEM+15:Dimensionality-Reduction; @BWZ16:Optimal-Principal-STOC]. This line of research exploits a variety of tricks to obtain algorithms that, theoretically, attain better asymptotic upper bounds on computational resource usage. contains a representative selection of these methods and their guarantees.
### Added in Press
When we wrote this paper, the literature did not contain sketching methods tailored for symmetric or positive-semidefinite matrix approximation. A theoretical paper [@CW17:Low-Rank-PSD] on algorithms for low-rank approximation of a sparse psd matrix was released after our work appeared.
### Error Bounds
Almost all previous papers in this area have centered on the following problem. Let ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ be an input matrix, let $r$ be a target rank, and let ${\varepsilon}> 0$ be an error tolerance. Given a randomized linear sketch \[eqn:lin-sketch\] of the input matrix, produce a rank-$r$ approximation $\hat{{\bm{A}}}_{\mathrm{eps}}$ that satisfies $$\label{eqn:eps-subopt}
{{{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\mathrm{eps}} \Vert}_{\mathrm{F}}}^2}
\leq (1 + {\varepsilon}) \cdot \min_{{\operatorname{rank}}( {\bm{B}} ) \leq r} {{{\Vert {\bm{A}} - {\bm{B}} \Vert}_{\mathrm{F}}}^2}
\quad\text{with high probability}.$$ To achieve \[eqn:eps-subopt\] for a general input, the sketch must have dimension $\Omega(r(m+n)/{\varepsilon})$ [@CW09:Numerical-Linear Thm. 4.10]. Furthermore, the analogous error bound for the spectral norm cannot be achieved for all input matrices under the sketching model [@Woo14:Sketching-Tool Ch. 6.2]. Nevertheless, Gu [@Gu15:Subspace-Iteration Thm. 3.4] has observed that \[eqn:eps-subopt\] implies a weak error bound in the spectral norm.
Li et al. [@LLS+17:Algorithm-971 App.] caution that the guarantee \[eqn:eps-subopt\] is often vacuous. For example, we frequently encounter matrices for which the Frobenius-norm error of an optimal rank-$r$ approximation is larger than the Frobenius norm of the approximation itself. In other settings, it may be necessary to compute an approximation with very high accuracy. Either way, ${\varepsilon}$ must be tiny before the bound \[eqn:eps-subopt\] sufficiently constrains the approximation error. For a general input matrix, to achieve a small value of ${\varepsilon}$, the sketch size must be exorbitant. We tackle this issue by providing alternative error estimates (e.g., \[thm:err-frob\]) that yield big improvements for most examples.
### Questions...
Our aim is to address questions that arise when one attempts to use sketching algorithms in practice. For instance, how do we implement these methods? Are they numerically stable? How should algorithm parameters depend on the input matrix? What is the right way to preserve structural properties? Which methods produce the best approximations in practice? How small an approximation error can we actually achieve? Does existing theoretical analysis predict performance? Can we obtain error bounds that are more illuminating than \[eqn:eps-subopt\]? These questions have often been neglected in the literature.
Our empirical study (\[sec:experiments\]) highlights the importance of this inquiry. Surprisingly, numerical experiments reveal that the pursuit of theoretical metrics has been counterproductive. More recent algorithms often perform worse in practice, even though—in principle—they offer better performance guarantees.
Background
==========
In this section, we collect notation and conventions, as well as some background on random matrices.
Notation and Conventions
------------------------
We write ${\mathbb{F}}$ for the scalar field, which is either ${\mathbb{R}}$ or ${\mathbb{C}}$. The letter ${\mathbf{I}}$ signifies the identity matrix; its dimensions are determined by context. The star ${}^*$ refers to the (conjugate) transpose operation on vectors and matrices. The dagger ${}^\dagger$ is the Moore–Penrose pseudoinverse. The symbol ${{\Vert \cdot \Vert}_{\mathrm{F}}}$ denotes the Frobenius norm.
The expression “${\bm{M}}$ has rank $r$” and its variants mean that the rank of ${\bm{M}}$ does not exceed $r$. The symbol ${\llbracket {{\bm{M}}} \rrbracket_{r}}$ represents an optimal rank-$r$ approximation of ${\bm{M}}$ with respect to Frobenius norm; this approximation need not be unique [@Hig89:Matrix-Nearness Sec. 6].
It is valuable to introduce notation for the error incurred by a best rank-$r$ approximation in the Frobenius norm. For each natural number $j$, we define the *$j$th tail energy* $$\label{eqn:tail-energy}
\tau_{j}^2({\bm{A}}) := \min_{{\operatorname{rank}}({\bm{B}}) < j} {{{\Vert {\bm{A}} - {\bm{B}} \Vert}_{\mathrm{F}}}^2}
= \sum\nolimits_{i \geq j} \sigma_i^2({\bm{A}}).$$ We have written $\sigma_i({\bm{A}})$ for the $i$th largest singular value of ${\bm{A}}$. The equality follows from the Eckart–Young Theorem; for example, see [@Hig89:Matrix-Nearness Sec. 6].
The symbol ${\operatorname{\mathbb{E}}}$ denotes expectation with respect to all random variables. For a given random variable $Z$, we write ${\operatorname{\mathbb{E}}}_{Z}$ to denote expectation with respect to the randomness in $Z$ only. Nonlinear functions bind before the expectation.
In the description of algorithms in the text, we primarily use standard mathematical notation. In the pseudocode, we rely on some <span style="font-variant:small-caps;">Matlab R2017a</span> functions in an effort to make the presentation more concise.
We use the computer science interpretation of $\Theta(\cdot)$ to refer to the class of functions whose growth is bounded above and below up to a constant.
Standard Normal Matrices
------------------------
Let us define an ensemble of random matrices that plays a central role in this work.
\[def:std-normal-mtx\] A matrix ${\bm{G}} \in {\mathbb{R}}^{m \times n}$ has the real standard normal distribution if the entries form an independent family of standard normal random variables (i.e., Gaussian with mean zero and variance one).
A matrix ${\bm{G}} \in {\mathbb{C}}^{m \times n}$ has the complex standard normal distribution if it has the form ${\bm{G}} = {\bm{G}}_1 + \mathrm{i} {\bm{G}}_2$ where ${\bm{G}}_1$ and ${\bm{G}}_2$ are independent, real standard normal matrices.
Standard normal matrices are also known as Gaussian matrices.
We introduce numbers $\alpha$ and $\beta$ that reflect the field over which the random matrix is defined: $$\label{eqn:alpha-parameter}
\alpha := \alpha({\mathbb{F}}) := \begin{cases} 1, & {\mathbb{F}}= {\mathbb{R}}\\ 0, & {\mathbb{F}}= {\mathbb{C}}\end{cases}
\quad\text{and}\quad
\beta := \beta({\mathbb{F}}) := \begin{cases} 1, & {\mathbb{F}}= {\mathbb{R}}\\ 2, & {\mathbb{F}}= {\mathbb{C}}\end{cases}.$$ This notation allows us to treat the real and complex case simultaneously. The number $\beta$ is a standard parameter in random matrix theory.
Last, we introduce notation to help make our theorem statements more succinct: $$\label{eqn:f-intro}
f(s, t) := \frac{s}{t - s - \alpha}
\quad\text{for integers that satisfy $t > s + \alpha > \alpha$.}$$ Observe that the function $f(s, \cdot)$ is decreasing, with range $(0, s]$.
Sketching the Input Matrix {#sec:mult-sketching}
==========================
First, we discuss how to collect enough data about an input matrix to compute a low-rank approximation. We summarize the matrix by multiplying it on the right and the left by random test matrices. The dimension and distribution of these random test matrices together determine the potential accuracy of the approximation.
The Input Matrix
----------------
Let ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ be a matrix that we wish to approximate. Our algorithms work regardless of the relative dimensions of ${\bm{A}}$, but there may sometimes be small benefits if we apply them to ${\bm{A}}^*$ instead.
The Target Rank
---------------
Let $r$ be a target rank parameter with $1 \leq r \leq \min\{m,n\}$. We aim to construct a low-rank approximation of ${\bm{A}}$ whose error is close to the optimal rank-$r$ error. We explicitly allow approximations with rank somewhat larger than $r$ because they may be significantly more accurate.
Under the sketching model \[eqn:lin-sketch\], the practitioner must use prior knowledge about the input matrix ${\bm{A}}$ to determine a target rank $r$ that will result in satisfactory error guarantees. This decision is outside the scope of our work.
Parameters for the Sketch
-------------------------
The sketch consists of two parts: a summary of the range of ${\bm{A}}$ and a summary of the co-range. The parameter $k$ controls the size of the range sketch, and the parameter $\ell$ controls the size of the co-range sketch. They should satisfy the conditions $$\label{eqn:param-assumption}
r \leq k \leq \ell
\quad\text{and}\quad
k \leq n
\quad\text{and}\quad
\ell \leq m.$$ We often choose $k \approx r$ and $\ell \approx k$. See \[eqn:my-param-choice,sec:best-parameters\] below.
The parameters $k$ and $\ell$ do not play symmetrical roles. We need $\ell \geq k$ to ensure that a certain $\ell \times k$ matrix has full column rank. Larger values of both $k$ and $\ell$ result in better approximations at the cost of more storage and arithmetic. These tradeoffs are quantified in the sequel.
The Test Matrices
-----------------
To form the sketch of the input matrix, we draw and fix two (random) test matrices: $$\label{eqn:test-matrices}
{\bm{\Omega}} \in {\mathbb{F}}^{n \times k}
\quad\text{and}\quad
{\bm{\Psi}} \in {\mathbb{F}}^{\ell \times m}.$$ This paper contains a detailed analysis of the case where the test matrices are statistically independent and follow the standard normal distribution. describes other potential distributions for the test matrices. We always state when we are making distributional assumptions on the test matrices.
The Sketch {#sec:sketch}
----------
The sketch of ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ consists of two matrices: $$\label{eqn:sketches}
{\bm{Y}} := {\bm{A}} {\bm{\Omega}} \in {\mathbb{F}}^{m \times k}
\quad\text{and}\quad
{\bm{W}} := {\bm{\Psi}} {\bm{A}} \in {\mathbb{F}}^{\ell \times n}.$$ The matrix ${\bm{Y}}$ collects information about the action of ${\bm{A}}$, while the matrix ${\bm{W}}$ collects information about the action of ${\bm{A}}^*$. Both parts are necessary.
The matrix sketching algorithms that appear in [@WLRT08:Fast-Randomized Sec. 5.2] and [@CW09:Numerical-Linear Thm. 4.9] and [@HMT11:Finding-Structure Sec. 5.5] and [@Woo14:Sketching-Tool Thm. 4.3] all involve a sketch of the form \[eqn:sketches\]. In contrast, the most recent approaches ([@BWZ16:Optimal-Principal-STOC Sec. 6.1.2] and [@Upa16:Fast-Space-Optimal Sec. 3]) use more complicated sketches; see \[sec:optimal-alg\].
The Sketch as an Abstract Data Type
-----------------------------------
We present the sketch as an abstract data type using ideas from object-oriented programming. <span style="font-variant:small-caps;">Sketch</span> is an object that contains information about a specific matrix ${\bm{A}}$. The test matrices $({\bm{\Omega}}, {\bm{\Psi}})$ and the sketch matrices $({\bm{Y}}, {\bm{W}})$ are private variables that are only accessible to the <span style="font-variant:small-caps;">Sketch</span> methods. A user interacts with the <span style="font-variant:small-caps;">Sketch</span> object by initializing it with a specific matrix and by applying linear updates. The user can query the <span style="font-variant:small-caps;">Sketch</span> object to obtain an approximation of the matrix ${\bm{A}}$ with specific properties. The individual algorithms described in this paper are all methods that belong to the <span style="font-variant:small-caps;">Sketch</span> object.
Initializing the Sketch and its Costs
-------------------------------------
See \[alg:sketch\] for pseudocode that implements the sketching procedure \[eqn:test-matrices\] and \[eqn:sketches\] with either standard normal test matrices (default) or random orthonormal test matrices (optional steps). Note that the orthogonalization step requires additional arithmetic and communication.
The storage cost for the sketch $({\bm{Y}}, {\bm{W}})$ is $mk + \ell n$ floating-point numbers in the field ${\mathbb{F}}$. The storage cost for two standard normal test matrices is $nk + \ell m$ floating point numbers in ${\mathbb{F}}$. Some other types of test matrices $({\bm{\Omega}}, {\bm{\Psi}})$ have lower storage costs, but the sketch $({\bm{Y}}, {\bm{W}})$ remains the same size.
For standard normal test matrices, the arithmetic cost of forming the sketch \[eqn:sketches\] is $\Theta((k+\ell)mn)$ flops when ${\bm{A}}$ is dense. If ${\bm{A}}$ is sparse, the cost is proportional to the number $\texttt{nnz}({\bm{A}})$ of nonzero entries: $\Theta((k + \ell) \, \texttt{nnz}({\bm{A}}))$ flops. Other types of test matrices sometimes yield lower arithmetic costs.
**private:** ${\bm{\Omega}}, {\bm{\Psi}}, {\bm{Y}}, {\bm{W}}$ ${\bm{\Omega}} \gets \texttt{randn}(n, k)$ ${\bm{\Psi}} \gets \texttt{randn}(\ell, m)$ ${\bm{\Omega}} \gets \texttt{randn}(n, k) + {\rm i} \, \texttt{randn}(n, k)$ ${\bm{\Psi}} \gets \texttt{randn}(\ell, m) + {\rm i} \, \texttt{randn}(\ell, m)$ ${\bm{\Omega}} \gets \texttt{orth}({\bm{\Omega}})$ ${\bm{\Psi}}^* \gets \texttt{orth}({\bm{\Psi}}^*)$ ${\bm{Y}} \gets {\bm{A\Omega}}$ ${\bm{W}} \gets {\bm{\Psi A}}$
${\bm{Y}} \gets \theta {\bm{Y}} + \eta {\bm{H \Omega}}$ ${\bm{W}} \gets \theta {\bm{W}} + \eta {\bm{\Psi H}}$
Processing Linear Updates {#sec:updates}
-------------------------
The sketching model \[eqn:sketches\] supports a linear update that is more general than \[eqn:additive-update\]. Suppose the input matrix ${\bm{A}}$ is modified as $${\bm{A}} \gets \theta {\bm{A}} + \eta {\bm{H}}
\quad\text{where $\theta, \eta \in {\mathbb{F}}$.}$$ Then we update the sketch \[eqn:sketches\] via the rule $$\label{eqn:linear-update}
{\bm{Y}} \gets \theta {\bm{Y}} + \eta {\bm{H}}{\bm{\Omega}}
\quad\text{and}\quad
{\bm{W}} \gets \theta {\bm{W}} + \eta {\bm{\Psi}} {\bm{H}}.$$ The precise cost of the computation depends on the structure of ${\bm{H}}$. See \[alg:sketch-update\] for pseudocode. This type of update is crucial for certain applications [@YUTC16:Sketchy-Decisions].
Choosing the Distribution of the Test Matrices {#sec:distributions}
----------------------------------------------
Our analysis is specialized to the case where the test matrices ${\bm{\Omega}}$ and ${\bm{\Psi}}$ are standard normal so that we can obtain highly informative error bounds.
But there are potential benefits from implementing the sketch using test matrices drawn from another distribution. The choice of distribution leads to some tradeoffs in the range of permissible parameters; the costs of randomness, arithmetic, and communication to generate the test matrices; the storage costs for the test matrices and the sketch; the arithmetic costs for sketching and updates; the numerical stability of matrix approximation algorithms; and the quality of *a priori* error bounds.
Let us list some of the contending distributions along with background references. We have ranked these in decreasing order of reliability.
- **Orthonormal.** The optional steps in \[alg:sketch\] generate matrices ${\bm{\Omega}}$ and ${\bm{\Psi}}^*$ with orthonormal columns that span uniformly random subspaces of dimension $k$ and $\ell$. When $k$ and $\ell$ are very large, these matrices result in smaller errors and better numerical stability than Gaussians [@DDH07:Fast-Linear; @HMT11:Finding-Structure].
- **Gaussian.** Following [@MRT11:Randomized-Algorithm; @HMT11:Finding-Structure], this paper focuses on test matrices with the standard normal distribution. Benefits include excellent practical performance and accurate *a priori* error bounds.
- **Rademacher.** These test matrices have independent Rademacher[^6] entries. Their behavior is similar to Gaussian test matrices, but there are minor improvements in the cost of storage and arithmetic, as well as the amount of randomness required. For example, see [@CW09:Numerical-Linear].
- **Subsampled Randomized Fourier Transform (SRFT).** These test matrices take the form $$\label{eqn:srft}
{\bm{\Omega}} = {\bm{D}}_1 {\bm{F}}_1 {\bm{P}}_1
\quad\text{and}\quad
{\bm{\Psi}} = {\bm{P}}_2 {\bm{F}}_2^* {\bm{D}}_2$$ where ${\bm{D}}_1 \in {\mathbb{F}}^{n \times n}$ and ${\bm{D}}_2 \in {\mathbb{F}}^{m \times m}$ are diagonal matrices with independent Rademacher entries; ${\bm{F}}_1 \in {\mathbb{F}}^{n \times n}$ and ${\bm{F}}_2 \in {\mathbb{F}}^{m \times m}$ are discrete cosine transform (${\mathbb{F}}= {\mathbb{R}}$) or discrete Fourier transform $({\mathbb{F}}= {\mathbb{C}})$ matrices; and ${\bm{P}}_1 \in {\mathbb{F}}^{n \times k}$ and ${\bm{P}}_2 \in {\mathbb{F}}^{\ell \times m}$ are restrictions onto $k$ and $\ell$ coordinates, chosen uniformly at random. These matrices work well in practice; they require a modest amount of storage; and they support fast arithmetic. See [@AC06:Approximate-Nearest; @WLRT08:Fast-Randomized; @AC09:Fast-Johnson-Lindenstrauss; @HMT11:Finding-Structure; @Tro11:Improved-Analysis; @BG13:Improved-Matrix; @CNW16:Optimal-Approximate].
- **Ultra-Sparse Rademacher.** Let $s$ be a sparsity parameter. In each row of ${\bm{\Omega}}$ and column of ${\bm{\Psi}}$, we place independent Rademacher random variables in $s$ uniformly random locations; the remaining entries of the test matrices are zero. These matrices help control storage, arithmetic, and randomness costs. On the other hand, they are somewhat less reliable. For more details, see [@CW13:Low-Rank-Approximation; @NN13:OSNAP-Faster; @MM13:Low-Distortion-Subspace; @NN14:Lower-Bounds; @Woo14:Sketching-Tool; @BDN15:Toward-Unified; @Coh16:Nearly-Tight].
Except for ultra-sparse Rademacher matrices, these distributions often behave quite like a Gaussian distribution in practice [@HMT11:Finding-Structure Sec. 7.4]. An exhaustive comparison of distributions for the test matrices is outside the scope of this paper; see [@Lib09:Accelerated-Dense].
Low-Rank Approximation from the Sketch {#sec:low-rank-recon}
======================================
Suppose that we have acquired a sketch $({\bm{Y}}, {\bm{W}})$ of the input matrix ${\bm{A}}$, as in \[eqn:test-matrices\] and \[eqn:sketches\]. This section presents the most basic algorithm for computing a low-rank approximation of ${\bm{A}}$ from the data in the sketch. This simple approach is similar to earlier proposals; see [@WLRT08:Fast-Randomized Sec. 5.2], [@CW09:Numerical-Linear Thm. 4.7], [@HMT11:Finding-Structure Sec. 5.5], [@Woo14:Sketching-Tool Thm. 4.3, display 1].
We have obtained the first accurate error bound for this method. Our result shows how the spectrum of the input matrix affects the approximation quality. This analysis allows us to make parameter recommendations for specific input matrices.
In \[sec:structured\], we explain how to refine this algorithm to obtain approximations with additional structure. In \[sec:fixed-rank\], we describe modifications of the procedures that produce approximations with fixed rank and additional structure. Throughout, we maintain the notation of \[sec:mult-sketching\].
The Main Algorithm
------------------
Our goal is to produce a low-rank approximation of the input matrix ${\bm{A}}$ using only the knowledge of the test matrices $({\bm{\Omega}}, {\bm{\Psi}})$ and the sketch $({\bm{Y}}, {\bm{W}})$. Here is the basic method.
The first step in the procedure is to compute an orthobasis for the range of ${\bm{Y}}$ by means of an orthogonal–triangular factorization: $$\label{eqn:def-Q}
{\bm{Y}} =: {\bm{QR}}
\quad\text{where}\quad
{\bm{Q}} \in {\mathbb{F}}^{m \times k}.$$ The matrix ${\bm{Q}}$ has orthonormal columns; we discard the triangular matrix ${\bm{R}}$. The second step uses the co-range sketch ${\bm{W}}$ to form the matrix $$\label{eqn:def-X}
{\bm{X}} := ({\bm{\Psi}} {\bm{Q}})^{\dagger} {\bm{W}} \in {\mathbb{F}}^{k \times n}.$$ The random matrix ${\bm{\Psi}} {\bm{Q}} \in {\mathbb{F}}^{\ell \times k}$ is very well-conditioned when $\ell \gg k$, so we can perform this computation accurately by solving a least-squares problem. We report the rank-$k$ approximation $$\label{eqn:Ahat}
\hat{{\bm{A}}} := {\bm{QX}} \in {\mathbb{F}}^{m \times n}
\quad\text{where}\quad
{\bm{Q}} \in {\mathbb{F}}^{m \times k}
\quad\text{and}\quad
{\bm{X}} \in {\mathbb{F}}^{k \times n}.$$ The factors ${\bm{Q}}$ and ${\bm{X}}$ are defined in \[eqn:def-Q,eqn:def-X\].
The approximation $\hat{{\bm{A}}}$ is algebraically, but not numerically, equivalent with the approximation that appears in Clarkson & Woodruff [@CW09:Numerical-Linear Thm. 4.7]; see also [@Woo14:Sketching-Tool Thm. 4.3, display 1]. Our formulation improves on theirs by avoiding a badly conditioned least-squares problem.
Intuition {#sec:intuition}
---------
To motivate the algorithm, we recall a familiar heuristic [@HMT11:Finding-Structure Sec. 1] from randomized linear algebra, which states that $$\label{eqn:A-QQA}
{\bm{A}} \approx {\bm{QQ}}^* {\bm{A}}.$$ Although we would like to form the rank-$k$ approximation ${\bm{Q}} ({\bm{Q}}^* {\bm{A}})$, we cannot compute the factor ${\bm{Q}}^* {\bm{A}}$ without revisiting the input matrix ${\bm{A}}$. Instead, we exploit the information in the co-range sketch ${\bm{W}} = {\bm{\Psi}}{\bm{A}}$. Notice that $${\bm{W}} = {\bm{\Psi}}({\bm{Q}} {\bm{Q}}^*{\bm{A}}) + {\bm{\Psi}}({\bm{A}} - {\bm{QQ}}^* {\bm{A}})
\approx ({\bm{\Psi}} {\bm{Q}})({\bm{Q}}^*{\bm{A}}).$$ The heuristic \[eqn:A-QQA\] justifies dropping the second term. Multiplying on the left by the pseudoinverse $({\bm{\Psi}} {\bm{Q}})^{\dagger}$, we arrive at the relation $${\bm{X}} = ({\bm{\Psi}} {\bm{Q}})^{\dagger} {\bm{W}} \approx {\bm{Q}}^* {\bm{A}}.$$ These considerations suggest that $$\hat{{\bm{A}}} = {\bm{QX}} \approx {\bm{QQ}}^*{\bm{A}} \approx {\bm{A}}.$$ One of our contributions is to give substance to these nebulae.
This intuition is inspired by the discussion in [@HMT11:Finding-Structure Sec. 5.5], and it allows us to obtain sharp error bounds. Our approach is quite different from that of [@CW09:Numerical-Linear Thm. 4.7] or [@Woo14:Sketching-Tool Thm. 4.3].
Algorithm and Costs
-------------------
give pseudocode for computing the approximation \[eqn:Ahat\]. The first presentation uses <span style="font-variant:small-caps;">Matlab</span> functions to abbreviate some of the steps, while the second includes more implementation details. Note that the use of the $\texttt{orth}$ command may result in an approximation with rank $q$ for some $q \leq k$, but the quality of the approximation does not change.
${\bm{Q}} \gets \texttt{orth}( {\bm{Y}} )$ ${\bm{X}} \gets ({\bm{\Psi}} {\bm{Q}}) \backslash {\bm{W}}$
$({\bm{Q}}, \sim) \gets \texttt{qr}( {\bm{Y}}, \texttt{0} )$ \[line:dlr-qr1\] $({\bm{U}}, {\bm{T}}) \gets \texttt{qr}({\bm{\Psi}} {\bm{Q}}, \texttt{0})$ \[line:dlr-qr2\] ${\bm{X}} \gets {\bm{T}}^{\dagger} ({\bm{U}}^* {\bm{W}})$ \[line:dlr-ls\]
Let us summarize the costs of the approximation procedure \[eqn:def-Q,eqn:def-X,eqn:Ahat\], as implemented in \[alg:detailed-low-rank-recon\]. The algorithm has working storage of $\mathcal{O}(k(m + n))$ floating point numbers. The arithmetic cost is $\Theta(k\ell (m + n))$ flops, which is dominated by the matrix–matrix multiplications. The orthogonalization step and the back-substitution require $\Theta(k^2 (m+n))$ flops, which is almost as significant.
A Bound for the Frobenius-Norm Error
------------------------------------
We have established a very accurate error bound for the approximation \[eqn:Ahat\] that is implemented in \[alg:simple-low-rank-recon,alg:detailed-low-rank-recon\]. This analysis is one of the key contributions of this paper.
\[thm:err-frob\] Assume that the sketch size parameters satisfy $\ell > k + \alpha$. Draw random test matrices ${\bm{\Omega}} \in {\mathbb{F}}^{n \times k}$ and ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times m}$ independently from the standard normal distribution. Then the rank-$k$ approximation $\hat{{\bm{A}}}$ obtained from formula \[eqn:Ahat\] satisfies $$\label{eqn:err-frob}
\begin{aligned}
{\operatorname{\mathbb{E}}}{{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}}^2}
&\leq (1 + f(k,\ell)) \cdot \min_{\varrho < k - \alpha} (1 + f(\varrho,k)) \cdot \tau_{\varrho+1}^2({\bm{A}}) \\
&= \frac{k}{\ell - k - \alpha} \cdot \min_{\varrho < k - \alpha} \frac{k}{k - \varrho - \alpha}
\cdot \tau_{\varrho+1}^2({\bm{A}}).
\end{aligned}$$ The index $\varrho$ ranges over natural numbers. The quantity $\alpha({\mathbb{R}}) := 1$ and $\alpha({\mathbb{C}}) := 0$; the function $f(s, t) := s/(t-s-\alpha)$; the tail energy $\tau_j^2$ is defined in \[eqn:tail-energy\].
The proof of \[thm:err-frob\] appears below in \[sec:proof-low-rank-recon\].
To begin to understand \[thm:err-frob\], it is helpful to consider a specific parameter choice. Let $r$ be the target rank of the approximation, and select $$\label{eqn:my-param-choice}
k = 2r + \alpha
\quad\text{and}\quad
\ell = 2k + \alpha.$$ For these sketch size parameters, with $\varrho = r$, \[thm:err-frob\] implies that $${\operatorname{\mathbb{E}}}{{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}}^2}
\leq 4 \cdot \tau_{r+1}^2({\bm{A}}).$$ In other words, for $k \approx 2r$, we can construct a rank-$k$ approximation of ${\bm{A}}$ that has almost the same quality as a best rank-$r$ approximation. This parameter choice balances the sketch size against the quality of approximation.
But the true meaning of \[thm:err-frob\] lies deeper. The minimum in \[eqn:err-frob\] reveals that the approximation \[eqn:Ahat\] automatically takes advantage of decay in the tail energy. This fundamental fact explains the strong empirical performance of \[eqn:Ahat\] and other approximations derived from it. Our analysis is the first to identify this feature.
The analysis in [@CW09:Numerical-Linear Thm. 3.7] shows that $\hat{{\bm{A}}}$ achieves a bound of the form \[eqn:eps-subopt\] when the sketch size parameters scale as $k = \Theta(r/{\varepsilon})$ and $\ell = \Theta(k/{\varepsilon})$. A precise variant of the same statement follows from \[thm:err-frob\].
The expectation bound presented in \[thm:err-frob\] also describes the typical behavior of the approximation \[eqn:Ahat\] because of measure concentration effects. It is possible to develop a high-probability bound using the methods from [@HMT11:Finding-Structure Sec. 10.3].
It is also possible to develop bounds for the spectral-norm error incurred by the approximation \[eqn:Ahat\]. These results depend on the decay of both the singular values and the tail energies. See [@TYUC17:Randomized-Single-View-TR Thm. 4.2].
Theoretical Guidance on the Sketch Size {#sec:best-parameters}
---------------------------------------
is precise enough to predict the performance of the approximation \[eqn:Ahat\] for many types of input matrices. As a consequence, we can offer concrete guidance on the best sketch size parameters $(k, \ell)$ for various applications.
Observe that the storage cost of the sketch \[eqn:sketches\] is directly proportional to the sum $T := k + \ell$ of the sketch size parameters $k$ and $\ell$. In this section, we investigate the best way to apportion $k$ and $\ell$ when we fix the target rank $r$ and the total sketch size $T$. Throughout this discussion, we assume that $T \geq 2r + 3\alpha + 3$. See \[tab:theory-params\] for a summary of these rules; see \[sec:theory-params\] for an empirical evaluation.
Problem Regime Notation Equation
--------------------------- ----------------------------------- ---------------------------------------
General purpose $(k_{\natural}, \ell_{\natural})$ \[eqn:decay-params\]
Flat spectrum $(k_{\flat}, \ell_{\flat})$ \[eqn:flat-params,eqn:flat-params-R\]
Decaying spectrum $(k_{\natural}, \ell_{\natural})$ \[eqn:decay-params\]
Rapidly decaying spectrum $(k_{\sharp}, \ell_{\sharp})$ \[eqn:fast-decay-params\]
: **Theoretical Sketch Size Parameters.** This table summarizes how to choose the sketch size parameters $(k, \ell)$ to exploit prior information about the spectrum of the input matrix ${\bm{A}}$.[]{data-label="tab:theory-params"}
### Flat Spectrum
First, suppose that the singular values $\sigma_j({\bm{A}})$ of the input matrix ${\bm{A}}$ do not decay significantly for $j > r$. This situation occurs, for example, when the input is a rank-$r$ matrix plus white noise.
In this setting, the minimum in \[eqn:err-frob\] is likely to occur when $\varrho \approx r$. It is natural to set $\varrho = r$ and to minimize the resulting bound subject to the constraints $k + \ell = T$ and $k > r + \alpha$ and $\ell > k + \alpha$. For ${\mathbb{F}}= {\mathbb{C}}$, we obtain the parameter recommendations $$\label{eqn:flat-params}
k_{\flat} := \max\left\{ r + 1, \
\left \lfloor T \cdot \frac{\sqrt{r(T - r)} - r}{T - 2r} \right \rfloor \right\}
\quad\text{and}\quad
\ell_{\flat} := T - k_{\flat}.$$ In case ${\mathbb{F}}= {\mathbb{R}}$, we modify the formula \[eqn:flat-params\] so that $$\label{eqn:flat-params-R}
k_{\flat} := \max\left\{ r + 2, \left \lfloor (T-1) \cdot \frac{\sqrt{r(T - r - 2)(1 - 2/(T-1))} - (r-1)}{T - 2r - 1} \right \rfloor \right\}.$$ We omit the routine details behind these calculations.
### Decaying Spectrum or Spectral Gap
Suppose that the singular values $\sigma_j({\bm{A}})$ decay at a slow to moderate rate for $j > r$. Alternatively, we may suppose that there is a gap in the singular value spectrum at an index $j > r$.
In this setting, we want to exploit decay in the tail energy by setting $k \gg r$, but we need to ensure that the term $f(k, \ell)$ in \[eqn:err-frob\] remains small by setting $\ell \approx 2k + \alpha$. This intuition leads to the parameter recommendations $$\label{eqn:decay-params}
k_{\natural} := \max\{ r + \alpha + 1, \ \lfloor (T-\alpha)/3 \rfloor \}
\quad\text{and}\quad
\ell_{\natural} := T - k_{\natural}.$$ This is the best single choice for handling a range of examples. The parameter recommendation \[eqn:my-param-choice\] is an instance of \[eqn:decay-params\] with a minimal value of $T$.
### Rapidly Decaying Spectrum
Last, assume that the singular values $\sigma_j({\bm{A}})$ decay very quickly for $j > r$. This situation occurs in the application [@YUTC16:Sketchy-Decisions] that motivated us to write this paper.
In this setting, we want to exploit decay in the tail energy fully by setting $k$ as large as possible; the benefit outweighs the increase in $f(k, \ell)$ from choosing $\ell = k + \alpha + 1$, the minimum possible value. This intuition leads to the parameter recommendations $$\label{eqn:fast-decay-params}
k_{\sharp} := \lfloor (T-\alpha-1)/2 \rfloor
\quad\text{and}\quad
\ell_{\sharp} := T - k_{\sharp}.$$ Note that the choice is unwise unless the input matrix has sharp spectral decay.
Low-Rank Approximations with Convex Structure {#sec:structured}
=============================================
In many instances, we need to reconstruct an input matrix that has additional structure, such as symmetry or positive-semidefiniteness. The approximation formula \[eqn:Ahat\] from \[sec:low-rank-recon\] produces an approximation with no special properties aside from a bound on its rank. Therefore, we may have to reform our approximation to instill additional virtues.
In this section, we consider a class of problems where the input matrix belongs to a convex set and we seek an approximation that belongs to the same set. To accomplish this goal, we replace our initial approximation with the closest point in the convex set. This procedure always improves the Frobenius-norm error.
We address two specific examples: (i) the case where the input matrix is conjugate symmetric and (ii) the case where the input matrix is positive semidefinite. In both situations, we must design the algorithm carefully to avoid forming large matrices.
Projection onto a Convex Set {#sec:pocs}
----------------------------
Let $C$ be a closed and convex set of matrices in ${\mathbb{F}}^{m \times n}$. Define the projector ${\bm{\Pi}}_C$ onto the set $C$ to be the map $${\bm{\Pi}}_C : {\mathbb{F}}^{m \times n} \to C
\quad\text{where}\quad
{\bm{\Pi}}_C({\bm{M}}) := {\operatorname{arg\,min}}\big\{ {{{\Vert {\bm{C}} - {\bm{M}} \Vert}_{\mathrm{F}}}^2} : {\bm{C}} \in C \big\}.$$ The ${\operatorname{arg\,min}}$ operator returns the matrix ${\bm{C}}_{\star} \in C$ that solves the optimization problem. The solution ${\bm{C}}_{\star}$ is uniquely determined because the squared Frobenius norm is strictly convex and the constraint set $C$ is closed and convex.
Structure via Convex Projection
-------------------------------
Suppose that the input matrix ${\bm{A}}$ belongs to the closed, convex set $C \subset {\mathbb{F}}^{m \times n}$. Let $\hat{{\bm{A}}}_{\rm in} \in {\mathbb{F}}^{m \times n}$ be an initial approximation of ${\bm{A}}$. We can produce a new approximation ${\bm{\Pi}}_C(\hat{{\bm{A}}}_{\rm in})$ by projecting the initial approximation onto the constraint set. This procedure always improves the approximation quality in Frobenius norm.
\[fact:convex-structure\] Let $C \in {\mathbb{F}}^{m \times n}$ be a closed convex set, and suppose that ${\bm{A}} \in C$. For any initial approximation $\hat{{\bm{A}}}_{\rm in} \in {\mathbb{F}}^{m \times n}$, $$\label{eqn:convex-structure}
{{\Vert {\bm{A}} - {\bm{\Pi}}_C(\hat{{\bm{A}}}_{\rm in}) \Vert}_{\mathrm{F}}}
\leq {{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm in} \Vert}_{\mathrm{F}}}.$$
This result is well known in convex analysis. It follows directly from the first-order optimality conditions [@BV04:Convex-Optimization Sec. 4.2.3] for the Frobenius-norm projection of a matrix onto the set $C$. We omit the details.
does not hold if we replace the Frobenius norm by the spectral norm.
Low-Rank Approximation with Conjugate Symmetry
----------------------------------------------
When the input matrix is conjugate symmetric, it is often critical to produce a conjugate symmetric approximation. We can do so by combining the simple approximation from \[sec:low-rank-recon\] with the projection step outlined in \[sec:pocs\].
### Conjugate Symmetric Projection
Define the set $\mathbb{H}^n({\mathbb{F}})$ of conjugate symmetric matrices with dimension $n$ over the field ${\mathbb{F}}$: $$\mathbb{H}^n := \mathbb{H}^n({\mathbb{F}}) := \{ {\bm{C}} \in {\mathbb{F}}^{n \times n} : {\bm{C}} = {\bm{C}}^* \}.$$ The set $\mathbb{H}^n({\mathbb{F}})$ is convex because it forms a real-linear subspace in ${\mathbb{F}}^{n \times n}$. In the sequel, we omit the field ${\mathbb{F}}$ from the notation unless there is a possibility of confusion.
The projection ${\bm{M}}_{\rm sym}$ of a matrix ${\bm{M}} \in {\mathbb{F}}^{n \times n}$ onto the set $\mathbb{H}^n$ takes the form $$\label{eqn:sym-part}
{\bm{M}}_{\rm sym} := {\bm{\Pi}}_{\mathbb{H}^n}({\bm{M}}) = \frac{1}{2} ({\bm{M}} + {\bm{M}}^*).$$ For example, see [@Hig89:Matrix-Nearness Sec. 2].
### Computing a Conjugate Symmetric Approximation {#sec:sym-recon}
Assume that the input matrix ${\bm{A}} \in \mathbb{H}^n$ is conjugate symmetric. Let $\hat{{\bm{A}}} := {\bm{QX}}$ be an initial rank-$k$ approximation of ${\bm{A}}$ obtained from the approximation procedure \[eqn:Ahat\]. We can form a better Frobenius-norm approximation $\hat{{\bm{A}}}_{\rm sym}$ by projecting $\hat{{\bm{A}}}$ onto $\mathbb{H}^n$: $$\label{eqn:Ahat-sym}
\hat{{\bm{A}}}_{\rm sym} := {\bm{\Pi}}_{\mathbb{H}^n}(\hat{{\bm{A}}})
= \frac{1}{2}(\hat{{\bm{A}}} + \hat{{\bm{A}}}^*)
= \frac{1}{2}({\bm{QX}} + {\bm{X}}^* {\bm{Q}}^*).$$ The second relation follows from \[eqn:sym-part\].
In most cases, it is preferable to present the approximation \[eqn:Ahat-sym\] in factored form. To do so, we observe that $$\frac{1}{2}({\bm{QX}} + {\bm{X}}^* {\bm{Q}}^*) = \frac{1}{2}
\begin{bmatrix} {\bm{Q}} & {\bm{X}}^* \end{bmatrix}
\begin{bmatrix} {\bm{0}} & {\mathbf{I}}\\ {\mathbf{I}}& {\bm{0}} \end{bmatrix}
\begin{bmatrix} {\bm{Q}} & {\bm{X}}^* \end{bmatrix}^*.$$ Concatenate ${\bm{Q}}$ and ${\bm{X}}^*$, and compute the orthogonal–triangular factorization $$\label{eqn:qx-qr}
\begin{bmatrix} {\bm{Q}} & {\bm{X}}^* \end{bmatrix}
=: {\bm{U}} \begin{bmatrix} {\bm{T}}_1 & {\bm{T}}_2 \end{bmatrix}
\quad\text{where}\quad
{\bm{U}} \in {\mathbb{F}}^{n \times 2k}
\text{ and }
{\bm{T}}_1 \in {\mathbb{F}}^{2k \times k}.$$ Of course, we only need to orthogonalize the $k$ columns of ${\bm{X}}^*$, which permits some computational efficiencies. Next, introduce the matrix $$\label{eqn:def-S}
{\bm{S}} := \frac{1}{2} \begin{bmatrix} {\bm{T}}_1 & {\bm{T}}_2 \end{bmatrix}
\begin{bmatrix} {\bm{0}} & {\mathbf{I}}\\ {\mathbf{I}}& {\bm{0}} \end{bmatrix}
\begin{bmatrix} {\bm{T}}_1 & {\bm{T}}_2 \end{bmatrix}^*
= \frac{1}{2}( {\bm{T}}_1 {\bm{T}}_2^* + {\bm{T}}_2 {\bm{T}}_1^* )
\in {\mathbb{F}}^{2k \times 2k}.$$ Combine the last four displays to obtain the rank-$(2k)$ conjugate symmetric approximation $$\label{eqn:Ahat-sym-factored}
\hat{{\bm{A}}}_{\rm sym} = {\bm{USU}}^*.$$ From this expression, it is easy to obtain other types of factorizations, such as an eigenvalue decomposition, by further processing.
$({\bm{Q}}, {\bm{X}}) \gets \textsc{LowRankApprox}(\,)$ $({\bm{U}}, {\bm{T}}) \gets \texttt{qr}([{\bm{Q}}, {\bm{X}}^*], \texttt{0})$ \[algl:symm-qr\] ${\bm{T}}_1 \gets {\bm{T}}(\texttt{:}, 1\texttt{:}k)$ and ${\bm{T}}_2 \gets {\bm{T}}(\texttt{:}, (k+1)\texttt{:}(2k) )$ ${\bm{S}} \gets ({\bm{T}}_1 {\bm{T}}_2^* + {\bm{T}}_2 {\bm{T}}_1^*)/2$ \[algl:symm-symm\]
### Algorithm, Costs, and Error
contains pseudocode for producing a conjugate symmetric approximation of the form \[eqn:Ahat-sym-factored\] from a sketch of the input matrix. One can make this algorithm slightly more efficient by taking advantage of the fact that ${\bm{Q}}$ already has orthogonal columns; we omit the details.
For \[alg:symm-low-rank-recon\], the total working storage is $\Theta(kn)$ and the arithmetic cost is $\Theta( k \ell n )$. These costs are dominated by the call to <span style="font-variant:small-caps;">Sketch.LowRankApprox</span>.
Combining \[thm:err-frob\] with \[fact:convex-structure\], we have the following bound on the error of the symmetric approximation \[eqn:Ahat-sym-factored\], implemented in \[alg:symm-low-rank-recon\]. As a consequence, the parameter recommendations from \[sec:best-parameters\] are also valid here.
\[cor:symm-recon\] Assume that the input matrix ${\bm{A}} \in \mathbb{H}^n({\mathbb{F}})$ is conjugate symmetric, and assume that the sketch size parameters satisfy $\ell > k + \alpha$. Draw random test matrices ${\bm{\Omega}} \in {\mathbb{F}}^{n \times k}$ and ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times n}$ independently from the standard normal distribution. Then the rank-$(2k)$ conjugate symmetric approximation $\hat{{\bm{A}}}_{\rm sym}$ produced by \[eqn:Ahat-sym\] or \[eqn:Ahat-sym-factored\] satisfies $${\operatorname{\mathbb{E}}}{{{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm sym} \Vert}_{\mathrm{F}}}^2}
\leq (1 + f(k,\ell)) \cdot \min_{\varrho < k - \alpha} (1 + f(\varrho,k)) \cdot \tau_{\varrho+1}^2({\bm{A}}).$$ The index $\varrho$ ranges over natural numbers. The quantity $\alpha({\mathbb{R}}) := 1$ and $\alpha({\mathbb{C}}) := 0$; the function $f(s, t) := s/(t-s-\alpha)$; the tail energy $\tau_j^2$ is defined in \[eqn:tail-energy\].
Low-Rank Positive-Semidefinite Approximation
--------------------------------------------
We often encounter the problem of approximating a positive-semidefinite (psd) matrix. In many situations, it is important to produce an approximation that maintains positivity. Our approach combines the approximation \[eqn:Ahat\] from \[sec:low-rank-recon\] with the projection step from \[sec:pocs\].
### PSD Projection
We introduce the set $\mathbb{H}_+^n({\mathbb{F}})$ of psd matrices with dimension $n$ over the field ${\mathbb{F}}$: $$\mathbb{H}_+^n := \mathbb{H}_+^n({\mathbb{F}})
:= \big\{ {\bm{C}} \in \mathbb{H}^n : {\bm{z}}^* {\bm{C}} {\bm{z}} \geq 0
\text{ for each ${\bm{z}} \in {\mathbb{F}}^n$} \big\}.$$ The set $\mathbb{H}_+^n({\mathbb{F}})$ is convex because it is an intersection of halfspaces. In the sequel, we omit the field ${\mathbb{F}}$ from the notation unless there is a possibility for confusion.
Given a matrix ${\bm{M}} \in {\mathbb{F}}^{n \times n}$, we construct its projection onto the set $\mathbb{H}_+^n$ in three steps. First, form the projection ${\bm{M}}_{\rm sym} := {\bm{\Pi}}_{\mathbb{H}^n}({\bm{M}})$ onto the conjugate symmetric matrices, as in \[eqn:sym-part\]. Second, compute an eigenvalue decomposition ${\bm{M}}_{\rm sym} =: {\bm{VDV}}^*$. Third, form ${\bm{D}}_+$ by zeroing out the negative entries of ${\bm{D}}$. Then the projection ${\bm{M}}_+$ of the matrix ${\bm{M}}$ onto $\mathbb{H}^n_+$ takes the form $${\bm{M}}_+ := {\bm{\Pi}}_{\mathbb{H}_+^n}({\bm{M}}) = {\bm{V}} {\bm{D}}_+ {\bm{V}}^*.$$ For example, see [@Hig89:Matrix-Nearness Sec. 3].
### Computing a PSD Approximation {#sec:psd-recon}
Assume that the input matrix ${\bm{A}} \in \mathbb{H}_+^n$ is psd. Let $\hat{{\bm{A}}} := {\bm{QX}}$ be an initial approximation of ${\bm{A}}$ obtained from the approximation procedure \[eqn:Ahat\]. We can form a psd approximation $\hat{{\bm{A}}}_+$ by projecting $\hat{{\bm{A}}}$ onto the set $\mathbb{H}_+^n$.
To do so, we repeat the computations \[eqn:qx-qr\] and \[eqn:def-S\] to obtain the symmetric approximation $\hat{{\bm{A}}}_{\rm sym}$ presented in \[eqn:Ahat-sym-factored\]. Next, form an eigenvalue decomposition of the matrix ${\bm{S}}$ given by \[eqn:def-S\]: $${\bm{S}} =: {\bm{VDV}}^*.$$ In view of \[eqn:Ahat-sym-factored\], we obtain an eigenvalue decomposition of $\hat{{\bm{A}}}_{\rm sym}$: $$\hat{{\bm{A}}}_{\rm sym} = ({\bm{UV}}) {\bm{D}} ({\bm{UV}})^*.$$ To obtain the psd approximation $\hat{{\bm{A}}}_+$, we simply replace ${\bm{D}}$ by its nonnegative part ${\bm{D}}_+$ to arrive at the rank-$(2k)$ psd approximation $$\label{eqn:Ahat-psd-factored}
\hat{{\bm{A}}}_+ := {\bm{\Pi}}_{\mathbb{H}_+^n}(\hat{{\bm{A}}}) = ({\bm{UV}}) {\bm{D}}_+ ({\bm{UV}})^*.$$ This formula delivers an approximate eigenvalue decomposition of the input matrix.
$({\bm{U}},{\bm{S}}) \gets \textsc{LowRankSymApprox}(\,)$ \[algl:psd-orth\] $({\bm{V}}, {\bm{D}}) \gets \texttt{eig}({\bm{S}})$ \[algl:psd-eig\] ${\bm{U}} \gets {\bm{U}} {\bm{V}}$ \[algl:psd-consol\] ${\bm{D}} \gets \texttt{max}({\bm{D}}, \texttt{0})$
### Algorithm, Costs, and Error
contains pseudocode for producing a psd approximation of the form \[eqn:Ahat-psd-factored\] from a sketch of the input matrix. As in \[alg:symm-low-rank-recon\], some additional efficiencies are possible
The costs of \[alg:psd-low-rank-recon\] are similar with the symmetric approximation method, \[alg:symm-low-rank-recon\]. The working storage cost is $\Theta(kn)$, and the arithmetic cost is $\Theta(k \ell n)$.
Combining \[thm:err-frob,fact:convex-structure\], we obtain a bound on the approximation error identical to \[cor:symm-recon\]. We omit the details.
Fixed-Rank Approximations from the Sketch {#sec:fixed-rank}
=========================================
The algorithms in \[sec:low-rank-recon,sec:structured\] produce high-quality approximations with rank $k$, but we sometimes need to reduce the rank to match the target rank $r$. At the same time, we may have to impose additional structure. This section explains how to develop algorithms that produce a rank-$r$ structured approximation.
The technique is conceptually similar to the approach in \[sec:structured\]. We project an initial high-quality approximation onto the set of rank-$r$ matrices. This procedure preserves both conjugate symmetry and the psd property. The analysis in \[sec:pocs\] does not apply because the set of matrices with fixed rank is not convex. We present a general argument to show that the cost is negligible.
A General Error Bound for Fixed-Rank Approximation
--------------------------------------------------
If we have a good initial approximation of the input matrix, we can replace this initial approximation by a fixed-rank matrix without increasing the error significantly.
\[prop:fixed-rank-err\] Let ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ be a input matrix, and let $\hat{{\bm{A}}}_{\rm in} \in {\mathbb{F}}^{m \times n}$ be an approximation. For any rank parameter $r$, $$\label{eqn:fixed-rank-frob}
{{\Vert {\bm{A}} - {\llbracket {\hat{{\bm{A}}}_{\rm in}} \rrbracket_{r}} \Vert}_{\mathrm{F}}}
\leq \tau_{r+1}({\bm{A}}) + 2 {{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm in} \Vert}_{\mathrm{F}}}.$$ Recall that ${\llbracket {\cdot} \rrbracket_{r}}$ returns a best rank-$r$ approximation with respect to Frobenius norm.
Calculate that $$\begin{aligned}
{{\Vert {\bm{A}} - {\llbracket {\hat{{\bm{A}}}_{\rm in} } \rrbracket_{r}} \Vert}_{\mathrm{F}}}
&\leq {{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm in} \Vert}_{\mathrm{F}}} + {{\Vert \hat{{\bm{A}}}_{\rm in} - {\llbracket {\hat{{\bm{A}}}_{\rm in}} \rrbracket_{r}} \Vert}_{\mathrm{F}}} \\
&\leq {{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm in} \Vert}_{\mathrm{F}}} + {{\Vert \hat{{\bm{A}}}_{\rm in} - {\llbracket {{\bm{A}}} \rrbracket_{r}} \Vert}_{\mathrm{F}}} \\
&\leq 2 {{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm in} \Vert}_{\mathrm{F}}} + {{\Vert {\bm{A}} - {\llbracket {{\bm{A}}} \rrbracket_{r}} \Vert}_{\mathrm{F}}}.
\end{aligned}$$ The first and last relations are triangle inequalities. To reach the second line, note that ${\llbracket {\hat{{\bm{A}}}_{\rm in}} \rrbracket_{r}}$ is a best rank-$r$ approximation of $\hat{{\bm{A}}}_{\rm in}$, while ${\llbracket { {\bm{A}}} \rrbracket_{r}}$ is an undistinguished rank-$r$ matrix. Finally, identify the tail energy \[eqn:tail-energy\].
\[rem:fixed-rank-spec\] A result analogous to \[prop:fixed-rank-err\] also holds with respect to the spectral norm. The proof is the same.
Fixed-Rank Approximation from the Sketch
----------------------------------------
Suppose that we wish to compute a rank-$r$ approximation of the input matrix ${\bm{A}} \in {\mathbb{F}}^{m \times n}$. First, we form an initial approximation $\hat{{\bm{A}}} := {\bm{QX}}$ using the procedure \[eqn:Ahat\]. Then we obtain a rank-$r$ approximation ${\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}}$ of the input matrix by replacing $\hat{{\bm{A}}}$ with its best rank-$r$ approximation in Frobenius norm: $$\label{eqn:Ahat-fixed0}
{\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}} = {\llbracket { {\bm{Q}}{\bm{X}} } \rrbracket_{r}}.$$
We can complete this operation by working directly with the factors. Indeed, suppose that ${\bm{X}} = {\bm{U\Sigma V}}^*$ is an SVD of ${\bm{X}}$. Then ${\bm{QX}}$ has an SVD of the form $${\bm{QX}} = ({\bm{QU}}) {\bm{\Sigma}} {\bm{V}}^*.$$ As such, there is also a best rank-$r$ approximation of ${\bm{QX}}$ that satisfies $${\llbracket {{\bm{QX}}} \rrbracket_{r}} = ({\bm{QU}}) {\llbracket {{\bm{\Sigma}}} \rrbracket_{r}} {\bm{V}}^*
= {\bm{Q}} {\llbracket {{\bm{X}}} \rrbracket_{r}}.$$ Therefore, the desired rank-$r$ approximation \[eqn:Ahat-fixed0\] can also be expressed as $$\label{eqn:Ahat-fixed}
{\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}} = {\bm{Q}} {\llbracket {{\bm{X}}} \rrbracket_{r}}.$$ The formula \[eqn:Ahat-fixed\] is more computationally efficient than \[eqn:Ahat-fixed0\] because the factor ${\bm{X}} \in {\mathbb{F}}^{k \times n}$ is much smaller than the approximation $\hat{{\bm{A}}} \in {\mathbb{F}}^{m \times n}$.
The approximation ${\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}}$ is algebraically, but not numerically, equivalent to a formula proposed by Clarkson & Woodruff [@CW09:Numerical-Linear Thm. 4.8]. As above, our formulation improves on theirs by avoiding a badly conditioned least-squares problem.
### Algorithm and Costs
contains pseudocode for computing the fixed-rank approximation \[eqn:Ahat-fixed\].
The fixed-rank approximation in \[alg:fixed-rank-recon\] has storage and arithmetic costs on the same order as the simple low-rank approximation (\[alg:simple-low-rank-recon\]). Indeed, to compute the truncated SVD and perform the matrix–matrix multiplication, we expend only $\Theta(k^2 n)$ additional flops. Thus, the total working storage is $\Theta(k (m + n))$ numbers and the arithmetic cost is $\Theta(k\ell(m+n))$ flops.
$({\bm{Q}}, {\bm{X}}) \gets \textsc{LowRankApprox}(\,)$ $({\bm{U}}, {\bm{\Sigma}}, {\bm{V}}) \gets \texttt{svds}({\bm{X}}, r)$ \[line:fr-svd\] ${\bm{Q}} \gets {\bm{Q}} {\bm{U}}$ \[line:fr-consol\]
### A Bound for the Error
We can obtain an error bound for the rank-$r$ approximation \[eqn:Ahat-fixed\] by combining \[thm:err-frob,prop:fixed-rank-err\].
\[cor:fixed-rank-recon\] Assume the sketch size parameters satisfy $k > r + \alpha$ and $\ell > k + \alpha$. Draw random test matrices ${\bm{\Omega}} \in {\mathbb{F}}^{n \times k}$ and ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times m}$ independently from the standard normal distribution. Then the rank-$r$ approximation ${\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}}$ obtained from the formula \[eqn:Ahat-fixed\] satisfies $$\begin{gathered}
\label{eqn:fixed-rank-bound}
{\operatorname{\mathbb{E}}}{{\Vert {\bm{A}} - {\llbracket { \hat{{\bm{A}}} } \rrbracket_{r}} \Vert}_{\mathrm{F}}}
\leq \tau_{r+1}({\bm{A}})
+ 2 \sqrt{1+f(k,\ell)} \cdot \min_{\varrho < k - \alpha}
\sqrt{1 + f(\varrho, k)} \cdot \tau_{\varrho+1}({\bm{A}}).\end{gathered}$$ The index $\varrho$ ranges over natural numbers. The quantity $\alpha({\mathbb{R}}) := 1$ and $\alpha({\mathbb{C}}) := 0$; the function $f(s, t) := s/(t-s-\alpha)$; the tail energy $\tau_j^2$ is defined in \[eqn:tail-energy\].
This result indicates that the fixed-rank approximation ${\llbracket {\hat{{\bm{A}}}} \rrbracket_{r}}$ automatically exploits spectral decay in the input matrix ${\bm{A}}$. Moreover, we can still rely on the parameter recommendations from \[sec:best-parameters\]. Ours is the first theory to provide these benefits.
\[rem:fixed-rank-prior\] The analysis [@CW09:Numerical-Linear Thm. 4.8] of Clarkson & Woodruff implies that the approximation \[eqn:Ahat-fixed\] can achieve the bound \[eqn:eps-subopt\] for any ${\varepsilon}> 0$, provided that $k = \Theta(r/{\varepsilon}^2)$ and $\ell = \Theta(k/{\varepsilon}^2)$. It is possible to improve this scaling; see [@TYUC17:Randomized-Single-View-TR Thm. 5.1].
It is possible to obtain an error bound for the rank-$r$ approximation \[eqn:Ahat-fixed\] with respect to the spectral norm by combining [@TYUC17:Randomized-Single-View-TR Thm. 4.2] and \[rem:fixed-rank-spec\].
Fixed-Rank Conjugate Symmetric Approximation
--------------------------------------------
Assume that the input matrix ${\bm{A}} \in \mathbb{H}^n$ is conjugate symmetric and we wish to compute a rank-$r$ conjugate symmetric approximation. First, form an initial approximation $\hat{{\bm{A}}}_{\rm sym}$ using the procedure \[eqn:Ahat-sym-factored\] in \[sec:sym-recon\]. Then compute an $r$-truncated eigenvalue decomposition of the matrix ${\bm{S}}$ defined in \[eqn:def-S\]: $${\bm{S}} =: {\bm{V}} {\llbracket {{\bm{D}}} \rrbracket_{r}} {\bm{V}}^* \ +\ \textrm{approximation error}.$$ In view of the representation \[eqn:Ahat-sym-factored\], $$\label{eqn:Ahat-sym-fixed}
{\llbracket {\hat{{\bm{A}}}_{\rm sym}} \rrbracket_{r}} = ({\bm{UV}}) {\llbracket {{\bm{D}}} \rrbracket_{r}} ({\bm{UV}})^*.$$ contains pseudocode for the fixed-rank approximation \[eqn:Ahat-sym-fixed\]. The total working storage is $\Theta(k n)$, and the arithmetic cost is $\Theta(k \ell n)$.
If ${\bm{A}}$ is conjugate symmetric, then \[cor:symm-recon,prop:fixed-rank-err\] shows that ${\llbracket {\hat{{\bm{A}}}_{\rm sym}} \rrbracket_{r}}$ admits an error bound identical to \[cor:fixed-rank-recon\]. We omit the details.
$({\bm{U}}, {\bm{S}}) \gets \textsc{LowRankSymApprox}(\,)$ $({\bm{V}}, {\bm{D}}) \gets \texttt{eigs}({\bm{S}}, r, \texttt{'lm'})$ ${\bm{U}} \gets {\bm{U}} {\bm{V}}$
Fixed-Rank PSD Approximation
----------------------------
Assume that the input matrix ${\bm{A}} \in \mathbb{H}_+^n$ is psd, and we wish to compute a rank-$r$ psd approximation ${\llbracket {\hat{{\bm{A}}}_+} \rrbracket_{r}}$. First, form an initial approximation $\hat{{\bm{A}}}_{+}$ using the procedure \[eqn:Ahat-psd-factored\] in \[sec:psd-recon\]. Then compute an $r$-truncated positive eigenvalue decomposition of the matrix ${\bm{S}}$ defined in \[eqn:def-S\]: $${\bm{S}} =: {\bm{V}} {\llbracket {{\bm{D}}_+} \rrbracket_{r}} {\bm{V}}^*
\ +\ \textrm{approximation error}.$$ In view of the representation \[eqn:Ahat-psd-factored\], $$\label{eqn:Ahat-psd-fixed}
{\llbracket {\hat{{\bm{A}}}_{+}} \rrbracket_{r}} = ({\bm{UV}}) {\llbracket {{\bm{D}}_+} \rrbracket_{r}} ({\bm{UV}})^*.$$ contains pseudocode for the fixed-rank psd approximation \[eqn:Ahat-psd-fixed\]. The working storage is $\Theta(kn)$, and the arithmetic cost is $\Theta(k \ell n)$. If ${\bm{A}}$ is psd, then \[cor:symm-recon,prop:fixed-rank-err\] show that ${\llbracket {\hat{{\bm{A}}}_{\rm psd}} \rrbracket_{r}}$ satisfies an error bound identical to \[cor:fixed-rank-recon\]; we omit the details.
$({\bm{U}},{\bm{S}}) \gets \textsc{LowRankSymApprox}(\,)$ $({\bm{V}}, {\bm{D}}) \gets \texttt{eigs}({\bm{S}}, r, \texttt{'lr'})$ ${\bm{U}} \gets {\bm{U}} {\bm{V}}$ ${\bm{D}} \gets \texttt{max}({\bm{D}}, \texttt{0})$
Computational Experiments {#sec:experiments}
=========================
This section presents the results of some numerical tests designed to evaluate the empirical performance of our sketching algorithms for low-rank matrix approximation. We demonstrate that the approximation quality improves when we impose structure, and we show that our theoretical parameter choices are effective. The presentation also includes comparisons with several other algorithms from the literature.
Overview of Experimental Setup
------------------------------
For our numerical assessment, we work over the complex field (${\mathbb{F}}= {\mathbb{C}}$). Results for the real field (${\mathbb{F}}= {\mathbb{R}}$) are similar.
Let us summarize the procedure for studying the behavior of a specified approximation method on a given input matrix. Fix the input matrix ${\bm{A}}$ and the target rank $r$. Then select a pair $(k, \ell)$ of sketch size parameters where $k \geq r$ and $\ell \geq r$.
Each trial has the following form. We draw (complex) standard normal test matrices $({\bm{\Omega}}, {\bm{\Psi}})$ to form the sketch $({\bm{Y}}, {\bm{W}})$ of the input matrix. \[We do not use the optional orthogonalization steps in \[alg:sketch\].\] Next compute an approximation $\hat{{\bm{A}}}_{\rm out}$ of the matrix ${\bm{A}}$ by means of a specified approximation algorithm. Then calculate the error relative to the best rank-$r$ approximation: $$\label{eqn:relative-error}
\textrm{relative error} \quad:=\quad
\frac{{{\Vert {\bm{A}} - \hat{{\bm{A}}}_{\rm out} \Vert}_{\mathrm{F}}}}{ \tau_{r+1}({\bm{A}}) }
- 1.$$ The tail energy $\tau_j$ is defined in . If $\hat{{\bm{A}}}_{\rm out}$ is a rank-$r$ approximation of ${\bm{A}}$, the relative error is always nonnegative. To facilitate comparisons, our experiments only examine fixed-rank approximation methods.
To obtain each data point, we repeat the procedure from the last paragraph 20 times, each time with the same input matrix ${\bm{A}}$ and an independent draw of the test matrices $({\bm{\Omega}}, {\bm{\Psi}})$. Then we report the average relative error over the 20 trials.
We include our <span style="font-variant:small-caps;">Matlab</span> implementations in the supplementary materials for readers who seek more details on the methodology.
Classes of Input Matrices {#sec:input-matrix-examples}
-------------------------
We perform our numerical tests using several types of complex-valued input matrices. illustrates the singular spectrum of a matrix from each of the categories.
![**Spectra of input matrices.** These plots display the singular value spectrum for an input matrix from each of the classes (`LowRank`, `LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`, `PolyDecayFast`, `ExpDecaySlow`, `ExpDecayFast`, `Data`) described in \[sec:input-matrix-examples\].[]{data-label="fig:singVal"}](figures/spectrum-noise.pdf "fig:"){height="1.5in"} ![**Spectra of input matrices.** These plots display the singular value spectrum for an input matrix from each of the classes (`LowRank`, `LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`, `PolyDecayFast`, `ExpDecaySlow`, `ExpDecayFast`, `Data`) described in \[sec:input-matrix-examples\].[]{data-label="fig:singVal"}](figures/spectrum-decay "fig:"){height="1.5in"} ![**Spectra of input matrices.** These plots display the singular value spectrum for an input matrix from each of the classes (`LowRank`, `LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`, `PolyDecayFast`, `ExpDecaySlow`, `ExpDecayFast`, `Data`) described in \[sec:input-matrix-examples\].[]{data-label="fig:singVal"}](figures/spectrum-data.pdf "fig:"){height="1.5in"}
### Synthetic Examples
We fix a dimension parameter $n = 10^3$ and a parameter $R = 10$ that controls the rank of the “significant part” of the matrix. In our experiments, we compute approximations with target rank $r = 5$. Similar results hold when the parameter $R = 5$ and when $n = 10^4$.
We construct the following synthetic input matrices:
1. **Low-Rank + Noise:** These matrices take the form $$\begin{bmatrix} {\mathbf{I}}_R & {\bm{0}} \\ {\bm{0}} & {\bm{0}} \end{bmatrix}
\quad+\quad \sqrt{\frac{\gamma R}{2n^2}} ({\bm{G}} + {\bm{G}}^*)
\quad\in\quad {\mathbb{C}}^{n \times n}.$$ The matrix ${\bm{G}}$ is complex standard normal. The quantity $\gamma^{-2}$ can be interpreted as the signal-to-noise ratio (SNR). We consider three cases:
1. **No noise (`LowRank`):** $\gamma = 0$.
2. **Medium noise (`LowRankMedNoise`):** $\gamma = 10^{-2}$.
3. **High noise (`LowRankHiNoise`):** $\gamma = 1$.
For these models, all the experiments are performed on a single exemplar that is drawn at random and then fixed.
2. **Polynomially Decaying Spectrum:** These matrices take the form $${\operatorname{diag}}\big( \underbrace{1,\ \dots,\ 1}_R,\ 2^{-p},\ 3^{-p},\ 4^{-p},\ \dots,\ (n-R+1)^{-p} \big) \quad\in\quad {\mathbb{C}}^{n \times n},$$ where $p > 0$ controls the rate of decay. We consider two cases:
1. **Slow polynomial decay (`PolyDecaySlow`):** $p = 1$.
2. **Fast polynomial decay (`PolyDecayFast`):** $p = 2$.
3. **Exponentially Decaying Spectrum:** These matrices take the form $${\operatorname{diag}}\big( \underbrace{1,\ \dots,\ 1}_R,\ 10^{-q},\ 10^{-2q},\ 10^{-3q},\ \dots,\ 10^{- (n-R) q} \big) \quad\in\quad {\mathbb{C}}^{n \times n},$$ where $q > 0$ controls the rate of decay. We consider two cases:
1. **Slow exponential decay (`ExpDecaySlow`):** $q = 0.25$.
2. **Fast exponential decay (`ExpDecayFast`):** $q = 1$.
We can focus on diagonal matrices because of the rotational invariance of the test matrices $({\bm{\Omega}}, {\bm{\Psi}})$. Results for dense matrices are similar.
### A Matrix from an Application in Optimization {#eqn:cgm-matrix}
We also consider a dense, complex psd matrix (`Data`) obtained from a real-world phase retrieval application. This matrix has dimension $n = 25,921$ and exact rank 250. The first five singular values decrease from 1 to around 0.1; there is a large gap between the fifth and sixth singular value; the remaining nonzero singular values decay very fast. See our paper [@YUTC16:Sketchy-Decisions] for more details about the role of sketching in this context.
Alternative Sketching Algorithms for Matrix Approximation {#sec:other-algs}
---------------------------------------------------------
In addition to the algorithms we have presented, our numerical study comprises other methods that have appeared in the literature. We have modified all of these algorithms to improve their numerical stability and to streamline the computations. To the extent possible, we adopt the sketch \[eqn:sketches\] for all the algorithms to make their performance more comparable.
### Methods Based on the Sketch \[eqn:sketches\]
We begin with two additional methods that use the same sketch \[eqn:sketches\] as our algorithms.
First, let us describe a variant of a fixed-rank approximation scheme that was proposed by Woodruff [@Woo14:Sketching-Tool Thm. 4.3, display 2]. First, form a matrix product and compute its orthogonal–triangular factorization: ${\bm{\Psi Q}} =: {\bm{UT}}$ where ${\bm{U}} \in {\mathbb{F}}^{\ell \times k}$ has orthonormal columns. Then construct the rank-$r$ approximation $$\label{eqn:woodruff-fixed}
\hat{{\bm{A}}}_{\mathrm{woo}}
:= {\bm{Q}} {\bm{T}}^{\dagger} {\llbracket {{\bm{U}}^*{\bm{W}}} \rrbracket_{r}}.$$ Woodruff shows that $\hat{{\bm{A}}}_{\rm woo}$ satisfies \[eqn:eps-subopt\] when the sketch size scales as $k = \Theta(r/{\varepsilon})$ and $\ell = \Theta(k/{\varepsilon}^2)$. Compare this result with \[rem:fixed-rank-prior\].
Second, we outline a fixed-rank approximation method that is implicit in Cohen et al. [@CEM+15:Dimensionality-Reduction Sec. 10.1]. First, compute the $r$ dominant left singular vectors of the range sketch: $({\bm{V}}, \sim, \sim) := \texttt{svds}({\bm{Y}}, r)$. Form a matrix product and compute its orthogonal–triangular factorization: ${\bm{\Psi V}} =: {\bm{UT}}$ where ${\bm{U}} \in {\mathbb{F}}^{\ell \times r}$. Then form the rank-$r$ approximation $$\label{eqn:cohen-fixed}
\hat{{\bm{A}}}_{\mathrm{cemmp}}
:= {\bm{V}} {\bm{T}}^{\dagger} {\llbracket {{\bm{U}}^*{\bm{W}}} \rrbracket_{r}}.$$ The results in Cohen et al. imply that $\hat{{\bm{A}}}_{\mathrm{cemmp}}$ satisfies \[eqn:eps-subopt\] when the sketch size scales as $k = \Theta(r/{\varepsilon}^2)$ and $\ell = \Theta(r/{\varepsilon}^2)$.
The approximations \[eqn:woodruff-fixed,eqn:cohen-fixed\] both appear similar to our fixed-rank approximation, \[alg:fixed-rank-recon\]. Nevertheless, they are derived from other principles, and their behavior is noticeably different.
### A Method Based on an Extended Sketch {#sec:optimal-alg}
Next, we present a variant of a recent approach that requires a more complicated sketch and more elaborate computations. The following procedure is adapted from [@BWZ16:Optimal-Principal-STOC Thm. 12], using simplifications suggested in [@Upa16:Fast-Space-Optimal Sec. 3].
Let ${\bm{A}} \in {\mathbb{F}}^{m \times n}$ be an input matrix, and let $r$ be a target rank. Choose integer parameters $k$ and $s$ that satisfy $r \leq k \leq s \leq \min\{m,n\}$. For consistent notation, we also introduce a redundant parameter $\ell = k$. Draw and fix *four* test matrices: $$\label{eqn:fancy-test}
{\bm{\Psi}} \in {\mathbb{F}}^{k \times m}; \quad
{\bm{\Omega}} \in {\mathbb{F}}^{n \times \ell}; \quad
{\bm{\Phi}} \in {\mathbb{F}}^{s \times m}; \quad\text{and}\quad
{\bm{\Xi}} \in {\mathbb{F}}^{n \times s}.$$ The matrices $({\bm{\Psi}}, {\bm{\Omega}})$ are standard normal, while $({\bm{\Phi}}, {\bm{\Xi}})$ are SRFTs; see \[sec:distributions\]. The sketch now has *three* components: $$\label{eqn:fancy-sketch}
{\bm{W}} := {\bm{\Psi A}}; \quad
{\bm{Y}} := {\bm{A\Omega}}; \quad\text{and}\quad
{\bm{Z}} := {\bm{\Phi A \Xi}}.$$ To store the test matrices and the sketch, we need $(2k + 1)(m + n) + s(s+2)$ numbers.
To obtain a rank-$r$ approximation of the input matrix ${\bm{A}}$, first compute four thin orthogonal–triangular factorizations: $$\begin{aligned}
{\bm{Y}} &=: {\bm{Q}}_1 {\bm{R}}_1
&& \quad\text{and}\quad &
{\bm{W}} &=: {\bm{R}}_2^* {\bm{Q}}_2^*; \\
{\bm{\Phi Q}}_1 &=: {\bm{U}}_1 {\bm{T}}_1
&& \quad\text{and}\quad &
{\bm{Q}}_2^* {\bm{\Xi}} &=: {\bm{T}}_2^* {\bm{U}}_2^*.
\end{aligned}$$ Then construct the rank-$r$ approximation $$\label{eqn:bwz}
\hat{{\bm{A}}}_{\mathrm{bwz}} :=
{\bm{Q}}_1 {\bm{T}}_1^{\dagger} {\llbracket {{\bm{U}}_1^* {\bm{Z}} {\bm{U}}_2} \rrbracket_{r}}
({\bm{T}}_2^{*})^{\dagger} {\bm{Q}}_2^*.$$ By adapting and correcting [@BWZ16:Optimal-Principal-STOC Thm. 12], one can show that $\hat{{\bm{A}}}_{\mathrm{bwz}}$ achieves \[eqn:eps-subopt\] for sketch size parameters that satisfy $k = \Theta(r/{\varepsilon})$ and $s = \Theta((r\log(1+ r))^2 / {\varepsilon}^6)$. With this scaling, the total storage cost for the random matrices and the sketch is $\Theta((m+n)r/{\varepsilon}+ (r\log(1+r))^2 / {\varepsilon}^6)$.
The authors of [@BWZ16:Optimal-Principal-STOC] refer to their method as “optimal” because the scaling of the term $(m+n) r/{\varepsilon}$ in the storage cost cannot be improved [@CW09:Numerical-Linear Thm. 4.10]. Nevertheless, because of the ${\varepsilon}^{-6}$ term, the bound is incomparable with the storage costs achieved by other algorithms.
Performance with Oracle Parameter Choices {#sec:oracle-params}
-----------------------------------------
It is challenging to compare the relative performance of sketching algorithms for matrix approximation because of the theoretical nature of previous research. In particular, earlier work does not offer any practical guidance for selecting the sketch size parameters.
The only way to make a fair comparison is to study the *oracle performance* of the algorithms. That is, for each method, we fix the total storage, and we determine the minimum relative error that the algorithm can achieve. This approach allows us to see which techniques are most promising for further development. Nevertheless, we must emphasize that the oracle performance is not achievable in practice.
### Computing the Oracle Error {#sec:oracle-error}
It is straightforward to compare our fixed-rank approximation methods, \[alg:fixed-rank-recon,alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\], with the alternatives \[eqn:woodruff-fixed,eqn:cohen-fixed\] from the literature. In each case, the sketch \[eqn:sketches\] requires storage of $(k + \ell)(m + n)$ numbers, so we can parameterize the cost by $T := k + \ell$. For a given choice of $T$, we obtain the oracle performance by minimizing the empirical approximation error for each algorithm over all pairs $(k, \ell)$ where the sum $k + \ell = T$.
It is trickier to include the Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] method \[eqn:bwz\]. For a given $T$, we obtain the oracle performance of \[eqn:bwz\] by minimizing the empirical approximation error over pairs $(k, s)$ for which the storage cost of the sketch \[eqn:fancy-sketch\] matches the cost of the simple sketch \[eqn:sketches\]. That is, $(2k+1)(m+n) + s(s+2) \approx T(m + n)$.
### Numerical Comparison with Prior Work
For each input matrix described in \[sec:input-matrix-examples\], \[fig:oracle-performance\] compares the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against several alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. We make the following observations:
- For matrices that are well-approximated by a low-rank matrix (`LowRank`, `PolyDecayFast`, `ExpDecaySlow`, `ExpDecayFast`, `Data`), our fixed-rank approximation, \[alg:fixed-rank-recon\], dominates all other methods when the storage budget is adequate. In particular, for the rank-1 approximation of the matrix `Data`, our approach achieves relative errors 3–6 orders of magnitude better than any competitor.
- When we consider matrices that are poorly approximated by a low-rank matrix (`LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`), the recent method \[eqn:bwz\] of Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] has the best performance, especially when the storage budget is small. But see \[sec:structured-approx\] for more texture.
- Our method, \[alg:fixed-rank-recon\], performs reliably for all of the input matrices, and it is the only method that can achieve high accuracy for the matrix `Data`. Its behavior is less impressive for matrices that have poor low-rank approximations (`LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`), but it is still competitive for these examples.
- The method \[eqn:bwz\] of Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] offers mediocre performance for matrices with good low-rank approximations (`LowRank`, `ExpDecaySlow`, `ExpDecayFast`, `Data`). Strikingly, this approach fails to produce a high-accuracy rank-5 approximation of the rank-10 matrix `LowRank`, even with a large storage budget.
- The method \[eqn:woodruff-fixed\] of Woodruff [@Woo14:Sketching-Tool Thm. 4.3, display 2] is competitive for most synthetic examples, but it performs rather poorly on the matrix `Data`.
- The method \[eqn:cohen-fixed\] of Cohen et al. [@CEM+15:Dimensionality-Reduction Sec. 10.1] has the worst performance for almost all the examples.
In summary, \[alg:fixed-rank-recon\] has the best all-around behavior, while the Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] method \[eqn:bwz\] works best for matrices that have a poor low-rank approximation. See \[sec:recommendations\] for more discussion.
### Structured Approximations {#sec:structured-approx}
In this section, we investigate the effect of imposing structure on the low-rank approximations. compares the oracle performance of our fixed-rank approximation methods, \[alg:fixed-rank-recon,alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\]. We make the following observations:
- The symmetric approximation method, \[alg:sym-fixed-rank-recon\], and the psd approximation method, \[alg:psd-fixed-rank-recon\], are very similar to each other for all examples.
- The structured approximations, \[alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\], always improve on the unstructured approximation, \[alg:fixed-rank-recon\]. The benefit is most significant for matrices that have a poor low-rank approximation (`LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`).
- match or exceed the performance of the Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] method \[eqn:bwz\] for all examples.
In summary, if we know that the input matrix has structure, we can achieve a decisive advantage by enforcing the structure in the approximation.
Performance with Theoretical Parameter Choices {#sec:theory-params}
----------------------------------------------
It remains to understand how closely we can match the oracle performance of \[alg:fixed-rank-recon,alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\] in practice. To that end, we must choose the sketch size parameters *a priori* using only the knowledge of the target rank $r$ and the total sketch size $T$. In some instances, we may also have insight about the spectral decay of the input matrix. shows how the fixed-rank approximation method, \[alg:fixed-rank-recon\], performs with the theoretical parameter choices outlined in \[sec:best-parameters\]. We make the following observations:
- The parameter recommendation \[eqn:flat-params\], designed for a matrix with a flat spectral tail, works well for the matrices `LowRankMedNoise`, `LowRankHiNoise`, and `PolyDecaySlow`. We also learn that this parameter choice should not be used for matrices with spectral decay.
- The parameter recommendation \[eqn:decay-params\], for a matrix with a slowly decaying spectrum, is suited to the examples `LowRankMedNoise`, `LowRankHiNoise`, `PolyDecaySlow`, and `PolyDecayFast`. This parameter choice is effective for the remaining examples as well.
- The parameter recommendation \[eqn:fast-decay-params\], for a matrix with a rapidly decaying spectrum, is appropriate for the examples `PolyDecayFast`, `ExpDecaySlow`, `ExpDecayFast`, and `Data`. This choice must not be used unless the spectrum decays quickly.
- We have observed that the same parameter recommendations allow us to achieve near-oracle performance for the structured matrix approximations, \[alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\]. As in the unstructured case, it helps if we tune the parameter choice to the type of input matrix.
In summary, we always achieve reasonably good performance using the parameter choice \[eqn:decay-params\]. Furthermore, if we match the parameter selection \[eqn:flat-params,eqn:decay-params,eqn:fast-decay-params\] to the spectral properties of the input matrix, we can almost achieve the oracle performance in practice.
Recommendations {#sec:recommendations}
---------------
Among the fixed-rank approximation methods that we studied, the most effective are \[alg:fixed-rank-recon,alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\] and the Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] method \[eqn:bwz\]. Let us make some final observations based on our numerical experience.
are superior to methods from the literature for input matrices that have good low-rank approximations. Although \[alg:fixed-rank-recon\] suffers when the input matrix has a poor low-rank approximation, the structured variants, \[alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\], match or exceed other algorithms for all the examples we tested. We have also established that we can attain near-oracle performance for our methods using the *a priori* parameter recommendations from \[sec:best-parameters\]. Finally, our methods are simple and easy to implement.
The Boutsidis et al. [@BWZ16:Optimal-Principal-STOC Thm. 12] method \[eqn:bwz\] exhibits the best performance for matrices that have very poor low-rank approximations when the storage budget is very small. This benefit is diminished by its mediocre performance for matrices that do admit good low-rank approximations. The method \[eqn:bwz\] requires more complicated sketches and additional computation. Unfortunately, the analysis in [@BWZ16:Optimal-Principal-STOC] does not provide guidance on implementation.
In conclusion, we recommend the sketching methods, \[alg:fixed-rank-recon,alg:sym-fixed-rank-recon,alg:psd-fixed-rank-recon\], for computing structured low-rank approximations. In future research, we will try to design new methods that simultaneously dominate our algorithms and \[eqn:bwz\].
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/LR_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/MED_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/HI_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/PSLOW_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/PFAST_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/ESLOW_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/EFAST_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/DATA1_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of our fixed-rank approximation, \[alg:fixed-rank-recon\], against alternative methods \[eqn:woodruff-fixed,eqn:cohen-fixed,eqn:bwz\] from the literature. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-performance"}](figures/DATA5_Frobenius.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/LR_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/MED_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/HI_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/PSLOW_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/PFAST_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/ESLOW_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/EFAST_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/DATA1_Alg789.pdf){height="1.5in"}
[.325]{}
![**Oracle performance of sketching algorithms for structured fixed-rank matrix approximation as a function of storage cost.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the unstructured approximation (\[alg:fixed-rank-recon\]), the conjugate symmetric approximation (\[alg:sym-fixed-rank-recon\]), and the positive-semidefinite approximation (\[alg:psd-fixed-rank-recon\]). The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. Each data series displays the best relative error \[eqn:relative-error\] that the specified algorithm can achieve with storage $T$. See \[sec:oracle-error\] for details.[]{data-label="fig:oracle-structured"}](figures/DATA5_Alg789.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/MED.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/HI.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/PSLOW.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/PFAST.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/ESLOW.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/EFAST.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/DATA1.pdf){height="1.5in"}
[.325]{}
![**Performance of a sketching algorithm for fixed-rank matrix approximation with a priori parameter choices.** For each of the input matrices described in \[sec:input-matrix-examples\], we compare the oracle performance of the fixed-rank approximation, \[alg:fixed-rank-recon\], against its performance at theoretically motivated parameter choices. The matrix dimensions are $m = n = 10^3$ for the synthetic examples and $m = n = 25,921$ for the matrix `Data` from the phase retrieval application. Each approximation has rank $r = 5$, unless otherwise stated. The variable $T$ on the horizontal axis is (proportional to) the total storage used by each sketching method. The oracle performance is drawn from \[fig:oracle-performance\]. Each data series displays the relative error \[eqn:relative-error\] that \[alg:fixed-rank-recon\] achieves for a specific parameter selection. The parameter choice `FLAT` \[eqn:flat-params\] is designed for matrices with a flat spectral tail; `DECAY` \[eqn:decay-params\] is for a slowly decaying spectrum; `RAPID` \[eqn:fast-decay-params\] is for a rapidly decaying spectrum. See \[sec:theory-params\] for details.[]{data-label="fig:theory-params"}](figures/DATA5.pdf){height="1.5in"}
Analysis of the Low-Rank Approximation
======================================
In this appendix, we develop theoretical results on the performance of the basic low-rank approximation \[eqn:Ahat\] implemented in \[alg:simple-low-rank-recon,alg:detailed-low-rank-recon\].
Facts about Random Matrices
---------------------------
Our arguments require classical formulae for the expectations of functions of a standard normal matrix. In the real case, these results are [@HMT11:Finding-Structure Prop. A.1 and A.6]. The complex case follows from the same principles, so we omit the details.
\[fact:expect-gauss-frob\] Let ${\bm{G}} \in {\mathbb{F}}^{t \times s}$ be a standard normal matrix. For all matrices ${\bm{B}}$ and ${\bm{C}}$ with conforming dimensions, $$\label{eqn:expect-gauss-frob}
{\operatorname{\mathbb{E}}}{{{\Vert {\bm{BGC}} \Vert}_{\mathrm{F}}}^2} = \beta {{{\Vert {\bm{B}} \Vert}_{\mathrm{F}}}^2} {{{\Vert {\bm{C}} \Vert}_{\mathrm{F}}}^2}.$$ Furthermore, if $t > s + \alpha$, $$\label{eqn:expect-gauss-pinv-frob}
{\operatorname{\mathbb{E}}}{{{\Vert {\bm{G}}^\dagger \Vert}_{\mathrm{F}}}^2} = \frac{1}{\beta} \cdot \frac{s}{t - s - \alpha}
= \frac{1}{\beta} \cdot f(s, t).$$ The numbers $\alpha$ and $\beta$ are given by \[eqn:alpha-parameter\]; the function $f$ is introduced in \[eqn:f-intro\].
Results from Randomized Linear Algebra
--------------------------------------
Our arguments also depend heavily on the analysis of randomized low-rank approximation developed in [@HMT11:Finding-Structure Sec. 10]. We state these results using the familiar notation from \[sec:mult-sketching,sec:low-rank-recon\].
\[fact:hmt-err\] Fix ${\bm{A}} \in {\mathbb{F}}^{m \times n}$. Let $\varrho$ be a natural number such that $\varrho < k - \alpha$. Draw the random test matrix ${\bm{\Omega}} \in {\mathbb{F}}^{k \times n}$ from the standard normal distribution. Then the matrix ${\bm{Q}}$ computed by \[eqn:def-Q\] satisfies $${\operatorname{\mathbb{E}}}_{{\bm{\Omega}}} {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}
\leq (1 + f(\varrho, k)) \cdot \tau_{\varrho+1}^2({\bm{A}}).$$ The number $\alpha$ is given by \[eqn:alpha-parameter\]; the function $f$ is introduced in \[eqn:f-intro\].
This result follows immediately from the proof of [@HMT11:Finding-Structure Thm. 10.5] using \[fact:expect-gauss-frob\] to handle both the real and complex case simultaneously.
Proof of \[thm:err-frob\]: Frobenius Error Bound {#sec:proof-low-rank-recon}
------------------------------------------------
In this section, we establish a second Frobenius-norm error bound for the low-rank approximation \[eqn:Ahat\]. We maintain the notation from \[sec:mult-sketching,sec:low-rank-recon\], and we state explicitly when we are making distributional assumptions on the test matrices.
### Decomposition of the Approximation Error
formalizes the intuition that ${\bm{A}} \approx {\bm{Q}}({\bm{Q}}^* {\bm{A}})$. The main object of the proof is to demonstrate that ${\bm{X}} \approx {\bm{Q}}^* {\bm{A}}$. The first step in the argument is to break down the approximation error into these two parts.
\[lem:err-decomp\] Let ${\bm{A}}$ be an input matrix, and let $\hat{{\bm{A}}} = {\bm{QX}}$ be the approximation defined in \[eqn:Ahat\]. The approximation error decomposes as $${{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}}^2}
= {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2} + {{{\Vert {\bm{X}} - {\bm{Q}}^*{\bm{A}} \Vert}_{\mathrm{F}}}^2}.$$
We omit the proof, which is essentially just the Pythagorean theorem.
### Approximating the Second Factor
Next, we develop an explicit expression for the error in the approximation ${\bm{X}} \approx {\bm{Q}}^* {\bm{A}}$. It is convenient to construct a matrix ${\bm{P}} \in {\mathbb{F}}^{n \times (n-k)}$ with orthonormal columns that satisfies $$\label{eqn:def-P}
{\bm{PP}}^* = {\mathbf{I}}- {\bm{QQ}}^*.$$ Introduce the matrices $$\label{eqn:Psis}
{\bm{\Psi}}_1 := {\bm{\Psi}} {\bm{P}} \in {\mathbb{F}}^{\ell \times (n-k)}
\quad\text{and}\quad
{\bm{\Psi}}_2 := {\bm{\Psi}} {\bm{Q}} \in {\mathbb{F}}^{\ell \times k}.$$ We are now prepared to state the result.
\[lem:subspace-err\] Assume that the matrix ${\bm{\Psi}}_2$ has full column-rank. Then $$\label{eqn:B-Q*A}
{\bm{X}} - {\bm{Q}}^* {\bm{A}} = {\bm{\Psi}}_2^\dagger {\bm{\Psi}}_1 ({\bm{P}}^*{\bm{A}}) .$$ The matrices ${\bm{\Psi}}_1$ and ${\bm{\Psi}}_2$ are defined in \[eqn:Psis\].
Recall that ${\bm{W}} = {\bm{\Psi}} {\bm{A}}$, and calculate that $$\begin{aligned}
{\bm{W}} = {\bm{\Psi}} {\bm{A}}
= {\bm{\Psi}}{\bm{PP}}^* {\bm{A}} + {\bm{\Psi}} {\bm{QQ}}^* {\bm{A}}
= {\bm{\Psi}}_1 ({\bm{P}}^*{\bm{A}}) + {\bm{\Psi}}_2 ({\bm{Q}}^* {\bm{A}}).
\end{aligned}$$ The second relation holds because ${\bm{PP}}^* + {\bm{QQ}}^* = {\mathbf{I}}$. Then we use \[eqn:Psis\] to identify ${\bm{\Psi}}_1$ and ${\bm{\Psi}}_2$. By hypothesis, the matrix ${\bm{\Psi}}_2$ has full column-rank, so we can left-multiply the last display by ${\bm{\Psi}}_2^\dagger$ to attain $${\bm{\Psi}}_2^\dagger {\bm{W}} = {\bm{\Psi}}_2^\dagger {\bm{\Psi}}_1 ({\bm{P}}^* {\bm{A}}) + {\bm{Q}}^* {\bm{A}}.$$ Turning back to \[eqn:def-X\], we identify ${\bm{X}} = {\bm{\Psi}}_2^\dagger {\bm{W}}$.
### The Expected Frobenius-Norm Error in the Second Factor
We are now prepared to compute the average Frobenius-norm error in approximating ${\bm{Q}}^*{\bm{A}}$ by means of the matrix ${\bm{X}}$. In contrast to the previous steps, this part of the argument relies on distributional assumptions on the test matrix ${\bm{\Psi}}$. Remarkably, for a Gaussian test matrix, ${\bm{X}}$ is even an unbiased estimator of the factor ${\bm{Q}}^*{\bm{A}}.$
\[lem:avg-subspace-err\] Assume that ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times n}$ is a standard normal matrix that is independent from ${\bm{\Omega}}$. Then $${\operatorname{\mathbb{E}}}_{{\bm{\Psi}}}[ {\bm{X}} - {\bm{Q}}^* {\bm{A}} ] = {\bm{0}}.$$ Furthermore, $${\operatorname{\mathbb{E}}}_{{\bm{\Psi}}} {{{\Vert {\bm{X}} - {\bm{Q}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}
= f(k,\ell) \cdot {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}.$$
Observe that ${\bm{P}}$ and ${\bm{Q}}$ are partial isometries with orthogonal ranges. Owing to the marginal property of the standard normal distribution, the random matrices ${\bm{\Psi}}_1$ and ${\bm{\Psi}}_2$ are statistically independent standard normal matrices. In particular, ${\bm{\Psi}}_2 \in {\mathbb{F}}^{\ell \times k}$ almost surely has full column-rank because \[eqn:param-assumption\] requires that $\ell \geq k$.
First, take the expectation of the identity \[eqn:B-Q\*A\] to see that $${\operatorname{\mathbb{E}}}_{{\bm{\Psi}}}[ {\bm{X}} - {\bm{Q}}^* {\bm{A}} ]
= {\operatorname{\mathbb{E}}}_{{\bm{\Psi}}_2}{\operatorname{\mathbb{E}}}_{{\bm{\Psi}}_1} [ {\bm{\Psi}}_2^\dagger {\bm{\Psi}}_1 {\bm{P}}^* {\bm{A}} ]
= {\bm{0}}.$$ In the first relation, we use the statistical independence of ${\bm{\Psi}}_1$ and ${\bm{\Psi}}_2$ to write the expectation as an iterated expectation. Then we observe that ${\bm{\Psi}}_1$ is a matrix with zero mean.
Next, take the expected squared Frobenius norm of \[eqn:B-Q\*A\] to see that $$\begin{aligned}
{\operatorname{\mathbb{E}}}_{{\bm{\Psi}}} {{{\Vert {\bm{X}} - {\bm{Q}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}
&= {\operatorname{\mathbb{E}}}_{{\bm{\Psi}}_2} {\operatorname{\mathbb{E}}}_{{\bm{\Psi}}_1} {{{\Vert {\bm{\Psi}}_2^\dagger {\bm{\Psi}}_1 ({\bm{P}}^* {\bm{A}}) \Vert}_{\mathrm{F}}}^2} \\
&= \beta \cdot {\operatorname{\mathbb{E}}}_{{\bm{\Psi}}_2} \big[ {{{\Vert {\bm{\Psi}}_2^\dagger \Vert}_{\mathrm{F}}}^2} \cdot {{{\Vert {\bm{P}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2} \big]
= f(k, \ell) \cdot {{{\Vert {\bm{P}}^*{\bm{A}} \Vert}_{\mathrm{F}}}^2}.
\end{aligned}$$ The last two identities follow from \[eqn:expect-gauss-frob\] and \[eqn:expect-gauss-pinv-frob\] respectively, where we use the fact that ${\bm{\Psi}}_2 \in {\mathbb{F}}^{\ell \times k}$. To conclude, note that $${{{\Vert {\bm{P}}^*{\bm{A}} \Vert}_{\mathrm{F}}}^2} = {{{\Vert {\bm{PP}}^*{\bm{A}} \Vert}_{\mathrm{F}}}^2}
= {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}.$$ The first relation holds because ${\bm{P}}$ is a partial isometry and the Frobenius norm is unitarily invariant. Last, we apply the definition \[eqn:def-P\] of ${\bm{P}}$.
### Proof of \[thm:err-frob\]
We are now prepared to complete the proof of the Frobenius-norm error bound stated in \[thm:err-frob\]. For this argument, we assume that the test matrices ${\bm{\Omega}}\in {\mathbb{F}}^{n \times k}$ and ${\bm{\Psi}} \in {\mathbb{F}}^{\ell \times m}$ are drawn independently from the standard normal distribution.
According to \[lem:err-decomp\], $${{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}}^2}
= {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2} + {{{\Vert {\bm{X}} - {\bm{Q}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}.$$ Take the expectation of the last display to reach $$\begin{aligned}
{\operatorname{\mathbb{E}}}{{{\Vert {\bm{A}} - \hat{{\bm{A}}} \Vert}_{\mathrm{F}}}^2}
&= {\operatorname{\mathbb{E}}}_{{\bm{\Omega}}} {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2}
+ {\operatorname{\mathbb{E}}}_{{\bm{\Omega}}} {\operatorname{\mathbb{E}}}_{{\bm{\Psi}}} {{{\Vert {\bm{X}} - {\bm{Q}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2} \\
&= (1+f(k,\ell)) \cdot {\operatorname{\mathbb{E}}}_{{\bm{\Omega}}} {{{\Vert {\bm{A}} - {\bm{QQ}}^* {\bm{A}} \Vert}_{\mathrm{F}}}^2} \\
&\leq (1+ f(k,\ell)) \cdot (1+f(\varrho, k)) \cdot \tau_{\varrho+1}^2({\bm{A}}).
\end{aligned}$$ In the first line, we use the independence of the two random matrices to write the expectation as an iterated expectation. To reach the second line, we apply \[lem:avg-subspace-err\] to the second term. Invoke the randomized linear algebra result, \[fact:hmt-err\]. Finally, minimize over eligible indices $\varrho < k - \alpha$.
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[^1]: California Institute of Technology, Pasadena, CA ().
[^2]: [É]{}cole Polytechnique F[é]{}d[é]{}ral de Lausanne, Lausanne, Switzerland ().
[^3]: Cornell University, Ithaca, NY ().
[^4]: [É]{}cole Polytechnique F[é]{}d[é]{}ral de Lausanne, Lausanne, Switzerland ().
[^5]: Dated 30 August 2016. Revised 13 January 2017 and 6 June 2017 and 4 September 2017.
[^6]: A Rademacher random variable takes the values $\pm 1$ with equal probability.
|
---
abstract: 'Recent work on a lepton mass matrix model based on an SU(3) flavor symmetry which is broken into S$_4$ is reviewed. The flavor structures of the masses and mixing are caused by VEVs of SU(2)$_L$-singlet scalars $\phi$ which are nonets ([**8**]{}+[**1**]{}) of the SU(3) flavor symmetry, and which are broken into ${\bf 2}+{\bf 3}+{\bf 3}''$ and ${\bf 1}$ of S$_4$. If we require the invariance under the transformation $(\phi^{(8)},\phi^{(1)}) \rightarrow (-\phi^{(8)},+\phi^{(1)})$ for the superpotential of the nonet field $\phi^{(8+1)}$, the model leads to a beautiful relation for the charged lepton masses. The observed tribimaximal neutrino mixing is understood by assuming two SU(3) singlet right-handed neutrinos $\nu_R^{(\pm)}$ and an SU(3) triplet scalar $\chi$.'
---
[**Broken SU(3) Flavor Symmetry**]{}
[**and Tribimaximal Neutrino Mixing** ]{}[^1]
[*IHERP, Osaka University,\
1-16 Machikaneyama, Toyonaka, Osaka 560-0043, Japan\
E-mail address: koide@het.phys.sci.osaka-u.ac.jp*]{}
[**1 Introduction**]{}
The observed mass spectra and mixings of the fundamental particles will provide promising clues to unified understanding of the quarks and leptons, especially, to the understanding of the “flavor". In the present paper, we notice the following observed characteristic features in the lepton sector [@PDG06]:
\(i) The observed charged lepton masses $(m_e, m_\mu, m_\tau)$ satisfy the relation [@Koidemass; @Koide90] $$m_e+m_\mu+m_\tau=\frac{2}{3}(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2 ,
\eqno(1.1)$$ with remarkable precision;
\(ii) The observed neutrino mixing $U_{\nu}$ is approximately given by the so-called tribimaximal mixing [@tribi] $$U_{TB}=\left(
\begin{array}{ccc}
\frac{2}{\sqrt6} & \frac{1}{\sqrt3} & 0 \\
-\frac{1}{\sqrt6} & \frac{1}{\sqrt3} & -\frac{1}{\sqrt2} \\
-\frac{1}{\sqrt6} & \frac{1}{\sqrt3} & \frac{1}{\sqrt2}
\end{array} \right) ,
\eqno(1.2)$$ which suggests that the mixing can be described by Clebsh-Gordan-like coefficients. Therefore, for a start, in the present paper, we investigate the lepton masses and mixings. The purpose of the present paper is to review a recent attempt [@Koide0705] to investigate the mass relation (1.1) and the tribimaximal mixing.
In order to understand the relation (1.1), for example, we assume that there are three scalars $\phi_i$ ($i=1,2,3$), and the values of the charged lepton masses $m_{ei}$ are proportional to the square of the vacuum expectation values (VEVs) $v_i =\langle\phi_i\rangle$ of the scalars $\phi_i$, $m_{ei} = k v_i^2$ (in the Ref.[@Koide90; @KF96; @KT96], for instance, a seesaw type model $(M_e)_{ij} = \delta_{ij} v_i (M_E)^{-1} v_j$ has been assumed). We define singlet $\phi_\sigma$ and doublet $(\phi_\pi, \phi_\eta)$ of a permutation symmetry S$_3$ [@S3] by $$\left(
\begin{array}{c}
\phi_\pi \\
\phi_\eta \\
\phi_\sigma
\end{array}
\right) =
\left(
\begin{array}{ccc}
0 & -\frac{1}{\sqrt2} & \frac{1}{\sqrt2} \\
\frac{2}{\sqrt6} & -\frac{1}{\sqrt6} & -\frac{1}{\sqrt6} \\
\frac{1}{\sqrt3} & \frac{1}{\sqrt3} & \frac{1}{\sqrt3}
\end{array} \right)
\left(
\begin{array}{c}
\phi_1 \\
\phi_2 \\
\phi_3
\end{array} \right) ,
\eqno(1.3)$$ from the three objects $(\phi_1, \phi_2, \phi_3)$, and we consider the following S$_3$ invariant scalar potential $V(\phi)$ [@Koide90; @Koide99; @Koide06]: $$V(\phi) = m^2 (\phi_\pi^2 +\phi_\eta^2 +\phi_\sigma^2)
+\lambda_1 (\phi_\pi^2 +\phi_\eta^2 +\phi_\sigma^2)^2
+\lambda_2 \phi_\sigma^2 (\phi_\pi^2 + \phi_\eta^2).
\eqno(1.4)$$ The minimizing condition of the potential (1.4) leads to the relation $$v_\pi^2+v_\eta^2=v_\sigma^2 .
\eqno(1.5)$$ The relation (1.5) means $$v_1^2+v_2^2+v_3^2=\frac{2}{3}(v_1+v_2+v_3)^2 ,
\eqno(1.6)$$ because $$v_1^2+v_2^2+v_3^2=v_\pi^2+v_\eta^2+v_\sigma^2=2v_\sigma^2
= 2 \left( \frac{v_1+v_2+v_3}{\sqrt{3}}\right)^2 .
\eqno(1.7)$$ Therefore, we can obtain the mass relation (1.1) from the assumption $m_{ei} \propto v_i^2$. Here, note that although the scalar potential (1.4) is invariant under the S$_3$ symmetry, but it is not a general one of the S$_3$ invariant form. As pointed out in Ref. [@Koide06], the scalar potential with a general form cannot lead to the relation (1.5). For the derivation of the VEV relation (1.5), it is essential to choose the specific form (1.4) of the S$_3$ invariant terms. Similar formulation is also possible for other discrete symmetries A$_4$ [@Koide0701] and S$_4$ (see below). However, in such a symmetry, we still need an additional specific selection rule. What is the meaning of such a specific selection? In the present paper, we investigate this problem by assuming that the S$_4$ flavor symmetry is embedded into SU(3).
On the other hand, the observed tribimaximal mixing suggests the following scenario: From the definition (1.2), we can denote the fields $(\psi_1, \psi_2, \psi_3)$ as $$\left(
\begin{array}{c}
\psi_1 \\
\psi_2 \\
\psi_3
\end{array} \right)
= U_{TB}
\left(
\begin{array}{c}
\psi_\eta \\
\psi_\sigma \\
\psi_\pi
\end{array}
\right).
\eqno(1.8)$$ The observed neutrino mixing (1.2) means that when the mass eigenstates of the charged leptons are given by the $(\psi_1, \psi_2, \psi_3)$ basis, the mass eigenstates of the neutrinos are given by the $(\psi_\eta, \psi_\sigma, \psi_\pi)$ basis. Therefore, the problem is to find a model where the charged lepton mass eigenstates are $(e_1, e_2, e_3)$, while the neutrino mass eigenstates are given by $(\nu_\eta, \nu_\sigma, \nu_\pi)$ with the mass hierarchy $m_\eta^2 < m_\sigma^2 \ll m_\pi^2$ (or $m_\pi^2 \ll m_\eta^2 < m_\sigma^2 $). In the S$_4$, we can define the same relation as (1.8).
Thus, the characteristic features (1.1) and (1.2) in the lepton sector may be understood from the language of S$_4$ (also S$_3$ or A$_4$). However, as seen from the above review, the characteristic features (1.1) and (1.2) cannot be understood from the S$_4$ symmetry only. We need some additional assumptions. In the present model, we will investigate these problems under an assumption that the present S$_4$ symmetry is embedded into an SU(3) symmetry [@Mohapatra-S4]. In the next section, the singlet $\phi_\sigma$ and doublet $(\phi_\pi, \phi_\eta)$ will be understood as members of a nonet scalar $\phi$ \[[**1**]{}+[**8**]{} of SU(3)\], and the VEV relation (1.5) will be derived by requiring that $W(\phi)$ is invariant under a Z$_2$ symmetry. In Sec.3, in order to give the charged lepton masses and tribimaximal neutrino mixing, we will discuss the effective Hamiltonian by assuming an Froggatt-Nelsen [@Froggatt] type model. Finally, Sec.4 will be devoted to the summary and concluding remarks.
[**2 VEVs of SU(3) nonet scalars**]{}
The goal in the present section is to obtain the VEV relation (1.6) \[i.e. (1.5)\]. As seen in the previous section, in order to obtain the desirable results (1.5), we need assume an equal weight between the doublet and singlet terms of S$_4$. In the present paper, we assume that the S$_4$ symmetry is embedded into an SU(3) symmetry. The doublet $(\phi_\pi, \phi_\eta)$ and singlet $\phi_\sigma$ of S$_4$ are embedded in the [**6**]{} and $({\bf 8}+{\bf 1})$ of SU(3) [@Mohapatra-S4]. In the present model [@Koide0705], we assume that the doublet $(\phi_\pi, \phi_\eta)$ and singlet $\phi_\sigma$ originate in SU(3) octet and singlet, respectively. The essential assumption in the present paper is that the fields $\phi_u$ and $\phi_d$ always appear in the theory with the form of the nonet of U(3): $$\phi = \left(
\begin{array}{ccc}
\phi_1^1 & \ast & \ast \\
\ast & \phi_2^2 & \ast \\
\ast & \ast & \phi_3^3
\end{array} \right) ,
\eqno(2.1)$$ where $$\begin{array}{l}
\phi_1^1 = \frac{1}{\sqrt{3}} \phi_\sigma + \frac{2}{\sqrt{6}} \phi_\eta , \\
\phi_2^2 = \frac{1}{\sqrt{3}} \phi_\sigma - \frac{1}{\sqrt{6}} \phi_\eta
- \frac{1}{\sqrt{2}} \phi_\pi , \\
\phi_3^3 = \frac{1}{\sqrt{3}} \phi_\sigma - \frac{1}{\sqrt{6}} \phi_\eta
+ \frac{1}{\sqrt{2}} \phi_\pi ,
\end{array}
\eqno(2.2)$$ and the index $f$ ($f=u,d$) has been dropped.
Although we have obtained the VEV (1.5) relation from the scalar potential (1.4) by calculating $\partial V/\partial \phi_a$ ($a=\pi, \eta, \sigma$), in the present paper, we will obtain the relation (1.5) from an SU(3) invariant superpotential $W$ (The derivation of the VEV relation (1.5) from a superpotential $W$ has first been attempted by Ma [@Ma0612]): The SU(3) invariant superpotential for the nonet fields $\phi_f$ ($f=u,d$) are given by $$W(\phi_f) = \frac{1}{2} m_f {\rm Tr}(\phi_f\phi_f) +
\frac{1}{2\sqrt3}\lambda_f {\rm Tr}(\phi_f\phi_f\phi_f).
\eqno(2.3)$$ Since, in the next section, we want to assign charges +1 and $-1$ of a Z$_3$ symmetry to the fields $\phi_u$ and $\phi_d$, respectively, we also assign the Z$_3$ charges +1 and $-1$ to the mass parameters $m_u$ and $m_d$ in Eq.(2.3), respectively. However, the U(3) invariant superpotential (2.3) cannot give the relation (1.5). As we show below, only when we drop the ${\rm Tr}[(\phi^{(8)})^3]$-term in the cubic terms ${\rm Tr}(\phi^3)$, we can obtain the VEV relation (1.5). Therefore, we introduce a Z$_2$ symmetry, and we assign the Z$_2$ parities $-1$ and $+1$ (the Z$_2$ charge +1 and 0) for the octet part $\phi^{(8)}$ and singlet part $\phi^{(1)}$ of the nonet field $\phi$, respectively. The symmetry Z$_2$ breaks U(3) into SU(3). (In other words, in the present model, the flavor symmetry U(3) is explicitly broken from the beginning by the Z$_2$ symmetry. ) Under the requirement of the Z$_2$ invariance, i.e. the invariance under the transformation $$(\phi^{(8)},\phi^{(1)}) \rightarrow (-\phi^{(8)}, +\phi^{(1)}) ,
\eqno(2.4)$$ the terms ${\rm Tr}(\phi^{(8)}\phi^{(8)} \phi^{(8)})$ are forbidden. Thus, the superpotential (2.3) with the Z$_2$ invariance leads to $$W(\phi)=
\frac{1}{2} m \left[ {\rm Tr}(\phi^{(8)}\phi^{(8)})
+\phi_{\sigma}^2 \right]
+ \frac{1}{2} \lambda \phi_{\sigma} \left[
{\rm Tr}(\phi^{(8)}\phi^{(8)}) + \frac{1}{3} \phi_{\sigma}^2
\right]$$ $$=\frac{1}{2}m\left( \phi_{\sigma}^2+ \phi_\pi^2+\phi_\eta^2 \right)
+\frac{1}{2} \lambda \left[( \phi_{\pi}^2+ \phi_\eta^2)\phi_\sigma
+\frac{1}{3}\phi_\sigma^3 \right] + \cdots .
\eqno(2.5)$$ From the superpotential (2.5) with the Z$_2$ invariance, we obtain the VEV relation (1.5) as follows: From the condition $$\frac{\partial W}{\partial (\phi^{(8)})_i^j} =
m (\phi^{(8)})_j^i + \lambda \phi_{\sigma}
(\phi^{(8)})_j^i =0 ,
\eqno(2.6)$$ we obtain $$m + \lambda \phi_{\sigma} =0,
\eqno(2.7)$$ for $(\phi^{(8)})_i^j \neq 0$. By eliminating $m$ from Eq.(2.7) and the condition $$\frac{\partial W}{\partial \phi_{\sigma}} =
m \phi_{\sigma}
+\frac{1}{2} \lambda \left[ {\rm Tr}(\phi^{(8)}\phi^{(8)})
+ \phi_{\sigma}^2 \right] =0 ,
\eqno(2.8)$$ we obtain the relation $$\phi_{\sigma}^2 = {\rm Tr}(\phi^{(8)}\phi^{(8)}) =
\phi_{\pi}^2 + \phi_{\eta}^2 + \cdots ,
\eqno(2.9)$$ where “$\cdots$" denotes the contributions of ${\bf 3}$ and ${\bf 3}'$ of S$_4$.
The result (2.9) is still not our goal, because the relation contains the VEVs of the ${\bf 3}$ and ${\bf 3}'$ of S$_4$. So far, we have not discussed the splitting among the S$_4$ multiplets. Now, we bring a soft symmetry breaking of SU(3) into S$_4$ with an infinitesimal parameter $\varepsilon$ into the mass term of $W(\phi)$ as $${\rm Tr}(\phi^{(8)}\phi^{(8)}) \Rightarrow
\phi_{\pi} \phi_{\pi} + \phi_{\eta} \phi_{\eta}
+(1+\varepsilon) \sum_{i\neq j} (\phi^{(8)})_i^j
(\phi^{(8)})_j^i ,
\eqno(2.10)$$ by hand. Recall that when we obtain the relation (2.7), we have assumed $(\phi^{(8)})_i^j \neq 0$. Now, the conditions (2.6) are modified into the following conditions: $$\left[(1+\varepsilon) m + \lambda \phi_{\sigma} \right]
(\phi^{(8)})_j^i =0 \ \ \ (i\neq j),
\eqno(2.11)$$ $$\left(m + \lambda \phi_{\sigma} \right)
\phi_a =0 \ \ \ (a=\pi , \eta),
\eqno(2.12)$$ Therefore, we must take either $(\phi^{(8)})_j^i =0$ ($i\neq j$) or $\phi_a =0$ ($a=\pi , \eta$) for $\varepsilon \neq 0$. When we choose the solution $$\langle (\phi^{(8)})_j^i \rangle = 0 \ \ \ (i\neq j) ,
\eqno(2.13)$$ we can obtain the desirable relation (1.5). (However, it is possible that we can also take another solution with $\phi_\pi=\phi_\eta=0$ and $(\phi^{(8)})_j^i \neq 0$. The VEV solutions are not unique. The result (1.5) is merely one of the possible solutions.)
Thus, we have obtained not only the desirable VEV relation (1.5), but also the results (2.13). It should be worthwhile noticing that if we have assume the superpotential (2.3) without requiring the Z$_2$ invariance, we could obtain neither (1.5) nor (2.13).
By the way, we know that the three masses in any sectors of quarks and leptons are completely different among them. Therefore, if we assume a flavor symmetry, the symmetry must finally be broken completely. Usually, a relation which we derive in the exact symmetry limit is only approximately satisfied under the symmetry breaking. Although we derive the VEV relation (1.5) under the S$_4$ symmetry, the problem is whether the VEV relation (1.5) which is obtained under the S$_4$ symmetry is spoiled or not when we introduce such a symmetry breaking. In Ref.[@Koide0705], we will find that such a symmetry breaking term without spoiling the relation (1.5) is indeed possible.
[**3 Effective Hamiltonian**]{}
If we regard the scalars $\phi_u$ and $\phi_d$ as SU(2)$_L$ doublets, such a model with multi-Higgs doublets causes a flavor changing neutral current (FCNC) problem. Therefore, we must consider that the fields $\phi_u$ and $\phi_d$ are SU(2)$_L$ singlets. In the present paper, we assume a Froggatt-Nielsen [@Froggatt] type model $$H^{eff}=y_e \overline{\ell}_L H_L^d \frac{\phi_d}{\Lambda}
\frac{\phi_d}{\Lambda} \frac{\xi}{\Lambda} e_R
+y_\nu \overline{\ell}_L H_L^u \frac{\phi_u}{\Lambda}
\frac{\chi}{\Lambda} \nu_R
+y_R \overline{\nu}_R \Phi_R \nu_R^\ast ,
\eqno(3.1)$$ where $\ell_{iL}$ are SU(2)$_L$ doublet leptons $\ell_{iL}=(\nu_{iL}, e_{iL})$, $H_L^d$ and $H_L^u$ are conventional SU(2)$_L$ doublet Higgs scalars, $\phi_f$ ($f=u,d$), $\xi$ and $\chi$ are SU(2)$_L$ singlet scalars, and $\Lambda$ is a scale of the effective theory. We consider that $\langle \phi_f \rangle/\Lambda$, $\langle \xi \rangle/\Lambda$ and $\langle \chi \rangle/\Lambda$ are of the order of 1. The scalar $\Phi_R$ has been introduced in order to generate the Majorana mass $M_R$ of the right-handed neutrino $\nu_R$. As we note later, in the present model, the right-handed neutrinos $\nu_R=(\nu_R^{(+)} +\nu_R^{(-)})/\sqrt2$ are singlets of the SU(3) flavor. The role of $\xi=(\xi^{(+)}+\xi^{(-)})/\sqrt2$ and $\chi$ will be explained later. In order to understand the appearance of the combinations $H_L^d \phi_d \phi_d \xi$ and $H_L^u \phi_u \chi$, we assume two Z$_3$ symmetries (Z$_3$ and Z$'_3$ in Table 1). Those quantum number assignments are given in Table 1. However, even with those quantum numbers, we cannot distinguish the state $\phi_f^\dagger$ from $\phi_f \phi_f$. For example, the interaction $\bar{\ell}_L H_d \phi_d^\dagger \xi e_R$ is possible in addition to the interaction $\bar{\ell}_L H_d \phi_d \phi_d \xi e_R$. Although we have started from an SUSY scenario in the previous section, now, we have adopted an effective Hamiltonian which is not renormalizable. Therefore, in principle, the interaction $\bar{\ell}_L H_d \phi_d^\dagger \xi e_R$ cannot be ruled out. For the moment, in order to forbid such an undesirable term, we assume that the fields which can appear in the effective Hamiltonian are confined to holomorphic ones.
[**Table 1**]{} SU(3) and S$_4$ assignments of the fields
Fields SU(2)$_L$ SU(3) S$_4$ Z$_3$ Z$_3^{\prime}$ Z$_2$
----------------- ----------- --------------------- --------------------------------------- ------- ---------------- -------
$\ell_L$ [**2**]{} [**3**]{} ${\bf 3}'$ 0 0 0
$e_R$ [**1**]{} [**3**]{} $ {\bf 3}'$ 0 0 0
$\nu_R^{(\pm)}$ [**1**]{} [**1**]{} [**1**]{} 0 0 0/+1
$\phi_u$ [**1**]{} [**1**]{}+[**8**]{} $ {\bf 1}+({\bf 2}+{\bf 3}+{\bf 3}')$ +1 +1 0/+1
$\phi_d$ [**1**]{} [**1**]{}+[**8**]{} ${\bf 1}+({\bf 2}+{\bf 3}+{\bf 3}')$ $-1$ $-1$ 0/+1
$\xi^{(\pm)}$ [**1**]{} [**1**]{} ${\bf 1}$ 0 $-1$ 0/+1
$\chi$ [**1**]{} [**3**]{} ${\bf 3}'$ +1 $-1$ 0
$H_L^u$ [**2**]{} [**1**]{} [**1**]{} +1 0 0
$H_L^d$ [**2**]{} [**1**]{} [**1**]{} $-1$ 0 0
$\Phi_R$ [**1**]{} [**1**]{} [**1**]{} 0 0 0
[**(a) Charged lepton sector**]{}
Recall that we have already assumed the invariance of the superpotential under the Z$_2$ transformation (2.4) in order to drop the cubic part of the octet $\phi^{(8)}$. Therefore, the term $\phi\phi$ means $\phi^{(8)}\phi^{(8)} +\phi^{(1)}\phi^{(1)}$ under the Z$_2$ invariance. However, in order to give $m_{ei} \propto \langle \phi_i^i \rangle^2$, what we want is not $\phi^{(8)}\phi^{(8)} +\phi^{(1)}\phi^{(1)}$, but $\phi^{(8)}\phi^{(8)} +\phi^{(1)}\phi^{(1)}
+\phi^{(8)}\phi^{(1)} +\phi^{(1)}\phi^{(8)}$. In order to evade this problem, we introduce additional fields $\xi^{(+)}$ and $\xi^{(-)}$ whose Z$_2$ parity are $+1$ and $-1$, respectively. The effective interactions in the charged lepton sector are given by $$H_e^{eff} = \frac{y_e}{\sqrt2} \bar{e}_L^i (\phi_d)_i^j (\phi_d)_j^k
(\xi^{(+)} +\xi^{(-)}) e_{Rk} ,
\eqno(3.2)$$ where we have dropped the Higgs scalar $H_L^d$ since we discuss flavor structure only. The expression (3.2) becomes $$H_e^{eff} = \frac{y_e}{\sqrt2}
\bar{e}_L [ (\phi_d^{(8)}\phi_d^{(8)}
+ \phi_d^{(1)}\phi_d^{(1)})\xi^{(+)} +
(\phi_d^{(8)}\phi_d^{(1)}+ \phi_d^{(1)}\phi_d^{(8)}) \xi^{(-)}] e_{R}.
\eqno(3.3)$$ Since we have assumed that $\xi^{(+)}$ and $\xi^{(-)}$ appear symmetrically in the theory, we also assume $$\langle\xi^{(+)}\rangle=\langle\xi^{(-)}\rangle \equiv v_\xi .
\eqno(3.4)$$ Then, we obtain the effective Hamiltonian for the charged leptons $$H_e^{eff} = \frac{y_e v_d v_\xi }{\sqrt2 \Lambda^3}
\sum_i \bar{e}_L^i \langle (\phi_d^{(8+1)})_i^i\rangle^2 e_{Ri} ,
\eqno(3.5)$$ where $v_d =\langle H_L^{d0}\rangle$. Since the fields $(\phi_d)_i^i$ are defined by Eq.(2.2), we can obtain the charged lepton mass relation (1.1) from the VEV relation (1.6).
[**(b) Neutrino sector**]{}
In the present model, the right-handed neutrinos $\nu^{(\pm)}$ are singlets of SU(3). Therefore, in the neutrino seesaw mass matrix $M_\nu = m_L^\nu M_R^{-1} (m_L^\nu)^T$, $M_R$ is a $1\times 1$ matrix and $m_L^\nu$ is a $3\times 1$ matrix. In order to compensate for the absence of the conventional triplet neutrinos $\nu_R$, a new scalar $\chi$ which is a triplet of SU(3) has been introduced. The neutrino Dirac mass terms are given by the following effective Hamiltonian $$H_{Dirac}^{eff} =y_\nu \frac{v_u}{\Lambda^2} \bar{\nu}_L^i
\langle (\phi_u)_i^j\rangle \langle\chi_j \rangle (\nu_R^{(+)}
+\nu_R^{(-)}) ,
\eqno(3.6)$$ where $v_u= \langle H_L^{u0}\rangle$. It is likely that the scalar potential $V(\chi)$ for the SU(3) triplet $\chi$ has a specific VEV solution $$\langle \chi_{1}\rangle =\langle \chi_{2}\rangle =
\langle \chi_{3}\rangle \equiv v_\chi .
\eqno(3.7)$$ When we assume the VEVs (3.7), we obtain $$H_{Dirac}^{eff} =y_\nu \frac{v_u v_\chi}{\sqrt2 \Lambda^2}
(\bar{\nu}_\eta\ \bar{\nu}_\sigma \ \bar{\nu}_\pi)_L
\left[
\left( \begin{array}{c}
v_{\eta} \\
0 \\
v_{\pi}
\end{array} \right) \nu_R^{(-)}
+\left( \begin{array}{c}
0 \\
v_{\sigma} \\
0
\end{array} \right) \nu_R^{(+)} \right] ,
\eqno(3.8)$$ where $v_a = \langle \phi_{ua}\rangle$ ($a=\pi, \eta, \sigma$) (for convenience, we have dropped the index $u$). Therefore, we obtain the effective neutrino mass matrix on the $(\eta, \sigma, \pi)$ basis, $$U_{TB}^T M_\nu U_{TB} \equiv M_\nu^{(\eta\sigma\pi)} = \frac{1}{M_R^{(-)}}
\left( \begin{array}{ccc}
v_\eta^2 & 0 & v_\pi v_\eta \\
0 & 0 & 0 \\
v_\pi v_\eta & 0 & v_\pi^2
\end{array} \right)
+ \frac{1}{M_R^{(+)}}
\left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & v_\sigma^2 & 0 \\
0 & 0 & 0
\end{array} \right) ,
\eqno(3.9)$$ where $M_R^{(\pm)} = y_R^{(\pm)}
\langle \Phi_R \rangle$, and we have dropped the common factors $(y_\nu v_u v_\chi/\sqrt2 \Lambda^2)^2$. By the way, the ratio $v_\pi/v_\eta$ cannot be determined from the potential (2.6), and the ratio is determined by a soft S$_4$ symmetry breaking term $W_{SB}$ which has been discussed in the previous section. We can choose a solution $v_\pi =0$ in the superpotential $W(\phi_u)$ by adjusting the parameter $\beta$ in $W_{SB}$, differently from the case of $W(\phi_d)$. Then, the neutrino mass matrix (3.9) becomes a diagonal form $D_\nu =(1/M_R^{(-)}){\rm diag}( v_\eta^2, 0, 0)+
(1/M_R^{(+)}){\rm diag}( 0, v_\sigma^2, 0)$. Since the mass matrix $M_\nu$ on the $(\nu_1,\nu_2,\nu_3)
=(\nu_e,\nu_\mu,\nu_\tau)$ basis is given by $$M_\nu =U_{TB} M_\nu^{(\eta\sigma\pi)} U_{TB}^T =U_{TB} D_\nu U_{TB}^T ,
\eqno(3.10)$$ we can obtain the tribimaximal mixing $$U_{\nu}=U_{TB} ,
\eqno(3.11)$$ and the neutrino masses $$m_{\nu 1}=k v_\eta^2 , \ \ m_{\nu 2}=k v_\sigma^2 , \ \
m_{\nu 3}=0,
\eqno(3.12)$$ for the case of $M_R^{(+)}=M_R^{(-)}\equiv M_R$, where $k=(y_\nu v_u v_\chi)^2/2M_R \Lambda^4$ and $(\nu_\eta, \nu_\sigma,\nu_\pi)$ has been renamed $(\nu_1,\nu_2,\nu_3)$ according to the conventional naming. However, since we have taken $v_\pi=0$, the value of $v_\eta$ satisfies $v_\eta^2=v_\sigma^2$ from the relation (1.5), so that the result (3.12) gives $m_{\nu 1}=m_{\nu 2}$. The observed value [@solar] $\Delta m^2_{solar}$ is small, but it is not zero. Therefore, we must consider a small deviation between the first and second terms in (3.9) (i.e. $M_R^{(+)} \neq M_R^{(-)}$). Since the value $M_R^{(-)}/M_R^{(+)}$ is free in the present model, we cannot predict an explicit value of the ratio $\Delta m^2_{solar}/\Delta m^2_{atm}$.
Since the present model gives an inverse hierarchy of the neutrino masses, the predicted effective electron neutrino mass $$\langle m_{\nu_e}\rangle =\left|\sum_i U_{ei}^2 m_{\nu i}\right|
\simeq |m_{\nu 1}|
\simeq |m_{\nu 2}| \simeq \sqrt{\Delta m_{atm}^2}
=5.23^{+0.25}_{-0.40} \times 10^{-2}\, {\rm eV},
\eqno(3.13)$$ where we have used the value [@atm] $\Delta m_{atm}^2 = 2.74^{+0.44}_{-0.26}\times 10^{-3}\, {\rm eV}^2$. This value (3.13) is sufficiently sensitive to the next generation experiments of the neutrinoless double beta decay.
[**4 Summary**]{}
In conclusion, on the basis of the S$_4$ symmetry which is embedded into SU(3), we have investigated a lepton mass model with the effective Hamiltonian of the Froggatt-Nielsen type (3.1). We have assumed that the singlet and doublet of S$_4$ originate in the singlet and octet of SU(3), and we have obtained the VEV relation (1.5). In the derivation of the VEV relation (1.5), the essential assumptions for the superpotential $W(\phi_f)$ are the following two: (i) the scalar fields $\phi_f$ always appear in terms of the nonet form (2.1) of U(3); (ii) the superpotential $W(\phi_f)$ is invariant under the Z$_2$ transformation (2.4). Then, we have obtained not only the VEV relation (1.5), but also $\langle (\phi^{(8)})_i^j \rangle =0$ ($i\neq j$) for the other components of $\phi^{(8)}$ (i.e. $\langle {\bf 3}\rangle =\langle {\bf 3}' \rangle =0$).
In the charged lepton sector, we have assumed the Frogatt-Nielsen-type effective Hamiltonian $\bar{e}_L^i (\phi_d)_i^j (\phi_d)_j^k e_{Rj}$. Since we have obtained the VEV of $\phi_i^j$ $$\langle \phi_d \rangle = {\rm diag}\left(
\frac{1}{\sqrt3}v_\sigma +\frac{2}{\sqrt6}v_\eta ,
\frac{1}{\sqrt3}v_\sigma -\frac{1}{\sqrt6}v_\eta -\frac{1}{\sqrt2}v_\pi ,
\frac{1}{\sqrt3}v_\sigma -\frac{1}{\sqrt6}v_\eta +\frac{1}{\sqrt2}v_\pi
\right) ,
\eqno(4.1)$$ we can obtain $$m_{e i} \propto \langle (\phi_d)_i^i \rangle^2 \equiv (v_i^i)^2 ,
\eqno(4.2)$$ so that the charged lepton masses $m_{ei}$ satisfy the relation (1.1) under the definition (2.2). Here, we would like to emphasize that the result $\langle\phi_i^j\rangle =0$ for $i\neq j$, Eq.(2.13), is essential to obtain Eq.(4.2) in the Froggatt-Nielsen-type model. In the Froggatt-Nielsen-type model, if we had considered a triplet scalar $\phi$, we could obtain the result $m_{ei} \propto v_i^2$, because the effective interaction $\bar{e}_L \phi \phi e_R$ gives $\bar{e}_{Li} \phi_i \phi_j e_{Rj}$. On the other hand, if we had adopted a seesaw-type model, by assuming a triplet scalar $\phi$ whose effective interaction is given by $$H_e =\sum_i\left( \bar{e}_{Li} \phi_i E_{Ri} + \bar{E}_{Li} \phi_i e_{Ri}
+M_E \bar{E}_{Li} E_{Ri} \right) ,
\eqno(4.3)$$ we could automatically obtain a form $$H_L^{eff} = \sum_i \frac{1}{M_E} \bar{e}_{Li} \phi_i^2 e_{Ri} ,
\eqno(4.4)$$ through a seesaw mechanism [@seesaw]. However, it is not easy to obtain the VEV relation (1.6) for the triplet scalar $\phi_i$. This is the main motive for introducing the nonet (not triplet) scalar $\phi$.
For the neutrino sector, we have obtained the tribimaximal mixing (1.2) by introducing an SU(3) triplet scalar $\chi$ and the two SU(3) singlet right-handed neutrinos $\nu_R^{(\pm)}$ in addition to the nonet scalar $\phi_u$. In the present model, the right-handed neutrinos $\nu_R^{(\pm)}$ are singlets of SU(3), the Majorana neutrino mass matrices $M_R^{(\pm)}$ have no flavor structure, (i.e. $M_R$ are $1\time 1$ matrices). Also note that the neutrino Dirac mass matrix $m_L^\nu$ is a $3\times 1$ matrix. This plays an essential role to derivation of the tribimaximal mixing. For the neutrino mass spectrum, since the model gives $m_{\nu 1}=m_{\nu 2}$ in the limit of $M_R^{(+)}= M_R^{(-)}$, we must consider a small deviation $M_R^{(+)} \neq M_R^{(-)}$. Since the value of $M_R^{(-)}/ M_R^{(+)}$ is a free parameter in the present model, we cannot predict the value $\Delta m^2_{solar}/\Delta m^2_{atm}$ at present, although the smallness of the ratio $\Delta m^2_{solar}/\Delta m^2_{atm}$ can be understood. Since the present model gives an inverse hierarchy of the neutrino masses, we can predict the effective electron neutrino mass $\langle m_{\nu_e}\rangle \simeq 0.05$ eV, which is sufficiently sensitive to the next generation experiments of the neutrinoless double beta decay.
The present model seems to provide suggestive hints on seeking for a model which leads to the tribimaximal mixing (1.2) and the charged lepton mass relation (1.1), although the model has still many points which should be improved. We summarize some of the future tasks:
\(i) Seeking for a more natural mechanism which can provide $m_{ei} \propto v_i^2$, apart from a Froggatt-Nielsen-type model.
\(ii) Seeking for a model which is completely predictable neutrino mass spectrum (in the present model, $\Delta m_{21}^2$ was not predictable) together with the nearly tribimaximal mixing.
\(iii) In the present model, the right-handed neutrino $\nu_R$ was a singlet of SU(3). However, it is likely that $\nu_R$ still a triplet.
\(iv) Seeking for the origin of the symmetry breaking of SU(3) into S$_4$.
We hope that the present model will also provide a promising clue to the unified mass matrix model of the quarks and leptons.
**Acknowledgments**
The author would like to thank E. Takasugi, H. Fusaoka and N. Haba for helpful conversations. This work is supported by the Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, Japan (No.18540284).
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[^1]: Contributed paper to XXIII International Symposium on Lepton and Photon Interactions at High Energy (Lepton-Photon 2007), Aug 13-18, 2007, Daegu, Korea.
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abstract: 'We propose a generic topological insulator bilayer (TIB) system to study the excitonic condensation with self-consistent mean-field (SCMF) theory. We show that the TIB system presents the crossover behavior from the Bardeen-Cooper-Schrieffer (BCS) limit to Bose-Einstein condensation (BEC) limit. Moreover, by comparison with traditional semiconductor systems, we find that for the present system the superfluid property in the BEC phase is more sensitive to electron-hole density imbalance and the BCS phase is more robust. Applying this TIB model into Bi$_{2}$Se$_{3}$-family material, we find that the BEC phase is most probable to be observed in experiment. We also calculate the critical temperature for Bi$_{2}$Se$_{3}$-family TIB system, which is $\mathtt{\sim}100$ K. More interestingly, we can expect this relative high-temperature excitonic condensation since our calculated SCMF critical temperature is approximately equal to the Kosterlitz-Thouless transition temperature.'
author:
- Zhigang Wang
- Ningning Hao
- 'Zhen-Guo Fu'
- Ping Zhang
title: Excitonic condensation for the surface states of topological insulator bilayers
---
[^1]
Introduction
============
Recent technological advances in microfabrication bring growing interests in studying exciton condensation in different bilayer physical systems such as the semiconductor electron-hole bilayers [@Snoke; @Sahin; @Zhu] and graphene bilayers [@CHZhang; @MacDonald]. A number of novel physical phenomena are obtained in these systems, such as the BCS-BEC crossover [@Comte] as well as the subtle phase transition in the crossover region induced by the density imbalance [@Strinati], the dark and bright excitonic condensation under spin-orbit coupling [@Can], anomalous exciton condensation in high Landau levels in magnetic field [@MacDonald], room-temperature superfluidity in graphene bilayers [@Mac], etc. The conventional electron-hole bilayers are fabricated with semiconductor heterostructures such as GaAs/AlGaAs/GaAs. The character of the semiconductor electron-hole bilayers is that the electron and hole bands are quadratic ones with different effective masses, which means missing particle-hole symmetry in these kinds of systems and small superfluid density. Hence, in semiconductor electron-hole bilayers, the excitonic condensation needs very low temperature. Another better candidate for electron-hole bilayers is graphene, which has a two-dimensional (2D) massless linear Dirac-band structure in low energy limit. However, the coupling between different Dirac-cone structures in the same Brilliouin zone brings flaw to graphene to fabricate electron-hole bilayers [@Franz].
On the other hand, another growing interest in condensed matter physics is the very recent theoretical prediction [@Bernevig] and experimental verification [@Konig] of the topological insulators [@Kane] (TIs) with strong spin-orbit interaction. Several three-dimensional (3D) solids, such as Bi$_{1-x}$Sb$_{x}$ alloys, Bi$_{2}$Se$_{3}$-family crystals, have been identified [@Fu; @Hsieh; @HJZhang; @Xia; @Chen] to be strong TIs possessing anomalous band structures. The energy scale for the surface states of these 3D TIs is dominated by the $k$-linear spin-orbit interaction. Especially, the strong TIs surface has single Dirac-cone band structure which is also different from graphene. As a result, it is expected that the excitonic condensate of these topological surface states probably have new characters.
![(Color online) Left panel: Schematic structure of double-well topological insulators in $x$-$y$ plane. The external gates can independently tune the electron and hole densities. Right panel: The linear energy dispersion around the Dirac point of the electrons and holes. []{data-label="f1"}](fig1.eps){width="0.6\linewidth"}
Inspired by this expectation, in this paper we propose a topological insulator bilayer (TIB) model analogous to Ref. [@Franz1], a gated double TI layers separated by an insulating spacer. Using this TIB model, we numerically study the excitonic condensation of TI surface states. We find that the system also presents BCS-BEC crossover along with the change in carrier densities in zero temperature limit. However, there are two characters different from those of conventional excitonic condensation in semiconductor bilayer systems. The first is that the BCS phase of TIB is more robust than that of the semiconductor bilayer systems; the second is that the superfluidity of the TIB is more sensitive to the electron-hole density imbalance than that of the semiconductor bilayer systems. These two characters physically root in the $k$-linear band dispersion of the TIB. Moreover, by putting this TIB model in Bi$_{2}$Se$_{3}$-family material, we investigate the excitonic condensation and only find the BEC phase occurring due to the values of the parameters of the material. The critical temperature of excitonic condensation in Bi$_{2}$Se$_{3}$-family TIB is also calculated in the self-consistent mean-field (SCMF) approximation ($\sim$ $100$ K), which is found to be higher than that in the traditional semiconductor electron-hole bilayers. More interestingly, we can expect this relative high-temperature excitonic condensation since our calculated SCMF critical temperature is approximately equal to the Kosterlitz-Thouless (KT) transition temperature.
The TIB Model
=============
The TIB system is schematically illustrated in the left panel in Fig. \[f1\]. Two TI films are separated by an insulating spacer of thickness $d$, and the electron (hole) density can be independently tuned by the external gate voltage $V_{1}$ ($V_{2}$). The linear dispersions of the TIs around Dirac point are cartoonishly depicted in the right panel in Fig. \[f1\]. The grand-canonical Hamiltonian describing this TIB system can be written as $$\begin{aligned}
H & =-\sum_{p,\mathbf{k,}\sigma}\mu_{p}\hat{p}_{\mathbf{k}\sigma}^{\dag}\hat{p}_{\mathbf{k}\sigma}+\sum_{p,\mathbf{k}}\hslash v_{F}^{p}\left(
k_{x}-ik_{y}\right) \hat{p}_{\mathbf{k}\uparrow}^{\dag}\hat{p}_{\mathbf{k}\downarrow}+h.c.\nonumber\\
& +\frac{1}{2\Omega}\sum_{p,p^{\prime}}\sum_{\mathbf{k},\mathbf{k}^{\prime
},\mathbf{q,}\sigma,\sigma^{\prime}}V_{\mathbf{q}}^{pp^{\prime}}\hat
{p}_{\mathbf{k}+\mathbf{q}\sigma}^{\dag}\hat{p}_{\mathbf{k}^{\prime
}-\mathbf{q}\sigma^{\prime}}^{\prime\dag}\hat{p}_{\mathbf{k}^{\prime}\sigma^{\prime}}^{\prime}\hat{p}_{\mathbf{k}\sigma}. \label{formula1}$$ Here, $\mathbf{k}$, $\mathbf{k}^{\prime}$, and $\mathbf{q}$ are 2D wave vectors in the layers, $\Omega$ is the quantization volume. $\mu_{p}$ is the chemical potential for electron layer ($p$=$e$) or hole layer ($p$=$h$). $\hat{p}_{\mathbf{k}\sigma}$ indicates the annihilation operator of electron at the wave vector $\mathbf{k}$ and spin $\sigma$ (=$\uparrow,\downarrow$) for electron layer ($p$=$e$), and hole layer ($p$=$h$). Note that $v_{F}^{e}$=$v_{F}$ and $v_{F}^{h}$=$-v_{F}.$The surface states of the strong TI film have the linear dispersion: $\epsilon_{\mathbf{k}e,h}$=$\pm\hslash
v_{F}|\mathbf{k}|$. $V_{\mathbf{q}}^{pp^{\prime}}$ is the Fourier transform of the Coulomb interaction: the intralayer Coulomb repulsive interaction $V_{\mathbf{q}}^{ee}$($V_{\mathbf{q}}^{hh}$)=$2\pi e^{2}/\left(
q\varepsilon\right) $, and the interlayer Coulomb attractive interaction [@Balatsky; @Shim] $V_{\mathbf{q}}^{eh}$=$-2\pi e^{2}\exp\left( -qd\right)
/\left( q\varepsilon\right) $, which indicates that on the one hand, in the limit of $d\rightarrow0$, the interaction between electron and hole becomes that in monolayer; on the other hand, in the large thickness limit $d\rightarrow\infty$, the interactions between the electrons in upper layer and holes in lower layer should vanish. Here, $\varepsilon$ is the background dielectric constant. Furthermore, for the present TIB system, the two TI films are separated by an inulating spacer such as SiO$_{2}$, and the spin-orbit interaction in the spacer is obviously negligible. Thus that it can be expected that our model is appropriate in neglectering the interlayer hopping coupling.
In the basis $(\hat{e}_{\uparrow},\hat{e}_{\downarrow},\hat{h}_{\uparrow},\hat{h}_{\downarrow})^{T}$, the Hamiltonian (\[formula1\]) can be decoupled to $H_{MF}$ under the mean-field approximation: $\Delta_{\sigma\sigma^{\prime
}}(\mathbf{k})$=$\sum_{\mathbf{q}}V^{eh}(\mathbf{q})\langle\hat{e}_{\mathbf{k}+\mathbf{q},\sigma}^{\dag}\hat{h}_{\mathbf{k}+\mathbf{q},\sigma^{\prime}}\rangle$, $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$=$-\sum_{\mathbf{q}}V^{pp}(\mathbf{q})\langle\hat{p}_{\mathbf{k}+\mathbf{q},\sigma}^{\dag}\hat{p}_{\mathbf{k}+\mathbf{q},\sigma^{\prime}}\rangle$. Then, $H_{MF}$ can be diagonalized with a 4$\times$4 unitary matirx $U(\mathbf{k})$, $U^{\dag}(\mathbf{k})H_{MF}(\mathbf{k})U(\mathbf{k})$=$\operatorname{diag}(E_{1}(\mathbf{k}),E_{2}(\mathbf{k}),E_{3}(\mathbf{k}),E_{4}(\mathbf{k}))$. The unitary matrix $U(\mathbf{k})$ is construsted by the normalized eigenfunctions of the Hamiltonian (\[formula1\]), which can be numerically calculated by diagonalizing the Hamiltonian matrix (\[formula1\]) in the basis $(\hat{e},\hat{e},\hat
{h},\hat{h})^{T}$. Explicitly, the elements $U_{ij}(\mathbf{k})$ denotes the $i$-th component of the eigenfunction corresponding to the eigenvalue $E_{j}$. The relevant mean-field equations to be solved for the variables $\mu_{e}$, $\mu_{h}$, and the gap functions $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ and self energies $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$ are $$\Delta_{jl}(\mathbf{k})=-\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{eh}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula2}$$$$\begin{aligned}
\Sigma_{jl}^{(e)}(\mathbf{k}) & =\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{ee}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula3}\\
\Sigma_{jl}^{(h)}(\mathbf{k}) & =\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{hh}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula4}$$$$\begin{aligned}
n_{e} & =\frac{1}{\Omega}\sum_{i=1}^{2}\sum_{j=2}^{3}\sum_{\mathbf{k}}\left\vert U_{ij}(\mathbf{k})\right\vert ^{2}f(E_{j}(\mathbf{k})),\label{formula5}\\
n_{h} & =\frac{1}{\Omega}\sum_{i=3}^{4}\sum_{j=2}^{3}\sum_{\mathbf{k}}\left[
1-\left\vert U_{ij}(\mathbf{k})\right\vert ^{2}f(E_{j}(\mathbf{k}))\right]
,\label{formula6}$$ where $f\left( E_{i}(\mathbf{k})\right) $=$1/(1+e^{E_{i}(\mathbf{k})/k_{B}T})$ is the Fermi distribution function and $E_{i}(\mathbf{k})$ ($i=1,...,4$) are the eigen-energies of $H_{MF}(\mathbf{k})$. In Table I we give an explicit correspondence between $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$, $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$ and $\Delta_{jl}(\mathbf{k})$, $\Sigma_{jl}^{(p)}(\mathbf{k})$. $$\overset{\text{TABLE I. The correspondence between }\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})\text{, }\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})\text{ and }\Delta_{jl}(\mathbf{k})\text{, }\Sigma
_{jl}^{(p)}(\mathbf{k})}{\begin{tabular}
[c]{|cc|cc|cc|}\hline\hline
$\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ & $\Delta_{jl}(\mathbf{k})$ &
$\Sigma_{\sigma\sigma^{\prime}}^{(e)}(\mathbf{k})$ & $\Sigma_{jl}^{(e)}(\mathbf{k})$ & $\Sigma_{\sigma\sigma^{\prime}}^{(h)}(\mathbf{k})$ &
$\Sigma_{jl}^{(h)}(\mathbf{k})$\\\hline
$\sigma\sigma^{\prime}$ & $jl$ & $\sigma\sigma^{\prime}$ & $jl$ &
$\sigma\sigma^{\prime}$ & $jl$\\\hline
$\uparrow\uparrow$ & 13 & $\uparrow\uparrow$ & 11 & $\uparrow\uparrow$ & 33\\
$\uparrow\downarrow$ & 14 & $\uparrow\downarrow$ & 12 & $\uparrow\downarrow$ &
34\\
$\downarrow\uparrow$ & 23 & $\downarrow\uparrow$ & 21 & $\downarrow\uparrow$ &
43\\
$\downarrow\downarrow$ & 24 & $\downarrow\downarrow$ & 22 & $\downarrow
\downarrow$ & 44\\\hline
\end{tabular}
\ \ }$$ In addition, for the present 2D case the average interparticle spacing is given by [@Strinati] $$r_{s}=\frac{1}{\sqrt{\frac{\pi}{2}\left( n_{e}+n_{h}\right) }}.
\label{formula7}$$
Many meaningful physical quantities, including the order parameters, can be obtained by self-consistently solving four-band Eqs. (\[formula2\])-(\[formula6\]) with the confinement of the electron and hole number densities. We numerically calculate the exciton’s energy spectrum and the order parameters under different exciton number densities: $r_{s}$=$1.5$, $\alpha$=$0$ and $5.0$, $\alpha$=$0$. Here the density imbalance parameter $\alpha$ is defined as $\alpha\mathtt{\equiv}\left( n_{e}\mathtt{-}n_{h}\right) /\left( n_{e}\mathtt{+}n_{h}\right) $. The calculated results are correspondingly shown by solid lines in Fig. \[f4\](a) and the inset in Fig. \[f2\].
Because the main goal of this paper is to focus on the general properties of the order parameters and neglect the other spin-dependent physical conditions, such as the effect of the Rashba-type spin-orbit coupling by surface inversion asymmetry, we plan to simplify our TIB model, i.e., to define a single order parameter $\Delta(\mathbf{k})$, which can approximately replace the four $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$. Similar to that in the semiconductor case [@Strinati], the corresponding simplified grand-canonical Hamiltonian describing this TIB system can be approximately written as$$\begin{aligned}
H & =\sum_{\mathbf{k},p}\left( \epsilon_{\mathbf{k}p}-\mu_{p}\right)
c_{\mathbf{k}p}^{\dag}c_{\mathbf{k}p}+\frac{1}{2\Omega}\label{formula8}\\
& \times\sum_{\substack{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}\\p,p^{\prime}}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{pp^{\prime}}c_{\mathbf{k}+\mathbf{q}/2p}^{\dag}c_{-\mathbf{k}+\mathbf{q}/2p^{\prime}}^{\dag}c_{-\mathbf{k}^{\prime}+\mathbf{q}/2p^{\prime}}c_{\mathbf{k}^{\prime
}+\mathbf{q}/2p}.\nonumber\end{aligned}$$ With the SCMF theory, Eq. (\[formula8\]) can be rewritten in a $2\mathtt{\times}2$ matrix in the basis $(e,h)^{T}$, the relevant mean-field equations to be solved for the variables $\mu_{e}$, $\mu_{h}$, and the gap function $\Delta_{\mathbf{k}}$ are $$\Delta_{\mathbf{k}}=-\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{eh}\frac{\Delta_{\mathbf{k}^{\prime}}}{2E_{\mathbf{k}^{\prime}}}\left[ f(E_{\mathbf{k}^{\prime}}^{+})-f(E_{\mathbf{k}^{\prime}}^{-})\right] , \label{formula9}$$$$\begin{aligned}
\Sigma_{\mathbf{k}}^{e} & =\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{ee}\left[ u_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{+})+v_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{-})\right] ,\label{formula10}\\
\Sigma_{\mathbf{k}}^{h} & =\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{hh}\left[ v_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{+})+u_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{-})\right] , \label{formula11}$$$$\begin{aligned}
n_{e} & =\frac{1}{\Omega}\sum_{\mathbf{k}}\left\{ u_{\mathbf{k}}^{2}f(E_{\mathbf{k}}^{+})+v_{\mathbf{k}}^{2}\left[ f(E_{\mathbf{k}}^{-})\right] \right\} ,\label{formula12}\\
n_{h} & =\frac{1}{\Omega}\sum_{\mathbf{k}}\left\{ u_{\mathbf{k}}^{2}[1-f(E_{\mathbf{k}}^{-})\}+v_{\mathbf{k}}^{2}\left[ 1-f(E_{\mathbf{k}}^{+})\right] \right\} , \label{formula13}$$ where $u_{\mathbf{k}}^{2}$=$1\mathtt{-}v_{\mathbf{k}}^{2}$=$\frac{1}{2}\left(
1\mathtt{+}\xi_{\mathbf{k}}/E_{\mathbf{k}}\right) $, and $E_{\mathbf{k}}^{\pm}$=$\delta\xi_{\mathbf{k}}\mathtt{\pm}E_{\mathbf{k}}$ with $\delta
\xi_{\mathbf{k}}$=$\frac{1}{2}\left( \xi_{\mathbf{k}e}\mathtt{+}\xi_{\mathbf{k}h}\right) $ and $E_{\mathbf{k}}$=$\sqrt{\xi_{\mathbf{k}}^{2}\mathtt{+}\Delta_{\mathbf{k}}^{2}}$ that are given by $\xi_{\mathbf{k}p}$=$\epsilon_{\mathbf{k}p}\mathtt{-}\mu_{p}\mathtt{+}\Sigma_{\mathbf{k}}^{p}$ ($p$=$e,h$).
![(Color online) (a) Exciton’s energy spectrum with $r_{s}$=$1$.$5$ and $\alpha$=$0$. The solid and dashed lines correspond to Eqs. (\[formula1\]) and (\[formula8\]), respectively; (b) Exciton density of the states with $r_{s}$=$5$ and $\alpha$=$0$. []{data-label="f4"}](fig2.eps){width="0.5\linewidth"}
![(Color online) Wave-vector dependence of the gap function $\Delta(\mathbf{k})$ for $\alpha$=$0$ and several values of $r_{s}$. Inset: calculated $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ from original Hamiltonian (\[formula1\]) at $r_{s}$=$1.5$ and $5.0$. The solid and dashed lines are corresponding to $\Delta_{\uparrow\uparrow}$ (=$\Delta
_{\downarrow\downarrow}$) and $\Delta_{\uparrow\downarrow}$ (=$\Delta
_{\downarrow\uparrow}$), respectively. Comparing with $\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})$ in the inset, we can approximately use $\Delta(\mathbf{k})$ replacing $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ to study the general properties of the order parameters without other spin-dependent interactions.[]{data-label="f2"}](fig3.eps){width="0.6\linewidth"}
We also self-consistently calculate the exciton’s energy spectrum from two-band Eqs. (\[formula9\])-(\[formula13\]). The result, for comparison with the original exact four-band results from Eqs. (\[formula2\])-(\[formula6\]), is plotted in Fig. \[f4\](a) with red dashed lines under the same parameters as used in four-band calculations. The corresponding density of states is shown in Fig. \[f4\](b). From Fig. \[f4\](a) one can clearly find that the exciton energy spectrum within the two-band approximation is wonderfully consistent with that within the exct four-band formalism. Another character found from Figs. \[f4\](a) and \[f4\](b) is that there is an evident stable energy gap protecting the excitonic condensation. In addition, we would like to point out that the parity of the linear dispersion relations of the particles and holes is odd, while the parity of the particle-particle and hole-hole Coulomb interaction is even. This parity asymmetry results in the energy-shift in Fig. \[f4\](a) and the corresponding DOS asymmetry in Fig. \[f4\](b) as well as the asymmetry in Fig. \[f3a\] below. In the following of this paper, all the numerical results except for those shown in the inset of Fig. 3 are calculated from two-band SCMF Eqs. (\[formula9\])-(\[formula13\]).
Numerical results and application to the Bi$_{2}$Se$_{3}$-family material
=========================================================================
First, we calculate the wave-vector dependence of $\Delta(\mathbf{k})$ for equal densities ($\alpha$=$0$) and several values of $r_{s}$. The results are shown in Fig. \[f2\]. We can find the generic feature of the BCS-BEC crossover behavior similar to that in the semiconductor bilayers. However, the striking character in the TI bilayers is that the maximum value of $\Delta(\mathbf{k})$ in the BCS limit is much larger than that in the traditional semiconductor electron-hole bilayers [@Strinati]. This prominent difference means that the BCS phase of TIB is more robust than that of the semiconductor bilayer for equal-density case. Also shown in Fig. 3 (inset) are the calculated four-band gap functions $\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})$ at $r_{s}$=1.5 and 5.0, which show the approximate coincidence in amplitude with the two-band result of $\Delta(\mathbf{k})$.
Because there are no obvious interface between BCS and BEC regimes in terms of the density, we plot in Fig. \[f33\] the calculated momentum magnitude $k$ at which the order parameter takes its maximum value $\Delta_{\max}$ versus $r_{s}$ at $\alpha$=$0$. From this figure, one can see that as $r_{s}\mathtt{\longrightarrow}0$, the number density $n_{e}$ ($n_{h}$) and $k_{\Delta_{\max}}$ tend to infinity, the exciton’s phase is in the BCS regime. On the other hand, as $r_{s}\mathtt{\longrightarrow}\infty$, the number density $n_{e}$ ($n_{h}$) and $k_{\Delta_{\max}}$ tend to $0$, and the exciton’s phase is now in the BEC regime. As $r_{s}$ takes a moderate value, the system is in a mixed regime.
![(Color online) BCS-BEC phase transform: The momentum magnitude $k$ at which the order parameter takes its maximum value $\Delta_{\max}$ versus $r_{s}$ (or number density) at $\alpha$=$0$. The dots are the calculated data, while the solid line is to guide the eyes.[]{data-label="f33"}](fig4.eps){width="0.6\linewidth"}
The effect of $\alpha$ on $\Delta_{\max}$ is shown in Fig. \[f3a\], where $\Delta_{\max}$=$\max\left\{ \Delta_{\mathbf{k}}\right\} $. It is evident to find that the density imbalance actually suppresses $\Delta_{\max}$ and it has different effects on two sides of the crossover. In the BEC regime, the main effect of the density imbalance is to reduce the number of electron-hole pairs, which results in that the superfluid properties are less sensitive to density imbalance. In the BCS regime, the density imbalance leads to the mismatch of the Fermi surfaces of electrons and holes and the finite momentum pairing, which is easier to be broken. However, comparing with that in the traditional semiconductor bilayers, we find that the superfluid property in the BEC phase in our case is more sensitive to electron-hole density imbalance. As an example, for $r_{s}$=$20$ the maximum of gap function $\Delta_{\max}$ for TIB disappears as $\alpha$ takes a value smaller than $0.5$, while it always takes finite values at $\alpha$ varies in the whole zone $\left( -1,1\right) $ for the traditional semiconductor electron-hole bilayers [@Strinati].
Now we apply this TIB model to study the condensation of electron-hole pairs for the topological surface states of the Bi$_{2}$Se$_{3}$-family material. The two TI films in the left panel of Fig. \[f1\] now are two ultrathin TI Bi$_{2}$Se$_{3}$-family films [@Hasegawa] (about $80$ Å thick). With the adopted experimental [@Nakajima] lattice constants $a$=$4.143$ Å and $c$=$28.636$ Å, we calculate the first-principles surface band structure of Bi$_{2}$Se$_{3}$-family [@Wang] by a simple supercell approach with spin-orbit coupling included and obtain the approximate Hamiltonian form describing the gapless surface states of Bi$_{2}$Se$_{3}$-family as follows:$$H(\mathbf{k})=\gamma k^{2}+\hslash v_{F}\left( k_{x}\sigma_{y}-k_{y}\sigma_{x}\right) . \label{formula14}$$ Although this Hamiltonian has the same form as that of the conventional two-dimensional electron gas (2DEG) system with Rashba spin-orbit coupling, the intrinsic difference between these two kinds of systems is that the $k$-linear spin-orbit interaction is primary to the TI surface states, while the parabolic term is dominant in the conventional 2DEG. By fitting the first-principles results, the parameters in Eq. (\[formula14\]) are given as $\gamma$=$0.21$ eV nm$^{2}$ and $\hslash v_{F}$=$0.2$ eV nm (namely, $v_{F}$=$3.04\times10^{5}$m/s). That means the energy dispersion around the Dirac point can be accurately described by $\epsilon_{\mathbf{k}}$=$\pm\hslash
v_{F}|\mathbf{k}|$ when the wave-vector $|\mathbf{k}|$ is much smaller than $1.0$ nm$^{-1}$. For numerical calculation, we choose nm as the length unit and $0.2$ eV as the energy unit in the following discussion. The dielectric constant $\varepsilon$=$1$ and the spacer width $d$=$10$ Å. In fact, the condition that the wave-vector $|\mathbf{k}|$ is much smaller than $1.0$ nm$^{-1}$ requires that only for $r_{s}\geq5$, then the TIB model is valid for Bi$_{2}$Se$_{3}$-family material. This means that the BEC phase is most possible to emerge in Bi$_{2}$Se$_{3}$-family bilayer system.
![(Color online) Maximum value $\Delta_{\max}$=$\max\left\{
\Delta_{\mathbf{k}}\right\} $ as a function of $\alpha$ for $d$=$1$ and several values of $r_{s}$. []{data-label="f3a"}](fig5.eps){width="0.6\linewidth"}
Now, let us discuss the critical temperature of this TIB system. The relation between the $\Delta_{\max}$ and temperature $T$ is respectively shown in Fig. \[f5\](a) for $d$=$1$, $\alpha$=$0$, and several values of $r_{s}$, and \[f5\](b) for $d$=$1$, $r_{s}$=$5$.$0$, and several values of $\alpha$. From Fig. \[f5\](a), we can find that the critical temperature $T_{c}$ decreases as $r_{s}$ increases (i.e., as the particle density decreases). For the Bi$_{2}$Se$_{3}$-family bilayer at $r_{s}$=$5$.$0$, the critical temperature $T_{c}$ is calculated as 0.05 in unit of 0.2 eV. That means the critical temperature $T_{c}$ is about $8\mathtt{\sim}10$ meV ($100$ K), which is much higher than that in the traditional semiconductor electron-hole bilayers. Although the Bi$_{2}$Se$_{3}$-family TIB system is in the BEC phases ($r_{s}$=$5$.$0$, $20.0$), the numerical calculated results shown in Fig. \[f5\](a) are consistent with the general relation of BCS superconductor,$$\frac{2\Delta(0)}{T_{c}}=2\pi e^{-\gamma}\approx3.53, \label{formula15}$$ where $\Delta(0)$ is the energy gap at zero temperature. The introduced electron-hole density imbalance ($\alpha\mathtt{\neq}0$) can reduce the critical temperature. This character is clearly shown in Fig. \[f5\](b): by increasing the density imbalance $\alpha$, the critical temperature $T_{c}$ decreases.
![(Color online) (a) Maximum value $\Delta_{\max}$=$\max\left\{
\Delta_{\mathbf{k}}\right\} $ vs temperature $T$ for $d$=$1$, $\alpha$=$0$, and several values of $r_{s}$. (b) $\Delta_{\max}$ as a function of the temperature $T$ at $r_{s}$=$5$ and several values of $\alpha$.[]{data-label="f5"}](fig6.eps){width="0.7\linewidth"}
As it is known that in 2D superfluids, the critical temperature is often substantially overestimated by mean-filed theory. It is ultimately limited by entropically driven vortex and antivortex proliferation at the Kosterlitz-Thouless (KT) transition temperature $T_{\text{KT}}$=$\frac{\pi}{2}\rho_{s}(T_{\text{KT}})$ with $\rho_{s}(T)$ being the superfluid density (the phase stiffness). Ref. [@Mac] gives an approximate formula to calculate the counterflow current, which is read as $$\rho_{s}(T)\approx\frac{v^{2}\hslash^{2}}{16\pi T}\int kdk\left[ \sec
\text{h}^{2}\left( \frac{\Delta^{z}}{2T}\right) -\sec\text{h}^{2}\left(
\frac{\Delta}{2T}\right) \right] , \label{formula16}$$ where $\Delta^{z}=\frac{-\mu_{e}+\mu_{h}+\Sigma_{\mathbf{k}}^{e}-\Sigma_{\mathbf{k}}^{h}}{2}$, and $\Delta=\sqrt{\left( \Delta^{z}\right)
^{2}}+\sqrt{\Delta_{\mathbf{k}}^{2}}$. We adopt this formula to calculate the superfluid density. The temperature dependence of superfluid density is shown in Fig. \[f6\] at $r_{s}$=$5$ and $\alpha$=$0$. From Fig. \[f6\], it is evident to estimate that the KT transition temperature $T_{\text{KT}}$ is about $0.05$ in unit of $0.2$ eV. Comparing with the critical temperature $T_{c}$ in Fig. \[f5\] at $r_{s}$=$5$ and $\alpha=0$, the striking conclusion is reached: $T_{c}\mathtt{\approx}$ $T_{\text{KT}}$, which means that high-temperature ($\mathtt{\sim}$100 K) excitonic condensation may occur in the Bi$_{2}$Se$_{3}$-family TIB system. On the other hand, we can estimate the KT temperature with the zero-temperature phase stiffness $\rho_{s}(T$=$0)\mathtt{\approx}E_{F}/4\pi$ which is similar to the graphene bilayers [@Mac]. Considering the case shown in Fig. \[f4\], the Fermi energy $E_{F}$ can be numerically calculated and is given to be $\mathtt{\sim}0.4$ (in unit of $0.2$ eV). Hence, the KT temperature is estimated as $T_{\text{KT}}\mathtt{\approx}E_{F}/8\mathtt{\approx}0.05$ in unit of $0.2$ eV. This means that the two estimated methods are consistent and the high-temperature excitonic condensation can emerge in the Bi$_{2}$Se$_{3}$-family TIB system.
![(Color online) The calculated $T_{KT}$ at $r_{s}$=$5$ and $\alpha
$=$0$.[]{data-label="f6"}](fig7.eps){width="0.5\linewidth"}
Summary and conclusions
=======================
In summary, we have performed a generic TIB model to study the excitonic condensation with the SCMF theory for the topological surface states. Similar to the traditional semiconductor electron-hole bilayers, the TIB system presents the crossover behavior from BCS limit to BEC limit by changing the exciton’s density. However, two prominent novel characters different from the traditional semiconductor electron-hole bilayers are found. One is that the superfluid property in the BEC phase is more sensitive to electron-hole density imbalance. The other is that the BCS phase is more robust than that of the semiconductor bilayer. Applying this TIB model to Bi$_{2}$Se$_{3}$-family material, we find that the BEC phase is most possibly observed in experiment. Moveover, we theoretically estimate the critical temperature for the Bi$_{2}$Se$_{3}$-family TIB system and find that it is much higher than that in the traditional semiconductor electron-hole bilayers. For example, at $r_{s}$=$5$ and $\alpha$=$0$, the critical temperature $T_{c}$ is obtained as about $100$ K. We have also studied the phase stiffness and find that the KT transition doesn’t suppress the critical temperature for Bi$_{2}$Se$_{3}$-family in SCMF approximation.
This work was supported by NSFC under Grants No. 90921003 and No. 10904005, and by the National Basic Research Program of China (973 Program) under Grant No. 2009CB929103.
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[^1]: Corresponding author. Email address: zhang\_ping@iapcm.ac.cn
|
---
author:
- 'Sidiney G. Alves'
- 'Tiago J. Oliveira'
- 'Silvio C. Ferreira [^1]'
title: Universal fluctuations in radial growth models belonging to the KPZ universality class
---
Growth phenomena remain a topic of great interest in nonequilibrium Statistical Physics, mainly because of the self-similarity and universality emerging from dynamical local processes of different systems. In this context, one of the most important examples is the Kardar-Parisi-Zhang (KPZ) universality class introduced by equation [@KPZ]: $$\frac{\partial h}{\partial
t} = \nu \nabla^2 h+\lambda |\nabla h|^2+\eta,$$ where $\eta$ is a Gaussian noise. This universality class was observed in a large number of models [@barabasi; @meakin] and a few experimental systems [@miettinen; @TakeSano; @TakeuchiSP].
Former works on surface dynamics were mainly concerned with the scaling properties of the surface fluctuations by means of scaling exponents [@barabasi; @meakin; @krug]. However, there is a number of other universal quantities that are also suitable for determining universality in [surfaces [@TakeuchiSP]]{}. Examples include the stationary distributions of the global interface width and the extremal height [@tiago1; @tiago2], and the height distributions during the transient regime [@johansson; @PraSpo1] that precedes saturation of the interface width.
The distributions during the transient (growth) regime were computed exactly for some models in the KPZ class, in $1+1$ dimension [@johansson; @PraSpo1; @krugrev; @Ferrari]. Among the most relevant cases, the height distribution (HD) of the single step model [@barabasi], determined analytically by Johansson [@johansson], has the Tracy-Widom distribution of the largest eigenvalue of the Gaussian unitary ensemble (GUE) of the random matrix theory [@TW1] as a limit solution. Prähofer and Spohn [@PraSpo1; @PraSpo2] also obtained analytically the scaling form of the KPZ universality class and a Tracy-Widom distribution for the HD of the polynuclear growth model (PNG). Furthermore, they showed that the HD for the growth from flat substrates is given by the Gaussian orthogonal ensemble (GOE), while growth from a single seed (radial growth) leads to a GUE distribution for the radii [@PraSpo1; @PraSpo2]. Recently, Sasamoto and Spohn [@SasaSpo1] found a solution of the one-dimensional KPZ equation with an initial condition that induces the growth of curved surfaces and the GUE distribution was confirmed as the limit for the radius fluctuations. Subsequently, a numerical evaluation of this solution corroborated these analytical results [@Prolhac]. Prähofer and Spohn [@PraSpo3] also obtained an analytical solution for the limiting process describing the surface fluctuations in the PNG model as the so-called Airy$_2$ process.
Experimentally, the GOE and GUE distributions were obtained in a few experiments exhibiting KPZ scaling. An evidence of the KPZ exponents was found in the slow combustion of paper sheets [@miettinen]. The burning fronts evolve from a flat initial condition and the obtained HD has a reasonable agreement with GOE. A recent experiment on electroconvection of turbulent liquid crystal films [@TakeSano] allowed to investigate an isotropic radial growth with high accuracy. It was shown that this system belongs to KPZ universality class exhibiting radius distribution (RD) in excellent agreement with GUE, including the cumulants from second to fourth order. However, the mean is shifted and tends to the GUE value as a power law $t^{-1/3}$. The same result was found by Sasamoto and Spohn [@SasaSpo1] in a solution of the KPZ equation in $1+1$ dimensions with an initial edge condition indicating a universal behaviour of the exponent 1/3. However, Ferrari and Frings [@Ferrari] have recently shown that the scaling law featuring this approach to GUE mean is not universal. Indeed, they have shown that the mean shift decays as $t^{-1/3}$ for the totally asymmetric simple exclusion process (ASEP) whereas no correction (up order $\mathcal{O}(t^{-2/3})$) is found for the weakly ASEP, both models belonging to the KPZ class. The liquid crystal film setup was also used to induce a front growth from a flat surface, and a good agreement with GOE was obtained for the HDs [@TakeuchiSP].
The theoretical and experimental evidences above mentioned strongly suggest that the GUE distribution is an universal feature of one-dimensional growth with radial symmetry in the KPZ class. However, this conjecture is based on a limited number of models with exact results [@PraSpo1; @PraSpo2; @SasaSpo1; @Ferrari] and a single experimental work [@TakeSano]. Numerical confirmation of the GUE distributions are, up to this moment, missing. In order to fill this gap, we investigate the RD of large radial clusters (larger than $3\times 10^9$ particles) generated with different versions of the Eden model [@eden; @LetSil; @SilSid1]. We show that the RDs exhibit very good agreement with GUE distribution. We also observed that the cumulants of the distribution converge to the GUE values, as previously observed in other systems [@TakeSano; @SasaSpo1]. The two-point correlation function also converges to the Airy$_2$ process, as predicted by the conjecture [@PraSpo3].
The Eden model [@eden] consists in adding new particles in the empty neighbourhood of a growing cluster. If the growth starts with a single particle, the model yields asymptotically spherical clusters with a self-affine surface exhibiting the scaling exponents of the KPZ universality class [@SilSid1]. We simulated off-lattice clusters in two-dimensions with the usual algorithm [@SilSid1; @BJP]: a particle in the active (growing) zone[^2] is selected at random and a new particle is added in a random position chosen in the empty neighbourhood of the selected particle. The procedure is repeated while the cluster does not reach $N$ particles. With suitable optimizations [@SilSid1; @BJP], we were able to grow clusters with up to $3\times 10^7$ particles. Since we randomly pick up a particle from a constantly updated list containing $N_s$ surface sites, the time step is simply $\Delta
t=1/N_s$. A total number of up to $10^3$ off-lattice clusters were grown in order to perform statistical averages.
We also have simulated Eden models on a square lattice using an algorithm proposed in Ref. [@LetSil] that removes the lattice anisotropy effects. The method consists in accepting a given growth step with probability $p_j=(n_j/4)^\nu$, where $n_j=1,2,3,4$ is the number of occupied nearest-neighbours (NN) of a selected growth site $j$ and $\nu$ is an adjustable parameter. This method allows to generate isotropic clusters containing more than $4\times 10^{9}$ particles. We investigated two algorithms for lattice simulations. In the version Eden A, one site is randomly selected among all $N_g$ growth sites (empty NNs of the cluster) and then occupied. In the version Eden B, one of the $N_b$ sites in the cluster border and one of its empty NN are randomly chosen and the empty one is occupied. The values of the parameter $\nu$ that produce isotropic clusters are $\nu_A=1.72$ and $\nu_B=1$ for Eden A and B, respectively [@LetSil]. At each attempt, the time is increased by $1/N_g$ and $1/N_b$ for Eden A and B, respectively, independently of the growth success. The numbers of clusters used for statistics were up to $10^4$ for Eden A and up to $10^3$ for Eden B. Notice that statistical fluctuations are much stronger in Eden A due to the higher amount of overhangs in the surface.
![(Color online) Radius distributions of on- and off-lattice Eden models rescaled to a null mean and a unitary variance. The mean radius of the aggregates are approximately $2500$ for the off-lattice model and $3.2 \times 10^{4}$ for lattice models. The solid line is the rescaled GUE distribution. In this plot, $R^*\equiv(R-\langle R\rangle)/\sigma_{R}$.[]{data-label="dist_tiago"}](Fig1.pdf){width="7.5cm"}
We start the analysis with a comparison among the RD of the Eden models and the GUE distribution, both suitably rescaled and shifted to have a null mean and a unitary variance. We assume a scaling form $$P(R) = \frac{1}{\sigma_{R}} G \left( \frac{R- \langle R\rangle }{\sigma_{R}} \right),
\label{eqRET}$$ where $\sigma_{R}^2$ is the variance of the RD and $G(x)$ is a normalized scaling function. This scaling form reduces finite size corrections, what has improved data collapses in other analyses, as interface width distributions [@tiago1], for example. In Fig. \[dist\_tiago\], we compare the rescaled RDs for the three Eden models with the rescaled GUE distribution. An excellent collapse of all curves upon a single curve $G(x)$ was obtained. Similar results hold for different growth times. This data collapse confirms that, a part of corrections to scaling in the cumulants described below, the RDs of the all investigated models agree with the GUE distribution, as conjectured by Prähofer and Spohn [@PraSpo1].
In radial growth belonging to the KPZ universality class, the radii are stochastic variables evolving in time as [@PraSpo1; @PraSpo2]: $$R(t) \simeq \lambda t + {\left( A^{2} \lambda t/2 \right) } ^{1/3} \chi_{GUE},
\label{eqPS}$$ where $\lambda$ and $A$ are two non-universal (model dependent) parameters. The random variable $\chi_{GUE}$ is distributed according to the GUE Tracy-Widom distribution [@TW1]. Therefore, $\lambda$ is the asymptotic radial growth rate obtained from $\lambda \simeq d\langle{R}\rangle/dt + a_{v} t^{-2/3}$ in the limit $t \rightarrow \infty$ [@krug1]. In Fig. \[hwl\](a), we show the average radius growth rate against a power of time. The extrapolated asymptotic values are $\lambda \approx 1.1843(2)$ for off-lattice, $\lambda
\simeq 0.2639(2)$ for Eden A, and $\lambda\simeq 0.4807(2)$ for Eden B, where the uncertainties obtained in the regressions are shown in parenthesis.
The parameter $A$ was estimated in two independent ways. We can use the local squared surface roughness, in a window of size $\epsilon$, defined as $$w^{2}(\epsilon,t) =\langle [R(x,t)]^2 \rangle_{\epsilon} -\langle R(x,t) \rangle_{\epsilon}^{2} ,$$ where $\left\langle\dots \right\rangle_\epsilon$ denotes an average within several windows in the interface. Alternatively, we can estimate the $A$ value using the height-height correlation function $$c(\epsilon,t) = \langle [ R(x+\epsilon,t) - R(x,t)]^{2}\rangle_\epsilon.
\label{eqcor}$$ For long times, theoretical arguments predict that $w^{2}\simeq A \epsilon /6$ and $c \simeq A \epsilon$ [@krug1]. Curves for $6 w^{2}/\epsilon$ as functions of $\epsilon$ are shown in Fig. \[hwl\](b). Well-defined plateaus are observed for both on-lattice models, except for short scales, when a small deviation is observed. Since off-lattice simulations are much smaller, the plateau is not so evident as in the on-lattice case, but the data also tend to a constant value. For sake of comparison, we measured the local roughness exponent, defined as $w(\epsilon)\sim \epsilon^\alpha$, and found $\alpha\approx
0.43$ for our largest off-lattice simulations. This value is considerably smaller than the expected exponent of the KPZ class $\alpha=1/2$, confirming the presence of strong finite time effects in the exponents. Roughness exponent for the on-lattice models are very close to the value $\alpha=1/2$. The estimates of parameter $A$ are represented by dashed lines in Figs. \[hwl\](b) and (c). The estimates using local interfaces width are: $A\approx1.46$ for off-lattice, $A\approx1.32$ for Eden A, and $A\approx0.71$ for Eden B models. The correlation function estimates are slightly larger: $A\approx1.55$ for off-lattice and A models, and $A\approx0.84$ for Eden B.
![[(Color online) Probability distribution of $q = (R - \lambda
t)/(A^2 \lambda t/2)^{1/3}$ for our largest-time simulations: Off-lattice Eden model with cluster mean radius $\bar{r}=2700$; Eden A and B models with mean radius $\bar{r} = 32000$. The solid line is the GUE distribution.]{}[]{data-label="eden_gue"}](Fig3.pdf){width="6.6cm"}
In agreement to Eq. (\[eqPS\]), the quantity $$q = \frac{R - \lambda t}{(A^2 \lambda t/2)^{1/3}}
\label{eq_q}$$ is a random variable given by a GUE distribution. The RDs shown in Fig. \[eden\_gue\] were obtained with the values $\lambda = 1.1842$ and $A=1.45$ for off-lattice simulations, $\lambda = 0.263887$ and $A=1.43$ for Eden A, and $\lambda = 0.4806$ and $A=0.805$ for Eden B. As predicted by radial KPZ conjecture, a very well collapse for different models (and different times) is observed. The agreement with the GUE distribution is noticeable, except by a shift in $q$, that vanishes as $t\rightarrow\infty$. The shift is more evident for Eden B as can also be seen in Fig. \[cumul\]. Our results confirm the limiting scaling form conjectured by Phähoffer and Spohn [@PraSpo1; @PraSpo2], as previously observed in theoretical [@krugrev; @PraSpo1; @PraSpo2; @SasaSpo1] and experimental [@TakeSano] systems.
In order to quantify the agreement between RDs and GUE distributions, we investigate the $n$th order cumulants of the probability distribution $P(q)$ denoted by $\kappa_n^q$. The differences between the cumulants obtained for off-lattice simulations and the GUE values are shown, as function of time, in Fig. \[cumul\](a). As expected, all cumulants converge to the GUE values. As observed experimentally by Takeuchi and Sano [@TakeSano], the first moment decreases towards the GUE value while the higher order cumulants increases towards GUE values. The same happens for Eden A, as can be seen in Fig. \[cumul\](b). Differently, the mean for Eden B increases (more slowly than the others) towards GUE and the higher order cumulants converges more quickly to the theoretical values. This negative amplitude of the difference between simulation and GUE mean was also observed in a solution of the KPZ equation [@SasaSpo1]. Indeed, the law describing the convergence of the mean have recently attracted great interest [@TakeSano; @SasaSpo1; @Ferrari]. Our off-lattice and Eden B simulations are very well described by the $t^{-1/3}$ law previously reported [@TakeSano; @SasaSpo1] while the simulation of Eden A are only consistent with this approach since a long power law regime was not observed.
A further evidence of the agreement between Eden growth and radial KPZ conjecture is yielded by the two-point correlation function given by $C_{2}(\epsilon,t)
= \langle R(x+\epsilon,t)R(x,t)\rangle-\langle R\rangle^2$, that, in agreement with the conjecture [@PraSpo3], scales at long times as $C_2(\epsilon,t) \simeq (A^2 \lambda t/2)^{2/3}
g_{2}(u)$ with $u = (A \epsilon/2)/(A^2 \lambda t/2)^{2/3}$, where $g_{2}(u)$ is the covariance of the Airy$_{2}$ process [@Bornemann]. In Fig. \[covar\], we show the correlation function $C_2$ for different models. A very good agreement between the scaled $C_{2}$ for all Eden models and $g_{2}$ function is obtained, showing that the models are well described by the Airy$_2$ process. It is worth mentioning that scaled $C_2$ approaches $g_2$ for long times and that this approach is slower for on-lattice models.
![(Color online) Scaled two-point correlation function for the Eden models. The average radius of the clusters are 2700 for off-lattice and 32000 for A and B models, respectively. The solid curve is the covariance of the Airy$_2$ process. In this plot, $\widetilde{C}_2 = (A^2 \lambda t/2)^{-2/3}C_2 $ and $u=(A \epsilon/2)(A^2 \lambda t/2)^{-2/3}$.[]{data-label="covar"}](Fig5.pdf){width="7.0cm"}
In summary, the KPZ universality class in radial growth has been subject of recent analytical [@SasaSpo1] and experimental [@TakeSano] investigations that agreed with the conjecture proposed by Prähofer and Spohn [@PraSpo1; @PraSpo2], where interface fluctuations in systems belonging to the KPZ universality class are described by well-known universal distributions. However, a computational verification in growth models was lacking until the present work. In the present Letter, we have investigated the radius distributions in clusters obtained with Eden growth models [@SilSid1; @LetSil] starting from a single particle.
The radius distributions obtained for all models exhibit an excellent agreement with the scaling ansatz given by Eq. (\[eqPS\]), that associates the radius fluctuations with the Tracy-Widom [@TW1] distribution of the Gaussian Unitary Ensemble. The cumulants of order $n\ge 1$ associated to RD converge to the corresponding GUE cumulants for all investigated models. A finite time correction of order $t^{-1/3}$ in the first moment was clearly observed for off-lattice and Eden B simulations, in agreement with other systems [@TakeSano; @SasaSpo1]. Finally, a correlation function in accordance with the so-called Airy$_2$ process yields a further strong evidence that the radius fluctuations in all investigated growth models are in agreement with the KPZ conjecture.
As a final remark, notice that the the small exponent $\alpha \approx
0.43$ obtained for the largest off-lattice simulations shows that scaling exponents undergo strong finite time effects. Therefore, the RD analysis may be more reliable than scaling exponents to determine the universality class of a system. In particular, the scaling form given by Eq. (\[eqRET\]) is simpler than the analysis with Eq. (\[eqPS\]), since fit procedures are not required in the first approach and it does not have finite time corrections.
In conclusion, the Prähofer-Spohn conjecture is now fully verified in the three general branches of Statistical Physics: experimental, theoretical, and computational.
This work was partially supported by the Brazilian agencies CNPq, FAPEMIG, and CAPES. Authors thank F. Bornemann by kindly providing the covariance of the Airy$_2$ process. We also thank the former discussions with Herbert Spohn that motivated the beginning of this work. SCF thanks the kind hospitality at the Departament de Física i Enginyeria Nuclear/UPC.
[10]{} url\#1[`#1`]{}
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(Cambridge University Press, Cambridge, England) 1995.
(Cambridge University Press, Cambridge, England) 1998.
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in proc. of , edited by Vol. 4 (University of California Press, Berkeley,California) 1961 pp. 223–239.
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[^1]: On leave at Departament de Física i Enginyeria Nuclear, Universitat Politécnica de Catalunya, Barcelona, Spain.
[^2]: The active zone consists of those particles having sufficient empty space in its neighbourhood to add at least a new particle.
|
---
author:
- |
Timothy B. Armstrong[^1]\
Yale University
- |
Michal Kolesár[^2]\
Princeton University
bibliography:
- '../../np-testing-library.bib'
title: 'Supplemental Materials for “Finite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness”'
---
=1
Proofs of auxiliary Lemmas and additional details
=================================================
Proof of Lemma \[lemma:linear-bias\]
------------------------------------
We will show that the constraint $f^{*}(x_{i},1)\leq f^{*}(x_{j},1)+\normx{x_{i}-x_{j}}$ holds for all $i, j\in\{1,\dotsc, n\}$. The argument that $f^{*}(x_{i},0)\leq f^{*}(x_{j},0)+\normx{x_{i}-x_{j}}$ holds for all $i, j\in\{1,\dotsc, n\}$ is similar and omitted. We assume, without loss of generality, that the observations are ordered so that $d_{j}=0$ for $j=1,\dotsc, n_{0}$ and $d_{i}=1$ for $i=n_{0}+1,\dotsc, n$. Observe that the bias can be written as $$\begin{gathered}
\sum_{i=n_{0}+1}^{n}(k(x_{i},1)-w(1))f(x_{i},1)
-\sum_{j=1}^{n_{0}}w(0)f(x_{j},1) \\
+\sum_{j=1}^{n_{0}}(k(x_{j},0)+w(0))f(x_{j},0)+
\sum_{i=n_{0}+1}^{n}w(1)f(x_{i},0).\end{gathered}$$ If $k(x_{i},1)=w(1)$ for $i\in\{n_{0}+1,\dotsc, n\}$, we can set $f^{*}(x_{i},1)
=\min_{j\in\{1,\dotsc, n_{0}\}}\{f^{*}(x_{j},1)+\normx{x_{i}-x_{j}}\}$ without affecting the bias, so that we can without loss of generality assume that
holds for all $i\in\{n_{0}+1,\dotsc, n\}$ and all $j\in\{1,\dotsc, n_{0}\}$.
If $w(0)=0$, then the assumptions on $k$ imply $k(x_{i},1)=w(1)$ for $i>n_{0}$, and the value of $f(\cdot,1)$ doesn’t affect the bias. If $w(0)>0$, then for each $j\in\{1,\dotsc, n_{0}\}$, at least one of the constraints $f^{*}(x_{i},1)\leq f^{*}(x_{j},1)+ \normx{x_{i}-x_{j}}$, $i\in\{n_{0}+1,\dotsc, n\}$, must bind, otherwise we could decrease $f^{*}(x_{j},1)$ and increase the value of the objective function. Let $i(j)$ denote the index of one of the binding constraints (picked arbitrarily), so that $f^{*}(x_{i(j)},1)=f^{*}(x_{j},1)+\normx{x_{i(j)}-x_{j}}$. We need to show that the constraints $$\begin{aligned}
f^{*}(x_{i},1)&\leq f^{*}(x_{i'},1)+ \normx{x_{i}-x_{i'}}& i, i'\in\{n_{0}+1,\dotsc, n\},\label{eq:fii-l1}\\
f^{*}(x_{j},1)&\leq f^{*}(x_{j'},1)+ \normx{x_{j}-x_{j'}}& j, j'\in\{1,\dotsc, n_{0}\},\label{eq:fjj-l1}\\
f^{*}(x_{j},1)&\leq f^{*}(x_{i},1)+ \normx{x_{i}-x_{j}}& j\in\{1,\dotsc, n_{0}\}, \;i\in\{n_{0}+1,\dotsc, n\}\label{eq:fij-l1}.\end{aligned}$$ are all satisfied. If doesn’t hold for some $(i, i')$, then by triangle inequality, for all $j\in\{1,\dotsc, n_{0}\}$, $$f^{*}(x_{i'},1)+\normx{x_{i}-x_{i'}}<f(x_{i},1)\leq f^{*}(x_{j},1)+\normx{x_{i}-x_{j}}
\leq f^{*}(x_{j},1)+\normx{x_{i}-x_{i'}}+\normx{x_{i'}-x_{j}},$$ so that $f^{*}(x_{i'},1)<f^{*}(x_{j},1)+\normx{x_{i'}-x_{j}}$. But then it is possible to increase the bias by increasing $f^{*}(x_{i'},1)$, which cannot be the case at the optimum. If doesn’t hold for some $(j, j')$, then by triangle inequality, for all $i$, $$\begin{gathered}
f^{*}(x_{j},1)+\normx{x_{i}-x_{j}}>f^{*}(x_{j'},1)
+ \normx{x_{i}-x_{j}}
+\normx{x_{j}-x_{j'}}\\
\geq
f^{*}(x_{j'},1)+\normx{x_{i}-x_{j'}}\geq f^{*}(x_{i},1).\end{gathered}$$ But this contradicts the assertion that for each $j$, at least one of the constraints $f(x_{i},1)\leq f(x_{j},1)+\normx{x_{i}-x_{j}}$ binds. Finally, suppose that doesn’t hold for some $(i, j)$. Then by triangle inequality, $$\begin{gathered}
f^{*}(x_{i},1)+\normx{x_{i}-x_{i(j)}}\leq
f^{*}(x_{i},1)+ \normx{x_{i}-x_{j}}+\normx{x_{i(j)}-x_{j}}\\
< f^{*}(x_{j},1)+\normx{x_{i(j)}-x_{j}} = f^{*}(x_{i(j)},1),\end{gathered}$$ which violates .
Proof of Lemma \[lemma:subset-constraints-optimal\]
---------------------------------------------------
We will show that
hold at the optimum for $d_{i}, d_{i'}=1$ and $d_{j}, d_{j'}=0$. The argument that they hold for $d_{i}, d_{i'}=0$ and $d_{j}, d_{j'}=1$ is similar and omitted. The first-order conditions associated with the Lagrangian
are $$\begin{aligned}
\label{eq:foc0}
m_{j}/\sigma^{2}(0)&=\mu w(0)+\sum_{i=1}^{n_{1}}\Lambda^{0}_{ij},
& \mu w(0)&=\sum_{i=1}^{n_{1}}\Lambda^{1}_{ij}&j&=1,\dotsc, n_{0}, \\
m_{i+n_{0}}/\sigma^{2}(1)&=\mu w(1)+\sum_{j=1}^{n_{0}}\Lambda^{1}_{ij},
& \mu w(1)&=\sum_{j=1}^{n_{0}}\Lambda^{0}_{ij}&i&=1,\dotsc, n_{1}.\label{eq:foc1}\end{aligned}$$ If $w(0)=0$, the first-order conditions together with the dual feasibility condition $\Lambda^{1}_{ij}\geq 0$ implies that $m_{i+n_{0}}=\mu w(1)\sigma^{2}(1)$, and the assertion of the lemma holds trivially, since $r_{j}=\mu w(1)\sigma^{2}(1)$ for $j=1,\dotsc, n$ achieves the optimum. Suppose, therefore, that $w(0)>0$. Then $\sum_{i=1}^{n_{1}}\Lambda^{1}_{ij}>0$, so that at least one of the constraints associated with $\Lambda^{1}_{ij}$ must bind for each $j$. Let $i(j)$ denote the index of one of the binding constraints (picked arbitrarily if it is not unique), so that $r_{j}=m_{i(j)+n_{0}}+\normx{x_{i(j)+n_{0}}-x_{j}}$. Suppose
didn’t hold, so that for some $j, j'\in\{1,\dotsc, n_{0}\}$, $r_{j}>r_{j'}+\normx{x_{j}-x_{j'}}$. Then by triangle inequality $$r_{j}>r_{j'}+\normx{x_{j}-x_{j'}}=m_{i(j')+n_{0}}+\normx{x_{i(j')+n_{0}}-x_{j'}}+
\normx{x_{j}-x_{j'}}\geq
m_{i(j')+n_{0}}+\normx{x_{i(j')+n_{0}}-x_{j}},$$ which violates the constraint associated with $\Lambda^{1}_{i(j')j}$. Next, if
didn’t hold, so that for some $i, i'\in\{1,\dotsc, n_{1}\}$, $m_{i+n_{0}}>m_{i'+n_{0}}+\normx{x_{i+n_{0}}-x_{i'+n_{0}}}$, then for all $j\in\{1,\dotsc, n_{0}\}$, $$r_{j}\leq m_{i'+n_{0}}+\normx{x_{i'+n_{0}}-x_{j}}
\leq m_{i'+n_{0}}+\normx{x_{i'+n_{0}}-x_{i+n_{0}}}+\normx{x_{i+n_{0}}-x_{j}}
< m_{i+n_{0}}+\normx{x_{i+n_{0}}-x_{j}},$$ The complementary slackness condition $\Lambda^{1}_{ij}(r_{j}-m_{i+n_{0}}-\normx{x_{i+n_{0}}-x_{j}})=0$ then implies that $\sum_{j}\Lambda^{1}_{ij}=0$, and it follows from the first-order condition that $m_{i+n_{0}}/\sigma^{2}(1)=\mu w(1)\leq m_{i'+n_{0}}/\sigma^{2}(1)$, which contradicts the assertion that $m_{i+n_{0}}>m_{i'+n_{0}}+\normx{x_{i+n_{0}}-x_{i'+n_{0}}}$. Finally, if
didn’t hold, so that $m_{i+n_{0}}>r_{j}+\normx{x_{i+n_{0}}-x_{j}}$ for some $i\in\{1,\dotsc, n_{1}\}$ and $j\in\{1,\dotsc, n_{0}\}$, then by triangle inequality $$m_{i+n_{0}}>r_{j}+\normx{x_{i+n_{0}}-x_{j}} =m_{i(j)}+\normx{x_{i(j)+n_{0}}-x_{j}}+\normx{x_{i+n_{0}}-x_{j}}
\geq m_{i(j)}+\normx{x_{i(j)+n_{0}}-x_{i+n_{0}}},$$ which contradicts
.
Derivation of algorithm for solution path
-----------------------------------------
Observe that $\Lambda^{0}_{ij}=0$ unless for some $k$, $i\in\mathcal{R}^{0}_{k}$ and $j\in\mathcal{M}^{0}_{k}$, and similarly $\Lambda^{1}_{ij}=0$ unless for some $k$, $j\in\mathcal{R}^{1}_{k}$ and $i\in\mathcal{M}^{1}_{k}$. Therefore, the first-order conditions and can equivalently be written as $$\begin{aligned}
\label{eq:foc0a}
m_{j}/\sigma^{2}(0)&=\mu w(0)+\sum_{i\in\mathcal{R}_{k}^{0}}\Lambda^{0}_{ij}
&j&\in\mathcal{M}_{k}^{0},
& \mu w(1)&=\sum_{j\in\mathcal{M}_{k}^{0}}\Lambda^{0}_{ij}&i&\in\mathcal{R}_{k}^{0}, \\
m_{i+n_{0}}/\sigma^{2}(1)&=\mu w(1)+\sum_{j\in\mathcal{R}^{1}_{k}}\Lambda^{1}_{ij}&i&\in\mathcal{M}_{k}^{1},
&\mu w(0)&=\sum_{i\in\mathcal{M}_{k}^{1}}\Lambda^{1}_{ij}&j&\in\mathcal{R}_{k}^{1}.\label{eq:foc1a}\end{aligned}$$ Summing up these conditions then yields $$\begin{aligned}
\sum_{j\in\mathcal{M}_{k}^{0}}m_{j}/\sigma^{2}(0)
&=\mu w(0)\cdot
\#{\mathcal{M}_{k}^{0}}+\sum_{j\in\mathcal{M}_{k}^{0}}\sum_{i\in\mathcal{R}_{k}^{0}}\Lambda^{0}_{ij}
=\#{\mathcal{M}_{k}^{0}}\cdot \mu w(0)+ \#\mathcal{R}_{k}^{0}\cdot \mu w(1), \\
\sum_{i\in\mathcal{M}_{k}^{1}}m_{i+n_{0}}/\sigma^{2}(1)
&=\mu w(1)\cdot\#\mathcal{M}_{k}^{1}
+\sum_{i\in\mathcal{M}_{k}^{1}}\sum_{j\in\mathcal{R}^{1}_{k}}\Lambda^{1}_{ij}=
\#\mathcal{M}_{k}^{1}\cdot\mu w(1)+\#\mathcal{R}_{k}^{1}\cdot\mu w(0).\end{aligned}$$ Following the argument in @osborne00 [Section 4], by continuity of the solution path, for a small enough perturbation $s$, ${N}^{d}(\mu+s)={N}^{d}(\mu)$, so long as the elements of $\Lambda^{d}(\mu)$ associated with the active constraints are strictly positive. In other words, the set of active constraints doesn’t change for small enough changes in $\mu$. Hence, the partition $\mathcal{M}_{k}^{d}$ remains the same for small enough changes in $\mu$ and the solution path is differentiable. Differentiating the preceding display yields $$\begin{aligned}
\frac{1}{\sigma^{2}(0)} \sum_{j\in\mathcal{M}_{k}^{0}}\frac{\partial
m_{j}(\mu)}{\partial \mu}
&=\#{\mathcal{M}_{k}^{0}}\cdot w(0)+ \#\mathcal{R}_{k}^{0}\cdot w(1), \\
\frac{1}{\sigma^{2}(1)} \sum_{i\in\mathcal{M}_{k}^{1}}\frac{\partial
m_{i+n_{0}}(\mu)}{\partial \mu}
&= \#\mathcal{M}_{k}^{1}\cdot w(1)+\#\mathcal{R}_{k}^{1}\cdot w(0).\end{aligned}$$
If $j\in\mathcal{M}^{0}_{k}$, then there exists a $j'$ and $i$ such that the constraints associated with $\Lambda_{ij}^{0}$ and $\Lambda_{ij'}^{0}$ are both active, so that $m_{j}+\normx{x_{i+n_{0}}-x_{j}}=r_{i+n_{0}}=m_{j'}+\normx{x_{i+n_{0}}-x_{j'}}$, which implies that $\partial m_{j}(\mu)/\partial \mu=\partial m_{j'}(\mu)/\partial \mu$. Since all elements in $\mathcal{M}^{0}_{k}$ are connected, it follows that the derivative $\partial m_{j}(\mu)/\partial \mu$ is the same for all $j$ in $\mathcal{M}_{k}^{0}$. Similarly, $\partial m_{j}(\mu)/\partial \mu$ is the same for all $j$ in $\mathcal{M}_{k}^{1}$. Combining these observations with the preceding display implies $$\begin{aligned}
\frac{1}{\sigma^{2}(0)} \frac{\partial
m_{j}(\mu)}{\partial \mu}
&= w(0)+ \frac{\#\mathcal{R}_{k(j)}^{0}}{\#{\mathcal{M}_{k(j)}^{0}}} w(1), &
\frac{1}{\sigma^{2}(1)} \frac{\partial
m_{i+n_{0}}(\mu)}{\partial \mu}
&= w(1)+\frac{\#\mathcal{R}_{k(i)}^{1}}{\#\mathcal{M}_{k(i)}^{1}} w(0),\end{aligned}$$ where $k(i)$ and $k(j)$ are the partitions that $i$ and $j$ belong to. Differentiating the first-order conditions and and combining them with the restriction that $\partial \Lambda_{ij}^{d}(\mu)/\partial \mu=0$ if $N^{d}_{ij}(\mu)=0$ then yields the following set of linear equations for $\partial\Lambda^{d}(\mu)/\partial\mu$: $$\begin{aligned}
\frac{\#\mathcal{R}^{0}_{k}}{\#{\mathcal{M}_{k}^{0}}}w(1)
& =\sum_{i\in\mathcal{R}_{k}^{0}}\frac{\partial \Lambda^{0}_{ij}(\mu)}{\partial\mu},
& w(1)&=\sum_{j\in\mathcal{M}_{k}^{0}}\frac{\partial\Lambda^{0}_{ij}(\mu)}{\partial\mu},
\\
\frac{\#\mathcal{R}^{1}_{k}}{\#{\mathcal{M}^{1}_{k}}}w(0)
& = \sum_{j\in\mathcal{R}_{k}^{1}}\frac{\partial\Lambda^{1}_{ij}(\mu)}{\partial\mu},
& w(0)&=\sum_{i\in\mathcal{M}_{k}^{1}}\frac{\partial\Lambda^{1}_{ij}(\mu)}{\partial\mu},
&\frac{\partial\Lambda^{d}_{ij}(\mu)}{\partial\mu}=0\qquad \text{if $N_{ij}^{d}(\mu)=0$}.\end{aligned}$$ Therefore, $m(\mu)$, $\Lambda^{0}(\mu)$, and $\Lambda^{1}(\mu)$ are all piecewise linear in $\mu$. Furthermore, since for $i\in\mathcal{R}^{0}_{k}$, $r_{i+n_{0}}(\mu)=m_{j}(\mu)+\normx{x_{i+n_{0}}-x_{j}}$ where $j\in\mathcal{M}_{k}^{0}$, it follows that $$\frac{\partial r_{i+n_{0}}(\mu)}{\partial \mu}
= \frac{\partial m_{j}(\mu)}{\partial \mu}
=\sigma^{2}(0)\left[w(0)+ \frac{\#\mathcal{R}_{k}^{0}}{\#{\mathcal{M}_{k}^{0}}}
w(1)\right].$$ Similarly, since for $j\in\mathcal{R}^{1}_{k}$, and $i\in\mathcal{M}^{1}_{k}$ $r_{j}(\mu) =m_{i+n_{0}}(\mu)+\normx{x_{i+n_{0}}-x_{j}}$, where $j\in\mathcal{M}^{0}_{k}$, we have $$\qquad
\frac{\partial r_{j}(\mu)}{\partial \mu}
= \frac{\partial m_{i+n_{0}}(\mu)}{\partial \mu}
=\sigma^{2}(1)\left[
w(1)+\frac{\#\mathcal{R}_{k}^{1}}{\#\mathcal{M}_{k}^{1}} w(0)\right].$$ Thus, $r(\mu)$ is also piecewise linear in $\mu$.
Differentiability of $m$ and $\Lambda^{d}$ is violated if the condition that the elements of $\Lambda^{d}$ associated with the active constraints are all strictly positive is violated. This happens if one of the non-zero elements of $\Lambda^{d}(\mu)$ decreases to zero, or else if a non-active constraint becomes active, so that for some $i$ and $j$ with $N^{0}_{ij}(\mu)=0$, $r_{i+n_{0}}(\mu)=m_{j}(\mu)+\normx{x_{i+n_{0}}-x_{j}}$, or for some $i$ and $j$ with $N^{1}_{ij}(\mu)=0$, $r_{j}(\mu)=m_{i+n_{0}}(\mu)+\normx{x_{i+n_{0}}-x_{j}}$. This determines the step size $s$ in the algorithm.
Proof of Lemma \[lemma:nn-consistency\]
---------------------------------------
For ease of notation, let $f_{i}=f(x_{i}, d_{i})$, $\sigma^{2}_{i}=\sigma^{2}(x_{i}, d_{i})$, and let $\overline{f}_{i}=J^{-1}\sum_{j=1}^{J}f_{\ell_{j}(i)}$ and $\overline{u}_{i}=J^{-1}\sum_{j=1}^{J}u_{\ell_{j}(i)}$. Then we can decompose $$\begin{gathered}
\frac{J+1}{J}(\hat{u}_{i}^{2}-u_{i}^{2})=[
f_{i}-\overline{f}_{i}+u_{i}-\overline{u}_{i}
]^{2}-\frac{J+1}{J}u_{i}^{2}\\
=[(f_{i}-\overline{f}_{i})^{2}
+2(u_{i}-\overline{u}_{i})(f_{i}-\overline{f}_{i})] -2\overline{u}_{i}u_{i} +
\frac{2}{J^{2}} \sum_{j=1}^{J} \sum_{k=1}^{j-1}
u_{\ell_{j}(i)}u_{\ell_{k}(i)}
+\frac{1}{J^{2}}\sum_{j=1}^{J}(u_{\ell_{j}(i)}^{2}-u_{i}^{2})\\
=T_{1i}+2T_{2i}+2T_{3i}+T_{4i}+T_{5i}+
\frac{1}{J^{2}}\sum_{j=1}^{J}(\sigma_{\ell_{j}(i)}^{2}-\sigma_{i}^{2}),\end{gathered}$$ where $$\begin{aligned}
T_{1i}
&=[(f_{i}-\overline{f}_{i})^{2}
+2(u_{i}-\overline{u}_{i})(f_{i}-\overline{f}_{i})],
& T_{2i}&=\overline{u}_{i}u_{i}\\
T_{3i}&=\frac{1}{J^{2}} \sum_{j=1}^{J}
\sum_{k=1}^{j-1} u_{\ell_{j}(i)}u_{\ell_{k}(i)},
& T_{4i} &=\frac{1}{J^{2}}\sum_{j=1}^{J}(u_{\ell_{j}(i)}^{2}-\sigma_{\ell_{j}(i)}^{2}),
& T_{5i}&= \sigma_{i}^{2}-u_{i}^{2}.\end{aligned}$$ Since $\max_{i}\norm{x_{\ell_{J}(i)}-x_{i}}\to 0$ and since $\sigma^{2}(\cdot, d)$ is uniformly continuous, it follows that $$\max_{i}\max_{1\leq j\leq J} \abs{\sigma^{2}_{\ell_{j}(i)}-\sigma^{2}_{i}}\to
0,$$ and hence that $\abs{\sum_{i=1}^{n}a_{ni}J^{-1}\sum_{j=1}^{J}
(\sigma_{\ell_{j}(i)}^{2}-\sigma_{i}^{2})}\leq \max_{i} \max_{j=1,\dotsc, J}
(\sigma_{\ell_{j}(i)}^{2}-\sigma_{i}^{2})\sum_{i=1}^{n}a_{ni}\to 0$. To prove the lemma, it therefore suffices to show that the sums $\sum_{i=1}^{n}a_{ni}T_{qi}$ all converge to zero.
To that end, $$E\abs{\sum_{i}a_{ni}T_{1i}}\leq \max_{i}(f_{i}-\overline{f}_{i})^{2}
\sum_{i}a_{ni}
+2\max_{i}\abs{f_{i}-\overline{f}_{i}}
\sum_{i}a_{ni}E\abs{u_{i}-\overline{u}_{i}},$$ which converges to zero since $ \max_{i}\abs{f_{i}-\overline{f}_{i}} \leq
\max_{i}\max_{j=1,\dotsc, J}(f_{i}-f_{\ell_{j}(i)})\leq
C_{n}\max_{i}\normx{x_{i}-x_{\ell_{J}(i)}}\to 0$. Next, by the von Bahr-Esseen inequality, $$E\abs{\sum_{i=1}^{n}a_{ni}T_{5i}}^{1+1/2K}\leq
2\sum_{i=1}^{n}a_{ni}^{1+1/2K}E\abs{T_{5i}}^{1+1/2K}\leq
2\max_{i}a_{ni}^{1/2K}\max_{j}E
\abs{T_{5j}}^{1+1/2K} \sum_{k=1}^{n}a_{nk}\to 0.$$ Let $\mathcal{I}_{j}$ denote the set of observations for which an observation $j$ is used as a match. To show that the remaining terms converge to zero, let we use the fact $\#\mathcal{I}_{j}$ is bounded by $J\overline{L}$, where $\overline{L}$ is the kissing number, defined as the maximum number of non-overlapping unit balls that can be arranged such that they each touch a common unit ball ([@mttv97 Lemma 3.2.1]; see also [@AbIm08cv]). $\overline{L}$ is a finite constant that depends only on the dimension of the covariates (for example, $\overline{L}=2$ if $\dim(x_{i})=1$). Now, $$\sum_{i}a_{ni}T_{4i}=\frac{1}{J^{2}}
\sum_{j=1}^{n}(u_{j}-\sigma^{2}_{j})\sum_{i\in \mathcal{I}_{j}}a_{ni},$$ and so by the von Bahr-Esseen inequality, $$\begin{gathered}
E\abs{ \sum_{i}a_{ni}T_{4i}}^{1+1/2K}\leq \frac{2}{J^{2+1/K}}
\sum_{j=1}^{n}E\abs{u_{j}-\sigma^{2}_{j}}^{1+1/2K}
\left(\sum_{i\in \mathcal{I}_{j}}a_{ni}\right)^{1+1/2K}\\
\leq \frac{(J\overline{L})^{1/2K}}{J^{2+1/K}} \max_{k}E\abs{u_{k}-\sigma^{2}_{k}}^{1+1/2K}
\max_{i}a_{ni}^{1+1/2K}
\sum_{j=1}^{n}\sum_{i\in \mathcal{I}_{j}}a_{ni},\end{gathered}$$ which is bounded by a constant times $\max_{i}a_{ni}^{1+1/2K} \sum_{j=1}^{n}\sum_{i\in \mathcal{I}_{j}}a_{ni}=
\max_{i}a_{ni}^{1+1/2K} J\sum_{i}a_{ni}\to 0$. Next, since $E[u_{i}u_{i'}u_{\ell_{j}(i)}u_{\ell_{k}(i')}]$ is non-zero only if either $i=i'$ and $\ell_{j}(i)=\ell_{k}(i')$, or else if $i=\ell_{k}(i')$ and $i'=\ell_{j}(i)$, we have $\sum_{i'=1}^{n}a_{ni'}E[u_{i}u_{i'}u_{\ell_{j}(i)}u_{\ell_{k}(i')}]\leq\max_{i'}a_{ni'}\left(
\sigma^{2}_{i}\sigma^{2}_{\ell_{j}(i)}+
\sigma_{\ell_{j}(i)}^{2}\sigma^{2}_{i}\right)$, so that $$\var(\sum_{i}a_{ni}T_{2i})=
\frac{1}{J^{2}}
\sum_{i, j,k, i'}a_{ni}a_{ni'}E[u_{i}u_{\ell_{k}(i')}
u_{i'}u_{\ell_{j}(i)}]\leq 2K^{2}\max_{i'}a_{ni'}
\sum_{i}a_{ni}
\to 0.$$ Similarly for $j\neq k$ and $j'\neq k$, $\sum_{i'=1}^{n}a_{ni'}
E[u_{\ell_{j}(i)}u_{\ell_{k}(i)}u_{\ell_{j'}(i')}u_{\ell_{k'}(i')}]\leq \max_{i'}
2\sigma^{2}_{\ell_{j}(i)}\sigma^{2}_{\ell_{k}(i)}$, so that $$\begin{gathered}
\var\Big(\sum_{i}a_{ni}T_{3i}\Big)\\
=\frac{1}{J^{4}}\sum_{i, i', j,j'}\sum_{k=1}^{j-1}\sum_{k'=1}^{j'-1}
a_{ni}a_{ni'}E[u_{\ell_{j}(i)}u_{\ell_{k}(i)}
u_{\ell_{j'}(i')}u_{\ell_{k'}(i')}]\leq
2K^{2}\max_{i'}a_{ni'}\sum_{i}a_{ni}\to 0.\end{gathered}$$
[^1]: email: `timothy.armstrong@yale.edu`
[^2]: email: `mkolesar@princeton.edu`
|
---
abstract: 'Device-to-Device (D2D) communication can support the operation of cellular systems by reducing the traffic in the network infrastructure. In this paper, the benefits of D2D communication are investigated in the context of a Fog-Radio Access Network (F-RAN) that leverages edge caching and fronthaul connectivity for the purpose of content delivery. Assuming offline caching, out-of-band D2D communication, and an F-RAN with two edge nodes and two user equipments, an information-theoretically optimal caching and delivery strategy is presented that minimizes the delivery time in the high signal-to-noise ratio regime. The delivery time accounts for the latency caused by fronthaul, downlink, and D2D transmissions. The proposed optimal strategy is based on a novel scheme for an X-channel with receiver cooperation that leverages tools from real interference alignment. Insights are provided on the regimes in which D2D communication is beneficial.'
author:
- 'Roy Karasik, Osvaldo Simeone, and Shlomo Shamai (Shitz) [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'bib/myBib.bib'
title: 'Fundamental Latency Limits for D2D-Aided Content Delivery in Fog Wireless Networks'
---
D2D, F-RAN, edge caching, latency.
Introduction
============
Device-to-Device (D2D) communication is a main enabler of novel applications such as mission critical communication, video sharing, and proximity-aware gaming and social networking. Furthermore, it can enhance conventional cellular services, including content delivery, by reducing the traffic at the cellular network infrastructure. D2D communication in cellular networks can be either out-of-band, whereby direct communication between the users takes place over frequency resources that are orthogonal with respect to the spectrum used for cellular transmission; or in-band, in which case the same frequency band is used for both D2D and cellular transmissions [@asadi2014survey].
In this paper, we study the benefits of out-of-band D2D communications for the modern cellular architecture of a Fog-Radio Access Network (F-RAN) by focusing on content delivery [@hung2015architecture; @tandon2016harnessing]. As illustrated in Fig \[fig\_model\], in an F-RAN, content delivery leverages both edge caching and fronthaul connectivity to a Cloud Processor (CP). In this work, we characterize the potential latency reduction that can be achieved by utilizing D2D links in an F-RAN, while properly accounting for the latency overhead associated with D2D communications.
![Illustration of the D2D-aided F-RAN model under study.[]{data-label="fig_model"}](F-RAN_with_D2D_2X2){width="3.5in"}
**Related Work:** The cache-aided interference channel was first studied in [@maddah2015cache], where an upper bound on the minimum delivery latency in the high signal-to-noise ratio (SNR) regime was derived for a system with three users. A lower bound on the Normalized Delivery Time (NDT), which measures the high-SNR worst-case latency relative to an ideal system with unlimited caching capability, was presented in [@sengupta2016cache] for any number of Edge Nodes (ENs) and User Equipments (UEs), and it was shown to be tight for the setting of two ENs and two UEs. Lower and upper bounds for arbitrary numbers of ENs and UEs, where both ENs and UEs have caching capabilities, were presented in [@naderializadeh2017fundamental] under the constraint of linear precoders at the ENs. The NDT of a general F-RAN system with fronthaul links was studied in [@sengupta2016fog], where the proposed schemes were shown to achieve the minimum NDT to within a factor of 2, and the minimum NDT was completely characterized for two ENs and two UEs, as well as for other special cases. In [@huang2009degrees], it was shown that, for the interference channel with in-band cooperation, transmitter or receiver cooperation cannot increase the high-SNR performance in terms of sum Degrees of Freedom (DoF). The interference channel with out-of-band receiver cooperation was studied in [@wang2011interference], where receiver cooperation was shown to increase the Generalized DoF metric. Importantly, reference [@wang2011interference] only imposes a rate constraint on the D2D links, hence not accounting for the latency overhead caused by D2D communications, which is of central interest in this work.
**Main Contributions:** In this paper, we study the D2D-aided F-RAN system with two ENs and two UEs in Fig. \[fig\_model\], and put forth the following main contributions. First, in Sec. \[sec:X\_channel\], we present a novel scheme that improves the NDT achievable on an X-channel with out-of-band D2D receiver cooperation. The proposed scheme enables interference cancellation at the receiver’s side with minimal overhead on the D2D links. Second, in Sec. \[sec:minimum\_NDT\], we characterize the minimum NDT of the D2D-aided F-RAN illustrated in Fig. \[fig\_model\]. The minimum NDT is used to identify the conditions under which D2D communication is beneficial, and to provide insights on the interplay between fronthaul and D2D resources.
System Model {#sec:sys_model}
============
We consider the F-RAN system with Device-to-Device (D2D) links depicted in Fig. \[fig\_model\], where two single-antenna User Equipments (UEs) are served by two single-antenna Edge Nodes (ENs) over a downlink wireless channel. The UEs are connected by two orthogonal out-of-band D2D links of capacity $C_D$ bits per symbol. The model generalizes the set-up studied in [@tandon2016cloud] by including D2D communications. Each EN is connected to a Cloud Processor (CP) by a fronthaul link of capacity $C_F$ bits per symbol. Throughout this paper, a symbol refers to a channel use of the downlink wireless channel.
Let $\mathcal F$ denote a library of $N\geq 2$ files, $\mathcal F=\{f_1,\ldots,f_N\}$, each of size $L$ bits. The library is fixed for the considered time interval. The entire library is available at the CP, while the ENs can only store up to $\mu NL$ bits each, where $0\leq\mu\leq 1$ is the fractional cache size. During the placement phase, contents are proactively cached at the ENs, subject to the mentioned cache capacity constraints.
After the placement phase, the system enters the delivery phase, which is organized in Transmission Intervals (TIs). In every TI, each UE arbitrarily requests one of the $N$ files from the library. The UEs’ requests in a given TI are denoted by the demand vector $\mathbf d\triangleq (d_1,d_2)\in[N]^2$, where for any positive integer $a$, we define the set $[a]\triangleq\{1,2,\ldots,a\}$. This vector is known at the beginning of a TI at the CP and ENs. The goal is to deliver the requested files to the UEs within the lowest possible delivery latency by leveraging fronthaul links, downlink channel and D2D links.
For a given TI, let $T_E$ denote the duration of the transmission on the wireless downlink channel. At time $t\in[T_E]$, each UE $k\in[2]$ receives a channel output given by
[rCl]{}\[eq:wireless\_channel\] y\_k(t)&=&\_[m=1]{}\^[2]{}h\_[km]{}x\_m(t)+z\_k(t),
where $x_m(t)\in\mathbb C$ is the baseband symbol transmitted from EN $m\in[2]$ at time $t$, which is subject to the average power constraint $\mathbb E |x_m(t)|^2\leq P$ for some $P>0$; coefficient $h_{km}\in\mathbb C$ denotes the quasi-static flat-fading channel between EN $m$ to UE $k$, which is assumed to remain constant during each TI; and $z_k(t)$ is an additive white Gaussian noise, such that $z_k(t)\sim\mathcal C\mathcal N(0,1)$ is independent and identically distributed (i.i.d.) across time and UEs. The Channel State Information (CSI) $\mathbf{H}\triangleq\{h_{km}:k\in[2],m\in[2]\}$ is assumed to be drawn i.i.d. from a continuous distribution, and known to all nodes.
Caching, Delivery and D2D Transmission
--------------------------------------
The operation of the system is defined by the following policies that perform caching, as well as delivery via fronthaul, edge and D2D communication resources.
### Caching Policy
During the placement phase, for EN $m$, $m\in[2]$, the caching policy is defined by functions $\pi^m_{c,n}(\cdot)$ that map each file $f_n$ to its cached content $s_{m,n}$ as
[rCl’l]{}\[eq:caching\_policy\] s\_[m,n]{}&& \_[c,n]{}\^m(f\_n),&n.
Note that, as per , we consider policies where only coding within each file is allowed, i.e., no inter-file coding is permitted. We have the cache capacity constraint $H(s_{m,n})\leq \mu L$. The overall cache content at EN $m$ is given by $s_m\triangleq(s_{m,1},s_{m,2}\ldots,s_{m,N})$.
### Fronthaul Policy
In each TI of the delivery phase, for EN $m$, $m\in[2]$, the CP maps the library, $\mathcal F$, the demand vector $\mathbf{d}$ and CSI $\mathbf{H}$ to the fronthaul message
[rCl]{} \_m=(u\_m\[1\],u\_m\[2\],…,u\_m\[T\_F\])=\_f\^m(F,s\_m,,),
where $T_F$ is the duration of the fronthaul message. Note that the fronthaul message cannot exceed $T_FC_F$ bits, i.e., $H(\mathbf{u}_m)\leq T_FC_F$.
### Edge Transmission Policies
After fronthaul transmission, in each TI, the ENs transmit using a function $\pi_{e}^m(\cdot)$ that maps the local cache content, $s_m$, the received fronthaul message $\mathbf{u}_m$, the demand vector $\mathbf{d}$ and the global CSI $\mathbf{H}$, to the output codeword
[rClCl]{} \_m&=&(x\_m\[1\],x\_m\[2\],…,x\_m\[T\_E\])&=&\_[e]{}\^m(s\_m,\_m,,).
### D2D Interactive Communication Policies
After receiving the signals over $T_E$ symbols, in any TI, the UEs use a D2D conferencing policy. For each UE $k\in[2]$, this is defined by the interactive functions $\pi^k_{\text{D2D},i}(\cdot)$ that map the received signal $\mathbf{y}_k\triangleq (y_k[1],\ldots,y_k[T_E])$, the global CSI and the previously received D2D message from UE $k'\neq k\in[2]$ to the D2D message
[l]{}\[eq:d2d\_policy\] v\_k\[i\]=\
\_[,i]{}\^k(\_k,,v\_[k’]{}\[1\],…,v\_[k’]{}\[i-1\],v\_[k]{}\[1\],…,v\_[k]{}\[i-1\]),
where $i=1,\ldots,T_D$, with $T_D$ being the duration of the D2D communication. The total size of each D2D message cannot exceed $T_DC_D$ bits. i.e., $H(v_k[1],\ldots,v_k[T_D])\leq T_DC_D$.
### Decoding Policy
After D2D communication, each UE $k\in[2]$ implements a decoding policy $\pi_d^k(\cdot)$ that maps the channel outputs, the D2D messages from UE $k'\neq k\in[2]$, the UE demand and the global CSI to an estimate of the requested file $f_{d_k}$ given as
[rCl]{} \_[d\_k]{}&=&\_d\^k(\_k,\_[k’]{},d\_k,).
The probability of error is defined as
[rCl]{} P\_e&=&\_\_[k]{}(\_[d\_k]{}f\_[d\_k]{}),
which is the worst-case probability of decoding error measured over all possible demand vectors $\mathbf{d}$ and over all users $k\in[2]$. A sequence of policies, indexed by the file size $L$, is said to be feasible if, for almost all channel realization $\mathbf{H}$, we have $P_e\rightarrow0$ when $L\rightarrow\infty$.
Performance Metric
------------------
As discussed, in each TI, the CP first sends the fronthaul messages to the ENs for a total time of $T_F$ symbols; then, the ENs transmit on the wireless shared channel for a total time of $T_E$ symbols; and, finally, the UEs use the out-of-band D2D links for a total time of $T_D$ symbols. For any sequence of feasible policies, the delivery time per bit $\Delta(\mu,C_F,C_D,P)$ is hence defined as the limit
[rCl]{}\[eq:del\_time\_per\_bit\] (,C\_F,C\_D,P)&&\_[L]{}.
The notation emphasizes the dependence on the fractional cache size $\mu$, the fronthaul and D2D capacities $C_F$ and $C_D$, respectively, and the average power constraint $P$.
We adopt the Normalized Delivery time (NDT), introduced in [@sengupta2016fog], as the performance metric of interest. To this end, we evaluate the performance in the high-SNR regime by parameterizing fronthaul and D2D capacities as $C_F=r_F\log(P)$ and $C_D=r_D\log(P)$. With this parametrization, the fronthaul rate $r_F\geq 0$ represents the ratio between the fronthaul capacity and the high-SNR capacity of each EN-to-UE wireless link in the absence of interference; and a similar interpretation holds for the D2D rate $r_D\geq0$.
For any given tuple $(\mu,r_F,r_D)$, the NDT of a sequence of achievable policies is defined as
[rCl]{}\[eq:NDT\_def\] (,r\_F,r\_D)&&\_[P]{}.
The factor $1/\log(P)$, used for normalizing the delivery time in , represents the minimal time to deliver one bit over an EN-to-UE wireless link in the high-SNR regime and in the absence of interference. The minimum NDT is finally defined as the minimum over all achievable policies
[l]{}\[eq:minimum\_ndt\_def\] \^\*(,r\_F,r\_D)\
{(,r\_F,r\_D):(,r\_F,r\_D)}.
By construction, we have the lower bound $\delta^*(\mu,r_F,r_D)\geq 1$. Furthermore, the minimum NDT can be proved by means of time- and memory-sharing arguments to be convex in $\mu$ for any fixed values of $r_F$ and $r_D$ [@tandon2016cloud Remark 1].
The Two-User X-Channel with Receiver Cooperation {#sec:X_channel}
================================================
In this section, we present a result of independent interest that will be used in Sec. \[sec:minimum\_NDT\] to derive the minimum NDT . Specifically, we develop a new delivery scheme for the special case in which no fronthaul communication is enabled, i.e., $r_F=0$, and the fractional cache size is $\mu=1/2$. In this regime, each EN can only store half of each file in the library. Under the mentioned caching strategy, in the worst-case scenario in which the UEs request different files, the set-up is equivalent to a two-user Gaussian X-channel with receiver cooperation. In this channel, as illustrated in Fig. \[fig:X\_chan\], each UE needs to download half of the requested file from one EN and the other half from the second EN.
![X-channel with receiver cooperation studied in Sec. \[sec:X\_channel\], which represents an F-RAN system with no fronthaul, i.e., with $r_F=0$, and fractional cache size $\mu=1/2$.[]{data-label="fig:X_chan"}](X_Channel-cropped){width="2.5in"}
The proposed scheme achieves the NDT detailed in the following Proposition.
\[prop:x\_ch\_ub\] For $\mu=1/2$, $r_F=0$ and $r_D\geq 0$, the minimum NDT is upper bounded as
[rCl]{}\[eq:x\_ch\_ub\] \^\*\_X1+.
Proposition \[prop:x\_ch\_ub\] is proved in the next two subsections by first proposing a novel scheme for the deterministic X-channel, and then adapting it for the Gaussian counterpart. The scheme is based on layered transmission and successive interference cancellation at the receivers.
As compared to existing schemes that are applicable for $\mu=1/2$ and $r_F=0$, real interference alignment [@motahari2014real] achieves an NDT of $3/2$ without using the D2D links [@tandon2016cloud]. Therefore, the proposed D2D-based scheme of Proposition \[prop:x\_ch\_ub\] is useful only when the D2D capacity is sufficiently large, i.e., when $r_D>1$. Furthermore, the scheme in [@wang2011interference] has an NDT lower bounded by 2, since the latency due to D2D communications equals the transmission time on the downlink channel. Hence, the scheme is not advantageous in terms of NDT. Finally, as an alternative policy, one could have each UE compress and forward the received signal to the other UE, allowing each UE to carry out Zero Forcing (ZF) linear equalization. By quantizing with a rate equal to $\log P$, one can ensure that the SNR scales linearly with $P$, and that the approach achieves an NDT equal to $1+1/r_D>\delta_X$ (see [@sengupta2016fog][@tandon2016cloud] for similar arguments).
The Deterministic Approach {#subsec:deterministic_approach}
--------------------------
We start by considering a deterministic approximation of the X-channel in order to facilitate the explanation of the main ideas behind the proposed scheme. We recall that, according to [@huang2012interference], in high SNR, the channel is approximated by the deterministic model
[rCl]{}\[eq:deterministic\_channel\] y\_1(t)&=&x\_1(t)+S\^[n\_d-n\_c]{}x\_2(t)\
y\_2(t)&=&S\^[n\_d-n\_c]{}x\_1(t)+x\_2(t),
where summations and multiplications are over the binary field $\mathbb F_2$; $n_d$ and $n_c$ represent the number of direct and cross signal levels, respectively, with $n_d>n_c$; $x_i(t)$ and $y_i(t)$ $\in\mathbb F_2^{n_d}$ for $i\in[2]$ are the binary vectors representing the inputs and outputs of the deterministic channel, respectively; and $S$ is the $n_d\times n_d$ shift matrix with all zeros except in the first lower diagonal, which contains all ones. This channel is illustrated in Fig. \[fig:det\_chan\].
![Deterministic X-channel with $n_d$ direct signal levels and $n_c=n_d-1$ cross signal levels considered in Sec. \[subsec:deterministic\_approach\].[]{data-label="fig:det_chan"}](deterministic_model-cropped){width="2in"}
The number of levels is selected as $n_d=\ceil{\log P}$, while $n_c$ will be taken to satisfy the limit $n_c/n_d\rightarrow 1$ when $n_d\rightarrow\infty$ in order to approximate the high-SNR behavior of the assumed channel model , as explained in [@huang2012interference Appendix B]. Following this model, we set the D2D link capacity $C_D=r_D\log P$ to equal $r_Dn_d$ signal levels between the UEs.
Consider, without loss of generality, the case where $n_c=n_d-1$ and $n_d$ is odd, as illustrated in Fig. \[fig:det\_chan\]. EN 1 and EN 2 at each time $t$ transmit independent bits $x_1=[a_1,\ldots,a_{n_d}]^T$ and $x_2=[b_1,\ldots,b_{n_d}]^T$ on the $n_d$ levels, where we have dropped the dependence on $t$. By , the received signals at the UEs are $y_1=[a_1,a_2\oplus b_1,\ldots,a_{n_d}\oplus b_{n_d-1}]^T$ and $y_2=[b_1,b_2\oplus a_1,\ldots,b_{n_d}\oplus a_{n_d-1}]^T$. UE 2 uses its D2D link to convey the bits received on the even-numbered levels
[rCl]{}\[eq:det\_d2d\_msg\] v\_2&=&
to UE 1, which consists of $(n_d-1)/2$ bits. UE 1 is thus able to decode the bits $\{a_1,b_2,a_3,b_4,a_5,\ldots,b_{n_d-1},a_{n_d}\}$ from $\{y_1,v_2\}$ by means of successive interference cancellation. To this end, it starts by decoding $a_1$ from $y_{1,1}=a_1$; then, it uses $a_1$ together with $b_2\oplus a_1$ in to decode $b_2$; next, it uses $b_2$ and $y_{1,3}=a_3\oplus b_2$ to decode $a_3$; and so on, until all the desired bits are decoded. Similarly, UE 2 is able to decode bits $\{b_1,a_2,b_3,a_4,b_5,\ldots,a_{n_d-1},b_{n_d}\}$ from $y_2$ and $v_1=\{a_2\oplus b_1,a_4\oplus b_3,\ldots,a_{n_d-1}\oplus b_{n_d-2}\}$. The number of channel uses required on the downlink channel to satisfy the UEs’ demands is $L/(n_d-1)$. For each channel use, each UE has to convey $(n_d-1)/2$ bits using a D2D link of capacity $r_Dn_d$. Therefore, the resulting NDT , if we let the number $n_d$ of levels be arbitrary, is
[rCl]{} \_[n\_d]{}=\_X.
Next, we show how to achieve the same NDT for the original model .
Real Interference Alignment with Receiver Cooperation {#sec:real_IA}
-----------------------------------------------------
In order to convert the proposed scheme from the deterministic model to the X-channel , we follow the *real interference alignment* approach of [@motahari2014real]. Accordingly, in a manner similar to the deterministic model, each transmitter uses $n_d$ signal layers, where $n_d$ is odd. The signal transmitted by the ENs at each symbol can be written as
[c]{}\[eq:layers\] x\_1=\_[i=1]{}\^[n\_d]{} g\_[1,i]{}a\_ix\_2=\_[i=1]{}\^[n\_d]{} g\_[2,i]{}b\_i,
where $\{g_{m,i}\}$, with $m\in[2]$ and $i\in[n_d]$, are precoder gains, and the values $a_i$ and $b_i$ are chosen from a discrete constellation, so that we have $a_i,b_i\in A\mathbb Z_Q\triangleq \{0,A,2A,\ldots,A(Q-1)\}$. Each layer $i$ is coded using random coding with rate $R$ bits per symbol. Appendix \[sec:appendixA\] shows that, by choosing parameters $\{g_{m,i}\}$, $A$, $Q$ and $R$ properly, UE 1 can decode the symbols $\{a_1,a_2+b_1,\ldots,a_{n_d}+b_{n_d-1},b_{n_d}\}$, while UE 2 decodes $\{b_1,b_2+a_1,\ldots,b_{n_d}+a_{n_d-1},a_{n_d}\}$. The UEs now exchange the even-numbered layers as in the deterministic model, so that UE 1 transmits the message $v_1=\{a_2+b_1,a_4+b_3,\ldots,a_{n_d-1}+b_{n_d-2}\}$ to UE 2, while UE 2 transmits $v_2=\{b_2+a_1,b_4+a_3,\ldots,b_{n_d-1}+a_{n_d-2}\}$ to UE 1. As a result, UE 1 can decode $\{a_1,b_2,a_3,b_4,\ldots,a_{n_d},b_{n_d}\}$ while UE 2 decodes $\{b_1,a_2,b_3,a_4,\ldots,b_{n_d},a_{n_d}\}$.
As also shown in Appendix \[sec:appendixA\], for high SNR, the rate can be selected as $R\approx\log Q\approx \log P/(n_d+1)$. Since each UE decodes $(n_d+1)/2$ layers from one EN and $(n_d-1)/2$ from the other, the number of channel uses required to satisfy the UEs’ demands on the edge channel is
[rCl]{} T\_E==.
For each channel use, the message $v_k$, $k\in[2]$, conveyed over the D2D link, comprises $(n_d-1)/2$ elements, each taking one of $2Q$ values. Therefore, for high SNR, the delay due to D2D transmissions is given as
[rCl]{} T\_D=T\_ET\_E,
and the resulting NDT is
[rCl]{}\[eq:fading\_X\_NDT\] \_[n\_d]{} .
Similar to the deterministic channel, by increasing the number of layers $n_d$, we have the limit $\lim_{n_d\rightarrow\infty}\delta_{n_d}=\delta_X$.
Minimum NDT {#sec:minimum_NDT}
===========
In this section, we derive the minimum NDT by presenting a novel achievable scheme and an information-theoretic lower bound. The achievable scheme leverages the D2D cooperative strategy introduced above along with the scheme in [@tandon2016cloud], which is optimal in the absence of D2D links.
\[th:minimum\_NDT\] The minimum NDT for the $2\times 2$ F-RAN system with number of files $N\geq 2$, fractional cache size $\mu\geq0$, fronthaul rate $r_F\geq 0$ and D2D rate $r_D\geq 0$ is given as
[ll]{}\[eq:minimum\_NDT\] &\^\*=\
&
[ll]{} &0r\_F,r\_D1\
1+& r\_F\
& r\_D>.
.
Before sketching the proof, we use the result in Theorem \[th:minimum\_NDT\] in order to draw conclusions on the role of D2D cooperation in improving the delivery latency. We start by observing that, for $r_D\leq\max\{1,r_F\}$, the minimum NDT is identical to the minimum NDT without D2D links derived in [@tandon2016cloud Theorem 1]. Therefore, D2D communication provides a latency reduction only when we have $r_D>\max\{1,r_F\}$. This is illustrated in Fig. \[fig:minimum\_NDT\], where we plot the minimum NDT as a function of the fractional cache size $\mu$ for fixed fronthaul rate $r_F$ and D2D rate $r_D$. For any $r_D\leq\max\{1,r_F\}$, the minimum NDT is not affected by the value of $r_D$, whereas a larger $r_D$ yields a reduced minimum NDT.
![Minimum NDT for the $2\times 2$ F-RAN with D2D links as a function of $\mu$: when $r_D\leq\max\cb{1,r_F}$, D2D communication cannot reduce the delivery latency, while a reduction of the NDT is obtained when $r_D>\max\cb{1,r_F}$.[]{data-label="fig:minimum_NDT"}](rD_greater_1_greater_rF-cropped){width="3in"}
The minimum useful value $\max\{1,r_F\}$ for the D2D rate $r_D$ increases with fronthaul rate $r_F$. This demonstrates that there exists a trade-off between fronthaul and D2D resources for the purpose of interference management, although their role is not symmetric. The use of fronthaul links is in fact necessary to obtain a finite NDT when the library is not fully available at the ENs, i.e., when $\mu<1/2$. D2D links can instead only reduce the NDT in regimes where fronthaul and edge resources would already be sufficient for content delivery with a finite NDT. In particular, as summarized in Fig. \[fig:minimum\_NDT\], when $r_D>\max\{1,r_F\}$, D2D communication reduces the minimum NDT for all values $0<\mu<1$. Furthermore, when $\mu>1/2$, irrespective of the value of $r_F$, the minimum NDT is achieved by leveraging only edge caching and D2D links, without having to rely on fronthaul resources, thus reducing the traffic at the network infrastructure. This is in contrast to the case $r_D\leq\max\{1,r_F\}$, where, by [@tandon2016cloud], fronthaul transmission is needed to obtain the minimum NDT unless $r_F\leq 1$.
*Achievability:* The strategy that achieves is based on time- and memory-sharing [@tandon2016cloud Remark 1] between the policies for the corner points $\mu=0,1/2$ and $1$. For $\mu=1$, we apply cache-aided cooperative ZF at the ENs by leveraging the fact that the ENs can both store the entire library of files. This achieves the NDT $\delta^*(\mu=1,r_F,r_D)=1$ [@tandon2016cloud Sec. IV.A]. For $\mu=0$, we apply the cloud-aided soft-transfer scheme of [@sengupta2016fog] and [@tandon2016cloud], which uses fronthaul links to convey quantized ZF-precoded signals, achieving the NDT $1+1/r_F$. Finally, for $\mu=1/2$, we use one of the following three schemes: *(i)* EN coordination via interference alignment, which results in an NDT of 3/2 without using either fronthaul or D2D links [@tandon2016cloud Sec. IV.A]; *(ii)* time- and memory-sharing between cloud-aided soft transfer and cache-aided cooperative ZF, which leverages fronthaul and cache resources, and results in an NDT of $1+1/(2r_F)$ [@tandon2016cloud Theorem 1]; *(iii)* the proposed D2D-based delivery scheme, which results in an NDT of $\delta_X$ by leveraging edge and D2D resources.
*Converse*: The proof of the lower bound can be found in Appendix \[sec:appendix\_lb\]. The proof leverages the approach of [@tandon2016cloud], which is based on a variation of cut-set arguments. Accordingly, subsets of information resources are identified, from which, in the high-SNR regime, the requested files can be reliably decoded when a feasible policy is implemented. In particular, the first subset, $\{s_1,s_2,\mathbf{u}_1,\mathbf{u}_2\}$, yields a lower bound on $T_F$ as a function of $\mu$ and $r_F$; the seconds subset, $\{\mathbf{y}_1,\mathbf{y}_2\}$, yields a lower bound on $T_E$; and the third subset, $\{s_1,\mathbf{u}_1,\mathbf{y}_1,\mathbf{v}_2\}$, where $\mathbf{v}_2\triangleq(v_2[1],\ldots,v_2[T_D])$, yields a lower bound on a linear combination of $(T_F,T_E,T_D)$ as a function of $\mu$, $r_F$ and $r_D$. Note that, only the latter bound differs with respect to [@tandon2016cloud]. These bounds are then linearly combined according to the values of the fronthaul rate $r_F$ and D2D rate $r_D$ to show that is a lower bound on the minimum NDT.
*Remark:* Although the definition of D2D conferencing policy allows for *interactive communication* [@willems1983discrete], the optimal scheme described above uses *simultaneous monologues*, whereby the D2D messages of the UEs are based only on the CSI and the respective own received signal.
Conclusions
===========
In this work, fundamental insights were provided on the benefits of D2D communication for content delivery in an F-RAN. Considering the Normalized Delivery Time (NDT) metric, an optimal strategy for utilizing the fronthaul and D2D links, as well as the downlink wireless channel, was presented. This strategy is based on a novel scheme for the X-channel with receiver cooperation. It was demonstrated that, for sufficiently large D2D and cache capacities, D2D communication can reduce the traffic on the fronthaul links, and hence help reducing the load on the network infrastructure. Among possible extensions of this work, we mention the generalization of the results to more than two users and ENs; the study of pipelined transmission, where fronthaul or D2D links can be used simultaneously with the downlink channel [@sengupta2016fog]; the evaluation of the impact of imperfect CSI; and the characterization of the optimal D2D strategy under linear precoding and hard-transfer constraints [@zhang2017fundamental].
Proof of Proposition \[prop:x\_ch\_ub\] {#sec:appendixA}
=======================================
In the layered transmitted signals , the precoder gains $\{g_{m,i}\}$ are chosen as
[rCl]{} g\_[m,i]{}&=&
[ll]{} \^\^&i\
\^\^h\_[m’,m’]{}h\_[m,m’]{}&i,
.\
where $m,m'\in[2]$ and $m'\neq m$. This choice results in layers $a_i$ and $b_{i-1}$ being summed at UE 1, while layers $b_i$ and $a_{i-1}$ are summed at UE 2, $i\in\{2,3,\ldots,n_d\}$. This is in the sense that, given the equalities $h_{11}g_{1,i}=h_{12}g_{2,i-1}$ and $h_{22}g_{2,i}=h_{21}g_{1,i-1}$, the received signals $y_1$ and $y_2$ can be written as
[rCl]{}\[eq:x\_ch\_rec\_ue1\] y\_1&=&h\_[11]{}g\_[1,1]{}a\_1+\_[i=2]{}\^[n\_d]{}h\_[11]{}g\_[1,i]{}\
&&+h\_[12]{}g\_[2,n\_d]{}b\_[n\_d]{}+z\_1,
and
[rCl]{}\[eq:x\_ch\_rec\_ue2\] y\_2&=&h\_[22]{}g\_[2,1]{}b\_1+\_[i=2]{}\^[n\_d]{}h\_[22]{}g\_[2,i]{}\
&&+h\_[21]{}g\_[1,n\_d]{}a\_[n\_d]{}+z\_2.
Since symbols $a_i$ and $b_i$ are selected from $A\mathbb Z_Q$, the noiseless received signal $y_k-z_k$ takes values from a discrete set as well. Let $d_{\text{min},k}$ denote the minimum distance between the element of this set, and define $d_\text{min}\triangleq\min\cb{d_{\text{min},1},d_{\text{min},2}}$. Almost surely, the effective channels $\cb{h_{11}g_{1,i},h_{12}g_{2,n_d}}_{i=1}^{n_d}$ are *rationally independent*[^3], and analogously for $\cb{h_{22}g_{2,i},h_{21}g_{1,n_d}}_{i=1}^{n_d}$. Therefore, for each layer, by [@motahari2014real Theorem 3], there exists a code, of rate
[rCl]{} R=Q-1
such that UE 1 can decode layers $\cb{a_1,a_2+b_1,\ldots,a_{n_d}+b_{n_d-1},b_{n_d}}$, while UE 2 decodes $\{b_1,b_2+a_1,\ldots,b_{n_d}+a_{n_d-1},a_{n_d}\}$. Furthermore, it follows from [@maddah2010degrees Theorem 4] that, for almost all CSI $\mathbf{H}$, the minimum distance $d_\text{min}$ satisfies the inequality
[rCl]{} d\_&>&,
for every $\epsilon>0$.
Next, we choose parameters $A$ and $Q$ such that, on the one hand, the average power constraint is satisfied, and, on the other hand, the minimum distance $d_{\text{min}}$ grows with $P$. Let $A=Q^{\frac{n_d-1}{2}+\epsilon'}$ and $Q=\rho(\mathbf{H},n_d)P^{\frac{1}{n_d+1+2\epsilon'}}$, with $\epsilon'>\epsilon$ and with a constant $0<\rho(\mathbf{H},n_d)<1$ that depends only on the CSI and the number of layers. Note that, since $Q$ should be an integer, additional rounding is required, but this does not affect the high-SNR analysis and is therefore omitted. With these choices, we have
[rCl]{} d\_&>&,
and
[rCl]{} (AQ)\^2=\^2=(,n\_d)\^[n\_d+1+2’]{}P,
and hence the role of $\rho(\mathbf{H},n_d)$ is to maintain the average power constraint by accounting for the precoder gains and the number of layers. It follows that, for high SNR, we have the limit
[rCl]{} \_[P]{}=\_[P]{}=.
As was described in Sec. \[sec:real\_IA\], UE 1 transmits the message $v_1=\{a_2+b_1,a_4+b_3,\ldots,a_{n_d-1}+b_{n_d-2}\}$ to UE 2, while UE 2 transmits $v_2=\{b_2+a_1,b_4+a_3,\ldots,b_{n_d-1}+a_{n_d-2}\}$ to UE 1, over the D2D links. These include $(n_d-1)/2$ layers, each taking one of $2Q$ values. As a result, UE 1 is able to decode $\{a_1,b_2,a_3,b_4,\ldots,a_{n_d},b_{n_d}\}$ while UE 2 decodes $\{b_1,a_2,b_3,a_4,\ldots,b_{n_d},a_{n_d}\}$, at the expense of the D2D latency
[rCl]{} T\_D=T\_E.
Since each UE decodes $(n_d+1)/2$ layers from one EN and $(n_d-1)/2$ from the other, the number of channels uses required to satisfy the UEs’ demands is
[rCl]{} T\_E=.
Thus, the resulting NDT is
[rCl]{} (=1/2,r\_F,r\_D)&=&\_[P]{}\_[L]{}\
&=&\_[P]{}\
&=&,
and for sufficiently small $\epsilon'$ and sufficiently large $n_d$, we can get arbitrarily close to $\delta_X$ .
Lower bound on the minimum MDT {#sec:appendix_lb}
==============================
Here we prove the converse result that demonstrates the optimality of the proposed scheme. To this end, without loss of generality, assume that UE 1 and UE 2 request files $f_1$ and $f_2$, respectively. Define $\mathbf{f}_n^N\triangleq \rb{f_n,f_{n+1},\ldots,f_N}$. The lower bounds are obtained by identifying subsets of information resources from which, for high-SNR, both files should be reliably decoded when a feasible policy is implemented. As discussed, we specifically consider in turn the subsets $\cb{s_1,s_2,\mathbf{u}_{1},\mathbf{u}_{2}}$, $\cb{\mathbf{y}_1,\mathbf{y}_2}$ and $\cb{s_1,\mathbf{u}_{1},\mathbf{y}_1,\mathbf{v}_2}$.
Let $\epsilon_L$ be a function of $L$, independent of $P$, such that $\epsilon_L\rightarrow 0$ as $L\rightarrow\infty$. Since the D2D messages $\mathbf{v}_1$ and $\mathbf{v}_2$ are functions of the downlink received signals $\mathbf{y}_1$ and $\mathbf{y}_2$, then we have
[rCl]{} H&=&H\
&&H+H\
&& 2\_L,
where the last step follows from Fano’s inequality since each file $f_k$ can be decoded from $\{\mathbf{y}_k,\mathbf{v}_{k'}\}$, $k,k'\in[2]$, $k'\neq k$, by the definition of feasible policies. Therefore, when considering subsets $\cb{s_1,s_2,\mathbf{u}_{1},\mathbf{u}_{2}}$ and $\cb{\mathbf{y}_1,\mathbf{y}_2}$, we can readily use the derivations of [@tandon2016cloud] to get *Inequality 2* and *Inequality 3* of [@tandon2016cloud Appendix A], that is,
[rCl]{} T\_F(P)-,
and
[rCl]{} T\_E(P)L-L\_L.
For the subset $\cb{s_1,\mathbf{u}_{1},\mathbf{y}_1,\mathbf{v}_2}$, we consider the following equality:
[rCl]{}\[eq:subset3\] 2L&=&H\
&=&I+H.
The first term in can be bounded as
[rCl]{} I &=&I+I\[eq:chain\_rule\]\
&&+I\
&&T\_E+I\[eq:lemma5\]\
&&+I\
&&T\_E+I\
&&+I\
&=&T\_E+I\
&&+I+I\
&&T\_E+L\_L\
&&+I+I\[eq:fano3\],
where follows from the chain rule for mutual information; inequality follows from [@sengupta2016fog Lemma 5] with $\lambda\triangleq\max_{k\in[2]}|h_{k1}+h_{k2}|^2$; and inequality follows from Fano’s inequality since $I\AgivenB{f_1,f_2;f_1}{\mathbf{y}_1,\mathbf{v}_2,\mathbf{f}_3^N}\leq H\AgivenB{f_1}{\mathbf{y}_1,\mathbf{v}_2}$. The mutual information $I\AgivenB{f_1,f_2;s_1,\mathbf{u}_1}{\mathbf{y}_1,\mathbf{v}_2,f_1,\mathbf{f}_3^N}$ in can be bounded as
[rCl]{} I&&H\
&&H\
&=&H\[eq:eq\_def\_s1\]\
&=&H\[eq:eq\_cache\]\
&&H\
&&H+H\
&&L+r\_FT\_F(P),
where follows from the definition of $s_1$; and equality follows from the fact that $s_{1,n}$ is a function of $f_n$ . Finally, the term $I\AgivenB{f_1,f_2;\mathbf{v}_2}{\mathbf{y}_1,\mathbf{f}_3^N}$ in can be bounded as
[rCl]{} I&=&H-H\
&&H\
&&H\
&&r\_DT\_D(P).\[eq:bound\_v\_given\_y\]
Therefore, the first term in is bounded as
[rCl]{}\[eq:I\_subset3\] I &&T\_E+L\_L+r\_DT\_D(P)+L+r\_FT\_F(P).
Next, the second term in can be bounded as
[rCl]{}\[eq:bound\_s\_u\_v\] H &=&H+I\
&&H+I\
&&L\_L+I\[eq:fano2\]\
&=&L\_L+h-h\
&=&L\_L+h-h\[eq:x\_func\_u\_s\]\
&& L\_L+h-h,\[eq:cond\_red\_ent\]
where follows from Fano’s inequality; equality follows because the transmitted signal $\mathbf{x}_1$ is a function of $s_1$ and $\mathbf{u}_1$ as well as from the fact that given all the files $\mathbf{f}_1^N$, the channel outputs $\mathbf{y}_2$ are a function of the channel noises $\mathbf{z}_2$ only; and inequality follows the conditioning reduces entropy property. Since the channel matrix $\mathbf{H}$ is drawn from a continuous distribution, then almost surely the received signals can be related as
[rCl]{} \_2&=&\_1-\_1-\_1+\_2,
and hence $\mathbf{y}_2+\tilde{\mathbf{z}}_2$, where $\tilde{\mathbf{z}}_2\triangleq (h_{22}/h_{12})\mathbf{z}_1-\mathbf{z}_2$, is almost surely given by
[rCl]{} \_2+\_2&=&\_1-\_1,
that is, the signal $\mathbf{y}_2+\tilde{\mathbf{z}}_2$ is a function of $\mathbf{y}_1$ and $\mathbf{x}_1$. Therefore, continuing from , we have
[rCl]{}\[eq:H\_subset3\] H&&L\_L+h-h\
&=&L\_L+h-h\
&&L\_L+h-h\
&=&L\_L+h-h\
&=&L\_L+h-h\
&&L\_L+h-h\
&=&L\_L+T\_E.
Substituting and in leads to the inequality
[rCl]{} 2L&&T\_E+L\_L+r\_DT\_D(P)+L+r\_FT\_F(P)+L\_L+T\_E.
To summarize, we have the following three inequalities:
[rCl]{} (P)&&L-2L\_L-T\_E-T\_E,\[eq:lb\_inequality1\]\
T\_F(P)&&-,\[eq:lb\_inequality2\]\
T\_E(P)&& L-L\_L\[eq:lb\_inequality3\].
For $0\leq r_F,r_D\leq 1$, we use to get the lower bound
[rCl]{} \^\*(,r\_F,r\_D)&=&\_[P]{}\_[L]{}\
&&\_[P]{}\_[L]{}\
&&2-.
Furthermore, by multiplying by $(1-r_F)$ and adding the resulting bound to , we get the lower bound
[rCl]{} \^\*(,r\_F,r\_D)&=&\_[P]{}\_[L]{}\
&&\_[P]{}\_[L]{}\
&=& \_[P]{}\_[L]{}\
&&2-+\
&=&1++.
Thus, for $0\leq r_F,r_D\leq 1$, we have
[rCl]{} \^\*(,r\_F,r\_D)&&\[eq:rD<1\_lb1\].
For $r_F>\max\cb{1,r_D}$, i.e., $r_F>1$ and $0\leq r_D\leq r_F$, by multiplying by $\rb{r_F-1}$ and adding the resulting bound to , we get
[rCl]{} (P)&& L-L\_L\
&&-T\_E-T\_E.\[eq:lb\_calc1\]
Moreover, for $r_D<r_F$, we have
[rCl]{} r\_F(P)&&(P),
which, together with , results in the lower bound
[rCl]{} \^\*(,r\_F,r\_D)&&1+\[eq:rD<1\_lb2\].
Next, for $r_D>\max\cb{1,r_F}$, i.e., $r_D>1$ and $0\leq r_F< r_D$, similar to -, by multiplying by $\rb{r_D-1}$ and adding the resulting bound to , we get the lower bound
[rCl]{} \^\*(,r\_F,r\_D)&&1+.
Moreover, we use $\eqref{eq:lb_inequality1} + \rb{r_D-r_F} \times \eqref{eq:lb_inequality2} + \rb{r_D-1} \times \eqref{eq:lb_inequality3}$ to get the inequality
[rCl]{} r\_D(P)&&L\
&&-L\_L\
&&-T\_Er\_D-T\_E.
Hence, we have an additional lower bound
[rCl]{} \^\*(,r\_F,r\_D)&&1++\
&=&1++.
Overall, for $r_D>\max\cb{1,r_F}$, the lower bound is given by
[rCl]{} \^\*(,r\_F,r\_D)&&,
which completes the proof.
[^1]: R. Karasik and S. Shamai are with the Faculty of Electrical Engineering, Technion - Israel Institute of Technology, Haifa, Israel (e-mails: royk@campus.technion.ac.il, sshlomo@ee.technion.ac.il). O. Simeone is with the Department of Informatics, King’s College of London, London, UK (e-mail: osvaldo.simeone@kcl.ac.uk).
[^2]: This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement Nos. 694630 and 725731).
[^3]: A set of $n$ complex numbers $\{c_1,\ldots,c_n\}\subset\mathbb C^n$ is said to be rationally independent if the only solution to the equation $\sum_{i=1}^{n}p_ic_i=0$ with integer coefficients $\{p_1,\ldots,p_n\}\in\mathbb Z^n$ is the trivial solution $p_i=0$ for all $i\in[n]$.
|
---
abstract: 'We study properties of topological phases by calculating the ground state degeneracy (GSD) of the $2$d Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always non-degenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.'
author:
- Yuting Hu
- 'Spencer D. Stirling'
- 'Yong-Shi Wu'
title: 'Ground State Degeneracy in the Levin-Wen Model for Topological Phases'
---
Introduction
============
In recent years two-dimensional topological phases have received growing attention from the science community. They represent a novel class of quantum matter at zero temperature whose bulk properties are robust against weak interactions and disorders. Topological phases may be divided into two families: *doubled* (with time-reversal symmetry, or TRS, preserved), and *chiral* ( with TRS broken). Either type may be exploited to do fault-tolerant (or topological) quantum computing [@Kitaev; @FKLW; @NSSFD; @Wang].
*Chiral* topological phases were first discovered in integer and fractional quantum Hall (IQH and FQH) liquids. Mathematically, their effective low-energy description is given by Chern-Simons theory [@ZhHK] or (more generally) topological quantum field theory (TQFT) [@Witten89]. One characteristic property of FQH states is ground state degeneracy (GSD), which depends only on the spatial topology of the system [@TW84; @NTW85; @WN90] and is closely related to fractionization [@HKW90; @HKW91; @SKW06] of quasiparticle quantum numbers, including fractional (braiding) statistics [@Wilczek82; @Wu84]. In some cases the GSD has been computed in refs. [@WZ91; @SKW06].
Chern-Simons theories are formulated in the continuum and have no lattice counterpart. Doubled topological phases, on the other hand, do admit a discrete description. The first known example was Kitaev’s toric code model [@Kitaev].
More recently, Levin and Wen (LW) [@LW] constructed a discrete model to describe a large class of doubled phases. Their original motivation was to generate ground states that exhibit the phenomenon of string-net condensation [@Wen03] as a physical mechanism for topological phases. The LW model is defined on a trivalent lattice (or graph) with an exactly soluble Hamiltonian. The ground states in this model can be viewed as the fixed-point states of some renormalization group flow [@CGW; @WEN11]. These fixed-point states look the same at all length scales and have no local degrees of freedom.
The LW model is believed to be a Hamiltonian version of the Turaev-Viro topological quantum field theory (TQFT) in three dimensional spacetime [@Turaev94; @KMR; @Wang] and, in particular cases, discretized version of *doubled* Chern-Simons theory [@FNSWW; @Simon]. Like Kitaev’s toric code model [@Kitaev], we expect that the subspace of degenerate ground states in the LW model can be used as a fault-tolerant code for quantum computation.
In this paper we report the results of a recent study on the GSD of the LW model formulated on a (discretized) closed oriented surface $M$. Usually the GSD is examined as a topological invariant[@Turaev94; @KMR; @Simon] of the 3-manifold $S^1\times M$. In a Hamiltonian approach accessible to physicists, we will explicitly demonstrate that the GSD in the LW model depends only on the topology of $M$ on which the system lives and, therefore, is a topological invariant of the surface $M$. We also show that the ground state of any LW Hamiltonian on a sphere is always non-degenerate. Moreover, we examine the LW model associated with quantum group $SU_k(2)$, which is conjectured to be equivalent to the doubled Chern-Simons theory with gauge group $SU(2)$ at level $k$, and compute the GSD on a torus. Indeed we find an agreement with that in the corresponding doubled Chern-Simons theory [@Witten89; @RT]. This supports the above-mentioned conjectured equivalence between the doubled Chern-Simons theory and the LW model, at least in this particular case.
The paper is organized as follows. In Section II we present the basics of the LW model, easy to read for newcomers. In Section III topological properties of the ground states are studied, and the topological invariance of their degeneracy is shown explicitly. In section IV we demonstrate how to calculate the GSD in a general way. In section V we provide examples for the calculation particularly on a torus. Section VI is devoted to summary and discussions. The detailed computation of the GSD is presented in the appendices.
The Levin-Wen model
===================
Start with a fixed (connected and directed) trivalent graph $\Gamma$ which discretizes a closed oriented surface $M$ (such as a torus). To each edge in the graph we assign a string type $j$, which runs over a finite set $j=0,1,...,N$. Each string type $j$ has a “conjugate” $j^*$ that describes the effect of reversing the edge direction. For example $j$ may be an irreducible representation of a finite group or (more generally) a quantum group [@Kassel].
Let us associate to each string type $j$ a quantum dimension $d_j$, which is a positive number for the Hamiltonian we define later to be hermitian. To each triple of strings $\{i,j,k\}$ we associate a *branching rule* $\delta_{ijk}$ that equals $1$ if the triple is “allowed” to meet at a vertex, $0$ if not (in representation language the tensor product $i\otimes j\otimes k$ either contains the trivial representation or not). This data must satisfy (here $D=\sum_{j}d^2_j$) $$\begin{aligned}
\label{dimcond}
\sum_{k}d_{k}\delta_{ijk^{*}}=d_{i}d_{j}\nonumber\\
\sum_{ij}d_{i}d_{j}\delta_{ijk^{*}}=d_{k}D\end{aligned}$$ $j=0$ is the unique “trivial” string type, satisfying $0^*=0$ and $\delta_{0jj^*}=1, \delta_{0ji^*}=0$ if $i\neq{j}$.
The Hilbert space is spanned by all configurations of all possible string types $j$ on edges. The Hamiltonian is a sum of some mutually-commuting projectors $H:=-\sum_{v}\hat{Q}_v-\sum_p\hat{B}_p$ (one for each vertex $v$ and each plaquette $p$). Here each projector $\hat{Q}_v=\delta_{ijk}$ with $i,j,k$ on the edges incoming to the vertex $v$. $\hat{Q}_v=1$ enforces the branching rule on $v$. Throughout the paper we work on the subspace of states in which $\hat{Q}_v=1$ for all vertices. Each projector $\hat{B}_p$ is a sum $D^{-1}\sum_{s}d_{s}\hat{B}^s_p$ of operators that have matrix elements (on a hexagonal plaquette for example) $$\begin{aligned}
\label{Bps}
&\Biggl\langle{\begin{matrix}}\includegraphics[height=0.6in]{plaq2.eps}{\end{matrix}}\Biggr|
\hat{B}_p^s
\Biggl|{\begin{matrix}}\includegraphics[height=0.6in]{plaq1.eps}{\end{matrix}}\Biggr\rangle\nonumber\\
&=
v_{j_1}v_{j_2}v_{j_3}v_{j_4}v_{j_5}v_{j_6}v_{j'_1}v_{j'_2}v_{j'_3}v_{j'_4}v_{j'_5}v_{j'_6}\\
&G^{{j_7}{j^*_1}{j_6}}_{{s^*}{j'_6}{j'^*_1}}G^{{j_8}{j^*_2}{j_1}}_{{s^*}{j'_1}{j'^*_2}}
G^{{j_9}{j^*_3}{j_2}}_{{s^*}{j'_2}{j'^*_3}}G^{{j_{10}}{j^*_4}{j_3}}_{{s^*}{j'_3}{j'^*_4}}
G^{{j_{11}}{j^*_5}{j_4}}_{{s^*}{j'_4}{j'^*_5}}G^{{j_{12}}{j^*_6}{j_5}}_{{s^*}{j'_5}{j'^*_6}}
\nonumber\end{aligned}$$ Here $v_j=\sqrt{d_j}$ is real. The symmetrized $6j$ symbols[@WEN11] $G$ are complex numbers that satisfy $$\begin{aligned}
\label{6jcond}
&\text{symmetry:}&G^{ijm}_{kln}=G^{mij}_{nk^{*}l^{*}}
=G^{klm^{*}}_{ijn^{*}}=(G^{j^*i^*m^*}_{l^*k^*n})^*\nonumber\\
&\text{pentagon id:}
&\sum_{n}{d_{n}}G^{mlq}_{kp^{*}n}G^{jip}_{mns^{*}}G^{js^{*}n}_{lkr^{*}}
=G^{jip}_{q^{*}kr^{*}}G^{riq^{*}}_{mls^{*}}\nonumber\\
&\text{orthogonality:}
&\sum_{n}{d_{n}}G^{mlq}_{kp^{*}n}G^{l^{*}m^{*}i^{*}}_{pk^{*}n}
=\frac{\delta_{iq}}{d_{i}}\delta_{mlq}\delta_{k^{*}ip}\end{aligned}$$
For example, these conditions are known to be satisfied [@LW] if we take the string types $j$ to be all irreducible representations of a finite group, $d_j$ to be the dimension of corresponding representation space, and $G$ to be the symmetrized Racah $6j$ symbols for the group. In this case the LW model can be mapped [@BA] to Kitaev’s quantum double model [@Kitaev]. More general sets of data $\{G,d,\delta\}$ can be derived from quantum groups (or Hopf algebras) [@Kassel]. We will discuss such a case later using the quantum group $SU_k(2)$ ($k$ being the level).
Ground states
=============
Any ground state $|\Phi\rangle$ (there may be many) must be a simultaneous $+1$ eigenvector for all projectors $\hat{Q}_v$ and $\hat{B}_p$. In this section we demonstrate the topological properties of the ground states on a closed surface with non-trivial topology.
Let us begin with *any two* arbitrary trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ discretizing the same surface (e.g., a torus). If we compare the LW models based on these two graphs, respectively, then immediately we see that the Hilbert spaces are quite different from each other (they have different sizes in general).
However, we may mutate between any two given trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ by a composition of the following elementary moves [@Pachner] (see also Fig \[fig:reducedgraphsidea\] ): $$\begin{aligned}
&f_1. {\begin{matrix}}\includegraphics[height=0.4in]{def1.eps}{\end{matrix}}\Rightarrow
{\begin{matrix}}\includegraphics[height=0.4in]{def1rot.eps}{\end{matrix}}\text{, for any edge;}
\nonumber\\
&f_2. {\begin{matrix}}\includegraphics[height=0.4in]{def2a.eps}{\end{matrix}}\Rightarrow
{\begin{matrix}}\includegraphics[height=0.4in]{def2b.eps}{\end{matrix}}\text{, for any vertex.}
\nonumber\\
&f_3. {\begin{matrix}}\includegraphics[height=0.4in]{def2b.eps}{\end{matrix}}\Rightarrow
{\begin{matrix}}\includegraphics[height=0.4in]{def2a.eps}
{\end{matrix}}\text{, for any triangle structure.}
\nonumber\end{aligned}$$
![Given any two trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ discretizing the same surface, we can always mutate $\Gamma^{(1)}$ to $\Gamma^{(2)}$ by a composition of elementary $f$ moves. In general $\Gamma^{(1)}$ and $\Gamma^{(2)}$ are not required to be regular lattices. These diagrams happen to be the same as [@Gu09], but in a slightly different context.[]{data-label="fig:reducedgraphsidea"}](graph1.eps "fig:"){height="0.75in"} ![Given any two trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ discretizing the same surface, we can always mutate $\Gamma^{(1)}$ to $\Gamma^{(2)}$ by a composition of elementary $f$ moves. In general $\Gamma^{(1)}$ and $\Gamma^{(2)}$ are not required to be regular lattices. These diagrams happen to be the same as [@Gu09], but in a slightly different context.[]{data-label="fig:reducedgraphsidea"}](graph12.eps "fig:"){height="0.75in"} ![Given any two trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ discretizing the same surface, we can always mutate $\Gamma^{(1)}$ to $\Gamma^{(2)}$ by a composition of elementary $f$ moves. In general $\Gamma^{(1)}$ and $\Gamma^{(2)}$ are not required to be regular lattices. These diagrams happen to be the same as [@Gu09], but in a slightly different context.[]{data-label="fig:reducedgraphsidea"}](graph2.eps "fig:"){height="0.75in"} $\Gamma^{(1)}\qquad\qquad\qquad\qquad\Rightarrow\qquad\qquad\qquad
\qquad\Gamma^{(2)}$
Suppose we are given a sequence of elementary $f$ moves that connects two graphs $\Gamma^{(1)}\rightarrow\Gamma^{(2)}$. We now construct a linear transformation $\mathcal{H}^{(1)}\rightarrow\mathcal{H}^{(2)}$ between the two Hilbert spaces. This is defined by associating linear maps to each elementary $f$ move: $$\begin{aligned}
\label{T1T2T3}
&\hat{T}_1:\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr\rangle\rightarrow\sum_{j'_{5}}v_{j_5}v_{j'_5}G^{j_{1}j_{2}j_{5}}_{j_{3}j_{4}j'_{5}}
\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{X2.eps}{\end{matrix}}\Biggr\rangle\nonumber\\
&\hat{T}_2:\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{Y1.eps}{\end{matrix}}\Biggr\rangle\rightarrow\sum_{j_{4}j_{5}j_{6}}\frac{v_{j_4}v_{j_5}v_{j_6}}
{\sqrt{D}}G^{j_{2}j_{3}j_{1}}_{j^*_{6}j_{4}j^*_{5}}
\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{Y2.eps}{\end{matrix}}\Biggr\rangle\nonumber\\
&\hat{T}_3:\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{Y2.eps}
{\end{matrix}}\Biggr\rangle\rightarrow
\frac{v_{j_4}v_{j_5}v_{j_6}}{\sqrt{D}}
G^{j^*_{3}j^*_{2}j^*_{1}}_{j^*_{4}j_{6}j^*_{5}}
\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{Y1.eps}{\end{matrix}}\Biggr\rangle\end{aligned}$$
The mutation transformations between $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ are constructed by a composition of these elementary maps. As a special example, the operator $\hat{B}_p=D^{-1}\sum_{s}d_s\hat{B}_p^s$ is such a transformation. In fact, on the particular triangle plaquette $p$ as in , we have $\hat{B}_{p=\triangledown}=\hat{T}_2\hat{T}_3$, by using the pentagon identity in .
Mutation transformations are unitary on the ground states. To see this, we only need to check that the elementary maps $\hat{T}_1$, $\hat{T}_2$, and $\hat{T}_3$ are unitary. First note that the following relations hold: $\hat{T}_1^\dagger=\hat{T}_1$, $\hat{T}_2^\dagger=\hat{T}_3$, and $\hat{T}_3^\dagger=\hat{T}_2$. We emphasize that these are maps between the Hilbert spaces on two different graphs. For example, we check $\hat{T}_1^\dagger=\hat{T}_1$ by comparing matrix elements $$\begin{aligned}
\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr\vert
\hat{T}_1^\dagger
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{X2.eps}{\end{matrix}}\Biggr\rangle
\equiv
&\Biggl(\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{X2.eps}{\end{matrix}}\Biggr\vert
\hat{T}_1
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr\rangle\Biggr)^*
\nonumber\\
=&v_{j_5}v_{j'_5}\left(G^{j_1 j_2 j_5}_{j_3 j_4 j'_5}\right)^*
\nonumber\\
=&v_{j'_5}v_{j_5}G^{j_4 j_1 j'_5}_{j_2 j_3 j^*_5}
\nonumber\\
=&\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr\vert
\hat{T}_1
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{X2.eps}{\end{matrix}}\Biggr\rangle\end{aligned}$$ where in the third equality we used the symmetry condition in .
Similarly, for $\hat{T}_2^\dagger=\hat{T}_3$ (or $\hat{T}_3^\dagger=\hat{T}_2$), we have $$\begin{aligned}
\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{Y1.eps}{\end{matrix}}\Biggr\vert
\hat{T}_2^\dagger
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{Y2.eps}{\end{matrix}}\Biggr\rangle
\equiv
&\Biggl(\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{Y2.eps}{\end{matrix}}\Biggr\vert
\hat{T}_2
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{Y1.eps}{\end{matrix}}\Biggr\rangle\Biggr)^*
\nonumber\\
=&\frac{v_{j_4}v_{j_5}v_{j_6}}
{\sqrt{D}}\left(G^{j_{2}j_{3}j_{1}}_{j^*_{6}j_{4}j^*_{5}}\right)^*
\nonumber\\
=&\frac{v_{j_4}v_{j_5}v_{j_6}}{\sqrt{D}}G^{j^*_{3}j^*_{2}j^*_{1}}_{j^*_{4}j_{6}j^*_{5}}
\nonumber\\
=&\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{Y1.eps}{\end{matrix}}\Biggr\vert
\hat{T}_3
\Biggl\vert{\begin{matrix}}\includegraphics[height=0.4in]{Y2.eps}{\end{matrix}}\Biggr\rangle\end{aligned}$$ Now we verify unitary. First, $\hat{T}_1^\dagger\hat{T}_1=\text{id}$ and $\hat{T}_2^\dagger\hat{T}_2=\hat{T}_3\hat{T}_2=\text{id}$ by the orthogonality condition in (note that, since we have not used any information about the ground states in this argument, $\hat{T}_1$ and $\hat{T}_2$ are unitary on the entire Hilbert space). For unitary of $\hat{T}_3$ we check $\hat{T}_3^\dagger\hat{T}_3=\hat{T}_2\hat{T}_3=1$. The last equality only holds on the ground states since we have already seen that $\hat{T}_2\hat{T}_3=\hat{B}_{p=\triangledown}$ and $\hat{B}_{p=\triangledown}=1$ only on the ground states.
As another consequence of the above relations, the Hamiltonian is hermitian since all $\hat{B}_p$’s consist of elementary $\hat{T}_1$, $\hat{T}_2$, and $\hat{T}_3$ maps. Particularly, on a triangle plaquette, we have $\hat{B}_{p=\triangledown}^\dagger=(\hat{T}_2\hat{T}_3)^\dagger=
\hat{T}_3^\dagger\hat{T}_2^\dagger=\hat{T}_2\hat{T}_3=\hat{B}_{p=\triangledown}$.
The mutation transformations serve as the symmetry transformations in the ground states. If $\vert\Phi\rangle$ is a ground state then $\hat{T}\vert\Phi\rangle$ is also a ground state, where $\hat{T}$ is a composition of $\hat{T}_i$’s associated with elementary $f$ moves from $\Gamma^{(1)}$ to $\Gamma^{(2)}$. This is equivalent to the condition $\hat{T}(\prod_p \hat{B}_p)=(\prod_{p'}
\hat{B}_{p'})\hat{T}$, which can be verified by the conditions in . (Here $p$ and $p'$ run over the plaquettes on $\Gamma^{(1)}$ and $\Gamma^{(2)}$, respectively. Also note that the $\hat{B}_p$’s are mutually-commuting projectors, i.e., $\hat{B}_p\hat{B}_p=\hat{B}_p$, and thus $\prod_p\hat{B}_p$ is the projector that projects onto the ground states.)
These symmetry transformations look a little different from the usual ones since they may transform between the Hilbert spaces $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ on two different graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$. In general, $\Gamma^{(1)}$ and $\Gamma^{(2)}$ do not have the same number of vertices and edges. And thus $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ have different sizes. However, if we restrict to the ground-state subspaces $\mathcal{H}^{(1)}_{0}$ and $\mathcal{H}^{(2)}_{0}$, mutation transformations are invertible. In fact, they are unitary as we have just shown.
The tensor equations on the $6j$ symbols in give rise to a simple result: each mutation that preserves the spatial topology of the two graphs induces a unitary symmetry transformation. During the mutations, local structures of the graphs are destroyed, while the spatial topology of the graphs is not changed. Correspondingly, the local information of the ground states may be lost, while the topological feature of the ground states is preserved. In fact, any topological feature can be specified by a topological observable $\hat{O}$ that is invariant under all mutation transformations $\hat{T}$ from $\mathcal{H}^{(1)}$ to $\mathcal{H}^{(2)}$: $\hat{O}'\hat{T}=\hat{T}\hat{O}$ (where $\hat{O}$ is defined on the graph $\Gamma^{(1)}$ and $\hat{O}'$ on $\Gamma^{(2)}$).
The symmetry transformations provides a way to characterize the topological phase by a topological observable. In the next section we will investigate the GSD as such an observable.
Let us end this section by remarking on uniqueness of the mutation transformations. There may be many ways to mutate $\Gamma^{(1)}$ to $\Gamma^{(2)}$ using $f_1$, $f_2$ and $f_3$ moves. Each way determines a corresponding transformation between the Hilbert spaces of ground states, $\mathcal{H}^{(1)}_{0}$ and $\mathcal{H}^{(2)}_{0}$. It turns out that all these transformations are actually the same if the initial and final graphs $\Gamma^{(1)}$ to $\Gamma^{(2)}$ are fixed, i.e., independent of which way we choose to mutate the graph $\Gamma^{(1)}$ to $\Gamma^{(2)}$. This means that the ground state Hilbert spaces on different graphs can be identified (up to a mutation transformation) and all graphs are equally good.
One consequence of the uniqueness of the mutation tranformation is that the degrees of freedom in the ground states do not depend on the specific structure of the graph. In this sense, the LW model is the Hamiltonian version of some discrete TQFT (actually, Turaev-Viro type TQFT, see[@KMR]). The fact that the degrees of freedom of the ground states depend only on the topology of the closed surface $M$ is a typical characteristic of topological phases [@TW84; @NTW85; @WN90; @WZ91; @SKW06].
Ground state degeneracy
========================
In this section we investigate the simplest nontrivial topological observable, the GSD. Since $\prod_p\hat{B}_p$ is the projector that projects onto the ground states, taking a trace computes $\text{GSD}=\text{tr}(\prod_{p}\hat{B}_p)$.
We can show that GSD is a topological invariant. Namely, in the previous section we mentioned that, by using , $\prod_{p}\hat{B}_p$ is invariant under any mutation $\hat{T}$ between the Hilbert spaces $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ : $\hat{T}^\dagger(\prod_{p'}\hat{B}_{p'})\hat{T}=\prod_{p}\hat{B}_{p}$. Taking a trace of both sides leads to $\text{tr}^\prime(\prod_{p'}\hat{B}_{p'})=\text{tr}(\prod_{p}\hat{B}_p)$, where the traces are evaluated on $\mathcal{H}^{(2)}$ and $\mathcal{H}^{(1)}$ respectively.
The independence of the GSD on the local structure of the graphs provides a practical algorithm for computing the $\text{GSD}$, since we may always use the simplest graph (see Fig \[fig:reducedgraphs\] and examples in the next section).
Expanding the GSD explicitly in terms of $6j$ symbols using we obtain $$\begin{aligned}
\label{GSD}
&\text{GSD}=\sum_{{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}...}\Biggl\langle{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr|(\prod_{p}\hat{B}_p)\Biggl|{\begin{matrix}}\includegraphics[height=0.4in]{X1.eps}{\end{matrix}}\Biggr\rangle\nonumber\nonumber\\
&=D^{-P}\sum_{{s_1}{s_2}{s_3}{s_4}...}d_{s_1}d_{s_2}d_{s_3}d_{s_4}...\nonumber\\
&\quad\sum_{{j^\prime_1}{j^\prime_2}{j^\prime_3}{j^\prime_4}{j^\prime_5}...}
d_{j^\prime_1}d_{j^\prime_2}d_{j^\prime_3}d_{j^\prime_4}d_{j^\prime_5}...
\sum_{{j_1}{j_2}{j_3}{j_4}{j_5}...}d_{j_1}d_{j_2}d_{j_3}d_{j_4}d_{j_5}...\nonumber\\
&\quad
\left({G}^{{j_2}{j_5}{j_1}}_{{s^*_1}{j^\prime_1}{j^\prime_5}}
G^{{j^\prime_1}{j_2}{j^\prime_5}}_{{s^*_2}{j_5}{j^\prime_2}}
G^{{j_5}{j^\prime_1}{j^\prime_2}}_{{s^*_3}{j_2}{j_1}}\right)
\left(G^{{j_3}{j_4}{j^*_5}}_{{s^*_1}{j^{\prime*}_5}{j^\prime_4}}
G^{{j^\prime_4}{j^{\prime*}_5}{j_3}}_{{s^*_2}{j^\prime_3}{j^*_5}}
G^{{j^*_5}{j^\prime_3}{j^\prime_4}}_{{s^*_4}{j_4}{j_3}}\right)...\end{aligned}$$ The formula needs some explanation. $P$ is the total number of plaquettes of the graph. Each plaquette $p$ contributes a summation over $s_p$ together with a factor of $\frac{d_{s_p}}{D}$. In the picture in the top plaquette is being operated on first by $\hat{B}^{s_1}_{p_1}$, next the bottom plaquette by $\hat{B}^{s_2}_{p_2}$, third the left plaquette by $\hat{B}^{s_3}_{p_3}$, and finally the right plaquette by $\hat{B}^{s_4}_{p_4}$. Although ordering of the $\hat{B}^{s}_{p}$ operators is not important (since all $\hat{B}_p$’s commute with each other), it is important to make an ordering choice (for all plaquettes on the graph) *once and for all*.
Each edge $e$ contributes a summation over $j_e$ and $j^\prime_e$ together with a factor of $d_{j_e}d_{j^\prime_e}$. Each vertex contributes three $6j$ symbols.
The indices on the $6j$ symbols work as follows: since each vertex borders three plaquettes where $\hat{B}^{s}_{p}$’s are being applied, we pick up a $6j$ symbol for each corner. However, ordering is important: because we have an overall ordering of $\hat{B}^{s}_{p}$’s, at each vertex we get an induced ordering for the $6j$ symbols. Starting with the $6j$ symbol furthest left we have no primes on the top row. The bottom two indices pick up primes. All of these variables (primed or not) are fed into the next $6j$ symbol and the same rule applies: the bottom two indices pick up a prime with the convention $()^{\prime\prime}=()$.
(a)![All trivalent graphs can be reduced to their simplest structures by compositions of elementary $f$ moves. (a) on a sphere: 2 vertices, 3 edges, and 3 plaquettes. (b) on a torus: 2 vertices, 3 edges, and 1 plaquette.[]{data-label="fig:reducedgraphs"}](spheretheta.eps "fig:"){height="0.8in"}![All trivalent graphs can be reduced to their simplest structures by compositions of elementary $f$ moves. (a) on a sphere: 2 vertices, 3 edges, and 3 plaquettes. (b) on a torus: 2 vertices, 3 edges, and 1 plaquette.[]{data-label="fig:reducedgraphs"}](torus.eps "fig:"){height="0.8in"}
By the calculation of the GSD, we have characterized a topological property of the phase using local quantities living on a graph discretizing $M$ of nontrivial topology.
Examples
========
[*(1) On a sphere.*]{} To calculate the GSD, we need to input the data $\{G^{ijm}_{kln},d_j,\delta_{ijm}\}$ and evaluate the trace in . We start by computing the GSD in the simplest case of a sphere.
Let’s consider the simplest graph as in Fig. \[fig:reducedgraphs\](a). We show in Appendix A that the ground state is non-degenerate on the sphere without referring to any specific structure in the model: $\text{GSD}^\text{sphere}=1$. In fact, for more general graphs one can write down [@Gu09] the ground state as $\prod_{p}\hat{B}_{p}|0\rangle$ up to a normalization factor, where in $|0{\rangle}$ all edges are labeled by string type 0.
We notice that the GSD on the open disk (which is topologically the same as the $2$d plane) can be studied using the same technique. This is because the open disk can be obtained by puncturing the sphere in Fig \[fig:reducedgraphs\](a) at the bottom. Although this destroys the bottom plaquette, we notice that the constraint $\hat{B}_p=1$ from the bottom plaquette is automatically satisfied as a consequence of the same constraint on all other plaquettes. The fact that $\text{GSD}^{\text{sphere}}(=\text{GSD}^{\text{disk}})=1$ indicates the non-chiral topological order in the LW model.
[*(2) Quantum double model.*]{} When the data are determined by representations of a finite group $G$, the LW model is mapped to Kitaev’s quantum double model[@Kitaev; @BA]. The ground states corresponds one-to-one to the flat $G$-connections[@Kitaev]. The GSD is $$\label{GSDQD}
\text{GSD}_{\text{QD}}=\left|\frac{\text{Hom}(\pi_{1}(M),G)}{G}\right|$$ where $\text{Hom}(\pi_{1}(\mathcal{M}),G)$ is the space of homomorphisms from the fundamental group $\pi_1(M)$ to $G$, and $G$ in the quotient acts on this space by conjugation.
In particular, the GSD on a torus is $$\label{torusGSDQD}
\text{GSD}^{\text{torus}}_{\text{QD}}
=\left|\{(a,b)|a,b\in{G};aba^{-1}b^{-1}=e\}/\sim\right|$$ where $\sim$ in the quotient is the equivalence by conjugation, $$(a,b)\sim (hah^{-1},hbh^{-1}) \quad \text{for all } h\in{G}
\nonumber$$
The number is also the total number of irreducible representations[@DPR] of the quantum double $D(G)$ of the group $G$. On the other hand, the quasiparticles in the model are classified[@Kitaev] by the quantum double $D(G)$. Thus the GSD on a torus is equal to the number of particle species in this example.
[*(3) $SU_k(2)$ structure on a torus.*]{} More generally, on a torus any trivalent graph can be reduced to the simplest one with two vertices and three edges, as in Fig \[fig:reducedgraphs\](b). On this graph the GSD consists of six local $6j$ symbols. $$\begin{aligned}
\label{torusGSD}
&\text{GSD}=D^{-1}\sum_{sj_{1}j_{2}j_{3}j'_{1}j'_{2}j'_{3}}
d_{s}d_{j_{1}}d_{j_{2}}d_{j_{3}}d_{j'_{1}}d_{j'_{2}}d_{j'_{3}}\nonumber\\
&
\left(G^{j_{1}j_{2}j^{*}_{3}}_{{s}j'^{*}_{3}j'_{2}}
G^{{j'^*_3}{j_1}{j'_2}}_{{s}{j_2}{j'_1}}
G^{{j_2}{j'^*_3}{j'_1}}_{{s}{j_1}{j^*_3}}\right)
\left(G^{{j^*_2}{j_3}{j^*_1}}_{{s}{j'^*_1}{j'_3}}
G^{{j'_3}{j'^*_1}{j^*_2}}_{{s}{j'^*_2}{j^*_1}}
G^{{j^*_1}{j'^*_2}{j'_3}}_{{s}{j_3}{j^*_2}}
\right)\end{aligned}$$
Now let us take the example using the quantum group $SU_k(2)$. It is known that $SU_k(2)$ has $k+1$ irreducible representations, and thus the GSD we calculate is finite. We take the string types to be these representations, labeled as ${0,1,...,k}$, and the data $\{G^{ijm}_{kln},d_j,\delta_{ijm}\}$ to be determined by these representations (for more details, see[@RT; @KR; @MV]).
In Appendix B we show that in this case (for the LW model on a torus with string types given by irreps of $SU_k(2)$) we have $\text{GSD}=(k+1)^2$. We argue this both analytically and numerically.
On the other hand, it is widely believed that when the string types in the LW model are irreps from a quantum group at level $k$, then the associated TQFT is given by doubled Chern-Simons theory associated with the corresponding Lie group at level ${\pm}k$[@Witten88; @RT]. This equivalence tells us that in this case the LW model can be viewed as a Hamiltonian realization of the doubled Chern-Simons theory on a lattice, and it provides an explicit picture of how the LW model describes doubled topological phases.
Along these lines, our result is consistent [@WZ] with the result $\text{GSD}_{CS}=k+1$ for Chern-Simons $SU(2)$ theory at level $k$ on a torus. This can be seen since the Hilbert space associated to doubled Chern-Simons should be the tensor product of two copies of Chern-Simons theory at level $\pm{k}$.
Summary and Discussions
=======================
In this paper, we studied the LW model that describes 2d topological phases which do not break time-reversal symmetry. By examining the 2d (trivalent) graphs with same topology which are related to each other by a given finite set of operations (Pachner moves), we developed techniques to deal with topological properties of the ground states. Using them, we have been able to show explicitly that the GSD is determined only by the topology of the surface the system lives on, which is a typical feature of topological phases. We also demonstrated how to obtain the GSD from local data in a general way. We explicitly showed that the ground state of any LW Hamiltonian on a sphere is non-degenerate. Moreover, the LW model associated with quantum group $SU_k(2)$ was studied, and our result for the GSD on a torus is consistent with the conjecture that the LW model associated with quantum group is the realization of a doubled Chern-Simons theory on a lattice or discrete graph.
Finally, let us indicate possible extension of the results to more general cases. First, more generally in the LW model, an extra discrete degree of freedom, labelled by an index $\alpha$, may be put on the vertices. Then the branching rule $\delta_{ijk}^{\alpha}$, when its value is $1$, may carry an extra index $\alpha$. (In representation language this implies that given irreducible representations $i$,$j$ and $k$, there may be multiple inequivalent ways to obtain the trivial representation from the tensor product of $i\otimes j\otimes k$. The index $\alpha$ just labels these different ways.) The $6j$ symbols accordingly carry more indices. (For more details see the first Appendix in the original paper [@LW] of the LW model.) The expression for GSD is expected to be generalizable to these cases. Secondly, the spatial manifold (e.g. a torus) on which the graph is defined may carry non-trivial charge, e.g. labelled by $i\bar{i}$ in the $SU_k(2)$ case. This corresponds to having a so-called fluxon excitation (of type $i\bar{i}$) above the original LW ground states. The lowest states of this subsector in the LW model coincide with the ground states for the Hamiltonian obtained by replacing the plaquette projector $\hat{B}_p = D^{-1} \sum_j d_j \hat{B}_p^j$ with $\hat{B}_p =
D^{-1} \sum_j s_{ij} \hat{B}_p^j$, where $s_{ij}$ is the modular $S$-matrix. (See Appendix B.) The GSD in this case is computable too, but we leave this for a future paper [@HSW2].
YH thanks Department of Physics, Fudan University for warm hospitality he received during a visit in summer 2010. YSW was supported in part by US NSF through grant No. PHY-0756958, No. PHY-1068558 and by FQXi.
$\text{GSD}=1$ on a sphere
==========================
In appendix, we derive $\text{GSD}=1$ on a sphere for a general Levin-Wen model, without referring to any specific structure of the data $\{d,\delta,G\}$. All we will use in the derivation are the general properties in eq. and eq. .
The simplest trivalent graph on a sphere has three plaquettes and three edges, as illustrated in Fig. \[fig:reducedgraphs\](a). Following the standard procedure as in , the GSD is expanded as $$\begin{aligned}
\label{GSDsphere}
&\text{GSD}^{\text{sphere}}
=\sum_{j_1j_2j_3}
\Biggl\langle{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\vert
\hat{B}_{p_2}\hat{B}_{p_3}\hat{B}_{p_1}
\Biggl\vert{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\rangle
\nonumber\\
=&\sum_{j_1j_2j_3}
\Biggl\langle{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\vert
\frac{1}{D}
\sum_{t}d_{t}\hat{B}_{p_2}^t
\nonumber\\
&\quad\qquad
\frac{1}{D}\sum_{s}d_{s}\hat{B}_{p_3}^s
\frac{1}{D}\sum_{r}d_{r}\hat{B}_{p_1}^r
\Biggl\vert{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\rangle
\nonumber\\
=&\sum_{j_1j_2j_3j'_1j'_2j'_3}
\frac{1}{D}\sum_{r}d_r
v_{j_1}v_{j_3}v_{j'_1}v_{j'_3}
G_{r^*{j'_1}^*j'_3}^{j_2^*j_3j_1^*}G_{r^*{j'_3}^*{j'_1}}^{j_2j_1j_3^*}
\nonumber\\
&\qquad\qquad\frac{1}{D}\sum_{s}d_s
v_{j'_1}v_{j_2}v_{j_1}v_{j'_2}
G^{{j'_3}{j'_1}^*j_2^*}_{s^*{j'_2}^*j_1^*}G^{{j'_3}^*j_2{j'_1}}_{s^*j_1{j'_2}}
\nonumber\\
&\qquad\qquad\frac{1}{D}\sum_{t}d_t
v_{j'_2}v_{j'_3}v_{j_2}v_{j_3}
G^{j_1^*{j'_2}^*j'_3}_{t^*j_3j_2^*}G^{j_1{j'_3}^*{j'_2}}_{t^*j_2j_3^*}\end{aligned}$$ where $\hat{B}_{p_1}$ is acting on the top bubble plaquette, $\hat{B}_{p_2}$ on the bottom bubble plaquette, and $\hat{B}_{p_3}$ on the rest plaquette outside the two bubbles.
All $6j$ symbols can be eliminated by using the orthogonality condition in eq. three times, $$\begin{aligned}
&\sum_{r}d_r
G_{r^*{j'_1}^*j'_3}^{j_2^*j_3j_1^*}G_{r^*{j'_3}^*{j'_1}}^{j_2j_1j_3^*}
=\frac{1}{d_{j_2}}\delta_{j'_1j_2{j'_3}^*}\delta_{j_1j_2j_3^*}
\nonumber\\
&\sum_{s}d_s
G^{{j'_3}{j'_1}^*j_2^*}_{s^*{j'_2}^*j_1^*}G^{{j'_3}^*j_2{j'_1}}_{s^*j_1{j'_2}}
=\frac{1}{d_{j'_3}}\delta_{j'_1j_2{j'_3}^*}\delta_{j_1j'_2{j'_3}^*}
\nonumber\\
&\sum_{t}d_t
G^{j_1^*{j'_2}^*j'_3}_{t^*j_3j_2^*}G^{j_1{j'_3}^*{j'_2}}_{t^*j_2j_3}
=\frac{1}{d_{j_1}}\delta_{j_1j_2j_3^*}\delta_{j_1j'_2{j'_3}^*}\end{aligned}$$
and the GSD is a summation in terms of $\{d,\delta\}$: $$\begin{aligned}
\text{GSD}^{\text{sphere}}
=\frac{1}{D^3}\sum_{j_1j_2j_3j'_1j'_2j'_3}d_{j'_1}d_{j'_2}d_{j_3}
\delta_{j_1j_2j_3^*}\delta_{j'_1j_2{j'_3}^*}\delta_{j_1j'_2{j'_3}^*}\end{aligned}$$
Summing over $j'_1$, $j'_2$, and $j_3$ using finally leads to $\text{GSD}^{\text{sphere}}=1$.
GSD on a torus for $SU_k(2)$
============================
Let us consider the example associated with the quantum group $SU_k(2)$ (with the level $k$ an positive integer) and calculate the GSD on a torus.
There are $k+1$ string types, labeled as $j=0,1,2,...,k$. They are the irreducible representations of $SU_k(2)$. The quantum dimensions $d_j$ are required to be positive for all $j$, in order that the Hamiltonian is hermitian. Explicitly, they are $$\begin{aligned}
\label{dD}
& d_j=\frac{\sin{\frac{(j+1)\pi}{k+2}}}{\sin{\frac{\pi}{k+2}}}
\nonumber\\
& D=\sum_{j=0}^{k}{d_j^2}=\frac{k+2}{2\sin^2{\frac{\pi}{k+2}}}\end{aligned}$$
The branching rule is $\delta_{rst}=1$ if $$\begin{aligned}
\label{fusionrule} \Biggl\{
\begin{array}{l}
r+s+t\text{ is even} \\
r+s\geq{t}, s+t\geq{r}, t+r\geq{s} \\
r+s+t\leq{2k}
\end{array}
\Biggr.\end{aligned}$$ and $\delta_{rst}=0$ otherwise. The explicit formula for the $6j$ symbol can be found in[@KR; @MV]. However, we do not need the detailed data of the $6j$ symbol in the following computation of the GSD.
Let us start with formula in , and reorder the $6j$ symbols, $$\begin{aligned}
\label{SU2GSDReOrganized}
\text{GSD}=&D^{-1}\sum_{sj_{1}j_{2}j_{3}j'_{1}j'_{2}j'_{3}} d_{s}
\left( v_{j_{1}}v_{j_3}v_{j'_1}v_{j'_3}
G^{{j^*_2}{j_3}{j^*_1}}_{{s^*}{j'^*_1}{j'_3}}
G^{{j_2}{j'^*_3}{j'_1}}_{{s^*}{j_1}{j^*_3}}\right)
\nonumber\\
&\quad\qquad\left( v_{j'_1}v_{j_2}v_{j_1}v_{j'_2}
G^{{j'_3}{j'^*_1}{j^*_2}}_{{s^*}{j'^*_2}{j^*_1}}
G^{{j'^*_3}{j_1}{j'_2}}_{{s^*}{j_2}{j'_1}}\right)
\nonumber\\
&\quad\qquad\left( v_{j'_2}v_{j'_3}v_{j_2}v_{j_3}
G^{{j^*_1}{j'^*_2}{j'_3}}_{{s^*}{j_3}{j^*_2}}
G^{j_{1}j_{2}j^{*}_{3}}_{{s^*}j'^{*}_{3}j'_{2}}\right)
\nonumber\\
=&D^{-1}\sum_{sj_{1}j_{2}j_{3}j'_{1}j'_{2}j'_{3}} d_{s} \left(
v_{j_{1}}v_{j_3}v_{j'_1}v_{j'_3}
G^{{j^*_2}{j_3}{j^*_1}}_{{s^*}{j'^*_1}{j'_3}}
G^{j_2^*j_1^*j_3}_{sj'_3{j'_1}^*}\right)
\nonumber\\
&\quad\qquad\left( v_{j'_1}v_{j_2}v_{j_1}v_{j'_2}
G^{{j'_3}{j'^*_1}{j^*_2}}_{{s^*}{j'^*_2}{j^*_1}}
G^{j'_3j_2^*{j'_1}^*}_{sj_1^*{j'_2}^*}\right)
\nonumber\\
&\quad\qquad\left( v_{j'_2}v_{j'_3}v_{j_2}v_{j_3}
G^{{j^*_1}{j'^*_2}{j'_3}}_{{s^*}{j_3}{j^*_2}}
G^{j_1^*{j'_3}{j'_2}^*}_{sj_2^*j_3}\right)\end{aligned}$$ where the symmetry condition in was used in the second equality.
Let us compare the formula in with that in . We set $j=j^*$ for all $j$ and drop all stars, since all irreducible representations of $SU_k(2)$ are self-dual. Then we find that the summation has the same form as the trace of $D^{-1}\sum_{s}d_{s}\hat{B}_{p_2}^s\hat{B}_{p_3}^s\hat{B}_{p_1}^s$ on the graph on a sphere as in ,
$$\begin{aligned}
\label{trBBB}
&\text{tr}^{\text{torus}}(\frac{1}{D}\sum_{s}d_s\hat{B}_p^s)
\nonumber\\
=&\sum_{j_1j_2j_3}
\Biggl\langle{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\vert
\frac{1}{D}
\sum_{s}d_{s}\hat{B}_{p_2}^s
\hat{B}_{p_3}^s
\hat{B}_{p_1}^s
\Biggl\vert{\begin{matrix}}{\scalefont{0.6}
\begin{tikzpicture}[scale=0.6]
\draw [<-,>=latex] (2,0) arc (0:180:1 and 0.8);
\draw [->,>=latex] (0,0) arc (180:360:1 and 0.8);
\draw [->,>=latex] (2,0) -- (0,0);
\node at(1,1.) {$j_1$};
\node at(1,-0.6) {$j_2$};
\node at(1,0.2) {$j_3$};
\end{tikzpicture}
}
{\end{matrix}}\Biggr\rangle
\nonumber\\
=&\text{tr}^{\text{sphere}}(\frac{1}{D}\sum_{s}d_s\hat{B}_{p_2}^s
\hat{B}_{p_3}^s\hat{B}_{p_1}^s)\end{aligned}$$
where $\hat{B}_p^s$ is defined on the only plaquette $p$ on the torus (see Fig. \[fig:reducedgraphs\](b)), while $\hat{B}_{p_1}^s\hat{B}_{p_2}^s\hat{B}_{p_3}^s$ is defined on the same graph on a sphere as in (see Fig. \[fig:reducedgraphs\](a)).
The GSD on a torus becomes a trace on a sphere. The latter is easer to deal with since the ground state on a sphere is non-degenerate. The counting of ground states on a torus turns into a problem dealing with excitations on the sphere.
In the following we evaluate the summation in the representation of elementary excitations. let us introduce a new set of operators $\{\hat{n}_p^r\}$ by a transformation, $$\begin{aligned}
\label{transformationnB}
\hat{n}_p^r=\sum_{s}s_{r0}s_{rs}\hat{B}_p^s,
\quad\hat{B}_p^s=\sum_{r}\frac{s_{rs}}{s_{r0}}\hat{n}_p^r\end{aligned}$$ Here $s_{rs}$ is a symmetric matrix (referred to as the modular $S$-matrix for $SU_k(2)$), $$\label{Smatrix}
s_{rs}=\frac{1}{\sqrt{D}}\frac{\sin{\frac{(r+1)(s+1)\pi}{k+2}}}
{\sin{\frac{\pi}{k+2}}}$$ and has the properties $$\begin{aligned}
\label{Sproperty} s_{rs}=s_{sr},{\quad}s_{r0}=d_r/\sqrt{D}
\nonumber\\
\sum_{s}s_{rs}s_{st}=\delta_{rt}
\nonumber\\
\sum_{w}\frac{s_{wr}s_{ws}s_{wt}}{s_{w0}}=\delta_{rst}\end{aligned}$$
Eq. can be viewed as a finite discrete Fourier transformation between $\{\hat{n}_p^r\}$ and $\{\hat{B}_p^s\}$. By properties , we see that $\{\hat{n}_p^r\}$ are mutually orthonormal projectors, and they form a resolution of the identity: $$\hat{n}_p^r\hat{n}_p^s=\delta_{rs}\hat{n}_p^r, \quad
\sum_r{\hat{n}_p^r}=\text{id}$$
In particular, $\hat{n}_p^0=\frac{1}{D}\sum_s{d_s}\hat{B}_p^s$ is the operator $\hat{B}_p$ in the Hamiltonian. The operator $\hat{n}_p^r$ projects onto the states with a quasiparticle (labeled by $r$ type) occupying the plaquette $p$. Expressed as common eigenvectors of $\{\hat{n}_p^r\}$, the elementary excitations are classified by the configuration of these quasiparticles.
Particularly, on the graph on a sphere as in , the Hilbert space has a basis of $\{\left|r_1,r_2,r_3\right\rangle\}$, where only those $r_1$, $r_2$, and $r_3$ that satisfy $\delta_{r_1r_2r_3}=1$ are allowed. Each basis vector $\left|r_1,r_2,r_3\right\rangle$ is an elementary excitation with the quasiparticles labeled by $r_1$, $r_2$, and $r_3$ occupying the plaquettes $p_1$, $p_2$, and $p_3$. The configuration of quasiparticles are globally constrained by $\delta_{r_1r_2r_3}=1$[@HSW2]. Therefore, tracing opertors $\{\hat{n}_p^r\}$ leads to $$\label{nnn}
\text{tr}(\hat{n}_{p_2}^{r_2}\hat{n}_{p_3}^{r_3}
\hat{n}_{p_1}^{r_1})=\delta_{r_2r_3r_1}$$
Applying this rule reduces the summation to $$\begin{aligned}
&\text{tr} (\frac{1}{D}
\sum_{s}d_{s}\hat{B}_{p_2}^s
\hat{B}_{p_3}^s
\hat{B}_{p_1}^s)
\nonumber\\
=&\text{tr}(
\frac{1}{D}\sum_{s}d_s
\sum_{r_1r_2r_3}
\frac{s_{sr_1}s_{sr_2}s_{sr_3}}{s_{r_10}s_{r_20}s_{r_30}}
\hat{n}_{p_2}^{r_2}\hat{n}_{p_3}^{r_3}\hat{n}_{p_1}^{r_1})
\nonumber\\
=&\sum_{r_1r_2r_3}\frac{1}{D}\sum_{s}d_s
\frac{s_{sr_1}s_{sr_2}s_{sr_3}}{s_{r_10}s_{r_20}s_{r_30}}
\delta_{r_1r_2r_3}\end{aligned}$$ Then we substitute , and in and obtain
$$\begin{aligned}
\text{GSD}^{\text{torus}}_{SU_k(2)}
=&\sum_{r_1,r_2,r_3=0}^{k}
\frac{\sin{\frac{\pi}{k+2}}\delta_{r_1+r_2+r_3,2k}}
{\sin{\frac{(r_1+1)\pi}{k+2}}\sin{\frac{(r_2+1)\pi}{k+2}}\sin{\frac{(r_3+1)\pi}{k+2}}}
\nonumber\\
=&\sum_{r=0}^{k}\sum_{s=0}^{r}\frac{\sin{\frac{\pi}{k+2}}}
{\sin{\frac{(r+1)\pi}{k+2}}\sin{\frac{(s+1)\pi}{k+2}}\sin{\frac{(r-s+1)\pi}{k+2}}}
\nonumber\\
=&(k+1)^2 .\end{aligned}$$
(Here we omit a rigorous proof of the last equality.)
We can also verify $\text{GSD}=(k+1)^2$ by a direct numerical computation. We take the approach in [@MV] to construct the numerical data of $6j$ symbols. The construction depends on a parameter, the Kauffman variable $A$ (in the same convention as in [@MV]), which is specialized to roots of unity. We make the following choice: $$\begin{aligned}
\label{Achoice} \Biggl\{
\begin{array}{ll}
A = \exp( \pi i /3)&\text{at }k=1\\
A = \exp( 3 \pi i/8)&\text{at }k=2\\
A = \exp( 3\pi i/5)&\text{at }k=3\\
\end{array}
\Biggr.\end{aligned}$$
By this choice, the quantum dimensions $d_j$ take the values as in , and the $6j$ symbols satisfy the self-consistent conditions in . Using such data of quantum dimensions $d_j$ and $6j$ symbols, We compute the summation at $$\begin{aligned}
\label{NumericalGSD} \Biggl\{
\begin{array}{ll}
\text{GSD} = 4 &\text{at }k=1\\
\text{GSD} = 9 &\text{at }k=2\\
\text{GSD} = 16 &\text{at }k=3\\
\end{array}
\Biggr.\end{aligned}$$ which verifies $\text{GSD}=(k+1)^2$ in the particular cases.
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---
abstract: 'The Formation of metastable molecules (Feshbach resonances) at the collision of two atoms and subsequent stimulated transition to a lower unbound electronic molecular state, with emission of a photon of the laser radiation has been investigated. This can develop, in particular, for $Rb_2$ molecules due to resonance scattering of two $Rb$ atoms. The considered process is a basis for the creation of excimer lasers. Expressions for the cross sections of elastic and inelastic resonance scattering and the intensity of the stimulated emission of the photons have been obtained.'
address: 'Institute for Physical Research, NAS of Armenia, Ashtarak-2, 0203, Armenia'
author:
- 'E. Gazazyan, A. Gazazyan'
title: 'Formation and stimulated photodissociation of metastable molecules with emission of photon at the collision of two atoms in a laser radiation field.'
---
November 2016
Introduction
============
Metastable molecules are formed due to collision of two atoms, when the energy of a bound molecular state in a closed channel is close to the energy of two atoms in the center of mass in an open channel. The weak coupling between the channels leads to the strong mixing of them, and the resulting metastable molecular states, which called the Feshbach resonances [@1], will have a finite lifetime and will decay both in the initial channel and in the other channels. The process of formation if intermediate metastable molecular states can be controlled, in particular, by the external magnetic field by the change of the resonance detuning, when the bound state in a closed channel has the different hyperfine state then the incoming atoms in the open channel. In this case, the difference between the hyperfine states, which is caused by the difference in the Zeeman shifts, can be controlled by the detuning of the Feshbach resonance by means of the magnetic field [@2; @3; @4; @5]. In case where magnetically tunable resonances are absent, an alternative method for control of Feshbach resonance is the optical method \[5\] (optical Feshbach resonance). Optical control of Feshbach resonances by means of quantum interference was proposed in work [@6; @7; @8].
In the theory of resonant collision, in addition to Feshbach method [@1], which has been developed for studies of nuclear reactions and is successfully applied for collision atoms in BEC, there is an alternative Fano approach [@9] exists exploiting the configuration interaction in multielectron atoms. Both approaches assume appearance of resonance phenomena when discrete states are coupled with continuum. Fano technique is usually associated with asymmetry of shapes of resonance lines which is known in atomic physics as “Fano profile”. Similar interference phenomena of asymmetry of resonance line shape are also observed in nuclear reactions [@10]. Fano technique is, however, used not only in atomic physics for, e.g., studies of autoionization and Rydberg states \[9\], resonance ionization of atoms [@11; @12], and laser induced continuum structures (LICS) [@13]. The Fano technique is also used for considerations of resonant collision \[9\] including those of electrons with atoms with formation of negative ions and interference phenomena in the field of laser radiation [@14; @15]. Fano technique is widely used in also other fields of physics. It is, e.g., used for explanation of asymmetry in the absorption impurity ions in crystals, which is caused by for formation of excitonic resonances [@16; @17]. With use of these resonances works [@18; @19], study the phenomenon of storage and reconstruction of quantum information in solids. The asymmetric form of the Fano resonance important in considerations of nanoscale structures of interacting quantum systems [@20; @21].
In the present work we consider collision of atoms with formation of the metastable molecules (Feshbach resonances) and the subsequent stimulated transition to the lower unbound molecular electronic states with emission of photon of the laser radiation. Expressions for the cross sections of the elastic and inelastic resonance scattering and the intensity of the stimulated emission of the photons have been obtained.
Formation of Feshbach Resonance
===============================
Consider elastic and inelastic collision of two atoms with formation of Feshbach resonance (Fig.1). $U$ the interaction which couples electronic states in the open and closed channel. Laser radiation with frequency $\omega$ couples the upper molecular quasi bound state with the lower uncoupled molecular state with interaction $\Omega_E$.
![Diagram of formation of metastable molecules (Feshbach resonances) at the collision of two atoms and subsequent stimulated transition to a lower unbound electronic molecular state, with emission of a photon of the laser radiation. Here $B^*$ is excited atom B.[]{data-label="fig:MolDiag"}](fig1.eps){width="0.6\linewidth"}
The Hamiltonian for the considered process (Fig.1) has the following form:
$$\begin{aligned}
\label{eq:Hamiltonian}
H=E_e|e\rangle\langle e|+\int dE E\bigg(|E\rangle_{11}\langle E|+|E^{-\omega}\rangle_{22}\langle E^{-\omega}|\bigg)+\nonumber \\ \int dE \bigg(U_e|e\rangle _1\langle E|+U^*_E|E\rangle\langle e|\bigg)+ \nonumber \\ \int dE \bigg(\Omega_{E-\omega}(t)|e\rangle _2\langle E-\omega|+\Omega^*_{E-\omega}(t)|E-\omega\rangle_2\langle e|\bigg) \end{aligned}$$
We represent the solution of the Schrödinger equation with the Hamiltonian (\[eq:Hamiltonian\]) as: $$\label{eq:Shrodinger}
|\Psi(t)\rangle = C_e(t)|e\rangle e^{-iE_et}+ \int dE e^{-iEt}(b_{1E}(t)|E\rangle _1+e^{i\omega t}b_{2,E-\omega}(t)|E-\omega\rangle _2)$$ Substituting expression (\[eq:Shrodinger\]) for the wave vector and using that $\Omega(t)=\Omega e^{-i\omega t}$, where $\omega$ is the frequency of laser radiation, into the Shrödinger equation we obtain the following system of differential equations for the coefficients of the expansion (\[eq:Shrodinger\]):
$$\begin{aligned}
i\frac{dC_e(t)}{dt}=\int dE e^{i(E_e-E)t}(b_{1,E}(t)U_E+b_{2,E-\omega}(t)\Omega_{E-\omega}) \nonumber \\
i\frac{db_{1,E}(t)}{dt}=c_e(t)e^{-i(E_e-E)t}U^*_E\\
i\frac{db_{2,E-\omega}(t)}{dt}=c_e(t)e^{-i(E_e-E)t}\Omega^*_{E-\omega}\nonumber\end{aligned}$$
After the Fourier transform for the coefficient in the formula (\[eq:Shrodinger\]) $$\begin{aligned}
\label{eq_4:Coefficient}
C_e(t)=\int d\lambda e^{-i(\lambda-E_e)t}C_E(\lambda) \nonumber \\
b_{1,E}(t)=\int d\lambda e^{-i(\lambda-E)t}b_{1,E}(\lambda) \\
b_{2,E-\omega}(t)=\int d\lambda e^{-i(\lambda-E)t}b_{2,E-\omega}(\lambda)\nonumber \end{aligned}$$ for the Fourier components we obtain for the Fourier components of the expansion coefficient the following system equations:
$$\begin{aligned}
\label{eq:5a}
(\lambda-E_e)C_e(\lambda)=\int dE(U_Eb_{1,E}(\lambda)+\Omega_{E-\omega}b_{2,E-\omega}(\lambda))
\end{aligned}$$
$$\label{eq:5b}
(\lambda -E)b_{1,E}(\lambda)=U^*_EC_e(\lambda)$$
$$\label{eq:5c}
(\lambda -E )b_{2,E-\omega}(\lambda)=\Omega^*_{E-\omega}C_e(\lambda)$$
Now we obtain from the equation (\[eq:5b\]),(\[eq:5c\]) [@10]:
$$\begin{aligned}
\label{eq:8}
b_{1,E}(\lambda)=[\frac{P}{\lambda-E}+Z(\lambda)\delta(\lambda-E)]U^*_EC_e(\lambda)\end{aligned}$$
$$\begin{aligned}
\label{eq:9}
b_{2,E-\omega}(\lambda)=[\frac{P}{\lambda-E}+Z(\lambda)\delta(\lambda-E)]\Omega^*_{E-\omega}C_e(\lambda),\end{aligned}$$
where
$$\label{eq7}
Z(\lambda)=\frac{\lambda-E_e-\Delta(\lambda)}{\Gamma(\lambda)}$$
In expression (\[eq7\]) $\Delta(\lambda)$ and $\Gamma(\lambda)$ are the full resonant shifts and width and $P$ denotes the principle value: $$\begin{aligned}
\Delta(\lambda)=\Delta_F(\lambda)+\Delta_L(\lambda) \nonumber \\
\Delta_F(\label)=P\int dE\frac{|U_E|^2}{\lambda-E} \nonumber \\
\Delta_L(\label)=P\int dE\frac{|\Omega_{E-\omega}|^2}{\lambda-E} \nonumber\\
\Gamma(\lambda)=\Gamma_F(\lambda)+\Gamma_L(\lambda) \nonumber\\
\Gamma_F(\lambda)=2\pi|U_{\lambda}|^2 \nonumber\\
\Gamma_L(\lambda)=2\pi |\Omega_{\lambda-\omega}|^2 \nonumber\end{aligned}$$ By substituting expressions (\[eq:8\]),(\[eq:9\]) into formula (\[eq:Shrodinger\]) from the ortonormalization condition we obtain the following expression for the $C_e(\lambda)$: $$\label{eq8}
C_e(\lambda)=\sqrt{\frac{2\pi}{\Gamma(\lambda)}\frac{1}{z^2(\lambda)+\pi^2}}$$ For the first solution obtain the following expression:
$$\begin{aligned}
\label{eq9}
|\Phi^{(1)}_\lambda (t)\rangle=c_e(\lambda) \bigg[ |e\rangle +\int dE \bigg( \frac{P}{\lambda-E}+z(\lambda)\delta(\lambda-E) \bigg)*\nonumber \\ \bigg(U^*_E|E\rangle _1 +e^{i\omega t}\Omega^*_{E-\omega}|E-\omega \rangle _2 \bigg) \bigg]\end{aligned}$$
For the second ortonormalized solution in the case of $c_e(\lambda)=0$, we have the following expression $$\label{eq10}
|\Phi^{(2)}_\lambda (t)\rangle =\sqrt{\frac{2\pi}{\Gamma (\lambda)}} \bigg(\Omega_{\lambda-\omega}|\lambda\rangle _1 - e^{i\omega t}U_\lambda|\lambda-\omega\rangle _2\bigg)$$ This solutions (\[eq9\]) and (\[eq10\]) are provide the ortonormalization condition for quasienergy functions: $$\label{eq11}
\langle \Phi^{(j')}_{\lambda '}(t)| \Phi^{(j)}_{\lambda}(t)\rangle=\delta_{j',j}\delta(\lambda '-\lambda)$$
Cross sections of elastic and inelastic scattering and intensity of emission radiation.
=======================================================================================
Asymptotic ($r\rightarrow \infty $) expressions for continuous-spectrum wave functions with orbital angular momentum l are known to have the following appearance: $$\begin{aligned}
\label{eq12a}
|E\rangle^{(l)}_1\propto \frac{1}{k_1r_1}\sin\bigg(k_1r_1+\delta_1-\frac{1}{2}\pi l_1\bigg)P_{l_1}(\cos\Theta_1), \hspace{1cm} k_1=k(E) \\
|E-\omega\rangle^{(l)}_2\propto \frac{1}{k_2r_2}\sin\bigg(k_2r_2+\delta_2-\frac{1}{2}\pi l_2\bigg)P_{l_2}(\cos\Theta_2), \hspace{0.3cm} k_2=k(E-\omega)
\end{aligned}$$ with $P_l(\cos\Theta)$ begin the Legendre polynomials. Taking into account that the wave functions of bound states of atoms vanish asymptotically ($|e\rangle=0$) at large distance ($r\rightarrow \infty $) we can write the quasienergy wave functions (\[eq9\])(\[eq10\]) as follows:
$$\begin{aligned}
\label{eq13}
|\Phi^{(1)}_\lambda(t)\rangle=-\frac{\pi c_e(\lambda)}{\sin\eta}\bigg(\frac{U^*_\lambda}{k(\lambda)r_1}\sin\bigg(k(\lambda)r_1+\eta+\delta_1-\frac{l_1\pi}{2}\bigg)P_{l_1}(\cos\Theta_1)+\nonumber\\
e^{i\omega t}\frac{\Omega^*_{\lambda-\omega}}{k(\lambda-\omega)r_2}\sin \bigg(k(\lambda-\omega)r_2+\eta+\delta_2-\frac{l_2\pi}{2}\bigg)P_{l_2}(\cos\Theta_2)\bigg)\end{aligned}$$
$$\begin{aligned}
\label{eq14}
|\Phi^{(1)}_\lambda(t)\rangle=\sqrt{\frac{2\pi }{\Gamma(\lambda)}}\bigg(\frac{\Omega_{\lambda-\omega}}{k(\lambda)r_1}\sin\bigg(k(\lambda)r_1+\delta_1-\frac{l_1\pi}{2}\bigg)P_{l_1}(\cos\Theta_1)-\nonumber \\
e^{i\omega t}\frac{u_\lambda}{k(\lambda-\omega)r_2}\sin \bigg(k(\lambda-\omega)r_2+\delta_2-\frac{l_2\pi}{2}\bigg)P_{l_2}(\cos\Theta_2)\bigg)\end{aligned}$$
where $\delta_l$ is the phase of non-resonant scattering and $\eta$ is the phase caused by resonant Feshbach scattering. $$\label{eq:15}
\tan\eta=-\frac{\pi}{z(\lambda)}$$ We now represent the scattering state vector $|\Phi_\lambda(1\rightarrow1,2)\rangle$ at $r\rightarrow \infty$ as superposition of quasienergy function (\[eq13\]),(\[eq14\]) $$\label{eq16}
|\Phi_\lambda(1\rightarrow1,2)\rangle=\sum_jA_j|\Phi_\lambda^{(j)}(t)\rangle$$ with $$\label{eq17}
\sum_j|A_j|^2=1$$ then, if we require the presents of incoming and outgoing waves in the first, elastic, channel and the absence of incoming wave in the second, inelastic, channel we can write for the expansion coefficients in (\[eq16\]) $A_j$ the following: $$\begin{aligned}
\label{eq18}
A_1=\frac{U_\lambda}{\sqrt{|U_\lambda|^2+|\Omega_{\lambda-\omega}|^2}},\hspace{1cm} A_2=\frac{\Omega^*_{\lambda-\omega}e^{-i\eta}}{\sqrt{|U_\lambda|^2+|\Omega_{\lambda-\omega}|^2}}\end{aligned}$$ From expressions (\[eq18\]) we can obtain for scattering state vector (\[eq16\]): $$\begin{aligned}
\label{eq19}
|\Phi_\lambda(1\rightarrow 1,2)\rangle=\frac{e^{-i(\delta_1+\eta)}}{k(\lambda)r_1}\bigg[\sin(k(\lambda)r_1-\frac{l_1\pi}{2})-\frac{e^{2i\delta_1}}{2i} \bigg(\frac{\Gamma_F(\lambda)}{\Gamma(\lambda)}\bigg(1-e^{2i\eta }\bigg)+ \nonumber \\
+e^{-2i\delta_1}-1\bigg) e^{i(k(\lambda)r_1-\frac{l_1\pi}{2})}\bigg] P_{l_1}(\cos\Theta_1)-e^{i\omega t} \frac{\sin\eta}{k(\lambda-\omega)r_2} \frac{\sqrt{\Gamma_F(\lambda)\Gamma_L(\lambda)}}{\Gamma(\lambda)} \nonumber \\ e^{i\delta_2}e^{-i(k(\lambda-\omega))r_2-\frac{l_2\pi}{2}}P_{l_2}(\cos\Theta_2)\end{aligned}$$ From expression (\[eq19\]) we can obtain the formulas for corresponding cross sections
$$\begin{aligned}
\label{eq20a}
\sigma(1\rightarrow 1)=\frac{4\pi(2l+1)}{k^2(\lambda)}\frac{1}{4}\bigg|\frac{\Gamma_F(\lambda)}{\Gamma(\lambda)}(1-e^{2i\eta})+e^{-2i\delta_l}-1\bigg|^2\\
\sigma(1\rightarrow 2)= \frac{4\pi(2l+1)}{k^2(\lambda-\omega)}\frac{\Gamma_F(\lambda)\Gamma_L(\lambda)}{\Gamma^2(\lambda)}\sin^2\eta
\end{aligned}$$
Let as separate in (\[eq20a\]) for elastic scattering the resonant state and write the potential part of scattering cross sections in the form $$\label{eq:22}
\sigma_{pot}=\frac{\pi(2l+1)}{k^2(\lambda)}4sin^2\delta_l$$ The total cross section of elastic scattering we write as: $$\label{eq23}
\sigma^{el}_{tot}=\sigma_{pot}+\sigma^{el}_{res}$$ where $$\sigma^{el}_{res}=\frac{\pi(2l+1)}{k^2(\lambda)}\frac{\Gamma^2_F(\lambda)}{\Gamma^2(\lambda)}\bigg|1-e^{2i\eta}\bigg|^2$$
From expressions (\[eq7\]) and (\[eq:15\]) for the resonant elastic and inelastic cross sections, we obtain:
$$\begin{aligned}
\label{eq25a}
\sigma^{el}_{res}=\frac{\pi(2l+1)}{4k^2(\lambda)}\frac{\Gamma^2_F(\lambda)}{(\lambda-E_e-\Delta(\lambda))^2+\frac{\Gamma^2(\lambda)}{4}} \\
\sigma^{inel}_{res}=\frac{\pi(2l+1)}{k^2(\lambda)}\frac{\Gamma_F(\lambda)\Gamma_L(\lambda)}{(\lambda+\omega-E_e-\Delta(\lambda))^2+\frac{\Gamma^2(\lambda)}{4}}
\end{aligned}$$
![Cross section for elastic collisions vs detuning ($x=\lambda - E_e-\Delta(\lambda))$[]{data-label="fig:Elastic"}](fig2){width="0.6\linewidth"}
![Cross section for inelastic collisions vs detuning ($x=\lambda +\omega- E_e-\Delta(\lambda))$[]{data-label="fig:Inelastic"}](fig3.eps){width="0.6\linewidth"}
Distribution for the spectral intensity of the emission radiation at the stimulated transition to the lower unbound molecular state has the following form [@22]: $$\label{eq26}
I(\omega)=I_0\frac{\Gamma}{2\pi}\frac{1}{(\omega+\lambda-E_e-\Delta(\lambda))^2+\frac{\Gamma^2}{4}}$$
Where $I_0$ is a full intensity of incident laser radiation field. $$I_0=\int^{\infty}_{-\infty}I(\omega)d\omega$$ The spectral intensity of the stimulated emission of photons coming from unity volume of the gas are obtained: $$\label{eq27}
I(\omega)=N^*I_0\frac{\Gamma}{2\pi}\frac{1}{(\omega+\lambda-E_e-\Delta(\lambda))^2+\frac{\Gamma^2}{4}}$$
![The spectral intensity of the stimulated emission of photons vs detuning ($x=\omega+\lambda - E_e-\Delta(\lambda))$ []{data-label="fig:Inten"}](fig4.eps){width="0.6\linewidth"}
In figures \[fig:Elastic\],\[fig:Inelastic\],\[fig:Inten\], shows the curves corresponding to the formulas (29,30,32), for the values $\Gamma_f=10^4Hz$ and $\Gamma_L=10\Gamma_f$, where $N^*$ is the density of excited atoms in the gase. It is seen from expression (\[eq27\]) at hight concentration of a excited atoms in the gas we have an intense stimulated radiation.
It should be noted that in the case of dense gases had to consider the cooperative effects, and the collision of atoms. These phenomena, we will consider in future studies.
Conclusion
==========
We consider collision of two atoms with formation of the metastable molecules (Feshbach resonance) and the subsequent stimulated transition under the influence of the laser radiation to a lower unbound molecular electronic state with emission of photon of the laser radiation field. This a situation can develop realized, in particular, for $Rb_2$ molecules due to resonance scattering of two $Rb$ atoms in states $5s$ and $5p$ with formation of metastable molecular state $^3\Pi_u$, with subsequent stimulated transition to the lower $1a^3\Sigma^+_u$ unbound molecular electronic state, with emission of photons of the laser radiation. Expressions for the cross sections of the elastic and inelastic resonance scattering and the intensity of the stimulated emission of photons coming from unity volume of the gas are obtained.
The typical atomic gas density is $10^{14}-10^{17}cm^{-3}$. If the density of excited atoms is $0.001\%$, then the intensity of emitted photons during the simulated transition is $10^9-10^{12}$ times larger. As a result, we have a source of high radiation, which serves as an example of an excimer laser. From the expression (32), the peak of the spectral intensity in the graph at $\omega=E_e+\Delta (\lambda)-\lambda$ is $I(\omega)=N^*I_0\frac{2}{\pi\Gamma}$
Acknowledgments
===============
We are very grateful to Professor A.V. Papoyan for fruitful discussion. Work was supported by the Ministry of Education and Science of Armenia (MESA) project 15T-1C066
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|
---
abstract: 'Ontologies provide conceptual abstractions over data, in domains such as the Internet of Things, in a way that sensor data can be harvested and interpreted by people and applications. The Semantic Sensor Network (SSN) ontology is the de-facto standard for semantic representation of sensor observations and metadata, and it is used at the core of the open source platform for the Internet of Things, OpenIoT. In this paper we present a Schema Editor that provides an intuitive web interface for defining new types of sensors, and concrete instances of them, using the SSN ontology as the core model. This editor is fully integrated with the OpenIoT platform for generating virtual sensor descriptions and automating their semantic annotation and registration process.'
author:
- Prem Prakash Jayaraman
- 'Jean-Paul Calbimonte'
- Hoan Nguyen Mau Quoc
bibliography:
- 'rsp.bib'
title: The Schema Editor of OpenIoT for Semantic Sensor Networks
---
Motivation {#sec:motivation}
==========
The Internet of Things (IoT) paradigm is expected to dramatically change the way we produce, transmit and process data. IoT makes it possible for devices, objects, people, and *things*, to observe, collect and send all sorts of data in different domains, ranging from environmental sensing to health monitoring or smart cities. As a result, a large number of highly heterogeneous interconnected objects will contribute to the Web of Data, challenging IoT systems to effectively exploit and make use of this data. One way to deal with this heterogeneity is through semantic models that provide explicit meaning about the data that is represented. Semantic technologies such as OWL and RDF are standards for modeling and defining concepts and relationships in arbitrary domains of use, and constitute a promising solution to help coping with this problem. Based on these well-founded semantic technologies, the OpenIoT open-source platform for IoT (<http://openiot.eu/>) provides a flexible cloud-based architecture that helps manage the life cycle of IoT services and applications. The OpenIoT architecture includes, among others, modules that manage the sensor data acquisition, namely X-GSN, the semantic data provision and querying (Linked Sensor Middleware-Light, namely LSM-Light), as well as front-end tools for data discovery and analytics (e.g. Request Definition, and Request Presentation). The integration of all these modules is possible thanks to the use of the OpenIoT ontology, which is based on the SSN ontology [@compton2012]. However, these core ontology models are not specific to any domain, and therefore require to be extended or complemented with other vocabularies in order to be used in practice.
General purpose ontology editors (e.g. Protégé [@knublauch2004]) are suitable for defining, modifying and customizing ontologies, but they require users to be familiar with ontology modeling and the basics of description logics. Considering that users of IoT platforms are usually not well-versed in ontological engineering, this can represent an overkill for IoT system administrators/users who simply need to add a new sensor or a type of sensor. Moreover, the general purpose editors are not integrated into the workflow of an IoT system (e.g. as OpenIoT) in such a way that sensor descriptions generated are automatically published as Linked Data, and ready to be discovered, queried and re-used. Hence, it is vital to provide simple and intuitive tools that allow IoT users to perform tasks such as add a new sensor or a sensor type intuitively while preserving the ontological foundations of the model. The Sensor Schema Editor of OpenIoT that we present in this paper aims at providing a solution to this problem. In this first evolution of the editor, we provide the means to: (i) define/modify new sensor types, and (ii) create new sensor instances. A novel feature of the Sensor Schema Editor compared to other UI-based ontology editors [@armin2] is that it is a fully functional, implemented prototype completely integrated with the OpenIoT system. The extensions to the ontology generated by the creation of new sensor types are linked dynamically to the OpenIoT ontology using the LSM-Light component. Hence, the extensions to the ontology created are accessible and visible to other system components.
Sensor Schema Editor {#sec:editor}
====================
The Sensor Schema Editor supports the average user in annotating sensors and sensor-related data using the OpenIoT ontology and Linked Data principles. The interface automates the generation of RDF descriptions for sensor node information submitted by the users. As an example, let us consider an IoT deployment where dozens of `WeatherStation` sensors are deployed in a determined geographical area. In OpenIoT [@soldatos2015], all sensors and observations are represented in terms of ontological concepts. For example, Figure \[fig:sensortype\_definition\] depicts a description of a sensor type following the SSN-based OpenIoT ontology. A sensor (e.g. `WeatherStation`) measures air temperature and humidity, and has some pre-defined *accuracy* and *frequency* parameters, typically defined by the vendor specification or configuration. This sensor type constitutes an extension of the ontology for this particular use case. Based on this new type of sensor, we are able to create instances with user provided descriptions that represent deployed sensors of that type. The LSM-Light component will then semantically annotate and publish the sensor type and instance descriptions as Linked Data, making it searchable and discoverable through SPARQL queries. Figure \[fig:sensorinstance\] illustrates an overview of how the sensor instance is annotated and published in Linked Data format based on the new sensor type (e.g. `WeatherStation`) created. The annotation process strictly follows the OpenIoT ontology which is an extension of SSN ontology.
![Description of Sensor Types in the OpenIoT Ontology[]{data-label="fig:sensortype_definition"}](sensorType.png)
![Description of a Sensor Instance in the OpenIoT System[]{data-label="fig:sensorinstance"}](sensor_instance.png)
Sensor Schema Editor Implementation {#sec:editorimplementation}
===================================
The Sensor Schema Editor[^1] has two components: 1) a web-based interface (Sensor Type and Instance Editors) and 2) a back-end server. The web interface is developed in Java using the JSF framework. The back-end is also developed in Java and employs the Restlet framework (http://restlet.org/). The current implementation of the Sensor Schema Editor is capable of generating new sensor types and instances based on the OpenIoT ontology.
Sensor Type Editor
------------------
Figure \[fig:editortype\] presents the Sensor Type Editor, an easy to use intuitive interface allowing novice users to define new sensor types. It supports the following concepts to define a new sensor type.
![OpenIoT Sensor Schema Editor: Sensor Type Interface[]{data-label="fig:editortype"}](Editor1.png)
*Sensor Type Name/id*: A human friendly name for the new sensor type.
*Observed Property:* A property that is observed by the new sensor type. The *observes* relation is used to define the relation between a *sensor* and its *property*. The editor allows a sensor to be associated with multiple observed properties.
*MeasuringCapability:* Collects together measurement properties, in particular the *accuracy* and *frequency*. *Accuracy* is the closeness of agreement between the value of an observation and the true value of the observed quality. *Frequency* is the smallest possible time between one observation and the next.
*Register:* The sensor type editor also provides means to generate the RDF description of the sensor and register it with the OpenIoT LSM-Light service. This allows the sensor type to be discovered, queried and re-used by user communities.
In the example depicted in Figure \[fig:editortype\], we define a sensor type `WeatherStation`. This sensor observes two properties namely `AirTemperature` and `Humidity` (URIs). Each of these properties has an associated measurement capability (*accuracy* and *frequency*) that can be defined by the user depending on the datasheet provided by the sensor manufacturer.
![OpenIoT Sensor Schema Editor: Sensor Instance Interface[]{data-label="fig:editorinstance"}](Editor2.png){width="80.00000%"}
Sensor Instance Editor
----------------------
The OpenIoT sensor instance editor uses the sensor type definition created earlier, to generate a concrete (deployed) sensor instance. The instance is a representation of the actual physical/virtual sensor. Figure \[fig:editorinstance\] provides a screenshot of the sensor instance editor. The instance includes the following information:
*Sensor Name*: the identification of the deployed sensor, e.g. `demo-weatherstation`
*Owner/Description*: Provides sensor description including who owns it.
*Location:* The physical location of the sensor (based on a Map).
*Feature of Interest:* This is used within the OpenIoT ontology to dynamically link the sensor instance to a domain ontology, e.g. `demo-weatherstation` points to the observed feature of interest `crop-growth`.
*Observed Properties:* These are fetched from the sensor type definition. The user specifically can define the unit of measurement (e.g. *Kelvin* or *Celsius* for temperature) and the mapping of the ontology observed property field to the X-GSN component (responsible to stream data from sensors). The mapping allows X-GSN to semantically annotate incoming sensor data streams with the sensor instance and type description.
*Generate Metadata:* This function registers the sensor instance with the LSM-Light component and also provides the user with a metadata configuration file required for the functioning of X-GSN.
Conclusions {#sec:conclusions}
===========
In this paper we have presented a web-based Sensor Schema Editor that assists users defining new types of sensors, thus extending the underlying ontology; and creating instances of them in the form of Linked Data, using the SSN ontology as its core model. This editor is part of the OpenIoT open-source platform for IoT development and deployment, and it bridges the gap between the know-how of IoT system administrators, and the SSN-based ontology model that governs the components of OpenIoT. In the future we plan to include customizing other parameters of the sensor description (e.g. custom measurement capabilities) and adding more complex validation mechanisms that alert the user if the produced schemas may produce conflicts in the ontology model. Furthermore, we plan to allow bulk generation of instances for the cases where large numbers of sensor instances need to be created.
### Acknowledgments {#acknowledgments .unnumbered}
Supported by the SNSF Nano-Tera OpenSense2 project.
[^1]: Available as part of OpenIoT on Github: <https://github.com/OpenIotOrg/openiot>
|
---
abstract: 'ALMA Cycle 2 observations of the long wavelength dust emission in 145 star-forming galaxies are used to probe the evolution of star-forming ISM. We also develop the physical basis and empirical calibration (with 72 low-z and z $\sim$2 galaxies) for using the dust continuum as a quantitative probe of interstellar medium (ISM) masses. The galaxies with highest star formation rates (SFRs) at $\rm <z>$ = 2.2 and 4.4 have gas masses up to 100 times that of the Milky Way and gas mass fractions reaching 50 to 80%, i.e. gas masses 1 - 4 $\times$ their stellar masses. We find a single high-z star formation law: $\rm SFR = 35~ M_{\rm mol}^{0.89} \times (1+z)_{z=2}^{0.95} \times (sSFR)_{MS}^{0.23}$ yr$^{-1}$ – an [**approximately linear dependence on the ISM mass and an increased star formation efficiency per unit gas mass at higher redshift**]{}. Galaxies above the Main Sequence (MS) have larger gas masses but are converting their ISM into stars on a timescale only slightly shorter than those on the MS – thus these ’starbursts’ are largely the result of having greatly increased gas masses rather than and increased efficiency for converting gas to stars. At z $> 1$, the entire population of star-forming galaxies has $\sim$ 2 - 5 times shorter gas depletion times than low-z galaxies. These [**shorter depletion times indicate a different mode of star formation in the early universe**]{} – most likely dynamically driven by compressive, high-dispersion gas motions – a natural consequence of the high gas accretion rates.'
author:
- 'N. Scoville, K. Sheth, H. Aussel, P. Vanden Bout, P. Capak, A. Bongiorno, C. M. Casey, L. Murchikova, J. Koda, J. Álvarez-Márquez, N. Lee, C. Laigle, H. J. McCracken, O. Ilbert, A. Pope, D. Sanders, J. Chu, S. Toft, R.J. Ivison and S. Manohar'
bibliography:
- 'scoville\_dust.bib'
title: |
ISM Masses and the Star Formation Law at z = 1 to 6\
ALMA Observations of Dust Continuum in 145 Galaxies\
in the COSMOS Survey Field
---
Introduction {#intro}
============
For star forming galaxies there exists a Main Sequence (MS) with galaxy star formation rates (SFRs) varying nearly linearly with stellar mass [@noe07]. At z $\sim$ 2 approximately 2% of the star forming galaxies have SFRs more than 4 times higher than the MS, contributing $\sim10$% of the total star formation [@rod11]. These are often identified as the starburst galaxy population [@elb11; @sar12].
The specific star formation rate, (sSFR $\equiv$ SFR/M$_{stellar}$), is roughly constant along the MS at each cosmic epoch but increases 20-fold out to z $\sim 2.5$, consistent with the overall increase in the cosmic star formation rate [@hop06; @kar11; @whi12; @lee15]. Understanding the cause of the MS evolution and the nature of galaxies above the MS is fundamental to understanding the cosmic evolution of star formation.
The interstellar medium (ISM) fuels the activities of both galactic star formation and galactic nuclei – in both cases, peaking at z $\sim$ 2. Is the cosmic evolution of these activities simply due to galaxies having larger ISM masses (M$_{\rm ISM}$) at earlier epochs, or are they forming stars with a higher efficiency ($\epsilon \equiv 1/\tau_{SF}$ = SFR/M$_{\rm ISM}$)? Specifically: 1) is the 20-fold increase in the SFR of the MS from z = 0 to 2 due to proportionally increased gas contents at early epochs or due to a higher frequency of starburst activity? and 2) are galaxies above the MS converting their gas to stars with higher efficiency or do they simply have more gas than those on the MS? Measurements of galactic ISM gas contents are critical to answering these very basic questions.
Over the last decade, the rotational transitions of CO have been used to probe the molecular ISM of high redshift galaxies [@sol05; @cop09; @tac10; @cas11; @bot13; @tac13; @car13]. Here, we employ an alternative approach – using the long wavelength dust continuum to probe ISM masses, specifically, *molecular ISM masses, at high redshift [@sco12; @eal12; @mag12]. This dust emission is optically thin. For high stellar mass star-forming galaxies, the dust continuum can also be detected by ALMA in just a few minutes of observing, whereas for the same galaxies, CO would require an hour or more with ALMA.*
Here, we use a sample of 70 galaxies (28 local star forming galaxies, 12 low redshift ULIRGs and 30 SMGs at z $\sim$ 2) to *empirically calibrate* the ratio of long wavelength dust emission to CO (1-0) line luminosity and, hence, molecular gas mass. Without making any corrections for variable dust temperatures, galaxy metallicity or CO excitation variations, the ratio of long wavelength dust luminosity to CO (1-0) luminosity and molecular gas mass is found to be remarkably constant across this sample, which includes normal star forming galaxies, starburst galaxies at low redshift and massive sub millimeter galaxies (SMGs) at z = 2 - 3 (see Figure \[empir\_cal\]).
It is particularly significant that the local ULIRGs exhibit the same proportionality between the long wavelength dust continuum and the CO (1-0) luminosity. (This argues against a different CO-to-H2 conversion factor in ULIRGs since the physical dependences (on density and $T_D$ or $T_K$) of the mass to dust and CO emission fluxes are different – to be discussed in a future work).
In both the calibration samples and the sample of high redshift galaxies we observed here with ALMA, we have intentionally restricted the samples to objects with high stellar mass ($M_{stellar} = 2\times10^{10} - 4\times10^{11}$); thus we are not probing lower metallicity systems where the dust-to-gas abundance ratio is likely to drop or where there could be significant molecular gas without CO [see @bol13].
The sample of galaxies at high redshift observed with ALMA is described in Section \[sample\]. The stellar mass, SFR, submm flux and estimated gas mass for each individual galaxy are tabulated in the tables in Appendix \[source\_app\]. Average flux measurements for subsamples of galaxies are presented in Section \[obs\] and the derived gas masses and gas mass fractions in Section \[mass\]. The implications for the evolution of ISM and star formation at the peak of cosmic activity are discussed in Section \[discuss\].
Long Wavelength Dust Continuum as a Gas Mass Tracer {#basis}
===================================================
At long wavelengths, the dust emission is optically thin and the observed flux density is proportional to the mass of dust, the dust opacity coefficient and the mean temperature of dust contributing emission at these wavelengths. Here we briefly summarize the foundation for using the dust continuum as a quantitative probe of ISM masses; in Appendix \[dust\_app\] we provide a thorough exposition.
To obviate the need to know explicitly the dust opacity and the dust-to-gas abundance ratio, we empirically calibrate the ratio of the specific luminosity at rest frame 850$\mu$m to ISM molecular gas mass using samples of observed galaxies – thus absorbing the opacity curve, abundance ratio and dust temperature into a single empirical constant $ \alpha_{850\mu \rm m} = L_{\nu_{850\mu \rm m}} / M_{\rm mol} $. This procedure was initially done by [@sco13] with three galaxy samples.
In Appendix \[dust\_app\], we have redone the calibration of the mass determination from the submm-wavelength dust continuum. The sample of calibration galaxies is greatly extended and we use Herschel SPIRE 500$\mu$m imaging instead of SCUBA 850$\mu$m. The SPIRE observations recover more accurately the extended flux components of nearby galaxies than the SCUBA measurements which were used in [@sco13]. (SCUBA observations use beam chopping to remove sky backgrounds and this can compromise the extended flux components.) For the molecular masses, we use CO(1-0) data which is homogeneously calibrated for the local galaxies. The empirical calibration samples now consist of 28 local star-forming galaxies (Table \[tab:local\_gal\]), 12 low-z ULIRGs (Table \[tab:ulirg\]) and 30 z = 1.4 – 3 SMGs (Table \[tab:smg\]).
All three samples exhibit the same linear correlation between CO(1-0) luminosity $L'_{CO}$ and $L_{\nu_{850\mu \rm m}}$ as shown in Figure \[empir\_cal\]-Left. To convert the CO luminosities to molecular gas masses, we then use a single CO-to-H$_2$ conversion constant for all objects (Galactic: $X_{CO} = 3\times10^{20}$ N(H$_2$) cm$^{-2}$ (K km s$^{-1})^{-1}$) and the resultant masses are shown rated to $L_{\nu_{850\mu \rm m}}$ in Figure \[empir\_cal\]-Right. We find a single calibration constant $\alpha_{850\mu \rm m} $= 6.7 $\times10^{19} \rm ~ergs~ sec^{-1} Hz^{-1} {\mbox{$\rm M_{\odot}$}}^{-1}$ (Equation \[alpha\]). \[This value for $\alpha_{850\mu \rm m} $ is in excellent agreement with that obtained from Planck data for the Galaxy (6.2$\times10^{19} \rm ~ergs~ sec^{-1} Hz^{-1} {\mbox{$\rm M_{\odot}$}}^{-1}$, see Section \[pla\]). The earlier value used by [@sco13] was 1$\times10^{20} \rm ~ergs~ sec^{-1} Hz^{-1} {\mbox{$\rm M_{\odot}$}}^{-1}$.\]
The long wavelength dust emissivity index which is needed to translate observations at different rest frame wavelengths is taken to be $\beta = 1.8 \pm 0.1$, based on the extensive Planck data in the Galaxy [@pla11b]. The mass of molecular gas is then derived from the observed flux density using Equation \[mass\_eq\], which gives the expected flux density at observed frequency $\nu_{obs}$ for high-z galaxies. Figure \[alma\_obs\] shows the expected flux for a fiducial ISM mass of 10$^{10}$ as a function of redshift for ALMA Bands 3, 4, 6 and 7. These curves can be used to translate our observed fluxes into ISM masses.
Dust Temperatures
-----------------
Although the submm flux will vary linearly with dust temperature (Equation \[fnu\]), the range of [**mass-weighted**]{} $<T_D>_M$ will be small, except in very localized regions. For radiatively heated dust, $T_D$ will vary as the 1/5 - 1/6 power of the ambient radiation energy density, implying a 30-fold increase in energy density to double the temperature. Extensive surveys of nearby galaxies with Herschel find a range of $T_D \sim 15 - 30$ K [@dun11; @dal12; @aul13]. Our three calibrations yielding similar values of $\alpha_{850\mu \rm m}$ including normal star-forming and starbursting systems lay a solid foundation for using the Rayleigh-Jeans (RJ) dust emission to probe global ISM masses without introducing a variable dust temperature (see Section \[temp\]). Here we advocate adoption of a single value $<T_D>_M = 25$K (see Section \[temp\]).
In fact, it would be wrong to use a variable temperature correction based on fitting to the overall spectral energy distribution (SED) since the temperature thus derived is a luminosity-weighted $<T_D>_L$. The difference is easily understood by looking at nearby star-forming Giant Molecular Clouds (GMCs) where spatially resolved far infrared imaging indicates $<T_D>_L\ \sim 40 - 60$K (dominated by the active star-forming centers) whereas the overall mass-weighted $<T_D>_M\ \simeq 20$K (dominated by the more extended cloud envelopes. A good illustration of this might be taken from spatially resolved far infrared imaging of nearby GMCs. In the Orion, W3, and Auriga Giant Molecular Clouds (GMCs) the far infrared luminosity weighted dust temperature is $\sim50$K and most of that luminosity originates in the few parsec regions associated with high mass SF (e.g. M42 and the Kleinmann-Low nebula in the case of Orion). On the other hand, most of the cloud mass is in the extended GMC of 30 - 40 pc diameter and far infrared color temperature $\sim15 - 25$K [e.g. @mot10; @har13; @riv15].
Within galaxies, there will of course be localized regions where $T_D$ is significantly elevated – an extreme example is the central 100 pc of Arp 220. There, the dust temperatures reach 100 - 200 K [@wil14; @sco15]; nevertheless, measurements of the whole of Arp 220 are still consistent with the canonical value of $ \alpha_{850\mu \rm m}$ adopted here [see Figure 1 in @sco14].
[lccccccccccc]{}\[ht\]
\
**[$\rm {\bf <z>}$ = 1.15]{}\
\
**[lowz cell 1]{} & 6 & 0.11$\pm$ 0.05 & 0.14$\pm$ 0.05 & 2.96 & 1.06 & 0.34 & 22. & 0.79 & 0.66 & 0.30 & 0.16$\pm$0.054\
**[lowz cell 2]{} & 4 & 0.13$\pm$ 0.07 & 0.30$\pm$ 0.08 & 3.96 & 1.15 & 0.60 & 26. & 0.70 & 1.41 & 0.54 & 0.19$\pm$0.048\
**[lowz cell 4]{} & 4 & 0.16$\pm$ 0.08 & 0.73$\pm$ 0.08 & 9.20 & 1.14 & 0.39 & 51. & 1.54 & 3.43 & 0.67 & 0.47$\pm$0.051\
**[lowz cell 5]{} & 8 & 0.92$\pm$ 0.17 & 0.27$\pm$ 0.05 & 4.99 & 1.15 & 0.69 & 67. & 1.72 & 4.34 & 0.65 & 0.39$\pm$0.070\
**[lowz cell 6]{} & 8 & 0.81$\pm$ 0.12 & 0.40$\pm$ 0.04 & 8.98 & 1.21 & 1.75 & 91. & 1.86 & 3.93 & 0.43 & 0.18$\pm$0.027\
**[lowz cell 7]{} & 5 & 0.86$\pm$ 0.16 & 0.39$\pm$ 0.06 & 6.47 & 1.12 & 0.35 & 105. & 3.37 & 4.07 & 0.39 & 0.53$\pm$0.101\
**[lowz cell 8]{} & 12 & 1.17$\pm$ 0.14 & 0.43$\pm$ 0.04 & 11.26 & 1.18 & 0.75 & 154. & 3.74 & 5.61 & 0.37 & 0.43$\pm$0.052\
**[lowz cell 9]{} & 6 & 1.94$\pm$ 0.23 & 0.42$\pm$ 0.06 & 7.47 & 1.23 & 1.54 & 178. & 3.60 & 9.46 & 0.53 & 0.38$\pm$0.046\
**[lowz cell 10]{} & 2 & 0.85$\pm$ 0.34 & 0.79$\pm$ 0.09 & 8.96 & 1.23 & 0.31 & 221. & 6.65 & 4.17 & 0.19 & 0.57$\pm$0.230\
**[lowz cell 11]{} & 2 & 2.07$\pm$ 0.20 & 0.73$\pm$ 0.09 & 8.19 & 1.25 & 0.50 & 390. & 9.61 & 10.06 & 0.26 & 0.67$\pm$0.063\
**[lowz cell 12]{} & 1 & 2.11$\pm$ 0.50 & 1.46$\pm$ 0.15 & 9.64 & 1.20 & 2.06 & 300. & 6.15 & 10.19 & 0.34 & 0.33$\pm$0.079\
\
**[$\rm {\bf <z>}$ = 2.2]{}\
\
**[midz cell 1]{} & 6 & 0.15$\pm$ 0.05 & 0.19$\pm$ 0.06 & 3.07 & 2.20 & 0.28 & 64. & 0.67 & 1.05 & 0.16 & 0.27$\pm$0.088\
**[midz cell 2]{} & 3 & 0.27$\pm$ 0.08 & 0.26$\pm$ 0.05 & 5.41 & 2.73 & 0.71 & 115. & 0.71 & 2.26 & 0.20 & 0.24$\pm$0.074\
**[midz cell 3]{} & 2 & 0.39$\pm$ 0.05 & 0.31$\pm$ 0.05 & 5.67 & 2.66 & 1.21 & 117. & 0.64 & 3.78 & 0.32 & 0.24$\pm$0.031\
**[midz cell 4]{} & 6 & 0.38$\pm$ 0.02 & 0.40$\pm$ 0.04 & 9.30 & 2.20 & 0.28 & 173. & 1.82 & 2.80 & 0.16 & 0.50$\pm$0.054\
**[midz cell 5]{} & 11 & 0.75$\pm$ 0.11 & 0.69$\pm$ 0.05 & 15.24 & 2.24 & 0.68 & 191. & 1.48 & 4.09 & 0.21 & 0.37$\pm$0.056\
**[midz cell 6]{} & 9 & 1.05$\pm$ 0.06 & 1.22$\pm$ 0.05 & 26.79 & 2.25 & 2.01 & 266. & 1.75 & 6.70 & 0.25 & 0.25$\pm$0.009\
**[midz cell 7]{} & 1 & 1.66$\pm$ 0.55 & 0.90$\pm$ 0.15 & 6.01 & 2.34 & 0.31 & 585. & 5.19 & 9.17 & 0.16 & 0.75$\pm$0.246\
**[midz cell 8]{} & 11 & 0.87$\pm$ 0.03 & 0.94$\pm$ 0.04 & 26.12 & 2.43 & 0.74 & 608. & 4.02 & 5.90 & 0.10 & 0.44$\pm$0.017\
**[midz cell 9]{} & 12 & 2.56$\pm$ 0.04 & 3.06$\pm$ 0.04 & 70.57 & 2.30 & 1.82 & 559. & 3.56 & 16.83 & 0.30 & 0.48$\pm$0.007\
**[midz cell 12]{} & 3 & 1.79$\pm$ 0.14 & 2.60$\pm$ 0.08 & 34.13 & 1.94 & 1.60 & 885. & 7.56 & 13.99 & 0.16 & 0.47$\pm$0.014\
\
**[$\rm {\bf <z>}$ = 4.4]{}\
\
**[highz cell 1]{} & 3 & 0.11$\pm$ 0.05 & 0.15$\pm$ 0.04 & 3.99 & 4.71 & 0.27 & 70. & 0.57 & 1.84 & 0.26 & 0.40$\pm$0.100\
**[highz cell 2]{} & 2 & 0.08$\pm$ 0.04 & 0.12$\pm$ 0.04 & 2.75 & 4.23 & 0.51 & 97. & 0.63 & 1.50 & 0.15 & 0.23$\pm$0.082\
**[highz cell 3]{} & 1 & 0.55$\pm$ 0.11 & 0.20$\pm$ 0.06 & 3.54 & 4.04 & 1.03 & 137. & 0.75 & 6.90 & 0.50 & 0.40$\pm$0.078\
**[highz cell 4]{} & 5 & 0.12$\pm$ 0.06 & 0.10$\pm$ 0.03 & 3.72 & 4.63 & 0.22 & 211. & 1.90 & 1.52 & 0.07 & 0.40$\pm$0.191\
**[highz cell 5]{} & 1 & 0.13$\pm$ 0.06 & 0.16$\pm$ 0.06 & 2.59 & 5.59 & 0.45 & 352. & 2.35 & 2.00 & 0.06 & 0.31$\pm$0.119\
**[highz cell 6]{} & 2 & 1.17$\pm$ 0.26 & 0.67$\pm$ 0.04 & 15.59 & 4.52 & 2.21 & 321. & 1.68 & 14.33 & 0.45 & 0.39$\pm$0.088\
**[highz cell 7]{} & 3 & 0.27$\pm$ 0.11 & 0.15$\pm$ 0.04 & 4.13 & 4.25 & 0.19 & 328. & 3.44 & 3.40 & 0.10 & 0.64$\pm$0.262\
**[highz cell 8]{} & 1 & 2.24$\pm$ 0.40 & 1.33$\pm$ 0.06 & 23.27 & 3.54 & 0.50 & 788. & 5.10 & 28.95 & 0.37 & 0.85$\pm$0.152\
**[highz cell 9]{} & 3 & 1.37$\pm$ 0.08 & 0.94$\pm$ 0.04 & 23.98 & 4.03 & 2.24 & 696. & 3.54 & 17.16 & 0.25 & 0.43$\pm$0.025\
**[highz cell 10]{} & 1 & 0.60$\pm$ 0.14 & 0.24$\pm$ 0.07 & 3.60 & 4.18 & 0.14 & 807. & 9.40 & 7.45 & 0.09 & 0.84$\pm$0.201\
**[highz cell 11]{} & 1 & 3.84$\pm$ 0.35 & 2.49$\pm$ 0.06 & 39.14 & 4.64 & 0.84 & 1114. & 6.35 & 47.03 & 0.42 & 0.85$\pm$0.077\
\
\[stacks\]**********************************************************************
Galaxy Samples for ALMA {#sample}
=======================
Our sample of 145 galaxies is taken from the COSMOS 2 deg$^2$ survey [@sco_ove]. This survey field has excellent photometric redshifts [@ilb13; @lai15] derived from deep 34 band (UV-Mid IR) photometry. The galaxies were selected to sample stellar masses $M_{stellar}$ in the range $0.2- 4\times10^{11}$ and the range of SFRs at each stellar mass. This is not a representative sampling of the galaxy population but rather meant to cover the range of galaxy properties. Here, 55% of the galaxies are within a factor 2.5 of the SFR on the MS, whereas for the overall population of SF galaxies at z $\sim$ 2, there is a much larger fraction. Forty-eight have spectroscopic redshifts and 120 have at least a single band detection in the infrared with Spitzer MIPS-24$\mu$m or Herschel PACS and SPIRE; 65 had two or more band detections with Herschel PACS/SPIRE.
The photometric redshifts and stellar masses of the galaxies are from [@mcc12; @ilb13; @lai15]. Preference was given to the most recent photometric redshift catalog [@lai15] which makes use of deep Spitzer SPASH IRAC imaging [@ste14] and the latest release of COSMOS UltraVista. The SFRs assume a Chabrier stellar initial mass function (IMF); they are derived from the rest frame UV continuum and infrared using Herschel PACS and SPIRE data as detailed in [@ilb13] and in [@lee15] . For sources with detections in at least at least two of the five available Herschel bands, $L_{IR}$ is estimated by fitting far-infrared photometry to a coupled, modified greybody plus mid-infrared power law, as in [@cas12]. The mid-infrared power-law slope and dust emissivity are fixed at $\alpha$ = 2.0 and $\beta$ = 1.5, respectively.
The original sample of galaxies observed with ALMA had 180 objects. However, subsequent to the ALMA observations, new photometric redshifts [@lai15] and analysis of the Hershcel PACs and SPIRE measurements in COSMOS [@lee15] became available. We have made use of those new ancillary data to refine the sample, including only those objects with most reliable redshifts, stellar masses and SFRs (as judged from the photometric redshift fitting uncertainties). We also required that the derived stellar masses agree within a factor 2 between the two most recent photometric redshift catalogs [@ilb13; @lai15]. The individual objects are tabulated in Appendix \[source\_app\]. There, the adopted redshifts, stellar masses and SFRs for each of the individual galaxies are tabulated in Tables \[lowz\] - \[highz\].
For each galaxy, we also list the specific star formation rate relative to that of the MS at the same redshift and stellar mass (sSFR$_{MS}$ / sSFR$_{MS}$). In recent years, there have been numerous works specifying the MS evolution [@noe07; @rod11; @bet12; @spe14; @lee15; @sch15]. The last two works have similar specification of the MS as a function of stellar mass at low redshift. Here, we use [@lee15] with no evolution of the MS beyond z = 2.5 (i.e. MS(z $>$ 2.5) = MS(z = 2.5). The [@lee15] MS was adopted here since the infrared-based SFRs were also taken from [@lee15]; thus the SFRs will have the same calibration. The three sub-samples with 59, 63 and 23 galaxies at $\rm < z > \sim$ 1.15, 2.2 and 4.4, respectively, probe SFRs from the MS up to $10\times sSFR_{MS}$ with $\rm M_{stellar} = 0.2 - 4\times10^{11} {\mbox{$\rm M_{\odot}$}}$ (Figure \[sample\_fig\] and Table \[stacks\]).
Observations and Flux Measurements {#obs}
==================================
The ALMA Cycle 2 observations (\#2013.1.00034.S) were obtained in 2014-2015. The z = 1.15 & 2.2 samples were observed in Band 7 (345 GHz), the z = 4.4 sample in Band 6 (240 GHz). On-source integration times were $\sim2$ minutes per galaxy and average rms sensitivities were 0.152 (Band 7) and 0.065 mJy beam$^{-1}$ (Band 6). Synthesized beam sizes were $\simeq0.6 - 1$. Data were calibrated and imaged with natural weighting using CASA.
The detection rates are summarized in Figure \[detection\_rates\] as a function of flux (Left Panel) and ISM mass (Right panel) and in Figure \[detection\_rates1\] as a function of M$_{stellar}$ and sSFR. The detection rates for individual galaxies are $\sim$70, 85 and 50% at z = 1.15, 2.2 and 4.4 respectively. All flux measures are restricted to within 1.5 on the galaxy position. For detections, we searched for significant peaks or aperture-integrated flux within the central 3 surrounding each program source. We required a 2$\sigma$ detection in S$\rm_{tot}$ or 3.6$\sigma$ in S$\rm_{peak}$, in order that the detection be classified as real. These limits ensure that there would be less than one spurious detection in the145 objects. Noise estimates for the integrated flux measures were derived from the dispersion in the aperture-integrated fluxes for 100 equivalent apertures, displaced off-source in the same image.
Stacked Samples
===============
Here we focus on results derived from stacking the images of subsamples of galaxies in cells of M$_{stellar}$ and sSFR (Figure \[detection\_rates1\]). The galaxy images of all galaxies in each cell were both median- and average-stacked. Given the small numbers of galaxies in many of the sub-samples (see Figure \[detection\_rates1\]), we used the average stack rather than the median; for such small samples the median can have higher dispersion. Flux and mass measurements for the stacked subsamples of galaxies are given in Table \[stacks\] along with the mean sSFR and M$_{\rm stellar}$ of each cell.
Gas Masses
----------
Figure \[stack\_results\] shows derived mean gas masses and gas mass fractions of each cell for the three redshifts. The values for $M\rm_{\rm mol}$ and the gas mass fraction (M$\rm_{\rm mol}$ / (M$\rm_{\rm mol}$ + M$\rm_{stellar}$)) are given by the large numbers in each cell and the statistical significance is given by the smaller number in the upper right of each cell.
Figure \[stack\_results\]-Top shows a large increase in the ISM masses from z = 1.15 to z = 2.2 for galaxies with stellar mass $\geq10^{11}$ and for galaxies with sSFR/sSFR$_{MS} \geq 4$ (i.e. galaxies in the upper right of the diagrams). For lower mass galaxies and galaxies at or below the MS, less evolutionary change is seen, although the MS is itself evolving upwards in sSFR. From z = 2.2 to 4.4, there is milder evolution since approximately equal numbers of cells have higher and lower M$_{\rm mol}$ at z = 4.4 compared with z = 2.2 and the differences don’t appear strongly correlated with sSFR and M$\rm_{stellar}$.
Gas Mass Fractions
------------------
The gas mass fractions shown in Figure \[stack\_results\]-Bottom range from $\sim0.2 - 0.5$ on the MS (bottom two rows of cells) up to 0.5 - 0.8 for the highest sSFR cells.
For perspective, we note that the Milky Way galaxy has a stellar mass $\simeq6\times10^{10}$[@mcm11], M$_{\rm ISM} \sim3 - 6\times10^9$ (approximately equally contributed by HI and H$_2$) and SFR $\sim1 - 2$ yr$^{-1}$. Thus for the Galaxy, sSFR $\simeq 0.015 - 0.030$ Gyr$^{-1}$ and gas mass fraction is $\sim$0.055. Lastly, to place the Milky Way in context at low z, the main sequence parameters given by [@bet12] and [@lee15] yield SFR = 4.2 and 3.8 yr$^{-1}$ and sSFR = 0.07 and 0.063 Gyr$^{-1}$ for the Milky Way’s stellar mass; thus, the Galaxy has a SFR $\sim$2 times below the z = 0 MS but is still classified as a MS galaxy.
Compared to the Galaxy, the MS galaxies at z $= 1 - 6$ therefore have $\sim5 - 10$ times higher gas mass fractions for the same stellar mass and $\sim100$ times higher gas masses in the highest stellar mass galaxies. At low redshift, such massive galaxies ($M_{stellar} = 4\times10^{11}$ ) would have largely evolved to become passive (non-star forming) red galaxies with much lower ISM masses.
The trends in gas masses, SFRs and gas mass fractions can be represented adequately by quite simple analytic functions. Using the IDL implementation of the Levenberg-Marquardt algorithm for non-linear least squares fitting (LMFIT) of the data shown in Figure \[stack\_results\] and Table \[stacks\], we obtain: $$\begin{aligned}
{\rm M_{\rm mol} \over {M_{\rm mol} + M_{stellar}}} &=& ~(0.30\pm 0.02)~\left({\rm M_{\rm stellar} \over 10^{11}{\mbox{$\rm M_{\odot}$}}}\right)^{-0.02 \pm 0.02} ~ \nonumber \\
&& \times \left({\rm 1+z \over 3}\right)^{0.44 \pm 0.05} \rm \left({\rm sSFR \over sSFR_{MS}}\right)^{0.32 \pm 0.02} . \label{gas-fraction_law}
\end{aligned}$$
; the parameter uncertainties in Equation \[gas-fraction\_law\] are those obtained from the Levenberg-Marquardt algorithm. We also attempted fitting the gas mass fractions with a sSFR / sSFR$_{MS}$ term; this did not improve the fit and we therefore kept the simpler, un-normalized SFR term.
The gas mass fractions derived here are quite consistent with values derived in a number of other studies from observations of CO (mostly 2-1 and 3-2 line). [@tac10] obtained a range of 0.2 - 0.5 at z $\sim 1.1$ and 0.3 -0.8 at z $\sim 2.3$. [@dad10] estimated a gas mass fraction $\sim 0.6$ for 6 galaxies at z = 1.5 and [@mag12a] measured 0.36 in a Lyman break galaxy at z = 3.2. Later, more extensive studies were done by: [@tac13] with 52 galaxies and mean gas fractions of 0.33 and 0.47 at z $\sim 1.2$ and 2.2, respectively; [@san14]
Several cells in the sSFR-M$_{stellar}$ plane have gas mass fractions 50 - 80%, implying gas masses 1 – 4 times the stellar masses. These galaxies have the highest sSFRs at z = 2.2 and 4.4. Clearly, [**such galaxies can not be made from the merging of two main sequence galaxies**]{} having gas mass fractions $\sim 40$%, since in a merger the resultant gas mass fraction would remain constant or even decrease (if there is significant conversion of gas to stars in a starburst).
These gas-dominated galaxies with very high sSFR galaxies must therefore indicate a different aspect of galaxy evolution – perhaps either [**nascent galaxies**]{} – having M$_{\rm mol} \rm > M_{stellar}$ (yet clearly having prior star formation given their large stellar masses and the presence of metal enriched ISM), or galaxies in environments yielding very high IGM accretion rates. These galaxies share the gas-rich properties of the submillimeter galaxies, yet the ones seen here were selected first in the optical-NIR without pre-selection for dust emission. The gas masses of these systems reach $4\times10^{11}$ – they are very likely the progenitors of the present epoch massive elliptical galaxies [@tof14].
Star Formation Law at High Redshift
-----------------------------------
Using measurements from the stacking in cells of sSFR and M$_{stellar}$ (Table \[stacks\]), we obtained a least-squares fit for the SFR dependence on gas mass, redshift and elevation above the main sequence:
$$\begin{aligned}
\rm SFR &=& (35\pm 16) ~ \left({\rm M_{\rm mol} \over 10^{10}~{\mbox{$\rm M_{\odot}$}}}\right)^{0.89 \pm 0.12} ~\times \nonumber \\
&& \left({\rm1+z\over3}\right)^{0.95\pm0.28} \left({\rm sSFR\over sSFR_{MS}}\right)^{0.23\pm0.15} {\mbox{$\rm M_{\odot}$}}\rm yr^{-1} . \label{sfr_law}
\end{aligned}$$
; the parameter uncertainties were obtained from the Levenberg-Marquardt procedure.
Equation \[sfr\_law\] indicates a high redshift SF law with an approximately linear dependence on gas mass and an increasing SF efficiency (SFR per unit gas mass) at higher redshift. The dependence on sSFR relative to that on the MS is relatively weak (0.23 power; see also Section \[mass\]). No significant dependence (less than 1 $\sigma$) on stellar mass was found so we omitted the stellar mass term from the fitting for Equation \[sfr\_law\]. Inversion of Equation \[sfr\_law\] yields an expression for the gas mass in terms of the observed SFRs.
For low redshift galaxies, [@ler13] derive a very similar power law index (0.95 - 1) for the SF as a function of molecular gas mass but with a lower SF rate per unit ISM mass; this result was also seen in earlier surveys of nearby galaxies in CO [see @you91].
In Figure \[sfr\_stacks\], we compare the SF law given in Equation \[sfr\_law\] with the stacking results of [@bet15] and [@alv15]. Their 1.1 mm stacked fluxes were translated into gas masses using the procedure developed here in Appendix \[dust\_app\]. [@bet15] had samples of galaxies at z $>$ 1 both on the MS and at higher SFRs; the [@alv15] sample has Lyman Break galaxies at z $\sim$ 3, presumably mostly MS galaxies. The 1.1 mm measurements on which the mass estimates in Figure \[sfr\_stacks\] are based used very large beams and therefore had many sources within the beam; this confusion was removed statistically [see @bet15; @alv15]. (We do not use their SPIRE 500$\mu$m stacked fluxes since for most of these redshifts $\lambda_{rest}$ will be less than 250$\mu$m and thus not safely on the RJ tail.) Figure \[sfr\_stacks\] indicates reasonable agreement between the ALMA results presented here and the 1.1 mm stacking results; both show an approximately linear dependence of the SFRs on the estimated gas masses.
Gas Depletion Times
-------------------
The existence of a [**single ‘linear’ relation between the available gas mass and the SFR**]{} for [**all**]{} our galaxies (independent of redshift at z $>$ 1, both on and above the MS) is a result with fundamental implications. The characteristic ISM depletion time $\rm \tau = M_{\rm mol} / SFR \simeq 2 - 7\times10^8$ yrs (Figure \[depletion\_time\]) is approximately constant for galaxies both on and above the MS. These are broadly consistent which previously determined typical values. [@tac13] found a mean gas depletion time of $\sim 7\times10^8$ yrs for a sample of 53 galaxies with CO at z = 1 to 2.5 and [@san14] found $\sim 1 - 3\times10^8$ yrs for a large sample using dust continuum measurements from Herschel (see their Figure 7).
The gas depletion time determined here does show evolution with redshift, having shorter timescales at z = 2.2 and shorter still at z = 4.4 compared to z = 1.1. This timescale is short compared with the $\sim2$ Gyr time differences between z = 4.4 and 2.2 and z = 2.2 to 1.15, implying that there must be substantial accretion of fresh gas to replace that being absorbed into stars.
For the nearby galaxies, the gas depletion times are $\sim1.5$ Gyr [@you91; @you95; @ler13]. The shorter depletion times at high redshift and the universality of these short timescales imply that star formation in the early universe is driven by very different processes than those in present day galaxies having low star formation efficiencies. This different [**star formation mode, dominant in high redshift gas-rich galaxies, is quite plausibly the same dynamically driven SF occurring in low-z galactic spiral arms, bars and merging systems**]{}. However, at high redshift the higher SF efficiencies occur throughout the SF galaxy population. At high redshift, dispersive gas motions (as opposed to ordered rotation) and/or galaxy interactions will lead to compression in the highly dissipative ISM, enhancing the SFRs per unit gas mass. Clearly, such motions will be damped on a galaxy crossing timescale ($\sim2\times10^8$ yr), but this is also the timescale that is implied for replenishment of the high-z gas masses in order to maintain the observed SFRs. The accreted gas from the intergalactic medium or galaxy merging should have typical infall/free-fall velocities of a few$\times100$ ; i.e. sufficient to maintain the dispersive ISM velocities which then can drive the higher SFRs, elevated relative to z = 0 quiescent galaxies.
On versus above the MS {#mass}
======================
In Table \[summary\_table\], we include average properties and gas masses for samples of galaxies on and above the MS for the three redshift ranges. At all three redshifts, the rate of SF per unit gas mass is similar for the MS galaxies and for galaxies with sSFR greater than 2.5 times above the MS (last column in Table \[summary\_table\]). In the samples above the MS, the sSFRs are typically 3 - 4 times above the MS samples, yet the gas depletion times are less than a factor 2 different. And for the complete sample at all redshifts (last three rows in Table \[summary\_table\]) the depletion times differ by only $\sim$ 20%.
Comparing the gas masses and depletion times on and above the MS in Table \[summary\_table\], it is clear that the higher SFRs for galaxies with elevated sSFRs are mostly due to those galaxies having much higher gas masses, rather than an increased efficiency, or rate of converting gas to stars. This conclusion is substantiated by fitting function results (Equation \[sfr\_law\]), giving a power-law dependence of 0.89 on the gas mass and only 0.23 on the sSFR relative to the MS. Thus, the so-called starburst population is largely a population of very gas-rich galaxies rather than galaxies converting gas to stars more efficiently. In fact, [@mag12] similarly find that the vertical spread of the MS band is due to variations in the gas mass fraction rather than variations in the SF efficiency (see their Section 6.3). On the other hand, [@sil15] reach a different conclusion with CO (2-1) observations for seven z = 1.6 galaxies having sSFR approximately 4 times above the MS. They attribute their lower CO to far infrared luminosity ratios to a higher star formation efficiency relative to galaxies on the MS. However, the offset shown in their Figure 3a is really very small – less than a 30% departure from the CO/FIR ratios occurring on the MS.
[ccccccccc]{}
$< z > = 1.1$\
MS & 19 & 1.16 & 0.92 & 64.20 & 1.48 & 6.3$\pm$0.8 & 0.42$\pm$0.04 & 1.09$\pm$0.13\
above MS & 25 & 1.20 & 0.81 & 169.15 & 4.49 & 10.6$\pm$0.9 & 0.55$\pm$0.04 & 0.65$\pm$0.07\
all & 44 & 1.19 & 0.88 & 123.69 & 3.15 & 9.0$\pm$0.7 & 0.50$\pm$0.03 & 0.84$\pm$0.07\
\
$< z > = 2.2$\
MS & 29 & 2.24 & 0.99 & 181.86 & 1.46 & 10.8$\pm$1.3 & 0.52$\pm$0.03 & 0.61$\pm$0.05\
above MS & 26 & 2.28 & 1.25 & 571.21 & 4.24 & 29.3$\pm$3.3 & 0.67$\pm$0.02 & 0.51$\pm$0.05\
all & 55 & 2.27 & 1.15 & 367.32 & 2.75 & 19.5$\pm$2.2 & 0.59$\pm$0.02 & 0.56$\pm$0.04\
\
$< z > = 4.4$\
MS & 6 & 4.28 & 0.35 & 117.36 & 1.19 & 4.3$\pm$0.6 & 0.58$\pm$0.05 & 0.42$\pm$0.04\
above MS & 9 & 4.07 & 0.66 & 583.19 & 4.59 & 13.4$\pm$2.4 & 0.68$\pm$0.06 & 0.24$\pm$0.03\
all & 15 & 4.20 & 0.60 & 399.54 & 3.55 & 10.6$\pm$2.2 & 0.64$\pm$0.04 & 0.31$\pm$0.04\
\
**[ all z]{}\
MS & 54 & 2.07 & 0.94 & 138.43 & 1.46 & 8.8$\pm$0.8 & 0.49$\pm$0.02 & 0.76$\pm$0.05\
above MS & 60 & 2.10 & 1.06 & 422.55 & 4.30 & 18.9$\pm$2.3 & 0.62$\pm$0.02 & 0.53$\pm$0.04\
all & 114 & 2.10 & 1.02 & 288.46 & 2.98 & 14.2$\pm$1.3 & 0.56$\pm$0.02 & 0.64$\pm$0.04\
\
\
\[summary\_table\]**
Summary and Comments {#discuss}
====================
We have provided a thorough physical and empirical foundation for the use of submm flux measurements as a probe of the interstellar medium gas mass in galaxies (Appendix \[dust\_app\]). We find that a single empirical scaling exists between the specific luminosity of the RJ dust continuum ($L_{\nu_{850\mu m}}$) and the mass as determined from CO(1-0) measurements over 3 orders of magnitude in $L_{\nu_{850\mu m}}$ (see Figure \[empir\_cal\]).[^1]
The ALMA Band 6 & 7 observations with typically only a few minutes of integration detect a majority (79%) of the 145 galaxies in our sample at z = 1 - 6 (see Figures \[detection\_rates\] and \[detection\_rates1\]). Thus, with ALMA, this technique immediately enables surveys of large numbers of objects. Using the RJ dust continuum, one also avoids the uncertainties of CO excitation variations which enter when translating higher rotational line measurements into equivalent CO(1-0) line luminosities and hence molecular gas masses.
The appropriate temperature characterizing the RJ dust emission is a mass-weighted $T_d$ and, from basic understanding of the dust emission, it is clear that a luminosity-weighted $T_D$ determined from SED fitting in each individual source should not be used in this technique – rather it is more appropriate to simply adopt a constant value with $T_D \simeq 25$ K as done here. This statement applies to global measurements of the dust continuum, not instances where small regions of a galaxy are resolved and have locally enhanced mass-weighted $T_D$ [e.g. the compact nuclei in Arp220, @sco15]. The galaxies used in our empirical calibration and those observed by us using ALMA are all fairly massive ($M_{stellar} > 2\times10^{10}$ ) so we are not exploring low metallicity systems where the calibration may depart from constancy due to variations in the dust-to-gas abundance ratio or the dust properties.
The results presented here suggest that:
- At high redshift, the primary difference between galaxies with sSFR above the MS and those on the MS is simply increased gas contents of the former, not higher efficiency for conversion of gas to stars.
However,
- the shorter ($\sim5\times$) gas depletion times at high redshift of [**all star forming galaxies, both on and above the MS**]{}, imply a more efficient mode for star formation from existing gas supplies. This is naturally a result of highly dispersive gas motions (due to prodigious on-going accretion needed to replenish gas contents and to galaxy interactions) for all high redshift galaxies – those on and above the MS.
Our result of a single SF law at high redshift is very different from some prior studies. [@dad10] and [@gen10] obtain different SF laws for normal SF galaxies and starburst/SMG galaxies; however, in both cases, their ISM masses at high redshift are derived from higher-J CO transitions and they use different high-J to J = 0 line ratios and CO-to-H$_2$ conversion factors for the two classes of galaxies [see also @sar14]. [@gen15] compared CO and dust continuum results in order to constrain the variations in the conversion factor between the MS and starburst population, obtaining general agreement with their earlier results. That their SF laws differ for normal and starburst/SMG galaxies is a result of their use of different CO-to-H$_2$ conversion factors, which we argue is inappropriate for global ISM measures (Appendix \[dust\_app\]). The technique developed by us here avoids the additional uncertainty introduced when observing higher J CO transitions at high redshift and global variations in the mass-weighted $T_D$ are likely to be small.
We thank Zara Scoville for proof reading the manuscript and Sue Madden for suggesting use of the SPIRE fluxes for calibration. This paper makes use of the following ALMA data: ADS/JAO.ALMA\# 2013.1.00034.S. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2013.1.00111.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. RJI acknowledges support from ERC in the form of the Advanced Investigator Programme, 321302, COSMICISM.
Long Wavelength Dust Continuum as an ISM Mass Tracer {#dust_app}
====================================================
Here, we summarize the physical and empirical basis for using long wavelength dust emission as a probe of ISM mass. The empirical calibration is obtained from: 1) a sample of 30 local star forming galaxies; 2) 12 low-z Ultraluminous Infrared Galaxies (ULIRGs); and 3) 30 z $\sim$ 2 submm galaxies (SMGs). We have completely redone the analysis of these three galaxy samples. We use Herschel SPIRE 350 and 500$\mu$m data which provide more reliable total submm fluxes for the extended objects at low redshift than were available from the SCUBA 850$\mu$m observations used in [@sco14].
The major differences, compared to the empirical calibration presented in [@sco14], are that here we empirically calibrate the submm fluxes relative to the molecular gas masses rather than HI plus H$_2$ (since HI is not well measured in many ULIRGs and not at all in the z = 2 SMGs). We also go back to the original sources for the CO (1-0) luminosities and use a single CO to H$_2$ conversion factor for all objects; we expand the samples when we find additional global CO and SPIRE flux measurements and we remove a couple objects where we find errors from SINGs/KINGFISH surveys [@dra07b]. All of these calibrations yield a very similar rest frame 850$\mu$m luminosity per unit molecular gas mass with small dispersions. The mean value is also within 10% of the value obtained by Planck for Milky Way molecular gas.
Rayleigh-Jeans Dust Continuum – Analytics
-----------------------------------------
The far infrared-submm emission from galaxies is dominated by dust re-emission of the luminosity from stars and active galactic nuclei (AGN). The luminosity at the peak of the FIR is often used to estimate the luminosity of obscured star formation or AGN. Equally important (but not often stressed) is the fact that the long-wavelength RJ tail of dust emission is nearly always optically thin, thus providing a direct probe of the total dust and, hence, the ISM mass – provided the dust emissivity per unit mass and the dust-to-gas abundance ratio can be constrained. Here, we take the approach of [**empirically calibrating the appropriate combination of these quantities rather than requiring determination of each one independently.**]{}
The observed flux density from a source at luminosity distance d$_L$ is $$\begin{aligned}
S_{\nu_{obs}} &=& { (1 - e^{- \tau_{d}(\nu_{rest})}) ~B_{\nu_{rest} }(T_d) (1+z) \over{ d_L^2}}
\end{aligned}$$
where B$_{\nu_{rest}}$ is the Planck function in the rest frame and $\tau_{d}(\nu)$ is the source optical depth at the emitted frequency. (The factor 1 + z accounts for the compression of frequency space in the observer’s frame if the source is at significant redshift.) The source optical depth is given by $\tau_{d}(\nu) = \kappa(\nu) \times 1.36 N_H m_H = \kappa(\nu) \times M_{gas}$, where $\kappa(\nu)$ is the absorption coefficient of the dust per unit [*total mass of gas*]{} (i.e. the effective area per unit mass of gas), N$_H$ is the column density of H nuclei and the factor 1.36 accounts for the mass contribution of heavier atoms (mostly He at 8% by number). Often the dust opacity coefficient is specified per unit mass of dust. However, here we are empirically calibrating the dust opacity relative to the ISM molecular gas mass, so it is convenient to use the above definition, avoiding a separate specification of the dust opacity coefficient per mass of dust and the dust-to-gas ratio.
At long wavelengths where the dust is optically thin, the flux density is then $$S_{\nu_{obs}} = {M_{mol} \kappa(\nu_{rest}) B_{\nu_{rest} } (1+z) \over{ d_L^2}} .
\label{fnu_1}$$
Because $\kappa$ is per unit gas mass, the gas-to-dust ratio is absorbed in $\kappa$ and it therefore does not appear explicitly in Equation A2. Written with a Rayleigh-Jeans $\nu ^2$ dependence, appropriate at long wavelengths, Equation A2 becomes $$S_{\nu_{obs}} = {M_{mol} \kappa(\nu_{rest}) 2kT_{\rm d} (\nu_{rest}/c)^2 {\it{\Gamma}_{RJ}(T_d,\nu_{obs},z) (1+z)}\over{d_L^2}}
\label{fnu}$$
where $\Gamma_{\rm RJ}$ is the correction for departure in the rest frame of the Planck function from Rayleigh-Jeans (i.e. $B_{\nu_{rest}} / RJ_{\nu_{rest}}$). $\Gamma_{\rm RJ}$ is given by $$\begin{aligned}
\it{\Gamma}_{\rm RJ}(T_d,\nu_{obs}, z) &=& {h \nu_{obs} (1+z) / k T_d \over{e^{h \nu_{obs} (1+z) / k T_d} -1 }} ~.
\end{aligned}$$
Equation \[fnu\] can be rewritten for the specific luminosity ($L_{\nu_{rest}}$) in the rest frame of the galaxy, $$\begin{aligned}
L_{\nu_{rest}} &=& S_{\nu_{obs}} 4\pi d_L^2 / (1+z) \nonumber \\
&=& \kappa(\nu_{rest}) 8\pi kT_{d} (\nu_{rest}/c)^2 \it{\Gamma}_{RJ} M_{mol} ~.\end{aligned}$$
The long wavelength dust opacity can be approximated by a power-law in wavelength: $$\kappa (\nu) = \kappa (\nu_{850\mu \rm m}) (\lambda/850\mu \rm m)^{-\beta} .$$\[kappa\]We adopt $\lambda = 850\mu$m ($\nu = 353$ GHz) as the fiducial wavelength since it corresponds to most of the high z SCUBA observations and is the optimum for ALMA (i.e. Band 7). We will use a spectral index $\beta \simeq 1.8$ (see Section \[beta\]).
The rest frame luminosity-to-mass ratio at the fiducial wavelength is given by $$\begin{aligned}
{L_{\nu_{850\mu m}} \over {M_{mol}}} &=& \kappa (\nu_{850\mu m}) {{8\pi k {\nu^2}}\over{c^2}} T_{ d} \it{\Gamma_{RJ}} ~~~~~\rm and ~ we ~ define~ \nonumber \\
\alpha_{\nu_{850\mu m}} &\equiv& { L_{\nu_{850\mu m}} \over{ M_{mol}}} = {{8\pi k {\nu^2}}\over{c^2}} \kappa (\nu_{850\mu m}) T_{d} \it{\Gamma_{RJ}} ~.\end{aligned}$$ In Section \[empirical\] we show that this luminosity-to-mass ratio ($ \alpha_{\nu_{850\mu m}}$) is relatively constant under a wide range of conditions in normal star-forming and starburst galaxies and at both low and high redshift. Then, once this constant is empirically calibrated, we use measurements of the RJ flux density and, hence the luminosity to estimate gas masses. **We note that this result is equivalent to a constant molecular gas mass to dust mass ratio over a wide range of redshifts for high stellar mass galaxies.**
Mass-weighted T$_d$ {#temp}
-------------------
It is important to recognize that the dust temperature relevant to the RJ emission tail is a [**mass-weighted**]{} $<T_d>_{M}$. This is definitely not the same as the luminosity-weighted $<T_d>_{L}$ which might be derived by fitting the IR SED (determined largely by the wavelength of peak IR emission). The former is a linear weighting with $T_d$; the latter is weighted as $\sim T^{5-6}$ depending on $\beta$. In dust clouds with temperature gradients, these temperatures are likely to differ by a factor of a few (depending on the optical depths and mass distributions).
In local star-forming galaxies, the mass-weighted $<T_d>_{M} \sim 15 -- 35$K, and even in the most vigorous starbursts like Arp 220, the mass-weighted dust temperature is probably less than 45 K if one considers the entire galaxy; in contrast, the luminosity-weighted $<T_d>_{M} \sim 50 - 200$ K for Arp 220, depending on the size of the region. It is incorrect then to do an SED fit and use the derived temperature for estimation of the masses.
In fact, variations in the effective dust temperature are probably small on galactic scales since theoretically one expects that the mass-weighted $<T_d>_M$ should depend on the $\sim 1/6$’th power of mean radiation energy density. The observed submm fluxes therefore directly probe the total mass of dust and depend only linearly on $<T_d>_{M}$ which varies very little. [@mag14] investigated the variations in $T_D$ derived from SED fits for stacks of galaxies at z = 1 to 2.3 from the MS to a factor 10 above the MS. The temperatures were found to increase from $\sim25$ to 33 K going to sSFR 10 times above the MS. Once again, we emphasize that those $T_D$ are luminosity-weighted, not mass-weighted, but even so, they do not indicate very large variations. [@gen15] have also advocated the use of a variable $T_D$ to reduce apparent scatter in the relationship between high J CO lines and the dust continuum; however, much of this scatter is likely due to CO excitation variations which enter from use of higher J transitions, so we do not follow this route.
In practice, it will be difficult or impossible to determine $<T_d>_{M}$ in most sources since the observed SEDs are not of sufficient accuracy to measure the small secondary peak due to the cold dust on the RJ tail of the SED. Moreover, this peak is unlikely to be discrete since there will be a range of temperatures in the cold component. In the Galaxy, the Planck data show $<T_d>_{M} = 15 - 22$ K [@pla11a]. Recognizing that most of the galaxies with higher SF rates at high redshift are likely to have slightly elevated dust temperatures, we adopt $\rm <T_d>_M = 25~\rm K$ for numerical estimates when necessary and might reasonably expect a range of 20 - 35 K. Since the mass estimates vary as $T_d^{-1}$, this range implies less than 25 - 30% variation associated with the expected range of [**global**]{} mass-weighted dust temperatures.
Figure \[rj\] shows the $\Gamma_{RJ}$ correction factor for dust temperatures of 25 and 35 K. \[$\Gamma_0$ is the value of $\Gamma$ appropriate to the z = 0, $T_d=25$ K and $\lambda= 850\mu \rm m$ used to calibrate $\alpha_{850\mu \rm m}$; $\Gamma_0 = 0.71$ (see Fig. \[rj\]).\]
The Dust Submm Spectral Index – $\beta$ {#beta}
---------------------------------------
In order to relate submm flux measurements of galaxies at different redshifts (i.e. different rest frame wavelengths), to an empirically constrained mass-light ratio at rest frame 850$\mu$m, one needs to know the spectral index of the RJ dust emission. The overall spectral slope of the rest frame submm dust emission flux density (Equation \[fnu\]) is observed to vary as $S_{\nu} \propto \nu^{\alpha}$ with $\alpha = 3 - 4$. Two powers of $\nu$ are from the RJ dependence; the remainder is due to the frequency variation in $\kappa(\nu) \propto \nu^{\beta}$. Most theoretical models for the dust have opacity spectral indices of $\beta$ = 1.5 – 2 [@dra11]. Empirical fits to the observed long wavelength SEDs suggest $\beta$ = 1.5 – 2 (Dunne & Eales 2001, Clements, Dunne & Eales 2009) for local galaxies. Probably the best determination at high redshift is that of [@cha09] who used their $\lambda =1.1$ survey to find $< \beta > = 1.75$ for 29 SMGs with a median z = 2.7.
Planck has provided a robust Galactic determination of $\beta$ using 7 submm bands (at $\lambda$ = 3 mm to 100 $\mu$m). For the Taurus cloud complex, $\beta = 1.78 \pm 0.08$ for both atomic and molecular ISM regions [@pla11a]. And for the Galaxy, [@pla11b] finds $\beta = 1.8\pm 0.1$ with no significant difference between the HI and H$_2$-dominant regions. We therefore adopt ${\beta = 1.8}$ when needed in the analysis below.
Empirical Calibration from Local Galaxies, Low-z ULIRGs and z = 2 SMGs {#empirical}
----------------------------------------------------------------------
In order to empirically calibrate the 850$\mu$m dust opacity per unit gas mass we make use of galaxy samples for which both the submm dust emission and molecular gas masses are well-determined globally for the whole galaxy. Our local sample includes 28 star forming galaxies from the Herschel KINGFISH survey [@dal12] and 12 ULIRGs from the Herschel VNGS and GOALS surveys [@coo12 Chu in prep.]. These have full imaging in the SPIRE 350 and 500$\mu$m bands.[^2]
For estimation of the associated gas masses we have used exclusively CO (1-0) data for which there are also global CO luminosity measures [@you95; @san91; @sol97; @san89] with consistent single-dish calibrations. Although some of these galaxies have mm-interferometric imaging, those data often resolve out larger spatial components and therefore often recover less than 50% of the single dish line fluxes. For the high redshift SMGs, there are no HI measurements so we restrict our empirical calibration entirely to molecular gas masses.
At high redshift, we make use of a sample of 30 SMGs for which there exist good SNR measurements of CO (1-0) from JVLA. For this sample we use SCUBA 850$\mu$m fluxes since the longer wavelength (compared to SPIRE 500$\mu$m) is needed to stay on the RJ tail and the sources are also quite compact. Most of these objects are at z $< 2.5$. Based on the high submm fluxes, it is clear that many of these SMGs are strongly lensed. This means that the *[apparent]{} $L_{\nu_{850\mu \rm m}}$ and M$_{mol}$ are large over-estimates of their true values. However, it is reasonable to assume that the magnifications are similar for the dust and the gas emission since they both arise in cold ISM. Thus, the $L_{\nu_{850\mu \rm m}}/M_{mol}$ will provide a consistency check (at high SNR) as to whether the relevant combination of the dust-to-gas mass ratio, the dust opacity function and the mass-weighted $<T_d>_M$ are similar to that in the low z calibrators.*
Our restriction to calibration samples with good CO (1-0) line measurements is very important. Only the CO(1-0) line luminosities have been well correlated with virial masses from large Galactic samples of self-gravitating GMCs [@sco87; @sol87]. The higher CO transitions have excitation-dependent flux ratios relative to the 1-0 emission luminosities both in Galactic GMCs [@san93] and in high z galaxies [@car13 see Figure 4]. For high redshift galaxies, this necessarily restricts calibration samples to those observed at high signal-to-noise ratio with JVLA or possibly GBT.
The observed SPIRE 500$\mu$m (local galaxies and ULIRGs: Tables \[tab:local\_gal\] and \[tab:ulirg\]) and SCUBA 850$\mu$m (SMGs: Table \[tab:smg\]) fluxes were converted to $850\mu$m specific luminosity $L_{\nu(850\mu \rm m)}$ of the assumed 25K dust using $$\begin{aligned}
L_{\nu(850\mu \rm m)} & =& 1.19\times10^{27}~S_{\nu}[{\rm Jy}]~{\left(\nu(850\mu \rm m) \over{\nu_{obs} (1+z)}\right)^{3.8}} ~~{(\it d_{L}[\rm Mpc])^2 \over {1+z}} ~~
{\it \Gamma_{RJ}(25,\nu_{850\mu m},0) \over{\it \Gamma_{RJ}(25,\nu_{obs},z)}}~\rm [ergs~ sec^{-1} Hz^{-1}] .
\label{lnu_eqn}\end{aligned}$$ (The term with ratios of $\Gamma_{RJ}$ is necessary since we wish to estimate $L_{\nu(850\mu \rm m)}$ which would be associated with 25K dust producing the same $\rm L_{\nu}$ at $\nu=\nu_{obs}\times(1+z)$ in the observed galaxy rest frame.)
Figure \[empir\_cal\]-Left shows the CO(1-0) luminosities for the three samples of galaxies plotted as a function of their specific continuum luminosities at $\lambda = 850\mu$m. All three sets of these very diverse galaxies (normal star-forming, ultraluminous starbursts and high redshift starbursts) fall on the same 1:1 line and this provides the empirical basis for using the RJ continuum as a tracer of ISM molecular gas mass. We note that many of these SMGs are likely lensed, but the high magnifications allow extension of the calibration to high redshift with excellent signal-to-noise ratios. The dust continuum and CO(1-0) emission experiences the same lensing magnifications since the SMGs lie on the same linear correlation as the unmagnified objects at low redshift.
Molecular gas masses were computed from the CO(1-0) integrated fluxes (S$\Delta v$) or line luminosities (L$^{\prime}_{CO}$) using the relations [@sol05; @bol13]:
$$\begin{aligned}
L^{\prime}_{CO}[\rm K~km~s^{-1} pc^2]&=&3.25\times10^{7}~(S\Delta v[\rm Jy~km~s^{-1}]) (\nu_{rest}[GHz])^{-2} (1+z)^{-1} (\it{d}_{L}[Mpc])^2 \\
M_{mol}[{\mbox{$\rm M_{\odot}$}}] &=& 6.5~ L^{\prime}_{CO}[\rm K~km~s^{-1} pc^2] . \label{co}\end{aligned}$$
The constant ($\alpha_{CO} = 6.5 ~{\mbox{$\rm M_{\odot}$}}/ \rm{K~km~s^{-1} pc^2}$) in Equation A10 is based on a standard Galactic conversion factor $X_{CO} = 3\times10^{20}$ N(H$_2$) cm$^{-2}$ (K km s$^{-1})^{-1}$ (see below) and it includes a factor 1.36 to account for the associated mass of heavy elements (mostly He at 8% by number). For the local SF galaxies, the integrated CO (1-0) fluxes were all taken from [@you95] so they have consistent calibration and technique for integrating over extended galaxies. (In the course of this work, we found that the molecular gas masses used in the KINGFISH papers [@dra07b; @dal12] are actually based on the same [@you95] CO survey even though they reference [@ken03] for the molecular masses. [@dra07b] argues for (and used) a higher $X_{CO}= 4\times10^{20}$, so their H$_2$ masses are larger (they don’t explicitly include the He contribution).
We note that [@bol13] have advocated a value of $X_{CO}=2\times10^{20}$ cm$^{-2}$ (K km s$^{-1})^{-1}$ based largely on Galactic $\gamma$-ray emission analysis. However, this relies on low angular resolution $\gamma$-ray (CosB, Egret, CGRAO and Fermi) and CO (Columbia Survey) datasets which highly weights gas in the solar neighborhood (mostly HI), rather than the molecular ring in the inner galaxy; it also does not resolve the distant GMCs. The $\gamma$-ray approach also relies on the questionable assumption that the high energy cosmic rays which produce the $\gamma$-rays are approximately constant in the Galactic disk, and that these particles penetrate fully the dense molecular clouds. It is noteworthy that there are also large variations in the $\gamma$-ray-based X$_{CO}$ values between the different analyses and as a function of Galactic radius [@bol13]. In contrast, extensive GMC surveys (with samples of more than 500 resolved GMCs in the inner Galactic plane, independently measured and analyzed) yielded X$_{CO} =
3.6$ and 3.0$\times10^{20}$ cm$^{-2}$ (K km s$^{-1})^{-1}$ [respectively: @sco87; @sol87 corrected to $R_0 = 8.5$ kpc].
We have chosen to use a single conversion factor ($\alpha_{CO}$ or X$_{CO}$) for all galaxies. A several times smaller conversion factor is often used for ULIRGs and SMGs in analyzing the CO transitions seen from the hot, dense nuclear regions of these merger systems. Similar to the argument we have given above for a uniform T$_D$, a smaller value of $\alpha_{CO}$ is inappropriate for the globally distributed molecular gas although it will sometimes be appropriate for high resolution observations which isolate the nuclear regions.
[lrrcrrrrcc]{}\[ht\] Antennae & 21.4 & 2000 & 9.35 & 10.16 & 14.06 & 4.69 & 0.803 & 0.440 & 0.30\
IC342 & 3.4 & 29220 & 8.92 & 9.73 & 247.95 & 96.90 & 16.886 & 0.233 & 0.44\
NGC0628 & 11.4 & 2160 & 8.83 & 9.65 & 29.07 & 12.64 & 2.185 & 0.340 & 0.76\
NGC1482 & 22.0 & 560 & 8.82 & 9.63 & 6.03 & 2.10 & 0.359 & 0.208 & 0.48\
NGC2146 & 15.0 & 2840 & 9.19 & 10.01 & 22.13 & 7.08 & 1.220 & 0.328 & 0.32\
NGC2798 & 24.7 & 440 & 8.82 & 9.63 & 2.76 & 1.03 & 0.175 & 0.128 & 0.30\
NGC2841 & 9.8 & 1870 & 8.64 & 9.45 & 15.20 & 6.66 & 1.153 & 0.132 & 0.47\
NGC2976 & 3.6 & 610 & 7.27 & 8.09 & 11.11 & 4.55 & 0.793 & 0.012 & 0.98\
NGC3184 & 8.1 & 1120 & 8.25 & 9.07 & 14.53 & 6.39 & 1.109 & 0.087 & 0.75\
NGC3351 & 9.3 & 700 & 8.17 & 8.98 & 13.01 & 5.05 & 0.876 & 0.091 & 0.95\
NGC3521 & 9.0 & 4920 & 8.99 & 9.80 & 44.84 & 18.43 & 3.194 & 0.309 & 0.49\
NGC3627 & 8.9 & 4660 & 8.95 & 9.77 & 35.72 & 13.68 & 2.371 & 0.225 & 0.38\
NGC3938 & 14.0 & 1750 & 8.92 & 9.73 & 9.78 & 4.12 & 0.711 & 0.167 & 0.31\
NGC4254 & 20.0 & 3000 & 9.47 & 10.28 & 25.27 & 8.70 & 1.493 & 0.714 & 0.38\
NGC4321 & 20.0 & 3340 & 9.51 & 10.33 & 26.50 & 10.26 & 1.760 & 0.842 & 0.40\
NGC4536 & 25.0 & 740 & 9.05 & 9.86 & 11.97 & 5.25 & 0.897 & 0.670 & 0.92\
NGC4569 & 20.0 & 1500 & 9.16 & 9.98 & 8.94 & 3.49 & 0.598 & 0.286 & 0.30\
NGC4579 & 20.0 & 910 & 8.95 & 9.76 & 8.43 & 3.36 & 0.577 & 0.276 & 0.48\
NGC4631 & 9.0 & 1740 & 8.54 & 9.35 & 51.77 & 22.80 & 3.952 & 0.383 & 1.72\
NGC4725 & 17.1 & 1950 & 9.14 & 9.96 & 15.77 & 7.53 & 1.296 & 0.453 & 0.50\
NGC4736 & 5.3 & 2560 & 8.24 & 9.06 & 26.60 & 11.21 & 1.950 & 0.066 & 0.58\
NGC4826 & 5.6 & 2170 & 8.22 & 9.03 & 15.58 & 5.99 & 1.041 & 0.039 & 0.36\
NGC5055 & 8.2 & 5670 & 8.97 & 9.78 & 60.99 & 24.80 & 4.301 & 0.346 & 0.57\
NGC5194 & 8.2 & 9210 & 9.18 & 9.99 & 62.60 & 21.28 & 3.691 & 0.297 & 0.30\
NGC5713 & 26.6 & 680 & 9.07 & 9.88 & 6.07 & 2.18 & 0.372 & 0.315 & 0.41\
NGC5866 & 12.5 & 250 & 7.98 & 8.79 & 2.98 & 1.08 & 0.187 & 0.035 & 0.57\
NGC6946 & 5.5 & 12370 & 8.96 & 9.77 & 103.55 & 40.66 & 7.071 & 0.256 & 0.43\
NGC7331 & 14.7 & 4160 & 9.34 & 10.15 & 38.57 & 15.68 & 2.702 & 0.700 & 0.49\
\[tab:local\_gal\]
[lrrcrrrrcc]{}\[ht\] 1ZW107 & 170.0 & 9.62 & 10.44 & 1 & 0.72 & 0.20 & 0.029 & 1.014 & 0.37\
Arp148 & 143.0 & 9.54 & 10.35 & 1 & 0.92 & 0.27 & 0.040 & 0.988 & 0.44\
Arp220 & 79.0 & 9.76 & 10.57 & 1,2 & 10.89 & 3.60 & 0.584 & 4.359 & 1.16\
IRASF05189-2 & 168.0 & 9.72 & 10.53 & 1 & 0.59 & 0.16 & 0.023 & 0.790 & 0.23\
IRASF08572+3 & 232.0 & 9.13 & 9.94 & 3,2 & 0.16 & 0.05 & 0.007 & 0.429 & 0.49\
IRASF10565+2 & 176.0 & 9.73 & 10.54 & 1,2 & 1.15 & 0.32 & 0.047 & 1.749 & 0.50\
IRASF12112+0 & 292.0 & 9.96 & 10.77 & 1 & 0.65 & 0.19 & 0.024 & 2.489 & 0.42\
IRASF14348-1 & 330.0 & 10.12 & 10.93 & 1 & 0.61 & 0.17 & 0.022 & 2.834 & 0.33\
IRASF22491-1 & 301.0 & 9.77 & 10.58 & 1 & 0.24 & 0.06 & 0.008 & 0.866 & 0.23\
Mrk231 & 174.0 & 9.72 & 10.53 & 1,2 & 1.83 & 0.51 & 0.076 & 2.741 & 0.80\
Mrk273 & 153.0 & 9.67 & 10.49 & 2,3 & 1.36 & 0.37 & 0.055 & 1.551 & 0.51\
\[tab:ulirg\]
[lrccccrrcrcc]{}\[ht\] EROJ164502+4 & 1.44 & 10.6 & 8 & 10.83 & 11.65 & 4.89(850) & 16.362 & 0.37\
H-ATLASJ0903 & 2.31 & 18.9 & 1 & 11.42 & 12.24 & 54.70(880) & 214.455 & 1.24\
H-ATLASJ0913 & 2.63 & 22.2 & 1 & 11.40 & 12.21 & 36.70(880) & 145.710 & 0.89\
H-ATLASJ0918 & 2.58 & 21.7 & 1 & 11.53 & 12.34 & 18.80(880) & 74.504 & 0.34\
H-ATLASJ1132 & 2.58 & 21.7 & 1 & 11.33 & 12.14 & 106.00(500) & 179.479 & 1.30\
H-ATLASJ1158 & 2.19 & 17.8 & 1 & 11.26 & 12.07 & 107.00(500) & 150.507 & 1.29\
H-ATLASJ1336 & 2.20 & 17.9 & 1 & 11.36 & 12.17 & 36.80(880) & 143.699 & 0.97\
H-ATLASJ1344 & 2.30 & 18.9 & 1 & 11.86 & 12.67 & 73.10(880) & 286.544 & 0.61\
H-ATLASJ1413 & 2.48 & 20.7 & 1 & 11.65 & 12.46 & 33.30(880) & 131.429 & 0.46\
HATLASJ08493 & 2.41 & 20.0 & 11 & 10.36 & 11.17 & 4.60(870) & 17.642 & 1.19\
HATLASJ08493 & 2.42 & 20.1 & 11 & 10.22 & 11.03 & 6.90(870) & 26.468 & 2.48\
HATLASJ08493 & 2.41 & 20.0 & 11 & 11.21 & 12.02 & 19.00(870) & 72.855 & 0.70\
HATLASJ08493 & 2.41 & 20.0 & 11 & 11.15 & 11.96 & 25.00(870) & 95.851 & 1.05\
HLSW-01 & 2.96 & 25.6 & 2 & 11.66 & 12.48 & 52.80(880) & 212.973 & 0.71\
HXMM01 & 2.31 & 19.0 & 12 & 11.66 & 12.48 & 27.00(880) & 105.868 & 0.35\
SMMJ02399-01 & 2.81 & 24.1 & 9 & 11.35 & 12.16 & 23.00(850) & 85.752 & 0.59\
SMMJ04135+10 & 2.85 & 24.5 & 6 & 11.39 & 12.20 & 25.00(850) & 93.462 & 0.59\
SMMJ04431+02 & 2.51 & 21.0 & 10 & 10.90 & 11.72 & 7.20(850) & 26.346 & 0.51\
SMMJ123549.4 & 2.20 & 17.9 & 6 & 10.89 & 11.71 & 8.30(850) & 29.839 & 0.59\
SMMJ123707.2 & 2.49 & 20.8 & 6 & 11.44 & 12.25 & 10.70(850) & 39.105 & 0.22\
SMMJ14009+02 & 2.93 & 25.3 & 9 & 11.09 & 11.91 & 15.60(850) & 58.678 & 0.73\
SMMJ14011+02 & 2.57 & 21.6 & 10 & 11.11 & 11.92 & 12.30(850) & 45.161 & 0.54\
SMMJ163550.9 & 2.52 & 21.1 & 9 & 10.97 & 11.78 & 8.40(850) & 30.754 & 0.51\
SMMJ163554.2 & 2.52 & 21.1 & 9 & 11.09 & 11.90 & 15.90(850) & 58.212 & 0.72\
SMMJ163555.2 & 2.52 & 21.1 & 9 & 10.83 & 11.65 & 12.50(850) & 45.765 & 1.04\
SMMJ163650.4 & 2.38 & 19.7 & 6 & 10.98 & 11.79 & 8.20(850) & 29.793 & 0.48\
SMMJ163658.1 & 2.45 & 20.4 & 6 & 11.04 & 11.85 & 10.70(850) & 39.022 & 0.55\
SMMJ2135-010 & 2.33 & 19.2 & 4 & 11.78 & 12.60 & 106.00(870) & 404.977 & 1.03\
SPT-S053816- & 2.79 & 23.8 & 5 & 11.64 & 12.46 & 125.00(870) & 488.049 & 1.71\
SPT-S233227- & 2.73 & 23.2 & 5 & 11.78 & 12.59 & 150.00(870) & 583.833 & 1.49\
\[tab:smg\]
In Figure \[empir\_cal\] the ratios L$_{\nu_{850\mu \rm m}} /\rm {M_{ mol}}$ are plotted as a function of L$_{\nu_{850\mu \rm m}}$ for the three samples of galaxies listed in Tables \[tab:local\_gal\] - \[tab:smg\]. The galaxies in all three samples clearly overlap in the luminosity-to-mass ratios and their mean ratios are indeed very similar. The mean of the local star-forming galaxies, ULIRGs and SMGs is
$$\begin{aligned}
\alpha_{\nu} \equiv < L_{\nu_{850\mu \rm m}} /M_{\rm mol}> = 6.7\pm1.7\times 10^{19} \rm erg ~sec^{-1} Hz^{-1} {{\mbox{$\rm M_{\odot}$}}}^{-1} \end{aligned}$$\[alpha\]
and we adopt this value in the analysis below.
Planck Measurements for HI and H$_2$ in the Galaxy {#pla}
--------------------------------------------------
The Planck measurements of the submm emission from the Galaxy provide both very high photometric accuracy and the ability to probe variations in the opacity to mass ratio between atomic and molecular phases, and with Galactic radius. (The latter could possibly provide a probe of metallicity dependence.)
In the Taurus complex, the [@pla11b] obtained resolved observations of the HI and H$_2$ ISM components with best fit ratios of $\tau_{250\mu \rm m} / N_{\rm H} = 1.1 \pm 0.2 ~\rm ~and~ 2.32 \pm 0.3 \times 10^{-25}$ cm$^{2}$ for the atomic and molecular phases. The HI column densities were derived from the optically thin 21cm emission with a small correction of 25% for optically thick 21 cm emission. The H$_2$ column densities were taken from [@pin10] who used NIR extinction measures as a primary measure of molecular gas column densities. (CO column densities were also obtained from a non-LTE radiative transfer analysis but these were not used for the Planck analysis). The mean dust temperature from the Planck observations was 18K derived in Taurus and the mean $<\beta> = 1.8$. We translate the value given above for $\tau_{250\mu \rm m} / N_{\rm H} $ in the molecular phase into a specific luminosity per unit mass of ISM (using $M_{\rm mol}$ = 1.36 $M_{\rm H_2}$ to account to He):
$$\begin{aligned}
{L_{\nu_{850\mu \rm m}} \over M_{\rm H_2}} &=& \left[ \tau_{250\mu \rm m} / N_{\rm H} \right] \left({\nu_{850\mu \rm m} \over \nu_{250\mu \rm m}}\right)^{\beta} {4 \pi B_{\nu} (T_d) \over m_H} \nonumber \\
&=& 8.4\times 10^{19} \rm ergs/sec/Hz/{\mbox{$\rm M_{\odot}$}}\nonumber \\
\alpha_{850\mu \rm m} &=& {L_{\nu_{850\mu \rm m}} \over M_{\rm mol}} = 6.2\times 10^{19} \rm ergs/sec/Hz/{\mbox{$\rm M_{\odot}$}}~.
\label{planck_alpha}\end{aligned}$$
This value for $\alpha_{850\mu \rm m}$ obtained from the Planck data in Taurus is remarkably similar to that found above (Equation \[alpha\]) in the samples of nearby star forming galaxies, ULIRGs and z $\sim 2$ SMGs. Using Planck data from the Galaxy, [@pla11a] found $\tau_{250\mu \rm m} / N_{\rm H} = 0.92 \pm 0.05 \times 10^{-25}$ cm$^{2}$ near the solar circle. This determination at low angular resolution and covering a large range of galactic latitude is strongly weighted toward the HI phase in the solar neighborhood. Hence it is not surprising that it agrees better with the value found in Taurus for the atomic gas.
Expected Submm Fluxes as a function of Redshift
-----------------------------------------------
Combining Equations \[lnu\_eqn\] and A4, the expected flux density at observed frequency $\nu_{obs}$ is given by
$$\begin{aligned}
S_{\nu_{obs}} &=& 0.563 ~ {M_{\rm mol} \over 10^{10}{\mbox{$\rm M_{\odot}$}}} ~ (1+z)^{4.8} ~ \left({\nu_{obs} \over{ \nu_{850\mu \rm m}}}\right)^{3.8} (d_{L} [\rm Gpc])^{-2} \nonumber \\
&& \times ~\left\{{\alpha_{850} \over{6.7\times10^{19} }} \right\} ~{\it{\Gamma_{RJ}} \over{\it{\Gamma_{0}}}}~ ~\rm mJy \\
\rm for && \lambda_{rest} \gtrsim 250 ~\mu \rm m \nonumber .
\label{snu_alpha}
\end{aligned}$$
We note that the empirical calibration of $\alpha_{850}$ was obtained from z $\simeq 0$ galaxies which have a non-negligible RJ departure ($\Gamma_{0} \sim 0.7$) which must be normalized out (i.e. the y-axis intercept in Figure \[rj\]). This is the term $\Gamma_0 = \Gamma_{RJ} (0,T_d,\nu_{850})$ in the equation above.
The restriction $ \lambda_{rest} \gtrsim 250 ~\mu \rm m$ is intended to ensure that one is on the RJ tail and that the dust is likely to be optically thin. If the dust is extremely cold one might need to be more restrictive and in the case of the most extreme ULIRGs the dust is probably optically thick to even longer wavelengths. Analogous expressions are readily obtained for the other ALMA bands.
Figure \[alma\_obs\] shows the expected flux as a function of redshift for the ALMA bands at 100, 145, 240 and 350 GHz (Bands 3, 4, 6 and 7). At low z, the increasing luminosity distance leads to reduced flux as z increases. However, above z = 1 the well known “negative k-correction" causes the flux per unit ISM mass to increase at higher z as one moves up the far infrared SED towards the peak at $\lambda \sim 100 \mu$m. Figure \[alma\_obs\] shows that the 350 GHz flux density plateaus at z = 1 and then decreases above z = 2. The latter is due to the fact that at higher redshift observed frame 350 GHz is approaching the rest frame far infrared peak (and no longer on the $\nu^2$ RJ tail). This is the factor $\Gamma_{RJ}$ coming in for 25 K dust.
At redshifts above 2.5, Figure \[alma\_obs\] indicates that one needs to shift to a lower frequency band, e.g. 240, 145 or 100 GHz, in order to avoid the large and uncertain $\Gamma_{RJ}$ corrections. Since future studies similar to that pursued here will push to higher redshifts, we have included the lower frequency bands in Figures \[rj\] and \[alma\_obs\].
Inverting Equation A13, the estimation of masses from observed flux densities can be done using $$\begin{aligned}
M_{\rm mol} &=& 1.78 ~S_{\nu_{obs}}[\rm mJy] ~ (1+z)^{-4.8} ~ \left({ \nu_{850\mu \rm m}\over{\nu_{obs} }}\right)^{3.8} ({\it{d}_L \rm{[Gpc ]}})^{2} \nonumber \\
&& \times ~\left\{{6.7\times10^{19}\over{\alpha_{850} }} \right\} ~{\it{\Gamma_{0}} \over{\it{\Gamma_{RJ}}}}~ ~10^{10}{\mbox{$\rm M_{\odot}$}}~~ for~~ \lambda_{rest} > 250 \mu m ~. \label{mass_eq}
\end{aligned}$$ The restriction $ \lambda_{rest} \gtrsim 250 ~\mu \rm m$ is intended to ensure that one stays on the Rayleigh-Jeans tail.
In the above analysis we used a single (standard) Galactic conversion factor $\alpha_{CO}$ to convert observed CO(1-0) luminosity to gas mass. As discussed in [@sol05], low-z studies of ULIRGs have led to the suggestion that the conversion factor could be several times smaller [@dow93; @bry99]. This can arise in the ULIRGs if the gas is concentrated in the nuclear regions (as a result of dissipative galaxy merging) and the molecular emission linewidths can be broadened by the galactic dynamics associated with the stellar mass – not just the self-gravitating gas mass as in individual GMCs in which the standard conversion factor was derived. In addition, the mean gas temperature and density ($\rho$) may be different in the ULIRG nuclei as a result of the intense star formation activity, and the $\alpha_{CO}$ should vary as $<\rho>^{1/2}/T_k$ [@dic86; @sco12 Equation 8.5].
Given the results obtained here which clearly show a quite similar $\alpha_{850\mu \rm m}$ in all three samples of galaxies, it would appear that there is little basis for using different $\alpha_{CO}$ for normal and star bursting galaxies – at least when considering global measurements. For the high z SMGs, it is not obvious that the lower $\alpha_{CO}$ (often used in low-z ULIRGs) is appropriate since it is uncertain that the bulk of the molecular gas in the SMGs is similarly concentrated. Our restriction to CO(1-0) in the above sample was specifically intended to avoid sensitivity to the presence of high excitation gas, and to sample the larger, presumably extended masses of cold gas. Indeed, the ratio of dust emission to gas mass is similar to that obtained in low z galaxies.
Summary – an approximately constant RJ mass-to-light ratio
----------------------------------------------------------
In the preceding sections, we have presented the physical explanation and, more importantly, strong empirical justification for using the long wavelength RJ dust emission in galaxies as a linear probe of ISM mass. The most substantial determination of the dust RJ spectral slope is that obtained by Planck from observations of the Galaxy [@pla11a; @pla11b], indicating a dust emissivity index $\beta = 1.8 \pm 0.1$ with no strong evidence of variation in Galactic radius or between atomic and molecular regions. Secondly, both the Planck data and measurements for nearby local galaxies, including both normal star forming and star bursting systems, indicate a similar constant of proportionality $\alpha_{850}$ for the dust emission at rest frame 850$\mu$m per unit mass of ISM. Lastly, we find that for a large sample of SMGs at z = 1.4 to 3, their ratio of rest frame 850$\mu$m per unit mass of ISM is essentially identical to that obtained for local galaxies. The similarities of these values of $\alpha_{850}$ argue strongly that for global ISM masses: the dust emissivity at long wavelengths, the dust-to-gas mass ratio and the mass-weighted dust temperatures vary little.
The submm flux to dust mass ratio is expected to vary linearly with dust temperature. In practice, the overall range of $T_d$ for the bulk of the mass of ISM is very small, since it requires very large increases in radiative heating to increase the dust temperatures ($T_d$ varies approximately as the 1/5 - 1/6 power of the radiation energy density). As noted above, the extensive surveys of local galaxies using Herschel find a range of $T_d \sim 15 - 30$ K [@dun11; @dal12; @aul13]. Where we have needed to specify a dust temperature (e.g. for the R-J correction) we have adopted 25 K, so we expect the uncertainties in the derived masses averaged on galaxy scales will be less than $\sim 25$%.
These calibrations include normal to star bursting systems and low to moderate redshift; they lay a solid foundation for using measurements of the RJ dust emission to probe galactic ISM masses. ALMA enables this technique for high redshift surveys, providing high sensitivity and the requisite angular resolution to avoid source confusion.
Cautions
--------
It is important to recognize that even for those objects detected in SPIRE, the SPIRE data can not be used to reliably estimate ISM masses (along the lines as done here) for the z = 1 and 2 samples. For those redshifts, the SPIRE data will be probing near the rest frame far infrared luminosity peak – [*not safely on the RJ tail and not necessarily optically thin*]{}. The longest wavelength channel (500$\mu$m or 600 GHz) will be probing rest frame 170$\mu$m for z = 2; for such measurements, so there will be substantial uncertainty in the mass estimate, depending on the assumed value of the dust temperature (see Figure \[rj\]). (In addition, the 500$\mu$m SPIRE data has relatively high source confusion on account of the large beam size.)
Often, the far infrared SEDs are analyzed by fitting either modified black body curves or libraries of dust SEDs to the observed SEDs [e.g. @dra07b; @dac11; @mag12; @mag12a]. In essentially all instances the intrinsic SEDs used for fitting are taken to be optical thin. They thus do not include the attenuation expected near the far infrared peak associated with optically thick dust, instead attributing the drop at short wavelengths to a lack of high temperature grains. The $T_D$ determined in these cases is not even a luminosity-weighted $T_D$ of all the dust but just the dust above $\tau \sim 1$.
*Fitting the observed spectral energy distribution (SED) to derive an effective dust temperature is [**not**]{} a reliable approach* – near the far infrared peak, the temperature characterizing the emission is ’luminosity-weighted’ (i.e. grains undergoing strong radiative heating) rather than mass-weighted. Hence, the derived $T_d$ will not reflect the temperature appropriate to the bulk of the ISM mass. Or, put another way, the flux measured near the peak is simply a measure of luminosity – not mass. At high redshifts, the large SPIRE beam at $500\mu$m results in severe source confusion at the expected flux levels; hence reliable flux measurements for individual galaxies are difficult. At z $> 2$ ALMA resolution and sensitivity are required and one must observe at $\nu \leq 350$ GHz to be on the RJ tail of the dust emission.
One might be concerned that some of the correlation between the SPIRE 500$\mu$m continuum and the CO(1-0) in our local galaxy samples was due to emission lines contributing substantially to the continuum flux in the SPIRE data. However, scaling the CO(1-0) fluxes (given in the above tables) to the frequencies covered in the 500$\mu$m filter (having width $\lambda / \Delta \lambda$ = 2.5) indicates that the CO lines will contribute less than $10^{-3}$ of the total continuum flux. At rest wavelengths longer than 2mm, the line contamination becomes an issue since the line fluxes decrease less rapidly than $\lambda ^{-2}$ while the dust continuum decreases as $\sim \lambda ^{-3.8}$. In fact, the Planck imaging of nearby molecular clouds show positive excess residuals at the level of 10% relative to the dust emission in the 100 GHz band [@pla11b attributed to CO(1-0) emission].
Lastly, we reiterate the caution that the calibration samples are intentionally restricted to objects with high stellar mass ($M_{stellar} > 5\times10^{10}$); thus we are not probing lower metallicity systems where the dust-to-gas abundance ratio is likely to drop or where there could be significant molecular gas without CO [see @bol13]. A similar calibration at lower metallicities will be more difficult given the lower CO and continuum fluxes and in fact, at very low metallicities it is quite likely that this technique will not be so robust.
Individual Galaxies and their Fluxes {#source_app}
====================================
In Tables \[lowz\] - \[highz\] we list the individual flux measurements, and galaxy properties are summarized for all 145 galaxies in our survey. The objects are taken from the COSMOS survey field [@sco_ove] and the galaxy properties are from the latest photometric redshift catalog [@ilb13]. This catalog has high accuracy photometric redshifts based on very deep 34 band photometry, including near infrared photometry from the Ultra-Vista survey. See [@ilb13] and [@lai15] for discussion of the accuracy of the redshifts and the stellar masses of the galaxies. The SFRs in Column 9 are from [@lee15] where there are two band Herschel detections and from [@lai15] using the UV continuum and optical/UV continuum SEDs.
The galaxy ID \# given in column 1 is taken from the most up to date COSMOS photometric redshift catalog [@lai15]. Columns 5 and 7 in each table list integrated and peak flux measurements for apertures of up to 2.5(Low-z and Mid-z) and 2(High-z) radius centered on the galaxy position. The aperture sizes are intended to include most of a galactic disk ($\sim 10$ kpc). The noise estimate in both cases is from the measured dispersion in the integrated and peak flux measurements obtained for 100 displaced off-center apertures of the same size in each individual image. The signal-to-noise ratio given in Column 7 is the better of those obtained from the integrated or peak flux measurement; it is the ratio of the signal in Columns 5 and 7 to the measured noise given in Columns 6 and 8. Columns 10 - 12 give the galaxy stellar mass, SFR and sSFR (relative to the Main Sequence at the same redshift and $M_{stellar}$ with the Main Sequence taken from [@lee15]. In the last column, the derived ISM molecular mass is given. Limits on the masses are at 2$\sigma$ and 3.6$\sigma$, depending on whether the better SNR (column 9) was obtained for the integrated or peak flux measurement. The detection thresholds of 2 and 3.6 $\sigma$ are chosen such that the chance of a spurious detection across the entirety of each sample is less than $\sim10$% (based on the measured noise in each individual image).
[rrrrrrrrrcccr]{} 890088 & 149.8157 & 2.6503 & 1.02 & 0.38 & 0.19 & 0.71 & 0.17 & 4.15 & 0.28 & 1.19 & 0.65 & 3.22$\pm$0.78\
846385 & 149.5421 & 2.5816 & 1.01 & 0.37 & 0.18 & 0.62 & 0.18 & 2.00 & 0.39 & 1.43 & 1.00 & $<$ 2.41\
405136 & 150.7642 & 1.9077 & 1.02 & 0.26 & 0.13 & 0.33 & 0.10 & 2.00 & 0.28 & 1.35 & 0.92 & $<$ 1.40\
471515 & 149.8180 & 2.0132 & 1.04 & 0.24 & 0.12 & 0.37 & 0.11 & 2.00 & 0.40 & 1.40 & 0.88 & $<$ 1.46\
257225 & 149.9700 & 1.6731 & 1.01 & 0.20 & 0.10 & 0.29 & 0.10 & 2.00 & 0.32 & 1.11 & 0.51 & $<$ 1.39\
354566 & 149.9792 & 1.8278 & 1.19 & 0.83 & 0.32 & 0.36 & 0.11 & 2.55 & 0.38 & 1.47 & 0.85 & 3.99$\pm$1.57\
737603 & 150.0182 & 2.4177 & 1.12 & 0.31 & 0.16 & 0.51 & 0.15 & 2.00 & 0.52 & 1.20 & 0.46 & $<$ 2.08\
787724 & 149.8601 & 2.4920 & 1.19 & 0.25 & 0.12 & 0.58 & 0.15 & 3.80 & 0.52 & 1.48 & 0.78 & 2.72$\pm$0.72\
543756 & 150.0946 & 2.1280 & 1.22 & 0.28 & 0.14 & 0.62 & 0.14 & 4.28 & 0.60 & 1.44 & 0.66 & 2.99$\pm$0.70\
805480 & 150.0256 & 2.5211 & 1.03 & 0.30 & 0.15 & 0.46 & 0.16 & 2.00 & 0.79 & 1.51 & 0.96 & $<$ 2.24\
687914 & 149.7395 & 2.3413 & 1.01 & 0.33 & 0.17 & 0.63 & 0.16 & 3.96 & 0.38 & 1.47 & 1.10 & 2.86$\pm$0.72\
711951 & 149.8388 & 2.3775 & 1.21 & 0.30 & 0.15 & 2.63 & 0.16 & 16.47 & 0.39 & 1.84 & 1.90 & 12.78$\pm$0.78\
800281 & 150.0747 & 2.5130 & 1.26 & 0.31 & 0.16 & 0.47 & 0.15 & 2.00 & 0.39 & 1.78 & 1.60 & $<$ 2.18\
873708 & 150.1164 & 2.6253 & 1.02 & 0.29 & 0.14 & 0.57 & 0.18 & 2.00 & 0.39 & 1.63 & 1.54 & $<$ 2.38\
599889 & 150.3524 & 2.2101 & 1.23 & 2.48 & 0.50 & 0.45 & 0.15 & 5.02 & 0.99 & 2.03 & 2.28 & 12.10$\pm$2.41\
671444 & 150.2712 & 2.3184 & 1.16 & 1.21 & 0.48 & 0.63 & 0.15 & 2.53 & 0.85 & 1.88 & 1.84 & 5.80$\pm$2.29\
918541 & 149.8367 & 2.6918 & 1.01 & 1.07 & 0.46 & 0.64 & 0.18 & 2.35 & 0.54 & 1.60 & 1.32 & 4.82$\pm$2.05\
984597 & 150.1099 & 2.7994 & 1.09 & 0.38 & 0.19 & 0.50 & 0.18 & 2.00 & 0.44 & 1.64 & 1.38 & $<$ 2.53\
802829 & 149.7721 & 2.5171 & 1.14 & 2.62 & 0.40 & 0.61 & 0.17 & 6.60 & 0.72 & 1.75 & 1.45 & 12.47$\pm$1.89\
806656 & 150.2180 & 2.5217 & 1.20 & 2.00 & 0.41 & 0.60 & 0.16 & 4.91 & 0.74 & 1.88 & 1.77 & 9.66$\pm$1.97\
842212 & 150.0628 & 2.5741 & 1.01 & 0.44 & 0.17 & 0.66 & 0.17 & 3.98 & 0.95 & 1.80 & 1.84 & 1.99$\pm$0.50\
483264 & 150.2705 & 2.0338 & 1.20 & 0.27 & 0.13 & 0.32 & 0.11 & 2.00 & 0.41 & 1.77 & 1.63 & $<$ 1.66\
541203 & 149.6961 & 2.1223 & 1.37 & 3.02 & 0.56 & 0.55 & 0.15 & 5.40 & 1.28 & 2.11 & 2.21 & 15.18$\pm$2.81\
914622 & 150.0537 & 2.6861 & 1.03 & 0.37 & 0.18 & 0.57 & 0.17 & 2.00 & 2.88 & 1.94 & 2.21 & $<$ 2.37\
983395 & 150.6615 & 2.7970 & 1.18 & 1.26 & 0.63 & 0.73 & 0.18 & 2.00 & 1.05 & 1.95 & 2.00 & 6.04$\pm$3.02\
985857 & 150.3477 & 2.8015 & 1.25 & 1.10 & 0.49 & 0.54 & 0.17 & 3.15 & 1.39 & 1.91 & 1.60 & 5.37$\pm$1.70\
422455 & 149.8490 & 1.9337 & 1.24 & 0.74 & 0.24 & 0.28 & 0.11 & 3.10 & 2.12 & 1.87 & 1.44 & 3.60$\pm$1.16\
434350 & 150.3446 & 1.9540 & 1.24 & 0.86 & 0.41 & 0.34 & 0.11 & 2.11 & 1.86 & 1.77 & 1.15 & 4.22$\pm$2.00\
505822 & 149.6633 & 2.0667 & 1.24 & 1.75 & 0.39 & 0.43 & 0.10 & 4.51 & 1.66 & 2.09 & 2.43 & 8.58$\pm$1.90\
340442 & 150.1541 & 1.8042 & 1.11 & 2.28 & 1.13 & 1.04 & 0.11 & 2.01 & 1.59 & 1.96 & 2.15 & 10.70$\pm$5.32\
868539 & 149.8634 & 2.6178 & 1.32 & 3.87 & 0.79 & 1.81 & 0.17 & 4.89 & 0.35 & 2.04 & 2.74 & 19.31$\pm$3.95\
288391 & 150.4103 & 2.6264 & 1.15 & 0.32 & 0.16 & 2.24 & 0.18 & 12.61 & 0.28 & 1.96 & 3.14 & 10.68$\pm$0.85\
960580 & 149.9191 & 2.7605 & 1.16 & 2.66 & 0.50 & 0.58 & 0.17 & 5.36 & 0.35 & 2.07 & 3.67 & 12.69$\pm$2.37\
409265 & 149.7843 & 1.9128 & 1.16 & 0.69 & 0.27 & 0.33 & 0.11 & 2.60 & 0.39 & 2.11 & 3.84 & 3.31$\pm$1.27\
351159 & 150.0908 & 1.8211 & 1.00 & 0.20 & 0.10 & 0.35 & 0.10 & 2.00 & 0.35 & 1.91 & 3.14 & $<$ 1.39\
601886 & 149.8117 & 2.2123 & 1.22 & 0.30 & 0.15 & 0.50 & 0.16 & 2.00 & 0.95 & 2.35 & 4.86 & $<$ 2.32\
237348 & 149.4016 & 2.4509 & 1.19 & 3.08 & 0.50 & 1.85 & 0.16 & 6.18 & 0.76 & 2.24 & 4.09 & 14.89$\pm$2.41\
781580 & 149.9086 & 2.4833 & 1.25 & 0.52 & 0.15 & 0.56 & 0.15 & 3.40 & 0.75 & 2.12 & 2.91 & 2.53$\pm$0.74\
134318 & 150.2174 & 2.1141 & 1.13 & 0.90 & 0.45 & 0.80 & 0.15 & 2.00 & 0.44 & 2.22 & 4.90 & 4.24$\pm$2.12\
560724 & 150.1236 & 2.1498 & 1.17 & 1.83 & 0.42 & 0.41 & 0.15 & 4.37 & 0.83 & 2.15 & 3.40 & 8.78$\pm$2.01\
585275 & 150.3987 & 2.1885 & 1.04 & 2.98 & 0.39 & 2.02 & 0.15 & 7.67 & 0.78 & 2.18 & 4.43 & 13.56$\pm$1.77\
969105 & 150.5307 & 2.7755 & 1.36 & 0.89 & 0.35 & 0.65 & 0.18 & 2.55 & 0.96 & 2.33 & 3.83 & 4.47$\pm$1.75\
831023 & 150.4179 & 2.5585 & 1.21 & 2.55 & 0.98 & 0.54 & 0.16 & 3.25 & 0.71 & 2.28 & 4.46 & 12.35$\pm$3.80\
6496 & 149.9741 & 1.6435 & 1.04 & 1.23 & 0.40 & 0.47 & 0.10 & 3.05 & 0.76 & 2.16 & 4.24 & 5.60$\pm$1.84\
418048 & 150.3503 & 1.9282 & 1.20 & 2.14 & 0.44 & 0.46 & 0.11 & 4.81 & 0.83 & 2.15 & 3.21 & 10.35$\pm$2.15\
108065 & 150.0087 & 2.0257 & 1.19 & 1.45 & 0.65 & 0.35 & 0.10 & 2.23 & 0.76 & 2.09 & 2.93 & 7.02$\pm$3.14\
269311 & 149.4324 & 1.6924 & 1.26 & 2.86 & 0.54 & 1.30 & 0.11 & 5.32 & 0.61 & 2.16 & 3.24 & 14.08$\pm$2.65\
160476 & 149.5986 & 2.2004 & 1.20 & 3.94 & 0.95 & 1.05 & 0.15 & 4.17 & 2.24 & 2.44 & 5.65 & 19.06$\pm$4.57\
872762 & 150.1296 & 2.6214 & 1.41 & 3.57 & 1.46 & 0.96 & 0.16 & 2.44 & 2.31 & 2.28 & 2.91 & 18.11$\pm$7.42\
916658 & 150.2231 & 2.6903 & 1.30 & 0.39 & 0.19 & 0.52 & 0.17 & 2.00 & 1.24 & 2.42 & 5.01 & $<$ 2.60\
811432 & 150.0425 & 2.5266 & 1.18 & 2.56 & 0.36 & 0.77 & 0.16 & 4.72 & 1.85 & 2.13 & 2.85 & 12.30$\pm$2.61\
485345 & 150.1895 & 2.0370 & 1.18 & 2.31 & 0.35 & 0.75 & 0.11 & 6.52 & 1.12 & 2.19 & 3.44 & 11.11$\pm$1.70\
370733 & 150.4130 & 1.8517 & 1.20 & 1.72 & 0.21 & 0.35 & 0.11 & 8.33 & 1.26 & 2.11 & 2.79 & 8.32$\pm$1.00\
627524 & 149.9813 & 2.2536 & 1.36 & 0.27 & 0.14 & 1.28 & 0.15 & 8.38 & 0.22 & 2.36 & 6.77 & 6.42$\pm$0.77\
344653 & 150.5036 & 1.8118 & 1.16 & 1.95 & 0.27 & 1.16 & 0.11 & 7.11 & 0.36 & 2.34 & 6.58 & 9.35$\pm$1.31\
504172 & 149.8325 & 2.0660 & 1.15 & 2.84 & 0.28 & 1.72 & 0.15 & 10.22 & 0.66 & 2.44 & 7.06 & 13.51$\pm$1.32\
9254 & 150.0120 & 1.6521 & 1.30 & 1.44 & 0.29 & 0.69 & 0.11 & 6.38 & 0.42 & 2.65 & 10.92 & 7.16$\pm$1.12\
570293 & 150.0981 & 2.1658 & 1.20 & 2.11 & 0.50 & 1.46 & 0.15 & 9.64 & 2.06 & 2.48 & 6.15 & 10.19$\pm$1.06\
\
\[lowz\]
[rrrrrrrrrcccr]{}
399465 & 150.4691 & 1.8996 & 2.20 & 0.26 & 0.13 & 0.34 & 0.14 & 2.00 & 0.35 & 1.86 & 0.69 & $<$ 2.29\
479473 & 150.0251 & 2.0288 & 2.17 & 0.24 & 0.12 & 0.42 & 0.14 & 2.00 & 0.26 & 1.70 & 0.56 & $<$ 2.25\
35012 & 149.9635 & 1.7615 & 2.00 & 0.40 & 0.15 & 0.39 & 0.15 & 2.55 & 0.32 & 1.81 & 0.78 & 2.18$\pm$0.86\
829041 & 149.7756 & 2.5571 & 2.03 & 1.10 & 0.24 & 0.36 & 0.12 & 4.58 & 0.27 & 1.82 & 0.83 & 5.97$\pm$1.30\
612589 & 149.9561 & 2.2301 & 2.43 & 0.26 & 0.13 & 0.30 & 0.12 & 2.00 & 0.21 & 1.88 & 0.73 & $<$ 1.97\
708203 & 150.7118 & 2.3724 & 2.28 & 0.26 & 0.13 & 0.39 & 0.12 & 2.00 & 0.31 & 1.72 & 0.49 & $<$ 1.93\
306429 & 150.1298 & 1.7536 & 2.31 & 1.13 & 0.52 & 0.60 & 0.15 & 4.11 & 0.50 & 1.93 & 0.66 & 6.23$\pm$1.52\
777598 & 149.9130 & 2.4806 & 2.89 & 0.48 & 0.20 & 0.25 & 0.06 & 2.43 & 0.72 & 2.09 & 0.72 & 6.55$\pm$2.70\
524944 & 150.0317 & 2.0987 & 2.34 & 1.63 & 0.49 & 0.59 & 0.12 & 3.32 & 0.82 & 2.01 & 0.68 & 8.97$\pm$2.70\
348260 & 150.2028 & 1.8191 & 2.39 & 1.74 & 0.37 & 0.65 & 0.14 & 4.55 & 1.59 & 2.22 & 0.96 & 9.63$\pm$2.11\
715833 & 150.5795 & 2.3850 & 2.71 & 0.46 & 0.20 & 0.29 & 0.06 & 2.29 & 1.15 & 2.04 & 0.59 & 6.44$\pm$2.81\
323041 & 149.8165 & 1.7798 & 2.11 & 1.54 & 0.47 & 0.73 & 0.14 & 5.18 & 0.33 & 2.29 & 2.08 & 8.37$\pm$1.62\
961356 & 149.6007 & 2.7629 & 2.37 & 0.27 & 0.14 & 0.42 & 0.13 & 2.00 & 0.16 & 2.29 & 2.34 & $<$ 2.09\
374178 & 149.7167 & 1.8609 & 2.28 & 0.89 & 0.12 & 0.77 & 0.07 & 7.10 & 0.32 & 2.34 & 2.01 & 13.04$\pm$1.84\
608918 & 150.1193 & 2.2241 & 2.25 & 1.66 & 0.33 & 0.46 & 0.12 & 4.99 & 0.17 & 1.95 & 1.14 & 9.12$\pm$1.83\
759305 & 150.3527 & 2.4511 & 1.91 & 0.26 & 0.13 & 0.35 & 0.12 & 2.00 & 0.24 & 2.09 & 1.82 & $<$ 1.92\
516419 & 149.7464 & 2.0845 & 2.14 & 0.22 & 0.11 & 0.36 & 0.12 & 2.00 & 0.38 & 2.10 & 1.25 & $<$ 1.94\
401783 & 149.9235 & 1.9038 & 2.16 & 3.08 & 0.50 & 1.55 & 0.14 & 6.16 & 0.87 & 2.40 & 1.94 & 16.80$\pm$2.73\
414218 & 149.9098 & 1.9234 & 2.07 & 1.46 & 0.03 & 0.57 & 0.13 & 42.61 & 0.40 & 2.01 & 1.04 & 7.91$\pm$0.19\
444936 & 149.7446 & 1.9724 & 2.30 & 1.41 & 0.44 & 0.66 & 0.15 & 3.23 & 0.78 & 2.30 & 1.37 & 7.76$\pm$2.41\
95500 & 149.8838 & 1.9812 & 2.13 & 2.31 & 0.60 & 0.74 & 0.15 & 3.83 & 0.79 & 2.25 & 1.41 & 12.60$\pm$3.29\
254938 & 150.3699 & 1.6707 & 2.01 & 0.77 & 0.24 & 0.46 & 0.13 & 3.43 & 0.79 & 2.09 & 1.11 & 4.16$\pm$1.21\
818426 & 150.7220 & 2.5419 & 2.30 & 0.49 & 0.14 & 0.47 & 0.12 & 3.42 & 0.75 & 2.51 & 2.26 & 2.70$\pm$0.79\
482039 & 150.1189 & 2.0321 & 2.15 & 0.23 & 0.12 & 0.39 & 0.12 & 2.00 & 0.84 & 2.11 & 1.00 & $<$ 1.92\
575173 & 149.9899 & 2.1741 & 2.01 & 1.03 & 0.50 & 0.46 & 0.12 & 2.05 & 0.54 & 2.28 & 1.91 & 5.58$\pm$2.72\
672025 & 150.0164 & 2.3210 & 2.33 & 2.73 & 0.47 & 1.51 & 0.12 & 5.77 & 0.52 & 2.33 & 1.60 & 15.03$\pm$2.60\
254150 & 150.0934 & 2.5073 & 2.22 & 3.74 & 0.49 & 2.21 & 0.12 & 7.58 & 0.76 & 2.21 & 1.21 & 20.48$\pm$2.70\
514900 & 150.4552 & 2.0835 & 2.78 & 1.41 & 0.25 & 0.70 & 0.12 & 5.75 & 0.56 & 2.33 & 1.33 & 7.95$\pm$1.38\
283400 & 149.6042 & 1.7164 & 2.05 & 4.18 & 0.02 & 2.95 & 0.15 & 174.99 & 2.37 & 2.41 & 1.96 & 22.67$\pm$0.13\
311139 & 149.7768 & 1.7610 & 2.30 & 1.86 & 0.54 & 1.14 & 0.15 & 7.85 & 1.28 & 2.47 & 1.87 & 10.23$\pm$1.30\
277716 & 149.4757 & 2.5882 & 2.04 & 2.14 & 0.78 & 0.93 & 0.13 & 2.73 & 2.63 & 2.36 & 1.72 & 11.58$\pm$4.24\
909889 & 149.6367 & 2.6824 & 2.23 & 0.43 & 0.16 & 0.44 & 0.13 & 2.77 & 1.10 & 2.23 & 1.18 & 2.36$\pm$0.85\
919588 & 150.1266 & 2.6961 & 2.22 & 4.62 & 1.22 & 3.25 & 0.13 & 3.79 & 2.32 & 2.45 & 1.82 & 25.26$\pm$6.66\
932436 & 150.3178 & 2.7165 & 2.58 & 4.83 & 0.47 & 2.05 & 0.12 & 10.21 & 2.80 & 2.46 & 1.43 & 26.97$\pm$2.64\
969701 & 149.5260 & 2.7769 & 2.09 & 1.42 & 0.30 & 0.64 & 0.12 & 4.75 & 1.29 & 2.38 & 1.82 & 7.70$\pm$1.62\
830116 & 150.5398 & 2.5586 & 1.84 & 2.75 & 1.24 & 1.14 & 0.13 & 2.22 & 2.64 & 2.30 & 1.87 & 14.72$\pm$6.64\
561437 & 150.5270 & 2.1541 & 2.73 & 4.66 & 0.34 & 1.50 & 0.12 & 13.84 & 1.64 & 2.62 & 2.18 & 26.23$\pm$1.90\
421924 & 150.3745 & 1.9364 & 2.34 & 1.66 & 0.55 & 0.90 & 0.15 & 6.01 & 0.31 & 2.77 & 5.19 & 9.17$\pm$1.52\
464593 & 150.3526 & 2.0048 & 2.21 & 2.47 & 0.58 & 2.15 & 0.14 & 4.23 & 0.65 & 2.70 & 3.94 & 13.51$\pm$3.19\
287250 & 149.6534 & 1.7231 & 2.84 & 3.07 & 0.60 & 2.20 & 0.15 & 14.97 & 0.71 & 2.66 & 2.67 & 17.40$\pm$1.16\
903144 & 150.4592 & 2.6714 & 2.04 & 1.21 & 0.41 & 0.59 & 0.13 & 2.98 & 0.54 & 2.54 & 3.31 & 6.56$\pm$2.20\
917423 & 149.9921 & 2.6934 & 2.12 & 2.98 & 1.20 & 1.44 & 0.13 & 2.49 & 0.72 & 2.82 & 5.47 & 16.20$\pm$6.51\
953800 & 150.1102 & 2.7516 & 2.30 & 8.66 & 0.65 & 4.05 & 0.13 & 13.35 & 1.00 & 2.75 & 3.75 & 47.57$\pm$3.56\
821753 & 150.2927 & 2.5466 & 2.60 & 1.31 & 0.52 & 0.61 & 0.12 & 2.53 & 0.57 & 2.67 & 2.89 & 7.31$\pm$2.90\
274938 & 149.9980 & 2.5782 & 2.35 & 1.89 & 0.34 & 1.17 & 0.15 & 5.59 & 0.72 & 2.94 & 5.85 & 10.43$\pm$1.87\
476581 & 150.3899 & 2.0247 & 2.73 & 1.53 & 0.30 & 1.31 & 0.06 & 5.08 & 0.86 & 2.95 & 5.05 & 21.37$\pm$4.20\
562990 & 150.1608 & 2.1547 & 2.30 & 2.08 & 0.33 & 0.45 & 0.12 & 6.40 & 0.49 & 2.56 & 2.81 & 11.44$\pm$1.79\
122443 & 149.6588 & 2.0720 & 2.29 & 5.42 & 0.56 & 2.36 & 0.12 & 9.62 & 0.72 & 2.68 & 3.34 & 29.78$\pm$3.09\
514719 & 149.5358 & 2.0825 & 2.17 & 0.25 & 0.13 & 0.40 & 0.12 & 2.00 & 0.84 & 2.54 & 2.65 & $<$ 1.97\
338500 & 150.2649 & 1.8029 & 2.36 & 9.17 & 0.67 & 6.11 & 0.16 & 13.79 & 1.69 & 2.68 & 2.78 & 50.54$\pm$3.67\
372039 & 150.4384 & 1.8561 & 2.58 & 9.46 & 0.61 & 7.01 & 0.15 & 15.60 & 1.23 & 2.82 & 3.54 & 52.75$\pm$3.38\
427827 & 150.3416 & 1.9456 & 2.78 & 3.47 & 0.29 & 2.67 & 0.18 & 15.08 & 3.32 & 2.73 & 2.65 & 19.58$\pm$1.30\
842737 & 150.6338 & 2.5783 & 2.67 & 8.24 & 0.73 & 5.29 & 0.12 & 11.23 & 1.32 & 2.87 & 3.92 & 46.21$\pm$4.11\
932331 & 149.6556 & 2.7162 & 2.11 & 7.13 & 1.08 & 3.47 & 0.13 & 6.63 & 2.83 & 2.58 & 2.66 & 38.77$\pm$5.85\
942076 & 150.1471 & 2.7315 & 2.42 & 14.33 & 0.39 & 5.73 & 0.13 & 36.79 & 1.44 & 2.82 & 3.70 & 79.24$\pm$2.15\
264030 & 150.7057 & 2.5404 & 2.15 & 4.34 & 0.69 & 2.19 & 0.12 & 6.26 & 1.95 & 2.72 & 3.68 & 23.64$\pm$3.77\
495704 & 149.9889 & 2.0533 & 1.92 & 5.90 & 0.61 & 2.06 & 0.12 & 9.61 & 2.04 & 2.76 & 4.98 & 31.73$\pm$3.30\
723263 & 149.8893 & 2.3964 & 2.18 & 5.26 & 0.61 & 2.69 & 0.13 & 8.56 & 1.06 & 2.59 & 2.85 & 28.71$\pm$3.36\
126711 & 149.6679 & 2.0874 & 2.30 & 11.42 & 0.59 & 4.96 & 0.13 & 19.47 & 1.74 & 2.91 & 4.94 & 62.76$\pm$3.22\
518250 & 150.1799 & 2.0886 & 2.32 & 5.40 & 0.31 & 3.00 & 0.13 & 17.62 & 2.65 & 2.67 & 2.71 & 29.71$\pm$1.69\
135052 & 150.4957 & 2.1162 & 2.21 & 4.89 & 0.67 & 2.30 & 0.12 & 7.29 & 1.23 & 2.70 & 3.44 & 26.73$\pm$3.66\
408649 & 149.6658 & 1.9139 & 1.93 & 3.33 & 0.49 & 2.51 & 0.14 & 18.62 & 1.46 & 2.85 & 6.26 & 17.91$\pm$0.96\
980250 & 150.0161 & 2.7924 & 1.80 & 7.13 & 1.20 & 2.99 & 0.13 & 5.96 & 1.76 & 2.84 & 6.98 & 37.96$\pm$6.37\
815012 & 150.6034 & 2.5366 & 2.10 & 5.38 & 0.74 & 3.55 & 0.13 & 7.32 & 1.57 & 3.10 & 9.36 & 29.24$\pm$4.00\
\
\[midz\]
[rrrrrrrrrcccr]{}
566428 & 150.0300 & 2.1627 & 5.89 & 0.14 & 0.07 & 0.22 & 0.07 & 2.00 & 0.25 & 1.74 & 0.46 & $<$ 2.51\
457406 & 150.3921 & 1.9937 & 4.00 & 0.16 & 0.08 & 0.19 & 0.06 & 2.10 & 0.26 & 1.83 & 0.56 & 2.03$\pm$0.97\
286380 & 150.0598 & 1.7217 & 4.35 & 0.12 & 0.06 & 0.31 & 0.07 & 4.69 & 0.31 & 1.94 & 0.67 & 3.77$\pm$0.80\
477614 & 150.3071 & 2.0261 & 4.30 & 0.13 & 0.07 & 0.18 & 0.07 & 2.00 & 0.45 & 2.07 & 0.79 & $<$ 2.41\
249399 & 150.1373 & 2.4902 & 4.16 & 0.09 & 0.05 & 0.18 & 0.06 & 2.00 & 0.56 & 1.90 & 0.50 & $<$ 2.16\
735699 & 150.6181 & 2.4158 & 4.04 & 0.55 & 0.11 & 0.20 & 0.06 & 3.54 & 1.03 & 2.14 & 0.75 & 6.90$\pm$1.95\
608706 & 150.5920 & 2.2251 & 4.85 & 0.11 & 0.05 & 0.21 & 0.07 & 2.00 & 0.16 & 2.35 & 2.37 & $<$ 2.41\
972851 & 149.9827 & 2.7821 & 4.82 & 0.10 & 0.05 & 0.14 & 0.06 & 2.00 & 0.24 & 2.30 & 1.72 & $<$ 2.07\
284164 & 150.5189 & 2.6097 & 4.21 & 0.13 & 0.07 & 0.14 & 0.06 & 2.00 & 0.33 & 2.48 & 2.27 & $<$ 2.23\
386988 & 150.1413 & 1.8805 & 4.71 & 0.30 & 0.11 & 0.17 & 0.06 & 2.89 & 0.18 & 2.04 & 1.08 & 3.71$\pm$1.28\
256965 & 150.3371 & 1.6746 & 4.59 & 0.39 & 0.17 & 0.26 & 0.06 & 4.21 & 0.18 & 2.32 & 2.09 & 4.81$\pm$1.14\
41128 & 149.3435 & 1.7836 & 5.59 & 0.13 & 0.06 & 0.16 & 0.06 & 2.00 & 0.45 & 2.55 & 2.35 & $<$ 2.32\
582526 & 149.8712 & 2.1871 & 4.55 & 0.12 & 0.06 & 0.21 & 0.06 & 2.00 & 3.66 & 2.48 & 1.48 & $<$ 2.34\
331108 & 150.4637 & 2.7859 & 4.49 & 1.98 & 0.35 & 1.12 & 0.06 & 19.54 & 1.02 & 2.53 & 1.85 & 24.34$\pm$1.25\
901851 & 150.4011 & 2.6707 & 4.14 & 0.27 & 0.07 & 0.23 & 0.06 & 3.77 & 0.11 & 2.40 & 3.38 & 3.33$\pm$0.88\
302769 & 150.0692 & 1.7477 & 4.33 & 0.58 & 0.15 & 0.27 & 0.06 & 3.88 & 0.12 & 2.46 & 3.65 & 7.12$\pm$1.84\
307139 & 150.1546 & 1.7550 & 4.30 & 0.46 & 0.05 & 0.21 & 0.06 & 3.35 & 0.36 & 2.66 & 3.30 & 5.72$\pm$1.71\
881017 & 150.4657 & 2.6361 & 3.54 & 2.24 & 0.40 & 1.33 & 0.06 & 23.27 & 0.50 & 2.90 & 5.10 & 28.95$\pm$1.24\
468591 & 150.5352 & 2.0115 & 4.13 & 1.43 & 0.15 & 0.84 & 0.07 & 9.54 & 1.06 & 2.73 & 2.95 & 17.85$\pm$1.87\
536066 & 150.4204 & 2.1177 & 3.96 & 1.34 & 0.11 & 0.99 & 0.07 & 11.91 & 2.48 & 2.89 & 3.86 & 16.87$\pm$1.42\
564267 & 150.2446 & 2.1597 & 4.00 & 1.39 & 0.35 & 1.02 & 0.07 & 15.65 & 3.16 & 2.89 & 3.82 & 17.46$\pm$1.12\
480666 & 149.4872 & 2.0303 & 4.18 & 0.60 & 0.14 & 0.24 & 0.07 & 3.60 & 0.14 & 2.91 & 9.40 & 7.45$\pm$2.07\
315797 & 149.9304 & 1.7687 & 4.64 & 3.84 & 0.35 & 2.49 & 0.06 & 39.14 & 0.84 & 3.05 & 6.35 & 47.03$\pm$1.20\
\
\[highz\]
[^1]: The SMG galaxies are probably gravitationally lensed so we do not include their luminosities in this estimate of the dynamic range.
[^2]: These SPIRE data are far superior to the earlier SCUBA imaging for extended galaxies. The ground-based SCUBA observations, taken in beam chopping mode to remove atmospheric background, can cancel extended emission components.
|
---
abstract: 'Attention is an important cognition process of humans, which helps humans concentrate on critical information during their perception and learning. However, although many machine learning models can remember information of data, they have no the attention mechanism. For example, the long short-term memory (LSTM) network is able to remember sequential information, but it cannot pay special attention to part of the sequences. In this paper, we present a novel model called long short-term attention (LSTA), which seamlessly integrates the attention mechanism into the inner cell of LSTM. More than processing long short term dependencies, LSTA can focus on important information of the sequences with the attention mechanism. Extensive experiments demonstrate that LSTA outperforms LSTM and related models on the sequence learning tasks.'
author:
- Guoqiang Zhong
- Xin Lin
- Kang Chen
- Qingyang Li
- Kaizhu Huang
bibliography:
- 'bicsbib2.bib'
title: 'Long Short-Term Attention'
---
Introduction
============
With the attention mechanism, human can naturally focus on vital information and ignore irrelevant information during one’s perception and cognition [@cognitive; @show]. Based on this fact, many brain-inspired learning models have been deeply studied and widely applied in recent years [@Brain-Inspired; @CognitiveComputation; @Modelling; @BiologicallyInspired; @VisualAttention; @Where; @bottom; @videosatt]. However, although many machine learning models can learn effective representations of data and memorize the data information, they cannot pay attention to important part of the data. For instance, long short-term memory (LSTM) [@lstmone] is a widely used model for sequence learning. However, it lacks the attention mechanism. To address this problem, some work tries to apply the attention mechanism to LSTM. Nevertheless, most of these models only add the attention mechanisms outside the LSTM cells and have not thoroughly solved the issue that LSTM have no the attention mechanism itself [@show; @effective; @self-attentive].
In this paper, we propose a novel model called long short term attention (LSTA), which seamlessly integrates the attention mechanism into the inner cell of LSTM. In this case, LSTA can simultaneously remember historical information and notice crucial details in the sequences. In the experiments for sequence learning, we demonstrate the advantage of LSTA over LSTM.
The rest of this paper is organized as follows. In Section \[Related Work\], we introduce some previous work related to LSTA, including LSTM and some models using the attention mechanism. In Section \[Architecture\], we present LSTA in detail. In Section \[Experiments\], we report the experimental results on two sequence learning tasks, i.e. image classification and sentiment analysis. Section \[Conclusion\] concludes this paper.
Related Work {#Related Work}
============
In this section, we review some previous work related to LSTA, including LSTM and several models using the attention mechanism.
LSTM {#Normative}
----
LSTM is a powerful learning model for sequential data and has been widely applied in many areas, such as speech recognition and handwritten character recognition [@LSTM; @Bidirectional]. The cell of LSTM includes an input gate, a forget gate and an output gate. These gate and the state of the cell can be updated as follows: $$\label{equation1}
% \begin{aligned}
\boldsymbol{{f}_{t}=\sigma({W}_{f}[h_{t-1},x_t]+{b}_{f})},$$ $$\boldsymbol{{i}_{t}=\sigma({W}_{i}[h_{t-1},x_t]+{b}_{i})},\\$$ $$\boldsymbol{\tilde{C}_{t}=\tanh({W}_{\tilde{c}}[h_{t-1},x_t]+{b}_{\tilde{c}})},\\$$ $$\boldsymbol{{C}_{t}={f}_{t}\ast {C}_{t-1}+{i}_{t}\ast\tilde{C}_{t}},\\$$ $$\boldsymbol{{o}_{t}=\sigma({W}_{o}[h_{t-1},x_t]+{b}_{o})},\\$$ $$\boldsymbol{{h}_{t}={o}_{t}\ast\tanh({C}_{t})}.\\
% \end{aligned}$$ Here, $\boldsymbol{{W}_{f},{W}_{i},{W}_{\tilde{c}},{W}_{o}}$ are the weight parameters and $\boldsymbol{{b}_{f},{b}_{i},{b}_{\tilde{c}},{b}_{o}}$ are biases. The forget gate $\boldsymbol{{f}_{t}}$ primarily controls the cell state by forgetting the previous moment information. In a similar way, the input gate $\boldsymbol{{i}_{t}}$ and output gate $\boldsymbol{{o}_{t}}$ control the information that will be input to the LSTM cell and output at the current moment, respectively. These three gates are crucial parts of the LSTM cell, which is used to update the current state of the LSTM cell $\boldsymbol{{C}_{t}}$ and obtain new cell output $\boldsymbol{{h}_{t}}$. In order to optimize the performance of LSTM, many extensions of LSTM have been proposed recently [@speech; @wider; @Sequentially; @phased]. In [@predrnn], the spatiotemporal LSTM (ST-LSTM) units are designed for memorizing both spatial and temporal information. [@convolutional] introduces a convolutional LSTM (ConvLSTM), which extends the fully connected LSTM to have convolutional architectures in both the input-to-gate and gate-to-gate transitions. In addition, [@wider] introduces a tensorized LSTM model, which represent the hidden states with tensors.
As discussed above, LSTM and most of its extensions mainly focus on processing the sequential data, but cannot pay attention to the important information in the sequences. In this paper, we present a model that can integrate the attention mechanism into the inner-cell of LSTM.
Models Using the Attention Mechanism
------------------------------------
The primary function of the attention mechanism is selection and allocation [@cognitive; @Where]. It leads to quick processing of information, with an efficient information choice and concentration of the computing power on the crucial tasks [@cognitive]. [@control] introduces the attention mechanism in the human cognitive system, with which human pays attention to the noteworthy information and ignores irrespective information [@show; @control; @AttentionBased]. In the cognitive computation area, the attention mechanism has been widely applied, such as the work to resolve the human visual neural computational problem [@CognitiveAttention] and that to model the retrieval mechanism of associations from the associative memory [@TheRole].
In particular, a large amount of attention based deep learning models have been proposed in recent years. For example, [@structured] presents the structured attention networks, which incorporate graphical models to generalize simple attention. Alternatively, [@attention] introduces a self-attention mechanism model, which is applied to replace the common recurrent and convolutional models. It relies entirely on the attention mechanism to compute representations of its input and output. Moreover, in [@Modelling], the selective attention for identification model (SAIM) is applied to visual search applications. The SAIM simulates the human ability to complete translation invariant recognition of multiple scenes. Additionally, in [@end], a recurrent attention mechanism network is proposed. It is an end-to-end memory learning model used on several language modeling tasks.
As mentioned above, many attention based methods have been proposed to address visual or language processing problems. However, rare work has integrated the attention mechanism into the cell of LSTM to improve its performance in sequence learning.
Long Short-Term Attention {#Architecture}
=========================
In this section, we introduce the proposed long short-term attention (LSTA) model in detail, which seamlessly integrates the attention mechanism into the cell of LSTM. For clarity, we first introduce the added attention gate in Sec. \[AttentionBlock\], and then the architecture and learning of LSTA in Sec. \[LSTA\].
![An illustration of the attention gate.[]{data-label="Attentionblock"}](Attentiongate.png){width="3in"}
The Attention Gate {#AttentionBlock}
------------------
Fig. \[Attentionblock\] shows the structure of the attention gate of LSTA, which accepts the inputs from the input gate and the forget gate. Eq. (\[equation4\]) is the update formula of the attention gate: $$\label{equation4}
\boldsymbol{{A}_{t}}=\boldsymbol{\psi(\hat{A}_{t}[f_t, i_t], \tilde{A}_{t}[f_t, i_t])}=\boldsymbol{\hat{A}_{t}\bigotimes\tilde{A}_{t}},$$ where $\boldsymbol{\hat{A}_{t}}$ and $\boldsymbol{\tilde{A}_{t}}$ are defined as follows: $$\label{equation23}
%\begin{aligned}
\boldsymbol{\hat{A}_{t}=\sigma({W}_{\hat{a}}[f_t, i_t]+{b}_{\hat{a}})},\\$$ $$\boldsymbol{\tilde{A}_{t}=\tanh({W}_{\tilde{a}}[f_t, i_t]+{b}_{\tilde{a}})}.
%\end{aligned}$$
Here, $\boldsymbol{{W}_{\tilde{a}}}$ and $\boldsymbol{{W}_{\hat{a}}}$ are weight parameters, while $\boldsymbol{{b}_{\tilde{a}}}$ and $\boldsymbol{{b}_{\hat{a}}}$ are biases. The sigmoid function $\boldsymbol{\sigma}$ is employed to compute $\boldsymbol{\hat{A}_{t}}$, which indicates the ratios of the attention elements as shown in Eq. (\[equation4\]). Similarly, the $\boldsymbol{\tanh}$ function is used to get the candidate attention values $\boldsymbol{\tilde{A}_{t}}$, which can be positive or negative.
In Eq. (\[equation4\]), $\bigotimes$ represents the element-wise multiplication. We multiply the elements between $\boldsymbol{\tilde{A}_{t}}$ and $\boldsymbol{\hat{A}_{t}}$ to obtain the output of the attention gate $\boldsymbol{{A}_{t}}$. The attention gate determines the attention distribution on the information at the current cell. In the following, we introduce how it can be seamlessly integrated into the cell of LSTM.
LSTA {#LSTA}
----
In order to endow the attention mechanism to LSTM, we propose the LSTA model which integrates the attention gate introduced above inside the LSTM cell. Fig. \[figure2\] is a diagram of the LSTA cell. Particularly, LSTA can pay attention to important information in the sequences during its learning process.
![The LSTA cell. The module with red color is the attention gate.[]{data-label="figure2"}](LSTA.png){width="2.5in"}
LSTA inherits the three gates of LSTM. For the update of its cell state, we can compute it as $$\label{equation5}
\begin{split}
\boldsymbol{\hat{C}_{t}}&=\boldsymbol{{C}_{t}+{A}_{t}}\\
&=\boldsymbol{{f}_{t}\ast{C}_{t-1}+{i}_{t}\ast\tilde{C}_{t}+{A}_{t}}\\
&=\boldsymbol{{f}_{t}\ast{C}_{t-1}+{i}_{t}\ast\tilde{C}_{t}+\boldsymbol{\tilde{A}_{t}\bigotimes\hat{A}_{t}}}.\\
\end{split}$$ Here, $\boldsymbol{{A}_{t}}$ is the output of the attention gate and $\boldsymbol{{C}_{t}}$ is the original LSTM cell state. In this case, we integrate the attention mechanism into LSTM unit, such that the new model, LSTA, can not only memorize the sequential information, but also pay attention to important information in the sequences.
Accordingly, the output gate of LSTA can be updated as $$\begin{aligned}
\label{equationh}
&\boldsymbol{{h}_{t}={o}_{t}\ast\tanh(\hat{C}_{t})}.\end{aligned}$$
Note that, in LSTA, the attention mechanise is applied inside the LSTM cell, unlike previous attention based LSTM models, in which attention is added after the whole sequence has been handled by all the LSTM cells. Therefore, LSTA is quite different from LSTM and most of its attention based variants. LSTA enables the sequence learning to focus on important parts of the input data and automatically ignore irrelevant parts, so as to improve its performance.
Experiments {#Experiments}
===========
To evaluate the proposed model LSTA, we have conducted extensive experiments on two sequence learning tasks, image classification and semantic analysis. In the following, we report the experimental settings and results.
Experiments on the Image Classification Task {#Image}
--------------------------------------------
In this section, we used the MNIST and Fashion-MNIST data sets to test the performance of LSTA. MNIST is a handwritten digit data set. It contains seventy thousands of $28 \times 28$ gray scale images, which belong to 10 classes. For all the images, 60,000 are used for training and the other 10,000 for test [@mnist]. Alternatively, Fashion-MNIST is an image data set, while its image format and number are both the same as the MNIST data set [@fashion]. In our experiments, we considered the rows of an image as sequential data to perform image classification.
For testing the performance of LSTA, we compared it with some relevant models. As LSTA integrates the attention mechanism into the LSTM cell, the most closely related model to LSTA is LSTM. Hence, we set LSTM as our baseline. Furthermore, we also compared LSTA with gated recurrent unit (GRU) [@ChungGCB14], bidirectional LSTM (Bi-LSTM) [@graves2005framewise] and nested LSTM (NLSTM) [@NLSTM] in our experiments. Note that, although there are many attention based variants of LSTM, they are quite different from LSTA. We can also apply the attention mechanism outside the LSTA cell as same as them. Hence, we have not compared with them in our work.
Table \[tableFS\] shows the image classification results obtained by LSTA and the compared models on the MNIST and Fashion-MNIST data sets. As we can see, LSTA outperforms all the compared models consistently. This demonstrate the advantage of LSTA over LSTM and its variants and the importance of the attention mechanism during sequence learning.
\[tableFS\]
To further analyze the advantage of LSTA over LSTM, we draw the learning curves of LSTM and LSTA obtained on the Fashion-MNIST data set in Fig. \[QualitativeF\]. Note that, we used the same loss function for LSTM and LSTA in our experiments. Fig. \[QualitativeF\] (a) shows the accuracy curves against the training steps, while Fig. \[QualitativeF\] (b) shows the loss curves against the training steps. As can be seen, due to the attention mechanism, LSTA consistently performs better, and converges faster than LSTM.
Experiments on the Sentiment Analysis Task {#sentimentanalysis}
------------------------------------------
Sentiment analysis is an interesting and important learning task [@SentimentAnalysis; @SemEval14; @Twitter]. In order to evaluate the performance of LSTA, we conducted experiments on both the classical sentiment analysis and aspect based sentiment analysis.
### Classical Sentiment Analysis {#IMDB}
In this experiment, we used the internet movie review database (IMDB) [@SentimentAnalysis] to test LSTA on classical sentiment analysis. IMDB is a crawler data set about the internet movie reviews. Based on the emotion of the reviews, it divides all the film reviews into the positive and negative categories.
In our work, we compared LSTA with LSTM and hybrid deep belief network (HDBN) [@yan2015learning], which is an effective deep network for sentiment analysis. The error rate and the running time of LSTA and the compared models are depicted in Fig. \[figureimdb\] and \[figureruntime\], respectively. As we can see, among the compared model, LSTA obtained the best classification accuracy and used the least running time.
### Aspect Based Sentiment Analysis {#aspect}
Aspect based sentiment analysis is one of the important tasks of semantic analysis [@Cabasc]. In order to verify the effect of LSTA on aspect based sentiment analysis, we conducted experiments on two data sets. One was the SemEval-2014 Task 4 (SemEval14) data set[@SemEval14], which contains two domains (Restaurant and Laptop). The other was the Twitter data set collected by Dong et al. [@Twitter]. In these two data sets, the aspect terms of each review are labeled by three sentiment polarities, which are positive, neutral and negative, respectively. For example, about an aspect term *fajitas*, when it is in a sentence *“I loved their fajitas, but the service is horrible."*, its polarity is positive, but for aspect term *service*, its polarity is negative. Concretely, the statistics of the two data sets are provided in Table \[tablesem\].
\[tablesem\]
In this experiment, we used Accuracy and Macro-averaged F-measure (Macro-F1) as the metrics to evaluate the effect of LSTA and the compared models [@DuyuTangSentiment; @Twitter]. The experimental results obtained by LSTA and the compared methods are shown in Table \[tablesemre\]. In this table, “Cabasc" is a content attention model for aspect based sentiment analysis [@Cabasc]. “ATAE-LSTM" is an attention-based LSTM with aspect embedding, which can focus on the parts of a sentence when several aspects are taken as input [@ATAE-LSTM].
From Table \[tablesemre\], we can see that LSTA performs best among the compared models. That is, LSTA outperforms both LSTM and previous attention models, including that apply the attention mechanism outside the cell of LSTM.
\[tablesemre\]
Conclusion {#Conclusion}
==========
In this paper, we present a novel LSTA model to alleviate the problem that LSTM lacks the attention mechanism. The key idea behind this model is to seamlessly integrate the attention mechanism into the cell of LSTM. Experiments demonstrate that LSTA performs better than LSTM, many variants of LSTM and related attention models. Hence, LSTA can be seen as a substitute of LSTM in the sequence learning tasks.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the National Key R&D Program of China under Grant No. 2016YFC1401004, the National Natural Science Foundation of China (NSFC) under Grant No. 41706010, and No. 61876155, the Science and Technology Program of Qingdao under Grant No. 17-3-3-20-nsh, the CERNET Innovation Project under Grant No. NGII20170416, and the Fundamental Research Funds for the Central Universities of China.
|
---
abstract: 'We make an important observation regarding iterated symmetric extensions which allows us to easily iterate certain counterexamples to the axiom of choice. We use this to improve previous results and to obtain a model where: (1) Every partially ordered set can be embedded into the cardinals; and (2) for every non-empty set $X$ there is a set $A$ which is the countable union of countable sets, and $\mathcal P(A)$ has a Dedekind-finite subset that can be mapped onto $X$. We also provide an outline for a construction of a model where every field admits a non-trivial vector space whose endomorphisms are all scalar multiplications. This method can be applied to similar “$\forall\exists$-$\exists\forall$ problems”.'
address: 'School of Mathematics, University of East Anglia. Norwich, NR4 7TJ, UK '
author:
- Asaf Karagila
date: 'November 26, 2019'
title: Iterated failures of choice
---
Introduction
============
Cohen proved that the axiom of choice does not follow from the axioms of the Zermelo–Fraenkel set theory in 1963 by introducing the technique of forcing and mimicking the known Fraenkel–Mostowski–Specker constructions which assume the existence of atoms (or urelements). This opened the flood gates for new results and during the 1960s and 1970s many proofs were written which explored the many ways a universe of set theory can be an extremely counterintuitive place for doing mathematics in the absence of the axiom of choice.
Most of the proofs were local, in the sense that for a given set we can build a specific extension of the universe where the axiom of choice fails and we have a certain witness for a certain failure of choice related to our set (e.g. have a Dedekind-finite set which maps onto that set [@Monro:1975]). But there were not that many successful attempts in constructing global results, namely extending the universe *once*, so that for every set the counterexample can be found in that extension. This is what we call a $\forall\exists$-$\exists\forall$ problem: *assuming for every set there is an extension where the set has a certain property, is there an extension in which all the sets have the property?*
One of the obvious solutions for a $\forall\exists$-$\exists\forall$ problem is to iterate our construction, and each step solve it for more and more sets (including new sets). But an iteration method for symmetric extensions—the main tool to construct models where the axiom of choice fails—did not exist until very recently, and forcing over models of ${{\mathsf{ZF}}}$ is a task rife with difficulties. Some results are obtained by products, but those only cover the ground model sets, or in general sets which can be well-ordered. We can solve this by showing that the resulting models might have some particular property that it is enough to work out the solution for ground model sets, but this is no an easy task either.
Another significant problem, to which is what we provide a partial solution in this paper, is the fact that forcing over models of ${{\mathsf{ZF}}}$ may introduce back the axiom of choice to initial segment of the model. For example, adding a subset to $\omega_1$ by countable approximation will either add new subsets of $\omega$, or a well-ordering of $\mathcal P(\omega)$ (the ground model’s version of this set, of course). This means that unlike the case in ${{\mathsf{ZFC}}}$ where we can just iterate forcings which add more and more information, in ${{\mathsf{ZF}}}$ it is a lot less trivial to ensure that (1) no sets of low rank are added; and (2) no well-orderings of old sets are added.
In this paper we point out a known observation can be applied to iterating of symmetric extensions to provide a framework for dealing with some $\forall\exists$-$\exists\forall$ problems.
We show that a minor modification (in line with the observation) of the previous work of the author in [@Karagila:Morris] (which was exactly a partial solution that covered all ground model sets, but not necessarily all sets) we can in fact obtain a general framework for $\exists\forall$ solutions, and we use it to prove that every partial order can be embedded into the cardinals of a model (extending [@Jech:ORD1966] and [@Tahakashi:1968] which obtained a local solution, and [@Karagila:Embeddings] where a global solution for ground model is shown).[^1] In addition we prove that in this model every set can be the surjective image of a Dedekind-finite set (extending the local, and global for ground model solutions of [@Monro:1975]).
We also outline the construction of a model in which every field has a vector space whose endomorphisms are only scalar multiplications (this extends the work of the author in [@Karagila:MSc], which is based on [@Lauchli:1963].)
The technical framework for iterating symmetric extensions will not be described here in full, since we circumvent its full power, but we hope that this work will encourage others to study it, as it can certainly be improved.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author is indebted to Boban Veličković for his invitation to Paris and for the opportunity to talk about related work, as well as to the other set theorists of the IMJ–PRG, University of Paris, and to Matteo Viale for asking many hard hitting questions and pushing the author to make these improvements.
Preliminaries
=============
Our treatment of forcing is standard. We say that ${\mathbb P}$ is a notion of forcing if ${\mathbb P}$ is a preordered set with a maximum, $1_{\mathbb P}$. We write $q\leq p$ to mean that $q$ is a stronger condition. Two conditions are compatible if they have a common extension, and are incompatible otherwise. If $\{\dot x_i\mid i\in I\}$ is a collection of ${\mathbb P}$-names, we write $\{\dot x_i\mid i\in I\}^\bullet$ to denote “the obvious ${\mathbb P}$-name” is creates, namely $\{{\langle1_{\mathbb P},\dot x_i\rangle}\mid i\in I\}^\bullet$. This notation extends to ordered pairs, sequences, etc. This also somewhat simplifies the canonical names for ground model sets, as we can now write $\check x=\{\check y\mid y\in x\}^\bullet$.
We will use the following group theoretic notion of the wreath product. If $A$ and $B$ are two sets, and $G\subseteq S_A$, $H\subseteq S_B$, where $S_X$ is the group of all permutations of $X$, then $G\wr H$ is a group of permutations of $A\times B$. If $\pi\in G\wr H$, then there are $\pi_A\in G$ and ${\langle\pi_a\mid a\in A\rangle}$ such that $\pi_a\in H$, and $\pi(a,b)={\langle\pi_A(a),f_a(b)\rangle}$.
Symmetric extensions and (some) iterations thereof
--------------------------------------------------
To violate the axiom of choice we cannot use forcing on its own, as forcing preserves the axiom of choice in the ground model. In order to violate the axiom of choice we need to pass from the generic extension, $V[G]$, to an inner model $M$ where it fails. The method of symmetric extensions allows us to identify an appropriate class of names which define such model $M$. Iterating symmetric extensions was developed by the author in [@Karagila:Iterations], and while the full theory is not trivial at all, we will only need a smaller fraction of it here. We start by defining symmetric extensions.
Let ${\mathbb P}$ be a notion of forcing, and let $\pi$ be an automorphism of ${\mathbb P}$. We can extend $\pi$ to act on ${\mathbb P}$-names by recursion,$$\pi\dot x=\{{\langle\pi p,\pi\dot y\rangle}\mid{\langlep,\dot y\rangle}\in\dot x\}.$$ As the forcing relation is defined from the order of ${\mathbb P}$, the following lemma should not be surprising.
$p{\mathrel{\Vdash}}\varphi(\dot x)\iff\pi p{\mathrel{\Vdash}}\varphi(\pi\dot x)$.
Fix ${\mathscr G}\subseteq{\mathbb P}$, and denote by $\operatorname{sym}_{\mathscr G}(\dot x)$ the subgroup $\{\pi\in{\mathscr G}\mid\pi\dot x=\dot x\}$. This is sometimes called the *stabiliser* of $\dot x$. We want to have a way to say that a name is stable under “most” of the automorphisms (in ${\mathscr G}$). And so we need a suitable notion of a filter.
We say that ${\mathscr F}$ is a filter of subgroups of ${\mathscr G}$ if it is a filter on the lattice of subgroups. Namely, it is a non-empty family of subgroups of ${\mathscr G}$ which is closed under finite intersections and supergroups. We say that ${\mathscr F}$ is *normal* if for every $H\in{\mathscr F}$ and every $\pi\in{\mathscr G}$, $\pi H\pi^{-1}\in{\mathscr G}$. We will say that ${\langle{\mathbb P},{\mathscr G},{\mathscr F}\rangle}$ is a *symmetric system* if ${\mathbb P}$ is a notion of forcing, ${\mathscr G}$ is a group of automorphisms of ${\mathbb P}$, and ${\mathscr F}$ is a normal filter of subgroups of ${\mathscr G}$. It is easier to assume only the case where ${\mathscr F}$ is a base for a normal filter, since this is preserved when extending the universe (and perhaps adding new subgroups to ${\mathscr G}$) and we will do so implicitly.
If $\operatorname{sym}_{\mathscr G}(\dot x)\in{\mathscr F}$, we say that $\dot x$ is *${\mathscr F}$-symmetric*, and if the condition holds hereditarily to names which are in $\dot x$, we say that it is *hereditarily ${\mathscr F}$-symmetric*. We write ${{\mathsf{HS}}}_{\mathscr F}$ to denote the class of hereditarily ${\mathscr F}$-symmetric names.
Let ${\langle{\mathbb P},{\mathscr G},{\mathscr F}\rangle}$ be a symmetric system, and let $G\subseteq{\mathbb P}$ be a $V$-generic filter. The class $M={{\mathsf{HS}}}_{\mathscr F}^G=\{\dot x^G\mid\dot x\in{{\mathsf{HS}}}_{\mathscr F}\}$ is a transitive class model of ${{\mathsf{ZF}}}$ inside $V[G]$ which contains $V$.
This model, $M$, is called a *symmetric extension*. The forcing relation relativises to ${{\mathsf{HS}}}_{\mathscr F}$, namely $p{\mathrel{\Vdash}}^{{\mathsf{HS}}}\varphi(\dot x)$ when $\dot x\in{{\mathsf{HS}}}_{\mathscr F}$ and $p{\mathrel{\Vdash}}\varphi^{{\mathsf{HS}}}(\dot x)$. The usual truth lemma holds for ${\mathrel{\Vdash}}^{{\mathsf{HS}}}$ and the Symmetry Lemma holds as well, assuming $\pi\in{\mathscr G}$.
The next step, after taking one symmetric extension, is to take many. This can be done simultaneously by a product or iteratively by an iteration (which is a general case of a product also in the “usual” context of forcing). We will not cover the whole apparatus for iterating symmetric extensions in this paper, but give a very informal account of the idea behind it.
If ${\langle{\mathbb Q}_0,{\mathscr G}_0,{\mathscr F}_0\rangle}$ is a symmetric system, and $M_0$ is the symmetric extension it defines after fixing some $V$-generic filter $G_0\subseteq{\mathbb Q}_0$, we want to take a symmetric extension of $M_0$. Say ${\langle{\mathbb Q}_1,{\mathscr G}_1,{\mathscr F}_1\rangle}$ is the second symmetric system. By the definition of $M_0$ there is a name ${\langle\dot{\mathbb Q}_1,\dot{\mathscr G}_1,\dot{\mathscr F}_1\rangle}^\bullet\in{{\mathsf{HS}}}_{{\mathscr F}_0}$ which is interpreted as the symmetric system, and there is some condition $p\in G_0$ forcing that this is a name of a symmetric system. Our goal is to identify a class of ${\mathbb Q}_0\ast\dot{\mathbb Q}_1$-names which will predict the second symmetric extension, as well as understand the conditions necessary for this process to continue in a coherent way.
So when is a name going to be interpreted in this intermediate model? It has to project to a ${\mathbb Q}_0$-name which is in ${{\mathsf{HS}}}_{{\mathscr F}_0}$ and be forced to be a $\dot{\mathbb Q}_1$-name that satisfies the property of being in ${{\mathsf{HS}}}_{{\mathscr F}_1}^\bullet$. In particular, that means there is a group $H_0\in{\mathscr F}_0$ and a name for a group $\dot H_1$ forced to be in $\dot{\mathscr F}_1$, such that automorphisms which live in the groups preserve the name at each step.
We can use this to derive a direct definition which looks a bit like that of a symmetric extension. We first observe the following: if $\dot x$ is a ${\mathbb Q}_0$-name and $p{\mathrel{\Vdash}}``\dot x$ is a $\dot{\mathbb Q}_1$-name”, then if $\pi$ is an automorphism, $\pi p{\mathrel{\Vdash}}``\pi\dot x$ is a $\pi\dot{\mathbb Q}_1$-name”. This leads us to the following definition which is a necessary condition for the apparatus to run smoothly.
Let ${\mathbb P}$ be a forcing, $\pi\in\operatorname{Aut}({\mathbb P})$ and $\dot A$ a ${\mathbb P}$-name. We say that $\pi$ *respects* $\dot A$ if $1{\mathrel{\Vdash}}\pi\dot A=\dot A$. If $\dot A$ has an implicit structure (e.g. it is a name for a forcing or a symmetric system) then we also implicitly require that the structure is respected.
If every $\pi\in{\mathscr G}$ respects the name for $\dot{\mathbb Q}_1$, then ${\langleq_0,\dot q_1\rangle}\mapsto{\langle\pi q_0,\pi\dot q_1\rangle}$ is indeed an automorphism of ${\mathbb Q}_0\ast\dot{\mathbb Q}_1$. Moreover, if $\dot\pi$ is a name for an automorphism of $\dot{\mathbb Q}_1$, then ${\langleq_0,\dot q_1\rangle}\mapsto{\langleq_0,\dot\pi\dot q_1\rangle}$ is also an automorphism of ${\mathbb Q}_0\ast\dot{\mathbb Q}_1$.
We can therefore combine ${\mathscr G}_0$ and $\dot{\mathscr G}_1$ to an automorphism group of ${\mathbb Q}_0\ast\dot{\mathbb Q}_1$. To simplify our statements in this section, we will set up the context: ${\langle{\mathbb Q}_0,{\mathscr G}_0,{\mathscr F}_0\rangle}$ is a symmetric system and ${\langle\dot{\mathbb Q}_1,\dot{\mathscr G}_1,\dot{\mathscr F}_1\rangle}^\bullet\in{{\mathsf{HS}}}_{{\mathscr F}_0}$ is a name forced to be a symmetric system which is also respected by ${\mathscr G}_0$.
If ${\langle\pi_0,\dot\pi_1\rangle}$ is a pair such that $\pi_0\in{\mathscr G}_0$ and ${\mathrel{\Vdash}}_{{\mathbb Q}_0}\dot\pi_0\in\dot{\mathscr G}_0$, then we define the automorphism ${{\textstyle\int_{{\langle\pi_0,\dot\pi_1\rangle}}}}$ of ${\mathbb Q}_0\ast\dot{\mathbb Q}_1$ as follows: $${{\textstyle\int_{{\langle\pi_0,\dot\pi_1\rangle}}}}{\langleq_0,\dot q_1\rangle}={\langle\pi_0 q_0,\pi_0(\dot\pi_1\dot q_1)\rangle}={\langle\pi_0,\pi_0(\dot\pi_1)(\pi_0\dot q_1)\rangle}.$$ We let ${\mathcal G}_1={\mathscr G}_0\ast\dot{\mathscr G}_1$ denote the group of all such automorphisms, and we call it the *generic semi-direct product*.
Next we need to handle the filters of groups. If $\dot x$ is a name which is to identify a set in the iterated symmetric extension, we essentially have $H_0\in{\mathscr F}_0$ and some name $\dot H_1$ forced to be in $\dot{\mathscr F}_1$ such that whenever $\pi_0\in H_0$ and ${\mathrel{\Vdash}}_{{\mathbb Q}_0}\dot\pi_1\in\dot H_1$ (which we will abbreviate as ${\mathrel{\Vdash}}{\langle\pi_0,\dot\pi_1\rangle}\in{\langleH_0,\dot H_1\rangle}$), then ${{\textstyle\int_{{\langle\pi_0,\dot\pi_1\rangle}}}}$ respects $\dot x$. We will write ${\mathcal F}_1={\mathscr F}_0\ast\dot{\mathscr F}_1$ to denote the collection of these pairs which we call *${\mathcal F}_1$-supports*.
${\langleH_0,\dot H_1\rangle}$ is an ${\mathcal F}_1$-support of $\dot x$ if whenever $p{\mathrel{\Vdash}}{\langle\pi_0,\dot\pi_1\rangle}\in{\langleH_0,\dot H_1\rangle}$, then $p{\mathrel{\Vdash}}{{\textstyle\int_{{\langle\pi_0,\dot\pi_1\rangle}}}}\dot x=\dot x$. In this case we say that $\dot x$ is *${\mathcal F}_1$-respected*, and if the property holds hereditarily we say that it is *hereditarily ${\mathcal F}_1$-respected*. We denote by ${{\mathsf{IS}}}_1$ the class of all hereditarily ${\mathcal F}_1$-respected names.
We can now extend this to any length with a finite support iteration.
Suppose that ${\langle\dot{\mathbb Q}_\alpha,\dot{\mathscr G}_\alpha,\dot{\mathscr F}_\alpha\mid\alpha<\delta\rangle}$ and ${\langle{\mathbb P}_\alpha,{\mathcal G}_\alpha,{\mathcal F}_\alpha\mid\alpha\leq\delta\rangle}$ are sequences satisfying the following:
1. ${\mathbb P}_\alpha$ is the finite support iteration of $\dot{\mathbb Q}_\alpha$.
2. ${\mathrel{\Vdash}}_\alpha{\langle\dot{\mathbb Q}_\alpha,\dot{\mathscr G}_\alpha,\dot{\mathscr F}_\alpha\rangle}^\bullet$ is a symmetric system.
3. ${\mathcal G}_0=\{\operatorname{id}\}$, ${\mathcal G}_{\alpha+1}={\mathcal G}_\alpha\ast\dot{\mathscr G}_\alpha$, and if $\alpha$ is a limit, then ${\mathcal G}_\alpha$ is the direct limit of ${\mathcal G}_\beta$ for $\beta<\alpha$.
4. ${\mathcal F}_0=\{{\mathcal G}_0\}$, ${\mathcal F}_{\alpha+1}={\mathcal F}_\alpha\ast\dot{\mathscr F}_\alpha$, and if $\alpha$ is limit, then ${\mathcal F}_\alpha$ is the direct limit of ${\mathcal F}_\beta$ for $\beta<\alpha$, i.e. the collection of sequences ${\langle\dot H_\beta\mid\beta<\alpha\rangle}$ such that ${\mathrel{\Vdash}}_\beta\dot H_\beta\in\dot{\mathscr F}_\beta$ and for all but finitely many $\beta<\alpha$, ${\mathrel{\Vdash}}_\beta\dot H_\beta=\dot{\mathscr G}_\beta$.
5. For each $\alpha$, ${\langle\dot{\mathbb Q}_\alpha,\dot{\mathscr G}_\alpha,\dot{\mathscr F}_\alpha\rangle}^\bullet$ is hereditarily ${\mathcal F}_\alpha$-respected, and the name itself is respected by any automorphism in ${\mathcal G}_\alpha$.
Then the class ${{\mathsf{IS}}}_\alpha$ for $\alpha\leq\delta$ is a class of ${\mathbb P}_\alpha$-names which predicts the iteration of symmetric extensions. Moreover, ${{\mathsf{IS}}}_{\alpha+1}$ will be a symmetric extension of ${{\mathsf{IS}}}_\alpha$ by ${\langle\dot{\mathbb Q}_\alpha,\dot{\mathscr G}_\alpha,\dot{\mathscr F}_\alpha\rangle}^\bullet$.
We also have a relativised forcing relation, ${\mathrel{\Vdash}}^{{\mathsf{IS}}}$, which is defined similarly as ${\mathrel{\Vdash}}^{{\mathsf{HS}}}$.
\[thm:preservation\] Let ${\langle\dot{\mathbb Q}_\alpha,\dot{\mathscr G}_\alpha,\dot{\mathscr F}_\alpha\mid\alpha<\delta\rangle}$ be a symmetric iteration such that for all $\alpha$, ${\mathrel{\Vdash}}_\alpha``\dot{\mathscr G}_\alpha$ witnesses the homogeneity of $\dot{\mathbb Q}_\alpha$” and let $\eta$ be some ordinal such that there is $\alpha<\delta$ such that for all $\beta\in[\alpha,\delta)$,$${\mathrel{\Vdash}}_\beta^{{\mathsf{IS}}}{\langle\dot{\mathbb Q}_\beta,\dot{\mathscr G}_\beta,\dot{\mathscr F}_\beta\rangle}^\bullet\text{ does not add new sets of rank }<\check\eta,$$ and let $G\subseteq{\mathbb P}_\delta$ be a $V$-generic filter. Then $V_\eta^{{{\mathsf{IS}}}_\alpha^{G{\mathbin\upharpoonright}\alpha}}=V_\eta^{{{\mathsf{IS}}}_\delta^G}$.
In other words, if each symmetric extension is homogeneous, and we do not add sets of rank $\eta$ on a tail below $\delta$, then we do not add such sets at the $\delta$th stage either. This is very important, as non-trivial forcing will tend to add Cohen reals at limit steps, or even collapse cardinals if we are not careful about our chain conditions.
Almost as importantly, this means that if we guarantee increasing distributivity and homogeneity, then we may iterate even class length, while preserving ${{\mathsf{ZF}}}$ in the final model.
We diverge from [@Karagila:Iterations] in the definition of supports, as we do not discuss where the names $\dot H_\alpha$ come from, and seem to hint that they are ${\mathbb P}_\alpha$-names. This is fine, and due to the finite support nature of the iteration, it is also equivalent to the definition given in the paper. The paper, however, points out that we utilise mixing over antichains to define the automorphisms and the iteration anyway, and there is something to be gained by allowing $\dot H_\alpha$ to be, in fact, a ${\mathbb P}_\delta$-name for a member of $\dot{\mathscr F}_\alpha$.
In other words, we are allowed to hold off on choosing our pointwise groups until much later in the iteration. This has a certain elegance to it, and it is certainly useful in smoothing out the definition (although causing bumps elsewhere). Nevertheless, we do not really care for this here, since our situation is going to be quite specific.
Iterated generic copies
=======================
Let $V$ be a model of ${{\mathsf{ZF}}}$. Suppose that ${\mathbb Q}$ is a forcing which does not add sets of rank $<\alpha$. Let ${\mathbb P}$ be a finite support product of ${\mathbb Q}$ indexed by $I$, and let ${\mathscr G}$ be a subgroup of $S_I$ satisfying that for every finite $E,F,F'\subseteq I$ there is a function $\pi\in{\mathscr G}$ such that $\pi{\mathbin\upharpoonright}E=\operatorname{id}$ and $\pi``(F\setminus E)\cap F'=\varnothing$. We define the action of ${\mathscr G}$ on ${\mathbb P}$ in the natural way: $\pi p(\pi i)=p(i)$. And let ${\mathscr F}$ be the filter generated by $\operatorname{fix}(E)=\{\pi\in{\mathscr G}\mid\pi{\mathbin\upharpoonright}E=\operatorname{id}\}$ for a finite $E$.
If $\dot x\in{{\mathsf{HS}}}$ such that ${\mathrel{\Vdash}}\operatorname{rank}(\dot x)<\check\alpha$, then for every $p$ there is $q\leq p$ and $y\in V_\alpha$ such that $q{\mathrel{\Vdash}}\dot x=\check y$.
We prove this by induction on $\dot x$, so we may assume that if $\dot u$ appears in $\dot x$, then it $\check u$ for some $u\in V_\alpha$. Now suppose that $p{\mathrel{\Vdash}}\check u\in\dot x$, for some $u\in V_\alpha$, then we claim $p{\mathbin\upharpoonright}E{\mathrel{\Vdash}}\check u\in\dot x$.
To see why, simply note that if $p'\leq p{\mathbin\upharpoonright}E$, then by the condition on ${\mathscr G}$ there is some $\pi\in\operatorname{fix}(E)$ such that $\pi$ moves $\operatorname{dom}p\setminus E$ to be disjoint from $\operatorname{dom}p'\setminus E$. Since $\pi p{\mathrel{\Vdash}}\check u\in\dot x$, it follows that $p'$ cannot force the opposite. In particular this means that we may replace $\dot x$ by the name $\{{\langlep{\mathbin\upharpoonright}E,\check u\rangle}\mid p{\mathrel{\Vdash}}\check u\in\dot x\}$ which is a name in a finite product of ${\mathbb Q}$. Therefore, by the assumption on ${\mathbb Q}$, there is some $q\leq p$ and $y\in V_\alpha$ such that $q{\mathrel{\Vdash}}\dot x=\check y$.
We now observe that if we iterate symmetric extensions starting from a model of ${{\mathsf{ZFC}}}+{{\mathsf{GCH}}}$, we may use $\prod_{i\in I\times\omega}\operatorname{Add}(\kappa,1)^V$ (for the appropriate $\kappa$) with the permutation group $\{\operatorname{id}\}\wr S_{<\omega}$ (acting on $I\times\omega$), where $S_{<\omega}$ is the group of permutations moving only finite many integers. And this will not add bounded sets to the universe, as every finite product commutes with the iteration as a whole.
This allows for class-length iterations where we add “generic copies” of sets which were not in the ground model, and we can do so uniformly as long as $I$ has a name which is respected by all permutations. This method can now be applied to a wide variety of $\forall\exists$-$\exists\forall$ problems. This can now be applied in a way similar to what is described in Theorem 4.3 of [@Karagila:DC].
We note, however, that the key point is not quite that we have to use something that looks like $\operatorname{Add}(\kappa,1)^V$. But rather that as long as the “finite parts” are sufficiently well-behaved, this reflects to the entire product, and thus to the iteration as a whole. Indeed, in neither examples below we use this exact formulation.
The Morris iteration
====================
We start with a model $V$ satisfying ${{\mathsf{ZFC}}}+{{\mathsf{GCH}}}$. While the assumption of ${{\mathsf{GCH}}}$ can be eliminated by allowing gaps between the “active iterands” of the forcing it is much easier to ignore these problems altogether in favour of a harmless axiom.
We define an iteration of symmetric extensions by working in tandem: first we force with a ground model partial order, adding some Cohen subsets to a regular cardinal $\kappa$, which will not add sets of rank $\eta$, for some appropriately chosen $\eta$, and then we take a symmetric extension over the *whole* model in such way that ensures that we are still not adding sets of rank $\eta$. If we are careful, and ensure that the sequence of these $\eta$s is indeed increasing from each pair of systems to the next, this will guarantee the wanted result.
Previously, in [@Karagila:Morris], the second step of each tandem was defined relative to the model $V[G]$ where $G$ was $V$-generic for the Cohen step. We then took a product of those two-steps iterations. This worked fine for a global solution for ground model sets, but we can now use sets from the full iterated model up to the stage of the tandem for the second step of the forcing. We also point out that an even simpler form of this idea can be found in §5.2 of [@Karagila:Fodor] where the first step did not include any symmetries at all, and was thus relegated to a preparatory forcing.
It will be easy, if so, to talk about the symmetric extensions as ${\mathbb Q}_{\alpha,0}$ and ${\mathbb Q}_{\alpha,1}$, after having constructed the iteration ${\mathbb P}_\alpha$ of all previous pairs. As we first describe the local construction of each of the two steps, we omit $\alpha$ and simply write ${\mathbb P},{\mathbb Q}_0,{\mathbb Q}_1$ (and similarly for ${\mathscr G}_0$ and ${\mathscr G}_1$, etc.). Nevertheless, it is a good place to point out that by the time we force with these two symmetric systems, we have constructed a mode of ${{\mathsf{ZF}}}$, denoted by $M$, between $V$ and $V[G]$ which is the full generic extension by ${\mathbb P}$. To avoid confusion, and to emphasise this, we will refer to the symmetric names in the iteration of ${\mathbb P}\ast{\mathbb Q}_0$ as ${{\mathsf{IS}}}\ast{{\mathsf{HS}}}$.
The first step, locally
-----------------------
Let $\kappa$ be a suitable regular cardinal, which means a large enough cardinal such that $V_\alpha$ of whatever intermediate step we reached this far satisfies that $|V_\alpha|<\kappa$ in the full generic extension, and as we started with ${{\mathsf{GCH}}}$, we may also include in this choice that $\kappa^{<\kappa}=\kappa$ still holds in the full generic extension as well.
The first step is similar to the first step of the local construction in [@Karagila:Morris]. We take ${\mathbb Q}_0=\operatorname{Add}(\kappa,\omega\times\omega\times\kappa)^V$, with ${\mathscr G}_0$ the group $\{\operatorname{id}\}\wr S_\omega\wr S_\kappa$ (as computed in $V$). In other words, if $\pi\in{\mathscr G}_0$ its action on $\omega\times\omega\times\kappa$ is given by the following process:
1. For each $n<\omega$, let $\pi^*_n\in S_\omega$ be a permutation,
2. for each $n,m<\omega$ let $\pi_{n,m}\in S_\kappa$ be a permutation,
3. map ${\langlen,m,\alpha\rangle}$ to ${\langlen,\pi^*_n(m),\pi_{n,m}(\alpha)\rangle}$.
The action on ${\mathbb Q}_0$ is defined by the equation $\pi p(\pi(n,m,\alpha),\beta)=p(n,m,\alpha,\beta)$.
Finally, ${\mathscr F}_0$ is generated by $\operatorname{fix}^*(E)$ for $E\in[\omega\times\omega\times\kappa]^{<\omega}$, where $$\operatorname{fix}^*(E)=\{\pi\in{\mathscr G}\mid\pi{\mathbin\upharpoonright}E=\operatorname{id}\land\forall{\langlen,m,\alpha\rangle}\in E,\pi^*_n=\operatorname{id}\}.$$
We need to verify that the conditions for iterating hold, e.g. that ${\mathbb Q}_0,{\mathscr G}_0$, and ${\mathscr F}_0$ have names respected by previous automorphisms, etc. but since these are all coming from $V$ this is trivial, as no previous automorphism moves canonical ground model names. In particular, we may ignore the fact we are iterating symmetric extensions.
We now define the following names:
1. $\dot x_{n,m,\alpha}=\{{\langlep,\check\beta\rangle}\mid p(n,m,\alpha,\beta)=1\}$.
2. $\dot a_{n,m}=\{\dot x_{n,m,\alpha}\mid\alpha<\kappa\}^\bullet$.
3. $\dot A_n=\{\dot a_n\mid n<\omega\}^\bullet$.
4. $\vec A={\langle\dot A_n\mid n<\omega\rangle}^\bullet$.
5. $\dot A=\{\dot a_{n,m}\mid n,m<\omega\}^\bullet=\bigcup_{n<\omega}\dot A_n$.
$\pi\dot x_{n,m,\alpha}=\dot x_{\pi(n,m,\alpha)}$.
As an immediate corollary, all of the names are in ${{\mathsf{HS}}}$.
${\mathrel{\Vdash}}^{{\mathsf{HS}}}\forall n<\omega,|\dot A_n|=\aleph_0$ and $|\dot A|>\aleph_0$.
The following theorem is an immediate corollary of the following general fact (which appears as Lemma 2.3 in [@Karagila:Fodor], with a proof given by Yair Hayut).
Let $\kappa$ be a regular cardinal. Suppose that ${\mathbb P}$ has $\kappa$-c.c. and ${\mathbb Q}$ is a $\kappa$-distributive forcing such that ${\mathrel{\Vdash}}_{\mathbb Q}\check{\mathbb P}$ has $\kappa$-c.c. Then ${\mathrel{\Vdash}}_{\mathbb P}\check{\mathbb Q}$ is $\kappa$-distributive.
Suppose that $\kappa$ is large enough that the ${\mathbb P}$ has $\kappa$-c.c., then ${\mathrel{\Vdash}}_{\mathbb P}\check{\mathbb Q}_0$ is $\kappa$-distributive. In particular, forcing over $M$ with ${\mathbb Q}_0$ adds no sets of rank $\eta$ if ${\mathrel{\Vdash}}_{\mathbb P}|V_\eta|<\kappa$.
The second step, semi-locally
-----------------------------
Let $N$ denote the model defined by the symmetric iteration on ${\mathbb P}\ast{\mathbb Q}_0$, which is the symmetric extension of $M$ by the first step above. Let $x_{n,m,\alpha}$, $a_{n,m}$, $A_n$ and $A$ denote the interpretation of the names above. Let $T$ denote the forcing $\bigcup_{n<\omega}\prod_{k<n}A_k$, namely the forcing which adds a choice function from the $A_n$’s. We let ${\mathbb Q}_1$ be the forcing whose conditions are sequences in the finite support product of $V_\alpha\times\omega$ copies of $T$, and here $V_\alpha$ is computed in the iterated model (as opposed to [@Karagila:Morris] where it is computed in the symmetric extension of $V$ by ${\mathbb Q}_0$).
For ${\langlex,n\rangle}\in V_\alpha\times\omega$ and $t\in{\mathbb Q}_1$ we write $t_{{\langlex,n\rangle}}$ to denote the ${\langlex,n\rangle}$th coordinate of $t$ which is a condition in $T$. We denote by $\operatorname{supp}t$ the set of ${\langlex,n\rangle}$ on which $t_x\neq\varnothing$, and this is a finite set by the definition of ${\mathbb Q}_1$. If $E\subseteq V_\alpha\times\omega$, we write $t{\mathbin\upharpoonright}E$ to denote the condition $t'$ for which $t'_x=t_x$ when $x\in E$ and otherwise $t'_x=\varnothing$, that is the condition obtained by restricting the support of $t$ into $E$.
We define the order on ${\mathbb Q}_1$ as follows:
1. $s\leq t$ if $\operatorname{supp}t\subseteq\operatorname{supp}s$, and for all $x\in\operatorname{supp}t$, $t_x\subseteq s_x$.
2. If $x,y\in\operatorname{supp}t$ and $n\notin\operatorname{dom}t_x\cap\operatorname{dom}t_y$, then $s_x(n)\neq s_y(n)$ whenever $s_x(n)$ and $s_y(n)$ are both defined. In other words, any extension in $s$ must be pairwise disjoint.
The group ${\mathscr G}_1$ is given by finitary permutations in $\{\operatorname{id}\}\wr S_\omega$. For $\pi\in{\mathscr G}_1$ we denote by $\pi_x$ the permutation of $\omega$ such that $\pi(x,n)={\langlex,\pi_xn\rangle}$, and by the finitary requirement, for only finitely many $x$’s we have $\pi_x\neq\operatorname{id}$ and for all but finitely many $n$’s, $\pi_x n\neq n$. The permutations are acting on ${\mathbb Q}_1$ by the same principle as before, $\pi t_{{\langlex,n\rangle}}=t_{{\langlex,\pi_xn\rangle}}$. To finish off the definition of the symmetric system, ${\mathscr F}_1$ is generated by $\operatorname{fix}(E)=\{\pi\in{\mathscr G}_1\mid\pi{\mathbin\upharpoonright}E=\operatorname{id}\}$ for $E\in[V_\alpha\times\omega]^{<\omega}$.
Unlike before, this time we cannot ignore the previous iterations, since are using $V_\alpha$ as computed in the iterated model, rather than the extension of $V$ by ${\mathbb Q}_0$. First we observe, as in [@Karagila:Morris], that if $s\in\omega^{<\omega}$, then $\dot f_s={\langle\dot a_{s(n)}\mid n\in\operatorname{dom}s\rangle}^\bullet$ is a canonical name for an element of $T$, and in fact $T=\{\dot f_s\mid s\in\omega^{<\omega}\}^\bullet$, so every name for a condition in $T$ can be extended to a canonical name.
This means that from the point of $N$, every condition in ${\mathbb Q}_1$ has a canonical name given by a finitary function from $V_\alpha\times\omega\to\omega^{<\omega}$. This, in turn, translates to a canonical name for ${\mathbb Q}_1$ in ${{\mathsf{IS}}}\ast{{\mathsf{HS}}}$: $\{{\langle\dot x,\check n,\dot f_s\rangle}^\bullet\mid\operatorname{rank}(\dot x)<\alpha,n<\omega,s\in\omega^{<\omega}\}$. Moreover, since automorphisms preserve rank, this means that ${\mathbb Q}_1$ is stable under all the automorphisms of ${\mathbb P}\ast{\mathbb Q}_0$ aggregated so far in the iteration. The same observation holds for ${\mathscr G}_1$ and ${\mathscr F}_1$, meaning that indeed we are not to worry regarding continuing our iteration.
Let $\dot b_{x,n}=\{{\langlet,\check a\rangle}\mid\exists m:t_x(m)=a\}$ and let $\dot B_x=\{\dot b_{x,n}\mid n<\omega\}^\bullet$. It is not hard to see that both are in ${{\mathsf{HS}}}_{{\mathscr F}_1}$.
If $\dot x\in{{\mathsf{HS}}}_{{\mathscr F}_1}$ is a name for a subset of $M$ and all the names appearing in $\dot x$ are of the form $\check y$, then there is some $t\in{\mathbb Q}_1$ and $x\in N$ such that $t{\mathrel{\Vdash}}\dot x=\check x$.
The proof here is essentially the same proof as in [@Karagila:Morris] of the same fact (here we combine Prop. 3.6 and Lemma 3.8 into a single proof).
Let $\dot x\in{{\mathsf{HS}}}$ be such a name. First we note that by the fact that ${\mathscr G}_1$ witnesses the homogeneity of ${\mathbb Q}_1$, if $E\subseteq V_\alpha\times\omega$ is a finite set such that $\operatorname{fix}(E)\subseteq\operatorname{sym}(\dot x)$, then $t{\mathrel{\Vdash}}\check y\in\dot x$ if and only if $t{\mathbin\upharpoonright}E{\mathrel{\Vdash}}\check y\in\dot x$. If $t'\leq_{{\mathbb Q}_1} t{\mathbin\upharpoonright}E$, then there is $\pi\in\operatorname{fix}(E)$ such that $\operatorname{supp}(\pi t')\setminus E$ is disjoint from $\operatorname{supp}t$, which means that the two are compatible. As $\pi\in\operatorname{fix}(E)$ we have $\pi\dot x=\dot x$ and $\pi\check y=\check y$. We may assume, therefore that $\operatorname{supp}t=E$.
We write $[\dot x]$ to denote a ${\mathbb P}\ast{\mathbb Q}_0$-name in ${{\mathsf{IS}}}\ast{{\mathsf{HS}}}$ for $\dot x$. Let $n$ be large enough such that if $\vec H^\frown\operatorname{fix}^*(e)$ is an ${\mathcal F}\ast{\mathscr F}$-support for $[\dot x]$, then $e=n\times e_1\times e_2$ where $e_1$ and $e_2$ are finite subsets of $\omega$ and $\kappa$ respectively.
Assume without loss of generality that $n$ also satisfies that $\operatorname{dom}t_x=n$ for all $x\in\operatorname{supp}t$. Let $s,s'\leq_{{\mathbb Q}_1}t$ such that $\operatorname{supp}s=\operatorname{supp}s'=E$ as well and without loss of generality $\operatorname{dom}s_x=\operatorname{dom}s'_x$ for all $x\in E$. Suppose that $s{\mathrel{\Vdash}}_{{\mathbb Q}_1}\check y\in\dot x$.
Let ${\langlep,q\rangle}\in{\mathbb P}\ast{\mathbb Q}_0$ be a condition such that ${\langlep,q\rangle}{\mathrel{\Vdash}}^{{\mathsf{IS}}}\dot s{\mathrel{\Vdash}}_{{\mathbb Q}_1}\dot y\in[\dot x]$, such that:
1. $\dot t,\dot s$ are canonical names for $t,s$, we will also assume that $\dot s'$ is a canonical name that ${\langlep,q\rangle}$ decides will be $s'$,
2. $\dot y$ is a ${\mathbb P}$-name for the canonical name for $y$, and
3. $[\dot x]$ is the name for the name $\dot x$.
By the condition that $\operatorname{supp}s=\operatorname{supp}t$, we know that for all $i\geq n$, $$|\{s_x(i)\mid x\in E\}|=|E|=|\{s'_x(i)\mid x\in E\}|.$$ We can therefore find a permutation $\pi\in\operatorname{fix}^*(n\times\{0\}\times\{0\})$ which satisfies the following conditions:
1. $\pi[\dot x]=[\dot x]$,
2. $\pi\dot t=\dot t$,
3. $\pi\dot a_{i,s(i)}=\dot a_{i,s'(i)}$ for all $i\geq n$,
4. $\pi\dot y=\dot y$, as it is a ${\mathbb P}$-name, and
5. $\pi q$ is compatible with $q$.
The last condition is obtainable by noting that we can choose $\pi_{n,i}$ as a permutation of $\kappa$ which moves all the $\alpha<\kappa$ which appear anywhere in $q$ to a disjoint subset of $\kappa$.
This means that ${\langlep,\pi q\rangle}{\mathrel{\Vdash}}^{{\mathsf{IS}}}\dot s'{\mathrel{\Vdash}}_{{\mathbb Q}_1}\dot y\in[\dot x]$, but since ${\langlep,q\rangle}$ and ${\langlep,\pi q\rangle}$ are compatible, and $\dot s'$ is a canonical name, we get that ${\langlep,q\rangle}$ cannot force a contradictory statement, that $s'{\mathrel{\not{\Vdash}}}_{{\mathbb Q}_1}\check y\in\dot x$.
In particular, that means that ${\langlep,q\rangle}{\mathrel{\Vdash}}^{{\mathsf{IS}}}\dot t{\mathrel{\Vdash}}_{{\mathbb Q}_1}\dot y\in[\dot x]$. Therefore any new subsets of $M$ was added by ${\mathbb Q}_0$, and is in $N$.
\[cor:pres\] If $\eta$ was such that $V_\eta^M=V_\eta^N$, then the symmetric extension of $N$ given by ${\langle{\mathbb Q}_1,{\mathscr G}_1,{\mathscr F}_1\rangle}$ have the same $V_\eta$ as well.
Dedekind-finiteness of the preimage
-----------------------------------
Very easily now, $b_{x,n}\mapsto b_x$ is a surjection onto $V_\alpha$ which extends to a surjection from the full power set of $A$. But we claim that $\bigcup_{x\in V_\alpha}B_x$ is in fact Dedekind-finite in this symmetric extension, and thus we prove the following proposition.
$N\models$ There is a Dedekind-finite set that can be mapped onto $V_\alpha$.
We work in $M$. Let $\dot B=\{\dot b_{x,n}\mid x\in V_\alpha,n<\omega\}^\bullet$ be the name for $\bigcup_{x\in V_\alpha}B_x$, easily $\dot B\in{{\mathsf{HS}}}_{{\mathscr F}_1}$. Suppose that $\dot f\in{{\mathsf{HS}}}_{{\mathscr F}_1}$ and $s\in{\mathbb Q}_1$ such that $s{\mathrel{\Vdash}}\dot f\colon\check\omega\to{\mathcal B}$, then there is a finite subset $E\subseteq V_\alpha\times\omega$ such that $\operatorname{fix}(E)\subseteq\operatorname{sym}(\dot f)$ and $\operatorname{supp}s\subseteq E$.
Let ${\langlex,n\rangle}\notin E$ and assume towards contradiction that for some $m<\omega$ there is $t\leq s$ such that $t{\mathrel{\Vdash}}\dot f(\check m)=\dot b_{x,n}$. Let $n'<\omega$ such that:
- $n\neq n'$,
- ${\langlex,n'\rangle}\notin E$,
- ${\langlex,n'\rangle}\notin\operatorname{supp}t$.
Easily, the permutation $\pi\in\operatorname{fix}(E)$ which satisfies $\pi_x=(n\ n')$ and the identity elsewhere satisfies that $\pi t$ is compatible with $t$. By virtue of being in $\operatorname{fix}(E)$ we have that $\pi s=s$ and $\pi\dot f=\dot f$. But $\pi t{\mathrel{\Vdash}}\dot f(\check m)=\dot b_{x,n'}$ and therefore $$t\cup\pi t{\mathrel{\Vdash}}\dot b_{x,n}=\dot f(\check m)=\dot b_{x,n'}\neq\dot b_{x,n}.$$ Therefore $s{\mathrel{\Vdash}}\dot f(\check m)\in\{\dot b_{x,n}\mid{\langlex,n\rangle}\in E\}^\bullet$ for all $m<\omega$, and in particular $s{\mathrel{\Vdash}}\dot f$ is not injective.
Embedding orders into the cardinals
-----------------------------------
We work in $N$, for $u\in V_{\alpha+1}$, i.e. $u\subseteq V_\alpha$, let $\dot S_u=\bigcup_{x\in u}\dot B_x$. It is easily seen that each $\dot S_u\in{{\mathsf{HS}}}$ and that $\pi\dot S_u=\dot S_u$. We repeat the argument from [@Karagila:Embeddings].
${\mathrel{\Vdash}}^{{\mathsf{HS}}}|\dot S_u|\leq^*|\dot S_v|\iff u\subseteq b$.
If $u\subseteq v$, then there is nothing to check since the names satisfy $\dot S_u\subseteq\dot S_v$ and therefore there is an injection, and in particular a surjection (or $u=\varnothing$, in which case $\dot S_u=\varnothing$ as well).
Suppose that $u\nsubseteq v$ and let $\dot f\in{{\mathsf{HS}}}$ such that $t{\mathrel{\Vdash}}^{{\mathsf{HS}}}\dot f\colon S_v\to S_u$, and let $E\in[V_\alpha\times\omega]$ be such that $\operatorname{fix}(E)\subseteq\operatorname{sym}(\dot f)$. We aim to show that $t{\mathrel{\Vdash}}^{{\mathsf{HS}}}\dot f$ is not surjective.
Let $x\in u\setminus v$ and let $n<\omega$ such that ${\langlex,n\rangle}\notin E$. Assume that $s\leq t$ is a condition such that $s{\mathrel{\Vdash}}^{{\mathsf{HS}}}\dot f(\dot b_{y,m})=\dot b_{x,n}$ for some $y\in v$ and $m<\omega$. Then there is some $k<\omega$ for which ${\langlex,k\rangle}\notin E\cup\operatorname{supp}s$, since $E\cup\operatorname{supp}s$ is finite.
Let $\pi$ be the permutation in ${\mathscr G}_1$ such that $\pi_x=(n\ k)$ and otherwise the identity. Easily, $\pi\in\operatorname{fix}(E)$ and $\pi s$ is compatible with $s$. However $\pi\dot b_{x,n}=\dot b_{x,k}$, so we have that $$s\cup\pi s{\mathrel{\Vdash}}^{{\mathsf{HS}}}\dot b_{x,n}=\dot f(\dot b_{y,m})=\dot b_{x,k}\neq\dot b_{x,n}.\qedhere$$
We can, in fact, use any subset of $V_\alpha$ in the symmetric extension by defining $\dot S_u=\{{\langlet,\dot b_{x,n}\rangle}\mid t{\mathrel{\Vdash}}\check x\in\dot u,n<\omega\}$ and showing that the same argument holds. However due to the nature of the iteration all those will be captured by the next pair anyway.
Tying it all together
---------------------
We now iterate through all the ordinals, the pair defined above, where for the $\alpha$th pair we pick $\kappa$ to be large enough such that $V_{\alpha+57}$ of the intermediate model is smaller than $\kappa$ in the full generic extension.[^2]
From we obtain that the resulting model satisfy ${{\mathsf{ZF}}}$. Moreover, $V_\alpha^M$ is the same as the one obtained by some initial segment of the iteration. If $x\in M$ is not empty, then it is in some $V_\alpha$ which is the same as the $V_\alpha$ of the $\eta$th model in the iteration, and therefore by the $\eta$th iterands we will have added a set which is a countable union of countable sets and can be mapped onto $V_\eta$, and thus on $x$. Moreover Dedekind-finite is preserved, so we get that the that every set is in fact the image of a Dedekind-finite set.
And finally, if ${\langleI,\leq\rangle}$ is any partial order in $M$, then it embeds into some ${\langleV_\eta,\subseteq\rangle}$, and therefore into the cardinals of $M$.
Every field has a Läuchli vector space
======================================
We say that a vector space $X$ is a *Läuchli space (over a field $F$)* if $\dim_F X>1$ and yet every endomorphism of $X$ as an $F$-vector space is a scalar multiplication.
If $X$ is a Läuchli space over $F$, then there are no two proper subspaces $X_0,X_1$ such that $X_0\cap X_1=\{0\}$ and $X_0+X_1=X$. Therefore $X$ admits no linear functionals except the $0$ functional. Therefore $X$ does not have a basis.
It is consistent that there is a Läuchli space such that every linearly independent subset is finite.
The author’s master thesis focused mainly on showing that if $V$ satisfies ${{\mathsf{ZFC}}}$, then any field $F$ has a Läuchli space in some symmetric extension (see [@Karagila:MSc]). While Läuchli originally worked over models of ${{\mathsf{ZF}}}$ with atoms, and used a countable field, the construction translates to symmetric extensions in a fairly direct way to any field (see [@Karagila:DC] for an outline of the technique).
It is consistent with the statement that every field has a proper class of Läuchli spaces.[^3]
At each step, let $\kappa_\alpha$ be large enough such that $\operatorname{Add}(\kappa_\alpha,1)^V$ does not add subsets of rank $<\alpha$. And for each field $F\in V_\alpha$, take $\prod_{v\in F^{<\omega}}\operatorname{Add}(\kappa_\alpha,\omega)^V$. For the group of automorphisms let $G_F$ denote the linear automorphisms of $F^{<\omega}$ which are the identity outside a finite dimensional subspace, and let $S_{<\omega}$ denote the finitary permutations of $\omega$. Then $G_F\wr S_{<\omega}$ has a natural action on the forcing. Finally, ${\mathscr F}$ is the filter of subgroups given by fixing pointwise finite subsets of $F^{<\omega}\times\omega$. Following the proof in [@Karagila:MSc], it is not hard to show that this indeed generates a Läuchli space over $F$.
Finally, we take the finite support product of all the symmetric systems described in the previous paragraph, for $F\in V_\alpha$ a field.
Where should we go now?
=======================
We only have a framework for finite supports. As we saw in [@Karagila:Morris] and [@Karagila:DC], we have a natural appeal for $\kappa$-support products, especially if each step is $\kappa$-closed, or so. As this will ensure the preservation of ${{\mathsf{DC}}}_{<\kappa}$.
In fact, working out this would serve as a good intermediate step for developing a more general theory of iterating symmetric extensions with $\kappa$-support iterations. Combining this with the results from [@Karagila:Aspero] which ensure the preservation of ${{\mathsf{DC}}}$ under proper forcing is sure to be very useful to the theory of the reals without the axiom of choice.
This is important since we have several candidates for using these style of iterations. For example, in [@Karagila:Schlicht] we provide a criteria for certain forcings to not add sets of ordinals. This could be extended in the style of [@Monro:1973] (which was extended in some sense by the author in [@Karagila:Iterations] and later by [@Shani:2018]). Another example, which perhaps illuminates why working with “locally well-ordered forcing” is nice in the choiceless context is the following proposition.
Suppose that ${\mathbb P}$ is a well-orderable forcing, then ${\mathrel{\Vdash}}_{\mathbb P}``\check A$ admits a choice function” if and only if $A$ already admits a choice function in the ground model.
Fix a well-ordering of ${\mathbb P}$ as $\{p_\alpha\mid\alpha<\lambda\}$. Suppose that $\dot f$ is a ${\mathbb P}$-name for a choice function. For every $a\in A$, define $F(a)=x$ if and only if the first $p_\beta$ which decides the value of $\dot f(\check a)$ decides it to be $\check x$.
As a consequence of this we immediately have that when iterating symmetric extensions such that every new subset is added by a well-ordered forcing, we are guaranteed to not well-order any $V_\alpha$ that was not well-ordered to begin with. In the case of class-length iterations we still need to verify the conditions of , so more is needed.
Other important questions would be how does the choice of symmetric systems reflect to the iteration. Namely, we choose at each step a “nice forcing” and take a symmetric extension based on products of this forcing. Can we provide conditions for the entire iteration to preserve the Boolean Prime Ideal theorem, or other weak choice principles?
Of course, this is not the ultimate result that we want to have. Ideally we want to find a “nicely definable forcing” which mimics $\kappa$-Cohen subsets in ${{\mathsf{ZFC}}}$ by adding subsets to an arbitrary set satisfying certain conditions (i.e. the “right definition” for a regular cardinal) without adding “bounded subsets”.
[10]{}
David [Asper[ó]{}]{} and Asaf [Karagila]{}, *[Dependent Choice, Properness, and Generic Absoluteness]{}*, ArXiv e-prints (2018), arXiv:1806.04077.
Eilon Bilinsky, *A realization of partial orders by orders on cardinals*, In preparation.
T. Jech, *On ordering of cardinalities*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. **14** (1966), 293–296 (loose addendum). [MR ]{}[0201319 (34 \#1203)]{}
A. Karagila, *Fodor’s lemma can fail everywhere*, Acta Math. Hungar. **154** (2018), no. 1, 231–242. [MR ]{}[3746534]{}
Asaf Karagila, *Vector spaces and antichains of cardinals in models of set theory*, Master’s thesis, Ben-Gurion University of the Negev, 2012.
Asaf Karagila, *Embedding orders into the cardinals with [$\mathsf{DC}_\kappa$]{}*, Fund. Math. **226** (2014), no. 2, 143–156. [MR ]{}[3224118]{}
[to3em]{}, *Iterating symmetric extensions*, J. Symb. Log. **84** (2019), no. 1, 123–159. [MR ]{}[3922788]{}
[to3em]{}, *The [M]{}orris model*, Proc. Amer. Math. Soc. **Online edition** (2019), 1–13.
[to3em]{}, *Preserving dependent choice*, Bull. Pol. Acad. Sci. Math. **67** (2019), no. 1, 19–29. [MR ]{}[3947821]{}
Asaf Karagila and Philipp Schlicht, *How to have more things by forgetting how to count them*, arXiv **1910.14480** (2019), Submitted.
H. L[ä]{}uchli, *Auswahlaxiom in der [A]{}lgebra*, Comment. Math. Helv. **37** (1962/63), 1–18. [MR ]{}[0143705 (26 \#1258)]{}
G. P. Monro, *Models of [${\rm ZF}$]{} with the same sets of sets of ordinals*, Fund. Math. **80** (1973), no. 2, 105–110. [MR ]{}[0347602]{}
G. P. Monro, *Independence results concerning [D]{}edekind-finite sets*, J. Austral. Math. Soc. **19** (1975), 35–46. [MR ]{}[0360268 (50 \#12718)]{}
Assaf Shani, *Borel reducibility and symmetric models*, arXiv **1810.06722** (2018).
Moto-o Takahashi, *On incomparable cardinals*, Comment. Math. Univ. St. Paul. **16** (1968), 129–142. [MR ]{}[0227014 (37 \#2599)]{}
[^1]: Eilon Bilinsky independently obtained a similar global result on partial orders using a different technique, see [@Bilinsky].
[^2]: This can probably be computed accurately, and it is most likely $\omega_{\alpha+1}$ or so, but we leave this as an exercise in futility to the interested graduate student. The same can be said about $\alpha+57$ which is used here to ensure that all the cardinal structure added by ${\mathbb Q}_1$ above is preserved, and in all likelihood, this can be reduced to $\alpha+1$ or so.
[^3]: Here a proper class means that for every set of vector spaces over the field there is a Läuchli space not isomorphic to any of them.
|
---
abstract: 'We present a multiwavelength study of the central part of the Carina Nebula, including Trumpler 16 and part of Trumpler 14. Analysis of the [*Chandra X-ray Observatory*]{} archival data led to the identification of nearly 450 X-ray sources. These were then cross-identified with optical photometric and spectroscopic information available from literature, and with deep near-infrared ($JHK_s$) imaging observations. A total of 38 known OB stars are found to be X-ray emitters. All the O stars and early B stars show the nominal relation between the X-ray and bolometric luminosities, $L_{\rm X} \sim 10^{-7} L_{\rm bol}$. A few mid- to late-type B stars are found to be associated with X-ray emission, likely attributable to T Tauri companions. We discovered 17 OB star candidates which suffer large extinction in the optical wavebands. Some 300 sources have X-ray and infrared characteristics of late-type pre-main sequence stars. Our sample presents the most comprehensive census of the young stellar population in the Carina Nebula and will be useful for the study of the star-formation history of this massive star-forming region. We also report the finding of a compact ($5\arcmin \times 4\arcmin$) group of 7 X-ray sources, all of which highly reddened in near-infrared and most X-ray bright. The group is spatially coincident with the dark ’V’ shaped dust lane bisecting the Carina Nebula, and may be part of an embedded association. The distribution of the young stellar groups surrounding the region associated with Trumpler 16 is consistent with a triggering process of star formation by the collect-and-collapse scenario.'
author:
- 'Kaushar Sanchawala, Wen-Ping Chen, Hsu-Tai Lee'
- 'Yasuhi Nakajima, Motohide Tamura'
- 'Daisuke Baba, Shuji Sato'
- 'You-Hua Chu'
title: 'X-RAY EMITTING YOUNG STARS IN THE CARINA NEBULA'
---
INTRODUCTION
============
Massive stars have a profound influence on neighboring molecular clouds. On the one hand, the powerful stellar radiation and wind from even a single such star would sweep away nearby clouds and henceforth prevent subsequent star formation. On the other hand, the massive star may provide “just the touch” to prompt the collapse of a molecular cloud which otherwise may not contract spontaneously. Whether massive stars play a destructive or a promotive role in cluster formation conceivably depends on the availability of cloud material within the range of action, though the details have not been fully understood. If massive stars by and large suppress star formation, low-mass stars could exist in the immediate surroundings only if the low-mass stars predated massive star formation. In both the Orion and Lacerta OB associations, @lee05 and @leechen have found an evidence of triggered star formation by massive stars. The UV photons from massive stars appear to have ionized adjacent molecular clouds and the implosive pressure then compresses the clouds to form next-generation stars of various masses, often in groups, with high star formation efficiencies. The process is self-sustaining and an entire OB association may be formed as a result.
The Carina Nebula, also known as NGC3372, is a remarkable star-forming region where the most massive stars known in the Milky-Way Galaxy co-exist. The Nebula, which occupies about 4 square degree area on the sky, contains at least a dozen known star clusters [@feinstein95]. The clusters with photometric and spectroscopic data are : Bochum (Bo) 10 and 11, Trumpler (Tr) 14, 15 and 16, Collinder (Cr) 228, NGC3293 and NGC3324. Tr14 and Tr16 are the most populous and youngest star clusters and located in the central region of the Nebula. The distance modulus for Tr16, quoted from the literature, ranges from 11.8 [@levato] to 12.55 [@mj93 MJ93 hereafter] and for Tr14, from 12.20 [@feinstein83] to 12.99 [@morrell]. @walborn derived a distance of $2.5$ kpc for Tr16 using $R =3.5$. @crowther derived a distance of $2.6$ kpc for Tr14. @walborn73 and @morrell concluded that the two clusters are at slightly different distance, whereas @turner and @mj93 concluded the two clusters are at the same distance. A distance of 2.5 kpc is adopted for our study. All the clusters listed above, contain a total of 64 known O-type stars, the largest number for any region in the Milky Way [@feinstein95]. Tr14 and Tr16 include six exceedingly rare main-sequence O3 stars. The presence of these very young stars indicates that the two clusters are extremely young. The two clusters also contain two Wolf-Rayet stars which are believed to be evolved from even more massive progenitors than the O3 stars (MJ93). Furthermore Tr16 is the parent cluster to the famous luminous blue variable (LBV), $\eta$ Carinae, which is arguably the most massive star of our Galaxy (MJ93). With such a plethora of unusually massive stars, the Carina Nebula is a unique laboratory to study the massive star formation process, and the interplay among massive stars, interstellar media and low-mass star formation.
In recent years, X-ray surveys have been very successful in defining the pre-main sequence population of young star clusters [@fei02]. X-ray emission has been detected from deeply embedded class I Young Stellar Objects (YSOs) to low-mass pre-main sequence (PMS) stars of T Tauri types, and intermediate-mass pre-main sequence stars of Herbig Ae/Be types to the zero-age main-sequence stars. For late type main-sequence stars, starting from late A to K and M dwarfs, the X rays are produced in the very high temperature gas in the corona, which is thought to be heated due to the dynamo magnetic fields [@maggio]. Massive stars, of O and early B types, on the other hand emit X rays which are produced in the shocks due to hydrodynamic instabilities in their radiatively driven strong stellar winds [@lucy82]. The X-ray emission from the classical T Tauri stars (CTTSs) or weak-lined T Tauri stars (WTTSs) is believed to be thermal emission from the gas rapidly heated to temperatures of the order $10^7$ K due to magnetic reconnection events similar to the solar magnetic flares, but elevated by a factor of $10^1$ –$10^4$ [@fei99]. A recent work by @preibisch05 presents the correlation of the X-ray properties with different stellar parameters, for a nearly complete sample of late-type PMS stars in the Orion Nebula Cluster. They concluded that the origin of X-ray emission in T Tauri stars seems to be either a turbulent dynamo working in stellar convection zone, or a solar like $\alpha$-$\Omega$ dynamo at the base of the convection zone if T Tauri stars are not fully convective. Among the existing methods to identify the young stellar populations in a young star cluster, the use of X-ray emission, which is nearly independent of the amount of circumstellar material around the young stars [@fei02], is the least biased, especially in selection of weak-lined T Tauri stars which lack the standard signatures of pre-main sequence stars such as the infrared excess or strong $H\alpha$ emission lines.
In this paper we used the [*Chandra X-ray Observatory*]{} archival data of the Carina Nebula to select the young stellar populations of the region. We then made use of the optical photometric and spectroscopic information available in the literature to identify the counterparts of the X-ray sources. We found that more than 2/3 of the X-ray sources do not have any optical counterparts. To characterize these sources further, we used the Simultaneous InfraRed Imager for Unbiased Survey (SIRIUS) camera, mounted on the Infrared Survey Facility, South Africa to carry out $J$, $H$, and $K_{s}$ band imaging observations. Figure 1[^1] shows the optical image of the Nebula from Digitized Sky Survey ($\sim$ 25 $\times$ 25), with the $Chandra$ field marked by a square centered on Tr 16 and covering part of Tr 14, which is in the north west of Tr 16. The field observed in the near infrared is about the same as the field of the optical DSS image. We discuss the X-ray and NIR properties of the known OB stars of the region. We discovered 17 massive star candidates on the basis of their NIR and X-ray properties similar to those of the known OB stars in the region. These candidate OB stars probably escaped previous detection because of their large extinction in the optical wavelengths. Furthermore, we identified some 300 CTTSs and WTTSs candidates, again on the basis of their X-ray and NIR properties. Our study therefore produces the most comprehensive young star sample in the Carina Nebula, which allows us to delineate the star formation history in this seemingly devastating environment. In particular we report the discovery of an embedded ($A_{\rm V} \sim 15$ mag) young stellar group located to the south-east of Tr16, and sandwiched between two dense molecular clouds. Similar patterns of newly formed stars in between clouds seem to encompass the Carina Nebula, a manifestation of the triggered star formation by the collect-and-collapse process [@deh05].
The paper is organized as follows. §2 describes the $Chandra$ and the NIR observations and the data analysis. In §3, we present the cross-identification of $Chandra$ sources with the optical spectroscopic information (available in the literature) and with our NIR sample. We discuss the results and implication of our study in §4. §5 summarizes our results.
OBSERVATIONS AND DATA REDUCTION
===============================
X-ray data — $Chandra$
-----------------------
The Carina Nebula was observed by the ACIS$-$I detector of [*Chandra X-ray Observatory*]{}. There were two observations in 1999 September 6, observation ID 50 and 1249 (Table 1). We began our data analysis with the Level 1 processed event and filtered cosmic-ray afterglows, hot pixels, $ASCA$ grades (0, 2, 3, 4, 6) and status bits. Charge transfer inefficiency (CTI) and time-dependent gain corrections were not applied, because the focal plane temperature of these two observations was not -120 C. Because of the background flaring at the beginning of observation of obs ID 50, we used a reduced exposure time of 8.5 ks. Therefore, the total exposure time of the two combined observations is 18120 s. The filtering process was done using the $Chandra$ Interactive Analysis of Observations (CIAO) package and following the Science Threads from $Chandra$ X-Ray Center. We also restricted the energy range from 0.4 to 6.0 keV. This would optimize the detection of the PMS stars and reduce spurious detections. Finally, we merged two observations to one image (Figure 2), which was used for source detection.
WAVDETECT program within CIAO was utilized to detect sources in the merged image. We ran wavelet scales ranging from 1 to 16 pixels in steps of $\sqrt{2}$ with a source significance threshold of 3$\times$10$^{-6}$. Removing some spurious detections, e.g. some sources around partial shell of X-ray emission surrounding $\eta$ Carinae [@sew01] and along the trailed line due to $\eta$ Carinae itself, we got 454 sources eventually. By using the merged image for source detection, we detected more than double the number of sources than reported by @evans.
We extracted the count of each source from the circular region centered on the WAVDETECT source position with a 95% encircled energy radii ($R$(95%EE)) [@fei02]. For the background determination, an annulus around each source between 1.2 and 1.5 $R$(95%EE) was used. Before extracting the source counts from each observation, exposure and background maps were created. An exposure map was computed to take into account vignetting and chip gaps, and an energy range of 1.2 keV was used for generating the exposure map. To avoid any sources within the background annuli for a given source, a background map was created excluding the sources in $R$(95%EE). This background map was used to obtain the source counts. We utilized DMEXTRACT tool of CIAO to extract source counts for each of the two observations. The total count of each source was then computed by combining the two observations. Finally the count rates were calculated for a total exposure time of 18120 s. The typical background count across the $Chandra$ field had a 3-$\sigma$ error of $\sim 1$ count.
Near-Infrared Data — SIRIUS
---------------------------
We carried out near-infrared imaging observations toward the Carina Nebula using the SIRIUS (Simultaneous InfraRed Imager for Unbiased Survey) camera mounted on the Infrared Survey Facility (IRSF) 1.4 m telescope, in Sutherland, South Africa. The SIRIUS camera [@nagayama] is equipped with three HAWAII arrays of $1024 \times
1024$ pixels and provides simultaneous observations in the three bands, $J$(1.25 $\mathrm{\mu m} $), $H$(1.63 $\mathrm{\mu m}$) and $K_s$(2.14 $\mathrm{\mu m}$) using dichroic mirrors. It offers a field of view of $7.\arcmin8 \times 7.\arcmin8$ with a plate scale of $0.\arcsec45$ $\mathrm{pixel^{-1}}$. In April 2003 nine pointings ($3 \times 3$) were observed covering effectively $22\arcmin \times 22\arcmin$ and including the $Chandra$ field. The central coordinates of the observed fields are $R.A.
= 10^h45^m05^s$ and $Dec. = -59\arcdeg 38\arcmin 52\arcsec$. For each pointing, 30 dithered frames were observed, with an integration time of 30 s each, giving a total integration time of 900 s. Two pointings (\#5 and \#6) of the April 2003 data which suffered weather fluctuations were re-observed in January 2005, for which 45 dithered frames were observed with an integration time of 20 s, yielding a total integration time of 900 s for each pointing. The typical seeing during our observations ranged from $1.\arcsec0$–$1.\arcsec4$ and the airmass from 1.2 to 1.5. The standard stars No. 9144 and 9146 from @persson were observed for photometric calibration.
We used the IRAF (NOAO’s Image Reduction and Analysis Facility) package to reduce the SIRIUS data. The standard procedures for image reduction, including dark current subtraction, sky subtraction and flat field correction were applied. The images in each band were then average-combined for each pointing to achieve a higher signal-to-noise ratio. We performed photometry on the reduced images using IRAF’s DAOPHOT package [@stetson]. Since the field is crowded, we performed PSF (point spread function) photometry in order to avoid source confusion. To construct the PSF for a given image, we chose about 15 bright stars, well isolated from neighboring stars, located away from the nebulosity and not on the edge of the image. The ALLSTAR task of DAOPHOT was then used to apply the average PSF of the 15 PSF stars to all the stars in the image, from which the instrumental magnitude of each star was derived. The instrumental magnitudes were then calibrated against the standard stars observed on each night.
X-RAY SOURCES AND STELLAR COUNTERPARTS
======================================
The optical spectroscopy of the stars in Tr14 and Tr16 has been done by several groups, eg., @walborn73 [@walborn82; @levato; @fitzgerald]. The latest work by MJ93 lists the brightest and the bluest stars of the two clusters. We have used this list (Table 4 in MJ93) to find the counterparts of our X-ray sources. Within a $3\arcsec$ search radius, our cross-identification resulted in 30 OB stars from MJ93 as counterparts of our X-ray sources. Apart from MJ93, we also checked for any possible counterparts using SIMBAD[^2]. This resulted into another 8 OB stars of the region [@tapia], the spectral types of which were determined by the photometric Q method [@json].
We also used our NIR data to search for the counterparts of the X-ray sources. Again with a $3\arcsec $ search radius, we found counterparts for 432 sources. Thus, more than 95% of the X-ray sources have NIR counterparts. For 51 cases out of 432 sources, the NIR photometric errors are larger than 0.1 mag in one or more bands. Most of these large photometric error cases are for stars located in pointing 5, which is the Tr16 region. The NIR photometry in this pointing is affected because of a large number of bright stars and nebulosity around $\eta$ Carinae. Since we are using the NIR colors of the sources to delineate their young stellar nature, an uncertainty larger than 0.1 mag cannot serve the purpose. Hence, in our analysis we consider only those cases for which the photometric uncertainties are smaller than 0.1 mag in all the three bands, which leaves us with 384 sources. For our analysis, we have converted the NIR photometry into California Institute of Technology (CIT) system using the color transformations between the SIRIUS and CIT systems as given in @nakajima.
RESULTS AND DISCUSSION
======================
Known OB stars
--------------
Table 2 lists the X-ray sources cross-identified with known OB stars. The coordinates of each X-ray source are listed in columns (1) and (2), followed by the identifier of the optical counterpart of the X-ray source, listed in column (3). The optical $B$, and $V$ magnitudes, and the spectral type, listed in columns (4)–(6), were adopted in most cases from MJ93 and in others from @tapia. The color excess of each source, $E(B-V)$, given in column (7), was also taken from MJ93, in which photometry and spectroscopy were used to estimate the intrinsic stellar $(B-V)_0$ [@fitzgerald70]. The bolometric magnitude, $M_{\rm bol}$, in column (8), was taken from @massey01. For a small number of cases, where the spectral types were taken from [@tapia], the color excesses as well as the bolometric magnitudes were estimated using their spectral types. Columns (9)–(11) list the IRSF NIR $J$, $H$ and $K_s$ magnitudes of the counterpart. Column (12) lists the X-ray counts of the sources derived by the DMEXTRACT tool of the CIAO software, as described in §3. We used the WebPIMMS[^3] to derive the unabsorbed X-ray flux of the sources. To convert an X-ray count to the flux, the Raymond Smith Plasma model with temperature $\log~T=6.65$, equivalent to $0.384 {\rm~keV}$, was adopted. For the extinction correction, the color excess, $E(B-V)$ of each source was used to estimate the neutral hydrogen column density, $N_H$. The X-ray flux is given in column (13), and the X-ray luminosity, computed by adopting a distance of 2.5 kpc, is in column (14). The last column (15) contains the logarithmic ratio of the X-ray luminosity to the stellar bolometric luminosity, where the latter was derived from the bolometric magnitude, i.e., column (8).
Figure 3 shows the distribution of X-ray luminosities of the known OB stars in our field. Most OB stars have $\log L_{\rm X} \ga 31{\rm ~ergs~s^{-1}}$ with the distribution peaking $\sim \log L_{\rm X} = 31.7 {\rm~ergs~s^{-1}}$. The Wolf-Rayet star (HD93162) is the brightest X-ray source in the sample, with $\log L_{\rm X} = 34.12 {\rm ~ergs~s^{-1}}$. This star has been known to be unusually bright in X rays as compared to other W-R stars in the region [@evans]. Though it has been thought to be a single star, a recent W-R catalog by @hucht lists it as a possible binary (see discussions in @evans).
Among the 38 X-ray OB stars, there are 3 B3-type, 3 B5-type and 1 B7-type stars. Mid- to late-B type stars are supposed to be X-ray quiet as they have neither strong enough stellar winds as in the case of O or early B stars, nor the convective zones to power the chromospheric/coronal activities as in the case of late-type stars. However, mid- to late-B type stars have been found to be X-ray emitters in earlier studies, e.g., @cohen. The X-ray luminosities of the mid- to late-type B stars in our sample are comparable to those of T Tauri candidates in the same sample. Although this seems to provide circumstantial evidence of CTTS or Herbig Ae/Be companions to account for the X-ray emission, it does not rule out the possibility of a so far unknown emission mechanism intrinsic to mid- to late-B stars.
The X-ray luminosities of the OB stars are known to satisfy the relation with the stellar bolometric luminosities, namely, $L_{\rm X} \propto
10^{-7} L_{\rm {bol}}$. All but a few stars in our sample satisfy this relation (Fig. 4). Among the outliers, labeled on the figure by their spectral types, only HD93162 (a W-R star), and Tr16$-$22 (an O8.5V star) are early type stars and hence their high $L_{\rm X}/L_{\rm
{bol}}$ is unusual. Tr16$-$22 is among the brightest X-ray sources in our sample, with $\log L_\mathrm{X} = 32.83$ ergs s$^{-1}$. It is brighter in X rays by a factor of 5–20 compared to other O8.5V stars and even brighter than the two O3 stars in the sample. @evans present a list of known binaries among the massive stars and discuss the X-ray luminosities against their single or binary status. A massive companion may enhance the X-ray production by colliding winds. No binary companion is known to exist for either HD93162 or Tr16$-$22 [@evans] to account for their high X-ray luminosities and high $L_{\rm X}/L_{\rm {bol}}$ ratios. The rest of the X-ray sources which do not satisfy the correlation are mid-B or late-B type stars. A study by @berghofer about the X-ray properties of OB stars using the $ROSAT$ database showed that the $L_{\rm X}/L_{\rm bol}$ relation extends to as early as the spectral type B1.5, and inferred this as a possibly different X-ray emission mechanism for the mid- or late-B type stars as compared to the O or early-B stars. In our sample, there are 3 B3 type stars which seem to satisfy this relation and all the stars later than B3 deviate significantly from the mean value of $L_{\rm X}/L_{\rm {bol}}$ ratio for O and early-B stars.
Candidate OB stars
------------------
There are 17 anonymous stars in our sample which have similar NIR and X-ray properties as the known OB stars in the region. These stars appear to be massive stars of O or B types, but we could not find their spectral type information in the literature, e.g., MJ93 or SIMBAD. These candidate OB stars, with their optical and NIR magnitudes along with their X-ray counts and X-ray luminosities are listed in Table 3. To determine their X-ray fluxes from counts, we made use of WebPIIMS. For extinction correction, we used an average $E(B-V) = 0.52$ based on the Table 4 of MJ93, as we did not have the spectral class information to determine their individual color excesses. Other parameters to obtain the X-ray fluxes from the X-ray counts remain the same as for the known OB stars. We found that the use of an average value of $E(B-V)$, rather than the individual $E(B-V)$ values, in case of the known OB stars would make a difference of a factor of two or less in the derived X-ray luminosities. Likewise for the temperature, using a $\log T$ between 6.4 to 7.1 also would make a difference of a factor of two or less in the X-ray luminosities. Hence the use of an average color excess for candidate OB stars should not affect much our results.
Figure 5 shows the NIR color-color diagram of the known OB stars and the candidate OB stars. The solid curve represents the dwarf and giant loci [@bb], and the parallel dashed lines represent the reddening vectors, with $A_J/A_V = 0.282$, $A_H/A_V = 0.272$, and $A_K/A_V = 0.112$ [@rieke]. The dotted line indicates the locus for dereddened classical T Tauri stars [@meyer]. It can be seen that the candidate OB stars are either intermixed with or redder than the known OB stars. Figure 6 shows the NIR color-magnitude diagram of the known and candidate OB stars. The solid line represents the unreddened main sequence [@koornneef] at 2.5 kpc. Some candidate OB stars are very bright in NIR, with a few even brighter than $K_s = 8$ mag. In contrast, the candidate OB stars are fainter and redder than the known OB stars in the optical wavelengths (Figure 7), indicative of the effect of dust extinction, while both samples show a comparable range in X-ray luminosities (compare Figure 8 with Figure 2). Thus, it appears that these candidate OB stars have escaped earlier optical spectroscopic studies because of their large optical extinction. Addition of these massive stars expands substantially the known list of luminous stars and thus contributes significantly to the stellar energy budget of the region.
PMS candidates
--------------
Figures 9 and 10 show the NIR color-color and color-magnitude diagrams of all the 380 X-ray sources with NIR photometric errors less than 0.1 mag. By using the criteria given in @meyer, we find about 180 stars as CTTS candidates. Apart from the CTTS candidates, the NIR colors suggest quite a many possible weak-lined T Tauri star (WTTS) candidates. The X-ray and NIR data together hence turn up a large population of low-mass pre-main sequence candidates. The T Tauri candidates in our sample (CTTS plus WTTS) should be a fairly secure T Tauri population, given their X-ray emission and their NIR color characteristics. Although much work has been done on the massive stellar content in Tr14 and Tr16, a comprehensive sample of the T Tauri population has not been obtained so far. @tapia03 presented $UBVRIJHK$ photometry of Tr14, Tr16 and two other clusters in the region, Tr15 and Cr232, and noticed some stars with NIR excess in Tr14 and Tr16. They estimated the ages of Tr14 and Tr16 to be 1–6 million years. To our knowledge, our sample represents the most comprehensive sample of the young stellar population in Tr14 and Tr16. The distribution of X-ray luminosities of the CTTS candidates is shown in Figure 11. Comparison with Figure 2 shows that the X-ray luminosities of the T Tauri candidates are on the average lower and hence consistent with the notion that late-type stars have weaker X-ray emission.
@fei05 pointed out that the X-ray luminosity functions (XLFs) of young stellar clusters show two remarkable characteristics. First, the shapes of the XLFs of different young stellar clusters are very similar to each other after the tail of the high luminosity O stars $(\log L_{\rm X} >
31.5 ~{\rm ergs~s^{-1}})$ is omitted. Secondly, the shape of this ’universal’ XLF in the 0.5–8.0 keV energy range resembles a lognormal distribution with the mean, $\log L_{\rm X} \approx 29.5 ~{\rm ergs~s^{-1}}$ and the standard deviation $\sigma(\log L_{\rm X}) \approx 0.9$ (see Figure 2 in @fei05). The $Chandra$ observations we used in this work include only part of Tr14. For Tr16, we can make an estimate of the total stellar population in reference to the XLF of the Orion Nebula Cluster (ONC) derived from the $Chandra$ Orion Ultradeep Project [@getman05]. The limiting X-ray luminosity of our sample is $L_{\rm X} \sim 30.5~{\rm
ergs~s^{-1}}$. Excluding the high X-ray luminosity tail, i.e., $L_{\rm
X} > 31.5~{\rm ergs~s^{-1}}$, which includes about 30 known OB stars and candidate OB stars described earlier, the slope of the Tr16 XLF is consistent with that of the ONC in the X-ray luminosity range of our sample. This suggests that our sample represents about 20% of the X-ray members in the cluster. We hence estimate that the total stellar population of Tr16 should be $\sim$ 1000–1300. Furthermore, the X-ray luminosities are known to be correlated with stellar masses, as found in the $ROSAT$ data [@fei93] and also in the $Chandra$ studies of the ONC [@flaccomio03; @preibisch05]. Comparing the XLF of Tr16 with the ONC XLF versus stellar mass (Figure 5 in @fei05b), we infer that our sample is about 60% complete for the stars with masses larger than 1 [${M_{\odot}}$]{}, and 40% complete between 0.3–1 [${M_{\odot}}$]{}. Our deep NIR data covering clusters Tr14, Tr16 and Cr232 would probe even lower mass end of the stellar population. The analysis of the complete NIR results will be presented elsewhere.
A compact embedded X-ray group
------------------------------
We notice a group of 7 X-ray sources concentrated in a field of 5$\times$ 4, located south-east of Tr16 and coincident with the prominent dark ’V’ shaped dust lane which bisects the Carina Nebula. Adopting a distance of 2.5 kpc, the physical size of this star group is about 4 pc. Each of these 7 sources has an NIR counterpart, listed in Table 6 with their coordinates, $J$, $H$, and $K_s$ magnitudes and X-ray counts. We use the star identification number in column 1 of Table 6 in further discussion. The NIR colors have been used to estimate the neutral hydrogen column density. Stars in this group are bright and suffer large amounts of reddening, as seen in the NIR color-color and color-magnitude diagrams (Figures 12 and 13). The brightest sources (stars 4, 6 and 7) in NIR, with $K_s \sim$ 8.5–10.5 mag, are also X-ray bright, with $L_{\rm X} \sim 10^{32}$ $\mathrm{ergs~s^{-1}}$. Star 4 is a known O4 star [@rgsmith]. Our NIR magnitudes for this star match with those reported by @rgsmith. Apart from star 4, we could not find any optical photometric or spectroscopic information in the literature for the others sources. The bright NIR and X-ray stars 4, 5, and 6 can be clearly seen in the optical Digitized Sky Survey (DSS) image (Figure 14), whereas the other sources which are highly extincted even in NIR are not visible at all. Figure 15 shows the IRSF $K_s$ image with the sources marked. Stars 1 and 5 are peculiar because they are highly extincted ($A_V \sim$ 15–25 mag estimated from their NIR colors), yet both are X-ray bright with $L_{\rm X} \sim 10^{33}$ $\mathrm{ergs ~ s^{-1}}$. What could be the nature of these sources? Their NIR fluxes and colors, along with their non-detection in the DSS image, imply that they could be reddened T Tauri or class I objects. But their X-ray luminosities are much higher than observed in typical T Tauri stars ($ < 10^{32} \mathrm{ergs~s^{-1}}$). One possibility is that they are heavily embedded massive stars. The rest two sources of the group, stars 2 and 3 are relatively faint in both NIR and X rays, thus appear to be reddened T Tauri stars.
It is worth noting that the above mentioned group is spatially close, $\sim 7\arcmin$, to the deeply embedded object, IRAS10430$-$5931. With $\mathrm{~^{12}CO(2-1)}$ and $\mathrm{~^{13}CO(1-0)}$ observations, @megeath found this $IRAS$ source associated with a bright-rimmed globule with a mass of $\sim 67$ [${M_{\odot}}$]{}. They also found sources with NIR excess around this IRAS object and provided the first indication of star-formation activity in the Carina region. More recently, a mid-infrared study by @nsmith discovered several clumps along the edge of the dark cloud east of $\eta$ Carinae, including the clump associated with IRAS10430$-$5931. They noted that each of these clumps is a potential site of triggered star-formation due to their location at the periphery of the Nebula behind the ionization fronts. We compared the spatial distribution of this group with the $\mathrm{~^{12}CO(1-0)}$ observations by @brooks in Figure 16. The group of young stars is ’sandwiched’ between two cloud peaks. It is not clear whether the group is continuation of Tr16 but obstructed by the dark dust lane, or is a separate OB group/association still embedded in the cloud.
Figure 17 shows all the $Chandra$ X-ray sources overlaid with the $\mathrm{~^{12}CO(1-0)}$ image [@brooks]. One sees immediately a general paucity of stars with respect to molecular clouds. Tr16 is “sandwiched" between the north-west and south-east cloud complexes. All the X-ray sources (i.e., young stars) associated with these clouds in turn are seen either intervening between clouds or situated near the cloud surfaces facing Tr16. The morphology of young stellar groups and molecular clouds peripheral to an region (i.e., Tr16) fits well the description of the collect-and-collapse mechanism for massive star formation, first proposed by @elmegreen77 and recently demonstrated observationally by @deh05 [@zav06]. The expanding ionization fronts from an region compress the outer layer of a nearby cloud until the gas and dust accumulate to reach the critical density for gravitational collapse to form next-generation stars, which subsequently cast out their own cavities. This collect-collapse-clear process may continue as long as massive stars are produced in the sequence and there is sufficient cloud material in the vicinity.
SUMMARY
=======
We detected 454 X-ray sources in the $Chandra$ image of the Carina Nebula observed in September 1999. About 1/3 of the X-ray sources have optical counterparts in the literature, including 38 known OB stars in the region. In comparison our NIR observations detect counterparts for more than 95% of the X-ray sources. The X-ray luminosities of the known OB stars range in $\sim 10^{31}$–$10^{34}~{\rm ergs~s^{-1}}$, with the Wolf-Rayet star, HD 93162, being the strongest X-ray source with $\log L_{\rm X} = 34.12 \mathrm{~ergs~s^{-1}}$. The W-R star also has a very high $L_{\rm X}/L_{\rm bol}$ ratio, $\sim -5.39$. The only other early-type star with a high $L_{\rm X}/L_{\rm bol}$ ratio is an O8.5V type star, Tr16$-$22, which also has a very high X-ray luminosity of $L_{\rm X} = 32.83$ for its spectral type. All other O and early B (up to B3 type) stars satisfy the canonical relation, $L_{\rm X} \sim ~10^{-7}~L_{\rm {bol}}$. There are several mid- to late-B type stars emitting X rays with X-ray luminosities comparable with those typical for T Tauri stars. Hence, it is possible that the X-ray emission from these mid- and late-B stars is coming from T Tauri companions. We discovered 17 candidate OB stars which have escaped detection in previous optical studies because of the larger dust extinction they suffer. These candidate OB stars have the same characteristics as known OB stars in terms of X-ray luminosities and NIR fluxes and colors. If most of them turn out to be bona fide OB stars, this will be already half the number of the known OB stars found as X-ray emitters in the region and would add significantly to the stellar energy budget of the region. The NIR colors of the X-ray counterparts show a large population of low-mass pre-main sequence stars of the classical T Tauri type or the weak-lined T Tauri type. Some 180 classical T Tauri candidates are identified, whose X-ray luminosities range between $10^{30}$ to $10^{32}~{\rm ergs~s^{-1}}$, lower than those for OB stars. Comparison of the X-ray luminosity function of Tr16—which is about 60% complete for stars with masses 1–3 [${M_{\odot}}$]{}and 40% complete for 0.3–1 [${M_{\odot}}$]{}—with that of typical young star clusters suggests a total stellar population $\sim$ 1000–1300 in Tr16. A compact group of highly reddened, X-ray bright and NIR bright sources is found to the south-east of Tr16. The group is associated with an $IRAS$ source and coincident with the dust lane where many mid-IR sources have been predicted to be the potential sites of triggered star-formation. This star group is “sandwiched" between two peaks of the $\mathrm{~^{12}CO(1-0)}$ emission. Such star-cloud morphology is also seen in the peripheries of the complex in Tr16, a manifestation of the collect-and-collapse triggering process to account for the formation of massive stars.
This publication makes use of the $Chandra$ observations of the Carina Nebula made in September 1999. We made use of the SIMBAD Astronomical Database to search the optical counterparts for the $Chandra$ X-ray sources. We thank Kate Brooks for providing us with the $\mathrm{~^{12}CO(1-0)}$ data of the Carina which was obtained with the Mopra Antenna, operated by the Australia Telescope National Facility, CSIRO during 1996-1997. KS, WPC and HTL acknowledge the financial support of the grant NSC94-2112-M-008-017 of the National Science Council of Taiwan.
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[^1]: Figures with better resolution can be obtained from http://cepheus.astro.ncu.edu.tw/kaushar.html
[^2]: http://simbad.u-strasbg.fr/sim-fid.pl
[^3]: http://http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html
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Introduction
============
There are many experiments planned which hope to reap the rich harvest of information available in the anisotropies of the Cosmic Microwave Background (CMB) [@BenTurWhi]. For the purposes of estimating cosmological parameters or constraining models it is important to know how well an experiment can constrain the angular power spectrum of CMB anisotropies.
It is common practice to estimate the errors on the angular power spectrum which could be obtained by an experiment with a given angular resolution and noise level. Such an estimate can give us insight into what regions of the angular power spectrum one could constrain, and what are the limiting factors in an experiment designed e.g. to constrain cosmological parameters. In this report we point out some simple limiting cases which allow one to gain intuition about the effect of multiple observing frequencies with differing beam sizes in the presence of (somewhat idealized) foregrounds.
If we write the angular power spectrum of the anisotropy as $C_\ell$ and the noise power spectrum as $N_\ell$ (typically $N_\ell=4\pi f_{\rm sky}\sigma^2/N_{\rm pix}$, where $f_{\rm sky}$ is the fraction of sky covered, $\sigma$ is the rms pixel noise and $N_{\rm pix}$ is the number of pixels) then $$\delta C_\ell \simeq \sqrt{ {2\over (2\ell+1) f_{\rm sky}} }
\left( C_\ell + {N_\ell\over W_\ell} \right)
\label{eqn:dcl}$$ where $W_\ell$ is the window function of the experiment and we have assumed that the noise is gaussian. For a gaussian beam of width $\theta_b$, $W_\ell=\exp(-\ell^2\theta_b^2)$. Eq. (\[eqn:dcl\]) assumes that the only sources of variance in an experiment are the anisotropy signal and the receiver noise, i.e. it neglects foregrounds. We have also implicitly assumed that $\delta C_\ell$ is a gaussian error, thus we imagine we are working at reasonable high-$\ell$ and will bin our power spectrum estimates into finite width bins in $\ell$. Eq. (\[eqn:dcl\]) has been widely used to estimate how well upcoming satellite experiments could constrain cosmological parameters [@Param].
Several authors have also addressed the question of foreground subtraction using multi-frequency observations [@Brandt; @Dod; @TegEfs; @Teg]. With multi-frequency observations it appears possible to separate the desired anisotropy signal from the foregrounds with encouraging precision. The authors of [@Brandt; @Dod] have shown that one can regard foreground subtraction as an enhancement of the noise in a foreground free experiment. The noise enhancement factor has been called the [*foreground degradation factor*]{} (FDF) by Dodelson [@Dod] who gave a simple expression for it. A method of foreground subtraction using both frequency and spatial information was proposed in Ref. [@TegEfs]. Various methods of foreground subtraction have been compared recently by Tegmark [@Teg].
Note that the “noise term” in Eq. (\[eqn:dcl\]) depends exponentially on $\ell$ once $\ell > \theta_b^{-1}$. In multi-frequency observations the angular resolution of the different frequency channels is rarely the same, leading to the question of which beam size to use. In this report we discuss a simple heuristic “effective” beam size and noise level. Since how well we can take out foregrounds depends on the angular structure of the foregrounds, the “noise” will be a function of the foreground angular power spectrum.
The formalism is directly comparable to that in [@TegEfs; @Teg] in that we use a minimum variance estimator of the CMB anisotropy power spectrum. An extension to “real world foregrounds” [@Teg] is straightforward, but the purpose of this report is to gain intuition through simple examples so we do not pursue this line of development.
Formalism
=========
As noted by Tegmark [@Teg], one can consider foregrounds as an additional noise component which is highly correlated among different frequency channels. If we label the frequency channels by a greek subscript we can define a noise correlation matrix $$N^\ell_{\alpha\beta} = {4\pi\over N_{\rm pix}} \sum_i
W_{\ell\alpha}^{1/2}
\left\langle f^i_{\ell\alpha} f^i_{\ell\beta}\right\rangle
W_{\ell\beta}^{1/2}
+ {4\pi\over N_{\rm pix}} \sigma^2_\alpha \delta_{\alpha\beta}$$ which has contributions from the pixel noise (assumed uncorrelated between channels here for simplicity) and foregrounds labelled by superscript $i$. Here $\langle f^i_{\ell\alpha} f^i_{\ell\beta}\rangle$ is the correlation matrix of foreground $i$ in channels $\alpha$ and $\beta$, with the $4\pi/N_{\rm pix}$ inserted for later convenience. If the foregrounds are 100% correlated between the channels then $\langle f^i_{\ell\alpha}f^i_{\ell\beta}\rangle=f^i_{\ell\alpha}f^i_{\ell\beta}$ where $f^i_{\ell\alpha}$ is the rms intensity as a function of frequency. We shall assume this case from now on, but see Ref. [@Teg].
It is straightforward to derive the minimum variance estimator of $C_\ell$, as a linear combination of measurements at different frequencies. If we have measured multipole moments $a^{\alpha}_{\ell m}$ at frequency $\alpha$ then we write the estimate of the CMB component as $\theta_{\ell m}=\sum_\alpha F_{\alpha} a^{\alpha}_{\ell m}$. Imagine that we can write the observed signal $a^{\alpha}_{\ell m}=t_{\ell m} W_{\ell\alpha}^{1/2}+n_{\ell m\alpha}$, where $t_{\ell m}$ is the cosmological signal of interest (the same in each frequency channel) and $n_{\ell m\alpha}$ is the sum of the noise and foregrounds (i.e. the non-CMB components). We now minimize the variance of our estimator $\theta_{\ell m}$ minus the “real” underlying sky ($t_{\ell m}$) with respect to the weighting matrix $F_{\alpha}$. We find that our minimum variance $C_\ell$ estimator is (averaging over $m$) $$\widehat{C}_\ell = \sum_{\alpha\beta}
{\theta_{\ell}^2 - F_{\alpha} N^\ell_{\alpha\beta} F_{\beta} \over
W^{1/2}_{\ell\alpha} F_{\alpha} \ W^{1/2}_{\ell\beta} F_{\beta} }
\label{eqn:clest}$$ where $\theta_{\ell}^2$ is the average of $\theta_{\ell m}^2$ over $m$, $$F_{\alpha} = \sum_\beta C_\ell W_{\ell\beta}^{1/2} \ \left(
W_{\ell\beta}^{1/2} C_\ell W_{\ell\alpha}^{1/2} +
N^{\ell}_{\alpha\beta} \right)^{-1}$$ and we have left the $\ell$-dependence of $F_\alpha$ implicit for notational convenience. The vector $F_\alpha$ projects out the $\ell$th CMB multipole moment from the signal in each channel in a minimum variance sense.
If we assume that this estimator is Gaussian then we can replace Eq. (\[eqn:dcl\]) by $$\delta C_\ell = \sqrt{ {2\over (2\ell+1)} }
\left( C_\ell + \sum_{\alpha\beta}
{(F_{\alpha} N_{\alpha\beta} F_{\beta})_\ell \over
(W_{\ell\alpha}^{1/2} F_{\alpha})^2 } \right)
\label{eqn:dcl2}$$ where we have set $f_{\rm sky}=1$ for simplicity (the scaling with $f_{\rm sky}$ is given in Eq. (\[eqn:dcl\])). Note that in the limit of one frequency channel and no foregrounds, the sums over $\alpha$ and $\beta$ are trivial, the noise term is independent of $F$ and we recover Eq. (\[eqn:dcl\]).
Eq. (\[eqn:dcl2\]) is the general result, and we consider several examples to gain intuition in the next section.
Examples
========
Let us consider various limits of Eq. (\[eqn:dcl2\]). We will focus on the noise term, since the part of the error proportional to $C_\ell$ simply reflects cosmic plus sample variance [@ScoSreWhi]. In the signal dominated limit ($C_\ell\gg N_\ell$) $F_\alpha$ is just the inverse square root of $W_{\ell\alpha}$ and the noise term in Eq. (\[eqn:dcl2\]) reduces to $$\left( {N_\ell\over W_\ell} \right)_{\rm eff} =
{4\pi\over N_{\rm pix}W_\ell^2} \left[
\sum_\alpha \sigma_\alpha^2 W_{\ell\alpha} +
\left(\sum_\alpha f_{\ell\alpha}W_{\ell\alpha}^{1/2}\right)^2 \right]
\label{eqn:signaldominated}$$ where $W_\ell=\sum_\alpha W_{\ell\alpha}$. Thus, in the absence of foregrounds the noise is the sum of the noises in those channels able to resolve features with multipole number $\ell$, divided by the number of such channels squared. The addition of foregrounds increases the variance, but by assumption both contributions to $\delta C_\ell$ are sub-dominant. We expect this regime to occur at low-$\ell$, where we are cosmic and sample variance limited.
Now let us consider the opposite limit. First imagine there are no foregrounds. In this limit ($N_\ell\gg C_\ell$) the noise term in Eq. (\[eqn:dcl2\]) reduces to $$\left( {N_\ell\over W_\ell} \right)^{-1}_{\rm eff} =
{N_{\rm pix}\over 4\pi} \sum_\alpha {W_{\ell\alpha}\over \sigma_\alpha^2 }
\label{eqn:noisedominated}$$ which would be the obvious way to combine the channels to obtain the effective noise: recall the error on a weighted mean is $\sigma^{-2}=\sum_i \sigma_i^{-2}$.
Now we can enhance this last example by the addition of foregrounds. For simplicity imagine a two channel experiment with 1 foreground $f_{\ell\alpha}$ in addition to uncorrelated noise $\sigma_\alpha$, with $\alpha=1,2$. For definiteness imagine that channel 1 is a low frequency channel with a “large” beam, while channel 2 is a high frequency channel with a “small” beam. Some trivial matrix algebra allows us to write $(N_\ell/W_\ell)_{\rm eff}$ as $${4\pi\over N_{\rm pix}}
\ {\sigma_1^2\sigma_2^2 + \sigma_1^2 f_{\ell 2}^2 W_{\ell 2} +
\sigma_2^2 f_{\ell 1}^2 W_{\ell 1} \over
\sigma_1^2 W_{\ell 2} + \sigma_2^2 W_{\ell1} +
[f_{\ell 1}-f_{\ell 2}]^2 W_{\ell 1}W_{\ell 2} }$$ which reduces to Eq. (\[eqn:noisedominated\]) as $f_\alpha\to0$.
To bring out the essential details let us take $\sigma_1=\sigma_2$ and ignore the different resolutions by working at low enough $\ell$ that $W_{\ell\alpha}\to1$, $$\left( {N_\ell\over W_\ell} \right)_{\rm eff}
\to {4\pi\sigma^2\over N_{\rm pix}}
\left( {\sigma^2 + f_{\ell 1}^2+f_{\ell 2}^2 \over
2\sigma^2 + (f_{\ell 1}-f_{\ell 2})^2 } \right) \quad .
\label{eqn:fdfgeneral}$$
The term in parentheses is the increase in the noise over the one channel result. As we decrease $\sigma^2$, this goes from ${1\over 2}$ (the case of no foregrounds: co-add the channels) to $(1+x^2)(1-x)^{-2}$ where $x=f_{\ell 2}/f_{\ell 1}$. For this case, the increase in the noise is just the FDF defined by Dodelson [@Dod], written in a slightly different notation. The point $x=1$ is when the frequency dependence of the foreground is the same as the CMB. Generically the minimum variance estimator Eq. (\[eqn:clest\]) does better than the FDF would indicate, as has been pointed out in Ref. [@Teg].
Now let us reinstate the window functions, but imagine we are at intermediate $\ell$, where $W_{\ell 1}\ll 1$. In this regime $$\left( {N_\ell\over W_\ell} \right)_{\rm eff} =
{4\pi\sigma_2^2\over N_{\rm pix}W_{\ell 2}}
\ \left( 1 + { f_{\ell 2}^2 W_{\ell 2}\over \sigma_2^2} \right) \quad .$$ So the appropriate resolution is that of channel 2, and the noise is the channel 2 noise [*plus*]{} a contribution from the foreground at the higher frequency. Note that the properties of channel 1 do not enter the expression, as expected. The foreground contribution declines as one approaches the resolution of channel 2, so that asymptotically the error is just $4\pi\sigma_2^2/N_{\rm pix}W_{\ell 2}$.
Finally note that if our foregrounds have a “steep” power spectrum, i.e. fall rapidly with $\ell$, then at high-$\ell$ we have the limit $f_{\ell\alpha}\to0$ and reproduce Eq. (\[eqn:noisedominated\]). At low-$\ell$ the foreground should dominate over the noise which is the case discussed below Eq. (\[eqn:fdfgeneral\]). This shows that for foregrounds without a lot of small-scale structure there is little effect at high-$\ell$ from removing the foregrounds (in this simplified example where the foreground properties are assumed known exactly).
Summary
=======
Let us summarize our results with $\ell$ increasing from the signal- to the noise-dominated limits. Under our assumptions, the error on the angular power spectrum starts dominated by cosmic and sample variance at low-$\ell$. Moving to higher $\ell$, if the noise or foregrounds start to dominate on scales which all channels can resolve, the effect of a foreground looks like an increase in the noise by a factor which is always less than Dodelson’s FDF [@Dod]. If the noise or foregrounds dominate on scales smaller than the size of the widest beam, but larger than the size of the smallest beam, the noise is that of the channel with the smaller beam size, increased by the variance of the foreground at that frequency. At $\ell>\theta_b^{-1}$ for even the highest frequency channel the error is simply that obtained from the noise in the highest frequency channel, regardless of foregrounds. If the foregrounds are always negligible compared to the noise, the appropriate noise level is the weighted sum of the noises in each channel which can resolve the given $\ell$, as shown in Eq. (\[eqn:noisedominated\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Douglas Scott for useful comments on the manuscript.
[99]{} C. Bennett, M.S. Turner, M. White, Physics Today, [**50**]{}, 32 (1997) L. Knox, Phys. Rev. [**D52**]{}, 4307 (1995); D. Scott, M. White, General Relativity and Gravitation, [**27**]{}, 1023 (1995); G. Jungman et al., Phys. Rev. Lett. [**76**]{}, 1007 (1996); M. Zaldarriaga, D. Spergel, U. Seljak, Astrophys. J., [**488**]{}, 1 (1997); J. Bond, G. Efstathiou, M. Tegmark, Mon. Not. R. Astron. Soc. in press \[astro-ph/9702100\]. W.N. Brandt et al., ApJ [**424**]{}, 1 (1994). S. Dodelson, ApJ [**482**]{}, 577 (1997). M. Tegmark, G. Efstathiou, MNRAS, [**281**]{}, 1297 (1996). M. Tegmark, preprint astro-ph/9712038 D. Scott, M. Srednicki, M. White, Astrophys. J., [**421**]{}, L5 (1994).
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[Numerical simulations of impacts involving porous bodies:\
I. Implementing sub-resolution porosity in a 3D SPH Hydrocode\
]{} TEL: (+41) 31 631 4057\
FAX: (+41 31 631 4405)\
E-MAIL: jutzi@space.unibe.ch\
**Length:\
** 60 manuscript pages\
2 Tables\
13 Figures
**Running Title:\
** Impact simulations with porosity\
\
**Corresponding author:\
** Martin Jutzi\
Physikalisches Institut\
University of Bern\
Sidlerstrasse 5\
CH-3012 Bern\
Switzerland\
TEL: (+41) 31 631 4057\
FAX: (+41) 31 631 4405\
E-MAIL: jutzi@space.unibe.ch\
[Abstract\
]{}
In this paper, we extend our Smooth Particle Hydrodynamics (SPH) impact code to include the effect of porosity at a sub-resolution scale by adapting the so-called $P-alpha$ model. Many small bodies in the different populations of asteroids and comets are believed to contain a high degree of porosity and the determination of both their collisional evolution and the outcome of their disruption requires that the effect of porosity is taken into account in the computation of those processes. Here, we present our model and show how porosity interfaces with the elastic-perfectly plastic material description and the brittle fracture model generally used to simulate the fragmentation of non-porous rocky bodies. We investigate various compaction models and discuss their suitability to simulate the compaction of (highly) porous material. Then, we perform simple test cases where we compare results of the simulations to the theoretical solutions. We also present a Deep Impact-like simulation to show the effect of porosity on the outcome of an impact. Detailed validation tests will be presented in a next paper by comparison with high-velocity laboratory experiments on porous materials (Jutzi et al., in preparation). Once validated at small scales, our new impact code can then be used at larger scales to study impacts and collisions involving brittle solids including porosity, such as the parent bodies of C-type asteroid families or cometary materials, both in the strength- and in the gravity-dominated regime.\
\
**Key Words:\
** Asteroids, Composition - Collisional Physics - Impact Processes
Introduction
============
The collisional process plays a key role in all stages of a planetary system history, from the phase of planetary formation through collisional accretion to the late phases where the populations of small bodies evolve collisionally in a disruptive way. So far, these problems have been addressed using models appropriate to solid bodies represented as brittle rocky materials in which porosity was neglected or modeled at a macroscopic (i.e. resolved) level. Numerical codes, called [*hydrocodes*]{} (see, e.g., Benz and Asphaug, 1994) have been developed to compute the impact induced fragmentation of such solid bodies by solving the elastic-plastic conservation equations associated with a model of brittle failure to account for the fracture of the solid material. This already allowed to improve our understanding of the impact response of small bodies such as basalt-like material and asteroids belonging to the taxonomic class S, supposed to be composed of material with negligible porosity. However, several evidence point to the presence of a high degree of porosity in some small body classes. Asteroids belonging to the C taxonomic class are now believed to be highly porous, as indicated by the low bulk density ($\approx 1.3$ g$/$cm$^3$) estimated for some of them, such as the asteroid 253 Mathilde encountered by the NEAR Shoemaker spacecraft (Yeomans et al., 1997), and as inferred from meteorite analysis (Britt et al., 2006). Small body populations evolving at larger distances from the Sun, i.e. the Jupiter-Family Comets, the Kuiper Belt Objects, and the other classes of comets contain also a high level of porosity, as indicated by the estimated low bulk densities (below $1$ g$/$cm$^3$, e.g., Rickman, 1998) and the analysis of interplanetary dust particles collected on Earth.
In parallel, the dissipative properties of porous media are invoked more and more as playing an important role in the formation of early planetesimals (e.g., Wurm et al., 2005). Hence, porosity emerges slowly as playing a major role from the time of the formation of the planets to the collisional evolution of the present day Solar System.
Despite the growing focus on porosity, our ability to model its effect on the outcome of impacts and collisions remains limited. To address the question of modeling porosity in this context it is necessary to first define its scale in comparison with the other relevant dimensions involved in the problem such as the size of the projectile and/or the crater, etc. In particular we define microscopic porosity as a type of porosity characterized by pores sufficiently small that one can reasonably assume that they are distributed uniformly and isotropically over these relevant scales. Macroscopic porosity on the other hand would be characterized by pores with sizes such that the medium can no longer be assumed to have homogeneous and isotropic characteristics over the scales of interest. In this case, pores have to be modeled explicitly and the current hydrocodes developed for the modeling of non-porous brittle solids can still be used. The presence of these large macroscopic voids will only affect the transfer efficiency and the geometry of the shock wave resulting from the impact which can be computed using this existing software. On the other hand, a body containing microporosity (i.e. porosity on a scale much smaller than can be numerically resolved) can be crushable: cratering on a microporous asteroid might be an event involving compaction rather than ejection (Housen et al., 1999). In an impact in microporous material, a part of kinetic energy is dissipated by compaction which leads to less ejection and lower velocities of the ejected material. These effects cannot be reproduced using hydrocodes developed for the modeling of non-porous solids. Therefore, a model is needed which takes pore compaction into account.
Sirono (2004) proposed a model which can be used to study low velocity collisions of porous aggregates. Using this model, the author showed that the energy dissipation by compaction can lead to the sticking of dust aggregates. However, this model is not appropriate for high velocity impacts because of the simplicity of the equation of state (by construction, the pressure only depends on density).
Recently, Wuennemann et al. (2006) proposed a so-called $\epsilon-alpha$ model suitable to model porous material and for the use in hydrocodes. In this paper, we propose an alternative approach based on the so-called $P-alpha$ model (Herrmann, 1969) which we found to be more appropriate to use with our numerical method (see Sec. \[sec:altmod\]). Our 3D Smooth Particle Hydrodynamics (SPH) code, originally developed by Benz and Asphaug (1994), includes a model of brittle failure of non-porous material and successfully reproduced impact experiments on non-porous basalt targets. Moreover, associated to the N-body code [*pkdgrav*]{} to account for the effect of gravity (see e.g., Richardson et al., 2000), it reproduced successfully for the first time the formation of S-type asteroid families resulting from the catastrophic disruption of non-porous parent bodies (see, e.g., Michel et al., 2001; Michel et al. 2003). A first implementation of a porosity model has been made in this code, by Benz and Jutzi (2006) using a simplified version of the $P-alpha$ model (the density $\rho$ was used instead of the pressure $P$) but was found to be inappropriate for materials with a high degree of porosity (see Sec. \[sec:highpor\]). Here, we improve on this first work by implementing the full $P-alpha$ model which offers several advantages and has a more general application.
In the following, we begin by describing our porosity model and comparing it to alternative models. We discuss some (counter intuitive) effects which occur when dealing with highly porous material. In the case of a high porosity, the energy dissipation during compaction leads to a large thermal pressure which can even become large enough to cause a decrease of the matrix density (Sec. \[sec:highpor\]). We show that our model is suitable to model these effects. In Sec. \[sec:nummod\], we recall the equations used in our modeling of brittle solids and then we show how porosity is related to these equations (Sec. \[sec:distsolid\]). In Sec. \[sec:tests\], simple test cases are presented: first, a simulation of a 1D shock wave in porous media is compared to the analytical solution; in the second test, we simulate the compression of porous pumice and measure the so-called crush-curve. As a first application, we perform a simulation of a Deep Impact-like impact and compare simulations with and without porosity model. We also compare different compaction models using a porous basalt target. Conclusions are then exposed in Sec. \[sec:concl\]. A detailed comparison of numerical simulations with high-velocity impact experiments on porous material is presented in a forthcoming paper (Jutzi et al., in preparation).
Porosity model {#sec:porosity}
==============
While porosity at large scales can be modeled explicitly by introducing macroscopic voids, porosity on a scale much smaller than the numerical resolution has to be modeled through a different approach. Our porosity model is based on the $P-alpha$ model originally proposed by Herrmann (1969) and later modified by Carroll and Holt (1972). The model provides a description of microscopic porosity with pore-sizes beneath the spatial resolution of our numerical scheme (sub-resolution porosity) and which is homogeneous and isotropic.
$P-alpha$ model
---------------
The basic idea underlying the $P-alpha$ model consists in separating the volume change in a porous material into two parts: the pore collapse on one hand and the compression of the material composing the matrix on the other hand. This separation can be achieved by introducing the so-called distention parameter $\alpha$ defined as
$$\alpha\equiv\frac{\rho_s}{\rho}
\label{defalpha}$$
where $\rho$ is the bulk density of the porous material and $\rho_s$ is the density of the corresponding solid (matrix) material. Distention is related to porosity as $1-1/\alpha$. According to its definition, the distention varies in the range $\alpha_0>\alpha>1$, where $\alpha_0$ is the initial distention.
Using the distention parameter $\alpha$, the equation of state (EOS) can be written in the general form: $$P=P(\rho,E,\alpha)
\label{geos}$$ According to Caroll and Holt (1972), the EOS of a porous material can explicitly be written as: $$P=\frac{1}{\alpha} P_s(\rho_s,E_s) = \frac{1}{\alpha} P_s(\alpha
\rho, E)
\label{meos}$$ where $P_s(\rho_s,E_s)$ represents the EOS of the solid phase of the material (the matrix). A crucial assumption in this model is that the pressure depends on the density of the matrix material. The pore space is modeled as empty voids and the internal energy $E$ is assumed to be the same in the porous and the solid material ($E=E_s$), which implies that the surface energy of the pores is neglected. The factor $1/\alpha$ in Eq. (\[meos\]) was introduced by Carroll and Holt (1972) to take into account that the volume average of the stress in the matrix is given by $$P_s=\alpha P$$ where $P$ is the applied pressure.
In the $P-alpha$ model, the distention is solely a function of the pressure $P$: $$\alpha=\alpha(P)
\label{eq:alphap}$$ where $P = P(\rho,E,\alpha)$. The relation between distention and pressure is often divided in an elastic regime ($P<P_e$) and a plastic regime ($P>P_e$), where $P_e$ is the pressure at which the transition between the two regimes occurs. In the elastic regime, the change of $\alpha$ with pressure is reversible. According to Herrmann (1969), the relation between distention and pressure can be defined as $$\label{dadpe}
[\frac{d\alpha}{dP}]_{elastic} =
\frac{\alpha^2}{K_0} [1-(\frac{1}{h(\alpha)^2})]$$ where $K_0=c_0^2\rho_0$ and $$h(\alpha)=1+(\alpha-1) \frac{c_e-c_0}{c_0(\alpha_e-1)}$$ where $\alpha_e=\alpha(P=P_e)$. This equation follows from the assumption that the elastic wave velocity changes linearly from the initial value $c_e$ to the bulk sound speed $c_0$ in the solid state. Using Eq. (\[dadpe\]) leads to a small change of the distention (from $\alpha_0$ to $\alpha_e$) in the elastic regime and an elastic wave velocity which is smaller than in the case of a constant $\alpha$. Since the distention changes only very little, it is often assumed that $d\alpha/dP=0$ in the elastic regime and therefore $c_e=c_0$.
In the plastic regime, the following quadratic form is often used to define the function $\alpha=\alpha(P)$: $$\alpha = 1 + (\alpha_e-1)
\frac{(P_s-P)^2}{(P_s-P_e)^2}.
\label{alphaqudratic}$$ where $P_e$ and $P_s$ are constant.
This is obviously a very simple model, but it is appropriate enough for many applications. A more realistic relation can be obtained experimentally by means of a one dimensional static compression of a sample, during which the actual distention $\alpha_m$ is measured as a function of the applied pressure $P_m$. The resulting crush-curve $\alpha_m(P_m)$ then provides the required relation between distention and pressure for the material. In Sec. \[sec:tests\], we present test simulations where we use such a measured relation to define $\alpha(P)$.
Alternative models {#sec:altmod}
------------------
According to the $P-alpha$ model, distention is a function of pressure $\alpha=\alpha(P)$. However, distention can also be defined as a function of other state variables. Wuennemann et al. (2006) for example use the volumetric strain ($\epsilon-alpha$ model). Another possibility is to define distention directly as a function of density. We will refer to that approach as $\rho-alpha$ model. Note that in each model, the pressure is computed according to Eq. (\[meos\]).
### $\epsilon-alpha$ versus $\rho-alpha$ model
According to Wuennemann et al. (2006), the volumetric strain can be expressed as $$\epsilon_v=\int_{V_0}^V \frac{V'}{dV'} = ln(V/V_0)$$ where $V_0$ is the initial volume and $V$ the actual volume. Assuming that the volume of the matrix is kept constant ($V_s=V_{s0}$) volumetric strain and distention can be related as $\epsilon_v=ln(\alpha/\alpha_0$), which leads to the compaction function $$\alpha=\alpha_0 e^{\epsilon_v}
\label{eq:ae}$$ Another (very similar) way to define the evolution of the distention follows from its definition: $\alpha=\rho_s/\rho$. If we again assume a constant matrix volume and therefore a constant matrix density ($\rho_s=\rho_{s0}$), we get the compaction function $$\alpha=\frac{\rho_{s0}}{\rho}.
\label{eq:arho}$$ This relation also follows from Eqs. (\[eq:ae\]) by replacing $\epsilon_v$ by $$\epsilon_v=ln(\frac{V}{V_0})= ln(\frac{\rho_0}{\rho})
\label{eq:evrho}$$ where $\rho_0=\rho_{s0}/\alpha_0$ is the initial density.
Both equations, (\[eq:ae\]) and (\[eq:arho\]), describe the fasted possible way to decrease distention as a function of volume (density) change under the assumption that the matrix density is constant. However, as Wuennemann et al. (2006) point out, during the compression of a porous material, not only pore space is compacted, but also the matrix is slightly compressed leading to an increase of the pressure. Wuennemann et al. (2006) take this into account by introducing a parameter $\kappa$ to control the compaction rate: $$\alpha=\alpha_0 e^{\kappa (\epsilon_v-\epsilon_e)}
\label{eq:aemod}$$ where $\epsilon_e$ is the critical strain where the compaction starts. Using Eq. (\[eq:evrho\]) and defining $\epsilon_e \equiv ln(\rho_0/\rho_e)$, we can write this equation in terms of density $$\alpha=\alpha_0 (\frac{\rho_e}{\rho})^\kappa.
\label{eq:arhomod}$$ Wuennemann et al. (2006) also use a power-law compaction regime at a certain volumetric strain. In this way, they are able to reproduce experimental compaction data obtained by a uniaxial compression of a porous material. In a similar way, one could also modify Eq. (\[eq:arho\]) to obtain similar results.
In both models, the time evolution of the distention parameter has a very simple form. In the $\epsilon-alpha$ model, it is given by $$\dot\alpha=\frac{d\alpha}{d\epsilon}\dot\epsilon
\label{dadededt}$$ and for the $\rho-alpha$ model we get $$\dot\alpha=\frac{d\alpha}{d\rho}\dot\rho
\label{dadrdrdt}$$ The comparison above shows that the $\epsilon-alpha$ model and the $\rho-alpha$ model are very similar and the only difference is the parameter which is chosen to measure the actual volume or density, respectively.
### $\rho-alpha$ versus $P-alpha$ model {#sec:rapa}
In the $P-alpha$ model, the distention depends on the density via the pressure given by Eq. (\[meos\]) but contrary to the two models described above, it is also a function of the internal energy. However, for small initial porosities, the energy contribution to the pressure remains small as long as $\alpha>1$, and can even be neglected in most cases. Therefore, the pressure can be approximated by $P\simeq P(\rho,\alpha)$ and consequently, we can transform the function $\alpha(P[\rho,\alpha])$ in $\alpha=\alpha(\rho)$. In this way, we can define a function $\alpha=\alpha(\rho)$ which, instead of mimicking the behavior of the $\epsilon-alpha$ model as in Eq. (\[eq:arhomod\]), approximately corresponds to the relation $\alpha=\alpha(P)$, but neglects the thermal contribution to the pressure. Test simulations aimed at comparing $\alpha=\alpha(P)$ and $\alpha=\alpha(\rho)$ show that for low porosities this assumption is valid and the difference between the two models is rather small (see Sec. \[sec:tests\]). However, for high porosities this assumption is not valid and we observed some problems using this form of the $\rho-alpha$ model.
Problems with high porosities {#sec:highpor}
-----------------------------
As we described above, the energy contribution to the pressure is very small in the porous regime ($\alpha > 1$). However, this is only true for small initial porosities. In highly porous material, the thermal pressure can become large enough that the density in the compressed state ($\alpha=1$) is below the initial density of the matrix (Zel’dovich and Raizer, 1967), while in a less porous material the density at $\alpha=1$ would be at least as high as the initial density of the matrix. Then, when the compressed state is reached, the density does not increase further with increasing pressure but rather decreases, and the volume increases accordingly. This behavior can lead to an anomalous but nonetheless actual Hugoniot curve (see Fig. \[fig:hugprho\], top). It is important to stress that this anomalous behavior of highly porous material is not only a theoretical concept but can actually be observed in experiments.
In the fully compressed state, by definition the material is only composed of the matrix. Since in this state the density of highly porous material is smaller than the initial value of the matrix density, this implies that the matrix density in the fully compacted state has decreased from its initial value. Such a decrease must occur even before this fully compressed state has been reached (see Fig. \[fig:hugprho\], bottom). As an important consequence, in highly porous materials the distention can be decreased faster (as a function of volume change) than in the case of a constant matrix density (Eqs. (\[eq:ae\]) and (\[eq:arho\])). This behavior occurs independently of the actual functional form of the distention. It is actually caused by the huge difference between the density of the porous material and the matrix density.
It is important to note that even in highly porous material, the change of the matrix density during the compaction is very small compared to the change of the bulk density (see Fig. \[fig:hugprho\]). Nevertheless, it has a great effect since according to Eq. (\[meos\]), the matrix density is used to compute the pressure.
To correctly simulate the compaction of highly porous materials, therefore, the compaction functions must allow the density of the matrix material to decrease. For the $\rho - alpha$ or $\epsilon - alpha$ models this implies that, above some threshold pressure, the distension must decrease faster (as a function of volume change) than defined by Eq. (\[eq:ae\]) or (\[eq:arho\]), which follow from the assumption of a constant matrix density. This could be achieved by a modification of Eq. (\[eq:aemod\]) or (\[eq:arhomod\]). On the other hand, defining distension as a function of pressure offers a simple prescription for compaction that allows for matrix expansion. The $P - alpha$ model, without modification, can therefore simulate the compaction of highly porous materials given suitable model parameters.
For illustration purposes, we show the pressure-distention relation in a unidimensional shockwave under three different assumptions of compaction behavior, and for two initial porosities (Fig. \[fig:hugcomp\]). In the first case we use the $P-alpha$ model which assumes a quadratic function (Eq. \[alphaqudratic\]) for $\alpha=\alpha(P)$ with $P_e$=8$\times$10$^8$ dyn/cm$^2$ and $P_s$=7$\times$10$^9$ dyn$/$ cm$^2$, and that $\alpha_e = 0$ in the elastic regime. The values of $P_e$ and $P_s$ approximately correspond to the ones used by Herrmann (1969) to study a (low porosity) compaction wave in porous aluminium. For illustrative purposes, we assume that $P_e$ and $P_s$ do not depend on the initial porosity. In the second case we assume a constant matrix density during compaction by using Eq. (\[eq:ae\]), with $\kappa = 1$. The density $\rho_e$ is chosen so that the compaction starts at $P = P_e$. As a third case, we also show the results obtained using a function $\alpha=\alpha(\rho)$ which follows from the assumption that the thermal contribution to the pressure can be neglected and that the transformation of $P \simeq P(\rho,\alpha)$ holds, as described in the previous section.
Since we only want to study the compaction behavior which takes place at moderate pressures ($P \lesssim P_s$) we use a simplified version of the Tillotson equation (Tillotson 1962, Melosh 1989) for this illustration. This simplified EOS provides a reasonable approximation of the full Tillotson equation in the considered regime. Furthermore, it allows us to examine the sensitivity of the pressure-distention relations on the equation of state (EOS) parameters. The following equation is used: $$\label{eostpcmod}
P = c \rho E + A\mu$$ where $c=a+b$, $\mu=\eta-1$ and $\eta=\rho / \rho_0$ and $A$, $a$ and $b$ are the usual Tillotson parameters. The assumptions made to obtain this equation are discussed in the appendix.
The pressure-distention relation is finally computed using Eq. (\[meos\]), the simplified EOS (Eq. \[eostpcmod\]) and the Hugoniot equation $$\label{henergycons}
E-E_0=(P+P_0)(V_0-V)/2$$ where $V_0=1/\rho_0$ and $V=1/\rho$ and $P_0$ = 0 and $E_0$ = 0 are used as initial values. We further use $c$ = 2 and $A$ = 7.5$\times$10$^{11}$ dyn/cm$^2$ (these values approximately correspond to the usual aluminium parameters) for the illustration (Fig. \[fig:hugcomp\]).
We have to point out the the following considerations are only valid in a moderate pressure regime where the assumptions leading to the simplified EOS are reasonable.
We show two cases with a different initial distention, $\alpha_0$ = 1.275 and $\alpha_0$ = 3.0. In the low porosity case, full compaction is reached with all three models at similar pressures. Of course, the curve obtained by using the $P - alpha$ model corresponds to the quadratic relation used to define $\alpha(P)$. The curve denoted by $\rho - alpha$ shows the results obtained using the approximation $P(\rho,\alpha,E)\simeq P(\rho,\alpha)$ followed by the transformation $\alpha(P) = \alpha(P[\rho,\alpha]) \rightarrow \alpha = \alpha(\rho)$, which assumes that the thermal contribution to the pressure can be neglected. As expected for low porosities, it has a similar shape as the $P - alpha$ curve because in this case the thermal component of the pressure is small. The difference in pressure between the two curves is equivalent to the thermal contribution to the pressure at a given level of compaction. Note also that the $P - alpha$ curve is less steep than the constant matrix-density curve ($\epsilon - alpha$ model curve with $\kappa = 1$) because in this case the $P - alpha$ relationship implies some compression of the matrix material during compaction.
In the high porosity case, on the other hand, full compaction is achieved in the considered regime only with the $P - alpha$ model. The curves of the other two models do not reach $\alpha$ = 1 until a much higher pressure is reached. Even the assumption of constant matrix density does not lead to full compaction. As described above, the difficulty in reaching complete compaction is caused by the fact that the matrix density can actually decrease with increasing pressure (due to the large thermal component) for highly porous materials. In the case of the constant matrix density, it can be shown that the minimal value that the distention can reach is given by $$\label{eq:aminc}
\alpha_{min}= \lim \limits_{P \to \infty} \alpha(\epsilon[P]) = \alpha_0\frac{\rho_e}{\rho_0}\frac{c}{c+2} \simeq \alpha_0\frac{c}{c+2}$$ We have to point out that this result follows using the simplified EOS (Eq. \[eostpcmod\]) which is of course not valid for infinite pressures. Using the full Tillotson equation of state, the contribution of the term $\mu^2 B$ and the reduction of the effective value of $c$ at very high pressures (much higher than 10$^{10}$ dyn/cm$2$) can still lead to full compaction.
As Eq. (\[eq:aminc\]) shows, the parameter $c$ determines whether or not full compaction is possible at moderate pressures. For illustration, we once again show the curves which follow from the constant matrix-density assumption but now for different values of $c$ (again for two cases with $\alpha_0$ = 1.275 and $\alpha_0$ = 3.0). For small porosities, changing $c$ from 0.5 to 2 only slightly changes the slope of the curves (see Fig. \[fig:hugcompevc\], top). On the other hand, in the high porosity case, the parameter $c$ has a great influence on the crushing behavior (see Fig. \[fig:hugcompevc\], bottom). For $c$ = 0.5, full compaction is reached at $P\simeq$ 5$\times$10$^9$ dyn/cm$^2$. However, for $c$ = 1 and $c$ = 2 we get an $\alpha_{min}$ of 1.0 and 1.5, respectively. The corresponding curves asymptotically approach these values. Again, using the full Tillotson equation, the behavior would be different for very high pressures and full compaction would be achieved even for $c$ = 2. Nevertheless, we think that the estimation of $\alpha_{min}$ (Eq. \[eq:aminc\]) can be used at least as a first guess of value of a critical distention $$\label{eq:acrit}
\alpha_{0crit} \simeq \frac{c+2}{c}$$ which follows from $\alpha_{min} = 1$. For an initial distention higher than this value ($\alpha_0 > \alpha_{0crit}$), anomalous effects can occur and models which do not allow matrix expansion fail to reach full compaction at moderate pressures. Obviously, this procedure to determine $\alpha_{0crit}$ only works for EOS where a parameter $c$ can be identified. In any case, the results indicate that using models where the distention is decoupled from pressure ($\epsilon - alpha$ and $\rho - alpha$ model), the compaction behavior can strongly depend on the EOS parameters which relate energy and pressure. On the other hand, using a model where the distention is a direct function of the pressure ($P - alpha$ model), the compaction behavior is not sensitive to these parameters.
In Sec. \[sec:deepimp\], results of impact simulations using the $P - alpha$, $\rho - alpha$ and $\epsilon - alpha$ model are compared.
Our actual model
----------------
Although the use of the $\rho-alpha$ model (as an analogue to the $\epsilon-alpha$ model) has some advantages, especially the simple form of the time evolution $\dot\alpha=\frac{d\alpha}{d\rho}\dot\rho$, we found that for our numerical scheme, the most appropriate variable to define a functional form of the distention is pressure: $\alpha=\alpha(P)$. The following reasons support this choice:
- The $P-alpha$ model can be used *without modification* to simulate the compaction of highly porous material.
- The relation between distention and pressure, which is used as an input in our model, can directly be obtained from the experimental crush-curve for the material considered.
- We found that using our numerical scheme, no iteration is needed to implicitly solve for pressure and distention.
For the actual form of the relation between distention and pressure we either use a quadratic relation (\[alphaqudratic\]) or, if available, we directly use the experimentally measured crush-curve to define the function $\alpha(P)$. By definition, the pressure distention relation ($\alpha(P)$) we actually use in our code does not depend on the strain rate. Using a crush-curve which was measured under quasi static conditions therefore assumes that the same curve would be obtained at high strain rates. Experiments (Nakamura et al., in preparation) show that there actually is a (small) strain rate dependence of the crushcurve. However, we found (Jutzi et al., in preparation) that the results of impact simulations (i.e., the fragment mass distribution) do not strongly depend on the exact shape of the $P-alpha$ relation. Therefore, we think that it is not problematic to use a low strain rate crush-curve as input in our code to simulate high strain rate events.
The following function allows a good fit to a wide range of experimental crush-curves: $$\alpha(P) = \begin{cases}
(\alpha_e-\alpha_t) \frac{(P_t-P)^{n1}}{(P_t-P_e)^{n1}} + (\alpha_t-1) \frac{(P_s-P)^{n2}}{(P_s-P_e)^{n2}}
+ 1 & \text{if } P_e<P<P_t \\
(\alpha_t-1) \frac{(P_s-P)^{n2}}{(P_s-P_e)^{n2}} + 1 & \text{if } P_t<P<P_s \\
1 & \text{if } P_s < P \\
\end{cases}
\label{alpha2reg}$$ where $P_s,P_e$ and $\alpha_e$ have the same meaning as in Eq. (\[alphaqudratic\]), and $P_e<P_t<P_s$ and $1<\alpha_t<\alpha_0$ are parameters indicating a transition pressure and distention, respectively. The function (\[alpha2reg\]) and its first derivative are smooth by definition, which allows the existence of two regimes of $\alpha(P)$, each with a individual slope ($n_1$ and $n_2$).
In the elastic region ($P < P_e$) we either use Eq. (\[dadpe\]) to define $[d\alpha/dP]_{elastic}$ or, to simplify matters, we assume that the distention is constant, i.e. $\alpha_e=\alpha_0$. For all simulations presented in this paper (except the one in Sec. \[sec:1dwave\]), we use $[d\alpha/dP]_{elastic}=0$ in the elastic regime.
The derivative $d\alpha/dP$ which is used to compute the time evolution of the distention is now given by $$\label{dadpdef}
d\alpha/dP =
\begin{cases}
[d\alpha/dP]_{elastic} & \text{if } P<P_e\\
[d\alpha/dP]_{plastic}& \text{otherwise}
\end{cases}$$ where $[d\alpha/dP]_{plastic}$ follows from Eq. (\[alpha2reg\]). We further assume that unloading (from a partially compacted) state is elastic. Consequently, we define $$\label{qconstraint}
d\alpha/dP =
\begin{cases}
d\alpha/dP & \text{if } dP>0\\
[d\alpha/dP]_{elastic} & \text{otherwise.}
\end{cases}$$ As discussed above, $[d\alpha/dP]_{elastic}$ is either computed using Eq. (\[dadpe\]) or assumed to be zero.
The time evolution of the distention parameter can be written as $$\dot\alpha=\frac{d\alpha}{dP}\frac{dP}{dt}$$ Using Eq. (\[meos\]) we finally get $$\dot \alpha(t)= \frac{\dot E \left(\frac{\partial P_s}{\partial E_s}\right) + \alpha \dot \rho
\left(\frac{\partial P_s}{\partial \rho_s}\right)}{\alpha + \frac{d\alpha}{dP} \left[P - \rho
\left(\frac{\partial P_s}{\partial \rho_s}\right)\right]}\cdot \frac{d\alpha}{dP}
\label{dadt}$$
The equations (\[meos\]) and (\[alpha2reg\] - \[qconstraint\]) define the constitutive equation which describes the compaction behavior of a porous material. In the original work of Herrmann (1969) and Carroll and Holt (1972), the $P-alpha$ model was intended to be a first order theory in which shear strength effects are considered secondary and consequently, the stress tensor was assumed to be diagonal. In this work, we use the full stress tensor and therefore, we extend the original model with a relation between the distention and the deviatoric stress tensor (Sec. \[sec:diststrength\] and \[pord\]).
Model equations and implementation {#sec:nummod}
==================================
Our numerical technique is based on the Lagrangian Smooth Particle Hydrodynamic (SPH) method. Since the basic method has already been described in many papers (see for examples reviews by Benz, 1990; Monaghan, 1992) we refer the interested reader to these earlier papers.
The standard gas dynamics SPH approach was extended (see for example Libersky and Petschek, 1991) to include an elastic-perfectly plastic material description and a fracture model based on the one of Grady and Kipp (1980) in order to model the behavior of brittle solids (Benz and Asphaug, 1994). As our porosity model interfaces with this material description, we begin with a short review of this previous approach. Note that the following equations describe non-porous material. Porosity is introduced in Sec. \[sec:distsolid\].
Elastic perfectly plastic strength model
----------------------------------------
The equations to be solved are the well-known conservation equations of elasto-dynamics; they can be found in most standard textbooks. The mass conservation can be written as: $$\frac{d\rho^{\kappa}}{dt}+\rho\frac{\partial v^{\kappa\lambda}}{\partial
x^ {\lambda}}=0
\label{eq:massconv}$$ where $d/dt$ is the Lagrangian time derivative, $\rho$ the density, $v$ the velocity and $x$ the position. The conservation of momentum has the following form: $$\frac{dv^{\kappa}}{dt}=\frac{1}{\rho}\frac{\partial\sigma^{\kappa\lambda}}
{\partial x^{\lambda}}$$ where $\sigma^{\kappa\lambda}$ is the stress tensor given by $$\sigma^{\kappa\lambda}=S^{\kappa\lambda}-P\delta^{\kappa\lambda}$$ where $P$ is the hydrostatic pressure, $\delta^{\kappa\lambda}$ is the Kroneker symbol and $S^{\kappa\lambda}$ is the (traceless) deviatoric stress tensor. Finally, the conservation of energy is given by the equation $$\frac{dE}{dt}=-\frac{P}{\rho}\frac{\partial}{\partial x^{\kappa}}v^{\kappa}+\frac{1}{\rho} S^{\kappa\lambda}
\dot\epsilon^{\kappa\lambda}$$ where $\dot\epsilon$ is the strain rate tensor given by $$\dot\epsilon^{\kappa\lambda}=\frac{1}{2}\left(\frac{\partial v^{\kappa}}{\partial x^{\lambda}}+\frac{\partial
v^{\lambda}}{\partial x^{\kappa}}\right).$$ In order to specify the time evolution of the deviatoric stress tensor $S^{\kappa\lambda}$ we adopt Hooke’s law and define the time evolution of the deviatoric stress tensor as: $$\label{eq:ds}
\frac{dS^{\kappa\lambda}}{dt}=2\mu\left(\dot\epsilon^{\kappa\lambda}-\frac{1}{3}\delta^{\kappa\lambda}
\dot\epsilon^{\nu\nu}\right)+S^{\kappa\lambda}\Omega^{\lambda\nu}+S^{\lambda\nu}\Omega^{\kappa\nu}$$ where $\mu$ is the shear modulus, and $\Omega$ is the rotation rate tensor: $$\Omega^{\kappa\lambda}=\frac{1}{2}\left(\frac{\partial v^{\kappa}}{\partial x^{\lambda}}-\frac{\partial
v^{\lambda}}{\partial x^{\kappa}}\right).$$ Finally, plasticity is treated using the von Mises yielding criterion.
In order to solve this set of equations, an equation of state has to be specified which relates density, energy and pressure:
$$P=P(\rho,E)$$
For the simulations presented in this paper we use the so-called Tillotson equation of state (e.g., Tillotson 1962, Melosh 1989).
Fracture
--------
Brittle materials cannot be modeled using elasticity and plasticity alone because these materials fail under tension or shear stress. To take this behavior into account, we use a fracture model introduced by Grady and Kipp (1980) and based on the presence of incipient flaws in the material and on crack propagation under increasing strain. This model has been introduced using an explicit distribution of incipient flaws by Benz and Asphaug (1994, 1995) in their SPH code. In this model it is assumed that the number density of active flaws at strain $\epsilon$ is given by a Weibull distribution (Weibull, 1939) $$n(\epsilon)=k\epsilon^m
\label{wbd}$$ where $k$ and $m$ are the material dependent Weibull parameters. When the local tensile strain has reached the activation threshold of a flaw, a crack is allowed to grow at a constant velocity $c_g$ which is some fraction of the local sound speed. The half length of a growing crack is therefore $$a=c_g(t-t')
\label{crackgr}$$ where $t'$ is the crack activation time.
Crack growth leads to a release of local stresses. To model this behavior, we follow Benz and Asphaug (1994, 1995) and introduce a state variable $D$ (for damage) which expresses the reduction in strength under tensile loading: $$\label{eq:sd}
\sigma_D=\sigma(1-D)$$ where $\sigma$ is the elastic stress in the absence of damage and $\sigma_D$ is the damage-relieved stress. The state variable $D$ is defined locally as the fractional volume that is relieved of stress by local growing cracks $$D=\frac{\frac{4}{3}\pi a^3}{V}
\label{defD}$$ where $V=4/3\pi R^3_s$ is the volume in which a crack of half - length $R_s$ is growing. Using Eqs. (\[crackgr\]) and (\[defD\]) we get the following equation for the damage growth $$\label{Dgr}
\frac{dD^{1/3}}{dt}=\frac{c_g}{R_s}$$ Damage accumulates at a rate given by Eq. (\[Dgr\]) when the local tensile strain $\epsilon_i$ reached the activation threshold of a flaw. Note that $\epsilon_i$ is obtained from the maximum tensile stress $\sigma^t_i$ after a principal axis transformation $$\epsilon_i=\frac{\sigma^t_i}{(1-D_i)E}
\label{eq:epsi}$$ where $D_i$ is the local value of the damage and $E$ is the Young modulus.
Interfacing porosity with the solid model {#sec:distsolid}
=========================================
So far, we only described how porosity (i.e. the distention parameter $\alpha$) is used to modify the pressure: $$P(\rho,E) \rightarrow \frac{1}{\alpha} P_s(\rho \alpha,E).$$ In this section, we show how distention interfaces with the material model exposed in the previous section, which is used to describe the behavior of solids under strain increase.
Distention and strength {#sec:diststrength}
-----------------------
As we have discussed in Sec. \[sec:porosity\], the pressure $P$ is calculated using the matrix density $\rho_s$ instead of $\rho$. Consequently, the deviatoric stress tensor has to be modified as well. In order to compute the time evolution of $S^{\kappa\lambda}$ as a function of the matrix variables, we introduce the following factor: $$\label{eq:fdef}
f \equiv \frac{[\vec\nabla \vec v]_s}{ [\vec\nabla \vec v]}$$ This factor relates the velocity divergence of the matrix and of the porous material. Using the continuity equation for the matrix $$\dot\rho_s = -\rho_s [\vec \nabla \vec v]_s$$ and for the porous material $$\dot\rho = -\rho [\vec \nabla \vec v]$$ we can write the factor $f$ as $$f=\frac{\dot\rho_s}{\rho_s}\frac{\rho}{\dot\rho}=\frac{\dot\rho_s}{\alpha \dot\rho}
\label{f}$$ Using $\dot \rho_s=\alpha \dot \rho + \dot \alpha \rho$ we finally get $$f = 1 + \frac{\dot\alpha \rho}{\alpha\dot\rho}
\label{fc}$$ The factor $f$ is then used to compute the time evolution of $S^{\kappa\lambda}$ for the porous material: $$\frac{dS^{\kappa\lambda}}{dt} \rightarrow f\frac{dS^{\kappa\lambda}}{dt}
\label{fdS}$$ The multiplication by the factor $f$ is motivated by the fact that both, the velocity divergence $$\vec\nabla \vec v = \frac{\partial v_1}{\partial x_1} + \frac{\partial v_2}{\partial x_2} + \frac{\partial v_3}{\partial x_3}$$ and the time derivative of the deviatoric stress tensor (Eq. \[eq:ds\]) are linear combinations of the spatial derivative of the components of the velocity vector (the linearity of Eq. (\[eq:ds\]) follows from Hooke’s law). Since according to Eq. (\[eq:fdef\]), the velocity divergence of the matrix is given by $$[\vec\nabla \vec v]_s = f [\vec\nabla \vec v],$$ we obtain $$\left[\frac{dS^{\kappa\lambda}}{dt}\right]_s = f \left[\frac{dS^{\kappa\lambda}}{dt}\right].$$ In addition to the multiplication by $f$, the deviatoric stress tensor $S^{\kappa\lambda}$ is multiplied by $\alpha^{-1}$ as it is done with the hydrostatic pressure $P$. We finally write the time evolution of $S^{\kappa\lambda}$ in the following form: $$\label{fdSa}
\frac{d}{dt}\left[\frac{1}{\alpha}S^{\kappa\lambda}\right]=\frac{1}{\alpha}\frac{dS^{\kappa\lambda}}
{dt}-\frac{1}{\alpha^2}S^{\kappa\lambda}\frac{d\alpha}{dt}$$ where $dS^{\kappa\lambda}/dt$ is modified according to Eq. (\[fdS\]).
The computation of the factor $f$ can fail for small $\dot \rho$. This is the main reason why the $\rho-alpha$ model was used in our first implementation (Benz and Jutzi, 2006) as in this case, this factor can be computed using a simpler relation: $f=(\rho/\alpha)(d\alpha/d\rho)$. For several reasons (see Sec. 2.4), the $P-alpha$ model is actually more appropriate. Therefore, we have worked out a functional form for the factor $f$ that does not lead to difficulties for small $\dot\rho$. For this, we replace $\dot E$ in Eq. (\[dadt\]) by $\dot E = P / \rho^2 \cdot \dot\rho$ and we rewrite Eq. (\[dadt\]) as $$\dot\alpha= \frac{P/\rho^2 \left(\frac{\partial P_s}{\partial E_s}\right) + \alpha \left(\frac{\partial P_s}
{\partial \rho_s}\right)}{\alpha + \frac{d\alpha}{dP} \left[P - \rho \left(\frac{\partial P_s}{\partial
\rho_s}\right)\right]}\cdot \frac{d\alpha}{dP} \cdot \frac{d\rho}{dt}$$ Defining $$\frac{d \alpha}{d\rho} \equiv \frac{P/\rho^2 \left(\frac{\partial P_s}{\partial E_s}\right) + \alpha
\left(\frac{\partial P_s}{\partial \rho_s}\right)}{\alpha + \frac{d\alpha}{dP} \left[P - \rho
\left(\frac{\partial P_s}{\partial \rho_s}\right)\right]}\cdot \frac{d\alpha}{dP}
\label{dadrhom}$$ the derivative of $\alpha$ can be written as $$\dot\alpha = \frac{d\alpha}{d \rho}\dot\rho$$ and we finally compute the correction factor in the following form: $$f = 1 + \frac{\dot\alpha \rho}{\alpha\dot\rho} = 1 + \frac{d\alpha}{d\rho}\frac{\rho}{\alpha}
\label{fcm}$$ The time evolution of the deviatoric stress tensor is then computed using Eq. (\[fdSa\]) and (\[fcm\]). In this way, the deviatoric stress is a function of the distention and and also of the used $P-alpha$ relation. On the other hand, we do not explicitly relate the yield strength $Y$ (used for the von Mises yielding criterion) and distention.
Distention and damage {#pord}
---------------------
Porosity does not only affect the stress behavior. It also has to be taken into account to compute the state variable damage.
Compression of a porous material is accompanied, if significant enough, by the breaking of cell walls. Our model takes into account this crushing behavior relating distention with the state variable damage. Since both damage $D$ and distention $\alpha$ are defined as volume ratios (Eqs. (\[defD\]) and (\[defalpha\]), respectively), we assume for simplicity a linear relation between $D$ and $\alpha$ (other forms will be investigated in the future). The conditions $D=0$ at $\alpha=\alpha_0$, and $D=1$ when all pores have been crushed ($\alpha=1$), lead to the following expression: $$D = 1 - \frac{(\alpha-1)}{(\alpha_0-1)}.
\label{defad1}$$ The time evolution of $D^{1/3}(\alpha)$ is then given by $$\frac{dD^{1/3}}{dt}=\frac{dD^{1/3}}{d\alpha}\frac{d\alpha}{dt}
\label{tevD1}$$ and using Eq. (\[defad1\]) we obtain $$\frac{dD^{1/3}}{dt}=-\frac{1}{3}\left[1-\frac{\alpha-1}{\alpha_0-1}\right]^{-\frac{2}{3}} \frac{1}{\alpha_0-1}\frac{d\alpha}{dt}.
\label{tevDtmp}$$ A close examination of Eq. (\[tevDtmp\]) reveals a problem since for $\alpha=\alpha_0$, the derivative $D^{1/3}/dt$ becomes infinite. In order to avoid this problem we add the small quantity $\delta D$ to the linear relation (\[defad1\]) and normalize it so that the conditions $D(\alpha_0)=0$ and $D(1)=1$ are still satisfied: $$D^{1/3} \rightarrow \frac{\left(D + \delta D\right)^{1/3} -(\delta D)^{1/3}} {\left(1+\delta D\right)^{1/3}-
(\delta D)^{1/3}}$$ Using Eqs. (\[tevD1\]) and (\[defad1\]) we finally get $$\label{tevD2}
\frac{dD^{1/3}}{dt}=-\frac{1}{3}\frac{\left[1-\frac{(\alpha-1)}{(\alpha_0-1)} + \delta D
\right]^{-\frac{2}{3}}}{\left(1+\delta D\right)^{1/3}-(\delta D)^{1/3}}\frac{1}{(\alpha_0-1)}
\frac{d\alpha}{dt}$$ where we set $\delta D=0.01$. The actual relation between distention and damage is shown in Figure \[fig:distdam\].
We now have two equations describing damage growth: the first treats damage under tension (\[Dgr\]) while the second (\[tevDtmp\]) is related to the compression of the (porous) material. Note that we do not *explicitly* model damage increase due to shear deformation. However, since the local scalar strain is computed from the maximum negative stress after principal axis transformation (Eq. \[eq:epsi\]), shear fracturing is implicitly accounted for in our model.
In order to get the total growth of damage, we build the sum of the two differential equations: $$\left[\frac{dD^{1/3}}{dt}\right]_{total}=\left[\frac{dD^{1/3}}{dt}\right]_{tension}+\left[\frac{dD^{1/3}}{dt}
\right]_{compression}$$ We now use this equation instead of Eq. (\[Dgr\]) to compute damage $D$ which varies between 0 and 1. According to Eq. (\[eq:sd\]), damage leads to a reduction of both tensile and shear strength by a factor of $(1-D)$.
According to our model, damage can grow only. We do not include any restoration of damage as Sirono (2004) proposed for low density grain aggregates.
Material parameters
-------------------
All parameters used by our porosity model are material parameters which can in principle be measured experimentally. Some of these parameters, such as the crush-curve, can be measured more easily than others (such as Weibull parameters, shear and tensile strengths). Unfortunately, such measurements are rarely done in practice and we plan to carry out some of them for several porous materials in future works.
The lack of an experimentally determined reliable database of relevant material parameters is actually one of the most limiting factor in our model. In particular, the thorough testing of the model by comparison with experiments is rendered particularly difficult if all material properties have not been measured properly. Freely choosing the missing values so as to match an experiment is not a satisfactory approach for an ab initio method such as ours. Unfortunately, this is often the only alternative we have. However, recent experiments have been performed on some porous materials, and material properties have been measured. A detailed comparison between results of simulations using our model with these experiments will be the subject of a next publication.
Tests {#sec:tests}
=====
In this section we present some simple test cases where we compare our numerical model to expected theoretical solutions. We also present a Deep Impact-like simulation to show the effect of porosity on the outcome of an impact. Further, different compaction models are compared by performing an impact in (highly) porous basalt.
The present tests aim at a first actual verification of the model by showing that it is consistent and correctly implemented in our code. A more detailed validation by comparison with actual impact experiments on porous material will be presented in a forthcoming paper.
1D compaction wave {#sec:1dwave}
------------------
As a first numerical test we consider a (plane) shock wave travelling in only one spatial dimension. We compare the simulation with the analytical results obtained by solving the corresponding Hugoniot equations.
In this simulation, we use porous aluminium with an initial distention of $\alpha_0=1.275$. We use a $P-alpha$ relation with both an elastic and a plastic regime. In the elastic regime $P<P_e$, the distention is computed using Eq. (\[dadpe\]). Using this equation instead of $d\alpha/dP=0$ leads to a smaller velocity of the elastic wave. We actually use Eq. (\[dadpe\]) in this case because it provides an additional test (properties of the elastic wave). However, for all other simulations presented in this paper we assume that $\alpha_e=\alpha_0$ and therefore $c_e=c_0$.
For the plastic regime we use the quadratic relation (\[alphaqudratic\]). The porosity parameters used in this simulation are given in Table \[porosityalu\].
As for the equation of state, we use the (full) Tillotson equation with aluminum parameters (Melosh, 1989). Since we are dealing with a purely one dimensional problem, strength is not included in this simulation and only the diagonal part of the stress tensor (pressure) is considered.
The analytical solution of this problem can be obtained by solving the Hugoniot equations together with Eq. (\[meos\]) and the $P-alpha$ relation defined above.
To carry out the simulation we use a cylinder aligned along the z-axis with a radius $r=0.2$ cm and a height $h=2$ cm. Since we want to study a one dimensional case, the forces acting on the particles in the x-y plane are set to zero: $$\left(\frac{dv_x}{dt}\right)_i=\left(\frac{dv_y}{dt}\right)_i = 0$$ The shock wave is then produced by moving the particles of the first layer (at the top of the cylinder) with a constant velocity $v_z$ = -45.8 $\times$ 10$^{3}$ cm/s in the z-direction. The last layer (at the bottom of the cylinder) is fixed. We use $5.6 \times 10^5$ particles in this simulation. However, in order to reduce boundary effects, only particles in a cylinder of $0.05$ cm radius are used for the comparison with the theoretical solution.
Figure \[fig:zpza\] (top) shows the compaction wave travelling in the z-direction at time $t$ = 3.5 $\mu$s. As expected, there are two waves: first, there is a so-called elastic precursor with an amplitude equal to the elastic pressure $P_e$. The elastic precursor is followed by a plastic compaction (shock) wave. The distention is only slightly changed due to the elastic wave and is decreased to $\alpha=1$ by the compaction wave (see Fig. \[fig:zpza\], bottom).
There is a good agreement between the theoretical solution (solid line) and the velocities of the two waves (the elastic precursor and the shock wave) obtained by the simulation. Moreover the corresponding pressure amplitudes have the expected value. The oscillations behind the compaction wave are due to the moving layer of particles (wall). The smoothed shape of the wave (especially of the elastic precursor) is caused by the artificial viscosity and the smoothing due to the SPH interpolation. No effort was made to fine tune the artificial viscosity to achieve a less diffusive solution.
Crush-curve simulation
----------------------
As we have discussed in Sec. \[sec:porosity\], an advantage of defining the distention as a function of pressure is that this relation can be directly determined experimentally by compressing a porous sample. Such an experiment (Nakamura et al., in preparation) was performed by A.M. Nakamura and K. Hiraoka at Kobe University (Japan). In this experiment, a cylindrical sample of porous pumice, confined within a steel cylinder, is compressed in one dimension. The applied force $F$ and the corresponding displacement $e$ of the penetrating piston (which defines the actual length $l$ of the sample) are measured. From these quantities, the applied pressure $P_m$ is given by the force $F$ per unit area and the actual distention can be determined in the following way $$\label{eq:alpham}
\alpha_m = \alpha_0 \frac{l}{l_0}$$ were $l_0$ is the initial length of the sample and $l=l_0-e$ is the actual length. In this way, we can obtain the relation $\alpha_m(P_m)$. Using the function (\[alpha2reg\]), we fit the curve $\alpha_m(P_m)$ and get the required analytical relation $\alpha=\alpha(P)$. The fitting parameters are given in Table \[porositypumice\].
As a further test, we use this relation in our porosity model and we simulate the compaction experiment. The simulation is performed moving the first layer of particles of a cylinder $A$ aligned along the $z$-axis with the velocity $v_z$. As in the experiment, the walls and the bottom of the cylinder are fixed. In order to determine the actual (macroscopic) distention and the applied pressure, we define a test cylinder $B$ within $A$ with a radius $r$ and an initial length of $l_0$ corresponding to the sample in the experiment. Since it is a quasi–static compression, we assume that the applied pressure corresponds to the pressure in the sample and we compute $P$ as the average pressure in the cylinder $B$. As in the experiment, we define the actual distention using equation Eq. (\[eq:alpham\]).
Figure \[fig:ccsim\] illustrates the setup of the simulation which has been performed in 3D. On this 2D slice, the dark particles correspond to the cylinder $B$ which represents the sample. The simulation is shown at the initial state (top) and during the compression (bottom). In order to control the uniformity of the compaction, we compute the standard deviation of the distention in the sample (particles within $B$) at certain timesteps. During the whole simulation, the standard deviation is of the order of 1% which indicates that the compaction proceeds uniformly.
In Figure \[fig:cccomp\], the crush-curve obtained by the simulation is compared to the experimental crush-curve. There is a very good agreement between the two curves. Of course, using the measured crush-curve as an input, one expects to obtain the same curve by the simulation. However, we have to point out that the pressure - distention relation used as input in the code is not necessarily the same as the pressure - distention relation measured in the simulation. In the $\alpha(P)$ relation used in the code, distention is defined as $\alpha=\rho_s/\rho$. In the relation obtained by the simulation, distention is defined according to Eq. (\[eq:alpham\]), which can be written as: $$\alpha_m = \alpha_0 \frac{l}{l_0} = \frac{\rho}{\rho_{s0}}$$ This definition therefore assumes a constant matrix density ($\rho_s=\rho_{s0}$). As we discussed in Sec. \[sec:highpor\], the matrix density changes during the compaction. However, the change of $\rho_s$ is only small (typically 1%) leading to an error of $\alpha_m$ which is in the same order. The fact that the two curves (simulated crush-curve and the crush-curve used as input) are in a very good agreement indicates that the change of the matrix denstiy is indeed very small. This result therefore suggests that the distention defined by $\alpha_m=\rho_{s0}/\rho$ is a reasonable approximation of the true distention $\alpha=\rho_s/\rho$ (even for highly porous material). However, it is important to note that even though the matrix density changes only very little, it can still have a great effect (see Sec. \[sec:highpor\]).
In a quasi static compression, the energy increase ($P dV$ work) is generally lower than in the case of shock compression, leading to a smaller thermal pressure. In the simulation above we assumed that the system is thermally insulated (i.e., the energy radiation is not taken into account). Therefore, the thermal pressure can still not be ignored. In Fig. \[fig:cccomp\], the crush-curve obtained by a simulation using a relation $\alpha=\alpha(\rho)$ which follows from the $\alpha = \alpha(P)$ curve neglecting the thermal pressure ($\rho - alpha$ model) is shown. To reach a given distention, a higher pressure is needed using the $\rho - alpha$ model instead of the $P - alpha$ one. This difference corresponds to the thermal pressure.
Impact simulations
------------------
In this section, we show several impact simulations using different material types (ice and basalt) and compaction models (no porosity model; $P - alpha$, $\rho - alpha$ and $\epsilon - alpha$ model). Note that the following results were obtained using a pre-damaged (i.e. strengthless) material. Therefore, we are in the gravity regime where the final crater size and the total amount of ejected material depend on the gravity acceleration. Due to the high resolution and the resulting small timestep (which is much smaller than the crater formation time), we only simulate the first phase of the crater formation.
### Deep-impact like impact {#sec:deepimp}
As a first application of our model, we show the effect of porosity on the outcome of an impact event by comparing simulations using porous and non-porous targets. Since we want to model a realistic case, we simulate a Deep-Impact-like impact, inspired from the Deep Impact Space mission (A’Hearn and Combi, 2007).
We model the target in two different ways. In the first case, we use an initial distention of $\alpha_0=3.0$ and we explicitly model porosity. To simplify matters we use a quadratic relation (Eq. \[alphaqudratic\]) for $\alpha=\alpha(P)$ with $P_e =$ 1$\times$10$^7$ dyn/cm$^2$ and $P_s =$ 2$\times$10$^9$ dyn/cm$^2$. These values are chosen rather arbitrarily, however, the resulting $\alpha(P)$ relation looks reasonable compared to measured crush-curves of other materials (see for expample Fig. \[fig:cccomp\]). In the second case, we use the same initial density ($\rho_0=\rho_{s0}/\alpha_0$), but the target is modeled as a solid (without porosity model). As target material we use pre-damaged (strengthless) ice. Only a small part of the target (comet Tempel-1) is modeled (half sphere with a radius of 28 m). In order to have a reasonable resolution in the region of interest (crater), we use a rather high number of particles for the target ($N_t$ = 5.2$\times$10$^6$). The resulting mass per SPH particle is $m_p = 2.6$ kg and the smoothing length is $h = 23$ cm. The impactor is modeled as a 370 kg aluminium sphere impacting at an angle of 30 degrees (from horizontal) with a velocity of 10 km/s. We use 140 SPH particles for the impactor in order to have the same mass per particle as in the target. The Tillotson equation of state (without simplifications) is used for these simulations.
Figure \[fig:diplot\] shows the outcome of the simulation after 50 ms in two dimensional slices of the three dimensional target. The dark particles in these plots have vertical velocities greater than 2 m/s (which is about the escape velocity of Tempel-1). Obviously, there is much less material ejected in the case where we explicitly model porosity (top) than in the solid target simulation (bottom). In fact, the difference of ejected mass is about a factor ten. Figure \[fig:divel\] shows the amount of mass ejected with a velocity higher than a certain velocity in both cases.
For comparison, the results obtained by using the $\rho-alpha$ model (instead of $P-alpha$) are also plotted in Fig. \[fig:divel\]. For this, we use a relation $\alpha=\alpha(\rho)$ which follows from $\alpha=\alpha(P)$ by neglecting the thermal component of the pressure (see Sec. \[sec:rapa\]). There is only a very small difference between the results of the two models. This is consistent with the estimation (Eq. \[eq:acrit\]) of $\alpha_{0crit} \simeq 6$ for $c$ = 0.4 (ice). Since $\alpha_0 = 3 < \alpha_{0crit}$, full compaction is possible and since $c$ is very small, we do not expect big differences. However, the situation looks different if we use basalt instead of ice as target material.
### Comparison between different compaction models
For the typically used basalt parameters we get $c \simeq 2$ and therefore $\alpha_{0crit} \simeq 2$. Consequentely, full compaction (at moderate pressures) is not possible for $\alpha_0 = 3 > \alpha_{0crit}$ using a compaction model which does not allow an expansion of the matrix.
To illustrate the different compaction behaviors using different compaction models, we simulate an impact in a basalt target with $\alpha_0=3.0$. For this simulation, we use the same initial conditions as above but we use a smaller target and the impact is now head on. We investigate three cases using the follwing compaction models:
1. $P - alpha$ with the same parameters as above.
2. $\rho - alpha$, using the relation $\alpha=\alpha(\rho)$ which follows from $\alpha=\alpha(P)$ neglecting the thermal pressure
3. $\epsilon - alpha$, $\kappa = 1$ (constant matrix density assumption)
In Fig. \[fig:dibasalt\], the degree of compaction (i.e., the distention $\alpha$) of the target material due to the impact is shown for simulations using the model 1 (top), model 2 (middle) and model 3 (bottom). On these plots, the black color corresponds to a distention of $\alpha=\alpha_0$ and the white color to full compaction $\alpha=1$. The results are shown at a time (20 ms) where compaction is finished and the distention $\alpha$ does not change anymore.
As it can be seen, only in the simulation using model 1 there is a small fully compacted zone below the crater. Fig. \[fig:diprofc\] shows the corresponding distention profiles. The minimal distention reached by model 2 and 3 is in both cases about $\alpha$ = 1.8. This indicates that also the maximal density is about the same in both cases, because $\alpha = \rho_s/\rho$ and $\rho_s$ changes only very little (model 2) or is even constant (model 3). The total volume which is compacted (derived from the actual distention) is 62.3/61.9/82.9 m$^3$ for the models 1/2/3. This means that even though model 3 does not lead to full compaction, the total volume of compacted material is higher than using model 1. The reason is that the distention profile resulting from model 3 is much less steep than using model 1 (Fig. \[fig:diprofc\]).
In Fig. \[fig:divvzc\] the cumulative volume of ejected material as a function of the vertical velocity is shown. The horizontal lines in the plot represent the total compacted volume. Interestingly, using model 1, there is less ejection than using model 3 (even though model 3 leads to more compaction). This can be explained by the fact that a given pressure amplitude leads to a higher degree of compaction using model 1 than using model 3, and therefore, more energy is dissipated. Using model 2, we get the same amount of ejected volume at high velocities as with model 3, but there is more material ejected at low velocities.
In order to study the dependence of the compacted and ejected volume on the compaction parameters, we perform a simulation using model 1 with $P_s =$ 1$\times$10$^8$ dyn/cm$^2$ instead of $P_s =$ 2$\times$10$^9$ dyn/cm$^2$. The other parameters are the same as above. Fig. \[fig:divvzcps\] shows the compacted volume and the cumulative volume of ejected material for the two simulations using model 1 (each with a different $P_s$). Clearly, using a very low value for $P_s$ (1$\times$10$^8$ dyn/cm$^2$) leads to much more compaction (95.2 m$^3$) and less ejection than using $P_s =$ 2$\times$10$^9$ dyn/cm$^2$. There is even more compaction than from model 3 which assumes a constant matrix density.
Conclusions {#sec:concl}
===========
In this paper, we have presented a new approach to model sub-resolution porosity in brittle solids that can be coupled to a 3D SPH hydrocode in order to simulate impacts and collisions involving porous bodies. Such bodies are believed to be present in all the populations of small bodies in our Solar System. Therefore, understanding their impact response is crucial to determine the collisional evolution of those populations during all stages, and to assess the accretion efficiency of small bodies during planetary formation. Moreover, this can help defining efficient mitigation strategies against the impact of a porous body on Earth.
In practice, the implementation of our model does not consume excessive CPU time and is easily implemented in a parallel code (porosity is a local property) so that simulations involving multi-million particles can readily be performed. In fact, extensive testing has shown that high resolution is really needed to obtain converged solutions in the case of simulations involving fracturing and/or porosity.
We presented two test cases. In the first test, we compared the simulation of a 1D compaction wave in porous material with the theoretical solution given by the Hugoniot equations. The amplitude and velocity of the resulting waves (elastic precursor and shock) in the simulation agree with the theoretical solution. The second test is a simulation of an experiment consisting in compressing a porous sample. We show that an advantage of the $P-alpha$ model is that an experimentally measured crush-curve can be directly used to define the relation $\alpha(P)$. Therefore, if a crush-curve has been measured for a given material, we can then simulate an impact on this material using its very crush-curve and not an arbitrary one which may lead to different outcomes.
As an application of our model, we presented the simulation of a Deep Impact-like impact. The main conclusion of this first application is that to model the behavior of a porous material during an impact, it is not enough to use the small bulk density expected for a porous material and compute its response from an impact by using a classical model of brittle failure of non-porous rock material, as one might be tempted to do. Actually, the cratering or disruption of porous material involves different processes than the ones involved in a non-porous brittle material (e.g. the crushing of pores), and this makes a huge difference in the outcome.
We also investigated the compaction behavior in an impact in basalt using different compaction models. We showed that using the $\rho-alpha$ or $\epsilon-alpha$ models (without modifications) can lead to difficulties to reach full compaction ($\alpha=1$) in the case of highly porous material. Using these models, the compaction behavior depends much on the EOS parameters which relate energy and pressure. On the other hand, the $P-alpha$ model can be used without modifications to simulate the compaction of highly porous material and the compaction behavior is not sensitive to these EOS parameters.
Our next step will be to validate our code in more details and in a context adapted to its future applications by confronting it with real high-velocity impact experiments on porous targets. Such a validation will mainly consist of reproducing the size distribution of the fragments (and their velocities when they are measured), using as inputs all measured material parameters. Indeed, it is important that matching the results does not rely on a fine tuning of all parameters but rather on the validity of the physical model. This will be the subject of a next paper. Comparisons with impact experiments in laboratory will have to be performed in the long run for a wide variety of materials (porous and non porous) to improve our understanding of the impact process as a function of material properties and impact conditions. The formation and evolution of planetary systems involves bodies with a wide range of properties and colliding with each other in different regimes of impact energies, leading either to their accretion and the formation of planets, or to their disruption as in the current stage of our Solar System. It is then important to have the possibility to simulate the collisional process in these different regimes and between bodies with different degrees of porosity. Collisional evolution models, which need constraints to characterize the outcomes of collisions, will certainly benefit from these investigations.
Appendix {#appendix .unnumbered}
========
Simplified Tillotson EOS {#simplified-tillotson-eos .unnumbered}
------------------------
First, we assume that the energy involved does not exceed the energy of incipient vaporization ($E<E_{iv}$). In this state, the EOS has the following form: $$\label{eostpc}
P = \left(a+\frac{b}{E/\left[E_0\eta^2\right]+1}\right)\rho E + A\mu +B\mu^2$$ where $\mu=\eta-1$ and $\eta=\rho / \rho_0$ and $a$, $b$, $A$ and $B$ are Tillotson parameters. We further assume that $\mu$ remains small and therefore $\mu^2<<\mu$. This assumption is motivated by the fact that according to Eq. (\[meos\]) $\mu$ is given by $\mu=1-\rho_s/\rho_{s0}$ and even though the matrix density is not constant (as discussed above), the variation remains small as long $P<P_s$ (about 1% in Fig. \[fig:hugprho\]). In fact, it can be shown that if $P_s << A$ then $(\rho_s - \rho_{s0}) / \rho_{s0} << 1$. According to Eq. (\[meos\]), the pressure is computed using the matrix density ($\mu=1-\rho_s/\rho_{s0}$), and therefore $\mu^2\simeq 0$ is a reasonable assumption.
Finally, we only want to consider cases where the energy $E$ remains small compared to the parameter $E_0$. The energy at $P=P_s$ can be estimated by: $$E = \frac{1}{2} P_s (\frac{1}{\rho_0}-\frac{1}{\rho}) \simeq \frac{1}{2} P_s (\frac{\alpha_0}{\rho_{s0}}-\frac{1}{\rho_{s0}}) = \frac{1}{2} P_s \frac{\alpha_0-1}{\rho_{s0}}$$ and therefore it is required that $$E \simeq \frac{1}{2} P_s \frac{\alpha_0-1}{\rho_s0} << E_0.$$ For $P_s$=7$\times$10$^9$ dyn/cm$^2$, $\rho_{s0}$=2.7 g/cm$^3$ and $\alpha_0$=3.0 we get $E\simeq$ 2.6$\times$10$^9$ erg/g which is indeed small compared to a typical value of $E_0$ (e.g. 5$\times$10$^{10}$ erg/g for aluminium) and so this condition is fullfilled.
Using the above assumptions, we can rewrite Eq. (\[eostpc\]) in the very simple form $$P = c \rho E + A\mu$$ where $c=a+b \simeq constant$.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are grateful to A.M. Nakamura and K. Hiraoka for providing us the crush-curve of pumice material that they measured in Kobe University. We also thank G. Collins and S. Sirono for their constructive reviews. M.J. and W.B. gratefully acknowledge partial support from the Swiss National Science Foundation and from the Rectors’ Conference of the Swiss Universities. M.J. acknowledges support from Kobe University (Japan) through the 21st Century COE (Center of Excellence) Program “Origin and Evolution of Planetary Systems”. P.M. acknowledges support from the french Programme National de Planétologie, from the Japanese Society for the Promotion of Science (JSPS) Invitation Fellowship for Research in Japan 2007, and with M.J. from the CNRS-JSPS cooperation program 2008-2009.
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==========
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------- -- ----------------
$P_e$ 8e8 dyn/cm$^2$
$P_s$ 7e9 dyn/cm$^2$
$c_0$ 5.35e5 cm/s
$c_e$ 4.11e5 cm/s
------- -- ----------------
: Parameters used in our porosity model for porous aluminium with an initial distention $\alpha_0=1.275$. Definitions of parameters are given in the text.[]{data-label="porosityalu"}
------------ -- -------------------
$\alpha_0$ 4.64
$\alpha_t$ 1.90
$P_e$ 1.00e7 dyn/cm$^2$
$P_t$ 6.80e9 dyn/cm$^2$
$P_s$ 2.13e9 dyn/cm$^2$
------------ -- -------------------
: Parameters used to fit the crush-curve of pumice.[]{data-label="porositypumice"}
[**Figure 1:**]{} Pressure versus density (top) and pressure versus matrix-density (bottom) for a material with low ( $\alpha_0=1.275$) and high ($\alpha_0=3.0$) porosity. In the case of a high porosity, the density does never reach the initial matrix density (dashed horizontal line) and the matrix density itself decreases at a certain point, before $\alpha=1$ (dashed vertical line).
[**Figure 2:**]{} Comparison of the pressure-distention relationship for different compaction models. The $P-alpha$ model assumes a quadratic function for $\alpha=\alpha(P)$. For the $\rho-alpha$ model we use a function $\alpha=\alpha(\rho)$ which follows from $\alpha=\alpha(P)$ by neglecting the thermal pressure. In the $\epsilon-alpha$ model we assume a constant matrix density ($\kappa=1$). In the low porosity case (top), full compaction is reached with all three models. In the high porosity case (bottom), only the $P-alpha$ model leads to full compaction. The minimal distention obtained using the constant matrix density assumption is about $\alpha$=1.5.
[**Figure 3:**]{} Pressure-distention relationship for different values of $c\simeq a + b$ (Tillotson parameters) using the constant matrix density assumption. For low porosities, the influence of $c$ is rather small. In the high velocity case, the value of $c$ determines whether or not full compaction is possible at moderate pressures.
[**Figure 4:**]{} Relation between distention and damage.
[**Figure 5:**]{} SPH Simulation of a 1D compaction wave in a 3D cylinder composed of porous aluminium. There are two waves (top): an elastic precursor followed by the compaction (shock) wave. Bottom: the main decrease of the distention is due to the compaction wave.
[**Figure 6:**]{} Simulation of compression of a porous sample shown when $\alpha=\alpha_0=4.64$ (top) and during the compression (bottom) when $\alpha=2.2$. The sample is represented by the dark particles.
[**Figure 7:**]{} Crush-curve of porous pumice measured by A.M. Nakamura and K. Hiraoka at Kobe University and obtained by a simulation using the $P-alpha$ and $\rho-alpha$ model.
[**Figure 8:**]{} Simulation of a Deep Impact-like impact in porous ice with $\alpha_0=3.0$ (top) and non porous ice with the same initial density (bottom). Dark particles have a vertical velocity greater than 2 m/s.
[**Figure 9:**]{} Cumulated mass of ejecta as a function of the ejection velocity as a result of a Deep Impact-like impact. The same initial density (i.e., the same mass) was used for the porous and non porous cases.
[**Figure 10:**]{} Impact in a basalt target with an initial distention of $\alpha_0=3.0$ (black). Full compaction ($\alpha=1$, white) is only reached using 1 model (top). Using model 2 (middle) or 3 (bottom), there are no fully compacted particles. For the definition of the models, see text.
[**Figure 11:**]{} Distention as a function of distance (negative z-direction) obtained by model 1/2/3.
[**Figure 12:**]{} Cumulative volume of ejected material for the models 1/2/3. For comparison, the total volume which was compacted is also shown for the three cases (horizontal lines).
[**Figure 13:**]{} Cumulative volume of ejected material for model 1 using $P_s$ = 2$\times$10$^9$ dyn/cm$^2$ and $P_s$ = 1$\times$10$^8$ dyn/cm$^2$. The horizontal lines indicate the total volume which was compacted in each case.
|
---
abstract: 'Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y''=A\,Y$ be a linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where $\mathfrak{g}$ is the Lie algebra of $[A]$ and $\bar{\mathbf{k}}$ denotes the algebraic closure of $\mathbf{k}$. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic [@Ko71a]. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system $X$. Using a previous result [@ApWea], we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order $m+1$ if the variational equations at order $m$ are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of $X$. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in [@WeNo63a]). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the Hénon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the Morales-Ramis-Simó theorem.'
address:
- 'XLIM, Université de Limoges, France'
- 'XLIM, Université de Limoges, France'
author:
- '[A. [Aparicio Monforte]{}]{}'
- '[ J.-A. [Weil]{}]{}'
date: 'June 2010 and, in revised form, Oct 12, 2010.'
title: A Reduction Method for Higher Order Variational Equations of Hamiltonian Systems
---
[^1]
Introduction
============
Let $(\mathbf{k}\,,\, '\,)$ be a differential field and let $[A]: \; Y'=AY$ be a linear differential system with $A\in \mathcal{M}_{n}(\mathbf{k})$. We say that the system is in [*reduced form*]{} if its matrix can be decomposed as $A=\sum^{d}_{i=1} \alpha_i A_i$ where $\alpha_i \in \mathbf{k}$ and $A_i\in Lie(Y'=AY)$, the Lie algebra of the differential Galois group of $[A]$.
This notion of reduced form was introduced in [@Ko71a] and subsequently used (for instance [@MiSi96a] and [@MiSi96b]) to study the inverse problem. It has been revived, with a constructive emphasis, in [@ApWea]. It is a powerful tool in various aspects of linear differential systems. The main contribution of this work lies in the context of Hamiltonian mechanics and Ziglin-Morales-Ramis theory [@MoRaSi07a]: reduced forms provide a new and powerful effective method to obtain (non-)abelianity and integrability obstructions from higher variational differential equations.
This article is structured in the following way. First we lay down the background on Hamiltonian systems, differential Galois theory, integrability and Morales-Ramis-Simó theorem. In section \[section: reduced forms\], we define precisely the notions of reduced form and Wei-Norman decomposition and the link between them. Section \[section: reduced VEm\] contains the theoretical core of this work: we focus on the application of reduced forms to the study of the meromorphical integrability of Hamiltonian systems. We introduce a reduction method for block lower triangular linear differential systems and apply it to higher variational equations, in particular when the Lie algebra of the diagonal blocks is abelian and of dimension 1. In section \[section: new proof\], we demonstrate the use of this method, coupled with our reduction algorithm for matrices in $\mathfrak{sp}(2,\mathbf{k})$ [@ApWea] by giving a new, effective and self-contained Galoisian non-integrability proof of the degenerate Hénon-Heiles system ([@Mo99a] ,[@MoRaSi07a], [@MaSi09a]) which has long served as a key example in this field.
Background
==========
Hamiltonian Systems
-------------------
Let $(M\,,\,\omega)$ be a complex analytic symplectic manifold of complex dimension $2n$ with $n\in\mathbb{N}$. Since $M$ is locally isomorphic to an open domain $U\subset\mathbb{C}^{2n}$, Darboux’s theorem allows us to choose a set of local coordinates $(q\,,\,p)=(q_1 \,\ldots q_n\,,\, p_1\ldots p_n)$ in which the symplectic form $\omega$ is expressed as $J:=\tiny\left[\begin{array}{cc}0 & I_n \\-I_n & 0\end{array}\right]$. In these coordinates, given a function $H\in C^{2}(U)\,:\,U\,\longrightarrow\,\mathbb{C}$ (the Hamiltonian) we define a Hamiltonian system over $U\in\mathbb{C}^{2n}$, as the differential equation given by the vector field $X_H:= J\nabla H$: $$\label{(1)}
\begin{array}{cccc}
\dot{q}_i = \frac{\partial H }{\partial p_i}(q\,,\,p) &,& \dot{p}_i = -\frac{\partial H }{\partial q_i}(q\,,\,p)& \text{for} \,\, i=1\ldots n
\end{array}$$
The Hamiltonian $H$ is constant over the integral curves of (\[(1)\]) because $X_H\cdot H:=\langle \nabla H\,,\, X_H\rangle = \langle \nabla H \,,\, J\nabla H\rangle =0$. Therefore, integral curves lie on the energy levels of $H$. A function $F\,:\, U\, \longrightarrow \,\mathbb{C}$ meromorphic over $U$ is called a *meromorphic first integral of* (\[(1)\]) if it is constant over the integral curves of (\[(1)\]) (equivalently $X_H \cdot F =0$). Observe that the Hamiltonian is a first integral of (\[(1)\]).
The Poisson bracket $\lbrace \,,\,\rbrace$ of two meromorphic functions $f, g$ defined over a symplectic manifold, is defined by $\lbrace f \,,\, g \rbrace:=\langle \nabla f \,,\, J\nabla g\rangle$; in the Darboux coordinates its expression is $\lbrace f \,,\, g \rbrace = \sum^{n}_{i=1} \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}$. The Poisson bracket endows the set of first integrals with a structure of Lie algebra. A function $F$ is a first integral of (\[(1)\]) if and only if $\lbrace F\,,\, H\rbrace=0$ (i.e $H$ and $F$ are *in involution*).
A Hamiltonian system with $n$ degrees of freedom, is called *meromorphically Liouville integrable* if it possesses $n$ first integrals (including the Hamiltonian) meromorphic over $U$ which are functionally independent and in pairwise involution.
Variational equations {#subsection:variational equations}
---------------------
Among the various approaches to the study of meromorphic integrability of complex Hamiltonian systems, we choose a Ziglin-Morales-Ramis type of approach. Concretely, our starting points are the Morales-Ramis [@Mo99a] Theorem and its generalization, the Morales-Ramis-Simó Theorem [@MoRaSi07a]. These two results give necessary conditions for the meromorphic integrability of Hamiltonian systems. We need to introduce here the notion of variational equation of order $m\in\mathbb{N}$ along a non punctual integral curve of (\[(1)\]).
Let $\phi(z,t)$ be the flow defined by the equation (\[(1)\]). For $z_0 \in \Gamma$, we let $\phi_{0}(t):=\phi(z_0 \,,\, t)$ denote a temporal parametrization of a non punctual integral curve $\Gamma$ of (\[(1)\]) such that $z_0 = \phi(w_0 , t_0)$. We define $\mathrm{(VE^{m}_{\phi_0})}$ the *$m^{th}$ variational equation* of (\[(1)\]) along $\Gamma$ as the differential equation satisfied by the $\xi_{j}:=\frac{\partial^{j}\phi(z\,,\,t)}{\partial z^j}$ for $j\leq m$. For instance, $\mathrm{(VE^{3}_{\phi_0})}$ is given by (see [@Mo99a] and [@MoRaSi07a]): $$\begin{aligned}
\nonumber \dot{\xi}_1 &=& d_{\phi_0} X_H \xi_1\\
\nonumber \dot{\xi}_2 &=& d^{2}_{\phi_0}X_H(\xi_1\,,\, \xi_1) + d_{\phi_0}X_H \xi_2\\
\nonumber \dot{\xi}_2 &=& d^{3}_{\phi_0}X_H(\xi_1\,,\, \xi_1\,,\, \xi_1) + 2 d^2_{\phi_0}X_H (\xi_1\,,\,\xi_2) + d_{\phi_0} X_H \xi_3.\end{aligned}$$ For $m=1$, the equation $\mathrm{(VE^{1}_{\phi_0})}$ is a linear differential equation $$\dot{\xi}_1 = A_{1} \xi_1\text{ where }A_{1}:= d_{\phi_0} X_H= J\cdot Hess_{\phi_0}(H)\in\mathfrak{sp}(n\,,\, \mathbf{k})\text{ and }
\mathbf{k}:=\mathbb{C}\langle \phi_0(t) \rangle.$$ Higher order variational equations are not linear in general for $m\geq 2$. However, taking symmetric products, one can give for every $\mathrm{(VE^{m}_{\phi_0})}$ an equivalent linear differential system $\mathrm{(LVE^{m}_{\phi_0})}$ called the [*linearized*]{} $m^{th}$ [*variational equation*]{} (see [@MoRaSi07a]).
Since the $\mathrm{(LVE^{m}_{\phi_0})}$ are linear differential systems, we can consider them under the light of differential Galois theory ([@PuSi03a; @Mo99a]). We take as base field the differential field $\mathbf{k} := \mathbb{C}\langle \phi_{0} \rangle$ generated by the coefficients of $\phi_{0}$ and their derivatives. Let $K_m$ be a Picard Vessiot extension of $\mathrm{(LVE^{m}_{\phi_0})}$ for $m\geq 1$. The differential Galois group $G_m := \text{Gal}(K_m /\mathrm{k} )$ of $\mathrm{(LVE^{m}_{\phi_0})}$ is the group of all differential automorphisms of $K_m$ that leave the elements of $\mathbf{k}$ fixed.
As $G_m$ is isomorphic to a algebraic linear group over $\mathbb{C}$, it is in particular an algebraic manifold and we can define its Lie algebra $\mathfrak{g}_m:=T_{I_{d_m}} G^{\circ}_m$, the tangent space of $G_m$ at $I_{d_m}$ (with $ d_m= \tiny\sum^{m}_{i=1} \binom{n+i-1}{n-1}$ the size of $\mathrm{(LVE^{m}_{\phi_0})}$). The Lie algebra $\mathfrak{g}_m$ is a complex vector space of square matrices of size $d_m$ whose Lie bracket is given by the commutator of matrices $[M\,,\,N] = M\cdot N - N \cdot M$. We say that $\mathfrak{g}_m$ is abelian if $[\mathfrak{g}_m\,,\, \mathfrak{g}_m] = 0$.
Following the notations above, we can finally give the Morales-Ramis-Simó theorem:
\[MRS\]([@MoRaSi07a]): If the Hamiltonian system (\[(1)\]) is meromorphically Liouville integrable then the $\mathfrak{g}_m$ are abelian for all $m\in \mathbb{N}^{\star}$.
Partial effective versions of this theorem have been proposed. In [@MoRaSi07a] (and already [@Mo99a]), a local criterion is given for the case when the first variational equation has Weierstrass functions as coefficients ; in [@MaSi09a], a powerful approach using certified numerical computations is proposed. In the case of Hamiltonian systems with a homogeneous potential, yet another approach is given in [@CaDuMaPr10a].\
Our aim is to propose an alternative (algorithmic) method using a (constructive) notion of reduced form for the variational equation. This strategy should supply new criteria of non-integrability as well as some kind of “normal form along a solution”. We will now explain this notion of reduced form (which we started investigating in [@ApWea]) and show how to apply it. We will then apply our reduction method in detail on the well-known degenerated case of the Henon-Heiles system proposed in [@MoRaSi07a].
Reduced Forms {#section: reduced forms}
=============
Let $(\mathbf{k}\,,\,'\,)$ be a differential field with field of constants $C$ and let $Y'=AY$ be a linear differential system with $A=(a_{i j})\in \mathcal{M}_{n}(\mathbf{k})$. Let $G$ be the differential Galois group of this system and $\mathfrak{g}$ the Lie algebra of $G$. We sometimes use the slight notational abuse $\mathfrak{g}=Lie(Y'=AY)$.\
Let $a_{1},\ldots,a_{r}$ denote a basis of the $C$-vector space spanned by the entries $a_{i,j}\in k$ of $A$. Then we have $$A:=\sum^{r}_{i=1} a_{i}(x) M_i ,\quad M_i \in\mathcal{M}_{n}(C).$$ This decomposition appears (slightly differently) in [@WeNo63a], we call it a *Wei-Norman decomposition* of $A$. Although this decomposition is not unique (it depends on the choice of the basis $(a_{i})$), the $C-$vector space generated by the $M_i$ is unique.
With these notations, the Lie algebra generated by $M_1 ,\ldots , M_r$ and their iterated Lie brackets is called *the Lie algebra associated to $A$*, and will be denoted as $Lie(A)$.
Consider the matrix $$A_1:=\left[\begin{array}{cccc} 0 & 0 & 2/x & 0 \\ 0 & 0 & 0 & 2/x\\
\frac{2(x^4 - 10 x^2 + 1 )}{x(x^2 + 1)^2} & 0 & 0 & 0\\ 0 & -\frac{12 x }{(x^2 + 1)^2 } & 0 & 0\end{array}\right].$$ Expanding the fraction $\frac{2(x^4 - 10 x^2 + 1 )}{x(x^2 + 1)^2}$ gives a Wei-Norman decomposition as $$A_{1}=\frac2{x} M_{1}
-\frac{12 x }{(x^2 + 1)^2 } M_{2},$$ where $$M_{1}= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right],\,
M_{2}=\left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\
2 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\end{array}\right]$$ and $Lie(A_{1})$ has dimension $6$.
A celebrated theorem of Kovacic (and/or Kolchin) states that $\mathfrak{g}\subset Lie(A)$. This motivates the following definition :
We say that $A$ is in *reduced form* if $Lie(A)=\mathfrak{g}$.
A [*gauge transformation*]{} is a change of variable $Y=PZ$ with $P\in \mathrm{GL}(n\,,\, \mathbf{k}) $. Then $Z'=RZ$ where $R:=P^{-1} (AP-P')$. In what follows, we adopt the notation $P[A]:=P^{-1} (AP-P')$ for the system obtained after the gauge transformation $Y=PZ$.\
The following theorem due to Kovacic (and/or Kolchin) ensures the existence of a gauge transformation $P\in \mathrm{GL}(n\,,\, \bar{\mathbf{k}}) $ such that $P[A]\in\mathfrak{g}(\bar{\mathbf{k}})$ when $ {\mathbf{k}} $ is a $C_1$-field[^2]
\[Kovacic\] Let $k$ be a differential $C_1$-field. Let $A\in\mathcal{M}_n (k)$ and assume that the differential Galois Group $G$ of the system $Y'=AY$ is connected. Let $\mathfrak{g}$ be the Lie algebra of $G$. Let $H$ be a connected algebraic group such that its Lie algebra $\mathfrak{h}$ satisfies $A\in\mathfrak{h}(k)$. Then $G\subset H$ and there exists $P\in H(k)$ such that the equivalent differential equation $F'=\tilde{A}F$, with $Y=PF$ and $\tilde{A}=P[A]=P^{-1}AP-P^{-1}P'$, satisfies $\tilde{A}\in \mathfrak{g}(k)$.
We say that a matrix $P\in\mathrm{GL}_n (\mathbf{k})$ is a *reduction matrix* if $P[A]\in\mathfrak{g}(\mathbf{k})$, i.e $P[A]$ is in reduced form. We say that a matrix $Q\in\mathrm{GL}_n (\mathbf{k})$ is a *partial reduction matrix* when $Q[A]\in\mathfrak{h}(\mathbf{k})$ with $\mathfrak{g}\subsetneq\mathfrak{h}\subsetneq Lie(A)$. The general method used to put $A$ in a reduced form consists in performing successive partial reductions until a reduced form is reached.\
In our paper [@ApWea], we provide a reduction algorithm that computes a reduction matrix $P_1 \in\mathrm{Sp}(2,\mathbf{k})$ for $4\times 4$ linear differential systems $Y'=A_1Y$ with $A_1\in\mathfrak{sp}(2,\mathbf{k})$ (and also for $2\times 2$ systems). The first variational equation of a Hamiltonian system with $n=2$ degrees of freedom belongs to this class of systems. If $P_1$ is a reduction matrix for $A_1$ then $Sym^{m} P_1$ is a reduction matrix for $sym^{m} A_1$ because $Sym^m $ is a group morphism (see [@PuSi03a], chapter 2 or [@FuHa91a] appendix B2).\
In what follows, we will assume that we have reduced the first variational equation, that its Lie algebra is abelian (so that the Morales-Ramis theorem gives no obstruction to integrability), and use this to start reducing higher variational systems.\
We will follow the philosophy of Kovacic’s theorem \[Kovacic\] and look for reduction matrices inside $\exp(Lie(A))$. We remark that, in the context of Lie-Vessiot systems, an analog of the above Kolchin-Kovacic reduction theorem is given by Blazquez and Morales ([@BlaMor10], section 5, in particular theorems 5.3 and 5.8) in relation to Lie reduction.\
The notion of a reduced form is useful in many contexts, such as: inverse problems (where the notion was first studied), the computation of the transcendence degree of Picard Vessiot extensions, fast resolution of linear differential systems with an abelian Lie algebra and to implement the Wei-Norman method for solving linear differential systems with a solvable Lie algebra (using the Campbell-Hausdorff formula) [@WeNo63a]. Reduced forms are also a new and powerful tool that provides (non-)abelianity and integrability obstructions for (variational) (see Theorem \[MRS\]) linear differential equations arising from Hamiltonian mechanics, as we will now see.
Reduced Forms for Higher Variational Equations {#section: reduced VEm}
==============================================
Preliminary results {#preliminary}
-------------------
Let $(\mathbf{k}\,,\, ' )$ be a differential field and let $d\in\mathbb{N}$. Consider a linear differential system $Y' = AY$ whose matrix $A\in {{\mathcal M}}_{d}(\mathbf{k})$ is block lower triangular as follows: $$A:=\left[\begin{array}{cc}A_{1} & 0 \\ A_{3} & A_{2}\end{array}\right]= A_{diag} + A_{sub} \text{ where } A_{diag}=\left[\begin{array}{cc} A_1 & 0 \\ 0 & A_2\end{array}\right] \text{ and } A_{sub}=\left[\begin{array}{cc} 0 & 0 \\ A_3 & 0\end{array}\right].$$ The submatrices satisfy $ A_{1}\in {{\mathcal M}}_{d_1}(\mathbf{k})$, $ A_{2}\in {{\mathcal M}}_{d_2}(\mathbf{k})$, $A_3\in {{\mathcal M}}_{d_2 \times d_1} (\mathbf{k})$ and their dimensions add-up $d=d_1 + d_2$.
Let $${{{\mathcal M}}}_{diag}:=\left\{\left[\begin{array}{cc}A_{1} & 0 \\ 0 & A_{2}\end{array}\right], A_{i}\in{{\mathcal M}}_{d_{i}}(\mathbf{k})\right\}$$ and $${{{\mathcal M}}}_{sub}:=\left\{\left[\begin{array}{cc}0 & 0 \\ B_{1} & 0\end{array}\right], B_{1}\in{{\mathcal M}}_{d_{2}\times d_{1}}(\mathbf{k})\right\}$$
\[diagsub\] Let $M_{1},M_{2}\in {{{\mathcal M}}}_{diag}$ and $N_{1},N_{2}\in {{{\mathcal M}}}_{sub}$. Then $M_{1}.M_{2}\in {{{\mathcal M}}}_{diag}$, $N_{1}.N_{2}=0$ (so that $N_{1}^{2}=0$ and $\exp(N_{1})=Id + N_{1}$), and $[M_{1},N_{1}]\in {{{\mathcal M}}}_{sub}$.
The proof is a simple linear algebra exercise.\
Let $\mathfrak{g}:=Lie(Y'=AY)$ be the Lie algebra of the Galois group of $Y'=AY$ and let $\mathfrak{h}:=Lie(A)$ denote the Lie algebra associated to $A$. We write $\mathfrak{h}_{diag}:=\mathfrak{h} \cap {{{\mathcal M}}}_{diag}$ and $\mathfrak{h}_{sub}:=\mathfrak{h} \cap {{{\mathcal M}}}_{sub}$. The lemma shows that they are both Lie subalgebras (with $\mathfrak{h}_{sub}$ abelian) and $\mathfrak{h}=\mathfrak{h}_{diag}\oplus \mathfrak{h}_{sub}$. Furthermore, $[\mathfrak{h}_{diag},\mathfrak{h}_{sub}]\subset \mathfrak{h}_{sub}$ (i.e $\mathfrak{h}_{sub}$ is an ideal in $\mathfrak{h}$). When $\mathfrak{h}_{diag}$ is abelian, obstructions to the abelianity of $\mathfrak{h}$ only lie in the brackets $[\mathfrak{h}_{diag},\mathfrak{h}_{sub}]$.
A first partial reduction for higher variational equations {#subsection: first partial reduction}
----------------------------------------------------------
Using the algorithm of [@ApWea], we may assume that the first variational equation has been put into a reduced form. We further assume that the first variational equation has an abelian Lie algebra (so that there is no obstruction to integrability at that level).\
As stated in section \[subsection:variational equations\], each $\mathrm{(VE^{m}_{\phi_0})}$ is equivalent to a linear differential system $\mathrm{(LVE^{m}_{\phi_0})}$ whose matrix we denote by $A_m$. The structure of the $A_m$ is block lower triangular , to wit $$A_m :=\left[\begin{array}{cc} sym^{m}(A_1) & 0 \\ B_m & A_{m-1}\end{array}\right]\in M_{d_m} (\mathbf{k})$$ where $A_1$ is the matrix of $\mathrm{(LVE^{1}_{\phi_0})}$. Assume that $A_{m-1}$ has been put in reduced form by a reduction matrix $P_{m-1}$. Then the matrix $Q_m \in \mathrm{GL}(d_m\,,\,\mathbf{k})$ defined by $$Q_m:=\left[\begin{array}{cc} Sym^{m}(P_1) & 0 \\ 0 & P_{m-1}\end{array}\right]$$ puts the diagonal blocks of the matrix $A_m$ into a reduced form (i.e the system would be in reduced form if there were no $B_{m}$) and preserves the block lower triangular structure. Indeed, $$Q_m [A_m] = \left[ \begin{array}{cc} Sym^m(P_1)[sym^m A_1] & 0 \\ \tilde{B}_{m} & P_{m-1}[A_{m-1}]\end{array}\right]$$ where $$\tilde{B}_{m}:=P^{-1}_{m-1} B_m Sym^{m}(P_1).$$ Applying the notations of the previous section to $\tilde{A}:=Q_m [A_m]$, we see that $Lie(\tilde{A})_{diag}$ and $Lie(\tilde{A})_{sub}$ are abelian. Obstructions to integrability stem from brackets between the diagonal and subdiagonal blocks. To aim at a reduced form, we need transformations which “remove” as many subdiagonal terms as possible while preserving the (already reduced) diagonal part. Recalling Kovacic’s theorem \[Kovacic\], our partial reduction matrices will arise as exponentials from subdiagonal elements.
Reduction tools for higher variational equations
------------------------------------------------
\[partial reduction\] Let $A:=Q_m [A_m]$ as above be the matrix of the $m$-th variational equation $Y'=AY$ after reduction of the diagonal part. Write $A=A_{diag} + \sum^{d_{sub}}_{i=1} \beta_i B_i$ with $\beta_i \in\mathbf{k}$, where the $B_{i}$ form a basis of $Lie(A)_{sub}$ (in the notations of section \[preliminary\]).\
Let $[A_{diag}\,,\, B_1]=\sum_{i=1}^{{d_{sub}}} \gamma_{i} B_{i}$, $\gamma_{i} \in\mathbf{k}$. Assume that the equation $y'=\gamma_{1}y+\beta_{1}$ has a solution $g_{1}\in k$. Set $P:=\exp(g_{1}B_{1})=(Id+g_{1}B_{1})$. Then $$P[A] = A_{diag} + \sum_{i=\bf{2}}^{d_{sub}} \left[\beta_{i}+g_{1}\gamma_{i}\right]B_{i},$$ i.e $P[A]$ no-longer has any terms in $B_{1}$.
Recall that $P[A] = P^{-1}(AP-P')$ and let $P=Id + g_1 B_1$. We have $P'=g'_1 B_1$ whence $$AP=(A_{diag} + \sum^{d_{sub}}_{i=1} \beta_i B_i)(I+g_{1} B_1) = A_{diag} + \sum_{i\geq 1} \beta_i B_i + g_{1} A_{diag} B_1$$ since $B_i B_j = 0$. Therefore we have $AP-P' = A_{diag}+ g_{1} A_{diag} B_1 + (\beta_{1}-g_{1}')B_{1} + \sum^{d}_{i=2}\beta_i B_i$ which implies $$\begin{aligned}
\nonumber P^{-1}(AP-P') &=& (Id-g_{1} B_1) \left[A_{diag}+ g_{1} A_{diag} B_1 + (\beta_{1}-g_{1}')B_{1} + \sum^{d}_{i=2}\beta_i B_i \right]\\
\nonumber&=&A_{diag} + g_{1}[A_{diag}\,,\, B_1] + (\beta_{1}-g_{1}')B_{1} + \sum^{d}_{i=2}\beta_i B_i\end{aligned}$$ because $B_1 A_{diag} B_1 = B_1 [A_{diag} \,,\, B_1] + A_{diag} B_{1} B_{1} = B_{1}\left[\sum \gamma_i B_i\right]=0$. So, as $g_{1}'=\gamma_{1}g_{1}+\beta_{1}$, we obtain $$P[A] = A_{diag} + \sum_{i=\bf{2}}^{d_{sub}} \left[\beta_{i}+g_{1}\gamma_{i}\right]B_{i}.$$
If $\gamma_{1}=0$ then we simply have $g_{1}=\int\beta_{1}$. In that case, suppose that $\mathbf{k}=\mathbb{C}(x)$ and that $\beta_1 = R'_1 + L_1$ where $R_1 \in \mathbb{C}(x)$ and $L_1 \in \mathbb{C}(x)$ has only simple poles, then $\int \beta_1 \notin \mathbb{C}(x)$. However, if we apply proposition \[partial reduction\] with the change of variable $Y= (I + R_1 B_1) Z$ a term in $B_1$ will be left that will only contain simple poles.
This proposition gives a nice formula for reduction. However, it is hard to iterate unless $Lie(A)$ has additional properties (solvable, nilpotent, etc) because the next iteration may “re-introduce” $B_{1}$ in the matrix (because of the expression of the brackets). This proposition provides a reduction strategy when the map $[A_{diag},.]$ admits a triangular representation.\
To achieve this, we specialize to the case when the Lie algebra $\mathfrak{g}_{diag}$ has dimension (at most) $1$. Then we have $A_{diag} = \beta_{0} A_{0}$ where $\beta_{0}\in k$ and $A_{0}$ is a constant matrix. The above proposition specializes nicely :
If $A_{diag}=\beta_0 A_0$ with $\beta_0\in\mathbf{k}$, $A_{0}\in\mathcal{M}_{n}(\mathbb{C})$ and $[A_0\,,\, B_1] = \lambda B_1$ for some constant eigenvalue $\lambda \neq 0$ then the change of variable $Y=PZ$ with $P:=(Id + g B_1)$, with $g'= \lambda g\beta_0 + \beta_1$, satisfies $P[A] = \beta_0 A_0 + \sum^{d_{sub}}_{i\geq 2} \beta_i B_i$.
To implement this (and obtain a general reduction method), we let $\Psi_{0} : \mathfrak{h}_{sub} \rightarrow \mathfrak{h}_{sub}$, $B\mapsto [A_{0},B]$. This is now an endomorphism of a finite dimensional vector space ; up to conjugation, we may assume the basis $(B_{i})$ to be the basis in which the matrix of $\Psi_{0}$ is in Jordan form. We are then in position to apply the proposition iteratively (see the example below for details on the process).
Not that $A_0$ needs not be diagonal. The calculations of lemma \[partial reduction\] and subsequent proofs remain valid when $A_0$ is block lower triangular.
We have currently implemented this in Maple for the case when $A_{diag}$ is monogenous, i.e. its associated Lie algebra has dimension $1$. We will show the power of this method and of the implementation by giving a new proof of non-integrability of the degenerate Henon-Heiles system whose first two variational equations are abelian but which is not integrable.
A new proof of the non integrability of a degenerate Hénon-Heiles system {#section: new proof}
========================================================================
In this section we consider the following Hénon Heiles Hamiltonian [@Mo99a], [@MoRaSi07a], $$\label{HH}
H:=\frac{1}{2}(p^2_1 + p^2_2) + \frac{1}{2}(q^2_1 + q^2_2) + \frac{1}{3}q^3_1 + \frac{1}{2}q_1 q^2_2$$ as given in [@Mo99a]. This Hamiltonian’s meromorphic non integrability was proved in [@MoRaSi07a]. The Hamiltonian field is $$\dot{q}_1 = p_1 \,,\, \dot{q}_2 = p_2 \,,\, \dot{p}_1 = -q_1 (1+q_1) - \frac{1}{2}q^2_2 \,,\,\dot{p}_2 = -q_2 (1+ q_1 ).$$ This degenerate Hénon Heiles system was an important test case which motivated [@MoRaSi07a]. Its non integrability was reproved in [@MaSi09a] to showcase the method used by the authors. We follow in this tradition by giving yet another proof using our systematic method. Our reduction provides a kind of “normal form along $\phi$” in addition to a non integrability proof. The readers wishing to reproduce the detail of the calculations will find a Maple file at the url
Êhttp://www.unilim.fr/pages_perso/jacques-arthur.weil/charris/
It contains the commands needed to carry on the reduction of the $\mathrm{(LVE^{m}_{\phi})}$ for $i=1\ldots 3$. The reduction of $\mathrm{(LVE^{3}_{\phi})}$ may take several minutes to complete.
Reduction of $\mathrm{(VE^{1}_{\phi})}$
---------------------------------------
On the invariant manifold $\lbrace q_2 = 0 \,,\, p_2 =0 \rbrace$ we consider the non punctual particular solution $$\phi(t) = \left(\,\frac{3}{2}\frac{1}{\cosh(t/2)^2} - 1 \,,\, 0\,,\, -\frac{3}{2}\frac{\sinh(t/2)}{\cosh(t/2)^3}\,,\,0\right).$$ and the base field is $\mathbf{k}=\mathbb{C}\langle \phi \rangle = \mathbb{C}(e^{t/2})$. Performing the change of independent variable $x=\mathrm{e}^{t/2}$, we obtain an equivalent system with coefficients in $\mathbb{C}(x)$ given by $$A_1:=\left[\begin{array}{cccc} 0 & 0 & 2/x & 0 \\ 0 & 0 & 0 & 2/x\\
\frac{2(x^4 - 10 x^2 + 1 )}{x(x^2 + 1)^2} & 0 & 0 & 0\\ 0 & -\frac{12 x }{(x^2 + 1)^2 } & 0 & 0\end{array}\right].$$\
Applying the reduction algorithm from [@ApWea] we obtain the reduction matrix $$P_1:=\left[\begin{array}{cccc} -\frac{6(x-1)(x+1)x^2}{(x^2 + 1)^3} & 0 & -\frac{x^{10} + 15 x^8 - 16 x^6 - 144x^4+15x^2+1}{12x^2 (x^2 +1)^3} & 0\\ 0 &\frac{x^4-4x^2 + 1}{(x^2 + 1)^2} & 0 & -\frac{5 x^4 + 16 x^2 - 13}{3(x^2 + 1 )^2}\\ \frac{6 x^2 (x^4 - 4x^2 + 1)}{(x^2 + 1) ^4} & 0 & -\frac{x^{12} + 4 x^{10} + 121 x^8 + 256 x^6 - 249 x^4 - 4 x^2 - 1}{12 x^2 (x^2 + 1) ^4} & 0 \\ 0 & \frac{6(x^2 - 1)x^2}{(x^2 + 1)^3} & 0 & \frac{x^6 - x^4 - 17 x^2 + 1}{(x^2 + 1)^3} \end{array}\right]$$ that yields the reduced form $$A_{1,R}=\frac{5}{3 x}\left[\begin{array}{cccc}0 &0 & 1 & 0\\ 0 & 0 & 0& 6/5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right].$$ We see that $\mathrm{dim}_{\mathbb{C}}\left(Lie(A_{1,R})\right) =1$ and since $\frac{5}{3 x}$ has one single pole, we cannot further reduce without extending the base field $\mathbf{k}$. We find, $$\mathfrak{g}_1 = \mathrm{span}_{\mathbb{C}}\left\lbrace \tilde{D}_1 := \tiny\left[\begin{array}{cccc} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 6/5 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right]\right\rbrace$$ which is trivially abelian and therefore doesn’t give any obstruction to integrability.
Reduction of $\mathrm{(LVE^{2}_{\phi})}$
----------------------------------------
We want now to put the matrix $A_2$ of $\mathrm{(LVE^{2}_{\phi})}$ into a reduced form. First we reduce the diagonal blocks as indicated in section \[subsection: first partial reduction\] using the partial reduction matrix $Q_2:=\tiny\left[\begin{array}{cccc}Sym^2 P_1 & 0 \\ 0 & P_1\end{array}\right]$ so that we obtain a partially reduced matrix (its diagonal blocks are reduced whereas its subdiagonal block is not): $$Q_2[A_2]:=\left[\begin{array}{cccc} sym^{2} A_{1,R} & 0 \\ \tilde{B}_2 & A_{1,R}\end{array}\right] \text{ with } \left\lbrace\begin{array}{ccc} Q_2[A_2]_{diag} & = & \tiny\left[\begin{array}{cc} sym^2 A_{1,R} & 0 \\ 0 & A_{1,R}\end{array}\right]\\
Q_2[A_2]_{sub} & = & \tiny\left[\begin{array}{cc} 0 & 0 \\ \tilde{B}_2 & 0\end{array}\right]\end{array}\right\rbrace$$ We compute a Wei-Norman decomposition and we obtain an associated Lie algebra $Lie(Q_2[A_2])$ of dimension $11$ such that:
- On one hand we obtain $Lie(Q_2[A_2])_{diag} = \mathrm{span}_{\mathbb{C}}\left\lbrace D_{2,0}:=\left[\begin{array}{cc} sym^2\tilde{D}_1 & 0 \\ 0 & \tilde{D}_1 \end{array}\right]\right\rbrace$ with coefficient $\beta_0:= \frac{5}{3 x}$.
- On the other hand, $Lie(Q_2[A_2])_{sub} = \mathrm{span}_{\mathbb{C}}\lbrace \mathcal{B}_2 \rbrace$ where $$\mathcal{B}_2 := \lbrace B_i := {\tiny\left[\begin{array}{cc} 0 & 0 \\ \tilde{B}_i & 0\end{array}\right] , i=1\ldots 10\rbrace}\text{ and }Q_2[A_2]_{diag} = \sum^{10}_{i=1} \beta_{2,i} B_{2,i}\text{ with }\beta_i \in \mathbf{k}.$$
The matrix of the application $$\Psi_{2,0}\,:\, Lie(Q_2[A_2])_{sub}\,\longrightarrow\, Lie(Q_2[A_2])_{sub} \,,\, B_j \,\mapsto\, [D_{2,0}\,,\, B_j]$$ expressed in the base $\mathcal{B}_2$ takes the following form: $$\Psi_{2,0}:=\tiny \left[ \begin {array}{cccccccccc} 0&0&0&0&0&0&0&0&1&0
\\\noalign{\medskip}-2&0&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0
&0&0&0&1\\\noalign{\medskip}0&0&-6/5&0&0&0&-1&0&0&0
\\\noalign{\medskip}0&-3&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&-{
\frac {12}{5}}&0&0&0&-1&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&6/5
\\\noalign{\medskip}0&0&0&0&0&0&-{\frac {12}{5}}&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0
&0&0&0&0\end {array} \right].$$ We denote by $J_{\Psi_{2,0}}$ the matrix of $\Psi_{2,0}$ expressed in its Jordan basis, given by the matrices $C_{2,i} =\tiny\left[\begin{array}{cc} 0 & 0 \\ \tilde{C}_{2,i} & 0 \end{array}\right]$ and their coefficients $\gamma_{2,i}$ with $i=1\ldots 10$. So the Jordan form is $$J_{\Psi_{2,0}}=\tiny \left[ \begin {array}{cccccccccc} 0&1&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&0&1&0&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&1&0&0
&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0
&0&0&0&1&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&1&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&1&0&0\\\noalign{\medskip}0&0&0&0&0&0
&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&1\\\noalign{\medskip}0&0
&0&0&0&0&0&0&0&0\end {array} \right].$$ To perform reduction we will use the Jordan basis $\mathcal{C}_2 := \lbrace C_{2,i} \,,\, i=1\ldots 10\rbrace$. The decomposition given by the Jordan basis $\mathcal{C}_2$ is $Q_2[A_2]:=D_0 + \sum^{10}_{i=1} \gamma_i C_i$ with $\gamma_i \in\mathbf{k}$ , $i=1\ldots10$. We notice that $J_{\Psi_0}$ is made of three Jordan blocks
- two blocks of dimension $4$ : $$\lbrace C_{2,4}\,,\,C_{2,3}\,,\,C_{2,2}\,,\,C_{2,1} \rbrace$$ and $$\lbrace C_{2,8}\,,\,C_{2,7}\,,\,C_{2,6}\,,\,C_{2,5} \rbrace$$
- and one block of dimension $2$ : $\lbrace C_{2,10}\,,\,C_{2,9} \rbrace$
The hypothesis of the first section of Proposition \[partial reduction\] are satisfied. Therefore the partial reduction of $Q_2[A_2]$ is done in the following way:
- Choose a Jordan block of dimension $d$ : $\lbrace C_{2,i}\,\ldots\, C_{2,i+d-1}\rbrace$. It satisfies $\Psi_{2,0}(C_{2,i+s}) = C_{2,i+s-1}$ for $s=1\ldots d-1$. Set $\tilde{A}_{2}:=Q_2[A_2]$ and set $s:=d-1$.
- For $s$ from $d-1$ to $1$, compute the decomposition $\gamma_{2,i+s}= R'_{2,i+s} + L_{2,i+s}$ where $R_{2,i+s}\,,\, L_{2,i+s}\in\mathbf{k}$ and $L_{2,i+s}$ has only simple poles.\
Take the change of variable $P_{2,i+s}=Id + R_{2,i+s} C_{2,i+s}$ and perform the gauge transformation $P_{2,i+s}[\tilde{A}_{2}]$.\
If $L_{2,i+s}=0$ then the Wei-Normal decomposition of $P_{2,i+s}[\tilde{A}_{2}]$ does not contain $C_{2,i+s}$ so $C_{2,i+s}\notin\mathfrak{g}_2$.\
Set $\tilde{A}_{2} := P_{2,i+s}[\tilde{A}_{2}]$ and set $s:=s-1$. Repeat this procedure recursively until $s=1$.
- Choose a Jordan block that has not been treated. Repeat until there are no more Jordan blocks left untreated.
In this way, only will be left in the subdiagonal block the $C_{2,i}$ that have coefficients $L_{2,i}$ (after the procedure) containing only simple poles. In our case, we obtain a reduced matrix for $\mathrm{(LVE^{2}_{\phi})}$: $A_{2,R}:=\frac{1}{x} \tilde{C}_0$ and
$$\tilde{C}_0:=\tiny\left[ \begin {array}{cccccccccccccc} 0&0&\frac53\, &0&0&0&0&0&0&0
&0&0&0&0\\\noalign{\medskip}0&0&0&2\, &0&\frac53\, &0&0&0&0&0
&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&\frac{10}{3}\, &0&0&0&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&0&\frac53\, &0&0&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&2\, &0&0&0&0&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&0&2\, &0&0&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&4\, &0&0&0&0
\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0
&0&0&0&0&0&0&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0
\\\noalign{\medskip}0&0&-\frac{10}{3}\, &0&0&0&2\, &\frac{95}{18} &0&-\frac{20}{3}\, &0&0&\frac53\, &0
\\\noalign{\medskip}0&0&0&0&0&2\, &0&0&-\frac{20}{3}&0&0&0&0&2\, \\\noalign{\medskip}0&0&0&0&0&0&0&\frac{10}{3}\,
&0&0&0&0&0&0\\\noalign{\medskip}0&0&0&0&0&0&0&0&-2\, &0&0&0&0&0
\end {array} \right]$$
As in the case of $A_{1,R}$, this matrix $A_{2,R}$ is in a reduced form because $Lie(A_{2,R})$ is monogenous and $\frac{1}{x}$ only has simple poles. Therefore $Lie(A_{2,R}) =\mathfrak{g}_2$ and $\mathfrak{g}_2$ is once more abelian bringing in no obstruction to integrability. We have then to look at $\mathrm{(LVE^{3}_{\phi})}$.
Reduction of $\mathrm{(LVE^{3}_{\phi})}$
----------------------------------------
We denote $P_2$ the reduction matrix of $A_2$. Once more we build a partial reduction matrix $Q_3 := \tiny\left[\begin{array}{cc} Sym^3 P_1 & 0 \\ 0 & P_2\end{array}\right]$ that puts the diagonal blocks of matrix $A_3$ into a reduced form and we obtain the partially reduced matrix $Q_3[A_3] := \tiny\left[\begin{array}{cc} sym^3 A_{1,R} & 0 \\ \tilde{B}_3 & A_{2,R}\end{array}\right]$. In this case we have a Wei-Norman decomposition of $Q_3[A_3]$ of dimension $18$, and $\mathrm{dim}_{\mathbb{C}}(Lie(Q_3[A_3]))=38$.
We thus have
- $\mathrm{dim}_{\mathbb{C}}(Lie(Q_3[A_3])_{diag})=1$ where $$Lie(Q_3[A_3])_{diag}=\mathrm{span}_{\mathbb{C}} \lbrace D_{3,0}:=\tiny\left[\begin{array}{cc} Sym^3 \tilde{D}_1 & 0 \\ 0 & \tilde{C}_{2,0}\end{array}\right]\rbrace$$
- and $\mathrm{dim}_{\mathbb{C}}(Lie(Q_3[A_3])_{sub})=37$ and $Lie(Q_3[A_3])_{sub} =\mathrm{span}_{\mathbb{C}}(\mathcal{B}_3)$ with $$\mathcal{B}_3=\tiny\lbrace B_{3,i}=\left[\begin{array}{cc} 0 & 0 \\ \tilde{B}_{3,i} & 0\end{array}\right] \,,\,{\tiny i=1\ldots 38} \rbrace$$ a base of generators of $Lie(Q_3[A_3])_{sub}$.
We define $\Psi_{3,0} \, : \, \mathfrak{h}_{3,sub}\,\longrightarrow \, \mathfrak{h}_{3,sub}\,,\,
B \, \mapsto \, [D_{3,0} \,,\, B]$. It is nilpotent and its Jordan basis will satisfy the conditions of the first section of Proposition \[partial reduction\]. In the Jordan basis $\mathcal{C}_{3}:=\lbrace C_{3,i}\,,\, i=1\ldots 37\rbrace$, the Jordan form of $J_{\Psi_{3,0}}$ is formed by the following Jordan blocks:
1. three Jordan blocks of dimension $5$ corresponding to : $\lbrace C_{3,5},\ldots , C_{3,1}\rbrace,$ $\lbrace C_{3,11},\ldots , C_{3,6}\rbrace,$ $\lbrace C_{3,17},\ldots , C_{3,12}\rbrace$
2. three Jordan blocks of dimension $4$: $\lbrace C_{3,18},\ldots , C_{3,21}\rbrace$ , $\lbrace C_{3,22},\ldots , C_{3,26}\rbrace$ and $\lbrace C_{3,31},\ldots , C_{3,27}\rbrace,$
3. and two Jordan blocks of dimension $2$: $$\lbrace C_{3,34},\ldots , C_{3,32}\rbrace \text{ and }\lbrace C_{3,37},\ldots , C_{3,35}\rbrace.$$
In the basis $\mathcal{C}_3$, a Wei-Norman decomposition is $$Q_3[A_3]= \beta_{0} D_{3,0} + \sum^{37}_{i=1} \gamma_{3,i} C_{3,i}.$$
We proceed blockwise as in the case of the second variational equation. This time, possible obstructions to integrability appear when handling the Jordan block $\lbrace C_{3,31}\,,\ldots \,,\, C_{3,27}\rbrace$. By decomposition $\gamma_{3,i}=R'_{3,i} + L_{3,i}$ (with $i=27\ldots 31$), we see that in particular $L_{3,30}$ and $L_{3,29}$ are non zero (and have “new poles”, i.e not the pole zero of the coefficient of the reduced form of $(VE_2)$) and therefore we suspect that $C_{3,29} , C_{3,30}$ (or some linear combination) lie in $\mathfrak{g}_{3}$. Since neither $C_{3,30}$ nor $C_{3,29}$ commute with $D_{3,0}$ that would suggest that $\mathfrak{g}_{3}$ is not abelian and therefore, intuitively, the Hamiltonian (\[HH\]) would be non integrable. We prove this rigorously in the following subsection.
Proof of non-integrability
--------------------------
After performing the partial reduction recursively for all blocks, we obtain the matrix $\tilde{A}_{3,R}$. It has a Wei-Norman decomposition $\tilde{A}_{3,R} =a_1 M_{3,1} + a_2 M_{3,2}$ where $M_{3,1}, M_{3,2}\in\mathcal{M}_{34}(\mathbb{C})$, $a_1 :=\frac{1}{x}$, $a_2:=\frac{x}{x^2 +1}$. The matrix $M_{3,1}$ is lower block triangular and $M_{3,2}\in Lie(\tilde{A}_{3,R})_{sub}$. We let $M_{3,3}:=[M_{3,1}\,,\, M_{3,2}]$, $M_{3,4}:=[M_{3,1}\,,\, M_{3,3}]$, $M_{3,5}:=[M_{3,1}\,,\, M_{3,4}]$ and check that $[M_{3,i}\,,\, M_{3,j}]=0$ otherwise. So $Lie(\tilde{A}_{3,R})$ has dimension $5$ and is generated by the $M_{3,i}$. Note that $M_{3,i}\in \mathcal{M}_{34,sub}(\mathbb{C})$ for $i\geq 2$. Again we let $$\Psi\,:\, Lie(\tilde{A}_{3,R})\,\longrightarrow\, Lie(\tilde{A}_{3,R})\quad,\quad M\mapsto [M_{3,1}\,,\, M].$$ By construction, the matrix of $\Psi$ is $\tiny\left[\begin{array}{ccccc} 0& 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 &0\end{array}\right]$.
$\tilde{A}_{3,R}$ is a reduced form for $\mathrm{(LVE^3_{\phi})}$ and $\mathfrak{g}_{3}$ is not abelian so the degenerate Hénon-Heiles Hamiltonian (\[HH\]) is not meromorphically integrable.
We know that $Lie(\tilde{A}_{3,R})$ is non abelian so we just need to prove that $\tilde{A}_{3,R}$ is a reduced form. To achieve this we will construct a Picard Vessiot extension $K_{3}$ still using our “reduction” philosophy and we prove that it has transcendence degree $5$: as $\mathfrak{g}_3\subset Lie(\tilde{A}_{3,R})$ and $\mathrm{dim}_{\mathbb{C}}(Lie(\tilde{A}_{3,R})) = 5$ this will show that $\mathfrak{g}_{3}=Lie(\tilde{A}_{3,R})$ because $\mathrm{dim}_{\mathbb{C}}(\mathfrak{g}_{3})=\mathrm{dtr}(K_3/ \mathbf{k})$ (see [@PuSi03a] Chap. 1.).
We apply proposition \[partial reduction\] to $\tilde{A}_{3,R}$. Apply the partial reduction $P_{1} = (Id + \int a_1 M_{3,1}) = Id+\ln(x) M_{3,1}$: $P_1[\tilde{A}_{3,R}]$ contains no terms in $M_{3,2}$ and $ P_{1}[\tilde{A}_{3,R}] = a_1 M_{3,1} + \left(a_1 \int a_2\right) M_{3,3} $; we call $I_2 =\int ( a_1 \int a_2) = Li_{2}(x^2)$ where $Li_2$ denotes the classical dilogarithm (see e.g [@Ca02a]). Similarly we obtain $I_3$ and $I_4$ as coefficients of successive changes of variable. We are left with a system $Y'=a_1 M Y$, the Picard-Vessiot extension is $$K_3= \mathbb{C}(x)(\ln(x)\,,\, \ln(1+x^2)\,,\, Li_{2}(x^2)\,,\, Li_{3}(x^2)\,,\, Li_{4}(x^2))$$
It is known to specialists that $\mathrm{dtr}(K_3/\mathbf{k})=5$ (and reproved for convenience below).
A self-contained proof of $\mathrm{dtr}(K_3/\mathbf{k}) =5$ {#section: appendix}
-----------------------------------------------------------
To remain self-contained we propose a differential Galois theory proof of the following classical fact (see [@Ca02a] for instance). The proof is simple and beautifully consistent with our approach. To simplify the notations, we write the proof in the case of the classical iterated dilogarithms $Li_{j}(-x)$ but, of course, it applies mutatis mutandis to our case of $Li_{j}(x^2)$.
\[appendix\] Let $K_3= \mathbb{C}(x)(\ln(x)\,,\, -\ln(1-x)\,,\, Li_{2}(-x)\,,\, Li_{3}(-x)\,,\, Li_{4}(-x))$, then $\mathrm{dtr}(K_3/\mathbf{k}) =5$
Let us prove that the functions $$x\,,\,\ln(x)\,,\, -\ln(1-x)\,,\, Li_{2}(-x)\,,\, Li_{3}(-x)\,,\, Li_{4}(-x)$$ are algebraically independent using a differential Galois theory argument. That $\ln(x)$ and $-\ln(1-x)$ are transcendent and algebraically independent over $\mathbb{C}(x)$ is a classical easy fact. We focus in proving the transcendence and algebraic independence of $Li_{2}(-x)\,,\, Li_{3}(-x)$ and $Li_{4}(-x)$. Set the following relations, $$Li_0(-x) := \frac{x}{1-x},\quad Li_{1}(-x) := -\ln(1-x),\quad Li_{2}(-x):=\int\frac{Li_{1}(-x)}{x} dx ,$$ $$Li_{3}(-x):=\int\frac{Li_{2}(-x)}{x} dx ,\quad Li_{4}(-x):=\int\frac{Li_{3}(-x)}{x} dx$$ and therefore $K_3 = \mathbb{C}(x)(\ln(x)\,,\,Li_{0}(-x),\ldots ,Li_{4}(-x))$ is a differential field (with $Li'_{i}(-x) = \frac{Li_{i-1}(-x)}{x}$). Of course, $\mathrm{dtr}(K_3/\mathbf{k})\leq 5$. Let us define $$V := \mathrm{span}_{\mathbb{C}}\left\lbrace 1\,,\,\ln(x)\,,\, \frac{\ln(x)^2}{2}\,,\, \frac{\ln(x)^3}{6}\,,\, Li_1(-x)\,,\, Li_2 (-x)\,,\, Li_3 (-x)\,,\, Li_4 (-x)\right\rbrace$$ and consider and element $\sigma \in Gal(K_3/\mathbf{k})$. As $\sigma(\ln'(x)) = \sigma(\frac{1}{x}) = \frac{1}{x}=\ln'(x) $ there exists a constant $c_0\in\mathbb{C}$ such that $\sigma(\ln(x)) = c_0$. Similarly, we obtain that $\sigma(\ln(x)^2 /2) = \ln(x)^2 / 2 + c_0\ln(x)+c^2_0$ and $\sigma(\ln(x)^3 /6) = \ln(x)^3 / 6 + c^2_0\ln(x)/2 +c_0\ln(x)^2 /2 c^3_0$. Since $Li'_{1}(-x) =\frac{x}{x^2 +1}\in\mathbf{k}$ we have that $\sigma(Li'_{1}(-x)) = Li'_{1}(-x)$ and therefore there exists $c_1\in\mathbb{C}$ such that $\sigma(Li_{1}(-x)) =Li_{1}(-x) + c_1$. As $Li'_{2}(-x) =\frac{Li_{1}(-1)}{x}$ we have that $\sigma(Li'_{2}(-x) ) =\sigma(\frac{Li_{1}(-1)}{x}) =\frac{Li_{1}(-x) }{x} + \frac{c_1}{x}$ and there exists $c_2\in\mathbb{C}$ such that $\sigma(Li_{2}(-x))=Li_{2}(-x) + c_1 \ln(x) +c_2$. We prove similarly the existence of $ c_3 , c_4 \in\mathbb{C}$ such that $$\begin{aligned}
\nonumber\sigma(Li_{3}(-x))&=&Li_{3}(-x) +c_1 \frac{\ln(x)^2}{2} + c_2 \ln(x) + c_3\\
\nonumber\sigma(Li_{4}(-x)) &=&Li_{4}(-x) + c_1\frac{\ln(x)^3}{6} + c_2 \frac{\ln(x)^2}{2} + c_3 \ln(x) + c_4.\end{aligned}$$ We see that $V$ is stable under the action of $Gal(K_3/\mathbf{k})$ and hence is the solution space of a differential operator $L\in\mathbf{k}[\frac{d}{dx}]$ of order $8$. Therefore, in this basis the matrix of the action of $\sigma$ on $V$: $$M_{\sigma}:=\tiny\left[\begin{array}{ccccccccc} 1 & c_0 & c^2_0 /2 & c^3_0/6 & c_1 & c_2 & c_ 3 & c_ 4 \\
0 & 1 &c_0 & c^2_0 & 0 & c_1 & c_2 & c_3\\
0 & 0 & 1 & c_0 & 0 & 0 & c_1 & c_2\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & c_1\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$ As $\ln(x)$ and $\ln(1-x)$ are transcendent (and algebraically independent) we know that $c_0$ and $c_1$ span $\mathbb{C}$. It follows that $\mathfrak{g}_3$ contains at least $$m_0 :={\tiny \left[ \begin{array}{cccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]} \quad \text{and}\quad
m_1 :={\tiny \left[ \begin{array}{cccccccc}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 &1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]}.$$ Since $m_0$ and $m_1$ do not commute, we know that the Lie algebra generated by the iterated Lie brackets has dimension at least $3$. Iterating the brackets of $m_0$ and $m_1$ we obtain a subalgebra of $\mathfrak{g}_3$ of dimension $5$. Therefore we have $\mathrm{dtr}(K_{3}/ \mathbf{k}) \geq 5$ and since we know that $\mathrm{dtr}(K_{3}/ \mathbf{k})\leq 5$ we obtain the equality and the result follows.
Horozov and Stoyanova [@HS07] make use of the properties of the dilogarithm in order to prove the non-integrability of some subfamilies of Painlevé VI equations: namely, they prove the non-abelianity of $\mathfrak{g}_2$, the Lie algebra of its second variational equation.
Conclusion
==========
The reduction method proposed here is systematic (and we have implemented it in Maple). Although it is currently limited to the case when $Lie(\mathrm{(VE^1_{\phi})})$ is one-dimensional, extensions to higher dimensional cases along the same guidelines are in progress and will appear in subsequent work. In work in progress with S. Simon, we will show another use of reduced forms, namely the expression of taylor expansions of first integrals along $\phi$ are then greatly simplified.\
We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm : assuming that $\mathrm{(LVE^m_{\phi_0})}$ is reduced (with an abelian Lie algebra), we believe that the output $\tilde{A}_{m+1,R}$ of our reduction procedure of sections 4 and 5 will always be a reduced form. In the context of complex Hamiltonian systems, this would mean that our method would lead to an effective version of the Morales-Ramis-Simó theorem.
[A]{}
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[^1]: The first author was supported by a Grant from the Region Limousin (France).
[^2]: A field $k$ is called a $C_1$-field (or cohomologically trivial) if any homogeneous polynomial $P\in k[X_1,\ldots,X_n]_{=d}$ of degree $d$ has a non-trivial zero in $k^n$ when $n>d$, i.e the number of variables is bigger than the degree. All differential fields of coefficients considered in this article will belong to the $C_1$ class.
|
---
abstract: 'The Travelling Salesman Problem (TSP) is a well known and challenging combinatorial optimisation problem. Its computational intractability has attracted a number of heuristic approaches to generate satisfactory, if not optimal, candidate solutions. Some methods take their inspiration from natural systems, extracting the salient features of such systems for use in classical computer algorithms. In this paper we demonstrate a simple unconventional computation method to approximate the Euclidean TSP using a shrinking blob of virtual material. The collective morphological adaptation behaviour of the material emerges from the low-level interactions of a population of particles moving within a diffusive lattice. A ‘blob’ of this material is placed over a set of data points projected into the lattice, representing TSP city locations, and the blob is reduced in size over time. As the blob shrinks it morphologically adapts to the configuration of the cities. The shrinkage process automatically stops when the blob no longer completely covers all cities. By manually tracing the perimeter of the blob a path between cities is elicited corresponding to a TSP tour. Over 6 runs on 20 randomly generated datasets of 20 cities this simple and unguided method found tours with a mean best tour length of 1.04, mean average tour length of 1.07 and mean worst tour length of 1.09 when expressed as a fraction of the minimal tour computed by an exact TSP solver. We examine the insertion mechanism by which the blob constructs a tour, note limitations of performance with problem scaling, and the relationship between the TSP and other structures which group points on the plane. The method is notable for its simplicity rather than its performance and may provide clues to the mechanism of human performance on TSP problems and raises questions about how to select the best fit between the natural behaviour of a physical system and its desired computational use.'
author:
- Jeff Jones and Andrew Adamatzky
bibliography:
- 'references.bib'
title: Computation of the Travelling Salesman Problem by a Shrinking Blob
---
[**Keywords:**]{} Travelling salesman problem, multi-agent, virtual material, unconventional computation, material computation
Introduction
============
The Travelling Salesman Problem (TSP) is a combinatorial optimisation problem well studied in computer science, operations research and mathematics. In the most famous variant of the problem a hypothetical salesman has to visit a number of cities, visiting each city only once, before ending the journey at the original starting city. The shortest path, or tour, of cities, amongst all possible tours is the solution to the problem. The problem is of particular interest since the number of candidate solutions increases greatly as $n$, the number of cities, increases. The number of possible tours can be stated as $(n-1)!/2$ which, for large numbers of $n$, renders assessment of every possible candidate tour computationally intractable. Besides being of theoretical interest efficient solutions to the TSP have practical applications such as in vehicle routing, tool path length minimisation, and efficient warehouse storage and retrieval.
The intractable nature of the TSP has led to the development of a number of heuristic approaches which can produce very short — but not guaranteed minimal — tours. A number of heuristic approaches are inspired by mechanisms seen in natural and biological systems. These methods attempt to efficiently traverse the candidate search space whilst avoiding only locally minimal solutions and include neural network approaches (most famously in [@hopfield1986computing]), evolutionary algorithms ??, simulated annealing methods ??, the elastic network approaches prompted in [@durbin1987analogue], ant colony optimisation [@dorigoa2000ant], living [@aono_neurophys] and virtual [@Jones2011reconfig] slime mould based approaches, and bumblebee foraging [@lihoreau2010travel].
Human performance on the TSP has also been studied in both naive and tutored subjects (see, for example, [@macgregor1996human] and [@macgregor2011human]). This is of particular interest for a number of reasons. Unlike many nature inspired approaches, the human computation of TSP is by an individual and not based on population methods which evaluate a number of candidate solutions. Human performance on the TSP is also, for a limited number of cities at least, comparable in performance with heuristic approaches ??, ??. Although there are a number of competing theories as to how exactly humans approximate the TSP (??, ??, ??), discovery of the methods employed may be useful as an insight into the mechanisms underlying complex perceptual and cognitive processes and as a practical aid for the development of computational algorithms.
In this paper we adopt a material-based, minimum complexity approach. We show how a spatially represented non-classical, or unconventional, computational mechanism can be used to approximate the TSP. Taking inspiration from the non-neural, material-based computational behaviour of slime mould, we employ a sheet, or ‘blob’ of virtual material which is placed over a spatial map of cities. By shrinking this blob over time, it conforms and adapts to the arrangement of cities and a tour of the TSP is formed. We give an overview of the inspiration for the method in Section \[sec:inspiration\]. The shrinking blob method is described in Section \[sec:blobdesc\]. Examples of the performance of the method compared to exact solutions generated by a TSP solver are given in Section \[sec:results\], along with an analysis of the underlying mechanism and factors affecting the performance of the approach. We conclude in Section \[sec:discussion\] by summarising the approach and its contribution in terms of simplicity. We examine similarities between the underlying mechanism of the shrinking blob method and proposed models of TSP tour perception and construction in studies of human performance on the TSP. We suggest further research aimed at improving the method and possible related applications for the approach.
Slime Mould Inspired Computation of the TSP {#sec:inspiration}
===========================================
The giant single-celled amoeboid organism, true slime mould *Physarum polycephalum*, has recently been of interest as a candidate organism for the study of non-neural distributed computation. In the vegetative plasmodium stage of its complex life cycle the organism forages towards, engulfs and consumes micro-organisms growing on vegetative matter. When presented with a spatial configuration of nutrient sources the plasmodium forms a network of protoplasmic tubes connecting the nutrients. This is achieved without recourse to any specialised neural tissue. The organism dynamically adapts its morphology to form efficient paths (in terms of a trade-off between overall distance and resilience to random damage) between the food sources [@NakagakiT04MultFoodSrc],[@nakagaki2007intelligent],[@nakagaki2007effects].
Research into the computational abilities of *Physarum* was prompted by Nakagaki, Yamada and Toth, who reported the ability of the *Physarum* plasmodium to solve a simple maze problem [@NakagakiT00MazeSolve]. It has since been demonstrated that the plasmodium successfully approximates spatial representations of various graph problems [@NakagakiT04MultFoodSrc; @shirakawa2009planedivision; @adamatzky_toussaint; @jones2010influences], combinatorial optimisation problems [@AonoM07PhysarumNeuroComp; @aono2008spontaneous], construction of logic gates and adding circuits [@TsudaS04PhysarumComp; @jones2010towards; @adamatzky_physarumgate], and spatially represented logical machines [@adamatzky2007kum; @adamatzky_jones_NC].
Although *Physarum* has been previously been used in the approximation of TSP [@aono_neurophys], this was achieved by an indirect encoding of the problem representation to enable it to be presented to a confined plasmodium in a controlled environment. In the work by Aono et. al. it was shown that the morphology of the plasmodium confined in a stellate chamber could be dynamically controlled by light irradiation of its boundary. When coupled to a feedback mechanism using an analysis method (to assess the presence of plasmodium at the extremities of the chamber), combined with Hopfield-Tank type neural network rules [@hopfield1986computing], the plasmodium was used to generate candidate solutions to simple instances of the TSP [@AonoM07PhysarumNeuroComp; @aono2008spontaneous].In its natural propagative state, however, *Physarum* does not approximate area representations of a set of points, including the Convex Hull, Concave Hull [@adamatzky2011planarshape] and the TSP. This is because the material comprising the plasmodium spontaneously forms networks spanning the nutrient sources. Even when the plasmodium is arranged initially as a solid sheet of material, the sheet is soon transformed into a network structure by competitive flux of material within the sheet [@NakagakiT04MultFoodSrc]. It is physically impractical to force a freely foraging plasmodium to conform to a TSP network structure during its nutrient foraging, as shown in Fig. \[fig:physarum\_attempt\].
Nevertheless, the material computation of *Physarum* presents interesting possibilities towards generating novel spatially represented methods of computation. An *in-silico* attempt at reproducing the pattern formation and adaptation behaviour of *Physarum* was introduced in [@jones2010emergence] and its pattern formation abilities characterised in [@jones_alife_2010]. Attempts at encouraging these multi-agent transport networks to conform to TSP-like requirements (degree of connectivity 2, no crossed paths) by dynamically adjusting the concentration of simulated nutrient attractants, using a feedback mechanism based on the current configuration of the network, were presented in [@Jones2011reconfig]. This resulted in extremely complex transitions of network dynamics and partial success in constructing TSP tours. In the approach outlined in this paper we attempt a simpler approach which utilises a larger aggregate mass of the same multi-agent collective which behaves as a morphologically adaptive cohesive ‘blob’ of virtual material.
A Material Approach to TSP by a Shrinking Blob {#sec:blobdesc}
==============================================
In the shrinking blob method we use a piece, or ‘blob’ of a virtual plasmodium material to approximate the TSP. The material is composed of thousands of simple mobile multi-agent particles interacting together in a 2D diffusive lattice. Each particle senses the concentration of a generic ‘chemoattractant’ diffusing within the lattice and each agent also deposits the same substance within the lattice upon successful forward movement. The multi-agent population collectively exhibits emergent properties of cohesion and shape minimisation as a results of the low-level particle interactions. The pattern formation and network adaptation properties of small populations of the material were discussed in [@jones_alife_2010] and were found to reproduce a wide range of Turing-type reaction-diffusion patterning. In this paper we use a relatively large population of particles which collectively behaves as a sheet of deformable virtual material. A full description of the virtual material method is given in the Appendix and an overview of the method follows.
Shrinkage Process
-----------------
We initialise a sheet of the virtual material around a set of data points corresponding to TSP city nodes (Fig. \[fig:shrink\_method\]a). Chemoattractant is projected into the diffusive lattice at node locations, however, projection is reduced at regions which are covered by the blob sheet. The initial shape of the sheet corresponds to the Convex Hull of the data points. We then shrink the material by systematically removing some of its constituent particle components. The city nodes act as attractants to the material, effectively ‘snagging’ the material at the locations of uncovered nodes and affecting its subsequent morphological adaptation. As the material continues to shrink its innate minimising properties conform to the locations of the city nodes and the area occupied by the material is reduced, becoming a concave area covering the nodes (Fig. \[fig:shrink\_method\]b-e). The shrinkage is stopped when all of the nodes are partially uncovered by the sheet (Fig. \[fig:shrink\_method\]f). The reader is encouraged to view the supplementary video recordings of the shrinkage process at <http://uncomp.uwe.ac.uk/jeff/material_tsp.htm>. The adaptation of the blob to the data stimuli is not entirely smooth, the video recordings show that the blob sheet adapts to the changing stimuli as data nodes are temporarily uncovered and re-covered by the blob. When the shrinkage is halted the area of the sheet corresponds to the area enclosed by a tour of the Euclidean Travelling Salesman Problem. The exact tour formed by the blob can be elucidated by tracking along the perimeter of the blob, adding a city to the tour list when it is first encountered. The tour is complete when the start city is re-encountered. The approach is simple, making use of the innate adaptive emergent properties of the material. Despite being completely unguided and containing no heuristic optimisation strategies the approach yields efficient tours. The separate stages of the approach will now be described in detail.
Halting the Computation
-----------------------
It is important to halt the shrinkage of the blob at the right time. If the shrinking is stopped too early an incomplete tour will be formed (i.e. only a partial subset of the nodes will be included in the tour if not all of the nodes are uncovered). Unlike guided heuristic methods a set of candidate tours is not initially formed and subsequently modified. Only a single tour is formed and the shrinking blob approach is akin to the ‘instance machines’ (as opposed to universal machines) proposed by Zauner and Conrad [@zauner1996parallel]. To automatically halt the computation we use a so-called ‘traffic light’ system. At the start of the method the sheet covers the entire set of nodes. Only the outer nodes are partially covered by the blob. To measure whether a node is covered by the sheet we assess the number of particles in a $5x5$ window around each node. If the number of particles is $<15$ then the node is classified as uncovered and the node indicator is set to green. Otherwise the node is classified as covered and the node indicator is set to red. At each scheduler step the indicators of all nodes are checked. When all nodes are set to green, all nodes underneath the blob are partially uncovered and the shrinkage is stopped.
Reading the Result of the Computation
-------------------------------------
To trace the path of cities in the tour discovered by the blob a manual process is used. The collection of partially uncovered nodes and blob shape may be interpreted as an island shape with the nodes representing cities on the coastline of the island (Fig. \[fig:trace\_method\]). We begin by selecting the city at the top of the arena. If more than one city is at this $y$ location the left-most city at this $y$ location is selected. This city is the start city of the tour and is added to the tour list $\textbf{T}$. Moving in a clockwise direction we trace the perimeter of the blob (walking around the shore of the island …). Each time we encounter a city, it is added to $\textbf{T}$. If a city is subsequently re-encountered (as in the case of narrow peninsula structures as described below) it is ignored. When the path reaches the starting city the tour is complete and the list in $\textbf{T}$ represents the tour of the TSP found by the shrinking blob.
![Elucidation of TSP tour by tracking perimeter of blob area. (a) Tracking is initialised at the top most node. Perimeter of blob is traced in a clockwise direction. Each time a node is encountered for the first time it is added to the tour. The tour is completed when the start node is re-encountered.[]{data-label="fig:trace_method"}](figs/trace_set_16.png){width="60.00000%"}
Some special cases in the tracking process must be noted in the case where a city lies on a narrow ‘peninsula’ of the blob as indicated in Fig. \[fig:peninsula\_cases\]. In Fig. \[fig:peninsula\_cases\]a the city nearest position $x$ in the path lies close to one side of a narrow peninsula. However the side at which the city is located can be deduced by a small convex bulge on the left side of the blob. In this case the city is not added until it is encountered on the left side of the peninsula. In the case of Fig. \[fig:peninsula\_cases\]b, however, the city at $x$ is located exactly in the middle of a peninsula and its closest side cannot be discerned. In this instance two interpretations are possible and the subsequent differences in possible tour paths are indicated by the dotted lines in Fig. \[fig:peninsula\_cases\]b, i) and ii). In interpretation i) the city is added to $\textbf{T}$ immediately and in ii) it is not added until it is encountered on its opposite side. If this situation occurs during the tracking process we add the city to $\textbf{T}$ when it is first encountered.
Results {#sec:results}
=======
We assessed the shrinking blob method by generating 20 datasets, each consisting of 20 randomly generated nodes within a circular arena in a $200x200$ lattice. To aid the manual tracking process we added the condition that points must have a separation distance of at least 25 pixels. For each run a population of particles was generated and initialised within the confines of the convex hull (algorithmically generated) of the point set. Any particles migrating out of the convex hull area were removed. As the shrinkage process started the cohesion of the blob emerged and, as shrinkage progressed, the blob adapted to the shape of the city nodes. Six experimental runs were performed on each dataset and the resulting blob shape was recorded and tracked by the manual tracking process to reveal the tour. The best, worst and mean performance over the 6 runs for each 20 datasets was recorded and these results were aggregated over the 20 datasets and shown in Fig. \[fig:results\]. Results of the shrinking blob method are expressed as a fraction of the shortest exact tour found by the Concorde TSP solver [@applegate2006concorde].
![Results of shrinking blob method over 6 runs on each of 20 randomly generated datasets of 20 points compared to exact results from the Concorde TSP solver.[]{data-label="fig:results"}](figs/table_results.png){width="95.00000%"}
Construction of Tour by Concave Insertion Process
-------------------------------------------------
Although the final tour list is read off by tracking the perimeter of the shrunken blob, the construction of the tour actually occurs by an insertion process as the blob shrinks. The blob is initially patterned with the shape of the convex hull. This is only a partial tour, since only the peripheral nodes which are part of the Convex Hull are included. By recording the stages by which nodes are uncovered and added during the shrinkage process, the method of construction can be elucidated. Fig. \[fig:conv\_conc\_tsp\] shows the visual deformation of the Convex Hull structure as the blob shrinks and new city nodes are added to the list. Note that the blob shrinks simultaneously from all directions and the order of insertion is related to both the proximity of the point from the periphery of the blob and the distance between two outer stimuli at the current periphery of the blob where a concavity forms (discussed further in Section \[concavity\]). The actual order of insertion of cities to this example is given in Fig. \[fig:tour\_construct\].
$$\begin{array}{cccccccccccccccccccc}
M & & & T & & D & K & & & G & & & S & N & Q & & & & & \\
M & & & T & & D & K & & & G & & & S & N & Q & & & & & \textbf{R} \\
M & \textbf{I} & & T & & D & K & & & G & & & S & N & Q & & & & & R \\
M & I & & T & & D & K & & & G & \textbf{J} & & S & N & Q & & & & & R \\
M & I & \textbf{H} & T & & D & K & & & G & J & & S & N & Q & & & & & R \\
M & I & H & T & & D & K & & \textbf{C} & G & J & & S & N & Q & & & & & R \\
M & I & H & T & & D & K & & C & G & J & \textbf{B} & S & N & Q & & & & & R \\
M & I & H & T & \textbf{E} & D & K & & C & G & J & B & S & N & Q & & & & & R \\
M & I & H & T & E & D & K & & C & G & J & B & S & N & Q & \textbf{F} & & & & R \\
M & I & H & T & E & D & K & & C & G & J & B & S & N & Q & F & & & \textbf{L} & R \\
M & I & H & T & E & D & K & \textbf{P} & C & G & J & B & S & N & Q & F & & & L & R \\
M & I & H & T & E & D & K & P & C & G & J & B & S & N & Q & F & & \textbf{A} & L & R \\
M & I & H & T & E & D & K & P & C & G & J & B & S & N & Q & F & \textbf{O} & A & L & R \\
\end{array}$$
As the blob shrinks, concavities form in the periphery of the blob which move inwards to the centre of the blob shape. The concave deformation is a transformation of the Convex Hull ($\mathbf{CH}$) into a Concave Hull ($\mathbf{OH}$). The Concave Hull, the area occupied by — or the ‘shape’ of — a set of points is not as simple to define as its convex hull. It is commonly used in Geographical Information Systems (GIS) as the minimum region (or footprint [@galton2006region]) occupied by a set of points, which cannot, in some cases, be represented correctly by the convex hull [@duckham2008efficient]. The Concave Hull is related to the structures known as $\alpha$-shapes [@edelsbrunner1983shape]. The $\alpha$-shape of a set of points, $P$, is an intersection of the complement of all closed discs of radius $1/\alpha$ that includes no points of $P$. An $\alpha$-shape is a convex hull when $\alpha \rightarrow \infty$. When decreasing $\alpha$, the shapes may shrink, develop holes and become disconnected, collapsing to $P$ when $\alpha \rightarrow 0$. A concave hull is non-convex polygon representing area occupied by $P$ and the concave hull is a connected $\alpha$-shape without holes. In contrast to $\alpha$-shapes, the blob (more specifically, the set of points which it covers) does not become disconnected as it shrinks. As the blob adapts its morphology from Convex Hull to TSP is demonstrates increased concavity with decreased area. Although the shrinkage process is automatically stopped when a TSP tour is formed, the process could indeed continue past the TSP. If shrinkage continues then the blob (now adopting a network shape) will approximate the Steiner minimum tree (SMT), the minimum path between all nodes. As demonstrated in [@jones2010influences] the additional Steiner nodes in the SMT may be removed by increasing the attractant projection from the data nodes. The material adapts to the increased attractant concentration by removing the Steiner nodes to approximate the Minimum Spanning Tree (MST).
Blob TSP Tour as a Waypoint in the Transition From Convex Hull to Spanning Tree
-------------------------------------------------------------------------------
The insertion process of adding nodes to the Convex Hull reveals an orderly transition to the TSP which continues after further shrinkage, leading to the following finding.
The evolution of the blob shape by morphological adaptation is a transition from $\mathbf{CH}$ to $\mathbf{OH}$ to $\mathbf{TSP}$ to $\mathbf{MST}$ to $\mathbf{SMT}$.
We do not explicitly include $\alpha$-shapes in this transition since $\alpha$-shapes can include holes and disconnected structures, which do not form in a defect-free shrinking blob. This transition is based on increasing concavity and decreasing area, and encompasses the a blob TSP tour $\mathbf{bTSP}$ as part of the hierarchy. Note that the blob tour $\mathbf{bTSP}$ is only one instance of the set of possible TSP tours $\mathbf{TSP}$ and is not guaranteed to be the minimal tour. The blob TSP tour is only a transient structure — a waypoint — in the natural shrinkage process (we halt the computation at this point merely because we are interested for the purposes of this report).
It is known from Toussaint that there is a hierarchy of proximity graphs (graphs where edges between points are linked depending on measures of neighbourhood and closeness) [@toussaint_1980]. Each member of the hierarchy adds edges and subsumes the edges of lower stages in the hierarchy, and some common graphs (see Fig. \[fig:compare\_hierarchy\]a-e) include the Delaunay triangulation $\mathbf{DTN}$ to Gabriel Graph $\mathbf{GG}$ to Relative Neighbourhood Graph $\mathbf{RNG}$ to Minimum Spanning Tree $\mathbf{MST}$. Also shown is the shortest possible tree between all nodes formed by adding extra Steiner nodes (Fig. \[fig:compare\_hierarchy\]f). It was found in [@adamatzky_toussaint] that *Physarum* approximates the Toussaint hierarchy of proximity graphs as it constructs transport networks during its foraging and it was demonstrated in [@jones2010influences] that multi-agent transport networks mimicking the behaviour of *Physarum* also minimise these proximity graphs by following this hierarchy in its downwards direction. From a biological perspective traversing the Toussaint hierarchy suggests a mechanism by which *Physarum* can exploit the trade-off between foraging efficiency (many network links) and transport efficiency (fewer but fault tolerant transport links). This mechanism, may also be present in terms of maximising foraging area searched (exploration) and minimising area for efficient transport (exploitation), as suggested in [@gunji2011adaptive]. We suggest that the hierarchy we observed in the shrinking blob from $\mathbf{CH}$ to $\mathbf{OH}$ to $\mathbf{TSP}$ to $\mathbf{MST}$ to $\mathbf{SMT}$ may encompass such an area-based exploration-exploitation mechanism (Fig. \[fig:compare\_hierarchy\]g-l). It is notable that there is some overlap between the Toussaint hierarchy and the shrinking blob hierarchy where deepening concavities in the blob hierarchy appear to correspond to the deletion of outer edges in the Toussaint hierarchy, suggesting that there may be some formal relationship between the two. This possible relationship may suggest further studies.
Variations in Performance of the Shrinking Blob Method {#concavity}
------------------------------------------------------
The results of the shrinking blob method show variations in performance from very good approximations of close-to-minimum tours (Fig. \[fig:results\_good\]) to less favourable fractions of the minimal tour (Fig. \[fig:results\_not\_good\]). What is the reason for the disparity in performance on these datasets? If we examine the tour paths we can glean some clues as to the difference in performance. In the ‘good’ results examples the major concave regions of the tour formed by the blob closely match the concavities in the exact computed TSP tour (e.g. Fig. \[fig:results\_good\]a and b). However in the ‘poor’ approximation results we can see that the major concave regions of the blob tour do not match the major concavities in the respective exact computed tours (e.g. Fig. \[fig:results\_not\_good\]a and b). Given that these concave regions are formed from the deformation of the initial Convex Hull we can see that the concavities in the blob tour appear to be formed, and ‘deepened’ where there are larger distances between the cities on the initial Convex Hull.
To explore the role of distance on concavity formation we patterned a blob into a square shape by placing regularly placed stimuli around the border of a square (Fig. \[fig:gap\_distance\]a, stimuli positions, 20 pixels apart, indicated by crosses). When shrinkage of the blob was initiated there is no difference between the stimuli distances. All regions between stimuli initially show small concavities (the ‘perforations’ in Fig. \[fig:gap\_distance\]b) until one gradually predominates and extends inwards. Also of note is the fact that when one concave region predominates, the other concavities shrink (Fig. \[fig:gap\_distance\]c-e). The position of the initial dominating concavity is different in each run (presumably due to stochastic influences on the collective material properties of the blob) and this may explain the small differences in performance on separate runs using the same datasets.
When there is a larger gap between stimuli points the predominating concavity forms more quickly and is larger. The the concave region also deepens more quickly. This is shown in Fig. \[fig:gap\_distance\]f-j which has a gap of 30 pixels between two stimulus points on the right side of the square and in Fig. \[fig:gap\_distance\]k-o which has a gap of 60 pixels between stimulus points. When there are two large distances between stimuli there is competition between the competing concavities and the larger region predominates. This is demonstrated in Fig. \[fig:gap\_distance\]p-t which has a distance gap of 40 pixels on the left side of the square and 60 pixels on the right side. Although two concave regions are formed, the larger deepens whilst the smaller concave region actually shrinks as the blob adapts its shape.
The synthetic examples illustrate the influence of city distance on concavity formation and evolution and these effects are more complex when irregular arrangements of city nodes are used. This is because arrangements of cities present stimuli to the blob sheet when partially uncovered, acting to anchor the blob at these regions, and the morphological adaptation of the blob is thus affected by the spatial configuration of city nodes. In the examples of relatively poor approximation of the minimum tour (Fig. \[fig:results\_not\_good\]) the initial incorrect selection of concavities are subsequently deepened by the shrinking process, resulting in tours which differ significantly in both their visual shape and in the city order. In the examples of good comparative performance with the exact solver the blob tours differ only in a small number of nodes.
Although outright performance is not the focus of this report, we tested the blob method on a randomly generated dataset of 50 nodes in a preliminary assessment of scalability. We found that over 6 runs the blob method found tours with a best of 1.064, worst of 1.099 and mean of 1.076 compared as a fraction of tour length to the exact minimum computed by the TSP solver, Fig. \[fig:50\_nodes\]. However, these results may also be subject to the variability in performance seen in the 20 node examples.
Discussion {#sec:discussion}
==========
We have presented a simple material approach to computation of the Travelling Salesman Problem using a shrinking blob. To the authors’ best knowledge the shrinking blob method is the first spatially represented non-classical computational method of approximating the TSP. We should again emphasise that the method is notable for its simplicity rather than its performance. Indeed the performance, when compared to exact TSP solvers or leading heuristic methods, compares rather unfavourably in terms of absolute tour distance or scaling ability. The method does, however, contain a number of properties that are intriguing.... MORE ON THIS
Acknowledgments
===============
This authors AA and JJ were supported by the EU research project “Physarum Chip: Growing Computers from Slime Mould” (FP7 ICT Ref 316366) and JJ was supported by the SPUR grant award for the project “Developing non-neural models of material computation in cellular tissues” from UWE.
Appendix: Shrinking Blob Particle Model Description {#appendix}
===================================================
The multi-agent particle approach to generate the behaviour of the virtual material blob uses a population of coupled mobile particles with very simple behaviours, residing within a 2D diffusive lattice. Lattice size was $200 \times 200$ pixels. The lattice stores particle positions and the concentration of a local diffusive factor referred to generically as chemoattractant. Collective particle positions represent the global pattern of the blob. The particles act independently and iteration of the particle population is performed randomly to avoid any artifacts from sequential ordering.
Generation of Emergent Blob Cohesion and Shape Adaptation
---------------------------------------------------------
The behaviour of the particles occurs in two distinct stages, the sensory stage and the motor stage. In the sensory stage, the particles sample their local environment using three forward biased sensors whose angle from the forwards position (the sensor angle parameter, SA), and distance (sensor offset, SO) may be parametrically adjusted (Fig. \[fig\_particle\_layout\]a). The offset sensors generate local coupling of sensory inputs and movement to generate the cohesion of the blob. The SO distance is measured in pixels and a minimum distance of 3 pixels is required for strong local coupling to occur. During the sensory stage each particle changes its orientation to rotate (via the parameter rotation angle, RA) towards the strongest local source of chemoattractant (Fig. \[fig\_particle\_layout\]b). After the sensory stage, each particle executes the motor stage and attempts to move forwards in its current orientation (an angle from 0–360 degrees) by a single pixel forwards. Each lattice site may only store a single particle and particles deposit chemoattractant into the lattice only in the event of a successful forwards movement. If the next chosen site is already occupied by another particle move is abandoned and the particle selects a new randomly chosen direction.
TSP Problem Representation
--------------------------
Twenty datasets were generated, each consisting of 20 randomly chosen data points within a circular arena. A condition was added that a minimum distance of 25 pixels must exist between data points. This gives a more uniform distribution of node points, preventing clustering of node points often found in real-world TSP instances, for example those based on real city locations. This was done partly to aid visual tracking of the tour path but also because a less clustered node distribution is known to provide more challenging problem instances (e.g. in human performance on TSP [@hirtle1992heuristic]) since there is less likelihood of providing pre-existing cues to intuitive solutions (for example nearest neighbour grouping). Each dataset was saved to a text file. TSP city data points for each dataset were loaded from the text files and were represented by projection of chemoattractant to the diffusion lattice at locations corresponding to a $3 \times 3$ window centred about their $x,y$ position. The projection concentration was 1.275 units per scheduler step. If a node was covered by a portion of the blob (i.e. if the number of agents in a $3 \times 3$ window surrounding the node was $>0$) the projection was reduced to 0.01275 units. Suppression of projection from covered sites was necessary to ensure a uniform concentration within the blob at internal data points. Uncovering of the data points by the shrinking blob acted to increase concentration at exposed nodes, causing the blob to be anchored by the nodes. Diffusion in the lattice was implemented at each scheduler step and at every site in the lattice via a simple mean filter of kernel size $3 \times 3$. Damping of the diffusion distance, which limits the distance of chemoattractant gradient diffusion, was achieved by multiplying the mean kernel value by $0.95$ per scheduler step.
The blob was initialised by creating a population of particles and inoculating the population within the bounds of a Convex Hull formed by the data points. The exact population size differed depending on the area of the Convex Hull but was typically between 10000 and 15000 particles. The convex hull was computed at the start of the experiment by a conventional algorithm. Particles were given random initial positions within these confines and random initial orientations. Any particles migrating out of the bounds of the Convex Hull region were deleted. Particle sensor offset (SO) was 7 pixels. Angle of rotation (RA) and sensor angle (SA) were both set to 60 degrees in all experiments. Agent forward displacement was 1 pixel per scheduler step and particles moving forwards successfully deposited 5 units of chemoattractant into the diffusion lattice. Both data projection stimuli and agent particle trails were represented by the same chemoattractant ensuring that the particles were attracted to both data stimuli and other agents’ trails. The collective behaviour of the particle population was cohesion and morphological adaptation to the configuration of stimuli.
![Architecture of a single component of the shrinking blob and its sensory algorithm. (a) Morphology showing agent position ‘C’ and offset sensor positions (FL, F, FR), (b) Algorithm for particle sensory stage.[]{data-label="fig_particle_layout"}](figs/appendix/morphology_algorithm.png){width="80.00000%"}
Shrinkage Mechanism
-------------------
Adaptation of the blob size is implemented via tests at regular intervals as follows. If there are 1 to 10 particles in a $9 \times 9$ neighbourhood of a particle, and the particle has moved forwards successfully, the particle attempts to divide into two if there is a space available at a randomly selected empty location in the immediate $3 \times 3$ neighbourhood surrounding the particle. If there are 0 to 80 particles in a $9 \times 9$ neigbourhood of a particle the particle survives, otherwise it is deleted. Deletion of a particle leaves a vacant space at this location which is filled by nearby particles, causing the blob to shrink slightly. As the process continues the blob shrinks and adapts to the stimuli provided by the configuration of city data points. The frequency at which the growth and shrinkage of the population is executed determines a turnover rate for the particles. The frequency of testing for particle division was every 5 scheduler steps and the frequency for testing for particle removal was every 10 scheduler steps. Since the shrinking blob method is only concerned with the reduction in size of the population it might be asked as to why there are tests for particle division at all. The particle division mechanism is present to ensure that the adaptation of the blob sheet is uniform across the sheet to prevent ‘tears’ or holes forming within the blob sheet, particularly at the start of an experiment when flux within the blob is initially established.
Halting Mechanism
-----------------
The shrinkage of the blob was halted when all data points were partially uncovered by using the following calculation. At each data point the number of particles surrounding the point in a $5 \times 5$ window was sampled. If the number of particles in this window was $< 15$ the node was classified as partially uncovered. When all nodes are uncovered the model is halted and a greyscale representation of the diffusive lattice is saved to disk to read the result of the blob tour. The calculated tour path was saved in a text file and tour distance was calculated. The exact minimum tour path for each dataset was calculated for comparison by loading the data point configuration files into the Concorde TSP solver [@applegate2006concorde].
|
---
author:
- |
Robert Brooks[^1]\
Department of Mathematics\
Technion– Israel Institute of Technology\
Haifa, Israel
- |
Orit Davidovich\
Department of Mathematics\
Technion– Israel Institute of Technology\
Haifa, Israel
date: ' [January, 2002]{}'
title: Isoscattering on Surfaces
---
\#1[[**[\[\#1\]]{}**]{}]{}
\[section\] \[section\] \[section\] \[section\]
\[section\]
\[section\] \[section\]
\#1
In this paper, we give a number of examples of pairs of non-compact surfaces $S_1$ and $S_2$ which are isoscattering, to be defined below. Our basic construction is based on a version of Sunada’s Theorem [@Su], which has been refined using the technique of transplantation ([@Be], [@Zel]) so as to be applicable to isoscattering. See [@BGP] and [@BP] for this approach, which is reviewed below.
Our aim here is to present a number of examples which are exceptionally simple in one or more senses. Thus, the present paper can be seen as an extension of [@BP], where the aim was to construct isoscattering surfaces with precisely one end. We will show:
\[genus\]
[(a)]{} There exist surfaces $S_1$ and $S_2$ of genus 0 with eight ends which are isoscattering.
[(b)]{} There exist surfaces $S_1$ and $S_2$ of constant curvature $-1$ which are of genus 0 and have fifteen ends.
[(c)]{} There exist surfaces $S_1$ and $S_2$ of genus $1$ with five ends, or genus $2$ with three ends, which are isoscattering.
[(d)]{} There exist surfaces $S_1$ and $S_2$ of constant curvature $-1$ which are of genus 1 with thirteen ends, or genus 2 with five ends, or genus 3 with three ends, which are isoscattering.
[(e)]{} There exist surfaces $S_1$ and $S_2$ of genus 3 with one end, or constant curvature of genus 4 with one end, which are isoscattering.
Part (e) is just a statement of the results of [@BP], and is recorded here for the sake of completeness. It will not be discussed further in this paper.
The nature of the ends in Theorem \[genus\] is not too important. In the cases where the curvature is variable, they can be taken to be hyperbolic funnels or Euclidean cones, or to be hyperbolic finite-area cusps. In the constant curvature $-1$ cases, they can be taken either to be infinite-area funnels or finite-area cusps.
Recall that a surface $S$ is called a [*congruence surface*]{} if $S=
{{{\Bbb{H}}}}^2/\Gamma$, where $\Gamma$ is contained in $PSL(2, {{\Bbb{Z}}})$ and contains a subgroup $$\Gamma_k = \left\{ {\left( \begin{array}{cc}}a&b \\ c & d { \end{array}\right)}\equiv \pm {\left( \begin{array}{cc}}1 &0\\ 0&1
{ \end{array}\right)}({{\hbox{mod}}\ }k) \right\}$$ for some $k$. In other words, the group $\Gamma$ is the inverse image of a subgroup of $PSL(2, {{\Bbb{Z}}}/k)$ under the natural map $$PSL(2, {{\Bbb{Z}}}) \to PSL(2, {{\Bbb{Z}}}/k).$$
We then have:
\[cong\] There exist two congruence surfaces $S_1$ and $S_2$ which are isoscattering.
Theorem \[cong\] answers a question which was raised to us by Victor Guillemin. The point here is that congruence surfaces have a particularly rich structure of eigenvalues embedded in the continuous spectrum. On the other hand, subgroups of $PSL(2, {{\Bbb{Z}}}/k)$ have a very rigid structure [@Di], and it is not [*a priori*]{} clear that the finite group theory is rich enough to support the Sunada method.
A version of Theorem \[genus\] was announced without proof in an appendix to [@BJP].
Theorems \[s4\] and \[D\] are certainly well-known to finite-group theorists. We hope that the explicit treatment given here will be useful to spectral geometers.
The first author would like to thank MIT for its warm hospitality and the Technion for its sabbatical support for the period in which this paper was written. He would also like to thank Victor Guillemin for his interest, and Peter Perry for his suggestion to pursue these questions in the context of the paper [@BJP].
Transplantation and Isoscattering
=================================
We recall the approach to the Sunada Theorem given in [@BGP]. Recall that a Sunada triple $(G, H_1, H_2)$ consists of a finite group $G$ and two subgroups $H_1$ and $H_2$ of $G$ satisfying
$$\label{dagger}
{\hbox{for all}}\ g \in G, \#([g] \cap H_1 ) = \#([g] \cap H_2),$$
where $[g]$ denotes the conjugacy class of $g$ in $G$.
\[sun\] Suppose that $M$ is a manifold and $\phi: \pi_1(M) \to G$ a surjective homomorphism.
Let $M^{H_1}$ and $M^{H_2}$ be the coverings of $M$ with fundamental groups $\phi^{-1}(H_1)$ and $\phi^{-1}(H_2)$ respectively.
Then there is a linear isomorphism $${{\cal{T}}}: C^{\infty}(M^{H_1}) \to
C^{\infty}(M^{H_2})$$ such that ${{\cal{T}}}$ and ${{\cal{T}}}^{-1}$ commute with the Laplacian.
We first remark that condition (\[dagger\]) is equivalent to the following: if we denote by $L^2(G/H_i)$ the $G$-module of functions on the cosets $G/H_i$, then
$$L^2(G/H_1)\ {\hbox{is isomorphic to}}\ L^2(G/H_2)\ {\hbox{as}}\
G-{\hbox{modules}}.$$
We may further rewrite this by noting that $L^2(G)$ has two $G$-actions, on the left and on the right, so that we may write $$L^2(G/H_i) \equiv (L^2(G))^{H_i},$$ where the equivariance under $H_i$ is taken with respect to the left $G$-action, and $G$-equivariance is taken with respect to the right $G$-action. Equation (\[dagger\]) is then equivalent to
$$\label{dags}
(L^2(G))^{H_1}\ {\hbox{is $G$-isomorphic to}}\ (L^2(G))^{H_2}.$$
We may further rewrite this equation as saying that there is a $G$-equivariant map $T: L^2(G) \to L^2(G)$ which induces an isomorphism $(L^2(G))^{H_1} \to (L^2(G))^{H_2}$.
Now any $G$-map is determined by its value on the delta function, which in turn can be described by a function $$c: G \to R,$$ so that the $G$-module map is given by $$T(f)(x) = \sum_{g \in G} c(g) f(g\cdot x).$$
The requirement that the image of this map lies in $(L^2(G))^{H_2}$ can be expressed in terms of $c$ by the condition that $$\label{star}
c(gh)= c(g)\ \quad {\hbox{for}}\ h \in H_2.$$ We may therefore express the condition (\[dagger\]) as the existence of a function $c$ on $G$ which satisfies (\[star\]), and which furthermore induces an isomorphism as in (\[dags\]).
Given such a function $c$, we may then write out the function ${{\cal{T}}}$ as follows: let $M^{id}$ be the covering of $M$ whose fundamental group is $\phi^{-1}(id)$. Then we may identify $C^{\infty}(M^{H_i})$ with $(C^{\infty}(M^{id}))^{H_i}$. The desired expression for ${{\cal{T}}}$ is then given by
$${{\cal{T}}}(f)(x) = \sum_g c(g) f(g\cdot x).$$
We emphasize that all of this makes perfectly good sense for any function $c$ satisfying (\[star\]). The condition that it induces an isomorphism is the crucial property we need.
Clearly, ${{\cal{T}}}$ and its inverse take smooth functions to smooth functions, and also commute with the Laplacian, since both statements are true of the action by $g$ and taking linear combinations.
This establishes the theorem.
We now consider the case when the manifold $M$ is complete and non-compact. We will discuss here the case where $M$ is hyperbolic outside of a compact set, the case of Euclidean ends having been discussed in [@BP].
We begin with a complete surface $M_0$, and consider a conformal compactification of $M_0$, consisting of one circle for each funnel and a point for each cusp. We also pick a [ *defining function*]{} $\rho$ on $M_0$, that is, a function which is positive on $M$ and vanishes to first order on the boundary of $M_0$.
If $0 < \lambda <1/4$, then we choose real $s$ so that $$\lambda = (s)(1-s).$$ Then, if $f \in C^{\infty}({\partial}M_0)$, there exist unique functions $u$ on $M_0$ and ${{\cal{S}}}_s(f) \in C^{\infty}({\partial}M_0) $ such that
[(i)]{} $ {\Delta}(u) = \lambda u.$
[(ii)]{} $u \sim (\rho)^s {{\cal{S}}}_s(f) + (\rho)^{1-s}f + {{\cal{O}}}(\rho)$ as $\rho \to 0$.
The operator ${{\cal{S}}}_s$ is the [*scattering operator*]{} for $s$, and continues for all $s$ to be a meromorphic operator. Two surfaces $M_0$ and $M_1$ will be [*isoscattering*]{} if they have poles of the same multiplicity at the same values of $s$.
We now pick $M$ as in Theorem \[sun\], and lift the defining function $\rho$ on $M$ to defining functions on $M^{H_1}$, $M^{H_2}$, and $M^{id}$. Note that we may identify ${{\cal{S}}}_s$ on $M^{H_i}$ with the operator ${{\cal{S}}}_s$ on the $H_1$-invariant part of $M^{id}$. We then have:
The surfaces $M^{H_1}$ and $M^{H_2}$ are isoscattering.
[[**[Proof]{}**]{}:]{}If we are given $f \in C^{\infty}({\partial}M^{id})$, then clearly $${{\cal{T}}}(u) \sim (\rho)^s {{\cal{T}}}(S_f(f)) + (\rho)^{1-s} {{\cal{T}}}(f) +
{{\cal{O}}}(\rho),$$ or, in other words,
$${{\cal{T}}}({{\cal{S}}}_s(f)) = {{\cal{S}}}_s({{\cal{T}}}(f)).$$
Thus, ${{\cal{T}}}$ intertwines ${{\cal{S}}}_s$ for all $s$, and hence ${{\cal{S}}}_s$ on $M^{H_1}$ and $M^{H_2}$ have poles (with multiplicities) at the same values.
This completes the proof.
The Group $PSL(3, {{\Bbb{Z}}}/2)$
=================================
It is a rather remarkable fact that most of the examples of isospectral surfaces ([@BT]) as well as all of the examples of Theorem \[genus\], can be constructed from one Sunada triple. This is the triple $(G, H_1, H_2)$ , where $$G= PSL(3, {{\Bbb{Z}}}/2),$$ and $$H_1 = {\left( \begin{array}{ccc}}{}* & {}* & {}* \\ 0 & {}* & {}* \\ 0 & {}* & {}* {{ \end{array}\right)}}\quad
H_2 = {\left( \begin{array}{ccc}}{}* & 0 & 0\\ {}* & {}* & {}* \\ {}* & {}* & {}* {{ \end{array}\right)}}.$$
Note that the outer automorphism $$A \to (A^{-1})^t$$ takes $H_1$ to $H_2$, and also takes elements of $H_1$ to conjugate elements. This is enough to show that $(G, H_1, H_2)$ is a Sunada triple.
In this section, we will present the necessary algebraic facts to prove Theorem \[genus\]. Many of these facts are proved easily by noting the isomorphism $$PSL(3, {{\Bbb{Z}}}/2) \cong PSL(2, {{\Bbb{Z}}}/7).$$ It is somewhat difficult to see the subgroups $H_1$ and $H_2$ in $PSL(2, {{\Bbb{Z}}}/7)$. The outer automorphism which takes $H_1$ to $H_2$ is, however, easy to describe. It is the automorphism $${\left( \begin{array}{cc}}a& b\\ c& d { \end{array}\right)}\to {\left( \begin{array}{cc}}a & -b\\ -c & d { \end{array}\right)}=
{\left( \begin{array}{cc}}-1 & 0 \\ 0 & 1 { \end{array}\right)}{\left( \begin{array}{cc}}a & b \\ c & d { \end{array}\right)}{\left( \begin{array}{cc}}-1 & 0 \\
0 & 1 { \end{array}\right)}.$$ The fact that this cannot be made an inner automorphism follows from the fact that $-1$ is not a square $({{\hbox{mod}}\ }7)$.
We now describe the conjugacy classes of $PSL(2, {{\Bbb{Z}}}/7)$:
Every element of $PSL(2, {{\Bbb{Z}}}/7)$ is of order $1, 2, 3, 4$, or $7$.
[(a)]{} The only element of order 1 is the identity.
[(b)]{} Every element of order $2$ is conjugate to ${\left( \begin{array}{cc}}0 & 1\\
-1 & 0 { \end{array}\right)}$.
[(c)]{} Every element of order $3$ is conjugate to ${\left( \begin{array}{cc}}1 & 1\\
-1 & 0 { \end{array}\right)}$.
[(d)]{} Every element of order $4$ is conjugate to ${\left( \begin{array}{cc}}2 & 1 \\
1 & 1 { \end{array}\right)}$.
[(e)]{} Every element of order $7$ is conjugate to either ${\left( \begin{array}{cc}}1
& 1\\ 0 & 1 { \end{array}\right)}$ or ${\left( \begin{array}{cc}}1 & -1 \\ 0 & 1 { \end{array}\right)}$.
Translating back into the group $PSL(3, {{\Bbb{Z}}}/2)$ gives
The elements of $PSL(3, {{\Bbb{Z}}}/2)$ satisfy:
\[con\]
[(a)]{} Every element of order $2$ is conjugate to $${\left( \begin{array}{ccc}}1 & 1 &
0\\ 0&1&0 \\ 0&0&1 {{ \end{array}\right)}}.$$
[(b)]{} Every element of order $3$ is conjugate to $${\left( \begin{array}{ccc}}0&1&0 \\ 0&0&1\\ 1&0&0 {{ \end{array}\right)}}.$$
[(c)]{} Every element of order $4$ is conjugate to $${\left( \begin{array}{ccc}}1&1&0 \\ 0&1&1 \\ 0&0&1 {{ \end{array}\right)}}.$$
[(d)]{} Every element of order $7$ is conjugate to either $${\left( \begin{array}{ccc}}1 & 1 & 1 \\ 1& 1&0 \\ 0 & 1 & 1 {{ \end{array}\right)}}$$ or $${\left( \begin{array}{ccc}}1&0&1 \\ 1&1&1\\ 1&1& 0 {{ \end{array}\right)}}.$$
[[**[Proof]{}**]{}:]{}It suffices to check that each matrix has the order indicated. We remark that a simple criterion for an element to be of order $7$ is that adding 1 to the diagonal entries produces a non-singular matrix.
To check that the two matrices in (d) above are not conjugate, we observe that their characteristic polynomials are distinct.
Identifying $G/H_1$ as non-zero row vectors, we now may calculate the action of an element of $G$ on $G/H_1$ as a permutation representation. We will be interested in the cycle structure of this representation, which clearly only depends on the conjugacy class of the element. It follows from the above that the same calculation is also valid for the permutation representation on $G/H_2$.
Let $g \in PSL(3, {{\Bbb{Z}}}/2)$. Then the cycle structure of the permutation representation of $g$ on $G/H_1$ and $G/H_2$ is given by:
[(a)]{} If $g$ is of order $2$, then $g$ acts as the product of two cycles of order 2 and three 1-cycles.
[(b)]{} If $g$ is of order $3$, then $g$ acts as two $3$-cycles and a $1$-cycle.
[(c)]{} If $g$ is of order $4$, then $g$ acts as a $4$-cycle, a $2$-cycle, and a $1$-cycle.
[(d)]{} If $g$ is of order $7$, then $g$ acts as a $7$-cycle.
The proof is just an evaluation in each case of the representatives in Lemma \[con\].
Proof of Theorem \[genus\]
==========================
In this section, we will prove Theorem \[genus\]. Our method will be to find an orbifold surface $M$ and a surjective homomorphism from $\pi_1(M)$ to a Sunada triple $(G, H_1, H_2)$. We will take $G$ to be $PSL(3, {{\Bbb{Z}}}/2) \cong PSL(2, {{\Bbb{Z}}}/7)$, and $H_1$ and $H_2$ the corresponding subgroups. We then would like to study the corresponding coverings $M^{H_1}$ and $M^{H_2}$.
Let $x$ be a singular point of $M$, and let $g_x$ be the element of $\pi_1(M)$ corresponding to going one around $x$, which is well-defined up to conjugacy. Then the points of $M^{H_1}$ (resp. $M^{H_2}$) lying over $x$ are in 1-to1 correspondence with the cycle decomposition of $g_x$ on $G/H_1$ (resp. $G/H_2$). If $g_x$ acts freely on the cosets, then we may choose the orbifold singularity at $x$ so that it smooths out to a regular (i.e. nonsingular) point in $G/H_1$ (resp. $G/H_2)$. By Theorem \[con\], this will happen only if $g_x$ is the identity or of order $7$, in which case there will be precisely one point lying over $x$.
We wish to calculate the genus of $M^{H_1}$ and $M^{H_2}$ as topological surfaces (that is, forgetting the orbifold structure). Our strategy will be the following: we remove all the singular points of $M$ and the points lying over them, so that the covering is now a regular (i.e. non-orbifold) covering. We then multiply the Euler characteristic of $M$ with the points removed by $7$, the index $[G:H_1] = [G:H_2]$. We then add back in the points lying over the singular points. By Theorem \[con\], there will be five such points if $g_x$ is of order 2, three such points if $g_x$ is of order 3 or order 4, and one if $g_x$ is of order $7$. We may now compute the genus of $M^{H_1}$ (resp. $M^{H_2}$) from the Euler characteristic.
We now have to calculate the number of ends. We must throw out those singular points lying over a singular points of oreder 2 (there are five of these), order 3, or order 4 (there are three of these in both cases). We need not throw out the point lying over a singular point of order $7$.
We then have to worry about whether $M^{H_1}$ and $M^{H_2}$ are distinct. As argued, for instance, in [@Su] or [@BGP], that if we choose a variable metric on $M$, then for a generic choice of such metrics $M^{H_1}$ and $M^{H_2}$ will be non-isometric. If we want constant curvature metrics, this will in general fail if $M$ is a sphere with three singular points, as examples in [@BT] show, but if $M$ is a sphere with $n$ singular points, $n \ge 4$, or a surface of higher genus with an arbirary number of singular points, then choosing a generic conformal structure on $M$ and generically placed points will produce $M^{H_1}$ and $M^{H_2}$ distinct.
Let us first take the case where $M$ is a sphere with three singular points.
Choosing these singular points to be of order $2, 3$, and $7$, we must find matrices $A, B,$ and $C$ in $PSL(2, {{\Bbb{Z}}}/7)$ such that:
[(i)]{} $A$ is of order 2, $B$ is of order $3$, and $C$ is of order $7$.
[(ii)]{} $ABC= {\hbox{id}}$.
[(iii)]{} $A$, $B$, and $C$ generate $PSL(2, {{\Bbb{Z}}}/7)$.
A simple choice is $$A = {\left( \begin{array}{cc}}0&1\\ -1 & 0 { \end{array}\right)}\quad B= {\left( \begin{array}{cc}}1 & 1\\ -1 & 0 { \end{array}\right)}$$ $$C= {\left( \begin{array}{cc}}1&0 \\ -1& 1 { \end{array}\right)}.$$
The computation of the genus of $M^{H_1}$ (resp. $M^{H_2}$) proceeds as follows: the thrice-punctured sphere has Euler characteristic $\chi
= -1$. Hence $M^{H_i}$ without the singular points has $\chi =
-7$. Putting in the five singular points lying over the singular point of order $2$, the three singular points lying over the point of order $3$, and the point lying over the point of order $7$ adds $5 + 3 +
1=9$ to this, yielding an Euler characteristic of $2$. Hence $M^{H_1}$ and $M^{H_2}$ are of genus $0$. We must make ends out of the points lying over the singular points of orders $2$ and $3$, to give a total of eight ends.
This establishes Theorem \[genus\] (a).
If we had used two singular points of order $7$ and one singular point of order $2$ (resp. 3 or 4), we would obtain for $M^{H_1}$ and $M^{H_2}$ surfaces of genus 1 (resp. 2) with five (resp. 3) ends, provided we can find the corresponding generators. But
$$B = {\left( \begin{array}{cc}}1 & k\\ 0 & 1 { \end{array}\right)}\quad {\hbox{and}}\ C= {\left( \begin{array}{cc}}1 & 0\\ l &
1 { \end{array}\right)}$$ generate $PSL(2,{{\Bbb{Z}}}/7)$ for any choice of $k,l$ prime to $7$, and their product has trace $2+ kl$. Hence, for appropriate choice of $k$ and $l$, we may find $A$ of order $2, 3,$ or $4$.
This establishes (c).
We now investigate what happens when we choose $M$ to be a sphere with four singular points.
Choosing the singular points to be of order $2, 2, 2$, and $7$ respectively, we calculate the Euler characteristic of $M^{H_i}$ as $$\chi = 7(-2) + 3\cdot 5 + 1 = 2,$$ so again the $M^{H_i}$ have genus $0$, now with fifteen ends, provided we can find matrices $A, B, C,$ and $D$ of these orders which generate $PSL(2, {{\Bbb{Z}}}/7)$ and whose product is 1. To do this, we first observe that we may find two matrices $B'$ and $C'$ of order $2$ such that their product is of order $3$. One choice is $$B'= {\left( \begin{array}{cc}}0 & 1 \\ -1 & 0 { \end{array}\right)}, \quad C'= {\left( \begin{array}{cc}}0 & 2 \\ 3 & 0
{ \end{array}\right)}.$$
We then conjugate $B'$ and $C'$ so that their product is ${\left( \begin{array}{cc}}1 & 1
\\ -1 & 0 { \end{array}\right)}$. We may then choose $A$ and $D$ to be ${\left( \begin{array}{cc}}0 & 1\\
-1 & 0 { \end{array}\right)}$ and ${\left( \begin{array}{cc}}1 & 0 \\ -1 & 1 { \end{array}\right)}$ as above.
This establishes (b).
To establish the first part of (d), we search for matrices of order $2,2,3$, and $7$ whose product is 1. Choosing $$C = {\left( \begin{array}{cc}}1 & 1 \\ -1 & 0 { \end{array}\right)}, \quad D= {\left( \begin{array}{cc}}1 & 0\\ k & 1 { \end{array}\right)},$$ we have that $C$ and $D$ generate $PSL(2, {{\Bbb{Z}}}/7)$, and for an appropriate choice of $k$ the product is of order $3$. We may then choose $A$ and $B$ as above to be two matrices of order two whose product of order $3$ is the inverse of this matrix.
To establish the second and third parts of (d), we proceed differently. The base surface $M$ will be of genus 1 with one singular point. If the singular point is of order $2$, the resulting $M^{H_i}$ will be of genus $2$, with five ends. If the singular point is of order $3$ or $4$, the resulting surface is of genus $3$, with three ends.
We therefore seek matrices $A$ and $B$ which generate $PSL(2, {{\Bbb{Z}}}/7)$, such that their commutator is of order $2$ (resp. $3$ or $4$).
Choosing $$A= {\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}, \quad B = {\left( \begin{array}{cc}}1 & 0\\ k& 1 { \end{array}\right)},$$ we get $$\left[A,B\right]= {\left( \begin{array}{cc}}1 + k^2 & -k\\ -k & 1{ \end{array}\right)},$$ and $A$ and $B$ generate $PSL(2,{{\Bbb{Z}}}/7)$. Choosing, for instance, $k=2$ gives a commutator of order $3$.
Choosing $$A = {\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}\quad B= {\left( \begin{array}{cc}}4& 1\\ 0 & 2 { \end{array}\right)}$$ gives $$\left[A, B\right] = {\left( \begin{array}{cc}}3 & 2\\ 2&4 { \end{array}\right)},$$ which is of order $2$.
We see no elegant way of seeing that $A$ and $B$ generate $PSL(2,
{{\Bbb{Z}}}/7)$, but $$A(BAB^2)(\left[A,B\right])= {\left( \begin{array}{cc}}1& 0\\ 2 & 1 { \end{array}\right)},$$ and this matrix and $A$ generate.
This concludes the proof of the second and third parts of (d), and hence Theorem \[genus\].
Proof of Theorem \[cong\]
=========================
In this section, we construct congruence surfaces which are isoscattering.
There are several difficulties in this setting which are not present in the general setting. First of all, congruence surfaces are constructed out of subgroups of the finite groups $PSL(2, {{\Bbb{Z}}}/k)$, and such groups are rather special. The subgroups of $PSL(2, {{\Bbb{Z}}}/p)$ have been classified, and are given in Dickson’s List [@Di]. It is not [*a priori*]{} evident, for instance, that $PSL(2,{{\Bbb{Z}}}/p)$ contains Sunada triples for general $p$. Of course, the case of $PSL(3, {{\Bbb{Z}}}/2)
\cong PSL(2, {{\Bbb{Z}}}/7)$ occurs as a very special example, but we will need a richer collection of examples.
Secondly, given such a Sunada triple $(G, H_1,H_2)$, we do not have the freedom of choosing a homomorphism $\pi_1(M) \to G$ as we did previously. It is given to us canonically.
Finally, we must worry about “extra isometries,” since we do not have the freedom to change parameters to guarantee that $M^{H_1}$ and $M^{H_2}$ will be distinct.
We begin our discussion with the group $G= PSL(2, {{\Bbb{Z}}}/7)$, and $H_1$ and $H_2$ the two subgroups as above. Taking $$\Gamma= PSL(2, {{\Bbb{Z}}}),$$ and considering the natural projection $\phi: \Gamma \to G$, we first notice that the $\phi^{-1}(H_i)$ contain torsion elements, so that the ${{{\Bbb{H}}}}^2/\phi^{-1}(H_i)$ are singular surfaces.
To remedy this problem, and also to introduce a technique we will use later, we note that for $k=k_1 k_2$, with $k_1$ and $k_2$ relatively prime, we have $$PSL(2, {{\Bbb{Z}}}/k)= P( SL(2, {{\Bbb{Z}}}/k_1) \times SL(2, {{\Bbb{Z}}}/k_2)).$$ Choosing $k=14$, we see that $$PSL(2, {{\Bbb{Z}}}/14)= PSL(2,{{\Bbb{Z}}}/2)\times PSL(2,{{\Bbb{Z}}}/7),$$ noting that the “$P$” in $PSL(2, {{\Bbb{Z}}}/2)$ is trivial.
Furthermore, the kernel $\Gamma_2$ of $PSL(2, {{\Bbb{Z}}}/2)$ satisfies that ${{{\Bbb{H}}}}^2/\Gamma_2$ is a (non-singular) thrice-punctured sphere. Hence, we resolve the issue of singularities by restricting to $\Gamma_2$.
Now let $S^{i}= {{{\Bbb{H}}}}^2/\Gamma^{H_i},$ where $$\Gamma^{H_i} = \Gamma_2 \cap \phi^{-1}(H_i).$$ Then, as before, $S^{1}$ and $S^{2}$ are isoscattering. They are, however, also isometric. This can be seen by noting that $H_1$ and $H_2$ are conjugate under the automorphism $$\tau: {\left( \begin{array}{cc}}a& b\\ c& d { \end{array}\right)}\to {\left( \begin{array}{cc}}a & -b \\ -c & d { \end{array}\right)}.$$ Furthermore, this $\tau$ induces an orientation-reversing isometry of $H^2/\Gamma$ which is reflection in the line ${\hbox{Re}}(z)=0$ in the usual fundamental domain for ${{{\Bbb{H}}}}^2/\Gamma$, and therefore lifts to an orientation-reversing isometry of $S^1$ to $S^2$. See the genus 3 example of [@BT], where a similar problem occurs.
We will handle this problem in the following way: let us assume for some $p$ different from $2$ or $7$, there is a subgroup $K$ of $SL(2,
{{\Bbb{Z}}})$ such that $K$ and $\tau(K)$ are not conjugate in $SL(2,
{{\Bbb{Z}}}/p)$. We may now choose our subgroups
$$\begin{array}{ll}
G &= P({\hbox{id}}\times SL(2, {{\Bbb{Z}}}/7) \times SL(2, {{\Bbb{Z}}}/p))\\
H_1 &= P({\hbox{id}}\times H_1 \times K)\\
H_2 &= P({\hbox{id}}\times H_2 \times K),
\end{array}$$ and let $\widetilde{S}^1$ and $\widetilde{S}^2$ be the coverings of ${{{\Bbb{H}}}}^2/PSL(2,{{\Bbb{Z}}})$ corresponding to $\phi^{-1}(H_1)$ and $\phi^{-1}(H_2)$.
$\widetilde{S}^1$ and $\widetilde{S}^2$ are isoscattering, but we want to show that they are not isometric. Any such isometry between them must be given by conjugation by some matrix $C$, by the involution $\tau$, or by a composition of the two.
The first possibility cannot obtain, because restricting to $PSL(2,
{{\Bbb{Z}}}/7)$, $C$ will give a conjugacy of $H_1$ to $H_2$. The second and third possibilities also cannot hold, since if such a matrix $C'$ exists, restricting to the $PSL(2, {{\Bbb{Z}}}/p)$ factor, it will give a conjugacy from $K$ to $\tau(K)$.
Thus, Theorem \[cong\] will follow once we find such a $p$ and $K$.
We now examine Dickson’s list [@Di] for likely subgroups $K$ of $PSL(2, {{\Bbb{Z}}}/p)$ for which $K$ is not conjugate to $\tau(K)$. We remark that $\tau$ is given by the outer automorphism $${\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}\to {\left( \begin{array}{cc}}-1 & 0 \\ 0 &1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d
{ \end{array}\right)}{\left( \begin{array}{cc}}-1 & 0 \\ 0 & 1 { \end{array}\right)},$$ and for $p \equiv 1\ ({{\hbox{mod}}\ }4),$ we have that $-1$ is a square root $({{\hbox{mod}}\ }p)$, so $\tau$ is actually an inner automorphism for such $p$.
To understand our choice of $K$, we observe that the subgroups $H_1$ and $H_2$ are isomorphic to the symmetric group $S(4)$ on four elements (a more detailed discussion will be given below). We will show:
\[s4\] For $ p \equiv 7\ ({{\hbox{mod}}\ }8)$, there exist subgroups $K$ of $PSL(2, {{\Bbb{Z}}}/p)$ isomorphic to $S(4)$, such that $K$ is not conjugate to $\tau(K)$ in $PSL(2, {{\Bbb{Z}}}/p)$.
[[**[Proof]{}**]{}:]{}We first discuss the restriction $p \equiv 7\ ({{\hbox{mod}}\ }8)$.
Indeed, $S(4)$ contains the cyclic subgroup ${{\Bbb{Z}}}/4$. In order for $PSL(2, {{\Bbb{Z}}}/p)$ to contain a cyclic subgroup of order $4$, we must have $p \equiv \pm 1 ({{\hbox{mod}}\ }8)$. The plus sign is ruled out by the condition $p \not\equiv 1({{\hbox{mod}}\ }4)$.
To construct a subgroup $K$ isomorphic to $S(4)$ in $PSL(2, {{\Bbb{Z}}}/p)$, we first note that $S(4)$ contains the dihedral group $$\{ A, D: A^4 =1, \quad D^2=1, \quad DAD= A^{-1} \},$$ given by $A= (1,2,3,4)$, $D=(1,2)(3,4)$.
We now seek such a subgroup in $PSL(2, {{\Bbb{Z}}}/p)$. A convenient choice for $A$ is $$A = {\left( \begin{array}{cc}}\alpha & \alpha\\ -\alpha & \alpha { \end{array}\right)},\quad \alpha^2
\equiv 1/2\ ({{\hbox{mod}}\ }p).$$
Note that $2$ is a square $({{\hbox{mod}}\ }p)$ by the condition that $ p \equiv
7 ({{\hbox{mod}}\ }8)$ and quadratic reciprocity. Note also that $$A^2= C_1 = {\left( \begin{array}{cc}}0&1\\- 1&0 { \end{array}\right)}.$$ We now seek a matrix $D$ such that $A$ and $D$ generate a dihedral group of order 8. That means that
$$\label{dihed}
D^2 = {\left( \begin{array}{cc}}1 &0 \\ 0&1 { \end{array}\right)}, \quad DAD= A^{-1}.$$
The second relation implies that $D$ commutes with $C_1$. We observe that any matrix $Z$ commuting with $C_1$ must either be of the form $$\label{ccomm1}
Z = {\left( \begin{array}{cc}}x&y \\-y & x { \end{array}\right)}, \quad x^2 + y^2 =1$$ or $$\label{ccomm2}
Z= {\left( \begin{array}{cc}}\beta & \gamma\\ \gamma & -\beta { \end{array}\right)}, \quad \beta^2 +
\gamma^2 = -1.$$ Since any matrix of the form (\[ccomm1\]) commutes with $A$, $D$ must be of the form (\[ccomm2\]). Furthermore, any matrix of the form (\[ccomm2\]) will satisfy (\[dihed\]).
For later reference, we remark that neither $\beta$ nor $\gamma$ can be zero, since $-1$ is not a square $({{\hbox{mod}}\ }p)$, and we may choose $\beta + \gamma -1 \not \equiv 0 ({{\hbox{mod}}\ }p)$, by changing the sign of $\gamma$ if necessary.
We now set $$C_2 = C_1 D = {\left( \begin{array}{cc}}-\gamma & \beta \\ \beta & \gamma { \end{array}\right)}.$$ We may embed this dihedral group in $S(4)$ by setting $$\begin{array}{ll}
A &\to (1,2,3,4)\\
C_1 &\to(1,3)(2,4)\\
D&\to (1,2)(3,4)\\
C_2 &\to (1,4)(2,3).
\end{array}$$
We now seek an element $E$ of $PSL(2, {{\Bbb{Z}}}/p)$ corresponding to the element $(1,2,3)$. Thus, $E$ must satisfy the conditions
$$\begin{array}{ll}
E^3 &=1\\
EC_1E^{-1} &= D\\
EDE^{-1} &= C_2\\
EC_2E^{-1}&= C_1\\
AEA^{-1} &= E^{-1}C_1.
\end{array}$$
To do this, let us tentatively set $$\widetilde{E} = {\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}.$$ We will want $\widetilde{E}$ to send the fixed points ${{\hbox{Fix}}}(C_1)$ of $C_1$ (viewed as a linear fractional tranformation) to ${{\hbox{Fix}}}(D)$, the fixed points ${{\hbox{Fix}}}(D)$ of $D$ to ${{\hbox{Fix}}}(C_2)$, and ${{\hbox{Fix}}}(C_2)$ to ${{\hbox{Fix}}}(C_1)$.
We calculate:
$$\begin{array}{ll}
{{\hbox{Fix}}}(C_1) &= \pm i,\\
{{\hbox{Fix}}}(D) &= \frac{\beta \pm i}{\gamma}\\
{{\hbox{Fix}}}(C_2) &= \frac{ -\gamma \pm i}{\beta},
\end{array}$$ where we have denoted by $i$ a square root of $-1$ in the field ${{\Bbb{F}}}_{p^2}$.
Choosing the plus sign in each case, we seek $\widetilde{E}$ satisfying
$$\begin{array}{ll}
\widetilde{E}(i) = \frac{\beta + i}{\gamma}\\
\widetilde{E}( \frac{\beta + i}{\gamma}) = \frac{-\gamma + i}{\beta}\\
\widetilde{E}(\frac{-\gamma+ i}{\beta}) = i.
\end{array}$$
After some tedious linear algebra, we find
$$\widetilde{E} = {\left( \begin{array}{cc}}\frac{\gamma + \beta^2}{\beta + \gamma - 1}
& \frac{-\beta + \beta \gamma - \gamma}{\beta + \gamma -1}\\
\frac{ \beta \gamma - 1}{\beta + \gamma -1} & \frac{\beta +
\gamma^2}{\beta + \gamma -1} { \end{array}\right)}.$$
Note that with these choices of $a,b,c,$ and $d$, we have $$a+d =1, \quad c-b=1, \quad ad-bc=1.$$
The first relation assures that $\widetilde{E}$ is of order $3$, and it is easily checked that the conjugacies of $C_1$, $C_2$, and $D$ by $\widetilde{E}$ are as desired.
It remains to check the condition $$A\widetilde{E}A^{-1} = \widetilde{E}^{-1}C_1.$$ This unappetizing calculation can be carried out by writing
$$A\widetilde{E} A^{-1} \widetilde{E} = {\left( \begin{array}{cc}}\alpha^2 & 0\\ 0&
\alpha^2 { \end{array}\right)}{\left( \begin{array}{cc}}1& 1\\ -1 &1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}{\left( \begin{array}{cc}}1&-1\\ 1&1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)},$$ using $a +d = 1$ and $c-b =1$ to eliminate $c$ and $d$, and $ad-bc =1$ to replace the quadratic terms $a^2 + b^2$ by $a-b-1$. We find that $$A\widetilde{E}A^{-1}\widetilde{E} =C_1$$ as desired.
Setting $E= \widetilde{E}$, we now have constructed a subgroup $G$ of $PSL(2, {{\Bbb{Z}}}/p)$ isomorphic to $S(4)$.
We now must show that $\tau(G)$ is not conjugate to $G$ in $PSL(2,{{\Bbb{Z}}}/p)$.
But if $\psi: G \to G$ is any isomorphism, we may assume, by replacing $\psi$ by a conjugate, that $$\psi(A) = A.$$
$\psi(C_1)$ must then be $C_1$, and after conjugating by $A$ if necessary, we must have $$\psi(D)= D, \quad \psi(C_2)= C_2.$$ It then follows that $$\psi(E) =E.$$ We now show that under these assumptions, we cannot have $$\psi(X) = Z\tau(X) Z^{-1}, \quad X \in G.$$ Noting that $\tau(C_1)= C_1$, this gives us $$C_1 = Z C_1 Z^{-1},$$ so that $Z$ must be of the form (\[ccomm1\]) or(\[ccomm2\]). Noting that $\tau(A)= A^{-1}$, this gives $$A = Z A^{-1}Z^{-1},$$ so that $Z$ must be of the form $$Z= {\left( \begin{array}{cc}}x & y\\ y & -x { \end{array}\right)}, \quad x^2 + y^2 = -1.$$
We now consider the equation $$D= Z \tau(D) Z^{-1},$$ which we write out as $$\pm {\left( \begin{array}{cc}}\beta & \gamma \\ \gamma& -\beta { \end{array}\right)}= {\left( \begin{array}{cc}}x& y\\
y&-x { \end{array}\right)}{\left( \begin{array}{cc}}-\beta & \gamma\\ \gamma & \beta { \end{array}\right)}{\left( \begin{array}{cc}}x&y\\
y&-x { \end{array}\right)}.$$
The term on the right is computed to be $${\left( \begin{array}{cc}}-x^2\beta + 2xy\gamma + y^2 \beta & -2xy\beta + y^2 \gamma -
x^2\gamma\\ -2xy\beta -x^2\gamma + y^2 \gamma & -y^2\beta -2xy\gamma +
x^2 \beta { \end{array}\right)}.$$
Taking first the plus sign, we get from the upper-left entry the equation $$-x^2 \beta + 2xy\gamma + y^2\beta = \beta = \beta(-x^2 -y^2),$$ or $$2xy\gamma + 2y^2\beta=0.$$
Since $y \neq 0$, this gives $$x\gamma + y\beta =0.$$ Similarly, the lower-left entry gives $$-x\beta + y\gamma=0.$$ Solving these equations gives $$\frac{x}{\gamma}(\gamma^2 + \beta^2)=0.$$ But this implies $x=0$, a contradiction.
Now we take the minus sign. We get the equations $$-x^2 \beta + 2xy\gamma + y^2\beta = -\beta = \beta(x^2 + y^2)$$ and $$-x^2 \gamma -2xy\beta + y^2 \gamma = \gamma(x^2 + y^2).$$ These become the two equations $$y\gamma = x\beta$$ and $$-y\beta = x\gamma,$$ which again yields a contradiction.
This contradiction proves Theorem \[s4\], and hence also Theorem \[cong\].
Another approach to Theorem \[cong\] may be based on $p \equiv 1 \
({{\hbox{mod}}\ }4)$. For ${{\cal{D}}}\in {{\Bbb{Z}}}/p$, let $${\tau_{{{\cal{D}}}}}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}= {\left( \begin{array}{cc}}{{\cal{D}}}& 0\\ 0& 1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}{\left( \begin{array}{cc}}{{\cal{D}}}^{-1} & 0\\ 0&1 { \end{array}\right)}= {\left( \begin{array}{cc}}a& {{\cal{D}}}b\\
\frac{1}{{{\cal{D}}}} c & d { \end{array}\right)}.$$
When ${{\cal{D}}}$ is a square $({{\hbox{mod}}\ }p)$, ${\tau_{{{\cal{D}}}}}$ is inner. We will show:
\[D\] Let $p \equiv 1\ ({{\hbox{mod}}\ }4)$, and let ${{\cal{D}}}$ be a non-square $({{\hbox{mod}}\ }p)$.
Then there exists a subgroup $K$ of $PSL(2,{{\Bbb{Z}}}/p)$ isomorphic to $S(4)$, such that $K$ and ${\tau_{{{\cal{D}}}}}(K)$ are not conjugate in $PSL(2,{{\Bbb{Z}}}/p)$.
Given Theorem \[D\], we set $$\Gamma^1 = \Gamma_2 \cap \phi^{-1}(K)$$ and $$\Gamma^2= \Gamma_2 \cap \phi^{-1}({\tau_{{{\cal{D}}}}}(K)).$$ Then $S^1 = {{{\Bbb{H}}}}^2/\Gamma^1$ and $S^2={{{\Bbb{H}}}}^2/\Gamma^2$ are isoscattering. The orientation-reversing isometry $\tau$ of ${{{\Bbb{H}}}}^2/PSL(2,{{\Bbb{Z}}})$ lifts to an isometry of $S^i$ to itself, $i=1, 2$, since $-1$ is a square $({{\hbox{mod}}\ }p)$, so Theorem \[D\] suffices to show that they are not isometric.
To prove Theorem \[D\], we set $$A = {\left( \begin{array}{cc}}\alpha(1+i) &0\\ 0 & \alpha(1-i) { \end{array}\right)}\quad C_1 = {\left( \begin{array}{cc}}i&0\\ 0&-i { \end{array}\right)},$$ and $$D= {\left( \begin{array}{cc}}0&1\\ -1 & 0 { \end{array}\right)},$$ where $i$ is a square root of $-1$ $({{\hbox{mod}}\ }p)$, and $\alpha^2 =
\frac{1}{2}\ ({{\hbox{mod}}\ }p)$. We may then solve for $E$ as above, to find $$E= {\left( \begin{array}{cc}}\frac{1-i}{2} & \frac{1-i}{2}\\ -\frac{(1+i)}{2} &
\frac{1+i}{2} { \end{array}\right)}.$$
Then $${\tau_{{{\cal{D}}}}}(A)= A, \quad {\tau_{{{\cal{D}}}}}(C_1)= C_1,$$ and ${\tau_{{{\cal{D}}}}}(D) = {\left( \begin{array}{cc}}0 & {{\cal{D}}}\\ - \frac{1}{{{\cal{D}}}} & 0 { \end{array}\right)}.$
We seek $Z$ such that $$ZAZ^{-1}=A,\quad ZC_1Z^{-1}= C_1,$$ and $$Z{\left( \begin{array}{cc}}0 & {{\cal{D}}}\\ -\frac{1}{{{\cal{D}}}} & 0 { \end{array}\right)}Z^{-1} = {\left( \begin{array}{cc}}0&1\\
-1 & 0 { \end{array}\right)}.$$
But $Z$ must be of the form $$Z= {\left( \begin{array}{cc}}x&0\\ 0 & \frac{1}{x} { \end{array}\right)},$$ so that $$Z {\left( \begin{array}{cc}}0 & \cal{D}\\ -\frac{1}{{{\cal{D}}}} &0 { \end{array}\right)}Z^{-1} = {\left( \begin{array}{cc}}0 &
x^2 {{\cal{D}}}\\ -\frac{1}{x^2{{\cal{D}}}} & 0 { \end{array}\right)},$$ which can’t be made equal to $C_1$.
**REFERENCES**
[Be]{} P. Berard, “Transplantation et Isospectralité,” Math.Ann. 292 (1992), pp. 547-560.
[BGP]{} R. Brooks, R. Gornet, and P. Perry, “Isoscattering Schottky Manifolds,” GAFA 10 (2000), pp. 307-326.
[BJP]{} D. Borthwick, C. Judge, and P. Perry, “Determinants of Laplacians and Isopolar Metrics on Surfaces of Infinite Area,” preprint.
[BP]{} R. Brooks and P. Perry, “Isophasal Scattering Manifolds in Two Dimensions,” Comm. Math. Phys. 223 (2001), pp. 465-474.
[BT]{} R. Brooks and R. Tse, “Isospectral Surfaces of Small Genus,” Nagoya Math. J. 107 (1987), pp. 13-24.
[Di]{} L. E. Dickson, [*Linear Groups,*]{} 1901.
[Su]{} T. Sunada, “Riemannian Coverings and Isospectral Manifolds,” Ann. Math. 121 (1985), pp. 169-186.
[Zel]{} S. Zelditch, “Kuznecov Sum Formulae and Szegö Limit Formulas on Manifolds,” Comm. P. D. E. 17 (1992), pp. 221-260.
[^1]: Partially supported by the Israel Science Foundation and the Fund for the Promotion of Research at the Technion
|
---
abstract: |
We calculate the R-R zero-norm states of type II string spectrum. To fit these states into the right symmetry charge parameters of the gauge transformations of the R-R tensor forms, one is forced to T-dualize some type I open string space-time coordinates and thus to introduce D-branes into the theory. We also demonstrate that the constant T-dual R-R 0-form zero-norm state, together with the NS-NS singlet zero-norm state are responsible for the SL(2,Z) S-duality symmetry of the type II B string theory.
PACS:11.25.-w
Keywords: String; D-branes.
address: |
Department of Electrophysics, National Chiao Tung University, Hsinchu 30050,\
Taiwan, R. O. C.
author:
- 'Jen-Chi Lee[^1]'
title: 'T-dual R-R zero-norm states, D-branes and S-duality of type II string theory'
---
Introduction {#sec:intro}
============
It has been pointed out for a long time that the complete space-time symmetry[@1] of string theory is related to the zero-norm state (a physical state that is orthogonal to all physical states including itself) in the old covariant quantization of the string spectrum.[@2] This observation had made it possible to explicitly construct many stringy ($%
\alpha ^{\prime }\rightarrow \infty $) massive symmetry of the theory. This includes the $w_{\infty }$ symmetry of the toy 2D string[@3] and the [*discrete*]{} massless and massive T-duality symmetry of closed bosonic string.[@4] The authors of [@5] show that, in string theory, some target space mirror symmetry of N=2 backgrounds on group manifolds is a Kac-Moody gaugy symmetry. Thus, like T-duality, it should be related to the zero-norm states. On the other hand, the massless and massive SUSY, and some new enlarged spacetime boson-fermion symmetries induced by zero-norm states were also discussed in [@6]. It is thus of interest to study the R-R zero-norm state and its relation to D-brane which was recently shown by Polchinski to be the symmetry charge carrier of the propagating R-R forms.[@7]
Presumably, there should be no R-R zero-norm state in the type II string spectrum since the fundamental string does not interact with the R-R forms. However, to our surprise, it was discovered that there do exist both massless and massive R-R zero-norm states in the type II string spectrum.[@6] It was then realized that the degree of freedom of massless R-R zero-norm states does not fit into that of the symmetry parameters of the propagating R-R forms and thus resolved the seeming inconsistency. This observation gives us another justification of the well-known wisdom that perturbative string does not carry the massless R-R charges, although the existence of these R-R zero-norm states remain mysterious.
In this paper, we will show that the T-dual R-R zero-norm states serve as the right symmetry parameters of the gauge transformations of the R-R propagating forms. Also one is forced to introduce Type I open string and D-branes into the type II string theory to incorporate these T-dual R-R zero-norm states. Our [*spacetime*]{} zero-norm states argument here is in complementary with the [*worldsheet*]{} string vertex operator argument first given by Binnchi, Pradisi and Sagnotti.[@8] They considered R-R one-point function in the (-$\frac{1}{2}$,-$\frac{3}{2}$) ghost picture on the [*disk*]{} and resulted in a conclusion which was consistent with D-brane as R-R charge carrier[@7]. As an important application, we demonstrate that the constant T-dual R-R 0-form zero-norm state, together with the NS-NS singlet zero-norm state which was always neglected in the previous discussions, are responsible for the [*discrete*]{} SL(2,Z) S-duality symmetry of the type II B string theory.[@9] This discovery suggests that not only stringy ($\alpha ^{\prime }\rightarrow \infty $) symmetry but also strong-weak ($g_{s}\rightarrow \infty $) duality symmetry are related to the existence of zero-norm states of the spectrum.
T-dual R-R zero-norm states
===========================
The massless physical NS state of the open superstring is (we use the notation in Ref [@10])
$$\varepsilon _{\mu }b_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle ;\text{ }%
k\cdot \varepsilon =0,\text{ }k^{2}=0$$
In addition, there is a singlet zero-norm state
$$k_{\mu }b_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle ;\text{ }k^{2}=0\text{
.}$$
The NS-NS symmetries of graviton and antisymmetry tensor of type II string were derived through the following two zero-norm states
$$\begin{aligned}
&&\varepsilon _{\mu }b_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle \otimes
k_{\mu }\stackrel{\sim }{b}_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle ,
\nonumber \\
&&k_{\mu }b_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle \otimes \varepsilon
_{\mu }\stackrel{\sim }{b}_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle\end{aligned}$$
in the first order weak field approximation (WFA)[@11]. The remaining interesting singlet zero-norm state
$$k_{\mu }b_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle \otimes k_{\mu }%
\stackrel{\sim }{b}_{-\frac{1}{2}}^{\mu }\left| 0,k\right\rangle$$
will be discussed in the next section.
We now discuss the massless R state. The only propagating spinor is
$$\left| \stackrel{\rightharpoonup }{S},k\right\rangle u_{\stackrel{%
\rightharpoonup }{s}};\text{ }F_{0}\left| \stackrel{\rightharpoonup }{S}%
,k\right\rangle u_{\stackrel{\rightharpoonup }{s}}=0\text{ .}$$
The GSO operator in the massless limit reduces to the chirality operator, and only one of the chiral spinor $8_{s}($or $8_{c})$ will be projected out. In addition, there is a massless fermionic zero-norm state
$$k_{\mu }\Gamma _{\stackrel{\rightharpoonup }{s},\stackrel{\rightharpoonup }{s%
}}^{\mu }\left| \stackrel{\rightharpoonup }{S},k\right\rangle \theta _{%
\stackrel{\rightharpoonup }{s}}\text{ .}$$
Eq.(6) is the only massless solution of the following
$$F_{0}\left| \psi \right\rangle ,\text{where }F_{1}\left| \psi \right\rangle
=L_{0}\left| \psi \right\rangle =0\text{.}$$
The state in equation (6) is crucial in the discussion of this paper. Note that $k\cdot \Gamma \left| \stackrel{\rightharpoonup }{S},k\right\rangle
\theta _{\stackrel{\rightharpoonup }{s}}$ is left-handed if $\left|
\stackrel{\rightharpoonup }{S},k\right\rangle \theta _{\stackrel{%
\rightharpoonup }{s}}$ is right-handed and both spinors have exactly the same degree of freedom. The massless propagating R-R states of type II string consist of tensor forms
$$G_{\alpha \beta }=\sum_{k=0}^{10}\frac{i^{k}}{k!}G_{\mu _{1}\mu _{2}...\mu
_{k}}(\Gamma ^{\mu _{1}\mu _{2}...\mu _{k}})_{\alpha \beta }\text{,}$$
where $\Gamma ^{\mu _{1}\mu _{2}...\mu _{k}}$ are the antisymmetric products of gamma-matrix, and $\alpha ,\beta $ are spinor indices. There is a duality relation which reduces the number of independent tensor components to up to k=5 form. The on-shell conditions, or two massless Dirac equations, imply G is indeed a field strength and can be written as
$$G_{(k)}=dA_{(k-1)}$$
which means perturbative string states do not carry the [*massless*]{} R-R symmetry charges. We are now in a position to discuss the R-related symmetry charges. Let’s first introduce the NS-R (R-NS) SUSY zero-norm states[@6]
$$\text{ \qquad }k\cdot b_{-\frac{1}{2}}\left| 0,k\right\rangle \otimes \left|
\stackrel{\rightharpoonup }{S},k\right\rangle \overline{u}_{\stackrel{%
\rightharpoonup }{s}}\text{and }\left| \stackrel{\rightharpoonup }{S}%
,k\right\rangle u_{\stackrel{\rightharpoonup }{s}}\otimes k\cdot \widetilde{b%
}_{-\frac{1}{2}}\left| 0,k\right\rangle$$
for the II A theory and a trivial modification for the II B theory. The corresponding worldsheet vertex operator in the ($0,-\frac{1}{2}$) picture for say the first state in equation (10) is
$$\begin{aligned}
&&k_{\mu }(\partial x^{\mu }(z)+ik\cdot \psi \psi ^{\mu })e^{ik\cdot
x(z)}u_{\alpha }\stackrel{\sim }{S}^{\alpha }(\overline{z})e^{-\frac{1}{2}%
\stackrel{\sim }{\phi }}e^{ik\cdot x(\overline{z})} \nonumber \\
&=&\partial e^{ik\cdot x(z)}u_{\alpha }\stackrel{\sim }{S}^{\alpha }e^{-%
\frac{1}{2}\stackrel{\sim }{\phi }}e^{ik\cdot x(\overline{z})},\end{aligned}$$
which is a worldsheet total derivative and, as in the case of bosonic sector[@2], one can introduce a worldsheet generator and deduce the SUSY current to be
$$\stackrel{\sim }{Q}_{\alpha ,-\frac{1}{2}}=\stackrel{\sim }{S}^{\alpha }e^{-%
\frac{1}{2}\stackrel{\sim }{\phi }},$$
where $\stackrel{\sim }{S}^{\alpha }$and $\stackrel{\sim }{\phi }$ are the right-moving spin field and the bosonized superconformal ghost respectively. This [*zero-norm state derivation* ]{}is consistent with the original approach.[@12] The advantage of our approach is that one can generalize to derive the enlarged stringy boson-fermion symmetry by using the massive fermion zero-norm state of the spectrum. We give one example here. There exists a m=2 NS-R zero-norm state
$$\left[ 2\theta _{\mu \nu }\alpha _{-1}^{\mu }b_{-\frac{1}{2}}^{\nu
}+k_{[\lambda }\theta _{\mu \nu ]}b_{-\frac{1}{2}}^{\lambda }b_{-\frac{1}{2}%
}^{\mu }b_{-\frac{1}{2}}^{\nu }\right] \left| 0,k\right\rangle \otimes
\widetilde{\alpha }_{-1}^{\lambda }\left| \stackrel{\rightharpoonup }{S}%
,k\right\rangle u_{\lambda ,\stackrel{\rightharpoonup }{s}}$$
with $\theta _{\mu \nu }=-\theta _{\nu \mu },$ $k^{\mu }\theta _{\mu \nu }=0$ and
$$\begin{aligned}
\lbrack (k\cdot d_{0})\alpha _{-1}^{\mu }+d_{-1}^{\mu }]u_{\mu ,\stackrel{%
\rightharpoonup }{s}} &=&0, \nonumber \\
d_{0}^{\mu }u_{\mu ,\stackrel{\rightharpoonup }{s}} &=&0.\end{aligned}$$
The corresponding vertex operator is calculated to be
$$\begin{aligned}
&&[2\theta _{[\mu \nu ],\lambda \alpha }(\partial x^{\mu }\partial x^{\nu
}-\psi ^{\mu }\partial \psi ^{\nu }+ik\cdot \psi \psi ^{\mu }\partial x^{\nu
})+k_{[\delta }\theta _{\mu \nu ],\lambda \alpha }(3\partial x^{\mu
}+ik\cdot \psi \psi ^{\mu }) \nonumber \\
&&\psi ^{\nu }\psi ^{\delta }]\overline{\partial }x^{\lambda }k\cdot
\overline{\psi }e^{-\frac{1}{2}\stackrel{\sim }{\phi }}\stackrel{\sim }{S}%
^{\alpha }e^{ik\cdot x(z,\overline{z})}\end{aligned}$$
where $\theta _{\mu \nu ,\lambda \alpha }\equiv \theta _{\mu \nu }\cdot
u_{\lambda \alpha }.$ It is straight-forward to construct the corresponding ward identity although the symmetry transformation law of the background fields is not easy to write down at this point.
We now turn to discuss the R-R zero-norm states. For the massless level, we have the following zero-norm states
$$k_{\mu }\Gamma _{\stackrel{\rightharpoonup }{s}^{\prime }\stackrel{%
\rightharpoonup }{s}}^{\mu }\left| \stackrel{\rightharpoonup }{S}%
,k\right\rangle \theta _{\stackrel{\rightharpoonup }{s}}\otimes \left|
\stackrel{\rightharpoonup }{S},k\right\rangle u_{\stackrel{\rightharpoonup }{%
s}}\text{ \qquad (II A)}$$
and
$$k_{\mu }\Gamma _{\stackrel{\rightharpoonup }{s}^{\prime }\stackrel{%
\rightharpoonup }{s}}^{\mu }\left| \stackrel{\rightharpoonup }{S}%
,k\right\rangle \overline{\theta _{\stackrel{\rightharpoonup }{s}}}\otimes
\left| \stackrel{\rightharpoonup }{S},k\right\rangle u_{\stackrel{%
\rightharpoonup }{s}}\text{ \qquad (II B).}$$
These are tensor forms as in equation (8). The on-shell condition on the right mover together with the trivial identity ($k\cdot \Gamma $)$^{2}\left|
\stackrel{\rightharpoonup }{S},k\right\rangle \theta _{\stackrel{%
\rightharpoonup }{s}}=0$ on the left mover imply, as in equation (9), that
$$F_{(k)}=d\omega _{(k-1)}.$$
Note that, for the II A (II B) theory, $\omega _{(p)}$ in eq(18) does not fit into the gauge symmetry parameters of $A_{(p)}$ forms of II A (II B) theory in eq(9) since they share the same tensor index structures. In fact, for a $p+1$ form $A_{(p+1)}$, one needs a $p$ form $\widetilde{\omega _{(p)}}
$ symmetry parameters, as can be seen from its spacetime coupling to D-brane
$$\int_{\text{world vol of D-brane}}A_{(p+1)}\equiv \int d^{p+1}\xi A_{\mu
_{1}\mu _{2}...\mu _{p+1}}(x)\partial _{1}x^{\mu _{1}}\cdot \cdot \cdot
\partial _{p+1}x^{\mu _{p+1}},$$
which implies a space-time gauge symmetry
$$A_{(p+1)}\rightarrow A_{(p+1)}+d\widetilde{\omega }_{(p)}.$$
This justifies that no perturbative type II string state carries the R-R charge. On the other hand, it is well-known that each time we T-dualize in an additional direction the dimension of the D-branes goes down by one and the R-R forms lose an index. To include the right closed string zero-norm state $\widetilde{\omega }_{(p)}$, one is thus forced to introduce the type I [*unoriented*]{} open string and T-dualizes k = odd (even) numbers of space-time coordinates and then takes the noncompact limit $R\rightarrow 0$ for each compatified radius. For k = odd (even), one has type II A (II B) string states [*in the bulk far away from D-branes*]{}. The right $%
\widetilde{\omega }_{(p)}\equiv \omega _{(p+1)}^{(T)}$ state, the T-dual R-R zero-norm state, is thus attached to the D p-brane for p= even (odd) in II A (II B) theory. Note that, near the D-branes, the orientation projection of the Type I theory leaves only one linear combination of two SUSY charges(SUSY zero-norm states in eq.(10)) of the Type II theory in the bulk. It is $Q_{\alpha }^{^{\prime }}+(\Pi _{m}^{k}\beta ^{m}\stackrel{\sim }{%
Q^{\prime }})_{\alpha }$ with $\beta ^{m}\equiv \Gamma ^{m}\Gamma .$ The T-dual R-R zero-norm states attached in the boundary of the open string 1-loop diagram with D-branes are the T-dual version of the R-R zero-norm states in the bulk of the closed string tree diagram. Our argument resolves the puzzle of seeming unwanted R-R zero-norm states in perturbative type II string spectrum and simultaneously [*motivates the introduction of D-branes into the theory* ]{}which is complementary to the argument in Ref[@7]. The space-time T-dual R-R zero-norm state has an interesting analogy from worldsheet vertex operator point of view. The authors of [@8] considered one point function of R-R vertex operator on the [*disk*]{}. Since the total right + left ghost charge number must add up to $-2$, one is forced to change the vertex operator in the conventional $(-\frac{1}{2},-%
\frac{1}{2})$ picture to either $(-\frac{1}{2},-\frac{3}{2})$ or $(-\frac{3}{%
2},-\frac{1}{2})$ picture. This inverse picture changing involves, among other unrelated things, a factor of $k\cdot \Gamma $ , which shifts the field strength to the potential and gives a strong hint that D-brane carries the R-R charge. On the other hand, our space-time T-dual R-R zero-norm states do contain this important $k\cdot \Gamma $ factor as can be seen from eqs (16) and (17). This again gives a strong support of our space-time T-dual R-R zero-norm state approach. In the next section, we will see an even more interesting application of these states.
Dilaton-Axion Symmetries and SL(2,Z) S-duality
==============================================
According to section II, for II A theory, we have $A_{(1)},$ $A_{(3)}$ potentials with $d\omega _{(0)}^{(T)}$, $d\omega _{(2)}^{(T)}$ T-dual zero-norm states and, for II B theory, we have $A_{(2)}$, $A_{(4)}$ potential with $d\omega _{(1)}^{(T)}$, $d\omega _{(3)}^{(T)}$ T-dual zero-norm states for their symmetry charge parameters. For completeness we have, in addition, an axion $A_{(0)}\equiv \chi $ in the II B theory. The corresponding T-dual zero-norm state is naturally identified to be the constant 0-form $F_{(0)}^{(T)}$, which is Poincare dual to the [*constant*]{} 10-form $F_{(10)}^{(T)}\equiv d\omega _{(9)}^{(T)}$. So we have the ”symmetry”
$$\chi \rightarrow \chi +F_{(0)}^{(T)}.$$
Note that, in eq. (8), there is a constant 10-form field strength $%
G_{(10)}=dA_{(9)}$ which is Poincare dual to the constant 0-form field strength in II A theory as well. This non-propagating degree of freedom can be included in the massive type II A supergravity and was conjectured to be related to the cosmological constant. See the interesting discussion of this 9-form potential $A_{(9)}$ by Polchinski in Ref.[@7]. Equation (21) is consistent with the fact that the axion $\chi $ is defined up to a constant. The interesting new result here is that we naturally identify this constant to be $F_{(0)}^{(T)}$.
We now turn to the discussion of NS-NS dilaton $\phi $. Remember we have a Remaining NS-NS singlet zero-norm state in equation (4). The physical meaning of this state will be discussed in the following. In reference [@11] each space-time symmetry of the bosonic background field in the first order WFA can be constructed through a superconformal deformation
$$(T^{(1)}=\overline{T}^{(1)},T_{F}^{(1)},\overline{T}_{F}^{(1)})$$
corresponding to a spacetime zero-norm state.
In equation (22), $T_{{}}^{(1)}$($T_{F}^{(1)}$) is the upper component (lower component) of deformation of the superstress tensor in the first order WFA when the background field is turned on. $\overline{T}^{(1)},(%
\overline{T}_{F}^{(1)})$ is its anti-holomorphic counterpart. It was shown that superconformal deformations constructed from zero-norm states in eq.(3) give the symmetries of graviton and antisymmetric tensor. The superconformal deformation constructed from the zero-norm state in eq.(4), which was neglected in the previous discussion., is calculated to be
$$\begin{aligned}
T^{(1)} &=&\overline{T}^{(1)}=\partial _{\mu }\partial _{\nu }\theta
(\partial x^{\mu }+\overleftarrow{\partial _{\lambda }}\psi ^{\lambda }\psi
^{\mu })(\overline{\partial }x^{\nu }+\overleftarrow{\partial _{r}}\overline{%
\psi }^{r}\overline{\psi }^{\nu }) \nonumber \\
&=&\partial _{\mu }\partial _{\nu }\theta \partial x^{\mu }\overline{%
\partial }x^{\nu }, \eqnum{23a}\end{aligned}$$
$$T_{F}^{(1)}=\frac{1}{2}\partial _{\mu }\partial _{\nu }\theta \overline{%
\partial }x^{\nu }\psi ^{\mu }, \eqnum{23b}$$
$$\overline{T}_{F}^{(1)}=\frac{1}{2}\partial _{\mu }\partial _{\nu }\theta
\partial x^{\nu }\overline{\psi }^{\mu } \eqnum{23c}$$
with condition $\Box \theta =constant$, $\Box \equiv \partial ^{\mu
}\partial _{\mu }.$ $\theta (x)$ in eq(23) is the background field corresponding to the singlet zero-norm state of eq(4). The induced ”symmetry” is calculated to be
$$\phi \rightarrow \phi +\Box \theta \eqnum{24}$$
and
$$h_{\mu \nu }\rightarrow h_{\mu \nu }+\partial _{\mu }\partial _{\nu }\theta .
\eqnum{25}$$
Equation (25) is merely a change of gauge in the linearized graviton and can be absorbed to the symmetry of the linearized graviton. Equation (24) is the ”symmetry” of the dilaton. The result that $\Box \theta $ is a constant is consistent with the fact that $\phi $ appears in the effecive equation of motion, constructed from vanishing $\sigma -$model $\beta -$function[@13]$,$ in an overall factor $e^{-2\phi }$ other than differentiated. The interesting result here is that we identify the constant $\square \theta $ to be the zero-norm state in equation (4). This completes the physical effects of all massless zero-norm states in the type II string spectrum. The ”symmetries” presented in equations (21) and (24) were derived in the first order WFA. They can be broken in the higher order correction. However, if one tries to generalize the superconformal deformation to [*second*]{} order in the WFA, one immediately meets the difficulty of nonperturbative nonnormalizibility of the 2d $\sigma -%
%TCIMACRO{\func{mod}}
%BeginExpansion
\mathop{\rm mod}%
%EndExpansion
$el$,$ and is forced to introduce counterterms which consist of an infinite number of massive tensor fields.[@14] This higher order effect is related to the stringy physics ($\alpha ^{\prime }\rightarrow \infty $) of string theory instead of point particle field theory. In fact, it was known that there exist important stringy bound states called (p,q) string which consists of p F-strings and q D-strings in the II B theory.[@15] The coupling of axion $\chi $ to (p,q) string, which is an higher order effect and so can not be seen in our first order WFA, breaks the symmetry in equation (21) down to integer shifts. On the other hand, it was known that the symmetry in eq.(24) was broken down to the discrete $\Box \theta \equiv
-2\left\langle \phi \right\rangle $ from the type II B supergravity. If we define
$$\rho =\chi +ie^{-\phi },\left\langle \rho \right\rangle \equiv \frac{\theta
}{2\pi }+\frac{i}{g_{s}}\equiv \tau , \eqnum{26}$$
these two discrete symmetries combine to form the well-known SL(2,Z) S-duality symmetry of II B string. Note that the nonlinearity of SL(2,Z) does not appear in our linear WFA. This is a generic feature of WFA in contract to the usual $\sigma $-model loop($\alpha ^{\prime }$) expansion. The former contains stringy phenomena (e. g. high energy symmetries) which can not be derived in the loop expansion scheme, while the latter is convenient to obtain the low energy effective field theory of the superstring. This will become clear when one considers the massive states of the string, which are crucial to make string theory different from the usual quantum field theory. An immediate application of this Type II B S-duality is the N=4, d=4 SUSY Yang-Mills S-duality[@16], where dyon with the electric charge p and magnetic charge q can be interpreted as the end points of the (p,q) string on the D3-brane. The $\tau $ parameter in this SUSY gauge theory is defined to be
$$\tau =\frac{\theta _{YM}}{2\pi }+\frac{i}{g_{YM}^{2}}, \eqnum{27}$$
and is interpreted to be the constant $\rho $ field of II B string in equation (26) associated to a stack of D3-branes.
Conclusion
==========
T-dual R-R zero-norm states motivate the introduction of D-branes into Type II string theory[*.* ]{}They serve as symmetry charge parameters of R-R tensor forms. The study in this paper reveals again that all space-time symmetries, including the [*discrete* ]{}T-duality and S-duality, are related to the zero-norm states in the spectrum. The unified description of S and T dualities makes one to speculate that they are all geometric symmetries (due to the redefinition of string backgrounds) and to conjecture the existence of a bigger discrete U-duality symmetry[@9],[@17] and its relation to the zero-norm state. In fact the SL(2,Z) S-duality of II B string led Vafa[@18] to propose a 12d F-theory, where $\tau $ is the geometric complex structure modulus of torus T$^{2}$. One can even generalize this zero-norm state idea to construct new stringy massive symmetries of string theory. In particular, the existence of some massive R-R zero-norm states and other evidences make us speculate that string may carry some massive R-R charges[@6]. Another interesting issue is the identification of D-brane charges with elements of K-theory groups.[@19] How T-dual R-R zero-norm states relate to K-theory groups is an interesting question to study.
Acknowledgments
===============
I would like to thank Pei-Ming Ho and Miao Li for comments. This research is supported by National Science Council of Taiwan, under grant number NSC 89-2112-M-009-006.
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[^1]: e-mail: jcclee@cc.nctu.edu.tw
|
---
abstract: |
We have identified a sample of 41 low-mass high–oxygen abundance outliers from the mass–metallicity relation of star-forming galaxies measured by @tremonti04. These galaxies, which have $8.6 <
12+\log(\mbox{O}/\mbox{H}) < 9.3$ over a range of $-14.4 > M_B > -19.1$ and $7.4 < \log (\mbox{M}_{\star}/\mbox{M}_{\odot}) < 10$, are surprisingly non-pathological. They have typical specific star formation rates, are fairly isolated and, with few exceptions, have no obvious companions. Morphologically, they are similar to dwarf spheroidal or dwarf elliptical galaxies. We predict that their observed high oxygen abundances are due to relatively low gas fractions, concluding that these are transitional dwarf galaxies nearing the end of their star formation activity.
author:
- 'Molly S. Peeples, Richard W. Pogge, & K. Z. Stanek'
title: 'Outliers from the Mass–Metallicity Relation I: A Sample of Metal-Rich Dwarf Galaxies from SDSS'
---
Introduction {#sec:intro}
============
There is a well-known positive correlation between galaxy luminosity and metallicity [@lequeux79; @garnett87]. @tremonti04 measured this relation for 53400 star-forming galaxies from the Sloan Digital Sky Survey (SDSS) Data Release 4 (DR4, @adelman06), estimating the gas phase oxygen abundance, [$12+\log(\mbox{O}/\mbox{H})$]{}, from region emission lines as the surrogate for “metallicity.” Using estimates of the galaxy stellar masses derived from the techniques of @kauffmann03, @tremonti04 showed that metallicity is better correlated with galaxy mass than with luminosity, which suggests that the observed relation arises from the fact that lower mass galaxies have lower escape velocities than higher mass galaxies and so lose metals more easily via, e.g., supernova winds [@larson74]. However, @dalcanton07 showed that even if a low-mass galaxy can selectively remove its metals via winds, subsequent star formation can essentially erase the effects on the gas-phase metallicity; both a low star formation rate and a large gas fraction are needed to retain low metal abundances. Other proposed origins for the mass–metallicity relation include lower star formation rates in less massive galaxies giving rise to fewer massive stars and therefore less metal production [@koppen07].
Despite the locus of galaxies in the mass–metallicity plane having intrinsically high scatter (e.g., a 1-$\sigma$ spread of $\pm 0.15$ dex in [$12+\log(\mbox{O}/\mbox{H})$]{} at $9.5 < \log(\mbox{M}_{\star}/\mbox{M}_{\odot}) < 9.6$), in the absence of good quality spectra it has become an increasingly common practice to deduce or assign metallicities to galaxies based simply on their luminosities. In particular, the idea persists that dwarf galaxies are always “metal-poor,” certainly a fair assumption in the Local Group. In this study however, we will show that there is a significant population of low-mass galaxies with high gas-phase oxygen abundances relative to the mass–metallicity relation defined by @tremonti04; high-mass low-metallicity outliers from the mass–metallicity relation will be the focus of an upcoming paper. As shown in Figure \[fig:ohmb\], we find 24 galaxies with $-17 \gtrsim
M_B \gtrsim -19$ mag and $12 + \log(\mbox{O}/\mbox{H} \gtrsim 9$ and 17 high-metallicity outliers with $M_B > -17$ mag. We explain how we selected this sample in §\[sec:outliers\] and verified the galaxies’ high oxygen abundances in §\[sec:metal\]. In §\[sec:disc\] we discuss some of the possible explanations for the existence of such a population of galaxies and explain why we predict that these galaxies should have relatively low gas masses. Specifically, in §\[sec:trans\] we discuss the evidence for our galaxies being so-called “transition” dwarf galaxies. Finally, we summarize our results and conclusions in §\[sec:conc\].
Finding Outliers from the Mass–Metallicity Relation {#sec:outliers}
===================================================
We began with the sample of $\sim 110000$ star-forming galaxies with measured gas-phase oxygen abundances and stellar masses from @tremonti04. High-metallicity outliers from the mass–metallicity locus can be due to main three causes, and our cuts were taken with these in mind. First, some galaxies can spuriously appear to be underluminous for their mass. Examples include highly inclined galaxies subject to strong internal extinction and regions in larger galaxies that SDSS has mistakenly flagged as a galaxy. Among relatively nearby galaxies (e.g., those in the Virgo cluster) some have large peculiar velocities relative to the Hubble flow, leading to a misestimation of their distance modulus and thus absolute magnitude. Secondly, rogue outliers can have spuriously high estimated metallicities; the galaxy, for example, may not be at high enough redshift to have the strongly constraining \[\]$\lambda
3727,9\,$Å emission line in the SDSS bandpass. Also, the strengths of the \[\] and \[\] lines paradoxically [*decrease*]{} with oxygen abundance, so at low signal-to-noise, the metallicity can be overestimated due to underestimated line fluxes. Finally, the object can be an honest outlier; these are the objects we want in our final sample. We found that a large number of cuts in different parameter spaces is useful for automatically rejecting many objects which would otherwise have to be thrown out by visual inspection. We ran two searches on the full @tremonti04 sample for high-metallicity outliers. For the first, we were very conservative with our cuts so as to be certain that the remaining galaxies are both statistically significant and not spurious. However, due to significant contamination at low luminosities, this “main” sample has no members with $\log\mbox{M}_{\star} < 9.15$. We therefore did a separate, less stringent search for the lowest luminosity outliers; to distinguish it from the main sample, we refer to this sample of lower mass galaxies as the “very low mass” sample.
Figure \[fig:ohmb\] shows where our sample falls in the [$12+\log(\mbox{O}/\mbox{H})$]{}–$M_B$ plane; all galaxy images are shown in Figure \[fig:images\] and summary information is presented in Table \[tbl:sample\]. For reference, in Figure \[fig:ohmb\] we plot the Small and Large Magellanic Clouds and Milky Way using $M_B$ from @karachentsev05. We recalculated [$12+\log(\mbox{O}/\mbox{H})$]{} for the SMC and LMC using the line ratios reported by @russell90 of six regions in the SMC and four regions in the LMC and the relation $12 +
\log(\mbox{O}/\mbox{H}) = 9.37 + 2.03\times \mbox{N}2 +
1.26\times\mbox{N}2^2 + 0.32\times\mbox{N}2^3$, where $\mbox{N}2 \equiv
\log([\mbox{\ion{N}{2}}]\lambda6584/\mbox{H}\alpha)$ [@pettini04; @kewley08]. We then transformed these abundances onto the @tremonti04 scale using the formulae given by @kewley08, resulting in average metallicities of 8.21 and 8.39 for the SMC and LMC, respectively. The Milky Way is plotted also plotted for reference at the Solar oxygen abundance of 8.86 [@delahaye06].[^1]
[rrclcrl]{} 193.9056 & $-1.32986$ & 8.68 & $-14.46$ & 7.65 & 0.0029 & blue core, IC 225\
180.4589 & $55.14507$ & 8.73 & $-14.56$ & 7.73 & 0.0035 &\
190.2097 & $4.52583$ & 9.06 & $-14.64$ & 7.39 & 0.0025 & blue core, VCC 1855\
179.0295 & $64.35073$ & 8.83 & $-14.89$ & 8.06 & 0.0047 &\
228.0340 & $1.58571$ & 8.76 & $-15.14$ & 8.21 & 0.0065 &\
126.6407 & $25.49979$ & 8.69 & $-15.23$ & 8.07 & 0.0072 &\
227.2679 & $0.82197$ & 8.85 & $-15.35$ & 8.17 & 0.0055 &\
193.6735 & $2.10447$ & 9.12 & $-15.50$ & 8.40 & 0.0029 & bright core\
126.6633 & $25.59821$ & 8.86 & $-15.87$ & 8.59 & 0.0078 &\
208.3607 & $5.20778$ & 8.94 & $-16.19$ & 8.64 & 0.0027 & blue core\
128.5841 & $50.45248$ & 8.95 & $-16.36$ & 8.69 & 0.0114 &\
40.3408 & $0.05813$ & 8.90 & $-16.56$ & 8.70 & 0.0227 &\
212.9768 & $53.93956$ & 8.85 & $-16.57$ & 8.81 & 0.0064 & blue core\
190.5886 & $2.06662$ & 9.00 & $-16.73$ & 8.40 & 0.0044 & bright core\
36.6179 & $1.16053$ & 8.82 & $-16.86$ & 7.92 & 0.0051 & blue core\
117.1775 & $26.53979$ & 8.94 & $-16.88$ & 9.08 & 0.0155 &\
182.7831 & $0.95682$ & 8.95 & $-16.98$ & 9.13 & 0.0206 &\
133.8883 & $31.21168$ & 9.10 & $-16.99$ & 9.29 & 0.0068 &\
25.7673 & $14.52434$ & 9.12 & $-17.16$ & 9.25 & 0.0284 &\
225.5670 & $38.80631$ & 9.12 & $-17.32$ & 9.45 & 0.0147 & bright core\
142.9797 & $39.27156$ & 9.07 & $-17.36$ & 9.35 & 0.0275 &\
139.3797 & $33.47552$ & 9.15 & $-17.52$ & 9.46 & 0.0221 &\
202.2088 & $-0.90846$ & 9.14 & $-17.58$ & 9.26 & 0.0217 &\
39.0487 & $-7.73400$ & 9.12 & $-17.66$ & 9.16 & 0.0314 &\
143.4695 & $41.07965$ & 9.14 & $-17.92$ & 9.54 & 0.0145 & bright core\
187.1760 & $44.09813$ & 9.13 & $-17.93$ & 9.35 & 0.0239 &\
182.8574 & $44.43604$ & 9.14 & $-17.97$ & 9.52 & 0.0231 &\
211.3199 & $54.20347$ & 9.13 & $-18.15$ & 9.54 & 0.0417 &\
258.4301 & $57.18840$ & 9.16 & $-18.18$ & 9.61 & 0.0290 &\
208.9341 & $4.24367$ & 9.14 & $-18.19$ & 9.68 & 0.0295 &\
227.6711 & $41.16220$ & 9.13 & $-18.20$ & 9.61 & 0.0316 &\
352.8635 & $13.90885$ & 9.14 & $-18.34$ & 9.74 & 0.0324 &\
206.4402 & $63.85402$ & 9.14 & $-18.36$ & 9.58 & 0.0313 &\
153.2177 & $12.34451$ & 9.20 & $-18.39$ & 9.57 & 0.0315 &\
177.7870 & $49.69415$ & 9.14 & $-18.41$ & 9.41 & 0.0482 &\
144.1848 & $33.91996$ & 9.14 & $-18.45$ & 9.71 & 0.0426 &\
162.6325 & $0.36061$ & 9.16 & $-18.60$ & 9.68 & 0.0384 &\
195.6390 & $-3.33894$ & 9.26 & $-18.66$ & 9.55 & 0.0471 &\
235.5660 & $51.73211$ & 9.16 & $-18.87$ & 9.83 & 0.0425 &\
163.2191 & $43.42840$ & 9.27 & $-18.92$ & 9.90 & 0.0242 &\
245.3673 & $40.20360$ & 9.25 & $-19.05$ & 9.91 & 0.0285 & blue core\
Main Sample Selection {#sec:main}
---------------------
Table \[tbl:cuts\] summarizes our selection of the main sample of 24 metal-rich low-mass galaxies. Spiral galaxies are known to have radial metallicity gradients such that the nuclear abundances are higher than averages over whole galaxies. Thus, if only a small fraction of the galaxy is within the SDSS spectroscopic $3\arcsec$ diameter fibers, the measured metallicity can appear to be artificially high relative to other galaxies. Following @tremonti04, we make an initial cut by requiring that the fraction of the galaxy covered by the fiber to be greater than 10%. (In our final cut, we follow @michel08 and change this lower bound to 20%, which only eliminates two of the galaxies contained in the penultimate sample listed in Table \[tbl:cuts\].) We wanted the sample to be statistically significant, so outliers were then determined via a series of cuts in the [$12+\log(\mbox{O}/\mbox{H})$]{} versus absolute $B$-band magnitude ($M_B$), $g$-band ($M_g$), and stellar mass (M$_{\star}$) planes, as demonstrated with M$_{\star}$ in Figure \[fig:cuts\]. For example, we divided the 52477 objects with SDSS magnitude errors $< 0.1$ mag in all bands into bins of $M_B$ of width $\Delta M_B = 0.4$ mag; in each bin we kept the 2.5% with the highest [$12+\log(\mbox{O}/\mbox{H})$]{}. We likewise took bins of [$12+\log(\mbox{O}/\mbox{H})$]{} of width 0.1 dex and kept 2.5% of the objects with the faintest $M_B$. Similar cuts were made with $M_g$ (binsize $\Delta M_g = 0.4$ mag) and $\log\mbox{M}_{\star}$ (binsize of $\Delta
\log\mbox{M}_{\star}=0.1$ dex). $M_g$ was calculated from the SDSS $g$-band magnitude and the spectroscopic redshift, $M_B$ was calculated using the low-resolution spectral templates of @assef08, and the stellar masses from the @tremonti04 sample were measured using a combination of SDSS colors and spectra as described by @kauffmann03. Only 58 objects survived this series of cuts.
At this point, we made a redshift cut: galaxies must have $z > 0.024$ in order to have the \[\] $\lambda\lambda 3727,9$Å emission line pair in their SDSS spectrum, and as this is a highly constraining line for the metallicity, we determined that it should be in the spectrum in order to remove a possible source of systematics and so that we can measure the metallicity with a diagnostic that uses this line. Seven $z
< 0.024$ galaxies pass our subsequent [$12+\log(\mbox{O}/\mbox{H})$]{} error and visual inspection cuts; after studying their spectra, as discussed below, we chose to keep these galaxies in our main sample. We also forced the cited $\pm 1\,\sigma$ error in [$12+\log(\mbox{O}/\mbox{H})$]{} to be less than 0.05 dex; metallicities with large uncertainties are obviously more likely to be spurious than ones with small uncertainties. Unsurprisingly, the $z >
0.024$ redshift cut (after all of the other cuts) also removed any galaxies with fiber fractions $<0.1$ which would have survived to that stage. We find, however, that this large number of cuts in various parameter spaces dramatically reduced the number of objects that had to be removed “by eye.” The final visual-inspection cut removed only two objects with nearby potential photometric contaminations. Finally, we re-examined the fiber fraction cut; following @michel08 we allowed the lower-limit fiber fraction to be 20% (excluding two objects from the sample). The final exclusion was a barred galaxy with a diameter of $\sim 10$; the SDSS fiber size is $3\arcsec$, but this galaxy is labeled as having a fiber fraction of $\sim 60$%. This galaxy probably has spuriously low SDSS Petrosian magnitudes. The 24 galaxies in this main sample have a range of fiber fractions between 25% and 60% and a redshift range of $0.007 < z < 0.048$.
[ll]{} Fiber fraction $> 0.1$ & 107992\
$k$-corrected SDSS magnitude errors $< 0.1$ mag & 48327\
97.5% large (O/H) and small $M_B$ & 227\
97.5% large (O/H) and small $M_g$ & 202\
99% large (O/H) w.r.t. $M_B$ & 87\
99% large (O/H) w.r.t. M$_{\star}$ & 66\
97.5% small M$_{\star}$ w.r.t. (O/H) & 51\
redshift $> 0.024$ for \[\] & 34\
[$12+\log(\mbox{O}/\mbox{H})$]{} error $< 0.05$ dex & 22\
Visual inspection & 20\
Fiber fraction $> 0.2$ and photometry & 17\
redshift $z < 0.024$ but surviving subsequent cuts & $7$
Very Low Mass Sample Selection {#sec:lowm}
------------------------------
It is obvious from Figure \[fig:ohmb\] that the selection of our main sample artificially imposes a lower mass cutoff. This does not mean that there is not a statistically significant sample of interesting galaxies with $M_B > -17$; it just means that there are more contaminating objects with measured high metallicities in the low-luminosity regime than at brighter magnitudes. That is, the most extreme outliers with $M_B < -17$ in one parameter space (e.g., the [$12+\log(\mbox{O}/\mbox{H})$]{} vs. $M_B$ plane) are not also outliers in one of the others (e.g., the [$12+\log(\mbox{O}/\mbox{H})$]{} vs. $\log\mbox{M}_{\star}$ plane). The [*true*]{} very low luminosity mass–metallicity outliers therefore have lower abundances than these spurious outliers. We therefore did a separate search for high-metallicity galaxies at extremely low luminosities and masses, as summarized in Table \[tbl:lowmcuts\]. We chose to not apply a fiber fraction cut for this sample because lower luminosity galaxies are preferentially closer and therefore subtend a larger angle on the sky, making it more difficult for a substantial fraction of the galaxy to be within the 3 SDSS fiber diameter. We therefore began the selection with a magnitude error cut, like the one for the main sample. We also removed objects with fiber fractions $>0.2$ occupying the parameter space already excluded by the main sample: any objects with $M_B$ brighter than $-17$ mag, $\log\mbox{M}_{\star}>9.15$, and a high [$\log(\mbox{O}/\mbox{H})$]{} at the 95% level with respect to M$_{\star}$ were excluded. The cuts in [$12+\log(\mbox{O}/\mbox{H})$]{} relative to $M_B$, $M_g$ and M$_{\star}$ are all less stringent than for the main sample, but because for this sample we are interested in the [*very*]{} low mass objects, we took a strong (99% level) cut in M$_{\star}$ relative to [$12+\log(\mbox{O}/\mbox{H})$]{}. Six galaxies were excluded due to metallicity re-estimation considerations (as discussed below). Three of the remaining galaxies are potentially members of the Virgo cluster; of these, only two remain clear outliers from the $M_B$–metallicity locus when their distance moduli are shifted to the Virgo value of $30.74$ magnitudes [@ebeling98]. (We do not attempt to correct the stellar mass estimates due to the effects of peculiar velocity.) Finally, the 95% high (O/H) with respect to $M_g$ cut excludes the galaxy IC 225 from the sample, which is pointed out by @gu06 to have a relatively high metallicity and a compact blue core. As this galaxy passes all of the other cuts and is clearly interesting, we added it back to the sample, raising the total number of galaxies in the very low mass sample to 17 and the full sample to 41. The very low mass sample has fiber fractions ranging from 0.04 to 0.45; more than half of the 17 galaxies in this sample have fiber fractions below 0.1 and only 5 are above 0.2. Also, none of these galaxies would have survived the main sample redshift cut, as the very low mass galaxies occupy the redshift range $0.00249 < z < 0.0227$, with the most nearby objects being only $\sim 12$ Mpc away.
[ll]{} $k$-corrected SDSS magnitude errors $< 0.1$ mag & 52744\
84% high (O/H) w.r.t. $M_B$ & 8405\
$M_B > -17$ mag & 217\
$\log\mbox{M}_{\star}<9.15$ & 201\
84% high (O/H) w.r.t. M$_{\star}$ & 192\
[$12+\log(\mbox{O}/\mbox{H})$]{} error $< 0.05$ dex & 139\
99% low $\log\mbox{M}_{\star}$ w.r.t. (O/H) & 105\
[*exclude*]{} 95% high (O/H) w.r.t. M$_{\star}$, fiber fraction $>0.1$ & 78\
95% high (O/H) w.r.t. $M_g$ & 32\
Visual inspection & 23\
Metallicity comparison & 17\
Absolute magnitude correction & 16
Measuring High Oxygen Abundances {#sec:metal}
--------------------------------
Are these 41 galaxies true outliers from the mass–metallicity relation, or are they just the tail of the scatter of the @tremonti04 measurements? As a first check, the galaxies fall where expected on the standard @baldwin81 (BPT) diagrams for $\log(\mbox{[\ion{O}{3}]}\,\lambda 5007/\mbox{H}\beta)$ vs.$\log(\mbox{[\ion{N}{2}]}\,\lambda 6548/\mbox{H}\alpha)$ and $\log(\mbox{([\ion{S}{2}]}\,\lambda\lambda 6717+31)/\mbox{H}\alpha)$ [@baldwin81; @kewley06b]. We note that there are several complications with measuring high metallicities using visible wavelength spectra (see @bresolin06 for a thorough review). The main problem is that cooling is more efficient at high metallicities, which translates into lower nebular temperatures and hence weaker \[\] and \[\] visible-wavelength lines, with the primary cooling load shifting to the far-infrared fine structure emission lines that cannot be readily observed. The result is that at high abundances the visible wavelength spectra become increasingly insensitive to changes in [$\log(\mbox{O}/\mbox{H})$]{}. In fact, some diagnostics, like the traditional R$_{23}$ line index, effectively saturate at high metallicity [@kewley08], making it rather difficult to accurately measure even relative abundances. Because we would like to verify the high estimated oxygen abundances and we cannot reproduce the @tremonti04 abundance calculations, which are based on a Bayesian statistical method, we measured [$\log(\mbox{O}/\mbox{H})$]{} in our galaxies using two recommended methods from @kewley08.
Another possible source of systematic inaccuracy in the abundance estimates is the treatment of extinction corrections. The corrections traditionally applied use the Balmer decrement and assume a simple uniform foreground obscuring screen model like that used for stars to estimate $A_V$ and correct the other emission lines. It is expected, however, that extinction towards extended sources like regions is better described as a clumpy screen, for example the turbulent screen models of @fischera05. In these, use of a simple screen tends to systematically [*overestimate*]{} the extinction correction for the $\lambda\lambda$3727,29Å emission line, leading to a systematic [*underestimate*]{} of the gas-phase oxygen abundance. In all of our galaxies the Balmer decrement measurements are consistent with low $A_V$ ($\lesssim 0.5$) for a simple screen extinction model, so this is not a big effect compared to other sources of measurement error given the attenuation curves in @fischera05.
@kewley08 have measured the gas-phase metallicities in star forming galaxies from SDSS using ten different methods, including that of @tremonti04; they also provide average relations relating each pair of methods. While no one metallicity estimate is strictly believable—i.e., the [*true*]{} Oxygen-to-Hydrogen ratio—the [*relative*]{} measurements are generally robust [@kewley08]. Using the Data Release 6 SDSS spectra [@adelman08], we subtracted the underlying stellar continuum using the STARLIGHT program [@fernandes05]. We then calculated [$12+\log(\mbox{O}/\mbox{H})$]{} using the revised @kewley02 method given in Equation (A3) of @kewley08, $$\begin{aligned}
\label{eqn:kd02}
\nonumber \log(\mbox{[\ion{N}{2}]}/\mbox{[\ion{O}{2}]}) & = & 1106.8660
- 532.1451Z + 96.37326Z^2 \\
& & - 7.8106123Z^3 + 0.32928247Z^4,\end{aligned}$$ where $Z \equiv 12 + \log(\mbox{O}/\mbox{H})$. Like @kewley08, we found the roots of this equation using the `fz_roots` program in IDL. We also measured the metallicity using the “O3N2” method of @pettini04, as recommended by @kewley08, where $$\label{eqn:pp04}
12 + \log(\mbox{O}/\mbox{H}) = 8.73 - 0.32\times
\log\left(\frac{[\mbox{\ion{O}{3}}]\lambda 5007/\mbox{H}\beta}{\mbox{[\ion{N}{2}]}\,\lambda 6584/\mbox{H}\alpha} \right).$$ We choose to not use the R$_{23}$ diagnostic for measuring [$12+\log(\mbox{O}/\mbox{H})$]{}because it is known to be difficult to calibrate at these abundances [@kewley02]. In particular, while the seventeen $z > 0.024$ galaxies in our main sample span $\sim 0.2$ dex on the @tremonti04 scale, they span $\sim 0.6$ dex when their metallicities are calculated using the R$_{23}$ methods of either @mcgaugh91 or @zaritsky94, reflecting the fact that the R$_{23}$ parameter essentially saturates at high oxygen abundances [@bresolin07].
The oxygen abundances for our 41 galaxies plotted against these adopted metallicities in Figure \[fig:convZ\]. The dotted line indicates equal measurements; the other two lines are the empirically determined relations from @kewley08. The galaxies in our sample fall preferentially above the relation for the @pettini04 method, but surprisingly those galaxies for which it is measurable fall [*below*]{} the relation for the @kewley02 method. One possible explanation for this discrepancy is that because \[\] and \[\] are widely separated in wavelength, a misestimation of the extinction or a poorly fitted continuum (especially for the weak \[\] emission line) can lead to underestimated abundances. Specifically, there is often a degeneracy when fitting the stellar continuum between the extinction and the stellar population. While the continuum fitting for our spectra are fairly good (i.e., they exhibit low residuals after the stellar template is subtracted), we tested the sensitivity to the continuum level at \[\]$\lambda 3727$Åby calculating the @kewley02 metallicity when the continuum is over- and underestimated; the mean of these two extreme metallicities is still systematically larger than expected by the mean relation, implying that the shift is not due to continuum fitting error. We quantify the differences between the @pettini04 and @kewley02 methods by examining the product of the vertical displacement of each galaxy (for an optimally fit continuum) from the two @kewley08 relations; only one galaxy in the main sample has a significantly high @tremonti04 metallicity relative to both of the other indicators. However, this [$12+\log(\mbox{O}/\mbox{H})$]{} was still high enough that even when the @tremonti04 metallicity was replaced with the one expected from either relation, the galaxy still passes the cuts in both the $M_g$- and M$_{\star}$-metallicity planes. While the $z<0.024$ galaxies do not have measurable \[\] (and hence we cannot use the @kewley02 diagnostic for them), the low redshift sample occupies a similar part of the @tremonti04 versus @pettini04 parameter space as the galaxies in the main sample. For the very low-mass sample of galaxies, out of the 22 galaxies which passed our visual inspection test, we excluded 6 because they were more than $0.1$ dex above the @pettini04 relation from @kewley08; this is actually a more stringent cut than used for the main sample because several of the galaxies in the main sample with $>0.1$ dex deviations from the @pettini04 relation have compensating negative deviations relative to the @kewley02 scale. We interpret these results to mean that the 41 galaxies we have identified are true high-metallicity low-mass outliers from the mass–metallicity relation.
Discussion {#sec:disc}
==========
The 41 dwarf galaxies selected as described in §\[sec:outliers\] are surprisingly non-pathological. They have undisturbed stellar morphologies (see Figure \[fig:images\]). Furthermore, given that by selection these galaxies have both low luminosities and high metallicities and that there is a trend for redder, brighter galaxies to have higher oxygen abundances (see, e.g., @cooper08) these metal-rich dwarfs do not occupy unexpected region of the color–magnitude diagram (see Figure \[fig:cmd\] and also §\[sec:trans\]). How, then, did they come to have such high oxygen abundances? One obvious possibility is that these metallicities are due to an environmental effect, but as we describe in §\[sec:enviro\], the galaxies in our sample are non-interacting and rather isolated. Though several models predict that these galaxies should have high specific star formation rates, we show in §\[sec:sfr\], that these galaxies have normal star formation rates for their masses. We explain in §\[sec:gasfrac\] why we predict that these metal-rich dwarf galaxies should have relatively low gas fractions, and we discuss the implications of this prediction in the broader context of transition-type dwarf galaxies in §\[sec:trans\].
Environment {#sec:enviro}
-----------
While the origin of scatter in the mass–metallicity relation is unknown, one popular proposal is that environment affects metallicity. Recently, @michel08 studied close pairs of star-forming galaxies in SDSS with projected separations of $< 100\,$kpc and radial velocity separations of $< 350\,$kms$^{-1}$. Their results indicate that for minor mergers, the less massive galaxy is likely to be preferentially more metal rich than predicted by the mass–metallicity relation. Likewise, @cooper08, after accounting for correlations within the color–magnitude diagram, find a strong positive correlation between metallicity and overdensity for $0.05 < z < 0.15$ SDSS-selected star-forming galaxies that can account for up to $\sim 15$% of the scatter in the mass–metallicity relation. In light of these results, we searched for nearby neighbors of the seventeen $z > 0.024$ main sample metal-rich dwarf galaxies in a cylindrical volume of depth $\pm
1000$kms$^{-1}$ and projected radius of 1 Mpc. We find that our galaxies are relatively isolated; none of them would have made it into the @michel08 sample. Seven out of these seventeen galaxies in our sample have no neighbors within this volume. Of the 10 remaining galaxies, four have no neighbors within 500kms$^{-1}$ and 500kpc, and only two have any neighbors within 500kms$^{-1}$ and 100kpc. One of these two seems to be on the outskirts of a nearby cluster, but it is unclear whether or not it is physically associated with the cluster (and there is only one small, faint, non-star-forming galaxy within the @michel08 volume). The other galaxy with a nearby neighbor is $\sim 400$kms$^{-1}$ and 85kpc from a less-massive relatively metal-poor ($12 + \log[\mbox{O}/\mbox{H}]
\approx 8.6$) star-forming galaxy. We therefore conclude that interactions with neighbors do not explain the observed high abundances of our main sample of $\log\mbox{M}_{\star}\sim 9.5$ galaxies.
The galaxies in the very low mass sample (as well the $z < 0.024$ galaxies in the main sample), however, are generally at low enough redshift that an automated search for neighbors in velocity and projected distance space is difficult. Like the galaxies in the main sample, none of the lower-mass galaxies are in obviously interacting systems. Only one has an clear companion; it is a $\log\mbox{M}_{\star}
= 9.13$ galaxy which appears to be a satellite of a non-starforming companion 33kpc and $\sim 120$kms$^{-1}$ away. Also, while two of the seventeen galaxies in the very low mass sample may be in the Virgo cluster, clearly a rich environment cannot explain the high oxygen abundances observed in all of these galaxies.
Star Formation Rates {#sec:sfr}
--------------------
@dalcanton07 finds that for a galaxy to have a low metallicity, it must have both a low star formation rate and a high gas fraction; she argues that the observed low star formation efficiency in galaxies with circular velocities $\lesssim 120$kms$^{-1}$ can explain the mass–metallicity relation. @koppen07 similarly suggest that the mass–metallicity relation could be due to a mass–star formation rate relation: a galaxy with a lower rate of star formation will have relatively fewer massive stars, and therefore a lower oxygen abundance. In either of these scenarios, we might expect the mass–metallicity outliers to have high star formation rates for their masses. On the other hand, @ellison08a find that at low stellar masses, galaxies with higher specific star formation rates tend to have [*lower*]{} oxygen abundances. As Figure \[fig:sfrm\] shows, the galaxies in the main sample do not have preferentially high or low instantaneous specific star formation rates, while typical star formation rates for the the very low mass sample galaxies are $\sim 0.3$dex [*lower*]{} than expected given their masses.
Effective Yields and Gas Fractions {#sec:gasfrac}
----------------------------------
So why do these galaxies have such high oxygen abundances? Consider a closed box star forming system (i.e., a system with no gas inflow or outflow). The metallicity $Z\equiv\;$(mass of metals in gas phase)/(total gas mass) is $$\label{eqn:closedbox}
Z = y\ln\left(\frac{1}{f_{\mbox{\scriptsize gas}}}\right),$$ where $f_{\mbox{\scriptsize gas}}\equiv M_{\mbox{\scriptsize
gas}}/(M_{\star}+M_{\mbox{\scriptsize gas}})$ is the gas fraction and $y\equiv\;$(mass of metals in gas)/(mass of metals in stellar remnants and main sequence stars) is the metal yield. An immediately striking aspect of Equation \[eqn:closedbox\] is that there is no implicit dependence on the total galaxy mass; the deviations from a universal metallicity[^2] observed via the mass–metallicity relation are presumably then due to the fact that galaxies are not scaled closed box versions of one another: variations of the yield with mass, a dependence on the gas fraction with mass, or a combination of these effects plays a role. For a galaxy to have a higher metallicity than other galaxies of the same mass, Equation \[eqn:closedbox\] tells us that it must have either a relatively high yield or a relatively low gas fraction. Observationally, the yield—which depends on both star formation physics as well as gas inflow and outflow—is a difficult quantity to measure; one popular way to address this problem is to define an effective yield, $$\label{eqn:yeff}
y_{\mbox{\scriptsize eff}} \equiv \left[\frac{Z_{\mbox{\scriptsize
gas}}}{\ln (1/f{\mbox{\scriptsize gas}})}\right],$$ as the yield the galaxy [*would*]{} have were it actually a closed system. Given the galaxy’s metallicity and gas fraction, one can then easily calculate $y_{\mbox{\scriptsize eff}}$. A common explanation for the mass–metallicity relation is that the effective yield is positively correlated with galaxy mass, so that lower mass galaxies are able to preferentially lose metals to the intergalactic medium via winds because of their relatively shallow potential wells [see e.g., @larson74; @finlator08 and references therein]. In particular, @tremonti04 tested this idea by comparing effective yields and baryonic masses. However, because when estimating the effective yield from Equation \[eqn:yeff\], @tremonti04 use the measured [$12+\log(\mbox{O}/\mbox{H})$]{} values, outliers in the mass–metallicity relation are practically guaranteed to be outliers in the effective yield–baryonic mass relation.
Regardless, it is obvious from Equation \[eqn:closedbox\] that our metal-rich dwarfs must either have unusually high yields or unusually low gas fractions for their masses. Galaxy yields can be affected by three processes: star formation, metal-deficient gas inflows, and metal-rich gas outflows. As shown in Figure \[fig:sfrm\] and discussed in §\[sec:sfr\], we find that the star formation rates for this population of galaxies are consistent with those of other galaxies of similar masses. @dalcanton07 has shown that metal-poor gas inflow is insufficient to explain the typically low metal abundances of low-mass galaxies; we therefore conclude that a lower inflow rate is insufficient to explain the higher abundances of our galaxies. It is possible that these galaxies are less effective at driving outflows than other dwarfs. For example, @ellison08a find that, at fixed mass, galaxies with smaller half-light radii tend to have higher abundances; this picture is consistent with the idea that it is more difficult to drive winds from deeper potential wells. (@tremonti04, on the other hand, find no correlation with how concentrated a galaxy’s light is and its oxygen abundance.) While the galaxies in our main sample do tend to have small radii for their masses (half-light radii of less than 2kpc), the typical [$12+\log(\mbox{O}/\mbox{H})$]{} for galaxies of similar masses and radii is still $\sim 0.3$dex lower (about $2\sigma$) than that of our galaxies. Furthermore, the very low mass sample galaxies do not have preferentially small radii for their masses, leading us to conclude that while small radii may be a contributing factor to why some of these galaxies have been able to retain their metals, size alone does not tell the whole story.
@dalcanton07 calculated that enriched gas outflows can only severely decrease a galaxy’s effective yield if the gas fraction is sufficiently high; that is, for low gas fractions, even a very strong outflow cannot drastically decrease the effective yield—and thus measured abundance. Likewise, if a galaxy has a relatively low gas fraction, then only a small amount of pollution is needed to enrich the gas and cause the measured abundance to be high. We therefore predict that these mass–metallicity outliers have anomalously low gas masses relative to other isolated galaxies of similar luminosities and star-formation rates. There are unfortunately no data in the literature for our galaxies that we can call upon to lend observational support to this prediction. However, @lee06 have found that, in a sample of 27 nearby dwarf irregular galaxies, the gas-phase oxygen abundance is negatively correlated with the -measured gas-to-stellar mass ratio, which lends observational credence to our expectation.
Transitional Dwarf Galaxies {#sec:trans}
---------------------------
If these high-metallicity dwarf galaxies really do have relatively little gas, then they should be rapidly approaching the end of their star formation. Specifically, these galaxies are likely to be transitioning from gas-rich dwarf irregulars (dIrr) to gas-deficient dwarf spheroidals (dSph) or the more massive dwarf ellipticals (dE). In general, so-called dIrr/dSph transitional dwarfs have similar star-formation histories as their currently non-starforming dSph cousins: both galaxy types typically have a mix of old and intermediate-age stellar populations [@grebel03]. In a study of five nearby star-forming transitional dwarfs, @dellenbusch07 found that these relatively isolated galaxies seem to have unusually high oxygen abundances for their luminosities, much like our sample of mass–metallicity outliers.[^3] @grebel03 stress that the only difference between dSph galaxies and the transitional dIrr/dSph galaxies is the absence of star formation and of gas in dSph galaxies. While much of this conclusion is based on galaxies in higher-density environments than ours are (i.e., the @grebel03 dwarf galaxies are mostly in the Local Group), we find it likely that our galaxies are part of a similar transition population: morphologically, they have smooth, undisturbed profiles (see Figure \[fig:images\]), indicating a stronger relationship to dwarf spheroidals than to the dwarf irregulars. In particular, for all of our galaxies we note a lack of the irregular flocculent structures or spiral features often associated with dIrr galaxies.
In fact, the only remarkable morphology any of the galaxies in our sample displays are the bright—and often very blue—cores noticed in ten of the galaxies. These galaxies are shown in Figure \[fig:blue\] and noted in Table \[tbl:sample\]. Some of these cores appear to have merely much higher surface brightnesses than their surroundings, but some are decidedly blue: @gu06 measure a difference of $\Delta(g-r) \gtrsim 0.3$ mag arcsec$^{-2}$ from the center of IC 225[^4] relative to isophotes at $>5$. These bright cores are almost certainly not a signature of active galactic nuclei; as mentioned in §\[sec:metal\], these galaxies fall squarely within the star-forming locus of the normal BPT line-ratio diagnostic diagrams [@baldwin81; @kewley06b]. @boselli08 find that for similar transitional dwarf galaxies with blue centers in the Virgo cluster (including VCC 1855, the least massive galaxy in our sample with $\log [\mbox{M}_{\star}/\mbox{M}_{\odot}]\sim 7.4$), star formation is limited to these central blue nuclei. (@boselli08 attribute this centralized star formation to the effects of ram pressure stripping preferentially removing gas from the outer regions of the galaxies. However, not all of the galaxies in our sample with bright nuclei are in the kinds of rich environments like the Virgo cluster where ram pressure stripping expected to play an important role.) Intriguingly (yet unsurprisingly), most of our galaxies which appear to be approaching or on the red sequence (see Figure \[fig:cmd\]) also have bright and/or blue cores, implying that their colors are already dominated by their non-starforming regions. Furthermore, having star formation limited to a relatively small region of the galaxy supports the idea that there is relatively little fuel available for forming stars, and thus that these galaxies are on their way to becoming standard dwarf spheroidal or dwarf elliptical galaxies.
Conclusions {#sec:conc}
===========
We have identified a sample of 41 low-luminosity high–oxygen abundance outliers from the mass–metallicity relation.
1. These galaxies are fairly isolated and none show any signs of morphological disturbance. It is likely, however, that the handful of galaxies in this sample which are possible group or cluster members have disturbed [*gas*]{} morphologies, due to e.g., ram-pressure stripping. However, environmental effects cannot account for the large gas-phase oxygen abundances observed in all of these galaxies.
2. Based on the effective yield considerations of @dalcanton07 and various related observational results [@grebel03; @dellenbusch07; @lee06; @boselli08], we predict that these galaxies should have low gas fractions relative to other dwarf galaxies of similar stellar masses and star formation rates in similar environments but with more typical (i.e., lower) oxygen abundances. Furthermore, in this scenario, the [*stellar*]{} metallicities should be consistent with that of other dwarfs of similar luminosity, i.e., the stellar metallicities should be lower than the measured gas-phase abundances.
3. These mass–metallicity outliers appear to be “transitional” dwarf galaxies: because they are running out of star-forming fuel, they are nearing the end of their star formation and becoming typical isolated dwarf spheroidals. This conclusion is most sound for the $\mbox{M}_{\star} < 9.1 M_{\sun}$ galaxies which are observed to have low specific star formation rates. More data—specifically, gas fractions and stellar metallicities—are needed to help elucidate the situation for the higher mass galaxies.
We would like to thank Roberto Assef for adapting his low-resolution template code for calculating $M_B$ for the SDSS sample for us, Cayman Unterborn for letting us use his data and code for generating the color–magnitude diagram in Figure \[fig:cmd\], and Sara Ellison for giving us access to some of the data from @ellison08a. We are grateful to the anonymous referee, David Weinberg, Christy Tremonti, and Thorsten Lisker for helpful comments on the text. We would also like to thank José Luis Prieto, Jeff Newman, Mike Cooper, Kristian Finlator, John Moustakas, Henry Lee, Paul Martini, and Todd Thompson for useful discussions and suggestions.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is `http://www.sdss.org/`.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
The STARLIGHT project is supported by the Brazilian agencies CNPq, CAPES and FAPESP and by the France-Brazil CAPES/Cofecub program.
[33]{} natexlab\#1[\#1]{}
, J. K., [Ag[ü]{}eros]{}, M. A., [Allam]{}, S. S., [Allende Prieto]{}, C., [Anderson]{}, K. S. J., [Anderson]{}, S. F., [Annis]{}, J., [Bahcall]{}, N. A., [Bailer-Jones]{}, C. A. L., [Baldry]{}, I. K., [Barentine]{}, J. C., [Bassett]{}, B. A., [Becker]{}, A. C., [Beers]{}, T. C., [Bell]{}, E. F., [Berlind]{}, A. A., [Bernardi]{}, M., [Blanton]{}, M. R., [Bochanski]{}, J. J., [Boroski]{}, W. N., [Brinchmann]{}, J., [Brinkmann]{}, J., [Brunner]{}, R. J., [Budav[á]{}ri]{}, T., [Carliles]{}, S., [Carr]{}, M. A., [Castander]{}, F. J., [Cinabro]{}, D., [Cool]{}, R. J., [Covey]{}, K. R., [Csabai]{}, I., [Cunha]{}, C. E., [Davenport]{}, J. R. A., [Dilday]{}, B., [Doi]{}, M., [Eisenstein]{}, D. J., [Evans]{}, M. L., [Fan]{}, X., [Finkbeiner]{}, D. P., [Friedman]{}, S. D., [Frieman]{}, J. A., [Fukugita]{}, M., [G[ä]{}nsicke]{}, B. T., [Gates]{}, E., [Gillespie]{}, B., [Glazebrook]{}, K., [Gray]{}, J., [Grebel]{}, E. K., [Gunn]{}, J. E., [Gurbani]{}, V. K., [Hall]{}, P. B., [Harding]{}, P., [Harvanek]{}, M., [Hawley]{}, S. L., [Hayes]{}, J., [Heckman]{}, T. M., [Hendry]{}, J. S., [Hindsley]{}, R. B., [Hirata]{}, C. M., [Hogan]{}, C. J., [Hogg]{}, D. W., [Hyde]{}, J. B., [Ichikawa]{}, S.-i., [Ivezi[ć]{}]{}, [Ž]{}., [Jester]{}, S., [Johnson]{}, J. A., [Jorgensen]{}, A. M., [Juri[ć]{}]{}, M., [Kent]{}, S. M., [Kessler]{}, R., [Kleinman]{}, S. J., [Knapp]{}, G. R., [Kron]{}, R. G., [Krzesinski]{}, J., [Kuropatkin]{}, N., [Lamb]{}, D. Q., [Lampeitl]{}, H., [Lebedeva]{}, S., [Lee]{}, Y. S., [Leger]{}, R. F., [L[é]{}pine]{}, S., [Lima]{}, M., [Lin]{}, H., [Long]{}, D. C., [Loomis]{}, C. P., [Loveday]{}, J., [Lupton]{}, R. H., [Malanushenko]{}, O., [Malanushenko]{}, V., [Mandelbaum]{}, R., [Margon]{}, B., [Marriner]{}, J. P., [Mart[í]{}nez-Delgado]{}, D., [Matsubara]{}, T., [McGehee]{}, P. M., [McKay]{}, T. A., [Meiksin]{}, A., [Morrison]{}, H. L., [Munn]{}, J. A., [Nakajima]{}, R., [Neilsen]{}, Jr., E. H., [Newberg]{}, H. J., [Nichol]{}, R. C., [Nicinski]{}, T., [Nieto-Santisteban]{}, M., [Nitta]{}, A., [Okamura]{}, S., [Owen]{}, R., [Oyaizu]{}, H., [Padmanabhan]{}, N., [Pan]{}, K., [Park]{}, C., [Peoples]{}, J. J., [Pier]{}, J. R., [Pope]{}, A. C., [Purger]{}, N., [Raddick]{}, M. J., [Re Fiorentin]{}, P., [Richards]{}, G. T., [Richmond]{}, M. W., [Riess]{}, A. G., [Rix]{}, H.-W., [Rockosi]{}, C. M., [Sako]{}, M., [Schlegel]{}, D. J., [Schneider]{}, D. P., [Schreiber]{}, M. R., [Schwope]{}, A. D., [Seljak]{}, U., [Sesar]{}, B., [Sheldon]{}, E., [Shimasaku]{}, K., [Sivarani]{}, T., [Smith]{}, J. A., [Snedden]{}, S. A., [Steinmetz]{}, M., [Strauss]{}, M. A., [SubbaRao]{}, M., [Suto]{}, Y., [Szalay]{}, A. S., [Szapudi]{}, I., [Szkody]{}, P., [Tegmark]{}, M., [Thakar]{}, A. R., [Tremonti]{}, C. A., [Tucker]{}, D. L., [Uomoto]{}, A., [Vanden Berk]{}, D. E., [Vandenberg]{}, J., [Vidrih]{}, S., [Vogeley]{}, M. S., [Voges]{}, W., [Vogt]{}, N. P., [Wadadekar]{}, Y., [Weinberg]{}, D. H., [West]{}, A. A., [White]{}, S. D. M., [Wilhite]{}, B. C., [Yanny]{}, B., [Yocum]{}, D. R., [York]{}, D. G., [Zehavi]{}, I., & [Zucker]{}, D. B. 2008, , 175, 297
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[^1]: We note that, unlike the other abundances plotted in Figure \[fig:ohmb\], the Solar abundance is neither a nebular abundance nor the abundance within the central $\sim 5\,$kpc of the Galaxy.
[^2]: While the $Z$ in Equation \[eqn:closedbox\] is [*not*]{} the same as [$12+\log(\mbox{O}/\mbox{H})$]{}—it is a mass ratio of [*all*]{} metals rather than the abundance ratio of one element relative to Hydrogen—the same arguments still qualitatively hold for observed abundances.
[^3]: All five of the @dellenbusch07 galaxies are in the @tremonti04sample; four were either too high mass or too low metallicity (as measured by @tremonti04) to be significant outliers by our definitions. The final galaxy (IC 745) was excluded from our sample because its measured @pettini04 abundance when converted to the @tremonti04 scale is 0.22 dex above the @kewley08 relation (as discussed in §\[sec:metal\]), highlighting the point that while our sample is rather pure, it is almost certainly incomplete.
[^4]: As mentioned in §\[sec:lowm\], IC 255 did not pass all of the cuts for our very low mass sample, and so we reinserted it into the sample by hand.
|
---
abstract: |
The breaking of SO(10) to $\text{SU(3)}_C \times
\text{U(1)}_\text{EM}$ can be accomplished by just four Higgs fields: the symmetric rank-two tensor, $S(\mathsf{54})$; a pair of spinors, $C(\mathsf{16})$ and ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{16}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{16}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{16}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$; and a vector, $T(\mathsf{10})$. This setup is also able to generate realistic fermion masses. The heavy color triplets in the vector and spinor fields mediate proton decay via dimension-five operators. The experimental bounds on proton decay constrain the structure and size of the Yukawa operators.
---
[**Perturbative SO(10) GUT and the Minimal Higgs Sector**]{}\
[Sören Wiesenfeldt and Scott Willenbrock]{}\
SO(10) [@so10] is arguably the most natural grand-unified theory (GUT): both the standard model (SM) gauge and matter fields are unified, introducing only one additional matter particle, the right-handed neutrino. It is an anomaly-free theory and therefore explains the intricate cancellation of the anomalies in the standard model [@Georgi:1972bb]. Moreover, it contains $B-L$ as a local symmetry, where $B$ and $L$ are baryon and lepton number, respectively; the breaking of $B-L$ naturally provides light neutrino masses via the seesaw mechanism.
Despite these attractive features, the breaking of the GUT symmetry has remained a problem in model building. Generally, two avenues have been pursued. One makes use of large representations, $\Phi(\mathsf{210})$, $\Sigma(\mathsf{126})$, ${\settowidth{\textlength}{$\Sigma$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\Sigma$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\Sigma$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{126}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{126}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{126}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$, and $T(\mathsf{10})$ [@large]. This approach has the advantage of a fully renormalizable superpotential and automatic $R$ parity. The unified gauge coupling, however, diverges just above the GUT scale, indicating that new physics must enter. Since this new physics is close to the GUT scale, it potentially has a large effect on the predictions of the model. In addition, this scenario is insufficient to reproduce the fermion mass spectrum, have successful gauge unification, and fulfill the proton decay constraints at the same time [@Melfo:2007sr]. Hence, a realistic renormalizable model requires at least another Higgs field [@large-new].
Models with only small representations remain perturbative up to the Planck scale [@Chang:2004pb] and also have the potential to arise from string theory. They introduce a moderate number of new fields, yielding only small threshold corrections at $M_\text{GUT}$. In the supersymmetric version of this scenario, higher-dimensional operators, suppressed by powers of a more fundamental scale $M$ (such as the Planck scale, $M_\text{P} = \left(8\pi G_N\right)^{-1/2} = 2\cdot 10^{18}$ GeV, or the string scale in the weakly coupled heterotic string, $M_\text{S}
\approx 5\cdot 10^{17}$ GeV), are essential to achieve the breaking to the SM group, $\text{G}_\text{SM} = \text{SU(3)}_C \times \text{SU(2)}_L \times
\text{U(1)}_Y$ [@Barr:1997hq]. Moreover, these operators play an important role in fermion masses and mixings [@bpw; @so10-small]. First, they generate Majorana masses for the right-handed neutrinos in the desired range, $M_\text{GUT}^2/M\sim 10^{14}$ GeV. Thus neutrino masses require the vacuum expectation value (vev) of the $B-L$ breaking field to be of the order of $M_\text{GUT}$. Second, they naturally explain why certain relations, such as the bottom-tau unification, are only approximately realized. Finally, they alter the couplings of the matter fields to the Higgs color triplets relative to the weak doublets, such that the proton decay rate via dimension-five operators can be significantly reduced [@Emmanuel-Costa:2003pu].
Models with small representations usually use the antisymmetric second-rank tensor, $A(\mathsf{45})$, together with a pair of spinors, $C(\mathsf{16})$ and ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{16}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{16}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{16}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$. In order to give GUT-scale masses to the color-triplet Higgs fields, one must introduce two ten-dimensional Higgs fields because the term $TA\,T$ vanishes due to the antisymmetry of $A$. This setup can implement the Dimopoulos-Wilczek mechanism [@dw; @Babu:1993we], yielding automatic mass splitting of the (light) Higgs doublets and (heavy) color triplets. However, the realization of this mechanism requires a second pair of spinorial Higgs fields and an extensive set of global symmetries [@Barr:1997hq]. Thus it is important to investigate alternative scenarios.[^1]
Although the breaking of SO(10) to the standard model by the symmetric rank-two tensor, $S(\mathsf{54})$, and a spinorial, $B-L$ breaking field is a standard textbook example [@books], a realistic supersymmetric model has never been worked out.[^2] The reason might be twofold. First, this scenario does not allow for the Dimopoulos-Wilczek mechanism, so we have to accept fine-tuning to have one pair of Higgs doublets light. Second, $S(\mathsf{54})$ breaks SO(10) to the Pati-Salam group, $\text{G}_\text{PS} = \text{SU(4)}_C \times
\text{SU(2)}_L \times \text{SU(2)}_R$. Hence, in contrast to $A$, it does not break $\text{SU(4)}_C$ and therefore preserves the unification of down-quark and charged-lepton masses. Thus, with only small Higgs representations, it is more involved to generate realistic quark and lepton masses.
The use of $S(\mathsf{54})$, however, has advantages over both models described above. In contrast to the model with large representations, the gauge coupling remains perturbative up to $M_\text{P}$ and unlike the scenario with $A(\mathsf{45})$, it requires only one ten-dimensional Higgs field for the electroweak symmetry breaking.[^3] In this letter we demonstrate that SO(10) can be broken to $\text{SU(3)}_C \times
\text{U(1)}_\text{EM}$ by a set of four Higgs fields: $S(\mathsf{54})$, $C(\mathsf{16})$, ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{16}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{16}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{16}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$, and $T(\mathsf{10})$. Analyzing the higher-dimensional operators, we show that realistic fermion masses and mixings can be generated and a sufficiently low proton decay rate can be obtained.
#### Breaking of SO(10).
The SO(10) symmetry is broken by the vevs of $S(\mathsf{54})$, $C(\mathsf{16})$, and ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{16}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{16}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{16}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$ [@non-susy]. The superpotential can be split into three parts, $$\begin{aligned}
W & = W_S + W_C + W_{SC} \;.
\label{eq:superpotential}\end{aligned}$$ The first part yields the breaking to $\text{SO(6)} \times
\text{SO(4)} \simeq \text{G}_\text{PS}$, $$\begin{aligned}
\label{eq:superpotential-S}
W_S & = \tfrac{1}{2}\, M_S \operatorname{tr}S^2 + \tfrac{1}{3} \lambda_S \operatorname{tr}S^3
\;, \qquad {\left\langle S\right\rangle} = v_s \operatorname{diag}(2,2,2,2,2,2;-3,-3,-3,-3) \;, \qquad
v_s = \frac{M_S}{\lambda_S} \;.\end{aligned}$$ ${\left\langle S\right\rangle}$ is chosen such that the first six entries preserve $\text{SO(6)} \simeq \text{SU(4)}_C$, whereas the last preserve $\text{SO(4)} \simeq \text{SU(2)}_L \times \text{SU(2)}_R$. The second part of the superpotential describes the breaking of SO(10) to SU(5) by the vevs of the spinorial representations. It requires the inclusion of one dimension-five operator,[^4] $$\begin{aligned}
\label{eq:superpotential-C}
W_C & = M_C\, C {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} + \frac{\lambda_C}{2M} \left( C {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}
\right)^2 , \qquad {\left\langle C{\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}\right\rangle} = -\frac{M_C M}{\lambda_C} \equiv
v_c^2 \;.\end{aligned}$$ Together, these vevs break SO(10) to the intersection of SU(5) and $\text{G}_\text{PS}$, namely the standard model group $\text{G}_\text{SM}$.
At the renormalizable level, the Higgs fields do not couple in the potential. Therefore we have two independent global SO(10) symmetries and the total number of Goldstones is $\left(45-24\right) +
\left(45-21\right) = 45$. Only $45-12=33$ true Goldstones are eaten by gauge bosons, so there are 12 pseudo-Goldstones. These are a vectorial pair of quark-doublet fields, $Q(3,2,\frac{1}{6})+{\settowidth{\textlength}{$Q$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$Q$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$Q$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}(\bar{3},2,-\frac{1}{6})$, contained both in $C + {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ and in $S$. This extra global symmetry, however, is accidental, and is violated by the non-renormalizable interaction term
[r]{}[.42]{}
{width="\linewidth"}
between the different Higgs fields, $$\begin{aligned}
W_{SC} & = \frac{\xi_C}{2M}\, C {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} S^2 .
\label{eq:superpotential-SC}\end{aligned}$$ The pseudo-Goldstones acquire masses of order $v_s v_c/M$. Their presence changes the 1-loop coefficients of the $\beta$-function of the gauge couplings $\left(\alpha_1,\alpha_2,\alpha_3\right)$ by $\Delta
b=\left(\frac{1}{5},3,2\right)$ such that $b=\left(\frac{34}{5},4,-1\right)$ [@Barger:2006fm]. Hence, they mostly modify the running of $\alpha_2$ and $\alpha_3$. Fig. \[fig:running\] shows the impact of the pseudo-Goldstones, at a mass of $3 \cdot 10^{14}$ GeV, on the running of the gauge couplings in the MSSM at one loop. Since the gauge coupling unification is upset, it is clear that the pseudo-Goldstone masses must be close to the GUT scale in order to preserve gauge coupling unification. This can be achieved provided that $v_c > v_s$; if $\lambda_C \sim \frac{M_C}{M}$, the pseudo-Goldstones become as heavy as $v_s$. In this scenario, SO(10) is first broken to SU(5) at $v_c$, and then to the SM at $v_s$. A drawback to this scenario is that $v_c \sim M$, so even higher-dimensional operators obtained by adding powers of $\left(C{\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}\right)/M^2$, $\left(C/M\right)^4$, and $({\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}/M)^4$ are not necessarily suppressed.
The renormalizable part of the superpotential for the electroweak symmetry breaking reads $$\begin{aligned}
\label{eq:superpotential-T}
W_T & = \tfrac{1}{2}\, M_T\, T^2 + \tfrac{1}{2}\, \lambda_T T S T +
\xi_T\, C C T + \xi_T^\prime\, {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} T \; .\end{aligned}$$ The mass matrix for the weak doublets, $H_u(T)$, $H_d(T)$, $\widetilde{H}_u(\bar{C})$, and $\widetilde{H}_d(C)$, is such that without fine-tuning, all doublets have GUT-scale masses, $$\begin{aligned}
\label{eq:m2-matrix}
\begin{pmatrix}
H_u & \widetilde{H}_u
\end{pmatrix}
\begin{pmatrix}
M_T - 3 \lambda_T v_s & \xi_T v_c \cr \xi_T^\prime v_c & M_C +
\frac{\lambda_C}{M}\, v_c
\end{pmatrix}
\begin{pmatrix}
H_d \cr \widetilde{H}_d
\end{pmatrix}
.\end{aligned}$$ This is the well-known doublet-triplet splitting problem. In order to arrange for two light doublets, we have to impose that the determinant of this mass matrix vanishes, up to weak-scale terms. As a result, the two light doublets are combinations of the four doublets in $T$, $C$, and ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$, and all doublets acquire weak-scale vevs.[^5] This is crucial in order to derive a realistic fermion mass spectrum, as we will discuss in the next section [@bpw; @so10-small]. We will denote the SU(5)-singlet and SU(2)-doublet vevs of $C$ as ${\langle C^\text{GUT}\rangle}$ and ${\langle C^\text{EW}\rangle}$ and similarly the vevs of ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$.
For later purposes, note that due to the vev of $S$ in Eq. (\[eq:superpotential-S\]), the color triplets of any ten-dimensional representation occupy the first six entries, whereas the last four are for the weak doublets. For $T$, e.g., we may write , where $H_C$ and ${\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}_C$ are the proton-decay-mediating color triplets.
#### Fermion Masses.
There is only one renormalizable operator that generates masses for the matter fields $\mathsf{16}_i$, $i=1,2,3$, $$\begin{aligned}
\label{eq:yukawa-dim4}
W_Y^{(4)} & = h_T^{ij}\, \mathsf{16}_i \mathsf{16}_j T ;\end{aligned}$$ this term predicts Yukawa unification at $M_\text{GUT}$, $h_u=h_d=h_e=h_\nu^\text{D}$. To have a realistic mass pattern, we need to consider higher-dimensional operators. At dimension five, we have $$\begin{aligned}
\label{eq:yukawa-dim5}
W_Y^{(5)} & = \frac{1}{M} \left[ h_S^{ij}\, \mathsf{16}_i
\mathsf{16}_j T S + h_C^{ij}\, \mathsf{16}_i \mathsf{16}_j C C +
h_{\bar{C}}^{ij}\, \mathsf{16}_i \mathsf{16}_j {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}
\right] .\end{aligned}$$ These operators can be generated by integrating out heavy fields in various SO(10) representations, as indicated in Table \[tb:yukawa\].
The first term in Eq. (\[eq:yukawa-dim5\]) contributes equally to the mass matrices of quarks and leptons since the vev of $S$ conserves $\text{SU(4)}_C$. This is different from the scenario with the antisymmetric representation, which breaks SO(10) to the left-right symmetric group, $\text{SU(3)}_C \times \text{SU(2)}_L \times
\text{SU(2)}_R \times \text{U(1)}_{B-L}$. The second term contributes (in equal measure) to the masses of down quarks and charged leptons, whereas the last term generates Dirac masses for up quarks and neutrinos as well as Majorana masses for the right-handed neutrinos [@Sayre:2006wf]. These terms allow for a non-trivial CKM-matrix and degrade the Yukawa relations to $h_d=h_e$. The relation $h_u=h_\nu^\text{D}$ is violated by the operators , obtained by integrating out heavy fields in either the singlet or adjoint representation, as shown in Table \[tb:yukawa\] [@Babu:2000ei].
In order to alter the unification of down quark and charged lepton masses, we have to go further and consider terms containing both $S$ and $C$ fields. The lowest operator is of dimension six, $$\begin{aligned}
\label{eq:yukawa-dim6}
W_Y^{(6)} & = \frac{h_{SC}^{ij}}{M^2}\, \mathsf{16}_i \mathsf{16}_j
C C S \;,\end{aligned}$$ suppressed by $v_c v_s/M^2$. For $M=M_\text{P}$, this suppression factor is of similar order of magnitude as the ratio of the strange quark mass and the weak scale, namely $10^{-3}$. Thus this term is large enough to account for the difference of strange-quark and muon masses. We will now show that this term does indeed violate the equality of down-quark and charged-lepton masses.
Actually, there are two distinct operators in Eq. (\[eq:yukawa-dim6\]). To see that, let us recall that the Higgs field $\mathsf{120}$ contains two pairs of weak doublets. The first pair is that of the SU(5)-fields $\text{5} + {\settowidth{\textlength}{$\text{5}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\text{5}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\text{5}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ of SU(5) and $\left(1,2,2\right)$ of $\text{G}_\text{PS}$, and it couples equally to down quarks and charged fermions. (The same applies to the pair of Higgs doublets in $T$.) The second pair of doublets is contained in $\text{45} + {\settowidth{\textlength}{$\text{45}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\text{45}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\text{45}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ of SU(5) and $(15,2,2)$ of $\text{G}_\text{PS}$. Here the couplings to the fermion fields pick up a $B-L$ factor, modifying the Yukawa unification [@Ross:1985ai].
In Eq. (\[eq:yukawa-dim6\]), the fermions effectively couple to $\mathsf{10}$, $\mathsf{120}$, or ${\settowidth{\textlength}{$\mathsf{126}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{126}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{126}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ fields in the $\mathsf{16} \times \mathsf{16} \times \mathsf{54}$ decomposition. We expect two types of couplings to be present, one that yields Yukawa unification, and one that violates it. Let us demonstrate that the operator in Eq. (\[eq:yukawa-dim6\]) can indeed modify the Yukawa unification by integrating out heavy $\mathsf{10}_M=\left(D,D^c;L,L^c\right)_M$ fields. In this case, the two types of couplings are and . The first operator gives equal contributions to down quark and charged lepton masses, so let us study the second in detail.
The coupling of the matter fields $\mathsf{16}_i$ and $\mathsf{10}_M$ to $C^\text{EW}$ (where $\widetilde{H}_d(C)$ acquires its vev) yields $$\begin{aligned}
\label{eq:yukawa-spoil-ew}
\mathsf{16}_i \mathsf{10}_M C^\text{EW} \ni \left( d_i D^c_M + e^c_i
L_M + \nu^c_i L^c_M \right) \widetilde{H}_d ,\end{aligned}$$ in terms of SO(10) and SM fields. The coupling to $C^\text{GUT}$ (where the SM singlet component, $N(C)$, acquires its vev) gives $$\begin{aligned}
\label{eq:yukawa-spoil-gut}
\mathsf{16}_j \mathsf{10}_M C^\text{GUT} \ni \left( d^c_j D_M + L_j
L^c_M \right) N .\end{aligned}$$ As noted above, the color triplets of the SO(10) vector live in the first six entries, the doublets in the last four. Then we may write and $\left[ 16 {\left\langle C^\text{GUT}\right\rangle} \right]
= \left(d^c;L\right) v_c$, where ${\langle \widetilde{H}_d\rangle}=v_d$. Now it is straightforward to see that this operator indeed violates the Yukawa unification, $$\begin{aligned}
\label{eq:yukawa-spoil}
\frac{h_{SC}^{(B)\,ij}}{M^2} \left[ \left(d;e^c,0\right)_i
{\langle C^\text{EW}\rangle} \vphantom{D^j_j} \right] {\left\langle S\right\rangle} \left[
\left(d^c;L\right)_j {\langle C^\text{GUT}\rangle} \right] =
h_{SC}^{(B)\,ij} \left( 2\, d_i d^c_j - 3\, e^c_i e_j \right) v_d
\frac{v_c v_s}{M^2} \;.\end{aligned}$$
[llr|dddd|dddd]{} operator & & & & & & & & & &\
$h_T^{ij}\,\mathsf{16}_i \mathsf{16}_j T$ & & & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\
$h_S^{ij}\,\mathsf{16}_i \mathsf{16}_j T S$ & & & -3 & -3 & -3 & -3 & 2 & 2 & 2 & 2\
$h_C^{ij}\,\mathsf{16}_i \mathsf{16}_j C C$ & & & - & 1 & 1 & - & - & 1 & 1 & -\
$h_{\bar{C}}^{ij}\,\mathsf{16}_i \mathsf{16}_j {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ & & \[1\] & - & - & - & 1 & - & - & - & -\
& & \[10\] & 1 & - & - & 1 & 1 & - & - & 1\
& & \[45\] & 8 & - & - & 3 & 8 & - & - & 8\
$h_{SS}^{ij}\,\mathsf{16}_i \mathsf{16}_j T S S$ & & & 9 & 9 & 9 & 9 & 4 & 4 & 4 & 4\
$h_{SC}^{ij}\,\mathsf{16}_i \mathsf{16}_j C C S$ & $(A)$ & & - & -3 & -3 & - & - & 2 & 2 & -\
& $(B)$ & & - & 2 & -3 & - & - & -3 & 2 & -\
$h_{S\bar{C}}\,\mathsf{16}_i \mathsf{16}_j {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} S$ & & & -3 & - & - & -3 & 2 & - & - & 2\
The various operators up to dimension six and their contributions to the quark and lepton mass matrices are listed in the left part of Table \[tb:yukawa\]. Remarkably, the operator $\mathsf{16}_i
\mathsf{16}_j {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} S$ contributes in equal measure to $h_u$ and $h_\nu^\text{D}$, in contrast to the dimension-five operator $\mathsf{16}_i \mathsf{16}_j {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$. From Table \[tb:yukawa\], we read off the relations $$\begin{aligned}
\label{eq:mass-relations}
Y_\nu^\text{D} - Y_u & = h_{\bar{C}}^{[1]} - 5\, h_{\bar{C}}^{[45]}
, \qquad Y_e - Y_d = 5\, h_{SC}^{(B)} \;.\end{aligned}$$
#### Proton Decay.
The couplings of the fermions to color-triplet Higgs fields give rise to the proton decay operators of mass-dimension five [@dim5op] with $$\begin{aligned}
\Gamma & \propto \left| \frac{C_5}{M_{H_C}} \right|^2 , \qquad C_5^L
= Y_{qq} Y_{ql} \;, \qquad C_5^R = Y_{ud} Y_{ue} \;,\end{aligned}$$ where $\Gamma$ is the decay rate, $M_{H_C}$ is the mass of the color triplets, and the baryon and lepton number violating couplings are denoted as $$\begin{aligned}
\label{eq:pdecay-cplg}
\left( \tfrac{1}{2}\,Y_{qq}^{ij}\, Q_i Q_j + Y_{ue}^{ij}\, u^c_i
e^c_j \right) H_C + \left( Y_{ql}^{ij}\, Q_i L_j + Y_{ud}^{ij}\,
u^c_i d^c_j \right) {\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}_C \, .\end{aligned}$$
[r]{}[.37]{}
Decay diagrams $p \to K^+ \bar\nu$ via the two distinct operators $QQQL$ and $u^c d^c u^c e^c$ are sketched in the adjoining figure.
The determination of the baryon and lepton number violating couplings is important for the calculation of the decay amplitude. In SU(5), the impact of higher-dimensional Yukawa operators on these couplings, relative to the mass terms, is sufficient to reduce the decay rate by several orders of magnitude and make it consistent with the experimental upper bound [@Emmanuel-Costa:2003pu].
The dimension-five operators are generated by integrating out the heavy color-triplet Higgs fields. In addition to the standard couplings to $H_C(T)$ and ${\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}_C(T)$, we have those to $\widetilde{H}_C({\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$ and $\widetilde{{\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}}_C(C)$, via the higher-dimensional operators [@bpw]. The coefficients are listed in the right part of Table \[tb:yukawa\]; note that the first two rows express the couplings to $H_C$ and ${\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}_C$, the remaining rows those to $\widetilde{H}_C$ and $\widetilde{{\settowidth{\textlength}{$H$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$H$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$H$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}}_C$.
Due to the two pairs of color triplets, we cannot simply read off the relations between the baryon-number-violating couplings and the mass matrices, in contrast to SU(5). We find, however, relations among the four different couplings, namely $$\begin{aligned}
Y_{qq} & = Y_{ue}\end{aligned}$$ and the SU(5) relation [@Emmanuel-Costa:2003pu; @Bajc:2002pg] $$\begin{aligned}
Y_{ud} - Y_{ql} & = Y_d - Y_e \,.\end{aligned}$$ A detailed study of the various decay modes requires a numerical analysis of the fermion masses and mixings, which is beyond the scope of this letter. Note that the new decay operators due to the color triplets in $C$ and ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ can change the branching ratios significantly [@bpw]. Naïve choices such as $Y_u=Y_{qq}$ yield a decay rate which could be in conflict with the experimental bounds [@Kobayashi:2005pe], provided that the color triplets have GUT-scale masses. A study along the lines of Ref. [@Emmanuel-Costa:2003pu], however, would provide means to control the total decay rate. The experimental bounds on proton decay, together with the observed pattern of fermion masses and mixings will constrain the structure and size of the various couplings.
#### Concluding Remarks.
We have shown that SO(10) can be broken to $\text{SU(3)}_C \times
\text{U(1)}_\text{EM}$ by a set of four Higgs fields: $S(\mathsf{54})$, $C(\mathsf{16})$, ${\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{16}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{16}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{16}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$, and $T(\mathsf{10})$. This raises the question of whether this scenario is the minimal set of Higgs fields capable of breaking SO(10) to $\text{SU(3)}_C \times \text{U(1)}_\text{EM}$. If we replace the symmetric tensor $S(\mathsf{54})$ by the antisymmetric tensor, $A(\mathsf{45})$, will this system work as well?
In order for $A$ to acquire a vev, we need a quartic coupling, such as $\frac{1}{M} \operatorname{tr}A^4$. Even if it acquires a vev in the $B-L$ direction, the Dimopoulos-Wilczek form is destabilized when $A$ couples to the spinors via the operator $C A {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$ and the $I_{3R}$ component gets a vev as well [@Barr:1997hq; @Babu:1993we]. (If $A$ acquires a vev in the hypercharge direction, both the $B-L$ and $I_{3R}$ components of $A$ have a vev from the beginning.) Moreover, this coupling generates a splitting among the electroweak doublets and color triplets in $C$ and ${\settowidth{\textlength}{$C$}
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\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}$. Hence, although there is not a renormalizable coupling of $A$ to $T$, the doublet-triplet splitting emerges and fine-tuning allows to arrange for two light weak doublets, while the color triplets are heavy. Therefore this scenario is capable of breaking SO(10) to $\text{SU(3)}_C \times \text{U(1)}_\text{EM}$ as well.
Comparing the two scenarios with either $A(\mathsf{45})$ or $S(\mathsf{54})$, we notice that in order to reproduce fermion masses, we only need operators of dimension five in the former case. In the latter scenario, however, we have seen that the dimension-six operators are necessary, and have a significant impact. Dimension-six operators, although not necessary, could have a significant impact in the models with $A(\mathsf{45})$ as well. Furthermore, looking at Table \[tb:yukawa\], we note that several operators contribute identically to the fermion mass matrices. This reduces the number of free parameters in these matrices. Hence, using $S(\mathsf{54})$ is a promising alternative to the already established scenarios.
We are grateful for valuable discussions with K. Babu and to C. Albright for comments on the manuscript. This work was supported in part by the U. S. Department of Energy under contract No. DE-FG02-91ER40677.
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[^1]: One attempt has been to realize the GUT gauge symmetry in more than four space-time dimension and use GUT-symmetry breaking boundary conditions on an orbifold for the breaking to the SM. This scenario yields doublet-triplet splitting and avoids the dangerous dimension-five proton-decay operators [@orbifold].
[^2]: The Higgs sector for this setup was considered for the non-supersymmetric case [@non-susy] and also for the supersymmetric scenario, but with two ten-dimensional fields [@Zhao:1981me]. In several supersymmetric models, $S$ is used as an additional field to achieve the symmetry breaking [@45-54].
[^3]: Recently, it was shown that SO(10) can be broken to $\text{G}_\text{SM}$ by a single pair of vector-spinors, $\Upsilon_{(+)}(\mathsf{144}) + \Upsilon_{(-)}({\settowidth{\textlength}{$\mathsf{144}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{144}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{144}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$ [@Babu:2005gx]; however, in order to generate realistic fermion masses, one needs to introduce additional matter fields, such as $\text{10}_M$ and $\text{45}_M$ [@Babu:2006rp]. We will not consider this approach in this letter.
[^4]: Alternatively, we may introduce a singlet $X$ such that $$\begin{aligned}
W_C & = X \left( \lambda_C C {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} - M_C^2 \right) , \qquad
\frac{\partial W_C}{\partial X} = \lambda_C C {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} - M_C^2 \
\Rightarrow\ {\left\langle C{\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
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\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
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\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}\right\rangle} = \frac{M_C^2}{\lambda_C} \ , \qquad
\frac{\partial W_C}{\partial C} = \lambda_C X {\settowidth{\textlength}{$C$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$C$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$C$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}} \
\Rightarrow {\left\langle X\right\rangle} = 0 \;,
\end{aligned}$$ avoiding the dimension-five operator. The price we pay is to introduce a singlet which in general can couple to the other fields as well.
[^5]: This is similar to the scenario with large representations, where the doublets mix through the vev of $\Phi(\mathsf{210})$ and the light doublets are mixtures of those in $T(\mathsf{10})$, $\Sigma(\mathsf{126})$, ${\settowidth{\textlength}{$\Sigma$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\Sigma$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\Sigma$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}}({\settowidth{\textlength}{$\mathsf{126}$}
\setlength{\overlinelength}{0.1pt}
\addtolength{\overlinelength}{0.75\textlength}
\makebox[\textlength][s]{$\mathsf{126}$} \hspace{-.55\textlength}
\hspace{-\overlinelength}\hspace{.55\overlinelength}
\overline{\makebox[\overlinelength][s]{\vphantom{$\mathsf{126}$}}}
\hspace{-.55\overlinelength}\hspace{.55\textlength}})$, and $\Phi(\mathsf{210})$ [@large].
|
---
abstract: 'Coherent boundaries of Lagrangian vortices in fluid flows have recently been identified as closed orbits of line fields associated with the Cauchy–Green strain tensor. Here we develop a fully automated procedure for the detection of such closed orbits in large-scale velocity data sets. We illustrate the power of our method on ocean surface velocities derived from satellite altimetry.'
author:
- Daniel Karrasch
- Florian Huhn
- George Haller
- |
Institute of Mechanical Systems\
ETH Zürich, Leonhardstrasse 21\
8092 Zürich, Switzerland
title: 'Automated detection of coherent Lagrangian vortices in two-dimensional unsteady flows'
---
**Keywords:** Coherent Lagrangian vortices, Transport, Index theory, Line fields, Closed orbit detection, Ocean surface flows.
Introduction
============
Lagrangian coherent structures (LCS) are exceptional material surfaces that act as cores of observed tracer patterns in fluid flows (see [@Peacock2010] and [@Peacock2013] for reviews). For oceanic flows, the tracers of interest include salinity, temperature, contaminants, nutrients and plankton—quantities that play an important role in the ecosystem and even in climate. Fluxes of these quantities are typically dominated by advective transport over diffusion.
An important component of advective transport in the ocean is governed by mesoscale eddies, i.e., vortices of $100$–$200$km in diameter. While eddies also stir and mix surrounding water masses by their swirling motion, here we focus on eddies that trap and carry fluid in a coherent manner. Eddies of this kind include the Agulhas rings of the Southern Ocean. They are known to transport massive quantities of warm and salty water from the Indian Ocean into the Atlantic Ocean [@Ruijter1999]. Current limitations on computational power prevent that climate models resolve mesoscale eddies in their flow field. Since the effect of mesoscale eddies on the global circulation is significant [@Wolfe2009], the correct parameterization of eddy transport is crucial for the reliability of these models. As a consequence, there is a rising interest in systematic and accurate eddy detection and census in large global data sets, as well as in quantifying the average transport of trapped fluid by all eddies in a given region [@Dong2014; @Zhang2014; @Petersen2013].
This quantification requires (i) a rigorous method that provides specific coherent eddy boundaries, and (ii) a robust numerical implementation of the method on large velocity data sets.
A number of vortex definitions have been proposed in the literature, [@Haller2005; @Zhang2014a], most of which are of Eulerian type, i.e., use information from the instantaneous velocity field. Typical global eddy studies [@Chelton2007; @Dong2014; @Zhang2014; @Petersen2013] are based on such Eulerian approaches. Evolving eddy boundaries obtained from Eulerian approaches, however, do not encircle and transport the same body of water coherently [@Haller2013a; @Zhang2014a]. Instead, fluid initialized within an instantaneous Eulerian eddy boundary will generally stretch, fold and filament significantly. Yet only coherently transported scalars resist erosion by diffusion in a way that a sharp signature in the tracer field is maintained. All this suggests that coherent eddy transport should ideally be analysed via Lagrangian methods that take into account the evolution of trajectories in the flow, such as, e.g., [@Provenzale1999; @Allshouse2012; @Mendoza2010; @Rypina2011; @Tallapragada2013; @Prants2013]. Notably, however, none of these methods focuses on the detection of vortices and none provides an algorithm to extract exact eddy boundaries in unsteady velocity fields. Only recently have mathematical approaches emerged for the detection of coherent Lagrangian vortices. These include the geometric approach [@Haller2005; @Haller2013a] and the set-oriented approach [@Froyland2010a; @Froyland2013; @Froyland2012a]. Here, we follow the geometric approach to coherent Lagrangian vortices, which defines a coherent material vortex boundary as a closed stationary curve of the averaged material strain [@Haller2013a]. All solutions of this variational problem turn out to be closed material curves that stretch uniformly. Such curves are practically found as closed orbits of appropriate planar line fields [@Haller2013a].
In contrast to vector fields, line fields are special vector bundles over the plane. In their definition, only a one-dimensional subspace (line) is specified at each point, as opposed to a vector at each point. The importance of line field singularities in Lagrangian eddy detection has been recognized in [@Haller2013a], but has remained only partially exploited. Here, we point out a topological rule that enables a fully automated detection of coherent Lagrangian vortex boundaries based on line field singularities. This in turn makes automated Lagrangian eddy detection feasible for large ocean regions.
Based on the geometric approach, coherent Lagrangian vortices have so far been identified in oceanic data sets [@Beron-Vera2013; @Haller2013a], in a direct numerical simulation of the two-dimensional Navier–Stokes equations [@Farazmand2014], in a smooth area-preserving map [@Haller2012], in a kinematic model of an oceanic jet in [@Haller2012], and in a model of a double gyre flow [@Onu2014]. With the exception of [@Haller2013a], however, these studies did not utilize the topology of line field singularities. Furthermore, none of them offered an automated procedure for Lagrangian vortex detection.
The orbit structure of line fields has already received considerable attention in the scientific visualization community (see [@Delmarcelle1994a; @Weickert2006] for reviews). The problem of closed orbit detection has been posed in [@Delmarcelle1994a Section 5.2.3], and was considered by [@Wischgoll2006], building on [@Wischgoll2001]. In that approach, numerical line field integration is used to identify cell chains that may contain a closed orbit. Then, the conditions of the Poincaré–Bendixson theorem are verified to conclude the existence of a closed orbit for the line field. This approach, however, does not offer a systematic way to search for closed orbits in large data sets arising in geophysical applications.
This paper is organized as follows. In , we recall the index theory of planar vector fields. In , we review available results on indices for planar line fields, and deduce a topological rule for generic singularities inside closed orbits of such fields. Next, in , we present an algorithm for the automated detection of closed line field orbits. We then discuss related numerical results on ocean data, before presenting our concluding remarks in .
Index theory for planar vector fields\[sec:Index\_vf\]
======================================================
Here, we recall the definition and properties of the index of a planar vector field [@Needham2000]. We denote the unit circle of the plane by $\mathcal{S}^{1}$, parametrized by the mapping $(\cos2\pi s,\sin2\pi s)\in\mathcal{S}^{1}\subset\mathds{R}^{2}$, $s\in\left[0,1\right]$. In our notation, we do not distinguish between a curve $\gamma\colon[a,b]\to\mathds{R}^{2}$ as a function and its image as a subset of $\mathds{R}^{2}$.
\[def:index\_vf\] For a continuous, piecewise differentiable planar vector field $\mathbf{v}\colon D\subseteq\mathds{R}^{2}\to\mathds{R}^{2}$ and a simple closed curve $\gamma\colon\mathcal{S}^{1}\to\mathds{R}^{2}$, let $\theta\colon\left[0,1\right]\to\mathds{R}$ be a continuous function such that $\theta(s)$ is the angle between the $x$-axis and $\mathbf{v}(\gamma(s))$. Then, the *index* (or *winding number*) *of $\mathbf{v}$ along $\gamma$* is defined as $$\operatorname{ind}_{\gamma}(\mathbf{v})\coloneqq\frac{1}{2\pi}\left(\theta(1)-\theta(0)\right),$$ i.e., the number of turns of $\mathbf{v}$ during one anticlockwise revolution along $\gamma$. Clearly, $\theta$ is well-defined only if there is no *critical point* of $\mathbf{v}$ along $\gamma$, i.e., no point at which $\mathbf{v}$ vanishes.
The index defined in has two important properties [@Perko2001]:
1. *Decomposition property*: $$\operatorname{ind}_{\gamma}(\mathbf{v})=\operatorname{ind}_{\gamma_{1}}\left(\mathbf{v}\right)+\operatorname{ind}_{\gamma_{2}}\left(\mathbf{v}\right),$$ whenever $\gamma=\gamma_{1}\cup\gamma_{2}\setminus(\gamma_{1}\cap\gamma_{2})$, and $\operatorname{ind}_{\gamma_{i}}\left(\mathbf{v}\right)$ are well-defined.
2. *Homotopy invariance*: $$\operatorname{ind}_{\gamma}\left(\mathbf{v}\right)=\operatorname{ind}_{\tilde{\gamma}}\left(\mathbf{v}\right),$$ whenever $\tilde{\gamma}$ can be obtained from $\gamma$ by a continuous deformation (homotopy).
If $\gamma$ encloses exactly one critical point $p$ of $\mathbf{v}$, then the *index of $p$ with respect to $\mathbf{v}$*, $$\operatorname{ind}\left(p,\mathbf{v}\right)\coloneqq\operatorname{ind}_{\gamma}\left(\mathbf{v}\right)$$ is well-defined, because its definition does not depend on the particular choice of the enclosing curve by homotopy invariance. Furthermore, the index of $\gamma$ equals the sum over the indices of all enclosed critical points, i.e., $$\operatorname{ind}_{\gamma}\left(\mathbf{v}\right)=\sum_{i}\operatorname{ind}(p_{i},\mathbf{v}),$$ provided all $p_{i}$ are isolated critical points. Finally, the index of a closed orbit $\Gamma$ of the vector field $\mathbf{v}$ is equal to $1$, because the vector field turns once along $\Gamma$. Therefore, closed orbits of planar vector fields necessarily enclose critical points.
Index theory for planar line fields\[sec:Index\_lf\]
====================================================
We now recall an extension of index theory from vector fields to line fields [@Spivak1999]. Let $\mathbb{P}^{1}$ be the set of one-dimensional subspaces of $\mathds{R}^{2}$, i.e., the set of lines through the origin $0\in\mathds{R}^{2}$. $\mathbb{P}^{1}$ is sometimes also called the *projective line*, which can be endowed with the structure of a one-dimensional smooth manifold [@Lee2012]. This is achieved by parametrizing the lines via the $x$-coordinate at which they intersect the horizontal line $y=1$. The horizontal line $y=0$ is assigned the value $\infty$.
Equivalently, elements of $\mathbb{P}^{1}$ can be parametrized by their intersection with the upper semi-circle, denoted $\mathcal{S}_{+}^{1}$, with its right and left endpoints identified. This means that lines through the origin are represented by a unique normalized vector, pointing in the upper half-plane and parametrized by the angle between the representative vector and the $x$-axis (Fig. \[fig:circle-semi-circle\]). A *planar line field* is then defined as a mapping $\mathbf{l}\colon D\subseteq\mathds{R}^{2}\to\mathbb{P}^{1}$, with its differentiability defined with the help of the manifold structure of $\mathbb{P}^{1}$.
![The geometry of the projective line and its parametrization. The double-headed arrows represent one-dimensional subspaces of the plane, i.e., elements of $\mathbb{P}^{1}$. The upper semi-circle $\mathcal{S}_{+}^{1}$ is shown in purple, its end-points in cyan, and the unit circle $\mathcal{S}^1$ in dashed magenta. The black points represent intersections of the lines with $y=1$ and with the unit circle, respectively.[]{data-label="fig:circle-semi-circle"}](circle-semi-circle){width="45.00000%"}
Line fields arise in the computation of eigenvector fields for symmetric, second-order tensor fields [@Delmarcelle1994; @Tricoche2006]. Eigenvectors have no intrinsic sign or length: only eigenspaces are well-defined at each point of the plane. Their orientation depends smoothly on their base point if the tensor field is smooth and has simple eigenvalues at that point. At repeated eigenvalues, isolated one-dimensional eigenspaces (and hence the corresponding values of the line field) become undefined.
Points to which a line field cannot be extended continuously are called *singularities*. These points are analogous to critical points of vector fields. Away from singularities, any smooth line field can locally be endowed with a smooth orientation. This implies the local existence of a normalized smooth vector field, which pointwise spans the respective line. Conversely, away from critical points, smooth vector fields induce smooth line fields when one takes their linear span pointwise.
Based on the index for planar vector fields, we introduce a notion of index for planar line fields following [@Spivak1999]. First, for some differentiable line field $\mathbf{l}$ and along some closed curve $\gamma\colon\mathcal{S}^{1}\to\mathds{R}^{2}$, pick at each point $\gamma(t)$ the representative upper half-plane vector from $\mathbf{l}(\gamma(t))$. This choice yields a normalized vector field along $\gamma$ which is as smooth as $\mathbf{l}$, except where $\mathbf{l}\circ\gamma$ crosses the horizontal subspace. At such a point, there is a jump-discontinuity in the representative vector from right to left or vice versa. To remove this discontinuity, the representative vectors are turned counter-clockwise by $\alpha\colon\mathcal{S}_{+}^{1}\to\mathcal{S}^{1}$, $\left(\cos2\pi s\sin2\pi s\right)\mapsto\left(\cos4\pi s,\sin4\pi s\right)$, $s\in[0,1/2]$, i.e., the parametrizing angle is doubled. Thereby, the left end-point with angle $\pi$ is mapped onto the right end-point with angle $0$. This representation $\alpha\circ\mathbf{l}$ permits the extension of the notion of index to planar line fields as follows.
\[def:index\_lf\] For a continuous, piecewise differentiable planar line field $\mathbf{l}\colon D\subseteq\mathds{R}^{2}\to\mathbb{P}^{1}$ and a simple closed curve $\gamma\colon\mathcal{S}^{1}\to\mathds{R}^{2}$, we define the *index of $\mathbf{l}$ along $\gamma$* as $$\operatorname{ind}_{\gamma}(\mathbf{l})\coloneqq\frac{1}{2}\operatorname{ind}_{\gamma}(\alpha\circ\mathbf{l}).$$
The coefficient $1/2$ in this definition is needed to correct the doubling effect of $\alpha$. It also makes the index for a line field, generated by a vector field in the interior of $\gamma$, equal to the index of that vector field. Since refers to , the additional definitions and properties described in for vector fields carry over to line fields.
We call a curve $\gamma$ an *orbit* of $\mathbf{l}$, if it is everywhere tangent to $\mathbf{l}$. The scientific visualization community refers to orbits of lines fields arising from the eigenvectors of a symmetric tensor as *tensor (field) lines* or *hyperorbit (trajectories*) [@Delmarcelle1994; @Tricoche2006; @Wischgoll2006].
By definition, the index of singularities of line fields can be a half integer, as opposed to the vector field case, where only integer indices are possible. Also, two new types of singularities emerge in the line field case: *wedges* (type $W$) of index $+1/2$, and *trisectors* (type $T$) of index $-1/2$ [@Delmarcelle1994; @Tricoche2006]. The geometry near these singularities is shown in Fig. \[fig:deg\_points\].
![Orbit topologies in the vicinity of the two generic line field singularity types: trisector (left) and wedge (right). All lines represent orbits, the solid lines correspond to boundaries of hyperbolic sectors.[]{data-label="fig:deg_points"}](trisector-wedge){width="50.00000%"}
Node, centre, focus and saddle singularities also exist for line fields, but these singularities turn out to be structurally unstable with respect to small perturbations to the line field [@Delmarcelle1994].
In this paper, we assume that only *isolated* singularities of the *generic* wedge and trisector types are present in the line field of interest. In that case, we obtain the following topological constraint on closed orbits of the line field.
\[thm:topo\_line\] Let $\mathbf{l}$ be a continuous, piecewise differentiable line field with only structurally stable singularities. Let $\Gamma$ be a closed orbit of $\mathbf{l}$, and let $D$ denote the interior of $\Gamma$. We then have $$W=T+2,\label{eq:toporule_line}$$ where $W$ and $T$ denote the number of wedges and trisectors, respectively, in $D$.
First, $\Gamma$ has index $1$ with respect to $\mathbf{l}$, i.e., $\operatorname{ind}_{\Gamma}\left(\mathbf{l}\right)=1$. Second, its index equals the sum over all enclosed singularities, i.e., $$\sum_{i}\operatorname{ind}_{\Gamma}(p_{i},\mathbf{l})=\operatorname{ind}_{\Gamma}(\mathbf{l})=1.\label{eq:sum}$$ Since we consider structurally stable singularities only, these are isolated and of either wedge or trisector type. From , we then obtain the equality $$\frac{1}{2}\left(W-T\right)=1,$$ from which Eq. follows.
Consequently, in the interior of any closed orbit of a structurally stable line field, there are at least two singularities of wedge type, and exactly two more wedges than trisectors. Thus, a closed orbit necessarily encircles a wedge pair, and hence the existence of such a pair serves as a necessary condition in an automated search for closed orbits in line fields. In Fig. \[fig:topo\_cycle\], we sketch two possible line field geometries in the interior of a closed orbit.
![Possible topologies inside closed orbits: the $\left(W,T\right)=(2,0)$ configuration (left) and the $(3,1)$ configuration (right). In practice, we have only observed the simpler $(2,0)$ configuration.[]{data-label="fig:topo_cycle"}](topology){width="50.00000%"}
Application to coherent Lagrangian vortex detection {#sec:Application}
===================================================
Finding closed orbits in planar line fields is the decisive step in the detection of coherent Lagrangian vortices in a frame-invariant fashion [@Haller2012; @Beron-Vera2013; @Haller2013a]. Before describing the algorithmic scheme and showing results on ocean data, we briefly introduce the necessary background and notation for coherent Lagrangian vortices.
Flow map, Cauchy–Green strain tensor and lambda–line field
----------------------------------------------------------
We consider an unsteady, smooth, incompressible planar velocity field $\mathbf{u}(t,\mathbf{x})$ given on a finite time interval $\left[t_{0},t_{0}+T\right]$, and the corresponding equation of motion for the fluid, $$\dot{\mathbf{x}}=\mathbf{u}(t,\mathbf{x}).$$ We denote the associated flow map by $\mathbf{F}_{t_{0}}^{t_{0}+T}$, which maps initial values $\mathbf{x}_{0}$ at time $t_{0}$ to their respective position at time $t_{0}+T$. Recall that the flow map is as smooth as the velocity field $\mathbf{u}$. Its linearisation can be used to define the *Cauchy–Green strain tensor field* $$\mathbf{C}_{t_{0}}^{t_{0}+T}\coloneqq\left(D\mathbf{F}_{t_{0}}^{t_{0}+T}\right)^{\top}D\mathbf{F}_{t_{0}}^{t_{0}+T},$$ which is symmetric and positive-definite at each initial value. The eigenvalues and eigenvectors of $\mathbf{C}_{t_{0}}^{t_{0}+T}$ characterize the magnitude and directions of maximal and minimal stretching locally in the flow. We refer to these positive eigenvalues as $\lambda_{1}\leq\lambda_{2}$, with the associated eigenspaces spanned by the normalized eigenvectors $\xi_{1}$ and $\xi_{2}$.
As argued by [@Haller2013a], the positions of coherent Lagrangian vortex boundaries at time $t_{0}$ are closed stationary curves of the averaged tangential strain functional computed from $\mathbf{C}_{t_{0}}^{t_{0}+T}$. All stationary curves of this functional turn out to be uniformly stretched by a factor of $\lambda>0$ under the flow map $\mathbf{F}_{t_{0}}^{t_{0}+T}$. These stationary curves can be computed as closed orbits of the *$\lambda$–line fields* $\eta_{\lambda}^{\pm}$, spanned by the representing vector fields $$\label{eq:lambda-line-field}
\eta_{\lambda}^{\pm}\coloneqq\sqrt{\frac{\lambda_{2}-\lambda^{2}}{\lambda_{2}-\lambda_{1}}}\xi_{1}\pm\sqrt{\frac{\lambda^{2}-\lambda_{1}}{\lambda_{2}-\lambda_{1}}}\xi_{2}.$$
We refer to orbits of $\eta_{\lambda}^{\pm}$ as *$\lambda$–lines*. In the special case of $\lambda=1$, the line field $\eta_{1}^{\pm}$ coincides with the *shear line field* defined in [@Haller2012], provided that the fluid velocity field $\mathbf{u}(t,\mathbf{x})$ is incompressible.
We refer to points at which the Cauchy–Green strain tensor is isotropic (i.e., equals a constant multiple of the identity tensor) as *Cauchy–Green singularities*. For incompressible flows, only $\mathbf{C}_{t_{0}}^{t_{0}+T}=\mathbf{I}$ is possible at Cauchy–Green singularities, implying $\lambda_{1}=\lambda_{2}=1$ at these points. The associated eigenspace fields, $\xi_{1}$ and $\xi_{2}$, are ill-defined as line fields at Cauchy–Green singularities, thus generically the line fields $\xi_{1}$, $\xi_{2}$ and $\eta_{1}^{\pm}$ have singularities at these points. Conversely, the singularities of the line fields $\xi_{1}$, $\xi_{2}$ and $\eta_{1}^{\pm}$ are necessarily Cauchy–Green singularities, as seen from the local vector field representation in Eq. .
Following [@Haller2012; @Haller2013a], we define an *elliptic Lagrangian Coherent Structure (LCS)* as a structurally stable closed orbit of $\eta_{\lambda}^{\pm}$ for some choice of the $\pm$ sign, and for some value of the parameter $\lambda$. We then define a *(coherent Lagrangian) vortex boundary* as the locally outermost elliptic LCS over all choices of $\lambda$.
Index theory for lambda–line fields
-----------------------------------
In regions where $\lambda_{1}<\lambda^{2}<\lambda_{2}$ is not satisfied, $\eta_{\lambda}^{\pm}$ is undefined. Such open regions necessarily arise around Cauchy–Green singularities, and hence $\eta_{\lambda}^{\pm}$ does not admit isolated point-singularities. Consequently, the index theory presented in does not immediately apply to the $\lambda$–line field. We show below, however, that Cauchy–Green singularities are still necessary indicators of closed orbits of $\eta_{\lambda}^{\pm}$ for arbitrary $\lambda$.
For $\lambda>1$, the set $D_{\lambda}^{2}=\left\{ \lambda_{2}<\lambda^{2}\right\}$, on which $\eta_{\lambda}^{\pm}$ is undefined, consists of open connected components. All Cauchy–Green singularities are contained in some of these $D_{\lambda}^{2}$-components. A priori, however, there may exist $D_{\lambda}^{2}$-components that do not contain Cauchy–Green singularities.
On the boundary $\partial D_{\lambda}^{2}$, we have $\lambda_{2}=\lambda^{2}$, and hence $\eta_{\lambda}^{\pm}$ coincides with $\xi_{2}$ on $\partial D_{\lambda}^{2}$, as shown in Fig. \[fig:continuation\]. Therefore, we may extend $\eta_{\lambda}^{\pm}$ into $D_{\lambda}^{2}$ by letting $\eta_{\lambda}^{\pm}(\mathbf{x})\coloneqq\xi_{2}(\mathbf{x})$ for all $\mathbf{x}\in D_{\lambda}^{2}$, thereby obtaining a continuous, piecewise differentiable line field, whose singularity positions coincide with those of the $\xi_{2}$-singularities.
applies directly to the continuation of the line field $\eta_{\lambda}^{\pm}$, and enables the detection of closed orbits lying outside the open set $D_{\lambda}^{2}$. In the case $\lambda<1$, the line field $\eta_{\lambda}^{\pm}$ can similarly be extended in a continuous fashion into the interior of the set $D_{\lambda}^{1}=\left\{ \lambda_{1}>\lambda^{2}\right\} ,$ through the definition $\eta_{\lambda}^{\pm}(\mathbf{x})\coloneqq\xi_{1}(\mathbf{x})$ for all $\mathbf{x}\in D_{\lambda}^{1}$.
![The original domain of definition of $\eta_{\lambda}^{\pm}$ (grey), and the domain $D_{\lambda}^{2}$ (white), to which $\eta_{\lambda}^{\pm}$ can be continuously extended via the line field $\xi_2$. Also shown is a point $\mathbf{p}$ denoting a Cauchy–Green singularity.[]{data-label="fig:continuation"}](continuation){width="45.00000%"}
After its extension into the set $D_{\lambda}=D_{\lambda}^{1}\cup D_{\lambda}^{2}$, the line field $\eta_{\lambda}^{\pm}$ inherits each Cauchy–Green singularity either from $\xi_{2}$ or from $\xi_{1}$. A priori, the same Cauchy–Green singularity may have different topological types in the $\xi_{1}$ and $\xi_{2}$ line fields. By [@Delmarcelle1994a Theorem 11], however, this is not the case: corresponding generic singularities of $\xi_{2}$ and $\xi_{1}$ share the same index and have the same number of hyperbolic sectors. Furthermore, the separatrices of the $\xi_{2}$-singularity are obtained from the separatrices of the $\xi_{1}$-singularity by reflection with respect to the singularity. In summary, $\xi_{1}$-wedges correspond exactly to $\xi_{2}$-wedges, and the same holds for trisectors. For the singularity type classification for $\eta_{\lambda}^{\pm}$, $\lambda\neq1$, we may therefore pick $\xi_{2}$, irrespective of the sign of $\lambda-1$.
The singularity-type correspondence extends also to the limit case $\lambda=1$, i.e., to $\eta_{1}^{\pm}$, as follows. Consider the one-parameter family of line-field extensions $\eta_{\lambda}^{\pm}$. By construction, the locations of $\eta_{\lambda}^{\pm}$ point singularities coincide with those of the $\xi_{2}$-singularities for any $\lambda$. Variations of $\lambda$ correspond to continuous line-field perturbations, which leave the types of structurally stable singularities unchanged. Hence, the types of $\eta_{1}^{\pm}$-singularities must match the types of corresponding $\eta_{\lambda}^{\pm}$-singularities, or equivalently of corresponding $\xi_{2}$-singularities. To summarize, we obtain the following conclusion.
\[prop:eta-field\] Any closed orbit of a structurally stable $\eta_\lambda^\pm$ field necessarily encircles Cauchy–Green singularities satisfying Eq. .
A simple example: coherent Lagrangian vortex in the double gyre flow
--------------------------------------------------------------------
We consider the left vortex of the double gyre flow [@Shadden2005], defined on the spatial domain $[0,1]\times[0,1]$ by the ODE $$\begin{aligned}
\dot{x} & =-\pi A\sin(\pi f(x))\cos(\pi y),\\
\dot{y} & =\pi A\cos(\pi f(x))\sin(\pi y)\partial_{x}f(t,x),\end{aligned}$$ where $$f(t,x)=\varepsilon\sin(\omega t)x^{2}+\left(1-2\varepsilon\sin(\omega t)\right)x.$$ We choose the parameters of the flow model as $A=0.2$, $\varepsilon=0.2$, $\omega=\pi/5$, $t_{0}=0$, and $T=5\pi/2$.
In the $\lambda$–line field shown in Fig. \[fig:double\_gyre\](a), we identify a pair of wedge singularities. Any closed $\lambda$–line must necessarily enclose this pair by . This prompts us to define a Poincaré section through the midpoint of the connecting line between the two wedges. For computational simplicity, we select the Poincaré section as horizontal. Performing a parameter sweep over $\lambda$–values, we obtain the outermost closed orbit shown in Fig. \[fig:double\_gyre\](a) for a uniform stretching rate of $\lambda=0.975$. Other non-closing orbits and the $\lambda$–line field are also shown for illustration. In addition, we show the line field topology around the wedge pair in the vortex core in Fig. \[fig:double\_gyre\](b).
In this simple example, the vortex location is known, and hence a Poincaré section could manually be set for closed orbit detection in the $\lambda$–line fields. In more complex flows, however, the vortex locations are a priori unknown, making a manual search unfeasible.
![(a) Vortex boundary ($\lambda=0.975$) for the left vortex of the double gyre flow. In the centre, the pair of wedge singularities (red) determines the topology of the $\lambda$–line field $\eta_{\lambda}^{-}$ (grey) and therefore indicates a candidate region for closed orbits. The $\lambda$–lines (black) are launched from the Poincaré section (black crosses) to find the outermost closed orbit (green). (b) A blow-up of the centre of the vortex with the detailed circular topology of the $\lambda$–line field $\eta_{\lambda}^{-}$ in the vicinity of the $(2,0)$ wedge pair configuration (cf. Fig. \[fig:topo\_cycle\]).[]{data-label="fig:double_gyre"}](double_gyre){width=".95\textwidth"}
Implementation for vortex census in large-scale ocean data {#sub:implementation}
----------------------------------------------------------
Our automated Lagrangian vortex-detection scheme relies on , identifying candidate regions in which Poincaré maps for closed $\lambda$–line detection should be set up. In several tests on ocean data, we only found the $(W,T)=(2,0)$ singularity configuration inside closed $\lambda$–lines. This is consistent with our previous genericity considerations. Consequently, we focus on finding candidate regions for closed $\lambda$–lines as regions with isolated pairs of wedges in the $\xi_{2}$ field. In the following, we describe the procedure for an automated detection of closed $\lambda$–lines.
#### 1. Locate singularities. {#locate-singularities. .unnumbered}
Recall that Cauchy–Green singularities are points where$\mathbf{C}_{t_{0}}^{t_{0}+T}=\mathbf{I}$. We find such points at subgrid-resolution as intersections of the zero level sets of the functions $c_{1}\coloneqq C_{11}-C_{22}$ and $c_{2}\coloneqq C_{12}=C_{21}$, where $C_{ij}$ denote the entries of the Cauchy–Green strain tensor.
#### 2. Select relevant singularities. {#select-relevant-singularities. .unnumbered}
We focus on generic singularities, which are isolated and are of wedge or trisector type. We discard tightly clustered groups of singularities, which indicate non-elliptic behavior in that region. Effectively, the clustering of singularities prevents the reliable determination of their singularity type. To this end, we select a minimum distance threshold between admissible singularities as $2\Delta x$, where $\Delta x$ denotes the grid size used in the computation of $\mathbf{C}_{t_{0}}^{t_{0}+T}$. We obtain the distances between closest neighbours from a Delaunay triangulation procedure.
#### 3. Determine singularity type. {#determine-singularity-type. .unnumbered}
Singularities are classified as trisectors or wedges, following the approach developed in [@Farazmand2013a]. Specifically, a circular neighbourhood of radius $r>0$ is selected around a singularity, so that no other singularity is contained in this neighbourhood. With a rotating radius vector $\mathbf{r}$ of length $r$, we compute the absolute value of the cosine of the angle enclosed by $\mathbf{r}$ and $\xi_{2}$, i.e., $\cos\left(\angle\left(\mathbf{r},\xi_{2}\right)\right)=\left|\mathbf{r}\cdot\xi_{2}\right|/r$, with the eigenvector field $\xi_{2}$ interpolated linearly to 1000 positions on the radius $r$ circle around the singularity. The singularity is classified as a trisector, if $\mathbf{r}$ is orthogonal to $\xi_{2}$ at exactly three points of the circle, and parallel to $\xi_{2}$ at three other points, which mark separatrices of the trisector (Fig. \[fig:deg\_points\]). Singularities not passing this test for trisectors are classified as wedges. Other approaches to singularity classification can be found in [@Delmarcelle1994] and [@Bazen2002], which we have found too sensitive for oceanic data sets.
#### 4. Filter {#filter .unnumbered}
We discard wedge points whose closest neighbour is of trisector-type, because these wedge points cannot be part of an isolated wedge pair. We further discard single wedges whose distance to the closest wedge point is larger than the typical mesoscale distance of $2\,^{\circ}\approx200$km. The remaining wedge pairs mark candidate regions for elliptic LCS (Fig. \[fig:coherent\_eddies\](a)).
#### 5. Launch $\lambda$–lines from a Poincaré-section {#launch-lambdalines-from-a-poincaré-section .unnumbered}
We set up Poincaré sections that span from the midpoint of a wedge pair to a point $1.5\,^{\circ}$ apart in longitudinal direction (Fig. \[fig:coherent\_eddies\](b)). This choice of length for the Poincaré section captures eddies up to a diameter of $3\,^{\circ}\approx300$km, an upper bound on the accepted size for mesoscale eddies. For a fixed $\lambda$–value, $\lambda$–lines are launched from 100 initial positions on the Poincaré section, and the return distance $P(x)-x$ is computed. Zero crossings of the return distance function correspond to closed $\lambda$–lines. The position of zeros is subsequently refined on the Poincaré section through the bisection method. The outermost zero crossing of the return distance marks the largest closed $\lambda$–line for the chosen $\lambda$–value. To find the outermost closed $\lambda$–line over all $\lambda$–values, we vary $\lambda$ from $0.85$ to $1.15$ in $0.01$ steps, and pick the outermost closed orbit as the Lagrangian eddy boundary. During this process, we make sure that eddy boundaries so obtained do enclose the two wedge singularities used in the construction, but no other singularities.
The runtime of our algorithm is dominated by the fifth step, the integration of $\lambda$–lines, as illustrated in Table \[tab:run-time\] for the ocean data example in the next section. This is the reason why our investment in the selection, classification and filtering of singularities before the actual $\lambda$–line integration pays off.
Coherent Lagrangian vortices in an ocean surface flow
-----------------------------------------------------
We now apply the method summarized in steps 1-5 above to two-dimensional unsteady velocity data obtained from AVISO satellite altimetry measurements. The domain of the data set is the Agulhas leakage in the Southern Ocean, represented by large coherent eddies that pinch off from the Agulhas current of the Indian Ocean.
Under the assumption of a geostrophic flow, the sea surface height $h$ serves as a streamfunction for the surface velocity field. In longitude-latitude coordinates $(\varphi,\theta)$, particle trajectories are then solutions of $$\begin{aligned}
\dot{\varphi} & =-\frac{\mathrm{g}}{\mathrm{R}^{2}f(\theta)\cos\theta}\partial_{\theta}h(\varphi,\theta,t), & \dot{\theta} & =\frac{\mathrm{g}}{\mathrm{R}^{2}f(\theta)\cos\theta}\partial_{\varphi}h(\varphi,\theta,t),\end{aligned}$$ where $\mathrm{g}$ is the constant of gravity, $\mathrm{R}$ is the mean radius of the Earth, and $f(\theta)\coloneqq 2\Omega\sin\theta$ is the Coriolis parameter, with $\Omega$ denoting the Earth’s mean angular velocity. For comparison, we choose the same spatial domain and time interval as in [@Beron-Vera2013; @Haller2013a]. The integration time $T$ is also set to $90$ days.
Fig. \[fig:coherent\_eddies\] illustrates the steps of the eddy detection algorithm. From all singularities of the Cauchy–Green strain tensor, isolated wedge pairs are extracted (Fig. \[fig:coherent\_eddies\](a)) and closed orbits are found by launching $\lambda$–lines from Poincaré sections anchored at those wedge pairs (Fig. \[fig:coherent\_eddies\](b)). Altogether, 14 out of the selected wedge pairs are encircled by closed orbits and, hence, by coherent Lagrangian eddy boundaries (Fig. \[fig:coherent\_eddies\](c)). The reduction to candidate regions consistent with leads to significant gain in computational speed. This is because the computationally expensive integration of the $\lambda$–line field is only carried out in these regions (Table \[tab:run-time\]). For comparison, the computational cost on a single Poincaré section is already higher than the cost of identifying the candidate regions. Note also that two regions contain three wedges, which constitute two admissible wedge pairs. This explains how 78 wedges constitute 40 wedge pairs altogether.
![Visualization of the eddy detection algorithm for an ocean surface flow. (a) Singularities of the Cauchy–Green strain tensor of trisector type (red triangles) and wedge type (green circles: kept, red dots: discarded). Wedge pairs are candidate cores of coherent eddies. A total of 40 wedge pairs were finally selected for further analysis out of all singularities (black crosses) by the procedure described in Section \[sub:implementation\]. (b) Poincaré sections anchored at the centre of the selected wedge pairs. Coherent vortex boundaries are found as closed $\lambda$–lines intersecting these Poincaré sections. (c) Boundaries of 14 coherent eddies on November 24, 2006. The $\log_{10}\lambda_{2}$ field is shown in the background as an illustration of the stretching distribution in the flow.[]{data-label="fig:coherent_eddies"}](ocean_eddies){height="0.75\textheight"}
Runtime Number of points
-------------------- ------------------------------------------- ----------------------
1\. Localization $11.0$s 14,211 singularities
2\. Selection $12.8$s 912 singularities
3\. Classification $85.9$s 414 wedges
4\. Filtering $0.5$s 78 wedges
5\. Integration $\sim200$s / wedge pair / $\lambda$–value 40 wedge pairs
End result — 14 eddies
: Runtime of the algorithm for the Agulhas data set on a CPU with 2.2GHz and 32GB RAM. Since the integration of $\lambda$–lines is the computationally most expensive part, the reduction of the number of candidate regions to only 40 by application of index theory yields a significant computational advantage.[]{data-label="tab:run-time"}
Conclusion {#sec:Conclusions}
==========
We have discussed the use of index theory in the detection of closed orbits in planar line fields. Combined with physically motivated filtering criteria, index-based elliptic LCS detection provides an automated implementation of the variational results of [@Haller2013a] on coherent Lagrangian vortex boundaries. Our results further enhance the power of LCS detection algorithms already available in the Matlab toolbox <span style="font-variant:small-caps;">LCS TOOL</span> (cf. [@Onu2014]).
Our approach can be extended to three-dimensional flows, where line fields arise in the computation of intersections of elliptic LCS with two-dimensional planes [@Blazevski2014]. Applied over several such planes, our approach allows for an automated detection of coherent Lagrangian eddies in three-dimensional unsteady velocity fields.
Automated detection of Lagrangian coherent vortices should lead to precise estimates on the volume of water coherently carried by mesoscale eddies, thereby revealing the contribution of coherent eddy transport to the total flux of volume, heat and salinity in the ocean. Related work is in progress.
Acknowledgment {#acknowledgment .unnumbered}
==============
The altimeter products used in this work are produced by SSALTO/DUACS and distributed by AVISO, with support from CNES (<http://www.aviso.oceanobs.com>). We would like to thank Bert Hesselink for providing Ref. [@Delmarcelle1994a], Xavier Tricoche for pointing out Refs. [@Wischgoll2001; @Wischgoll2006], and Ulrich Koschorke and Francisco Beron-Vera for useful comments.
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---
abstract: 'In this paper we report the first LDA+DMFT results (method combining Local Density Approximation with Dynamical Mean-Field Theory) for spectral properties of superconductor LaFePO. Calculated [**k**]{}-resolved spectral functions reproduce recent angle-resolved photoemission spectroscopy (ARPES) data \[D. H. Lu [*et al*]{}., Nature [**455**]{}, 81 (2008)\]. Obtained effective electron mass enhancement values $m^{*}/m\approx$ 1.9 – 2.2 are in good agreement with infrared and optical studies \[M. M. Qazilbash [*et al*]{}., Nature Phys. [**5**]{}, 647 (2009)\], de Haas–van Alphen, electrical resistivity, and electronic specific heat measurements results, that unambiguously evidence for moderate correlations strength in LaFePO. Similar values of $m^{*}/m$ were found in the other Fe-based superconductors with substantially different superconducting transition temperatures. Thus, the dynamical correlation effects are essential in the Fe-based superconductors, but the strength of electronic correlations does not determine the value of superconducting transition temperature.'
author:
- 'S. L. Skornyakov,$^{1,2}$ N. A. Skorikov,$^{1}$ A. V. Lukoyanov,$^{1,2}$ A. O. Shorikov,$^{1,2}$ and V. I. Anisimov$^{1,2}$'
title: LDA+DMFT spectral functions and effective electron mass enhancement in superconductor LaFePO
---
Introduction.
=============
The discovery of superconductivity with the transition temperature $T_{c}\approx$ 4 K in LaFePO[@Kamihara06] and $T_{c}\approx$ 26–55 K in RO$_{1-x}$F$_x$FeAs (R = La, Sm)[@Kamihara08] has generated great interest to the new class of Fe-based superconductors. [@review] While the microscopic mechanism of superconductivity in LaFePO is not yet clear, [@Kuroki09; @Yamashita09] its electronic properties have been studied extensively. Various experiments revealed the presence of electronic correlations in LaFePO. Analyzing angle-resolved photoemission spectroscopy (ARPES) data, D. H. Lu [*et al*]{}. (Ref. ) demonstrated that the DFT band structure should be renormalized by a factor of 2.2 to fit the experimental angle-resolved photoemission spectra. From infrared and optical conductivity data the authors of Ref. made a conclusion that the effective electron mass renormalization is about 2 in LaFePO. Similarly, the electron mass renormalization obtained from de Haas–van Alphen study is $m^{*}/m\approx$ 1.7 – 2.1 (Ref. ) Comparison of the experimental electronic specific heat coefficient $\gamma_n$ = 10.1 mJ/mol K$^2$ for LaFePO[@Suzuki09] with the DFT-value 5.9 mJ/mol K$^2$ (Ref. ) gives the value of 1.7 for electron mass enhancement. Also the electrical resistivity at low temperatures has $T^2$-dependence[@Suzuki09; @Yamashita09] showing importance of correlation effects. However, the authors of Ref. from analysis of x-ray absorption (XAS) and resonant inelastic x-ray scattering (RIXS) data for several iron pnictide compounds (SmO$_{0.85}$FeAs, BaFe$_2$As$_2$, LaFe$_2$P$_2$) arrived at a conclusion that correlations are not very strong here.
So far only a few electronic structure calculations for LaFePO by the DFT-based first-principles methods without any account for electronic correlations have been reported[@Lu; @Lebegue; @Vildosola; @Che08; @KamiPRB] thus making the problem of correlation effects study for this material very timely.
The combination of Density Functional Theory (DFT) and Dynamical Mean-Field Theory (DMFT) called LDA+DMFT method[@LDA+DMFT] is presently recognized state-of-the-art many-particle method to study correlation effects in real compounds. LDA+DMFT calculations for the FeAs-based superconductors[@Haule; @jpcm; @bafe; @Aichhorn] lead to diverse conclusions on the strength of electron correlations in these materials. The authors of Ref. have proposed an extended classification scheme of the electronic correlation strength in the Fe-pnictides based on analysis of several relevant quantities: a ratio of the Coulomb parameter $U$ and the band width $W$ ($U$/$W$), quasiparticle mass enhancement $m^{*}/m$, ${\bf k}$-resolved and ${\bf k}$-integrated spectral functions $A({\bf k},\omega)$ and $A(\omega)$. Applying this scheme to LDA+DMFT results for BaFe$_{2}$As$_{2}$ they came to conclusion that this material should be regarded as a [*moderately*]{} correlated metal. In this work we report the results of LDA+DMFT study for electronic correlation effects in LaFePO. For this purpose we have calculated spectral functions $A({\bf k}, \omega)$, effective electron mass enhancement $m^{*}/m$ and compare our results with the available measurements for LaFePO finding a very good agreement between calculated and experimental data. A moderate spectral functions renormalization corresponding to $m^{*}/m\approx$ 2 was found in LaFePO similar to values obtained for BaFe$_{2}$As$_{2}$[@bafe] while superconducting transition temperature in those materials could be an order of magnitude different.
Method.
=======
The LDA+DMFT scheme is constructed in the following way: First, a Hamiltonian $\hat H_{LDA}$ is produced using converged LDA results for the system under investigation, then the many-body Hamiltonian is set up, and finally the corresponding self-consistent DMFT equations are solved. By projecting onto Wannier functions, [@projection] we obtain an effective 22-band Hamiltonian which incorporates five Fe [*d*]{}, three O [*p*]{}, and three P [*p*]{} orbitals per formula unit. In the present study we construct Wannier states for an energy window including both [*p*]{} and [*d*]{} bands. Thereby hybridization effects between [*p*]{} and [*d*]{} electrons were explicitly taken into account and eigenvalues of the Wannier functions Hamiltonian $\hat H_{LDA}$ exactly correspond to the 22 Fe, O, and P bands from LDA. The LDA calculations were performed with the experimentally determined crystal structure[@Zimmer] using the Elk full-potential linearized augmented plane-wave (FP-LAPW) code. [@Elk] Parameters controlling the LAPW basis were kept to their default values. The calculated LDA band structure $\epsilon_{LDA}({\bf k})$ was found to be in good agreement with that of Lebègue *et al.* (Ref. ).
The many-body Hamiltonian to be solved by DMFT has the form $$\hat H= \hat H_{LDA}- \hat H_{dc}+\frac{1}{2}\sum_{i,\alpha,\beta,\sigma,\sigma^{\prime}}
U^{\sigma\sigma^{\prime}}_{\alpha\beta}\hat n^{d}_{i\alpha\sigma}\hat n^{d}_{i\beta\sigma^{\prime}},$$ where $U^{\sigma\sigma^{\prime}}_{\alpha\beta}$ is the Coulomb interaction matrix, $\hat n^d_{i\alpha\sigma}$ is the occupation number operator for the $d$ electron with orbital $\alpha$ or $\beta$ and spin indices $\sigma$ or $\sigma^{\prime}$ in the $i$-th site. The term $\hat H_{dc}$ stands for the [*d*]{}-[*d*]{} interaction already accounted in LDA, so called double-counting correction. The double-counting has the form $\hat H_{dc}=\bar{U}(n_{\rm dmft}-\frac{1}{2})\hat{I}$ where $n_{\rm dmft}$ is the total self-consistent number of [*d*]{} electrons obtained within the LDA+DMFT and $\bar{U}$ is the average Coulomb parameter for the [*d*]{} shell.
The DMFT self-consistency equations were solved iteratively for imaginary Matsubara frequencies. The auxiliary impurity problem was solved by the hybridization function expansion Continuous-Time Quantum Monte-Carlo (CTQMC) method. [@QMC] In the present implementation of the CTQMC impurity solver the Coulomb interaction is taken into account in density-density form. The elements of $U_{\alpha\beta}^{\sigma\sigma'}$ matrix were parameterized by $U$ and $J$ according to procedure described in Ref. . We used interaction parameters $\bar{U}$ = 3.1 eV and $J$ = 1 eV similar to the values calculated by the constrained LDA method for Wannier functions[@Korotin] in Fe-pnictides. [@jpcm] Calculations were performed in the paramagnetic state at the inverse temperature $\beta=1/T$ = 20 eV$^{-1}$. The real-axis self-energy needed to calculate spectral functions was obtained by the Padé approximant[@Pade] (see Appendix).
Results and discussion.
=======================
![(Color online) Orbitally resolved Fe 3[*d*]{}, O [*p*]{} and P [*p*]{} normalized spectral functions of LaFePO obtained within LDA+DMFT (upper panel) are compared with the LDA results (lower panel).[]{data-label="DMFTvsLDA"}](fig1.eps){width="0.95\linewidth"}
The orbitally resolved Fe 3$d$, O [*p*]{} and P [*p*]{} spectral functions computed within LDA and LDA+DMFT, respectively, are compared in Fig. \[DMFTvsLDA\]. Within the LDA all five Fe $d$ orbitals form a common band in the energy range ($-$2.5, $+$2.0) eV relative to the Fermi level (band width $W\approx$ 4.5 eV). There is a significant hybridization of the Fe $3d$ orbitals with the P $p$ and O $p$ orbitals, leading to appearance of Fe $d$ states contribution in the energy interval ($-$5.5, $-$2.5) eV where the P $p$ band is located. The corresponding features of LDA+DMFT spectral functions (upper panel in Fig. \[DMFTvsLDA\]) in the energy area ($-$5.5, $-$2.5) eV should not be mistaken for Hubbard bands because the same peaks are present in non-correlated LDA bands (lower panel in Fig. \[DMFTvsLDA\]). Correlation effects do not result in Hubbard bands appearance but lead to significant renormalization of the spectral function around the Fermi energy: “compressing” of energy scale so that separation between peaks of LDA+DMFT curves becomes $\approx$2 times smaller than in corresponding non-correlated spectra.
It is instructive to plot energy dependence of real part of self-energy Re$\Sigma(\omega)$ (see Fig. \[Sigmax2my2\]). Peaks in spectral function $A({\bf k}, \omega)$ are determined by the poles of $(\omega-\epsilon({\bf k})-\Sigma(\omega))^{-1}$ function or the energy values $\omega=\epsilon({\bf k})+\mbox{Re}\Sigma(\omega)$ (here $\epsilon({\bf k})$ is non-correlated band dispersion). In Fig. \[Sigmax2my2\] together with Re$\Sigma(\omega)$ a function $\omega+(H_{dc})_{ii}$ is plotted as a stripe having the width of non-correlated band $\epsilon({\bf k})$. The peaks of spectral function $A({\bf k}, \omega)$ correspond to energy area where this stripe crosses Re$\Sigma(\omega)$ curve. As one can see such crossing happens only once in the energy interval around the Fermi level so that spectral functions will have only poles corresponding to quasiparticle bands and no Hubbard band poles will be observed.
![(Color online) The real part of the self-energy $\Sigma(\omega)$ (black line) depicted together with $\omega+(H_{dc})_{ii}$ (light stripe, see text).[]{data-label="Sigmax2my2"}](fig2.eps){width="0.885\linewidth"}
A quantitative measure of the electron correlation strength is provided by the quasiparticle renormalization factor $Z=(1-\frac{\partial \Sigma}{\partial\omega}|_{\omega=0})^{-1}$ which gives an effective mass enhancement $m^{*}/m=Z^{-1}$. In general, the self-energy is a matrix, leading to different effective masses for different bands. The calculated $m^{*}/m$ values for every [*d*]{}-orbital are presented in Table 1. The $d_{x^{2}-y^{2}}$ orbital has the smallest effective mass renormalization $m^{*}/m$ = 1.942. The other [*d*]{} orbitals have approximately the same value $m^{*}/m\approx$ 2.2.
Orbitals $d_{xy}$ $d_{yz,xz}$ $d_{3z^{2}-r^{2}}$ $d_{x^{2}-y^{2}}$
----------- ---------- ------------- -------------------- -------------------
$m^{*}/m$ 2.189 2.152 2.193 1.942
: Effective mass renormalization $m^{*}/m$ of quasiparticles in LaFePO for different orbitals of the Fe [*d*]{} shell from the LDA+DMFT calculation.
The calculated effective mass enhancement $m^{*}/m\approx$ 1.9 – 2.2 in LaFePO agrees very well with the de Haas–van Alphen experiments[@dHvA] where it was found to range from 1.7 to 2.1, and with the estimations of the effective mass renormalization of a factor of 2 from optical conductivity data[@Qazi] and specific heat measurements. [@Suzuki09]
We now calculate the [**k**]{}-resolved spectral function $$A({\bf k}, \omega)=-{\rm Im}\frac{1}{\pi} Tr[(\omega+\mu)\hat{I}-\hat h_{\bf k}-\hat{\Sigma} (\omega)]^{-1}.
\label{SFunction}$$ Here $\hat h_{\bf k} = \hat H_{LDA}- \hat H_{dc}$ is the 22$\times$22 Hamiltonian matrix on a mesh of [**k**]{}-points and $\mu$ is the self-consistently determined chemical potential. In Fig. \[Ake\_contour\] we compare our results with ARPES data of Lu [*et al*]{}. (Ref. ). Both theory and experiment show dispersive bands crossing the Fermi level near the $\Gamma$ and M points. In addition, two bands can be seen at –0.2 eV and in the region from –0.3 to –0.4 eV near the $\Gamma$ point. The calculated shape and size of the hole and electron pockets centered at the $\Gamma$ and M points, respectively, are in good agreement with the ARPES, see Fig. \[Ake\_contour\] (lower panel), and de Haas–van Alphen[@dHvA] data.
![(Color online) The [**k**]{}-resolved total spectral function A([**k**]{}, $\omega$) of LaFePO along the $\Gamma-X-\Gamma$ and $\Gamma-M$ lines in the Brillouin zone is depicted as a contour plot. Upper panel: The LDA+DMFT spectral function. Lower panel: The corresponding experimental ARPES intensity map of Lu [*et al*]{}. (Ref. ).[]{data-label="Ake_contour"}](fig3.eps){width="0.95\linewidth"}
The correlated band structure $\epsilon_{DMFT}({\bf k})$ is also shown in Fig. \[Ake\_contour\] (upper panel). Near the Fermi energy, i.e., in the energy range from –0.2 eV to zero where quasiparticles are well defined (as expressed by a linear behavior of Re$ \Sigma(\omega)$, see Fig. \[Sigmax2my2\]), this dispersion is very well represented by the scaling relation $\epsilon_{DMFT}({\bf k})=\epsilon_{LDA}({\bf k})/(m^*/m)$, with $m^*/m$ taken as the computed mass enhancement from Table 1.
The present results mean that the band structure in LaFePO is renormalized by the correlations and consists of the quasi-two-dimensional Fermi surface sheets observed in the experiments. At the same time, there is no substantial spectral weight transfer from quasiparticle bands near the Fermi energy to Hubbard bands and the system is far away from metal-insulator transition. Such spectral function behavior does not allow to classify LaFePO either as [*strongly*]{} or [*weakly*]{} correlated material. According to classification proposed in Ref. this material can be regarded as a [*moderately*]{} correlated metal, like BaFe$_{2}$As$_{2}$. [@bafe]
In other iron pnictide materials the electron mass enhancement was also reported to be close to value of $\approx$2. For example, from the de Haas–van Alphen experiments $m^{*}/m$ = 1.13 – 3.41 (Ref. ) was found for SrFe$_{2}$P$_{2}$, isostructural analogue of the superconducting compounds Sr$_{1-x}$K$_{x}$Fe$_{2}$As$_{2}$ ($T_{c}$ = 37 K). For the series of isostructural compounds Ba(Fe$_{1-x}$Co$_{x}$)$_2$As$_{2}$, $x$ = 0 – 0.3 the authors of Ref. evaluated $m^{*}/m$ = 2 – 4 from the ARPES data. Also from the analysis of ARPES an electron mass renormalization by 2.7 was evaluated in Ba$_{0.6}$K$_{0.4}$Fe$_{2}$As$_{2}$ ($T_{c}$ = 37 K)[@YiLu] as comparing with the DFT-band structure. Further, the DFT-calculated plasma frequencies are by a factor of 1.5 to 2 larger than the experimental values, [@Drechsler] showing the electron mass enhancement of the same strength for LaFePO, LaFeAsO, SrFe$_{2}$As$_{2}$, BaFe$_{2}$As$_{2}$, K$_{0.45}$Ba$_{0.55}$Fe$_{2}$As$_{2}$ ($T_{c}\approx$ 30 K), and LaO$_{0.9}$F$_{0.1}$FeAs ($T_{c}\approx$ 26 K). From the reported data, it follows that the electronic correlation strength in the FeAs-based superconductors with the superconducting transition temperatures up to 37 K and their parent compounds is the same as in superconductor LaFePO with $T_{c}\approx$ 4 K. Thus, the strength of electronic correlations in the Fe-based superconductors seems to be not intrinsically connected with the superconducting transition temperature.
Conclusion.
===========
By employing the LDA+DMFT method we have calculated the spectral functions and single-particle [**k**]{}-resolved spectrum of superconductor LaFePO for the first time. Very good agreement with the ARPES data was found. In the spectral functions we observed no substantial spectral weight transfer. The obtained effective electron mass enhancement values $m^{*}/m\approx$ 1.9 – 2.2 are in good agreement with infrared and optical studies, de Haas–van Alphen, and specific heat results. The electronic correlation strength in LaFePO with small value of superconducting temperature 4 K is similar to the other Fe-pnictide superconductors with the transition temperatures up to 37 K.
Acknowledgments.
================
The authors thank D. Vollhardt for useful discussions, J. Kuneš for providing DMFT computer code used in our calculations, P. Werner for the CT-QMC impurity solver, D. H. Lu and Z.-X. Shen for their ARPES data. This work was supported by the Russian Foundation for Basic Research (Projects Nos. 10-02-00046a, 09-02-00431a, and 10-02-00546a), the Dynasty Foundation, the fund of the President of the Russian Federation for the support of scientific schools NSH 4711.2010.2, the Program of the Russian Academy of Science Presidium “Quantum microphysics of condensed matter” N7, Russian Federal Agency for Science and Innovations (Program “Scientific and Scientific-Pedagogical Trained of the Innovating Russia” for 2009-2010 years), grant No. 02.740.11.0217. S.L.S. and V.I.A. are grateful to the Center for Electronic Correlations and Magnetism, University of Augsburg, Germany for the hospitality and support of the Deutsche Forschungsgemeinschaft through SFB 484.
Appendix: self-energy on the real axis
======================================
![(Color online) The imaginary part of the Fe $d_{yz}$ self-energy $\Sigma(i\omega_{n})$ calculated within LDA+DMFT (black circles) is compared with the corresponding Padé approximant (red curve).[]{data-label="DMFTvsPade"}](fig4.eps){width="0.88\linewidth"}
In the DMFT method temperature (Matsubara) Green functions formalism is used with arguments in the form of imaginary time $\tau$ or corresponding Matsubara imaginary energies $i\omega_n=i(2n+1)\pi/\beta$. In order to calculate a spectral function one needs to have self-energy as a function of real energy $\Sigma(\omega)$ that means to perform analytical continuation of the function $\Sigma(i\omega_n)$ to real axis. One of the usual algorithms for analytical continuation is Padé approximant, [@Pade] and it was successfully used in earlier LDA+DMFT calculations where effective impurity problem was solved by Iterative Perturbation Theory (IPT) method. [@LDA+DMFT_JPCM] However, when impurity problem is solved by stochastic Quantum Monte Carlo (QMC) method numerical noise appears in calculated $\Sigma(i\omega_n)$, see Fig. \[DMFTvsPade\]. Attempts to apply the Padé approximant method to such noisy data result in completely wrong widely oscillating real axis function.
In order to solve this problem Maximum Entropy method (MEM) method[@jarrell96] was proposed. In this method spectral function $A(\omega)$ corresponding to imaginary time Green function $G(\tau)$ from QMC calculation is found as a best approximation to solution of the integral equation: $$G(\tau)=-\int_{-\infty}^{\infty}d\omega\frac{e^{-\tau\omega}}{1+e^{-\beta\omega}}A(\omega)
\ ,
\label{ME8}$$ with the condition of maximization of effective entropy functional that gives a smooth spectral function. The resulting spectral function $A(\omega)$ is identified then with the [**k**]{}-integrated analogue of Eq. \[SFunction\] that gives equations for unknown self-energy $\Sigma(\omega)$. For many-orbital case that gives a set of equations with the corresponding number of unknown variables $\Sigma_{i}(\omega)$. Solution of such a set of equations can be a rather difficult problem. In addition to that the MEM method smears out all high-energy features in $A(\omega)$ due to the factor $e^{-\beta\omega}$ in the kernel of integral equation .
![(Color online) Orbitally resolved Fe 3[*d*]{} spectral functions of LaFePO from the Maximum Entropy method (green shaded areas) and the real-axis self-energy $\Sigma(\omega)$ obtained with the use of Padé approximation (blue curves).[]{data-label="MEvsPade"}](fig5.eps){width="0.8\linewidth"}
In the present work we have used a modified version of the Padé approximant method. To make the analytical continuation procedure of the noisy self-energy $\Sigma(i\omega)$ numerically stable, we construct the approximant using only those frequencies values where the self-energy is a smooth function. Practically, that means to use a few first $i\omega_n$ at the lowest frequencies and the data at the large frequencies where $\Sigma(i\omega)$ approaches asymptotic behavior. In the result we obtain a smooth function that has correct analytical behavior at two limits: small energies $\omega\rightarrow 0$ and large energies where it obeys known asymptotic.
In Fig. \[DMFTvsPade\] we compare the Fe $d_{yz}$ self-energy $\Sigma(i\omega_{n})$ obtained within Padé approximation with the corresponding numerical data from QMC solution of DMFT equations. The approximant has accurate derivatives in vicinity of zeroth Matsubara frequency, that guaranties correct analytical properties near the Fermi level and correct asymptotic behavior. At the same time it approximates the noisy region with a smooth curve.
In Fig. \[MEvsPade\] the Fe 3[*d*]{} spectral functions obtained with Padé real-axis self-energy are compared with the MEM curves. The results for energies near the Fermi level are in very good agreement with each other. However, going to the higher and lower energies MEM curve very soon becomes smeared and nearly featureless while the curve obtained with Padé real-axis self-energy has much better resolved peaks and shoulders. For example, all the peaks in the energy region ($-$6, $-$2) eV are completely missed in MEM curve. This is due to exponential nature of the MEM kernel that suppresses all high-energy features.
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|
---
author:
- Stefano De Leo
- Pietro Rotelli
date: 'Submitted [*April, 2008*]{}'
title: 'FERMION-FERMION BOUND STATE CONDITION FOR SCALAR EXCHANGES'
---
I. INTRODUCTION {#i.-introduction .unnumbered}
===============
Since the experimental discovery of the mass of the neutrinos[@PDG] a legitimate question has been posed. Is there a possibility of a bound state between weakly interacting particles such as an electron and a neutrino? If the particles involved where nonrelativistic the answer would be immediate and negative. From numerical studies of the Schrödinger equation[@COHEN] the existence of a bound state produced by a Yukawa (attractive) coupling $$V_{\Y}(r)=-\,\frac{g_{\eff}^{\2}}{4\,\pi}\, \frac{\exp[-\,\mu
\,r]}{r} \,\,,$$ has been found to be $$\label{condg} \frac{g^{\2}_{\eff}}{4\,\pi} \ge 0.84\,
\frac{\mu}{m}\,\,,$$ where $\mu$ is the exchanged particle mass and $m$ is the reduced mass. A related derivation is the use of a surrogate to the Yukawa potential, the Hulthen potential, $V_{\H}$, which approximates the Yukawa potential for small $r$, $$V_{\H}(r)=-\,\frac{g_{\eff}^{\2}}{4\,\pi}\,
\frac{2\,\mu}{\exp[2\,\mu\, r] -1}\,\,.$$ Our choice of $V_{\H}$ is made so that the terms $r^{\mi \1}$ and $r^{\0}$ in a series development about $r=0$ are identical to the Yukawa potential. The Schrödinger equation with the Hulthen potential can be solved [*analytically*]{}[@FLU] and the existence of a bound state yields a similar result to that above, i.e. $$\label{hc}
\frac{g^{\2}_{\eff}}{4\,\pi} \ge
\frac{\mu}{m}\,\,.$$ High mass exchanges would necessitate extremely strong couplings, obviously unphysical for weak interactions where $\mu/ m > 10^{\1
\1}$[@ZUB]. However, if one considers the relativistic corrections to the Schrödinger equation one encounters the well known Darwin term[@SAK] $$\frac{1}{8\,m}\,\nabla^{^{2}}V(\boldsymbol{r})\,\,,$$ which for a Yukawa potential $V_{\Y}(r)$ yields $$\frac{1}{8\,m^{\2}}\,\left[\,\mu^{\2}\,V_{\Y}(r) +
4\,\pi\,g^{\2}_{\eff}\,\delta(\boldsymbol{r})\,\right]\,\,.$$ The first term above can be summed with the potential contribution to yield an overall amplification factor $$\left(\,1 + \,\frac{\mu^{\2}}{8\,m^{\2}}\,\right)\,V_{\Y}(r)\,\,.$$ This is what has been called Yukawa coupling amplification[@AYC]. Here the effect must be small to comply with the very nature of correction terms. However, if one where so bold as to assume this amplification for high $\mu/m$ one would invert the resonance condition (\[hc\]), i.e. $$\frac{g^{\2}_{\eff}}{4\,\pi} \ge
\frac{8\,m}{\mu}\,\,,$$ which allows bound states even for the weak interactions. The problem, theoretically, now shifts to determining the resonance condition for high mass exchanges in a more rigorous manner. A method has been introduced and applied in field theory[@G69; @GROSS]. It consists of confronting the lowest ladder contributions (box and crossed) to the scattering amplitude at rest, with the tree diagram contribution (also in the rest frame). The requirement that the sum of the former be equal or greater than the tree contribution reproduces exactly the Hulthen condition for low $\mu/m$ within a scalar-scalar model with scalar particle exchanges. In this paper it will also be shown to be also valid in the case of fermion-fermion (f-f) interacting with scalar exchanges. More recently[@EPJC], the scalar model calculation was extended to the high $\mu/m$ limit (in either limit, approximations or numerical calculations are needed). The high $\mu/m$ result was even more restrictive than the Hulthen inequality (\[hc\]), i.e. it required even larger $g^{\2}_{\eff}$, specifically $$\frac{g^{\2}_{\eff}}{2\,\pi^{^{\2}}} \geq
\frac{\mu^{\2}}{m_{\1}m_{\2}} \,\mbox{\large /}\,\left(
\,\ln\frac{\mu^{\2}}{m_{\1}m_{\2}} +
\frac{1+\rho^{\2}}{1-\rho^{\2}} \ln \rho\,\right)\,\,,$$ with $\rho=m_{\1}/m_{\2}$. However, it was noted that since the Klein-Gordon equation lacks a Darwin term correction there is no reason to expect Yukawa amplification. In this paper, we essentially repeat our low and high $\mu/m$ limits for f-f interacting via scalar exchange. This case does contain a Darwin term identical to that of the well known electrostatic case although some additional corrections also exist.
In the next Section, we illustrate the model and reduce the first order ladder contributions to a single integral in $\mbox{d}|\boldsymbol{k}|=\mbox{d}k$. In Section III, we perform the small $\mu/m$ limit and reproduce the Hulthen inequality (\[hc\]). In Section IV, we perform the high $\mu/m$ limit. We propose a phenomenological expression for the $k$ integral based upon numerical simulations. In Section V, we draw our conclusions.
II. THE FERMIONIC MODEL {#ii.-the-fermionic-model .unnumbered}
=======================
In the center of mass system and for forward scattering (see Fig.1), the Feynman rules[@ZUB] for the amplitudes of the box $(\square)$ and crossed box $(\times)$ diagram yield $$\begin{aligned}
\mathcal{M}^{\square}(\boldsymbol{p})& =&
i\,g_{\1}^{\2}g_{\2}^{\2} \int\frac{\mbox{d}^{\4}k}{(2\pi)^{\4}}\,
\frac{\bar{u}_{\1}^{(r)}(-\boldsymbol{p})\left\{\left[
E_{\1}(\boldsymbol{p})+E_{\2}(\boldsymbol{p})\right]\gamma_{\0}
-k\hspace*{-.2cm}\slash+m_{\1}
\right\}u_{\1}^{(r')}(-\boldsymbol{p})\,\bar{u}_{\2}^{(s)}(\boldsymbol{p})\left(
k\hspace*{-.2cm}\slash + m_{\2}
\right)u_{\2}^{(s')}(\boldsymbol{p})
}{D^{\square}_{\1}(\boldsymbol{p})D_{\2}(\boldsymbol{p})
D^{^{\2}}_{\0}(\boldsymbol{p})}
\nonumber \\
& \nonumber \\
& \\
& \nonumber \\
\mathcal{M}^{\times}(\boldsymbol{p})& =&i\,g_{\1}^{\2}g_{\2}^{\2}
\int\frac{\mbox{d}^{\4}k}{(2\pi)^{\4}}\,
\frac{\bar{u}_{\1}^{(r)}(-\boldsymbol{p})\left\{k\hspace*{-.2cm}\slash
+m_{\1} + \left[
E_{\1}(\boldsymbol{p})-E_{\2}(\boldsymbol{p})\right]\gamma_{\0}
\right\}u_{\1}^{(r')}(-\boldsymbol{p})\,\bar{u}_{\2}^{(s)}(\boldsymbol{p})\left(
k\hspace*{-.2cm}\slash + m_{\2} \right)u_{\2}^{(s')}(\boldsymbol{p})
}{D^{\times}_{\1}(\boldsymbol{p})D_{\2}(\boldsymbol{p})D^{^{\2}}_{\0}(\boldsymbol{p})}
\nonumber\end{aligned}$$ with $$u^{(s)}_{\1,\2}(\boldsymbol{q}) = \sqrt{E_{\1,\2}(\boldsymbol{q})+m_{{\1,\2}}}\,
\left( \begin{array}{c} \chi_{s}\\ \\
\displaystyle{\frac{\boldsymbol{\sigma}\cdot
\boldsymbol{q}}{E_{\1,\2}(\boldsymbol{q})+m_{\1,\2}}}\,\chi_s\end{array}
\right)\,\,\,\,\,\,\,\mbox{\small $(s=1,2$)}\,\,,\,\,\,\,\,
\chi_{\1}=\left( \begin{array}{c}1\\0
\end{array} \right)\,\,\,,\,\,\,\,\,\,\,
\chi_{\2}=\left( \begin{array}{c}0\\1 \end{array} \right)\,\,.$$ The denominators factors are, $$\begin{aligned}
D_{\1}^{\square}(\boldsymbol{p}) & = & E^{^{\2}}_{\1}(\boldsymbol{k})-
\left[\, k_{\0} - E_{\1}(\boldsymbol{p})-E_{\2}(\boldsymbol{p})
\right]^{^{\2}} -i\epsilon\,\,,
\nonumber \\
D_{\1}^{\times}(\boldsymbol{p}) & = &
E^{^{\2}}_{\1}(\boldsymbol{k})- \left[\, k_{\0} +
E_{\1}(\boldsymbol{p})-E_{\2}(\boldsymbol{p}) \right]^{^{\2}}
-i\epsilon\,\,,
\nonumber \\
D_{\2}(\boldsymbol{p}) & = & E^{^{\2}}_{\2}(\boldsymbol{k})-
k^{\2}_{\0}-i\epsilon\,\,, \nonumber \\
D_{\0}(\boldsymbol{p}) & = &
E^{^{\2}}_{\0}(\boldsymbol{k}-\boldsymbol{p})-\left[\, k_{\0}
-E_{\2}(\boldsymbol{p}) \right]^{^{\2}} -i\epsilon\,\,,\end{aligned}$$ where $$E_{\1,\2}(\boldsymbol{q})=\sqrt{\boldsymbol{q}^{\2}+m_{\1,\2}^{\2}}\,\,\,,
\,\,\,\,\,E_{\0}(\boldsymbol{q})=\sqrt{\boldsymbol{q}^{\2}+\mu^{\2}}\,\, .$$ At threshold ($\boldsymbol{p}\approx
\boldsymbol{0}$), $$\begin{aligned}
\mathcal{M}^{\square}(\boldsymbol{0})& =&
i\,\left(\,2\,g_{\1}g_{\2}\sqrt{m_{\1}m_{\2}}\,\right)^{\2}\,\delta_{rr'}\,\delta_{ss'}
\int\frac{\mbox{d}^{\4}k}{(2\pi)^{\4}}\,
\frac{(k_{\0}+m_{\2})(2\,m_{\1}+m_{\2} -
k_{\0})}{D^{\square}_{\1}(\boldsymbol{0})D_{\2}(\boldsymbol{0})
D^{^{\2}}_{\0}(\boldsymbol{0})}\, \, ,
\nonumber \\
& \\
\mathcal{M}^{\times}(\boldsymbol{0}) & =&
i\,\left(\,2\,g_{\1}g_{\2}\sqrt{m_{\1}m_{\2}}\,\right)^{\2}\,\delta_{rr'}\,\delta_{ss'}
\int\frac{\mbox{d}^{\4}k}{(2\pi)^{\4}}\,
\frac{\left(k_{\0}+2\,m_{\1}-m_{\2}\right)\left(k_{\0}+m_{\2}\right)}
{D^{\times}_{\1}(\boldsymbol{0})D_{\2}(\boldsymbol{0})
D^{^{\2}}_{\0}(\boldsymbol{0})}\,\, . \nonumber\end{aligned}$$ The poles in the lower half complex $k_{\0}$ plane are at $$k_{\0,\1}^{\square} = E_{\1}(\boldsymbol{k}) + m_{\1}+
m_{\2}\,\,,\,\,\,\,\, k_{\0,\1}^{\times} = E_{\1}(\boldsymbol{k})
- m_{\1}+ m_{\2}\,\,, \,\,\,\,\, k_{\0,\2} =
E_{\2}(\boldsymbol{k}) \,\,, \,\,\,\,\, k_{\0,\0} =
E_{\0}(\boldsymbol{k}) + m_{\2}\,\,.$$ The box and crossed box diagrams give the following fourth-order contribution to the invariant scattering amplitude $$\begin{aligned}
\label{integral}
\mathcal{M}^{\square}(\boldsymbol{0})+\,\mathcal{M}^{\times}(\boldsymbol{0})
&=&
\frac{\left(\,2\,g_{\1}g_{\2}\sqrt{m_{\1}m_{\2}}\,\right)^{\2}}{(2\pi)^{^{\3}}}\,
\delta_{rr'}\,\delta_{ss'} \int\,\mbox{d}^{\3}k \, \sum_{s=\0}^{\2}
\left[\,
R_{s}^{\square}(\boldsymbol{k})+R_{s}^{\times}(\boldsymbol{k})\,\right] \nonumber \\
& = &
\left(\,\frac{g_{\1}g_{\2}\sqrt{2\,m_{\1}m_{\2}}}{\pi}\,\right)^{\2}\,
\delta_{rr'}\,\delta_{ss'} \int_{\0}^{\infty}\mbox{d}k
\,\,k^{\2}\,\,\left[\, R^{\square}(k)+R^{\times}(k)\,\right]\,\,.\end{aligned}$$ Below by $E_s$ we intend $E_s(k)$ and by $W$ and $\Delta$ we intend $m_{\1}+m_{\2}$ and $m_{\2}-m_{\1}$ respectively. A simple calculation shows that the explicit formulas for the residues in the $k_{\0}$-plane for the box and the crossed box diagram are respectively $$\begin{aligned}
\label{residues}
R_{\1}^{\square}(k)
& = & \left[\,2\,m_{\2}(m_{\1}-E_{\1})-k^{\2} \right]\,/\,\left\{\,4\, W\,E_{\1}
\left(E_{\1}+m_{\1}\right)\left[\,\mu^{\2}-2\,m_{\1}\left(E_{\1}+m_{\1}\right)\,
\right]^{\2}\right\}\,\,, \nonumber \\
R_{\2}^{\square}(k)
& = &\left[k^{\2} - 2\, m_{\1}(E_{\2}+m_{\2})\right]\,/\,\left\{\,4\, W\,E_{\2}
\left(E_{\2}-m_{\2}\right)\left[\,\mu^{\2}+2\,m_{\2}\left(E_{\2}-m_{\2}\right)\,
\right]^{\2}\right\}\,\,,\nonumber \\
R_{\0}^{\square}(k) & = &\left[ 2\,(E_{\0}+2\,
m_{\2})(2\,m_{\1}-E_{\0})\right]
\left[\left(E_{\0}-m_{\1}\right)\,B\,C +
\left(E_{\0}+m_{\2}\right)\,A_{\square}\,C -
\,A_{\square}\,B\,\right]\,/\,
\left[\,A_{\square}^{\2}B^{\2}C^{\3}\,\right] \nonumber \\
& &
+\, 2\,\left(m_{\1}- m_{\2} -
E_{\0}\right)\,/\,\left[\,A_{\square}\,B\,C^{\2}\,\right]\,\,,\end{aligned}$$ with $A_{\square}=2\,E_{\0}\,m_{\1} - \mu^{\2}$, $B=-
2\,E_{\0}\,m_{\2} - \mu^{\2}$ and $C=2\,E_{\0}$, and $$\begin{aligned}
R_{\1}^{\times} & = & \left[\, k^{\2}+2\,m_{\2}
\left(E_{\1}+m_{\1} \right) \right]\,/\,\left\{\,4\,
\Delta\,E_{\1}
\left(E_{\1}-m_{\1}\right)\left[\,\mu^{\2}+2\,m_{\1}\left(E_{\1}-m_{\1}\right)\,
\right]^{\2} \right\}\,\,,\nonumber \\
R_{\2}^{\times}
& = & -\,\left[\, k^{\2}+2\,m_{\1}
\left(E_{\2}+m_{\2} \right) \right]\,/\,\left\{\,4\,
\Delta\,E_{\2} \left(E_{\2}-m_{\2}\right)
\,\left[\,\mu^{\2}+
2\,m_{\2}\left(E_{\2}-m_{\2}\right)\,
\right]^{\2}\right\}\,\,,\nonumber \\
R_{\0}^{\times} & = & 2\,(E_{\0}+2\, m_{\1})(E_{\0}+2\, m_{\2})
\left[\left(E_{\0}+m_{\1}\right)\,B\,C +
\left(E_{\0}+m_{\2}\right)\,A_{\times}\,C -
A_{\times}\,B\,\right]\,/\,\left[\,A_{\times}^{\2}B^{\2}C^{\3}\,\right]
\nonumber \\
& & +\,2\,\left(E_{\0}+W\right)\,/\,\left[\,A_{\times}\,B\,C^{\2}\,
\right]\,\,,\end{aligned}$$ with $A_{\times}=-\,\left(2\,E_{\0}\,m_{\1} + \mu^{\2}\right)$. It is to be noted, and can be used in calculation, that the residues of the box and crossed residues are related by $$R_{\1,\2,\0}^{\times} = - \, R_{\1,\2,\0}^{\square}[m_{1}\to
-\,m_{\1}]\,\,.$$ However, care must be used when applying this symmetry because for example $\sqrt{m^{\2}_{\1}}+m_{\1}=2\,m_{\1}$, while, under $m_{\1} \to -\,m_{\1}$, $\sqrt{(-m_{\1})^{\2}}-m_{\1}=0\neq -
\,2\,m_{\1}$. The rule of thumb is that square root factors should be left as such before applying such symmetries.
Before passing to the actual calculation of the small and large $\mu/ m$ results, we must discuss two important technical questions. The first is the question of the convergence of the $k$ integrals. The second is the feature of real pole contributions in some of these residue integrals.
$\bullet$ Convergence. {#bullet-convergence. .unnumbered}
----------------------
Individually, the leading residues terms yield divergent integrals, both linear and logarithmic. This was not the case for the scalar model[@GROSS]. However, when summed, the divergences cancel, specifically in the limit $k \to \infty$, $$\begin{array}{lclrcl}
& & &16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\, R_{\1}^{\square} & = &
-\,\displaystyle{\frac{m_{\2}^{\2}}{W}}
-\,\displaystyle{\frac{(\mu^{\2}+2\,m_{\1}m_{\2}-3\,m_{\1}^{\2})m_{\2}^{\2}}{W\,m_{\1}\,k}}
+ \, \mbox{O}\left(\frac{1}{k^{\2}}\right)\,\,,
\\
& & &16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\, R_{\2}^{\square} & = &
+\,\displaystyle{\frac{m_{\1}^{\2}}{W}} -
\,\displaystyle{\frac{(\mu^{\2}+2\,
m_{\1}m_{\2}-3\,m_{\2}^{\2})m_{\1}^{\2}}{W\,m_{\2}\,k}} + \,
\mbox{O}\left(\frac{1}{k^{\2}}\right)\,\,,\\
& & & 16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\,R_{\0}^{\square} & = &
+\,\Delta + \displaystyle{\frac{(\mu^{\2}+2\,
m_{\1}m_{\2})(m_{\1}^{\3}+m_{\2}^{\3})/W-3\,m_{\1}^{\2}m_{\2}^{\2}}{m_{\1}\,m_{\2}\,k}}
+ \, \mbox{O}\left(\frac{1}{k^{\2}}\right)\,\,.
\end{array}$$ Consequently, $$16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\, R^{\square}
=-\,\displaystyle{\frac{m_{\1}^{\2}m_{\2}^{\2}}{2\,k^{\3}}} +
\displaystyle{\mbox{O}\left(\frac{1}{k^{\5}}\right)}\,\,\,\,\,\mbox{and}\,\,\,\,\,
16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\, R^{\times}=
+\,\displaystyle{\frac{m_{\1}^{\2}m_{\2}^{\2}}{2\,k^{\3}}} +
\displaystyle{\mbox{O}\left(\frac{1}{k^{\5}}\right)}\,\,.$$ Both these results lead to convergent integrals, however, when summed, the leading terms again cancel and finally $$16\,m_{\1}^{\2}\,m_{\2}^{\2}\,k^{\2}\, (R^{\square}+R^{\times}) =
-\,\displaystyle{\frac{63\,m_{\1}^{\3}m_{\2}^{\3}}{4\,k^{\5}}} +
\displaystyle{\mbox{O}\left(\frac{1}{k^{\7}}\right)}\,\,,$$ which is a highly convergent integrand. Notice that this leading order result is symmetric under $m_{\1} \leftrightarrow m_{\2}$. We have not specified which mass, $m_{\1}$ or $m_{\2}$, is the lower mass and the Feynman diagrams are clearly symmetric under the interchange $m_{\1} \leftrightarrow m_{\2}$. [*Any final results must therefore be symmetric under this symmetry*]{}. This feature may be used as a test of all of the following results.
$\bullet$ Poles. {#bullet-poles. .unnumbered}
----------------
By explicit observation the quadratic term in the denominator of $R_{\1}^{\square}$ vanishes at $\mu^{\2}=2\,
m_{\1}\,(E_{\1}+m_{\1})$. Poles also occur in the expression for $R_{\0}^{\square}$ when $A_{\square}=0$, i.e. at $\mu^{\2}=2
m_{\1}\,E_{\0}$. Both of these conditions correspond to the [*same*]{} value of $k$, which we indicate by $k_p$, $$k_p^{\2} =\left(\frac{\mu^{\2}}{2\,m_{\1}}\right)^{\2} - \mu^{\2}$$ No other residues have poles. Thus, $R_{\1}^{\square}$ and $R_{\0}^{\square}$ exhibit double and single poles on the real axis at $k_p$. However, when summed [*all pole contributions cancel*]{}. This is demonstrated in some detail in the Appendix. The cancellation of the double pole is simple to show. That of the single pole which receives a contribution from $R_{\1}^{\square}$ and four contributions from $R_{\0}^{\square}$, one from each term in the last line of Eq.(\[residues\]), is more cumbersome to see. However, it must be proved since it would otherwise dominate the large $\mu/m$ calculation, and radically change our conclusions.
III. THE EXCHANGE OF SMALL MASS SCALARS {#iii.-the-exchange-of-small-mass-scalars .unnumbered}
=======================================
For incoming fermions with mass $m_{\1}$ and $m_{\2}$ interacting by the exchange of a scalar with mass $\mu \ll m_{\1,\2}$, $R^{\square}$ and $R^{\times}$ contribute to the invariant scattering amplitude only for value of $k\ll m_{\1,\2}$ (indeed of the order of $\mu$). In this small $\mu$ limit, we may use the approximation $$E_{\1,\2}=\sqrt{k^{^{2}}+m_{\1,\2}^{^{2}}}\approx
\, m_{\1,\2}+\frac{k^{\2}}{2\,m_{\1,\2}}\,\,.$$ We note, as an aside that for small $\mu$ ($\ll m_{\1,\2}$) there are no poles on the real axis. Now it is easy to show that $$R_{\1}^{\square}/R_{\2}^{\square} = \mbox{O}[(\mu/m)^{\8}]\ll 1\,\,.$$ Whence in the rest of this Section $R_{\1}^{\square}$ will be neglected. The other residue contributions yield $$\begin{aligned}
k^{\2}\,R_{\2}^{\square} & \approx &
-\,\frac{2\,m_{\1}m_{\2}}{W}\,\frac{1}{E_{\0}^{^4}}\,+\,
\frac{1}{2\,W}\,\frac{k^{\2}}{E_{\0}^{^4}}\,-\,
\frac{m_{\1}}{m_{\2}\,W}\,\frac{k^{\4}}{E_{\0}^{^6}}
\,\,,\\
k^{\2}\,R_{\2}^{\times} & \approx &
-\,\frac{2\,m_{\1}m_{\2}}{\Delta}\,\frac{1}{E_{\0}^{^4}}\,-\,
\frac{1}{2\,\Delta}\,\frac{k^{\2}}{E_{\0}^{^4}}\,-\,
\frac{m_{\1}}{m_{\2}\,\Delta}\,\frac{k^{\4}}{E_{\0}^{^6}}
\,\,, \\
k^{\2}\,R_{\1}^{\times} & \approx &
+\,\frac{2\,m_{\1}m_{\2}}{\Delta}\,\frac{1}{E_{\0}^{^4}}\,+\,
\frac{1}{2\,\Delta}\,\frac{k^{\2}}{E_{\0}^{^4}}\,+\,
\frac{m_{\2}}{m_{\1}\,\Delta}\,\frac{k^{\4}}{E_{\0}^{^6}}
\,\,, \\
k^{\2}\,R_{\0}^{\square} & \approx &
+\,\frac{3}{4}\,\frac{k^{\2}}{E_{\0}^{^5}}\,-\,
\frac{\Delta}{2\,m_{\1}m_{\2}}\,\frac{k^{\2}}{E_{\0}^{^4}}\,+\,
\frac{\mu^{\2}\Delta}{2\,m_{\1}\,m_{\2}}\,\frac{k^{\2}}{E_{\0}^{^6}}
\,\,, \\
k^{\2}\,R_{\0}^{\times} & \approx &
-\,\frac{3}{4}\,\frac{k^{\2}}{E_{\0}^{^5}}\,-\,
\frac{W}{2\,m_{\1}m_{\2}}\,\frac{k^{\2}}{E_{\0}^{^4}}\,+\,
\frac{\mu^{\2}W}{2\,m_{\1}\,m_{\2}}\,\frac{k^{\2}}{E_{\0}^{^6}}
\,\,.\end{aligned}$$ Thus, $$k^{\2}\, \left[ R^{\square}(k) + R^{\times}(k)\,\right] \approx
-\,2\,m\,\frac{1}{E_{\0}^{^4}}\,+\,
\left(\,\frac{1}{2\,W}\,-\,\frac{1}{m_{\1}}\,\right)\,\frac{k^{\2}}{E_{\0}^{^4}}\,+\,
\left(\,\frac{1}{W}\,+\,\frac{1}{m_{\1}}\,\right)\,\frac{k^{\4}}{E_{\0}^{^6}}\,+\,
\frac{\mu^{\2}}{m_{\1}}\,\frac{k^{\2}}{E_{\0}^{^6}}\,\,,$$ and, by making use of the elementary integrals $$\frac{4\,\mu^{\3}}{\pi}\,\int_{\0}^{\infty} \frac{\mbox{d}k}{E_{\0}^{^{4}}}
=\,\frac{4\,\mu}{\pi}\,\int_{\0}^{\infty} \frac{k^{\2}
\mbox{d}k}{E_{\0}^{^{4}}} =
\,\frac{16\,\mu}{3\,\pi}\,\int_{\0}^{\infty} \frac{k^{\4}
\mbox{d}k}{E_{\0}^{^{6}}} =
\,\frac{16\,\mu^{\3}}{\pi}\,\int_{\0}^{\infty} \frac{k^{\2}
\mbox{d}k}{E_{\0}^{^{4}}} = 1\,\,,$$ we find that $$\mathcal{M}^{\square} + \mathcal{M}^{\times} \approx 2\,
m_{\1}m_{\2}\,\left( \frac{g_{\1}g_{\2}}{\pi} \right)^{^{2}}\,
\left(\, -\,\frac{\pi}{2}\,\frac{m}{\mu^{\3}} \,
+\,\frac{5\,\pi}{16}\,\frac{1}{W\,\mu}\,\right) \,\,.$$ Comparing now this fourth-order total scattering amplitude, $$\mathcal{M}^{\square}+\mathcal{M}^{\times} \approx - \,
\frac{g_{\1}^{^{\2}}g_{\2}^{^{\2}}}{\pi}\,
\frac{m_{\1}^{\2}m_{\2}^{\2}}{W\,\mu^{\3}}\,\left(\,1
\,-\,\frac{5}{8}\,\frac{\mu^{\2}}{m_{\1}m_{\2}}\,\right)\,\,,$$ with the one boson exchange amplitude (tree diagram) $$-\,4\,m_{\1}m_{\2}\,\frac{g_{\1}g_{\2}}{\mu^{\2}}\,\, ,$$ we find that the fourth-order amplitude is greater or comparable to the second-order amplitude when $$\label{condsmall} \frac{g_{\1}g_{\2}}{4\,\pi} \geq
\frac{\mu}{m}\,\left(\,1
\,+\,\frac{5}{8}\,\frac{\mu^{\2}}{m_{\1}m_{\2}}\,\right)\,\,,$$ which, to the leading order, reproduces exactly the Hulthen inequality, where $g^{\2}_{\eff}=g_{\1}g_{\2}$. We have explicitly calculated and exhibited the correction term in the above inequality, and we will refer to this factor in our conclusions.
IV. THE EXCHANGE OF HIGH MASS SCALARS {#iv.-the-exchange-of-high-mass-scalars .unnumbered}
=====================================
The high $\mu/m$ limit is more difficult to treat and we rely upon numerical tests of the following expressions. We have three masses in our calculation of $\mathcal{M}^{\square,\times}$ so if we consider an adimensional expression, it can only be a function of $\mu/m_{\1}$ and $\mu/m_{\2}$ or alternatively of $$\omega=\frac{\mu^{\2}}{m_{\1}m_{\2}}\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,
\rho=\frac{m_{\1}}{m_{\2}}\,\,.$$ Indeed, $$-\,\mu^{\2}\,\int_{\0}^{\infty}\mbox{d}k \,\,k^{\2}\,\left[
R^{\square}(k)+R^{\times}(k)\right] =
F\left(\omega\,,\,\rho\right)\,\,,$$ and this can be tested numerically. Now we try to parameterize $\mathcal{M}^{\square,\times}$ by a form derived in the scalar model case. We write, $$-\,\mu^{\2}\,\int_{\0}^{\infty}\mbox{d}k \,\,k^{\2}\,\left[
R^{\square}(k)+R^{\times}(k)\right] = \frac{\alpha}{\omega}\,
\left( \,\ln\omega + \frac{1+\rho^{\2}}{1-\rho^{\2}} \ln
\rho\,\right)\,\,.$$ The value $\alpha=1$ reproduces the scalar model result. This phenomenological form has been tested for a wide but limited range of $\omega$ and $\rho$ values, specifically for $$\omega=10^{\6}\,,\,10^{\7}\,,\,10^{\8}\,\,\,\,\,\,\,
\mbox{and}\,\,\,\,\,\,\,\rho=2\,,\,10\,,\,50\,\,.$$ In the following Table $$\begin{array}{|c|c|c|}
\hline \,\,\,\omega\,\,\, & \,\,\,\rho\,\,\, &
\mbox{\,\,\,Phen/Num\,\,\,}
\\ \hline \hline
10^{\6} & \,\,\,2 & .995 \\
\hline
10^{\6} & 10 & .989 \\
\hline
10^{\6} & 50 & .978 \\
\hline \hline
10^{\7} & \,\,\,2 & 1.005\,\, \\
\hline
10^{\7} & 10 & 1.000\,\, \\
\hline
10^{\7} & 50 & .993 \\
\hline \hline
10^{\8} & \,\,\,2 & 1.012\,\, \\
\hline
10^{\8} & 10 & 1.008\,\, \\
\hline
10^{\8} & 50 & 1.003\,\, \\
\hline
\end{array}$$ we give the comparison of phenomenological/numerical (Phen/Num) results for a best fit value of $\alpha$, $$\alpha=0.663\,\,.$$ We see that to within a few per cent the agreement is good. We could of course improve the comparison if we included a constant term $\ln \beta$ in the brackets which could correspond to e renormalization of the logarithmic terms. However, we consider this an excessive finess. The important point is that the large $\mu/m$ behavior is similar to the scalar model result. The high $\mu$ resonance inequality thus reads $$\label{condition} \frac{g^{\2}_{\eff}}{2\,\pi^{^{\2}}} \geq
\frac{\mu^{\2}}{\alpha\, m_{\1}m_{\2}} \,\mbox{\large /}\,\left(
\,\ln\frac{\mu^{\2}}{m_{\1}m_{\2}} +
\frac{1+\rho^{\2}}{1-\rho^{\2}} \ln \rho\,\right)\,\,.$$
V. CONCLUSIONS {#v.-conclusions .unnumbered}
==============
We have applied in this paper a field theoretic approach to the determination of the coupling strengths needed for the existence of a fermion-fermion bound state via scalar boson exchanges. For low $\mu/m$, we again find the Hulthen inequality[@FLU; @GROSS] as seen in the scalar model. For high $\mu / m$, we obtain an even more restrictive condition (\[condition\]), a result again similar to the scalar field model[@EPJC]. The similarity between the scalar field model and this calculation suggests that the bound state inequality condition depends essentially upon the exchanged particles rather than the incoming ones. This was by no means obvious since the numerators of the residues are different in the two cases. Indeed at first sight the fermion-fermion model seemed to yield divergent results as a simple power count of the $\boldsymbol{k}$-integral suggests. We have shown in this paper that the individual divergence contributions cancel. We have also shown that the real pole contributions to $\mathcal{M}^{\square}$ also cancel both for the double and single poles. Again it is not clear if this would happen with say vector particle exchanges and it must be said that a contribution from a simple pole would completely alter our high $\mu / m$ results. For the existence of a relativistic bound state such a contribution could even be desirable.
There is however a problem with our results for small $\mu /m$ and the arguments based upon the relativistic corrections to the Schrödinger equation mentioned in the introduction. The Dirac equation with a scalar potential contains a Darwin term as does the better known electrostatic case[@SAK]. This lead us to expect, at least for small $\mu / m$ (nonrelativistic) a coupling amplification. We have purposefully kept the $\mbox{O}(\mu^{\2}/m^{\2})$ corrections in the small $\mu / m$ case and as can be seen in the result (\[condsmall\]) the corrections terms correspond to a coupling deamplification. The coupling constants must be somewhat [*increased*]{} to compensate the correction terms. This result is consistent with the tougher large $\mu / m$ inequality. We predict that the Hulthen inequality is a lower limit inequality for any $\mu /m$. Is this disagreement between our field theory calculation and the nonrelativistic reduced mass equation serious? This may well be a matter of opinion but some observations are in order:\
- The Hulthen inequality is [*not*]{} exactly in agreement with the Yukawa numeric inequality. So, we have a formal discrepancy even neglecting the relativistic correction terms;\
- The higher order Feynman diagrams cannot be parameterized by a simple Yukawa potential. However, the Coulomb potential works admirably well for Hydrogen like atoms except for one of the supreme successes of field theory, the Lamb shift. Unfortunately, we known of no direct way to derive the potential bound state spectrum from field theory;\
- It must also be remembered that not all the fourth order Feynman diagrams have been calculated.\
Nevertheless, we remain troubled by this result. At the very least, we must moderate any expectations for a weak interaction calculation in which intermediate vector particles are exchanged. We expect the same low $\mu /m$ inequality (except perhaps for the correction term) but hope for a very different high $\mu / m$ result.
Our results have one physical consequence, we predict that weak interacting fermion-fermion (or scalar-scalar) particles cannot produce a bound state simply by Higgs boson exchanges[@PDG]. It is our intention to tackle the full weak interaction case in the near future.
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J.J. Sakurai, [*Advanced quantum mechanics*]{}, Addison-Wesley, New York (1987).
S. De Leo and P. Rotelli, “Amplification of coupling for Yukawa potentials”, Phys. Rev. D [**69**]{}, 034006-5 (2004).
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APPENDIX: THE POLE CONTRIBUTIONS {#appendix-the-pole-contributions .unnumbered}
================================
In this Appendix, we calculate the pole contributions of $R_{\1}^{\square}$ and $R_{\0}^{\square}$ at $$k_p =\sqrt{\left(\frac{\mu^{\2}}{2\,m_{\1}}\right)^{\2} -
\mu^{\2}}\,\,.$$ For convenience, we define the functions $F(k)$, $G(k)$ and $\boldsymbol{H}(k)$ by $$\begin{aligned}
k^{\2}R_{\1}^{\square}(k) & = & F(k)\,\,,\\
k^{\2}R_{\0}^{\square}(k) & = &
G(k)\,+\,\sum_{n=\1}^{\3}H_n(k)\,\,,\end{aligned}$$ where $$\begin{aligned}
G(k) & = & 2\,k^{\2}\,(E_{\0}+2\, m_{\2})(2\,m_{\1}-E_{\0})
\left(E_{\0}-m_{\1}\right)\,/\,
\left(\,A_{\square}^{\2}\,B\,\,C^{\,\2}\,\right)\,\,,\\
H_{\1}(k) & = &2\,k^{\2}\,(E_{\0}+2\, m_{\2})(2\,m_{\1}-E_{\0})
\left(E_{\0}+m_{\2}\right)\,/\,
\left(\,A_{\square}\,B^{\2}C^{\,\2}\,\right)\,\,, \\
H_{\2}(k) & = &-\, 2\,k^{\2}\, (E_{\0}+2\,
m_{\2})(2\,m_{\1}-E_{\0})\,/\,
\left(\,A_{\square}\,B\,\,C^{\,\3}\,\right)\,\,, \\
H_{\3}(k) & = &2\,k^{\2}\,\left(m_{\1}- m_{\2} -
E_{\0}\right)\,/\,\left(\,A_{\square}\,B\,\,C^{\,\2}\,\right)\,\,.\end{aligned}$$ The first pole terms in the MacLaurin series of these functions are $$\left\{\, F(k)\,,\,G(k)\,,\,\boldsymbol{H}(k) \,\right\} =
\left\{\, \frac{F^{^{\m2}}(k_{p})}{(k-k_{p})^{^{2}}}\,+\,
\frac{F^{^{\min1}}(k_{p})}{k-k_{p}}\,,\,
\frac{G^{^{\m2}}(k_{p})}{(k-k_{p})^{^{2}}}\,+\,
\frac{G^{^{\min1}}(k_{p})}{k-k_{p}}\,,\,
\frac{\boldsymbol{H}^{^{\min1}}(k_{p})}{k-k_{p}}\,\right\}\, +\,
\mbox{O}\left(1\right)\,\,,$$ where $F^{^{\m2}}(k_{p})$ is the coefficient of $(k-k_{p})^{^{\mi
\2}}$ in $F(k)$ and so forth.
Now for the double pole, we find that the only two contributions are $$F^{^{\m2}}(k_{p})=-\,G^{^{\m2}}(k_{p})=
\frac{(2\,m_{\1}^{\2}-\mu^{\2})(4\,m_{\1}m_{\2}+\mu^{\2})}{16\,m_{\1}^{\2}W\,\mu^{\4}}
\,\,k_{p}^{\2}\,\,,$$ whence their sum cancels.
The single pole contributions can be written as $$\label{polec} \left\{\, F^{^{\min1}}(k_{p})\,,\,
G^{^{\min1}}(k_{p})\,,\,
\boldsymbol{H}^{^{\min1}}(k_{p})\,\right\} =
\frac{k_{p}}{32\,m_{\1}^{\2}W^{\2}\,\mu^{\6}}\,\left\{\,
f^{^{\min1}}_{p}\,,\, g^{^{\min1}}_{p}\,,\, \boldsymbol{h}^{^{
\min1}}_{p}\,\right\}\,\,,$$ and in the Table we list the factors in graph brackets above as a series in even powers of $\mu$. For example, $$f^{^{\min1}}_{p} = -\,2\,W\,\mu^{\6} -(m_{\2}+\Delta)\,W\,\frac{\mu^{\4}}{2\,m_{\1}}
-\, m_{\2}W\,\frac{\mu^{\2}}{8\,m_{\1}^{^{3}}}\,\,.$$ The important point is contained in the last line of the Table. All single pole contributions also cancel. Thus, in conclusion, there are no real axis poles in $k^{\2}\left(R^{\square}+R^{\times}\right)$.
$$\begin{array}{l|c|c|c|c}
& \mu^{\6} & \mu^{\4}\,/\,2\,m_{\1} & \mu^{\2}\,/ \, 8\,m_{\1}^{\3} &
\mu^{\0}\,/\,3\,m_{\1}^{\5}m_{\2} \\
\hspace*{1cm} & \hspace*{2.6cm} & \hspace*{3.8cm} &
\hspace*{2.6cm} & \hspace*{2.6cm}
\\ \hline
& & & & \\ f_{p}^{^{\min1}} & -2\,W&-(m_{\2}+\Delta)W &-m_{\2}W & 0 \\
& & & & \\ \hline & & & & \\ g_{p}^{^{\min1}} & m_{\1}+2\,W
&2\,\Delta W - 2\,m_{\1}^{\2}+3\,m_{\1}m_{\2} &
2\,m_{\2}\Delta & - (m_{\2}+W)\\
& & & & \\ \hline & & & & \\ h_{p,\1}^{^{\min1}} & - m_{\1} &
2\,m_{\1}^{\2} - 3\,m_{\1}m_{\2} &
m_{\2} (2\,m_{\1} - \Delta) & m_{\2}\\
& & & & \\ \hline
& & & & \\ h_{p,\2}^{^{\min1}} & 0 & - m_{\1} W & - \Delta W & W \\
& & & & \\ \hline
& & & & \\ h_{p,\3}^{^{\min1}} & 0 & 2\,m_{\1} W & \Delta W & 0 \\
& & & & \\ \hline \hline
& & & & \\ Sum & 0 & 0 & 0 & 0 \\
& & & & \\ \hline
\end{array}$$ [**Table.**]{} The coefficients of powers of $\mu^{\2}$ in the factors $f^{^{\min1}}_{p}$, $g^{^{\min1}}_{p}$ and $\boldsymbol{h}^{^{ \min1}}_{p}$ for the single pole contributions. These factors are defined in Eq.(\[polec\]).
{width="19cm" height="24cm"}
|
---
abstract: 'In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of $p$-adic fields in order to get some results towards algebraic independence of $p$-adic exponentials for almost all primes $p$.'
address: 'Lebanese University, Faculty of Sciences, Beirut, Lebanon'
author:
- Ali Bleybel
title: 'A Note on complex $p$-adic exponential fields'
---
Introduction
============
Let $\mathbb{Q}_p$ be the field of $p$-adic numbers, for $p$ a prime number. Given an algebraic closure $\mathbb{Q}_p^{\rm alg}$ of $\mathbb{Q}_p$, it comes naturally equipped with a norm ${\lvert\cdot\rvert}_p$, uniquely extending the usual norm on $\mathbb{Q}_p$. Recall that the standard normalization for ${\lvert\cdot\rvert}_p$ is ${\lvertp\rvert}_p=p^{-1}$.\
Denote by $\mathbb{C}_p$ the completion of $\mathbb{Q}_p^{\rm alg}$ with respect to the norm ${\lvert\cdot\rvert}_p$. Then $\mathbb{C}_p$ is also algebraically closed. It is called a complex $p$-adic field.
The $p$-adic exponential map $$\exp_p: E_p \to \mathbb{C}_p^{\times}, x \mapsto \sum_{n=0}^\infty \frac{x^n}{n!},$$ where $E_p$ is the set $E_p= \{ x \in \mathbb{C}_p: |x|_p < p^{-\frac{1}{p-1}} \}$ (the domain of convergence of the defining power series of the exponential) shares several properties with the complex exponential map $\exp$ (such as $\exp_p(x+y) = \exp_p (x) \exp_p(y)$, $(\exp_p(x))'=\exp_p(x)$ where $()'$ denotes the usual derivative).
There are important open problems regarding the exponential map over a non-archimedean valued field. One of these concerns the algebraic independence of the values of the exponential map at different arguments.\
Such issues are encapsulated in the following well-known conjecture ($p$-adic Schanuel’s conjecture)\
[**($p$-SC)**]{} Let $\bar{x} := (x_1,\dots, x_n) \in \mathbb{C}_p^n$ be an $n$-tuple of complex $p$-adic numbers satisfying the requirement $${\lvert\bar{x}\rvert}_p := \max_{1 \leq i \leq n}\{{\lvertx_i\rvert}_p \} < p^{-1/p-1}.$$ Assume that $x_1, \dots, x_n$ are $\mathbb{Q}$-linearly independent, then $$\textrm{td}_{\mathbb{Q}}(x_1, \dots, x_n, \exp_p(x_1), \dots, \exp_p(x_n)) \geq n,$$ where td$_{\mathbb{Q}}$ denotes the transcendence degree of the extension $$\mathbb{Q}(\bar{x}, \exp_p(\bar{x}))/\mathbb{Q}.$$
In the following we will denote by $\mathbb{G}$ the algebraic group $\mathbb{G}_a \times \mathbb{G}_m$, with $\mathbb{G}_a$ denoting the additive group of a field (say $\mathbb{C}_p$) and $\mathbb{G}_m$ its multiplicative group.\
In the above statement we used the abbreviation $f(\bar{x}):= (f(x_1), \dots, f(x_n))$ for any $n$-tuple $\bar{x}$.\
An equivalent statement to ($p$-SC) is the following:\
[**($p$-SC)’**]{} Let $\bar{x} := (x_1,\dots, x_n) \in \mathbb{C}_p^n$ be an $n$-tuple of complex $p$-adic numbers satisfying ${\lvert\bar{x}\rvert}_p < p^{-1/p-1}$. Assume that $$(\bar{x}, \exp_p(\bar{x})) \in V(\mathbb{C}_p),$$ for some subvariety $V$ of $\mathbb{G}^n$ defined over $\mathbb{Q}$ (i.e. a $\mathbb{Q}$-variety), which is furthermore of dimension $<n$. Then, $x_1, \dots, x_n$ are $\mathbb{Q}$-linearly dependent, i.e. $$m_1 x_1 + \dots + m_n x_n =0,$$ for some $m_1, \dots, m_n \in \mathbb{Q}$, not all zero.
In this paper we apply the ultraproduct construction and basic model theory in order to obtain some results in the above direction.\
The main Theorem can be obtained by applying Ax-Schanuel’s Theorem [@A] to a non-principal ultraproduct of $\mathbb{C}_p$, and it reads as:
\[main\] Let $V$ be a $\mathbb{Q}$-variety of dimension $n$ in an affine $2n$ space. Assume that for infinitely many primes $p$, $V$ has a $\mathbb{C}_p$-point of the form $(\bar{a}_p, \exp_p(\bar{a}_p))$, then there exist a finite set $S(V) \subset \mathbb{P}$, and a finite set of rational tuples $\bar{\alpha}_i, i \in I$, (where $I$ is a finite set) such that for all $p \in \mathbb{P}\setminus \!S(V)$, and for all $n$-tuples $\bar{x}_p \in E_p^n$ satisfying $(\bar{x}_p, \exp_p(\bar{x}_p)) \in V(\mathbb{C}_p)$, there is a rational linear dependence that holds for the tuple $\bar{x}_p$ of the form $$\alpha_{i,1} x_{p,1} + \dots + \alpha_{i,n} x_{p,n} =0,$$ for some $i \in I$.
An equivalent statement (with a geometrical flavor) is the following:\
*Let $V$ be a $\mathbb{Q}$-variety of dimension $n$ in a $2n$-space. If, for infinitely many primes $p$, $V$ has a $\mathbb{C}_p$-point of the form $(\bar{a}_p, \exp_p(\bar{a}_p))$, then there exist a finite set $S \subset \mathbb{P}$ and a finite set of hyperplanes $H_i \subset \mathbb{A}^{n}_\mathbb{Q}, i \in I$ such that for all $p \in \mathbb{P} \setminus \! S$, we have* $$\forall \bar{x}_p \in E_p^n, (\bar{x}_p, \exp_p(\bar{x}_p)) \in V(\mathbb{C}_p) \longrightarrow (\exists i \in I) (\bar{x}_p \in H_i(\mathbb{C}_p)).$$ In the above Theorem, the order of quantifiers is essential: for each variety $V$ as above, there is a set $S(V)$ of exceptional primes (i.e. primes $p$ for which the stated implication might not hold), and such set is only dependent on the variety $V$. By *almost all primes* we mean all except a finite set of primes. This is to distinguish from the notion of $\cU$-*almost all* (for a given ultrafilter $\cU$) which will be encountered later. A *uniform* rational linear dependence is a linear dependence of the form $$m_1 x_{1,p} + \dots + m_n x_{n,p} =0,$$ for some *fixed* rationals $m_1, \dots, m_n$, not all zero. The Theorem implies in particular that, for each family of $n$-tuples $(\bar{x})_p$ as above there exists a partition of $\mathbb{P} \setminus S(V)$ into finitely many sets, on each of which the obtained linear dependence is uniform.
The method of proof uses Ax’s result [@A] on Schanuel’s property for differential exponential fields.\
For each variety $V \subset \mathbb{G}^n$ of dimension $n$ as above, the conclusions of Theorem \[main\] hold for all but possibly finitely many primes belonging to some exceptional set $S(V)$. For a particular variety $V \subset \mathbb{G}^n$ having dimension $ \leq n$, and a given prime $p \notin S(V)$, the conclusion of Theorem \[main\] is strictly stronger than what is given by conjecture ($p$-SC) (or, more precisely, its equivalent ($p$-SC)’). That is, according to ($p$-SC), there might exist $\mathbb{Q}$-linearly independent tuples $\bar{x}_p$ (in the domain of $\exp_p$) for which $(\bar{x}_p, \exp_p(\bar{x}_p)) \in V(\mathbb{C}_p)$ if $V$ is of dimension $n$, while this is not the case for Theorem \[main\] whenever $p \notin S(V)$. This is due to the statement of Ax’s Theorem, in which the weak inequality in ($p$-SC) is replaced by a strict one.
Let $\cU$ be a non-principal ultrafilter over the set $\mathbb{P}$ of prime numbers. Consider the ultraproduct $$\mathbb{K}_{\cU} := \prod_{p \in \mathbb{P}} \mathbb{C}_p / \cU.$$ Then $\mathbb{K}_{\cU}$ is an algebraically closed valued field (whose valuation is induced by $p$-adic valuations on each $\mathbb{C}_p$). We may define a partial exponential map on $\mathbb{K}_{\cU}$, induced by the maps $\exp_p$. As explained in [@KMS], $\mathbb{K}_{\cU}$ can be embedded in a differential exponential valued field, to which it is possible to apply Ax-Schanuel’s Theorem.\
Then by an application of Łós’ Theorem on ultraproducts, we obtain the required result.
We will consider stronger versions of these results in a forthcoming paper. Theorem \[main\] will be proved in section \[Proofs\], after several preliminary sections, which contain reminders of known results concerning valued fields, ultraproducts of valued fields and other related concepts.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I am grateful to the anonymous referee for his careful reading of the manuscript, and for many valuable comments and suggestions.
Background
===========
In this section we introduce background results that will be needed in the rest of the paper.\
Recall that the field $\mathbb{C}_p$ is the completion (with respect to the norm ${\lvert\cdot\rvert}_p$) of an algebraic closure of $\mathbb{Q}_p$, the field of $p$-adic numbers. One may consider instead the additive valuation $\ordp$ defined on $\mathbb{C}_p$. This valuation is defined through the relation: $${\lvertz\rvert}_p = p^{-\ordp(z)}.$$
In [@A] J. Ax proved the following result, already conjectured by S. Schanuel:
\[Ax\] (Ax [@A]) Let $K$ be a differential field equipped with a derivation $D$, and let $C$ be its field of constants.\
Let $y_1,\dots, y_n, z_1, \dots, z_n \in K^{\times}$ be such that $Dy_i = Dz_i/z_i$.\
Assume that the $y_i,\; i=1, \dots, n$ are $\mathbb{Q}$-linearly independent modulo $C$, then $${\rm td}_{C} (y_1,\dots, y_n, z_1, \dots,z_n) \geq n+1.$$
Recall that a *derivation* over a (commutative) field $K$ is a map $D: K \to K$ satisfying additivity ($D(x+y) = Dx + Dy$) and Leibniz rule ($D(xy)=xDy +yDx$). The [*field of constants for $D$*]{} is the set of $x \in K$ for which $Dx=0$. Using additivity and Leibniz rule, one can see that $C$ is indeed a subfield of $K$.\
In [@KMS] this result was restated as follows:
\[Theorem A\] Let $y_1, \dots,y_n,z_1, \dots,z_n \in K^\times$ be such that $Dy_k = \frac{Dz_k}{z_k} \mbox{ for } k= 1,\dots,n$.\
If ${\rm td}_CC(y_1,\dots,y_n, z_1, \dots,z_n) \leq n$, then $\sum_{i=1}^n
m_iy_i\in C$ for some $m_1, \dots, m_n \in \mathbb{Q}$ not all zero.
A corollary of the above is given by (this is essentially Corollary 3 in [@KMS]):
\[ASP\] Let $(K,\exp)$ be a partial differential exponential field (that is, a field equipped with a partial exponential map $\exp$, satisfying $D\exp(x) = \exp(x)Dx$), with a field of constants $k$. Then, for any $n$-tuple $\bar{x} := (x_1,\dots,x_n) \in K^n$ of elements of $K$, where $x_1, \dots, x_n$ belong to the domain of the exponential map.\
If $(\bar{x}, \exp (\bar{x})) \in V(K)$ for some algebraic variety $V$ of dimension $n$ with rational coefficients, $V \subset \mathbb{G}^n$, then $\sum_{i=1}^n m_i x_i \in k$, for some $m_1, \dots, m_n \in \mathbb{Q}$ not all zero.
Language and Logical Setting {#lang}
----------------------------
Let $L=(+, -, \cdot, (\,)^{-1}, 1,0)$ be the language of fields, with the standard interpretation of the symbols involved. We consider the expansion $\cL$ of $L$: $$\cL = L \cup \{R, \textrm{Exp}\},$$ where $R$ is a unary predicate symbol while $\textrm{Exp}$ is a function symbol (to be interpreted as an exponential map $K \to K^\times$, with $K^\times$ being the set of invertible elements of $K$).\
Let $K$ be a differentially valued partial exponential field. Denote by val the valuation on $K$, $\Gamma$ its value group, and let $R$ be the valuation ring with maximal ideal $\cP$. As we will see below, in many cases of interest one can extend the partial exponential on $K$ to a total exponential map (which is not uniquely determined though). Notable exceptions are Laurent power series fields, as well as generalized power series fields (whose definition will be recalled below). It follows that by making the appropriate interpretation of each symbol of $\cL$, the field $K$ is then naturally an $\cL$-structure (with $\textrm{Exp}$ denoting the (extended) exponential map).\
Furthermore, the maximal ideal $\cP$ of $R$ can be defined as follows: $$x \in \cP \quad {\rm iff} \quad x \in R \; \& \; x^{-1} \notin R.$$ Assume now that $R$ is a discrete valuation ring, and let $\pi \in R$ be a uniformizer, i.e. val$(\pi) =1$. Let $\cL_\pi$ be the expansion $\cL \cup \{\pi\}$, with $\pi$ denoting a constant in $K$.\
Using $\pi$, the valuation val can be defined using the predicate $R$ in a standard way: ${\rm val}(x) \geq 0$ iff $R(x)$ ($x$ is in the valuation ring), and for all $x \in K$, val$(x) = n \in \mathbb{Z}$ iff val$(x/\pi^n) \geq 0 \; \&$ val$(x/\pi^n) \leq 0$.\
For any valued field $(K,{\rm val})$ (with a possibly non discrete, or even, a non-archimedean value group $G$ where we fix a (generally non-canonical) embedding $\mathbb{Z} \hookrightarrow G$ and identify $\mathbb{Z}$ with its image in $G$), one can still use the language $\cL_\pi$ (for some $\pi$ satisfying val$(\pi)=1 $), and in this case any set of the form $\{ x \in K \; | \; {\rm val}(x) > e\}, e \in \mathbb{Q}$ is $\cL_\pi$-definable. Explicitly, the above set is defined through the formula (below $e=n/m$): $$\varphi_e(x): R(\frac{x^m}{\pi^n}) \; \& \; \neg R(\frac{\pi^n}{x^m}).$$ The complex $p$-adic field $\mathbb{C}_p$ falls in particular in the above case: the value group of the standard valuation $\ordp$ is $\mathbb{Q}$, and any non-principal ultraproduct $\prod_p \mathbb{C}_p/\cU$ (see below) has a non-archimedean value group.\
From the above remarks, one can see that a formula of the form val$(x) = $ val$(y)$ is an abbreviation of $R(x/y) \; \& \; R(y/x)$. Note that the expressive power of the language $\cL$ falls short of defining every ball in $K$ (a set of the form val$(x-a) \geq g$ for $g \in G$ and some $a \in K$), since $g$ might be a non-standard element.
An $\cL$-structure is a tuple $(K, R, \textrm{Exp})$, where $K$ is a valued field, $R$ its valuation ring and $\textrm{Exp}$ is an exponential map $\textrm{Exp}: K \to K^\times$.
The field $\mathbb{K}_{\cU}$
----------------------------
Let $\mathbb{P}$ be the set of prime numbers, and let $\cU$ be a non-principal ultrafilter on $\mathbb{P}$.\
Here the predicate $R$ is interpreted as the set $\mathbb{C}^\circ_p$ of complex $p$-adic numbers with non-negative $p$-adic valuation.\
Define the field $\mathbb{K}_\cU$ as the ultraproduct of the fields $\mathbb{C}_p$: $$\mathbb{K}_\cU := \prod_{p \in \mathbb{P}} \mathbb{C}_p / \cU.$$ The field $\mathbb{K}_{\cU}$ becomes an $\cL$-structure upon interpreting the function and predicate symbols in the standard way, for instance $R([(x_p)_{p \in \mathbb{P}}])$ if and only if the set of $p \in \mathbb{P}$ for which $x_p \in \mathbb{C}_p^\circ$ is in $\cU$. In this case we say that $x_p \in \mathbb{C}^\circ_p$ for $\cU$-almost all primes.\
By application of Łoś Theorem on ultraproducts, $\mathbb{K}_{\cU}$ is shown to be an algebraically closed field equipped with the valuation induced by $\ordp$ (for $p$ running over $\mathbb{P}$). Equip $\mathbb{K}_{\cU}$ with the valuation val defined as: $${\rm val} ([x]) = [(\ordp(x_p))_{p \in \mathbb{P}}],$$ where we have used the notation $[x] := [(x_p)_{p \in \mathbb{P}}] \in \mathbb{K}_{\cU}$. The elements $ (\ordp(x_p))_{p \in \mathbb{P}}$ belong to the Cartesian product of value groups $\prod_{p \in \mathbb{P}} \mathbb{Q}$, and $ [(\ordp(x_p))_{p \in \mathbb{P}}]$ belongs to the ultrapower of $\mathbb{Q}$, i.e. $\mathbb{Q}^\cU$. It is immediate to verify that val is indeed a valuation on $\mathbb{K}_{\cU}^{\times}$.\
For more details about ultraproducts of valued fields (and ultraproducts in general), see, e.g. [@S].\
Let $k_\cU$ be the residue field, $k_\cU= R/P$, with $R$ and $P$ the valuation ring and its maximal ideal. It follows from Łoś Theorem that $k_\cU$ is an algebraically closed field of characteristic zero, hence $\mathbb{K}_\cU$ is an equicharacteristic valued field.
The exponential map
-------------------
Let $p$ be a prime number. Fix an extension ${\rm EXP}_p$ of the $p$-adic exponential $\exp_p$ such that ${\rm EXP}_p$ is an exponential map defined for all elements of $\mathbb{C}_p$, i.e. ${\rm EXP}_p: \mathbb{C}_p \to \mathbb{C}_p^{\times}$ and $$\begin{aligned}
\forall x \in \mathbb{C}_p, {\lvertx\rvert}_p <p^{-1/p-1}, {\rm EXP}_p(x) & = & \exp_p(x), \\
\forall x, y \in \mathbb{C}_p \;\; {\rm EXP}_p(x+y) & = & {\rm EXP}_p(x) {\rm EXP}_p(y).\end{aligned}$$ The existence of such an extension is guaranteed by Zorn Lemma (see [@R] chap. 5, section 4.4). However, it is not unique. It can be seen that ${\rm EXP}_p$ is a continuous homomorphism from the additive group $(\mathbb{C}_p, +)$ to the multiplicative group $(\mathbb{C}^{\times}_p, \cdot)$.\
For each prime $p$, the field $\mathbb{C}_p$ equipped with the exponential map EXP$_p: \mathbb{C}_p \to \mathbb{C}_p^{\times}$ is a structure for $\cL$.\
Note that the use of the extension EXP$_p$ (rather than just the standard $p$-adic exponential $\exp_p$) seems to be useful from the model-theoretic point of view, in view of the intended application. More precisely, since we are considering an ultraproduct of the $\mathbb{C}_p$’s, the map $E([x]) := [(\exp_p(x_p))_p]$ (see below) is defined on an open disc around the origin of radius $1 -\epsilon$, with $\epsilon > 0$ an infinitesimal, whereas the domain of $\exp_p$ is the open disc of radius $r_p:= p^{-1/(p-1)}$ as already observed. Using instead the maps EXP$_p$, allows us to have a uniform definition of the domain of the exponential map.
Ordered abelian groups
----------------------
Let $(G, +, \leq)$ be an ordered abelian group under the law $+$, where $\leq $ denotes the order relation on $G$. Let $G^{>0}$ be the semi-group of positive elements of $G$ (i.e. elements greater than $0$).\
Let $\Delta$ be the set of *archimedean classes* of $G^{>0}$ (see, e.g. [@Hah]). The archimedean class of an element $g \in G$ will be denoted by $[g]$.\
If $\Delta$ is not a singleton, we say that $G$ is non-archimedean. The set $\Delta$ comes equipped with the inherited order $\preceq$ defined as: $ \delta_1=[g_1] \preceq \delta_2=[g_2] \;\; {\rm iff} \; \; ({\lvertg_2\rvert} \leq {\lvertg_1\rvert}),$ for any $\delta_1, \delta_2 \in \Delta$. Obviously, we may define the induced relations $\prec$ and $\succ$ in a similar way. Let $[0]= \infty$. The order $\preceq$ can then be extended to $\Delta \cup \{\infty\}$ by setting $\delta \preceq \infty$ for all $\delta \in \Delta \cup \{\infty\}$.\
Denote by $v_1$ the map (called natural valuation) $v_1: G \to \Delta \cup \{\infty \}$ defined as $v_1(g) = [g]$.
### Hahn Embedding Theorem
A central result in the theory of linearly ordered abelian groups is the following:\
Let $G$ be a linearly ordered abelian group. Then there exists an embedding of ordered groups $i: G \hookrightarrow H(\Delta) \subset \mathbb{R}^{\Delta}$ where $\Delta$ is the set of archimedean classes of $G$, and $H(\Delta)$ (the [*Hahn group*]{} with respect to $\Delta $) is given by $$H(\Delta) := \{a= (a_\gamma)_{\gamma \in \Delta}: a_\gamma \in \mathbb{R} \; {\rm and} \; {\rm Supp} (a) {\rm \; is \; well \; ordered} \}.$$ Here $H(\Delta)$ is equipped with the lexicographic order, and Supp$(a)$ (for $a \in \mathbb{R}^{\Delta}$) is defined as $${\rm Supp}(a) := \{ \gamma \in \Delta: a_\gamma \neq 0\}.$$ Any element $g$ of $G$ can be written as $g = \sum_{\phi \in \Delta} g_\phi \mathbf{1}_\phi$ where $g_\phi \in \mathbb{R}$ and ${\mathbf{1}}_\phi, \phi \in \Delta$ the element of $\Gamma$ that corresponds through the embedding $i$ to $(a_\psi)_{\psi \in \Delta} \in \mathbb{R}^\Delta$, with $a_\phi=1$ $a_\psi=0$, for $\psi \neq \phi$. We have $ v_1(g) = \min(\rm Supp(g)) \in \Delta \cup \{\infty\}$.\
Let $\Gamma$ be the value group of $\mathbb{K}_\cU$, $\Gamma:= (\prod_{p \in \mathbb{P}} \mathbb{Q} /\cU, +)$.\
Observe that we have a canonical embedding $\mathbb{Q} \hookrightarrow \Gamma$, $r \mapsto [(r_p)_{p \in \mathbb{P}}]$ (with $r_p =r$ for all $p \in \mathbb{P}$). An element of $\Gamma$ is called standard if it is in the image of $\mathbb{Q}$ by this embedding.\
Let $\gamma:= [(g_p)_{p \in \mathbb{P}}]$ be an element of $\Gamma$ such that for any $\varepsilon >0$, there exists $p_0 \in \mathbb{P}$ for which $\forall p \in \mathbb{P}, \, p >p_0 \Rightarrow \! {\lvertg_p\rvert} < \varepsilon$. Then clearly, $\gamma$ is an infinitesimal element, since it is smaller (in absolute value) than any element of $\mathbb{Q}^{>0}$. Similarly, an element $[(g_p)_{p \in \mathbb{P}}]$ of $\Gamma$ is infinite iff it satisfies $$\forall A \in \mathbb{Q}^{>0}, \exists p_0 \in \mathbb{P} (p > p_0 \rightarrow {\lvertg_p\rvert} > A).$$ Note that the above definitions are not first-order, since we have no way of quantifying over standard positive rationals in the language. We have:
The group $\Gamma$ is an ordered abelian group. Furthermore, the set $\Delta$ of archimedean classes of $\Gamma$ is an infinite, unbounded, densely linearly ordered set having uncountable cofinality.
The first assertion follows using standard properties of ultraproducts, e.g. [@AP]. To see that $\Delta$ is unbounded we equip $\Gamma$ with the multiplicative operation (compatible with the order) induced by standard multiplication on $\mathbb{Q}$, endowing $\Gamma$ with an ordered field structure. Hence $\Delta$ acquires a group structure through $v_1(\alpha) + v_1(\beta)= v_1(\alpha \beta)$, for all $\alpha, \beta \in \Gamma^{>0}$, where $v_1: \Gamma \to \Delta \cup \{\infty \}, g \mapsto [g]$ as defined above.\
It follows that $\Delta$ is the value group for the natural valuation $v_1$. Now the required conclusion follows since $\Gamma$ is nonarchimedean.\
Let us now show that $\Delta$ has uncountable cofinality.\
Assume there exists a countable sequence $(\delta_n)_{n \in \mathbb{N}}, \delta_n \in \Gamma$ such that $$\forall \alpha \in \Gamma, \exists n_0 \in \mathbb{N}, \forall n > n_0, [\alpha] \preceq [\delta_n]. \qquad \qquad (\dagger)$$ Writing $\delta_n=[(\delta_{nk})_{k \in \mathbb{P}}]$, $\delta_{nk} \in \mathbb{Q}_+$, one can check that the double sequence $(\delta_{nk})_{n \geq 0, k \in \mathbb{P}}$ is strictly increasing (beyond some $n_0, k_0$). Now let $\alpha:= [(\alpha_k)_k]$ be defined by $\alpha_k = \delta_{kk}$.\
Then it can be checked that $(\dagger)$ does not apply for $\alpha$, contradiction. Since $\Gamma$ has cardinality $2^{\aleph_0}$, the cofinality of $\Delta$ is at most $2^{\aleph_0}$.\
Finally, to see that $\Delta$ is densely ordered, assume the contrary. It suffices then to observe that the induced order on the set $\Gamma^{\geq 0}$ is of type $>\omega$, on which there exists no possible cancellative semi-group structure compatible with the ordering. This contradiction proves the result.
### Kaplansky embedding theorem
Let $\Gamma$ be an ordered abelian group and $k$ a commutative field. Let $k((t^\Gamma))$ be the field of generalized power series with a well-ordered set of exponents in $\Gamma$ and coefficients in $k$. Denote by $v$ the $t$-adic valuation of $k((t^\Gamma))$. We are now able to apply the following:
\[K\] (Kaplansky [@K]) Let $(K, \textrm{val})$ be a valued field of zero equi-characteristic, with value group $\Gamma$ and algebraically closed residue field $k$. Then $(K, \textrm{val})$ is **analytically isomorphic** to a subfield of $(k((t^\Gamma)), v)$, i.e. there exists a value preserving embedding of fields $K \hookrightarrow k((t^\Gamma))$.
The original statement in [@K] is more general, allowing non-algebraically closed residue fields at the expense of introducing *factor sets* into the definition of the multiplicative operation of monomials in the power series field. By Theorem 7 of [@K], this turns out not to be necessary in the special case of an algebraically closed residue field.
A differential exponential valued field
=======================================
Now we consider again the field $\mathbb{K}_{\cU}$. Observe that the valuation ring of $\mathbb{K}_{\cU}$ is given by $$R = \{ x:=[(x_p)_{p \in \mathbb{P}}] : {\rm val}(x) \geq 0 \},$$ where the order relation (on the value group of $\mathbb{K}_{\cU}$) has already been explained in the previous section.\
We can easily show that the residue class field $k_\cU$ ($k_\cU = R/P$ where $P$ is the maximal ideal of $R$) is given by $$k_\cU = \prod_{p \in \mathbb{P}} \mathbb{F}_p^{\rm alg} /\cU,$$ where $\mathbb{F}_p^{\rm alg}$ is the algebraic closure of the finite field $\mathbb{F}_p$. By Lefshetz principle (see, e.g. Theorem 2.4.3 [@S]) we have $k_\cU \simeq \mathbb{C}$, since both are algebraically closed fields of cardinality $2^{\aleph_0}$ having characteristic zero.\
Applying Kaplansky’s result mentioned above, there exists an embedding of valued fields $\mathbb{K}_{\cU} \hookrightarrow \mathbb{L}_{\cU} := k_\cU((t^\Gamma))$. For each non-principal ultrafilter $\cU$ over $\mathbb{P}$, we fix an embedding $\iota_\cU$ $$\iota_\cU: \mathbb{K}_{\cU} \hookrightarrow \mathbb{L}_{\cU},$$ and we will denote by $v$ the canonical valuation on $\mathbb{L}_\cU$.\
The $p$-adic exponential map on each $\mathbb{C}_p$ can be used to introduce a total exponential map on $\mathbb{K}_{\cU}$.\
More precisely, one may show (using Łoś Theorem) that the map $\textrm{Exp}: [(x_p)_p] \mapsto [({\rm EXP}_p(x_p))_p]$ is indeed an exponential map, $\mathbb{K}_{\cU} \to \mathbb{K}_\cU^{\times}$ (satisfying $\textrm{Exp}(x+y) =\textrm{Exp}(x) \cdot \textrm{Exp}(y)$).
### An exponential differential field {#expdif}
In order to be able to apply Ax’s Theorem, we need to define an embedding of $\mathbb{K}_\cU$ into a (partial) exponential differentiable field, along the lines of [@KM] and [@KMS] (see also [@M] for a general survey). As will be seen, this embedding need not be an embedding of *differential fields*, neither this is assumed.\
First we define a right-shift map $\sigma: \Delta \to \Delta, \phi \mapsto \sigma(\phi)$ such that $\sigma(\phi) \succ \phi$ and $\sigma$ is order-preserving.\
Let $\delta$ be the archimedean class of some infinitesimal element of $\Gamma$. We set: $$\sigma: \Delta \to \Delta, \phi \mapsto \delta \cdot \phi.$$ Here by $\delta \cdot \phi$ we mean the archimedean class of any product of two elements in $\delta$ and $\phi$ respectively. It can be seen that this is independent of the choices, and that, indeed $\sigma(\phi) \succ \phi, \forall \phi \in \Delta$.\
Then, as in “Case 1” of Example (6) in [@KMS], one may define a derivation $D: \mathbb{L}_{\cU} \to \mathbb{L}_{\cU}$ with field of constants $k_\cU$, and which satisfies furthermore: $Dx= \frac{D(\exp(x))}{\exp x}$ since D is a series derivation (see [@H], Corollary (3.9)).\
Let us denote by $\mathbb{L}_{\cU}^{\circ}$ the ring of bounded elements of $\mathbb{L}_{\cU}$, and by $\mathbb{L}_{\cU}^{\circ \circ}$ its maximal ideal, i.e. the ideal of infinitesimal elements. Note that $\mathbb{L}_{\cU}^{\circ \circ} = k_{\cU}((t^{\Gamma^{>0}}))$ (the set of generalized power series with strictly positive support).\
Let $\cD_\cU$ be the set defined as: $$\cD_\cU := \bigg\{ x=[(x_p)_{p \in \mathbb{P}}] \in \mathbb{K}_{\cU} \; : \; \ordp(x_p) > \frac{1}{p-1} \; \textrm{for} \; \cU\!\!-\!\textrm{almost all} \; p \in \mathbb{P} \bigg\}.$$ Consider the map $E: \cD_\cU \rightarrow \mathbb{K}_{\cU}^{\times} \subset \mathbb{L}_{\cU}^{\times}$ defined by $$[(x_p)]_{p \in \mathbb{P}} \mapsto E( [(x_p)]_{p \in \mathbb{P}}) := [(\exp_p(x_p))_{p \in \mathbb{P}}].$$ Using Łós Theorem we see that $E$ is a partial exponential map on $\mathbb{L}_{\cU}$ (i.e. $E(x+y)= E(x) E(y)$).\
In what follows we note that, using Neumann Lemma (see [@N]), the series $\sum_{n \in \mathbb{N}} \frac{x^n}{n!}$ is summable for all $x \in k((t^{G^{>0}}))$, for any field $k$ and ordered abelian group $G$, and $\exp(x) \in k((t^G))$. In particular the map $\exp: \mathbb{L}_\cU^{\circ \circ} \to \mathbb{L}_\cU^\times, x \mapsto \exp(x):= \sum_{n \in \mathbb{N}} \frac{x^n}{n!}$ is well defined.
The map $E$ coincides with the map $x \mapsto \exp(x)= \sum_{n \in \mathbb{N}} \frac{x^n}{n!}$ on $\cD_\cU$, i.e. $E(x)$ is given by the Taylor formula for the standard exponential map. In other words, the embedding $\iota_\cU: \mathbb{K}_{\cU} \hookrightarrow \mathbb{L}_{\cU}$ commutes with the exponential, i.e. $\iota_\cU(E(x)) = \exp(\iota_\cU(x))$ for all $x \in \cD_\cU$.
Let $\alpha \in \cD_\cU$. Then $\alpha=[(\alpha_p)_p]$, and $\ordp(\alpha_p) > \frac{1}{p-1}$ for $\cU$-almost all $p$. Consider the fields $F_0= k_{\cU}(\alpha)$ and $F= k_\cU(\alpha, E(\alpha))$ and let $\Gamma_0$ be the divisible hull of the group val$(F^{\times})$. By ( [@AP] Theorem 3.4.3) $\Gamma_0$ is an ordered abelian group having finite dimension as a linear space over $\mathbb{Q}$. This will be shown directly below.\
From the embedding $F \hookrightarrow \mathbb{L}_{\cU}$ we get an embedding of valued fields $F \hookrightarrow k_\cU((t^{\Gamma_0}))$ such that the following diagram commutes $$\xymatrix{
F \ar[d] \ar[r] & \mathbb{L}_{\cU} \\
k_\cU((t^{\Gamma_0})) \ar[ur] &
}$$ obtained by identifying $F$ with its image in $\mathbb{L}_{\cU}$.\
Let $s_N(\alpha)$ be the partial sum $s_N(\alpha):= \sum_{n=0}^N \frac{\alpha^n}{n!}$. For every $p$, the sequence $(s_N(\alpha_p))_N$ is a Cauchy sequence in $\mathbb{C}_p$, hence in particular it is pseudo-Cauchy, and $\exp_p(\alpha_p)$ is a (pseudo-)limit. Thus we obtain using Łós Theorem that for all positive integers $N_1 < N_2 <N_3$ $$\mathbb{K}_{\cU} \models \textrm{val}(s_{N_3}(\alpha) -s_{N_2}(\alpha)) > \textrm{val}(s_{N_2}(\alpha) -s_{N_1}(\alpha)),$$ (where we used the abbreviation val defined in section \[lang\]) and $(s_N(\alpha))_N$ is pseudo-Cauchy in $\mathbb{K}_\cU$. In particular, $(s_N(\alpha))_N$ is pseudo-Cauchy in $F_0$. Also, we have $$\mathbb{K}_{\cU} \models \textrm{val}(E(\alpha) -s_{N}(\alpha)) = \textrm{val}(s_{M}(\alpha) -s_N(\alpha)),$$ (by Łós Theorem) for all sufficiently large $N$, and all $M>N$.\
It follows by [@K] (Theorems 2 and 3) that the extension $F_0
\hookrightarrow F_0(E(\alpha)) = F$ is an immediate valued field extension, and $\Gamma_0 = \mathbb{Q} \cdot \gamma$ where $\gamma= \textrm{val}(\alpha)$, hence $\Gamma_0$ has finite dimension as a linear space over $\mathbb{Q}$, as claimed. It follows in particular that the field $k_\cU((t^{\Gamma_0}))$ is Hausdorff and complete with respect to the topology induced by the valuation $v_{|k_\cU((t^{\Gamma_0}))}$.\
For any $x \in \cD_\cU$, one has $$\mathbb{C}_p \models \ordp(\exp_p(x_p) -s_N(x_p)) = \ordp\bigg(\frac{x_p^{N+1}}{(N+1)!}\bigg)$$ for all $N$, for $\cU$-almost all $p$. Hence, in this case we have: $$\mathbb{K}_\cU \models \textrm{val}(E(\alpha) - s_N(\alpha)) = \textrm{val}\bigg(\frac{\alpha^{N+1}}{(N+1)!}\bigg)$$ for all $N$.
From the above observation we see that the sequence val$(\alpha^N)$ is cofinal in $\Gamma_0$, hence $\alpha^N \to 0$ as $N \to \infty$ (in $F$) and the sequence $E(\alpha)-s_N(\alpha)$ converges to zero in $k_{\cU}((t^{\Gamma_0}))$. Also, in $k_\cU((t^{\Gamma_0}))$ we have that $s_N(\alpha) \rightarrow \exp(\alpha)$. Consequently, $E(\alpha)=\exp(\alpha)$ as required.
Using the above, we reach the following corollary:
For all $x \in \cD_\cU$, one has: $D(E(x))= E(x)Dx$.
This follows from $E(x) = \sum_{n \in \mathbb{N}} \frac{x^n}{n!}= \exp(x)$ and that $Dx= \frac{D(\exp(x))}{\exp(x)}$ as observed in subsection \[expdif\].
Proof of Theorem \[main\] {#Proofs}
-------------------------
In this section we apply the above considerations in order to obtain Theorem \[main\].\
Let $V$ be a variety of dimension $n$ in the affine $2n$-space $\mathbb{A}^{2n}_{\mathbb{Q}}$. For each prime $p$, denote by $W(\mathbb{C}_p)$ the set of tuples $\bar{a}_p \in E_p^n$ for which $(\bar{a}_p, \exp_p(\bar{a}_p)) \in V(\mathbb{C}_p)$.\
Let $S' \subset \mathbb{P}$ denote the set of primes $p$ for which $V$ has a $\mathbb{C}_p$-point of the form $(\bar{a}_p, \exp_p(\bar{a}_p))$ (hence, in particular, ${\lvert\bar{a}_p\rvert}_p < p^{-1/p-1}$). The following will be assumed throughout:\
$(\star)$ There are infinitely many primes $p$ such that $V$ has a $\mathbb{C}_p$-point of the form $(\bar{a}_p, \exp_p(\bar{a}_p))$. In other words, $S'$ is an infinite set.\
The proof of Theorem \[main\] proceeds by assuming the contrapositive. More precisely, let $(\dagger)$ denote the following statement:\
$(\dagger)$ There exists no [*uniform*]{} rational linear dependence that holds for $\bar{a}_p$, for infinitely many $p \in S'$.\
Let $\cU$ be a non-principal ultrafilter on $\mathbb{P}$, such that $S' \in \cU$. Define $\bar{x} =(x_1, \dots, x_n) = [(\bar{x}_p)_p] \in \mathbb{K}_\cU^n \subset \mathbb{L}_\cU^n$ as follows:\
If $p \in S'$, then $\bar{x}_p = \bar{a}_p$. Otherwise, we let $\bar{x}_p$ be an arbitrary $n$-tuple of complex $p$-adic numbers which lie in the domain of the corresponding $\exp_p$ (this last assumption, though harmless, is not strictly necessary). Then $\bar{x}$ is an $n$-tuple of elements of $\cD_\cU$.\
Applying Łós Theorem to $\mathbb{K}_\cU$ it follows that $(\bar{x}, \textrm{Exp}(\bar{x})) \in V(\mathbb{K}_\cU)$, consequently $(\bar{x}, E(\bar{x})) \in V(\mathbb{K}_\cU) \subset V(\mathbb{L}_\cU)$. Hence, applying Proposition \[ASP\] to the (partial) exponential valued field $\mathbb{L}_\cU$, it follows that $$m_1 x_1 + \dots + m_n x_n \in k_{\cU},$$ for some $m_1, \dots, m_n \in \mathbb{Q}$ (not all zero). Clearing denominators, we can assume that $m_1, \dots, m_n$ are integers.\
Furthermore, since the elements $x_1, \dots, x_n$ lie in the maximal ideal $\mathbb{L}_\cU^{\circ \circ }$, any $\mathbb{Z}$-linear combination of $x_1, \dots, x_n$ will necessarily be in $\mathbb{L}_\cU^{\circ \circ}$. Writing $x_i = [(x_{p,i})_p]$ for $i=1, \dots, n$, it follows $$m_1 x_{p,1} + \dots + m_n x_{p,n} =0, \qquad \qquad (*)$$ for $\cU$-almost all $p \in \mathbb{P}$, i.e., the set of primes $p$ for which $(*)$ holds belongs to the ultrafilter $\cU$. In particular, observing that the intersection of any two sets in $\cU$ is in $\cU$ (so it is an infinite set, $\cU$ being a non-principal ultrafilter), one has, for infinitely many primes $p$ $$m_1 a_{p,1} + \dots + m_n a_{p,n} =0, \qquad \qquad \qquad \qquad (\ddagger)$$ with fixed $m_1, \dots, m_n$ (not all zero), contradicting the hypothesis.\
In particular it follows from the above argument that the set of rational $n$-tuples $(m_1, \dots, m_n)$ (up to a non-zero multiplicative constant) for which $(\ddagger)$ holds uniformly for infinitely many primes $p \in S'$ is finite. Let us denote this set by $A$.\
In order to show the remaining statement of Theorem \[main\], let us define $\bar{\alpha}:= (\bar{\alpha}_p)_p$ (i.e. $\bar{\alpha}$ defines a family of $n$-tuples of complex $p$-adic numbers that belong to $W(\mathbb{C}_p)$ for infinitely many $p$’s) and denote by $S_{\bar{\alpha}}$ the set of primes for which $(\ddagger)$ does not hold for the tuple $\bar{\alpha}_p$ for any $(m_1, \dots, m_n) \in A$. By the above reasoning, $S_{\bar{\alpha}}$ is a finite set. Assume, for the sake of contradiction, that there is no finite set $S$ for which $S_{\bar{\alpha}} \subset S$ for all $\bar{\alpha}$. Choose a countable subset of these $\bar{\alpha}$ enumerated as $\bar{\alpha}_1, \bar{\alpha}_2, \dots$, such that $\bigcup_{i=1}^\infty S_{\bar{\alpha}_i} \subset \mathbb{P}$ is an infinite set. Without loss of generality the sets $S_{\bar{\alpha}_i}$ can be assumed disjoint. One can then construct an infinite family of tuples $(\bar{\beta}_p)_p$ which satisfy $(\dagger)$: set $\bar{\beta}_p = \bar{\alpha}_{i,p}$ for $p \in S_{\bar{\alpha}_i}$, and assign arbitrary values to $\bar{\beta}_p$ for $p \notin S_{\bar{\alpha}_i}, \forall i$. Repeating the above reasoning, we reach a contradiction. This proves the claim.\
Combining the above results, we see that Theorem \[main\] is now fully proved.
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[**** ]{}\
Qie He^1^, Junfeng Zhu^1^, David Dingli^2^, Jasmine Foo^3\*\*^, Kevin Leder^2\*^,\
**1** Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN, USA\
**2** Department of Hematology, Mayo Clinic, Rochester, MN, USA\
**3** Department of Mathematics, University of Minnesota, Minneapolis, MN\
\* lede0024@umn.edu@umn.edu, \*\*jyfoo@math.umn.edu
Abstract {#abstract .unnumbered}
========
Over the past decade, several targeted therapies (e.g. imatinib, dasatinib, nilotinib) have been developed to treat Chronic Myeloid Leukemia (CML). Despite an initial response to therapy, drug resistance remains a problem for some CML patients. Recent studies have shown that resistance mutations that preexist treatment can be detected in a substantial number of patients, and that this may be associated with eventual treatment failure. One proposed method to extend treatment efficacy is to use a combination of multiple targeted therapies. However, the design of such combination therapies (timing, sequence, etc.) remains an open challenge. In this work we mathematically model the dynamics of CML response to combination therapy and analyze the impact of combination treatment schedules on treatment efficacy in patients with preexisting resistance. We then propose an optimization problem to find the best schedule of multiple therapies based on the evolution of CML according to our ordinary differential equation model. This resulting optimization problem is nontrivial due to the presence of ordinary different equation constraints and integer variables. Our model also incorporates realistic drug toxicity constraints by tracking the dynamics of patient neutrophil counts in response to therapy. Using realistic parameter estimates, we determine optimal combination strategies that maximize time until treatment failure.
Author Summary {#author-summary .unnumbered}
==============
Targeted therapy using imatinib, nilotinib or dasatinib has become standard treatment for chronicle myeloid leukemia. A minority of patients, however, fail to respond to treatment or relapse due to drug resistance. One primary driving factor of drug resistance are point mutations within the driving oncogene. Laboratory studies have shown that different leukemic mutants respond differently to different drugs, so a promising way to improve treatment efficacy is to combine multiple targeted therapies. We build a mathematical model to predict the dynamics of different leukemic mutants with imatinib, nilotinib and dasatinib, and employ optimization techniques to find the best treatment schedule of combining the three drugs sequentially. Our study shows that the optimally designed combination therapy is more effective at controlling the leukemic cell burden than any monotherapy under a wide range of scenarios. The structure of the optimal schedule depends heavily on the mutant types present, growth kinetics of leukemic cells and drug toxicity parameters. Our methodology is an important step towards the design of personalized optimal therapeutic schedules for chronicle myeloid leukemia.
Introduction {#introduction .unnumbered}
============
Chronic Myeloid Leukemia (CML) is an acquired hematopoietic stem cell disorder leading to the over-proliferation of myeloid cells and an increase in cellular output from the bone marrow that is often associated with splenomegaly. The most common driving mutation in CML is a translocation between chromosomes 9 and 22 that produces a fusion gene known as BCR-ABL. The BCR-ABL protein promotes proliferation and inhibits cell apoptosis of myeloid progenitor cells and thereby drives expansion of this cell population. By targeting the BCR-ABL oncoprotein, imatinib (brand name Gleevec) is able to induce a complete cytogenetic remission in the majority of chronic phase CML patients. A minority of patients, however, either fail to respond or eventually develop resistance to treatment with imatinib [@druker2006five]. It is thought that a primary driver of this resistance to imatinib is point mutations within the BCR-ABL gene. A recent study utilizing sensitive detection methods demonstrated that a small subset of these mutations may exist before the initiation of therapy in a significant fraction of patients, and that this status is correlated with eventual treatment failure [@iqbal2013sensitive]. Second generation agents such as dasatinib and nilotinib have been developed and each has shown efficacy against various common mutant forms of BCR-ABL. This leads to the observation that the various mutant forms of BCR-ABL result in CML that have unique dynamics under therapy, and that combinations of these inhibitors may be necessary to effectively control a rapidly evolving CML population. Patients with CML often die due to transformation of the disease into an acute form of leukemia known as blast crisis. It has been shown that blast crisis is due to the accumulation of additional mutations in CML progenitor cells [@jamieson2004granulocyte].
The goal of this work is to leverage the differential responses of CML mutant strains to design novel sequential combination treatment schedules using dasatinib, imatinib and nilotinib that optimally control leukemic burden and delay treatment failure due to pre-existing resistance. We develop and parametrize a mathematical model for the evolution of both wild-type (WT) CML and mutated (resistant) CML cells in the presence of each therapy. Then we formulate the problem as a discrete optimization problem in which a sequence of monthly treatment decisions is optimized to identify the temporal sequence of imatinib, dasatinib, and nilotinib administration that minimizes the total CML cell population over a long time horizon.
There has been a significant amount of work done in the past to mathematically model CML. For example, in [@FoKeCl91] the authors developed a system of ordinary differential equations (ODEs) that model both the normal progression from stem cell to mature blood cells and abnormal progression of CML. A hierarchical system of differential equations was used to model the response of CML cells to imatinib therapy in [@MiIwHu05]; this model fit the biphasic nature of decline in BCR-ABL positive cells during imatinib treatment. In [@leder2011fitness] the authors investigated the number of different resistant strains present in a newly diagnosed chronic phase CML patient. An optimal control approach was utilized to optimize imatinib scheduling in [@AiBe10]. Particularly relevant to our work is [@komarova2009combination; @KaKo10] where the authors investigated simultaneous continuous administration of dasatinib, nilotinib and imatinib; in particular, they explored the minimal number of drugs necessary to prevent drug resistance. In the current work, we focus on understanding the optimal administration schedule of multiple therapies to prevent resistance, and studying the impact of toxicity constraints on optimal scheduling. Since several of the available tyrosine kinase inhibitors (TKI) share similar toxicities (in particular neutropenia, see e.g., [@quintas2004granulocyte; @guilhot2007dasatinib; @swords2009nilotinib]) combining them together can lead to elevated risk of adverse events. Thus we consider sequential combination therapies in which only one TKI may be administered at a time. Moreover, it has been shown that the risk of treatment failure and blast crisis are highest within the first 2 years from diagnosis [@druker2006five]. Therefore it is possible that optimized, sequential single agent therapy may be sufficient to minimize the risk of treatment failure. Allowing only one treatment at a time leads to a complex, time-dependent discrete optimization problem.
Another line of research closely related to the current work is the use of optimal control techniques in the design of optimal temporally continuous drug concentration profiles (see, e.g., review articles [@Swan90; @Shi14] and the textbook [@MartinTeo93]). In this field the tools of optimal control such as the Pontryagin principle and the Euler-Lagrange equations are used to find drug concentration profiles that result in minimal tumor cell populations under toxicity constraints. Particularly relevant to the current work is [@WeZeNo97] where the authors searched for optimal anti-HIV treatment strategies. They dealt with the similar problem of treating heterogeneous populations with multiple drugs. One major drawback of these works is the fact that it is nearly impossible to to achieve a specific optimal continuous drug-concentration profile in patients, since drug concentration over time is a combined result of a treatment schedule (e.g. sequence of discrete oral administrations) and pharmacokinetic processes in the body including metabolism, elimination, etc. Thus the clinical utility of an optimal continuous drug concentration profile is limited. In contrast to these previous works, here we model the optimization problem as a more clinically realistic sequence of monthly treatment decisions. Imposition of this fixed discrete set of decision times leads to a challenging optimization problem. Such dynamical systems are referred to as ‘switched nonlinear systems’ in the control community [@liberzon2012switching], and our problem additionally imposes fixed switching times. In this work we will leverage the system structure and tools from mixed-integer linear optimization [@nemhauser1999integer] to solve this problem numerically, resulting in optimal therapy schedules that are easy to implement in practice.
Computational framework {#computational-framework .unnumbered}
=======================
Model of CML dynamics {#sec:Model .unnumbered}
---------------------
We consider an ODE model of the differentiation hierarchy of hematopoietic cells, adapted from [@MiIwHu05; @Foo_PlosCBCML; @olshen2014dynamics]. Stem cells (SC) on top of the hierarchy give rise to progenitor cells (PC), which produce differentiated cells (DC), which in turn produce terminally differentiated cells (TC). This differentiation hierarchy applies to both normal and leukemic cells [@strife1988biology]. We consider in our model leukemic WT cells as well as pre-existing BCR-ABL mutant cell types. We use type 1, type 2, and type $i$ ($3\le i \le n$) cells to denote normal, leukemic WT, and $(n-2)$ leukemic mutant cells; layer 1, 2, 3, 4 cells to denote SC, PC, DC, and TC; and drug 0, 1, 2, 3 to denote a drug holiday, nilotinib, dasatinib, and imatinib, respectively. Let $x_{l,i}(t)$ denote the abundance of type $i$ cell at layer $l$ and time $t$, and $x(t)=(x_{l,i}(t))$ be the vector of all cell abundance at time $t$. If drug $j\in \{0,1,2,3\}$) is taken from month $m$ to month $m+1$, then the cell dynamics are modeled by the following set of ODEs.
\[eq:odeconcise:j\] $$\begin{aligned}
\dot{x}(t)=f^j(x(t)), \ & t\in [m{\Delta t}, (m+1){\Delta t}],\\
x(m{\Delta t}) = x^m, \ &\end{aligned}$$
for some function $f^j$, where ${\Delta t}=30$ days and $x^m$ is the cell abundance at the beginning of month $m$. The concrete form of function $f^j$ under drug $j$ is described as follows.
\[eq:hierarchical\] $$\begin{aligned}
\text{SC level} \qquad & \dot{x}_{1,i}=(b^j_{1,i}\phi_i - d^j_{1,i})x_{1,i}, \; i=1,\ldots,n\\
\text{PC level} \qquad & \dot{x}_{2,i}= b^j_{2,i} x_{1,i} - d^j_{2,i}x_{2,i}, \; i=1,\ldots,n \\
\text{DC level} \qquad & \dot{x}_{3,i}= b^j_{3,i} x_{2,i} - d^j_{3,i}x_{3,i}, \; i=1,\ldots,n \\
\text{TC level} \qquad & \dot{x}_{4,i}= b^j_{4,i} x_{3,i} - d^j_{4,i}x_{4,i}, \; i=1,\ldots,n.\end{aligned}$$
Here we describe the function of each parameter of this model. For a detailed discussion of how these parameters were estimated from biological data, please see section \[sec:PARAM\] of the Appendix. Type $i$ stem cells divide at rate $b^j_{1,i}$ per day under drug $j$. The production rates of type $i$ progenitors, differentiated cells, and terminally differentiated cells under drug $j$ are $b^j_{l,i}$ per day for $l=2,3,4$, respectively. The type $i$ cell at layer $l$ dies at rate $d^j_{l,i}$ per day under drug $j$, for each $i$, $l$ and $j$. The competition among normal and leukemic stem cells is modeled by the density dependence functions $\phi_i(t)$, where $\phi_i(t) =1/(1+p_i\sum_{i=1}^nx_{1,i}(t))$ for each $i$; these functions ensure that the normal and leukemic stem cell abundances remain the same once the system reaches a steady state. The parameter $p_1$ (resp. $p_2$) is computed from the equilibrium abundance of normal (resp. leukemic WT) stem cells assuming only normal (resp. leukemic WT) cells are present, and we set $p_i=p_2$ for each $i\ge 3$. In particular, $p_1=(b^0_{1,1}/d^0_{1,1}-1)/K_1$ and $p_2 = (b^0_{1,2}/d^0_{1,2}-1)/K_2$, with $K_1$ (resp. $K_2$) being the equilibrium abundance of normal (resp. leukemic WT) stem cells assuming only normal (resp. leukemic WT) cells are present.
Toxicity modeling {#subsec:ANC .unnumbered}
-----------------
One of the most common side effects of TKIs in CML is neutropenia, or the condition of abnormally low neutrophils in the blood. Neutropenia is defined in terms of the absolute neutrophil count (ANC). To incorporate toxicity constraints we develop a model of the dynamics of the patient’s ANC in response to each therapy schedule. We then constrain our optimization problem by considering only schedules during which the patient’s ANC stays above an acceptable threshold level ${L_{anc}}$. Typically, ANC at diagnosis is within normal limits (between $1500-8000/\mathrm{mm}^3$); thus we set each patient’s initial ANC to be $3000/\mathrm{mm}^3$. Treatment with imatinib, dasatinib and nilotinib all result in reduction of the ANC at varying rates. Neutropenia is defined as an ANC level below ${L_{anc}}= 1000/\mathrm{mm}^3$. If a patient’s ANC falls below the threshold, a drug holiday is required at the next monthly treatment decision stage. During a drug holiday, ANC level will recover back to safe levels.
To model this process, we assume the patient’s ANC decreases at rate $d_{anc,j}$ per month taking drug $j$, for $j=1,2,3$. During a drug holiday, ANC increases at rate $b_{anc}$ per month but never exceeds the normal level of ${U_{anc}}= 3000/\mathrm{mm}^3$. More specifically, let $y^m$ denote the ANC level at the beginning of month $m$ and the binary variable $z^{m,j}$ indicate whether drug $j$ is taken in month $m$ or not, i.e., $z^{m,j}=1$ (resp. $z^{m,j}=0$) indicates drug $j$ is (resp. not) taken in month $m$. The kinetics of ANC is modeled through the truncated linear function $$y^{m+1}=r(y^m, z^{m,0},z^{m,1}, z^{m,2}, z^{m,3})=\min\{y^m + {b_{anc}}z^{m,0} - \sum_{j\in \{1,2,3\}} d_{anc,j}z^{m,j}, {U_{anc}}\}.$$ For example, after a patient with ANC level $y^m$ takes a drug holiday in month $m$, her ANC level at the beginning of month $m+1$ becomes $y^m+{b_{anc}}$ if $y^m+{b_{anc}}$ is not higher than the normal level ${U_{anc}}$, or ${U_{anc}}$ if $y^m+{b_{anc}}$ exceeds ${U_{anc}}$. If the patient instead takes nilotinib in month $m$, then her ANC level at the beginning of month $m+1$ becomes $y^m - d_{anc,1}$. The parameters governing ANC rate of change are provided in section \[sec:PARAM\] of the Appendix.
Treatment optimization problem {#sec:optmodel .unnumbered}
==============================
Assume the initial population of each cell type is known. Our goal is to select a treatment plan to minimize the tumor size at the end of the planning horizon subject to certain toxicity constraints. We call this the Optimal Treatment Plan problem (OTP). Each treatment plan is completely characterized by a temporal sequence of monthly treatment decisions over a long time horizon. Between each monthly treatment decision, the dosing regimen is identical from day to day. The standard regimens for each drug, which we will utilize throughout the work, are $300$mg twice daily for nilotinib, $100$mg once daily for dasatinib, and $400$mg once daily for imatinib [@o2012chronic]. For example, let $1$ denote nilotinib, 2 - dasatinib, 3 - imatinib, and 0 - drug holiday. Then the sequence $\{1,1,\ldots,1\}$ represents that the patient takes the standard nilotinib regimen, $300$mg twice daily, every day, every month. The sequence $\{2, 0, 2, 0, \ldots \}$ represents that the patient alternates between the standard dasatinib regimen, $100$mg once daily, and a drug holiday on alternate months.
We introduce the binary decision variables $z^{m,j}$ to indicate whether drug $j$ is taken in month $m$ or not, for each $j=0,1,2,3$ and $m=0,1,\ldots,M-1$. An assignment of values to all $z^{m,j}$ variables that satisfy all constraints in the optimization model gives a feasible treatment plan.
The optimization problem {#the-optimization-problem .unnumbered}
------------------------
Note the total leukemic cell abundance at day $t$ is given by $\sum_{l \ge 1}\sum_{i\ge 2} x_{l,i}(t)$. The OTP can be formulated as the following mixed-integer optimization problem with ODE constraints.
\[eq:OTP\] $$\begin{aligned}
\min \ & \sum_{l \ge 1}\sum_{i\ge 2} x_{l,2}(M{\Delta t}) \label{eq:OTP:obj}\\
\text{s.t.} \ & \dot{x}(t) = \sum_{j=0}^3 z_{m,j}f^j(x(t)) , & t\in [m{\Delta t}, (m+1){\Delta t}], m=0,1,\ldots,M-1, \label{eq:OTP:ode}\\
& \sum_{j=0}^3 z^{m,j} = 1, &m=0,1,\ldots,M-1, \label{eq:OTP:card}\\
& y^{m+1} = r(y^m, z^{m,0}, z^{m,1}, z^{m,2}, z^{m,3}), &m=0,1,\ldots,M-1, \label{eq:OTP:tox1}\\
& y^m \ge {L_{anc}}, &m=0,1,\ldots,M, \label{eq:OTP:tox2}\\
& z^{m,j} \in \{0,1\}, & j=0,1,2,3, m=0,1,\ldots,M-1 \label{eq:OTP:binary}\\
& x(0) = x^0, \; y^0 \text{ is given}.\end{aligned}$$
To summarize the previous display, in equation we state that our objective is to minimize the leukemic cell population at the end of the treatment horizon. In equation we stipulate that the cell dynamics are governed by the system of differential equations given by . Together and stipulate that during each time period we administer either one drug or no drug. Equations and reflect the toxicity constraints described above.
The OTP problem is a mixed-integer nonlinear optimization problem, in which some constraints are specified by the solution to a nonlinear system of ODEs . This optimization problem is beyond the ability of state-of-the-art optimization software. However, if we assume the TKI therapies do not affect the stem cell compartment, then it is possible to handle the ODE constraints numerically. This is because the non-linearities in the ODE model are only present in the stem cell compartment, and the remaining compartments are modeled by linear differential equations. Thus we are able to build a refined linear approximation to the ODE constraints (see Section \[sec:LIN\_ODE\] of the Appendix), and recast the problem as a mixed-integer linear optimization problem (see Section \[sec:MILO\] of the Appendix).
A quick reference for notation {#a-quick-reference-for-notation .unnumbered}
------------------------------
Below we summarize our notation for the ease of the reader.
- ${\mathcal{I}}=\{1,2,\ldots, n\}$: the set of cell types. Type 1 denotes normal cells, type 2 denotes leukemic WT cells, and type $i$ ($3\le i\le n$) denotes one type of leukemic mutants.
- ${\mathcal{L}}=\{1,2,\ldots,L\}$: the set of cell layers. We have $L=4$, and layer 1, 2, 3 and 4 denotes SC, PC, DC, and TC, respectively.
- ${\mathcal{J}}=\{0,1,2,\ldots, J\}$: The set of drugs for CML. We have $J=3$, drug 0 refers to a drug holiday, and drug 1 to drug 3 refers to nilotinib, dasatinib, and imatinib, respectively.
- ${\mathcal{M}}=\{0,1,\ldots,M\}$: the set of months for treatment.
- ${\Delta t}$: the duration during which a patient takes one drug before deciding to switch to another drug or take a drug holiday. We set ${\Delta t}= 30$ days.
- $K_1$: the equilibrium abundance of normal stem cells when only normal cells are present.
- $K_2$: the equilibrium abundance of leukemic WT stem cells when only leukemic cells are present.
- $b^j_{l,i}$: the production rate of type $i$ cell at layer $l$ under drug $j$.
- $d^j_{l,i}$: the death rate of type $i$ cell at layer $l$ under drug $j$.
- ${b_{anc}}$: the average increase ($/\mathrm{mm}^3$) of the ANC in a patient without any drug after time ${\Delta t}$.
- ${d_{anc,j}}$: the average decrease ($/\mathrm{mm}^3$) of ANC in a patient under drug $j$ after time ${\Delta t}$.
- ${L_{anc}}$: The lower limit of the ANC. We assume that the patient develops neutropenia if the ANC is less than ${L_{anc}}$, at which a drug holiday needs to be taken.
- ${U_{anc}}$: The normal level of ANC.
Results {#results .unnumbered}
=======
In this work we consider the dynamics of CML response to single-agent and combination schedules utilizing the standard therapies imatinib, dasatinib and nilotinib.
Evolution of preexisting BCR-ABL mutants under standard monotherapy {#subsec:stand_sim .unnumbered}
-------------------------------------------------------------------
We first utilize the model to demonstrate the dynamics of CML populations with preexisting BCR-ABL mutations under monotherapy with the standard therapies imatinib, dasatinib and nilotinib. Recall that the standard dosing regimens are $300$mg twice daily for nilotinib, $100$mg once daily for dasatinib, and $400$mg once daily for imatinib [@o2012chronic]. Growth rate parameters for each cell type in the model are estimated using *in vitro* IC50 values reported in [@redaelli2009activity] for each drug. The initial cell populations at the start of therapy are derived by running the model starting from clonal expansion of a single leukemic cell in a healthy hematopoietic system at equilibrium [@foo2009eradication] until CML detection (when the total leukemic burden reaches approximately $10^{12}$ cells [@holyoake2002elucidating]). At this point the total cell burden is 2-3 times the normal cell burden in a healthy individual and thus the total leukemic cells make up approximately $77\%$ of the total cell population; this is consistent with clinical reports [@dingli2008chronic]. Details on deriving the initial cell abundances at diagnosis are provided in section \[sec:PARAM\] of the Appendix.
In the first example we consider a patient harboring a low level of the BCR-ABL mutant F317L (which is resistant to dasatinib) before the initiation of TKI therapy. The initial population conditions are given in Table \[tab:p1:F317L\] with the leukemic WT and F317L cells taking up $95\%$ and $5\%$ of the leukemic cells, respectively.
normal cell Wild-type F317L
---- ----------------------- ----------------------- -----------------------
SC $7.34 \times 10^{4}$ $2.80 \times 10^{5}$ $1.48 \times 10^{4}$
PC $1.61 \times 10^{7}$ $3.87 \times 10^{7}$ $2.04 \times 10^{6}$
DC $3.24 \times 10^{9}$ $1.03 \times 10^{10}$ $5.40 \times 10^{8}$
TC $3.24 \times 10^{11}$ $1.03 \times 10^{12}$ $5.40 \times 10^{10}$
: [**The initial cell abundance.**]{}
\[tab:p1:F317L\]
We plot in Fig \[fig:nilotinib:p2\] the cell dynamics over 120 months for four treatment plans: (1) nilotinib monotherapy (2) dasatinib monotherapy, (3) imatinib monotherapy, (4) no therapy - control. We observe that as predicted, the disease burden responds well to imatinib and nilotinib; the percentage of cancerous cells after a 24 month treatment drops to $0.19\%$ with nilotinib and $0.26\%$ with imatinib, respectively. However, the F317L mutant population is fairly resistant to dasatinib; we observe that the percentage of cancerous cells after 24 months is $58.1\%$ with dasatinib and $95.4\%$ with no treatment. Over the 120 month period dasatinib treatment provides only modest improvement over the ‘no drug’ option in controlling the F317L population; however, dasatinib remains quite effective in controlling the WT leukemic population. It is interesting to note that overall, nilotinib is the most effective in controlling both the WT and F317L leukemic populations. However, nilotinib also negatively impacts the healthy cell population more severely than imatinib, which is slightly less effective in controlling the leukemic populations. This suggests that some trade-offs between these drugs exist, and these trade-offs may be exploited in designing combination therapies.
![[**Long term dynamics of healthy, WT leukemic and F317L mutant leukemic cell populations under treatment with standard regimen monotherapy nilotinib (blue), dasatinib (yellow), imatinib (green) and no drug (orange).**]{} The dynamics of healthy normal cells with mono imatinib (green) and no drug (orange) coincide. Initial conditions are provided in Table \[tab:p1:F317L\] and parameter choices are provided in Appendix \[sec:PARAM\]. []{data-label="fig:nilotinib:p2"}](F317L.pdf)
In the next example we consider a patient with BCR-ABL mutant type M351T preexisting therapy. In contrast to the previous example, this commonly occurring mutant has been found to be partially sensitive in varying degrees to all three therapies. The initial conditions are given in Table \[tab:p1:M351T\]. Once again we have assumed that WT and M351T cells take up $95\%$ and $5\%$ of total leukemic cells, respectively.
Normal cell Wild-type M351T
---- ----------------------- ----------------------- -----------------------
SC $7.34 \times 10^{4}$ $2.80 \times 10^{5}$ $1.48 \times 10^{4}$
PC $1.61 \times 10^{7}$ $3.87 \times 10^{7}$ $2.04 \times 10^{6}$
DC $3.24 \times 10^{9}$ $1.03 \times 10^{10}$ $5.40 \times 10^{8}$
TC $3.24 \times 10^{11}$ $1.03 \times 10^{12}$ $5.40 \times 10^{10}$
: [**The initial cell abundance.**]{}[]{data-label="tab:p1:M351T"}
In Fig \[fig:nilotinib\] the cell dynamics over 120 months for the four standard treatment plans are plotted: (1) nilotinib monotherapy (2) dasatinib monotherapy, (3) imatinib monotherapy, (4) no therapy - control. Since the M351T mutant is responsive to each drug in contrast to the previous example, the percentage of cancerous cells after a 24 month treatment drops to $0.18\%$ with nilotinib, $0.18\%$ with dasatinib, and $0.25\%$ with imatinib, respectively. Without treatment, the percentage of cancerous cells after 24 months is $95.4\%$. Here, we observe that although nilotinib is more effective than dasatinib in controlling the total mutant M351T burden, the effect is reversed in the progenitor population. Higher levels of stem and progenitor populations will lead to faster rebound during treatment breaks, suggesting another trade-off to consider in the combination setting.
![[**Long term dynamics of healthy, WT leukemic and M351T mutant leukemic cell populations under treatment with standard regimen monotherapy nilotinib (blue), dasatinib (yellow), imatinib (green) and no drug (orange).**]{} The dynamics of healthy normal cells with mono imatinib (green) and no drug (orange) coincide. Initial conditions are provided in \[tab:p1:F317L\] and parameter choices are provided in Appendix \[sec:PARAM\]. []{data-label="fig:nilotinib"}](M351T.pdf)
Optimization of combination therapies {#optimization-of-combination-therapies .unnumbered}
-------------------------------------
We next solve the discrete optimization problem to identify sequential combination therapies utilizing imatinib, dasatinib and nilotinib to optimally treat CML patients with preexisting BCR-ABL mutations. We consider schedules in which a monthly treatment decision is made between one of four choices: imatinib, dasatinib, nilotinib, and drug holiday. During months in which one of the three drugs is administered, the dosing regimen is fixed at $300$mg twice daily for nilotinib, $100$mg once daily for dasatinib, and $400$mg once daily for imatinib. In the following we optimize over feasible treatment decision sequences that result in a minimal leukemic cell burden after 3 years. Each treatment plan is completely characterized by a temporal sequence of drugs over a long time horizon.
### Optimal therapy for preexisting M351T mutation, no toxicity constraints {#sec:M3NoTox .unnumbered}
In our first example we assume that the mutant M351T preexists therapy. For demonstration purposes no toxicity constraint is considered in this example. The initial cell populations are given in Table \[tab:p1:M351T\]. The remaining parameters are described in Section \[sec:PARAM\]. Note that WT and M351T leukemic cells comprise $95\%$ and $5\%$ of leukemic cells, respectively. The optimal schedule we obtain for this scenario is provided in Table \[tab:JF\_b\]. The proposed combination therapy is similar to the monotherapy using dasatinib, but switches to nilotinib towards the end of the 36 month time horizon. We note that the optimization result is robust to changes in the initial abundance of the leukemic mutant cells; increasing the frequency of initial M351T mutants to $50\%$ of the leukemic population results in an almost identical optimal schedule (data not shown).
[-0.5in]{}[0in]{}
Optimal combination 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1
--------------------- -------------------------------------------------------------------------
: [**Optimized treatment schedule for preexisting M351T.**]{}
Initial conditions are provided in Table \[tab:p1:M351T\]. No toxicity constraints. Recall that 0 - Drug Holiday, 1 - Nilotinib, 2 - Dasatinib, and 3 - Imatinib.
\[tab:JF\_b\]
In Fig \[fig:opt\] we compare the performance of 4 different schedules including the optimized schedule. Amongst the four schedules tested the optimal schedule provides the lowest leukemic cell burden at the 36 month mark. It is interesting to see that there is no single best drug. For monotherapy, it is best to use nilotinib if the treatment horizon is shorter than 12 months, and dasatinib if the treatment horizon longer than 12 months. The proposed combination therapy performs better than three monotherapies at the 36 month mark: the leukemic cell population at the end of 36 months is $2.75\times 10^7$ with the proposed combination therapy and $5.92 \times 10^7$ with dasatinib (the best monotherapy). We can see that the proposed optimal treatment schedule leads to more than $50\%$ reduction on final leukemic cell abundances over the best monotherapy. Figure \[fig:opt\] also shows that imatinib has less efficacy than nilotinib or dasatinib in reducing the leukemic cell burden when WT and M351T are present. An important question is, why does the optimal schedule take that specific form. In our parameter estimates (see Appendix \[sec:PARAM\]) we see that dasatinib is better at killing progenitors than nilotinib, while nilotinib is better at killing differentiated cells. Thus the optimal schedule uses dasatinib at first to bring down the progenitor cell population, and then switches to nilotinib near the end of the treatment horizon to decrease the population of differentiated cells.
![[**Plot of cell number versus time for three monotherapies and optimal combination therapy for preexisting M351T, with no toxicity constraints.**]{}[]{data-label="fig:opt"}](p1_M351T.pdf)
[*Optimal schedules robust to varying objective function and treatment length.*]{} We also consider an alternative objective function in which the goal is to minimize the average leukemic cell burden over the whole treatment horizon. Consider the scenario with M351T mutation preexisting at initiation of therapy with no toxicity constraints again. The altered objective function results in an optimal strategy of nilotinib monotherapy. To understand this, we note that for minimizing area under the cell population curve it is important to decrease the initial tumor population as quickly as possible. This tends to favor taking nilotinib the entire time since it leads to the quickest reduction in total tumor cell population, by reducing differentiated and therefore terminally differentiated cells. We also ran optimization experiments to evaluate the impact of varying the length of treatment between 35 and 38 months; these resulted in very similar optimal schedules.
### Optimal therapy for preexisting F317L mutation, no toxicity constraints {#optimal-therapy-for-preexisting-f317l-mutation-no-toxicity-constraints .unnumbered}
Next we consider a patient with preexisting mutant F317L instead of M351T. According to the *in vitro* IC50 value reported in [@redaelli2009activity], F317 is resistant to dasatinib, and moderately resistant to nilotinib and imatinib. The initial cell abundances are given in Table \[tab:p1:F317L\]; the mutant leukemic cells make up of $5\%$ of the total leukemic cells as in the baseline model except we replace mutant M351T with F317L. The proposed combination therapy is listed in Table \[tab:p1:F317L:opt\], and for comparison the optimal therapy for the previous example where M351T preexisted therapy is also provided. Note that dasatinib is used in the first 9 months and nilotinib is used in the next 27 months in the presence of F317L.
[-0.5in]{}[0in]{}
------------------- -------------------------------------------------------------------------
M351T preexisting 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
F317L preexisting 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
------------------- -------------------------------------------------------------------------
: [**Optimal combination schedules.**]{}
\[tab:p1:F317L:opt\]
In Fig \[fig:p3\] we show the comparison between proposed schedule and three different monotherapies. The final leukemic cell abundances are $7.46\times 10^7$ and $9.48\times 10^7$ under the propose schedule and monotherapy with nilotinib, respectively. The combination therapy performs better in reducing final leukemic cell population than three monotherapies, but the improvement is marginal in this case. Again the optimal schedule uses dasatinib to reduce the wild-type progenitor cell population, but switches to nilotinib much earlier to reduce the wild-type differentiated cell and F317L cell popoluations.
![[**Plot of cell number versus time for three monotherapies and optimal combination therapy for preexisting F317L, with no toxicity constraints.**]{}[]{data-label="fig:p3"}](p1_F317L.pdf)
### Incorporating toxicity constraints {#incorporating-toxicity-constraints .unnumbered}
We next study how drug toxicity affects the optimal therapy, in particular with the drug toxicity constraint introduced in the **Toxicity Modeling** subsection. Recall that the toxicity constraint prevents ANC from dipping below a threshold value ${L_{anc}}$. The ANC decreases at a constant rate each month under each drug, and increases at a constant rate without drug. The ANC never exceeds the normal level of ${U_{anc}}$. We first assume that nilotinib has a higher toxicity than dasatinib, and dasatinib has a higher toxicity than imatinib. In particular, the monthly decrease rates of ANC for nilotinib, dasatinib, and imatinib are $350/\mathrm{mm}^3$, $300/\mathrm{mm}^3$, and $250/\mathrm{mm}^3$, respectively, and ANC increases by $2,000/\mathrm{mm}^3$ with one month drug holiday.
We incorporated the toxicity constraints into the preexisting M351T mutant scenario described previously, i.e. initial cell populations are given in Table \[tab:p1:M351T\]. The three monotherapies and resulting optimal combination therapy are shown in Table \[tab:tox1\] below. Note that the proposed combination therapy is very close to the one described without toxicity constraints (i.e., Table \[tab:JF\_b\]), except now drug holidays are inserted to maintain the ANC level above ${L_{anc}}$.
[-0.5in]{}[0in]{}
------------- -------------------------------------------------------------------------
Nilotinib 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1
Dasatinib 2 2 2 2 2 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 2
Imatinib 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3 3 3 3 3 3 3 3 0 3
Combination 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 1 1 1 1
------------- -------------------------------------------------------------------------
: [**Treatment schedules with drug toxicities.**]{}
\[tab:tox1\]
The cell dynamics of three monotherapies and the proposed combination therapy are given in Fig \[fig:tox1\]. It can be seen that after drug holidays, the total leukemic cell population almost returns to the level at the beginning of treatment. This indicates that a one month drug holiday may be too long for the patient.
![[**Plot of cell number versus time for three monotherapies and optimal combination therapy for preexisting M351T, incorporating toxicity constraints.**]{}[]{data-label="fig:tox1"}](tox_ndi.pdf)
Since it is not clear whether nilotinib or dasatinib result in higher toxicity effects, we also switched the monthly ANC depletion rates to nilotinib - 300, dasatinib - 350, and imatinib - 250, so that dasatinib has the highest toxicity. Other conditions are kept the same. The recommended combination therapy is shown in Table \[tab:tox2\] below. Note that now imatinib is used more frequently, due to the increase in toxicity of dasatinib. We also compare the performance of the four different schedules in Fig \[fig:tox2\].
[-0.5in]{}[0in]{}
----------------------- -------------------------------------------------------------------------
Nilotinib$>$Dasatinib 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 1 1 1 1
Dasatinib$>$Nilotinib 2 0 2 2 2 2 2 0 3 2 2 2 2 2 0 3 2 2 2 2 2 0 3 2 2 2 2 2 0 3 3 1 1 1 1 1
----------------------- -------------------------------------------------------------------------
: [**The optimal treatment schedules with different drug toxicities.**]{}
\[tab:tox2\]
![[**Plot of cell number versus time for three monotherapies and optimal combination therapy for preexisting M351T, incorporating toxicity constraints.**]{} Here it is assumed that the ANC reduction rate during dasatinib treatment is higher than during nilotinib treatment.[]{data-label="fig:tox2"}](tox_dni.pdf)
### Multiple mutants preexisting before the initiation of therapy {#multiple-mutants-preexisting-before-the-initiation-of-therapy .unnumbered}
Lastly we investigate how much gain can be expected from combination therapy if more than one mutant type preexists before initiation of therapy. We again consider an optimization model over a 36 month horizon. We assume mutants M351T and F317L preexist therapy at a low level (each consists of $5\%$ of the total leukemic cell population); the initial conditions are given in Table \[tab:2mutants:init\].
[-0.5in]{}[0in]{}
Normal cell Wild-type M351T F317L
---- ----------------------- ----------------------- ----------------------- -----------------------
SC $7.34 \times 10^{4}$ $2.66 \times 10^{5}$ $1.48 \times 10^{4}$ $1.48 \times 10^{4}$
PC $1.61 \times 10^{7}$ $3.66 \times 10^{7}$ $2.04 \times 10^{6}$ $2.04 \times 10^{6}$
DC $3.24 \times 10^{9}$ $9.72 \times 10^{9}$ $5.40 \times 10^{8}$ $5.40 \times 10^{8}$
TC $3.24 \times 10^{11}$ $9.72 \times 10^{11}$ $5.40 \times 10^{10}$ $5.40 \times 10^{10}$
: [**The initial cell abundance.**]{}
\[tab:2mutants:init\]
The recommended combination therapy is the same as the recommended therapy when only one mutant F317L is present. The result is reasonable since the F317L has higher resistance to our therapies, and thus has a more significant impact on the structure of the optimal treatment schedule.
We now assume that the two mutants present are E255K and F317L. According to the *in vitro* IC50 value reported in [@redaelli2009activity], E255K is resistant to each drug. The recommended combination therapy is shown in Table \[tab:2mutants:therapy\] below. The combination therapy is different from the combination therapies proposed in the baseline model and the model with M351T and F317L, and is close to the monotherapy with dasatinib.
[-0.5in]{}[0in]{}
--------------- -------------------------------------------------------------------------
M351T 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1
F317L 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
M351T & F317L 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
F317L & E255K 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
--------------- -------------------------------------------------------------------------
: [**The optimal treatment schedules with two mutants.**]{}
\[tab:2mutants:therapy\]
We also compare the performance of the four different schedules in Fig \[fig:2mutants:therapy\]. The leukemic cell population is driven down in the first several months, but increases thereafter due to the increase of E255K population. Since E255K is resistant to each drug, even with the best therapy the leukemic cells still consist of over $73.5\%$ of the total cell population after 36 months. These results demonstrate that the optimal combination schedule is strongly dependent upon the specific type and combination of preexisting BCR-ABL mutants present at the start of therapy.
![[**The cell dynamics with F317L and E255K preexisting therapy.**]{}[]{data-label="fig:2mutants:therapy"}](p1_E255K_F317L.pdf)
Discussion {#discussion .unnumbered}
==========
In this work we have considered the problem of finding optimal treatment schedules for the administration of a variety of TKIs for treating chronic phase CML. We modeled the evolution of wild-type and mutant leukemic cell populations with a system of ordinary differential equations, and incorporated a dynamics model of patient ANC level to account for toxicity constraints. We then formulated an optimization problem to find the sequence of TKIs that lead to a minimal cancerous cell population at the end of a fixed time horizon of 36 months. The 36 month therapeutic horizon is clinically meaningful since it appears that the risk of therapeutic failure and disease progression to blast crisis is highest within the first two years from diagnosis [@druker2006five].
At first glance the optimization problem studied in this work (OTP) is quite challenging. It is a mixed-integer nonlinear optimization problem, where the nonlinear constraints are specified by the solution to a nonlinear system of differential equations. However, one factor mitigating the complexity of the problem is the assumption that the TKIs do not effect the stem cell compartment. This has the effect of making the evolution of the stem cell compartment independent of the TKI schedule chosen. In addition, the remaining layers in the cellular hierarchy are modeled by linear differential equations. We can thus numerically solve the differential equation governing the stem cell layer, and treat this function as an inhomogeneous forcing term in the linear differential equation governing the progenitor cells. This allows us to approximate the nonlinear constraints specified by the differential equations by linear constraints with high accuracy. Then the OTP problem can be approximated by a mixed-integer linear optimization problem, which we are able to solve with state-of-the-art optimization software CPLEX [@cplex126] within one hour.
[*Importance of minimizing progenitor cell population.*]{} We first aimed to minimize leukemic cell burden at 36 months after initiation of therapy, starting with an initial leukemic population of wild-type CML cells and either M351T (sensitive to all three therapies) or F317L (resistant to dasatinib) mutant leukemic cells. For both starting mutant populations, we observed that the optimal schedule involves initiating therapy with dasatinib and later switching to nilotinib, although the timing of the switch differed. To further understand this result, we noted that within this parameter regime, dasatinib is the most effective of the three TKIs at controlling leukemic progenitor cells, while nilotinib is the most effective at controlling the differentiated cells, which comprise most of the total leukemic burden. Thus, we note that controlling the leukemic progenitor cell population is important in long-term treatment outcome. This is further supported by the observation that blast crisis emerges due the acquisition of additional mutations in CML progenitor cells (not stem cells) [@jamieson2004granulocyte]. Our approach suggests that using combination TKI therapies may be a viable method of controlling this population. Our modeling suggests that it is best to reduce the progenitor cells early and then reduce the differentiated cells towards the end of the treatment planning horizon. An early reduction in progenitor cells pays off in later stages of the treatment planning horizon, since a small progenitor cell population results in a lower growth rate for differentiated cells which leads to a greater response to subsequent TKI therapy.
[*Effects of toxicity constraints.*]{} We also imposed a toxicity constraint on therapy optimization procedure by mandating that patient ANC levels stay above a given threshold that reduces the risk of infections. We observed that incorporating this toxicity constraint does impact the structure of the optimal schedules significantly, resulting in mandated treatment breaks as well as switching some months to imatinib therapy, which has a lower toxicity effect. We also noted that the choice of treatment breaks occurring also in one-month intervals may result in dangerous rebound of leukemic burden to levels close to pre-treatment, suggesting that shorter breaks to combat toxicity may be recommended. Although the model we have used for describing the dynamics of the ANC levels is simple, our findings demonstrate that incorporating a mechanistically modeled toxicity constraint into optimization of therapy scheduling is both feasible and important in determining optimal scheduling. [*Multiple preexisting mutant types.*]{} While some previous studies have suggested that the majority of CML patients are diagnosed with 0 or 1 preexisting BCR-ABL mutations, some patients do harbor multiple mutants at the start of therapy [@iqbal2013sensitive; @leder2011fitness]. Thus we also investigated the impact of having 2 mutant types present (M351T and F317, or E255K and F317L) at the start of therapy, on optimal schedules. We observed the number and specific combination of preexisting mutants present can significantly impact the optimization results. This points to the importance of determining which BCR-ABL mutations preexist in patients at diagnosis, before treatment planning is done.
Throughout this work we have observed that the structure of the optimal therapy depends heavily on model parameters, e.g., cellular growth rates and ANC decay rates. It is likely that each individual patients will have unique model parameters, and therefore a unique best schedule. An exciting application of this work would be the development of personalized optimal therapeutic schedules. Determination of (i) the mutant types (if any) present in a patient’s leukemic cell population, (ii) growth kinetics of their leukemic cell populations, and (iii) patient ANC level responses under various TKIs, would enable our optimization framework to build treatment schedules in a patient-specific setting.
Parameters {#sec:PARAM}
==========
In this section we describe the model parametrization for the examples shown above. A major source of our parameters is the work [@olshen2014dynamics] which statistically fit a hierarchical differential equation model (similar to ) to time series data of CML patents undergoing TKI therapy.
Stem cell kinetics {#appendix:sc}
------------------
- Density dependence parameters $\phi_i$ of type $i$ stem cell, for each $i$. We have $\phi_i =1/(1+p_i\sum_{i=1}^n x_{1,i}(t))$, with $p_1=(b^0_{1,1}/d^0_{1,1}-1)/K_1$, $p_2 = (b^0_{1,2}/d^0_{1,2}-1)/K_2$, and $p_i=p_2$ for $i\ge 3$. The values of $K_1$ and $K_2$ are given in Section \[appendix:initial\].
- The birth rates $b^j_{1,i}$. The estimates $b^j_{1,1}=0.008$ and $b^j_{1,i}=0.01$ for any cell type $i\ge 2$ and drug $j$. The value 0.01 is used in [@olshen2014dynamics] for the birth rate of leukemic stem cells without drug. We further assume that this value remains the same under any therapy, which is different from [@olshen2014dynamics].
- The death rates $d^j_{1,i}$. The estimate $d^j_{1,i}=0.0005$ for any $i$ and $j$, from [@olshen2014dynamics].
Progenitor cell kinetics
------------------------
- The death rates $d^j_{2,i}$. The estimates $d^1_{2,i}=0.0028$ and $d^2_{2,i}=0.0053$ for any $i$, from [@olshen2014dynamics]. The death rate of leukemic progenitor cells under high-dose imatinib ($800$ mg/day) is 0.0035 in [@olshen2014dynamics]. We consider imatinib with regular dose ($400$ mg/day) in this paper, so we set $d^3_{2,i}=0.0035/2=0.00175$ for any $i$. Note the death rates are the same across all cell types with the same therapy, but vary with different therapies. In addition, we set the death rate of normal progenitor cells $d^0_{2,i}=\min\{d^1_{2,i}, d^2_{2,i},d^3_{2,i}\}=0.00175$ for any $i$.
- The differentiation rates $b^j_{2,i}$.
- For normal cells, $b^j_{2,1}=0.35$ for any $j$.
- For wild type, $b^0_{2,2}=2b^j_{2,1} = 0.70$, $b^1_{2,2}=b^0_{2,2}/400=0.00175$, $b^2_{2,2}=b^0_{2,2}/200=0.0035$, and $b^3_{2,2}=b^0_{2,2}/400=0.00175$. All estimates are from [@olshen2014dynamics].
- For mutants, the differentiation rates are listed in Table \[table:mutant:rate\]. Since there are little *in vivo* data available in the literature related to leukemic mutant birth rate, our estimation is based on *in vitro* data for these mutants, in particular the IC50 values. We use a piecewise linear interpolation to estimate the differentiation rates, based on the relative IC50 values of mutants under nilotinib, dasatinib, and imatinib reported in [@redaelli2009activity]. For sensitive or moderately resistant mutants (the relative IC50 value is less than or equal to 4), the differentiation rate of mutant $i$ is estimated using the linear interpolation $$b^j_{2,i}=\text{relative IC50 value of mutant $i$ under drug $j$} \times b^j_{2,2}.$$ For resistant mutants (the relative IC50 value is between 4.01 and 10), the differentiation rate is estimated with the following linear interpolation: $$b^j_{2,i} = 0.9 b^{4}_{2,2} + \frac{0.1 b^4_{2.2}}{10-4.01} (\text{relative IC50 value of mutant $i$ under drug $j$} - 4.01).$$ Thus if the relative IC50 value for a resistant mutant is 4.01, then its differentiation rate is $90\%$ of the differentiation rate of the WT cell without any drug (0.7 per day); if the relative IC50 value for a resistant mutant is 10, then its differentiation rate is equal to the birth rate of the WT progenitor cell without drug. For highly resistant mutants (the relative IC50 is larger than 10), we set its differentiation rate to the differentiation rate of the WT progenitor cells without drug.
E255K E255V F317L M351T Y253F V299L
------------------------- -------- -------- --------- --------- --------- ---------
Nilotinib ($b^1_{2,i}$) 0.6614 0.7 0.00389 0.00077 0.00565 0.00235
Dasatinib ($b^2_{2,i}$) 0.6488 0.0120 0.6354 0.00308 0.00553 0.6843
Imatinib ($b^3_{2,i}$) 0.6536 0.7 0.00455 0.00308 0.00627 0.00270
: The differentiation rate of mutant progenitor cells under three drugs[]{data-label="table:mutant:rate"}
Differentiated cell kinetics
----------------------------
- The death rate $d^j_{3,i}$. The estimates $d^1_{3,i}=0.0442$ and $d^2_{3,i}=0.0394$ for any $i$, from [@olshen2014dynamics]. The death rate of leukemic differentiation cells under high-dose imatinib ($800$ mg/day) is $0.055$ in [@olshen2014dynamics]. We consider imatinib with regular dose ($400$ mg/day) in this paper, so we set $d^3_{3,i}=0.055/2=0.0275$ for any $i$. In addition, $d^0_{3,i}=\min\{d^1_{3,i}, d^2_{3,i},d^3_{3,i}\}=0.0275$ for any $i$.
- The differentiation rates $b^j_{3,i}$.
- For normal cells, $b^j_{3,1}=5.5$ for any $j$.
- For wild type, $b^0_{3,2}=1.5b^0_{3,1} = 8.25$, $b^1_{3,2}=b^0_{3,2}/600=0.01375$, $b^2_{3,2}=b^0_{3,2}/300=0.0275$, and $b^3_{3,2}=b^0_{3,2}/600=0.01375$. All estimates are from [@olshen2014dynamics].
- For the mutant, if it is sensitive or moderately resistant to drug $j$ (the relative IC50 value is less than or equal to 4), then $b^j_{3,3}= b^j_{2,3} \times b^j_{3,2}/b^j_{2,2}$, for $j=1,2,3$; otherwise $b^j_{3,3}= b^j_{2,3} \times b^4_{3,2}/b^4_{2,2}$, for $j=1,2,3$.
Terminally differentiated cell kinetics {#appendix:tc}
---------------------------------------
Using the estimates from [@olshen2014dynamics], we set the differentiation rates $b^j_{4,i}=100$ and death rates $d^j_{4,i}=1$ for any $i$ and $j$.
Initial cell populations at diagnosis {#appendix:initial}
-------------------------------------
The normal marrow output in an adult is approximately $3.5\times 10^{11}$ cells per day [@dingli2008chronic]. To achieve this equilibrium condition, we set $K_1 =8.75 \times 10^4$ in differential equations with parameters described in Sections \[appendix:sc\] to \[appendix:tc\] and in the absence of leukemic cells. To obtain an estimate of $K_2$, we assume that diagnosis of CML occurs once the leukemic cell burden reaches a threshold of $10^{12}$ cells [@holyoake2002elucidating], and that the differential equations have parameters described in Sections \[appendix:sc\] to \[appendix:tc\] and start with $K_1=8.75\times 10^4$ normal stem cells, one wild-type leukemic stem cell, and no other cells. We set $K_2=3\times 10^6$ so that the patient is diagnosed with CML around 78 months (6.5 years) after the first leukemic stem cell arises. At diagnosis, the normal stem cell, progenitor cell, differentiated cell, and terminally differentiated cell populations are $7.34\times 10^4$, $1.61\times 10^7$, $3.24\times 10^9$, and $3.24\times 10^{11}$, respectively; the leukemic stem cell, progenitor cell, differentiated cell, and terminally differentiated cell populations are $2.95\times 10^5$, $4.07\times 10^7$, $1.08\times 10^{10}$, and $1.08\times 10^{12}$ respectively. These are used as the initial cell populations for a patient diagnosed with CML.
ANC kinetics
------------
- We require the patient’s ANC cannot fall below ${L_{anc}}=1000/\mathrm{mm}^3$, the normal level of ANC is ${U_{anc}}=3000/\mathrm{mm}^3$, and the patient’s initial ANC is $3000/\mathrm{mm}^3$.
- It is observed that nilotinib has higher toxicity than imatinib [@cortes2010nilotinib]. We set the estimated monthly decrease rates of ANC to be $d_{anc,1}=350/\mathrm{mm}^3$ under nilotinib, $d_{anc,2}=300/\mathrm{mm}^3$ under dasatinib, and $d_{anc,3}=250/\mathrm{mm}^3$ under imatinib. The ANC of a patient increases by $b_{anc} = 2000/\mathrm{mm}^3$ during a drug holiday, before it reaches the normal level $3000/\mathrm{mm}^3$. We also investigate how optimal schedule is affected if dasatinib has a higher toxicity than nilotinib, with $d_{anc,1}=300/\mathrm{mm}^3$ and $d_{anc,2}=350/\mathrm{mm}^3$.
Method to solve the optimization model {#sec:MILO}
======================================
We describe the method to solve the optimization model introduced in Section \[sec:optmodel\]. Our strategy is to build a mixed-integer linear optimization model [@nemhauser1999integer] that approximates the optimization model , and then solve the approximation model to optimality numerically by off-the-shelf optimization software CPLEX [@cplex126]. The mixed-integer linear optimization model is built through two steps: (1) we first approximate the ODE constraints by bilinear constraints; (2) we then transform the bilinear constraints and nonlinear constraints into equivalent linear constraints, by adding auxiliary decision variables.
We first describe how to approximate the ODE constraints by bilinear constraints. Suppose patients take drug $j$ in month $m$. Since the cell dynamics are modeled by the following set of ODEs
\[eq:odeconcise\] $$\begin{aligned}
\dot{x}(t)=f^j(x(t)), \ & t\in [m{\Delta t}, (m+1){\Delta t}],\\
x(m{\Delta t}) = x^m, \ &\end{aligned}$$
the cell abundances in month $m+1$, $x^{m+1}$, are completely determined by the initial cell abundance $x^m$ and function $f^j$. Without loss of generality, we assume this relationship is described by $$\label{eq:odesol}
x^{m+1}_{l,i}=g^j_{l,i}(x^{m})$$ with some unknown nonlinear function $g^j_{l,i}:{\mathbb{R}}^{Ln} \rightarrow {\mathbb{R}}$, for each month $m$, layer $l$, and cell type $i$. Recall that $L$ is total number of cell layers ($L=4$), and $n$ is the total number of cell types. Then the ODE constraints are equivalent to the constraints below $$\label{eq:OTP:odenlp}
x^{m+1}_{l,i} = \sum_{j\in {\mathcal{J}}} z^{m,j}g^j_{l,i}(x^{m}), \text{ for each } m, l, i.$$ We will approximate the nonlinear function $g^j_{l,i}$ with an affine function $\hat{g}^{m,j}_{l,i}: {\mathbb{R}}^{Ln} \rightarrow {\mathbb{R}}$, for each $m,j,l$, and $i$. In particular, the function $$\label{eq:ode:affine}
\hat{g}^{m,j}_{l,i}(x)=a^{j,l,i}x+h^{m,j}_{l,i},$$ where $a^{j,l,i}$ is an $(Ln)$-dimensional vector and does not depend on $m$. Details of how $\hat{g}^{m,j}_{l,i}$ is constructed are provided in Section \[sec:LIN\_ODE\] of the Appendix. Let $a^{j,l,i}=[a^{j,l,i}_{1,1}, \ldots, a^{j,l,i}_{k,s}, \ldots, a^{j,l,i}_{L,n}]$. Then constraint can be approximated by the bilinear constraint $$\label{eq:OTP:bilinear}
x^{m+1}_{l,i} = \sum_{j\in {\mathcal{J}}} z^{m,j}\hat{g}^{m,j}_{l,i}(x^{m}) =\sum_{j\in {\mathcal{J}}} z^{m,j}(\sum_{k\in {\mathcal{L}}, s \in {\mathcal{I}}} a^{j,l,i}_{k,s} x^m_{k,s}+h^{m,j}_{l,i}),$$ for each type $i$ cell at layer $l$ in month $m$.
We now describe how to transform bilinear constraints and piecewise linear constraints into linear constraints. These are standard techniques in mixed-integer linear optimization [@nemhauser1999integer]. We introduce auxiliary continuous variables $v^{m,j}_{k,s}$, and set $v^{m,j}_{k,s}=z^{m,j}x^m_{k,s}$. Then bilinear constraints are transformed into the equivalent linear constraints below. $$\label{eq:OTP:odelin}
\begin{split}
x^{m+1}_{l,i} =& \sum_{j\in {\mathcal{J}}} (\sum_{k\in {\mathcal{L}}, s \in {\mathcal{I}}} a^{j,l,i}_{k,s} v^{m,j}_{k,s}+h^{m,j}_{l,i}z^{m,j})\\
0 \le &v^{m,j}_{k,s} \le U_{k,s}z^{m,j}, \\
0 \le & x^m_{k,s} - v^{m,j}_{k,s} \le U_{k,s}(1-z^{m,j}),
\end{split}$$ where $U_{k,s}$ is an upper bounds of cell abundance $x^m_{k,s}$ for each $m$. The value of $U_{k,s}$ can be obtained by taking the maximum value of layer $k$ type $s$ cell abundances over the whole planning horizon under all three monotherapies and no treatment. The piecewise linear constraints can be transformed into equivalent linear constraints below, by introducing auxiliary continuous variable $u^m$ and binary variable $q^m$ for each $m$.
\[eq:OTP:toxlin\] $$\begin{aligned}
u^{m+1} = y^{m} + {b_{anc}}- \sum_{j\in {\mathcal{J}}\setminus\{0\}} {d_{anc,j}}z^{m,j}, \\
y^{m+1} \ge u^{m+1} - b_{anc}q^{m+1},\\
y^{m+1} \ge {U_{anc}}- ({U_{anc}}- {L_{anc}}) (1-q^{m+1}), \\
y^{m+1} \le u^{m+1},\\
y^{m+1} \le {U_{anc}},\\
q^{m+1} \in \{0,1\}.\end{aligned}$$
Overall, the optimization model is approximated by the following mixed-integer linear optimization model.
\[eq:OTP:MILP\] $$\begin{aligned}
\min \ &\sum_{l \ge 1} \sum_{i\ge 2} x^M_{l,i}\\
\text{s.t.} \ & x^{m+1}_{l,i}= \sum_{j\in {\mathcal{J}}} \sum_{k\in {\mathcal{L}}, s \in {\mathcal{I}}} a^{j,l,i}_{k,s} v^{m,j}_{k,s}+\sum_{j\in {\mathcal{J}}}h^{m,j}_{l,i}z^{m,j}, &i\in {\mathcal{I}}, l\in {\mathcal{L}}, m \in{\mathcal{M}\setminus \{M\}}\\
& 0 \le v^{m,j}_{k,s} \le U_{k,s}z^{m,j}, &i\in {\mathcal{I}}, l\in {\mathcal{L}}, m \in{\mathcal{M}\setminus \{M\}}\\
& 0 \le x^m_{k,s} - v^{m,j}_{k,s} \le U_{k,s}(1-z^{m,j}), &i\in {\mathcal{I}}, l\in {\mathcal{L}}, m \in{\mathcal{M}\setminus \{M\}}\\
&u^{m+1} = y^{m} + {b_{anc}}- \sum_{j\in {\mathcal{J}}\setminus\{0\}} {d_{anc,j}}z^{m,j}, & m\in {\mathcal{M}\setminus \{M\}}\\
&y^{m+1} \ge u^{m+1} - b_{anc}q^{m+1}, & m \in {\mathcal{M}\setminus \{M\}}\\
&y^{m+1} \ge {U_{anc}}- ({U_{anc}}- {L_{anc}}) (1-q^{m+1}), & m\in {\mathcal{M}\setminus \{M\}}\\
&y^{m+1} \le u^{m+1}, & m\in {\mathcal{M}\setminus \{M\}}\\
&y^{m+1} \le {U_{anc}}, & m\in {\mathcal{M}\setminus \{M\}}\\
& y^m \ge {L_{anc}}, & m\in {\mathcal{M}}\\
& \sum_{j\in {\mathcal{J}}} z^{m,j} = 1, &m\in {\mathcal{M}\setminus \{M\}}\\
& z^{m,j}, q^{m+1}\in \{0,1\}, & m\in {\mathcal{M}\setminus \{M\}}, j\in {\mathcal{J}}\\
& x(0) = x^0, \; y^0 \text{ is given}.\end{aligned}$$
Linear approximation to the solutions of the ODEs {#sec:LIN_ODE}
=================================================
In this section, we describe how to construct the affine function $\hat{g}^{m,j}_{l,i}$ in in Section \[sec:MILO\] of the Appendix. If we assume that the drugs do not affect stem cells, we can compute the abundance of stem cells over the planning horizon numerically in advance, regardless of the treatment schedules. Thus we assume $x_{1,1}(t), \ldots, x_{1,n}(t)$ are given as data, for any $t$. We can first eliminate all the variables $x^m_{1,i}$ and constraints containing $x^m_{1,i}$, for each $m$ and $i$, in the optimization problem . The dynamics of wild-type leukemic cells and each mutant type have no impact on each other. We can decouple the ODEs into a series of linear ODEs as follows, each describing the dynamics for type $i$ cell from layer 2 to layer 4. $$\label{eq:ode:lineage}
\left[
\begin{array}{c}
\dot{x}_{2,i}(t) \\
\dot{x}_{3,i}(t) \\
\dot{x}_{4,i}(t)\\
\end{array}
\right]
=
\left[
\begin{array}{ccc}
-d^j_{2,i} & 0 & 0 \\
b^j_{3,i} & -d^j_{3,i} & 0\\
0 & b^j_{4,i} & -d^j_{4,i}\\
\end{array}
\right]
\left[
\begin{array}{c}
x_{2,i}(t) \\
x_{3,i}(t) \\
x_{4,i}(t)\\
\end{array}
\right]
+
\left[
\begin{array}{c}
b^j_{2,i} x_{1,i}(t) \\
0 \\
0\\
\end{array}
\right]$$
Write the above equations in the matrix form, we have $$\label{eq:ode:matrix}
\dot{v}_i(t) = W^j_i v_i(t) + w^j_i(t), \text{ for } t\in [m{\Delta t}, (m+1){\Delta t}],$$ where $v_{i}(t)=[x_{2,i}(t), x_{3,i}(t), x_{4,i}(t)]^{\top}$, $w^j_i(t)=[b^j_{2,i}x_{1,i}(t), 0,0]^{\top}$, and $W^j_i$ is the lower triangular matrix in .
We divide $(m{\Delta t}, (m+1){\Delta t})$ into ${\Delta t}=30$ one-day sub-intervals. Consider a sub-interval $(t_0, t_0+1)$. By assuming $w^j_i(t) = w^j_i(t_0)$ for any $t\in (t_0, t_0+1)$, we solve approximately and obtain $$~\label{eq:ode:oneday}
v_i(t_0+1) \approx e^{W^j_i}v_i(t_0) + (e^{W^j_i} - I)(W^j_i)^{-1} w^j_i(t_0).$$ By combining equations for $t_0=m{\Delta t}, m{\Delta t}+1,\ldots, (m+1){\Delta t}-1$, we have $$\label{eq:linapprox}
v_i((m+1){\Delta t}) = e^{W^j_i {\Delta t}}v_i(m {\Delta t}) + \sum_{d=0}^{{\Delta t}-1}e^{W^j_i({\Delta t}-1-d)}(e^{W^j_i} - I)(W^j_i)^{-1} w^j_i(m {\Delta t}+d).$$
Recall that $v_{i}(m{\Delta t})=[x^{m}_{2,i}, x^{m}_{3,i}, x^m_{4,i}]^{\top}$ for each $m$. Thus can be rewritten as $$\label{eq:linapprox:concise}
\left[
\begin{array}{c}
x^{m+1}_{2,i}\\
x^{m+1}_{3,i} \\
x^{m+1}_{4,i}
\end{array}
\right]
= A^{j,i}
\left[
\begin{array}{c}
x^{m}_{2,i}\\
x^{m}_{3,i} \\
x^{m}_{4,i}
\end{array}
\right]
+ h^{m,j}_i,$$ where $A^{j,i}=(e^{W^j_i})^{{\Delta t}}$ and $h^{m,j}_i= \sum_{d=0}^{{\Delta t}-1}(e^{W^j_i})^{{\Delta t}-1-d}(e^{W^j_i} - I)(W^j_i)^{-1} w^j_i(m{\Delta t}+d)$. Each equation in is used as the affine function $\hat{g}^{m,j}_{l,i}$ in , for each $m,j,i$, and $l=2,3,4$.
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|
---
abstract: |
Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information about $G$ is encoded in all the twisted group rings of $G$ over $R$.
We investigate this problem over fields. We start with abelian groups and show how the results depend on the characteristic of $R$. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when $R$ is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.
address:
- 'Departament of Mathematics, Free University of Brussels, 1050 Brussels, Belgium'
- 'Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel'
author:
- Leo Margolis
- Ofir Schnabel
bibliography:
- 'TGRIP2.bib'
title: The twisted group ring isomorphism problem over fields
---
[^1]
**2010 Mathematics Subject Classification:** 16S35, 20C25, 20K35.
Introduction {#Intro}
============
In [@MargolisSchnabel] we proposed a twisted version of the celebrated group ring isomorphism problem (GRIP), namely “the twisted group ring isomorphism problem”(TGRIP).
Recall that for a finite group $G$ and a commutative ring $R$, the group ring isomorphism problem asks whether the ring structure of $RG$ determines $G$ up to isomorphism. In other words, does, $RG\cong
RH$ imply $G\cong H$ for groups $G$ and $H$? Roughly speaking the twisted group ring isomorphism problem asks if for a group $G$ and a commutative ring $R$, the ring structure of all the twisted group rings of $G$ over $R$ determines the group $G$. The role twisted group rings of $G$ over $R$ play for the projective representation theory is in many ways the same played by the group ring $RG$ for the representation theory of $G$ over $R$, as it was shown in the ground laying work of I. Schur [@Schur]. In this sense the (TGRIP) can also be understood as a question on how strongly the projective representation theory of a group influences its structure. For results on the classical (GRIP) see [@RoggenkampScott; @Sehgal1993; @Hertweck] for the case $R = \mathbb{Z}$. Also questions on character degrees, as addressed e.g. in [@Isaacs; @Navarro], can be viewed as results for the case $R = \mathbb{C}$.
We denote by $R^*$ the unit group in a ring $R$. For a $2$-cocycle $\alpha \in Z^2(G, R^*)$ the twisted group ring $R^\alpha G$ of $G$ over $R$ with respect to $\alpha$ is the free $R$-module with basis $\{u_g\}_{g \in G}$ where the multiplication on the basis is defined via $$u_g u_h = \alpha(g, h) u_{gh} \ \ \text{for all} \ \ g,h \in G$$ and any $u_g$ commutes with the elements of $R$. Notice that if we consider $\alpha$ only as a function (not necessarily a $2$-cocycle) from $G\times G$ to $R^*$, then $R^\alpha G$ is associative if and only if $\alpha$ is a $2$-cocycle. The ring structure of $R^{\alpha}G$ depends only on the cohomology class $[\alpha]\in
H^2(G,R^*)$ of $\alpha$ and not on the particular $2$-cocycle. Notice that the ring $R$ is central in the twisted group ring $R^{\alpha}G$ and correspondingly the associated second cohomology group is with respect to a trivial action of $G$ on $R^*$. See [@KarpilovskyProjective Chapter 3] for details.
Let $G$ and $H$ be groups and let $R$ be a commutative ring. We define an equivalence relation which corresponds to the regular (GRIP) by $G \Delta_R H$ if and only if $RG \cong RH$, and the twisted problem is defined using a refinement of this relation as follows.
Let $R$ be a commutative ring and let $G$ and $H$ be finite groups. We say that $G \sim_R H$ if there exists a group isomorphism $$\psi :H^2(G,R^*)\rightarrow H^2(H,R^*)$$ such that for any $[\alpha] \in H^2(G,R^*)$, $$R^{\alpha}G\cong R^{\psi (\alpha)}H.$$
It is easy to see that $\sim_R$ is indeed a refinement of $\Delta_R$, cf. Corollary \[cor:1dimtrivcoho\]. The main problem we are interested in is the following.
For a given commutative ring $R$, determine the $\sim_R$-classes. Answer in particular, for which groups $G \sim_R H$ implies $G \cong H$.
In [@MargolisSchnabel] we investigated (TGRIP) over the complex numbers and gave some results for families of groups, e.g. abelian groups, $p$-groups, groups of central type and groups of cardinality $p^4$ and $p^2q^2$ for $p,q$ primes. In this paper we investigate (TGRIP) and related problems for fields other than $\mathbb{C}$. In particular, our main motivation is to explore:
1. The differences between the (TGRIP) and the (GRIP).
2. The differences between the (TGRIP) over $\mathbb{C}$ and the (TGRIP) over other fields.
For example we showed in [@MargolisSchnabel Lemma 1.2] that any abelian group is a $\sim_{\mathbb{C}}$-singleton which is clearly not true for $\Delta_{\mathbb{C}}$. We show that over other fields $F$, abelian groups are no longer necessarily $\sim_{F}$-singletons (see Example \[ex:C9\]). This is particularly interesting since, when $\text{char}(F)$ does not divide $|G|$, i.e. the semi-simple case, $G \Delta_{F} H$ implies $G \Delta_{\mathbb{C}} H$, while we show that $G \sim_{F} H$ not necessarily implies $G
\sim_{\mathbb{C}} H$. In this sense, $\mathbb{C}$ is no longer “the worst" field in distinguishing between groups in the semi-simple case.
A main result is related to the so called Dade’s Example. In [@Dade] E. Dade gave a family of examples of non-isomorphic groups $G$ and $H$ of order $p^3q^6$ for $p,q$ primes satisfying some arithmetic conditions, such that $FG \cong FH$ for any field $F$ while $\mathbb{Z}G\not \cong \mathbb{Z}H$. Consequently, the ring structure of all the group rings of a group over all fields is not sufficient to determine the group up to isomorphism. We prove:
Let $G$ and $H$ be the groups from Dade’s example of even order. Then there exists an infinite number of fields $F$ such that $G \not
\sim_{F} H$.
A key ingredient in the proof of Theorem 1, and in general for studying the ring structure of twisted group rings over fields, is a generalization of the Schur cover which we develop in Section \[UnSchur\]. This generalization exists also when the field is not algebraically closed. The idea for this kind of cover was introduced originally by Yamazaki [@YamazakiUnschur] and for this reason we call it a *Yamazaki cover*. This object generalizes the Schur cover of a group $G$ in the sense that over not necessarily algebraically closed fields, any projective representation of $G$ is projectively equivalent to a linear representation of its Yamazaki cover.
In Theorem \[th:YamazakiGT\] we give a group theoretical criterion how a Yamazaki cover of a group can be recognized. This mimics the theorem that for given $G$ any group containing a group of order $|H^2(G, \mathbb{C}^*)|$ in the intersection of the center and commutator subgroup is a Schur cover of $G$, but for the Yamazaki cover more conditions need to be checked. After the construction of Yamzaki covers for both groups from Dade’s example we prove Theorem 1.
As mentioned above, a Yamazaki cover may exist when the field $F$ is not necessarily algebraically closed. Throughout this paper, for a finite group $G$ and a field $F$, we will assume that $H^2(G,F^*)\cong H^2(G,t(F^*))$. It turns out that this is a sufficient (and necessary) condition for the existence of a Yamazaki cover of $G$ over $F$. Here, $t(F^*)$ denotes the torsion subgroup of $F^*$. It was shown by Yamazaki that this condition is equivalent to $F^* = (F^*)^{\exp(G/G')}t(F^*)$ [@YamazakiUnschur Proposition 3].
For example, for any finite group $G$, the field $F$ can be the complex numbers, the real numbers or any finite field. However, for any non-trivial $G$ we cannot choose $F$ to be the rational numbers.
The following problem is natural in view of Theorem 1.
Let $G$ and $H$ be groups such that $G \sim_{F} H$ for all fields $F$.
1. Is it true that $G$ and $H$ are necessarily isomorphic?
2. Find families of groups for which the answer to the question above is positive.
An example of such a family are the abelian groups. In fact, if two abelian groups $G$ and $H$ satisfy $\mathbb{C}G\cong
\mathbb{C}H$ and $H^2(G,\mathbb{C}^*)\cong H^2(H,\mathbb{C}^*)$ then $G \cong H$ (see [@MargolisSchnabel Lemma 1.2]). Moreover, it is clear that the above two conditions, namely isomorphic group rings and isomorphic second cohomology groups, are necessary for groups in order to be in the twisted relation. In [@MargolisSchnabel Examples 3.2, 3.5] we proved that for non-abelian groups the combination of these two conditions is not sufficient even to imply that $G \sim_{\mathbb{C}} H$. Here we prove the following
1. Let $G$ and $H$ be finite abelian groups. Assume there exists a field $F$ of characteristic zero which satisfies $FG\cong
FH$ and $H^2(G,t(F^*))\cong H^2(G,F^*)\cong H^2(H,F^*)\cong
H^2(H,t(F^*)).$ Then $G$ and $H$ are isomorphic.
2. There exist non-isomorphic abelian groups $G$ and $H$ and a finite field $F$ such that $FG$ is semisimple and $G \sim_{F} H$. In particular, $\text{char}(F) \nmid |G|$ does not imply that $\sim_{F}$ is a refinement of $\sim_{\mathbb{C}}$.
3. There exist abelian groups $G$ and $H$ and a finite field $F$ such that $FG \cong FH$ and $H^2(G,F^*) \cong H^2(H,F^*)$, but $G \not \sim_{F} H$.
The paper is organized as follows. Most of Section \[pre\] is devoted to well-known definitions and tools related to twisted group rings and the second cohomology group of a finite group. However, we also prove in Proposition \[prop:abcomesfromEXT\] an interesting result about simple commutative components of twisted group rings. In Section \[abeliangroups\] we deal with the twisted relation for abelian groups. In particular we prove Theorem 2. In Section \[UnSchur\] we introduce and construct the Yamazaki cover of a group which is a generalization of a Schur cover of a group which exists also when $F$ is not algebraically closed. Lastly, in Section \[Dade\] we prove Theorem 1 by constructing the Yamazaki covers for the groups from Dade’s example and then evaluating their Wedderburn decompositions.
Preliminaries {#pre}
=============
In this section we will recall some definitions and tools that will be useful later on. Recall that throughout this paper we will assume for a finite group $G$ and a field $F$ that $H^2(G,F^*)\cong H^2(G,t(F^*))$, although it is sometimes redundant.
Clearly two main objects that we need to understand in order to study the (TGRIP) are the ring structure of twisted group rings, and the structure of the second cohomology group of a finite group.
We use standard group theoretical notation. In particular we denote by $C_n$ a cyclic group of order $n$, by $\circ(g)$ the order of a group element $g$ in a group $G$, by $Z(G)$ the center and by $G'$ the commutator subgroup of $G$, by $\operatorname{exp}(G)$ the exponent of $G$, by $\operatorname{GL}(V)$ the general linear group acting on a vector space $V$ and by $\operatorname{PGL}(V)$ the projective general linear group, i.e. $\operatorname{GL}(V)/Z(\operatorname{GL}(V))$. Moreover for an abelian group $G$ we denote by $\text{rk}(G)$ the rank of $G$, i.e. the minimal number of generators of $G$. We denote by $\mathbb{F}_q$ a finite field of order $q$.
Projective representations and twisted group rings {#sec:ProjRep}
--------------------------------------------------
The theory presented here is standard and can be found e.g. in [@KarpilovskyProjective Chapter 3]. A [*projective representation*]{} of a group $G$ over a field $F$ is a map $$\eta: G\rightarrow GL(V),$$ where $V$ is an $F$-vector space, such that the composition of $\eta$ with the natural projection from $GL(V)$ to $PGL(V)$ is a group homomorphism. As in the ordinary case, two projective representations are equivalent if they differ by a basis change of $V$. A projective representation $\eta: G\rightarrow GL(V)$ is [*irreducible*]{} if $V$ admits no proper $G$-subspace. Two projective representations $\eta_1: G \rightarrow GL(V_1)$ and $\eta_2: G \rightarrow GL(V_2)$ are called *projectively equivalent* if there is a map $\mu: G \rightarrow F^*$ satisfying $\mu(1) = 1$ and a vector space isomorphism $f:V_1 \rightarrow V_2$ such that $$\eta_1(g) = \mu(g) f^{-1} \eta_2(g) f$$ for every $g \in G$.
With the above notation, we can define $\alpha \in Z^2(G,F^*)$ by $$\alpha(g_1,g_2)=\eta (g_1) \eta(g_2) \eta (g_1g_2)^{-1},$$ and refer to $\eta$ as an $\alpha$-representation of $G$. For a fixed 2-cocycle $\alpha$, the set of projective equivalence classes of irreducible $\alpha$-representations of $G$ is denoted by $\text{Irr}(G,\alpha)$. As in the ordinary case, there is a natural correspondence between projective representations of $G$ over $F$ with an associated 2-cocycle $[\alpha]$, and $F^\alpha G$-modules.
A projective representation $\eta: G\rightarrow GL(V)$ can be extended to a homomorphism of algebras $$\begin{array}{rcl}
\tilde{\eta}: F^\alpha G & \rightarrow &\operatorname{End}_F(V)\\
\sum_{g\in G} a_g u_g & \mapsto &\sum_{g\in G} a_g \eta (g).
\end{array}$$ For any ring $R$ and an irreducible $R$-module $M$, there is a surjective ring homomorphism $R\rightarrow \operatorname{End}_D M$ for $D=\operatorname{End}_R M$. A generalized Maschke’s theorem states that if $\text{char}(F) \nmid |G|$ then any twisted group algebra $F^{\alpha}G$ is semisimple. Therefore, with the above notations for any irreducible $\alpha$-representation $V$ of $G$, the ring $\operatorname{End}_D V$ can be identified with one of the components of the Artin-Wedderburn decomposition of the semisimple algebra $F^{\alpha}G$. In other words, $F^{\alpha}G$ admits a decomposition $$F^{\alpha}G=\bigoplus_{[W]\in\text{Irr}(G,\alpha)}\operatorname{End}_{D_W}(W),$$ where $D_W=\operatorname{End}_{F^{\alpha}G} W$. In particular, if $F$ is a finite field such that $\text{char}(F) \nmid |G|$ then $$F^{\alpha}G=\bigoplus_{[W]\in\text{Irr}(G,\alpha)}\operatorname{End}_{F_W}(W),$$ where here $F_W$ is a field extension of $F$ corresponding to $W$.
In some of our examples later on we will use the structure of the center of a twisted group algebra. Let $G$ be a finite group and let $\alpha \in Z^2(G,F^*)$. An element $g\in G$ is called $\alpha$-regular if $\alpha(g,h)=\alpha(h,g)$ for any $h\in G$ which commutes with $g$. Note that if $g$ is $\alpha$-regular and $\beta \in Z^2(G, F^*)$ such that $[\alpha] = [\beta]$ in $H^2(G,F^*)$ then $g$ is also $\beta$-regular. The following is well known (see e.g [@NauwelaertsVanOystayen Theorem 2.4]).
\[lemma:centeroftga\] Let $G$ be a finite group, let $\alpha \in Z^2(G,F^*)$, let $g\in
G$ be an $\alpha$-regular element and let $T$ be a transversal of the centralizer of $g$ in $G$. Then
1. The element $$S_g=\sum _{t\in T} u_tu_gu_t^{-1}$$ is a central element in $F^{\alpha}G$.
2. The elements $S_g$, where $g$ runs over all the $\alpha$-regular conjugacy classes in $G$, form an $F$-basis for the center of $F^{\alpha}G$.
The second cohomology group of a finite group
---------------------------------------------
The second cohomology group of a group $G$ over the complex numbers in denoted by $M(G)$ and is called the *Schur multiplier*. An important tool to understand $H^2(G,F^*)$ is the following exact sequence (see [@KarpilovskyVolII Theorem 11.5.2]) $$\label{eq:UCT}
1\rightarrow \operatorname{Ext}(G/G',F^*) \rightarrow
H^2(G,F^*)\rightarrow \operatorname{Hom}(M(G),F^*)\rightarrow 1.$$ Moreover, this sequence splits (not canonically). Here, for abelian groups $G,A$ $$\text{Ext}(G,A)=\{[\alpha]\in H^2(G,A) \ | \ \alpha \text{ is symmetric}\},$$ where a cocycle $\alpha \in Z^2(G,A)$ is called *symmetric* if $\alpha (x,y)=\alpha (y,x)$ for all $x,y\in G$ (see [@KarpilovskyProjective Chapter 2, §1]). Notice that $\operatorname{Ext}(G,A)$ corresponds to equivalence classes of abelian central extensions of a group $G$ by a group $A$. The map in from $\operatorname{Ext}(G/G',F^*)$ to $H^2(G,F^*)$ is the restriction of the inflation map hereby explained. Let $G$ be a finite group with normal subgroup $N$, let $A$ be an abelian group and let $\varphi: G\rightarrow G/N$ be the quotient map. Then, for any $\beta \in Z^2(G/N,A)$ we can define $\alpha \in Z^2(G,A)$ by $$\alpha (x,y)=\beta(\varphi (x),\varphi (y)).$$ The map from $Z^2(G/N,A)$ to $Z^2(G,A)$ sending $\beta$ to $\alpha$ induces a map $$\text{inf}:H^2(G/N,A)\rightarrow H^2(G,A)$$ which is called the *inflation map*. The map in from $\operatorname{Ext}(G/G',F^*)$ to $H^2(G,F^*)$ is the restriction to the subgroup $\operatorname{Ext}(G/G',F^*)$ of the inflation map from $H^2(G/G',F^*)$ to $H^2(G,F^*)$. In the sequel we will sometimes abuse notations and denote the image of this map in $H^2(G,F^*)$ as $\operatorname{Ext}(G/G',F^*)$ and its complement in $H^2(G,F^*)$ by $\operatorname{Hom}(M(G),F^*)$
For the sake of completeness and for later use, before going forward with the description of the second cohomology group, we would like to introduce a third map which is associated to the second cohomology group. Let $$1\rightarrow
N \rightarrow H\overset{\alpha}\rightarrow G\rightarrow 1$$ be a central extension, let $\mu$ be a section of $\alpha$ and define $f\in Z^2(G,N)$ by $f(x,y)=\mu (x) \mu (y) \mu (xy)^{-1}$. Then, for any abelian group $A$ and any $\chi \in \operatorname{Hom}(N,A)$ we have $\chi
\circ f \in Z^2(G,A)$ and the cohomology class $[\chi \circ f]$ does not depend on the choice of $\mu$.
\[def:Tra\] With the above notation, the map $\operatorname{Tra}: \operatorname{Hom}(N,A)\rightarrow
H^2(G,A)$ defined by $\chi \mapsto [\chi \circ f]$ is called the *transgression map*.
We like to point out that the three maps mentioned above, inflation, restriction and transgression, are connected to each other as demonstrated in the celebrated Hochschild and Serre exact sequence.
Now recall that (see e.g. [@KarpilovskyProjective Corollary 2.3.17]) for any natural numbers $n_1$,...,$n_r$ $$\label{eq:EXTdecomposition}
\operatorname{Ext}(\Pi _{i=1}^rC_{n_r}, F^*) \cong \Pi _{i=1}^r
\operatorname{Ext}(C_{n_r}, F^*).$$ Therefore, in order to understand $\operatorname{Ext}(G/G',F^*)$ it is sufficient to understand the description of $\operatorname{Ext}(C_n,F^*)\cong H^2(C_n,F^*)$. This is well known (see e.g. [@KarpilovskyProjective Theorem 1.3.1]): $$\label{eq:cohoofcyclic}
\operatorname{Ext}(C_n,F^*)\cong H^2(C_n,F^*)\cong F^*/(F^*)^n.$$ Notice that by our assumption that always $H^2(G,F^*)\cong
H^2(G,t(F^*))$, we deduce that $H^2(C_n,F^*)\cong F^*/(F^*)^n\cong
t(F^*)/t(F^*)^n$. This is a finite cyclic group for any field $F$ as any two elements $a,b \in t(F^*)$ generate a finite, and hence cyclic, group and so also $\langle a, b \rangle / \langle a,b \rangle^n$ is cyclic.
We will use the above to recall the known structure of the second cohomology group of abelian groups (see e.g. [@YamazakiAbelian Corollary in §2.2]).
Let $G$ be an abelian group. Then $G$ admits a decomposition $$\label{eq:decompofabeliangroupptok}
G=C_{n_1}\times C_{n_2}\times \ldots \times C_{n_r}\cong \langle
x_1 \rangle \times \langle x_2 \rangle\times \ldots \times \langle
x_r \rangle$$ such that $n_i$ is a divisor of $n_{i+1}$ for any $1\leq i\leq
r-1$. Clearly, $$\label{eq:STforAB}
\operatorname{Ext}(G/G',F^*) \cong \prod _{i=1}^r F^*/(F^*)^{n_i}.$$ We want to describe $\operatorname{Hom}(M(G),F^*)$. First notice, that if $g$ and $h$ are commuting elements in a group $G$ with orders $n$ and $m$ correspondingly, then $[u_g,u_h]=\lambda$ in the twisted group algebra $ F^{\alpha}G$, and $\lambda$ is a root of unity dividing $\gcd(m,n)$. This follows directly from the fact that for any $x\in G$ the element $u_x^{\circ (x)}$ is central in $
F^{\alpha}G$ and therefore $[u_g^{\circ (g)},u_h]=\lambda ^{\circ
(g)}=1$. Now, for any natural numbers $n$ and $m$ denote by $d(n,m,F)$ the maximal order of a root of unity in $F$ which divides the greatest common divisor of $m$ and $n$. If $m$ is a divisor of $n$, we denote $d(n,m,F)$ by $d(m,F)$. By the above, for $G$ as in , $$\label{eq:NAforA}
\operatorname{Hom}(M(G),F^*)\cong \prod
_{i=1}^{r-1}C_{d(n_i,F)}^{r-i},$$ generated by the tuple of functions $$\left(\alpha _{ij}\right)_{1\leq i<j\leq r},$$ where $\alpha _{ij}(x_i,x_j)$ is a primitive $d(n_i,F)$-th root of unity and $1$ elsewhere. From , and , for $G$ as in we have $$\label{eq:cohomologyofABgroup}
H^2(G,F^*)\cong \left(\prod _{i=1}^r F^*/(F^*)^{n_i}\right) \times
\left(\prod _{i=1}^{r-1}C_{d(n_i,F)}^{r-i}\right).$$ As a consequence of the above, over the complex numbers, non-isomorphic abelian groups of the same cardinality admit non-isomorphic cohomology groups (see [@Schur] or [@KarpilovskyProjective Corollary 2.3.16]).
Commutative components of twisted group rings
---------------------------------------------
In this section we study twisted group rings admitting a commutative component in their Wedderburn decomposition. We start with a straightforward result.
\[lemma:GRNOTGR\] Let $G$ be a group, $R$ a commutative ring and let $\alpha \in Z^2(G,R^*)$. If there exists an $\alpha$-projective representation of dimension $1$, then $\alpha$ is cohomologicaly trivial.
This is clear by the definition of co-boundary.
\[cor:1dimtrivcoho\] Let $G$ and $H$ be groups, let $R$ be a commutative ring and let $\alpha \in Z^2(G,R^*)$. Then $R^{\alpha}G$ admits a $1$-dimensional simple module if and only if $\alpha$ is cohomologically trivial. In particular, $\sim_R$ is a refinement of $\Delta_R$.
We wish to generalize this result to commutative components with dimension not necessarily $1$ over fields.
\[prop:abcomesfromEXT\] Let $G$ be a group, let $F$ be a field such that $\text{char}(F) \nmid |G|$ and let $[\alpha]\in H^2(G,F^*)$. Then $F^{\alpha}G$ admits a commutative simple component if and only if $[\alpha]$ is in the image of the inflation map from $\operatorname{Ext}(G/G',F^*)$ to $H^2(G,F^*)$ as defined in Section \[sec:ProjRep\].
Denote by $\bar{F}$ the algebraic closure of $F$. Consider the following commutative diagram related to the exact sequence in . Here the vertical maps are just obtained by understanding elements of $Z^2(G, F^*)$ as elements of $Z^2(G, \bar{F}^*)$.
(triv1F)[$1$]{}; (EXT\_F) \[right of=triv1F\] [$\operatorname{Ext}(G/G',F^*)$]{}; (H2F) \[right of=EXT\_F\] [$H^2(G,F^*)$]{}; (HOM\_F) \[right of=H2F\] [$\operatorname{Hom}(M(G),F^*)$]{}; (triv2F)\[right of=HOM\_F\][$1$]{}; (triv1Fclosed)\[below of=triv1F\] [$1$]{}; (EXT\_Fclosed) \[right of=triv1Fclosed\] [$\operatorname{Ext}(G/G',\bar{F}^*)$]{}; (H2Fclosed) \[right of=EXT\_Fclosed\] [$H^2(G,\bar{F}^*)$]{}; (HOM\_Fclosed) \[right of=H2Fclosed\] [$\operatorname{Hom}(M(G),\bar{F}^*)$]{}; (triv2Fclosed)\[right of=HOM\_Fclosed\][$1$]{};
(triv1F) to node \[above\] [$$]{} (EXT\_F); (EXT\_F) to node \[above\] [$\text{inf}$]{} (H2F); (H2F) to node \[above\] [$d$]{} (HOM\_F); (HOM\_F) to node \[above\] [$$]{} (triv2F);
(triv1Fclosed) to node \[above\] [$$]{} (EXT\_Fclosed); (EXT\_Fclosed) to node \[above\] [$\text{inf}$]{} (H2Fclosed); (H2Fclosed) to node \[above\] [$d$]{} (HOM\_Fclosed); (HOM\_Fclosed) to node \[above\] [$$]{} (triv2Fclosed);
(EXT\_F) to node \[left\] [$ $]{} (EXT\_Fclosed); (H2F) to node \[left\] [$ $]{} (H2Fclosed); (HOM\_F) to node \[left\] [$ $]{} (HOM\_Fclosed);
Assume first that $[\alpha]$ is in the image of the inflation map from $\operatorname{Ext}(G/G',F^*)$ to $H^2(G,F^*)$ and denote its (unique) pre image in $\operatorname{Ext}(G/G',F^*)$ by $[\beta]$. Then, since $\operatorname{Ext}(G/G',\bar{F}^*)$ is trivial, $[\beta]$ is also trivial as an element of $\operatorname{Ext}(G/G',\bar{F}^*)$ and therefore $\gamma :=\text{inf}( [\beta])$ is the trivial cohomology class in $H^2(G,\bar{F}^*)$. Hence $\bar{F}^{\gamma}G\cong \bar{F}G$ admits $\bar{F}$ as a simple component. Now, since $\bar{F}^{\gamma}G\cong F^{\alpha }G\otimes _F \bar{F}$ we conclude that $F^{\alpha }G$ admits a commutative simple component.
Conversely, assume that $F^{\alpha }G$ admits a commutative simple component. Let $[\gamma]$ be the cohomology class in $H^2(G, \bar{F}^*)$ obtained from $[\alpha]$. Then, $F^{\alpha }G \otimes _F \bar{F}\cong \bar{F}^{\gamma}G$ also admits a commutative simple component. However, since $\bar{F}$ is algebraically closed this component is $\bar{F}$ itself. Consequently, by Corollary \[cor:1dimtrivcoho\] $[\gamma]$ is the trivial cohomology class. Clearly from the diagram above $[\alpha]$ is in the image of the inflation map from $\operatorname{Ext}(G/G',F^*)$ to $H^2(G,F^*)$
Abelian groups {#abeliangroups}
==============
The main result of this section is Theorem $2$. The proof is done is three steps. In Theorem \[th:AbelianIso\] we prove Theorem 2(1), Example \[ex:C9\] shows Theorem 2(2) and lastly, Proposition \[prop:notinrelation\] gives Theorem 2(3).
In a way, the group ring isomorphism problem asks whether it is possible to distinguish groups by their group ring structure over a commutative ring $R$. For this purpose it is clear that the ring of integers is “the best” ring since for any commutative ring $R$ and finite groups $G$ and $H$ the isomorphism $\mathbb{Z}G\cong \mathbb{Z}H$ implies that $RG \cong RH$. Also, in a sense, in the semi-simple case, the field of complex numbers is “the worst” commutative domain in the sense that if $F$ is a commutative domain, $G$ and $H$ are finite groups such that $FG\cong FH$ is semi-simple then $\mathbb{C}G\cong \mathbb{C}H$. This follows from the fact that if $\bar{F}$ denotes the algebraic closure of the quotient field of $F$ then $\bar{F}G \cong \bar{F}
\otimes_F FG$ and the character theories over algebraically closed fields coincide in the semi-simple case [@CR1 Corollary 18.11]. We don’t know yet, if $\mathbb{Z}$ is also “best” in distinguishing groups in the twisted case, but it is clear that $\mathbb{C}$ is no longer the “worst” in the semi-simple case.
\[ex:C9\] Let $G=C_3\times C_3$, let $H=C_9$ and let $F=\mathbb{F}_{17}$. Then, $H^2(G,F^*)$ and $H^2(H,F^*)$ are trivial and $$FG\cong FH\cong F\oplus 4\mathbb{F}_{17^2}.$$ So $G\sim _F H$.
It is clear that $G \not \sim _{\mathbb{C}} H$, since these groups admit non-isomorphic Schur multipliers by (see also [@MargolisSchnabel Lemma 1.2]).
Notice, that for abelian groups $G$ and $H$, if $\mathbb{C}G\cong
\mathbb{C}H$ and $M(G)\cong M(H)$ then $G$ and $H$ are isomorphic. By the above example, this is not true in general over other fields. However, due to our example it is natural to ask the following.
\[Q:ab1\] Let $G$ and $H$ be finite abelian groups and let $F$ be a field such that $FG\cong FH$ and $H^2(G,F^*)\cong H^2(H,F^*)$. Is it true that $G\sim _F H$?
It turns out that over fields of characteristics $0$, the answer is yes, and even more the group algebra and the second cohology group together determine the group up to isomorphism. In fact the situation here is similar to the complex case, however the proof is more evolved. We will use the following lemma.
\[lemma:exp-m-isomorphic\] Let $G$ and $H$ be finite abelian $p$-groups for a prime $p$ such that $|G| = |H|$. Let $F$ be a field and let $p^m$ be the cardinality of the maximal $p$-subgroup of $F^*$ (here $m$ being infinity is allowed). If $H^2(G,F^*)\cong H^2(H,F^*)$ then the maximal subgroups of $G$ and $H$ of exponent dividing $p^m$ are isomorphic. In particular, for $m\geq 1$ the groups $G$ and $H$ have the same rank.
First, the lemma is clear for $m=0$, that is if $F$ contains no primitive $p$-th roots of unity. Second, if $F^*$ admits a $p$-subgroup of infinite order then $\operatorname{Ext}(G/G',F^*)$ and $\operatorname{Ext}(H/H',F^*)$ are trivial and hence by $$M(G)\cong H^2(G,F^*)\cong H^2(H,F^*)\cong M(H).$$ Consequently by [@MargolisSchnabel Lemma 1.2] $G$ and $H$ are isomorphic. We are left with the case $m$ is some natural number. Assume $$G=\langle \sigma _1 \rangle \times \langle \sigma _2 \rangle
\times \ldots \times \langle \sigma _k \rangle.$$ $$H=\langle \tau _1 \rangle \times \langle \tau _2 \rangle \times
\ldots \times \langle \tau _r \rangle.$$ Now define for $1\leq i \leq m$ $$\begin{array}{cc}
b_i(G)=
\left\{
\begin{array}{cc}
|\{ j:\circ (\sigma _j )=p^i \}|, & i<m . \\
|\{ j:\circ (\sigma _j )\geq p^m\}| , & i=m .
\end{array}
\right. &
\end{array}
b_i(H)=
\left\{
\begin{array}{cc}
|\{ j:\circ (\tau _j )=p^i\}| , & i<m . \\
|\{ j:\circ (\tau _j )\geq p^m\}| , & i=m .
\end{array}
\right.$$ By we have $$H^2(G,F^*)\cong H^2(H,F^*)\cong a_m C_{p^m}\times a_{m-1}C_{p^{m-1}}\times \ldots \times a_1 C_p$$ for some natural numbers $a_1$,...,$a_m$. Also by we can express the $a_i$ in terms of the $b_i$ such that $a_m$ only depends on $b_m$, $a_{m-1}$ only depends on $b_m$ and $b_{m-1}$ etc. Namely: $$b_i(G) + \binom{b_i(G)}{2} + b_i(G)\left(\sum_{j = i+1}^m b_j(G)\right) = a_i = b_i(H) + \binom{b_i(H)}{2} + b_i(H)\left(\sum_{j = i+1}^m b_j(H)\right).$$ This formula follows as, in the notation of , the first of the two products which appear as direct factors contributes $b_i(G)$ copies of $C_{p^i}$, the $b_i(G)$ cyclic groups of order exactly $C_{p^i}$ contribute $\binom{b_i(G)}{2}$, one for each choice of two such groups, and each cyclic group of order bigger than $p^i$ contributes $b_i(G)$ copies.
Consequently, $b_i(G)=b_i(H)$ for any $1\leq i \leq m$ and the result follows.
We are now ready do prove
\[th:AbelianIso\] Let $F$ be a field of characteristic zero and let $G$ and $H$ be abelian groups, such that $FG\cong FH$ and $H^2(G,F^*)\cong H^2(H,F^*)$. Then $G$ and $H$ are isomorphic.
Clearly it is sufficient to prove the theorem for abelian $p$-groups for primes $p$. Set $e_G = \log_p(\text{exp}(G))$ and $e_H = \log_p(\text{exp}(H))$. Let $p^m$ be the cardinality of the maximal $p$-subgroup of $F^*$ (here $m$ being infinity is allowed). If $m\geq \max \{e_G, e_H\}$ the result follows from Lemma \[lemma:exp-m-isomorphic\]. In the following argument we use the fact that the characteristic of $F$ is zero. Namely will use that if $\zeta$ is a primitive $p^n$-th root of unity with $n > m$ then $F[\zeta]^*$ contains no elements of order $p^{n+1}$. Assume $m<e_G$, then in the Artin-Wedderburn decomposition of $FG$ the maximal field extension appearing has a certain degree $\frac{\varphi(p^{e_G})}{\varphi(p^m)}$, where $\varphi$ denotes Euler’s totient function. Since $FG\cong FH$, the degree of the maximal field extension in the Artin-Wederbrun decomposition of $FH$ is also $\frac{\varphi(p^{e_G})}{\varphi(p^m)} =
\frac{\varphi(p^{e_H})}{\varphi(p^m)}$. Consequently, $e_G=e_H =:
e$. Since $FG \cong FC_{p^{e}} \otimes F(G/C_{p^{e}})$ and $FH
\cong FC_{p^{e}} \otimes F(G/C_{p^{e}})$ we conclude by induction that $FG \cong FH$.
However, it turns out that in general the answer to Question \[Q:ab1\] is negative.
\[prop:notinrelation\] Let $G\cong C_8\times C_2$ and let $H\cong C_4\times C_4$. Then
1. There exist fields $F$ such that $FG\cong FH$ and $H^2(G,F)\cong H^2(H,F)$. But
2. For any field $F$ the relation $G \sim_{F} H$ does not hold.
Let $\mathbb{F}_q$ be a finite field such that $q-1$ is divisible by $2$ but not divisible by $4$, that is $\mathbb{F}_q$ contains roots of unity of order $2$ but does not contain roots of unity of order $4$. In this case it follows from that $$H^2(G,\mathbb{F}_q^*)\cong H^2(H,\mathbb{F}_q^*)\cong C_2 \times C_2 \times C_2.$$ If additionally $q^2-1$ is divisible by $8$, then $$\mathbb{F}_q G\cong \mathbb{F}_q H\cong 4\mathbb{F}_q\oplus \mathbb{F}_{q^2}.$$ This concludes the first part of the proposition. We want to show that for $F$ any field, $G\not \sim _{F} H$. By Theorem \[th:AbelianIso\] this is clear for fields of characteristic zero. Let $F$ be a field of positive characteristic, $\text{char}(F)=p$. If $p=2$ then, since the modular isomorphism problem has a positive solution for abelian groups, $FG \not \cong
FH$ and therefore, $G\not \sim _{F} H$ [@Passmanp4 Corollary 5].
Consequently we may assume $p>2$. Let $$G\cong C_8\times C_2= \langle g_1 \rangle \times \langle g_2 \rangle, \quad H\cong C_4\times C_4=\langle h_1 \rangle \times \langle h_2 \rangle .$$ In the following arguments about the center of twisted group algebras we use Lemma \[lemma:centeroftga\]. If there exists a primitive 4-th root of unity $\zeta$ in $F$ then there exists a twisted group algebra over $H$ with a 1-dimensional center, determined by the relation $[u_{h_1}, u_{h_2}] = \zeta$. But all twisted group rings over $G$ admit a center of dimension at least $4$ spanned by $u_{g_1}^2$. We are left with the case $p>2$ and $p-1$ is not divisible by $4$. Consider $[\alpha]\in H^2(H,F^*)$ determined in $F^{\alpha}H$ by $$[\alpha]:\quad [u_{h_1},u_{h_2}]=-1,\quad u_{h_1}^4=u_{h_2}^4=1.$$ Then the center of $F^{\alpha}H$ is isomorphic to $F(C_2\times
C_2)\cong 4F$ and in particular, $4$ is not a divisor of the order of any central element of $F^{\alpha}H$. However, for any $[\beta]\in H^2(G,F^*)$ the element $u_{g_1}^2$ is central in $F^{\beta}G$ of order multiple of $4$. This completes the proof.
It is interesting to compare the situation in Proposition \[prop:notinrelation\] to the following example.
Let $G=C_{16}\times C_4$ and let $H=C_8 \times C_8$, then $G\sim
_{\mathbb{F}_{31}} H$.
Let $\mathbb{F}_{31}=F$ and $\mathbb{F}_{{31}^2}=K$. Assume $$G\cong C_{16}\times C_4= \langle g_1 \rangle \times \langle g_2 \rangle, \quad H\cong C_8\times C_8=\langle h_1 \rangle \times \langle h_2 \rangle .$$ Since, $F^*$ admits an element of order $2$ but no elements of order $4$, by $$H^2(G,F^*)\cong H^2(H,F^*)\cong C_2\times C_2\times C_2.$$ In order to prove that $G\sim _{\mathbb{F}_{31}} H$ we will need also to describe generators for the cohomology groups. For $H^2(G,F^*)$ we have the generators $$[\alpha _1]:\hspace{0.5cm}[u_{g_1},u_{g_2}]=-1,\hspace{0.5cm} u_{g_1}^{16}=1,\hspace{0.5cm} u_{g_2}^{4}=1,$$ $$[\alpha _2]:\hspace{0.5cm}[u_{g_1},u_{g_2}]=1,\hspace{0.5cm} u_{g_1}^{16}=-1,\hspace{0.5cm} u_{g_2}^{4}=1,$$ $$[\alpha _3]:\hspace{0.5cm}[u_{g_1},u_{g_2}]=1,\hspace{0.5cm} u_{g_1}^{16}=1,\hspace{0.5cm} u_{g_2}^{4}=-1.$$ For $H^2(H,F^*)$ we have the generators $$[\beta _1]:\hspace{0.5cm}[u_{h_1},u_{h_2}]=-1,\hspace{0.5cm} u_{h_1}^{8}=1,\hspace{0.5cm} u_{g_2}^{8}=1,$$ $$[\beta _2]:\hspace{0.5cm}[u_{h_1},u_{h_2}]=1,\hspace{0.5cm} u_{h_1}^{8}=-1,\hspace{0.5cm} u_{h_2}^{8}=1,$$ $$[\beta _3]:\hspace{0.5cm}[u_{h_1},u_{h_2}]=1,\hspace{0.5cm} u_{h_1}^{8}=1,\hspace{0.5cm} u_{h_2}^{8}=-1.$$ We claim that the isomorphism from $\psi :H^2(G,F^*)\rightarrow
H^2(H,F^*)$ sending $[\alpha _i]$ to $[\beta _i]$ induces a ring isomorphism $F^{\alpha}G\cong F^{\psi(\alpha)}H $ for any $[\alpha] \in H^2(G,F^*)$. The group rings $FG$ and $FH$ are clearly isomorphic, namely to $4F \oplus 30K$. Now, let $[\alpha]\in H^2(G,F^*)$ and $[\beta]\in H^2(H,F^*)$ be non-trivial cohomology classes such that $F^{\alpha}G$ and $F^{\beta}H$ are commutative. By Lemma \[lemma:GRNOTGR\] the twisted group rings admit no $1$-dimensional components (over $F$). And therefore, since $K ^*$ admits elements of order $32$ we conclude that $$F^{\alpha}G\cong F^{\beta}H \cong \oplus _{i=1}^{32} K.$$
A well known result says that for any group $G$, the order of a cohomology class $[\gamma]\in H^2(G,F^*)$ divides the dimension of each $\gamma$-projective representation of $G$ [@KarpilovskyProjective Proposition 6.2.6]. Therefore, by Proposition \[prop:abcomesfromEXT\] for any $[\alpha]\in
H^2(G,F^*)$ and $[\beta]\in H^2(H,F^*)$ such that $F^{\alpha}G$ and $F^{\beta}H$ are non-commutative, they are isomorphic to a direct sum of $2\times 2$-matrix rings over $F$ and $K$. Therefore they are isomorphic if and only their center is isomorphic.
By Lemma \[lemma:centeroftga\] for any $[\alpha]\in H^2(G,F^*)$ and $[\beta]\in H^2(H,F^*)$ such that $F^{\alpha}G$ and $F^{\beta}H$ are non-commutative, the center of $F^{\alpha}G$ is generated (as an algebra) by $u_{g_1}^2,u_{g_2}^2$ and similarly the center of $F^{\beta}H$ is generated (as an algebra) by $u_{h_1}^2,u_{h_2}^2$. Again, by Lemma \[lemma:GRNOTGR\], if the restriction of $\alpha$ (similarly $\beta$) to the subgroup generated by $g_1,g_2$ (similarly $h_1,h_2$) is non-trivial then $$Z(F^{\alpha}G)\cong Z(F^{\beta}H)\cong 8K.$$ This holds for the cohomology classes $$[\alpha _1 \alpha _2],[\alpha_1 \alpha _3],[\alpha _1 \alpha _2 \alpha _3]\in H^2(G,F^*), \quad
[\beta _1 \beta _2],[\beta_1 \beta _3],[\beta _1 \beta _2
\beta_3]\in H^2(H,F^*).$$ Finally, $$Z(F^{\alpha_1}G)\cong Z(F^{\beta_1}H)\cong 4F\oplus 2K.$$ This completes the proof.
The Yamazaki cover {#UnSchur}
==================
Let $p$ be prime, let $F$ be a field and let $\zeta$ be a primitive root of unity of order $p^k$ which is maximal in the sense that there are no primitive roots of unity in $F$ of order $p^{k+1}$. Then, by our assumption that $H^2(G,F^*)\cong
H^2(G,t(F^*))$, we may always assume that for a cyclic group $C_{p^r}$ with generator $\sigma$, the group $H^2(C_{p^r},F^*)$ is generated by a cohomology class which admits a $2$-cocycle which is determined by $u_{\sigma}^{p^r}=\zeta$ (see e.g. [@YamazakiUnschur p.31]). Notice that this does not necessarily hold without our assumption on the field. For example $H^2(C_2,\mathbb{Q}^*)$ is an infinite group.
Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $0$. Then there exists a group $G^*$ with an abelian normal subgroup $A\cong H^2(G,F^*)$ such that $$1 \rightarrow A \rightarrow G^* \rightarrow G \rightarrow 1$$ is a stem-extension, i.e. $A \leq Z(G^*) \cap (G^*)'$. This $G^*$ is called a representation group of $G$ or a Schur cover of $G$. Clearly, $|G^*|=|G||H^2(G,F^*)|$. In general the isomorphism type of a Schur cover is not unique, but each cover satisfies $$\label{eq:GRSCHUR}
FG^* \cong \oplus_{[\alpha] \in H^2(G,F^*)} F^\alpha G.$$ See [@KarpilovskyProjective Chapter 3, §3] for the details.
Different variations and generalizations of representation groups have been studied, see e.g. [@LassueurThevenaz; @Sambonet] for some of the most recent.
The following example demonstrates that over non-algebraically closed fields there is no Schur cover, and at the same time suggests how to find an analog, in a sense as in , in the non-algebraically closed case.
Let $G\cong C_2$ be generated by an element $g$ and let $F=\mathbb{F}_5$. We can define a $2$-cocycle $\beta \in Z^2(G,F^*)$ by $u_g^2=\zeta $ where $\zeta$ is of order $4$. Notice that $u_g$ is an element of order $8$ in $F^\beta G$. It is clear that $H^2(G,F^*)\cong C_2$ and therefore, if $G$ admits a Schur cover it is of order $4$. However, $FC_4\cong F(C_2\times C_2) \cong 4F$ and in particular it does not contain elements of order $8$. Consequently, is not satisfied and there is no Schur cover for $G$ over $F$. However, it is not hard to check that $$\mathbb{F}_5C_8 \cong 4\mathbb{F}_5\oplus 2\mathbb{F}_{25}=2\left(\mathbb{F}_5 C_2\oplus \mathbb{F}_5^\beta
C_2\right)=2 \left(\oplus_{[\alpha] \in H^2(C_2,\mathbb{F}_5^*)} \mathbb{F}_5^\alpha
C_2\right).$$
We wish to find a group $G^*$ which will play a similar role of the Schur cover over non-algebraically closed fields in the sense that any twisted group ring over $G$ will be a direct summand of the group ring over $G^*$. Since the construction of this group is based on a proof of Yamazaki [@YamazakiUnschur] we will give here the existence theorem with a sketch of the part of the proof which describes how to construct this object. Again, for a field $F$ we will denote by $t(F^*)$ the torsion part of $F^*$.
\[Yamazaki cover\][@YamazakiUnschur] (see also [@KarpilovskyProjective Theorem 3.3.2]) Let $G$ be a finite group and let $F$ be a field such that $H^2(G,F^*)=H^2(G,t(F^*))$. There exists a finite central extension $$\label{eq:Yamazaki1}
1 \rightarrow A \rightarrow G^* \rightarrow G \rightarrow 1,$$ such that any projective representation of $G$ is projectively equivalent to a linear representation of $G^*$.
**Construction of $G^*$.** First, we need to describe the group $A$ in . Since $H^2(G,F^*)$ is a finite abelian group we may write $$H^2(G,F^*)=\langle c_1 \rangle \times \langle c_2 \rangle \times \ldots \times \langle c_m \rangle.$$ Construct a new group as follows. Choose in any cohomology class $c_i$ a cocycle $\alpha _i$ of order $d_i$, let $A_i \cong C_{d_i}$ and let $$A=A_1\times A_2\times \ldots \times A_m.$$ Now, the group $G^*$ will be determined by a cohomology class $\beta \in H^2(G,A)$. This $\beta$ can be considered as $$(\beta _1, \beta _2, \ldots , \beta _m) \in H^2(G,A_1)\times H^2(G,A_2) \times \ldots \times H^2(G,A_m),$$ while the only restriction on $\beta _i$ is that $\tilde{\chi}_i(\beta _i)=c_i$ for the natural morphism $\tilde{\chi _i}:H^2(G,A_i)\rightarrow H^2(G,F^*).$
We will call the group $G^*$ in Theorem \[Yamazaki cover\] a *Yamazaki cover* and will denote a Yamazaki cover of a group $G$ over a field $F$ by $Y_F(G)$.
If there is no *proper* quotient of $G^*$ which is also a Yamazaki cover of $G$ we call $G^*$ a *minimal Yamazaki cover*.
The following remarks are in order.
With the notations above we have a surjective morphism $\psi :A\rightarrow H^2(G,F^*)$. In fact this is the well-known transgression map $\operatorname{Hom}(A, F^*) \rightarrow H^2(G, F^*)$, cf. Definition \[def:Tra\] or [@KarpilovskyProjective Theorem 3.2.9].
Notice that with the above notations, $A$ is not uniquely determined, and in fact even its cardinality is not uniquely determined, since there could be in $c_i$ cocycles $\alpha$ and $\alpha'$ of distinct order. Furthermore, like in the situation with the classical Schur cover, for a fixed $A$ different choices of $\beta$ can lead to non-isomorphic Yamazaki covers.
The existence of $Y_F(G)$ depends on the condition that $H^2(G, F^*) = H^2(G, t(F^*))$. This condition was also investigated by Yamazaki. He showed that $H^2(G, F^*) = H^2(G, t(F^*))$ if and only if $F^* =
(F^*)^{\operatorname{exp}(G/G')}t(F^*)$ [@YamazakiUnschur] (cf. also [@KarpilovskyProjective Corollary 3.3.4]). In particular over every finite field, the real and the complex numbers Yamazaki covers always exist.
The following is immediate now from Theorem \[Yamazaki cover\] and the construction of the Yamazaki cover.
Let $Y_F(G)$ be a Yamazaki cover of a group $G$ over a field $F$ which corresponds to . Then $$FY_F(G)\cong \frac{|A|}{|H^2(G,F^*)|} \oplus_{[\alpha] \in
H^2(G,F^*)} F^\alpha G.$$
For given groups $G$ and $H$ there is a well-known group theoretical condition how to determine whether $H$ is a Schur cover of $G$, assuming we know the order of $H^2(G, \mathbb{C}^*)$ [@KarpilovskyProjective Theorem 3.3.7]. For minimal Yamazaki covers we can provide a similar criterion which requires a few more things to check though. For an abelian group $A$ and a prime $p$ denote by $A_p$ the Sylow $p$-subgroup of $A$.
\[th:YamazakiGT\] Let $1 \rightarrow Z \rightarrow H \rightarrow G \rightarrow 1$ be a central extension of a finite group $G$ and $F$ a field such that $H^2(G,F^*)\cong H^2(G,t(F^*))$. Assume that this extension satisfies the following:
- $Z \cap H' \cong \operatorname{Hom}(M(G), F^*)$.
- $\operatorname{rk}(G/G') = \operatorname{rk}(H/H')$.
- For each prime $p$ we have the following: If $F^*$ contains a maximal finite $p$-subgroup and the order of this group is $p^m$ then $(Z/Z\cap H')_p$ is a direct product of $\operatorname{rk}(\operatorname{Ext}((G/G')_p, F^*))$ cyclic $p$-groups of order $p^m$.
- $H' \cap Z$ has a complement in $Z$, i.e. the short exact sequence $1 \rightarrow Z \cap H' \rightarrow Z \rightarrow Z/(Z \cap H') \rightarrow 1$ is split.
Then $H$ is a minimal Yamazaki cover of $G$ over $F$.
Short exact sequence illustrating the steps of the proof can be found in . Note that by assumption the exponent of $Z$ divides the exponent of $F^*$, so $Z \cong \operatorname{Hom}(Z, F^*)$. We need to show that the transgression map (see Definition \[def:Tra\]) ${{\operatorname{Tra}}}:
\operatorname{Hom}(Z, F^*) \rightarrow H^2(G, F^*)$ is surjective and moreover that this is not the case for any central extension $1
\rightarrow Z/\tilde{Z} \rightarrow H/\tilde{Z} \rightarrow G
\rightarrow 1$ for $\tilde{Z}$ a proper subgroup of $Z$.
Let $Z = (Z \cap H') \times C$ for a subgroup $C$ of $Z$ and identify $C$ and $Z/(Z \cap H')$. By our assumption that $Z \cap
H' \cong \operatorname{Hom}(M(G), F^*)$ and [@KarpilovskyVolII Lemma 11.5.1] it follows that the image of $\operatorname{Tra}|_{Z \cap H'}$ is isomorphic to $\operatorname{Hom}(M(G), F^*)$. Define $H^2_0(G, F^*)$ as in [@KarpilovskyProjective Definition before Theorem 2.2.9] to be the part of $H^2(G,F^*)$ which corresponds to all central extensions $1 \rightarrow A \rightarrow E \rightarrow G
\rightarrow 1$ with the property that $A' \cap E = 1$. Then [@KarpilovskyProjective Theorem 2.2.9] implies that $H^2_0(G,
F^*)$ is exactly the image of $\operatorname{Ext}(G/G', F^*)$ under the inflation map. In particular the transgression map ${{\operatorname{Tra}}}: C \rightarrow H^2(G, \mathbb{F}^*)$ related to the short exact sequence $$1 \rightarrow Z/(Z \cap H') \rightarrow H/(Z\cap H') \rightarrow G \rightarrow 1$$ has an image lying in $H^2_0(G, F^*)$. It remains to show that this is indeed the whole image and that this is not the case for any group smaller than $H/(Z \cap H')$. It is enough to show this for a non-trivial Sylow $p$-subgroup $P$ of $C$ for some fixed prime $p$ with respect to the Sylow $p$-subgroup of $H^2_0(G, (F^*)_p)$ as it follows for each Sylow subgroup of $C$ in the same way.
It follows from our second and third assumptions that ${{\operatorname{rk}}}((H/H')_p) = {{\operatorname{rk}}}((G/G')_p)$. Let $P = \langle a_1 \rangle \times \langle a_2 \rangle \times ... \times \langle a_r \rangle$ for some $a_1$,...,$a_r$. Then each $a_i$ has order $p^m$ by assumption and $r = {{\operatorname{rk}}}(\operatorname{Ext}((G/G')_p, (F^*)_p)) = {{\operatorname{rk}}}(H^2_0(G, (F^*)_p))$. Fix some $1 \leq i \leq r$. An abelian extension of $G/G'$ by $(F^*)_p$ corresponding to $a_i$ is not of the form $1 \rightarrow (F^*)_p \rightarrow (F^*)_p \times G/G' \rightarrow G/G' \rightarrow 1$, as ${{\operatorname{rk}}}((H/H')_p) = {{\operatorname{rk}}}((G/G')_p)$. So by [@KarpilovskyProjective Theorem 2.1.2 and Corollary 2.1.3] the coclass ${{\operatorname{Tra}}}(a_i)$ is not a coboundary for any $1 \leq i \leq r$. So ${{\operatorname{rk}}}({{\operatorname{Tra}}}(P)) = {{\operatorname{rk}}}(H^2_0(G, (F^*)_p))$.
Assume that ${{\operatorname{Tra}}}(P)$ is a proper subgroup of $H^2_0(G, (F^*)_p)$. Then there is an $1\leq i \leq r$ and a cocycle $b \in Z^2(G, (F^*)_p)$ such that $b^p = \operatorname{Tra}(a_i)$. But then the $b$ must have a value which is a $p^{m+1}$-th primitive root of unity in $F^*$, contradicting our choice of $m$.
Lastly, the minimality of $H$, follows from the fact that $a_i$ corresponds to an element in $\operatorname{Ext}(G/G', (F_p)^*)$, that is an abelian extension with kernel $C_{p^m}$ and hence $a_i$ must have order at least $p^m$.
$$\label{eq:picture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzset{thick arc/.style={->, black, fill=none, >=stealth,
text=black}} \tikzset{node distance=2cm, auto}
\node (triv_UPLEFT){$1$};
\node (Z_UP) [right of=triv_UPLEFT] {$Z$};
\tikzset{node distance=3cm, auto}
\node (H_UP) [right of=Z_UP] {$H$};
\node (G_UP) [right of=H_UP] {$G$};
\tikzset{node distance=2cm, auto}
\node (triv_UPRIGHT)[right of=G_UP]{$1$};
\tikzset{node distance=2cm, auto}
\node (triv_MIDDLELEFT)[below of=triv_UPLEFT] {$1$};
\node (Zmod_MIDDLE) [right of=triv_MIDDLELEFT] {$Z/(Z\cap H')$};
\tikzset{node distance=3cm, auto}
\node (Hmod_MIDDLE) [right of=Zmod_MIDDLE] {$H/(Z\cap H')$};
\node (G_MIDDLE) [right of=Hmod_MIDDLE] {$G$};
\tikzset{node distance=2cm, auto}
\node (triv_MIDDLERIGHT)[right of=G_MIDDLE]{$1$};
\tikzset{node distance=2cm, auto}
\node (triv_DOWNLEFT)[below of=triv_MIDDLELEFT] {$1$};
\node (Zmod_DOWN) [right of=triv_DOWNLEFT] {$Z/(Z\cap H')$};
\tikzset{node distance=3cm, auto}
\node (Hmod_DOWN) [right of=Zmod_DOWN] {$H/H'$};
\node (Gmod_DOWN) [right of=Hmod_DOWN] {$G/G'$};
\tikzset{node distance=2cm, auto}
\node (triv_DOWNRIGHT)[right of=Gmod_DOWN]{$1$};
\draw[thick arc, draw=black] (triv_UPLEFT) to node [above] {$$} (Z_UP);
\draw[thick arc, draw=black] (Z_UP) to node [above] {$$} (H_UP);
\draw[thick arc, draw=black] (H_UP) to node [above] {$$} (G_UP);
\draw[thick arc, draw=black] (G_UP) to node [above] {$$} (triv_UPRIGHT);
\draw[thick arc, draw=black] (triv_MIDDLELEFT) to node [above] {$$} (Zmod_MIDDLE);
\draw[thick arc, draw=black] (Zmod_MIDDLE) to node [above] {$$} (Hmod_MIDDLE);
\draw[thick arc, draw=black] (Hmod_MIDDLE) to node [above] {$$} (G_MIDDLE);
\draw[thick arc, draw=black] (G_MIDDLE) to node [above] {$$} (triv_MIDDLERIGHT);
\draw[thick arc, draw=black] (triv_DOWNLEFT) to node [above] {$$} (Zmod_DOWN);
\draw[thick arc, draw=black] (Zmod_DOWN) to node [above] {$$} (Hmod_DOWN);
\draw[thick arc, draw=black] (Hmod_DOWN) to node [above] {$$} (Gmod_DOWN);
\draw[thick arc, draw=black] (Gmod_DOWN) to node [above] {$$} (triv_DOWNRIGHT);
\draw[thick arc, draw=black] (Z_UP) to node [left] {$ $} (Zmod_MIDDLE);
\draw[thick arc, draw=black] (Zmod_MIDDLE) to node [left] {$ $} (Zmod_DOWN);
\draw[thick arc, draw=black] (H_UP) to node [left] {$ $} (Hmod_MIDDLE);
\draw[thick arc, draw=black] (Hmod_MIDDLE) to node [left] {$ $} (Hmod_DOWN);
\draw[thick arc, draw=black] (G_UP) to node [left] {$ $} (G_MIDDLE);
\draw[thick arc, draw=black] (G_MIDDLE) to node [left] {$ $} (Gmod_DOWN);
\end{tikzpicture}$$
We provide an example for $Y_{\mathbb{F}_3}(D_8)$ where $D_8$ denotes a dihedral group of order $8$. We also give an example that the last condition in Theorem \[th:YamazakiGT\] is necessary. Let $G = D_8$ and $F = \mathbb{F}_3$.
We have $G/G' \cong C_2 \times C_2$, so $\operatorname{Ext}(G/G', F^*) \cong C_2 \times C_2$. Moreover $M(G) \cong C_2$ [@KarpilovskyProjective Proposition 4.6.4]. A minimal Yamazaki cover of $G$ is given by $$\begin{aligned}
Y(G) = (\langle a \rangle \times \langle b \rangle ) \rtimes \langle c \rangle: \ \ a^2 = 1, \ b^8 = 1, \ c^4 = 1, \ a^c = a, \ b^c = ab^3.\end{aligned}$$ Then $Z(Y(G)) = \langle a, b^4, c^2 \rangle$ is an elementary abelian group of order $8$. Moreover $Y(G)' = \langle ab^2 \rangle$ is a cyclic group of order $4$. Setting $Z = Z(Y(G))$ we observe that all conditions from Theorem \[th:YamazakiGT\] are satisfied. Using the package `Wedderga` [@Wedderga] of the computer algebra system `GAP` [@GAP] we obtain moreover $$FY(G) \cong 4\mathbb{F}_3 \oplus 6\mathbb{F}_9 \oplus 8M_2(\mathbb{F}_3) \oplus 2M_2(\mathbb{F}_9).$$
We now exhibit an example that the last condition in Theorem \[th:YamazakiGT\] is necessary. Set $$\begin{aligned}
H = (\langle a \rangle \times \langle b \rangle ) \rtimes \langle c \rangle: \ \ a^4 = 1, \ b^4 = 1, \ c^4 = 1, \ a^c = a^{-1}, \ b^c = ab.\end{aligned}$$ Then we have that $Z(H) = \langle ab^2, c^2 \rangle \cong C_4 \times C_2$ and $H' = \langle a \rangle \cong C_4$. Set $Z = Z(H)$. Then $H/Z \cong G$, $Z \cap H' = \langle a^2 \rangle \cong C_2$, $\operatorname{rk}(Z/(Z \cap H')) = 2$, $Z/(Z \cap H') \cong C_2 \times C_2$ and $\operatorname{rk}(H/H') = 2$. So the extension $1 \rightarrow Z \rightarrow H \rightarrow G \rightarrow 1$ satisfies all the conditions of Theorem \[th:YamazakiGT\] except the last one. But $H$ is not a Yamazaki cover of $G$ as its group algebra over $F$ is not isomorphic with the group algebra of $Y(G)$ given above. Indeed, $$FH \cong 4\mathbb{F}_3 \oplus 6\mathbb{F}_9 \oplus 4M_2(\mathbb{F}_3) \oplus 4M_2(\mathbb{F}_9),$$ which again can be calculated using [@Wedderga].
The Dade example {#Dade}
================
In 1971 E. Dade, answering a question of R. Brauer [@Brauer63 Problem 2\*], provided a family of examples of non-isomorphic finite groups $G$ and $H$ such that the group algebras of $G$ and $H$ are isomorphic over any field $F$. We will show that for a subclass of Dade’s examples there are fields $F$ such that $G
\not\sim_F H$. Note that the groups of Dade are metabelian and hence have non-isomorphic group rings over the integers, a result due to Whitcomb already known at the time Dade solved Brauer’s problem [@Whitcomb].
We will first describe the groups given by Dade. Let $p$ and $q$ be primes such that $q \equiv 1 \mod p^2$ and let $w$ be an integer such that $w \not \equiv 1 \mod q^2$, but $w^p \equiv 1
\mod q^2$. Let $Q_1$ and $Q_2$ be the following two non-abelian groups of order $q^3$. $$\begin{aligned}
Q_1 &= (\langle \tau_1 \rangle \times \langle \sigma_1 \rangle) \rtimes \langle \rho_1 \rangle, \\
Q_2 &= \langle \sigma_2 \rangle \rtimes \langle \rho_2 \rangle, \\
\tau_1^q &= \sigma_1^q = \rho_1^q = \sigma_2^{q^2} = \rho_2^q = 1, \ \sigma_2^q =: \tau_2, \\
\tau_1^{\rho_1} &= \tau_1, \ \sigma_1^{\rho_1} = \tau_1\sigma_1, \ \sigma_2^{\rho_2} = \tau_2\sigma_2\end{aligned}$$ So $Q_1$ and $Q_2$ are just the two non-abelian groups of order $q^3$ such that $Q_1$ has exponent $q$ (aka the Heisenberg group).
Let $\langle \pi_1 \rangle \cong C_{p^2}$, $\langle \pi_2 \rangle \cong C_p$ and for $i,j \in \{1,2 \}$ let $$\rho_i^{\pi_j} = \rho_i, \ \sigma_i^{\pi_j} = \sigma_j^w, \ \tau_i^{\pi_j} = \tau_i^w.$$ Define two groups by $$\begin{aligned}
G &= (Q_1 \rtimes \langle \pi_1 \rangle) \times (Q_2 \rtimes \langle \pi_2 \rangle), \\
H &= (Q_1 \rtimes \langle \pi_2 \rangle) \times (Q_2 \rtimes \langle \pi_1 \rangle).\end{aligned}$$ These are the groups constructed by Dade as a counterexample to Brauer’s question.
Notice that $G=G_1\times G_2$ and $H=H_1\times H_2$ for $$G_1=Q_1 \rtimes \langle \pi_1 \rangle,\quad G_2=Q_2 \rtimes \langle \pi_2 \rangle,\quad
H_1=Q_1 \rtimes \langle \pi_2 \rangle \quad H_2=Q_2 \rtimes \langle \pi_1 \rangle.$$
The second cohomology groups of $G$ and $H$
-------------------------------------------
In order to calculate the Schur multipliers of $G$ and $H$ we will use a result of Schur [@Schur] about the Schur multiplier of direct products of groups (see also [@KarpilovskyProjective Corollary 2.3.14]). Define the tensor product of two finite groups $A$ and $B$ by $$A\otimes B=A/A' \otimes _{\mathbb{Z}} B/B'.$$
\[th:directproductSchur\] Let $A$ and $B$ be finite groups. Then $$ M(A \times B)=M(A)\times M(B)\times (A \otimes B).$$
Notice that (slightly abusing notation) $$G_1'=\langle \tau_1 \rangle \times \langle \sigma_1 \rangle=H_1' \text{ and }
G_2'= \langle \sigma_2 \rangle=H_2'.$$ Therefore $$G_1/G_1'\cong C_q\times C_p\cong H_2/H_2' \text{ and } G_2/G_2'\cong C_q\times C_{p^2}\cong H_1/H_1'.$$ Consequently $$G_1\otimes G_2\cong H_1\otimes H_2\cong C_q\times C_p.$$ We will use Theorem \[th:directproductSchur\] to compute the Schur multipliers of $G_1$, $G_2$, $H_1$ and $H_2$. Notice that $G_1$, $G_2$, $H_1$ and $H_2$ are written as semi-direct products of subgroups of coprime order. The following lemma will be of use.
(see [@KarpilovskySchur Corollary 2.2.6])\[lemma:semicoprimeSchur\] Let $N$ and $T$ be subgroups of a group $G$ of co-prime order and assume $G = N \rtimes T$. Then $$M(G)=M(T)\times M(N)^T.$$ Here $M(N)^T$ are the elements in $M(N)$ which are invariants under the $T$-action.
First, by [@KarpilovskyProjective Theorem 4.7.3], $M(Q_1)\cong C_q\times C_q$ and $Q_2$ admits a trivial Schur multiplier. Therefore, since a Schur multiplier of a cyclic group is trivial, we get by Lemma \[lemma:semicoprimeSchur\] that $$M(G_2)=M(H_2)=1.$$ We are left with the computation of $M(G_1)$ and $M(H_1)$. As written above $M(Q_1)\cong C_q\times C_q$. In fact, $M(Q_1)$ is generated by the cohomology classes $\alpha$ and $\beta$ which are determined by the following relations in the corresponding twisted group algebras $\mathbb{C}^{\alpha}Q_1$ and $\mathbb{C}^{\beta}Q_1$ $$\begin{aligned}
\alpha:& [u_{\tau},u_{\sigma}]=\zeta,\quad [u_{\tau},u_{\rho}]=1, \\
\beta:& [u_{\tau},u_{\sigma}]=1,\quad [u_{\tau},u_{\rho}]=\zeta.\end{aligned}$$ Here $\zeta$ denotes a primitive $q$-th roots of unity. Notice, that for $i=1,2$, $$[u_{\tau^{\pi_i}},u_{\rho^{\pi_i}}]=[u_{\tau^w},u_{\rho}]=[u_{\tau},u_{\rho}]^w.$$ Therefore, $\beta$ is not invariant under the action of $\langle \pi _i \rangle$ for $i=1,2$. We need to check whether $\alpha$ is invariant. It turns out that $\alpha$ is invariant if and only if $p=2$. Indeed, in $\mathbb{C}^{\alpha}Q_1$ $$[u_{\tau^{\pi_i}},u_{\sigma^{\pi_i}}]=[u_{\tau}^w,u_{\sigma}^w]=\zeta^{w^2}.$$ Therefore, $\alpha$ is invariant if and only if $w^2 \equiv 1 \mod q$ which happens if and only if $p=2$ because $w^p \equiv 1 \mod q^2$. As a consequence of the above we obtain the following.
\[prop\_SchurMultiplier\] With the above notations, if $p=2$ $$M(G) \cong M(H) \cong C_q\times C_q\times C_p,$$ and for $p>2$ $$M(G) \cong M(H) \cong C_q\times C_p.$$
We proceed to construct $H^2(G,F^*)$ using the exact sequence given in . Observe that $$G/G'\cong C_q\times C_q\times C_2\times C_4=\langle G'\rho_1 \rangle \times \langle G'\rho_2 \rangle \times \langle G'\pi_1 \rangle \times \langle G'\pi_2 \rangle.$$
Therefore, by equations and we get $$\operatorname{Ext}(G/G',F^*)\cong C_q\times C_q\times C_p\times C_{p^2}.$$
\[cor:H2Dade\] For $p = 2$ we have $$H^2(G,F^*)\cong \operatorname{Ext}(G/G',F^*)\times
\operatorname{Hom}(M(G),F^*)\cong \left(C_q\times C_q\times C_p\times C_{p^2}
\right)\times \left(C_q\times C_q\times C_p \right).$$ and for $p >2$ we get $$H^2(G,F^*)\cong \operatorname{Ext}(G/G',F^*)\times
\operatorname{Hom}(M(G),F^*)\cong \left(C_q\times C_q\times C_p \times C_{p^2}
\right)\times \left(C_q\times C_p \right).$$
Notice that all the arguments above about $G$ are true also for $H$.
The Yamazaki covers of $G$ and $H$
----------------------------------
From now on we will assume that $p=2$ and $q$ is any prime satisfying the relations in Dade’s groups. Note that we can then assume w.l.o.g. $w=-1$. Moreover we assume that $F=\mathbb{F}_r$ is a finite field such that
- $r-1$ is divisible by $q$ but not by $q^2$,
- $r-1$ is divisible by $4$ but not by $8$ and
- $r^2-1$ is divisible by $8$ but not by $16$.
There exist infinitely many such fields, e.g. by Dirichlet’s theorem on primes in arithmetic progressions.
This allows us to give the Yamazaki covers of $G$ and $H$ using less notation, though it is not hard to give them also in case $p>2$. But the difference observed between the Schur multipliers in Proposition \[prop\_SchurMultiplier\] turns out to be crucial for our arguments, so we concentrate on this case. See Remark \[rem:OddCase\] about the case $p>2$.
Let $\zeta$ be a primitive $q$-th and $\xi$ a primitive $4$-th root of unity in $F$. In order to construct the Yamazaki covers of $G$ and $H$ we will need to describe the group $A$ in the construction after Theorem \[Yamazaki cover\] as computed in the previous subsection and in particular in Corollary \[cor:H2Dade\]. Let $$\begin{aligned}
H^2(G,F^*) =& \operatorname{Hom}(M(G),F^*) \times \operatorname{Ext}(G/G',F^*) \\
=& (\langle \alpha \rangle \times \langle \beta \rangle \times \langle \gamma \rangle) \times (\langle \kappa \rangle \times \langle \lambda \rangle \times \langle \mu \rangle \times \langle \nu \rangle),\end{aligned}$$ where
- $\alpha$ is of order $q$, determined by $[u_{\rho_1},u_{\rho_2}]=\zeta$.
- $\beta$ is of order $2$, determined by $[u_{\pi_1},u_{\pi_2}]=-1$.
- $\gamma$ is of order $q$, determined by $[u_{\rho_1},u_{\sigma_1}]=\zeta$.
- $\kappa$ is of order $q$ determined by $u_{\rho_1}^q=\zeta$.
- $\lambda$ is of order $q$ determined by $u_{\rho_2}^q=\zeta$.
- $\mu$ is of order $4$ determined by $u_{\pi _1}^4=\xi$.
- $\nu$ is of order $2$ determined by $u_{\pi _2}^4=\xi$.
Notice, that from the above the only cohomology class in which the order of the cocycle is bigger than the order of the cohomology class is for $\nu$. Here the order of $\nu$ is $2$ and the order of the corresponding cocycle is $4$. Therefore we may consider the extending group $A$ to be like $H^2(G,F^*)$ with the only difference being that the $C_2$ generated by $\nu$ in $H^2(G,F^*)$ will have a representative cocycle $\bar{\nu}$ in $A$ which will generate a $C_4$.
Now in order to construct the Yamazaki cover we need to construct a cohomology class $\beta _{(G,A)}\in H^2(G,A)$ which will correspond to the central extension . Let $\{\tilde{g}\}_{g \in G}$ be a section of $G$ in $G^*$ corresponding to . Then, abusing notation, $\beta _{(G,A)}$ can be chosen to be the cohomology class determined by (compare with the classes given above) $$\begin{aligned}
&[u_{\rho_1}, u_{\rho_2}]=\zeta ,\quad [u_{\pi_1}, u_{\pi_2}]=-1 ,\quad [u_{\rho_1}, u_{\sigma _1}]=\zeta , \\
&u_{\rho_1}^q=\zeta, \quad u_{\rho_2}^q=\zeta , \quad u_{\pi _1}^4=\xi ,\quad u_{\pi _2}^2=\xi.\end{aligned}$$
This leads us also to the Yamazaki covers of $G$ and $H$ over $F$. Since from now on we will only work with these covers and their subgroups we will use the same notations for the elements as before in the “uncovered” groups. Here we will introduce cyclic subgroup $\langle x \rangle$, $\langle y \rangle$ and $\langle z \rangle$ corresponding to the cohomology classes $\alpha$, $\beta$ and $\gamma$ respectively. The orders of the other generators change according to the cohomology classes $\kappa$, $\lambda$, $\mu$ and $\nu$. We will construct both Yamazaki covers as the quotient of the same infinite group.
**Notation:** Let $Y$ be a group generated by elements $\sigma_1$, $\sigma_2$, $\rho_1$, $\rho_2$, $\tau_1$, $\pi_1$, $\pi_2$, $x$, $y$ and $z$ subject to the following relations:
$$\begin{aligned}
\ \sigma_2^{q^2} &= x^q = \rho_2^{q^2} = y^2 = \pi_2^8 = z^q = \tau_1^q = \rho_2^{q^2} = \sigma_1^q = \pi_1^{16} = 1, \ \sigma_2^q =: \tau_2, \\
\sigma_2^{\rho_2} &= \sigma_2 \tau_2, \ \rho_1^{\sigma_1} = \tau_1 \rho_1, \ \tau_1^{\sigma_1} = z\tau_1, \ \rho_2^{\rho_1} = x\rho_2. \\\end{aligned}$$
Moreover we have $x, y, z \in Z(Y)$ and unless otherwise specified in the relations above for $g,h \in \{ \sigma_1, \sigma_2, \rho_1, \rho_2, \tau_1\} $ we have $[g,h] = 1$ in $Y$.
\[lemma\_YamazakiCover\] Let $Y$ be the group described above. Let $Y(G)$ be the quotient of $Y$ in which $\pi_i$ commutes with $\sigma_j$, $\rho_j$, $\tau_j$ for $i \neq j$ and which is additionally subject to the following relations $$\sigma_1^{\pi_1} = \sigma_1^{-1}, \ \sigma_2^{\pi_2} = \sigma_2^{-1}, \ \tau_1^{\pi_1} = z\tau_1^{-1}, \ \ \pi_2^{\pi_1} = y\pi_2.$$ Let $Y(H)$ be the quotient of $Y$ in which $\pi_i$ commutes with $\sigma_i$, $\rho_i$, $\tau_i$ for $i \in \{1,2\}$ and where additionally we have the relations $$\sigma_2^{\pi_1} = \sigma_2^{-1}, \ \tau_1^{\pi_2} = z\tau_1^{-1}, \ \sigma_1^{\pi_2} = \sigma_1^{-1}, \ \pi_1^{\pi_2} = y\pi_1.$$
Then $Y(G)$ and $Y(H)$ are minimal Yamazaki covers of $G$ and $H$ respectively.
**Remark:** Using semi-direct products one can write: $$\begin{aligned}
Y(G) &= (( \langle \sigma_2 \rangle \rtimes (\langle x \rangle \times \langle \rho_2 \rangle )) \rtimes (\langle y \rangle \times \langle \pi_2 \rangle )) \rtimes ((( \langle z \rangle \times \langle \tau_1 \rangle \times \langle \rho_1 \rangle) \rtimes \langle \sigma_1 \rangle ) \rtimes \langle \pi_1 \rangle), \\
Y(H) &= (( \langle \sigma_2 \rangle \rtimes (\langle x \rangle \times \langle \rho_2 \rangle )) \rtimes (\langle y \rangle \times \langle \pi_1 \rangle )) \rtimes ((( \langle z \rangle \times \langle \tau_1 \rangle \times \langle \rho_1 \rangle) \rtimes \langle \sigma_1 \rangle ) \rtimes \langle \pi_2 \rangle).\end{aligned}$$ Note that the only difference when writing this way is an interchange between $\pi_1$ and $\pi_2$.
We will use Theorem \[th:YamazakiGT\] and Corollary \[cor:H2Dade\]. In the notation of Theorem \[th:YamazakiGT\] we have $$Z = \langle x \rangle \times \langle y \rangle \times \langle z \rangle \times \langle \rho_1^q \rangle \times \langle \rho_2^q \rangle \times \langle \pi_1^4 \rangle \times \langle \pi_2^4 \rangle.$$ Moreover $$Y(G)' = \langle x,y,z,\sigma_2,\tau_1,\sigma_1 \rangle.$$ So $Y(G)' \cap Z = \langle x \rangle \times \langle y \rangle \times \langle z \rangle \cong \text{Hom}(M(G), F^*)$. The other conditions are now easy to check.
The same statements hold for $Y(H)$, even using formally the same elements.
Proof of Theorem 1
------------------
We keep the assumptions from the previous subsection and we will show that in this case $G \not \sim_F H$. We will use the minimal Yamazaki covers $Y(G)$ and $Y(H)$ introduced in Lemma \[lemma\_YamazakiCover\] and explicit elements will refer to these groups.
To show that $G$ and $H$ are not in relation over $F$ we will work with Wedderburn decompositions of $FY(G)$ and $FY(H)$. The groups $Y(G)$ and $Y(H)$ are supersolvable as can be seen by their defining relations and hence both groups are monomial, i.e. each irreducible character of these groups is induced by a linear character of a subgroup. This holds over $\mathbb{C}$ by [@Isaacs Theorem 6.22] and over finite fields of characteristic not dividing $|G|$ by [@BrochedelRio Corollary 8].
Each Wedderburn component of $FY(G)$ and $FY(H)$ corresponds to a Wedderburn component of a twisted group algebra $F^\varphi G$ and $F^\varphi H$ respectively. Let $B$ be such a Wedderburn component. Then in fact we can easily determine $\varphi$ from the character $\chi$ corresponding to $B$. Namely if we view $\varphi$ as a product of powers of the generators $\alpha$, $\gamma$, $\kappa$, $\lambda$, $\beta$, $\mu$ and $\nu$, then we can read of $\varphi$ from the powers of $\zeta$, $-1$ and $\xi$ appearing in the values of $\varphi$ on $x$, $z$, $\rho_1^q$, $\rho_2^q$, $y$, $\pi_1^{4}$ and $\pi_2^{4}$ respectively. This follows from the natural correspondence between projective representations and 2-cocycles as explained in Section \[sec:ProjRep\] .
Denote by $F_2$ the field obtained from adjoining a primitive $8$-th root of unity to $F$ and by $F_4$ the field obtained from adjoining a primitive $16$-th root of unity to $F$. Note that these fields are different by our choice of $F$.
We will show that there is a cohomology class $\psi$ in $H^2(G,F^*)$ such that every Wedderburn component of $F^\psi G$ is a matrix ring over the field $F_4$, but there is no cohomology class $\varphi$ in $H^2(H, F^*)$ such that $F^\varphi H$ is the direct sum of matrix rings over $F_4$. This will be proven in the next two lemmas and clearly imply $G \not \sim_F H$.
For $\psi = \gamma \mu$ the Wedderburn decomposition of $F^\psi G$ is a direct sum of matrix rings over $F_4$.
Both $\gamma$ and $\mu$ only influence the subgroup $G_1 = Q_1 \rtimes \langle \pi_1 \rangle$, in the sense that we can choose a cocycle $\psi'$ representing $\psi$ such that $\psi'((g_1,g_2),(1,\tilde{g}_2)) = 1$ for every $g_1 \in G_1$ and $g_2, \tilde{g}_2 \in G_2$. So $k^\psi G = kG_2 \otimes k^{\psi_1}G_1$, where $\psi_1$ denotes the restriction of $\psi$ to $G_1$. It is hence sufficient to show that $k^{\psi_1}G_1$ is a direct sum of matrix rings over $F_4$. A minimal Yamazaki cover of $G_1$ over $F$ is given by $$Y(G_1) = (( \langle z \rangle \times \langle \tau_1 \rangle \times \langle \rho_1 \rangle) \rtimes \langle \sigma_1 \rangle ) \rtimes \langle \pi_1 \rangle$$ where the orders of the generators and the relations between them are exactly as in $Y(G)$.
The Wedderburn decompositions of $FY(G_1)$ can also be computed in positive characteristic as described in [@BrochedelRio]. In particular each Wedderburn component corresponds to a pair $(S,T)$ of subgroups in $Y(G_1)$ such that $S$ has a linear character $\chi$ with kernel $T$ and the induction $\operatorname{ind}_S^{Y(G_1)}(\chi)$ of $\chi$ to $Y(G_1)$ is irreducible. Moreover assume that $\operatorname{ind}_S^{Y(G_1)}(\chi)$ corresponds to some Wedderburn component of $F^{\psi_1}G_1$, i.e. we have $z, \pi_1^{8} \notin T$ and $\rho_1^q \in T$. Our claim will follow once we show that $S$ necessarily contains an element of order $16$ or equivalently:
*Claim:* Every irreducible character of $Y(G_1)$ whose kernel contains $\rho_1$, but not $z$ and $\pi_1^8$, has odd degree.
The claim is true over $F$ if and only if it is true over $\mathbb{C}$. To make the calculations a bit easier we use the bar-notation to denote the natural projection modulo $\langle \rho_1^q, \pi_1^8 \rangle$ and the reduction of $Y(G_1)$ and set $R = Y(G_1)/\langle \rho_1^q, \pi_1^8 \rangle$. We will prove that any irreducible character of $R$ whose kernel does not contain $z$ has odd degree which will imply the claim.
First of all observe that $\langle \bar{z}, \bar{\tau}_1,
\bar{\rho}_1 \rangle$ is an abelian normal subgroup of $R$ of index $2q$ and so Ito’s Theorem [@Isaacs Theorem 6.15] implies that the character degree of each irreducible character of $R$ divides $2q$. So each irreducible character of odd degree has degree $1$ or $q$. Note that the number of characters of degree $1$ of $R$ equals $|R/R'| = |R/\langle \bar{z}, \bar{\tau}_1,
\bar{\sigma}_1 \rangle| = 2q$. By [@Isaacs Theorem 13.26], a very special version of the McKay-conjecture, the number of irreducible characters of odd degree of $R$ is the same as that of $N_R(\langle \bar{\pi}_1 \rangle)$. Now $N_R(\langle \bar{\pi}_1 \rangle) = \langle
\bar{z}, \bar{\rho}_1, \bar{\pi}_1 \rangle$ is an abelian group of order $2q^2$ and has $2q^2$ irreducible characters of odd degree. Moreover $R/\langle \bar{z} \rangle$ has also $2q$ characters of degree $1$ and $\frac{q(q-1)}{2}$ irreducible characters of degree $2$ which are those having $\bar{\tau}_1$ in its kernel. This follows since $$R/\langle \bar{z}, \bar{\tau}_1 \rangle \cong \langle \bar{\rho}_1 \rangle \times (\langle \bar{\sigma}_1 \rangle \rtimes \langle \bar{\pi}_1 \rangle) \cong C_q \times D_{2q},$$ where $D_{2q}$ denotes a dihedral group of order $2q$, and $D_{2q}$ has exactly $\frac{q-1}{2}$ irreducible characters of degree $2$. Moreover the subgroup $\langle \bar{\tau}_1,
\bar{\rho}_1, \bar{\sigma}_1 \rangle$ of $R/ \langle \bar{z}
\rangle$, which is an extraspecial $q$-group, has $q-1$ irreducible characters of degree $q$, see e.g. [@DornhoffA Theorem 31.5]. The induction of each of these characters, which are all not real-valued, to $R/ \langle \bar{z} \rangle$ is irreducible, since it is real on the real conjugacy class of $\bar{\tau}_1$, and two of them induce the same character. So $R/\langle
\bar{z} \rangle$ has $\frac{q-1}{2}$ irreducible characters of degree $2q$. Summing the squares of the degrees of the irreducible characters of $R/\langle \bar{z} \rangle$ obtained so far we obtain $$2q\cdot1^2 + \frac{q(q-1)}{2}\cdot2^2 + \frac{q-1}{2}\cdot(2q)^2 = 2q^3.$$ So there are no further irreducible characters of $R/\langle \bar{z} \rangle$. In particular from all irreducible odd degree characters of $R$ only the $2q$ linear characters of $R$ have $\bar{z}$ in its kernel. But since any other irreducible odd degree character has degree $q$, there are $2q^2$ such characters and since $$(2q^2-2q)q^2 = 2q^4 - 2q^3 = |R| - |R/ \langle \bar{z} \rangle|$$ these are actually all irreducible characters of $R$ which do not contain $\bar{z}$ in its center. Hence the claim follows. This also finishes the proof of the lemma.
There is no $\varphi \in H^2(H,F^*)$ such that every direct summand of $F^\varphi H$ is a matrix algebra over $F_4$.
Since all groups involved are monomial a Wedderburn component of $F^\varphi H$ is determined by a pair $(S,T)$ of subgroups of $Y(H)$ which satisfy the following. $S$ has a linear character $\chi$ with kernel $T$ such that $\operatorname{ind}_S^{Y(H)}(\chi)$ is irreducible and $\chi$ has values on $x$, $y$, $z$, $\rho_1^q$, $\rho_2^q$, $\pi_1^{p^2}$ and $\pi_2^{p^2}$ which correspond to the powers of the natural generators $\alpha$, $\beta$, $\gamma$, $\kappa$, $\lambda$, $\mu$ and $\nu$ appearing in $\varphi$ respectively. The corresponding matrix algebra lies over $F_4$ if and only if $S$ contains an element of order $16$ none of whose powers lies in $T$. So it is sufficient to show that for any $\varphi \in H^2(H, F^*)$ there is a corresponding pair $(S,T)$ such that $S$ contains no element of order $16$. Instead of describing $\varphi$ we will distinguish the different $T$. For example the condition $x \in T$ means that in writing $\varphi$ in the natural generators the factor $\alpha$ does not appear. We will study some cases separately. Note that we can make assumptions only on $\langle x,y,z,\rho_1^q, \rho_2^q,
\pi_1^{p^2}, \pi_2^{p^2} \rangle \cap T$, since this fixes which natural generators appear in $\varphi$. The general goal in all cases will be to achieve $\sigma_2 \in S \setminus T$, because then an element of order $16$ does not commute with $S/T$, so there can be no element of order $16$ in $S$ which has no power in $T$. Set $Z = Z(Y(H)) = \langle x,\rho_2^q, y, \pi_1^2, z, \rho_1^q, \pi_2^2 \rangle$.
- $x,z \in T$.\
Let $S = \langle Z, \sigma_2, \rho_2, \tau_1, \rho_1, \sigma_1,
\pi_2 \rangle$. Then $S' = \langle \sigma_2^q, z, \tau_1, \sigma_1
\rangle$ and let $T$ be a subgroup of $S$ containing $S'$ such that $S/T$ is cyclic and $T$ does not contain $\sigma_2$. Let $\chi$ be a linear character of $S$ with kernel $T$. Then $\chi' =
\operatorname{ind}_S^{Y(H)} \chi$ is of degree $2$ and $\chi'(\sigma_2) = \chi(\sigma_2) + \chi(\sigma_2)^{-1} \neq 2$. Moreover $\chi'$ is irreducible, since otherwise it would decompose into two linear characters. But linear characters contain $\sigma_2$ in its kernel, since $\sigma_2 \in Y(H)'$, and then we would have $\chi'(\sigma_2) = 2$.
- $x \notin T$, $z \in T$.\
Set $S = \langle Z, \sigma_2, \rho_2, \tau_1, \sigma_1, \pi_2
\rangle$. Then $S' = \langle \sigma_2^q, z, \tau_1, \sigma_1
\rangle$. Let $T$ again be a subgroup of $S$ containing $S'$ such that $S/T$ is cyclic, $\sigma_2 \notin T$ and let $\chi$ and $\chi'$ be defined similarly as in Case 1. Then $\chi'$ is a character of degree 10 such that $\chi'(\sigma_2) =
5(\chi(\sigma_2)+\chi(\sigma_2)^{-1})$. This means that the restriction of $\chi'$ to $\langle \sigma_2, \pi_1 \rangle$ decomposes into five $2$-dimensional characters. So if $\chi'$ decomposes it decomposes into characters of even degree. But on the other hand its restriction to $(\langle x \rangle \times \langle \rho_2 \rangle )
\rtimes \langle \rho_1 \rangle$ has to decompose into characters of degree $5$, since these are the only characters of this group not having $x$ in the kernel.
- $x \in T$, $z \notin T$.\
Set $S = \langle Z, \sigma_2, \rho_2, \tau_1, \rho_1, \pi_2
\rangle$. Note that $\tau_1^{\pi_2} = z\tau_1^{-1} =
\tau_1(z\tau_1^3)$. So we have $S' = \langle \sigma_2^q, x,
z\tau_1^3 \rangle$. Again let $T$ be a normal subgroup of $S$ such that $S/T$ is cyclic, $\sigma_2 \notin T$ and let $\chi$ and $\chi'$ be defined as in the previous cases. If $\chi'$ decomposes then the summands have even degree, due to the value of $\chi'$ on $\sigma_2$ and its restriction to $\langle \sigma_2, \pi_1 \rangle$. But at the same time the degree of a summand would be divisible by $5$, due to its character value $0$ on $z$ and the character theory of the extraspecial $q$-group $\langle z, \tau_1, \sigma_1 \rangle $.
- $x,z \notin T$.\
Set $S= \langle Z, \sigma_2, \tau_1, \rho_1, \pi_2 \rangle$. Then $S' = \langle z\tau_1^3 \rangle$. Let again $T$, $\chi$ and $\chi'$ have analogues properties as before such that $\sigma_2^q
\notin T$. Note that $\tau_1 \not \in T$. By Frobenius reciprocity and Clifford theory we have, considering the scalar product of characters, $$\langle \chi' , \chi' \rangle_{Y(H)} = \sum_{g \in Y(H)/S} \langle \chi, \chi^g \rangle_S.$$ Now a system of coset representatives of $Y(H)/S$ is given by $$\{\pi_1^i \rho_2^j \sigma_1^k \ | \ 0 \leq i \leq 1, \ 0 \leq j,k \leq q-1 \}.$$ Set $a_{i,j,k} = \pi_1^i \rho_2^j \sigma_1^k$. Then $\sigma_2^{a_{i,j,k}} = \sigma^{(1+qj)\cdot (-1)^i}$ and $\tau_1^{a_{i,j,k}} = z^k\tau_1$. Since $\langle \tau_1, \sigma_2 \rangle \cap T = 1$ we hence have $\chi^{a_{i,j,k}} = \chi$ if and only if $i = j = k = 0$. So $\chi'$ is irreducible.
\[rem:OddCase\] The calculations of the cohomology groups for the groups $G$ and $H$ from Dade’s example suggest that if the groups are of odd order then it is very well possible that $G \sim_F H$ over any field $F$. In the words of Passman the “surprise” in the proof of Dade is the fact that $FG \cong FH$ for fields of characteristic $q$ and that “this isomorphism is so easily proved” [@Passman p. 664]. This proof relies on the fact that setting $e = \frac{1}{p}\sum_{i=0}^{p-1} \pi_1^{pi} $ in $FG$ and $FH$ respectively $FG$ and $FH$ are direct sums of algebras isomorphic to $eFG$ and $eFH$ respectively. As $eFG \cong
F(G/\langle \pi_1^p \rangle) \cong F(H/\langle \pi_1^p \rangle)
\cong eFH$ the isomorphism of $FG$ and $FH$ follows immediately.
It seems impossible to imitate this argument in the twisted case, since there is no natural idempotent in the twisted group ring of a cyclic group corresponding to $e$. For example $\mathbb{F}_5^\alpha C_4$ is a simple algebra isomorphic to $\mathbb{F}_{5^4}$ for $[\alpha] \in H^2(G, \mathbb{F}_5^*)$ of order $4$, so it has no quotients which “kill” exactly the cyclic group of order $2$. This is a special instance of the fact that a twisted group ring of $G$ has no “obvious homomorphism” [@Passman p. 14] to some twisted group ring of a given quotient of $G$. So though $G \sim_F H$ might still be true for any field $F$ the arguments to prove this would be different from the argument of Dade.
Also Yamazaki covers can not bring the whole solution as $H^2(G,
\mathbb{F}^*)$ can be infinite, e.g. for $F = \mathbb{Q}$, and then no Yamazaki cover exists.
The probably most famous example obtained in the study of the classical group ring isomorphism problem is Hertweck’s counterexample to the integral isomorphism problem [@Hertweck]. This counterexample consists of two non-isomorphic groups $G$ and $H$ of order $2^{21}\cdot 97^{28}$ such that $\mathbb{Z}G \cong \mathbb{Z}H$. It is not clear to us if there exists a ring $R$ such that $G \not\sim_R H$. But it clear that $RG\cong RH$ and $H^2(G,R^*)\cong H^2(H,R^*)$ for any commutative ring $R$. This follows from the fact that $RG \cong R
\otimes_\mathbb{Z} \mathbb{Z}G$ and the functorial definition of group cohomology, $H^n(G, M) \cong
\operatorname{Ext}^n_{\mathbb{Z}G}(\mathbb{Z}, G)$ for any $G$-module $M$ and where $\operatorname{Ext}$ denotes the $\operatorname{Ext}$-functor. So $H^2(G, M)$ depends only on the group ring $\mathbb{Z}G$ and not $G$ itself. It would be very interesting to determine if $G \sim_R H$ indeed holds independently of $R$.
**Acknowledgement:** We thank Yuval Ginosar for useful discussions.
[^1]: The first author is a postdoctoral researcher of the Research Foundation Flanders (FWO - Vlaanderen). We are grateful for the Technion - Israel Institute of Technology, for supporting the first author’s visit to Haifa
|
---
abstract: 'We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.'
address: |
Institute for Geometry and Topology\
University of Stuttgart\
Pfaffenwaldring 57\
70569 Stuttgart\
Germany
author:
- 'M. J. D. Hamilton'
title: The minimal genus problem for elliptic surfaces
---
Introduction
============
In their classical work from 1961, Kervaire and Milnor [@KM] showed that certain second homology classes in simply-connected 4-manifolds are not represented by embedded spheres. It therefore became an interesting question to find for a given homology class in a 4-manifold the minimal genus of an embedded closed connected oriented surface realizing that class. This question has been solved at least partly for rational and ruled surfaces and for 4-manifolds with a free circle action [@FV1; @FV2; @La; @Li; @LiLi1; @LiLi2; @LiLi3; @LiLi4; @R]. On symplectic 4-manifolds the question is related to the Thom conjecture [@KrMr; @MSzT; @OzSz]. In general, the adjunction inequality from Seiberg-Witten theory gives a lower bound on the genus of a surface representing a homology class in a closed oriented 4-manifold with a basic class and we can then ask if this lower bound is indeed realized. Usually the question is more tractable for classes of positive self-intersection and is still open in most situations in the case of negative self-intersections. In particular, it is still unknown whether there exist embedded spheres in the $K3$ surface of arbitrarily negative self-intersection.
An interesting class of 4-manifolds are elliptic surfaces. We will restrict to minimal simply-connected elliptic surfaces with $b_2^+> 1$, but generalizations should be possible. It is natural to consider these 4-manifolds, because they are the second case in the Enriques-Kodaira classification of minimal simply-connected complex surfaces after the complex projective plane and the Hirzebruch surfaces, for which the minimal genus problem has already been solved [@KrMr; @R]. The remaining third case are surfaces of general type.
Note that every orientation preserving self-diffeomorphism of a closed oriented 4-manifold induces an isometry of the intersection form on the second homology (modulo torsion). A very useful fact is that for elliptic surfaces the image of the orientation preserving diffeomorphism group in the orthogonal group of the intersection form is known. This is due to Borcea, Donaldson and Matumoto [@B; @D; @Mat] for the $K3$ surface and to Friedman-Morgan and Lönne in the general case [@FM; @Lo]. We will combine this knowledge with the work of Wall on the transitivity of the orthogonal groups of unimodular quadratic forms [@W]. Similar to the case of rational surfaces, this will allow us to reduce the problem of representing a homology class by a minimal genus surface to certain special classes. We cannot treat the minimal genus problem in full generality. Instead we will concentrate on the first interesting special cases that come to mind. To state one of the results, we will prove the following in the special case that the elliptic surface has no multiple fibres, i.e. is given by a surface $E(n)$ with $n\geq 2$:
Let $X$ be an elliptic surface diffeomorphic to $E(n)$ with $n\geq 2$. Suppose $A$ is a non-zero class in $H_2(X;\mathbb{Z})$ orthogonal to the canonical class $K$ and of self-intersection $A^2=2c-2$ with $c\geq 0$. Then $A$ is represented by a surface of genus $c$ in $X$. This is the minimal possible genus.
Note that the self-intersection number of classes orthogonal to the canonical class is always even, because the canonical class is characteristic. There is a similar, slightly more restrictive theorem in the case of elliptic surfaces with multiple fibres. We are also interested if we can realize homology classes by surfaces that are contained in certain nice neighbourhoods inside the elliptic surface, given by copies of an embedded Gompf nucleus $N(2)$.
Notations {#notations .unnumbered}
---------
In the following, $X$ will denote a minimal simply-connected elliptic surface with the complex orientation. By an elliptic surface we always mean a surface of this kind. Using the classification of elliptic surfaces [@GS], $X$ is diffeomorphic to $E(n)_{p,q}$, where the coprime indices denote logarithmic transformations. We restrict to the case $n\geq 2$ or equivalently $b_2^+>1$; see [@La] for a discussion of Dolgachev surfaces $E(1)_{p,q}$ with $b_2^+=1$. All self-diffeomorphisms of $X$ are orientation preserving. We often denote a closed oriented surface in $X$ and the homology class it represents by the same symbol.
Action of the diffeomorphism group
==================================
Let $H_2(X)$ denote the integral second homology of $X$ and $\text{Diff}^+(X)$ the group of orientation preserving self-diffeomorphisms of $X$. The intersection form on second homology induces a unimodular quadratic form on the lattice $H_2(X)$. We denote by $O$ the orthogonal group of all automorphisms of $H_2(X)$ that preserve the intersection form. The elements of this group are called automorphisms of the intersection form. The action of diffeomorphisms on homology defines a group homomorphism $\text{Diff}^+(X)\rightarrow O$.
There is a homomorphism $O\rightarrow\mathbb{Z}_2$, called the [*spinor norm*]{}, which is defined as follows. We can choose an orientation on all maximal positive definite linear subspaces of $H_2(X;\mathbb{R})$, cf. [@Ray]: Fix any such subspace $U_0$ and let $\pi\colon H_2(X;\mathbb{R})\rightarrow U_0$ denote the orthogonal projection. The restriction of $\pi$ to any maximal positive definite subspace $U$ is an isomorphism with $U_0$. Choosing an orientation for $U_0$ we get an orientation for all maximal positive definite subspaces $U$ via $\pi$. This orientation varies continuously with $U$. The spinor norm of an element $\phi\in O$ is defined to be $\pm 1$ depending on whether $\phi$ preserves or reverses the orientation when mapping a maximal positive definite subspace of $H_2(X;\mathbb{R})$ to another one. A deformation argument shows that this does not depend on the choice of such a subspace. The subgroup of $O$ of elements of spinor norm $1$ is denoted by $O'$.
We let $K$ denote the canonical class of $X$, which is minus the first Chern class. If $X$ is not the $K3$ surface and hence $K\neq 0$, let $k$ denote the Poincaré dual of $K$ divided by its divisibility. If $X$ is the $K3$ surface, let $k$ denote the class of a general fibre. In any case, $k$ is a primitive homology class of self-intersection zero. We also choose a second homology class $V$ such that $k\cdot V=1$. For example if $X$ has no multiple fibres we can choose for $V$ a section of an elliptic fibration. We denote by $O_k$ the automorphisms of the intersection form fixing $k$ and by $O_k'$ those of spinor norm $1$.
The next statement follows from Theorem 8 in [@Lo] due to Lönne.
\[thm Lonne\] The image of the diffeomorphism group $\text{Diff}^+(X)$ in $O$ is equal to $O'$ for the $K3$ surface and contains $O_k'$ for all other elliptic surfaces $X$.
We now consider integral unimodular quadratic forms in general. We let $H$ denote the even hyperbolic form of rank $2$ and $E_8$ the standard positive definite even form of rank $8$. A [*standard basis*]{} for $H$ is a basis $e,f$ such that $$e^2=0, f^2=0, e\cdot f=1.$$ Let $Q$ denote the quadratic form $Q=lH\oplus m(-E_8)$ with $l\geq 2$ and $m\in\mathbb{Z}$. In Theorem 6 in [@W], Wall proved the following.
The orthogonal group of $Q$ acts transitively on primitive elements of given square.
We want to deduce the following.
\[Prop subgroup spinor norm 1\] The subgroup of elements of spinor norm 1 in the orthogonal group of $Q$ acts transitively on primitive elements of given square.
We first prove the following lemma.
For any even number $2a$ there exist primitive elements $p$ and $q$ of square $2a$ and automorphisms of $Q$ of spinor norm $+1$ and $-1$ which map $p$ to $q$.
We consider $Q=lH\oplus m(-E_8)$ and let $e, f$ denote a standard basis for the first $H$ summand. Let $p=e+af$ and $q=-e-af$. Then $p$ and $q$ are primitive elements and $p^2=q^2=2a$. Consider the automorphism of $Q$ which is minus the identity on the first $H$ summand and the identity on all other summands and the automorphism which is minus the identity on the first two $H$ summands and the identity on all other summands. These automorphisms have spinor norm $-1$ and $+1$ and map $p$ to $q$.
We now prove Proposition \[Prop subgroup spinor norm 1\].
Let $x$ and $y$ be arbitrary primitive elements of square $2a$ and let $p$ and $q$ be the elements from the lemma of the same square. By Wall’s theorem there exist automorphisms in $O$ mapping $x$ to $p$ and $q$ to $y$. Choosing an automorphism that maps $p$ to $q$ of the correct spinor norm we get by composing an automorphism of spinor norm $+1$ mapping $x$ to $y$.
We now consider the elliptic surface $X$.
The self-intersection number $V^2$ is even if and only if $X$ is spin.
The intersection form on the span of $k$ and $V$ is unimodular, hence it is unimodular on the orthogonal complement. The intersection form on this complement is even, since the canonical class $K$ is characteristic. The claim now follows because $X$ is spin if and only if the intersection form on both summands is even.
Let $V^2=2b$ in the spin case and $V^2=2b+1$ in the non-spin case.
Define a homology class $W=V-bk$. Then the intersection form on the span of $k$ and $W$ is the form $H$ in the spin case and the form $H'$ given by $$H'=\left(\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array}\right)$$ in the non-spin case. Note that $H'$ is isomorphic to $\langle +1\rangle \oplus \langle -1 \rangle$.
The complete intersection form of $X$ is then given by $$\label{eqn int form}
Q_X=H\oplus lH\oplus m(-E_8)\text{ or } H'\oplus lH\oplus m(-E_8),$$ depending on whether $X$ is spin or non-spin, where $l\geq 2$ since $b_2^+\geq 3$ and $m>0$. We also want to choose a standard basis for the second $H$ summand in the intersection form. We first consider the case of the $K3$ surface: It is known that the $K3$ surface contains a rim torus $R$ of self-intersection zero and a vanishing sphere $S$ of self-intersection $-2$ such that $R$ and $S$ intersect transversely in one positive point; see page 73 in [@GS]. Both arise from the fibre sum construction $K3=E(1)\#_{F=F}E(1)$ along a general fibre $F$, given by $$K3=(E(1)\setminus\text{int}\,\nu F)\cup_\psi(E(1)\setminus\text{int}\,\nu F),$$ with fibred tubular neighbourhood $\nu F\cong S^1\times S^1\times D^2$ and gluing diffeomorphism $\psi$ on the boundary of the tubular neighbourhood. The gluing diffeomorphism preserves the splitting and is given by the identity on the torus and complex conjugation on $\partial D^2$. The rim torus in this construction is given by $$R=S^1\times \{\ast\}\times \partial D^2\subset \partial\nu F\subset K3.$$ The vanishing sphere is obtained by sewing together two vanishing disks of relative self-intersection $-1$ coming from elliptic Lefschetz fibrations on $E(1)$. These vanishing disks bound the vanishing cycles $$\{\ast\}\times S^1\times\{\ast\}\subset \partial\nu F$$ in each copy of $E(1)\setminus\text{int}\,\nu F$, that get identified under the gluing diffeomorphism. Recall the following definition from [@Gnuc]:
The [*nucleus $N(2)$*]{} is defined as the 4-manifold with boundary given by the neighbourhood of a cusp fibre and a section of self-intersection $-2$ in $K3$.
The nucleus contains also a smooth torus fibre homologous to the cusp. The second homology of the nucleus is isomorphic to $\mathbb{Z}^2$ and spanned by this torus and the section. In addition to the nucleus containing a fibre and a section given by the definition, the $K3$ surface contains two other embedded copies of $N(2)$, disjoint from the first one. The rim torus $R$ and the vanishing sphere $S$ are embedded in one such copy [@GM] and correspond to the fibre and the section. Since this nucleus is disjoint from a general fibre it is still contained in an arbitrary elliptic surface $X$ of the type above, because the fibre sums and the logarithmic transformations resulting in the manifold $X=E(n)_{p,q}$ are performed in the complement of the nucleus. We can also choose the surface representing the class $V$ to be disjoint from this nucleus.
Let $T$ denote the torus of self-intersection zero obtained by smoothing the intersection between $R$ and $S$. Then $T$ represents the class $R+S$ and the classes $R$ and $T$ are a standard basis for the second $H$ summand in the intersection form of the elliptic surface $X$.
Using Theorem \[thm Lonne\] and Proposition \[Prop subgroup spinor norm 1\] we have the following:
\[prop main\] Let $X$ be an elliptic surface and $B$ an arbitrary class in the subgroup $lH\oplus m(-E_8)$ of $H_2(X)$ as in equation . Then we can map $B$ to any other class in $lH\oplus m(-E_8)$ of the same square and divisibility by a self-diffeomorphism of $X$. In particular, we can map $B$ to a linear combination of the classes $R$ and $S$. This diffeomorphism is the identity on the other summand $H$ or $H'$ of $H_2(X)$, given by the span of $k$ and $W$. Suppose $X$ is the $K3$ surface. Then we can map any class in $H_2(X)$ to any other class in $H_2(X)$ of the same divisibility and square using a self-diffeomorphism of $X$.
The result for the $K3$ surface can also be found in [@KrMr0; @La]. As a final preparation, we consider the following theorem on the adjunction inequality from Seiberg-Witten theory [@GS; @KrMr; @OzSz]:
Let $Y$ be a closed oriented 4-manifold with $b_2^+>1$. Assume that $\Sigma$ is an embedded oriented connected surface in $Y$ of genus $g(\Sigma)$ with self-intersection $\Sigma^2\geq 0$, such that the class represented by $\Sigma$ is non-zero. Then for every Seiberg-Witten basic class $L$ we have $$2g(\Sigma)-2\geq \Sigma^2+|L\cdot\Sigma|.$$ If $Y$ is of simple type and $g(\Sigma)>0$, then the same inequality holds for $\Sigma\subset Y$ with arbitrary square $\Sigma^2$.
A basic class is a characteristic class in $H^2(Y;\mathbb{Z})$ with non-vanishing Seiberg-Witten invariant. If $L$ is a basic class, then $-L$ is also a basic class. The basic classes of the elliptic surfaces $X=E(n)_{p,q}$ are completely known [@FS]. They are given by the set $$\{rk\mid r\equiv npq-p-q\bmod 2,\, |r|\leq npq-p-q\},$$ where $k$ is the primitive class as above. The canonical class of the elliptic surface $E(n)_{p,q}$ is given by $$K=(npq-p-q)k.$$ It follows that the basic classes are certain multiples of the class $k$, where the maximal values at the end are given (up to sign) by the canonical class $K$. Hence we get:
The adjunction inequality for the elliptic surfaces $X$ reduces to the statement that $$2g(\Sigma)-2\geq \Sigma^2+|K\cdot\Sigma|$$ for every embedded surface $\Sigma$ of genus $g(\Sigma)$, representing a non-zero class with self-intersection $\Sigma^2\geq 0$.
Minimal genus problem for the $K3$ surface
==========================================
The minimal genus problem for classes of non-negative square in the $K3$ surface has already been solved [@La]. In this section we recall this solution as a preparation for the general case. The $K3$ surface has canonical class $K=0$. Hence the adjunction inequality implies for the genus of a smooth surface $\Sigma$ that $2g(\Sigma)-2\geq\Sigma^2$ if the homology class represented by this surface is non-zero.
[*The standard surface of genus $g$ embedded in the nucleus $N(2)$*]{} is by definition the section of self-intersection $-2$ ($g=0$), the general fibre of self-intersection $0$ ($g=1$) or the surface of genus $g\geq 2$ and self-intersection $2g-2$ obtained by smoothing the intersection points of the section and $g$ parallel copies of the general fibre. These surfaces represent primitive homology classes.
According to Proposition \[prop main\] we can map any primitive class in the $K3$ surface via a self-diffeomorphism to any other primitive class of the same square. Hence every primitive class of self-intersection $2c-2$ with $c\geq 0$ is represented by a surface of genus $c$ inside some nucleus in $K3$. This is the minimal possible genus according to the adjunction inequality. In particular, every primitive homology class in the $K3$ surface of square zero is represented by the standard torus in a nucleus $N(2)$ inside $K3$.
To solve the case of divisible classes with non-negative square we use Lemma 7.7 in [@KrMr1] due to Kronheimer-Mrowka (see also Lemma 14 in [@La]):
\[lem Lawson\] Suppose that $Y$ is a closed connected oriented 4-manifold. For an embedded surface $\Sigma$ let $a(\Sigma)=2g(\Sigma)-2-\Sigma^2$. If $h\in H_2(Y;\mathbb{Z})$ is a homology class with $h^2\geq 0$ and $\Sigma_h$ is a surface of genus $g$ representing $h$ and $g\geq 1$ when $h^2=0$, then for all $r>0$, the class $rh$ can be represented by an embedded surface $\Sigma_{rh}$ with $$a(\Sigma_{rh})=ra(\Sigma_h).$$
We can apply the construction of this lemma to divisible classes of non-negative square inside the nucleus $N(2)$ to get surfaces that represent these classes in the nucleus (the construction in the proof of this lemma works in a tubular neighbourhood of $\Sigma_h$ and does not need the assumption that $Y$ is closed). In this case $a(\Sigma_h)$ is zero, hence also $a(\Sigma_{rh})$ is zero. We have:
\[stand div N(2)\] Every non-zero class in $H_2(N(2))$, not necessarily primitive, which has self-intersection $2c-2$ with $c\geq 0$ is represented by an embedded surface of genus $c$ in $N(2)$.
We call the surfaces in the nucleus obtained by the construction preceding Corollary \[stand div N(2)\] [*standard*]{}.
The transitivity of the action of the diffeomorphism group then implies that every divisible class of non-negative square in $K3$ can also be represented by a standard surface inside a nucleus $N(2)$. Hence we get:
\[cor K3\] Consider the $K3$ surface. Every non-zero homology class of self-intersection $2c-2$ with $c\geq 0$ is represented by a surface of genus $c$. We can assume that it is embedded as the standard surface in a nucleus $N(2)$ inside $K3$. This is the minimal possible genus.
This result can be found in the paper [@La] due to Lawson, except for the observation that these surfaces of minimal genus can be realized inside an embedded nucleus $N(2)$ in the $K3$ surface.
Minimal genus problem for other elliptic surfaces {#section min genus other elliptic}
=================================================
We now consider the general case of minimal simply-connected elliptic surfaces $X$ with $b_2^+>1$. The adjunction inequality implies for surfaces $\Sigma$ orthogonal to $K$ again that $2g(\Sigma)-2\geq \Sigma^2$. The self-intersection of such a surface is even, because the canonical class is characteristic. Using Proposition \[prop main\] and Corollary \[stand div N(2)\] we get:
Let $X$ be an elliptic surface. Then every non-zero homology class $A$ of self-intersection $2c-2$ with $c\geq 0$ that is orthogonal to the classes $K$ and $V$ is represented by a surface of genus $c$. We can assume that it is embedded as the standard surface in a nucleus $N(2)$ inside the 4-manifold $X$. This is the minimal possible genus.
The assumptions imply that $A$ can be mapped via a diffeomorphism to $A'=\gamma R+\delta S$. Since $R$ and $S$ are constructed in a nucleus $N(2)$ the claim follows.
If we relax the assumption and only assume that $A$ is orthogonal to $K$, it seems that the surface is in general not contained in a nucleus $N(2)$. For example the general fibre is contained in a nucleus $N(n)_{p,q}$.
We can deal with the case $A^2=-2$ in a slightly more general situation:
Let $X$ be an elliptic surface. Then any homology class $A$ orthogonal to $K$ and of self-intersection $-2$ is represented by the standard sphere in a nucleus $N(2)$ in the 4-manifold $X$.
The assumptions imply that there exists a self-diffeomorphism of $X$ mapping $A$ to $$A'=\alpha k+S,$$ where $S$ is the vanishing sphere. Consider the following map $\phi$ on $H_2(X)$ which on the first two summands of the intersection form is given by $$\begin{aligned}
k&\mapsto k\\
W&\mapsto W+\alpha R\\
R&\mapsto R\\
S&\mapsto S-\alpha k\end{aligned}$$ and is the identity on all other summands. It is easy to check that $\phi$ is an isometry. Letting $\alpha$ be a real number and taking $\alpha\rightarrow 0$ we see that $\phi$ has spinor norm $+1$. Hence it is an element in $O'_k$ and therefore induced by a self-diffeomorphism. It maps $A'$ to $S$. This implies the claim.
This result should be compared to the fact that every class of square $-2$ in the complement of a general fibre in $X$ is represented by an embedded sphere [@FM; @Lo].
We now restrict to the case of elliptic surfaces without multiple fibres, i.e. $X=E(n)$ with $n\geq 2$, because the following arguments seem to work only in this case. The class $k$ is represented by a general fibre $F$. We also have the rim torus $R$. Proposition \[prop main\] implies:
If $A$ is a homology class orthogonal to $K$ and of self-intersection zero, then there exists a self-diffeomorphism of $X$ that maps $A$ to $$A'=\alpha F+\gamma R.$$
We want to show that $A'$ can be represented by an embedded torus. The construction involves the circle sum from [@LiLi4]. The idea is the following: Let $\Sigma_0$ and $\Sigma_1$ denote two disjoint connected embedded oriented surfaces in a 4-manifold $Y$. We can tube them together in the standard way to get a surface of genus $g(\Sigma_0)+g(\Sigma_1)$. Sometimes, however, we can perform a different surgery that results in a surface of smaller genus. Let $S^1_i\subset \Sigma_i$ denote embedded circles that represent non-trivial homology classes in the surfaces. In each surface we delete an annulus $S_i^1\times I$. We get two disjoint surfaces whose boundaries consist of two circles for each surface. We want to connect these circles by annuli embedded in $Y$. There are several ways to do this: One possibility is to connect the circles from the same surface. In this way we simply get back the surfaces $\Sigma_0$ and $\Sigma_1$. Another possibility is to connect the boundary circles from different surfaces. If this is possible we get an embedded connected surface of genus $g(\Sigma_0)+g(\Sigma_1)-1$ representing the class $[\Sigma_0]+[\Sigma_1]$.
The construction works if we can find an embedded annulus $\Delta$ in $Y$ that intersects the surfaces $\Sigma_0$ and $\Sigma_1$ precisely in the circles $S^1_0$ and $S^1_1$. We also need a nowhere vanishing normal vector field along $\Delta$ that at the ends of $\Delta$ is tangential to the surfaces $\Sigma_0$ and $\Sigma_1$. The annuli connecting the four boundary circles are then constructed as normal push-offs of the annulus $\Delta$.
There exists an embedded annulus $\Delta$ connecting the tori $F$ and $R$ that satisfies the necessary assumptions for the circle sum in [@LiLi4].
The elliptic surface $X=E(n)$ is obtained as a fibre sum of $E(n-1)$ and $E(1)$ along a general fibre. Let $S^1\times S^1\times D^2$ denote a tubular neighbourhood of the fibre in one of the summands. We think of $D^2$ as the unit disk in the complex plane and let $I$ denote the interval $[\frac {1}{2},1]$ along the real axis. In forming the fibre sum we delete the open tubular neighbourhood of radius $\frac{1}{4}$ of the general fibre in the centre of the tubular neighbourhood. The fibre $F$ in $X$ is realized as $S^1\times S^1\times \{\frac{1}{2}\}$ while the rim torus $R$ is $S^1\times \{\ast\}\times \partial D^2$. Consider the annulus $\Delta=S^1\times \{\ast\}\times I$. It intersects the tori $F$ and $R$ precisely in the circles $S_F^1=S^1\times\{\ast\}\times \{\frac{1}{2}\}$ and $S_R^1=S^1\times\{\ast\}\times \{1\}$. The tangent bundle of $S^1\times S^1\times I\times \partial D^2$ is canonically trivial. Let $v_F$ be a unit tangent vector to $S^1$ in the point $\ast$ and $v_R$ a unit tangent vector to $\partial D^2$ in $1$. Then $$e_F=(0,v_F,0,0)\quad\text{along $S^1_F$}$$ and $$e_R=(0,0,0,v_R)\quad\text{along $S^1_R$}$$ are framings of the circles $S_F^1$ and $S_R^1$ inside the tori. Consider the normal vector field along the annulus $\Delta$, given on $S^1\times \{\ast\}\times t$ by $$e=(0,(2-2t)v_F,0,(2t-1)v_R).$$ This is equal to the framings $e_F$ and $e_R$ on the boundary and is the required framing of the annulus.
This construction allows us to circle sum $F$ and $R$. A similar, but easier construction allows us to circle sum $|\alpha|$ parallel copies of $F$ and $|\gamma|$ parallel copies of $R$ with a suitable orientation to get embedded tori $\Sigma_0$ and $\Sigma_1$ representing the classes $\alpha F$ and $\gamma R$. The torus $\Sigma_0$ contains as an open subset a copy of the torus $F$ with an annulus deleted, and similarly for $\Sigma_1$. Circle summing $\Sigma_0$ and $\Sigma_1$ along these subsets we get an embedded torus representing the class $\alpha F+\gamma R$. This construction proves:
Let $X$ be an elliptic surface without multiple fibres. Then any homology class $A$ orthogonal to $K$ and of self-intersection zero is represented by an embedded torus.
This is clearly the minimal possible genus allowed by the adjunction inequality if the class $A$ is non-zero. The same method can be used to prove the following generalization:
\[main thm without multiple\] Let $X$ be an elliptic surface without multiple fibres. Suppose $A$ is a non-zero homology class orthogonal to $K$ such that $A^2=2c-2$ with $c\geq 0$. Then $A$ is represented by a surface of genus $c$ in $X$. This is the minimal possible genus.
The cases $c=0$ and $c=1$ have been proved above. We can assume that $c\geq 2$. The assumptions imply that there exists a self-diffeomorphism of $X$ mapping $A$ to $$A'=\alpha F+ \gamma R+ \delta T,$$ where $\gamma$ and $\delta$ are positive with $\gamma\delta=c-1$. We circle sum $|\alpha|$ parallel copies of $F$ with a suitable orientation to get a torus $\Sigma_0$ representing $\alpha F$. Taking circle sums of parallel copies of the tori $R$ and $T$ we get tori representing $\gamma R$ and $\delta T$ that intersect transversely in $\gamma\delta$ points. Smoothing these intersections we get a surface $\Sigma_1$ of genus $\gamma\delta +1=c$. This surface contains as an open subset a copy of the torus $R$ with an annulus and $\delta$ points deleted. We circle sum the surface $\Sigma_1$ to the torus $\Sigma_0$ to get an embedded surface of genus $c$ representing $A'$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank T. Jentsch and D. Kotschick for reading a preliminary version of this paper and the referee for helpful comments.
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abstract: 'The presence of a space charge region induces an internal electric field within the charged region that, in a ferroelectric material, would rotate the polarisations to align with the field. The strength of the induced field would therefore determine the domain patterns and polarisation switching properties of the material. Using a phase-field model, we investigate the effect of charged layers in fully and partially depleted [[BiFeO]{}[$_{\text{3}}$]{}]{} thin films in the rhombohedral phase. While the domain pattern in a charge-free [[BiFeO]{}[$_{\text{3}}$]{}]{} film consists of only two polarisation variants, we observe complex patterns with four coexisting variants that form within the charged layers at sufficiently high induced fields. These variants form a head-to-head configuration with an interface that is either wavy or planar depending on the internal field strength, which is determined by the charge density as well as the thickness of the charged layer. For depletion layers with sufficient thickness, there exists a range of charge density values for which the interface is wavy, while at high densities the interface becomes planar. We find that films with wavy interfaces exhibit enhanced susceptibilities with reduced hystereses compared to the charge-free film. The results of our work suggest that introducing space charge regions by careful selection of dopant density and electrode materials can engineer domain patterns that yield a higher response with a smaller hysteresis.'
author:
- Wei Li Cheah
- Nathaniel Ng
- Rajeev Ahluwalia
bibliography:
- 'elsarticle-template.bib'
title: Influence of space charge on domain patterns and susceptibility in a rhombohedral ferroelectric film
---
Introduction
============
When a metal electrode is placed in contact with a semiconductor material, there may be a flow of charge carriers from the semiconductor to the metal, depending on the work function of the metal [@Damjanovic1998]. An accumulation of charge carriers results at the electrode/semiconductor interface, inducing an electrostatic potential that depletes these carriers in adjacent regions [@Zubko2006; @Yang2012thesis]. This space charge region, or depletion layer, is typically formed near interfaces such as grain boundaries and interfaces between heterophases, as well as near surfaces. In ferroelectric materials which are regarded as wide band gap semiconductors, the presence of such regions may affect the properties of the material, in many cases detrimental for applications such as non-volatile memory devices and capacitors. It lowers the ferroelectric phase transition temperature and suppresses ferroelectricity in the space charge regions [@Bratkovsky2000]. Furthermore, the domain pattern that forms under the influence of space charge can substantially differ from those without [@Xiao2005; @Ng2012; @Ahluwalia2013; @Wang2012]. Consequently when such a sample is switched, the hysteresis loop constricts with a lowering of the coercive field, and depending on the extent of the depletion layer, the remnant polarisation as well [@Zubko2006; @Wang2012; @Baudry2005; @Warren1994]. If the distribution of charges in the sample is asymmetric, the resulting internal electric field leads to a preferred polarisation direction, thereby shifting the hysteresis loop along the electric field axis; this phenomena is called imprint [@Wang2012; @Misirlioglu2011effect].
Using a scanning probe microscope (SPM) tip to apply an electric field over a small region in a (100) [[BiFeO]{}[$_{\text{3}}$]{}]{} film, Vasudevan *et al.* observed the formation of domains with polarisations that form a closed loop (i.e. head-to-tail configuration of domains) at low applied fields, and domains with polarisations directed towards a central point (head-to-head configuration) at very high fields. [@Vasudevan2011]. Similar patterns may also be formed within space charge regions in ferroelectric films due to the internal electric field induced by the charges. Head-to-tail and 180 domains, whose walls are charge-neutral, are the more stable configurations in a charge-free material, whereas charged walls between head-to-head or tail-to-tail domains may stabilise in the presence of space charge [@Park1998; @Gureev2011; @Wu2006; @Zuo2014]. Usually in theoretical investigations concerning the latter type of domains, the models have flat interfaces between the domains [@Wu2006; @Zuo2014; @Sluka2012; @Eliseev2011]. Other studies, both experimental [@Randall1987; @Han2014; @Abplanalp1998] and theoretical [@Misirlioglu2012], report the formation of zigzagged domain walls. It has been suggested that the formation of such interfaces allows the electrostatic energy to be reduced, as locally at the domain walls the polarisations are rotated into head-to-tail configurations [@Randall1987]. Misirlioglu *et al.* find a critical thickness of the space charge layer above which this type of domain wall develops in their simulated BaTiO[$_{\text{3}}$]{} films, and below which the film remains as a single domain [@Misirlioglu2012].
It can therefore be deduced that an approach to engineer complex domain structures is to manipulate the space charge regions in ferroelectric materials.
The thickness of space charge layers in thin films is determined by the built-in voltage across the layer. For a given dopant concentration, the built-in voltage is a property that depends on the material of the electrode. The layer thickness, which determines whether the film is fully or partially depleted, is thus dependent on the selection of dopant concentration and the electrode in experimental set-ups. When simulating a charged layer within a film, the layer thickness and dopant concentration may be chosen as variables instead; this would be equivalent to specifying the built-in voltage across the layer.
In investigating the properties of ferroelectric materials, the phase-field method has been shown to be a powerful technique and has been widely used [@Chen2008]. A real-space approach with this technique takes into account long-range elastic and electrostatic interactions in the material of study, and is convenient for nanoscale systems in which free surfaces are important since it is straightforward to apply the boundary conditions [@Yang2012]. This method has been applied in simulating the domain patterns in free-standing nanostructures of tetragonal lead zirconate-titanate [@Ng2012domain].
Most of the studies of space charge effects in ferroelectric materials have been devoted to materials such as BaTiO[$_{\text{3}}$]{} [@Ng2012; @Zuo2014; @Sluka2012; @Abplanalp1998; @Misirlioglu2012], PbTiO[$_{\text{3}}$]{} [@Wang2012; @Park1998; @Wu2006] and PbZr[$_{\text{1-x}}$]{}Ti[$_{\text{x}}$]{}O[$_{\text{3}}$]{} (PZT) for $x> 0.47$ [@Warren1994; @Misirlioglu2011effect], all of which adopt the tetragonal ferroelectric phase. These systems have 90 and 180 domain walls, with only the former producing a ferroelastic configuration. In these works, the external electric field is usually applied parallel to the polar direction.
Few have investigated the effect of space charge on rhombohedral ferroelectric materials [@Vasudevan2011; @Randall1987]. Unlike tetragonal ferroelectric materials, the rhombohedral phase has a larger number of ferroelectric variants with two possible ferroelastic domain patterns of 71 and 109 domain walls. Thus it can have more complex domain patterns [@Yin2000], whose piezoelectric response may be enhanced especially when applying an electric field that is not along one of the polar directions [@Sluka2012; @Fu2000; @Wada1999; @Bellaiche2001; @Ahluwalia2005; @Wang2007; @Liang2010].
A rhombohedral ferroelectric material, [[BiFeO]{}[$_{\text{3}}$]{}]{}, has been attracting much interest recently because of its multiferroic [@Ramesh2007; @Catalan2009] and photoelectric [@Choi2009; @Yang2009] properties. It has potential applications in memory storage, solar energy storage, sensors and telecommunications devices [@Catalan2009; @Guo2013; @Gao2007; @Nuraje2013].
Therefore in this work, we select [[BiFeO]{}[$_{\text{3}}$]{}]{} with polarisations along the $\langle 111\rangle $ directions (with reference to the pseudocubic lattice) as our material of study and examine the domain structures that form in \[001\] thin films under the influence of space charge. We also investigate the effective polarisation-electric field behaviour when applying an electric field in the \[001\] direction.
Assuming static charges on the timescale of polarisation switching [@Ng2012] for the case where the space charge region is generated due to the presence of an electrode/film interface, what we find is a wealth of domain patterns that varies with the strength of the electric field induced by the charges.
Calculating the response in the direction parallel to the applied field, we note an optimum charge density with minimum charged layer thickness that produces a domain pattern with a wavy interface whose mobility is higher than the others. Our findings would be useful for applications that require materials with high piezoelectric response and small hystereses.
Computational details
=====================
The dynamics of the polarisation order parameter components $P_x$, $P_y$ and $P_z$ evolving with time $t$ is given by the time-dependent Ginzburg-Landau equation as $$\frac{\partial P_i}{\partial t}=-\Gamma \left[\frac{\delta F}{\delta P_i }-E_i\right]
\label{ginzburglandau}$$ where $\Gamma$ is a kinetic coefficient related to the domain wall mobility, $F$ is the free energy of the ferroelectric material and $E_i$ is the component $i$ of the electric field $\mathbf{E}$ in the sample. $F$ consists of the Landau free energy $F_l$, the gradient free energy associated with the domain walls $F_g$, the elastic free energy $F_e$ and the electrostatic free energy $F_{el}$: $$F= F_l+ F_g+ F_e + F_{el}$$ We express $F_l$ in terms of $P_x$, $P_y$, and $P_z$ up to the fourth order: $$\begin{aligned}
F_l=\int d\mathbf{r} & \Big[ \alpha_1 \left( P_x^2 + P_y^2 + P_z^2 \right)
+\alpha_{11} \left( P_x^4 + P_y^4 + P_z^4 \right) \nonumber\\
& + \alpha_{12} \left( P_x^2 P_y^2 + P_x^2 P_z^2 + P_y^2 P_z^2 \right) \Big]\end{aligned}$$ where $\alpha_1$, $\alpha_{11}$ and $\alpha_{12}$ are material parameters. $F_g$ can be expressed as: $$F_g = \int d\mathbf{r} \left( \frac{K}{2} \left[ \left( \nabla P_x \right) ^2 + \left( \nabla P_y \right) ^2 + \left( \nabla P_z \right) ^2 \right] \right)$$ where $K$ is a gradient coefficient, while the expression for $F_e$ is as below: $$\begin{split}
F_e = & \int d\mathbf{r} \bigg\{
\frac{C_{11}}{2} \left[ \left( \varepsilon_{xx} - \varepsilon^0_{xx} \right) ^2 + \left( \varepsilon_{yy} - \varepsilon^0_{yy} \right) ^2 \right. \\
&\left. + \left( \varepsilon_{zz} - \varepsilon^0_{zz} \right) ^2 \right] \\
&+ C_{12} \left[ \left( \varepsilon_{xx} - \varepsilon^0_{xx} \right) \left( \varepsilon_{yy} - \varepsilon^0_{yy} \right) \right. \\
&\left. + \left( \varepsilon_{xx} - \varepsilon^0_{xx} \right) \left( \varepsilon_{zz} - \varepsilon^0_{zz} \right) \right. \\
&\left. + \left( \varepsilon_{yy} - \varepsilon^0_{yy} \right) \left( \varepsilon_{zz} - \varepsilon^0_{zz} \right) \right] \\
&+ 2C_{44} \left[ \left( \varepsilon_{xy} - \varepsilon^0_{xy} \right) ^2 + \left( \varepsilon_{xz} - \varepsilon^0_{xz} \right) ^2 \right. \\
&\left. + \left( \varepsilon_{yz} - \varepsilon^0_{yz} \right) ^2 \right] \bigg\}
\end{split}$$ $C_{11}$, $C_{12}$ and $C_{44}$ are material constants. $\varepsilon_{ij}$ are the strain tensors having the usual definition in terms of displacements $u_i$ $$\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$$ and the electrostrictive strain tensors $\varepsilon^0_{ij}$ have the following expressions: $$\begin{aligned}
\varepsilon^0_{ii} = & Q_{11} P^2_i + Q_{12} \left( P^2_j + P^2_k \right) \\
\text{and } \varepsilon^0_{ij} = & Q_{44} P_i P_j \text{ for } i\neq j\end{aligned}$$ with $Q_{11}$, $Q_{12}$ and $Q_{44}$ being material-specific electrostrictive constants.
We determine the electric field $\mathbf{E}$ in the sample from the electrostatic potential $\phi$ using the relationship $$\mathbf{E} = - \mathbf{\nabla} \phi$$ as well as Gauss’s law relating the electric displacement **D**, **E** and polarisation **P**: $$\mathbf{\nabla} \cdot \mathbf{D} = \mathbf{\nabla} \cdot \left( \epsilon_0 \epsilon_{br} \mathbf{E} + \mathbf{P} \right)
= - \epsilon_0 \epsilon_{br} \nabla ^2 \phi + \mathbf{\nabla} \cdot \mathbf{P} = \rho \left( \mathbf{r} \right)$$ $\epsilon_0$ is the permittivity of free space, $\epsilon_{br}=10$ is a background permittivity [@Hlinka2006] and $\rho \left( \mathbf{r} \right)$ we have set as $$\rho \left( z \right) =
\begin{cases}
\rho _0 ,& L - z < w \\
0, & \text{otherwise}
\end{cases}$$ where $L$ is the thickness of the film and $w$ is the thickness of the space charge layer. We assume $\rho_0 = q N_D$ where $q$ is the charge of a carrier and $N_D$ is the defect (or carrier) density. In our simulations, we arbitrarily select $q=-e$ where $e$ is the elementary charge. We now have $F_{el}$, defined as $$F_{el} = \int d\mathbf{r} \left( \frac{\varepsilon_0\varepsilon_{br}}{2} E_i \cdot E_i \right)$$
The dissipative force balance equations give the displacement field dynamics: $$\bar{ \rho} \frac{\partial ^2 u_i}{\partial t^2} = \frac{\partial \sigma_{ij} }{\partial x_j} + \eta \nabla ^2 \frac{\partial u_i}{\partial t}
\label{mechanical}$$ where $\bar{\rho}$ is the density and $\eta$ is a viscous damping term. The latter functions to drive the system to mechanical equilibrium such that $\frac{\partial \sigma_{ij} }{\partial x_j} = 0$. The stress tensors $\sigma_{ij}$ are obtained as $$\begin{aligned}
\sigma_{ij} = \frac{\delta F}{\delta \varepsilon_{ij} }
\label{final}\end{aligned}$$
We consider a (001) epitaxial [[BiFeO]{}[$_{\text{3}}$]{}]{} film on top of a substrate, with an electrode deposited at the top surface of the film, and simulate the dynamics of the polarisations by numerically solving the model described in equations \[ginzburglandau\] to \[final\] using finite differences. Initialising $P_x$, $P_y$ and $P_z$ with small, random fluctuations about zero to simulate the paraelectric initial conditions, we examine the ferroelectric domain patterns that form in fully and partially depleted films. We employ a $128 \text{nm} \times 128\text{nm} \times 32 \text{nm}$ simulation cell with periodic boundary conditions for **P**, **u** and $\phi$ in the lateral directions. The top surface is stress-free with $\sigma_{ij} \cdot n = 0$, where $n$ is the surface normal, and the bottom surface is clamped ($u_x = u_y = u_z = 0$). The film is also laterally clamped as we have periodic boundary conditions on $u_x$ and $u_y$ laterally. Assuming that the top and bottom interface charges compensate exactly the respective bound interface charges, we impose $\frac{\partial P_z}{\partial z} = 0$ at $z = 0$ and $z = L = 32$ nm. Rescaling all lengths as $\mathbf{r} = \mathbf{r}^* \delta $, we choose $\delta=1$ nm as our grid spacing. We rescale the polarisation as $\mathbf{P} = P_0 \mathbf{P}^*$ where $P_0=0.9$ Cm$^{-2}$ is the spontaneous polarisation of [[BiFeO]{}[$_{\text{3}}$]{}]{}, the strains as $\varepsilon_{ij} = \varepsilon_0 \varepsilon_{ij}^*$ where $\varepsilon_0 = Q_{11} P_0^2$, and the time as $t=t_0t^*$ where $t_0 = (\Gamma \lvert \alpha_1 \rvert ) ^{-1}$. All other coefficients are consequently rescaled in the same way as reported in reference , and we use their values for rescaled coefficients of $\Gamma$, $K$, $\eta$ and $\bar{\rho}$. The material specific parameters are taken from reference and listed in Table \[param\]. We model films with values of $N_D$ between $10^{26}$ $\text{m}^{-3}$ to $10^{27}$ $ \text{m}^{-3}$, which are within the range of those for heavily doped [[BiFeO]{}[$_{\text{3}}$]{}]{} films [@Yang2009electric; @Hung2014]. The charges are uniformly distributed within space charge layers parallel and adjacent to the film/electrode interface with thicknesses $w$ of 0, 5, 10, 15 and 32 nm, with the latter thickness corresponding to the fully depleted case.
--------------- ------------------------------------
$\alpha_{1}$ $4.9(T-1103)\times 10^5$ V m/C
$\alpha_{11}$ $6.5 \times 10^8$ V m$^5$/C$^3$
$\alpha_{12}$ $1.0 \times 10^8$ V m$^5$/C$^3$
$C_{11}$ $3.00 \times 10^{11}$ N/m$^2$
$C_{12}$ $1.62 \times 10^{11}$ N$^9$/m$^2$
$C_{44}$ $0.691 \times 10^{11}$ N$^9$/m$^2$
$Q_{11}$ 0.032 m$^4$/C$^2$
$Q_{12}$ -0.016 m$^4$/C$^2$
$Q_{44}$ 0.02 m$^4$/C$^2$
--------------- ------------------------------------
: The material constants for rhombohedral [[BiFeO]{}[$_{\text{3}}$]{}]{} taken from reference . $T$ is the temperature in Kelvin. \[param\]
Results and discussions
=======================
The charge-free [[BiFeO]{}[$_{\text{3}}$]{}]{} thin film
--------------------------------------------------------
\
Bulk [[BiFeO]{}[$_{\text{3}}$]{}]{} in the rhombohedral phase has eight possible polarisation variants along the $\langle 111\rangle $ directions, which are commonly denoted as $r_1^+ = \left[ 1 1 1 \right]$, $r_2^+ = \left[ \bar{1} 1 1 \right]$, $r_3^+ = \left[ \bar{1} \bar{1} 1 \right]$, $r_4^+ = \left[ 1 \bar{1} 1 \right]$, $r_1^- = \left[ \bar{1} \bar{1} \bar{1} \right]$, $r_2^- = \left[ 1 \bar{1} \bar{1} \right]$, $r_3^- = \left[ 1 1 \bar{1} \right]$, $r_4^- = \left[ \bar{1} 1 \bar{1} \right]$ (with reference to the pseudocubic lattice). Without any presence of charges in the film, our [[BiFeO]{}[$_{\text{3}}$]{}]{} thin film evolves from the paraelectric state to form 71 twinned domains with domain walls at an angle greater than 45 to the film/substrate interface, consistent with results reported by Zhang *et al.* using nearly similar material constants in their phase-field model [@Zhang2008]. Figure \[uncharged\_domain\] shows $r_1^-/r_4^-$ domains that form in some of our simulations when steady state is reached. With different random seeds to initialise the polarisations in the paraelectric state, we have observed other twinned domain formations with {101}-type walls as suggested by Streiffer *et al.* for films with rhombohedral lattices [@Streiffer1998] such as $r_3^+/r_4^+$, $r_2^+/r_3^+$ and $r_1^-/r_2^-$, to name a few. They also noted that all domains in a film must have the same $P_z$ direction to maintain a charge-neutral film. In all of our simulated films, the domain walls deviate slightly from the ideal {101} orientation; Zhang *et al.* have shown that this deviation is due to the competition between the domain wall energy that minimises when the domain wall is perpendicular to the film/substrate interface, and the elastic energy that favours the ideal orientation at 45 to the interface [@Zhang2008].
We then apply a varying potential difference across the thickness of the film with $r_1^-/r_4^-$ domains to simulate the ferroelectric switching of the charge-free film, by varying the external electric field from 0 kV/cm to 3000 kV/cm, $-3000$ kV/cm and back to 0 kV/cm in $t^*=10^5$ timesteps. Computing the average $P_z$ as a function of the applied field $E_{app}$ we show the resulting curve in Figure \[uncharged\_loop\]. The $r_1^-/r_4^-$ domains are observed for the remnant state with $P_z < 0$, corresponding to the point labelled as (ii) in Figure \[uncharged\_loop\]. These domains switch fully to $r_3^+/r_2^+$ domains upon forward bias, and back to $r_1^-/r_4^-$ domains with reverse bias. The $r_3^+/r_2^+$ domain pattern, illustrated in Figure \[positive\_remnant\_domain\], is stable at the remnant state with $P_z > 0$ which is labelled as (i) in Figure \[uncharged\_loop\]. The $P_x$ and $P_y$ components of the polarisation in the film remain unchanged during the switching as we expect, with only the $P_z$ component switching, since we are applying the electric field in the \[001\] direction. As we switch between the two remnant states, there is no long-range migration of the domain wall; only the inclination of the wall is altered.
Domain formation in fully and partially depleted films
------------------------------------------------------
When we include uniformly distributed static charges throughout the film to model a fully depleted film, we find that instead of having only twinned $r_1^-/r_4^-$ domains in the film, the top of the film consists of a set of $[P_x,P_y,-P_z]$ and $[P_x,-P_y,-P_z]$ twinned domains and the bottom half, $[P_x,P_y,P_z]$ and $[P_x,-P_y,P_z]$ domains. Figure \[fullydepleteddomains\] illustrates these domain configurations forming in fully depleted films for different values of $N_D$, clearly demonstrating that the geometry of the domain wall between these sets of twins depends on the magnitude of the charge density. The film with a lower value of $N_D=3 \times 10^{26}$ m$^{-3}$ has a wavy domain wall in the lateral directions of the film separating the sets of twinned domains that meet in a head-to-head configuration (Figure \[domain\_rho0\_05\]). For high values of $N_D$, this wall has a planar geometry (Figures \[domain\_rho0\_1\] and \[domain\_rho0\_2\]).
![The profile of the internal field $E_z$ (in the paraelectric phase) induced by uniformly distributed charges in fully depleted films with $N_D$ values ranging from $3 \times 10^{26}$ m$^{-3}$ to $1.1 \times 10^{27}$ m$^{-3}$ across the thickness of the films. \[electricfield\]](fig3_efield.pdf){width="\columnwidth"}
To understand the formation of the domain configurations described above, we examine the internal electric field induced by the charges. A uniform charge density produces a radial electric field inside a charged body. For the thin film geometry considered in our work, the field varies only in the $z$ direction. This internal field may influence the domain pattern that forms when the sample is cooled from the paraelectric phase to below the ferroelectric transition temperature. Since the field is already present in the paraelectric phase, we perform additional simulations in the paraelectric phase for the same values of $N_D$ as before to determine the variation of the internal electric field within the films. As shown in Figure \[electricfield\], the internal electric field generated due to the uniform distribution of charges in a film is equal in magnitude but opposite in direction in the top and bottom halves of the film. Since we have introduced negative charges in the film, the electric field in the bottom half is positive while that in the top half is negative. The magnitude of the field is maximum at the top and bottom surfaces and decreases to zero towards the middle of the film.
The electric field will induce a rotation of the polarisations to align with the field locally if the magnitude of the field is large enough. For films with sufficiently high values of $N_D$ to generate large enough internal fields, the positive field in the bottom half of the charged region stabilises domains with $P_z > 0$ and the negative field in the top half, domains with $P_z <0$. In rhombohedral ferroelectrics, this means that only $r_1^+$, $r_2^+$, $r_3^+$ and $r_4^+$ may form in the bottom half and $r_1^-$, $r_2^-$, $r_3^-$ and $r_4^-$ in the top half. In our simulations, two of the four possible domains in each half are formed, where the domains in one half must each be twins of those in the other half through the (001) twin plane, giving rise to the four-domain pattern observed in Figure \[fullydepleteddomains\]. Thus it is the inhomogeneity in the internal electric field that is responsible for the formation of multiple domains within the charged region.
\
\
Comparing the variation of the internal field for films with different values of $N_D$ (Figure \[electricfield\]), we observe that the magnitude of the internal field within the charged region increases as $N_D$ increases. For low values of $N_D$, the small magnitude of the internal field is insufficient to produce a pattern of domains different from that in the charge-free film. For depleted films with intermediate values of $N_D$, the magnitude of the internal field which is larger in the near-surface regions, is sufficient to induce the formation of domains with polarisations that align with the field in these regions. Towards the middle of the film, however, the magnitude of the field is too small to force the polarisations to rotate with the field. In this case, the polarisations rotate into a head-to-tail configuration locally near the domain wall, accommodating the constraint imposed by the induced domains in the near-surface regions and at the same time minimising the electrostatic energy. Consequently the interface takes on a wavy geometry, as illustrated in Figure \[vector\_32rho0\_05\] showing the $P_z-P_y$ vectors in an $x-z$ plane for the fully depleted film with $N_D=3 \times 10^{26}$ m$^{-3}$.
When the value of $N_D$ is high, the magnitude of the electric field increases more rapidly from the middle of the film. The field is therefore large enough in the middle of the film to induce a head-to-head domain configuration with a planar interface (Figures \[Pvector\](b) and (c)). It also results in a rotation of the dipoles from pure rhombohedral orientations towards the \[001\] and $\left[ 0 0 \bar 1 \right]$ directions, such that for the fully depleted film with $N_D=1.1 \times 10^{27}$ m$^{-3}$, the domains approach the tetragonal phase. The increasing $P_z$ component in the near-surface regions of films with increasing $N_D$ can be observed from the $P_z-P_y$ vector maps in Figure \[Pvector\], and we also represent the degree of the polarisation rotation with the changing shades within the domains in Figure \[fullydepleteddomains\].
\
\
\
In partially depleted films, the presence of a charged layer shows a similar effect. For films with uniformly distributed charges of $N_D=3\times 10^{26}$ $\text{m}^{-3}$ and $N_D=6\times 10^{26}$ $\text{m}^{-3}$ within charged layers of width $w=5$ nm at the top surface of the films, the domain patterns resemble that of a charge-free film with twin $r_3^+/r_2^+$ domains as shown in Figures \[domains\](a) and (b), except for a slight decrease in $P_z$ near the top of the layer (indicated by the different shades in this region). When we increase $N_D$ to $1.1\times 10^{27}$ $\text{m}^{-3}$ for the same value of $w$, we observe the formation of the $r_1^-/r_4^-$ twinned set of domains in the top half of the charged layer and the $r_3^+/r_2^+$ set in the bottom half of the charged layer, with the latter set extending into the charge-free region (Figure \[domains\](c)).
We obtain a similar trend for films with $w=10$ nm, but with the appearance of the double set of twinned domains from lower values of $N_D$. Small regions of $r_1^-/r_4^-$ domains already form at the domain walls in the film with $N_D=3\times 10^{26}$ $\text{m}^{-3}$, represented as the dark-shaded regions in Figure \[domains\](d). Increasing $w$ to 15 nm with $N_D=3\times 10^{26}$ $\text{m}^{-3}$ leads to the appearance of such domains in a larger number (Figure \[domains\](g)), such that the domain pattern begins to resemble that of the fully depleted film having the same $N_D$ with the wavy domain interface (Figure \[domain\_rho0\_05\]). The $r_3^+/r_2^+$ set of domains from the lower half of the film extends up to the top surface of the film in some regions, as $w=15$ nm is not sufficiently large to allow the formation of a continuous wavy interface. For films with this value of $w$, and $N_D$ values of $6 \times 10^{26}$ $\text{m} ^{-3}$ and $1.1 \times 10^{27}$ $\text{m} ^{-3}$, the two sets of twinned domains form with planar interfaces, with rotation of the polarisations approaching the tetragonal phase as before even within the charge-free regions.
![The $E_z$ profile across the thickness of partially depleted films with $N_D=6 \times 10^{26}$ m$^{-3}$, for values of $w$ from 5 nm to 15 nm. \[efield\_rho0\_1\]](fig6_efield_rho0_1.pdf){width="45.00000%"}
Whether the set of $r_1^-/r_4^-$ domains forms within the charged layer of a partially depleted film depends on the strength of the internal electric field, determined by the values of $w$ and $N_D$. The field within the charged region, whose profile is shown in Figure \[efield\_rho0\_1\] for films with $N_D=6 \times 10^{26}$ m$^{-3}$, does not induce the formation of these domains in the top half of the charged layer if the magnitude of the field is too small. As the magnitude increases with increasing $w$ and $N_D$, the $r_1^-/r_4^-$ domains are observed near the top of charged layers with higher values of these parameters. Within the much larger charge-free region, the internal field is non-zero and positive, increasing in magnitude with increasing $N_D$ and $w$. Although the magnitude in this region is much smaller than the maximum magnitude attained at the top of the charged layer, it is sufficient to bias the formation of the $r_3^+/r_2^+$ domains within the charge-free region, with the domains approaching tetragonality with increasing magnitude of the internal field.
The appearance of the domain pattern with the wavy domain interface for $w \geq 15$ nm is consistent with the finding by Misirlioglu *et al.*. From their 2-dimensional Ginzburg-Landau-Devonshire model of depleted tetragonal BaTiO[$_{\text{3}}$]{} films, they predict a critical thickness above which head-to-head domains with a zigzagged interface develop [@Misirlioglu2012]. We also observe a critical value of $w$ associated with the strength of the induced electric field for the appearance of the domain pattern with planar horizontal interface that we find in films with high $N_D$. For films with $N_D= 6 \times 10^{26}$ $\text{m} ^{-3}$, this configuration appears when $w \geq 10$ nm (Figure \[domains\](e)), while films with $N_D= 1 \times 10^{27}$ $\text{m} ^{-3}$ need only $w \geq 5$ nm for the same configuration to form (Figure \[domains\](c)).
There is in addition a critical value of $N_D$ beyond which the wavy domain wall transitions to a planar interface. We have observed this progression for the fully depleted films, as well as for the partially depleted films with $w=10$ nm (not shown here) and 15 nm (Figures \[domains\](g-i)). For a film with a thinner depletion layer of $w=5$ nm, the charged layer is too thin to allow the formation of a wavy wall. Thus the twinned domain pattern resembling the charge-free film directly transitions to the four-domain pattern with planar interface when $N_D \geq 1.1 \times 10^{27} \text{m} ^{-3}$.
Various experimental studies have reported observations of domains in a head-to-head or tail-to-tail configuration with a wavy interface, one of which have recently been observed in a tetragonal PZT film [@Han2014]. The authors in that work attributed the domain shapes to the anisotropy of domain growth, and suggest opposite internal fields near the top and bottom surfaces of the film due to high oxygen vacancy concentration and electronic band bending at the film/substrate interface respectively. These are thought to induce domains with polarisations pointing down towards the middle of the film ($P_z<0$) near the top of the film, and domains with upwards polarisations ($P_z>0$) at the bottom of the film. Our results show that domains in a head-to-head configuration may be generated by a single negatively charged layer if the magnitude of the induced internal field is sufficiently high, with a residual field even within the charge-free regions. Furthermore, the geometry of the interface is dictated by the strength of the field. Having two oppositely charged layer in a film may produce the same domain configuration provided the induced field within the other positively charged layer is small enough in magnitude. If the magnitude is sufficiently high, additional domains in tail-to-tail configuration would result in the positively charged layer.
We observe that in both partially and fully depleted films that we have considered in our work, the induced internal fields are very large in magnitude. In principle, such large fields in the ferroelectric state may lead to charge transport which can modify the field and thus the domain patterns, particularly in films with large $N_D$. However, we expect the charge migration to be a slow process. For example, in a study of the ageing phenomenon in ferroelectrics, it has been shown that charge migration induced by a depolarisation field leads to a build-up of space charge over time scales ranging from 10$^5$ seconds to 10$^7$ seconds [@Genenko2008]. This is much slower than the time scales associated with domain reorientation, which is on the order of 10$^{-9}$ seconds. Thus we expect findings based on a model with static space charge to be valid for short times and fast switching frequencies. The fact that the resulting domain patterns, such as those with wavy interfaces, have indeed been observed in experiments for various systems [@Randall1987; @Han2014; @Abplanalp1998] gives us confidence that our observations will still hold in the long time limit, as the internal field may not be completely eliminated by mobile charges.
Polarisation switching and response of the films
------------------------------------------------
When an external electric field is applied to a film with depleted layers, the internal field produced by the presence of charges may oppose the effect of the externally applied field. Within the thin films that we have considered in this work, we have shown that the internal field along the $z$-direction is positive in the lower regions of the charged layers but negative in the upper regions in the absence of external bias. Therefore one of these regions will always oppose an applied field parallel to the internal field. Consequently, the domains will be pinned, requiring prohibitively large applied fields to completely switch to domains having the same polarity in $z$.
{width="95.00000%"}
\
\
Figure \[hys0\_05\] demonstrates these effects for the fully depleted film with a wavy domain wall ($N_D=3 \times 10^{26}$ $\text{m} ^{-3}$) under the same switching conditions that we have applied for the charge-free film in Figure \[uncharged\]. We also illustrate the domain patterns at maximum magnitude and zero applied fields in the same figure. The magnitude of the average $P_z$ increases with the magnitude of the applied field, and the wavy domain wall moves towards the top or bottom surface of the film. However, not all regions of the domains switch to align with the external field as evident from the presence of darker shaded $r_n^-$ domains remaining at the top surface of the film in Figure \[maxEdomain\] and lighter shaded $r_n^+$ domains at the bottom surface of the film in Figure \[minEdomain\]. The internal field induced by the charges at the top of the film opposes the external field under forward bias, and that at the bottom of the film when the field is reverse-biased, leading to incomplete switching of domains. The magnitude of average $P_z$ at maximum magnitude of applied field is thus less than the saturation polarisation of a charge-free film.
When the external field is removed, the internal field restores the four domains almost to the original state, where the average $P_z$ is close to zero. Therefore the remnant $P_z$ is close to zero and the magnitude of an applied field required to reduce the magnitude of $P_z$ to zero, i.e. the coercive field, is also close to zero; correspondingly, the hysteresis loop narrows. This constriction of the hysteresis loop, as well as the decrease in the remnant polarisation, has been observed in earlier works [@Zubko2006; @Warren1994].
\
The hysteresis loops for the fully depleted films become more constricted with increasing values of $N_D$, as shown in Figure \[loop\_full\]. This is consistent with the trend observed in other works [@Wang2012; @Baudry2005]. The increasing magnitude of the internal field in the films increasingly opposes the polarisation switching, such that for films with $N_D\geq 6 \times 10^{26} \text{ m} ^{-3}$ there is no switching of domains but only a slight enhancement in the magnitude of average $P_z$ with $E_{app}$. With very little movement of the domain walls in the $z$ direction, the hysteresis completely disappears. The curve becomes nearly horizontal and linear, reminiscent of that for a paraelectric material.
We also observe a similar trend for partially depleted films, as shown in Figure \[loop\_partial\] for films with $w=5$ nm. In particular, we find that for the partially depleted film with $N_D=1.1 \times 10^{27} \text{ m} ^{-3}$ which forms the four-domain pattern in the absence of external field (Figure \[domains\](c)), the $r_1^-/r_4^-$ domains at the top of the charged layer do not switch to $r_3^+/r_2^+$ domains under forward bias (Figure \[saturation\_forward\]), although complete switching of the $r_3^+/r_2^+$ domains to $r_1^-/r_4^-$ domains occurs with reverse bias (Figure \[saturation\_reverse\]). The field induced within the charged layer in this film follows the same trend as those shown in Figure \[efield\_rho0\_1\]: the internal field at the top of the charged layer, where $E_z<0$, is much larger in magnitude than elsewhere in the film. It is sufficiently large to impede the switching of the $r_1^-/r_4^-$ domains, whereas the induced field within the $r_3^+/r_2^+$ domains is too small to pin the polarisations. This incomplete switching of domains, which we also observe for other partially depleted films with the head-to-head domain configuration, leads to the observation of non-switchable regions in ferroelectrics [@Han2014].
\
There is in addition a shift of the hysteresis loop along the $E_{app}$ axis that is not present in that of a fully depleted film, commonly referred to as imprint. Due to the asymmetrical distribution of the charges, the upper region of the charged layer where the internal field $E_z < 0$, is much smaller than elsewhere in the film where $E_z > 0$ (Figure \[efield\_rho0\_1\]). In the absence of an external field, the average value of the internal field is positive in the $z$ direction despite the presence of the local negative field of much larger magnitude, and hence domains with $P_z > 0$ will be the preferred polarisation state. Consequently, a larger electric field is required to switch the domains such that an average $P_z < 0$ is produced in the film, compared to that for the reverse. The resulting hysteresis loop is thus left-shifted and becomes further left-shifted with increasing strength of the internal field, as can be observed for the film with $w=5$ nm in Figure \[loop\_partial\].
![$\chi/\chi_0$ of partially and fully depleted films as a function of $N_D$. \[susceptibility\]](fig10_susceptibility.pdf){width="\columnwidth"}
Having demonstrated the influence of charges on domain patterns and polarisation switching behaviour, we now consider the implications on the dielectric response. From the $P_z-E_{app}$ curve, we calculate $\frac{dP_z}{dE_{app}}$ at zero applied field for $P_z > 0$, which gives an indication of the susceptibility of the film in $z$, $\chi=\frac{1}{\epsilon_0}\frac{dP}{dE}$. Figure \[susceptibility\] shows this quantity relative to that of the charge-free film, $\chi/\chi_0$, as a function of $N_D$ for partially and fully depleted films.
We find that an initial increase in $N_D$ results in increasing enhancement in the susceptibility of the films. The magnitude of the internal field at top/bottom surfaces of a charged layer, when large enough, rotates and pins the polarisations from $P_z>0$ to $P_z<0$ in the top region of the charged layer, thus leading to the appearance of head-to-head domains that are stable even at remanence; in comparison, a charge-free film at remanence would have a single polarisation direction in $z$. Figure \[remanence\] illustrates the increasing volume fraction of $r_1^-$/$r_4^-$ domains at remanence with $N_D$. As 180 switching of the local polarisations $P_z$ in films with head-to-head domains produces a larger polarisation response compared to that of domains with a single polarisation direction in $z$, the susceptibility of the film therefore enhances with $N_D$. Additionally we observe that films with higher values of $w$, and thus having larger volume fractions of $r_1^-$/$r_4^-$ domains, show greater enhancement in susceptibility. The response of the fully depleted film exhibits an increment of almost fourfold when $N_D=2.3 \times 10^{26}$ m$ ^{-3}$, relative to that of the charge-free film. The partially depleted films with $w=15$ nm also show significantly enhanced response, though lower than that of the fully depleted film, of almost twofold when $N_D=4 \times 10^{26}$ m$ ^{-3}$. These films have in common wavy interfaces between domains in head-to-head configuration, with hysteresis loops that narrow without complete loss of hysteresis and without significant shifts along the $E_{app}$ axis.
Below these $N_D$ values, the internal field that acts to pin the domains in the head-to-head configuration is relatively weak in the vicinity of the domain wall, allowing switching of the polarisation with externally applied fields. However, there is a competing effect also as a result of increasing internal field strength with $N_D$ that inhibits the polarisation switching. The movement of the domain walls becomes more limited and the susceptibility then decreases with further increase in $N_D$, eventually falling to a value that is lower than that of the charge-free film, as can be observed in Figure \[susceptibility\].
Furthermore, the maximum in the susceptibility occurs at lower $N_D$ values with increasing $w$, as the magnitude of the internal field that inhibits domain switching also increases with $w$. For films with suppressed susceptibilities, we have observed that the interface becomes planar and the hysteresis loop becomes almost linear or further left-shifted.
Thus there exists a range of values for $N_D$ for a given value of $w$ that enhances the susceptibility of a film containing a depletion region, beyond which is detrimental to the susceptibility. This indicates that the presence of space charge in ferroelectric materials may be useful for applications that require high susceptibilities as well as small hystereses. The enhancement due to the presence of charges is in addition to the already known effect of poling along a non-polar direction in a ferroelectric material.
Conclusions
===========
We have investigated the influence of space charge on domain patterns in rhombohedral ferroelectric thin films, taking [[BiFeO]{}[$_{\text{3}}$]{}]{} as an example. Phase field simulations of domain patterns and polarisation switching have been performed for different charge densities and charged layer widths. We show that the introduction of a charged layer creates an internal electric field which may lead to the formation of double twinned domains in a head-to-head configuration. Depending on the magnitude of the internal field which we vary with charge density and layer width, we find a wavy or planar interface between the head-to-head domains. The former forms in films with intermediate values of $N_D$ above a critical value of $w$, while the latter is observed when $N_D$ is large.
We have also examined the polarisation switching behaviour of the fully depleted as well as the partially depleted films with the applied field along the \[001\] direction. The widely observed reduction in saturation and remnant polarisation, reduction in coercive field, constriction of the hysteresis loop and, for partially depleted films, its translation along the applied electric field axis, is explained in terms of the internal field that pins or imprints the domain structures as they evolve with the applied switching field. For films in which the charged layer induces a sufficiently large field, the pinning of the domains also results in regions with non-switchable polarisations. We find that the susceptibility of the films can be increased within a range of relatively low $N_D$ values that is high enough to pin domains in a head-to-head (or tail-to-tail) configurationthis also requires sufficiently thick depletion layer thicknessbut yet low enough to allow movement of domain walls with applied electric fields. In our films, this range of $N_D$ and $w$ coincides with the formation of a wavy interface between head-to-head domains. Our results suggest that space charge-induced domains may engineer small hysteresis, large susceptibility response in ferroelectric materials for device applications.
We are grateful to Khuong P. Ong for useful discussions.
|
---
abstract: 'We present essentially exact solutions of the Schrödinger equation for three fermions in two different spin states with zero-range $s$-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all $s$-wave scattering lengths.'
author:
- Seyed Ebrahim Gharashi
- 'K. M. Daily'
- 'D. Blume'
title: 'Three $s$-wave interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients'
---
Introduction {#sec_introduction}
============
There has been extensive interest in ultracold atom physics in the last decade [@review_blume; @review_giorgini; @review_bloch]. Ultracold atomic bosonic and fermionic gases are realized experimentally under varying external confinements. In these experiments, the number of particles and the scattering length of the two-body interactions are tunable [@chin_rmp]. Although the complete energy spectrum of the many-body system cannot, in general, be obtained from first principles, the energy spectra of selected few-body systems can, in some cases, be determined within a microscopic quantum mechanical framework [@busch; @calarco-r; @calarco; @kestner; @blume-greene_prl; @javier-greene_prl]. In some cases, the properties of the few-body system have then been used to predict the properties of the corresponding many-body system [@ho-1; @ho-2; @rupak; @drummond_prl; @drummond_pra; @drummond_2d; @salomon-2010; @zwierlein-2012; @daily_2012].
The behavior of atomic and molecular systems depends strongly on the dimensionality of the system [@olshanii1; @olshanii2; @petrov; @esslinger]. In three dimensions, e.g., weakly-bound two-body $s$-wave states exist when the $s$-wave scattering length is large and positive but not when it is negative. In strictly one- and two-dimensional geometries, in contrast, $s$-wave bound states exist for all values of the $s$-wave scattering length [@busch].
In ultracold atomic gases, the de Broglie wavelength of the atoms is much larger than the van der Waals length that characterizes the two-body interactions. This allows one to replace the van der Waals interaction potential in free-space low-energy scattering calculations by a zero-range $s$-wave pseudopotential [@fermi; @huang; @huang2]. If the particles are placed in an external trap, the validity of the pseudopotential treatment (at least if implemented without accounting for the energy-dependence of the coupling strength) requires that the van der Waals length is much smaller than the characteristic trap length [@blume_greene_pra; @bolda_pra]. In many cases, the use of pseudopotentials greatly simplifies the theoretical treatment. For example, the eigenequation for two particles interacting through a $s$-wave pseudopotential under harmonic confinement has been derived analytically for spherically symmetric, strictly one-dimensional, strictly two-dimensional and anisotropic harmonic potentials [@busch; @calarco-r; @calarco].
The $s$-wave pseudopotential has also been applied successfully to a wide range of three-body problems, either in free space or under confinement [@nielsen; @kartavtsev; @rittenhouse-2010; @mora; @mora-2005; @werner; @petrov3]. The present paper develops an efficient numerical framework for treating the three-body system under anisotropic harmonic confinement. The developed formalism allows us to study the dependence of the three-body properties on the dimensionality of the system. We focus on fermionic systems consisting of two identical spin-up atoms and one spin-down atom. The dimensional crossover of two-component Fermi gases has attracted a great deal of interest recently [@sommer-2012; @kohl-2012; @thomas-2012]. This paper considers the three-body analog within a microscopic quantum mechanical framework. We note that our framework readily generalizes to bosonic three-body systems. The study of the dimensional crossover of bosonic systems is interesting as it allows one to study how, under experimentally realizable conditions, Efimov trimers [@braaten] that are known to exist in three-dimensional space disappear as the confinement geometry is tuned to an effectively low-dimensional geometry [@nishidatan].
This paper generalizes the methods developed in Refs. [@mora; @kestner] for three equal-mass fermions in two different pseudospin states under spherically symmetric harmonic confinement to anisotropic harmonic confinement. We develop an efficient and highly accurate algorithm to calculate the eigenenergies and eigenstates of the system up to relatively high energies as functions of the interaction strength and aspect ratio of the trap. Several applications are considered: [*[(i)]{}*]{} The BCS-BEC crossover curve is analyzed throughout the dimensional crossover. [*[(ii)]{}*]{} For large and small aspect ratios, the energy spectra are analyzed in terms of strictly one-dimensional and strictly two-dimensional effective three-body Hamiltonian. [*[(iii)]{}*]{} The second- and third-order virial coefficients are analyzed as functions of the temperature, aspect ratio and scattering length. In particular, we show that the high-temperature limit of the third-order virial coefficient $b_3$ at unitarity is independent of the shape of the trap in agreement with expectations derived through use of the local density approximation. For finite scattering lengths, $b_2$ and $b_3$ for anisotropic harmonic confinement are well approximated by those for isotropic harmonic confinement.
The remainder of this paper is organized as follows. Section \[sec\_formalsolution\] presents a formal solution to the problem of three $s$-wave interacting fermions confined in an axially symmetric harmonic trap. We also consider the extreme cases of strictly one-dimensional and strictly two-dimensional confinement. Sections \[sec\_cigar\] and \[sec\_pancake\] apply the formal solution to cigar-shaped and pancake-shaped traps, respectively. We determine a large portion of the eigenspectrum as a function of the scattering length and discuss the transition to strictly one-dimensional and strictly two-dimensional geometries. Section \[sec\_virial\_coeff\] uses the two- and three-body eigenspectra to calculate the second- and third-order virial coefficients as a function of the temperature and the geometry of the confinement. Finally, Sec. \[sec\_conclusion\] concludes.
Formal solution {#sec_formalsolution}
===============
We consider a two-component Fermi gas consisting of two spin-up atoms and one spin-down atom with interspecies $s$-wave interactions under anisotropic harmonic confinement. We refer to the two spin-up atoms as particles 1 and 2, and to the spin-down atom as particle 3. We introduce the single-particle Hamiltonian $H_0({\bf{r}}_j,\mathcal{M})$ for the $j^{th}$ particle with mass $\mathcal{M}$ under harmonic confinement, $$\label{eq_H0}
H_0({\bf{r}}_j,\mathcal{M})=\frac{-\hbar^2}{2\mathcal{M}}{\boldsymbol{\nabla}}^2_{{\bf{r}}_j}+\frac{1}{2}\mathcal{M} (\omega^2_zz_j^2 + \omega^2_{\rho} \rho_j^2).$$ Here, ${\bf{r}}_j$ is measured with respect to the trap center, and in cylindrical coordinates we have ${\bf{r}}_j=(z_j, \rho_j, \phi_j)$. In Eq. (\[eq\_H0\]), $\omega_z$ and $\omega_{\rho}$ are the angular trapping frequencies in the $z$- and $\rho$-directions, respectively. The aspect ratio $\eta$ of the trap is defined through $\eta = \omega_{\rho} / \omega_z$. In this paper, we consider cigar-shaped traps with $\eta>1$ as well as pancake-shaped traps with $\eta<1$. Our three-particle Hamiltonian $H$ then reads $$H = \sum_{j=1}^3 H_0({\bf{r}}_{j}, \mathcal{M}) +V_{\rm{int}},
\label{eq_Hamiltonian}$$ where $V_{\rm{int}}$ accounts for the interspecies $s$-wave two-body interactions, $$\label{eq_v_int}
V_{\rm{int}} = V_{\rm{ps}}^{\rm{3D}}({{\bf{r}}_{31}}) + V_{\rm{ps}}^{\rm{3D}}({{\bf{r}}_{32}}).$$ The regularized pseudopotential $V_{\rm{ps}}^{\rm{3D}}$ is characterized by the three-dimensional $s$-wave scattering length $a^{\rm{3D}}$ [@fermi; @huang; @huang2], $$\label{eq_pseudopotential}
V_{\rm{ps}}^{\rm{3D}}({{\bf{r}}_{jk}}) = \frac{4 \pi \hbar^2 a^{\rm{3D}}}{\mathcal{M}} \delta ({\bf{r}}_{jk})\frac{\partial}{\partial r_{jk}}r_{jk},$$ where ${{\bf{r}}_{jk}}={{\bf{r}}_{j}}-{{\bf{r}}_{k}}$ and $r_{jk}=|{\bf{r}}_{jk}|$.
Since the trapping potential is quadratic, the relative and center of mass degrees of freedom separate and we rewrite the Hamiltonian $H$ in terms of the relative Hamiltonian $H_{\rm{rel}}$ and the center of mass Hamiltonian $H_{\rm{cm}}$, $H = H_{\rm{rel}} + H_{\rm{cm}}$. In the following, we obtain solutions to the relative three-body Schrödinger equation $H_{\rm{rel}} \Psi=E_{\rm{3b}} \Psi$, where $$\begin{aligned}
\label{eq_rel_Hamiltonian}
{H_{\rm{rel}}}= H_{\rm{rel},0} + V_{\rm{int}}\end{aligned}$$ with $$\begin{aligned}
\label{eq_rel_Hamiltonian_0}
H_{\rm{rel},0}=H_0({\bf{r}},\mu) + H_0({\bf{R}},\mu).\end{aligned}$$ In Eq. (\[eq\_rel\_Hamiltonian\_0\]), $\mu$ is the two-body reduced mass, $\mu=\mathcal{M}/2$, and the relative Jacobi coordinates ${\bf{r}}$ and ${\bf{R}}$ are defined through ${\bf{r}}={\bf{r}}_{31}$ and ${\bf{R}}=\frac{2}{\sqrt{3}} (\frac{{{\bf{r}}_{1}}+
{{\bf{r}}_{3}}}{2}-{\bf{r}}_{2})$. Depending on the context, we use either ${\bf{r}}$ and ${\bf{R}}$ or ${\bf{r}}_{31}$ and ${\bf{r}}_{32}$ to describe the relative degrees of freedom of the three-body system.
To determine the relative three-body wave function $\Psi({\bf{r}},{\bf{R}})$, we take advantage of the fact that the solutions to the “unperturbed” relative Hamiltonian $H_{\rm{rel},0}$ are known and consider the Lippmann-Schwinger equation (see, e.g., Ref. [@kestner]) $$\begin{aligned}
\label{eq_lippmann-schwinger}
\Psi({\bf{r}},{\bf{R}})= - \int G(E_{\rm{3b}};{\bf{r}},{\bf{R}};{\bf{r'}},{\bf{R'}})
V_{\rm{int}}({\bf{r'}},{\bf{R'}})
\Psi({\bf{r'}},{\bf{R'}})~d{\bf{r'}}d{\bf{R'}}.\end{aligned}$$ The Green’s function $G$ for the two “pseudoparticles” of mass $\mu$ associated with the Jacobi vectors ${\bf{r}}$ and ${\bf{R}}$ is defined in terms of the eigenstates $\Phi_{\boldsymbol\lambda_1}({\bf{r}}) \Phi_{\boldsymbol\lambda_2}({\bf{R}})$ and the eigenenergies $E_{\boldsymbol\lambda_1}+E_{\boldsymbol\lambda_2}$ of $H_{\rm{rel},0}$, $$\begin{aligned}
\label{eq_two_Green}
G(E_{\rm{3b}};{\bf{r}},{\bf{R}};{\bf{r'}},{\bf{R'}})=
\sum_{{\boldsymbol\lambda_1},{\boldsymbol\lambda_2}}
\frac{
\Phi_{\boldsymbol\lambda_1}^*({\bf{r'}}) \Phi_{\boldsymbol\lambda_2}^*({\bf{R'}})
\Phi_{\boldsymbol\lambda_1}({\bf{r}}) \Phi_{\boldsymbol\lambda_2}({\bf{R}})
}{
(E_{\boldsymbol\lambda_1}+E_{\boldsymbol\lambda_2})-E_{\rm{3b}}
}.\end{aligned}$$ Here, ${\boldsymbol\lambda}$ collectively denotes the quantum numbers needed to label the single-particle harmonic osillator states. In cylindrical coordinates, we have ${\boldsymbol\lambda}=(n_z,n_{\rho},m)$ with $n_z = 0, 1, 2, \cdots$, $n_{\rho} = 0, 1, 2, \cdots$, and $m = 0, \pm 1, \pm2, \cdots$. The single-particle harmonic oscillator eigenenergies and eigenstates read $$\label{eq_ond-b_energy}
E_{\boldsymbol \lambda}=
\left(n_z+\frac{1}{2}\right) \hbar \omega_z +
\left(2n_{\rho}+|m|+1\right)\eta \hbar \omega_z$$ and $$\label{eq_ond-b_wavefunction}
\Phi_{\boldsymbol\lambda}({\bf{r}})=
\varphi_{n_z}(z)R_{n_{\rho},m}(\rho)\frac{e^{i m \phi}}{\sqrt{2 \pi}},$$ where $$\label{eq_one-d_wavefunction}
\varphi_{n_z}(z)=
\sqrt{\frac{1}{a_z \sqrt{\pi}~2^{n_z} ~n_z!}}
\exp{\left(-\frac{z^2}{2 a_z^2}\right)} H_{n_z}(z/a_z)$$ and $$\label{eq_two-d_wavefunction}
R_{n_{\rho},m}(\rho)=\sqrt{\frac{2 ~ \eta ~{n_{\rho}!}}{a_z^2(n_{\rho}+ |m|)!}} \exp{\left(-\frac{ \eta \rho^2}{2 a_z^2}\right)}\left(\frac{\eta^{1/2} \rho}{a_z}\right)^{|m|}
L_{n_\rho}^{(|m|)}\left(\eta \rho^2/a_z^2\right).$$ In the last two equations, $H_{n_z}(z/a_z)$ and $L_{n_{\rho}}^{(|m|)}\left(\eta \rho^2/a_z^2\right)$ denote Hermite and associated Laguerre polynomials, respectively. Throughout most of Secs. \[sec\_formalsolution\]-\[sec\_pancake\], we use the oscillator energy $E_z$ and oscillator length $a_z$ \[$E_z=\hbar \omega_z$ and $a_z=\sqrt{\hbar/(\mu \omega_z)}$\] as our energy and length units.
In Eqs. (\[eq\_lippmann-schwinger\])-(\[eq\_two-d\_wavefunction\]), we employ cylindrical coordinates since this choice allows us to write the Green’s function $G$ compactly. However, the two-body $s$-wave interaction potential is most conveniently expressed in spherical coordinates \[see Eq. (\[eq\_pseudopotential\])\]. Since the pseudopotential $V_{\rm{ps}}^{\rm{3D}}({\bf{r}})$ acts only at a single point, namely at $r=0$, it imposes a boundary condition on the relative three-body wave function $\Psi({\bf{r}},{\bf{R}})$ (see, e.g., Ref. [@petrov3]), $$\label{eq_boundary}
\left. \Psi({\bf{r}},{\bf{R}})\right|_{r \rightarrow 0}
\approx
\frac{f({\bf{R}})}{4 \pi a_z^{3/2}}\left(\frac{a_z}{r} - \frac{a_z}{a^{\rm{3D}}}\right).$$ The unknown function $f({\bf{R}})$ can be interpreted as the relative wave function of the center of mass of the interacting pair and the third particle. Similarly, the pseudopotential $V_{\rm{ps}}^{\rm{3D}}({\bf{r}}_{32})$ imposes a boundary condition on the wave function $\Psi({\bf{r}},{\bf{R}})$ when $r_{32}\rightarrow 0$. Since the wave function $\Psi({\bf{r}},{\bf{R}})$ must be anti-symmetric under the exchange of the two identical fermions, i.e., $P_{12}\Psi({\bf{r}},{\bf{R}}) = -\Psi({\bf{r}},{\bf{R}})$, where $P_{12}$ exchanges particles 1 and 2, the properly anti-symmetrized boundary condition corresponding to $V_{\rm{ps}}^{\rm{3D}}({\bf{r}}_{32})$ reads $$\label{eq_boundary2}
\left. \Psi({\bf{r}}_{32},{\bf{R}}_{32})\right|_{r_{32} \rightarrow 0}
\approx -\frac{f({\bf{R}}_{32})}{4 \pi a_z^{3/2}}
\left(\frac{a_z}{r_{32}} - \frac{a_z}{a^{\rm{3D}}}\right).$$ Here, we defined ${\bf{R}}_{32} =
\frac{2}{\sqrt{3}}\left(\frac{{\bf{r}}_2+{\bf{r}}_3}{2}-{\bf{r}}_{1}\right).$
To simplify the right hand side of Eq. (\[eq\_lippmann-schwinger\]), we impose the limiting behaviors of $\Psi({\bf{r'}},{\bf{R'}})$ for $r'_{31}\rightarrow 0$ and $r'_{32}\rightarrow 0$, and expand $f({\bf{R'}})$ in terms of the non-interacting harmonic oscillator functions, $f({\bf{R'}})= \sum_{\boldsymbol\lambda'}f_{\boldsymbol\lambda'} \Phi_{\boldsymbol\lambda'}({\bf{R'}})$. Using Eq. (\[eq\_two\_Green\]) for $G$ and orthonormality of the single-particle harmonic oscillator functions, we find $$\begin{aligned}
\label{eq_wf_expansion}
\Psi({\bf{r}},{\bf{R}})=\frac{E_z a_z^{3/2}}{2}\sum_{\boldsymbol\lambda}f_{\boldsymbol\lambda}
\left[
\mathcal{G}^{\rm{3D}}\left(E_{\rm{3b}}-E_{\boldsymbol\lambda};{\bf{r}};{\bf{0}}\right) \Phi_{\boldsymbol\lambda}({\bf{R}}) -
\mathcal{G}^{\rm{3D}}\left(E_{\rm{3b}}-E_{\boldsymbol\lambda};\frac{{\bf{r}} + \sqrt{3} {\bf{R}}}{2};{\bf{0}}\right)
\Phi_{\boldsymbol\lambda}\left(\frac{\sqrt{3}{\bf{r}} - {\bf{R}}}{2}\right)
\right].\end{aligned}$$ Here, we used that ${\bf{r}}_{32}$ can be written as $({\bf{r}} + \sqrt{3}{\bf{R}})/2$ and introduced the one-body Green’s function $\mathcal{G}^{\rm{3D}}\left(E;{\bf{r}};{\bf{r'}}\right)$ for the pseudoparticle of mass $\mu$ that is associated with the relative distance vector ${\bf{r'}}$, $$\begin{aligned}
\label{eq_one_green}
\mathcal{G}^{\rm{3D}}\left(E;{\bf{r}};{\bf{r'}}\right) =
\sum_{\boldsymbol\lambda'}\frac{\Phi_{\boldsymbol\lambda'}^*({\bf{r'}})
\Phi_{\boldsymbol\lambda'}({\bf{r}})
}{E_{\boldsymbol\lambda'}-E}.\end{aligned}$$ The one-body Green’s function $\mathcal{G}^{\rm{3D}}\left(E_{\rm{2b}};{\bf{r}};{\bf{r'}}\right)$ with ${\bf{r'}}={\bf{0}}$ coincides with the solution to the relative Schrödinger equation for two particles under harmonic confinement interacting through the zero-range pseudopotential $V_{\rm{ps}}^{\rm{3D}}({\bf{r}})$ with $s$-wave scattering length $a^{\rm{3D}}$ and relative two-body energy $E_{\rm{2b}}$. $\mathcal{G}^{\rm{3D}}\left(E;{\bf{r}};{\bf{0}}\right)$ is known for all aspect ratios $\eta$ [@calarco-r; @calarco] (see also Secs. [\[sec\_cigar\]]{} and [\[sec\_pancake\]]{}).
To determine the expansion coefficients $f_{\boldsymbol\lambda}$, we apply the operation $\left. \frac{\partial}{\partial r}(r \cdot )\right|_{r \rightarrow 0}$ to the left hand side and the right hand side of Eq. (\[eq\_wf\_expansion\]), i.e., we multiply both sides of Eq. (\[eq\_wf\_expansion\]) by $r$, then apply the derivative operator and lastly take the limit $r \to 0$. Defining $$\begin{aligned}
\label{eq_f}
\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)= 2 \pi E_z a_z^3
\left. \frac{\partial}{\partial r}\left\{r\mathcal{G}^{\rm{3D}}\left([\epsilon_{\boldsymbol{\lambda}}+\eta+1/2]E_z;{\bf{r}};{\bf{0}}\right)\right\}\right|_{r \rightarrow 0}\end{aligned}$$ with $(\epsilon_{\boldsymbol{\lambda}}+\eta + 1/2)E_z=
E_{\rm{3b}}-E_{\boldsymbol{\lambda}}$, we find $$\begin{aligned}
\label{eq_eigen1}
& -\frac{a_z}{2 \pi a^{\rm{3D}}}\sum_{\boldsymbol{\lambda'}}
f_{\boldsymbol{\lambda'}}\Phi_{\boldsymbol\lambda'}({\bf{R}})=
\nonumber\\
&
\sum_{\boldsymbol{\lambda'}}f_{\boldsymbol{\lambda'}}
\left\{\frac{1}{2 \pi}\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}},\eta)
\Phi_{\boldsymbol\lambda'}\left({\bf{R}}\right)-
E_z a_z^3 \mathcal{G}^{\rm{3D}}\left([\epsilon_{\boldsymbol{\lambda'}}+\eta+1/2]E_z;\frac{\sqrt{3}}{2}{\bf{R}};{\bf{0}}\right)
\Phi_{\boldsymbol\lambda'}\left(\frac{-{\bf{R}}}{2}\right)\right\}.\end{aligned}$$ The quantity $\epsilon_{\boldsymbol{\lambda'}}$ can be interpreted as a non-integer quantum number associated with the interacting pair. If we multiply Eq. (\[eq\_eigen1\]) by $\Phi^*_{\boldsymbol\lambda}({\bf{R}})$ and integrate over ${\bf{R}}$, we find an implicit eigenequation for the relative three-body energy $E_{\rm{3b}}$ or equivalently, the non-integer quantum number $\epsilon_{\boldsymbol{\lambda}}$, $$\begin{aligned}
\label{eq_eigen2}
\sum_{\boldsymbol{\lambda'}}\left[
I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}
\left(\epsilon_{\boldsymbol{\lambda'}}\right)-
\mathcal{F}^{\rm{3D}}\left(\epsilon_{\boldsymbol{\lambda}},\eta\right)
\delta_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}\right]
f_{\boldsymbol{\lambda'}}=
\frac{a_z}{a^{\rm{3D}}}f_{\boldsymbol{\lambda}},\end{aligned}$$ where $$\begin{aligned}
\label{eq_Ilambda}
I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}
\left(\epsilon_{\boldsymbol{\lambda'}}\right)
= 2 \pi E_z a_z^3
\int
\mathcal{G}^{\rm{3D}}
\left(
[\epsilon_{\boldsymbol{\lambda'}}+\eta+1/2]E_z;
\frac{\sqrt{3}}{2}{\bf{R}};{\bf{0}}\right)
\Phi_{\boldsymbol\lambda'}\left(\frac{-{\bf{R}}}{2}\right)
\Phi_{\boldsymbol\lambda}^*({\bf{R}})~d{\bf{R}}\end{aligned}$$ and $ \delta_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}$ is the Kronecker delta symbol. The determination of $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ and $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}},\eta)$ for $\eta > 1$ and $\eta < 1$ is discussed in Secs. [\[sec\_cigar\]]{} and [\[sec\_pancake\]]{}, respectively.
Equation (\[eq\_eigen2\]) can be interpreted as a matrix equation with eigenvalues $a_z/a^{\rm{3D}}$ and eigenvectors $f_{\boldsymbol{\lambda}}$ [@kestner; @drummond_prl]. In practice, we first calculate the matrix elements $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ in Eq. (\[eq\_eigen2\]) for a given three-body energy $E_{\rm{3b}}$ and obtain the corresponding scattering lengths for this energy by diagonalizing the matrix with elements $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})-
\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta) \delta_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}$. This step is repeated for several three-body energies. Lastly, we invert $a^{\rm{3D}}(E_{\rm{3b}})$ to get $E_{\rm{3b}}(a^{\rm{3D}})$, i.e., to get the three-body energies as a function of the $s$-wave scattering length.
Equation (\[eq\_eigen2\]) has a simple physical interpretation. If the interaction between particles 2 and 3 is turned off, the matrix $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ vanishes and the solution reduces to that of an interacting pair (particles 1 and 3) and a non-interacting spectator particle (particle 2). The relative energy $(\epsilon_{\boldsymbol{\lambda'}}+\eta+1/2)E_z$ of the pair is determined by solving the relative two-body eigenequation $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}},\eta)=-a_z/a^{\rm{3D}}$. The matrix $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ thus arises from the fact that particle 3 not only interacts with particle 1 but also with particle 2. Correspondingly, the terms in Eq. (\[eq\_eigen2\]) that contain $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ can be interpreted as exchange terms that arise from exchanging particles 1 and 2 [@drummond_prl].
For $\eta = 1$, the function $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)$ is given in Table \[tab\_two\_scatt\] and the evaluation of $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ has been discussed in detail in Ref. [@drummond_pra]. The $\eta \neq 1$ cases are discussed in Secs. [\[sec\_cigar\]]{} and [\[sec\_pancake\]]{}.
$\rm{3D}$ $\rm{2D}$ $\rm{1D}$
----------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------
$V_{\rm{ps}}$ $g^{\rm{3D}}\delta^{(3)}({\bf{r}})\frac{\partial}{\partial r}r$ $g^{\rm{2D}}\delta^{(2)}({\boldsymbol{\rho}})\frac{\partial}{\partial \rho}\rho$ $g^{\rm{1D}}\delta(z)$
$g$ $2\pi \frac{\hbar^2}{\mu}a^{\rm{3D}}$ $\pi \frac{\hbar^2}{\mu} \left[\ln(\frac{\rho}{a^{\rm{2D}}})+1\right]^{-1}$ $ -\frac{\hbar^2}{\mu}\frac{1}{a^{\rm{1D}}}$
$ \left.\psi({\bf{q}}) \right|_{|{\bf{q}}| \to 0} $ $\propto \left(\frac{1}{r}-\frac{1}{a^{\rm{3D}}}\right)$ $\propto \left[\ln(a^{\rm{2D}})-\ln(\rho)\right]$ $\propto (z- a^{\rm{1D}})$
Bethe-Peierls B.C. $\left . \frac{\partial (r \psi)}{\partial r} \right|_{r \to 0} = \frac{-1}{a^{\rm{3D}}}\left . (r \psi) \right|_{r \to 0}$ $\left . \rho \frac{\partial \psi}{\partial \rho} \right|_{\rho \to 0} = \left . \frac{\psi}{\ln\left(\rho/a^{\rm{2D}}\right)} \right|_{\rho \to 0}$ $\left . \frac{d \psi}{d z} \right|_{z \to 0} = \frac{-1}{a^{\rm{1D}}}\left . \psi \right|_{z \to 0}$
$\mathcal{F}$ $\mathcal{F}^{\rm{3D}}(\epsilon,\eta)= 2 \pi E_z a_z^3$ $\mathcal{F}^{\rm{2D}}(\epsilon)= \pi E_{\rho} a_{\rho}^2$ $\mathcal{F}^{\rm{1D}}(\epsilon)= E_z a_z$
$ \left\{ \frac{\partial}{\partial r}\left[r\mathcal{G}^{\rm{3D}}\left([\epsilon+\eta+\frac{1}{2}]E_z;{\bf{r}};{\bf{0}}\right)\right]\right\}_{r \rightarrow 0}$ $\left\{ \mathcal{G}^{\rm{2D}}\left([\epsilon+1]E_\rho;{{\rho}};{{0}}\right) + \ln(\rho/a_{\rho})\right\}_{\rho \rightarrow 0}$ $\left\{ \mathcal{G}^{\rm{1D}}\left([\epsilon+\frac{1}{2}]E_z;z;0\right)\right\}_{z \rightarrow 0}$
$\mathcal{F}^{\rm{3D}}(\epsilon,1)=\frac{-2\Gamma(-\epsilon/2)}{\Gamma(-\epsilon/2-1/2)}$ $\mathcal{F}^{\rm{2D}}(\epsilon)=-\frac{1}{2}\psi_g(-\epsilon/2) - \gamma$ $\mathcal{F}^{\rm{1D}}(\epsilon)=\frac{\Gamma(-\epsilon/2)}{2 \Gamma(-\epsilon/2+ 1/2)}$
two-body energy $\mathcal{F}^{\rm{3D}}(\epsilon, \eta) = -a_z/a^{\rm{3D}}$ $\mathcal{F}^{\rm{2D}}(\epsilon) = \ln(a^{\rm{2D}}/a_{\rho})$ $\mathcal{F}^{\rm{1D}}(\epsilon) = a^{\rm{1D}}/a_z$
: Two-body properties in three dimensions ($s$-wave channel), two dimensions ($m=0$ channel), and one dimension (even parity channel). $\psi({\bf{q}})$ denotes the relative two-body wave function and $\mu$ the two-body reduced mass. ${\bf{q}}$ stands for ${\bf{r}}$, ${\boldsymbol{\rho}}$, and $z$ in three, two, and one dimensions, respectively. For 3D, 2D and 1D, we have $\epsilon=E_{\rm{2b}}/E_z-\eta-1/2$, $\epsilon=E_{\rm{2b}}/E_{\rho}-1$ and $\epsilon=E_{\rm{2b}}/E_z-1/2$, respectively, where $E_{\rm{2b}}$ denotes the relative two-body energy. $\psi_g$ denotes the digamma function and $\gamma$ the Euler constant, $\gamma \approx 0.577$.
\[tab\_two\_scatt\]
For a spherically symmetic system with $\eta = 1$, the total relative angular momentum quantum number $L$, the corresponding projection quantum number $M$ and the parity $\Pi$ are good quantum numbers, and the eigenvalue equation can be solved for each $L$ and $M$ combination separately using spherical coordinates [@kestner]. For a fixed $L$ and $M$, $\boldsymbol{\lambda}=(n,l,m)$ and $\boldsymbol{\lambda'}=(n',l',m')$ in Eq. (\[eq\_eigen2\]) are constrained by $l=l'=L$ and $m=m'=M$. The parity of the three-body system is given by $\Pi=(-1)^L$.
We emphasize that the outlined formalism makes no approximations, i.e., Eq. (\[eq\_eigen2\]) with $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}
\left(\epsilon_{\boldsymbol{\lambda'}}\right)$ given by Eq. (\[eq\_Ilambda\]) describes all eigenstates of $H_{\rm{rel}}$ \[see Eq. (\[eq\_rel\_Hamiltonian\])\] that are affected by the interactions. In particular, all “channel couplings” are accounted for. In practice, the construction of the matrix $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$ requires one to choose a maximum for ${\boldsymbol{\lambda}}$ and $\boldsymbol{\lambda'}$, or alternatively, a cutoff for the single-particle energy $E_{\boldsymbol{\lambda}}$. As has been shown in Ref. [@drummond_pra], good convergence is achieved for a relatively small number of “basis functions” for $\eta=1$. As we show below, good convergence is also obtained for anisotropic confinement geometries.
The formalism outlined can also be applied to strictly one-dimensional and strictly two-dimensional systems. Table \[tab\_two\_scatt\] defines the one-dimensional and two-dimensional pseudopotentials as well as a number of key properties of the corresponding relative two-body system. Making the appropriate changes in the outlined derivation and using the properties listed in Table \[tab\_two\_scatt\], we find for strictly one-dimensional systems $$\begin{aligned}
\label{eq_eigen_1d}
\sum_{n_z'=0}^{\infty}
\left[
I_{n_z,n_z'}^{\rm{1D}}\left(\epsilon_{n_z'}\right)
- \mathcal{F}^{\rm{1D}}\left(\epsilon_{n_z}\right)
\delta_{n_z,n_z'}
\right]
f_{n_z'}=
-\frac{a^{\rm{1D}}}{a_z}f_{n_z},\end{aligned}$$ where $\mathcal{F}^{\rm{1D}}$ is defined in Table \[tab\_two\_scatt\], $$\begin{aligned}
\label{eq_Inz_1d}
I_{n_z,n_z'}^{\rm{1D}}\left(\epsilon_{n_z'}\right)=
E_z a_z\int_{-\infty}^{\infty}
\mathcal{G}^{\rm{1D}}\left([\epsilon_{n_z'}+1/2]E_z;
\frac{\sqrt{3}}{2}z;0\right)
\varphi_{n_z'}\left(-\frac{z}{2}\right)\varphi_{n_z}^*(z) dz,\end{aligned}$$ and $E_{\rm{3b}} - E_{n_z'} = (\epsilon_{n_z'} + 1/2)E_z$. Here, $E_{n_z}$ denotes the single-particle energy of the one-dimensional system, $E_{n_z}=(n_z+1/2) E_z$, and $\mathcal{G}^{\rm{1D}}\left(E;z;z'\right)$ the one-dimensional even parity single-particle Green’s function, $$\begin{aligned}
\label{eq_one_green_1d}
\mathcal{G}^{\rm{1D}}\left(E;z;z'\right) =
\sum_{n_z'=0}^{\infty}\frac{\varphi_{2n_z'}^*(z')
\varphi_{2n_z'}(z)
}{E_{2n_z'}-E}.\end{aligned}$$ For $z'=0$, the single-particle Green’s function is given by $$\begin{aligned}
\label{eq_2-body_1d}
\mathcal{G}^{\rm{1D}}\left(E;z;0\right)
= \frac{1}{2\sqrt{\pi} E_z a_z}
\exp\left(-\frac{z^2}{2 a_z^2}\right)
\Gamma\left(-\frac{E/E_z-1/2}{2}\right)
U\left(-\frac{E/E_z-1/2}{2},\frac{1}{2},\frac{z^2}{a_z^2}\right),\end{aligned}$$ where $\Gamma(x)$ is the Gamma function and $U(a,b,z)$ the confluent hypergeometric function. The strictly one-dimensional relative three-body wave function $\Psi$ is characterized by the parity $\Pi_z$. For even parity states, i.e., for states with $\Pi_z=1$, $n_z$ and $n_z'$ in Eq. (\[eq\_eigen\_1d\]) have to be even. For odd parity states, i.e., for states with $\Pi_z=-1$, $n_z$ and $n_z'$ have to be odd.
Similarly, for strictly two-dimensional systems, expressed in units of $E_{\rho}$ and $a_{\rho}$ \[$E_{\rho}=\hbar \omega_{\rho}$ and $a_{\rho}=\sqrt{\hbar/(\mu \omega_{\rho})}$\], we find, in agreement with Ref. [@drummond_2d], $$\begin{aligned}
\label{eq_eigen_2d}
\sum_{n_{\rho}'=0}^{\infty}
\left[
I_{n_{\rho},n_{\rho}',m}^{\rm{2D}}(\epsilon_{n_{\rho}',m}) -
\mathcal{F}^{\rm{2D}}(\epsilon_{n_{\rho},m})
\delta_{n_{\rho},n_{\rho}'}
\right]
f_{n_{\rho}',m}=
\ln\left(\frac{a_{\rho}}{a^{\rm{2D}}}\right)f_{n_{\rho},m},\end{aligned}$$ where $\mathcal{F}^{\rm{2D}}$ is defined in Table \[tab\_two\_scatt\], $$\begin{aligned}
\label{eq_Inrho_2d}
I_{n_{\rho},n_{\rho}',m}^{\rm{2D}}(\epsilon_{n_{\rho}',m})=
(-1)^m ~\pi
E_{\rho} a_{\rho}^2
\int_0^{\infty}
\mathcal{G}^{\rm{2D}}
\left([\epsilon_{n_{\rho}',m}+1]E_{\rho};\frac{\sqrt{3}}{2}\rho;0\right)
R_{n_{\rho}',m}\left(\frac{\rho}{2}\right)
R_{n_{\rho},m}\left(\rho\right) \rho d \rho,\end{aligned}$$ and $E_{\rm{3b}}-E_{n_{\rho}',m}=(\epsilon_{n_{\rho}',m}+1) E_{\rho}$. Here, $E_{n_{\rho},m}$ denotes the single-particle energy of the two-dimensional system, $E_{n_{\rho},m}= (2 n_{\rho} + |m| +1)E_{\rho}$. The two-dimensional single-particle Green’s function $\mathcal{G}^{\rm{2D}}\left(E;{\rho}; {\rho}'\right)$ is defined analogously to the three- and one-dimensional counterparts \[see Eqs. (\[eq\_one\_green\]) and (\[eq\_one\_green\_1d\])\]. For $\rho'=0$ and states affected by the zero-range $s$-wave interactions [@calarco-r; @calarco], one finds $$\begin{aligned}
\label{eq_2-body_2d}
\mathcal{G}^{\rm{2D}}(E;{\rho};{0}) =
\frac{1}{2\pi E_{\rho} a_{\rho}^2}
\exp\left(-\frac{\rho^2}{2 a_{\rho}^2}\right)
\Gamma\left(-\frac{E/E_{\rho}-1}{2}\right)
U\left(-\frac{E/E_{\rho}-1}{2},1,\frac{\rho^2}{a_{\rho}^2}\right).\end{aligned}$$ The strictly two-dimensional relative three-body wave function is characterized by the projection quantum number $M$ and the parity $\Pi_{\boldsymbol{\rho}}$, $\Pi_{\boldsymbol{\rho}}=(-1)^{M}$. For a fixed $M$, $m$ in Eq. (\[eq\_eigen\_2d\]) is constrained to the value $m=M$. The next two sections analyze, utilizing our results for strictly one- and two-dimensional systems, Eq. (\[eq\_eigen2\]) for cigar- and pancake-shaped traps.
Cigar-shaped trap {#sec_cigar}
=================
To apply the formalism reviewed in Sec. [\[sec\_formalsolution\]]{} to axially symmetric traps, we need the explicit forms of the functions $\mathcal{G}^{\rm{3D}}\left([\epsilon_{\boldsymbol{\lambda}}+\eta+1/2]E_z;
{\bf{r}};{\bf{0}}\right)$ and $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)$, that is, the relative solutions to the trapped two-body system. For cigar-shaped traps ($\eta > 1$), it is convenient to write $\mathcal{G}^{\rm{3D}}$ as [@calarco-r; @calarco] $$\begin{aligned}
\label{eq_G_cigar}
\mathcal{G}^{\rm{3D}}\left([\epsilon_{\boldsymbol{\lambda}}+\eta+1/2]E_z;
{\bf{r}};{\bf{0}}\right)=
\frac{\eta}{\pi a_z^2}\exp\left(-\frac{ \eta \rho^2}{2 a_z^2}\right)
\sum_{j=0}^{\infty} L_j\left( \eta \rho^2 / a_z^2\right)
\mathcal{G}^{\rm{1D}}\left([\epsilon_{\boldsymbol\lambda}-2 \eta j+
1/2]E_z ;z;0\right),\end{aligned}$$ where $\mathcal{G}^{\rm{1D}}\left(E;z;0\right)$ is defined in Eq. (\[eq\_2-body\_1d\]). Using Eq. (\[eq\_G\_cigar\]) in Eq. (\[eq\_Ilambda\]), we obtain $$\begin{aligned}
\label{eq_Ilambda_cigar}
I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}
(\epsilon_{\boldsymbol{\lambda'}}) =
\sqrt{2\eta}(-1)^m\delta_{m,m'}\times \nonumber \\
\lim_{j_{\rm{max}} \to \infty}\sum_{j=0}^{j_{\rm{max}}}I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j) I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j),\end{aligned}$$ where $$\begin{aligned}
\label{eq_Iz_cigar}
I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)=
E_z a_z
\int_{-\infty}^{\infty}
\mathcal{G}^{\rm{1D}}\left([\epsilon_{\boldsymbol{\lambda'}}-2 \eta j+
1/2]E_z;\frac{\sqrt{3}}{2}z; 0\right)
\varphi_{n_z'}\left(-\frac{z}{2}\right)\varphi_{n_z}(z)~dz\end{aligned}$$ and $$\begin{aligned}
\label{eq_Irho_cigar}
& I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)=\nonumber \\
&a_z\int_0^{\infty} R_{j,0}\left(\frac{\sqrt{3}}{2}\rho\right) R_{n_{\rho}',m}\left(\frac{\rho}{2}\right) R_{n_{\rho},m}(\rho)~\rho d\rho.\end{aligned}$$ The evaluation of the integrals $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ and $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$ is discussed in Appendix \[appendix\]. The superscript “c” indicates that the integrals apply to cigar-shaped systems; for pancake-shaped systems (see Sec. \[sec\_pancake\]), we introduce the integrals $I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)$ and $I_{n_z,n_z'}^{\rm{p}}(j)$ instead.
Although it is possible to calculate $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)$ numerically for any trap aspect ratio $\eta$, we restrict ourselves to integer aspect ratios for simplicity. For traps with integer aspect ratio, an exact analytical expression for $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)$ is known [@calarco-r; @calarco], $$\begin{aligned}
\label{eq_f26_cigar}
\mathcal{F}^{\rm{3D}} \left( \epsilon_{\boldsymbol{\lambda}},\eta \right)=
-2 \frac
{\Gamma\left(-\frac{\epsilon_{\boldsymbol{\lambda}}}{2}\right)}
{\Gamma\left(-\frac{1}{2}-\frac{\epsilon_{\boldsymbol{\lambda}}}{2}\right)}
+ \frac
{\Gamma\left(-\frac{\epsilon_{\boldsymbol{\lambda}}}{2}\right)}
{\Gamma\left(\frac{1}{2}-\frac{\epsilon_{\boldsymbol{\lambda}}}{2}\right)}
\times \nonumber \\
\sum _{k=1}^{\eta -1} {}_2F_1 \left(1,-\frac{\epsilon_{\boldsymbol{\lambda}}}{2};\frac{1}{2}-\frac{\epsilon_{\boldsymbol{\lambda}}}{2};\exp
\left(\frac{2 \pi \imath k}{\eta} \right) \right),\end{aligned}$$ where $ {}_2F_1(a,b;c;z)$ is the hypergeometric function [@abramowitz]. Knowing $I_{\boldsymbol{\lambda},\boldsymbol{\lambda'}}^{\rm{3D}}
(\epsilon_{\boldsymbol{\lambda'}})$ and $\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}},\eta)$, Eq. (\[eq\_eigen2\]) can be diagonalized separately for each $(\Pi_z, M,\Pi_{\boldsymbol{\rho}})$ combination. We recall from Sec. \[sec\_formalsolution\] that $\boldsymbol{\lambda}=(n_z,n_{\rho},m)$ and $\boldsymbol{\lambda}'=(n_z',n_{\rho}',m')$. The $m$ and $m'$ values are constrained by $m=m'=M$. Moreover, for $\Pi_z=+1$ and $\Pi_z=-1$, we have $n_z=n_z'=even$ and $n_z=n_z'=odd$, respectively.
Figure \[fig\_cigar\]
![Relative three-body energies $E_{\rm{3b}}/E_z$ as a function of the inverse scattering length $a_z/a^{\rm{3D}}$ for a cigar-shaped trap with aspect ratio $\eta = 2$ and (a) $M=0$ and $\Pi_z = +1$, and (b) $M=0$ and $\Pi_z = -1$. []{data-label="fig_cigar"}](fig1_short.eps){width="70mm"}
shows the three-body relative energies $E_{\rm{3b}}/E_z$ for $\eta=2$ for states with (a) $M=0$ and $\Pi_z = +1$ and (b) $M=0$ and $\Pi_z = -1$ as a function of the inverse scattering length $a_z/a^{\rm{3D}}$. The non-interacting limit is approached when $(a^{\rm{3D}})^{-1} \to \pm \infty$, and the infinitely strongly-interacting regime for $(a^{\rm{3D}})^{-1}=0$ (center of the figure). For each fixed projection quantum number $M$, we include around $840$ basis functions. This corresponds to a cutoff of around $(82+2M)E_z$ for the single-particle energy $E_{\boldsymbol{\lambda}}$. We find that $j_{\rm{max}} \gtrsim 30$ yields converged values for $I_{{\boldsymbol{\lambda}},\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}})$, Eq. (\[eq\_Ilambda\_cigar\]). For small $|a^{\rm{3D}}/a_z|$ ($a^{\rm{3D}}$ positive and negative), our eigenenergies agree with those obtained within first-order perturbation theory. Our analysis shows that the energy of the ground state at unitarity has a relative error of the order of $10^{-5}$. The accuracy decreases with increasing energy. For example, for energies around $20 E_z$, the relative accuracy at unitarity is of the order of $10^{-4}$.
The eigenstates fall into one of two categories: atom-dimer states and atom-atom-atom states. The eigenenergies associated with the former are negative for large positive $a_z/a^{\rm{3D}}$ while those associated with the latter remain positive for large positive $a_z/a^{\rm{3D}}$. The energy spectra shown in Fig. \[fig\_cigar\] exhibit sequences of avoided crossings. To resolve these crossings, a fairly fine mesh in the three-body energy is needed. In the $(a^{\rm{3D}})^{-1} \to - \infty$ limit, the lowest $M=0$ state has negative parity in $z$, i.e., $\Pi_z = -1$. This is a direct consequence of the fact that the two identical fermions cannot occupy the same single particle state. In the $(a^{\rm{3D}})^{-1} \to + \infty$ limit, in contrast, the lowest $M=0$ state has positive parity in $z$, i.e., $\Pi_z = +1$. This is a direct consequence of the fact that the system consists, effectively, of a dimer and an atom.
The main part of Fig. \[fig\_cigar\_gr\_st\]
![(Color online) “Crossover curve” of the three-body system with $M=0$, shifted by $2 E_{\rho} = 2 \eta E_z$, for cigar-shaped traps with $\eta = 2$ (solid line), $4$ (dotted line), $6$ (dashed line), $8$ (dash-dot-dotted line), and $10$ (dash-dotted line) as a function of $a_z/a^{\rm{3D}}$. The scattering lengths at which the parity of the corresponding eigenstate changes from $\Pi_z = -1$ (“left side of the graph”) to $\Pi_z = +1$ (“right side of the graph”) are marked by asterisks. At these points, the derivative of the crossover curve is discontinuous; the discontinuities are not visible on the scale shown. The inset shows the (unshifted) crossover curve as a function of $a_{\rho}/a^{\rm{3D}}$.[]{data-label="fig_cigar_gr_st"}](fig2.eps){width="70mm"}
shows the relative energy of the energetically lowest-lying state, the so-called crossover curve, of the three-body system with $M=0$ for various aspect ratios of the trap ($\eta = 2,\cdots,10$) as a function of the inverse scattering length $a_z/a^{\rm{3D}}$. For comparative purposes, we subtract the ground state energy of $2\eta E_z$ of the strictly two-dimensional non-interacting system, that is, the energy that the system would have in the $\rho$-direction if the dynamics in the tight confinement direction were frozen, from the full three-dimensional energy. In Fig. \[fig\_cigar\_gr\_st\], asterisks mark the scattering lengths at which the eigenstate associated with the crossover curve changes from $\Pi_z = -1$ to $\Pi_z = +1$. With increasing $\eta$, the parity change occurs at larger $a_z / a^{\rm{3D}}$ (that is, smaller $a^{\rm{3D}}/a_z$). The inset of Fig. \[fig\_cigar\_gr\_st\] replots the crossover curves as a function of $a_{\rho}/a^{\rm{3D}}$.
We now discuss the large $\eta$ limit in more detail. Using the limiting behavior of $\mathcal{F}^{\rm{3D}}(\epsilon, \eta)$ for $\eta\gg 1$ and $\eta\gg |\epsilon|$ [@calarco-r; @calarco], $$\begin{aligned}
\label{eq_f_large_eta}
\left.
\mathcal{F}^{\rm{3D}}
(\epsilon, \eta)\right|_{\eta\gg 1}
\approx
2\eta \mathcal{F}^{\rm{1D}}(\epsilon)
+\sqrt{\eta} \zeta(1/2),\end{aligned}$$ the two-body eigenequation for the relative energy becomes [@calarco-r; @calarco] $$\begin{aligned}
\label{eq_two-b_1d}
\mathcal{F}^{\rm{1D}}(\epsilon)=\frac{a_{\rm{ren}}^{\rm{1D}}}{a_z},\end{aligned}$$ where the renormalized one-dimensional scattering length $a_{\rm{ren}}^{\rm{1D}}$ is given by [@olshanii1; @olshanii2] $$\begin{aligned}
\label{eq_olshanii}
\frac{a_{\rm{ren}}^{\rm{1D}}}{a_z}=\frac{1}{\sqrt{\eta}} \left[
-\frac{a_{\rho}}{2 a^{\rm{3D}}}-\frac{\zeta(1/2)}{2}
\right].\end{aligned}$$ Figure \[fig\_cigar\_quasi\_2-body\](a)
.[]{data-label="fig_cigar_quasi_2-body"}](fig3.eps){width="70mm"}
shows the relative two-body energies for a system with $\eta = 10$, $M=0$ and $\Pi_z=+1$ obtained by solving the eigenequation $\mathcal{F}^{\rm{3D}}(\epsilon, \eta=10)=-a_z/a^{\rm{3D}}$ \[see Eq. (\[eq\_f26\_cigar\]) for $\mathcal{F}^{\rm{3D}}(\epsilon, \eta)$\]. Figure \[fig\_cigar\_quasi\_2-body\](b) compares the full three-dimensional energy (solid line) with the energy obtained by solving the strictly one-dimensional eigenequation, Eq. (\[eq\_two-b\_1d\]), with renormalized one-dimensional scattering length $a_{\rm{ren}}^{\rm{1D}}$ (dotted line). To facilitate the comparison, we add the energy of the tight confinement direction to the energy of the one-dimensional system. The agreement is quite good for all scattering lengths. The inset of Fig. \[fig\_cigar\_quasi\_2-body\](b) shows the difference between the strictly one-dimensional energy and the full three-dimensional energy as a function of $a_z/a^{\rm{3D}}$. The maximum deviation occurs around unitarity and is of the order of $0.2 \%$.
Next, we discuss the behavior of the three-body system in the large $\eta$ limit. If we use Eqs. (\[eq\_f\_large\_eta\]) and (\[eq\_olshanii\]) in Eq. (\[eq\_eigen2\]), we find $$\begin{aligned}
\label{eq_eigen_large_eta}
\sum_{\boldsymbol{\lambda'}}\left[
\frac{1}{2\eta} I^{\rm{3D}}_{\boldsymbol{\lambda},\boldsymbol{\lambda'}}(\epsilon_{\boldsymbol{\lambda}'})-
\mathcal{F}^{\rm{1D}}(\epsilon_{\boldsymbol{\lambda}})
\delta_{\boldsymbol{\lambda},\boldsymbol{\lambda}'}
\right]f_{\boldsymbol{\lambda}'}
= - \frac{a_{\rm{ren}}^{\rm{1D}}}{a_z}f_{\boldsymbol{\lambda}}.\end{aligned}$$ A straightforward analysis shows that Eq. (\[eq\_eigen\_large\_eta\]) reduces to its strictly one-dimensional analog if [*[(i)]{}*]{} the sum over $\boldsymbol{\lambda}'$ is restricted to a sum over $n_z'$ \[$\boldsymbol{\lambda}'=(n_z',0,0)$\]; [*[(ii)]{}*]{} the index $\boldsymbol{\lambda}$ is restricted to $\boldsymbol{\lambda}=(n_z,0,0)$; [*[(iii)]{}*]{} the energy $E_{\rm{3b}}$ is replaced by $E_{\rm{3b}} -2\eta E_z$; and [*[(iv)]{}*]{} $j_{\rm{max}}$ in Eq. (\[eq\_Ilambda\_cigar\]) is set to zero. Under these assumptions, Eq. (\[eq\_eigen\_large\_eta\]) reduces to Eq. (\[eq\_eigen\_1d\]) with $a^{\rm{1D}}$ replaced by $a_{\rm{ren}}^{\rm{1D}}$. We emphasize that the assumption $\eta\gg |\epsilon_{\boldsymbol{\lambda}}|$ \[see discussion around Eq. (\[eq\_f\_large\_eta\])\] is not valid when two atoms form a tight molecule. In this limit, the three-dimensional $s$-wave scattering length, or the size of the dimer, is smaller than the harmonic oscillator length in the transverse direction, which implies that the strictly one-dimensional description is not valid.
Figure \[fig\_cigar\_quasi\_3-body\](a)
![(Color online) (a) Relative three-body energies $E_{\rm{3b}}/E_z$ as a function of the inverse scattering length $a_z/a^{\rm{3D}}$ for a cigar-shaped trap with aspect ratio $\eta = 10$, $M=0$ and $\Pi_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly one-dimensional eigenequation with renormalized one-dimensional scattering length $a_{\rm{ren}}^{\rm{1D}}$ (the energy of $2 E_{\rho}=2\eta E_z$ has been added to allow for a comparison with the full three-dimensional energy). For comparison, the dashed line shows one of the three-dimensional energy curves for $M=0$ and $\Pi_z=-1$ \[not shown in panel (a)\]; the corresponding strictly one-dimensional energy is shown by a dotted line. The difference between the full three-dimensional and strictly one-dimensional descriptions is hardly visible on the scale shown. Solid and dashed lines in the inset show the differences between the strictly one-dimensional energies \[dotted lines in panel (b)\] and the full three-dimensional energies \[solid and dashed lines in panel (b)\] as a function of the inverse scattering length $a_z/a^{\rm{3D}}$ for $\Pi_z=+1$ and $\Pi_z=-1$, respectively.[]{data-label="fig_cigar_quasi_3-body"}](fig4_short.eps){width="70mm"}
shows the relative three-body energies for states with $M=0$ and $\Pi_z=+1$ as a function of the inverse scattering length $a_z/a^{\rm{3D}}$ for a cigar-shaped trap with $\eta=10$. Figure \[fig\_cigar\_quasi\_3-body\](b) compares the energy of the energetically lowest-lying three-atom state with $\Pi_z=+1$ (solid line) \[see thick solid line in Fig. \[fig\_cigar\_quasi\_3-body\](a)\] with the corresponding state obtained by solving the strictly one-dimensional equation with renormalized one-dimensional scattering length $a_{\rm{ren}}^{\rm{1D}}$. We also include the energy of one of the eigenstates with $M=0$ and $\Pi_z= -1$ (dashed line). The inset shows the difference between the energy obtained within the strictly one-dimensional and the full three-dimensional frameworks. The maximum of the deviation occurs near unitarity. The agreement between the full three-dimensional and the strictly one-dimensional descriptions is good. Importantly, the deviations for the three-body system with $M=0$ and $\Pi_z=+1$ \[solid line in the inset of Fig. \[fig\_cigar\_quasi\_3-body\](b)\] are only slightly larger than those for the two-body system \[inset of Fig. \[fig\_cigar\_quasi\_2-body\](b)\], suggesting that the presence of the third atom does not, in a significant manner, reduce the applicability of the strictly one-dimensional framework—at least for states in the low-energy regime characterized as gas-like three-atom states.
Pancake-shaped trap {#sec_pancake}
===================
For pancake-shaped traps with $\eta < 1$, we use the following form of the Green’s function $\mathcal{G}^{\rm{3D}}$ [@calarco-r; @calarco], $$\begin{aligned}
\label{eq_G_pancake}
\mathcal{G}^{\rm{3D}}(
[\epsilon_{\boldsymbol\lambda}+\eta+1/2]E_z
;{\bf{r}};{\bf{0}})=
\frac{1}{\sqrt{\pi} E_z a_z^3}\exp\left(-\frac{ z^2}{2 a_z^2}\right)
\sum_{j=0}^{\infty}
\frac{(-1)^j}{2^{2j}j!}
H_{2j}(z/a_z)
\mathcal{G}^{\rm{2D}}
\left( \left[
\frac{\epsilon_{\boldsymbol\lambda}-2 j}{\eta}+1 \right]E_{\rho};
\rho, 0\right).\end{aligned}$$ This expression is equivalent to Eq. (\[eq\_G\_cigar\]) but converges faster for pancake-shaped traps than Eq. (\[eq\_G\_cigar\]). Using Eq. (\[eq\_G\_pancake\]) in Eq. (\[eq\_Ilambda\]), we obtain $$\begin{aligned}
\label{eq_Ilambda_pancake}
I_{{\boldsymbol{\lambda}},
\boldsymbol{\lambda'}}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda'}}) =
2 \sqrt{\pi}(-1)^m \delta_{m,m'}\times \nonumber \\
\lim_{j_{\rm{max}} \rightarrow \infty} \sum_{j=0}^{j_{\rm{max}}}
\frac{(-1)^j\sqrt{\pi^{1/2}(2j)!}}{2^{j} j!}
I_{n_z,n_z'}^{\rm{p}}(j)
I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j),\end{aligned}$$ where $$\begin{aligned}
\label{eq_Iz_pancake}
I_{n_z,n_z'}^{\rm{p}}(j)=
a_z^{1/2}
\int_{-\infty}^{\infty} \varphi_{2j}\left(\frac{\sqrt{3}}{2}z\right)
\varphi_{n_z'}\left(-\frac{z}{2}\right) \varphi_{n_z}(z)~dz\end{aligned}$$ and $$\begin{aligned}
\label{eq_Irho_pancake}
&I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)=
\nonumber \\
&E_z a_z^2 \int_0^{\infty}
\mathcal{G}^{\rm{2D}}\left(
\left[\frac{\epsilon_{\boldsymbol\lambda'}-2 j}{\eta}+1
\right]E_{\rho};
\frac{\sqrt{3}}{2}\rho; 0\right)
R_{n_{\rho}',m}\left(\frac{\rho}{2}\right)
R_{n_{\rho},m}(\rho)
~\rho d\rho.\end{aligned}$$ Details regarding the evaluation of the integrals are explained in Appendix \[appendix\]. In the following, we limit ourselves to cases where the reciprocal of the aspect ratio is an integer. In this case, we have [@calarco-r; @calarco] $$\begin{aligned}
\label{eq_f41_pancake}
\mathcal{F}^{\rm{3D}}(\epsilon_{\boldsymbol{\lambda}},\eta)=
-2 \eta \sum_{k=0}^{1/\eta-1}
\frac{\Gamma(-\frac{\epsilon_{\boldsymbol{\lambda}}}{2} + k \eta)}{\Gamma(-\frac{\epsilon_{\boldsymbol{\lambda}} + 1}{2} + k \eta)}.\end{aligned}$$
Figure \[fig\_pancake\]
![Relative three-body energies $E_{\rm{3b}}/E_{\rho}$ as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$ for a pancake-shaped trap with aspect ratio $\eta = 1/2$ and (a) $M=0$ and $\Pi_z=+1$, and (b) $M=\pm 1$ and $\Pi_z=+1$.[]{data-label="fig_pancake"}](fig5_short.eps){width="70mm"}
shows the relative three-body energies $E_{\rm{3b}}/E_{\rho}$ as a function of the inverse scattering length for $\eta = 1/2$, $\Pi_z=+1$, and (a) $M=0$ and (b) $M=\pm1$. In the $(a^{\rm{3D}})^{-1} \to -\infty$ limit, the ground state has $M=\pm 1$ and $\Pi_z=+1$ symmetry. In the $(a^{\rm{3D}})^{-1} \to +\infty$ limit, in contrast, the ground state has $M=0$ and $\Pi_z=+1$ symmetry.
Figure \[fig\_pancake\_gr\_st\]
![(Color online) “Crossover curve” of the three-body system with $\Pi_z=+1$, shifted by $E_z = E_{\rho} / \eta$, for various aspect ratios of the trap, $\eta = 1/2$ (solid line), $1/4$ (dotted line), $1/6$ (dashed line), $1/8$ (dash-dot-dotted line), and $1/10$ (dash-dotted line) as a function of $a_{\rho} / a^{\rm{3D}}$. The scattering lengths at which the $M$ quantum number of the corresponding eigenstate changes from $M = \pm 1$ (“left side of the graph”) to $M = 0$ (“right side of the graph”) are marked by asterisks. The inset shows the (unshifted) crossover curve as a function of $a_z / a^{\rm{3D}}$.[]{data-label="fig_pancake_gr_st"}](fig6.eps){width="70mm"}
shows the relative energy of the energetically lowest-lying state, the so-called crossover curve, of the three-body system with $\Pi_z=+1$ for various aspect ratios of the trap ($\eta = 1/2, \cdots, 1/10$) as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$. For comparative purposes, we subtract the relative ground state energy of $E_z$ of the non-interacting one-dimensional system, that is, the energy that the system would have in the $z$-direction if the dynamics in the tight confinement direction were frozen, from the full three-dimensional energy. The scattering lengths at which the symmetry of the corresponding eigenstate changes from $M = \pm 1$ to $M = 0$ are marked by asterisks. The symmetry change occurs around $a_z/a^{\rm{3D}} \approx 1$ (see inset).
It is instructive to compare Fig. \[fig\_pancake\_gr\_st\] (pancake-shaped trap) and Fig. \[fig\_cigar\_gr\_st\] (cigar-shaped trap). For both geometries, the crossover curve changes symmetry. The change of the symmetry is associated with the low-energy coordinate (the $\rho$-coordinate for pancake-shaped systems and the $z$-coordinate for cigar-shaped systems). For both geometries, the symmetry change occurs, for the aspect ratios considered, when $a^{\rm{3D}}$ is of the order of the oscillator length in the tight confinement direction.
Next, we consider the small $\eta$ limit in more detail. For $\eta\ll 1$ and $|\epsilon|\ll 1$, we have [@calarco-r; @calarco] $$\begin{aligned}
\label{eq_f_small_eta}
\left. \mathcal{F}^{\rm{3D}}(\epsilon, \eta)\right|_{\eta\ll 1}
\approx
\frac{1}{\sqrt{\pi}}
\left[ 2 \mathcal{F}^{\rm{2D}}(\epsilon)
-2 \ln (\mathcal{C})
- \ln(\eta)
\right],\end{aligned}$$ and the two-body eigenequation for the relative energy becomes $$\begin{aligned}
\label{eq_two-b_2d}
\mathcal{F}^{\rm{2D}}(\epsilon)= \ln(a^{\rm{2D}}_{\rm{ren}}/a_{\rho}),\end{aligned}$$ where the renormalized two-dimensional scattering length $a_{\rm{ren}}^{\rm{2D}}$ is given by [@petrov] $$\begin{aligned}
\label{eq_ren_a_2d}
\frac{a_{\rm{ren}}^{\rm{2D}}}{a_{\rho}}=
\sqrt{\eta}
{\mathcal{C}} \exp \left( -\frac{\sqrt{\pi}a_z}{2 a^{\rm{3D}}} \right)\end{aligned}$$ with ${\mathcal{C}} \approx 1.479$. Figure \[fig\_pancake\_quasi\_2-body\](a)
.[]{data-label="fig_pancake_quasi_2-body"}](fig7.eps){width="70mm"}
shows the relative two-body energies for a system with $\eta = 1/10$, $M=0$ and $\Pi_z=+1$ obtained by solving the eigenequation $\mathcal{F}^{\rm{3D}}(\epsilon, \eta=1/10)=-a_z/a^{\rm{3D}}$ \[see Eq. (\[eq\_f41\_pancake\]) for $\mathcal{F}^{\rm{3D}}(\epsilon, \eta)$\]. Figure \[fig\_pancake\_quasi\_2-body\](b) compares the full three-dimensional energy (solid line) with the energy obtained by solving the strictly two-dimensional eigenequation, Eq. (\[eq\_two-b\_2d\]), with renormalized two-dimensional scattering length $a_{\rm{ren}}^{\rm{2D}}$ (dotted line). For comparative purposes, we add the energy of the tight confinement direction to the energy of the strictly two-dimensional system. The inset of Fig. \[fig\_pancake\_quasi\_2-body\](b) shows the difference between the strictly two-dimensional energy and the full three-dimensional energy as a function of $a_{\rho}/a^{\rm{3D}}$. The maximum deviation occurs around unitarity and is of the order of $0.4 \%$.
To treat the three-body system in the small $\eta$ limit, we insert Eqs. (\[eq\_f\_small\_eta\]) and (\[eq\_ren\_a\_2d\]) into Eq. (\[eq\_eigen2\]). This yields $$\begin{aligned}
\label{eq_eigen_small_eta}
\sum_{\boldsymbol{\lambda'}}\left[
\frac{\sqrt{\pi}}{2} I^{\rm{3D}}_{\boldsymbol{\lambda},\boldsymbol{\lambda'}}
(\epsilon_{\boldsymbol{\lambda}'})-
\mathcal{F}^{\rm{2D}}(\epsilon_{\boldsymbol{\lambda}'})
\delta_{\boldsymbol{\lambda},\boldsymbol{\lambda}'}
\right]f_{\boldsymbol{\lambda}'}
= \ln \left(\frac{a_{\rho}}{ a_{\rm{ren}}^{\rm{2D}}} \right) f_{\boldsymbol{\lambda}}.\end{aligned}$$ For fixed $M$, Eq. (\[eq\_eigen\_small\_eta\]) reduces to the strictly two-dimensional eigenequation, Eq. (\[eq\_eigen\_2d\]), if [*[(i)]{}*]{} the sum over $\boldsymbol{\lambda}'$ is restricted to a sum over $n_{\rho}'$ \[i.e., if $\boldsymbol{\lambda}'=(0,n_{\rho}',m'=M)$\]; [*[(ii)]{}*]{} the index $\boldsymbol{\lambda}$ is restricted to $\boldsymbol{\lambda}=(0,n_{\rho},m=M)$; [*[(iii)]{}*]{} the energy $E_{\rm{3b}}$ is replaced by $E_{\rm{3b}} -E_z$; and [*[(iv)]{}*]{} $j_{\rm{max}}$ in Eq. (\[eq\_Ilambda\_pancake\]) is set to zero. Under these assumptions, Eq. (\[eq\_eigen\_small\_eta\]) reduces to Eq. (\[eq\_eigen\_2d\]) with $a^{\rm{2D}}$ replaced by $a_{\rm{ren}}^{\rm{2D}}$.
Figure \[fig\_pancake\_quasi\_3-body\](a)
![(Color online) (a) Relative three-body energies $E_{\rm{3b}}/E_{\rho}$ as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$ for a pancake-shaped trap with aspect ratio $\eta = 1/10$, $M=0$ and $\Pi_z=+1$. (b) The solid curve from panel (a) is replotted and compared with the energy obtained by solving the strictly two-dimensional eigenequation with renormalized two-dimensional scattering length $a_{\rm{ren}}^{\rm{2D}}$ (dotted line; the energy of $E_{z}$ has been added to allow for a comparison with the full three-dimensional energy). The inset shows the difference between the dotted and solid lines as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$.[]{data-label="fig_pancake_quasi_3-body"}](fig8_short.eps){width="70mm"}
shows the relative three-body energies for states with $M=0$ and $\Pi_z=+1$ as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$ for a pancake-shaped trap with $\eta=1/10$. Figure \[fig\_pancake\_quasi\_3-body\](b) compares the energy of the energetically lowest-lying three-atom state with $M=0$ (solid line) \[see thick solid line in Fig. \[fig\_pancake\_quasi\_3-body\](a)\] with the corresponding state obtained by solving the strictly two-dimensional equation with renormalized two-dimensional scattering length $a_{\rm{ren}}^{\rm{2D}}$ (dotted line). The inset shows the difference between the energies obtained within the strictly two-dimensional and the full three-dimensional frameworks. Similar to the one-dimensional case, the maximum of the deviation occurs near unitarity. Comparison of the insets of Figs. \[fig\_pancake\_quasi\_2-body\](b) and \[fig\_pancake\_quasi\_3-body\](b) suggests that, at least in this low-energy example, the presence of the third atom does not, in a significant manner, reduce the applicability of the strictly two-dimensional framework. For the same aspect ratio, the deviations are expected to increase with increasing energy.
Second- and third-order virial coefficients {#sec_virial_coeff}
===========================================
This section utilizes the two- and three-body energy spectra to determine the second- and third-order virial coefficients as functions of the $s$-wave scattering length $a^{\rm{3D}}$, aspect ratio $\eta$ and temperature $T$. The $n^{th}$-order virial coefficient $b_n$ enters into the high-temperature expansion of the grand-canonical thermodynamic potential $\Omega$ of the equal-mass two-component Fermi gas with interspecies $s$-wave interactions [@ho-1; @ho-2; @rupak; @drummond_prl; @drummond_pra; @drummond_2d; @salomon-2010; @zwierlein-2012; @daily_2012], $\Omega = \Omega^{(1)} + \Omega^{(2)} + \Omega^{(12)}$, where $\Omega^{(1)}$ and $\Omega^{(2)}$ denote the grand-canonical thermodynamic potential of the spin-up component and the spin-down component, respectively, and $\Omega^{(12)}$ accounts for the interspecies interactions, $$\begin{aligned}
\label{eq_omega-12}
\Omega^{(12)} = -k_{\rm{B}}T Q_{1,0} \sum_{n=2}^{\infty} b_n z^n.\end{aligned}$$ Here, $z$ is the fugacity, $z=\exp[\mu/(k_{\rm{B}}T)]$. In the high-temperature limit, $z$ is a small parameter and the expansion given in Eq. (\[eq\_omega-12\]) is expected to provide a good description if the sum is terminated at quadratic or cubic order. The coefficient $b_2$ of the $z^2$ term is determined by one- and two-body physics and the coefficient $b_3$ of the $z^3$ term is determined by one-, two- and three-body physics. As the temperature approaches the transition temperature from above, the de Broglie wave length increases and, correspondingly, the fugacity $z$ increases. It follows that the number of $b_n$ coefficients needed to accurately describe the thermodynamics increases with decreasing temperature. Comparison with experimental data has shown that the inclusion of $b_2$ and $b_3$ yields quite accurate descriptions of the high-temperature thermodynamics of $s$-wave interacting two-component Fermi gases (see, e.g., Refs. [@drummond_prl; @salomon-2010; @zwierlein-2012]).
In Eq. (\[eq\_omega-12\]), $Q_{1,0}$ denotes the canonical partition function of a single spin-up particle. We define the canonical partition function $Q_{n_1,n_2}$ of the system consisting of $n_1$ spin-up particles and $n_2$ spin-down particles through $$\begin{aligned}
\label{eq_can_parttion}
Q_{n_1,n_2} = \sum_j \exp \left( -\frac{E_j^{\rm{tot}}(n_1,n_2)}{ k_{\rm{B}}T} \right),\end{aligned}$$ where $E_j^{\rm{tot}}(n_1,n_2)$ denotes the total energy of the system (including the center-of-mass energy) and the summation over $j$ includes all quantum numbers allowed by the symmetry of the system. For equal-mass fermions, as considered throughout this paper, we have $Q_{1,0}=Q_{0,1}=Q_1$. The virial coefficients $b_2$ and $b_3$ can be expressed as [@drummond_prl; @footnote1] $$\begin{aligned}
\label{eq_b2_1}
b_2= \frac{Q_{1,1}-Q_1^2}{Q_1}\end{aligned}$$ and $$\begin{aligned}
\label{eq_b3_1}
b_3= 2~ \frac{Q_{2,1} - Q_{2,0} Q_1 - b_2 Q_1^2}{Q_1}.\end{aligned}$$ The virial coefficients $b_2$ and $b_3$ depend on the interspecies scattering length $a^{\rm{3D}}$, aspect ratio $\eta$ and temperature $T$. Once the thermodynamic potential is known, physical observables such as the pressure and the entropy can be calculated.
We now discuss the determination of $b_2$ and $b_3$ for equal-mass two-component Fermi gases under anisotropic harmonic confinement. In this case, the single-particle canonical partition function $Q_1$ can be determined analytically, $$\begin{aligned}
\label{eq_1-body_can_partition}
Q_1 = \frac
{ \exp \left(\left[1/2+\eta \right]\tilde \omega_z \right)}
{\left[\exp \left(\tilde\omega_z \right) - 1\right]
\left[\exp \left(\eta \tilde \omega_z \right) - 1\right]^2},\end{aligned}$$ where $\tilde \omega_z$ denotes the “inverse temperature” in units of $E_z$, $\tilde \omega_z = E_z / (k_{\rm{B}}T)$. Alternatively (see below), we express the inverse temperature in units of $E_{\rho}$ or $E_{\rm{ave}}$, $\tilde \omega_{\rho} = E_{\rho} / (k_{\rm{B}}T)$ and $\tilde \omega_{\rm{ave}}(\eta) = E_{\rm{ave}} / (k_{\rm{B}}T)$. The average energy $E_{\rm{ave}}$ is defined in terms of the root-mean-square or, in short, average angular frequency $\omega_{\rm{ave}}(\eta)$, $E_{\rm{ave}}=\hbar \omega_{\rm{ave}}(\eta)$, where $$\begin{aligned}
\label{eq_omega-ave}
\omega_{\rm{ave}}(\eta) =
\sqrt{\frac {2 \omega_{\rho}^2 + \omega_z^2}{3}}.\end{aligned}$$ We note that the average angular frequency coincides with the angular trapping frequency for $\eta=1$ but not for $\eta \ne 1$. Below, we frequently suppress the explicit dependence of $\tilde{\omega}_{\rm{ave}}$ on $\eta$. The partition functions $Q_{1,1}$ and $Q_{2,1}$ can be determined from the two- and three-body energy spectra (see Secs. \[sec\_cigar\] and \[sec\_pancake\]) for each $s$-wave scattering length $a^{\rm{3D}}$, aspect ratio $\eta$ and temperature $T$.
At unitarity, the high-temperature expansion of $b_n$ reads (for $\eta=1$, see Ref. [@drummond_prl]) $$\begin{aligned}
\label{eq_bn_taylor}
b_n \approx b_n^{(0)} + b_n^{(2)}(\eta) \tilde \omega_{z}^2
+ b_n^{(4)}(\eta) \tilde \omega_{z}^4 + \cdots.\end{aligned}$$ The coefficients $b_n^{(0)}$, that is, the high-temperature limits of the trapped virial coefficients $b_n$, are independent of the aspect ratio $\eta$. This has previously been shown to be the case for $n=2$ [@peng_2011]. Here, we extend the argument to all $n$. Application of the local density approximation [@menotti; @drummond_prl; @daily_2012] to axially symmetric confinement potentials shows that the virial coefficients $b_n^{\rm{hom}}$ of the homogeneous system, which have been shown to be temperature-independent at unitarity [@ho-1; @ho-2; @rupak], are related to $b_n^{(0)}$ through $$\begin{aligned}
\label{eq_bn_hom}
b_n^{\rm{hom}} = n^{3/2} b_n^{(0)}.\end{aligned}$$ Since Eq. (\[eq\_bn\_hom\]) holds for all $\eta$, $b_n^{(0)}$ has to be independent of $\eta$ for all $n$.
The expansion coefficients $b_n^{(k)}$, $k=2,4,\cdots$, parametrize “trap corrections”, that is, corrections that arise due to the fact that the harmonic confinement defines a meaningful (finite) length scale. In fact, for $\eta \neq 1$, the confinement defines two length scales, suggesting that the $b_n^{(k)}$, $k=2,4,\cdots$, depend on $\eta$. Equation (\[eq\_bn\_taylor\]) expresses the temperature dependence of $b_n$ in terms of the inverse temperature associated with the $z$-direction, regardless of whether $\eta$ is greater or smaller than 1. Interestingly, it was shown in Ref. [@peng_2011] that the dependence of $b_2^{(k)}$, $k=2,4,\cdots$, on the aspect ratio can be parametrized, to a good approximation, in terms of the average inverse temperature $\tilde \omega_{\rm{ave}}(\eta)$, $$\begin{aligned}
\label{eq_b2_approx}
b_2^{(k)}(\eta)\tilde \omega_z^{k} \approx
b_2^{(k)}(1) [\tilde \omega_{\rm{ave}}(\eta)]^k.\end{aligned}$$ Equation (\[eq\_b2\_approx\]) implies that the trap corrections for two-body systems with $\eta \ne 1$ can be parametrized in terms of the trap corrections for the spherically symmetric system if the inverse temperature is expressed in terms of the average trapping frequency that characterizes the anisotropic system.
We now illustrate that Eq. (\[eq\_b2\_approx\]) applies not only to $b_2$ but also to $b_3$. Figures \[fig\_virial-1\](a) and \[fig\_virial-1\](b)
![(Color online) (a) Third-order virial coefficient $b_3$ at unitarity as a function of the inverse temperature $\tilde \omega_z$ for $\eta = 1$ (solid line), $2$ (dashed line), and $3$ (dash-dotted line). (b) Third-order virial coefficient $b_3$ at unitarity as a function of the inverse temperature $\tilde \omega_{\rho}$ for $\eta = 1$ (solid line), $1/2$ (dashed line), and $1/3$ (dash-dotted line). For $\eta \neq 1$, $b_3$ terminates at the inverse temperature of about 0.25 since our calculations include a finite number of three-body energies; obtaining the behavior of $b_3$ in the high-temperature limit requires the inclusion of infinitely many three-body energies. For $\eta \ne 1$, dotted lines show $b_3$ for $\eta = 1$, calculated using the average frequency of the respective anisotropic system. This approximate description is quite good. The insets of panels (a) and (b) show the same data as the main figure, but now as a function of $\tilde{\omega}_{\rm{ave}}$ as opposed to $\tilde{\omega}_z$ and $\tilde{\omega}_{\rho}$. The insets show that the third-order virial coefficients for different $\eta$ collapse to a universal curve for all $\eta$ (deviations arise in the low-temperature regime, i.e., for $\tilde{\omega}_{\rm{ave}} \gtrsim 1$). []{data-label="fig_virial-1"}](fig9.eps){width="70mm"}
show the third-order virial coefficient at unitarity for systems with $\eta \ge 1$ and $\eta \le 1$, respectively. The virial coefficients are plotted as a function of the inverse temperature expressed in units of the weak confinement direction, i.e., in terms of $\omega_z$ for $\eta \ge 1$ and in terms of $\omega_{\rho}$ for $\eta \le 1$. In the high-temperature limit, $b_3$ approaches a constant, confirming that $b_3^{(0)}$ is independent of $\eta$. The dotted lines show the third-order virial coefficient for $\eta=1$, calculated using the average trapping frequency characteristic for the respective anisotropic system. Figure \[fig\_virial-1\] illustrates that the third-order virial coefficient for anisotropic traps is approximated well by that for $\eta=1$ with appropriately scaled angular frequency. The insets of Fig. \[fig\_virial-1\] show that the third-order virial coefficients of the anisotropic system collape, to a very good approximation, to a universal curve over a surprisingly large temperature regime, i.e., down to temperatures around $k_{\rm{B}} T \approx E_{\rm{ave}}/2$. We conjecture that $b_n^{(k)}(\eta)\tilde \omega_z^{k}$ can be approximated quite well by $b_n^{(k)}(1) [\tilde \omega_{\rm{ave}}(\eta)]^k$ for $n=4,5,\cdots$ as well, as long as $k$ is not too large, i.e, as long as the temperature is not too low.
Next, we discuss the behavior of $b_2$ for finite $s$-wave scattering lengths. Figure \[fig\_b2-diff-asc-3d\]
![Second-order virial coefficient $b_2$ for the two-body system under isotropic confinement as a function of the inverse scattering length $a_{\rm{ave}}/a^{\rm{3D}}$ ($a^{\rm{3D}}$ negative) and the inverse temperature $\tilde{\omega}_{\rm{ave}}$. The smallest $\tilde{\omega}_{\rm{ave}}$ considered is $0.0003$. In the $\tilde{\omega}_{\rm{ave}} \rightarrow 0$ limit, $b_2$ approaches $1/2$ for all $a_{\rm{ave}}/a^{\rm{3D}}$ ($a^{\rm{3D}}<0$; see text for further discussion).[]{data-label="fig_b2-diff-asc-3d"}](fig10.eps){width="70mm"}
shows a surface plot of $b_2$ for $\eta=1$ as a function of the inverse scattering length $a_{\rm{ave}}/a^{\rm{3D}}$ ($a^{\rm{3D}} \leq 0$) and the inverse temperature $\tilde{\omega}_{\rm{ave}}$. Here, $a_{\rm{ave}}$ denotes the oscillator length associated with the average trapping frequency, $a_{\rm{ave}}=\sqrt{\hbar/(\mu \omega_{\rm{ave}})}$. At unitarity, $b_2$ is only weakly-dependent on the temperature and approximately equal to $1/2$ (see discussion above). The smallest inverse temperature $\tilde{\omega}_{\rm{ave}}$ considered in Fig. \[fig\_b2-diff-asc-3d\] is $0.0003$. For this inverse temperature, $b_2$ is fairly close to $1/2$ for all $a_{\rm{ave}}/a^{3D}$ shown. Thus, Fig. \[fig\_b2-diff-asc-3d\] shows that the high-temperature limit of $b_2$ is nearly independent of the scattering length. This behavior can, as we now show, be understood from the two-body energy spectrum.
Figure \[fig\_2-body-E\_1\](a)
![(Color online) (a) Relative two-body energies $E_{\rm{2b}}/E_{\rm{ave}}$, shifted by $2(n-1)$, for isotropic confinement ($\eta=1$ and $s$-wave channel) as a function of the inverse scattering length $a_{\rm{ave}}/a^{\rm{3D}}$. (b) Relative two-body energies $E_{\rm{2b}}/E_{\rho}$, shifted by $2(n-1)$, for pancake-shaped confinement ($\eta=1/10, M=0$, and $\Pi_z=+1$) as a function of the inverse scattering length $a_{\rho}/a^{\rm{3D}}$. Solid, dotted and dashed lines show the energies for $n=1, 100$ and $10000$, respectively. []{data-label="fig_2-body-E_1"}](fig11.eps){width="70mm"}
shows selected relative two-body energies as a function of $a_{\rm{ave}}/a^{\rm{3D}}$ for the trapped system with $\eta=1$. In the low-energy regime (solid line), the two-body energy changes by nearly $2E_{\rm{ave}}$ for the scattering length range shown. As the energy increases (dashed and dotted lines show energies around $200 E_{\rm{ave}}$ and $20000 E_{\rm{ave}}$, respectively), the two-body energy undergoes less of a change and eventually becomes nearly flat over the scattering length region shown. This implies that the high-energy portion of the two-body spectrum looks like that of the unitary gas over an increasingly large region around unitarity. The behavior of the energy spectrum can be understood by expanding the transcendental two-body eigenequation $\mathcal{F}^{\rm{3D}}(\epsilon, 1)=-a_{\rm{ave}}/a^{\rm{3D}}$ around unitarity. Assuming $|a_{\rm{ave}}/a^{\rm{3D}}| \ll 1$, we find [@peng_2011; @footnotetypo] $$\begin{aligned}
\label{eq_euni_taylor}
\frac{E_{\rm{2b}}}{E_{\rm{ave}}} -
\left(2n+\frac{1}{2} \right)
\approx
-\frac{\Gamma(n+1/2)}
{\pi \Gamma(n+1)} \frac{a_{\rm{ave}}}{a^{\rm{3D}}}.\end{aligned}$$ Equation (\[eq\_euni\_taylor\]) provides a good description of the energies as long as the right hand side is small. Since the right hand side of Eq. (\[eq\_euni\_taylor\]) scales for large $n$ as $(a_{\rm{ave}}/a^{\rm{3D}}) / \sqrt{n}$, Eq. (\[eq\_euni\_taylor\]) provides, as $n$ increases, a good description for an increasingly large region around unitarity. That is, the energies vary approximately linearly with $a_{\rm{ave}}/a^{\rm{3D}}$, with a slope that approaches zero, as $n \to \infty$. This analysis rationalizes why $b_2$ approaches $1/2$ as $T \rightarrow \infty$, regardless of the value of the scattering length ($a^{\rm{3D}}<0$ and finite).
Figure \[fig\_b2-diff-asc-3d\] has been obtained for a spherically symmetric system, that is, for $\eta=1$. We now demonstrate that Fig. \[fig\_b2-diff-asc-3d\] applies, for experimentally relevant temperatures, to all aspect ratios and not just to $\eta=1$. Figure \[fig\_b2-diff-asc\]
![(Color online) Second-order virial coefficient $b_2$ of the trapped two-body system as a function of the inverse temperature $\tilde \omega_{\rm{ave}}$ for three different values of the inverse scattering length ($a_{\rm{ave}}/a^{\rm{3D}} = -1$, $0$ and $+1$; see labels) for $\eta=1$ (solid lines), $\eta = 1/5$ (dashed lines) and $\eta = 1/100$ (dotted lines).[]{data-label="fig_b2-diff-asc"}](fig12.eps){width="70mm"}
shows $b_2$ as a function of $\tilde{\omega}_{\rm{ave}}$ for three different aspect ratios, $\eta=1$ (solid lines), $\eta = 1/5$ (dashed lines) and $\eta = 1/100$ (dotted lines). Three different scattering lengths are considered: $a^{\rm{3D}} = -a_{\rm{ave}}$, $\infty$ and $a_{\rm{ave}}$. It can be seen that $b_2$ is, to a very good approximation, independent of $\eta$ in the high-temperature (small $\tilde{\omega}_{\rm{ave}}$) limit. We stress that the independence of $b_2$ of $\eta$ requires that the inverse temperature and scattering length are expressed in terms of the average oscillator units $E_{\rm{ave}}$ and $a_{\rm{ave}}$, respectively.
To understand the universality implied by Fig. \[fig\_b2-diff-asc\], that is, the fact that Fig. \[fig\_b2-diff-asc-3d\] applies to all aspect ratios and not just to $\eta=1$, we analyze the behavior of the high-lying part of the relative two-body spectra for $\eta \ne 1$. Figure \[fig\_2-body-E\_1\](b) exemplarily shows the relative energies for a pancake-shaped system with $\eta=1/10$. Comparison with Fig. \[fig\_2-body-E\_1\](a) shows that the qualitative behavior of the high-energy part of the spectrum is independent of $\eta$. This is confirmed by a more quantitative analysis that Taylor expands the implicit eigenequation for the anisotropic two-body system around unitarity. We conclude that two two-body systems with different aspect ratios but identical $a_{\rm{ave}}/a^{\rm{3D}}$ and $\tilde \omega_{\rm{ave}}$ are characterized by approximately the same $b_2$ value. Our analysis of the three-body energies for anisotropic confinement suggests that analogous conclusions hold for $b_3$. We speculate that the conclusions hold also for the virial coefficients with $n>3$.
To estimate the extent of the universal behavior, it is instructive to reexpress $\tilde{\omega}_{\rm{ave}}$ in terms of the Fermi temperature. For a spin-balanced system of $N$ fermions under spherically symmetric confinement, we use the semi-classical expression $k_{\rm{B}} T_{\rm{F}}=(3 N)^{1/3} \hbar \omega_{\rm{ave}}$, yielding $\tilde \omega_{\rm{ave}}=(3 N)^{-1/3}(T/T_{\rm{F}})^{-1}$. For $N=10^2$, $10^4$ and $10^6$, $T/T_{\rm{F}}=1$ corresponds to $\tilde \omega_{\rm{ave}} \approx 0.149$, $0.032$ and $0.007$, respectively. For these temperatures, the thermodynamic behavior is, according to our discussion above, expected to be essentially universal over a fairly wide range of scattering lengths. For $T/T_{\rm{F}}=1$, we estimate that the deviation of $b_2$ from the value $1/2$ approaches $5\%$ for $a_{\rm{ave}}/a^{\rm{3D}} \approx -0.16$, $-0.38$ and $-0.78$ for $N=10^2$, $10^4$ and $10^6$, respectively. This implies that uncertainties of the scattering length dependence on the magnetic field in recent experiments at unitarity [@salomon-2010; @zwierlein-2012] should have a negligibly small effect on the equation of state at unitarity. For all three cases considered above, the corresponding $(k_{\rm{F}} a^{\rm{3D}})^{-1}$ values is approximately $-0.03$.
The determination of the high-temperature behavior of the third-order virial coefficient for different scattering lengths and aspect ratios is much more demanding than that of the second-order virial coefficient. Although our analysis of $b_3$ is less exhaustive than that of $b_2$, it suggests that the conclusions drawn above for the second-order virial coefficient carry over to the third-order virial coefficient.
Conclusion {#sec_conclusion}
==========
This paper developed a Lippmann-Schwinger equation based approach to determine the energy spectrum and corresponding eigenstates of three-body systems with zero-range $s$-wave interactions under harmonic confinement with different transverse and longitudinal angular trapping frequencies. The formalism was applied to the equal-mass system consisting of two identical fermions and a third distinguishable particle in a different spin-state. The energy spectra were determined as a function of the interspecies $s$-wave scattering length for various aspect ratios $\eta$, $\eta>1$ (cigar-shaped trap) and $\eta<1$ (pancake-shaped trap). For $\eta \gg 1$, we showed that the low-energy portion of the energy spectra are reproduced well by an effective one-dimensional Hamiltonian with renormalized one-dimensional coupling constant. Similarly, for $\eta \ll 1$, we showed that the low-energy portion of the energy spectra are reproduced well by an effective two-dimensional Hamiltonian with renormalized two-dimensional coupling constant. As the energy increases, the description based on these effective low-dimensional Hamiltonian deteriorates.
The two- and three-body energy spectra were then used to determine the second- and third-order virial coefficients that determine the virial equation of state, applicable to two-component Fermi gases at temperatures above the Fermi temperature. Our key findings are:
[*[(i)]{}*]{} At unitarity, the second- and third-order virial coefficients $b_2$ and $b_3$ approach constants in the high-temperature regime. The constants (referred to as $b_2^{(0)}$ and $b_3^{(0)}$) are independent of $\eta$.
[*[(ii)]{}*]{} For finite scattering length $a^{\rm{3D}}$, we find that $b_2$ and $b_3$ collapse, to a very good approximation, to a single curve for all $\eta$ if the temperature and scattering length are expressed in terms of the average energy $E_{\rm{ave}}$ and the average oscillator length $a_{\rm{ave}}$, respectively. Deviations from the universal curve arise in the low-temperature regime where the virial equation of state is not applicable.
[*[(iii)]{}*]{} The virial coefficient $b_2$ is approximately equal to $1/2$ over a fairly large temperature and scattering length regime around unitarity.
The work presented in this paper is directly relevant to on-going cold atom experiments. The three-body spectra, e.g., can be measured experimentally through rf spectroscopy [@esslinger; @selim1; @selim2]. Moreover, the formalism can be employed to characterize the molecular states in more detail, quantifying the “perturbation” of the dimer due to the third particle throughout the dimensional crossover. The determination of the virial coefficients is of immediate relevance to cold atom experiments that study the dynamics of large fermionic clouds under low-dimensional confinement. The formalism developed in Secs. \[sec\_formalsolution\]-\[sec\_pancake\] of this paper can be extended fairly straightforwardly to three-boson and unequal-mass systems.
Acknowledgement
===============
We gratefully acknowledge support by the ARO and thank Krittika Goyal for contributions at the initial stage of this work. This work was additionally supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.
Calculation of integrals involved {#appendix}
=================================
This appendix presents the evaluation of the integrals defined in Secs. \[sec\_cigar\] and \[sec\_pancake\]. The integrals $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$ and $ I_{n_z,n_z'}^{\rm{p}}(j)$ are energy-independent. They are tabulated once and then used for each three-body energy considered. To evaluate the integral $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$, Eq. (\[eq\_Irho\_cigar\]), we expand each of the three radial two-dimensional harmonic oscillator functions $R$ \[see Eq. (\[eq\_two-d\_wavefunction\])\] in terms of a finite sum, i.e., we use the series expansion of the associated Laguerre polynomials [@abramowitz], $$\begin{aligned}
\label{eq_laguerre_series}
L_n^{(k)}(x)=
\sum_{j=0}^n(-1)^j \frac{(n+k)!}{(n-j)!(k+j)!j!} x^j.\end{aligned}$$ $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$ then becomes a finite triple sum that contains integrals of the form $$\begin{aligned}
\label{eq_gaussian_integral}
\int_0^{\infty}\exp \left(-\frac{\eta \rho^2}{a_z^2} \right)
\left(\frac{\rho}{a_z} \right)^k ~d\rho =
\frac{1}{2}
\left( \frac{1}{a_z} \right) ^k
\left( \frac{\eta}{a_z^2} \right)^{-\frac{1+k}{2}}
\Gamma\left(\frac{1+k}{2}\right).\end{aligned}$$ The finite sum is readily evaluated and $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$ is stored for each $n_{\rho}$, $n_{\rho}'$, $m$ and $j$ combination. To evaluate $ I_{n_z,n_z'}^{\rm{p}}(j)$, Eq. (\[eq\_Iz\_pancake\]), we rewrite each of the three one-dimensional harmonic oscillator functions $\varphi$ \[see Eq. (\[eq\_one-d\_wavefunction\])\] in terms of associated Laguerre polynomials instead of Hermite polynomials [@abramowitz_hermite], $$\begin{aligned}
\label{eq_laguerre_hermit_1}
H_{2n}(z/a_z)=(-1)^n2^{2n}n! L_n^{(-1/2)}(z^2/a_z^2)\end{aligned}$$ and $$\begin{aligned}
\label{eq_laguerre_hermit_2}
H_{2n+1}(z/a_z)=(-1)^n 2^{2n+1} n! (z/a_z) L_n^{(1/2)}(z^2/a_z^2).\end{aligned}$$ The evaluation of $ I_{n_z,n_z'}^{\rm{p}}(j)$ then proceeds analogously to that of $I_{n_{\rho},n_{\rho}',m}^{\rm{c}}(j)$.
The integrals $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ and $I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)$, Eqs. (\[eq\_Iz\_cigar\]) and (\[eq\_Irho\_pancake\]), are energy-dependent. To evaluate these integrals, we first “separate out” the energy dependence and then proceed along the lines discussed above. The energy dependence of $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ enters through $\mathcal{G}^{\rm{1D}}([\epsilon_{\boldsymbol\lambda'}-2 \eta j+1/2]E_z;
\sqrt{3}/2z; 0)$. Using the identity [@krittika_2007] $$\begin{aligned}
\label{eq_hyper_series}
\Gamma(a)U\left(a,1/2,x\right)=\sum_{k=0}^{\infty}\frac{L_{k}^{(-1/2)}(x)}{k+a},\end{aligned}$$ $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ can be written as an infinite series, $$\begin{aligned}
\label{eq_Iz_series_cigar}
I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)
= \lim_{k_{\rm{max}} \to \infty}
\frac{1}{2 \sqrt{2}}
\sum_{k=0}^{k_{\rm{max}}}\frac{1}{k-\frac{\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j}{2}}
C_{n_z,n_z'}^k,\end{aligned}$$ where the coefficients $C_{n_z,n_z'}^k$ are energy-independent, $$\begin{aligned}
\label{eq_Iz_cnnpk}
C_{n_z,n_z'}^k = \int_{-\infty}^{\infty}
\varphi_{n_z}(z) \varphi_{n_z'}(-z/2)
\exp{\left(-\frac{3z^2}{8a_z^2} \right) }
L_{k}^{(-1/2)}\left(\frac{3z^2}{4 a_z^2}\right)dz.\end{aligned}$$ The evaluation of the $C_{n_z,n_z'}^k$ now proceeds as above.
The series introduced in Eq. (\[eq\_Iz\_series\_cigar\]) diverges if $(\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2$ is a positive integer or zero, that is, in the non-interacting limit. In practice, this does not pose a constraint since the energy grid can be chosen such that it does not contain the non-interacting energies. To analyze the dependence of $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ on the upper summation index $k_{\rm{max}}$, Fig. \[fig\_conv\](a)
![Convergence study. (a) The dots show the coefficient $C_{n_z,n_z'}^k $, Eq. (\[eq\_Iz\_cnnpk\]), for $n_z=2$ and $n_z'=80$ as a function of $k$. (b) The circles and triangles show the sum $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$, Eq. (\[eq\_Iz\_series\_cigar\]), for $(\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2= 25.745$ and $40.234$, respectively, as a function of the cutoff $k_{\rm{max}}$. As in panel (a), we used $n_z=2$ and $n_z'=80$.[]{data-label="fig_conv"}](fig13.eps){width="70mm"}
shows the behavior of $C_{n_z,n_z'}^k$ for fixed $n_z$ and $n_z'$ as a function of $k$. In this example, $C_{n_z,n_z'}^k$ vanishes nearly identically for $k \gtrsim 30$. The coefficients $C_{n_z,n_z'}^k$ are multiplied by the $k$-dependent prefactor $[k-(\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2]^{-1}$. This prefactor is maximal for $k \approx (\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2$. Figure \[fig\_conv\](b) shows the quantity $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ as a function of $k_{\rm{max}}$ for the same $n_z$ and $n_z'$ as in Fig. \[fig\_conv\](a) but two different values of $(\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2$, that is, for $(\epsilon_{\boldsymbol{\lambda'}} - 2 \eta j)/2=25.745$ and $40.234$. In the first case, the absolute value of the prefactor takes its maximum at $k = 26$, where the coefficients $C_{n_z,n_z'}^k$ are non-zero. Correspondingly, $I_{n_z,n_z'}^{\rm{c}}$ shows a sharp peak near $k_{\rm{max}} = 26$ and then smoothly approaches its asymptotic value \[see circles in Fig. \[fig\_conv\](b)\]. In the second case, the absolute value of the prefactor takes its maximum at $k = 40$, where the $C_{n_z,n_z'}^k$ coefficients are very small. Correspondingly, the $C_{n_z,n_z'}^k$ coefficients suppress the maximum of the prefactor and the quantity $I_{n_z,n_z'}^{\rm{c}}$ approaches its asymptotic value for $k_{\rm{max}} \approx 30$ \[see triangles in Fig. \[fig\_conv\](b)\]. We find that the choice of $k_{\rm{max}} \gtrsim 2 \max(n_z, n_z')$ yields well converged values for $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ for all energies considered.
In an alternative approach, we determined $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$ through numerical integration for each $n_z$, $n_z'$, $n_{\rho}'$, $m$, $j$ and energy. We have found this approach to be computationally more time-consuming than the tabulation approach discussed above.
The evaluation of the integral $I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)$, Eq. (\[eq\_Irho\_pancake\]), proceeds analogously to that of $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$. We use [@krittika_2007] $$\begin{aligned}
\label{eq_hyper_series_2}
\Gamma(a)U(a,1,x)=\sum_{k=0}^{\infty}\frac{L_{k}(x)}{k+a}\end{aligned}$$ to separate out the energy-dependence that enters through $\mathcal{G}^{\rm{2D}}$. The integral is then written as $$\begin{aligned}
\label{eq_Irho_series_pancake}
I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)=
\lim_{k_{\rm{max}} \rightarrow \infty}
\frac{1}{2 \pi}\sum_{k=0}^{k_{\rm{max}}}
\frac{1}{k-\frac{\epsilon_{\boldsymbol{\lambda'}} - 2 j}{2\eta}}
C_{n_{\rho},n_{\rho}',m}^k,\end{aligned}$$ where $$\begin{aligned}
\label{eq_Irho_cnnpk}
C_{n_{\rho},n_{\rho}',m}^k =
&\int_0^{\infty}
R_{n_{\rho},m}(\rho)
R_{n_{\rho}',m}\left(\frac{\rho}{2}\right)
\exp \left( -\frac{3 \eta \rho^2}{8 a_z^2} \right)
L_{k}\left(\frac{3 \eta \rho^2}{4 a_z^2} \right)\rho d\rho.\end{aligned}$$ We tabulate $C_{n_{\rho},n_{\rho}',m}^k$ and calculate $I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)$ for each $\epsilon_{\boldsymbol{\lambda'}}$ “on the fly”. The convergence behavior of $I_{n_{\rho},n_{\rho}',m}^{\rm{p}}(\epsilon_{\boldsymbol{\lambda'}},j)$ is similar to that of $I_{n_z,n_z'}^{\rm{c}}(\epsilon_{\boldsymbol{\lambda'}},j)$.
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---
abstract: |
In this brief note we prove that there is a residual subset $\mathcal{R}$ of $\textrm{Diff}^1(M)$ such that for any $f\in\mathcal{R}$ and any partially hyperbolic homoclinic class $H(p,f)$ with one dimensional center direction, the set of central Lyapunov exponents associated to the ergodic with full support is an interval. Also, some remarks on the general case presented.
**Keywords:** Homoclinic class, partially hyperbolic, Lyapunov exponent, $C^1$-generic.
**AMS Subject Classification:** 37B20, 37C29, 37C50
**Received** April 19, 2013.
author:
- 'Abbas Fakhari$^{\natural\sharp}$'
title: Connectedness of the set of central Lyapunov exponents
---
**\
Introduction
============
By definition, hyperbolic dynamical systems have nonzero Lyapunov exponents. However, they are not generic in the space of all dynamical systems. This necessitated weakening the notion of hyperbolicity. A wider class is that of partially hyperbolic systems studied by many authors (for a complete review, see \[1-3\]). A partially hyperbolic set may admit zero Lyapunov exponents along their central directions.
**Question.** Is the set of central Lyapunov vectors associated to ergodic measures on a partially hyperbolic set is convex?
In general, the answer is negative. In [@lor], the authors have studied destroying horseshoes via heterodimensional cycles. These maps are partially hyperbolic with a one dimensional center direction. They proved that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has a gap. However, there are examples of open sets of partially hyperbolic maps whose central Lyapunov exponents form a convex set (see for instance [@gi]). As known, the existence of heterodimensional cycles isn’t a generic phenomenon. Now, a question may be arisen:
[***what can be said in generic mode?***]{}
There are partial answers to the question in the case of generic “locally maximal" or “Lyapunov stable" homoclinic classes [@abcdw; @csy]. In this note, we give a positive answer to the question in the case of one dimensional centeral direction. In addition, we will prove that the support of the ergodic measure fills a gap equals to the whole of the homoclinic class.
Let $M$ be a compact boundless Riemannian manifold and $f$ be a diffeomorphism on it. For a periodic point $p$ of $f$, we denote by $\pi(p)$ the period of $p$. For a hyperbolic periodic point $p$ of $f$ of period $\pi(p)$ the sets
$W^s(p)=\{x\in M : f^{\pi(p)n}(x)\to p ~\textit {as}~ n\to\infty\},~\mbox{and }$\
$W^u(p)=\{x\in M : f^{-\pi(p)n}(x)\to p ~\textit{ as}~
n\to\infty\};$
are injectively immersed submanifolds of $M$. A point $x\in
W^s(p)\,\cap\, W^u(p)$ is called a [*homoclinic point*]{} of $f$ associated to $p$, and it is called a [*transversal homoclinic point*]{} of $f$ if the above intersection is transversal at $x$. The closure of the transversal homoclinic points of $f$ associated to the orbit of $p$ is called the [*homoclinic class*]{} of $f$ associated to $p$, and denoted by $H(p,f)$. Two saddles $p$ and $q$ are called [*homoclinically related*]{}, and write $p\sim q$, if $$W^s(\mathcal{O}(p))\,{\overline{\pitchfork}}\,W^u(\mathcal{O}(q))\not=\phi\,\,\,\mbox{
and}\,\,\,W^u(\mathcal{O}(p))\,{\overline{\pitchfork}}\,W^s(\mathcal{O}(q))\not=\phi
.$$ Note that, by [*Smale’s transverse homoclinic point theorem*]{} , $H(p,f)$ coincides with the closure of the set of all hyperbolic periodic points $q$ with $p\sim q$.
A compact invariant set $\Lambda$ is [*partially hyperbolic*]{} if it admits a splitting $T_\Lambda M=E^{ss}\oplus E^c\oplus E^{uu}$ such that $$\|Df|_{E^{ss}}\|<\|Df|_{E^{c}}\|<\|Df|_{E^{uu}}\|,$$ furthermore, $E^{ss}$ is uniformly contraction and $E^{uu}$ is uniformly expansion; it means $$\|Df|_{E^{ss}}\|<1<\|Df|_{E^{uu}}\|.$$
The bundles $E^{ss}$ and $E^{uu}$ of a partially hyperbolic set are always uniquely integrable. Denote by $W^{ss}_{loc}(x)$ (resp. $W^{uu}_{loc}(x)$) the [*local strong stable*]{} (resp. [*unstable*]{}) manifold of $x$ tangent to $E^{ss}(x)$ (resp. $E^{uu}(x)$) at $x$ (\[1-3\]). Recall that if a diffeomorphism $f$ has a partially hyperbolic set $\Lambda$ then there is an open neighborhood $U$ of $\Lambda$ and a neighborhood $\mathcal{U}$ of $f$ such that the maximal invariant set $\tilde{\Lambda}_g$ of a diffeomorphism $g\in\mathcal{U}$ which is entirely contained in $U$ is partially hyperbolic.
Denote by $\mathcal{M}_f (M)$ the set of $f$-invariant probability measures endowed with its usual topology; i.e., the unique metrizable topology such that $\mu_n\to\mu$ if and only if $\int
f\,d\mu_n\to\int f\,d\mu$ for every continuous function $f:M\to\mathbb{R}$. The set of all ergodic elements of $\mathcal{M}_f(M)$ supported on an invariant set $\Lambda$ is denoted by $\mathcal{M}_e(f|_\Lambda)$.
A point $x$ in a partially hyperbolic set $\Lambda$ is called [*regular*]{} if the limit $$\lim_{n\to\infty}\frac{1}{n}\log\|Df^n(x)|_{E^c(x)}\|$$ exists. By Oseledec’s theorem, given any invariant measure $\mu$ the limits exist $\mu$-a.e.. The limits above are called [*central Lyapunov exponent*]{} of $x$. When $\mu$ is ergodic the limit doesn’t depend to initial point and is constant $\mu$-a.e.. We recall that an invariant set $\Lambda$ of a diffeomorphism $f$ is [*Lyapunov stable*]{} if for any neighborhood $U$ of $\Lambda$ there is a $V$ of it such that $f^ n(V )\subset U$, for any $n\geq 0$. Surly, for any hyperbolic periodic point $p$ in a Lyapunov stable invariant set $\Lambda$ we have $W^u(p)\subset \Lambda$. The set $\Lambda$ is [*bi-Lyapunov stable*]{} if it is Lyapunov stable for $f$ and $f^{-1}$.
**Theorem A.** For a $C^1$-generic diffeomorphisms $f$ and any partially hyperbolic homoclinic class $H(p,f)$ of $f$ with the splitting $E^s\oplus E^c\oplus E^u$ such that
- either $\textrm{dim}(E^c)=1$ or
- $H(p,f)$ is bi-Lyapunov stable and $E^c$ has a dominated splitting $E^c_1\oplus E^c_2$ into one dimensional bundles.
the central Lyapunov exponents associated to the ergodic measures with support equal to $H(p,f)$ forms a convex set.
**Problem 1.** In general, is the set of Lyapunov vectors associated to the set of ergodic measures on a generic homoclinic class convex?
Proof of Theorem A
==================
First, we state two following lemmas to guarantee the existence of hyperbolic periodic points with desired central Lyapunov exponents inside the homoclinic class.
(Pliss’s Lemma [@pl]) Given $\lambda_0<\lambda_1$ and $\mathcal{O}(p)\subset
\tilde{\Lambda}$ there is some natural number $m$ such that if $\prod_{j=0}^{t} \|Df|_{E^c(f^j(p_n))}\|\leq \lambda_0^t$ for some $t\geq m$ then there is a sequence $n_1<n_2<\cdots<n_k\leq t$ such that $\prod_{j=n_r}^{t} \|Df|_{E^c(f^j(p_n))}\|\leq
\lambda_1^{t-n_r}$, for any $1\leq r\leq k$.
([@ps]) For any $0<\lambda<1$, there is $\epsilon>0$ such that for $x\in
\tilde{\Lambda}$ which satisfies $\prod_{j=0}^{n-1}
\|Df|_{E^c(f^j(p_n))}\|\leq \lambda^n$, for all $n>0$, we have $\textrm{diam}(f^n(W^{cs}_\epsilon(x))\to 0$. In other word, the center stable manifold of $x$ with size $\epsilon$ is in fact stable manifold.
To prove Theorem A, we need to recall some $C^1$ generic statements. For a partially hyperbolic homoclinic class $H(p,f)$ and $\mu\in\mathcal{M}_e(f|_{H(p,f)})$, Let $\lambda^c_\mu$ be the Lyapunov exponent of $\mu$ along the central direction. For a hyperbolic periodic point $q\in H(p,f)$ of period $\pi(q)$, $\lambda^c_q=\log|\nu|/\pi(q)$ where $\nu=Df^{\pi(q)}(q)|_{E^c(q)}$. Put $$\mathcal{LE}^c(H(p,f))=\{\lambda^c_q;\,q\in H(p,f)\cap Per(f)\}$$
There is a residual subset $\mathcal{R}_0$ of $\textrm{Diff}^1(M)$ such that for any $f\in\mathcal{R}$ and any hyperbolic periodic point $p$ of $f$,
- $f$ is Kupka-Smale,
- the set $\mathcal{LE}^c(H(p,f))$ is connected ([@abcdw]),
- for any periodic point $q\in H(p,f)$ with $\textrm{index}(q)=\textrm{index}(p)$, we have $q\sim p$, ([@csy; @fa]),
- for any ergodic measure $\mu$ of $f$, there is a sequence of periodic point $p_n$ such that $\mu_{p_n}\to \mu$ in weak$^*$ topology and $\mathcal{O}(p_n)\to \textrm{Supp}(\mu)$ in Housdorff metric ([@abc]).
- If a homoclinic class $H(p,f)$ is bi-Lyapunov stable then for any sufficiently close $g$, $H(p_g,g)$ is bi-Lyapunov stable, where $p_g$ is the continuation of $p$.
Using arguments above we prove that any hyperbolic ergodic measure supported on a $C^1$-generic partially hyperbolic homoclinic class $H(p,f)$ can be approximated by periodic measures also supported on the homoclinic class. In fact, this is a special case of Bonatti’s conjecture.
**Bonatti’s Conjecture.** For a $C^1$-generic diffeomorphism $f$, any ergodic measure supported on a homoclinic class $H(p,f)$ of $f$ can be approximate in weak topology by periodic measures supported on the homoclinic class.
Lemma 2.4, below, gives a positive answer to the conjecture in the case of hyperbolic measures on a partially hyperbolic homoclinic class.
For any $f\in\mathcal{R}_0$, any partially hyperbolic homoclinic class $H(p,f)$, any hyperbolic measure $\mu\in\mathcal{M}_e(f|_{H(p,f)})$ and any $\epsilon>0$, there is a periodic point $q\in H(p,f)$ such that for any one dimensional center bundle $E^c$, $$|\lambda^c_p-\lambda^c_\mu|<\epsilon.$$
Choose $\lambda_0=\lambda^c_\mu+\epsilon<\lambda_1<0$. By Proposition 2.3, for any $\mu\in\mathcal{M}_e(f|_{H(p,f)})$, there is a is a sequence $\{p_n\}\subset \tilde{\Lambda}$ such that $\lambda^c_{p_n}\to\lambda^c_\mu$. Hence, for sufficiently large $n$, $\lambda^c_{p_n}<\lambda_0=\lambda^c_\mu+\epsilon<0$ and so $\prod_{j=0}^{\pi(p_n)}|Df|_{E^c(f^j(p_n))}|\leq
(\exp^{\lambda_0})^{\pi(p_n)}.$ Now, by Pliss Lemma, for large $n$, there is a point $q_n\in\mathcal{O}(p_n)$ such that $\prod_{j=0}^{t}|Df|_{E^c(f^j(q_n))}|\leq (\exp^{\lambda_1})^t$ for any $t\leq \pi(p_n)$. By Lemma 2.2, the center-stable manifolds of $q_n$’s are in fact stable manifolds and have uniform size. This implies that for large $n,m\in\mathbb{N}$, $p_n\sim p_m$ and $p_n\in
H(p,f)$.
In what follows, we first state a lemma which is a modification of some recent results on $C^1$-generic dynamics in our context ([@csy]). The second part of the Lemma helps us to give a positive answer again to Bonatti’s Conjecture in the case of ergodic measures, not necessarily hyperbolic, supported on bi-Lyapunov stable homoclinic class, Theorem 2.6 below. We recall that a periodic point $p$ of a diffeomorphism $f$ has [*weak Lyapunov exponent*]{} along a center bundle $E^c$ if for some small $\delta>0$, $|\lambda^c|<\delta.$.
There is a residual subset $\mathcal{R}_1$ of $\textrm{Diff}^1(M)$ such that for any $f\in\mathcal{R}_1$ and any homoclinic class $H(p,f)$ of $f$ having a partially hyperbolic splitting $E^s\oplus
E^c\oplus E^u$ with non-hyperbolic center bundle $E^c$, we have
- if $E^c$ is one-dimensional then there is a periodic point in the homoclinic class with weak Lyapunov exponent along $E^c$,
- if $E^c$ has a dominated splitting $E^c=E^c_1\oplus E^c_2$ into one dimensional subbundles then either $H(p,f)$ contains a periodic point whose index is equal to $\textrm{dim}(E^s)$ or it contains a periodic point with weak Lyapunov exponent along $E^c_1$.
For $C^1$-generic bi-Lyapunov stable homoclinic class $H(p,f)$, satisfying Proposition 2.3 and Lemma 2.5, any ergodic measure can be approximate by periodic measure inside the homoclinic class.
As before, if an ergodic measure is hyperbolic then nothing remains. Hence, suppose that $\mu$ is a non-hyperbolic ergodic measure supported on the homoclinic class $H(p,f)$. Using the forth part of Proposition 2.3, one can find a sequence $\{p_n\}$ of periodic point such that $\mathcal{O}(p_n)\to Supp(\mu)$ in Housdorff metric and also the periodic measures associated to $p_n$ tends toward $\mu$ in weak topology. Now, If the homoclinic class contains a periodic points $q$ of index $s=\textrm{dim}(E^s)$ then by the partial hyperbolicity, for large $n$, and positive iteretes $m_n$, $W^u(q)\cap W^s(f^{m_n}(p_n))\neq\emptyset$. Hence, by the Lyapunov stability, $p_n\in H(p,f)$. Thus, suppose that for any periodic point in the homoclinic class its index equals to $s+1$.
By Lemma 2.5, there is periodic point $q$ in the homoclinic class with weak Lyapunov exponents along the central direction $E^c_1$. We note that by the assumption $\textrm{index}(q)=\textrm{index}(p)=s+1$ and so $q\sim p$. Now, using Gourmelon’s perturbation lemma [@ga], which allows us to perturb the derivative controlling the invariant manifolds, we can produce a periodic point $q^\prime$ of index $s$ such that $W^s(p)\cap W^u(q^\prime)\neq\emptyset$. Again, by the Lyapunov stability, $q^\prime\in H(p,f)$. This is contradiction.
Now, we follow the approach suggested in [@gikn] to provide sufficient conditions for the existence of desired ergodic invariant measure as a limit of periodic ones.
a periodic orbit $X$ is a $(\gamma,\chi)$-good approximation of a periodic orbit $Y$ if the following holds
- for a subset $\Gamma$ of $X$ and a projection $\rho:\Gamma\to
Y$, $$d(f^j(x),f^j(\rho(x)))<\gamma$$ for any $x\in \Gamma$ and any $j=0,1,\ldots,\pi(Y)-1$;
- $card(\Gamma)/card(X)\geq \chi$;
- $card(\rho^{-1}(y)$ is the same for all $y\in Y$.
The next lemma is a key point in the proof of the ergodicity of a limit measure. This lemma was suggested by Yu. Ilyashenko [@gikn]. The following modified version, borrowed from [@bdg], determines the support of the obtained ergodic measure.
Let $\{X_n\}$ be a sequence of periodic orbits with increasing periods $\pi(X_n)$ of $f$. Assume that there are two sequences of numbers $\{\gamma_n\}$, $\gamma_n>0$ and $\{\chi_n\}$, $\chi_n\in
(0,1]$, such that
- for any $n\in\mathbb{N}$ the orbit $X_{n+1}$ is a $(\gamma_n,\chi_n)$-good approximation of $X_n$;
- $\sum _{n=1}^\infty \gamma_n<\infty$;
- $\prod_{n=1}^\infty \chi_n\in (0,1]$.
Then the sequence $\{\mu_{X_n}\}$ of atomic measures has a limit $\mu$ which is ergodic and $$\textrm{Supp}(\mu)=\bigcap_{k=1}^\infty\big(\overline{\bigcup_{l=k}^\infty
X_l}\big)$$
(lemma 3.5 in [@bdg]) There is a residual subset $\mathcal{R}_2$ of $\textrm{Diff}^1(M)$ such that for any $f\in\mathcal{R}_2$, any partially hyperbolic homoclinic class $H(p,f)$, any one dimensional center bundle $E^c$, any saddle $q$ in $H(p,f)$ and any $\epsilon>0$, $H(p,f)$ contains a saddle $q_1$ which is homoclinically related to $q$, its orbit is $\epsilon$-dense in the homoclinic class $H(p,f)$ and $$|\lambda^c_q-\lambda^c_{q_1}|<\epsilon.$$
**Proof of Theorem A.** We first proceed with the first case of the theorem. Let $f\in
\mathcal{R}_0\cap\mathcal{R}_1\cap\mathcal{R}_2$ and $H(p,f)$ be a partially hyperbolic homoclinic class of $f$ with one dimensional center direction. Put $$\lambda^c_{min}=\min
\lambda^c(\mu),\,\,\lambda^c_{max}=\max \lambda^c(\mu),$$ where “$\min$" and “$\max$" are given over all $f$-invariant ergodic measures supported on $H(P,f)$. We consider three possible cases.
\(i) $\lambda^c_{min}<0<\lambda^c_{max}$. Let $s\in(\lambda^c_{min},\lambda^c_{max})$. We use Lemma 2.6 to construct an ergodic measure $\mu$ with $Supp(\mu)=H(p,f)$ such that $\mu^c_\mu=s$. By Lemma 2.7, there are two saddles $p$ and $q_0$ such that $\lambda^c_p<0<s<\lambda^c_{q_0}$. Two cases may be occurred
$\bullet~s=0$. This case is deduced from [@bdg]. In fact, the authors in [@bdg] have proved that if a $C^1$-generic homoclinic class which has partial splitting with one dimensional center direction has periodic points with different index then it can admit a non-hyperbolic ergodic measure whose support equals to the whole of the homoclinic class.
$\bullet~s\neq 0$. Suppose that $s>0$. By the second item of Proposition 2.3, there is a periodic point $p_0\in H(p,f)$ such that $0<\lambda^c_{p_0}<s<\lambda^c_{q_0}$. Inductively, we find two sequences of $\{p_n\}$ and $\{q_n\}$ of the same index such that
- $0<\lambda^c_{p_{n-1}}<\lambda^c_{p_n}<\big(s+\lambda^c_{p_{n-1}}\big)/2
<s<\lambda^c_{q_n}<(s+\lambda^c_{q_{n-1}})/2<\lambda^c_{q_{n-1}}$,
- $q_n$ is $\epsilon/2^n$-dense in $H(p,f)$,
- $q_n$ is $(\epsilon/2^n,(1-c/2^n))$-good approximation of $q_{n-1}$, for some positive constant $c$.
Let $0<\lambda^c_{p_{n-1}}<s<\lambda^c_{q_{n-1}}$. By Lemma 2.7, there is a saddle $p_n\in H(p,f)$ with $\epsilon/2^n$-dense orbit such that $0<\lambda^c_{p_{n-1}}<\lambda^c_{p_n}<\big(s+\lambda^c_{p_{n-1}}\big)/2<s$. Since $p_n$ and $q_{n-1}$ have the same index they are homoclinically related. Let $$x\in
W^s(\mathcal{O}(q_{n-1}))\,\overline{\pitchfork}\,W^u(\mathcal{O}(p_{n}))
\,\,\,\textrm{and}\,\,\, y\in
W^u(\mathcal{O}(q_{n-1}))\,\overline{\pitchfork}\,W^s(\mathcal{O}(p_{n})).$$ Consider a locally maximal hyperbolic set $\Lambda_n$ containing $\mathcal{O}(p_n)\cup
\mathcal{O}(x)\cup\mathcal{O}(y)\cup\mathcal{O}(q_{n-1})$. For some suitable $m_n\in\mathbb{N}$, put $$\mathcal{PO}_n=\{\underbrace{\mathcal{O}(p_n)}_{t_n-\textrm{times}},f^{-m_m}(x),\ldots,f^{m_n}(x),
\underbrace{\mathcal{O}(q_{n-1})}_{s_n-\textrm{times}},f^{-m_n}(y),\ldots,f^{m_n}(y)\},$$ where $2m_nM/\big(t_n\pi(p_n)\lambda^c_{p_n}+s_n
\pi(q_{n-1})\lambda^c_{q_{n-1}}+2mM\big)<\epsilon/2^n$. If $\mathcal{PO}_n$ is $\epsilon/2^n$-shadowed by a saddle $q_n$ then $\pi(q_n)=t_n\pi(p_n)+s_n\pi(q_{n-1})+2m_n$ and for suitable $t_n$ and $s_n$ one has $$s<\lambda^c_{q_n}\thickapprox\frac{t_n\pi(p_n)\lambda^c_{p_n}+s_n
\pi(q_{n-1})\lambda^c_{q_{n-1}}+2m_nM}
{t_n\pi(p_n)+s_n\pi(q_{n-1})+2m_n}<\frac{s+\lambda^c_{q_{n-1}}}{2},$$ where $M=\max_{x\in M}\log\|Df(x)\|$. It is’t difficult to see that $$\chi_n=\frac{s_n\pi(p_n)}{t_n\pi(q_{n-1})+s_n\pi(p_n)+2m}=
1-\frac{t_n\pi(q_{n-1})+2m}{t_n\pi(q_{n-1})+s_m\pi(p_n)+2m}>1-c/2^n,$$ for some positive constant $c$. Now, the sequence $\{q_n\}$ satisfies in the assumption of Lemma 2.6. Hence, the weak$^*$ limit $\mu$ of the atomic measures $\mu_{q_n}$ is ergodic and $$\lambda^c_\mu=\int \|Df|_{E^c}\|\,d\mu=\lim_{n\to\infty} \int
\|Df|_{E^c}\|\,d\mu_{q_n}=\lim_{n\to\infty}\lambda^c_{q_n}=s.$$ Furthermore, by (1), $\textrm{Supp}(\mu)=H(p,f)$.
\(ii) $\lambda^c_{min}>0$ or $\lambda^c_{min}<0$. In this case, the central direction is hyperbolic ([@cao]).
\(iii) $\lambda^c_{min}=0$ or $\lambda^c_{max}=0$. In this case by the first part of Lemma 2.5, the problem is reduced to the previous ones.
In the second case of the theorem, one should notice that by Theorem 2.6, any ergodic measure can be approximated by periodic measures supported on the homoclinic class. By the second part of Proposition 2.3, we can suppose that these sequence of periodic orbits are coindex one. Hence, the lemma turns to the first case. $\hfill\Box$
Remarks in the case $E^c=E^c_1\oplus E^c_2$
===========================================
In general, we don’t know much about the topological properties of the set of ergodic measures supported on a homoclinic class. Since the central directions are one dimensional the Lyapunov exponents along them define a continues map with in weak-topology. As before, one can deduce that the closure of the set of central Lyapunov exponents associated to the hyperbolic ergodic measures forms a convex set. One should notice that by the domination, the Lyapunov exponent along $E^c_1$ is strictly less than the Lyapunov exponents along $E^c_2$. Hence, in the case of non-hyperbolic ergodic measures two disjoint cases may be occurred, either $\lambda^\mu_1<0=\lambda^\mu_2$ or $\lambda^\mu_1=0<\lambda^\mu_2$. The overall picture of the set of central Lyapunov exponents is illustrated in Figure 1.
$$\includegraphics[scale=.7]{INR2.eps}$$
The shadowed part is the closure of the set of central Lyapunov exponents associated to the hyperbolic ergodic measures supported on $H(p,f)$ which is in fact the set $\mathcal{LE}^c(H(p,f))$. Using the central model for chain transitive sets, S. Crovisier [*et al.*]{} have obtained the following version of the Mane’s ergodic closing lemma inside the homoclinic class ([@csy]).
$$\includegraphics[scale=.8]{INR2C.eps}$$
For any $C^1$-generic diffeomorphism $f$, let $H(p,f)$ be a partially hyperbolic homoclinic class with one dimensional central bundles. Let $i$ be the minimal stable dimension of its periodic orbits. If $H(p,f)$ contains periodic point with $i^{th}$ weak Lyapunov exponent, then for any ergodic measure supported on $H(p,f)$ whose $(i-1)^{th}$ Lyapunov exponent is zero, there exists periodic orbits contained in $H(p,f)$ whose associated measures converge for the weak topology towards the measure.
The lemma suggests two simplest situations showed in the Figure 2.
[0]{} Hasselblatt, H. and Pesin, Y. *Partially hyperbolic dynamical systems*, Handbook of dynamical systems. 1B, 1-55, Elsevier B. V., Amsterdam, (2006). Brin, M, and Pesin, Y., [*Partially hyperbolic dynamical systems*]{}, Izv. Akad. Nauk SSSR Ser. Mat. [**38**]{}, (1974), 170-212. Hertz, F.R., Hertz, M.A, and Ures, R., [*A survey on partially hyperbolic dynamics*]{}, http://arxiv.org/abs/math/0609362. Leplaideur, R., Oliveira, K. and Rios, I. [*Eeuilibrium states for partially hyperbolic horseshoes*]{}, Ergod. Th. & Dynam. Sys. (2011), [**31**]{}, 179–195. Gorodetski, A.S, and Yu. S. Ilyashenko, *Certain properties of skew products over a horseshoe and a solenoid*, Grigorchuk, R. I. (ed.), Dynamical systems, automata, and infinite groups. Proc. Steklov Inst. Math. 231, (2000), 90-112. Gorodetski, A.S., Ilyashenko, Yu., Kleptsyn, V. and Nalsky, M., [*Nonremovable zero Lyapunov esponents*]{}, Functional Analysis and Its Applications, [**39**]{}, (2005), 27–38. Abdenur, F., Bonatti, C., Crovisier, C., Diaz, L., and Wen, L., *Periodic points in homoclinic classes*, Ergod. The. and Dyn. Syst., [**27**]{}, (2007), 1-22. Crovisier, S., Sambarino, M., and Yang, D., [*Partial hyperbolicity and homoclinic tangency*]{}, Preprint. Gourmelon, N., [*Thése de Doctorat*]{}, Instabilité de la dynamique en lábsence de décomposition dominée Université de Bourgogne. Pliss, V., [*On a conjecture due to Smale*]{}, Diff. Uravnenija., [**8**]{}, (1972), 268-282. Pujals, E. and Sambarino, M., [*Homoclinic tangencies and hyperbolicity for surface diffeomorphisms*]{}, Ann. Math., [**151**]{}, (2000), 961-1023. Fakhari, A., [*Uniform Hyperbolicity Along Periodic Orbits*]{}, Proc. AMS., to appera. Abdenur, F., Bonatti, C., and Crovisier, S. *Non-uniform hyperbolicity for $C^1$-generic diffeomorphisms*, Israel J. of Math., to appera. Bonatti, C., Diaz, L., and Grodetski, A., [*Non-hyperbolic ergodic measures with large support*]{}, Nonlinearity, [**23**]{}, (2010), 687-705. Cao, Y., *Non-zero Lyapunov exponents and uniform hyperbolicity*, Nonlinearity [**16**]{}, (2003), 1473-1479.
|
\
WHAT DOES THE JOSEPHSON EFFECT TELL US\
ABOUT THE SUPERCONDUCTING STATE OF THE CUPRATES?\
\
R. Hlubina$^1$, M. Grajcar$^1$, and E. Il’ichev$^2$\
\
$^1$Department of Solid State Physics, Comenius University, SK-84248 Bratislava, Slovakia\
$^2$ Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany
1\. INTRODUCTION
The defining property of a superconductor is the stiffness of its phase. In conventional low-$T_c$ superconductors this stiffness is so large that in order to directly observe its finite value one has to create a region where it is locally suppressed, the so-called superconducting weak link. The resulting weak link region gives rise to a variety of Josephson effects (for a review, see [@Likharev79]), in which the superconducting phase becomes a measurable quantity.
In the cuprate superconductors the sensitivity of the Josephson effect to the phase has played a decisive role in establishing a phase-sensitive test of the (now well established) unconventional $d$-wave symmetry of their pairing state (for a review, see [@Tsuei00]).
The purpose of this paper which is based on the results of our measurements of the current-phase relation in the cuprates [@Ilichev99; @Ilichev01; @Komissinski02; @Grajcar02] is to argue that besides providing a phase-sensitive test of the pairing symmetry, a quantitative analysis of the Josephson effect in the cuprates provides new insights into the microscopic physics of the cuprates, in particular into the reduced phase stiffness of the cuprates.
Our experiments were carried out on two types of Josephson junctions: \[001\] tilt grain boundary junctions [@Ilichev99; @Ilichev01; @Grajcar02] and $c$-axis junctions between a cuprate and a low-$T_c$ superconductor [@Komissinski02]. Let us start with a brief discussion of these two types of junctions.
1\. Grain boundary junctions have been studied since the very beginning of the cuprate research (for a review see [@Hilgenkamp02]). In particular, it has been realized very early on that grain boundaries are ${\it the}$ current limiting element of polycrystalline materials and as such are to be avoided in large current applications. However, after having been used in the phase sensitive experiments [@Tsuei00], the situation changed completely when it was realized that cuprate grain boundary junctions might form the basis of a completely new type of superconducting electronics, based on the use of the so-called $\pi$-junctions.
In an idealized picture, a grain boundary junction can be thought of as a planar interface of two grains. The junction is characterized by two angles $\theta_1$ and $\theta_2$ between the normal to the interface and the crystallographic axes in the grains, see Fig. \[Fig:Angles\_def\].
It is an experimental fact that the junction transparency ${\cal D}$ depends predominantly on the misorientation angle $\theta=\theta_2-\theta_1$. The microstructure of the grain boundary is quite complicated and at the moment there exists no generally accepted picture of it. In this paper we adopt the point of view advocated by Hilgenkamp and Mannhart [@Hilgenkamp02], who view the junction as a locally underdoped (and insulating) region. The width of this insulating barrier is supposed to be proportional to $\theta$, therefore leading quite naturally to the experimentally observed scaling ${\cal D}(\theta)=\exp(-\theta/\theta_0)$ with $\theta_0\approx 5^\circ$. In this paper we shall be concerned mostly with grain boundary junctions with misorientation angle $\theta=45^\circ$ for which ${\cal D}\approx e^{-9}\approx 10^{-4}$.
![Schematic drawing of a \[001\] tilt grain boundary junction.[]{data-label="Fig:Angles_def"}](Angles_def.eps){width="10cm"}
We believe that grain boundary junctions with $\theta>\theta_0$ are to be described as featureless junctions in the tunnel limit. An alternative explanation of the very small transmission probability ${\cal D}(\theta)$ could invoke the presence of pinholes in the barrier, whose density would decrease exponentially with $\theta$. We believe that measurements [@Ilichev99a; @Grajcar02] of the relation between the superconducting current $I$ and the phase difference $\varphi$ across the junctions with $\theta=24^\circ$ and $36^\circ$ have ruled out this possibility, since the results were fully consistent with the tunnel limit prediction $I(\varphi)\propto\sin\varphi$ and no trace of higher harmonics indicative of small barrier weak links were found, see Fig. \[Fig:Comp\].
2\. The other type of junctions whose current-phase relation we have studied are the so-called $c$-axis junctions between the cuprates and low-$T_c$ superconductors. In this type of junctions the interface is parallel to the CuO$_2$ planes and the current flows along the $c$ axis of the cuprates. Because of the layered structure of the cuprates the interlayer forces are presumably much weaker than those within the layers, and therefore much cleaner and better defined interfaces are to be expected for this type of junctions, when compared with the grain boundary junctions.
![Current-phase relation at $T=15$ K for symmetric GBJJs with $\theta=24^\circ$ (dotted line), 36$^\circ$ (dashed line) and 45$^\circ$ (solid line). The inset shows the scaling $j_c\propto
\exp(-\theta/\theta_0)$ of the critical current density with $\theta_0=5.6^\circ$. Taken from [@Grajcar02].[]{data-label="Fig:Comp"}](Comp_SQUID01.eps){width="10cm"}
2\. THEORY
2.1. GRAIN BOUNDARY JUNCTIONS. Our main interest in this paper regards the current-phase relation $I(\varphi)$, which is on general grounds assumed to be a $2\pi$ periodic function. Moreover, in absence of magnetic fields we expect that $I(-\varphi)=-I(\varphi)$ and therefore we can write [@Likharev79] $$I(\varphi)=I_1\sin\varphi+I_2\sin 2\varphi+\ldots.
\label{eq:Fourier}$$ The $d$-wave symmetry of the pairing state in the cuprates requires that odd harmonics $I_1,I_3,\ldots$ change sign with $\theta_i\rightarrow
\theta_i+\pi/2$. If we keep only the lowest-order angular harmonics, we can therefore write [@Walker96] $$I_1=I_c\cos 2\theta_1\cos 2\theta_2+
I_s\sin 2\theta_1\sin 2\theta_2.
\label{eq:Walker_flat}$$ The coefficients $I_c,I_s$ are functions of the barrier strength, temperature $T$, etc. In particular, if the tunneling is allowed only for a narrow range of impact angles around normal incidence, only the $I_c$ term survives. This situation has been considered in the early paper by Sigrist and Rice [@Sigrist92], who suggested to use the first term of Eq. \[eq:Walker\_flat\] as a simple approximate formula which respects the symmetries of the Ambegaokar-Baratoff like expression (at $T=0$) for the critical current, $${\pi\Delta_k\Delta_q\over eR_N(|\Delta_k|+|\Delta_q|)}
K\left({|\Delta_k-\Delta_q|\over |\Delta_k|+|\Delta_q|}
\right) \rightarrow I_c\cos 2\theta_1\cos 2\theta_2,
\label{eq:ambegaokar}$$ where $R_N$ is the resistance of the junction in the normal state, $\Delta_k$, $\Delta_q$ are gap functions for the case of normal incidence in the two superconducting electrodes, and $K$ is the complete elliptic integral.
On the other hand, the even harmonics $I_2,I_4,\ldots$ are not forced by symmetry to change sign and therefore we neglect their weak angular dependence (except for the strong dependence on the barrier transparency ${\cal D}(\theta)$). This then implies (as first noted in [@Tanaka94]) that for the so-called asymmetric $45^\circ$ grain boundary junctions (i.e. $\theta_1=45^\circ$, $\theta_2=0^\circ$), the first harmonic $I_1$ vanishes by symmetry and the current-phase relation can be approximated by $I(\varphi)=I_2\sin 2\varphi$.
In real life the junction interface meanders around its mean (nominal) direction. In that case we wish to interpret the function $I_1(\theta_1,\theta_2)$ as a relation between the local critical current density and the local interface geometry. Structural studies of grain boundaries [@Mannhart96] show that the interface is typically faceted with a typical size of a facet $\approx 100$ nm. Since this length scale is much larger than both relevant electronic length scales, the Fermi wavelength and the coherence length, we believe that the macroscopic first harmonic can be estimated by a simple averaging of Eq. \[eq:Walker\_flat\] along the interface. Making use of such a procedure, if the local junction geometry is described by $\theta_i+\chi(x)$, $I_1$ of a rough interface is given by [@Grajcar02] $$I_1={1\over 2}(I_c+I_s)\cos 2\theta
+{x\over 2}(I_c-I_s)\cos 2(\theta_1+\theta_2),
\label{eq:Walker_rough}$$ where $x=\langle\cos 4\chi\rangle$ is an interface roughness parameter and $\langle\ldots\rangle$ denotes an average along the interface. Note that in the completely rough limit $x=0$, $I_1$ depends only on the misorientation angle $\theta$ and not on the individual angles $\theta_i$ [@Tsuei00], since in that limit the notion of the direction of the interface is meaningless.
On the other hand, since $I_2$ is not forced by symmetry to depend strongly on $\theta_i$ (except for the dependence through the barrier transmission on the misorientation angle $\theta$), we expect that $I_2$ will exhibit only a weak dependence on the surface roughness $x$.
Summarizing the above symmetry arguments, it follows that $d$-wave symmetry and sufficient interface roughness (small $x$) imply enhanced $I_2/I_1$ ratios for all $45^\circ$ junctions.
However, while very robust, the symmetry arguments do not provide estimates of the ratio $I_2/I_1$. In what follows we attempt a more quantitative analysis of the Josephson effect. The increased level of detail has its prize, however: we have to introduce a microscopic model of the junction. Two such models have been discussed in the literature: a perfectly flat interface which leads to the presence of midgap states in the case of an impenetrable barrier [@Hu94], and a model emphasizing the roughness of the interface and the subsequent development of a nontrivial pattern of currents flowing along the interface [@Millis94].
2.1.1. IDEALLY FLAT JUNCTIONS. Let us consider first the ‘flat’ scenario. The key role in this approach is played by the concept of the so-called midgap states, which should form at surfaces of anomalous superconductors. Theory predicts that the number of midgap states should be maximal for (110)-like surfaces of $d$-wave superconductors and no such states should form at (100) and (001)-like surfaces [@Hu94]. The presence (and the nontrivial surface orientation dependence) of such midgap states in the cuprates is well documented by now by STM spectroscopy [@Wei98] and by grain boundary tunneling spectroscopy [@Alff98]. It has been also suggested that such bound states are at the origin of the observed nonmonotonic temperature dependence of the penetration depth [@Walter98].
Let us turn now to the study of the influence of the midgap states on the Josephson effect in the cuprates, following the early proposal of Tanaka and Kashiwaya [@Tanaka96] (for reviews, see [@Kashiwaya00; @Lofwander01]). Within the simplest description, we consider a circular Fermi line, specular reflection at the interface, and a constant order parameter in the superconducting grains. Because of the translational symmetry along the junction, the scattering problem factorizes into a set of independent problems for each given impact angle.
The energy of the Andreev bound states is governed by the probability $D(\theta,\vartheta)$ of transmission for impact angle $\vartheta$. The function $D(\theta,\vartheta)$ depends on the details of the barrier, such as its width and height, none of which are known reliably in the present context. The only direct experimental measure of the barrier transmission is the resistance per square of the junction in the normal state, $R_\Box=(h/2e^2)(\pi d/k_F{\cal D})$, which provides us with one moment of the function $D(\theta,\vartheta)$, ${\cal
D}(\theta)=\int_0^{\pi/2}d\vartheta\cos\vartheta D(\theta,\vartheta)$.
![Trajectories of the particles for specular reflection on a flat junction interface. The impact angle $\vartheta$ is the same for both sides of the junction.[]{data-label="Fig:Angles"}](Angles.eps){width="10cm"}
Let us calculate the energy of the Andreev bound state corresponding to the incoming (outgoing) trajectories in the left and right grains denoted 1 and 2 (3 and 4), respectively (see Fig. \[Fig:Angles\]). To this end, let us denote the order parameter on the trajectory $i$ as $\Delta_i=|\Delta_i|S_i$, where $S_i=\pm 1$ is the sign. The qualitative character of the bound state depends on the relative value of the signs $S_i$.
Before proceeding, let us recall that for an impenetrable barrier with $D\rightarrow 0$, a midgap state is formed in the left (right) grain for $S_1=-S_3$ ($S_2=-S_4$), i.e. when the sign of $\Delta$ changes under reflection at the interface [@Hu94]. In the tunnel limit, the energy of the Andreev bound state can belong to one of the following three classes:
\(i) The signs of $\Delta$ change under reflection on both sides of the junction, $S_1=-S_3$ and $S_2=-S_4$. In that case the two midgap states on both sides of the junction are split by the finite value of $D$ [@Barash00]: $$\varepsilon(\theta,\vartheta,\varphi)=\pm
\sqrt{{4D(\theta,\vartheta)|\Delta_1\Delta_2\Delta_3\Delta_4|}\over
{(|\Delta_1|+|\Delta_3|)(|\Delta_2|+|\Delta_4|)}}
\sqrt{1-S_1S_2\cos\varphi\over 2}.
\label{eq:en_case1}$$ Note that $\varepsilon\propto \sqrt{D}$, as is usual for degenerate perturbation theory.
\(ii) The midgap state is formed only on one side of the junction, the left one for definiteness ($S_1=-S_3$, $S_2=S_4$). In that case there is only one anomalous Andreev bound state, [@Barash00]: $$\varepsilon(\theta,\vartheta,\varphi)=
D(\theta,\vartheta)
{|\Delta_1\Delta_3|\over{|\Delta_1|+|\Delta_3|}}
S_1S_2\sin\varphi.$$
\(iii) There are no midgap states on both sides of the junction, i.e. $S_1=S_3$ and $S_2=S_4$. In this case there are no amalous Andreev bound states and the junction resembles qualitatively that between two $s$-wave superconductors. Note that this type of processes is always realized for $\vartheta=0^\circ$.
Once the quasiparticle energies are known, the free energy due to Andreev levels per unit area of the junction reads $$F_\Box^A(\varphi)=-{k_FT\over 2\pi d}\sum_n\int_{-\pi/2}^{\pi/2}
d\vartheta\cos\vartheta
\ln\left[1+\exp\left(-{\varepsilon_n(\vartheta)\over T}\right)\right],
\label{eq:F_andreev}$$ where $d$ is the average distance between the CuO$_2$ planes and the index $n$ numerates all $\varphi$-dependent energy levels for the impact angle $\vartheta$. According to general principles, the current-phase relation $I(\varphi)$ can be determined from the $\varphi$-dependence of the total junction free energy, $I=(2\pi/\Phi_0)\partial F/\partial\varphi$ and the density of supercurrent across the junction reads $j(\varphi)=(2\pi/\Phi_0)\partial F_\Box/\partial\varphi$. We emphasize that it is the total $\varphi$-dependent free energy density $F_\Box(\varphi)$ (and not only the contribution of the Andreev bound states $F_\Box^A$) which determines $j(\varphi)$.
Before proceeding we should like to point out that, in general, a finite phase difference $\varphi$ across the junction leads to a nonzero current along the junction [@Huck97]. This current generates a magnetic field which has to be screened by the Meissner currents in the superconducting electrodes [@Sigrist98], thus leading to an additional term in the free energy, $F_\Box^M(\varphi)$.
The spontaneously generated current along the interface can be calculated as follows. Since all four momenta involved in interface scattering with impact angle $\vartheta$ (see Fig. \[Fig:Angles\]) have the same momentum $k_y$ along the junction, the $y$ component of the current density carried by the Andreev bound state with energy $\varepsilon(\vartheta)$ is $$j_y(\vartheta,x)={e\hbar k_y\over m}
\left(\left|u(x,\vartheta)\right|^2+|v(x,\vartheta)|^2\right),$$ where $u(x)$ and $v(x)$ are the two components of the Andreev level wavefunction. Since $u(x),v(x)$ decay exponentially on the distance $\sim\xi$ from the interface (where $\xi\sim \hbar v_F/\Delta$ is the coherence length), also the net current along the interface $$j_y(x)={k_F L\over 2\pi}\sum_n\int_{-\pi/2}^{\pi/2}
d\vartheta\cos\vartheta
j_y(\vartheta,x)f\left(\varepsilon_n(\vartheta)\right)
\label{eq:j_y}$$ is localized within $\xi$ from the interface. From Eq. \[eq:j\_y\] it follows that the linear density of the total current flowing along the grain boundary per CuO$_2$ layer, $J_0=\int_{-\infty}^\infty
j_y(x)$, is $$J_0={e\varepsilon_F\over \hbar d}\sum_n\int_{-\pi/2}^{\pi/2}
{d\vartheta\over 2\pi}
\sin 2\vartheta f(\varepsilon_n(\vartheta)),
\label{eq:J_0}$$ where $\varepsilon_F$ is the Fermi energy. In deriving Eq. \[eq:J\_0\] we have made use of the fact that the Andreev level wavefunctions are normalized, $\int_{-\infty}^\infty dx
(|u(x)|^2+|v(x)|^2)=(Ld)^{-1}$.
![Schematic drawings of asymmetric (a) and symmetric (b) $45^\circ$ grain boundary junctions. Shaded regions correspond to a positive order parameter. For asymmetric junctions, impact angles for which $S_1S_2=-1$ are denoted by the hatched regions. For symmetric junctions the hatched regions correspond to trajectories with midgap states on both sides of the junction.[]{data-label="Fig:Circles"}](Circles.eps){width="10cm"}
[*Asymmetric $45^\circ$ junctions.*]{} In what follows, let us apply the above formalism to the two special cases of interest to us, namely the asymmetric and symmetric $45^\circ$ junctions. For asymmetric junctions, the trajectories for all impact angles are of type (ii), see Fig. \[Fig:Circles\]. Since in this case $\Delta_1(\vartheta)=-\Delta_3(\vartheta)=\Delta\sin 2\vartheta$ and $\Delta_2(\vartheta)=\Delta_4(\vartheta)=\Delta\cos 2\vartheta$, the energy of the Andreev levels is $$\varepsilon(\vartheta)={\Delta\over 2}D(\vartheta)\:
|\sin 2\vartheta|\:{\rm sign}(\sin 4\vartheta)\sin\varphi.$$ For the sake of simplicity we consider a simple model for the barrier transmission function, $D(\vartheta)=D(0)$ for $|\vartheta|<\vartheta_0<\pi/4$ and $D(\pi/4)\ll D(0)$ otherwise. In this case ${\cal D}\approx D(0)\sin\vartheta_0$ and the typical energy of the bound states $\sim {\cal D}\Delta$ sets the energy scale of the problem. In the two extreme limits $T\ll{\cal D}\Delta$ and $T\gg
{\cal D}\Delta$, respectively, we find $$\begin{aligned}
F_\Box^A&=&
-{\vartheta_0\over 4\pi}{k_F{\cal D}\Delta\over d}
|\sin\varphi|,
\nonumber
\\
F_\Box^A&=&{\vartheta_0\over 48\pi}
{k_F{\cal D}^2\Delta^2\over Td}
\cos 2\varphi.
\label{eq:FA_asym}\end{aligned}$$ From Eq. \[eq:J\_0\] it follows that $J_0$ depends only on the population of the Andreev levels. Let us for definiteness consider the case $\varphi>0$. Depending on the impact angle $\vartheta$, the trajectories can be classified into two distinct groups: The hatched regions in Fig. \[Fig:Circles\] correspond to $S_1S_2=-1$, whereas for the remaining trajectories $S_1S_2=1$. Thus the energy of the Andreev levels in the hatched (not hatched) regions is negative (positive). Therefore at $T=0$ only the Andreev levels in the hatched regions are occupied and from Eq. \[eq:J\_0\] we find $J_0=0$. This effect has been called superscreening in the literature [@Amin01]. At finite temperatures $T\gg D(\pi/4)\Delta$ an explicit calculation leads to a finite current along the junction, $$J_0=-{\vartheta_0^2\over 4\pi}
{ek_Fv_F\over d}
{{\cal D} \Delta\sin\varphi\over
3T+{\cal D} \Delta |\sin\varphi|}.$$
The surface currents due to the Andreev bound states generate magnetic fields which have to be screened by the Meissner currents in the superconducting grains. As pointed out in [@Lofwander00], this leads to an increase of the total junction energy, whose magnitude we now estimate in the simple case of a two dimensional interface (in the $y$-$z$ plane) between 3D superconducting grains. Of course, most experiments are carried out on thin films and thus the 3D geometry is not fully appropriate. However, since the film thicknesses are typically $\sim 1000$ Å which is comparable to the bulk penetration depth, our estimates should remain qualitatively correct.
![The distribution of supercurrent density (a) in units of $j_0=J_0/\lambda$ and magnetic field (b) in units of $B_0=\mu_0J_0$ for symmetric $45^\circ$ grain boundary junctions. (c,d): the same for asymmetric $45^\circ$ junctions. In all figures we have used $\xi/\lambda=0.1$.[]{data-label="Fig:Mag"}](Mag.eps){width="15cm"}
Since the current due to Andreev bound states flows on scale $\xi$, we model its distribution as ${\bf j}=(0,j,0)$ with $j(x)=J_0\delta(x)$. Within the London theory (assuming bulk penetration depth $\lambda$), the distribution of the Meissner current and of the magnetic field can be readily calculated and the results are shown in Fig. \[Fig:Mag\]. The sum of the energy of the magnetic field and of the kinetic energy of the superflow (per unit area of the interface) is easily seen to be $F_\Box^M=\mu_0 J_0^2\lambda/4$.
Therefore the $\varphi$-dependent part of the total free energy density $F_\Box(\varphi)=F_\Box^A(\varphi)+F_\Box^M(\varphi)$ for $T\ll {\cal D}\Delta$ and $T\gg {\cal D}\Delta$ reads $$\begin{aligned}
F_\Box(\varphi)&=&{\vartheta_0\over 4\pi}{k_F\over d}\Delta
\left[-{\cal D} |\sin\varphi|+
{\vartheta_0^3\over 8}{\xi\over\lambda}
\left(
{\sin\varphi\over |\sin\varphi|+\alpha}
\right)^2
\right],
\nonumber\\
F_\Box(\varphi)&=&{\vartheta_0\over 48\pi}{k_F\over d}
{{\cal D}^2\Delta^2\over T}
\left[1-{\vartheta_0^3\over 12}{\hbar v_F/T\over\lambda}
\right]\cos 2\varphi,
\label{eq:F_asym}\end{aligned}$$ respectively, where $\xi=\hbar v_F/\Delta$ and $\alpha=3T/{\cal
D}\Delta$. Several points regarding Eq. \[eq:F\_asym\] are to be stressed. First, in the low temperature limit, our result agrees qualitatively with [@Lofwander00] in that the free energy is minimized for $\varphi=0$ or $\varphi=\pi$, if ${\cal D}\ll
\vartheta_0^3\xi/\lambda$ (i.e. the bound state energy gain is smaller than the Meissner energy loss). In the opposite limit when the bound state energy dominates $F_\Box$, the free energy is minimized for $\varphi=\pm\pi/2$.
From the normal state resistivity of $45^\circ$ junctions we estimate ${\cal D}\sim 10^{-4}$. If we take $\xi\approx 30$ Å and $\lambda\approx 1500$ Å relevant for the cuprates, we estimate that for not too strongly angle dependent tunneling, the Meissner energy should dominate $F_\Box$ at low temperatures. We should like to point out that by no means this implies trivial physics. Just on the contrary, as shown in Fig. \[Fig:Ipsteep\], dominant Meissner energy leads to a doubled periodicity of $I(\varphi)$. This is qualitatively similar to the well studied case when $F_\Box^A$ dominates. However, there are two major qualitative differences with respect to the case of dominant bound state energy: (i) the (degenerate) ground state corresponds to a state when no currents flow along the junction, and (ii) the current-phase relation becomes steep in the minima (not maxima) of the junction energy, see Fig. \[Fig:Ipsteep\].
![(a) Meissner energy $F^M$ of an asymmetric $45^\circ$ junction (in arbitrary units) as a function of the phase difference $\varphi$ across the junction. (b) Superconducting current $I=(2\pi/\Phi_0)\partial F^M/\partial\varphi$ (in arbitrary units) as a function of $\varphi$. In both figures $\alpha=1,2,5,10$ from top to bottom.[]{data-label="Fig:Ipsteep"}](Ip_steep.eps){width="9cm"}
At temperatures above the bound state energy ${\cal D}\Delta$ (which is presumably the case down to helium temperatures due to the small value of ${\cal D}$), we find $F_\Box\propto \cos
2\varphi$ leading to $$j(\varphi)R_\Box=-{\pi\vartheta_0\over 12}
{{\cal D}\Delta^2\over eT}
\left[1-{\vartheta_0^3\over 12}{\hbar v_F/T\over\lambda}
\right]\sin 2\varphi.
\label{eq:iphi_asym}$$ A pure second harmonic is seen to develop, in complete agreement with the symmetry analysis. Note that in this temperature range the Meissner contribution to $I(\varphi)$ becomes negligible. This is a result of the thermal fluctuations which reduce the surface current $J_0$ severely.
[*Symmetric $45^\circ$ junctions,*]{} i.e. junctions with $\theta_1=-\theta_2=22.5^\circ$. In this case, trajectories with impact angles in the hatched regions of Fig. \[Fig:Circles\] are of type (i), whereas the remaining ones are of type (iii). As a function of the impact angle, the gap functions can be written $\Delta_1(\vartheta)=\Delta_2(\vartheta)=\Delta\cos(2\vartheta-\pi/4)$ and $\Delta_3(\vartheta)=\Delta_4(\vartheta)=\Delta\cos(2\vartheta+\pi/4)$. Again we have to make an assumption about the dependence of the transparency on the impact angle. We start by considering the case of a weakly dependent $D(\vartheta)$. As a rough approximation, we approximate all Andreev levels of type (iii) by their value for $\vartheta=0^\circ$ [@Barash00], and all levels of type (i) by the $\vartheta=45^\circ$ case, $$\begin{aligned}
\varepsilon(0,\varphi)&=&\pm
2^{-1/2}\Delta\sqrt{1-D(0)\sin^2(\varphi/2)},
\\
\varepsilon(\pi/4,\varphi)&=&\pm
2^{-1/2}\Delta\sin(\varphi/2)\sqrt{D(\pi/4)}, \end{aligned}$$ where we assume that $D(\pi/4)<D(0)\ll 1$. Note that for every impact angle $\vartheta$ there exist two levels with energies $\pm\varepsilon(\vartheta)$. Therefore $\sum_n
f(\varepsilon_n(\vartheta))=1$ and the integral Eq. \[eq:J\_0\] vanishes. In other words, the total supercurrent along the interface vanishes by symmetry. This does not mean, however, that no currents are flowing along the interface. In order to estimate the Meissner contribution to the interface free energy $F_\Box^M$, we model the current distribution generated by the Andreev levels as $j(x)=J_0[\delta(x+\xi)-\delta(x-\xi)]$ where $\xi$ is the coherence length and $J_0=\int_0^\infty dx j_y(x)$. The resulting magnetic field and the Meissner screening currents can be calculated within the London theory in a similar way as for the asymmetric junctions (see Fig. \[Fig:Mag\]) and the resulting contribution to the junction free energy is $F_\Box^M=\mu_0 J_0^2\xi$. Note that $F_\Box^M$ is reduced with respect to the case of asymmetric junctions (if we assume the same value of $J_0$ in both cases) by a factor $4\xi/\lambda\ll
1$, since it arises from a higher order moment of the current distribution $j_y(x)$. Due to this small factor, $F_\Box^M/F_\Box^A\sim (\xi/\lambda)^2/D$ even if we take the maximal possible value of $J_0$, $J_0\sim ek_Fv_F/d$. Therefore the Meissner contribution to the interface free energy will be neglected and the Josephson current is calculated from $$j(\varphi)={2\pi\over\Phi_0}{\partial F_\Box^A\over\partial\varphi}=
{k_F\over\Phi_0 d}\sum_n\int_{-\pi/2}^{\pi/2} d\vartheta\cos\vartheta
f(\varepsilon_n(\vartheta)){\partial\varepsilon_n\over\partial\varphi}.
\label{eq:j_andreev}$$ This implies that $j(\varphi)=j_n(\varphi)+j_a(\varphi)$, where the normal and anomalous contributions are, respectively: $$\begin{aligned}
j_n(\varphi)&=&{\pi\over 8}{k_F\Delta\over\Phi_0d}
{D(0)\sin\varphi\over 2^{3/2}\sqrt{1-D(0)\sin^2(\varphi/2)}}
\tanh{\Delta\sqrt{1-D(0)\sin^2(\varphi/2)}\over 2^{3/2}T},
\nonumber
\\
j_a(\varphi)&=&-{\pi\over 8}{k_F\Delta\over\Phi_0d}
\cos(\varphi/2)\sqrt{D(\pi/4)}\tanh
{\Delta\sin(\varphi/2)\sqrt{D(\pi/4)}\over 2^{3/2}T}.
\label{eq:iphi_sym1}\end{aligned}$$ At intermediate temperatures $\Delta\sqrt{D(\pi/4)}\ll T\ll \Delta$ the anomalous (normal) Andreev levels are in the high (low) temperature limit. Therefore Eq. \[eq:iphi\_sym1\] simplifies considerably and the first two harmonics of $j(\varphi)$ can be written as $$\begin{aligned}
{j_1\over j_L}&=&D(0)-{\Delta\over 2T}D(\pi/4),
\nonumber\\
{j_2\over j_L}&=&-{1\over 8}D(0)^2-
{1\over 24}\left({\Delta\over 2T}\right)^3D(\pi/4)^2,
\label{eq:iphi_sym2}\end{aligned}$$ where $j_L=(\pi/16\sqrt{2})k_F\Delta/\Phi_0 d$ is comparable with the bulk depairing current density. Thus, for weakly angle-dependent tunneling and above the energy of anomalous Andreev levels, theory predicts a sign change of the first harmonic at a temperature $T^\ast=\Delta D(\pi/4)/2D(0)$ and a second harmonic monotonically increasing with decreasing temperature.
In the opposite limit of a peaked barrier transparency $D(\vartheta)=D(0)\exp(-\vartheta/\vartheta_0)$, the junction is equivalent to a symmetric junction between $s$-wave superconductors with order parameters $\Delta/\sqrt{2}$. Therefore the current-phase relation in the $T\rightarrow 0$ and $T\rightarrow T_c$ limits, respectively, reads $$\begin{aligned}
j(\varphi)R_\Box&=&{\pi\over 2\sqrt{2}}{\Delta\over e}
\left[\sin\varphi-{D(0)\over 16}\sin 2\varphi\right],
\nonumber\\
j(\varphi)R_\Box&=&{\pi\over 8}{\Delta^2\over eT}
\left[\sin\varphi-{D(0)\over 48}\left({\Delta\over 2T}\right)^2
\sin 2\varphi\right].
\label{eq:iphi_sym3}\end{aligned}$$ Unlike Eq. \[eq:iphi\_sym2\], the result Eq. \[eq:iphi\_sym3\] predicts a monotonically increasing first harmonic with decreasing temperature.
2.2. $c$-AXIS JUNCTIONS. Let us briefly discuss the supercurrent in $c$-axis junctions between YBa$_{2}$Cu$_{3}$O$_{x}$ (YBCO) and low-$T_c$ superconductors [@Komissinski02]. In YBCO the dominant component of the superconducting order parameter has $d$-wave symmetry [@Tsuei00]. However, due to the orthorhombic structure of YBCO, a finite component with $s$-wave symmetry is admixed to the dominant $d$-wave order parameter. The in-plane phase sensitive experiments imply that the $d$-wave component remains coherent through the whole sample [@Tsuei00], while an elegant $c$-axis tunneling experiment shows directly that the $s$-wave order parameter component does change sign across the twin boundary [@Kouznetsov97].
The above picture of the YBCO pairing state is challenged by the experimental observation of a finite $c$-axis Josephson current between heavily twinned YBCO and a Pb counterelectrode [@Sun96]. Namely, the contribution of the $s$-wave part of the YBCO order parameter to the Josephson coupling between a conventional superconductor (superconductor with a pure $s$-wave symmetry of the order parameter, for example Pb or Nb) and YBCO should average to zero for equal abundances of the two types of twins in YBCO. In other words, the macroscopic pairing symmetry of twinned YBCO samples should be a pure $d$-wave [@Walker96]. Tanaka has shown that a finite second order Josephson current obtains for a junction between the $s$-wave and $c$-axis oriented pure $d$-wave superconductors [@Tanaka94]. However, measurements of microwave induced steps at multiples of $hf/2e$ on the $I$-$V$ curves of Pb/Ag/YBCO tunnel junctions imply dominant first order tunneling [@Kleiner96]. Therefore the finite $c$-axis Josephson current has to result from a nonvanishing admixture of the $s$-wave component to the macroscopic order parameter of YBCO [@Walker96]. Two alternatives of how this can take place in the junctions based on twinned YBCO have been discussed in the literature:
\(i) Sigrist *et al.* have suggested that the phase of the $s$-wave component in YBCO does not simply jump from 0 to $\pi $ upon crossing the twin boundary, but rather changes in a smooth way, attaining the value of $\pi /2$ right at the twin boundary [@Sigrist96]. The twinned YBCO sample is thus assumed to exhibit a macroscopic $d+is$ pairing symmetry. A related picture has been proposed by Haslinger and Joynt, who suggest a $d+is$ surface state of YBCO [@Haslinger00].
\(ii) A difference in the abundances of the two types of twins implies a $d+s$ symmetry of the macroscopic pairing state [@ODonovan97]. It has been pointed out [@Komissinski02] that also structural peculiarities of other type (such as a lamellar structure in a preferred direction) may lead to the $d+s$ macroscopic pairing symmetry.
The primary motivation of our experiment [@Komissinski02] was to determine which of the above two scenaria is realized in YBCO. In what follows we will show that this question can be conveniently answered by measuring the relative sign of the first two harmonics of $I(\varphi)$. Let us consider first the $d+s$ scenario. As will be shown explicitly in Eqs. \[eq:j\_1\], \[eq:j\_2\], in this case the junction free energy can be written $$F(\varphi)=-F_1\cos\varphi+F_2\cos 2\varphi
\label{eq:sign1}$$ with $F_1,F_2>0$. As shown in Fig. \[Fig:Signs\], this choice of signs corresponds to a free energy which is flat (curved) close to the minima (maxima). In terms of the current phase relation, this case corresponds to the usual case when $I(\varphi)$ is steep in the vicinity of $\pm\pi$, as in the Kulik-Omelyanchuk theory [@Likharev79].
![(a,b): Schematic drawing of the contributions from the first (solid lines) and second (dashed lines) harmonics to the free energy of $c$-axis junctions between a conventional superconductor and YBCO for various macroscopic pairing states in YBCO. (a): $d+s$ pairing, (b): $d+is$ pairing. (c,d): Total free energy (c) and the current-phase relation (d) in the $d+s$ case (solid lines) and $d+is$ case (dashed lines).[]{data-label="Fig:Signs"}](Signs.eps){width="14cm"}
The case of $d+is$ pairing is more complicated. In fact, first order coupling between the conventional superconductor and the $s$-wave component leads to a term in free energy $-F_1\cos(\varphi_{\rm
conv}-\varphi_s)$, where $\varphi_{\rm conv}$ and $\varphi_s$ are the phases of the conventional superconductor and of the $s$-wave component in YBCO and $F_1>0$. On the other hand, the second order term is the same as in the $d+s$ scenario. Therefore, since $\varphi=\varphi_{\rm conv}-\varphi_d$ where $\varphi_d$ is the phase of the $d$-wave component in YBCO, and since $\varphi_s-\varphi_d=\pi/2$ in the $d+is$ scenario, the junction free energy can be written $$F(\varphi)=-F_1\cos(\varphi-\varphi_{\rm min})
-F_2\cos 2(\varphi-\varphi_{\rm min}),
\label{eq:sign2}$$ where $\varphi_{\rm min}=\pi/2$ is the equilibrium phase difference and $F_1,F_2>0$, see Fig. \[Fig:Signs\]. In the experiment it is a delicate issue to determine the absolute value of $\varphi_{\rm
min}$. Usually $\varphi_{\rm min}$ is set equal to zero in the free energy minimum and with such conventions the two cases Eqs. \[eq:sign1\],\[eq:sign2\] lead to a different relative sign of $I_1$ and $I_2$ for the $d+s$ and $d+is$ scenaria, respectively.
2.3. FACETED JUNCTIONS. In situations when the first harmonic $I_1$ vanishes by symmetry (i.e. for asymmetric $45^\circ$ grain boundary junctions or for $c$-axis tunneling between an $s$-wave and a $d$-wave superconductor), small local deviations from ideal geometry lead to a finite local coupling $\delta I_1$ across the junction. In such cases the junction can be viewed as a parallel series of Josephson junctions with local critical current whose magnitude and sign fluctuate along the interface. It is customary to call the junctions with a positive (negative) $\delta I_1$ as 0 and $\pi$ junctions, respectively. Let us note in passing that very recently it has become possible to prepare samples with a prescribed pattern of 0 and $\pi$ junctions, making use of the so-called zigzag junctions [@Smilde02].
An example of a series of 0 and $\pi$ junctions is shown in Fig. \[Fig:Facet\], where the geometry of an asymmetric $45^\circ$ grain boundary between two twinned YBCO samples is shown. Twinning in the grain with $\theta_1=0^\circ$ is not shown explicitly, since it is irrelevant for our argument. Another often considered source of deviations from ideal interface geometry is associated with the experimentally well documented meandering of interface [@Mannhart96].
![Schematic drawing of an asymmetric $45^\circ$ grain boundary junction between twinned YBCO superconducting grains. In the upper grain the sign of the dominant lobe of the order parameter changes between the neighboring twins, leading to a sequence of 0 and $\pi$ junctions. Also shown are the spontaneously generated currents along the interface.[]{data-label="Fig:Facet"}](Facet.eps){width="10cm"}
If the average interface geometry is close to ideal, the number of 0 and $\pi$ junctions is roughly the same. As noted first by Millis [@Millis94] (for an alternative formulation making use of the sine-Gordon equation, see e.g. [@Mannhart96; @Mints98]), this does not imply a vanishing Josephson coupling across the interface. The general idea is as follows: let the average phase difference across the junction is $\varphi$. The phase difference along the interface is modulated from this value towards $0$ and $\pi$ in the 0 and $\pi$ junction regions, respectively, by an amount $\sim\chi$. This leads to spontaneously generated currents along the interface which are schematically depicted in Fig. \[Fig:Facet\]. In this way, the junction gains Josephson energy $\propto\chi$, whereas a Meissner energy of only $\propto\chi^2$ is lost. The total energy gain is maximal for $\varphi=\pm\pi/2$ and this explains qualitatively the development of a second harmonic in such junctions.
![Schematic drawing of a $c$-axis junction between an $s$-wave superconductor and a twinned YBCO sample. The black and white regions (of typical size $a$) correspond to $0$ and $\pi$ junctions, respectively. Also shown are the penetration depths in the $s$-wave superconductor ($\lambda_R$) and in YBCO ($\lambda,\lambda_c$).[]{data-label="Fig:Checkboard"}](Checkboard.eps){width="10cm"}
In what follows, let us discuss the faceted scenario in more detail. For definiteness, we consider the case of a $c$-axis junction between an $s$-wave superconductor and a twinned YBCO sample, see Fig. \[Fig:Checkboard\]. For the sake of simplicity, we model the randomly twinned structure by a checkerboard distribution of 0 and $\pi$ junctions with a periodic distribution of the local critical current density $j_c({\bf x})=j_0 g({\bf x})$, where $g({\bf
x})=\sin(\pi x/a)\sin(\pi y/a)$ and for $a$ we take the typical size of the twins. Here we have assumed a coordinate system such that the interface lies in the $xy$ plane. According to [@Millis94], a spontaneous magnetic field is generated, which we now find by minimization of the total junction energy as a function of the phase difference $\varphi$ across the junction. The symmetry of the problem dictates that the magnetic field reads $$\begin{aligned}
B_x&=&\sum_n^\prime B_n \exp(-K_n|z|)
\sin(n\pi x/a) \cos(n\pi y/a),
\nonumber
\\
B_y&=&-\sum_n^\prime B_n \exp(-K_n|z|)
\cos(n\pi x/a) \sin(n\pi y/a),
\nonumber
\\
B_z&=&0,
\label{eq:b_millis}\end{aligned}$$ where the prime restricts the summation to odd values of $n$. Note that $\nabla\cdot {\bf B}=0$, as it should be. For $z>0$, i.e. in the conventional superconductor, the ${\bf B}$ field satisfies the equation $(\nabla^2-\lambda_R^{-2}){\bf B}=0$ if $K_n^2=\lambda_R^{-2}\left[1+2n^2\left(\pi\lambda_R/a\right)^2\right]$. On the other hand, in the anisotropic cuprates, ${\bf B}=(B_x,B_y,0)$ is governed by $B_{x,y}=\left[\lambda_c^2(\partial_x^2+\partial_y^2)+
\lambda^2\partial_z^2\right]B_{x,y}$ and therefore for $z<0$ we have to take $K_n^2=\lambda^{-2}\left[1+
2n^2\left(\pi\lambda_c/a\right)^2\right]$. From Eq. \[eq:b\_millis\], the distribution of Meissner screening currents can be calculated using $\nabla\times{\bf B}=\mu_0{\bf j}$. The total junction energy per unit area corresponding to the magnetic field described by Eq. \[eq:b\_millis\] is $$E_\Box={1\over 4\mu_0}\sum_n^\prime B_n^2\lambda_n
-{\Phi_0j_0\over 2\pi S}\int dS g({\bf x})\cos\theta({\bf x}),
\label{eq:energy_millis}$$ where $S$ is the junction area and $$\lambda_n=\lambda_R\sqrt{1+2n^2(\pi\lambda_R/a)^2}+
\lambda\sqrt{1+2n^2(\pi\lambda_c/a)^2}.$$ The local phase difference across the junction $\theta({\bf x})$ can be determined from the Josephson equations $$\begin{aligned}
{\partial\theta\over\partial x}&=&
{2\pi\mu_0\over\Phi_0}\left[\lambda^2j_x(z=0_{-})-
\lambda_R^2j_x(z=0_{+})\right]=
{2\pi\over\Phi_0}\sum_n^\prime
B_n\lambda_n\cos{n\pi x\over a}\sin{n\pi y\over a},
\nonumber
\\
{\partial\theta\over\partial y}&=&
{2\pi\mu_0\over\Phi_0}\left[\lambda^2j_y(z=0_{-})-
\lambda_R^2j_y(z=0_{+})\right]=
{2\pi\over\Phi_0}\sum_n^\prime
B_n\lambda_n\sin{n\pi x\over a}\cos{n\pi y\over a}.\end{aligned}$$ These equations are solved by $\theta({\bf x})=\varphi+\chi({\bf
x})$, where $\varphi$ is the average phase difference and $$\chi({\bf x})={2a\over\Phi_0}\sum_n^\prime
{B_n\lambda_n\over n}\sin{n\pi x\over a}\sin{n\pi y\over a}$$ is the spontaneously generated modulation of $\theta({\bf x})$.
The Fourier components $B_n$ have to be determined by minimization of the energy Eq. \[eq:energy\_millis\]. To this end let us assume now that $|\chi|\ll 1$, which assumption will be justified at the end of the calculation. The second term in Eq. \[eq:energy\_millis\] simplifies in this case to $(2\mu_0)^{-1}B_1B_{\rm
eff}\lambda_1\sin\varphi$, where $B_{\rm eff}=(2\pi)^{-1}\mu_0j_0a$. Note that Fourier components $B_n$ with $n\geq 3$ raise the energy and therefore vanish. This means that the junction energy is at the end a function only of $B_1$, $E_\Box=(\lambda_1/4\mu_0)\left[(B_1+B_{\rm
eff}\sin\varphi)^2-B_{\rm eff}^2\sin^2\varphi\right]$. The minimum is reached for $B_1=-B_{\rm eff}\sin\varphi$. The minimized value of the junction energy density and the resulting current-phase relation read $$\begin{aligned}
E_\Box(\varphi)&=&-{B_{\rm eff}^2\lambda_1\over 4\mu_0}
\sin^2\varphi,
\nonumber\\
j(\varphi)&=&-{\pi B_{\rm eff}^2\lambda_1\over 2\mu_0\Phi_0}
\sin 2\varphi.
\label{eq:j_millis}\end{aligned}$$ Note that $j\propto j_0^2\propto{\cal D}^2$. Therefore the second harmonic scales with the barrier transparency in the same way both within the flat (for $T\gg{\cal D}\Delta$) and the faceted scenaria. Finally, let us point out that the result Eq. \[eq:j\_millis\] is valid only for $|\chi|\ll 1$, or equivalently for $$B_{\rm eff}\ll{\Phi_0\over 2a\lambda_1}.
\label{eq:crit_millis}$$
2.4. THE JOSEPHSON PRODUCT. According to standard theory (for homogeneous featureless barriers), the Josephson product $I_1R_N$ (where $R_N$ is the junction resistance in the normal state) is independent of the junction area and of the barrier transparency, thus giving an intrinsic information about the superconducting banks. The measured Josephson product of cuprate grain boundary junctions [@Hilgenkamp02] can be well described by $I_1R_N=\alpha^2(\pi/4)(\Delta/e)\cos 2\theta$, where $\Delta$ is the maximal superconducting gap. The angular dependence of $I_1R_N$ is fully consistent (apart from the $\theta$-independent renormalization factor $\alpha^2\sim 10^{-1}$) with the BCS prediction Eq. \[eq:Walker\_rough\] for rough junctions between $d$-wave superconductors. This means that $$(I_c+I_s)R_N=\alpha^2(\pi/2)(\Delta/e),
\label{eq:Jos_prod}$$ much smaller than the theoretical prediction Eq. \[eq:ambegaokar\] for $I_cR_N$. Note that the interpretation of a small factor $\alpha^2$ as a result of a nearly complete cancellation of $I_c$ and $I_s$ is not plausible, since the cancellation would have to occur for all misorientation angles, whereas the physical origin of the $I_c$ and $I_s$ terms is quite different.
It is worth pointing out that Eq. \[eq:Jos\_prod\] applies (with the same $\alpha^2\sim
10^{-1}$) also for the break junctions (in which the misorientation angle $\theta=0$) [@Miyakawa99]. Moreover, in [@Miyakawa99] it has been shown that $\Delta$ is not depressed in the junction region, thus explicitly demonstrating the breakdown of the BCS prediction for $I_1R_N$ in the cuprates.
There exists no generally accepted explanation of the small renormalization factor $\alpha^2$. One of the reasons is that the microstructure of Josephson junctions is typically quite complicated. In fact, it is well known that small angle grain boundaries can be modelled by a sequence of edge dislocations, while at larger misorientation angles the dislocation cores start to overlap and no universal picture applies to the structure of the grain boundary. For large-angle grain boundaries, Halbritter has proposed [@Halbritter92] that the junction can be thought of as a nearly impenetrable barrier with randomly placed highly conductive channels across it. If due to strong Coulomb repulsion only the normal current (and no supercurrent) is supported by these channels, the small value of $I_1R_N$ follows quite naturally.
On the other hand, we have argued [@Hlubina02] that the smallness of the Josephson product does not follow from the particular properties of the barrier, but is rather an intrinsic property of the cuprates. Such a point of view has been first advocated in [@Deutscher99]. However, that paper did not consider alternative more conventional explanations. In order to support our point of view, in [@Hlubina02] we have discussed the Josephson product for intrinsic Josephson junctions in the $c$-axis direction. Such junctions can be viewed as an analogue of $ab$-plane break junctions (since the misorientation angle vanishes for both), but are preferable because of simpler geometry of the interface. Moreover, zero energy surface bound states which may develop at $ab$-plane surfaces because of the $d$-wave symmetry of the pairing state [@Hu94] do not form in the $c$-axis direction, simplifying the analysis of intrinsic Josephson junctions. By an analysis of the experimental data [@Latyshev99], we came to the conclusion that the experimental Josephson product is reduced with respect to theoretical predictions. Thus we have shown that although the barriers in grain boundaries and in intrinsic Josephson junctions are of very different nature, both types of junctions exhibit a suppressed Josephson product. Therefore we concluded that this suppression is not due to specific barrier properties as suggested in [@Halbritter92], but rather due to some intrinsic property of the high-$T_c$ superconductors.
3\. EXPERIMENT
As shown in the previous Section, theory predicts anomalous behavior of the Josephson current in the cuprates as a function of the phase difference $\varphi$ and temperature. One of the most striking predicted phenomena is a non-monotonic temperature dependence of the Josephson critical current, forced by the negative contribution of anomalous Andreev states. However, this has not been observed by standard means in spite of enormous efforts of many groups. The main obstacle seems to originate from the interface roughness which leads to a suppression of the anomalous Andreev states [@Matsumoto95; @Barash96]. Even the most promising type of Josephson junctions, grain boundary junctions, contain defects on a characteristic scale of 1 $\mu$m (Fig. 2 of [@Mannhart96]) coming from defects in the bicrystal substrate, see Fig. \[Fig:Photo\]. A similar problem is relevant also in $c$-axis Josephson junctions, where a large second harmonic of the Josephson current is expected. In that case, in order to avoid tunneling in the $ab$ plane direction and to achieve a high transparency of the interface, atomically flat surfaces are required.
![Schematic picture of the RF SQUID (taken from [@Ilichev01]). The YBCO thin film occupies the gray area. The inset shows an electron microscope image of the narrow grain boundary Josephson junction.[]{data-label="Fig:Photo"}](Photo.eps){width="10cm"}
Thus, in order to observe the predicted anomalies, one should avoid the structural defects at the interface. Following this strategy the $c$-axis Josephson junctions were prepared in situ by covering YBCO by thick Au layer preventing the degradation of the YBCO surface during processing. On the other hand, we have chosen the position of the grain boundary junctions in such a way that they were placed between the defects of the substrate shown in Fig. \[Fig:Photo\]. Therefore we were forced to reduce the size of the junction below 1 $\mu$m. However, for submicron $45^\circ$ junctions where the anomalies should be most pronounced, the Josephson coupling energy $E_J$ is comparable with $T$ in the temperature range of interest. This leads to large phase fluctuations and consequently to a suppression of the Josephson current. At first sight it seems that the Josephson current of submicron $45^\circ$ junctions cannot be measured. But this is not so.
3.1. EXPERIMENTAL METHOD. Phase fluctuations can be suppressed by connecting the Josephson junction into a ring (rf SQUID), since in that geometry the phase difference across the junction is controlled by the large phase stiffness of the superconducting ring. This technique therefore offers a unique possibility to study Josephson junctions at temperatures $T$ much higher than the junction energy. In addition, one can change the phase difference $\varphi$ on the Josephson junction by applying external magnetic flux $\Phi_{dc}$ to the ring $$\varphi=\varphi_{dc}-\frac{2\pi L_s}{\Phi_0}I(\varphi),
\label{Eq:phi}$$ where $\varphi_{dc}=2\pi\Phi_{dc}/\Phi_0$, $L_s$ is the inductance of the rf SQUID, and $I(\varphi )$ is the Josephson current. We have used the modified Rifkin-Deaver method [@Rifkin76; @Ilichev01a] to restore the current-phase dependence $I(\varphi)$. The method is simple: The rf SQUID is inductively coupled to a high-quality parallel resonance circuit driven at its resonant frequency $\omega_0$. The angular phase shift $\alpha$ between the rf driving current $I_{rf}$ and the voltage across the circuit is measured by a rf lock-in voltmeter as a function of the external magnetic flux. The flux is set by a dc current $I_{dc}$ through the coil of the resonant circuit with inductance $L_T$. Thus the total external magnetic flux can be expressed as $\varphi_e=\varphi_{dc}+\varphi_{rf}$, where $\varphi_{dc}=2\pi MI_{dc}/\Phi_0$ and $\varphi_{rf}=2\pi M
I_{rf}/\Phi_0$ and $M$ is the mutual inductance between the rf SQUID and the resonant circuit. The rf SQUID and the resonant circuit are characterized by quality factors $q=\omega_0L_s/R$ and $Q=R_T/\omega_0L_T$, respectively. The analysis of such a system can be considerably simplified in the adiabatic ($q\ll 1$, $Q\gg 1$), small signal regime ($\varphi_{rf}\ll 1+(2\pi
L_s/\Phi_0)dI(\varphi)/d\varphi$), when the rf SQUID follows adiabatically the signal from the resonant circuit and the internal flux can be expressed by Eq. \[Eq:phi\]. $I(\varphi_{dc} )$ is calculated from the experimental $\alpha(\varphi_{dc})$ data using $$I(\varphi_{dc})=\frac{L_TI_0^2}{2\pi
Q\Phi_0}\int_0^{\varphi_{dc}}\tan\alpha
(\varphi_{dc})d\varphi_{dc}, \label{Eq:Iphi}$$ where $I_0$ is the period of $\alpha(I_{dc})$ (i.e. $MI_0=\Phi_0$) and the quality factor $Q$ is measured independently from the width of the resonance curve of the parallel resonant circuit. Using Eqs. \[Eq:phi\],\[Eq:Iphi\] $I(\varphi)$ can be restored. This method, being differential with respect to $\varphi$, provides a high sensitivity of the current phase measurement.
Let us emphasize that the critical current determined by this method does not depend on the inductance of the rf SQUID. In fact, one can easily show that $I_c=I(\varphi_{dc}^0)$, where $\tan\alpha(\varphi^0_{dc})=0$ can be determined from experimental data by requiring $\int_0^\pi\tan\alpha(\varphi_{dc})
d\varphi_{dc}=0$. The last condition comes from the periodicity of $I(\varphi)$ and enables a subtraction of a constant phase shift coming from electronics and cables. Since the only sample characteristics entering Eq. \[Eq:Iphi\] is $I(\varphi)$, the measurement system can be calibrated using samples with known critical current and $I(\varphi)$. We have usually carried out the calibration making use of Nb rf SQUIDs. Typical values of the quantities used in our experimental setup are: $L_s=80$ pH, $L_t=0.27\ \mu$H, $Q=155$. Critical currents measured by the present method and by standard transport measurements were in good coincidence. As already explained, $I(\varphi)$ is measurable even if the thermal energy exceeds the Josephson coupling energy. In fact, critical currents down to 50 nA were recently detected at $T=4.2$ K [@Ilichev01] using cooled preamplifier placed near the parallel resonant circuit. All $I(\varphi)$ measurements presented in this paper have been performed in a gas-flow cryostat with a five-layer magnetic shielding in the temperature range $1.6\le T<90$ K.
3.2. GRAIN BOUNDARY JOSEPHSON JUNCTIONS. The rf SQUIDs were prepared at IPHT Jena by laser deposition of YBCO thin films of thickness 100 nm on bicrystal substrates. They were patterned in the shape of a square washer 3500 $\mu$m$\times
3500~\mu$m with a hole $50$ $\mu$m$\times 50$ $\mu$m by electron beam lithography (see Fig. \[Fig:Photo\]). The estimated Josephson penetration depth $\lambda_J$ is much smaller than the width of the wide junction, $w_l=1725$ $\mu$m, and larger than the width of the narrow junction, $w_s=1\ \mu$m and $w_s=0.7\ \mu$m for asymmetric and symmetric $45^\circ$ junctions, respectively. Thus the behavior of the rf SQUID is dictated by the narrow junction only. For symmetric $45^\circ$ junctions the submicron bridge was formed at a position between the defects of the substrate which are visible in Fig. \[Fig:Photo\]. The experimental results are shown in Fig. \[Fig:Alpha45\]. Local minima appear at low temperatures on the $\alpha (\varphi_{dc})$ curve close to $\varphi_{dc}=2\pi n$ where $n$ is integer. The existence of the local minima dictates $$\left. \frac{d^3I(\varphi)}{d\varphi^3}\right|_{\varphi=2\pi n}>0.
\label{eq:min}$$ Note that neither the conventional tunneling theory nor the I and II theories by Kulik and Omelyanchuk predict such local minima on the derivatives of CPR [@Likharev79].
![The phase angle $\alpha$ as a function of $\varphi_{dc}$ measured at different temperatures. (a) Asymmetric $45^\circ$ grain boundary [@Ilichev99]. Top to bottom curves correspond to $T=$40, 30, 20, 15, 10, and 4.2 K, respectively. (b) Symmetric $45^\circ$ grain boundary [@Ilichev01]. Top to bottom curves correspond to $T=$35, 30, 25, 20, 15, 11, 10, 5, and 1.6 K, respectively.[]{data-label="Fig:Alpha45"}](Alpha45.eps){width="8cm"}
To demonstrate that the anomalous behavior of the Josephson current is a peculiarity of $45^\circ$ junctions we have measured $I(\varphi)$ of symmetric YBCO grain boundary junctions with misorientations $\theta=24^\circ$ and $36^\circ$. The results of this study are shown in Fig. \[Fig:Comp\]. We have found that the critical current decreases exponentially with the misorientation angle $\theta$, $\exp(-\theta/\theta_0)$ with $\theta=5.6^\circ$, in good agreement with [@Hilgenkamp98]. Moreover, both for $\theta=24^\circ$ and $36^\circ$, no considerable deviation from sinusoidal dependence of $I(\varphi)$ was observed, implying tunneling as the dominant transport mechanism.
![Current-phase relation for asymmetric (a) and symmetric (b) $45^\circ$ grain boundary junctions (taken from [@Grajcar02]). (a) Top to bottom curves at $\varphi=0.75\pi$: $T=4.2$, 10, 15, 20, and 30 K. (b) Bottom to top curves at $\varphi=0.375\pi$: $T=1.6$, 5, 10, 11, 15, and 25 K.[]{data-label="Fig:Ip45"}](Ip_SQUID01.eps){width="8cm"}
Let us return to the case of $45^\circ$ junctions now. Theory predicts that the first harmonic should be suppressed for such junctions. If we in addition take into account that the barrier transmission is low, we can model $I(\varphi)$ of such junctions by neglecting terms of higher orders ($n>2$) in Eq. \[eq:Fourier\]. Therefore the condition (\[eq:min\]) for the existence of the local minima at $\varphi_{dc}=2\pi n$ dictates $I_2/I_1 < -1/8$. Thus we conclude that the $45^\circ$ junctions exhibit anomalously large second harmonic of the Josephson current which has opposite sign in comparison with $I_1$. The current-phase relations calculated from the experimental data in Fig. \[Fig:Alpha45\] are shown in Fig. \[Fig:Ip45\]. Note the anomalous form of $I(\varphi)$ at low temperatures. The temperature dependence of the first two harmonics $I_1$ and $I_2$, determined from a Fourier analysis of $I(\varphi)$, is shown in Fig. \[Fig:I1I2\]. For both asymmetric and symmetric $45^\circ$ junctions the second harmonic monotonically increases as the temperature decreases. The first harmonic is more or less constant for asymmetric junctions whereas for symmetric $45^\circ$ junctions the first harmonic starts to exhibit a downturn below 20 K. The most striking result is that for $T$=12 K, $I_1$ changes sign. In the same temperature region where $I_1$ starts to exhibit a downturn, the absolute value of $I_2$ rises from a negligible value at high temperatures to values comparable to $I_1$ at low temperature. This experimental fact suggests a common origin of both phenomena. Similar effects were theoretically predicted for Josephson junctions between $d$-wave superconductors (Section 2) and our experiments can be therefore regarded as an independent experimental test of the $d$-wave symmetry of pairing in YBCO.
![Critical current $I_c$ (triangles) and the first two harmonics $I_1$ (squares) and $I_2$ (circles) as a function of temperature for asymmetric (a) and symmetric (b) grain boundary $45^\circ$ junctions (taken from [@Grajcar02]). The lines are guides to the eye.[]{data-label="Fig:I1I2"}](I1I2_SQUID01.eps){width="8cm"}
3.2. $c$-AXIS JOSEPHSON JUNCTIONS. The $c$-axis Josephson junctions were fabricated and characterized jointly by the IRE Moscow and Chalmers groups following [@Komissinski99]. Epitaxial (001)-oriented YBCO thin films with thickness $150$ nm were obtained by laser deposition on (100) LaAlO$_{3}$ and (100) SrTiO$_{3}$ substrates and [*in situ*]{} covered by a $8\div 20$ nm thick Au layer, thus preventing the degradation of the YBCO surface during processing. Afterwards, $200$ nm thick Nb counterelectrodes were deposited by DC-magnetron sputtering. Junctions with dimensions $10\times 10\ \mu$m$^{2}$ were formed by photolithography and low energy ion milling techniques. The interface resistance per unit area $R_\Box=R_NS$ (where $R_N$ is the normal state resistance and $S$ is the junction area) was $R_\Box=10^{-5}\div 10^{-6}\
\Omega\cdot$cm$^{2}$. Details of the junction fabrication were reported elsewhere [@Komissinski99].
Surface quality of the YBCO films is very important when current transport in the $c$-direction is investigated. High-resolution atomic force microscopy reveals a smooth surface consisting of approximately 100 nm long islands with vertical peak-to-valley distance of $3\div 4$ nm [@Komissinski02]. We can exclude that substantial $ab$-plane tunnel currents flow between YBCO and Nb at the boundaries of these islands. In fact, theory predicts formation of midgap states at the surface of semi-infinite CuO$_2$ planes [@Hu94; @Tanaka95]. Therefore zero bias conductance peaks should be expected in the $I$-$V$ characteristics at temperatures larger than the critical temperature of Nb, if the contribution of $ab$-plane tunneling was nonnegligible. However, no such peaks have been observed for all fabricated Nb/Au/YBCO junctions. Moreover, from the size of the islands and from the vertical peak-to-valley distance we estimate that the area across which $ab$-plane tunneling might take place is only $\approx 6$% of the total junction area. However, since the interface resistances per square are of the same order of magnitude [@Sun96] for both, the $c$-axis and $ab$-plane junctions, we conclude that $ab$-plane tunneling from YBCO, if present, is negligibly small.
More than 20 junctions were characterized by transport measurements. At small voltages typical $I$-$V$ curves can be described by the resistively shunted junction model with a small capacitance [@Likharev79]. Typical critical current densities were $j_{c}=1\div 12$ A/cm$^2$ and $j_{c}R_\Box=10\div 90\ \mu$V. The differential resistance $vs.$ voltage dependence $R_{d}(V)$ exhibits a gap-like structure at $V\approx 1.2$ mV. This structure has a BCS-like temperature dependence and disappears at $T_{cR}\approx 9.1$ K, therefore we ascribe it to the superconducting energy gap of Nb.
![Phase shift $\alpha$ as a function of $\varphi_{dc}$ for a $c$-axis junction at $T$=1.7, 2.5, 3.5, 4.2, and 6.0 K (from bottom to top). Taken from [@Komissinski02].[]{data-label="Fig:Alphac"}](Alpha_epl01.eps){width="10cm"}
The current phase relation of the $c$-axis Josephson junctions was measured by closing the Nb/Au/(001)YBCO heterostructure into a superconducting ring with the same geometry as for $45^\circ$ grain boundary junctions. The experimental results are shown in Figs. \[Fig:Alphac\],\[Fig:Ipc\]. When compared with $45^\circ$ junctions, the second harmonic of the Josephson current was considerably smaller but still anomalously large, leading to local minima of $\alpha(\varphi_{dc})$. As follows from the analysis in Section 2.2, the opposite signs of the first and second harmonics of the Josephson current provide direct evidence that in our YBCO samples, pairing with a macroscopic $d+s$ symmetry is realized. The large first harmonic has to be due to an uncompensated $s$-wave component whose origin is discussed in detail in the next Section.
![The current-phase relation $I(\varphi )$ of the $c$-axis junction from Fig. \[Fig:Alphac\] at $T$=1.7, 2.5, 3.5, 4.2, and 6.0 K (from top to bottom). Inset: Temperature dependence of $I_1$ (squares) and $|I_2|$ (circles). Solid lines are fits to Eqs. (\[eq:j\_1\],\[eq:j\_2\]) using $\Delta_R(T)=\Delta_R(0)\tanh[\Delta_R(T)T_{cR}/\Delta_R(0)T]$. Taken from [@Komissinski02].[]{data-label="Fig:Ipc"}](Ip_epl01.eps){width="10cm"}
4\. DISCUSSION
4.1. GRAIN BOUNDARY JUNCTIONS. [*Asymmetric $45^\circ$ junctions.*]{} The large second harmonic observed in [@Ilichev99] confirms the naive expectations based on the symmetry of the pairing state in the cuprates. However, at the time of writing the paper [@Ilichev99], we were not able to explain the details of the shape of the current-phase relation. In particular, we could not find any mechanism leading to a current-phase relation which was steep in the minima and flat in the maxima of energy. For instance the Kulik-Omelyanchuk theory [@Likharev79] leads to an exactly opposite picture of jumps in $I(\varphi)$, which occur in the maxima of energy (at $\varphi=\pm\pi$). This is a consequence of the fact that for $\varphi=\pm\pi$ two locally stable branches of the junction energy cross, one centered around $\varphi=0$ and another one around $\varphi=\pm 2\pi$. We believe that in the present work we have found one physically plausible scenario for how the pattern of jumps of $I(\varphi)$ observed in [@Ilichev99] can be explained. In fact, Fig. \[Fig:Ipsteep\] shows that behavior qualitatively similar to the results of [@Ilichev99] obtains if the Meissner energy dominates the junction energy Eq. \[eq:F\_asym\]. A more detailed investigation of this idea is under way. A preliminary analysis indicates that in order to be applicable to [@Ilichev99], also the effect of a spatially fluctuating barrier similar to that discussed at the end of Section 4.2 needs to be taken into account.
Although consistent with naive expectations, under closer inspection, the experimentally found large value of the second harmonic remains mysterious:
In the flat scenario, the second harmonic is given by Eq. \[eq:iphi\_asym\]. On the other hand, by combining Eqs. \[eq:Walker\_rough\],\[eq:Jos\_prod\] we estimate that the first harmonic $j_1R_\Box\approx\pi\alpha^2\delta\Delta/e$, where $\delta$ is the deviation of the misorientation angle $\theta$ from the nominal value of $45^\circ$. Taking a realistic value of such a deviation at least $\delta \sim 10^{-2}$, by comparing to Eq. \[eq:iphi\_asym\] we find that the experimental value $j_2/j_1\sim 1$ can be explained, only if $\vartheta_0\sim 1$ (and therefore the tunneling has to be possible in a wide range of impact angles). In this estimate we have made use of the well established [@Hilgenkamp02] value ${\cal D}\sim 10^{-4}$ and we have taken $\Delta/T\sim 10^2$ at helium temperatures. Note that in order to obtain $j_2/j_1\sim 1$ we had to assume that only the first harmonic is renormalized by $\alpha^2\sim 10^{-1}$, while the second harmonic has to remain unrenormalized. This implies very peculiar microscopic physics.
Now let us consider the faceted scenario. As an order of magnitude estimate, we use the result Eq. \[eq:j\_millis\] for the second harmonic. Unlike in the $c$-axis case, in the present case all penetration depths have to be set equal to the in-plane penetration depth of the cuprates $\lambda\approx 0.15 \mu$m. This yields $j_2=(8\pi\Phi_0)^{-1}\mu_0j_0^2a^2\lambda_1$, where $\lambda_1=2\lambda\sqrt{1+2(\pi\lambda/a)^2}$. For $a$ we take the typical length scale of faceting, $a\approx 0.1 \mu$m, since the twin size is even smaller. This yields $\lambda_1\approx 2 \mu$m. The only unknown in the expression for $j_2$ is the local current density $j_0$. In what follows we determine $j_0$ from $j_0=\sqrt{8\pi
j_2\Phi_0/(\mu_0 a^2 \lambda_1)}$ where for $j_2$ we take the experimental critical current density $j_2\sim 10^4$ A/cm$^2$ [@Hilgenkamp02], and we obtain a reasonable value $j_0\sim 10^6$ A/cm$^2$. Note that since $B_{\rm
eff}=(2\pi)^{-1}\mu_0j_0a\approx 3\times 10^{-4}$ T and $\Phi_0/(2a\lambda_1)\approx 5\times 10^{-3}$ T, the criterion Eq. \[eq:crit\_millis\] for the applicability of Eq. \[eq:j\_millis\] is well satified. On the other hand, we estimate the first harmonic from ${j_1/j_0}\approx (a/L)^{1/2}$, which is a random walk-type formula, indicating that $j_1$ averages to zero in a sufficiently long junction. The experiment requires $j_2>j_1$. This is possible only for sufficiently long junctions, $L\sim 1$ mm, whereas in [@Ilichev99] much shorter junctions with $L\sim 1\mu$m were studied. Therefore within the faceted scenario it is not possible to explain the large measured $I_2/I_1$ ratio.
We conclude that both within the flat and faceted scenaria, the second harmonic measured in [@Ilichev99] appears to be anomalously large.
[*Symmetric $45^\circ$ junctions.*]{} The anomalous temperature dependence of the first harmonic and the large second harmonic observed in this type of junctions [@Ilichev01] is qualitatively consistent with the theoretical prediction Eq. \[eq:iphi\_sym2\] and not with Eq. \[eq:iphi\_sym3\]. The data seems to imply (as was the case for asymmetric junctions as well) that the impact angle dependence of the barrier transmission is weak.
In what follows we attempt a more quantitative discussion of the results found in [@Ilichev01]. According to Fig. \[Fig:I1I2\] the first harmonic changes sign at a temperature $T^\ast\approx
12$ K. This together with the estimate $\Delta\approx 20$ meV [@Alff98] requires $D(\pi/4)/D(0)\approx 0.1$, i.e. the barrier, although presumably quite thin, can’t be modelled by a delta function.
Now let us compare the relative magnitudes of the first and second harmonics. Theory predicts that for $T>T^\ast$, the maximal value of $j_1$ is $j_1^{\rm max}/j_L\approx D(0)$. On the other hand, from Eq. \[eq:iphi\_sym2\] it follows that at $T=T^\ast$ the second harmonic $|j_2(T^\ast)|/j_L\approx D(0)^3/24D(\pi/4)$. Therefore according to theory $|j_2(T^\ast)|/j_1^{\rm max}\approx
D(0)^2/24D(\pi/4)\approx D(0)/2$, whereas from Fig. \[Fig:I1I2\] we estimate $|j_2(T^\ast)|/j_1^{\rm max}\approx 1$. Thus theory can describe the experimental results only if $D(0)\sim 1$.
However, in what follows we show that $D(0)\ll 1$ and therefore the experimental second harmonic is again too large when compared with simple minded theory. In fact, in order that higher-order harmonics are negligible [@Ilichev01] even at the lowest studied temperatures $T_{\rm min}\approx$ 1.6 K, we require that the typical mid-gap state energy $\Delta\sqrt{D(\pi/4)/2}<2T_{\rm min}$, yielding $D(\pi/4)<4\times 10^{-4}$, and therefore $D(0)\approx 10
D(\pi/4)<4\times 10^{-3}$. Moreover, $D(0)\sim 10^{-3}$ seems to be consistent with the exponential decrease of $j_c$ with the misorientation angle $\theta$ [@Hilgenkamp98].
In [@Grajcar02] we have proposed that the first harmonic might by suppressed by interface roughness according to Eq. \[eq:Walker\_rough\]. This would require a very small roughness parameter $x\sim 10^{-3}$ and therefore the junction would have to be nearly completely rough. However, we have overlooked the fact that for such junctions it is hard to believe that the deviation $\delta$ from the misorientation angle $\theta=45^\circ$ can be sufficiently small in order to keep the first term in Eq. \[eq:Walker\_rough\] small.
In summary, the second harmonic of $45^\circ$ grain boundary junctions seems to be too large to be explicable by conventional theory.
4.2. $c$-AXIS JUNCTIONS. Let us start by estimating the transparency of the barrier between YBCO and Nb from the normal-state resistance per unit area $R_\Box$. According to the band-structure calculations (for a review, see [@Pickett89]), the hole Fermi surface of YBCO is a slightly warped barrel with an approximately circular in-plane cross-section (to be called Fermi line) with radius $k_F$. In what follows, we represent the electron wavevector $\mathbf{k}$ in cylindrical coordinates, $\mathbf{k}=(k,\theta,k_z)$. We estimate the uncertainty of the in-plane momentum as $\delta k\approx 2\pi/l$, where $l$ is the characteristic size of the islands on the YBCO surface (fig.1). We evaluate $R_\Box$ making use of the Landauer formula and note that only tunneling from a shell around the Fermi line with width $\delta k$ is kinematically allowed. The barrier transparency $D(\theta)$ depends on the details of the $c$-axis charge dynamics in YBCO, with maxima in those directions $\theta$, in which the YBCO $c$-axis Fermi velocity $w(\theta)$ is maximal. Since for $\theta=\pi/4$ and symmetry equivalent directions $w(\theta)$ is minimal [@Xiang96], we expect that there will be 8 maxima of $D(\theta)$ on the YBCO Fermi line where $D(\theta)\approx D$, which are situated at $\theta=\theta_0$ and symmetry equivalent directions. The modulation of the function $D(\theta)$ depends on the thickness of the barrier between YBCO and Nb [@Wolf85]. We consider two limiting distributions of the barrier transparency $D(\theta)$ along the YBCO Fermi line: (a) a featureless $D(\theta)\approx D$ and (b) a strongly peaked $D(\theta)$, roughly corresponding to thin and thick barriers, respectively [@Wolf85]. In the thick barrier limit the angular size of the maxima of $D(\theta)$ can be estimated as $\delta\theta\approx \delta
k/k_F$. With these assumptions we find $$R_\Box^{-1}={\frac{\langle D\rangle e}{\Phi_0}}A,
\label{eq:conductivity}$$ where $A$ measures the number of conduction channels and $\langle\ldots\rangle$ denotes an average over the junction area. In the thin and thick barrier limits, we find $A\approx k_F\delta k/\pi$ and $A\approx 2\delta k^2/\pi$, respectively. Taking $l\approx 100$ nm and $k_F\approx 0.6$ Å$^{-1}$ [@Shen95], the measured $R_\Box=6\times 10^{-5}$ $\Omega$cm$^2$ can be fitted with $\langle
D\rangle_{\mathrm{thin} }\approx 1.7\times 10^{-5}$ and $\langle
D\rangle_{\mathrm{thick}}\approx 8.3\times 10^{-4}$.
Now we can turn to the discussion of $I(\varphi)$. Since we have observed no midgap surface states in the $R_d(V)$ curves, we can neglect the surface roughness, and the Josephson current can be calculated from [@Zaitsev84] $$I(\varphi)={\frac{2e}{\hbar}}\sum_{k,\theta} k_BT\sum_{\omega} {\frac{
D\Delta_R\Delta_{\mathbf{k}}\sin\varphi}{2\Omega_R\Omega_{\mathbf{k}}+D\left[
\omega^2+\Omega_R\Omega_{\mathbf{k}} +\Delta_R\Delta_{\mathbf{k}}\cos\varphi
\right]}},$$ where the sum over $k,\theta$ is taken over the same regions with areas $A$ as in Eq. (\[eq:conductivity\]), $\Delta_R$ and $\Delta_{\mathbf{k}}$ are the Nb and YBCO gaps, respectively, and $\Omega_i=\sqrt{\omega^2+\Delta_i^2}$ with $i=R,\mathbf{k}$. Keeping only terms up to second order in the (small) junction transparency $D$, the first and second harmonics of the Josephson current densities for an untwinned YBCO sample read $$\begin{aligned}
j_0(T)R_\Box&\approx&{\frac{\Delta_s}{\Delta_d^\ast}}{\frac{\Delta_R(T)}{e}},
\label{eq:j_1} \\
j_2(T)R_\Box&\approx&-{\frac{\pi}{8}} {\frac{\langle D^2\rangle}{\langle
D\rangle}} {\frac{\Delta_R(T)}{e}}\tanh\left({\frac{\Delta_R(T)}{2k_BT}}
\right),
\label{eq:j_2}\end{aligned}$$ where $\Delta_d^\ast=\pi\Delta_d[2\ln(3.56\Delta_d/T_{cR})]^{-1}$ and $ \Delta_d^\ast=\Delta_d|\cos 2\theta_0|$ in the thin and thick barrier limits, respectively. In Eqs. (\[eq:j\_1\],\[eq:j\_2\]) we have assumed that the angular variation of the YBCO gap can be described as $\Delta(\theta)=\Delta_s+\Delta_d\cos 2\theta$, where $\Delta_d$ and $\Delta_s$ are the $d$-wave and $s$-wave gaps. We have assumed that $\Delta_d^\ast$ is larger than both, $\Delta_R$ and $\Delta_s$. The factor $\Delta_s/\Delta_d^\ast$ can be estimated from the measured $j_0R_\Box$ products for Josephson junctions between untwinned YBCO single crystals and Pb counterelectrodes. For such junctions $j_0(0)R_\Box\approx 0.5\div 1.6$ mV [@Sun96]. Using the Pb gap $\Delta_R=1.4$ meV in Eq. (\[eq:j\_1\]), we obtain $\Delta_s/\Delta_d^\ast\approx 0.36\div 1.1$.
From the relative sign of $I_{1}$ and $I_{2}$ we know that the finite first harmonic has to be due to the macroscopic $d+s$ symmetry of our YBCO sample. The simplest way how this can be realized is to assume that the numbers of the two types of twins are unequal. In fact, detailed structural studies show that this is the case for sufficiently thin YBCO films even if they are grown on the cubic substrate SrTiO$_{3}$[@Didier97]. If we denote the twin fractions as $(1+\delta )/2$ and $(1-\delta )/2$, then the measured first harmonic of the CPR, $j_{1}$, is proportional to the deviation from equal population of twins, $j_{1}=\delta j_{0}$ [@ODonovan97]. Using $\Delta _{R}=1.2$ meV determined from the $R_{d}(V)$ data and our estimate $\Delta _{s}/\Delta _{d}^{\ast
}\approx 0.36\div 1.1$, we find that the measured first harmonic $j_{1}$ can be fitted with $\delta \approx 0.07\div 0.21$, which is in qualitative agreement with [@Didier97], where $\delta \approx 0.14$ for 1000 Å thick YBCO films has been observed.
Fitting the measured $j_2R_\Box$ by Eq. (\[eq:j\_2\]), $\langle
D^2\rangle/\langle D\rangle\approx 3.2\times 10^{-2}$ was found, which is much larger than both $\langle D\rangle_{\mathrm{thin}}$ and $\langle D\rangle_{\mathrm{thick}}$. Similarly as in the case of grain boundary junctions, this means that the experimental second harmonic is too large when compared with naive estimates. In what follows we discuss the possible causes of such behavior.
Let us start by considering the faceted scenario. From the known values of $j_0R_\Box$ [@Sun96] and $R_\Box$ we estimate $j_0\approx 8\div 27$ A/cm$^2$. Taking for a typical twin size $a\approx 10$ nm, the effective magnetic field from Section 2.3 is estimated as $B_{\rm eff}\approx (1.6\div 5.4)\times
10^{-10}$ T. Moreover, since $\lambda_R\approx 39$ nm, $\lambda\approx
240$ nm, and $\lambda_c\approx 3$ $\mu$m (as an upper bound, for the cuprates we take numbers which are valid for underdoped YBCO [@Cooper94]), we find $\lambda_1\approx 320$ $\mu$m. Since this implies $\Phi_0/(2a\lambda_1)\approx 3\times 10^{-4}$ T, the criterion Eq. \[eq:crit\_millis\] is seen to be well satisfied and therefore the second harmonic can be calculated from Eq. \[eq:j\_millis\], yielding $|j_2|R_\Box\approx (3\div 35)\times 10^{-11}$ V, orders of magnitude smaller than the experimental value $|j_2|R_\Box\approx
15$ $\mu$V.
Thus we were led to look for alternative explanations of the large second harmonic. In [@Komissinski02] we came up with an explanation which assumed that the junction transparency $D$ is a fluctuating function of the position $\mathbf{r}$. Adopting the WKB description of tunneling [@Wolf85], we assumed that the local barrier transparency $D(s(\mathbf{r}))=\exp(-s_0-s(\mathbf{r}))$, where $s_0$ is the WKB tunneling exponent and $s(\mathbf{r})$ its local deviation from the mean. Assuming a Gaussian distribution of $s$ with a mean deviation $\eta$, $P(s)\propto\exp(-s^2/\eta^2)$, we estimated the spatial averages as $\langle
D^n\rangle=\int_{-s_0}^{s_0} ds P(s) D^n(s)$. In the thin barrier limit, the values $\langle D^2\rangle_{\mathrm{thin}}=8.6\times
10^{-7}$ and $\langle D\rangle_{\mathrm{thin}}=2.3\times 10^{-5}$ required to fit the experiments correspond to an average WKB exponent $s_0^{\mathrm{thin}}\approx 15.5$ with $\eta_{\mathrm{thin}}\approx
4.3$. In the thick barrier limit we obtain $s_0^{\mathrm{thick}}\approx 9.1$ and $\eta_{\mathrm{thick}}\approx
2.8$.
4.3. MICROSCOPIC IMPLICATIONS. Very recently it has been pointed out by one of us [@Hlubina02] that the two apparently unrelated experimental facts, namely the suppressed Josephson product $I_1R_N$ and the enhanced ratio $|I_2/I_1|$, can be explained by a single assumption that in the cuprates some mechanism is operative which leads to a suppression of $I_1$, while leaving $R_N$ and $I_2$ intact. In what follows we describe one such mechanism which we believe to be the most promising one. Namely, we suggest that at low temperatures the superconducting state of the cuprates supports fluctuations of the superconducting phase. Such fluctuations presumably do not affect $R_N$, while they do influence the Josephson current. In simplest terms, if we denote the phases of the superconducting grains forming the junction as $\phi_i$, then the fluctuations renormalize the first and second harmonics by the factors $\langle e^{i\phi_1}\rangle\langle e^{i\phi_2}\rangle$ and $\langle
e^{2i\phi_1}\rangle\langle e^{2i\phi_2}\rangle$, respectively, where $\langle\ldots\rangle$ denotes a ground-state expectation value. Thus experiment requires that the fluctuations have to be of such type that $|\langle e^{i\phi}\rangle|=\alpha\approx 0.3$ and $|\langle
e^{2i\phi}\rangle|\approx 1$. Precisely this behavior is expected if the $d$-wave order parameter fluctuates towards $s$-wave pairing (which pairing is expected to be locally stable within several microscopic models of the cuprates).
An independent check of our picture is provided by measurements of the Josephson product for junctions between the cuprates and low-$T_c$ superconductors. In such a case, we predict that the Josephson product should be renormalized by a factor $\alpha$ instead of $\alpha^2$ for junctions between two cuprates. Junctions of this type have been studied extensively in the past. In particular, $ab$-plane junctions between YBCO and Pb exhibit Josephson products of $0.2\div 1.2$ mV [@Sun96]. On the other hand, Eq. \[eq:ambegaokar\] predicts $I_cR_N\approx e^{-1}\Delta_{\rm Pb}\ln(4\Delta_{\rm YBCO}/\Delta_{\rm
Pb})\approx 5.7$ mV in this case, if we assume a (100) YBCO surface and take $\Delta_{\rm YBCO}\approx 20$ meV and $\Delta_{\rm Pb}\approx
1.4$ meV. We interpret the large experimental scatter of $I_cR_N$ as being due to varying $ab$-plane tunneling direction. Thus theory has to be compared with the largest experimental value, and the theoretical result has to be multiplied by a renormalization factor $\alpha\approx 0.2$ in order to bring it in agreement with experiment. This value is in semiquantitative agreement with $\alpha\approx 0.3$ which was determined from the Josephson product of grain boundary junctions. It is worth mentioning that the phase fluctuation picture also may be relevant for the experiment [@Komissinski02], where a large second harmonic has been found in a $c$-axis Josephson junction between YBCO and Nb.
5\. CONCLUSIONS
In this paper we have tried to argue that the Josephson effect can provide nontrivial information not only about the symmetry of the pairing state in the cuprates, but also about the fluctuations of the superconducting order paramater. The latter are expected to be quite large especially in the underdoped region, where charge fluctuations should be suppressed. A preliminary analysis of the Josephson product and of the current-phase relation in grain boundary and $c$-axis junctions indicates [@Hlubina02] that the phase fluctuations of the superconducting order parameter are quite large and of a very special type, favoring local fluctuations from $d$-wave towards extended-$s$ pairing.
Surprisingly, the potential of the Josephson effect as a test of the phase rigidity of the cuprates seems to have been missed in the past and therefore quite few studies have attempted quantitative analysis of the Josephson phenomena. Therefore there are still more questions than answers in this field. Some of the most pressing problems of the field (from our point of view) are listed below.
1\. The nature of the barrier in both grain boundary and $c$-axis junctions is unknown. In the better studied case of grain boundary junctions, we believe that the barrier is in the tunnel limit and for sufficiently large misorientation angles there are no pinholes in it. However, the typical barrier width and height don’t seem to be well known. Therefore also the impact angle dependence of the barrier transmission is not known a priori. On the other hand, in order to explain the current-phase relation of $45^\circ$ grain boundary junctions, we had to assume a weak impact angle dependence. It remains to be seen whether this agrees with the barrier properties determined independently, e.g. making use of the STM microscopy where the junction is viewed from above along a path crossing the grain boundary. With such a method, both the barrier height and width could be measurable.
![Free energy $F(\varphi)$ as a function of the phase difference across the weak link (taken from [@Ilichev01]). The zero of energy has been set so that $F(0)=0$. (a) Asymmetric $45^\circ$ grain boundary. Top to bottom curves correspond to $T=$ 40, 30, 20, 15, 10, and 4.2 K, respectively. (b) Symmetric $45^\circ$ grain boundary. Top to bottom curves correspond to $T=$20, 15, 11, 10, 5, and 1.6 K, respectively.[]{data-label="Fig:Ephi"}](Ep_PRL01.eps){width="10cm"}
2\. The role of various types of disorder in the junction region (interface roughness and faceting, spatially fluctuating barrier height and width, etc.) has to be studied systematically. In particular, disorder is believed to reduce the ability of $45^\circ$ grain boundary junctions to support mid-gap states [@Matsumoto95] and therefore diminishes the second harmonic as well. On the other hand, simple symmetry arguments predict a suppression of the first harmonic of $I(\varphi)$ by the surface roughness. Also an explicit calculation of the first harmonic in the presence of a finite barrier roughness supports this conclusion [@Barash96]. A reliable answer to the question about which of the above two effects of disorder dominates is therefore crucial in order to decide whether disorder increases the ratio of the second and first harmonics of the current-phase relation $|I_2/I_1|$ as suggested in Section 4, or it rather diminishes it. If the latter alternative is realized, then the experimental observation of a large second harmonic provides even stronger argument in favour of anomalous quantum phase fluctuations in the (bulk) cuprates.
3\. Interface roughness and disorder in the barrier region may be also responsible for the weak impact angle dependence of the barrier transmission, which is implied by the large second harmonics observed in $45^\circ$ grain boundary junctions.
4\. From the point of view of applications, especially the grain boundary Josephson junctions have attracted a lot of interest recently. In particular, junctions with sufficiently large second harmonics support doubly degenerate ground states (see Fig. \[Fig:Ephi\]) and it has been suggested [@Ioffe99] that this property might be exploited in the construction of a ‘quiet qubit’, i.e. of a two level system which couples only weakly to its environment and can be used as a basic element in a quantum computer. This is a fascinating proposal and a lot of efforts is being spent on its realization. Several nontrivial problems have to be solved before the final goal can be reached: on the technological side, a technique for a reproducible fabrication of well behaved submicron grain boundary junctions has to be developed. From the point of view of basic science, new routes to reducing the coupling of the junction to the environment are to be looked for, in order to make the junctions really quiet.
ACKNOWLEDGEMENTS
We thank M. H. S. Amin, A. Golubov, H. E. Hoenig, R. P. J. IJsselsteijn, Z. Ivanov, S. Kashiwaya, P. V. Komissinski, M. Yu. Kupriyanov, S. A. Kovtonyuk, H.-G. Meyer, A. N. Omelyanchouk, G. A. Ovsyannikov, V. Schultze, Y. Tanaka, N. Yoshida, A. M. Zagoskin, and V. Zakosarenko for collaborations on solving the problems discussed in this paper. We learned a lot also from discussions with Yu. S. Barash, V. Bezák, M. V. Fistul, H. Hilgenkamp, R. Kleiner, J. Mannhart, R. G. Mints, A. Plecenik, N. Schopohl, M. Sigrist, and A. Y. Tzalenchuk. R. H. and M. G. were supported by the Slovak Scientific Grant Agency under Grant No. VEGA-1/9177/02 and by the Slovak Science and Technology Assistance Agency under Grant No. APVT-51-021602. E. I. was supported by DFG (Ho461/3-1). The support by D-Wave Systems is also acknowledged.
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---
abstract: 'We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\nu=2/(2M+q)$ with integers $M$ and even(odd) integers $q$ for fermionic(bosonic) states. They are termed $Z_2 \times Z_2$ orbifold states, which have a topological order with a neutral sector described by the $c=1$ orbifold conformal field theory (CFT) at radius $R_{\rm orbifold}=\sqrt{p/2}$ with even integers $p$. When $p=2$, the state can be viewed as two decoupled layers of Moore-Read (MR) state, whose neutral sector is described by the Ising $\times$ Ising CFT and contains a $Z_2 \times Z_2$ fusion subalgebra. We demonstrate that orbifold states with $p>2$, also containing a $Z_2 \times Z_2$ fusion algebra, can be obtained by coupling together an array of MR$\times$MR wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the $Z_4$ orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer $p$ and contains a $Z_4$ fusion algebra.'
author:
- Pok Man Tam
- Yichen Hu
- 'Charles L. Kane'
bibliography:
- 'orbifoldQH.bib'
title: 'Coupled Wire Model of $Z_2 \times Z_2$ Orbifold Quantum Hall States'
---
\[sec1\]Introduction
====================
The fractional quantum Hall effect provides a fertile setting for studying topological electronic states with richly structured patterns of quantum entanglement [@01]. The low energy structure of a quantum Hall state is characterized by a 2+1D topological quantum field theory that characterizes the charge and statistics of its elementary quasiparticles [@02; @03; @04; @05; @06; @07]. A closely related 2D conformal field theory (CFT) characterizes the edge states[@08; @09; @10; @11; @20], where the primary fields of the CFT are associated with the quasiparticle types. A simple and explicit connection between the CFT and the TQFT is provided by the coupled wire construction [@8; @9; @3; @1], which takes strips of quantum Hall states with edges described by the 1+1D CFT, and couples them by electron tunneling to form a 2+1D topological phase.
The simplest Abelian fractional quantum Hall states are described by an Abelian 2+1D Chern-Simons theory, and have edge states that can be expressed in terms of free bosons and have an integer chiral central charge $c$ [@09]. The connection to conformal field theory enabled Moore and Read [@10], and later Read and Rezayi [@11] to envision non-Abelian states, which are a subject of intense interest because of their proposed application to quantum computation [@17; @15; @21]. Non-Abelian states feature non-Abelian anyons, along with topological ground state degeneracies. In general, their edge states have non-integer central charge, and can not be described in terms of free bosons compactified on a circle.
In this paper we consider a class of non-Abelian quantum Hall states derived from the CFT of a free boson defined on an orbifold, that is a circle in which angles $\varphi$ and $-\varphi$ are identified. This type of CFT was studied extensively in the 1980’s in the context of string theory [@2; @4; @12]. The free boson on an orbifold of radius $R_{\rm orbifold}$ and the free boson on a circle of radius $R_{\rm circle}$ define a continuous space of $c=1$ CFT’s as illustrated in Fig. \[c=1CFT\]. The two lines parametrizing the orbifold and circle theories intersect at a single point, $U(1)_8$, which can be described as either the circle at $R_{\rm circle} = \sqrt{2}$ or the orbifold at $R_{\rm orbifold} =1/ \sqrt{2}$. The orbifold CFT is almost the same as the circle CFT (which is equivalent to the theory of a conventional Luttinger liquid), except that it contains additional twist operators.
Orbifold quantum Hall states were first studied by Barkeshli and Wen [@5; @18; @19]. There is a sense in which they form the simplest of the non-Abelian states. They possess non-Abelian quasiparticles with associated topological ground state degeneracies, but the edge states have integer central charge $c$. Using effective field theory, slave-particle formulation and the “pattern of zeros" method, Barkeshli and Wen identified quantum Hall states associated with orbifolds with a discrete series of radii, $R_{\rm orbifold} = \sqrt{p/2}$, where $p$ is an integer. At these radii the orbifold CFT’s were fully characterized by Dijkgraaf, Vafa, Verlinde and Verlinde [@2], and it was shown that the two cases in which the integer $p$ is even or odd have different structures to their fusion algebras. The case in which $p$ is odd, which includes the special cases of $U(1)_8$ ($p=1$) and $Z_4$ parafermions ($p=3$), have a quasiparticle with a $Z_4$ fusion algebra, in which a cluster of four identical quasiparticles can fuse to the identity. In contrast, when $p$ is even, which includes the special cases of Ising $\times$ Ising ($p=2$) and 4-state Potts ($p=4$), there is a $Z_2\times Z_2$ structure, which includes two sets of quasiparticles for which a pair of identical quasiparticles fuse to the identity.
In a previous paper [@1], a coupled wire construction was introduced to show that the orbifold states with odd integer $p$ arise naturally in a theory in which electrons cluster into groups of four. These states form a generalization of the $Z_4$ parafermion Read-Rezayi state, whose wavefunction can also be interpreted in terms of the clustering of 4 electrons. However, this wire construction did not describe the states with even integer $p$, and it was clear that the $4e$-clustering was inconsistent with the $Z_2\times Z_2$ fusion algebra of the even-$p$ orbifold theory. If one instead consider the $2e$-clustering of electrons, that is to couple together Luttinger liquids in the paired state, a $Z_2$ fusion algebra can indeed arise. Such a construction has led to the Moore-Read (MR) state as discussed in Ref. [@3].
In this paper, we introduce a coupled wire construction for the even-$p$ orbifold states following the above reasoning. Our starting point, as depicted in Fig. \[wiremodel\], is a set of strips consisting of two layers of identical MR states, which are each described by the Ising TQFT. When the strips are glued together in the trivial way, a bilayer MR state is formed, whose quasiparticle structure in the neutral sector is described by the Ising $\times$ Ising theory, which can be identified with the $p=2$ point of the orbifold theory. We show that, by introducing local interactions within and between the strips, we can construct the sequence of orbifold quantum Hall states described by all even integers $p$. We initiate to name them as the $Z_2 \times Z_2$ orbifold states so as to reflect their fusion structure.
The rest of the paper is organized as follows. In Sec. \[sec3\], we provide a detailed study of a single wire of MR$\times$MR quantum Hall strip, first explaining how the orbifold theory arises in the neutral sector (Sec.\[sec3.1.0\]), as well as the associated fusion algebra that interests us (Sec.\[sec3.1.1\]). We then systematically construct physical operators in our theory and present their bosonized representation (Sec.\[sec3.1.2\]), which immediately allow us to demonstrate how the orbifold radius in a single wire can be tuned by local intra-wire interaction (Sec.\[sec3.2\]). Proceeding with an array of such wires in Sec. \[sec4\], we gap out the bulk charge sector by introducing charge-$2e$ inter-wire tunneling which leads to a bilayer Halperin state for strongly paired electrons. In Sec. \[sec5\], we further introduce charge-$e$ tunneling interaction to gap the bulk neutral sector, which leaves a pair of gapless chiral edges described by the orbifold CFT at $R_{\rm orbifold} = \sqrt{p/2}$ with even integer $p$. The completely gapped bulk would be described by a non-Abelian TQFT as requried by the bulk-boundary correspondence. The quasiparticle spectrum of the even-$p$ orbifold quantum Hall states, which is specific to the construction being presented here, is analyzed in Sec.\[sec7\]. Concluding remarks, together with comparisons to the orbifold states proposed by Barkeshli and Wen, would be given in Sec. \[sec6\].
![[]{data-label="c=1CFT"}](c1space.png){width="9cm" height="6cm"}
\[sec3\]Single Wire
===================
![[]{data-label="wiremodel"}](wiremodel.png){width="9.5cm" height="3.3cm"}
\[sec3.1\]Moore-Read $\times$ Moore-Read
----------------------------------------
A single wire in our model is a quantum Hall strip comprised of two layers (to be labeld by $\uparrow,\downarrow$ respectively) of Moore-Read Pfaffian state at filling factor $\nu_0=1/q$, with underlying electrons being fermionic (bosonic) when integer $q$ is even (odd). A coupled wire construction of a single layer of MR state has been discussed in Ref.[@3]. As is well-known, the edge CFT of Moore-Read Pfaffian state consists of a $c=1$ $U(1)$ chiral boson theory in the charge sector and a $c=1/2$ critical Ising theory in the neutral sector, so that the edge Hamiltonian of a single layer of MR can be written as $$\mathcal{H}_{\text{MR}} = \mathcal{H}^{\rho}_{\text{MR}}+\mathcal{H}^{\sigma}_{\text{MR}}$$ with
$$\begin{aligned}
\mathcal{H}^{\rho}_{\text{MR}} &= \frac{v_\rho}{2\pi} \sum_{r=R,L} (\partial_x \phi^\rho_r)^2, \\
\mathcal{H}^\sigma_{\text{MR}} &= iv_F (\gamma_R \partial_x \gamma_R - \gamma_L \partial_x \gamma_L).\end{aligned}$$
Here $\phi^\rho_{r}$ is the chiral bosonic charge mode, which is compactified on a circle such that $\phi^\rho_{r} \equiv \phi^\rho_{r} +2\pi$, and $\gamma_{R/L}$ is the chiral Majorana neutral mode. With two layers of MR in a single wire, the charge sector is simply composed of two decoupled $U(1)$ chiral boson theories, with the Hamiltonian $$\label{chargeMRHamiltonian}
\begin{split}
\mathcal{H}^{\rho}_j &= \frac{v_\rho}{2\pi} \sum_{a=\uparrow,\downarrow}[(\partial_x \phi^{\rho,a}_{j,R})^2+(\partial_x \phi^{\rho,a}_{j,L})^2]\\
&= \frac{v_\rho}{\pi} \sum_{a=\uparrow,\downarrow}[(\partial_x \varphi^{\rho,a}_{j})^2+(\partial_x \theta^{\rho,a}_{j})^2].
\end{split}$$ Here $j$ labels the wire and $a=\uparrow/\downarrow$ labels the layer. We also introduce the conjugate bosonic variables $\{\varphi,\theta\}$, which are related to the chiral fields by $\phi_{R/L} = \varphi \pm \theta$. The chiral charge fields obey the following commutation relation, $$\label{oldbosoncommutation}
[\phi^{\rho,a}_{j,r}(x),\phi^{\rho,b}_{j',r'}(x')] = iq\pi r\delta_{r,r'}\delta_{j,j'}\delta_{a,b}\text{ sgn}(x-x')$$ with $a,b=\uparrow/\downarrow$ and $r,r'=R/L=+1/-1$.
As for the neutral sector, one may simply view it as two decoupled critical Ising models. Alternatively, we want to take a more interesting perspective, by viewing the Ising $\times$ Ising CFT as a $c=1$ orbifold conformal field theory at radius $R_{\rm orbifold}=1$ [@2; @4]. This perspective motivates a generalization of the Moore-Read $\times$ Moore-Read quantum Hall state to a family of $Z_2 \times Z_2$ orbifold states, as we will explain below.
### \[sec3.1.0\] Orbifold theory in neutral sector
Combining two Majorana fermions into a Dirac fermion such that $\psi_{j,r} = \gamma^\uparrow_{j,r}+i\gamma^\downarrow_{j,r}$ (with $r=R,L$), the neutral sector Hamiltonian of the bilayer system can be written as $$\label{majoranahamiltonian}
\begin{split}
\mathcal{H}^\sigma_j &= iv_F \sum_{\substack{a=\uparrow,\downarrow}} (\gamma^a_{j,R} \partial_x \gamma^a_{j,R}-\gamma^a_{j,L} \partial_x \gamma^a_{j,L}) \\
& = iv_F (\psi_{j,R}^\dagger \partial_x \psi_{j,R}-\psi_{j,L}^\dagger \partial_x \psi_{j,L}).
\end{split}$$ It is important to note that, since the Ising model is dual to a Majorana fermion coupled to a dynamical $Z_2$ gauge field [@7], we have to take into account the effects of two copies of $Z_2$ gauge symmetry here, which leads to redundant labeling of physical states. To explore the consequences, we bosonize the complex fermion as $$\psi_{j,r} = \gamma^{\uparrow}_{j,r} + i\gamma^{\downarrow}_{j,r} \sim e^{i\phi^{\sigma}_{j,r}}.$$ where $\phi^\sigma_{j,r}$ is the chiral bosonic neutral mode. Applying $Z_2$ symmetry on both layers, which identifies $\psi_{j,r}$ with $-\psi_{j,r}$, we obtain the shift symmetry $$\label{radius1/2}
\phi^\sigma_{j,r} \mapsto \phi^\sigma_{j,r} + \pi.$$ Applying $Z_2$ symmetry on a single layer, which identifies $\psi_{j,r}$ with $\psi_{j,r}^\dagger$, we obtain the orbifold symmetry $$\phi^\sigma_{j,r} \mapsto -\phi^\sigma_{j,r}.$$ Equivalently, the conjugate bosonic fields have the following symmetry transformations,
\[bosonsym\] $$\begin{aligned}
&\varphi^\sigma_{j} \mapsto \varphi^\sigma_{j} +\pi, \quad\theta^\sigma_{j} \mapsto \theta^\sigma_{j}+2\pi.\\
&\varphi^\sigma_{j} \mapsto -\varphi^\sigma_{j}, \quad\quad\theta^\sigma_{j} \mapsto -\theta^\sigma_{j}.
\end{aligned}$$
In contrast to the $\pi$-shift symmetry of $\varphi^\sigma$ as effected by the $Z_2$ gauge symmetry, $\theta^\sigma$ retains its usual $2\pi$-shift symmetry since it is not affected in (\[radius1/2\]). The above transformation thus defines an orbifold theory of circle with radius $R_{\rm orbifold}=1$ [@13]. The corresponding Hamiltonian, bosonized from (\[majoranahamiltonian\]), takes the form $$\label{neutralMRHamiltonian}
\begin{split}
\mathcal{H}^{\sigma}_j &= \frac{v_\sigma}{2\pi}[(\partial_x \phi^{\sigma}_{j,R})^2+(\partial_x \phi^{\sigma}_{j,L})^2]\\
&= \frac{v_\sigma}{\pi}[(\partial_x \varphi^{\sigma}_{j})^2+(\partial_x \theta^{\sigma}_{j})^2].
\end{split}$$
It will be useful for latter discussion to mention that the chiral neutral fields are defined to obey the following commutation relation, $$\label{sigmacommutation}
[\phi^{\sigma}_{j,r}(x),\phi^\sigma_{j',r'}(x')] = i\pi \text{ sgn}(x_{j,r} - x_{j',r'}).$$ Particularly, the coordinates here are ordered in a “raster pattern", in which $... < x_{j,L} < x_{j,R} < x_{j+1,L} < x_{j+1,R} < ...$, by defining $x_{j,R}= L+x+2Lj$ and $x_{j,L} = L-x+2Lj$. Doing so, we can always ensure fermions on different wires to anticommute without the use of Klein factors.
To sum up, a single MR $\times$ MR wire is described by the following Hamiltonian $$\mathcal{H}_j = \mathcal{H}^{\rho}_{j}+\mathcal{H}^{\sigma}_{j}$$ with the charge and neutral sectors defined in (\[chargeMRHamiltonian\]) and (\[neutralMRHamiltonian\]) respectively. This is the starting point of our coupled wire construction. We will describe below the bosonization of local operators, such as the creation/annihilation operator for edge electrons and operators that scatter quasiparticles within the MR layer, so as to prepare ourselves for the study of allowed interactions between wires. But before that, let us first investigate into the orbifold theory just established. After all, our goal is to construct quantum Hall states whose neutral edge modes are described by such theories. The orbifold theories have non-trivial operator product structures, which translate into exotic fusions of non-Abelian quasiparticles in the bulk.
### \[sec3.1.1\]$Z_2 \times Z_2\;$ fusion algebra
Let us first understand better our starting point, where there are two decoupled Ising models in the neutral sector. For a single Ising model, we denote the vacuum, Majorana fermion and Ising spin operator as $\textbf{1}, \gamma$ and $\sigma$ respectively. The Ising $\times$ Ising primary fields are summarized in Table \[Ising2\]. The table also shows a correspondence, at the operator level, between Ising $\times$ Ising model and the $c=1$ orbifold conformal field theory at radius $R_{\rm orbifold}=1$. The latter model at a generic radius $R_{\rm orbifold}=\sqrt{p/2}$ ($p\in \mathbb{Z}$), also known as the $Z_2$ orbifold of $Z_{2p}$ Gaussian model, is first studied by Dijkgraaff *et al.* in Ref.[@2].
Notation in Ref.[@2] Dimension Operator ($p=2$)
------------------------- ----------- ----------------------------------------------------------------------------------------
$1$ $0$ $\textbf{1}^{\uparrow} \textbf{1}^{\downarrow}$
$j$ $1$ $\gamma^{\uparrow} \gamma^{\downarrow}$
$\phi^i_p\;(i=1,2)$ $p/4$ $\textbf{1}^{\uparrow} \gamma^{\downarrow},\;\gamma^{\uparrow}\textbf{1}^{\downarrow}$
$\phi_k\;(k=1,...,p-1)$ $k^2/4p$ $\sigma^{\uparrow}\sigma^{\downarrow}$
$\sigma_i\;(i=1,2)$ $1/16$ $\textbf{1}^{\uparrow} \sigma^{\downarrow},\;\sigma^{\uparrow}\textbf{1}^{\downarrow}$
$\tau_i\;(i=1,2)$ $9/16$ $\gamma^{\uparrow} \sigma^{\downarrow},\;\sigma^{\uparrow}\gamma^{\downarrow}$
: Primary fields of the orbifold conformal field theory at radius $R_{\rm orbifold}=\sqrt{p/2}$. The $p=2$ case, which corresponds to two decoupled Ising models in the neutral sector of MR $\times$ MR, is illustrated explicitly on the right.[]{data-label="Ising2"}
Following the well-known Ising fusion rules, we can easily check for $p=2$ that
\[Z2fusion\] $$\begin{aligned}
j \times j &=1, \\
\phi^i_p \times \phi^i_p &=1, \\
\phi^1_p \times \phi^2_p &= j,
$$
with $i=1,2$. Therefore, the operators $\{1, j, \phi^1_p, \phi^2_p\}$ form a $Z_2 \times Z_2$ fusion algebra. This turns out to be a general property of the $Z_2$ orbifold of $Z_{2p}$ Gaussian model for all even integers $p$. The non-Abelian nature of the theory can be seen by fusing the twist fields, which generate all the vertex operators through
\[nonAbelianfusion\] $$\begin{aligned}
\sigma_i \times \sigma_i &= 1+\phi^i_p+\sum_{\mathclap{k\in \text{ even}}} \phi_k,\\
\sigma_1 \times \sigma_2 &= \sum_{\mathclap{k\in \text{ odd}}} \phi_k,\\
j \times \sigma_i &= \tau_i.\end{aligned}$$
For a more elaborate discussion on the fusion algebra, we refer the interested readers to the standard reference [@2].
A family of non-Abelian fractional quantum Hall states with such edge theories are first proposed by Barkeshli and Wen from the effective field theory perspective, and are termed the orbifold quantum Hall states [@5]. In this paper, we will present a coupled wire construction for these states, so as to provide support to their (theoretical) existence at a more microscopic level. Based on the general property of fusion algebra in this family, we initiate to name them as the $Z_2 \times Z_2$ orbifold states, in order to distinguish them from the $Z_4$ orbifold states, which are associated to $Z_2$ orbifold of $Z_{2p}$ Gaussian model for odd integer $p$ and have been constructed in a coupled wire model already [@1].
### \[sec3.1.2\]Local operators and their bosonization
In order to couple together an array of MR $\times$ MR wires to create a gapped bulk, and for the gapless edges to be described by an orbifold theory at an arbitrary even integer $p$, we have to turn on interactions within and between wires. In principle, the allowed interactions (also known as local interactions) are those that can be written as products of fundamental electron operators. In a quantum Hall state, these include operators that create/annihilate edge electrons and those that scatter quasiparticles from one edge to another. The quasiparticle spectrum of the Moore-Read Pfaffian state is well-known [@10] and summarized in Table \[MRspectrum\]. This table will guide us to construct the useful local operators for our coupled wire construction.
We make use of two types of local operators: (1) ones that create/annihilate edge electrons, and change the charge on the edge by an integer; (2) ones that scatter fractionally charged quasiparticles from one edge to another. For the first case, we have the local electron operator, $$\label{edgeelectron}
\Psi_{e,r}^a \sim \gamma^a_r e^{i\phi^{\rho,a}_r}$$ with $a=\uparrow/\downarrow$ and $r=R/L=+1/-1$ (for simplicity, we suppress the wire label in this subsection, as all operators introduced here correspond to the same wire). As discussed earlier in Sec. \[sec3.1.0\], by combining Majorana fermions from different layers into a Dirac fermion we can bosonize as follows,
$$\begin{aligned}
\gamma^\uparrow_{r} &\sim \cos\phi^\sigma_{r} = \cos(\varphi^\sigma+r\theta^\sigma),\\
\gamma^\downarrow_{r} &\sim \sin\phi^\sigma_{r} = \sin(\varphi^\sigma+r\theta^\sigma),\end{aligned}$$
where the anticommutation of fermions is ensured by (\[sigmacommutation\]). Applying (\[edgeelectron\]) twice, one can annihilate a charge-$2e$ pair locally. Due to the fusion $\gamma \times \gamma =\textbf{1}$, the corresponding operator is trivial in the neutral sector, $$\label{bare2e}
\Phi_{2e,r}^a \sim e^{2i\phi^{\rho,a}_r}.$$
0 $e/4$ $e/2$ $3e/4$ $e$
---------- ----------- --------- --------- --------- -----------
**1** $\Circle$ $\odot$ $\odot$
$\sigma$ $\odot$ $\odot$
$\gamma$ $\odot$ $\odot$ $\CIRCLE$
: Particle spectrum of the Moore-Read Pfaffian state at filling factor $\nu_0=1/2$. The symbols $\odot$ indicate existing quasiparticles, $\CIRCLE$ indicates the electron and $\Circle$ indicates the vacuum. The generalized Moore-Read state at filling factor $\nu_0=1/q$ has a similar spectrum, which particularly contains the charge-$e/q$ Abelian quasiparticle and non-Abelian $\sigma$-particle of charge-$e/2q$. Local operators can be constructed by considering scattering processes of the above particles.[]{data-label="MRspectrum"}
Next, we look at local operators that scatter quasiparticles in the MR state at $\nu_0=1/q$. One that is trivial in the neutral sector describes backscattering of charge-$e/q$ Abelian quasiparticles, $$\mathcal{V}_{1}^a\sim (e^{\frac{i}{q}\phi^{\rho,a}_L})^\dagger e^{\frac{i}{q}\phi^{\rho,a}_R} \sim e^{\frac{2i}{q}\theta^{\rho,a}}.$$ By repeatedly applying $\mathcal{V}^{\uparrow/\downarrow}_{1}$, we further obtain $$\label{e/2scattering}
(\mathcal{V}_{1}^{\uparrow})^l (\mathcal{V}_{1}^{\downarrow})^k \sim e^{\frac{2i}{q}(l\theta^{\rho,\uparrow}+k\theta^{\rho,\downarrow})},$$ which is local as long as $l$ and $k$ are integers.
Local operators that are trivial in the charge sector can be constructed by scattering neutral fermions. The following two operators will be of particular importance to our wire construction,
$$\begin{aligned}
\mathcal{V}_{2\gamma} &= i(\gamma^{\uparrow}_R \gamma^{\uparrow}_L + \gamma^{\downarrow}_R \gamma^{\downarrow}_L) \sim \cos 2 \theta^\sigma,\\
\mathcal{V}_{4\gamma} &= \gamma^{\uparrow}_R \gamma^{\downarrow}_R \gamma^{\uparrow}_L \gamma^{\downarrow}_L \sim \partial_x \phi^\sigma_R\;\partial_x \phi^\sigma_L.\end{aligned}$$
Finally, we will also make use of the operator that backscatters a pair of charge-$e/2q$ non-Abelian $\sigma$-particles (one from each layer), which is non-trivial in both the charge and neutral sector, $$\label{4sigma}
\mathcal{V}_{4\sigma} = \sigma^{\uparrow}_{R}\sigma^{\uparrow}_{L}\sigma^{\downarrow}_{R}\sigma^{\downarrow}_{L} e^{\frac{i}{q}(\theta^{\rho,\uparrow}+\theta^{\rho,\downarrow})}.$$ Though it is not straightforward to bosonize the neutral sector, it should be noted that $\sigma^{\uparrow}_{R}\sigma^{\uparrow}_{L}\sigma^{\downarrow}_{R}\sigma^{\downarrow}_{L}$ acts as a Jordan-Wigner string to the Dirac fermion $\psi_{R/L}$[@3]. Thus its bosonized representation should contain only $e^{in\theta^\sigma}$ with odd integers $n$. In addition, for the local scattering interaction to be invariant under the orbifold symmetry $\theta^\sigma \rightarrow -\theta^\sigma$, the bosonized representation must be a linear combination of $\cos n \theta^{\sigma}$. Therefore, we can write $$\label{4sigmabosonized}
\mathcal{V}_{4\sigma} \sim \Big(\sum_{n\in \text{ odd}} B_n\cos n\theta^{\sigma}\Big)e^{\frac{i}{q}(\theta^{\rho,\uparrow}+\theta^{\rho,\downarrow})}.$$ The coefficients $B_n$ assume generic values, as the scattering of $\sigma$-particles can always be accompanied by scatterings of neutral fermions, through the $\mathcal{V}_{2\gamma}$ term.
\[sec3.2\]Changing the orbifold radius
--------------------------------------
With local operators and their bosonized representations established for a single MR $\times$ MR wire, we now demonstrate how the orbifold radius in the neutral sector can be adjusted, so as to obtain generalizations of the MR $\times$ MR quantum Hall states. We include the forward-scattering of Majorana fermions $\mathcal{V}_{4\gamma}$, so that the neutral sector Hamiltonian becomes $$\bar{\mathcal{H}}^{\sigma}_j = \frac{v_\sigma}{2\pi}[(\partial_x \phi^{\sigma}_{j,R})^2+2\kappa(\partial_x \phi^\sigma_{j,R})(\partial_x \phi^\sigma_{j,L})+ (\partial_x \phi^{\sigma}_{j,L})^2].$$ Adjusting the coupling $\kappa$ such that $p=2\sqrt{(1-\kappa)/(1+\kappa)}$ and introducing a new set of chiral fields,
\[neutralchiralfields\] $$\begin{aligned}
\bar{\phi}^{\sigma}_{j,R} &= \sqrt{\frac{2}{p}}\varphi^\sigma_j + \sqrt{\frac{p}{2}}\theta^\sigma_j,\\
\bar{\phi}^{\sigma}_{j,L} &= \sqrt{\frac{2}{p}}\varphi^\sigma_j - \sqrt{\frac{p}{2}}\theta^\sigma_j,\end{aligned}$$
the Hamiltonian can then be re-diagonalized as $$\label{neutralpHamiltonian}
\begin{split}
\bar{\mathcal{H}}^{\sigma}_j &= \frac{\bar{v}_\sigma}{2\pi}[(\partial_x \bar{\phi}^{\sigma}_{j,R})^2+(\partial_x \bar{\phi}^{\sigma}_{j,L})^2]\\
&=\frac{\bar{v}_\sigma}{\pi}[\frac{2}{p}(\partial_x \varphi^{\sigma}_{j})^2+\frac{p}{2}(\partial_x \theta^{\sigma}_{j})^2]
\end{split}$$ with $\bar{v}_\sigma = v_\sigma \sqrt{1-\kappa^2}$. Comparing with the $R_{\rm orbifold}=1$ theory in (\[neutralMRHamiltonian\]), it shows that the orbifold radius has been rescaled to $R_{\rm orbifold}=\sqrt{p/2}$. Therefore, starting from a MR $\times$ MR wire and turning on the local interaction $\mathcal{V}_{4\gamma}$, $R_{\rm orbifold}$ can indeed be adjusted, allowing the theory to move along the orbifold line as depicted in Fig. \[c=1CFT\].
At this stage, one can in principle choose any value for $\kappa$ and then obtain an orbifold theory with an arbitrary radius. However, as we want to construct quantum Hall states with a gapped bulk, locality of inter-wire interaction would require $p/2$ to be an integer. This will be clear when we engineer inter-wire couplings to gap the neutral sector in Sec. \[sec5\]. But before dealing with the more subtle issues there, let us first gap out the charge sector for an array of MR $\times$ MR wires, from which we can determine the sequence of filling fractions where these orbifold quantum Hall states would arise.\
![[]{data-label="2etunnelterm"}](2etunnelterm.png){width="9cm" height="5cm"}
\[sec4\]Coupled Wires: Charge Sector
====================================
We will now consider an array of MR $\times$ MR wires (Fig. \[wiremodel\]) coupled together by local interactions of electron tunneling and quasiparticle scattering. In this section we focus on the two charge sectors (labeled by $\uparrow,\downarrow)$, and illustrate how they can be gapped by inter-wire tunneling of pairs of electrons, accompanied by intra-wire scattering of charge-$e/q$ Abelian quasiparticles. We begin by explicitly constructing such operators.\
\[sec4.1\]Charge-$2e$ tunneling operator
----------------------------------------
Combining the bare charge-$2e$ operator (\[bare2e\]) with backscatterings of charge-$e/q$ Abelian quasiparticles (\[e/2scattering\]), we can define a local composite charge-$2e$ operator (on the $j$-th wire) for each layer $$\begin{split}
\Phi_{j,l k}^{\rho,\uparrow} &\sim e^{2i(\varphi^{\rho,\uparrow}_{j}+\frac{l}{q}\theta^{\rho,\uparrow}_j+\frac{k}{q}\theta^{\rho,\downarrow}_j)},\\
\Phi_{j,l k}^{\rho,\downarrow} &\sim e^{2i(\varphi^{\rho,\downarrow}_{j}+\frac{l}{q}\theta^{\rho,\downarrow}_j+\frac{k}{q}\theta^{\rho,\uparrow}_j)},
\end{split}$$ where $l,k \in \mathbb{Z}$. Here $l$ and $k$ are integers to ensure locality, as the number of scattering of Abelian quasiparticles in each layer is restricted to be an integer. To tunnel charge-$2e$ particles through the top or bottom layer, we consider the following terms $$\mathcal{H}_{2,j+1/2}^{\uparrow/\downarrow} \sim ( \Phi_{j+1,-l -k}^{\rho,\uparrow/\downarrow})^{\dagger} \Phi_{j,l k}^{\rho,\uparrow/\downarrow}+h.c.$$ where for simplicity $l, k$ are chosen to be the same for both layers, so that our construction is symmetric under layer-exchange. The tunneling and scattering processes involved in the above operator are shown explicitly in Fig. \[2etunnelterm\]. Upon bosonization, they can be written as
$$\begin{aligned}
\mathcal{H}_{2,j+1/2}^{\uparrow} &=-t_2^{\uparrow} \cos 2[ \varphi_j^{\rho,\uparrow}-\varphi_{j+1}^{\rho,\uparrow}+\frac{l}{q}(\theta_j^{\rho,\uparrow}+\theta_{j+1}^{\rho,\uparrow})+\frac{k}{q}(\theta_j^{\rho,\downarrow}+\theta_{j+1}^{\rho,\downarrow})],\\
\mathcal{H}_{2,j+1/2}^{\downarrow} &=-t_2^{\downarrow}\cos 2[ \varphi_j^{\rho,\downarrow}-\varphi_{j+1}^{\rho,\downarrow}+\frac{l}{q}(\theta_j^{\rho,\downarrow}+\theta_{j+1}^{\rho,\downarrow})+\frac{k}{q}(\theta_j^{\rho,\uparrow}+\theta_{j+1}^{\rho,\uparrow})].\end{aligned}$$
The above expressions motivate us to introduce a new set of chiral bosonic fields
\[newchargechiral\] $$\begin{aligned}
\bar{\phi}_{j,R}^{\rho,\uparrow} &= 2(\varphi_j^{\rho,\uparrow}+\frac{l}{q}\theta_{j}^{\rho,\uparrow}+\frac{k}{q}\theta_j^{\rho,\downarrow}),\\
\bar{\phi}_{j,L}^{\rho,\uparrow} &= 2(\varphi_j^{\rho,\uparrow}-\frac{l}{q}\theta_{j}^{\rho,\uparrow}-\frac{k}{q}\theta_j^{\rho,\downarrow}),\\
\bar{\phi}_{j,R}^{\rho,\downarrow} &= 2(\varphi_j^{\rho,\downarrow}+\frac{l}{q}\theta_{j}^{\rho,\downarrow}+\frac{k}{q}\theta_j^{\rho,\uparrow}),\\
\bar{\phi}_{j,L}^{\rho,\downarrow} &= 2(\varphi_j^{\rho,\downarrow}-\frac{l}{q}\theta_{j}^{\rho,\downarrow}-\frac{k}{q}\theta_j^{\rho,\uparrow}).\end{aligned}$$
As such, the charge-$2e$ operator becomes $e^{i\bar{\phi}_{j,R}^{\rho,\uparrow/\downarrow}}$ and the tunneling terms can be written as $\cos (\bar{\phi}_{j,R}^{\rho,\uparrow}-\bar{\phi}_{j+1,L}^{\rho,\uparrow})$ and $\cos (\bar{\phi}_{j,R}^{\rho,\downarrow}-\bar{\phi}_{j+1,L}^{\rho,\downarrow})$ respectively. Following the spirit of wire construction, we anticipate a gapped bulk when these tunnelings flow to strong coupling. According to (\[oldbosoncommutation\]), the new chiral fields obey [@6] $$\label{Kmatrix1}
[\partial_x \bar{\phi}^{\rho,a}_{j,r}(x), \bar{\phi}^{\rho,b}_{j',r'}(x')] = 2\pi i rK_{ab} \delta_{r,r'}\delta_{j,j'}\delta_{x,x'}$$ where $a,b=\uparrow/\downarrow$, $r=R/L=+1/-1$ and the $K$-matrix is $$\label{Kmatrix2}
K_{ab} = 4\begin{pmatrix}
\;\; l \; & \;k \;\;\; \\ \;\;k\; & \;l\;\;\;
\end{pmatrix}.$$ The charge density on the $j$-th wire is $$\label{Kmatrix3}
\rho_j =\frac{1}{2\pi} \sum_{a=\uparrow,\downarrow} t_a \partial_x(\bar{\phi}^{\rho,a}_{j,R}-\bar{\phi}^{\rho,a}_{j,L}),$$ with $$\label{Kmatrix4}
t_a = \frac{1}{2(l+k)} \begin{pmatrix}
1 \\ 1
\end{pmatrix}.$$
For an array of wires, it is more convenient to consider variables associated to the links. For the $(j+1/2)$-link that connects the $j$-th and the $(j+1)$-th wire, we define
$$\begin{aligned}
\tilde{\theta}_{j+1/2}^{\rho,a} &= (\bar{\phi}^{\rho,a}_{j,R} - \bar{\phi}^{\rho,a}_{j+1,L})/2,\\
\tilde{\varphi}_{j+1/2}^{\rho,a} &= (\bar{\phi}^{\rho,a}_{j,R} + \bar{\phi}^{\rho,a}_{j+1,L})/2,\end{aligned}$$
which satisfy $$\begin{split}
[\tilde{\theta}_{j+1/2}^{\rho,a}, \tilde{\theta}_{j'+1/2}^{\rho,b}] =[\tilde{\varphi}_{j+1/2}^{\rho,a} , \tilde{\varphi}_{j'+1/2}^{\rho,b} ]=0, \\
[\partial_x \tilde{\theta}_{j+1/2}^{\rho,a}, \tilde{\varphi}_{j'+1/2}^{\rho,b}] = i\pi K_{ab} \delta_{j,j'}\delta_{x,x'}.
\end{split}$$ In terms of these link variables, $2e$-tunneling terms are expressed as $$\label{2etunnel}
\mathcal{H}^{a}_{2, j+1/2}= -t_2^{a} \cos 2\tilde{\theta}^{\rho,a}_{j+1/2}\quad (a=\uparrow,\downarrow).$$ When $t_2^{\uparrow}$ and $t_2^{\downarrow}$ both flow to strong coupling, $\tilde{\theta}^{\rho,\uparrow}$ and $\tilde{\theta}^{\rho,\downarrow}$ will be locked simultaneously (as they commute) at integer multiples of $\pi$, and as such, the two bulk charge sectors will be gapped. Below, we will demonstrate how the tunneling terms can be ensured to flow to strong coupling, by incorporating a carefully designed inter-wire interaction in our model.
\[sec4.2\]Gapping the charge sector
-----------------------------------
Let us begin with the charge sector Hamiltonian in (\[chargeMRHamiltonian\]). We include additional density-density scattering of the form $\partial_x \phi^{\rho,a}_R \partial_x \phi^{\rho,b}_L$, so that the Hamiltonian can be diagonalized by the new set of chiral fields $\bar{\phi}^{\rho,a}_{R/L}$ defined in (\[newchargechiral\]), $$\bar{\mathcal{H}}^{\rho}_j = \frac{\bar{v}_\rho}{2\pi}\sum_{a=\uparrow,\downarrow}[(\partial_x \bar{\phi}_{j,R}^{\rho,a})^2+(\partial_x \bar{\phi}_{j,L}^{\rho,a})^2].$$ Setting up an array of $N$ such wires and before turning on any inter-wire tunneling, the bulk Hamiltonian can be written in terms of link variables as $$\bar{\mathcal{H}}^{\rho}_{0,\text{bulk}} =\frac{\bar{v}_\rho}{\pi} \sum_{j=1}^{N-1} \sum_{a=\uparrow,\downarrow} [(\partial_x\tilde{\theta}^{\rho,a}_{j+1/2})^2+(\partial_x\tilde{\varphi}^{\rho,a}_{j+1/2})^2].$$ At this stage, the charge $2e$-tunnelings described by (\[2etunnel\]) are not guaranteed to gap the charge sector. As one can easily see for the simpler case when $k=0$, the scaling dimension $\Delta_a$ of $\cos 2\tilde{\theta}^{\rho,a}_{j+1/2}$ is $$\Delta_a = 4l,$$ which implies that $t_2^a$ is perturbatively irrelevant for $l>0$.
In order to make $t_2^a$ relevant for generic $l$ and $k$, let us incorporate an inter-wire forward-scattering interaction so that the bulk Hamiltonian becomes $$\label{chargeinterwire}
\begin{split}
\tilde{\mathcal{H}}^{\rho}_{0,\text{bulk}} =\frac{\bar{v}_\rho}{2\pi} \sum_{j=1}^{N-1} \sum_{a=\uparrow,\downarrow}[(\partial_x \bar{\phi}^{\rho,a}_{j,R})^2
&+ 2\lambda_\rho(\partial_x\bar{\phi}^{\rho,a}_{j,R})(\partial_x \bar{\phi}^{\rho,a}_{j+1,L})\\ &+ (\partial_x \bar{\phi}^{\rho,a}_{j+1,L})^2],
\end{split}$$ with $\lambda_\rho$ controlling the strength of forward-scattering. Writing in terms of link variables again, we obtain $$\label{linkHamiltonian}
\tilde{\mathcal{H}}^{\rho}_{0,\text{bulk}} =\frac{\tilde{v}_\rho}{\pi} \sum_{j=1}^{N-1} \sum_{a=\uparrow,\downarrow} [\frac{1}{g_\rho}(\partial_x\tilde{\theta}^{\rho,a}_{j+1/2})^2+g_\rho(\partial_x\tilde{\varphi}^{\rho,a}_{j+1/2})^2],$$ where $g_\rho =\sqrt{(1+\lambda_\rho)/(1-\lambda_\rho)}$ and $\tilde{v}_\rho = \bar{v}_\rho \sqrt{1-\lambda_\rho^2}$. It is then clear that tuning $\lambda_\rho \rightarrow -1^-$ will make $g_\rho \rightarrow 0$ and ensure that $\Delta_{\uparrow/\downarrow} < 2 $ for arbitrary $l$ and $k$. Consequently, $t_2^{\uparrow}$ and $t_2^{\downarrow}$ all flow to strong coupling, and the array of wires with the bulk charge sector described by $$\mathcal{H}^{\rho}_{\text{bulk}} = \tilde{\mathcal{H}}^{\rho}_{0,\text{bulk}} + \sum_{j=1}^{N-1} \sum_{a=\uparrow,\downarrow} \mathcal{H}_{2, j+1/2}^a$$ will be gapped. The link variables $\tilde{\theta}^{\rho\uparrow}$ and $\tilde{\theta}^{\rho\downarrow}$ are then pinned at integer multiples of $\pi$.
In fact, if the neutral sector is gapped in a trivial way, namely the neutral sector in each wire is gapped individually by an intra-wire scattering of Majorana fermions (*i.e.* $\mathcal{V}_{2\gamma}$), the fate of the coupled wire model would be an Abelian bilayer quantum Hall state for strongly-paired electrons, characterized by the $K$-matrix in (\[Kmatrix2\]). Interestingly, as we will discuss in Sec. \[sec5\], there is alternative way to gap out the bulk neutral sector by tunneling a composite electron across neighboring wires, producing the non-Abelian $Z_2 \times Z_2$ orbifold quantum Hall states.
\[sec4.3\]Filling Fraction
--------------------------
By specifying the tunneling operators that gap the charge sector, namely fixing the values of $l$ and $k$, the filling fraction is fixed accordingly. The most direct approach is to apply the $K$-matrix formalism [@14]. Using (\[Kmatrix1\]-\[Kmatrix4\]), we obtain [@6] $$\nu= \textbf{t}^T K\;\textbf{t} = \frac{2}{l+k}.$$ Given the microscopic picture provided by the coupled wire model, the above result can also be obtained from a more physical perspective. As explained below, the filling fraction is determined by requiring tunneling and scattering processes to altogether conserve momentum in the presence of background magnetic fields.
Recall that our starting point is a single wire of MR $\times$ MR bilayer quantum Hall state (each layer at $\nu_0=1/q$), whose implementation already requires certain flux insertion. Let us denote the one-dimensional (along wire) flux density inserted within the wire as $b_{1}$ and that inserted between wires as $b_2$. Then, the change of momentum (with the effect of Lorentz force taken into account) when a right-moving edge electron is scattered to a left-moving one within the wire is $b_1$, and when the electron is tunneled to the next wire the change is $b_2$. Here, the tunneling term required to gap the bulk is $$( \Phi_{j+1,-l -k}^{\rho,\uparrow/\downarrow})^{\dagger} \Phi_{j,l k}^{\rho,\uparrow/\downarrow},$$ which is depicted in Fig. \[2etunnelterm\] and consists of the following processes: two right-moving edge electrons from the $j$-th wire are tunneled to the $(j+1)\text{-th}$ wire and become left-moving, and a total of $(k+l-q)$ charge-$e/q$ Abelian quasiparticles are scattered from right to left within each wire. Balancing the change of momenta in the above processes, we obtain $$b_2=\frac{b_1}{q}(k+l-q).$$ The filling fraction of this coupled wire construction is once again found to be $$\label{fillingfraction1}
\nu = \frac{2b_1/q}{b_1+b_2} =\frac{2}{k+l},$$ where we have used the fact that electron number density in a single wire is $b_1/q$ per Moore-Read layer.
Finally, as we shall see in the next section, gapping the neutral sector further demands $l$ and $k$ to have the same (opposite) parity when $q$ is even (odd), so the orbifold states constructed in this paper actually correspond to the following sequence of filling fractions, $$\label{fillingfraction2}
\nu= \frac{2}{2M+q}
$$ with $M$ being an integer. This sequence only depends on the fermionic/bosonic nature of the orbifold state. Particularly, as we shall see next, it is independent of the orbifold radius.
\[sec5\]Coupled Wires: Neutral Sector
=====================================
With the charge sector gapped, we now proceed to gap the neutral sector. We begin with an array of wires whose neutral sector is described by the Hamiltonian in (\[neutralpHamiltonian\]), *i.e.* with the orbifold radius already tuned to $R_{\rm orbifold}=\sqrt{p/2}$ after incorporating appropriate intra-wire scattering of neutral fermions. We have to assume $p/2$ to be an integer in order to construct local operators, to be discussed below, that gap the bulk neutral sector. At the end, we will be left with a gapless chiral neutral mode at each edge characterized by the orbifold theory.
Charge-$e$ tunneling operator
-----------------------------
![[]{data-label="tunnelterm"}](tunnelterm.png){width="9cm" height="5cm"}
To achieve our goal we need to construct a tunneling term of the form $\cos{\sqrt{\frac{p}{2}}(\bar{\phi}^{\sigma}_{j,R}-\bar{\phi}^{\sigma}_{j+1,L})}$, and ensure that this term flows to strong coupling. Depending on the parity of $p/2$, there are two distinct approaches to tunnel a single electron across neighboring wires in order to do this. A schematic of the tunneling process is provided in Fig. \[tunnelterm\]. We first discuss the case when $p/2$ is odd, in which the associated tunneling term is simpler in the sense that intra-wire scattering of non-Abelian $\sigma$-particles is absent.
### p = 2 mod 4
Let us consider tunneling the following composite electron $$\label{oddelectron}
\Psi_{j,r} \sim \Big(\sum_{\mathclap{n\in\text{ even}}} A_n\cos n\theta^{\sigma}_j \Big) \gamma^{\uparrow}_{j,r} e^{i(\varphi^{\rho,\uparrow}_{j}+r\frac{l}{q}\theta^{\rho,\uparrow}_j+r\frac{k}{q}\theta^{\rho,\downarrow}_j)}$$ with $r=R/L=+1/-1$. The composite electron is formed by dressing the bare edge-electron $\Psi_{e,r}^\uparrow$ with $\cos n\theta^{\sigma}$ terms ($n\in 2\mathbb{Z}$), which are generated from the scattering of neutral Majorana fermions $\mathcal{V}_{2\gamma}$, introduced in Sec. \[sec3.1.2\]. These terms are local by themselves without the charge sector. The electron is also dressed by intra-wire scattering of Abelian quasiparticles (\[e/2scattering\]). Then one can check that the composite electron operator is local only when $$\label{parityconstraint1}
\begin{cases}
\text{$l$ is even and $k$ is even,} &\text{for even $q$ (fermionic)}.\\
\text{$l$ is odd and $k$ is even,} &\text{for odd $q$ (bosonic)}.
\end{cases}$$ This constrains the allowed filling fractions from (\[fillingfraction1\]) to (\[fillingfraction2\]).
Let us now examine the tunneling term, by first separating the charge and neutral parts as follows, $$\label{e-tunnel}
\mathcal{H}_{1,j+1/2} = -t_1\Psi_{j+1,L}^{\dagger}\Psi_{j,R} +h.c. \equiv \mathcal{O}^{\sigma}\mathcal{O}^{\rho}+h.c.$$ The charge part is simply $\mathcal{O}^{\rho}= e^{i\tilde{\theta}^{\rho,\uparrow}_{j+1/2}}$. As already discussed in Sec. \[sec4.2\] , after gapping the charge sectors the link variable $\tilde{\theta}^{\rho,\uparrow}_{j+1/2}$ is already locked at integer multiples of $\pi$. We should then treat $\mathcal{O}^{\rho}$ as a pure number and absorb it into the coupling $t_1$. As for the neutral part, upon bosonizing $\gamma^{\uparrow}_r \sim \cos(\varphi^{\sigma}+r\theta^{\sigma})$, we obtain $$\label{oddneutral}
\begin{split}
&\mathcal{O}^{\sigma} =\sum_{\mathclap{\substack{r=R,L \\ n,m\in \text{ odd}}}} A^{r}_{n,m} \cos[\varphi^{\sigma}_j+r\varphi^{\sigma}_{j+1}-n\theta^{\sigma}_j-m\theta^{\sigma}_{j+1}]\\
&= \sum_{\mathclap{\substack{r=R,L \\ n,m\in \text{ even}}}} \tilde{A}^{r}_{n,m} \cos[\varphi^{\sigma}_j+r\varphi^{\sigma}_{j+1}+(\frac{p}{2}-n)\theta^{\sigma}_j+(\frac{p}{2}-m)\theta^{\sigma}_{j+1}].
\end{split}$$ For latter convenience, in the second equality we have shifted the dummy indices $n, m$ by $p/2$. Then the specific term with $r=L=-1$ and $n=m=0$ corresponds to $\cos{\sqrt{\frac{p}{2}}(\bar{\phi}^{\sigma}_{j,R}-\bar{\phi}^{\sigma}_{j+1,L})}$, which is exactly what we need. However, $\mathcal{O}^{\sigma}$ contains many other terms as well which potentially complicate the physics. Fortunately, just as in the charge sector, we can introduce further inter-wire interaction to render $\cos{\sqrt{\frac{p}{2}}(\bar{\phi}^{\sigma}_{j,R}-\bar{\phi}^{\sigma}_{j+1,L})}$ as the only relevant term. Before we demonstrate that, let us also consider the tunneling term appropriate for even $p/2$, which despite being constructed differently, contain a similar expression for $\mathcal{O}^\sigma$.
### p = 0 mod 4
In this case we need to dress the electron operator with $\mathcal{V}_{4\sigma}$, which scatter a charge-$e/2q$ $\sigma$-particle in each layer from the right edge to the left edge. Combining (\[4sigmabosonized\]) with (\[oddelectron\]), we obtain $$\label{evenelectron}
\Psi_{j,r} \sim \Big(\sum_{\mathclap{n\in \text{ odd}}} B_n\cos n\theta^{\sigma}_j\Big) \gamma^{\uparrow}_{j,r} e^{i(\varphi^{\rho,\uparrow}_{j}+r\frac{l}{q}\theta^{\rho,\uparrow}_j+r\frac{k}{q}\theta^{\rho,\downarrow}_j)}.$$ Due to the additional charge part of $\mathcal{V}_{4\sigma}$, the above composite electron operator is local only when $$\label{parityconstrait2}
\begin{cases}
\text{$l$ is odd and $k$ is odd,} &\text{for even $q$ (fermionic)}.\\
\text{$l$ is even and $k$ is odd,} &\text{for odd $q$ (bosonic)}.
\end{cases}$$ Again, this constrains the allowed filling fractions from (\[fillingfraction1\]) to (\[fillingfraction2\]). Thus, all even-$p$ orbifold states share the same sequence of filling fractions which depends only on the fermionic/bosonic nature of the system.
Tunneling the composite electron appropriate for even $p/2$, the charge part of the tunneling term (\[e-tunnel\]) is once again $\mathcal{O}^{\rho}= e^{i\tilde{\theta}^{\rho,\uparrow}_{j+1/2}}$, which can be absorbed into $t_1$ by gapping the charge sectors. What interests us more is the bonsonized representation of the neutral part $$\label{evenneutral}
\begin{split}
&\mathcal{O}^{\sigma} = \sum_{\mathclap{{\substack{r=R,L \\ n,m\in \text{ even}}}}} B^{r}_{n,m} \cos[\varphi^{\sigma}_j+r\varphi^{\sigma}_{j+1}-n\theta^{\sigma}_j-m\theta^{\sigma}_{j+1}]\\
&= \sum_{\mathclap{{\substack{r=R,L \\ n,m\in \text{ even}}}}} \tilde{B}^{r}_{n,m} \cos[\varphi^{\sigma}_j+r\varphi^{\sigma}_{j+1}+(\frac{p}{2}-n)\theta^{\sigma}_j+(\frac{p}{2}-m)\theta^{\sigma}_{j+1}].
\end{split}$$ Once again, the inter-wire tunneling term that we need appears in the second equality above when $r=L=-1$ and $n=m=0$.
Comparing (\[oddneutral\]) and (\[evenneutral\]), we see that although different constructions of the tunneling operator are adopted for different parities of $p/2$, the same bosonized form is reached. As we will see below, the exact values of the coefficients $\tilde{A}$ and $\tilde{B}$ are immaterial to us, so let us summarize the above two cases and express in terms of the neutral chiral fields (\[neutralchiralfields\]) as follows, $$\begin{split}
\mathcal{O}^{\sigma} = \sum_{\substack{r=R,L \\ n,m\in \mathbb{Z}}} &C^{r}_{n,m} \cos\sqrt{\frac{2}{p}}\;[\;n\bar{\phi}^{\sigma}_{j,L} + (\frac{p}{2}-n) \bar{\phi}^{\sigma}_{j,R} \\
&+(r\frac{p}{2}-m)\bar{\phi}^{\sigma}_{j+1,L}+ m\bar{\phi}^{\sigma}_{j+1,R}\;].
\end{split}$$ For the tunneling operator to gap the neutral sector, it is required that $C^{L}_{0,0}$ flows to strong coupling exclusively while all other $C^{r}_{n,m}$ vanish at low energy. This is achieved by an inter-wire scattering to be discussed next.
\[sec5.2\]Gapping the neutral sector
------------------------------------
Consider an inter-wire interaction analogous to the one in (\[chargeinterwire\]) so that the array of wires has the following bulk Hamiltonian in the neutral sector, $$\label{neutralinterwire}
\begin{split}
\tilde{\mathcal{H}}^{\sigma}_{0,\text{bulk}} =\frac{\bar{v}_\sigma}{2\pi} \sum_{j=1}^{N-1}[(\partial_x \bar{\phi}^{\sigma}_{j,R})^2
&+ 2\lambda_\sigma(\partial_x\bar{\phi}^{\sigma}_{j,R})(\partial_x \bar{\phi}^{\sigma}_{j+1,L})\\ &+ (\partial_x \bar{\phi}^{\sigma}_{j+1,L})^2].
\end{split}$$
While one might have noticed that the additional inter-wire term $\lambda_\sigma(\partial_x\bar{\phi}^{\sigma}_{j,R})(\partial_x \bar{\phi}^{\sigma}_{j+1,L})$ is not an allowed local operator, one can actually construct a local operator with this term in the neutral sector by attaching an appropriate charge-sector factor (see Appendix \[secappendixa\]). Upon gapping the charge sector, the charge variable is locked so that it can be absorbed into the coupling constant $\lambda_\sigma$. It is in this sense that (\[neutralinterwire\]) provides a well-defined local Hamiltonian for our analysis.
Proceeding with $\tilde{\mathcal{H}}^{\sigma}_{0,\text{bulk}}$, it can be diagonalized by the following link variables,
$$\begin{aligned}
\widetilde{\theta}^{\sigma}_{j+1/2} &= (\bar{\phi}^{\sigma}_{j,R}-\bar{\phi}^{\sigma}_{j+1,L})/2,\\
\widetilde{\varphi}^{\sigma}_{j+1/2} &=(\bar{\phi}^{\sigma}_{j,R}+\bar{\phi}^{\sigma}_{j+1,L})/2,\end{aligned}$$
which obey $$\begin{split}
[\tilde{\theta}_{j+1/2}^{\sigma}, \tilde{\theta}_{j'+1/2}^{\sigma}] =[\tilde{\varphi}_{j+1/2}^{\sigma} , \tilde{\varphi}_{j'+1/2}^{\sigma} ]=0, \\
[\partial_x \tilde{\theta}_{j+1/2}^{\sigma}, \tilde{\varphi}_{j'+1/2}^{\sigma}] =i\pi \delta_{j,j'}\delta_{x,x'}.
\end{split}$$ Then, analogous to (\[linkHamiltonian\]), the neutral sector Hamiltonian can be written as $$\tilde{\mathcal{H}}^{\sigma}_{0,\text{bulk}} =\frac{\tilde{v}_\sigma}{\pi} \sum_{j=1}^{N-1} [\frac{1}{g_\sigma}(\partial_x\tilde{\theta}^{\sigma}_{j+1/2})^2+g_\sigma(\partial_x\tilde{\varphi}^{\sigma}_{j+1/2})^2],$$ where $g_\sigma =\sqrt{(1+\lambda_\sigma)/(1-\lambda_\sigma)}$ and $\tilde{v}_\sigma = \bar{v}_\sigma \sqrt{1-\lambda_\sigma^2}$. Consequently, by tuning $\lambda_\sigma \rightarrow -1^-$, the scaling dimension of $e^{i n\widetilde{\varphi}^\sigma}$ will diverge, while that of $e^{in\widetilde{\theta}^\sigma}$ will vanish.
Equipped with this observation, let us re-write $\mathcal{O}^{\sigma}$ in terms of the link variables, so that the scaling dimension of each term can be easily deduced. We have
$$\mathcal{O}^{\sigma} = \sum_{\substack{r=R,L \\ n,m\in \mathbb{Z}}} C^{r}_{n,m} \cos\sqrt{\frac{2}{p}}\Big\{n(\tilde{\varphi}^\sigma_{j-1/2}-\tilde{\theta}^\sigma_{j-1/2})+[(1+r)\frac{p}{2}-n-m]\;\tilde{\varphi}^\sigma_{j+1/2}+[(1-r)\frac{p}{2}-n+m]\;\tilde{\theta}^\sigma_{j+1/2}+m(\tilde{\varphi}^\sigma_{j+3/2}+\tilde{\theta}^\sigma_{j+3/2})\Big\}.$$
Though the above expression looks complicated, our observation above suggests that any cosine terms containing $\tilde{\varphi}^{\sigma}$ would be rendered irrelevant at low energy. It is clear that the only relevant term, whose scaling dimension can be engineered to be arbitrarily small, corresponds to $C^{L}_{0,0,}$.
As such, considering the coupled wire model described by $$\mathcal{H}^\sigma_{\text{bulk}} = \tilde{\mathcal{H}}^\sigma_{0,\text{bulk}} + \sum_{j=1}^{N-1} \mathcal{H}_{1, j+1/2}\;,$$ the bulk neutral sector is gapped following the above arguments. $C^{L}_{0,0,}$ flows exclusively to strong coupling and leads to the locking of bulk variables $\tilde{\theta}^\sigma_{j+1/2}$. There are two edge modes, $\bar{\phi}^{\sigma}_{1,L}$ and $\bar{\phi}^\sigma_{N,R}$, being decoupled from the bulk and thus are left fluctuating freely. As we have seen in Sec. \[sec3.2\], these gapless edge modes are described by the orbifold conformal field theory at radius $R_{\rm orbifold}=\sqrt{p/2}$ with even integer $p$. This completes the coupled wire construction for the $Z_2 \times Z_2$ orbifold quantum Hall states.
\[sec7\]Spectrum of Quasiparticles
==================================
In this section, we briefly discuss the structure of quasiparticles in the $Z_2 \times Z_2$ orbifold states just constructed. We are not going to derive from our microscopic theory the topological $\mathcal{S}$ matrix, which already exists in the literature [@2] and does not depend on the exact implementation of the coupled wire model. Instead, we will focus on characterizing quasiparticle sectors by the amount of charge to which one can attach, which is model-dependent. This will determine the conformal dimension $h_\alpha$ of the allowed quasiparticles. Together with the central charge of our model, which is $c=3$ (as there are two charge sectors with $c=1$ each and one neutral sector with $c=1$), we can obtain the topological $\mathcal{T}$ matrix $$\mathcal{T}_{\alpha\beta} = \delta_{\alpha\beta} e^{2\pi i (h_\alpha-c/24)}$$ with $\alpha,\beta$ labeling quasiparticle sectors. The $\mathcal{S}$ and $\mathcal{T}$ matrices together encode all the topological properties, including braiding statistics, of the bulk quasiparticles.
Our reasoning relies on the edge theory, which is described by the chiral charge fields $\bar{\phi}^{\rho,\uparrow/\downarrow}_{R/L}$ and the chiral neutral fields $\bar{\phi}^{\sigma}_{R/L}$. Any allowed quasiparticle operator must be a combination of primary fields in the charge and neutral sectors, so that back-scattering of a quasiparticle is represented by $$\Omega_R^\sigma \Omega_L^\sigma\; e^{i[Q^\uparrow(\bar{\phi}^{\rho,\uparrow}_R-\bar{\phi}^{\rho,\uparrow}_L)+Q^\downarrow(\bar{\phi}^{\rho,\downarrow}_R-\bar{\phi}^{\rho,\downarrow}_L)]},$$ where $2eQ^{\uparrow/\downarrow}$ is the charge carried by the quasiparticle in the $\uparrow/\downarrow - $layer. Our task is to determine what combinations of $\Omega^\sigma$ and $(Q^\uparrow,Q^\downarrow)$ are allowed by locality, or in other words, expressible as a product of local operators of the original MR $\times$ MR state.
Quasiparticles that are trivial in the neutral sector can be obtained by considering the following operator already introduced in Sec. \[sec3.1.2\], $$\begin{split}
(\mathcal{V}_1^{a})^\frac{\chi^a}{2} &\sim e^{\frac{i\chi^a}{q}\theta^{\rho,a}} \\
&= \exp \{\frac{i\chi^a}{4(k^2-l^2)}[k(\bar{\phi}^{\rho,\bar{a}}_R-\bar{\phi}^{\rho,\bar{a}}_L)-l(\bar{\phi}^{\rho,a}_R-\bar{\phi}^{\rho,a}_L)]\}.
\end{split}$$ Here $a=\uparrow/\downarrow$ and $\bar{a}=\downarrow/\uparrow$, and the change of variables in (\[newchargechiral\]) has been used. The above operator is local as long as $\chi^a$ is an integer multiple of 2. In the context of orbifold states, it represents a transfer of quasiparticle from right to left, with net charge (adding up the electric charge contributed by the two charge sectors) $$e\cdot\frac{\chi^a}{2}(\frac{k}{k^2-l^2}-\frac{l}{k^2-l^2})= \frac{e\chi^a}{2(k+l)} = \frac{\nu e}{4}\chi^a .$$ The last equality is obtained using (\[fillingfraction1\]). Such a quasiparticle is trivial in the neutral sector, and can be attached to any quasiparticles without modifying their topological sectors. Motivated by this observation, we express the charge of a generic quasiparticle (in unit of $2e$) as follows, $$\label{layercharge}
(Q^\uparrow,Q^\downarrow) = \frac{\chi^\uparrow}{4(k^2-l^2)}(-l,\;k)+\frac{\chi^\downarrow}{4(k^2-l^2)}(k,\;-l).$$ By defining $\bm{Q} = (Q^\uparrow,Q^\downarrow)^T$ and $\bm{\chi}=(\chi^\uparrow,\chi^\downarrow)^T$, the above expression can be compactly written as $$\bm{Q} = K^{-1}\bm{\chi},$$ which is in the familiar form for a bilayer Halperin state characterized by the $K$-matrix in (\[Kmatrix2\]). As we will see next, the quasiparticle sectors in the $Z_2 \times Z_2$ orbifold states are classified by $\bm{\chi}$ mod 2.
The charge assignment can actually be inferred from the fusion rules in the orbifold theory. According to (\[nonAbelianfusion\]), quasiparticles in the $1,\phi_p^{1,2}$ and $\phi_{k=\rm even}$ sectors should have the same charge, up to modification by quasiparticles with a trivial neutral sector, and thus are all neutral in this sense. Then (\[Z2fusion\]) further suggests that the $j$ sector is neutral as well. The fusion rules also imply that the twisted sectors should carry half the charge of the untwisted sectors. Finally, by requiring that the charge assignment be compatible with the known spectrum of MR $\times$ MR, which is the starting point of our model, we are left with the following classification, $$\begin{alignedat}{3}
(\chi^\uparrow,\chi^\downarrow)&=(0,0) \text{ mod }2,\quad \alpha&&=1,\;j,\;\phi_p^{1,2},\;\phi_{k=\rm even}.\\
(\chi^\uparrow,\chi^\downarrow)&=(1,1) \text{ mod }2,\quad \alpha&&=\phi_{k=\rm odd}.\\
(\chi^\uparrow,\chi^\downarrow)&=(0,1) \text{ mod }2,\quad \alpha&&= \sigma_1,\;\tau_1.\\
(\chi^\uparrow,\chi^\downarrow)&=(1,0) \text{ mod }2,\quad \alpha&&= \sigma_2,\;\tau_2.
\end{alignedat}$$ With this result, the conformal dimension of quasiparticles $h_\alpha$, and subsequently the topological $\mathcal{T}$ matrix, can be determined.
From the experimental perspective, it is also interesting to characterize the spectrum by the net electric charge of quasiparticles, which is easier to measure than the exchange statistics. In terms of the net charge $$Q \equiv 2e(Q^\uparrow+Q^\downarrow) =\frac{\nu e}{4}\chi_{\rm total} ,$$ where we have defined $\chi_{\rm total} \equiv \chi^\uparrow+\chi^\downarrow$, we are left with the following charge spectrum, $$\begin{split}
\chi_{\rm total} &=0 \text{ mod 2}\;:\quad \alpha=1,\;\;j,\;\;\phi_p^{1,2},\;\;\phi_{k}.\\
\chi_{\rm total} &=1 \text{ mod 2}\;: \quad \alpha=\sigma_{1,2},\;\;\tau_{1,2}.
\end{split}$$
\[sec6\]Discussion
==================
In this paper we have presented a microscopic construction for the $Z_2 \times Z_2$ orbifold quantum Hall states by coupling wires of MR$\times$MR-bilayer. These are non-Abelian states with the neutral sector characterized by the $c=1$ orbifold conformal field theory at radius $R_{\rm orbifold} =\sqrt{p/2}$ with even integers $p$. The spectrum of quasiparticles have been studied, where quasiparticles can be classified into two groups according to their electric charge. These orbifold states are shown to exist at the following filling factors $$\nu=\begin{cases}
1/n, &\text{(for fermions)}\\
2/(2n+1), &\text{(for bosons)}
\end{cases}$$ with $n\in\mathbb{Z}$, for all even integers $p$. We would like to point out that the even-$p$ orbifold states constructed here, although motivated by the proposal in Ref.[@5] and thus share certain similarities in places like the fusion rules, are not identical to the orbifold states introduced by Barkeshli and Wen. Particularly, the even-$p$ orbifold states discussed there occur at filling $$\label{BWfilling}
\nu_{\rm BM}=\begin{cases}
1/2n, &\text{(for $p=2$ mod $4$, fermions)}\\
1/(2n+1), &\text{(for $p=0$ mod $4$, fermions)}\\
1/(2n+1), &\text{(for $p=2$ mod $4$, bosons)}\\
1/2n, &\text{(for $p=0$ mod $4$, bosons)}\\
\end{cases}$$ with $n\in\mathbb{Z}$. Therefore, with the underlying electrons being fermionic, orbifold states constructed in this paper actually occur at a more extensive set of filling factors than the ones introduced by Barkeshli and Wen. Another important difference lies in the central charge, as our model has $c=3$ (with two $c=1$ charge sectors and one $c=1$ orbifold neutral sector) while the model of Barkeshli and Wen has $c=2$. The $p=2$ case in our model describes two copies of Moore-Read Pfaffian states, while in their case it is a Pfaffian state with an extra copy of Ising model.
Hence, one possible extension of this work would be to consider a coupled wire model with wires of MR$\times$Ising-bilayer. If the bulk of this model can be gapped by local interactions, one can obtain orbifold states with central charge $c=2$. Following similar arguments as those presented in this paper, one can argue that such orbifold states would occur at filling $\nu=1/l$, where $l$ has to satisfy the parity constraints in either (\[parityconstraint1\]) or (\[parityconstrait2\]) depending on $p$ mod $4$. In this way, one can recover the sequence of filling factors in (\[BWfilling\]), as proposed by Barkeshli and Wen. However, it is a non-trivial task to introduce local interactions that sew together the MR$\times$Ising wires so as to obtain a gapped bulk and gapless edges described by the orbifold conformal field theory. Certain progress can be made by considering a Gross-Neveu-like interaction for two pairs of Majorana fields from consecutive wires. This is a local interaction that can sew an array of Ising wires together and give rise to a gapped two-dimensional bulk with a pair of gapless Ising edges, as demonstrated in Ref.[@16]. It is however unclear how to generalize this procedure to obtain orbifold states for $p>2$. Alternatively, one may consider an explicit implementation of the Ising model, through which local operators, similar to the ones adopted here for intra/inter-wire interactions, can be identified. We will leave this for future work.
The authors would like to thank Ady Stern for helpful discussions. This work is in part supported by the Croucher Scholarship for Doctoral Study from the Croucher Foundation (PMT), grant EP/S020527/1 from EPSRC (YH) and a Simons Investigator grant from the Simons Foundation (CLK).
\[secappendixa\]Repairing the Neutral Sector
============================================
In Sec. \[sec5.2\], we argue about how an inter-wire interaction of the form $\lambda_\sigma(\partial_x\bar{\phi}^{\sigma}_{j,R})(\partial_x \bar{\phi}^{\sigma}_{j+1,L})$ can render only the desired tunneling term $C^{L}_{0,0}$ relevant at low energy and lead to the gapping of neutral sector. However, it is not trivial to see why this term is actually allowed to appear in a Hamiltonian. Here, we will settle this issue by explicitly constructing a local interaction with such a form in the neutral part, and with a charge part that can be condensed by gapping charge sectors.
Let us begin with interactions that can be easily seen to be local, such as
$$\begin{aligned}
\Theta_R &= \Theta^\sigma_R \Theta^\rho= \gamma^\uparrow_{R}\gamma^\downarrow_{R}e^{\frac{i}{2}\bar{\phi}^{\rho, \uparrow}_{R}}e^{\frac{i}{2}\bar{\phi}^{\rho, \downarrow}_{R}},\\
\Theta_L &= \Theta^\sigma_L \Theta^\rho= \gamma^\uparrow_{L}\gamma^\downarrow_{L}e^{\frac{i}{2}\bar{\phi}^{\rho, \uparrow}_{R}}e^{\frac{i}{2}\bar{\phi}^{\rho, \downarrow}_{R}}.\end{aligned}$$
We first focus on operators in a single wire so the wire label $j$ is suppressed for simplicity. From the definition of the charged chiral fields in (\[newchargechiral\]), the charge part of the above operator can be written in two equivalent ways as $$\Theta^\rho \sim
\begin{cases}
e^{i\phi^{\rho,\uparrow}_R}e^{i\phi^{\rho,\downarrow}_R}e^{i\frac{l+k-q}{q}(\theta^{\rho,\uparrow}+\theta^{\rho, \downarrow})}, \quad\text{for }\Theta_R.\\
e^{i\phi^{\rho,\uparrow}_L}e^{i\phi^{\rho,\downarrow}_L}e^{i\frac{l+k+q}{q}(\theta^{\rho,\uparrow}+\theta^{\rho, \downarrow})}, \quad\text{for }\Theta_L.
\end{cases}$$ It is then clear that we can interpret $\Theta_{R/L}$ as annihilation of a MR edge electron and $(l+k\pm q)/2$ scatterings of charge-$e/q$ Abelian quasiparticles in each layer. Following the parity requirements in (\[parityconstraint1\]) or (\[parityconstrait2\]), $(l+k\pm q)/2$ is indeed an integer and hence $\Theta_{R/L}$ is local.
We then bosonize $i\gamma^{\uparrow}_R\gamma^{\downarrow}_R \sim \partial_x \phi^{\sigma}_R$ and $i\gamma^{\uparrow}_L\gamma^{\downarrow}_L \sim \partial_x \phi^{\sigma}_L$. It can be seen from (\[neutralchiralfields\]) that $\bar{\phi}^\sigma_{R}$ is a linear combination of $\phi^{\sigma}_R$ and $\phi^{\sigma}_L$, and hence by taking linear combination of two local operators $\Theta_R$ and $\Theta_L$, we conclude that the following operator is also local, $$\Xi_{R} \sim \partial_x \bar{\phi}^\sigma_R\;e^{\frac{i}{2}\bar{\phi}^{\rho, \uparrow}_{R}}e^{\frac{i}{2}\bar{\phi}^{\rho, \downarrow}_{R}}.$$ Analogously, one can argue that $$\Xi_{L} \sim \partial_x \bar{\phi}^\sigma_L\;e^{\frac{i}{2}\bar{\phi}^{\rho, \uparrow}_{L}}e^{\frac{i}{2}\bar{\phi}^{\rho, \downarrow}_{L}}$$ is an allowed local operator. Now we can put back the wire-labels and consider a product of the above two operators (on successive wires), $$\begin{split}
&\Xi_{j+1, L}^\dagger \Xi_{j,R} + h.c. \\
&\sim (\partial_x \bar{\phi}^\sigma_{j,R})(\partial_x \bar{\phi}^\sigma_{j+1,L}) \cos(\tilde{\theta}^{\rho,\uparrow}_{j+1/2}+\tilde{\theta}^{\rho,\downarrow}_{j+1/2}).
\end{split}$$ Upon gapping the charge sectors, link variables $\tilde{\theta}^{\rho,\uparrow/\downarrow}$ are pinned at integer multiples of $\pi$. The above cosine term then simply takes value of $\pm1$, and hence can be safely discarded. This justifies our argument in the main text, where we treat $(\partial_x \bar{\phi}^\sigma_{j,R})(\partial_x \bar{\phi}^\sigma_{j+1,L})$ as a local operator and allow it to appear in the bulk neutral sector Hamiltonian in (\[neutralinterwire\]).
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abstract: 'The NEMO 3 experiment searches for neutrinoless double beta decay and makes precision measurements of two-neutrino double beta decay in seven isotopes. The latest two-neutrino half-life results are presented, together with the limits on neutrinoless half-lives and the corresponding effective Majorana neutrino masses. Also given are the limits obtained on neutrinoless double beta decay mediated by $R_p$-violating SUSY, right-hand currents and different Majoron emission modes.'
author:
- Irina Nasteva on behalf of the NEMO collaboration
bibliography:
- 'supernemo.bib'
title: |
Neutrinoless double beta decay search\
with the NEMO 3 experiment
---
[ address=[Particle Physics Group, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK]{}, email=[Irina.Nasteva@manchester.ac.uk]{}]{}
Introduction
============
Neutrinoless double beta decay ($0\nu \beta \beta$) is a process in which two neutrons in a nucleus undergo simultaneous beta decays with the emission of two electrons. It violates lepton number and is therefore forbidden in the Standard Model. The observation of neutrinoless double beta decay would prove that neutrinos are Majorana particles and would provide access to their absolute mass scale.
The half-life of $0\nu \beta \beta$ is given by: $$\frac{1}{T^{0\nu}_{1/2}(A,Z)}=|M^{0\nu}(A,Z)|^2 G^{0\nu}(Q,Z) \langle m_{\beta\beta} \rangle ^2\,,
\label{eq1}$$ where $M^{0\nu}(A,Z)$ is the nuclear matrix element (NME) and $G^{0\nu}(Q,Z)$ is a known phase space factor that depends on the transition energy ($Q$-value) of the process. The effective Majorana neutrino mass, $\langle m_{\beta\beta} \rangle$, is a sum over the mass eigenstates, weighted by the squared elements, $U^2_{ei}$, of the PMNS neutrino mixing matrix: $$\langle m_{\beta\beta} \rangle=\sum_{i=1,2,3} U_{ei}^2 m_i \,.
\label{eq2}$$
In addition to light Majorana neutrino exchange (described by ), $0\nu \beta \beta$ decay could be mediated by other physics mechanisms such as a right-hand current admixture in the Lagrangian, Majoron emission or supersymmetric particle exchange. Measuring the energy spectrum and topology of the final state electrons could allow to distinguish between these underlying mechanisms.
The NEMO 3 experiment
=====================
NEMO 3 [@nemo3tdr] (Neutrino Ettore Majorana Observatory) has been taking data since 2003 at the Modane Underground Laboratory in France. The experiment is dedicated to searching for $0\nu \beta \beta$ decay and making precise lifetime measurements of $2\nu \beta \beta$ decay in seven isotopes. The main isotopes used for the $0\nu \beta \beta$ search are 7 kg of $^{100}$Mo and 1 kg of $^{82}$Se. There are smaller amounts of $^{116}$Cd, $^{130}$Te, $^{150}$Nd, $^{96}$Zr and $^{48}$Ca for $2\nu \beta \beta$ studies, and natural Te and Cu for background measurements.
NEMO 3 employs an experimental technique of calorimetry and tracking, in order to detect the two final state electrons. The detector is cylindrical and is radially segmented into 20 equal sectors, each housing a thin source foil placed in the middle of a tracking volume, which is surrounded by the calorimeter. The tracker consists of 6180 drift cells operated in Geiger mode. The calorimeter walls are made up of 1940 plastic scintillator blocks coupled to low-radioactivity PMTs. They achieve energy resolution in the range 14%–17% FWHM for 1 MeV electrons and timing resolution of 250 ps. A solenoid magnetic field of 25 G is applied to provide charge identification. The whole detector is enclosed in a radon-free air tent, and covered by two levels of external radiation shielding.
The NEMO 3 detector measures the individual particle trajectories and energies, thus reconstructing the final state topology and kinematics of the events. Through particle identification of $e^-$, $e^+$, $\alpha$ and $\gamma$ it achieves excellent background suppression, which is further enhanced by the time-of-flight measurement used to reject external particles crossing the detector.
Double beta decay events are selected by requiring two tracks with a negative curvature, originating from a common vertex in the source foil and associated to isolated scintillator energy deposits. The timing of the calorimeter hits is required to agree with the time-of-flight hypothesis of two electrons emitted from the foil at the same time. The remaining backgrounds in the two-electron signal sample are estimated by looking at control channels.
$2\nu\beta\beta$ results
========================
Two-neutrino double beta decay ($2\nu \beta \beta$) is a Standard Model weak interaction process occurring in nuclei for which beta decay is energetically forbidden or strongly suppressed. The importance of understanding the $2\nu\beta\beta$ process is because it forms the irreducible background to $0\nu\beta\beta$ decay. In addition, precise measurements of its half-life and event kinematics are used to constrain the nuclear models used to calculate the neutrinoless NME.
The NEMO 3 experiment has been performing high-statistics measurements of $2\nu\beta\beta$ decay in its seven isotopes. Figures \[ndsum\] and \[zrsum\] show the recent preliminary results for the two-electron energy sum distributions obtained from $^{150}$Nd and $^{96}$Zr, respectively. Table \[all2nu\] summarises the current half-life measurements, along with the isotope characteristics and the signal-to-background ratios (S/B), obtained from all isotopes.
![Energy sum distribution of the two electrons observed in $^{150}$Nd decays.[]{data-label="ndsum"}](esum_Nd_paper_all.pdf){height=".3\textheight"}
![Energy sum distribution of the two electrons observed in $^{96}$Zr decays.[]{data-label="zrsum"}](zr_etot.pdf){height=".23\textheight"}
---------------- ------ ------- ------ ---------------------------------------------------------------
$\rm ^{100}Mo$ 6914 3.034 40 0.711 $\rm\pm$ 0.002 (stat) $\pm$ 0.054 (syst) [@nemoresult1]
$\rm ^{82}Se$ 932 2.995 4 9.6 $\pm$ 0.3 (stat) $\pm$ 1.0 (syst) [@nemoresult1]
$\rm ^{130}Te$ 454 2.529 0.25 76 $\pm$ 15 (stat) $\pm$ 8 (syst) [@ssr]
$\rm ^{116}Cd$ 405 2.805 7.5 2.8 $\pm$ 0.1 (stat) $\pm$ 0.3 (syst) [@ssr]
$\rm ^{150}Nd$ 37.0 3.367 2.8 $0.920 ^{+0.025}_{-0.022} $(stat) $\pm$ 0.073 (syst)
$\rm ^{96}Zr$ 9.4 3.350 1.0 2.3 $\pm$ 0.2 (stat) $\pm$ 0.3 (syst)
$\rm ^{48}Ca$ 7.0 4.272 6.8 $4.4 ^{+0.5}_{-0.4} $(stat) $\pm$ 0.4 (syst)
---------------- ------ ------- ------ ---------------------------------------------------------------
: NEMO 3 results for $2\nu\beta\beta$ half-life measurements for seven isotopes.[]{data-label="all2nu"}
$0\nu \beta \beta$ search
=========================
Light neutrino exchange
-----------------------
![The endpoint of the $^{100}$Mo two-electron energy sum distribution. The line shows a simulation of a neutrinoless signal of half-life $10^{23}$ years.[]{data-label="mosum"}](mo0n.pdf){height=".28\textheight"}
---------------- ---------------------- ---------------------------------------------------
$\rm ^{100}Mo$ $>5.8\times 10^{23}$ $< 0.6-1.3$ [@kort75; @kort76; @Rodin; @Simkovic]
$\rm ^{82}Se$ $>2.1\times 10^{23}$ $<1.2-2.2$ [@kort75; @kort76; @Rodin; @Simkovic]
$\rm ^{150}Nd$ $>1.8\times 10^{22}$ $<1.7-2.4$ [@Rodin]
$<4.8-7.6$ [@Hirsch]
$\rm ^{96}Zr$ $>8.6\times 10^{21}$ $<7.4-20.1$ [@kort75; @kort76; @Rodin; @Simkovic]
$\rm ^{48}Ca$ $>1.3\times 10^{22}$ $<29.6$ [@Caurier]
---------------- ---------------------- ---------------------------------------------------
: NEMO 3 limits at 90% CL on the half-lives of $0\nu\beta\beta$ and the corresponding effective neutrino mass ranges.[]{data-label="all0nu"}
Neutrinoless double beta decay mediated by light Majorana neutrino exchange would lead to a peak at the endpoint energy $Q$ of the two electrons, smeared by the energy resolution of the detector. The endpoint of the energy distribution of $2\nu\beta\beta$ decay for $^{100}$Mo is shown on Fig. \[mosum\]. No excess of events was observed, therefore a lower limit on the $0\nu\beta\beta$ half-life was obtained, $T_{1/2}^{0\nu\beta\beta}>5.8\times 10^{23}$ years (90% CL). This translates into a range of upper limits on the effective Majorana neutrino mass, $\langle m_{\beta\beta} \rangle< 0.6-1.3$ eV, according to the most recent NME calculations [@kort75; @kort76; @Rodin; @Simkovic]. Table \[all0nu\] summarises the half-life and neutrino mass limits obtained by NEMO 3.
The $0\nu\beta\beta$ analysis of $^{100}$Mo and $^{82}$Se has now been blinded. The projected half-life sensitivities when unblinded in 2010 are $2\times 10^{24}$ years for $^{100}$Mo and $8\times 10^{23}$ years for $^{82}$Se, leading to neutrino mass reaches of $0.3-0.7$ eV and $0.6-1.1$ eV, respectively.
SUSY particle exchange
----------------------
Neutrinoless double beta decay could be mediated by the exchange of superparticles in $R_p$-violating SUSY [@rpvsusy]. The half-life of this process is inversely proportional to the SUSY lepton-number violating parameter $\eta^{11}_{(q)LR}$, which is related to the sum of $R_p$-violating trilinear couplings, $\lambda'_{11k}\lambda'_{1k1}$ ($k=1,2,3$). From the $^{100}$Mo half-life limit shown in Table \[all0nu\], a limit was obtained of $\eta^{11}_{(q)LR}<9.2\times 10^{-9}$ that corresponds to upper limits on the trilinear couplings of $\lambda'_{111}\lambda'_{111}<1.7\times10^{-5}$, $\lambda'_{112}\lambda'_{121}<8.7\times10^{-7}$ and $\lambda'_{113}\lambda'_{131}<3.6\times10^{-8}$ [@rpvsusy].
Other exotic mechanisms
-----------------------
--------------- ------------------------------------------------------------------ ----------------------- -- -- --
(V+A) current $>3.2\times {10^{23}} \tablenote{$\lambda < 1.8\times 10^{-6}$}$ $>1.2\times 10^{23}$
$n=1$ $>2.7\times {10^{22}} \tablenote{$g < (0.4-1.8)\times 10^{-4}$}$ $>1.5\times 10^{22} $
$n=2$ $>1.7\times 10^{22}$ $>6.0\times 10^{21}$
$n=3$ $>1.0\times 10^{22}$ $>3.1\times 10^{21}$
$n=7$ $>7.0\times 10^{19}$ $>5.0\times 10^{20}$
--------------- ------------------------------------------------------------------ ----------------------- -- -- --
: \[exotic\] Constraints at 90% CL from NEMO 3 data on the half-lives of exotic processes, and on the (V+A) Lagrangian parameter $\lambda$ and the Majoron to neutrino coupling strength $g$ [@nemoresult2].
Other exotic mechanisms such as right-hand ($V+A$) currents and Majoron emission could also contribute to $0\nu\beta\beta$ decay. They would lead to a distortion of the shape of the two-electron energy sum distribution. A maximum likelihood analysis of the deviation of the energy shape from the calculated $2\nu\beta\beta$ shape was performed [@nemoresult2]. The resulting limits on the half-lives and couplings for ($V+A$) currents and different Majoron spectral indices $n$ are shown in Table \[exotic\].
The SuperNEMO project
=====================
The next-generation $0\nu\beta\beta$ project SuperNEMO aims to extrapolate the successful NEMO 3 experimental technique to a detector with $\sim100$ kg of source isotopes. It will use the technology of calorimetry and tracking in a modular structure, whilst improving critical performance parameters such as energy resolution, acceptance and source purity. SuperNEMO aims to achieve a $0\nu\beta\beta$ half-life sensitivity of $T_{1/2}^{0\nu\beta\beta}>2\times 10^{26}$ years, corresponding to an effective neutrino mass reach of $\langle m_{\beta\beta} \rangle< 0.05-0.1$ eV.
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abstract: 'Assessing the model fit quality of statistical models for network data is an ongoing and under-examined topic in statistical network analysis. Traditional metrics for evaluating model fit on tabular data such as the Bayesian Information Criterion are not suitable for models specialized for network data. We propose a novel self-developed goodness of fit (GOF) measure, the “stratified-sampling cross-validation” (SCV) metric, that uses a procedure similar to traditional cross-validation via stratified-sampling to select dyads in the network’s adjacency matrix to be removed. SCV is capable of intuitively expressing different models’ ability to predict on missing dyads. Using SCV on real-world social networks, we identify the appropriate statistical models for different network structures and generalize such patterns. In particular, we focus on conditionally independent dyad (CID) models such as the Erdos Renyi model, the stochastic block model, the sender-receiver model, and the latent space model.'
author:
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Jun Hee Kim\
`junheek1@andrew.cmu.edu`
- |
Eun Kyung Kwon\
`eunkyunk@andrew.cmu.edu`
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Qian Sha\
`qsha@andrew.cmu.edu`
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Brian Junker\
`brian@stat.cmu.edu`
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Tracy Sweet\
`tsweet@umd.edu`
title: '**CID Models on Real-world Social Networks and Goodness of Fit Measurements** '
---
= 30pt = 0pt = 597pt
Introduction
============
Social networks are ubiquitous in our society, and they represent important interactions among individuals. The interdependency of entities makes social networks compelling, yet complicated and difficult to analyze. Many statistical models for network data have been proposed and their mathematical properties have been examined. For example, the Erdos-Renyi model assumes that every pair of nodes has an equal probability of forming an edge, while the sender-receiver model assumes each node has its own probability of sending an edge and that of receiving an edge. However, whether these models actually fit well on real-world network datasets remains an unsolved issue. In fact, quantitative goodness of fit (GOF) measures for statistical models on network data have not been considered to a great extent. Current GOF metrics on models for non-network data fail to capture features of network data. For example, the Akaike Information Criterion and the Bayesian Information Criterion penalize model complexity, but not only is it difficult to settle on suitable criteria for complexity but also undesired to penalize for an essential feature of social network data. Therefore, in this paper, we identify appropriate conditionally independent dyad (CID) models for different network structures and propose a metric of GOF assessment to evaluate the developed statistical models for network data.
This paper first introduces, in Section 2, basic concepts and properties of graphs as a data structure, specifically those frequently used throughout this study. In Section 3, four CID models for network data analysis are discussed: the Erdos-Renyi model, the stochastic block model, the sender-receiver model, and the latent space model. Section 4 describes our initial attempt of developing a GOF metric using cross-validation and its limitations, followed by our remedy, the “stratified-sampling cross-validation” (SCV) metric. Section 5 introduces a set of real-world social networks that were analyzed throughout the study. Section 6 discusses the SCV accuracy results for each network, and Section 7 reports the fit of the best model chosen according to the results in Section 6. Section 8 generalizes the conclusions of this study. Lastly, we propose potential future work in Section 9 considering the limitations in our research.
Graph Terminology
=================
In this section, we introduce a set of basic graph terminologies to help understand the applications mentioned in the following sections. A network is mathematically represented as a graph $G = (V, E)$, where $V$ is the set of vertices (nodes or actors) and $E$ is the set of edges (links or ties) that are either directed or undirected. In social networks, entities are usually represented as nodes, and a social relationship between any two entities is denoted as an edge connecting those two corresponding nodes. Note that an undirected edge $i-j$ can be viewed as two directed edges $i \rightarrow j$ and $j \rightarrow i$. Furthermore, the adjacency matrix, also known as incidence matrix or sociomatrix, is often used to represent a finite graph. It is defined as an $|V| x |V|$ matrix $A$ such that $A_{ij} = 1$ if there exists an edge from node $i$ to node $j$ and $0$ otherwise, where $A_{ij}$ denotes the $i^{th}$ row, $j^{th}$ column entry of matrix $A$. A diagonal entry represents whether a node has an edge to itself, which we assume to be not true in the scope of our study.
Reciprocity and density are two commonly used descriptive statistics for networks. Reciprocity is a measure of the likelihood that nodes in a directed network are mutually linked. The reciprocity of a network is computed as:
$$\text{Reciprocity} = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij} A_{ji}}{\sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij}}$$
where $n = |V|$. Note that in this paper, if no additional description is provided, $n$ always denotes $|V|$, the number of nodes in the network. Reciprocity is frequently used to study complex networks as it gives a sense of how likely the relationships are mutual rather than one-directional. By definition, an undirected graph has reciprocity of 1 since a pair of nodes having an undirected edge is equivalent to having two directed edges in both directions and thus the relationship is mutual.
Density measures how close the number of edges is to the maximum possible number of edges, which is determined by the number of nodes. The formula for graph density is:
$$\text{Density} = \left\{
\begin{array}{ll}
\frac{2 |E|}{|V|(|V|-1)} & \quad \text{undirected graph} \\
\frac{|E|}{|V|(|V|-1)} & \quad \text{directed graph}
\end{array}
\right.$$
The higher the density of a network, the more the nodes are likely to be linked to one another. With this measure we can not only assess how closely knitted a network is, but also easily compare networks against each other or at different regions within a single network.
Conditionally Independent Dyad (CID) Models
===========================================
In this section, we examine a family of statistical models specifically developed for network data, namely the conditionally independent dyad (CID) models where dyads, which are random variables for each pair of distinct nodes on whether or not there is an edge, are assumed to be conditionally independent given the model parameters. We denote the probability that an edge from node $i$ to node $j$ exists by $p_{ij}$. Again, throughout this study we assume that a node cannot have an edge to itself. That is: for any node $i$, $p_{ii} = 0$. Then the likelihood is:
$$P(A=A^{(obs)}| \theta) = \left\{
\begin{array}{ll}
\prod_{i < j} p_{ij}^{A_{ij}^{(obs)}} (1-p_{ij})^{1-A_{ij}^{(obs)}}
& \quad \text{undirected graph} \\
\prod_{i \ne j} p_{ij}^{A_{ij}^{(obs)}} (1-p_{ij})^{1-A_{ij}^{(obs)}}
& \quad \text{directed graph}
\end{array}
\right.$$
where $A^{(obs)}$ is the observed adjacency matrix, and $\theta$ denotes the set of model parameters (so each $p_{ij}$ depends on $\theta$).
Erdos-Renyi Model
-----------------
The Erdos-Renyi model (ER)$^{1}$ is the simplest CID model. The model assumes that each pair of distinct nodes in a network has equal probability $p$ of having an edge. That is: for every pair of nodes $i$ and $j$ where $i \ne j$, $$p_{ij} = p$$ When we fit the Erdos-Renyi model to network data, the optimal value of $p$, the only parameter, is estimated.
Stochastic Block Model
----------------------
The stochastic block model (SBM)$^{2}$ is commonly used for detecting block structures in networks. The SBM is defined by three components:
1. A number $k$, representing the number of blocks to be fitted
2. An $n$-dimensional vector $\vec{z}$, where the $i^{th}$ component $z_i$ denotes block assignment of node $i$
3. A $k \times k$ stochastic block matrix $M$, where $M_{ij}$ denotes the probability that there is an edge from a node assigned to block $i$ to a node assigned to block $j$
where $k$ is a hyperparameter that must be pre-specified, while $\vec{z}$ and $M$ are model parameters to be estimated.
The diagonal entries of $M$ are probabilities that nodes assigned to the same block have an edge, while the non-diagonal entries are probabilities that nodes assigned to different blocks have an edge. In most applications of the SBM, especially for community detection, it is desired that entries on the diagonal of $M$ are large and those on the off-diagonal blocks are small. In this case, nodes within the same block are more likely to be connected compared to those in different blocks. However, there are exceptions (opposite case) where nodes in the same block are not expected to be connected while those in different blocks are. In the scenario of describing marriage relationship between a group of men and a group of women, if each group is represented as a block in an SBM (so total $2$ blocks), then the opposite case holds. What the R package does when fitting an SBM is to first, for each node, compute the probability that this node should be assigned to each of the $k$ blocks (so total $k$ probabilities that sum to $1$), and then select the block with the highest probability as the final block assignment for that node.
Sender-Receiver Model
---------------------
The sender-receiver model (SR), which is a special case of the p1 models introduced in Holland & Leinhardt (1981)$^{3}$, takes into consideration each node’s tendency to send edges towards others and also that to receive edges from others. In social network context, these can be though of as how outgoing an entity is and how popular to other people an entity is, respectively. The model has two parameters associated to each node $i$: $\beta_{i}^{(send)}$ and $\beta_{i}^{(receive)}$, which represent how likely this node sends edges to others and receive from others, respectively. The model also has an intercept parameter $\beta_{0}$, so there are total $2n + 1$ model parameters to be estimated. There are two versions of the edge probability formula for the SR model:
$$p_{ij} =
\left\{
\begin{array}{ll}
\Phi(\beta_{0} + \beta_{i}^{(send)}+\beta_{j}^{(receive)}) & \text{probit version} \\
\frac{\text{exp}\{\beta_{0} + \beta_{i}^{(send)} + \beta_{j}^{(receive)}\}}{1 + \text{exp}\{\beta_{0} + \beta_{i}^{(send)} + \beta_{j}^{(receive)}\}} & \text{logistic version}
\end{array}
\right.$$
where $\Phi(\cdot)$ denotes the standard Gaussian CDF.
Latent Space Model
------------------
In some networks, the probability of an edge between two nodes may increase proportional to some measure of similarity. Thus, a subset of entities with high number of edges between them may be indicative of a group of nodes who have adjacent positions in the “social space” of characteristics.
The latent space model (LSM)$^{4}$ takes account of this factor, assigning each node $i$ to a position in the Euclidean space $z(i)$. The edge probability between nodes $i$ and $j$ depends on their positions in the (unobserved) latent space, and the potential strength of a dyad decreases proportionally to the distance between $z(i)$ and $z(j)$. The LSM also has two versions of edge probability formulas:
$$p_{ij} = \begin{cases} \Phi(\mu+X_{ij}+U_{ij}) &\mbox{probit version} \\
\frac{\textrm{exp}(\mu+X_{ij}+U_{ij})}{1+\textrm{exp}(\mu+X_{ij}+U_{ij})} &\mbox{logistic version}
\end{cases}$$
where $X_{ij}$ denotes covariates, $U_{ij} = -||z(i)-z(j)||$, and again $\Phi(\cdot)$ denotes the standard Gaussian CDF.
An advantage of this method is that it can provide visual (if the dimension of the latent space is $\le 3$) and interpretable model-based spatial representations. Note that the latent space’s dimension is a hyperparameter which should be pre-specified.
Suitability of Cross-validation for CID Models
----------------------------------------------
From CID models, we have an advantage of learning about how nodes relate to one another. For example, entity relations are modelled probabilistically in an LSM. That is, the observation of an edge from node $i$ to node $j$ and also from node $j$ to node $k$ suggests that nodes $i$ and $k$ are not too far apart in the social space, making them more likely to have an edge.
Likewise, the interdependence between edges is implicit in CID models. If a subset of edges in the graph are missing at random, we do not need further imputation to estimate the model parameters. Again referring to the LSM, we assume that each node $i$ has position $z(i)$ on the unobserved latent space. The edges in the network are presumed to be conditionally independent given the model parameters (latent space position of each node), and the probability of an edge connecting two entities can be modelled as a function of their positions. Without any additional computation, we can utilize this function to predict the missing entries in the adjacency matrix of a graph.
For this reason, cross-validation or out-of-sample prediction can be considered a natural approach to assess GOF for CID models. There are some previous work that introduce cross-validation methods for statistical models for network data. For example, Dabbs (2016)$^{5}$ introduces latinCV that assigns each edge indicator, or dyad, to one of the $K$ folds, and then run the traditional $K$-fold cross-validation, where for each fold, that fold acts as the test set and the others are used for training. The fold assignment in latinCV is done by:
1. Construct a fixed $n \times n$ fold assignment matrix where each row and column has an equal number of occurrences of each fold.
2. Then, permute the rows and columns randomly to get the finalized fold assignment matrix.
Under this motivation, we chose to base our GOF metric on a cross-validation approach, introduced in the following section.
GOF Measure: Stratified-sampling Cross-validation (SCV)
=======================================================
In this section, we propose a self-developed goodness of fit measure for statistical models on network data. As previously mentioned, defining a criterion for model fit on network data is a difficult problem.
First, we discuss a self-developed procedure that is similar to traditional $K$-fold cross-validation and its shortcomings. In this method, around $20\%$ of the entries of the adjacency matrix are randomly sampled and set to be missing entries (NA). Then, the model is fit on the remaining entries and estimates of the model parameters are obtained. Based on the probabilistic output of the fitted model, for each node, prediction on whether or not there should be an edge is performed using the edge probability. More specifically, for each missing entry, we compute the estimated edge probability of the corresponding pair of nodes using the parameter estimates and formula according to the model, and predict by running a Bernoulli trial using that probability. This 3-step procedure of deleting, fitting, and predicting is repeated $K$ times, and then the model’s prediction accuracy is computed as the average of the $K$ prediction accuracies.
This approach, which is similar to latinCV but does not necessarily assign each dyad to exactly one fold, is simple and intuitive, but it had a limitation. Since the majority of the collected data and real-world social networks in general are sparse, most entries of their adjacency matrices were $0$’s. To take account of this fact we compared the previously-mentioned prediction to zero imputation which simply always predicts as $0$, and observed that the latter yields a much higher prediction accuracy. One problem is that an extremely high proportion of the deleted entries were $0$’s, so a model fitted on an extremely sparse (sub)network inevitably does well in the prediction task. Another issue was that the density of the subnetwork is not reasonably similar to that of the original network since the sampled entries are randomly selected from a bucket where $0$’s are dominant over $1$’s. Hence the model fitted on the subnetwork is not a good approximation, or simulation, of the model being fitted on the original network.
Such limitations motivate our “stratified-sampling cross-validation” (SCV) metric, which incorporates an alteration in the sampling procedure. The main objective is to retain a balanced proportion of the $0$’s and $1$’s in the sample, so that density is reasonably preserved even for the subnetwork. For every iteration, we randomly select an equal proportion of edges (i.e. $1$’s) and non-edges (i.e. $0'$s) from the adjacency matrix, using approximately $20\%$ in our simulation. Again, the sampled entries are set to be missing values, and the model is fit on the remaining dyads. Prediction is performed in the same way as before of running a Bernoulli trial using the estimated edge probability for the corresponding pair of nodes. Using this method, the dominance in overall prediction accuracy of zero imputation was moderately controlled for. The procedure of SCV is shown in Figure 1.
To examine the large contrast between prediction on edges compared to non-edges, we recorded results separately for the two cases. Also, due to the randomness involved in the process of sampling the entries and fitting the models, the ranking of accuracies across models can differ among iterations. Therefore, we designed our metric in such a way that each trial runs $10$ iterations of this process, and the average of the $10$ accuracy values is computed as the final result of that trial. We use this prediction accuracy as a representative measure of how good the model fits the network. The results shown hereafter are based on five such trials, which contain total $50$ iterations.
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Network Data Description
========================
In the following section, we introduce the six real-world social networks analyzed in this study. Table 1 contains the summary statistics for each of the six networks. Three out of the six networks have reciprocity of $1$ and thus are undirected, while the other three are directed. Most of the networks are extremely sparse, and the Freshmen Network is the only graph that has density not below $0.15$. This feature is not surprising since the majority of real-world social networks have very low density. The following subsections report the contexts and exploratory data analyses of the networks.
Network Number of Nodes Number of Edges Density Reciprocity
----------------------------- ----------------- ----------------- --------- -------------
Highschool Network 70 366 0.076 0.503
Karate Club Network 34 78 0.139 1
Oxford Preschoolers Network 19 41 0.120 0.195
Freshmen Network 29 282 0.347 0.567
Dolphins Network 62 159 0.084 1
Twitter Retweet Network 96 117 0.026 1
: Summary Statistics of Each Social Network
Highschool Network
------------------
The Highschool Network$^{6}$ describes friendships among boys in a small high school at Illinois. Each of the 70 boys was asked in Fall 1957 and then later in Spring 1958 to report the boys whom he considers his friends. For each of the directed edges, the weight is $1$ if that report was made only once, and $2$ if that report was made in both surveys. This dataset does not specify, for the edges with weight $1$, whether the only-one time decision was in the first or the second report.
Figure 2 shows a plot of the network. Note that among the 366 edges, 226 edges have weight $1$, and the other 140 have weight $2$, so it is relatively rare for a relationship to exist in both surveys: that is, to last longer. Moreover, the network has a triangular structure in such a way that visually each node can be seen as being in one of the three vertices of the triangle.
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In the analyses discussed in the following sections, the edges of this network are treated as equally-weighted. That is, this network was assumed to be non-weighted.
Karate Club Network
-------------------
The Karate Club Network data$^{7}$ were collected by Wayne Zachary in 1977. They describe an absence or a presence of friendship outside of the university karate club activities between each pair of members. The 34 nodes represent the members of this karate club, and the 78 non-weighted edges represent the friendship between the corresponding two members.
By the nature of this network which aims to determine the existence of mutual friendship, each edge in this network is undirected. Figure 3 shows a plot of this network.
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Oxford Preschoolers Network
---------------------------
The Oxford Preschoolers Network$^{8}$ was obtained from a series of ethological observations over preschool children in Oxford, England. These data were derived from observing a group of 19 boys competing for a toy. Each competition occurred between two individuals, and results recorded a “loser” and a “winner” in each incidence, where the “winner” won the toy. This network contains multiple occurrences of such competitions. Every node is labelled by the corresponding child’s initials. Each of the 41 directed edges represents a dominance relationship between the two children who competed for a toy and has a weight showing how many times the same incidence (the competition between the same two kids with the same winner) happened. In the following sections, we assume the edges are not weighted.
The reciprocity of the network is also very low (see Table 1), meaning that relationships described in this network are barely mutual, so it is hardly likely that a child $i$ who has lost to another child $j$ in a competition has beat $j$ in a different competition. Hence, we believe the hierarchical structure for dominance is highly consolidated in this society; “winners” tended to remain their status and this relationship was not likely to be overturned. In Figure 4, we can also observe that some boys, NP and RC, decided not to engage in the competition at all.
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Freshmen Network
----------------
The Freshmen Network$^{9}$ describes friendship relationships among freshmen studying Sociology in the University of Groningen during 1998-1999. These social network data contain information about 34 freshmen and how intimate each of them considers the other 33 colleagues.
Each directed edge has weight defined as (When collecting the data, each student reported each relationship by choosing one of these five): $1$ for “Best friendship”, $2$ for “Friendship”, $3$ for “Friendly relationship”, $4$ for “Neutral relationship”, and $5$ for “Troubled relationship”.
Note that edges with weight $4$ or $5$ indicate non-intimate relationships that are neutral or hostile. So we only analyze the network only containing edges with weight $1$, $2$, or $3$, which all indicate intimacy of different levels. In this paper, the term “Freshmen Network” (including those in Table 1 and Figure 5) refers to this subnetwork only containing edges with one of those three weight values, and not the original network. Figure 5 shows a plot of the network. In the following analyses on this network, the edges are treated as equally-weighted.
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Dolphins Network
----------------
The Dolphins Network$^{10}$ describes the associations among 62 bottlenose dolphins. Each of the undirected and unweighted edges indicates frequent associations between the corresponding two dolphins, where for a pair of dolphins $a$ and $b$, the strength of their association is depicted by the half-weight index (HWI): $$\text{HWI}(a,b) = \frac{X}{X+0.5\times(Y_{a}+Y_{b})}$$ where $X$ is the number of schools where dolphins $a$ and $b$ were seen together, $Y_a$ is the number of schools where dolphin $a$ was sighted but not dolphin $b$, and $Y_b$ is the number of schools where dolphin $b$ was sighted but not dolphin $a$. More specifically, whether or not a pair of dolphins has an edge was determined by the following $3$-step process$^{11}$:
1. Randomly permute individuals within schools ($20000$ times), keeping the school size and the number of times each individual was seen constant.
2. After each permutation the HWI for each pair is calculated, the observed HWI is compared to the $20000$ expected HWI values.
3. If more than $95\%$ of the expected HWI values are smaller than the observed HWI, the pair of dolphins was defined as a preferred companionship and has an undirected edge. Otherwise, the pair does not have an edge.
Figure 6 shows a visualization of this network, in which the node size reflects the degree of each node in this network graph.
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Twitter Retweet Network
-----------------------
The Twitter Retweet Network$^{12}$ contains information about 96 Twitter users and 117 retweets. The network data were collected over social and political hashtags on a social networking site, Twitter. Each edge in this graph is undirected and unweighted, indicating that the corresponding two users have retweeted each other on posts.
One significant structural feature in the network are cliques (Figure 7), which are groups of users that have all mutually retweeted each other. They potentially represent an interest cartel, which can be important information for applications like personalized recommendations.
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Stratified-sampling Cross-validation Results
============================================
For each of the networks discussed in Section 5, four CID models were tested in SCV: the Erdos-Renyi model (ER), the stochastic block model (SBM), the sender-receiver model (SR) and the latent space model (LSM). Each fit had five SCV trials, where each trial runs $10$ iterations of the procedure of sampling, fitting, and predicting. As mentioned before, the ER model has only one parameter $p$ and hence is considered the simplest CID model. Therefore, it is expected that non-ER models, if appropriate for a network, have better prediction accuracy than the ER model’s since they should identify certain structures or patterns in the network that the ER model is not capable of capturing.
Figure 8 shows the SCV results for all six networks. Note that the red boxplots represent prediction accuracies on non-edges, and the blue boxplots describe those on edges. We observe a big difference in the prediction accuracies on edges from those on non-edges. Overall, there was a tendency of this discrepancy to decrease as the network is more dense. Considering the high sparsity of the examined networks, we find it more meaningful to examine prediction on edges. Moreover, our primary interest lies on the existence of interaction between nodes to understand how entities are related to each other. Therefore, we focus on the prediction accuracy on edges (i.e. $1$’s in the adjacency matrix) when choosing the final model to fit for each network.
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The top leftmost plot in Figure 8 displays the SCV results for the Karate Club Network. We firstly fit the SBM with both $2$ and $3$ blocks and visualized the block assignments for comparison. It turns out that the SBM with 3 blocks looked relatively more reasonable than than the SBM with $2$ blocks, which did not tell a sensible story about the formations of the relationships. Hence, $3$ was selected as the number of blocks for the SBM in the SCV for this network. The prediction accuracies on edges range from $0.15$ to $0.3$, with those for the SBM and the SR model overlapping with each other. The relatively low prediction accuracies yielded by the ER model suggests the existence of a clear pattern or structure in the network. It is also interesting that the ER model has the highest mean prediction accuracy on non-edges but the lowest on edges. Overall, both the SBM ($3$ blocks) and the SR model work similarly well compared to the other two. However, since the SR model has larger variation, we consider the SBM the winner model.
The Freshmen Network is the only network among the six analyzed in this study in which the density is greater than $0.3$. An interesting result is that the overall gap between the prediction accuracies on the non-edges (red) and those on the edges (blue) is considerably small compared to the other networks, which are much more sparse. This feature indicates that SCV tends to yield better prediction accuracies on edges for denser networks. Another important fact is that unlike most of the networks (except for the Highschool Network), this network has a clear winner according to the prediction accuracy on edges: the SR model. The SR model not only has the highest mean accuracy but also the smallest variation, suggesting that the parameter estimates are stable relative to other models. So compared to block structures (SBM) or similarity among nodes (LSM), personal tendencies of forming friendships with other entities have a huge influence on the formation of all the relationships within this social network.
In the SCV result plot for the Twitter Retweet Network, we observe that the mean prediction accuracy on edges is extremely low. Overall, the boxplots for all four models overlap each other, indicating that there is no significant difference between prediction performance across the models. Even so, we observe that the mean accuracy of the ER model is one of the lowest. This result suggests that the Twitter Retweet Network has a clearly defined network structure. The SBM fit on $2$ blocks also seems to perform as bad in terms of mean prediction accuracy, with larger variance and wider range. From this we can infer that there might not be a conspicuous community or block structure within this network. On the other hand, the LSM with $2$ dimensions and the SR model seem to perform relatively better. Although the mean prediction accuracy for these two models are moderately similar, we must take note that in the simulations, the LSM produced the largest variance. Not only is the spread wide within the interquartile range, but also the longer upper tail implies that the distribution of the prediction accuracies are skewed.
The bottom leftmost plot in Figure 8 shows the result for the Dolphins Network. The prediction accuracies on edges range from $0.08$ to $0.14$, excluding outliers. Since the LSM has lowest mean prediction accuracy on edges and largest variation for accuracies on non-edges, it fits badly on this network. The relatively high prediction accuracies of the ER model may suggest that there is no clear structure for this network. The SBM and the SR model have similar mean accuracies to the ER model’s. Since no model outperforms the ER model, we attempt to conclude that there is no clear structure in the Dolphin Network that the non-ER CID models can capture and produce good fits. But for the sake of selecting the final model, considering the relatively small variation for prediction accuracies on edges and the high prediction accuracies on non-edges of the SBM, we finally decide to fit an SBM ($2$ blocks) on the Dolphins Network.
Just like for the Freshmen Network, there is a clear winner model for the Highschool Network, at least according to the SCV accuracies: the SBM with $3$ blocks. As shown in the plot of the original network (Figure 2), there is a clear triangular structure within the graph, and thus it is not surprising that the $3$-block SBM outperforms the other models. Furthermore, the ER model has the lowest mean accuracy for prediction on edges. Such a result suggests that there is some clear structure within the interactions among the highschool students in this network, and as just mentioned, it turned out that the SBM captures that structure the best among the non-ER models. Another interesting fact is that this network is the only one where the SBM outperforms the SR model in prediction on edges. This network is an exception to the general trend of the SR model performing at least as well as the SBM for most cases, which will be discussed more later on.
For the Oxford Preschoolers Network, in general the prediction accuracies range from $0.1$ to $0.2$, except for the ER model’s. For this network, the ER’s performance is notably low compared to rest of the models. So, we may infer that structural components in the Oxford Preschoolers Network are considerably clear. Simulations of the SBM were conducted using $2$ blocks since we observed that doing so produces a better fit than using $3$ blocks on the original network. Nevertheless, the SBM does not show high performance when compared to the LSM and the SR model. Although the ranges of the three boxplots for the LSM, the SBM and the SR model overlap, we observe distinct difference in the mean prediction accuracy. The SR model performs second best, but the boxplot displays a large degree of variation which could be an indicator that model fit is actually bad and induces very unstable parameter estimates. The LSM with $2$ dimensions outperforms the rest of the models for the Oxford Preschoolers Network. It not only has the highest mean prediction accuracy, but also has the smallest variation within the computed accuracies although there is an outlier in the upper tail.
Model Fit Results
=================
For the non-ER models (i.e. SBM, SR, and LSM), given the parameter estimates, we display the model fit via the following plots:
- : We set a color for each block, and color each node using the color corresponding to the block it is assigned to. It is expected that nodes with the same colors are somehow clustered so that block divisions are clearly visible.
- : We draw two plots, where in the first plot the node size is proportional to the fitted sender parameters, and in the second plot the node size is proportional to the fitted receiver parameters. It is expected that each node has node size reasonably proportional to the number of incoming edges (receivers plot) or that of outgoing edges (senders plot).
- : We plot the latent space positions in two dimensions, and add a line between every pair of points that correspond to a pair of nodes that have an edge. It is expected that points close to each other in the latent space tend to be connected, while points far away are not.
Based on the SCV results displayed in the boxplots, we chose one winner model for each network. The final model fit results are shown in the following.
SBM (3 Blocks) Fit on Highschool Network
----------------------------------------
According to Figure 2, the Highschool Network has a reasonably clear cluster structure. The entire network has a shape similar to a right triangle, where nodes located near each of the three points of the triangle naturally form a community. Together with the SCV result of the SBM with $3$ blocks being the clear winner, we decided the $3$-block SBM as the final model for this network.
Unsurprisingly, an SBM with $k = 3$ blocks was a good choice: the fitted block assignments are similar to what we expected when we visually saw the original data, as shown in Figure 9 below. The three communities are formed near the three points of the triangle that we thought of, and more importantly, nodes in the same blocks are connected much more frequently compared to nodes in different blocks. One interesting result is that most of the 70 probabilistic block assignments are of probability $1$, so the block memberships were assigned with great levels of certainty.
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SBM (3 Blocks) Fit on Karate Club Network
-----------------------------------------
As mentioned in Section 6, the choice of $3$ as the number of blocks in the SBM for this network was only because the $3$-block fit looked *relatively* better than the $2$-block fit, not because the former captured clear clusters. The visualization denoted by $3$ different colors (Figure 10) does show a cluster pattern at the upper right corner (denoted by blue nodes); however, we fail to detect cluster patterns for pink and green nodes.
We attempt to conclude that, based on the model fitting plots and block assignments obtained so far, the SBM with $3$ blocks is not suitable for identifying clusters for the Karate Club Network. Instead, it identifies club members who are in the core of this network from those who are in the periphery. For example, within-group edge density for group 1 (pink nodes), group 2 (blue nodes) and group 3 (green nodes) are 1, 0.25 and 0.077, respectively. It is also interesting to look at the edge probabilities. In this $3$-Block SBM, the within-group edge probability for group 1 (1.00) is the highest, and that for group 2 and group 3 respectively decreases to 0.293 and 0.099. The lowest between-group edge probability exists between group 2 and group 3, $0.013$; while the highest between-group edge probability is for group 1 and group 3, with a probability of $0.318$. These quantitative measures well present the plot structure but with higher accuracy than visual estimation.
{width="6cm" height="6cm"}
LSM (2 Dimensions) Fit on Oxford Preschoolers Network
-----------------------------------------------------
After fitting an LSM with $2$ dimensions on the Oxford Preschoolers Network, we plotted the obtained models parameters: positions on dimension 1 and dimension 2. To enhance interpretability, arrows indicating directed edges in the original network were added to the plot (Figure 11).
One eye-catching trait in Figure 11 is that nodes placed close on the latent space tend to be actually connected in the original network. On the bottom right corner of the plot, we observe a cluster of nodes placed close to each other. Toward this corner of the plot, there is an increase in the number of observed edges. However, as nodes get placed further away from this corner, the number of edges decreases. Two nodes that have no connections in the network are indeed located furthest away from the rest of the preschoolers. Therefore, the LSM captures this network structure well and thus produces good model fit results.
{width="8cm" height="8cm"}
SR Fit on Freshmen Network
--------------------------
Figure 12 shows the two plots corresponding to the fit of the SR model on the Freshmen Network. A structural feature of this network is that most of the nodes are highly interconnected among one another at the central part of the plot, and the remaining few nodes such as Node 9 and Node 26 are in the outer boundary. Those few nodes in the outer boundary do not have much outgoing edges compared to the central nodes, meaning that they are less outgoing than the majority of the entities.
According to Figure 12, the sender parameter estimates of the central nodes are considerably large compared to the few outer nodes. On the other hand, the relative gap of the receiver parameter estimates is very small. These plots suggest that the central nodes are much more active in sending (i.e. request social relationships), while both the central and outer nodes are similarly active in receiving such requests. This implication can indeed be verified by the degrees of each node, and thus the SR model successfully conveys a story behind the formation of the interactions in this social network. Such a result is consistent to our common sense in the real world as well. This network deals with college freshmen in the same department, so it has not been long since these students met one another in school. Common sense indicates that at such an early time point, individual tendency of initially forming social relationships (which an SR model seeks) is more influential than clique structures among the students (which an SBM seeks) and common interests or similarities (which an LSM seeks).
{width="12cm" height="6cm"}
SBM (2 Blocks) Fit on Dolphins Network
--------------------------------------
As mentioned in section 6, we fit an SBM with $2$ blocks on the Dolphins Network. According to the initial plot, there is a two-cluster pattern in the Dolphins Network. When we fit a $2$-block SBM on the Dolphins Network, not surprisingly, the visualization (Figure 13) where the $2$ colors represent the block assignments for each node tells a sensible story about the distribution of blocks; namely, there does exist an obvious display of two clusters just as what we have assumed. In addition, the block assignment probabilities are mostly near $1$ so the model has a high confidence in choosing one group over the other one for each node and is highly confident in its model fit. We can therefore conclude that the SBM with $2$ blocks is a good model to fit on the Dolphins Network.
{width="6cm" height="6cm"}
LSM (2 Dimensions) Fit on Twitter Retweet Network
-------------------------------------------------
Figure 14 shows the estimated model parameters, which are the positions of the nodes in the $2$-dimensional Euclidean space, together with lines between nodes that have an edge. Since the Twitter Retweet Network is an undirected graph, each edge in the original network was represented as lines, not arrows, on the latent space.
Similarly to Figure 11, Figure 14 shows that nodes positioned close on the latent space were indeed connected in the original graph. This enforces the transitive relationship between the nodes. Each edge represents an instance of a retweet, for which we expect to occur between posts of similar content. Therefore, it is reasonable that users who posted similar content are placed closer on the latent space and are connected to each other since it becomes more likely form them to retweet each other’s posts. Another feature we observe is that there seems to be clusters of nodes that connect to a central node. We conjecture that this represents an influential popular user at the center and several followers who share the same interest at its periphery. Moreover, node connectivity increases at the center of Figure 14, and decreases toward the outskirts of the plot.
{width="8cm" height="8cm"}
Conclusions and Discussions
===========================
Based on our SCV procedures and final model fit results, we conclude the following major findings.
Considering how social relationships existent in a network were generated, the more there were clear patterns or factors that affected each relationship, the less it is likely the ER model performs well. That is, a model would produce a good fit on structured networks only if it is capable of capturing that pattern that generated the edges between nodes. For example, in the Highschool Network, the SBM was able to capture the three-block structure that led to the formations of the edges in the graph reasonably well. However, the ER model assumes that there is no such pattern in this network. Instead, it argues, or hopes, that every edge is simply a result of a random coin toss with a constant probability of heads, and thus it cannot capture the significant structures, if any, in a network. This result is verified by the SCV accuracies (Figure 8); there is no network such that the ER model and another non-ER model are the two winners. So if there is a non-ER model, which seeks to capture some type of structure (depending on the model definition), that performs reasonably well, then it is very unlikely that the ER model will do similarly good.
Another interesting pattern is that for most networks, the prediction accuracies on edges of the SR model are at least as good as those of the SBM. The only exception is the Highschool Network in which the SBM outperformed all other models. Recall that the SR model captures the individual tendency of sending and receiving requests for social relationships, and the SBM seeks to find an optimal group assignment for each of the entities. Note that the formations of groups among people are influenced by the personal tendencies in the first place. For example, if most of the people are outgoing (sender) and/or popular (receiver), it is more likely that groups will be actively formed. Therefore, we observe this pattern that if capturing personal tendencies can be done well, then so can capturing group formations among the entities.
Moreover, the more clear the definition of ’similarity’ of entities is, the higher the LSM’s prediction accuracies on edges tend to be. The LSM tries to map each node to a latent space position in such a way that the distance between two nodes is inversely proportional to how likely they are connected. Therefore, the model needs a clear notion of ’similarity’ between nodes that are connected by an edge to produce a good fit. In this study, two networks had good LSM fits: the Oxford Preschoolers Network and the Twitter Retweet Network. In the former, similarity can be represented as the common interest towards the same toy. That is: the fact that two nodes have a connecting edge means that the two kids competed for a single toy, which implies that both of them had a desire for the same toy. In the Twitter Retweet Network, the notion of similarity is even more explicit. The existence of an edge between two nodes implies that the corresponding to Twitter users retweeted each other’s posts, meaning that they have some common interest or thought about an issue.
We also observed two major limitations in our self-developed GOF measure. The first is that the SCV tends to yield higher prediction accuracies on edges when the network is more dense. This characteristic is not surprising since higher density implies that the model has more information about how the existent edges were formulated. However, most real-world social networks are (extremely) sparse. Realistically speaking, considering the numerous entities around a person, it is difficult for him or her to initialize and maintain social interactions with all (or the majority) of them. Therefore, in order for the SCV metric to be applied well in real-world social network datasets, it should be improved so that it works well even on sparse networks.
The second limitation is the variability in the SCV results. The boxplots in Figure 8 show that some model fits have very large difference in the mean prediction accuracy across different trials. One conjecture about this trend is that the particular model chosen for a certain network is not the optimal one and thus the fits themselves turn out to be unstable. In other words, the better the model fits on the network, the more stable the estimation on the parameters and the SCV results will be.
Future Work
===========
Based on the limitations of our study, we propose the following directions for future work. The first is to try more models and also more networks. For example, we chose the SBM with $3$ blocks as the final model to fit on the Karate Club Network. However, we can hardly conclude that the SBM is the best model. Rather, the reason why we chose this model is that it works better than other three models considering the higher variation and lower prediction accuracies of the results of other models. Trying more types of models, or even trying different hyperparameter values for the models already explored in this study, can help find a better one. The R package$^{13}$ currently supports the following other models: the covariate model$^{14}$, the hierarchical block model$^{15}$, the latent vector model$^{16}$ and the mixed membership stochastic block model$^{17}$. Another possible model is the degree-corrected stochastic block model$^{18}$, which is a combination of the SBM and the SR model. Similarly, performing these comparisons on more real-world social networks would give us much more insights. In this study, we analyzed six networks and did find some interesting patterns. But in terms of generalizing such patterns, results on more social networks would be helpful. Appendix A contains the network data descriptions of the four additional real-world social networks which we have explored but haven’t applied the SCV and model-fitting yet.
Another possible approach is to incorporate generative process in our GOF measure. Sample networks can be generated via the estimated parameter values from the best model based on the SCV accuracies. More specifically, for each pair of nodes in the original network, we compute the estimated edge probability using the parameter estimates and the formula defined by the model, and then run a Bernoulli trial to decide whether or not to put an edge. But if such newly-generated networks are considerably different from the original network, then either the model or the SCV itself as metric is problematic. Utilizing these generative capabilities could provide more insights. Note that we must specify a criterion of the difference between two networks in order to use this approach.
Finally, we suggest identifying appropriate prior distributions used in Bayesian inference for each network involved in estimation process. The R package performs Bayesian estimation on model parameters and requires a prior distribution to be pre-specified. Using Gibbs sampling, the *CID.Gibbs* function in the package performs Markov Chain Monte Carlo that generates a sample from the distribution that is expected to be close to that of the parameters and takes the mean as the parameter estimate. In our study, we mainly used the default prior distributions given in the package. We believe that finding more network-specific prior distributions will improve the estimation process and yield better model fit quality. In addition, we firmly believe in the need to lift the computational constraint on the time-consuming simulation process. Optimizing the current source code in the R package can make the model fitting process more efficient and quick so that we can examine multiple trials within a smaller time window.
Acknowledgements {#acknowledgements .unnumbered}
================
We sincerely thank Dr. Brian Junker and Dr. Nynke Niezink in the Department of Statistics & Data Science at Carnegie Mellon University for their guidance on the direction of this research project, their effort to improve our working progress, their time for holding weekly meetings with us and finally, their unconditional support and encouragement. We also express our gratitude to Dr. Tracy Sweet in the Department of Human Development and Quantitative Methodology at University of Maryland for providing the learning materials about the software used in this project.
This work was supported in part by the US Department of Education, Institute of Education Sciences, under grant \#R305D150045. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the granting agencies.
[99]{} Paul Erdos and Alfred Renyi. “On Random Graphs.” *Publications Mathematicae*, 6, 1959, pp. 290-297. Paul Holland, Kathryn Laskey, and Samuel Leinhardt. “Stochastic blockmodels: First steps.” *Social Networks*, 5, 1983, pp. 109-137. Paul Holland and Samuel Leinhardt. “An exponential family of probability distributions for directed graphs (with discussion).” *Journal of the American Statistical Association*. 76:373, 1981, pp. 33–65. Peter D Hoff, Adrian E Raftery, and Mark S Handcock. “Latent Space Approaches to Social Network Analysis.” *Journal of the American Statistical Association*, 97:460, 2002, pp. 1090-1098. Beau Dabbs. “Characteristics of Cross-Validation Methods for Model Selection in the Stochastic Block Model for Networks.” 2016. PhD thesis, Carnegie Mellon University, Pittsburgh PA. Highschool network dataset – KONECT, April 2017. <http://konect.uni-koblenz.de/networks/moreno_highschool> Wayne Zachary. “An information flow model for conflict and fission in small groups.” *Journal of Anthropological Research*, 33, 1977, pp. 452-473, <http://moreno.ss.uci.edu/data.html#zachary> William C. McGrew. *An Ethological Study of Children’s Behavior*. New York, Academic Press, 1972. <http://moreno.ss.uci.edu/data.html#kids2> Gerhard G. Van De Bunt, Marijtje A.J. Van Duijn, and Tom A.B. Snijders. “Friendship networks through time: An actor-oriented statistical network model.” *Computational and Mathematical Organization Theory*, 5, 1999, pp. 167-192. <http://www.stats.ox.ac.uk/~snijders/siena/Zeggelink_data.htm> David Lusseau, Karsten Schneider, Oliver J Boisseau, Patti Haase, Elisabeth Slooten, and Steve M Dawson. “The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations.” *Behavioral Ecology and Sociobiology* 54(4), 2003, pp. 396-405. <http://networkrepository.com/soc-dolphins.php> David Lusseau. “The emergent properties of a dolphin social network,” *Proceedings of the Royal Society of London-Series B* (suppl.) 270, 2003, pp. 186-188. Ryan A. Rossi, David F. Gleich, Assefaw H. Gebremedhin, and Mostofa A. Patwary. “What if CLIQUE were fast? Maximum Cliques in Information Networks and Strong Components in Temporal Networks.” *arXiv preprint arXiv:1210.5802*, 2012, pp. 1-11. <http://networkrepository.com/rt-retweet.php> Beau Dabbs, Samrachana Adhikari, Tracy Sweet, and Brian Junker. “Conditionally Independent Dyad Models.” Paper in preparation (2018). Anna Goldenberg, Alice Zheng, Stephen Fienberg, and Edoardo Airoldi. “A Survey of Statistical Network Models.” *Foundations and Trends in Machine Learning*, 2:2, 2009, pp. 129-233. Vince Lyzinski, Minh Tang, Avanti Athreya, Youngser Park, and Carey Priebe. “Community Detection and Classification in Hierarchical Stochastic Blockmodels.” *IEEE Transactions on Network Science and Engineering*, 4:1, 2017, pp. 13-26. Peter Hoff. “Multiplicative latent factor models for description and prediction of social networks.” *Computational and Mathematical Organization Theory*, 15, 2009, pp. 261-272. Edoardo Airoldi, David Blei, Stephen Fienberg, and Eric Xing. “Mixed membership stochastic blockmodels.” *Journal of Machine Learning Research*, 9, 2008, pp. 1981-2014. Brian Karrer and Mark Newman. “Stochastic blockmodels and community structure in networks.” *Physical Review E*, 83, 2011, 016107. Jie Tang, Tiancheng Lou, and Jon Kleinberg. “Inferring Social Ties across Heterogeneous Networks.” *Proceedings of the Fifth ACM International Conference on Web Search and Data Mining*, 2012, pp. 743-752. <https://aminer.org/data-sna#Enron> Michael Fire, Gilad Katz, Yuval Elovici, Bracha Shapira, and Lior Rokach. “Predicting student exam’s scores by analyzing social network data”. *Active Media Technology*. Springer Berlin Heidelberg, 2012, pp. 584-595. <http://proj.ise.bgu.ac.il/sns/students.html> Linton Clarke Freeman, Cynthia Marie Webster, and Deirdre M Kirke. “Exploring social structure using dynamic three-dimensional color images.” *Social Networks*, 20(2), 1998, pp. 109-118. <http://konect.uni-koblenz.de/networks/moreno_oz> <http://networkrepository.com/divorce.php>
Appendix A {#appendix-a .unnumbered}
==========
This appendix contains the network data descriptions of the four additional real-world social networks that we have explored but have not applied SCV and model-fitting yet. Table 2 shows the summary statistics of these four social networks.
Network Number of Nodes Number of Edges Density Reciprocity
------------------------------ ----------------- ----------------- --------- -------------
Enron Network 151 266 0.012 0.075
Students Cooperation Network 185 360 0.021 1
Residence Hall Network 209 900 0.0207 0.529
Divorce Network 59 225 0.132 1
: Summary Statistics of Each Social Network in Appendix A
1) Enron Network {#enron-network .unnumbered}
----------------
The Enron Network data$^{19}$ describe two types of social relationships among $151$ people who work in Enron company: manager-subordinate relationship and colleague relationship. There are total $266$ edges (i.e. relationships), where exactly half ($133$ edges) have the manager-subordinate relationship type, and the other $133$ edges have the colleague relationship type. The edges are directed specifically because of the nature of manager-subordinate relationships. Note that each colleague edge is also expressed as a directed edge, but we should interpret it as an undirected edge since a colleague relationship is naturally symmetric. For some reason, as depicted in the reciprocity, around $7.5$% of the edges are bi-directional. Figure 15 shows a plot of the network.
{width="7.5cm" height="6.5cm"}
2) Students Cooperation Network {#students-cooperation-network .unnumbered}
-------------------------------
The Students Cooperation Network data$^{20}$ were collected by the BGU (Ben-Gurion University) Social Networks Security Research Group. This social network contains information about $185$ participating students from two different departments. Each undirected edge represents a cooperation of the corresponding pair of students while they were working on there homework assignments. The network contains total $360$ edges, and each edge has one of the three types: “Partners”, “Computer”, and “Time”. A “Partners” edge represents the explicit connection between the students who submitted theoretical or coding assignments together as partners. A “Computer” edge is the implicit connection between students who used the same computer for solving an online assignment. Lastly, a “Time” edge is defined as the second implicit connection between students who probably solved the homework assignments together but submitted from different computers. Figure 16 shows a plot of the network.
{width="9cm" height="9cm"}
3) Residence Hall Network {#residence-hall-network .unnumbered}
-------------------------
The Residence Hall Network data$^{21}$ describes friendship relationships among $217$ students at a residence hall located on the Australian National University campus. This social network contains $2672$ edges each representing a friendship. Every edge is directed and has a weight value $\in \{1, \cdots , 5 \}$, which is proportional to the strength of the friendship. Due to such a large number of edges, if we were to visualize the original network, it will be extremely difficult to see the structures in the network. Therefore, we only plot the edges with weights of $4$ or $5$ since they represent more intimate friendships compared to the other edges. The term “Residence Hall Network” (including those in Table 2 and Figure 17) refers to this subnetwork, and not the original network. Figure 17 shows a plot of the subnetwork.
{width="7.5cm" height="8.5cm"}
4) Divorce Network {#divorce-network .unnumbered}
------------------
The Divorce Network data$^{22}$ contains information about $9$ types of divorce laws in the $50$ US states. This network is a legal basis for divorce in the states and is the only bipartite graph among the networks introduced in this paper. That is: the network contains a set of $50$ nodes corresponding to the states and another of $9$ nodes each representing a type of divorce law. Moreover, the $225$ unweighted edges, each corresponding to a pair of a state and a category, indicate that the state constitutes a divorce law in that category. The $9$ types of divorce laws included in this network are: “incompat” (incompatible of temperament), “cruelty” (cruelty), “desertn” (disertion), “nonsupp” (non-supplement), “alcohol” (alcohol), “felony” (felony), “impotenc” (impotence), “insanity” (insanity) and “separate” (separation). Figure 18 shows a plot of this bipartite network. Since it’s hard to see all the edges precisely, we have also included Tables 3 and 4 that show which of the laws each state has.
{width="17cm" height="20cm"}
State incompat cruelty desertn nonsupp alcohol felony impotenc insanity separate
--------------- ---------- --------- --------- --------- --------- -------- ---------- ---------- ----------
Alabama Yes Yes Yes Yes Yes Yes Yes Yes Yes
Alaska Yes Yes Yes No Yes Yes Yes Yes No
Arizona Yes No No No No No No No No
Arkansas No Yes Yes Yes Yes Yes Yes Yes Yes
California Yes No No No No No No Yes No
Colorado Yes No No No No No No No No
Connecticut Yes Yes Yes Yes Yes Yes No Yes Yes
Delaware Yes No No No No No No No Yes
Florida Yes No No No No No No Yes No
Georgia Yes Yes Yes No Yes Yes Yes Yes No
Hawaii Yes No No No No No No No Yes
Idaho Yes Yes Yes Yes Yes Yes No Yes Yes
Illinois No Yes Yes No Yes Yes Yes No No
Indiana Yes No No No No Yes Yes Yes No
Iowa Yes No No No No No No No No
Kansas Yes Yes Yes No Yes Yes Yes Yes No
Kentucky Yes No No No No No No No No
Louisiana No No No No No Yes No No Yes
Maine Yes Yes Yes Yes Yes No Yes Yes No
Maryland No Yes Yes No No Yes Yes Yes Yes
Massachusetts Yes Yes Yes Yes Yes Yes Yes No Yes
Michigan Yes No No No No No No No No
Minnesota Yes No No No No No No No No
Mississippi Yes Yes Yes No Yes Yes Yes Yes No
Missouri Yes No No No No No No No No
: Types of Laws that Each State Has in Divorce Network (1/2)
State incompat cruelty desertn nonsupp alcohol felony impotenc insanity separate
---------------- ---------- --------- --------- --------- --------- -------- ---------- ---------- ----------
Montana Yes No No No No No No No No
Nebraska Yes No No No No No No No No
Nevada Yes No No No No No No Yes Yes
New.Hampshire Yes Yes Yes Yes Yes Yes Yes No No
New.Jersey No Yes Yes No Yes Yes No Yes Yes
New.Mexico Yes Yes Yes No No No No No No
New.York No Yes Yes No No Yes No No Yes
North.Carolina No No No No No No Yes Yes Yes
North.Dakota Yes Yes Yes Yes Yes Yes Yes Yes No
Ohio Yes Yes Yes No Yes Yes Yes No Yes
Oklahoma Yes Yes Yes Yes Yes Yes Yes Yes No
Oregon Yes No No No No No No No No
Pennsylvania No Yes Yes No No Yes Yes Yes No
Rhode.Island Yes Yes Yes Yes Yes Yes Yes No Yes
South.Carolina No Yes Yes No Yes No No No Yes
South.Dakota No Yes Yes Yes Yes Yes No No No
Tennessee Yes Yes Yes Yes Yes Yes Yes No No
Texas Yes Yes Yes No No Yes No Yes Yes
Utah No Yes Yes Yes Yes Yes Yes Yes No
Vermont No Yes Yes Yes No Yes No Yes Yes
Virginia No Yes No No No Yes No No Yes
Washington Yes No No No No No No No Yes
West.Virginia Yes Yes Yes No Yes Yes No Yes Yes
Wisconsin Yes No No No No No No No Yes
Wyoming Yes No No No No No No Yes Yes
: Types of Laws that Each State Has in Divorce Network (2/2)
|
---
abstract: 'Machine translation is going through a radical revolution, driven by the explosive development of deep learning techniques using Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN). In this paper, we consider a special case in machine translation problems, targeting to convert natural language into Structured Query Language (SQL) for data retrieval over relational database. Although generic CNN and RNN learn the grammar structure of SQL when trained with sufficient samples, the accuracy and training efficiency of the model could be dramatically improved, when the translation model is deeply integrated with the grammar rules of SQL. We present a new encoder-decoder framework, with a suite of new approaches, including new semantic features fed into the encoder, grammar-aware states injected into the memory of decoder, as well as recursive state management for sub-queries. These techniques help the neural network better focus on understanding semantics of operations in natural language and save the efforts on SQL grammar learning. The empirical evaluation on real world database and queries show that our approach outperform state-of-the-art solution by a significant margin.'
author:
- |
Ruichu Cai$^1$, Boyan Xu$^1$, Zhenjie Zhang$^2$, Xiaoyan Yang$^2$, Zijian Li$^1$, Zhihao Liang$^1$\
$^1$ Faculty of Computer, Guangdong University of Technology, China\
$^2$ Singapore R&D, Yitu Technology Ltd.\
cairuichu@gmail.com, hpakyim@gmail.com, zhenjie.zhang@yitu-inc.com\
xiaoyan.yang@yitu-inc.com, leizigin@gmail.com, zhihaolzh95@gmail.com
bibliography:
- 'ijcai18-translation.bib'
title: 'An Encoder-Decoder Framework Translating Natural Language to Database Queries'
---
= 1
Introduction {#sec:intro}
============
Machine translation is known as one of the fundamental problems in machine learning, attracting extensive research efforts in the last few decades [@koncar1997natural; @castano1997machine]. In recent years, with the explosive development of deep learning techniques, the performance of machine translation is dramatically improved, by adopting convolutional neural network [@gehring2017convolutional] or recurrent neural network [@cho2014learning; @wu2016google; @zhou2016deep]. The growing demands of computer-human interaction in the big data era, however, is now looking for additional support from machine translation to convert human commands into actionable items understandable to database systems [@giordani2012translating; @li2014constructing; @mou2017coupling; @popescu2003towards; @rabinovich2017abstract; @yin2017syntactic], in order to ease the efforts of human users on learning and writing complicated Structured Query Language (SQL). Our problem is known to be more challenging than the traditional semantic parsing problem, e.g., latest SCONE dataset involving context-dependent parsing [@long2016simpler], because of the high complexity of database querying language. Given a complex real world database, e.g., Microsoft academic database [@roy2013microsoft], it contains dozens of tables and even more primary-foreign key column pairs. A short natural language question, such as “Find all IJCAI 2018 author names" must be converted into a SQL query with more than 10 lines, because the result query involves four tables.
Recently, a number of research works attempt to apply neural network approaches on data querying, such as [@neelakantan2016learning; @yin2016neural], which target to generate data processing results by directly linking records in data tables to the semantic meanings of the natural language questions. There are two major limitations rooted at the design of their solutions. First, such methods are not scalable to big data tables, since the computation complexity is almost linear to the cardinality of the target data tables. Second, the conversion results of such methods are not reusable when a database is updated. The original natural language queries must be recalculated from scratch, in order to generate results on a new table or newly incoming records. The key to a more scalable and extensible solution is to transform original natural language queries into SQL queries instead of query answers, such that the result SQL queries are simply reusable on all tables of arbitrary size at any time.
Technically, we opt to employ encoder-decoder framework as the underlying translation model, based on Recurrent Neural Network and Long Short Term Memory (LSTM) [@hochreiter1997long]. Basically, in the encoder phase, the neural network recognizes and maintains the semantic information of the natural language question. In the decoder phase, it outputs a new sequence in another language based on the information maintained in the hidden states of the neural network. Encoder-decoder framework has outperformed conventional approaches over generic translation tasks for various pairs of natural languages. When the output domain is a structured language, such as SQL, although encoder-decoder framework is supposed to learn the grammar structure of SQL when given sufficient training samples, the cost is generally too high to afford. It spends most of the computation power on grammar understanding, but only little effort on the semantical interpretation of original questions. Even given sufficient training data, the output of standard encoder-decoder may not fully comply with SQL standard, potentially ruining the utility of the result SQL queries on real databases.
In this paper, we propose a new approach smoothly combining deep learning techniques and traditional query parsing techniques. Different from recent studies with similar strategy [@iyer2017learning; @rabinovich2017abstract; @yin2017syntactic], we include a suite of new methods specially designed for structured language outputting. On the encoder phase, instead of directly feeding word representations into the neural network, we inject a few new bits into the memory of the neural network based on language-aware semantical labels over the input words, such as *table names* and *column names* in SQL. These additional dimensions are not directly learnable by language models, but explicitly recognizable based on the properties of the structured language. On the decoder phase, we insert additional hidden states in the memory layer, called grammatical states, which indicate the states of the translation output in terms of Backus-Naur Form (BNF) of SQL. To handle the complexity behind schema-relevant information, our system generates two types of dependency and masking mechanisms to better capture the constraints based on SQL grammar as well as database schema. We also allow the neural network to recursively track the grammatical state when diving into sub-queries, such that necessary information is properly maintained even when nested queries are finished.
The core contributions of the paper are summarized as follows: 1) we present an enhanced encoder-decoder framework deeply integrated with known grammar structure of SQL; 2) we discuss the new techniques included in encoder and decoder phases respectively on grammar-aware neural network processing; 3) we evaluate the usefulness of our new framework on synthetic workload of real world database for natural language querying.
Related Work {#sec:related}
============
The emergence of deep learning techniques, particularly recurrent neural network for sequential domain, enables the machine learning models to build such complicated dependencies, and greatly enhance the translation accuracy. Encoder-decoder framework is known as a typical RNN framework designed for machine translation [@cho2014learning; @sutskever2014sequence]. On the other hand, convolutional neural network models are recently recognized as an effective alternative to recurrent neural network model for machine translation tasks. In [@gehring2017convolutional], Gehring et al. show that convolution allows the machine learning system to better train translation model by using GPUs and other parallel computation techniques.
In last two years, researchers are turning to adopt recurrent neural network for automatic data querying and programming based on natural language inputs, which aims to translate original natural language into programs and data querying results. Semantic parsing, for example, is the problem of converting natural language into formal and executable logics. In last two years, sequence-to-sequence model is becoming state-of-the-art solution of semantic parsing [@xiao2016sequence; @dong2016language; @guu2017from]. While most of the existing studies exploit the availability of human intelligence for additional labels [@jia2016data; @liang2016learning], our approach learns the translation with input-output sample pairs only. While masking is proposed in the literature for symbolic parsing by storing key-variable pairs in the memory [@liang2017neural], the masking technique proposed in this paper supports more complex operations, covering both short-term and long-term dependencies. Moreover, we hereby emphasize that the grammar structure of SQL is known to be much more complicated than the logical forms used in semantic parsing.
Besides of semantic parsing, researchers are also attempting to generate executable logics by directly linking the semantic interpretation of the input natural language and the records in the database. Neural networks are employed to identify appropriate operators [@neelakantan2016learning], while distributed representations are used [@mou2017coupling; @yin2016neural] to columns, rows and records in the data table. As pointed out in the introduction, such approaches do not scale up in terms of the data size, and the outputs are not reusable over a new data table or updated table with new records.
Recently, [@iyer2017learning; @rabinovich2017abstract; @yin2017syntactic] try to integrate grammar structure into sequence-to-sequence model for data processing query generation. These studies share common idea of our paper on tracking grammar states of the output sequence. Our approach, however, differentiates on two major points. Firstly, we design consistent and systematic approach based on grammar rule (i.e., centered at non-terminal symbols in BNF) for both encoder and decoder phases. Secondly, we include both short-term and long-term dependency in output word screening based on grammar state, exploiting the information from the schemas of the databases. These features bring significant robustness improvement.
Overview {#sec:prelim}
========
![A running example of our new Encoder-Decoder framework. The encoder phase accepts new semantic labels of the input words based on text analysis. The decoder phase employs additional augmented memory controlled by grammatical state.[]{data-label="fig:motivation"}](overview-3){width="\columnwidth"}
In this paper, we formulate the translation process as a mapping from a natural language domain $\mathbb{N}$ to a structured language domain $\mathbb{S}$, i.e., $\mathbb{N}\mapsto\mathbb{S}$. The input from $\mathbb{N}$ is a natural language sentence, $N=(w_1,w_2,\ldots,w_{L_N})$ with every word $w_i$ from a known dictionary $D_{\mathbb{N}}$. Similarly, the output of the mapping is another text sequence, $S=(w_1,w_2,\ldots,w_{L_S})$, in a structured language domain, e.g., SQL on a relational database, with dictionary $D_{\mathbb{S}}$. The goal of the translation learning is to reconstruct the mapping, based on given samples of the translation, i.e., a training set with natural language and corresponding queries $T=\{(N_1,S_1),\ldots,(N_n,S_n)\}\subset\mathbb{N}\times\mathbb{S}$.
Encoder-decoder framework [@sutskever2014sequence] is the state-of-the-art solution to general machine translation problem between arbitrary language pairs. As is shown in Figure \[fig:motivation\], there are two phases in the transformation from an input sequence to output sequence, namely *encoder phase* and *decoder phase*. The encoder phase mainly processes the input sequence, extracts key information of the input sequence and appropriately maintains them in the hidden layer, or memory in another word, of the neural network. The decoder phase is responsible for output generation, which sequentially selects output words in its dictionary and updates the memory state accordingly.
In this paper, we propose a variant encoder-decoder model, with new features designed based on the purpose of converting natural language into executable and structured language. The general motivations of these new features are also presented in Figure \[fig:motivation\]. In the encoder phase, besides of the vectorized representations of the input words, we add a number of additional binary bits into the input vector to the neural network. These binary bits are used to indicate the possible semantical meaning of these words. In our example, the word “IJCAI” is labeled as string value expression and the word “Authors” is marked as a potential column name in the table. Note that such information is not directly inferrable by a distributed representation system, e.g., [@mikolov2013distributed]. In the decoder phase, we also add new binary bits to the hidden memory layer. These states are not manipulated by the neural network, but by certain external control logics. Given the history of the output words, the external logics calculate the grammatical status of the output sequence. These augmented grammatical status is further utilized to mask candidate words for outputting. as well as feedforward features to the hidden layer of neural network. This mechanism enables our system output executable SQL at any time and enhances the learnability of the neural network.
![BNF of grammar structure of selection queries in SQL-92 standard. In the derivation rules, we use $\langle\rangle$ to indicate a symbol, $[]$ to indicate an option and $\{\}$ to indicate a block of symbols.[]{data-label="fig:bnf"}](bnf){width="\linewidth"}
Different from recent studies [@rabinovich2017abstract; @yin2017syntactic], we utilize Backus Normal Form (BNF) to generate grammatical state tracking. A BNF specification of a language is a set of derivation rules, consisting of a group of *symbols* and *expressions*. There are two types of symbols, *terminal* symbols and *non-terminal symbols*. If a symbols is non-terminal, corresponding expression contains one or more sequences of symbols. These sequences are separated by the vertical bars, each of which is a possible substitution for the symbol on the left. Terminal symbols never appear on the left side of any expression. In Figure \[fig:bnf\], we present the BNF of SQL-92, with $\langle$query$\rangle$ as the root symbol. All colored symbols, e.g., $\langle$table expression$\rangle$, are non-terminal symbols, and symbols in black, e.g., $\langle$numeric value expression$\rangle$, are terminal symbols. Theoretically, the language is context-free, if it could be written in form of BNF, and therefore deterministically verified by a push-down automaton. Given the BNF of SQL in Figure \[fig:bnf\], parsers in relational database systems can easily track the grammatical correctness of an input SQL query by scanning the query from beginning to end. Although grammar tracking strategy is similar to [@rabinovich2017abstract; @yin2017syntactic], we employ short-term and long-term dependencies to accurately mask words based on both SQL grammar and database schema.
Techniques {#sec:tech}
==========
**Encoder Processing:** The key of encoder phase in the framework is to digest the original natural language input and put the most important information in the memory before proceeding to the decoder phase. In order to extract useful information from the words in the sentence, we propose to extract additional *semantic* features that link the original words to the semantics of the grammatical structure of the target language.
{width="\linewidth"}
We generate a group of labels based on the BNF of the target language $\mathbb{S}$. Specifically, each label corresponds to a terminal symbol in the BNF. Based on the BNF in Figure \[fig:bnf\], there are four terminal symbols with corresponding labels.
*$\langle$Derived column$\rangle$*: refers to words used to describe the columns specified in the database query, e.g., the word “name" in Figure \[fig:motivation\].
*$\langle$Table reference$\rangle$*: refers to words used to describe the tables specified in the database query, e.g., the word “author”.
*$\langle$Value expression$\rangle$*: refers to words containing numeric values used to describe the conditions in the database query.
*$\langle$String expression$\rangle$*: refers to words containing string values used to describe the conditions in the database query, e.g., the word “IJCAI" in Figure \[fig:motivation\].
Given a small group of samples, we manually label the words with these four label types and employ conditional random fields (CRFs) [@lafferty2001conditional] to build effective classifiers for these labels. **Decoder Processing:** We employ two different techniques in the decoder phase, including the embedding of grammar state in the hidden layer and the masking of word outputs.
Basically, given a particular word in the output sequence, the grammar state of the word is the last expression of BNF this word fits in. When a parser interprets a SQL query, it selects the candidate expression for the words based on the structure of BNF. In the example shown in Figure \[fig:eg\], the parser enters state *derived column* when it encounters word “name” in step 2. To facilitate grammar state tracking, we use a binary vector structure to represent all possible states. The length of the vector is identical to the number of expressions in the BNF of $\mathbb{S}$. Each binary bit in the vector denotes if a particular expression is active based on the parser. When reading a new word of the output of the decoder, the SQL parser updates the grammar vector to reflect the semantic meaning of the word. The grammar state is used not only for state tracking but also for the update of the memory of the neural network. Let $g_t$ denote the grammar status information at time $t$. To incorporate $g_t$ into the model, the memory of the neural network is updated as follows: $$\begin{aligned}
f_t &= \sigma_g(W_f x_t + U_f h_{t-1} + V_f g_{t-1} + b_f) \\
i_t &= \sigma_g(W_i x_t + U_i h_{t-1} + V_i g_{t-1} + b_i) \\
o_t &= \sigma_g(W_o x_t + U_o h_{t-1} + V_o g_{t-1} + b_o) \\
c_t &= f_t \otimes c_{t-1} + i_t \otimes \sigma_c(W_f x_t + U_f h_{t-1} + V_f g_{t-1} + b_f) \\
h_t &= o_t \otimes \sigma(c_t)\nonumber
\end{aligned}$$ where $\otimes$ indicates element-wise multiplication operation.
ID State Current Word/Symbol Next Word/Symbol
---- -------------------------------- ------------------------------------ ------------------------------------
S1 $\langle$query$\rangle$ SELECT $\langle$derived column$\rangle$
S2 $\langle$select list$\rangle$ $\langle$derived column$\rangle$ $\langle$comma$\rangle$,FROM
S3 $\langle$select list$\rangle$ $\langle$comma$\rangle$ $\langle$derived column$\rangle$
S4 $\langle$from clause$\rangle$ FROM $\langle$table name$\rangle$
S5 $\langle$from clause$\rangle$ $\langle$table name$\rangle$ WHERE, Stop\_symbol
S6 $\langle$where clause$\rangle$ WHERE $\langle$derived column$\rangle$
S7 $\langle$where clause$\rangle$ $\langle$derived column$\rangle$ $\langle$equals operator$\rangle$
S8 $\langle$where clause$\rangle$ $\langle$equals operator$\rangle$ $\langle$value expression$\rangle$
S9 $\langle$where clause$\rangle$ $\langle$value expression$\rangle$ Stop\_symbol
: Partial rules of *Short-term Dependencies*.[]{data-label="tb:short"}
ID Symbol Current Word Long Term Word Mask
---- ---------------------------------- -------------- ---------------------------------
L1 $\langle$derived column$\rangle$ name publication\_table/column, name
L2 $\langle$table name$\rangle$ author publication\_table/column
: Partial rules of *Long-term Dependencies*.[]{data-label="tb:long"}
**Output Words Masking:** In the decoder, there are two types of word masks used to filter out invalid words for outputing, which are mainly based on *short-term dependencies* and *long-term dependencies* respectively. At each step, the decoder chooses one rule from candidate short-term dependencies, e.g., rules in Table \[tb:short\], and possibly multiple rules from candidate long-term dependencies, e.g., rules in Table \[tb:long\]. The short-term dependency rule is updated according to the current grammar state as well as the last output word from the decoder. In Table \[tb:short\], the columns of “State" and “Current Word/Symbol" are used for rule matching, while the column “Next Word/Symbol" indicates all valid output words in next step of the decoder. Once the decoder identifies a matching rule, it generates a mask on the dictionary to block the output of words not allowed by the rule. Long-term dependencies are updated based on the active symbols chosen by the SQL parser, maintained in the grammar state vector. For each active symbol, the decoder includes a rule from all long-term dependency rules, e.g., Table \[tb:long\], by matching on “Symbol“ and ”Word". Given the rule, the decoder generates the output word mask accordingly. The rules for long-term dependencies are removed from the decoder, only when the corresponding symbol turns inactive.
We use a binary vector $s$ to indicate the masks generated by the single rule of short-term dependency, and $l_{i}$ for the $i$-th mask generated by the rule of long-term dependencies. Given these masks, the word selection process in the decoder is modified as: $$\begin{aligned}
y_t = \sigma_y(W_y h_t + b_y) \otimes s \otimes l_1 ... \otimes l_L,\end{aligned}$$ where $L$ is the number of active long-term dependency rules.
In Figure \[fig:eg\], we present a detailed running example on the evolution of the active rules and corresponding masks, to elaborate the effect of combining the neural network and the grammar state transition. Following the example in Figure \[fig:motivation\], the query attempts to retrieve names from the author table, with the grammar states and masks updated based on the descriptions above.
**Dependency Rule Generation:** The automatic generation of rules for short-term dependencies and long-term dependencies are different. Due to the limited space, we only provide a sketch of the generation methods in the current version.
For short-term dependency, the framework identifies the reachable terminal symbols for every pair of symbol and word. Consider S1 in Table \[tb:short\]. Given the symbol $\langle$query$\rangle$ and word output “SELECT", the only matching expression in BNF is $\langle$query$\rangle$ ::= SELECT $\langle$select list$\rangle$ $\langle$table expression$\rangle$. The following symbol is $\langle$select list$\rangle$. Since $\langle$select list$\rangle$ is not a terminal symbol, we iterate over the BNF to find the terminal symbols to generate in next step. In this case, we reach the terminal symbol $\langle$derived column$\rangle$ and thus insert it into the fourth column of S1 in Table \[tb:short\].
For long-term dependency, the framework must combine the BNF as well as the schema of the database. Currently, we only consider $\langle$derived column$\rangle$ and $\langle$table name$\rangle$, which forbid the adoption of non-relevant tables and columns in the rest of the SQL query.
Experiments {#sec:exp}
===========
**Workload Preparation:** We run our experiments on three databases, namely *Geo880*, *Academic* and *IMDB*. The workload on *Geo880* is generated by converging logical form outcomes to equivalent relational table and SQL queries. *Academic* database has 17 tables, collected by Microsoft Academic Search [@roy2013microsoft]. This database is employed in the experiments of [@li2014constructing]. *IMDB* has 3 tables, containing records of 3,654 movies, 4,370 actors and 1,659 directors. On *Academic* and *IMDB*, we generate SQL query workloads and ask volunteers to label the queries with natural language descriptions. Specifically, two types of workloads are generated, namely *Select* workload and *Join* workload. The queries in *Select* and *Join* (with 4 concrete aggregation operators for AGG, including Min, Max, Average and Count) are in the following two forms respectively:
SELECT <column_array>
FROM <table> WHERE <column> =/> <value>
SELECT AGG(<table_1.column_array>)
FROM <table_1> INNER JOIN <table_2>
ON <table_1.key> = <table_2.key>
WHERE <table_2.column> =/> <value>
Given the standard forms of the queries above, we generate concrete queries by randomly selecting the tables and columns without replacement. For each combination of tables and columns, we randomly select values for the conditions in the queries. By manually filtering out meaningless queries, we generate 35 queries on *Academic* database and 75 queries on *IMDB* database. Each query is manually labeled by at least 5 independent volunteers. Given a pair of natural language description label and query, we further generalize it to a group of variant queries, by modifying the search conditions in where clauses. Consequently, we get 1,456 (376 select query and 1,080 join query) pairs of samples on *Academic* database and 2,103 (1,082 select query and 1,021 join query) pairs of samples on *IMDB* database. We also build a *Mixed* workload, by simply combining all samples from both *Select* and *Join* workloads. We reuse the queries and natural language descriptions in *Geo880* database, and use the standard training/test split as in [@iyer2017learning].
**Baseline Approaches:** We employ two state-of-the-art and representative approaches as baseline in our experiments, including NLP translation model *NMT* [@wu2016google] and semantic parsing model with feedback *SPF* [@iyer2017learning]. Note that we do not compare against cell-based data querying approaches [@neelakantan2016learning; @yin2016neural], because they are only applicable to small tables while our testing databases contain way tens of thousands records. **Performance Metrics:** We examine the quality of translation using three types of metrics. First, we report the token-level BLEU following [@yin2017syntactic] to measure the quality of translation. Second, given the groundtruth SQL query $q$ and the predicted one $\widehat{q}$, we measure *query accuracy* as the fraction of queries with identical returned tuples. This is assessed by executing the predicted and groundtruth queries in the databases and examine their returned tuples. Third, we calculate the *tuple recall* and *tuple precision* of returned tuples of each $\widehat{q}$, by comparing these tuples against the outcomes from groundtruth $q$. The average precision and recall over all queries are reported. Note that the second metric focuses on query-level correctness, while the third metric evaluates individual tuples in query results. They are therefore numerically independent. All numbers reported in the experiments on *MAS* and *IMDB* are average of 5-fold cross validations.
**Model Training and Optimization:** In preprocessing, our approach uses NLTK to implement Conditional Random Fields [@lafferty2001conditional] (CRFs) to annotate the natural language queries. The overall accuracy of annotation result is over 99.5%. Therefore, the semantic features of the input words fed to the encoders are highly reliable. Our model is implemented in Tensorflow 1.2.0. The distributed representations of the words in the dictionary are automatically calculated and optimized by Tensorflow. We optimize the hyerparameters in all approaches and use the configuration with best results. The result hyperparameters are listed in Table \[tab:exp:parameter\].
Hyperparameter NMT SPF Ours
----------------------- ------- ------- -------
Batch Size 128 100 128
Hidden Layer Size 512 600 512
Encoder Layer 2 2 2
Decoder Layer 2 1 2
Optimizer ADAM ADAM ADAM
Learning Rate 0.001 0.001 0.001
Bidirectional Encoder Used Used Used
Encoder Dropout Rate 0.2 0.4 0.2
Decoder Dropout Rate 0.2 0.5 0.2
Beam Search Size - 5 -
: Hyerparameters of all approaches in experiments.[]{data-label="tab:exp:parameter"}
Metric NMT SPF Ours
----------------- ------ ------ ----------
BLEU 83.2 38.1 **85.2**
Query Accuracy 75.0 81.7 **82.8**
Tuple Recall 77.4 83.7 **84.1**
Tuple Precision 76.9 83.6 **83.7**
: Results on *Geo880* workload.[]{data-label="tab:exp:geo"}
Metric NMT SPF Ours
----------------- ------ ------ ----------
BLEU 82.6 82.8 **83.0**
Query Accuracy 43.8 45.5 **47.9**
Tuple Recall 62.7 64.6 **66.2**
Tuple Precision 63.8 65.1 **66.6**
: Results on *MAS* workload.[]{data-label="tab:exp:mas"}
Metric NMT SPF Ours
----------------- ------ ---------- ----------
BLEU 85.7 **86.7** 85.7
Query Accuracy 91.7 95.4 **97.2**
Tuple Recall 96.9 **97.8** 97.5
Tuple Precision 96.9 **97.5** **97.5**
: Results on *IMDB* workload.[]{data-label="tab:exp:imdb"}
Metric - Short - Long - State
----------------- --------- -------- ---------
BLEU 0.37 -0.21 0.60
Query Accuracy -1.82 -1.82 -0.91
Tuple Recall -2.15 -1.57 -1.58
Tuple Precision -1.97 -2.10 -1.52
: Evaluation of individual component on training set of *Geo880* workload using 5-fold cross-validation. Difference between the “simpler” model and our original one are reported.[]{data-label="tab:exp:component"}
**Experimental Results:** We report the experimental results on three databases in Tables \[tab:exp:geo\], \[tab:exp:mas\] and \[tab:exp:imdb\] resepctively. In terms of translation quality, our model achieves the highest BLEU on *Geo880* and *MAS* while SPF performs the best on IMDB. The BLEU of SPF on Geo880 (38.1) is much lower than that of the other methods. This is because SPF uses templates to enlarge the training data significantly. Thus it tends to generate queries following those templates, which although returns the identical results but contains redundant components in the predicted query. In terms of quality of returned tuples by predicted queries, our model achieves the highest query accuracy on all three databases, i.e., the highest percentage of predicted queries with identical returned tuples. It is a significant improvement over the existing methods. The system could return completely right answers to over 80% of the questions on *Geo880* and *IMDB* databases. Although the query accuracy of all approaches is below 50% on *MAS* database, due to the high complexity on both schema and content, the recall and precision of the outcomes are all above 60%. It implies that there remains certain utility even when the translation results contain errors. An interesting observation on the results over *IMDB* database is: although SPF achieves the highest BLEU, the accuracy on query results by SPF and our approach are almost identical. It shows that translation quality, as used as the golden standard in traditional machine translation tasks, may not be the best metric for our problem setting.
We conduct ablation studies on the training set of *Geo880* (Table \[tab:exp:component\]) and find that short/long-term dependencies (Short/Long) and grammar state (State) help improve quality of returned tuples in terms of query accuracy and tuple recall/precision. However, short-term dependencies and grammar state have negative effect on BLEU, i.e., the predicted queries are more similar to the groundtruth in the token level but are less accurate. This further implies that BLEU may not be the best metric for our problem setting.
Conclusion {#sec:conclu}
==========
In this paper, we present a new encoder-decode framework designed for translation from natural language to structured query language (SQL). The core idea is to deeply integrate the known grammar structure of SQL into the neural network structure used by the encoders and decoders. Our results show significant improvements over baseline approaches for standard machine translation, especially on the accuracy of outcomes by executing the SQL queries on real databases. It greatly improves the usefulness of natural language interface to relational databases.
Although our technique is designed for SQL outputs, the proposed techniques are generically applicable to other languages with BNF grammar structures. As future work, we will extend the usage to automatic programming, enabling machine learning systems to write programs, e.g., in C language, based on natural language inputs.
Acknowledgements {#sec:acknow .unnumbered}
================
This research was supported in part by NSFC-Guangdong Joint Found (U1501254), Natural Science Foundation of China (61472089), Guangdong High-level Personnel of Special Support Program (2015TQ01X140), Pearl River S&T Nova Program of Guangzhou (201610010101), Science and Technology Planning Project of Guangzhou (201604016075), Natural Science Foundation of Guangdong (2014A030306004, 2014A030308008), Science and Technology Planning Project of Guangdong (2015B010108006, 2015B010131015). And it was done when Zhang and Yang were with Advanced Digital Sciences Center, supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.
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---
abstract: 'The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d > 0$ such that there exists $C = C(H,d)$ for which every $H$-free graph $G$ with minimum degree at least $d |G|$ satisfies $\chi(G)\le C$. We prove that $$\delta_\chi(H) \, \in \, \left\{ \frac{r-3}{r-2}, \frac{2r-5}{2r-3}, \frac{r-2}{r-1} \right\}$$ for every graph $H$ with $\chi(H)=r\ge 3$. We moreover characterise the graphs $H$ with a given chromatic threshold, and thus determine $\delta_\chi(H)$ for every graph $H$. This answers a question of Erdős and Simonovits \[Discrete Math. 5 (1973), 323–334\], and confirms a conjecture of [Ł]{}uczak and Thomassé \[preprint (2010), 18pp\].'
address:
- ' Peter Allen, Julia Böttcher, Yoshiharu Kohayakawa Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508–090 São Paulo, Brasil. '
- ' Robert Morris, Simon Griffiths IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil '
author:
- Peter Allen
- Julia Böttcher
- Simon Griffiths
- Yoshiharu Kohayakawa
- Robert Morris
bibliography:
- 'ChromaticThreshold.bib'
title: The chromatic thresholds of graphs
---
Introduction {#sec:results}
============
Two central problems in Graph Theory involve understanding the structure of graphs which avoid certain subgraphs, and bounding the chromatic number of graphs in a given family. For more than sixty years, since Zykov [@Zykov] and Tutte [@Tutte] first constructed triangle-free graphs with arbitrarily large chromatic number, the interplay between these two problems has been an important area of study. The generalisation of Zykov’s result, by Erdős [@Erd59], to $H$-free graphs (for any non-acyclic $H$), was one of the first applications of the probabilistic method in combinatorics.
In 1973, Erdős and Simonovits [@ES] asked whether such constructions are still possible if one insists that the graph should have large minimum degree. As a way of investigating their problem, they implicitly defined what is now known as the *chromatic threshold* of a graph $H$ as follows (see [@ES Section 4]): $$\begin{gathered}
\delta_\chi(H) \; := \; \inf \Big\{ d \,:\, \exists\, C = C(H,d) \text{
such that if } G \textup{ is a graph on $n$ vertices, } \\ \text{ with }
\delta(G) \ge d n \text{ and } H \not\subset G, \textup{ then } \chi(G) \le C\Big\}.\end{gathered}$$ That is, for $d < \delta_\chi(H)$ the chromatic number of $H$-free graphs with minimum degree $d n$ may be arbitrarily large, while for $d >
\delta_\chi(H)$ it is necessarily bounded. In this paper we shall determine $\delta_\chi(H)$ for every graph $H$, and thus completely solve the problem of Erdős and Simonovits.
The chromatic threshold has been most extensively investigated for the triangle $H = K_3$, where in fact much more is known. Erdős and Simonovits [@ES] conjectured that $\delta_\chi(K_3) = \frac{1}{3}$, which was proven in 2002 by Thomassen [@Thomassen02], and that moreover if $G$ is triangle-free and $\delta(G) > n/3$ then $\chi(G) \le 3$. This stronger conjecture was disproved by Häggkvist [@Hagg], who found a $(10n/29)$-regular graph with chromatic number four. However, Brandt and Thomassé [@BraTho] recently showed that the conjecture holds with $\chi(G)\le 3$ replaced by $\chi(G)\le 4$. Hence the situation is now well-understood for triangle-free graphs $G$ (see [@AES; @Bra; @BraTho; @CJK; @Hagg; @Jin; @LT]), and can be summarised as follows:
-------------------------- -------- ---------- ------- -----------------
$\delta(G) >$ $2n/5$ $10n/29$ $n/3$ $(1/3 - \eps)n$
\[+0.6ex\]
\[-2.1ex\] $\chi(G) \le$ 2 3 4 $\infty$
-------------------------- -------- ---------- ------- -----------------
\
For bipartite graphs $H$, it follows trivially from the Kövári-Sós-Turán Theorem [@KST] that $\delta_\chi(H) = 0$, and for larger cliques, Goddard and Lyle [@GodLyl] determined the chromatic threshold, proving that $\delta_\chi(K_r) = \frac{2r-5}{2r-3}$ for every $r \ge
3$. Erdős and Simonovits also conjectured that $\delta_\chi(C_5) = 0$, which was proven (and generalised to all odd cycles) by Thomassen [@Thomassen07].
For graphs other than cliques and odd cycles, very little was known until the recent work of Lyle [@Lyle10] and the breakthrough of [Ł]{}uczak and Thomassé [@LT] who introduced a new technique which allows the study of $\delta_\chi(H)$ for more general classes of graphs. In order to motivate their results, let us summarise what was known previously for $3$-chromatic graphs $H$. We have seen that there are such graphs with chromatic threshold $0$ (the odd cycle $C_5$), and chromatic threshold $\frac{1}{3}$ (the triangle). A folklore result is that there are also $3$-chromatic graphs with chromatic threshold $\frac{1}{2}$, such as the *octahedron* $K_{2,2,2}$. Indeed, given a graph $H$ with $\chi(H)=r\ge 3$, the *decomposition family* $\cM(H)$ of $H$ is the set of bipartite graphs which are obtained from $H$ by deleting $r-2$ colour classes in some $r$-colouring of $H$. Observe that $K_{2,2,2}$ has the property that its decomposition family contains no forests (in fact, $\cM(K_{2,2,2})=\{C_4\}$). It is not difficult to show that whenever there are no forests in $\cM(H)$, the graph $H$ has chromatic threshold $\frac{1}{2}$ (see Proposition \[noforest\]).
Thus it remains to consider those $3$-chromatic graphs whose decomposition family does contain a forest; in other words, graphs which admit a partition into a forest and an independent set (such as all odd cycles). Lyle [@Lyle10] proved that these graphs have chromatic threshold strictly smaller than $\frac{1}{2}$. In fact, as we shall show, they have chromatic threshold at most $\frac{1}{3}$ (see Theorem \[mainthm\]). [Ł]{}uczak and Thomassé [@LT] described a large sub-family of these graphs with chromatic threshold strictly smaller than $\frac13$. More precisely they considered triangle-free graphs which admit a partition into a (not necessarily perfect) matching and an independent set; they called a graph (such as $C_5$) *near-bipartite* if it is of this form, and proved that if $H$ is near-bipartite then $\delta_\chi(H)=0$.
However, [Ł]{}uczak and Thomassé did not believe that the near-bipartite graphs are the only graphs with chromatic threshold zero. Generalising near-bipartite graphs, they defined $H$ to be *near-acyclic* if $\chi(H)=3$ and $H$ admits a partition into a forest $F$ and an independent set $S$ such that every odd cycle of $H$ meets $S$ in at least two vertices. Equivalently, for each tree $T$ in $F$ with colour classes $V_1(T)$ and $V_2(T)$, there is no vertex of $S$ with neighbours in both $V_1(T)$ and $V_2(T)$ (see also Figure \[fig:near-acyclic\]). Observe that the near-bipartite graphs are precisely the near-acyclic graphs in which every tree is a single edge or vertex. The first graph in Figure \[fig:examples\] is near-acyclic (as illustrated by the highlighted forest), but has no matching in its decomposition family and thus is not near-bipartite.
![An illustration of near-acyclic graphs with partition into an independent set $S$ and a forest $F$ consisting of two trees $T_1$ and $T_2$.[]{data-label="fig:near-acyclic"}](nearacyclic.eps){width="5cm"}
Lyle [@Lyle10] proved that $\delta_\chi(H)=0$ for a sub-family of near-acyclic graphs $H$ which are not necessarily near-bipartite, and [Ł]{}uczak and Thomassé gave a construction (see Section \[borsuksec\]) showing that every graph which is *not* near-acyclic has chromatic threshold at least $\frac{1}{3}$. They made the following conjecture.
\[LTconj\] Let $H$ be a graph with $\chi(H) = 3$. Then $\delta_\chi(H) = 0$ if and only if $H$ is near-acyclic.
We shall prove Conjecture \[LTconj\], and moreover determine $\delta_\chi(H)$ for *every* graph $H$. In this theorem, we use the following generalisation of near-acyclic graphs. We call a graph $H$ *$r$-near-acyclic* if $\chi(H) = r\ge 3$, and there exist $r - 3$ independent sets in $H$ whose removal yields a near-acyclic graph. Note in particular that if $H$ is $r$-near-acyclic, then there is a forest in $\cM(H)$. Our main theorem is as follows.
\[mainthm\] Let $H$ be a graph with $\chi(H) = r \ge 3$. Then $$\delta_\chi(H) \, \in \,
\left\{ \frac{r-3}{r-2}, \, \frac{2r-5}{2r-3}, \, \frac{r-2}{r-1} \right\}\,.$$ Moreover, $\delta_\chi(H) \neq \frac{r-2}{r-1}$ if and only if $H$ has a forest in its decomposition family, and $\delta_\chi(H) = \frac{r-3}{r-2}$ if and only if $H$ is $r$-near-acyclic.
![A near-acyclic graph that is not near-bipartite, the dodecahedron, and the icosahedron.[]{data-label="fig:examples"}](notnearbip.eps "fig:"){width="4cm"}![A near-acyclic graph that is not near-bipartite, the dodecahedron, and the icosahedron.[]{data-label="fig:examples"}](dodecahedron.eps "fig:"){width="4cm"}![A near-acyclic graph that is not near-bipartite, the dodecahedron, and the icosahedron.[]{data-label="fig:examples"}](icosahedron.eps "fig:"){width="4cm"}
For example, the *dodecahedron* is $3$-chromatic and near-bipartite, hence it has chromatic threshold $0$ (Figure \[fig:examples\] shows the dodecahedron together with a corresponding matching). The *icosahedron* on the other hand has chromatic threshold $\frac35$ because it is four-chromatic and has a forest in its decomposition family (a partition of the icosahedron into this forest and two independent sets is also displayed in Figure \[fig:examples\]), but is not $4$-near-acyclic.
For easier reference, given $H$ with $\chi(H)=r$, we define $$\theta(H) := \frac{r-3}{r-2}\,,\quad\lambda(H) := \frac{2r - 5}{2r-3}\quad\text{and}\quad\pi(H) := \frac{r-2}{r-1}\,.$$ Observe that $\pi(H)$ is precisely the Turán density of $H$, and therefore the Erdős-Stone Theorem [@ErdosStone1946] yields $\delta_\chi(H)\le\pi(H)$ for all $H$. Furthermore, the constructions giving the lower bounds in Theorem \[mainthm\] are straightforward extensions of those given in [@LT; @Lyle10]. It follows that our main challenge is to prove that $\delta_\chi(H) \le \lambda(H)$ when $\cM(H)$ contains a forest, and that $\delta_\chi(H) \le \theta(H)$ when $H$ is $r$-near-acyclic.
The recent results of both Lyle [@Lyle10] and [Ł]{}uczak and Thomassé [@LT] contain important new techniques, which we re-use and extend here. Most significantly, [Ł]{}uczak and Thomassé [@LT] introduced a concept which they termed *paired VC-dimension*, which is based on the classical Vapnik-Červonenkis dimension of a set-system [@VC]. Our proof of Conjecture \[LTconj\] relies on an extension of this technique (see Section \[VCsec\]), together with a new embedding lemma (see Section \[Zykovsec\]) which allows us to find a copy of $H$ in sufficiently many ‘well-structured’ copies of the ‘Zykov graph’, which is a universal near-bipartite graph.
Lyle [@Lyle10] introduced a novel graph partitioning method based on the celebrated Szemerédi Regularity Lemma. We shall use his partition in Section \[Lylesec\], together with averaging arguments similar to those in [@Allen10], to prove that $\delta_\chi(H)\le\lambda(H)$ for any graph $H$ such that $\cM(H)$ contains a forest. In Section \[proofsec\] we shall combine and extend both techniques in order to generalise Conjecture \[LTconj\] to arbitrary $r \ge 3$.
#### **Organisation**
In Section \[toolsec\] we state the Regularity Lemma in the form in which we shall use it, together with some auxiliary tools, and provide some notes on notation. In Section \[Lylesec\] we prove that $\delta_\chi(H)\le\lambda(H)$ for any graph $H$ such that $\cM(H)$ contains a forest. In addition we give a construction which shows that if $\cM(H)$ does not contain a forest, then $\delta_\chi(H)\ge\pi(H)$. In Section \[borsuksec\] we provide a construction (using the Borsuk-Ulam Theorem) which shows that for graphs $H$ which are not $r$-near-acyclic, we have $\delta_\chi(H)\ge\lambda(H)$. In Section \[Zykovsec\] we introduce a generalisation of the class of Zykov graphs, a class of universal near-bipartite graphs which were used in [@LT]. We show that for every near-acyclic graph $H$, if $G$ contains a suitably well-structured collection of Zykov graphs, then $G$ contains $H$. Complementing this, in Section \[VCsec\] we refine [Ł]{}uczak and Thomassé’s paired VC-dimension argument to show that every graph with linear minimum degree and sufficiently large chromatic number indeed contains such a well-structured collection of Zykov graphs. Completing the proof of Theorem \[mainthm\], in Section \[proofsec\] we give a construction which shows that for any $r$-chromatic graph $H$ we have $\delta_\chi(H)\ge\theta(H)$, and combine the results of Sections \[Zykovsec\] and \[VCsec\] with the Regularity Lemma in order to show that for $r$-near-acyclic graphs $H$, we have $\delta_\chi(H)\le\theta(H)$. Finally, in Section \[probsec\] we conclude with a collection of open problems.
Tools and the Regularity Lemma {#toolsec}
==============================
In this section we shall state some of the tools used in the proof of Theorem \[mainthm\]. In particular, we shall recall the Szemerédi Regularity Lemma, which is one of the most powerful and important results in Graph Theory. Introduced in the 1970s by Szemerédi [@SzRL] in order to prove that sets of positive density in the integers contain arbitrarily long arithmetic progressions (a result known as Szemerédi’s Theorem [@Sz73]), it says (roughly) that *any* graph can be approximated well by a bounded number of ‘quasi-random’ graphs. The lemma has turned out to have an enormous number of applications, and many important extensions and variations have been proved (see for example [@Gowers; @KRSS; @LovSze07; @RodSch10] and the references therein). The reader who is unfamiliar with the Regularity Lemma is encouraged to see the excellent surveys [@KS93; @KSSS].
We begin by stating the Regularity Lemma in the form in which we shall use it. Let $(A,B)$ be a pair of subsets of vertices of a graph $G$. We write $d(A,B) = \frac{e(A,B)}{|A||B|}$, and call $d(A,B)$ the *density* of the pair $(A,B)$. (Here $e(A,B)$ denotes the number of edges with one endpoint in $A$ and the other in $B$.) For each $\eps > 0$, we say that $(A,B)$ is *$\eps$-regular* if $| d(A,B) - d(X,Y) | < \eps$ for every $X \subset A$ and $Y \subset B$ with $|X| \ge \eps |A|$ and $|Y|
\ge \eps |B|$.
A partition $V_0 \cup V_1 \cup \ldots \cup V_k$ of $V(G)$ is said to be an *$\eps$-regular partition* (or sometimes a Szemerédi partition of $G$ for $\eps$) if $|V_0| \le \eps n$, $|V_1| = \ldots = |V_k|$, and all but at most $\eps k^2$ of the pairs $(V_i,V_j)$ are $\eps$-regular. We will often refer to the partition classes $V_1,\ldots,V_k$ as the *clusters* of the regular partition. In its simplest form, the Regularity Lemma is as follows.
For every $\eps > 0$ and every $k_0 \in {\mathbb{N}}$, there exists a constant $k_1 = k_1(k_0,\eps)$ such that the following holds. Every graph $G$ on at least $k_1$ vertices has an $\eps$-regular partition into $k$ parts, for some $k_0 \le k \le k_1$.
We shall in fact use a slight extension of the statement above, which follows easily from [@KS93 Theorem 1.10] (a proof can be found in, e.g., [@KOTPlanar Proposition 9]). Given $0 < d < 1$ and a pair $(A,B)$ of sets of vertices in a graph $G$, we say that $(A,B)$ is *$(\eps,d)$-regular* if it is $\eps$-regular and has density at least $d$.
Given an $\eps$-regular partition $V_0 \cup V_1\cup \dots \cup V_k$ of $V(G)$ and $0<d<1$, we define a graph $R$, called the *reduced graph* of the partition, as follows: $V(R) =[k]= \{1,\ldots,k\}$ and $ij \in E(R)$ if and only if $(V_i,V_j)$ is an $(\eps,d)$-regular pair. We shall occasionally omit the partition, and simply say that $G$ has *$(\eps,d)$-reduced graph* $R$.
Let $0 < \eps < d < \delta < 1$, and let $k_0 \in {\mathbb{N}}$. There exists a constant $k_1 = k_1(k_0,\eps,\delta,d)$ such that the following holds. Every graph $G$ on $n > k_1$ vertices, with minimum degree $\delta(G) \ge \delta n$, has an $(\eps,d)$-reduced graph $R$ on $k$ vertices, with $k_0 \le k \le k_1$ and $\delta(R) \ge \big( \delta - d - \eps \big)k$.
Thus the reduced graph $R$ of $G$ ‘inherits’ the high minimum degree of $G$. The main motivation for the definition of $\eps$-regularity is the following so-called ‘counting lemma’ (see [@KS93 Theorem 3.1], for example).
Let $G$ be a graph with $(\eps,d)$-reduced graph $R$ whose clusters contain $m$ vertices, and suppose that there is a homomorphism $\phi\colon H\to R$. Then $G$ contains at least $$\frac{1}{\big|{\textup{Aut}}(H)\big|}\big(d-\eps|H|\big)^{e(H)}m^{|H|}$$ copies of $H$, each with the property that every vertex $x\in V(H)$ lies in the cluster corresponding to the vertex $\phi(x)$ of $R$.
Note that since we count unlabelled copies of $H$, it is necessary to correct for the possibility that two different maps from $H$ to $G$ may yield the same copy of $H$ (precisely when they differ by some automorphism of $H$). In fact, such an automorphism of $H$ must also preserve $\phi$, but dividing by the number of elements of the full automorphism group ${\textup{Aut}}(H)$ provides a lower bound which is sufficient for our purposes. We state one more useful fact about subpairs of $(\eps,d)$-regular pairs.
\[prop:subpair\] Let $(U,W)$ be an $(\eps,d)$-regular pair and $U'\subset U$, $W'\subset W$ satisfy $|U'|\ge \alpha|U|$ and $|W'|\ge\alpha|W|$. Then $(U',W')$ is $(\eps/\alpha,d-\eps)$-regular.
We shall also use the following straightforward and well-known fact several times.
\[prop:forest\] Let $F$ be a forest and $G$ be a graph on $n\ge 1$ vertices. If $e(G) \ge |F|n$, then $F\subset G$.
Since $G$ has average degree at least $2|F|$, it contains a subgraph $G'$ with minimum degree at least $|F|$. It is easy to show that $G'$ contains $F$; for example, remove a leaf and apply induction.
Notation
--------
We finish this section by describing some of the notation which we shall use throughout the paper. Most is standard (see [@MGT], for example); we shall repeat non-standard definitions when they are first used.
For each $t \in {\mathbb{N}}$, let $[t] = \{1,\ldots,t\}$. We say that we *blow up a vertex $v \in V(G)$ to size $t$* if we replace $v$ by an independent set of size $t$, and replace each edge containing $v$ by a complete bipartite graph. Given disjoint sets $X$ and $Y$, we shall write $K[X,Y]$ for the edge set of the complete bipartite graph on $X \cup Y$, that is, the set of all pairs with one end in $X$ and the other in $Y$. We write $K_s(t)$ for the complete $s$-partite graph with $t$ vertices in each part: that is, the graph obtained from $K_s$ by blowing up each of its vertices to size $t$.
Given a graph $G$, we write $E(G)$ for the set of edges of $G$, and $e(G)$ for $|E(G)|$. We use both $|G|$ and $v(G)$ to denote the number of vertices of $G$. Given a set $X \subset V(G)$, we write $E(X)$ for the set of edges of $G$ with both ends in $X$, and $N(X)$ for the set of *common* neighbours of the vertices in $X$. If $D \subset E(G)$, then $N(D)$ denotes the set of common neighbours of $V(D)$, the set of the endpoints of edges in $D$. (In particular, if $e = xy$ is an edge then $N(e) = N\big(\{x,y\}\big)=N(x) \cap N(y)$.) Further, we let $G[D]$ denote the subgraph of $G$ with vertex set $V(D)$ and edge set $D$, and write ${\overline}{d}(D)$ for the average degree of $G[D]$ and $\delta(D)$ for the minimum degree of $G[D]$.
A graph $G$ is said to be *$C$-degenerate* if there exists an ordering $(v_1,\ldots,v_n)$ of $V(G)$ such that $v_{k+1}$ has at most $C$ neighbours in $\{v_1,\ldots,v_k\}$ for every $1\le k\le n-1$. Finally, if $e_1,\dots,e_\ell
\in E(G)$, then we shall write ${\mathbf{e}^{\ell}}$ for the tuple $(e_1,\dots,e_\ell)$.
Graphs with large chromatic threshold {#Lylesec}
=====================================
In this section we shall categorise the graphs $H$ with chromatic threshold greater than $\lambda(H)$. First observe that a trivial upper bound on $\delta_\chi(H)$ is given by the *Turán density* of $H$, $$\pi(H) \, = \, \lim_{n \to \infty} \frac{\operatorname{ex}(n,H)}{\binom{n}{2}} \, = \, \frac{\chi(H)-2}{\chi(H)-1},$$ since if $\delta(G)=\delta n$ with $\delta > \pi(H) n$ then $H \subset G$, by the Erdős-Stone Theorem [@ErdosStone1946]. Moreover, it is not hard to prove the following sufficient condition for equality, which can be found, for example, in [@Lyle10]. Recall that $\cM(H)$ denotes the decomposition family of $H$.
\[noforest\] Let $H$ be a graph with $\chi(H) = r \ge 3$. If $\cM(H)$ does not contain a forest, then $\delta_\chi(H) = \pi(H)=\frac{r-2}{r-1}$.
For each $k,\ell \in {\mathbb{N}}$, we shall call a graph $G$ a *$(k,\ell)$-Erdős graph* if it has chromatic number at least $k$, and girth (length of the shortest cycle) at least $\ell$. In one of the first applications of the probabilistic method, Erdős [@Erd59] proved that such graphs exist for every $k$ and $\ell$.
Let $H$ be a graph with $\chi(H)=r\ge 3$ such that $\cM(H)$ contains no forest, let $C \in {\mathbb{N}}$, and let $G'$ be a $(C,|H|+1)$-Erdős graph; that is, $\chi(G') \ge C$ and ${\textup{girth}}(G') \ge |H| + 1$. Let $G$ be the graph obtained from the complete, balanced $(r-1)$-partite graph on $(r-1)|G'|$ vertices by replacing one of its partition classes with $G'$. Then $\delta(G) = \frac{r-2}{r-1}v(G)$, $H
\not\subset G$, and $\chi(G) \ge C$.
We remark that the same construction, with the complete balanced $(r-1)$-partite graph replaced by a complete balanced $(r-2)$-partite graph, shows that, whatever the structure of $H$, its chromatic threshold is at least $\frac{r-3}{r-2}$ (see Proposition \[lower\]).
Lyle [@Lyle10] showed that the condition of Proposition \[noforest\] is necessary.
\[lylethm\] If $\chi(H) = r\ge 3$, then $\delta_\chi(H) < \pi(H)=\frac{r-2}{r-1}$ if and only if the decomposition family of $H$ contains a forest.
In this section we shall strengthen this result by proving that if $\delta_\chi(H) < \pi(H)$, then it is at most $\lambda(H)$.
\[thm:forest\] Let $H$ be a graph with $\chi(H) = r \ge 3$. If $\cM(H)$ contains a forest, then $$\delta_\chi(H) \, \le \, \frac{2r-5}{2r-3}.$$
The proof of Theorem \[thm:forest\] is roughly as follows. Let $\gamma > 0$, and let $G$ be a sufficiently large graph with $$\delta(G) \, \ge \, \left(
\frac{2r-5}{2r-3} + \gamma \right)v(G).$$ For some suitably small $\eps$ and $d$, let $V_0 \cup \ldots \cup V_k$ be the partition of $V(G)$ given by the minimum degree form of the Szemerédi Regularity Lemma, and $R$ be the $(\eps,d)$-reduced graph of this partition. Define, for each $I \subset
[k]$, $$X_I \, := \, \Big\{ v\in V(G) \,\colon\, i \in I \Leftrightarrow |N(v)
\cap V_i| \ge d|V_i| \Big\}.$$ We remark that this partition was used by Lyle [@Lyle10]. We show that $\chi\big(G[X_I]\big)$ is bounded if $H\not\subset G$. We distinguish two cases. If $|I| \ge (2r-4)/(2r-3)$, then it is straightforward to show that $R[I]$ contains a copy of $K_{r-1}$, and hence, by the Counting Lemma, that $N(x)$ contains ‘many’ (i.e., a positive density of) copies of $K_{r-1}(t)$ for every $x \in X_I$. We then use the pigeonhole principle (see Lemma \[pigeon\], below), to show that either $|X_I|$ is bounded, or $H \subset G$.
If $|I| \le (2r-4)/(2r-3)$, then set $V_I = \bigcup_{j \in I} V_j$, and observe that every pair $x,y \in X_I$ has ‘many’ common neighbours in $V_I$. We use a greedy algorithm (in the form of Lemma \[pigeon\]\[pigeon:a\]) to conclude that every edge is contained in a positive density of copies of $K_r$. Finally, we shall use a counting version of a lemma of Erdős (Lemma \[lem:erdos\]) together with the pigeonhole principle to show that when $H \not\subset G$, $\chi(G[X_I])$ is bounded (see Lemma \[lem:extend\]).
We begin with some preliminary lemmas. The following lemma from [@Allen10] will be an important tool in the proof; it is a counting version of a result of Erdős [@Erdos64].
\[lem:erdos\] For every $\alpha > 0$ and $s,t \in {\mathbb{N}}$ there is an $\alpha' =
\alpha'(\alpha,s,t) > 0$ such that the following holds. Let $G$ be a graph on $n$ vertices with at least $\alpha n^s$ copies of $K_s$. Then $G$ contains at least $\alpha'n^{st}$ copies of $K_s(t)$.
We shall also use the following easy lemma, which is just an application of a greedy algorithm and the pigeonhole principle. Let $G + H$ denote the graph obtained by taking disjoint copies of $G$ and $H$, and adding a complete bipartite graph between the two.
\[pigeon\] Let $\alpha,\delta > 0$ and $s,t \in {\mathbb{N}}$, let $F$ be a forest, and suppose that $H \subset F + K_s(t)$. Let $G$ be a graph on $n$ vertices, and $X \subset V(G)$.
1. \[pigeon:a\] If $\delta(G) \ge \delta n$ and $$|X| \ge \big(\alpha^{1/s} s + (1 - \delta)(s-1) \big) n\,,$$ then $G[X]$ contains at least $\alpha n^s$ copies of $K_s$.
2. \[pigeon:b\] If $G\big[N(x)\big]$ contains at least $\alpha
n^{(s+1)|H|}$ copies of $K_{s+1}(|H|)$ for every $x \in X$, then either $H \subset G$ or $|X| \le |H|/\alpha$.
For part \[pigeon:a\], we construct copies of $K_s$ in $G[X]$ using the following greedy algorithm: First choose an arbitrary vertex $x_1 \in
X$, then a vertex $x_2 \in X$ in the neighbourhood of $x_1$, then $x_3 \in
N(x_1) \cap N(x_2) \cap X$, and so on, until we find $x_s \in X$ in the common neighbourhood of $x_1,\dots,x_{s-1}$. Clearly, $G\big[\{x_1,\dots,x_s\}\big]$ is a copy of $K_s$.
Now we simply count: for choosing $x_i$ we have at least $$|X| - (i-1)(1-\delta)n \ge \alpha^{1/s}s \cdot n \ge (s!\alpha)^{1/s}
\cdot n$$ possibilities, so in total we have at least $s!\alpha n^s$ choices. Since the algorithm can construct a particular copy of $K_s$ at most in $s!$ different ways, we have found at least $\alpha n^s$ distinct $K_s$-copies in $G[X]$.
For part \[pigeon:b\], simply observe that, by the pigeonhole principle, there is a copy $T$ of $K_{s+1}(|H|)$ in $G$ such that $T
\subset G\big[N(x)\big]$ for at least $\alpha|X|$ vertices of $X$. Since $H \subset K_{s+2}(|H|)$ this implies that either $H \subset G$ or $\alpha|X| < |H|$.
The following result follows easily from Lemma \[lem:erdos\]. For a forest $F$ and $H\subset F+K_s(t)$, it will enable us to draw conclusions about the chromatic number of an $H$-free graph which contains many $K_{s+2}$-copies arranged in a suitable way.
\[lem:extend\] For every $\alpha > 0$ and $s,t \in {\mathbb{N}}$, there exists $\alpha' =
\alpha'(\alpha,s,t)$ such that for every forest $F$ and every graph $H \subset F
+ K_s(t)$, the following holds. Let $G$ be an $H$-free graph on $n$ vertices, and let $X \subset V(G)$ be such that every edge $xy \in E\big(G[X]\big)$ is contained in at least $\alpha n^s$ copies of $K_{s+2}$ in $G$.
Then $G[X]$ is $(2|F|/\alpha')$-degenerate, and hence $\chi\big(G[X]\big) \le
(2|F|/\alpha')+1$.
Given a subgraph $K$ of $G$, we say that an edge $e = xy$ of $G$ *extends to $e + K$* if $V(K)\subset N(x,y)$. We say $e$ *extends to a copy of $e+K_s(t)$* if $e$ extends to $e+K$ for some copy $K$ of $K_s(t)$ in $G$.
Let $\alpha' =
\alpha'(\alpha,s,t)$ be the constant provided by Lemma \[lem:erdos\], let $xy
\in E(G[X])$ and let $G' = G[N(x) \cap N(y)]$. Then, by our assumption, $G'$ contains at least $\alpha n^{s}$ copies of $K_s$. By Lemma \[lem:erdos\], it follows that $G'$ contains $\alpha' n^{st}$ copies of $K_s(t)$, so $xy$ extends to at least $\alpha' n^{st}$ copies of $e + K_s(t)$ in $G$.
Now, let $x_1,\dots,x_{|X|}$ be an ordering of the vertices of $X$ with the property that $x_i$ has minimum degree in $G_i := G[X \setminus
\{x_1,\dots,x_{i-1}\}]$ for each $1\le i\le |X|$. In order to show that $G[X]$ is $(2|F|/\alpha')$-degenerate, it suffices to prove that $$e(G_i) \, \le \,
\frac{|F|}{\alpha'} \cdot |G_i|,$$ since then $\delta(G_i) \le 2|F|/\alpha'$, as desired.
Since each edge $e \in E(G_i)$ extends to at least $\alpha' n^{st}$ copies of $e + K_s(t)$, it follows, by the pigeonhole principle, that there is a copy $K'$ of $K_s(t)$ and a set $E_i \subset E(G_i)$ with $|E_i| \ge \alpha' e(G_i)$, such that $e$ extends to $e + K'$ for every $e \in E_i$. Let $G^*_i$ be the graph with vertex set $V(G_i)$ and edge set $E_i$. Since $H \not \subset G$, it follows that $F \not\subset G^*_i$. Thus, by Fact \[prop:forest\], we have $$|F| \cdot |G_i| \, > \, e(G^*_i) \, \ge \, \alpha' e(G_i),$$ as required.
We are ready to prove Theorem \[thm:forest\]. We shall apply the minimum degree form of the Szemerédi Regularity Lemma, together with the Counting Lemma and Lemmas \[pigeon\] and \[lem:extend\].
Let $F$ be a forest, let $r \ge 3$, and let $H$ be a graph with $\chi(H) = r$ and $F \in \cM(H)$. Observe that we have $H \subset F + K_{r-2}(|H|)$. Let $\gamma > 0$, and let $G$ be an $H$-free graph with $$\delta(G) \,\ge\, \left(
\frac{2r-5}{2r-3} + 2\gamma \right) n,$$ where $n = |G|$. We shall show that $\chi(G)$ is bounded above by some constant $C = C(H,\gamma)$.
The first step is to apply the minimum degree form of Szemerédi’s Regularity Lemma to $G$, with $$\label{eq:forest:epsd}
d := \frac{\gamma}{2},\qquad k_0 := r^2\qquad\text{and}\qquad
\eps := \min\bigg\{\frac{\gamma}{2}, \,\frac{d^2}{2d+2|H|}\bigg\}\,.$$
We obtain a partition $V(G) = V_0 \cup V_1 \cup
\ldots \cup V_k$, where $k_0 \le k \le k_1 = k_1(\eps,d,k_0)$, with an $(\eps,d)$-reduced graph $R$ such that $$\delta(R) \; { { {\overset{\mbox{\tiny{\eqref{eq:forest:epsd}}}}{\ge}} } }
\; \left( \frac{2r-5}{2r-3} + \gamma \right) k.$$ We now partition the vertices of $V(G)$ depending upon the collection of the sets $V_i$ to which they send ‘many’ edges. More precisely, define $V(G) = \bigcup_{I\subset[k]} X_I$ by setting $$X_I:= \Big\{ v\in V(G) \,\colon\, i \in I \Leftrightarrow |N(v) \cap V_i| \ge d|V_i| \Big\},$$ for each $I \subset [k]$. We claim that $\chi\big(G[X_I]\big) \le
\max\{C_1,C_2+1\}$ for all $I\subset[k]$, where $C_1 = C_1(H,\gamma)$ and $C_2
= C_2(H,\gamma)$ are constants defined below. Since the $X_I$ form a partition, this implies that $\chi(G)\le2^k\max\{C_1,C_2+1\}\le 2^{k_1}\max\{C_1,C_2+1\}=
C(H,\gamma)$ as desired.
In order to establish this claim we distinguish two cases.
**Case 1**: $|I| \, \ge \, \left( {\displaystyle}\frac{2r-4}{2r-3} \right)
k$.
In this case we shall show that $|X_I|\le C_1$ (where $C_1$ is a constant defined below, and independent of $n$), and thus trivially $\chi\big(G[X_I]\big)\le C_1$. We first claim that $R[I]$ contains a copy of $K_{r-1}$. Indeed, by our minimum degree condition on $R$, we have $$\delta\big( R[I] \big) \; \ge \; \delta(R) -
\big( k - |I| \big) \; \ge \; |I| - \left( \frac{2}{2r-3} - \gamma \right) k \;
\ge \; \left( \frac{r-3}{r-2} + \gamma \right) |I|.$$ Thus, by Turán’s Theorem (or simply by proceeding greedily), $R[I]$ contains a copy of $K_{r-1}$, as claimed. Let $\{W_1,\dots,W_{r-1}\} \subset
\{V_1,\ldots,V_k\}$ be the set of parts corresponding to this copy of $K_{r-1}$.
Now let $x \in X_I$, set $W'_i = N(x) \cap W_i$ for each $i \in [r-1]$, and note that $|W'_i|\ge d|W_i|$, by the definition of $X_I$. By Fact \[prop:subpair\], each pair $(W'_i,W'_j)$, $i\neq j$, is $(\eps/d,d-\eps)$-regular. By the Counting Lemma and , it follows that $G[N(x)]$ contains at least $$\alpha_1
n^{(r-1)|H|}$$ copies of $K_{r-1}(|H|)$, for some $\alpha_1 = \alpha_1(H,\gamma) > 0$. Thus, by Lemma \[pigeon\]\[pigeon:b\] (applied with $\alpha = \alpha_1$, $s = r - 2$ and $X = X_I$), we have either $H \subset G$, a contradiction, or $|X_I| \le |H|/\alpha_1=C_1(H,\gamma)$, as claimed.
**Case 2**: $|I| \, \le \, \left( {\displaystyle}\frac{2r-4}{2r-3}
\right) k$.
In this case we shall show that $G[X_I]$ is $C_2$-degenerate (where $C_2$ is defined below and independent of $n$), using Lemma \[lem:extend\]. It follows that $\chi\big(G[X_I]\big)\le C_2+1$. First, we shall show that every edge of $G[X_I]$ is contained in at least $\gamma^{r-2} n^{r-2}$ copies of $K_r$.
Let $V_I:=\bigcup_{i \in I} V_i$ denote the set of vertices in clusters corresponding to $I$, and let $xy \in E(G[X_I])$. By the definition of $X_I$, $x$ and $y$ each have at most $(d+\eps)n$ neighbours outside $V_I$, and thus, since $d+\eps<\gamma$, they each have at least $\big(\frac{2r-5}{2r-3} +
\gamma \big)n$ neighbours in $V_I$. It follows that they have at least $2\big(\frac{2r-5}{2r-3}+\gamma\big)-|V_I|$ common neighbours in $V_I$. Finally, since $|I|\le\big(\frac{2r-4}{2r-3}\big)k$, we have $|V_I|\le\big(\frac{2r-4}{2r-3}\big)n$, and thus $x$ and $y$ have at least $$\left( \frac{2r-6}{2r-3} + \gamma \right) n$$ common neighbours in $V_I$.
Now, apply Lemma \[pigeon\]\[pigeon:a\] with $\alpha = \gamma^{r-2}$, $\delta = \frac{2r-5}{2r-3} +
\gamma$, $s = r - 2$, and $X = N(x) \cap N(y)$. We have $$\frac{2r-6}{2r-3} +
\gamma \; = \; (r-2) \gamma + (r-3)\left( 1 - \frac{2r-5}{2r-3} - \gamma
\right)\,,$$ and so $G[N(x) \cap N(y)]$ contains at least $\gamma^{r-2} n^{r-2}$ copies of $K_{r-2}$, i.e., every edge of $G[X_I]$ is contained in $\gamma^{r-2}
n^{r-2}$ copies of $K_r$. Hence, by Lemma \[lem:extend\] (applied with $\alpha=\gamma^{r-2}$, $s = r - 2$ and $t=|H|$), there exists $\alpha'=\alpha'(\gamma^{r-2},r-2,|H|)>0$ such that $G[X_I]$ is $2|H|/\alpha'=C_2$-degenerate, and so $\chi\big(G[X_I]\big) \le C_2+ 1$, as required.
Borsuk-Hajnal graphs {#borsuksec}
====================
In this section we shall describe the constructions (based on those in [@LT]) which provide the lower bounds on $\delta_\chi(H)$ in Theorem \[mainthm\]. One of the main building blocks in these constructions is a class of graphs which also mark the first (and most famous) application of algebraic topology in combinatorics: the *Kneser graphs* ${\textup{Kn}}(n,k)$, which are defined as follows. Given $n,k \in
{\mathbb{N}}$, let ${\textup{Kn}}(n,k)$ have vertex set $\binom{[n]}{k}$, the family of $k$-vertex subsets of $[n]$, and let $\{S,T\}$ be an edge if and only if $S$ and $T$ are disjoint. (For example, ${\textup{Kn}}(5,2)$ is the well-known Petersen graph.) These graphs were first studied by Kneser [@Kneser], who conjectured that $\chi\big( {\textup{Kn}}(n,k) \big) = n - 2k + 2$ for every $n$ and $k$. This problem stood open for 23 years, until it was solved by Lovász [@Lovasz], whose proof led eventually to a new area, known as Topological Combinatorics (see [@Mat03], for example). Hajnal (see [@ES]) used the Kneser graphs in order to give the first examples of dense triangle-free graphs with high chromatic number. Given $k,\ell,m \in {\mathbb{N}}$ such that $2m + k$ divides $\ell$, let the *Hajnal graph*, denoted $H(k,\ell,m)$, be the graph obtained as follows: first take a copy of ${\textup{Kn}}(2m+k,m)$, and a complete bipartite graph $K_{2\ell,\ell}$, with vertex set $A \cup B$, where $|A| = 2\ell$, and $|B|
= \ell$; next partition $A$ into $2m + k$ equally sized pieces $A_1,\ldots,A_{2m+k}$; finally, add an edge between $S \in V\big({\textup{Kn}}(2m+k,m)\big)$ and $y \in A_j$ whenever $j \in S$.
The following theorem, which appeared in [@ES], implies that $\delta_\chi(K_3) \ge 1/3$.
\[Hajnal\] For all $\nu>0$ and $k \in {\mathbb{N}}$ there exist integers $m$ and $\ell_0$ such that, for every $\ell \ge \ell_0$, the Hajnal graph $G=H(k,\ell,m)$ satisfies $v(G)=3\ell+\binom{2m+k}{m}$, $\chi(G)\ge k+2$ and $\delta(G)\ge(\frac13-\nu)v(G)$, and is triangle-free.
In order to generalise Theorem \[Hajnal\] from triangles to arbitrary 3-chromatic graphs which are not near-acyclic, [Ł]{}uczak and Thomassé [@LT] defined the so-called Borsuk-Hajnal graphs. We shall next describe their construction.
The *Borsuk graph* ${\textup{Bor}}(k,\eps)$ has vertex set $S^k$, the $k$-dimensional unit sphere, and edge set $\{xy\colon {\measuredangle}(x,y) \ge \pi - \eps\}$, where ${\measuredangle}(x,y)$ denotes the angle between the vectors $x$ and $y$. It follows from the Borsuk-Ulam Theorem (see [@Mat03], for example) that $\chi\big({\textup{Bor}}(k,\eps)\big) \ge k + 2$ for any $\eps > 0$.
In order to construct Borsuk-Hajnal graphs from Borsuk graphs, we also need the following theorem, which follows easily from a result of of Nešetřil and Zhu [@NesZhu].
\[HellNes\] Given $\ell \in {\mathbb{N}}$ and a graph $G$, there exists a graph $G'$ with girth at least $\ell$, such that $\chi(G') = \chi(G)$, and such that there exists a homomorphism $\phi$ from $G'$ to $G$.
Now, given $k \in {\mathbb{N}}$, a set $W \subset S^k$ with $|W|$ even, and $\eps,\delta > 0$, we define the Borsuk-Hajnal graph, ${\textup{BH}}={\textup{BH}}(W;k,\eps,\delta)$, as follows.
First, let $B = {\textup{Bor}}(k,\eps)$ be the Borsuk graph, and let $U \subset S^k = V(B)$ be a finite set, with $U$ chosen such that $\chi(B[U]) = k + 2$. (This is possible by the de Bruijn-Erdős Theorem [@BruErd51], which states that every infinite graph with chromatic number $k'$ has a finite subgraph with chromatic number $k'$.) Let $B'$ denote the graph given by Theorem \[HellNes\], applied with $G=B[U]$ and $\ell=k$, let $\phi$ be the corresponding homomorphism from $B'$ to $B[U]$, and let $U'$ be the vertex set of $B'$.
Let $X$ be a set of size $|W|/2$, and recall that $K[W,X]$ denotes the edge set of the complete bipartite graph with parts $W$and $X$.
\[def:BH\] Define ${\textup{BH}}={\textup{BH}}(W;k,\eps,\delta)$ to be the graph on vertex set $U' \dcup W
\dcup X$, where $U'$, $W$ and $X$ are pairwise disjoint and as described above, with the following edges: $$E(B') \cup K[W,X] \cup \Big\{
\{u,w \} : u \in U', \, w \in W \textup{ and } {\measuredangle}\big(\phi(u),w\big) <
\frac{\pi}{2} - \delta \Big\}\,.$$
Observe that $\chi({\textup{BH}}) \ge \chi\big(B[U]\big) > k$.
\[LTborsuk\] For every $k \in {\mathbb{N}}$ and $\nu>0$, there exist $\eps,\delta > 0$ and $W \subset S^k$, such that, setting ${\textup{BH}}={\textup{BH}}(W;k,\eps,\delta)$, we have $$\chi({\textup{BH}}) \,\ge\, k \qquad \text{ and } \qquad \delta({\textup{BH}}) \,\ge\, \left( \frac13 - \nu \right)v({\textup{BH}}).$$ Moreover every subgraph $H \subset {\textup{BH}}$ with $v(H) < k$ and $\chi(H) = 3$ is near-acyclic.
Hence, for any $H$ with $\chi(H)=3$ which is not near-acyclic, we have $\delta_\chi(H)\ge 1/3$.
We shall generalise the [Ł]{}uczak-Thomassé construction further, as follows, to give our claimed lower bound on $\delta_\chi(H)$ for $r$-chromatic $H$ which are not $r$-near-acyclic.
\[def:BHr\] Define ${\textup{BH}}_r(W;k,\eps,\delta)$ to be the graph obtained from the Borsuk-Hajnal graph ${\textup{BH}}= {\textup{BH}}(W;k,\eps,\delta)$ by adding $r-3$ independent sets $Y_1,\dots,Y_{r-3}$ of size $|Y_1| = \ldots = |Y_{r-3}| = |W|$, and the following edges: $$\bigcup_{1 \le i < j \le r-3} K[Y_i,Y_j] \,\cup\, \bigcup_{i = 1}^{r-3} K[Y_i,V({\textup{BH}})].$$ That is, we add the complete $(r-2)$-partite graph on $V({\textup{BH}}) \cup Y_1 \cup \ldots \cup Y_{r-3}$.
The following result extends Theorem \[LTborsuk\] to arbitrary $r \ge 3$.
\[thm:borsuk\] For every $r \ge 3$, $k \in {\mathbb{N}}$ and $\nu>0$, there exist $\eps,\delta > 0$ and $W \subset S^k$, such that, setting ${\textup{BH}}_r={\textup{BH}}_r(W;k,\eps,\delta)$, we have $$\chi({\textup{BH}}_r) \,\ge\, k \qquad \text{ and } \qquad \delta({\textup{BH}}_r) \,\ge\, \left( \frac{2r-5}{2r-3} - \nu \right)v({\textup{BH}}_r).$$ Moreover every subgraph $H \subset {\textup{BH}}_r$ with $v(H) < k$ and $\chi(H) = r$ is $r$-near-acyclic.
Hence, for any $H$ with $\chi(H)=r$ which is not $r$-near-acyclic, we have $\delta_\chi(H)\ge \frac{2r-5}{2r-3}$.
Theorem \[thm:borsuk\] follows easily from Łuczak and Thomassé’s argument for Theorem \[LTborsuk\]; for completeness, we shall provide a proof here.
As noted above, we have $\chi\big( {\textup{BH}}_r(W;k,\eps,\delta) \big) > k$ for every choice of $W$, $\eps$ and $\delta$. For the other properties, we shall choose $W$ randomly, and $\eps,\delta > 0$ as follows.
Let $r \ge 3$, $k \in {\mathbb{N}}$, and $\nu > 0$ be arbitrary, and choose $\delta >
0$ such that the spherical cap of $S^k$ (centred around the pole) with polar angle $\frac{\pi}{2}-\delta$ covers a $(\frac12-\frac\nu2)$-fraction of $S^k$. Set $\eps = \delta/(2k)$, $$u_0 \,:=\, v\big({\textup{BH}}_r(\emptyset;k,\eps,\delta)\big) \, = \, |U'|,$$ and choose $w_0$ sufficiently large such that $$\label{eq:borsuk:w} \exp\left(-\frac{\nu^2w_0}{4}\right)<\frac{1}{u_0}
\quad\text{and}\quad
\left(\frac{2r-5}{2}-\nu\right)w_0\ge
\left(\frac{2r-5}{2r-3}-\nu\right)\left(\frac{2r-3}{2}w_0+u_0\right),$$ which is possible because $(2r-3)/2 > 1$. Draw $w_0$ points uniformly at random from $S^k$, call the resulting set $W$ and consider the graph ${\textup{BH}}_r={\textup{BH}}_r(W;k,\eps,\delta)$.
We show first that, with positive probability, ${\textup{BH}}_r$ has the desired minimum degree. Let $Y = Y_1 \cup \ldots \cup Y_{r-3}$, and recall that ${\textup{BH}}_r$ has vertex set $U' \dcup W \dcup X \dcup Y$ and that $|U'| = u_0$.
\[cl:borsuk:degree\] With positive probability the following holds. For every $v\in V({\textup{BH}}_r)$, $$\deg_{{\textup{BH}}_r}(v) \,\ge\, \left( \frac{2r-5}{2} - \nu \right) |W|.$$
For $v \in W \cup X \cup Y$, it is easy to check that $\deg_{{\textup{BH}}_r}(v) \ge \left( \frac{2r-5}{2} \right)|W|$. Moreover, if $v \in U'$ then $Y \subset N(v)$ and $|Y| = (r-3)|W|$. Thus it will suffice to show that the following event $\sigma$ holds with positive probability: For every $v \in U'$ we have $\deg_{{\textup{BH}}_r}(v,W)\ge (\tfrac12-\nu)|W|$.
To this end observe that, for a given $v \in U'$, the value of $\deg_{{\textup{BH}}_r}(v,W)$ is a random variable $B$ with distribution ${\textup{Bin}}\big( |W|,\frac12-\frac\nu2 \big)$. This follows because $W$ was chosen uniformly at random from $S^k$, by our choice of $\delta$, and since by Definition \[def:BH\], $v$ is adjacent to $w\in W$ if and only if ${\measuredangle}(\phi(v),w)\le \frac\pi2 - \delta$. Thus, by Chernoff’s inequality (see, e.g., [@JaLuRu:Book Chapter 2]), $${\mathbb{P}}\Big(\deg_{{\textup{BH}}_r}(v,W) < \big( \tfrac12 - \nu \big)|W|\Big) \,\le\, \exp\Big( -\frac{\nu^2|W|}{4}\Big)
\,{ { {\overset{\mbox{\tiny{\eqref{eq:borsuk:w}}}}{<}} } }\, \frac{1}{|U'|}.$$ By the union bound, the event $\sigma$ holds with positive probability, as required.
Using , we have $$\left( \frac{2r-5}{2} - \nu \right) |W| \, \ge \, \left( \frac{2r-5}{2r-3} - \nu \right) v({\textup{BH}}_r),$$ and so the desired lower bound on $\delta({\textup{BH}}_r)$ follows immediately from the claim.
Finally, let us show that every subgraph $H \subset {\textup{BH}}_r$ with $v(H) < k$ and $\chi(H) = r$ is $r$-near-acyclic. We begin by showing that $H' := H[U'\cup W \cup X]$ is near-acyclic.
Observe first that $H'[W]$ is independent, and recall (from Definition \[def:BH\]) that ${\textup{BH}}_r[U']$ has girth at least $k >
v(H)$. Thus $H'[U' \cup X]$ is a forest, since all of its edges are contained in $U'$. It therefore suffices to prove the following claim.
\[cl:borsuk:path\] Every odd cycle in $H'$ contains at least two vertices of $W$.
Let $C$ be an odd cycle in $H'$. (Hence $v(C)<k$.) If $V(C)\cap X \neq
\emptyset$ then $|V(C)\cap W|\ge 2$ since $e(U',X)=0$ and $X$ is independent. Thus we may assume that $V(C) \cap X =
\emptyset$. Similarly, since $H'[U']$ is a forest, we must have $|V(C)
\cap W| \ge 1$.
Let $P = v_1\dots v_p$ be a path in $U'$ with $p < k$ and $p$ even. Recall that $\phi(v_1),\dots,\phi(v_p)$ are vectors from $S^k$ such that ${\measuredangle}\big(\phi(v_i),\phi(v_{i+1})\big)\ge\pi-\eps$ for all $i\in[p-1]$. We shall show that $N_{{\textup{BH}}_r}(v_1)\cap N_{{\textup{BH}}_r}(v_p)\cap
W = \emptyset$, i.e., that $\phi(v_1)$ and $\phi(v_p)$ do not lie in a common spherical cap with angle $\frac\pi2 - \delta$. Indeed, we have ${\measuredangle}\big(\phi(v_1),\phi(v_3)\big)\le 2\eps$, and, in general, ${\measuredangle}\big(\phi(v_1),\phi(v_{2j})\big)\ge\pi-2j\eps$ for every $j\in[p/2]$. Hence ${\measuredangle}\big(\phi(v_1),\phi(v_p)\big)\ge\pi-k\eps >
2(\frac\pi2-\delta)$, and so, by Definition \[def:BH\], $v_1$ and $v_p$ do not have a common neighbour in $W$, as required.
This implies that $V(C) \cap U'$ cannot be a path on $v(C)-1$ vertices and thus we conclude $|V(C) \cap W| \ge 2$.
Finally, note that as $H[Y_i]$ is an independent set for each $i \in [r-3]$, and $H'$ is obtained by removing these sets, $H$ is indeed $r$-near-acyclic, as required.
Zykov graphs {#Zykovsec}
============
In this section we shall prove a key result on Zykov graphs (see Definition \[def:Zykov\] and Proposition \[lem:rich-H\], below), which will be an important tool in our proof of Theorem \[mainthm\]. Let $G'$ be a graph with connected components $C_1,\ldots,C_m$, and let $G$ be the graph obtained from $G'$ by adding, for each $m$-tuple $\bfu = (u_1,\ldots,u_m) \in C_1 \times
\dots \times C_m$, a vertex $v_\bfu$ adjacent to each $u_j$. This construction was introduced by Zykov [@Zykov] in order to obtain triangle-free graphs with high chromatic number.
We shall use the following slight modification of Zykov’s construction. Recall that to blow up a vertex $v \in V(G)$ to size $t$ means to replace $v$ by an independent set of size $t$, and replace each edge containing $v$ by a complete bipartite graph, and that $K(v,X)$ denotes the set of pairs $\{ vx : x \in X\}$.
\[def:Zykov\] Let $T_1,\ldots,T_\ell$ be (disjoint) trees, and let $T_j$ have bipartition $A_j
\dcup B_j$. We define $Z_\ell(T_1,\ldots,T_\ell)$ to be the graph on vertex set $$V\big( Z_\ell(T_1,\ldots,T_\ell) \big) := \bigg( \bigcup_{j \in
[\ell]} A_j \cup B_j \bigg) \cup \big\{ u_I \colon I \subset [\ell] \big\}$$ and with edge set $$E\big( Z_\ell(T_1,\ldots,T_\ell) \big) \, := \, \bigcup_{j = 1}^\ell
\Bigg(E(T_j) \cup \bigcup_{j \in I \subset [\ell]} K\big( u_I,A_j \big) \cup
\bigcup_{j \not\in I \subset [\ell]} K\big( u_I,B_j \big) \Bigg).$$ For each $r \ge 3$ and $t \in {\mathbb{N}}$, the *modified Zykov graph* $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ is the graph obtained from $Z_\ell(T_1,\ldots,T_\ell)$ by performing the following two operations:
1. Add vertices $W = \{ w_1, \dots, w_{r-3} \}$, and all edges with an endpoint in $W$.
2. Blow up each vertex $u_I$ with $I \subset [\ell]$ and each vertex $w_j$ in $W$ to a set $S_I$ or $S'_j$, respectively, of size $t$.
Finally, we shall write $Z_\ell^{r,t}$ for the modified Zykov graph obtained when each tree $T_i$, $i\in[\ell]$, is a single edge; that is, $Z_\ell^{r,t} = Z_\ell^{r,t}(e_1,\ldots,e_\ell)$.
Note that $Z_\ell^{r,t}$ has $(2^\ell + r - 3)t + 2\ell$ vertices, and that, in the special case $r = 3$ and $t = 1$, the graph $Z_\ell^{r,t}$ coincides with that obtained by Zykov’s construction (described above) applied to a matching of size $\ell$.
The following observation motivates (and follows immediately from) Definition \[def:Zykov\].
\[acy=Zyk\] Let $\chi(H) = r$. Then $H$ is $r$-near-acyclic if and only if there exist trees $T_1,\ldots,T_\ell$ and $t \in {\mathbb{N}}$ such that $H$ is a subgraph of $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$.
Recall that $H$ is $r$-near-acyclic if and only if there exist $r-2$ independent sets $U_1,\ldots,U_{r-3},W$ such that $H \setminus \big(W\cup\bigcup_j U_j\big)$ is a forest $F$ whose components are trees $T_1,\ldots,T_\ell$ with the following property. For each $i\in[\ell]$, there is no vertex of $W$ adjacent to vertices in both partition classes of $T_i$. If $H\subset Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ then we can take $W =
\bigcup_{I \subset [\ell]}S_I$ and $U_1,\ldots,U_{r-3}$ to be the sets $S'_1,\ldots,S'_{r-3}$, and so $H$ is $r$-near-acyclic, as claimed. Conversely if $H$ is $r$-near-acyclic, then $H \subset Z_\ell^{r,t}(T_1,\ldots,T_\ell)$, where $t = |H|$ and $T_1, \ldots, T_\ell$ are the components of the forest $F$.
It will be convenient for us to provide a compact piece of notation for the adjacencies in $Z_\ell^{r,t}$. For this purpose, given a graph $G$ and a set $Y\subset
V(G)$, and integers $\ell,t \in {\mathbb{N}}$ and $r \ge 3$, define $\cG^{r,t}_\ell
(Y)$ to be the collection of functions $$S \,:\, 2^{[\ell]} \cup [r-3] \,\to\, \binom{Y}{t}\,.$$ It is natural to think of $S$ as a family $\{S_I : I \subset [\ell]\} \cup \{
S'_j : j \in [r-3]\}$ of subsets of $Y$ of size $t$. We say that $S \in
\cG_\ell^{r,t}(Y)$ is *proper* if these sets are pairwise disjoint and $E(G)$ contains all edges $xy$ with $x\in S_I\cup S'_j$ and $y\in S'_{j'}$, whenever $j\neq j'$. We shall write $\cF_\ell^{r,t}(Y)$ for the collection of proper functions in $\cG_\ell^{r,t}(Y)$. The idea behind this definition is that we will later want to consider a vertex set $Y\subset V(G)$ and a family of disjoint subsets $\{S_I\colon I\subset[\ell]\}\cup\{S'_j\colon j\in[r-3]\}$ of size $t$ in $Y$ that we want to extend to a copy of $Z_\ell^{r,t}$.
For an ordered pair $(x,y)$ of vertices of $G$, a function $S\in\cF_\ell^{r,t}(Y)$, and $i \in [\ell]$, we write $(x,y) \to_i S$, if $S'_j \subset N(x,y)$ for every $j \in [r-3]$ and $$\bigcup_{I \,:\, i \in I} S_I \subset N(x) \qquad \textup{and} \qquad
\bigcup_{I \,:\, i \not\in I} S_I \subset N(y)\,.$$ For an edge $e=xy\in
E(G)$, we write $e\to_i S$ if either $(x,y)\to_i S$ or $(y,x)\to_i S$. Recall that ${\mathbf{e}^{\ell}}$ denotes the $\ell$-tuple $(e_1,\ldots,e_\ell)$, with ${\mathbf{e}^{0}}$ the empty tuple. Define $${\mathbf{e}^{\ell}} \to S \qquad \Leftrightarrow \qquad e_i
\to_i S \quad \textup{for each $i \in [\ell]$}\,.$$ Observe that the graph $Z_\ell^{r,t}$ consists of a set of pairwise disjoint edges $e_1,\ldots,e_\ell$ and an $S \in \cF_\ell^{r,t}(Y)$ such that ${\mathbf{e}^{\ell}} \to S$. An advantage of this notation is that we can write ${\mathbf{e}^{\ell}}\to S$ even if the edges in ${\mathbf{e}^{\ell}}$ are not pairwise disjoint. This will greatly clarify our proofs.
In Section \[VCsec\], we shall show how to find a well-structured set of many copies of $Z_\ell^{r,t}$ inside a graph with high minimum degree and high chromatic number. The following definition (in which we shall make use of the compact notation just defined) makes the concept of ‘well-structured’ precise. Recall that, given $X \subset V(G)$, we write $E(X)$ for the edge set of $G[X]$, and that if $D \subset E(G)$, then $\delta(D)$ denotes the minimum degree of the graph $G[D]$.
\[def:rich\] Let $X$ and $Y$ be disjoint vertex sets in a graph $G$, let $C \in {\mathbb{N}}$ and $\alpha > 0$, and let $s := (2^\ell+ r - 3) t$. We say that $(X,Y)$ is *$(C,\alpha)$-rich* in copies of $Z_\ell^{r,t}$ if $$\begin{aligned}
& \exists \, D=D({\mathbf{e}^{0}})\subset E(X) \; \forall \, e_1\in D \; \exists \,
D({\mathbf{e}^{1}})\subset E(X) \; \forall \, e_2 \in D({\mathbf{e}^{1}}) \quad \dots \\ &
\hspace{2cm} \dots \quad \forall \, e_{\ell-1} \in D({\mathbf{e}^{\ell-2}}) \;
\exists \, D({\mathbf{e}^{\ell-1}}) \subset E(X) \; \forall \, e_\ell \in
D({\mathbf{e}^{\ell-1}})\end{aligned}$$ the following properties hold:
1. \[def:rich:a\] $\delta(D), \delta\big(D({\mathbf{e}^{1}})\big), \dots,
\delta\big(D({\mathbf{e}^{\ell-1}})\big) > C$, and
2. \[def:rich:b\] $\left|
\big\{ S \in \cF_\ell^{r,t}(Y) \,:\, {\mathbf{e}^{\ell}} \to S \big\} \right| \ge
\alpha |Y|^s$.
If $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$, then, for each $q \in [\ell]$, define $$\label{def:Dq}
\cD_q(X,Y) \, := \, \Big\{ {\mathbf{e}^{q}} \in E(X)^q \,:\, e_{j} \in D({\mathbf{e}^{j-1}})
\textup{ for each } j \in [q] \Big\},$$ where $D({\mathbf{e}^{0}}) := D$.
The aim of this section is to prove the following proposition, which says that if some pair $(X,Y)$ in $G$ is $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$ (where $\alpha > 0$ and $C$ is sufficiently large), then for any ‘small’ $T_1,\ldots,T_\ell$ we have $Z_\ell^{r,t}(T_1,\ldots,T_\ell) \subset G$, and hence (by Observation \[acy=Zyk\]) $G$ is not $H$-free for any $r$-near-acyclic graph $H$.
\[lem:rich-H\] Let $G$ be a graph, and let $X$ and $Y$ be disjoint subsets of its vertices. Let $r,\ell, t \in {\mathbb{N}}$, with $r \ge 3$, and let $\alpha > 0$. Let $T_1,\ldots,T_\ell$ be trees, and set $C := 2^{\ell+3} \alpha^{-1}
\sum_{i=1}^{\ell} |T_i|$. If $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$, then $Z_\ell^{r,t}(T_1,\ldots,T_\ell) \subset G$.
The proof of Proposition \[lem:rich-H\] uses a double counting argument and proceeds by induction. We shall find a set of functions $\cS
\subset \cF^{r,t}_\ell(Y)$ such that, for each $S \in \cS$, we can construct (one by one) a collection of subgraphs $E_1,\ldots,E_\ell$ of $G[X]$ with the following properties: each subgraph has large average degree, and for *any* choice $e_1 \in E_1,\ldots,e_\ell \in E_\ell$, we have ${\mathbf{e}^{\ell}} \to S$. Recall that this simply says that $e_i \to_i
S$ for every $e_i \in E_i$.
Let the graph $G$, disjoint subsets $X,Y \subset V(G)$, constants $\alpha > 0$ and $r, \ell, t \in {\mathbb{N}}$ with $r \ge 3$, and trees $T_1,\ldots,T_\ell$ be fixed for the rest of the section. Set $C := 2^{\ell+3} \alpha^{-1}
\sum_{i=1}^{\ell} |T_i|$ and $s := (2^\ell + r - 3)t$, and let $0 \le q \le
\ell$. For our induction hypothesis we use the following definition.
\[def:good\] A function $S \in \cF_\ell^{r,t}(Y)$ is *$(r,\ell,t,C,\alpha)$-good* for a tuple ${\mathbf{e}^{q}}$ and $(X,Y)$ if there exist sets $$E_{q+1},\ldots,E_\ell \subset E(X), \quad \textup{with} \quad
{\overline}{d}(E_j) \ge 2^{-\ell} \alpha C \quad \textup{ for each $q + 1 \le j
\le \ell$,}$$ such that for every $e_{q+1} \in E_{q+1}, \ldots, e_\ell
\in E_\ell$, we have ${\mathbf{e}^{\ell}} \to S$.
When the constants $(r,\ell,t,C,\alpha)$ and the sets $(X,Y)$ are clear from the context, we shall omit them. We shall abbreviate ‘$(r,\ell,t,C,\alpha)$-good for ${\mathbf{e}^{0}}$ and $(X,Y)$’ to ‘$(r,\ell,t,C,\alpha)$-good for $(X,Y)$’.
The pair $(X,Y)$ is *$(C,\alpha)$-dense* in copies of $Z_\ell^{r,t}$ if there exist at least $2^{-\ell} \alpha |Y|^s$ families $S \in \cF(Y)$ which are $(r,\ell,t,C,\alpha)$-good for $(X,Y)$.
The next lemma constitutes the inductive argument in the proof of Proposition \[lem:rich-H\]. The final assertion we shall also need in Section \[proofsec\].
\[q-induc\] For any $0\le q\le\ell$, if $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$, then $$\left| \big\{ S \in \cF_\ell^{r,t}(Y) \,:\, S \text{ is
good for } {\mathbf{e}^{q}} \big\} \right| \, \ge \, 2^{q-\ell} \alpha
|Y|^s$$ for every ${\mathbf{e}^{q}} \in \cD_q(X,Y)$.
In particular, if $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$, then $(X,Y)$ is $(C,\alpha)$-dense in copies of $Z_\ell^{r,t}$.
The proof is by induction on $\ell - q$. The base case, $q = \ell$, follows immediately from the definition of $(C,\alpha)$-rich. Indeed, a function $S$ is good for ${\mathbf{e}^{\ell}}$ if and only if ${\mathbf{e}^{\ell}}
\to S$, and by Property \[def:rich:b\] in Definition \[def:rich\], we have $\left| \big\{ S \in \cF_\ell^{r,t}(Y) \,:\, {\mathbf{e}^{\ell}} \to S
\big\} \right| \ge \alpha |Y|^s$.
So let $0 \le q < \ell$, assume that the lemma holds for $q + 1$, and let ${\mathbf{e}^{q}} \in \cD_q(X,Y)$. Set $\beta=\alpha 2^{q-\ell}$. By Definition \[def:rich\], there exists a set $D({\mathbf{e}^{q}}) \subset
E(X)$ with $\delta\big( D({\mathbf{e}^{q}}) \big) > C$, such that ${\mathbf{e}^{q+1}}
\in \cD_{q+1}(X,Y)$ for every $e_{q+1} \in D({\mathbf{e}^{q}})$. Thus, by the induction hypothesis, $$\label{eq:goodS}
\forall e_{q+1} \in D({\mathbf{e}^{q}})\text{ at least } 2 \beta
|Y|^s\text{ functions }S \in \cF_\ell^{r,t}(Y)\text{ are good for }
{\mathbf{e}^{q+1}}\,,$$ that is, for each such $S$ there exist sets $$E_{q+2},\ldots,E_\ell
\subset E(X), \quad \textup{with} \quad {\overline}{d}(E_j) \ge 2^{-\ell} \alpha
C \quad \textup{ for each $q + 2 \le j \le \ell$,}$$ such that for every $e_{q+2} \in E_{q+2}, \ldots, e_\ell \in E_\ell$, we have ${\mathbf{e}^{\ell}}
\to S$. It is crucial to observe that given $S$, if the edge sets $E_{q+2},\ldots,E_\ell$ have this last property for *some* $e_{q+1}\in D({\mathbf{e}^{q}})$ with ${\mathbf{e}^{q+1}}\to S$, then $E_{q+2},\ldots,E_{\ell}$ have this property for *all* $e_{q+1}\in
D({\mathbf{e}^{q}})$ with ${\mathbf{e}^{q+1}}\to S$.
The $S \in \cF_\ell^{r,t}(Y)$ that will be good for ${\mathbf{e}^{q}}$ are those which are good for many ${\mathbf{e}^{q+1}}$. More precisely, for each $S
\in \cF_\ell^{r,t}(Y)$, let $$W_S \, := \, \big\{ e_{q+1} \in
D({\mathbf{e}^{q}}) \,:\, S \textup{ is good for } {\mathbf{e}^{q+1}} \big\},$$ and let $Z = \big\{ S \in \cF_\ell^{r,t}(Y) \,:\, |W_S| \ge \beta
|D({\mathbf{e}^{q}})| \big\}$. By the number of pairs $(e_{q+1},S)$ with $e_{q+1}$ in $W_S$ is at least $|D({\mathbf{e}^{q}})| \cdot
2 \beta |Y|^s$. On the other hand, for every $S\in Z$ there are at most $|D({\mathbf{e}^{q}})|$ pairs $(e_{q+1},S)$ with $e_{q+1}$ in $W_S$, and for every $S\in\cF_\ell^{r,t}(Y)\setminus Z$, there are (by definition of $Z$) at most $\beta|D({\mathbf{e}^{q}})|$ such pairs. Putting these together, we obtain $$|D({\mathbf{e}^{q}})|\cdot 2\beta |Y|^s\le |D({\mathbf{e}^{q}})||Z|+\beta
|D({\mathbf{e}^{q}})||Y|^s$$ and hence $|Z| \ge \beta |Y|^s$.
We claim that every $S \in Z$ is good for ${\mathbf{e}^{q}}$. Indeed, fix $S\in
Z$. Set $E_{q+1} = W_S$, and let $E_{q+2},\ldots,E_\ell$ be the sets defined above (for any, and thus all, $e_{q+1}\in E_{q+1}$), i.e., those obtained by the induction hypothesis. Since $S \in Z$ we have $|W_S| \ge
\beta |D({\mathbf{e}^{q}})|$, so it follows from $\delta\big( D({\mathbf{e}^{q}})
\big) > C$ that ${\overline}{d}(E_{q+1}) \ge \beta C \ge 2^{-\ell} \alpha C$. Since $S$ is good for ${\mathbf{e}^{q+1}}$ for every $e_{q+1} \in E_{q+1}$, we have $e_i \to_i S$ for every $1 \le i \le q+1$, and by the induction hypothesis, we have $e_i \to_i S$ for every $e_i \in E_i$ and every $q+2
\le i \le \ell$. Thus ${\mathbf{e}^{\ell}} \to S$ for every such ${\mathbf{e}^{\ell}}$, as required. Since $|Z| \ge \beta |Y|^s$, this completes the induction step, and hence the proof of the lemma.
Lemma \[q-induc\] shows that richness in copies of $Z_\ell^{r,t}$ implies denseness in copies of $Z_\ell^{r,t}$. Observe that if $(X,Y)$ is dense in copies of $Z_\ell^{r,t}$, then in particular there is a function $S\in\cF_\ell^{r,t}(Y)$ which is good for $(X,Y)$. The next lemma now shows that in this case we have $Z_\ell^{r,t}(T_1,\ldots,T_\ell)\subset G$.
\[lem:denseembed\] Let $X$ and $Y$ be disjoint vertex sets in $G$. Given $r,\ell,t\in{\mathbb{N}}$, $\alpha>0$, and trees $T_1,\ldots,T_\ell$, if $C\ge 2^{\ell+3}\alpha^{-1}\sum_{i=1}^\ell |T_i|$ and $S\in\cF_\ell^{r,t}(Y)$ is $(r,\ell,t,C,\alpha)$-good for $(X,Y)$, then $Z_\ell^{r,t}(T_1,\ldots,T_\ell)\subset G$.
Let $S$ be $(r,\ell,t,C,\alpha)$-good for $(X,Y)$. Then there exist sets $$E_{1},\ldots,E_\ell \subset E(X), \quad \textup{with} \quad {\overline}{d}(E_j) \ge
2^{-\ell} \alpha C \quad \textup{ for each $1 \le j \le \ell$,}$$ such that for every $e_{1} \in E_{1}, \ldots, e_\ell \in E_\ell$, we have ${\mathbf{e}^{\ell}} \to S$.
For each $j \in [\ell]$ and each edge $e \in E_j$, let $e=xy$ be such that $(x,y) \to_j S$, and orient the edge $e$ from $x$ to $y$. Recall that $C \ge
2^{\ell+3} \alpha^{-1} \sum_{i=1}^{\ell} |T_i|$, and so ${\overline}{d}(E_j) \ge 8 \sum_{i=1}^{\ell} |T_i|$ for each $j \in [\ell]$. For each $j\in[\ell]$, by choosing a maximal bipartite subgraph of $E_j$, and then removing at most half the edges, we can find a set $E'_j\subset E_j$ such that,
1. $E'_j$ is bipartite, with bipartition $(A_j,B_j)$,
2. every edge $e \in E'_j$ is oriented from $A_j$ to $B_j$, and
3. $\smash{{\overline}{d}(E'_j) \ge 2 \sum_{i=1}^{\ell} |T_i|}$.
Thus, by Fact \[prop:forest\], there exists, for each $j \in [\ell]$, a copy $T'_j$ of $T_j$ in $$E'_j \,-\, \big( V(T'_1) \cup \ldots \cup V(T'_{j-1})
\big),$$ since removing a vertex can only decrease the average degree by at most two. These trees, together with $S$, form a copy of $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ in $G$, so we are done.
It is now easy to deduce Proposition \[lem:rich-H\] from Lemma \[q-induc\] and Lemma \[lem:denseembed\].
Let $(X,Y)$ be $(C,\alpha)$-rich in copies of $Z_\ell^{r,t}$, and apply Lemma \[q-induc\] to $(X,Y)$ with $q = 0$. Note that $ \cD_0(X,Y) = \{{\mathbf{e}^{0}} \}$ consists of the tuple of length zero, and let $\cS = \big\{ S \in \cF_\ell^{r,t}(Y) \,:\,
S \textup{ is good for } {\mathbf{e}^{0}} \big\}$. Then $|\cS| \ge \alpha 2^{-\ell} |Y|^s$, and so in particular $\cS$ is non-empty. Let $S\in\cS$, and apply Lemma \[lem:denseembed\] to obtain a copy of $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ in $G$.
The paired VC-dimension argument {#VCsec}
================================
In this section we shall modify and extend a technique which was introduced by [Ł]{}uczak and Thomassé [@LT], and used by them to prove Conjecture \[LTconj\] in the case where $H$ is near-bipartite. This technique is based on the concept of *paired VC-dimension*, which generalises the well-known Vapnik-Červonenkis dimension of a set-system (see [@Sauer; @VC]). We shall not state our proof in the abstract setting of paired VC-dimension, which is more general than that which we shall require, but we refer the interested reader to [@LT] for the definition and further details.
We shall use the paired VC-dimension (or ‘booster tree’) argument of [Ł]{}uczak and Thomassé in order to prove the following result, which may be thought of as a ‘counting version’ of Theorem 5 in [@LT]. The case $r = 3$ of Theorem \[mainthm\] will follow as an easy consequence of Propositions \[lem:VC\] and \[lem:rich-H\] (see Section \[proofsec\]).
\[lem:VC\] For every $\ell,t \in {\mathbb{N}}$ and $d > 0$, there exists $\alpha > 0$ such that, for every $C \in {\mathbb{N}}$, there exists $C' \in {\mathbb{N}}$ such that the following holds. Let $G$ be a graph and let $X$ and $Y$ be disjoint subsets of $V(G)$, such that $|N(x) \cap Y| \ge d|Y|$ for every $x \in X$.
Then either $\chi\big( G[X] \big) \le C'$, or $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$.
In order to prove Proposition \[lem:VC\], we shall break $X$ up into a bounded number of suitable pieces, $X_1,\ldots,X_m$, and show that either $G[X_j]$ has bounded chromatic number, or $(X_j,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$.
Let $d(x,Y):=d\big(\{x\},Y\big) = \big|N(x)\cap Y\big|/|Y|=
e\big(\{x\},Y\big)/|Y|$ be the density of the neighbours of $x$ in $Y$. The key definition, which will allow us to choose the sets $X_j$, is as follows.
\[def:boost\] Let $G$ be a graph, let $X$ and $Y$ be disjoint subsets of $V(G)$, and let $\eps
> 0$. We say that $x \in X$ is *$\eps$-boosted* by $Y' \subset Y$ if $d(x,Y') \ge (1 + \eps) d(x,Y)$.
Now let $C,p \in {\mathbb{N}}$, and let $\beta > 0$. Let $Y_1 \dcup
\ldots \dcup Y_p$ of $Y$ be a partition of $Y$, and $X_0\dcup\ldots\dcup X_p$ be a partition of $X$. We say that $(X_0,\emptyset),(X_1,Y_1),\ldots,(X_p,Y_p)$ is a *$(p,C,\eps,\beta)$-booster* of $(X,Y)$ if
1. \[def:boost:a\] $G[X_0]$ is $C$-degenerate.
2. \[def:boost:b\] Every $x \in X_j$ is $\eps$-boosted by $Y_j$, for each $j \in [p]$.
3. \[def:boost:c\] $|Y_j| \ge \beta |Y|$ for every $j \in [p]$.
We say that a partition $\{Y_1,\ldots,Y_p\}$ of $Y$ induces a $(p,C,\eps,\beta)$-booster of $(X,Y)$ if there exists a partition $X_0\dcup\ldots\dcup X_p$ of $X$ such that $(X_0,\emptyset),(X_1,Y_1),\ldots,(X_p,Y_p)$ is a *$(p,C,\eps,\beta)$-booster* of $(X,Y)$.
We remark that this is slightly different from the definition of a $p$-booster in Section 5 of [@LT], where Condition \[def:boost:a\] was replaced by ‘$G[X_0]$ is independent’, and Condition \[def:boost:c\] was missing.
Using Definition \[def:boost\], we can now state the second key definition.
\[def:boostree\] Let $C,p_0 \in {\mathbb{N}}$ and $\beta,\eps > 0$. A *$(p_0,C,\eps,\beta)$-booster tree* for $(X,Y)$ is an oriented rooted tree $\cT$, whose vertices are pairs $(X',Y')$ such that $X' \subset X$ and $Y' \subset Y$, and all of whose edges are oriented away from the root, such that the following conditions hold:
1. \[def:boostree:a\] The root of $\mathcal{T}$ is $(X,Y)$.
2. \[def:boostree:b\] No vertex of $\mathcal{T}$ has more than $p_0$ out-neighbours.
3. \[def:boostree:c\] The out-neighbourhood of each non-leaf $(X',Y')$ of $\mathcal{T}$ forms a $(p,C,\eps,\beta)$-booster of $(X',Y')$ for some $p\le p_0$.
4. \[def:boostree:d\] If $(X',Y')$ is a leaf of $\cT$, then either $G[X']$ is $C$-degenerate, or there does not exist a $(p,C,\eps,\beta)$-booster for $(X',Y')$ for any $p \le p_0$.
A vertex $(X',Y')$ of $\mathcal{T}$ is called $\emph{degenerate}$ if it is a leaf of $\mathcal{T}$ and $G[X']$ is $C$-degenerate.
The following lemma is immediate from the definitions.
\[exists-booster\] Let $C,p_0 \in {\mathbb{N}}$, and $\beta,\eps > 0$, let $G$ be a graph, and let $X$ and $Y$ be disjoint subsets of $V(G)$. If $d(x,Y) \ge d$ for every $x\in X$, then there exists a $(p_0,C,\eps,\beta)$-booster tree $\cT$ for $(X,Y)$ such that $|\cT|$ is bounded as a function of $\eps$, $d$ and $p_0$.
Moreover, if $(X',Y')$ is a non-degenerate vertex of $\mathcal{T}$, then $d(x,Y')\ge d$ for all $x\in X'$, and $|Y'| \ge \beta^{|\cT|}|Y|$.
We construct $\mathcal{T}$, with root $(X,Y)$, as follows: We simply repeatedly choose a $(p,C,\eps,\beta)$-booster $(X_0,\emptyset),(X_1,Y_1),\ldots,(X_p,Y_p)$ for each leaf $(X',Y')$ of $\mathcal{T}$ such that $G[X']$ is not $C$-degenerate, until this is no longer possible for any $p\le p_0$. We add to $\mathcal{T}$ the vertices $(X_0,\emptyset),(X_1,Y_1),\ldots,(X_p,Y_p)$ as out-neighbours of $(X',Y')$.
By the definition of ‘$\eps$-boosted’ and the construction of $\mathcal{T}$, if $(X',Y')$ is a non-degenerate vertex of $\mathcal{T}$ at distance $t$ from the root, we have $d(x,Y')\ge
(1+\eps)^t d$ for every $x\in X'$, and we have $|Y'|\ge\beta^t|Y|\ge\beta^{|\cT|}|Y|$. This both establishes that the height $h(\mathcal{T})$ of $\cT$ is bounded in terms of $\eps$ and $d$, and that we have $d(x,Y')\ge d$ for every $x\in
X'$. Since $\mathcal{T}$ has no vertex of out-degree greater than $p_0$ and $h(\cT)$ is bounded by a function of $\eps$ and $d$, it follows that $|\mathcal{T}|$ is bounded as a function of $\eps$, $d$ and $p_0$.
The following lemma is the key step in the proof of Proposition \[lem:VC\].
\[no-p-booster\] Let $\ell,t \in {\mathbb{N}}$ and $d > 0$. Let $\beta=\big(\tfrac{d}{4}\big)^\ell$ and $\eps=\beta/2$. There exists $\alpha > 0$ such that the following holds for every $C \in {\mathbb{N}}$. Let $G$ be a graph, let $X$ and $Y$ be disjoint subsets of $V(G)$, and suppose that $|N(x) \cap Y| \ge d|Y|$ for every $x \in X$.
If there does not exist a $(p,C,\eps,\beta)$-booster of $(X,Y)$ for any $p
\le 2^\ell$, then $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$.
Lemma \[no-p-booster\] is proved by repeating a fairly straightforward algorithm $\ell$ times, at each step $q\in[\ell]$ finding a set $D({\mathbf{e}^{q}})$ as in the definition of $(C,\alpha)$-richness (see Definition \[def:rich\]). In order to make the proof more transparent, we shall state a slightly more technical lemma, which is proved by induction on $q$, and from which Lemma \[no-p-booster\] follows immediately.
The following definition will simplify the statement. It is a slight strengthening of the concept of $(C,\alpha)$-richness in the case $r = 3$. Recall that ${\mathbf{e}^{\ell}}$ is just a shorthand for $(e_1,\ldots,e_\ell)$ and ${\mathbf{e}^{0}}$ is the empty tuple. Further, recall the definitions of $\cF_\ell^{r,t}$ and ${\mathbf{e}^{\ell}}\to S$ from Section \[Zykovsec\].
\[def:CaZykov\] Let $X$ and $Y$ be disjoint vertex sets in a graph $G$, let $C,\ell \in {\mathbb{N}}$ and $\alpha > 0$. We say that $(X,Y)$ is *$(C,\alpha,\ell)$-Zykov* if $$\begin{aligned}
& \exists \, D=D({\mathbf{e}^{0}}) \subset E(X) \; \forall \, e_1\in D \; \exists
\, D({\mathbf{e}^{1}})\subset E(X) \; \forall \, e_2 \in D({\mathbf{e}^{1}}) \quad
\dots \\ & \hspace{2cm} \dots \quad \forall \, e_{\ell-1} \in
D({\mathbf{e}^{\ell-2}}) \; \exists \, D({\mathbf{e}^{\ell-1}}) \subset E(X) \; \forall
\, e_\ell \in D({\mathbf{e}^{\ell-1}})
\end{aligned}$$ the following properties hold:
1. \[def:CaZykov:a\] $\delta\big( D \big),
\delta\big(D({\mathbf{e}^{1}})\big), \dots, \delta\big(D({\mathbf{e}^{\ell-1}})\big) >
C$, and
2. \[def:CaZykov:b\] $\exists \, S \in \cF^{3,\alpha |Y|}_\ell(Y)$ such that ${\mathbf{e}^{\ell}} \to S$.
We remark that the requirement \[def:rich:b\] of Definition \[def:rich\] that each tuple ${\mathbf{e}^{\ell}}$ should extend to *many* copies of $Z_\ell^{3,t}$ is replaced in this definition by the requirement that ${\mathbf{e}^{\ell}}$ should extend to *one* much bigger copy of $Z_\ell^{3,\alpha|Y|}$. In particular, if $(X,Y)$ is $(C,\alpha,\ell)$-Zykov, then, for any $t \in {\mathbb{N}}$, it is $(C,\alpha')$-rich in copies of $Z_\ell^{3,t}$, where $\alpha' = \left(
\frac{\alpha}{t} \right)^{2^\ell t}$ (this is shown in the proof of Lemma \[no-p-booster\]).
\[no-boost-tech\] Let $\ell \in {\mathbb{N}}$ and $d > 0$. For $\beta=\left( \frac{d}{4}
\right)^\ell$ and $\eps=\beta / 2$, the following holds for every $C \in {\mathbb{N}}$. Let $G$ be a graph, let $X$ and $Y$ be disjoint subsets of $V(G)$, and suppose that $|N(x) \cap Y| \ge d|Y|$ for every $x \in X$.
If there does not exist a $(p,C,\eps,\beta)$-booster of $(X,Y)$ for any $p
\le 2^\ell$, then $(X,Y)$ is $(C,\alpha_q,q)$-Zykov for every $q\in[\ell]$, where $\alpha_q = \left( \frac{d}{4} \right)^q$.
Let $C,\ell \in {\mathbb{N}}$ and $d > 0$, and let $\beta = \left( \frac{d}{4}
\right)^\ell$ and $\eps = \beta / 2$. Let $G$ and $X,Y$ be as described in the statement, and suppose that there does not exist a $(p,C,\eps,\beta)$-booster of $(X,Y)$ for any $p \le 2^\ell$. We proceed by induction.
We begin with the base case, $q = 1$. We are required to find a set $D =
D({\mathbf{e}^{0}}) \subset E(X)$, with $\delta(D) > C$, such that, for every $e_1=xy \in D$, there exists $S({\mathbf{e}^{1}})\in\cF_1^{3,\alpha_1|Y|}$ such that ${\mathbf{e}^{1}}\to S({\mathbf{e}^{1}})$; that is, there exist disjoint sets $S_\emptyset({\mathbf{e}^{1}})$ and $S_{\{1\}}({\mathbf{e}^{1}})$ in $Y$, both of size $\alpha_1|Y|=\frac{d}{4} |Y|$, such that $S_{\{1\}}({\mathbf{e}^{1}})\subset N(x)$ and $S_\emptyset({\mathbf{e}^{1}})\subset N(y)$. Since there is no $(1,C,\eps,\beta)$-booster of $(X,Y)$, it follows that $G[X]$ is not $C$-degenerate, and so there exists a subgraph $G_0 \subset G[X]$ with $\delta(G_0) > C$. We choose $D:=E(G_0)$. Now for each $e_1 = x y \in
D$, let $A_\emptyset({\mathbf{e}^{1}}): = N(y) \cap Y$ and $A_{\{1\}}({\mathbf{e}^{1}}): = N(x) \cap Y$. Since $|A_\emptyset({\mathbf{e}^{1}})|,|A_{\{1\}}({\mathbf{e}^{1}})| \ge d|Y|$ by the assumption of the lemma, there exist disjoint sets $S_\emptyset({\mathbf{e}^{1}}) \subset A_\emptyset({\mathbf{e}^{1}})$ and $S_{\{1\}}({\mathbf{e}^{1}}) \subset A_{\{1\}}({\mathbf{e}^{1}})$ with $|S_\emptyset({\mathbf{e}^{1}})|,|S_{\{1\}}({\mathbf{e}^{1}})| = \frac{d}{4} |Y|$, as required.
For the induction step, let $1 < q \le \ell$ and assume that the result holds for $q - 1$. By this induction hypothesis $$\begin{aligned}
& \exists D({\mathbf{e}^{0}}) \subset E(X) \forall \, e_1\in D({\mathbf{e}^{0}}) \; \exists \, D({\mathbf{e}^{1}})\subset E(X)
\; \forall \, e_2 \in D({\mathbf{e}^{1}}) \quad \dots \\
& \hspace{1cm} \dots \quad \forall \, e_{q-2} \in D({\mathbf{e}^{q-3}}) \;
\exists \, D({\mathbf{e}^{q-2}}) \subset E(X) \; \forall \, e_{q-1} \in
D({\mathbf{e}^{q-2}})
\end{aligned}$$ we have
1. \[no-boost-tech:a\] $\delta\big( D({\mathbf{e}^{0}}) \big),
\delta\big(D({\mathbf{e}^{1}})\big), \dots, \delta\big(D({\mathbf{e}^{q-2}})\big) >
C$, and
2. \[no-boost-tech:b\] $\exists \, S({\mathbf{e}^{q-1}}) \in
\cF^{3,\alpha_{q-1} |Y|}_{q-1}(Y)$ such that ${\mathbf{e}^{q-1}} \to
S({\mathbf{e}^{q-1}})$.
As in Definition \[def:rich\], set $$\cD_q(X,Y) \, := \, \Big\{ {\mathbf{e}^{q}} \in E(X)^q \,:\, e_{j} \in
D({\mathbf{e}^{j-1}}) \textup{ for each } j \in [q] \Big\}\,.$$ We shall show that for every ${\mathbf{e}^{q-1}}\in \cD_{q-1}(X,Y)$, there exists a set of edges $D({\mathbf{e}^{q-1}}) \subset E(X)$, with $\delta\big(D({\mathbf{e}^{q-1}})\big) > C$, such that for every $e_q \in
D({\mathbf{e}^{q-1}})$ there exists an $S({\mathbf{e}^{q}}) \in \cF^{3,\alpha_q
|Y|}_{q}(Y)$ such that ${\mathbf{e}^{q}} \to S({\mathbf{e}^{q}})$.
Indeed, given ${\mathbf{e}^{q-1}}\in\cD_{q-1}(X,Y)$, by \[no-boost-tech:b\] there exists $$\big\{ S_I({\mathbf{e}^{q-1}}) \subset Y : I \subset [q-1] \big\} \, = \, S({\mathbf{e}^{q-1}})\, \in \, \cF^{3,\alpha_{q-1} |Y|}_{q-1}(Y)$$ with ${\mathbf{e}^{q-1}} \to S({\mathbf{e}^{q-1}})$. In particular, note that by definition of $\cF^{3,\alpha_{q-1} |Y|}_{q-1}(Y)$, the sets $S_I({\mathbf{e}^{q-1}})$ are disjoint, and that $|S_I({\mathbf{e}^{q-1}}) | =
\alpha_{q-1} |Y|$ for every $I \subset [q-1]$. Let $R = Y \setminus \bigcup_I
S_I({\mathbf{e}^{q-1}})$, and recall that $q \le \ell$, and that there is no $(p,C,\eps,\beta)$-booster of $(X,Y)$ for any $p \le 2^\ell$. Thus the partition $S({\mathbf{e}^{q-1}}) \cup \{R\}$ of $Y$ does not induce a $(p,C,\eps,\beta)$-booster of $(X,Y)$.
By our choice of $\beta$, we have $\beta\le\alpha_{q-1}$, and thus $|S_I({\mathbf{e}^{q-1}})| \ge \beta|Y|$ for all $I \subset [q-1]$. Similarly, since $2^{q-1}\alpha_{q-1} = 2^{q-1}(d/4)^{q-1} \le 1/2 < 1 - \beta$, we have $|R|>\beta |Y|$. Let $X' \subset X$ be the set of vertices which are not $\eps$-boosted by any of the sets $S({\mathbf{e}^{q-1}})\cup\{R\}$. Since $S({\mathbf{e}^{q-1}}) \cup \{R\}$ does not induce a $(2^{q-1}+1,C,\eps,\beta)$-booster of $(X,Y)$, the graph $G[X']$ is not $C$-degenerate, and hence there exists a set of edges $D({\mathbf{e}^{q-1}})
\subset E(X')$ such that $\delta\big( D({\mathbf{e}^{q-1}}) \big) > C$. We claim that this is the set we are looking for.
In order to verify this, let $e_q = x y \in D({\mathbf{e}^{q-1}})$ be arbitrary. Our task is to show that there exists $S({\mathbf{e}^{q}})\in\cF_q^{3,\alpha_q|Y|}$ such that ${\mathbf{e}^{q}}\to
S({\mathbf{e}^{q}})$. Recall that $x$ and $y$ are not $\eps$-boosted by $S({\mathbf{e}^{q-1}}) \cup \{R\}$. Hence $d(x,U) < (1 + \eps)d(x,Y)$ for each $U \in S({\mathbf{e}^{q-1}}) \cup \{R\}$, and so, for every $I \subset [q-1]$, $$\begin{split}\label{eq:boostcalc}
e\big(x,S_I({\mathbf{e}^{q-1}})\big)&\,=\,e(x,Y)-e\big(x,Y\setminus
S_I({\mathbf{e}^{q-1}})\big) \\
&\,\ge \, e(x,Y) - \big(1 + \eps\big) \left(1 - \left(
\tfrac{d}{4} \right)^{q-1} \right) e(x,Y)
\,\ge \, \frac{1}{2} \left( \frac{d}{4}\right)^{q-1} e(x,Y)\,,
\end{split}$$ where we used $|S_I({\mathbf{e}^{q-1}})| = \alpha_{q-1}|Y|=\left( \frac{d}{4}
\right)^{q-1} |Y|$ for the first inequality, and $\eps=\tfrac{1}{2}\big(\tfrac{d}{4}\big)^\ell$ for the second. By the same argument, $y$ has at least $\frac{1}{2} \left( \frac{d}{4} \right)^{q-1}
e(y,Y)$ neighbours in $S_I({\mathbf{e}^{q-1}})$ for each $I \subset [q-1]$.
Define, for each $I \subset [q]$, the set $A_I({\mathbf{e}^{q}}) \subset Y$ as follows: $$\begin{split}\label{eq:defA}
A_I({\mathbf{e}^{q}}): = N(x) \cap S_{I \setminus
\{q\}}({\mathbf{e}^{q-1}}) \;\, \quad \; &
\textup{ if } q \in I\\
A_I({\mathbf{e}^{q}}): = N(y) \cap S_I({\mathbf{e}^{q-1}}) \quad \quad \,\, & \textup{ if } q
\not\in I.
\end{split}$$ Since $e(x,Y), e(y,Y) \ge d |Y|$, we conclude from , that we have $|A_I({\mathbf{e}^{q}})| \ge 2 \left( \frac{d}{4} \right)^q |Y|$ for every $I \subset [q]$. Moreover, the sets $A_I({\mathbf{e}^{q}})$ and $A_J({\mathbf{e}^{q}})$ are disjoint unless $I \setminus \{q\} = J \setminus \{q\}$. Hence we may choose disjoint sets $S_I({\mathbf{e}^{q}}) \subset A_I({\mathbf{e}^{q}})$ with $|S_I({\mathbf{e}^{q}})| =
\left( \frac{d}{4} \right)^q |Y|$ for each $I \subset [q]$.
Let $S({\mathbf{e}^{q}}) = \big\{ S_I({\mathbf{e}^{q}}) : I \subset [q] \big\}$. We claim that this is the desired family; that is, that $S({\mathbf{e}^{q}}) \in
\cF^{3,\alpha_q |Y|}_{q}(Y)$ and ${\mathbf{e}^{q}} \to S({\mathbf{e}^{q}})$. Indeed, the sets $S_I({\mathbf{e}^{q}})$ are disjoint, and $$|S_I({\mathbf{e}^{q}})|= \left( \frac{d}{4} \right)^q |Y|=\alpha_q |Y|$$ for each $I \subset [q]$, by construction. Finally, we prove that ${\mathbf{e}^{q}}
\to S({\mathbf{e}^{q}})$, i.e., that $e_i \to_i S({\mathbf{e}^{q}})$ for each $i \in
[q]$. For $i \le q - 1$, this follows because ${\mathbf{e}^{q-1}} \to
S({\mathbf{e}^{q-1}})$, and $$S_I({\mathbf{e}^{q}}) \cup S_{I \cup \{q\}}({\mathbf{e}^{q}}) \subset
S_I({\mathbf{e}^{q-1}})$$ for every $I \subset [q-1]$ by . For $i = q$, it follows since $S_I({\mathbf{e}^{q}}) \subset N(x)$ if $q \in I\subset[q]$ and $S_I({\mathbf{e}^{q}}) \subset N(y)$ if $q \not\in I\subset[q]$ by . Hence $ {\mathbf{e}^{q}} \to S({\mathbf{e}^{q}})$, as required. This completes the induction step, and hence the proof of the lemma.
We can now easily deduce Lemma \[no-p-booster\].
By Lemma \[no-boost-tech\] (applied with $q = \ell$), it suffices to show that if $(X,Y)$ is $(C,\alpha_\ell,\ell)$-Zykov, then it is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$, where $\alpha_\ell=\big(\tfrac{d}{4}\big)^\ell$ and $\alpha = \left(
\frac{\alpha_\ell}{t} \right)^{2^\ell t}$. In other words, we want to prove that if there exists $S \in \cF^{3,\alpha_\ell |Y|}_\ell(Y)$ with ${\mathbf{e}^{\ell}} \to S$, then $$\left| \big\{ S' \in \cF_\ell^{3,t}(Y) \,:\,
{\mathbf{e}^{\ell}} \to S' \big\} \right| \; \ge \; \alpha |Y|^s\,,$$ where $s = 2^\ell t$. Indeed, this is true because $|S_I|=\alpha_\ell|Y|$ for $I\subset[\ell]$, and the number of ways of choosing, for each $I\subset[\ell]$, a $t$-subset of $S_I$ is $$\prod_{I\subset[\ell]}\binom{|S_I|}{t} \,=\, \binom{\alpha_\ell |Y|}{t}^{2^\ell}
\ge \, \left( \frac{\alpha_\ell|Y|}{t} \right)^{2^\ell t} = \, \alpha
|Y|^s\,,$$ as claimed.
It is now straightforward to prove Proposition \[lem:VC\].
Let $C,\ell,t \in {\mathbb{N}}$ and $d > 0$, and set $\beta = \left( \frac{d}{4}
\right)^\ell$ and $\eps = \beta / 2$. Let $G$ and $(X,Y)$ be as described in the statement, so $|N(x) \cap Y| \ge d|Y|$ for every $x \in
X$. By Lemma \[exists-booster\] there exists a $(2^\ell,C,\eps,\beta)$-booster tree for $(X,Y)$, and moreover $|\cT|$ is bounded as a function of $d$, $\eps$ and $\ell$.
Recall that the leaves of $\cT$ correspond to a partition of $X$ (and a partition of $Y$). If every leaf $(X',Y')$ of $\cT$ is degenerate then $\chi(G[X]) \le |\cT|(C+1)=:C'$, where $C'$ depends only upon $C$, $\ell$ and $d$. So we may assume that some leaf $(X',Y') \in V(\cT)$ is not degenerate.
By the definition of a $(2^\ell,C,\eps,\beta)$-booster tree, it follows that there is no $(p,C,\eps,\beta)$-booster of $(X',Y')$ for any $p \le
2^\ell$. Further, $|N(x)\cap Y'|=d(x,Y')|Y'|\ge d|Y'|$ for every $x\in
X'$ by Lemma \[exists-booster\]. Then, by Lemma \[no-p-booster\] (applied with $\ell$, $t$ and $d$), $(X',Y')$ is $(C,\alpha')$-rich in copies of $Z_\ell^{3,t}$, for some $\alpha' = \alpha'(d,\ell,t) >
0$. Since (again by Lemma \[exists-booster\]) $|Y'| \ge \beta^{|\cT|}
|Y|$, it follows that $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$, where $\alpha =\alpha'\beta^{|T|s}$ is a constant depending only on $d$, $\ell$ and $t$, as required.
The proof of Theorem \[mainthm\] {#proofsec}
================================
In this section we shall complete the proof of Theorem \[mainthm\]. As a warm-up, we begin with the case $r = 3$, which is an almost immediate consequence of the results of the last four sections.
The following theorem proves Conjecture \[LTconj\]. The proof does not use the Regularity Lemma; it follows from Propositions \[lem:rich-H\] and \[lem:VC\].
\[nearacyclic\] If $H$ is a near-acyclic graph, then $\delta_\chi(H) = 0$.
Let $H$ be a near-acyclic graph (so in particular $\chi(H) = 3$), let $\gamma
> 0$ be arbitrary, and let $G$ be an $H$-free graph on $n$ vertices, with $\delta(G) \ge 2\gamma n$. We shall prove that the chromatic number of $G$ is at most $C'$, for some $C' = C'(H,\gamma)$.
First, using Observation \[acy=Zyk\], choose $t \in {\mathbb{N}}$ and a collection $T_1,\ldots,T_\ell$ of trees such that $H \subset
Z_\ell^{3,t}(T_1,\ldots,T_\ell)$. Choose a maximal bipartition $(X,Y)$ of $G$, assume without loss of generality that $\chi(G[X]) \ge \chi(G[Y])$, and note that $|N(x) \cap Y| \ge \gamma |Y|$ for every $x \in X$.
Let $\alpha > 0$ be given by Proposition \[lem:VC\] (applied with $\ell$, $t$ and $\gamma$), let $C := 2^{\ell+3} \alpha^{-1} \sum_{i=1}^{\ell}
|T_i|$, and apply Proposition \[lem:VC\]. We obtain a $C' = C'(H,\gamma) >
0$ such that either $\chi(G) \le 2 \chi\big( G[X] \big) \le 2C'$, or $(X,Y)$ is $(C,\alpha)$-rich in copies of $Z_\ell^{3,t}$.
In the former case we are done, and so let us assume the latter. By Proposition \[lem:rich-H\] and our choice of $C$, it follows that $Z_\ell^{3,t}(T_1,\ldots,T_\ell) \subset G$. But then $H \subset G$, which is a contradiction. Thus $\chi(G)$ is bounded, as claimed.
The case $r = 3$ of Theorem \[mainthm\] now follows from Proposition \[noforest\], and Theorems \[thm:forest\], \[LTborsuk\] and \[nearacyclic\].
Let $H$ be a graph with $\chi(H) = 3$, and recall that $\cM(H)$ denotes the decomposition family of $H$. By Proposition \[noforest\], if $\cM(H)$ does not contain a forest then $\delta_\chi(H) = \frac{1}{2}$, and by Theorem \[thm:forest\], if $\cM(H)$ does contain a forest then $\delta_\chi(H) \le \frac{1}{3}$.
Now, by Theorem \[LTborsuk\], if $H$ is not near-acyclic then $\delta_\chi(H)
\ge \frac{1}{3}$, and by Theorem \[nearacyclic\], if $H$ is near-acyclic then $\delta_\chi(H) = 0$. Thus $$\delta_\chi(H) \, \in \, \big\{ 0, \, 1/3, \, 1/2
\big\},$$ where $\delta_\chi(H) \neq \frac{1}{2}$ if and only if $H$ has a forest in its decomposition family, and $\delta_\chi(H) = 0$ if and only if $H$ is near-acyclic, as required.
The rest of this section is devoted to the proof of the following theorem, which generalises Theorem \[nearacyclic\] to arbitrary $r \ge 3$.
\[r-acyclic\] Let $H$ be a graph with $\chi(H) = r \ge 3$. If $H$ is $r$-near-acyclic, then $$\delta_\chi(H) \,=\, {\displaystyle}\frac{r-3}{r-2}\,.$$
We begin with the lower bound, which follows by essentially the same construction as in Proposition \[noforest\].
\[lower\] For any graph $H$ with $\chi(H) = r \ge 3$, we have $\delta_\chi(H) \ge
\frac{r-3}{r-2}$.
We claim that, for any such $H$, $n_0$ and $C$, there exist $H$-free graphs on $n\ge n_0$ vertices, with minimum degree $\frac{r-3}{r-2} n$, and chromatic number at least $C$. Recall that we call a graph a $(k,\ell)$-Erdős graph if it has chromatic number at least $k$ and girth at least $\ell$, and that such graphs exist for every $k,\ell \in {\mathbb{N}}$.
Let $G'$ be a $(C,|H|+1)$-Erdős graph on at least $n_0$ vertices, and let $G$ be the graph obtained from the complete, balanced $(r-2)$-partite graph on $(r-2)|G'|$ vertices by replacing one of its partition classes with $G'$. Then $G$ is $H$-free, since every $|H|$-vertex subgraph of $G$ has chromatic number at most $r-1$. Moreover, $\delta(G) = \frac{r-3}{r-2} n$ and $\chi(G) \ge C$, as required.
We now sketch the proof of the upper bound of Theorem \[r-acyclic\]. Let $G$ be an $n$-vertex, $H$-free graph with minimum degree $\big(\frac{2r-5}{2r-3} + 3\gamma\big) n$. Let $T_1,\ldots,T_\ell$ be such that $H\subset Z_\ell^{r,t}(T_1,\ldots,T_\ell)$. First, we take an $(\eps,d)$-regular partition, using the degree form of the Regularity Lemma (where $\eps$ and $d$ will be chosen sufficiently small given $\gamma$). We then construct a second partition $\cP$ of $V(G)$, similar to that used in the proof of Theorem \[thm:forest\]. Our aim is to show that $\chi(G[X])\le C'$ for each $X\in\cP$.
In the next step, we observe that the minimum degree condition guarantees that for each $X\in\cP$, there are clusters $Y$ and $Z_1,\ldots,Z_{r-3}$ of the $(\eps,d)$-regular partition with the following properties. First, for each $v\in X$ we have $d_Y(v)\ge\gamma |Y|$, and for each $i\in[r-3]$ we have $d_{Z_i}(v)\ge\big(\tfrac{1}{2}+\gamma\big)|Z_i|$. Second, $Y,Z_1,\ldots,Z_{r-3}$ forms a clique in the reduced graph of the $(\eps,d)$-regular partition.
Now recall that $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ contains independent sets $S_I$ for each $I\subset[\ell]$, and independent sets $S_i$ for each $i\in[r-3]$. The idea now is to show that if $\chi(G[X])\le C'$ does not hold, then we find a copy of $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ in which the trees $T_1,\ldots,T_\ell$ lie in $X$, the independent sets $S_I$ lie in $Y$, and $S_i$ lies in $Z_i$ for each $i\in[r-3]$, which contradicts the assumption that $G$ is $H$-free.
In order to achieve this, we work as follows. We apply the paired VC-dimension argument (Proposition \[lem:VC\]) to $(X,Y)$, with constants $\ell^*$ and $t^*$ which are much larger than $\ell$ and $t$, and a very large $C^*$. This yields our $C'$ and an $\alpha>0$ such that either $\chi(G[X])\le
C'$ (in which case we are done), or $(X,Y)$ is $(C^*,\alpha)$-rich in copies of $Z_{\ell^*}^{3,t^*}$.
In the latter case, we apply Lemma \[q-induc\] to conclude that $(X,Y)$ is $(C^*,\alpha)$-dense in copies of $Z_{\ell^*}^{3,t^*}$. The main work of this section then is to show (in Proposition \[3-to-r-rich\]) that this implies that there is an $S\in\cF_\ell^{r,t}(Y\cup Z_1\cup\cdots\cup Z_{r-3})$ such that $S$ is $(r,\ell,t,C,\alpha)$-good for $(X,Y\cup Z_1\cup\cdots\cup Z_{r-3})$. Finally, applying Lemma \[lem:denseembed\] we find that there is a copy of $Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ in $G$.
As just explained, the following proposition is the main missing tool for the proof of Theorem \[r-acyclic\].
\[3-to-r-rich\] For every $r>3$, $\ell,t \in {\mathbb{N}}$ and $d,\gamma > 0$ there exist $\ell^*,t^*\in{\mathbb{N}}$ such that for every $\alpha>0$ and $C\in{\mathbb{N}}$, there exist $\eps_1>0$ and $C^*\in{\mathbb{N}}$, such that for every $0<\eps<\eps_1$ the following holds.
Let $G$ be a graph, and let $X$, $Y$ and $Z_1,\ldots,Z_{r-3}$ be disjoint subsets of $V(G)$, with $|Y| = |Z_j|$ for each $j \in [r-3]$. Let $Z:=Z_1\cup\cdots\cup Z_{r-3}$. Suppose that $(Y,Z_j)$ and $(Z_i,Z_j)$ are $(\eps,d)$-regular for each $i \ne j$, and that $$|N(x) \cap Z_j| \ge \left( \frac{1}{2} + \gamma \right) |Z_j|$$ for every $x
\in X$ and $j \in [r-3]$.
If $(X,Y)$ is $(C^*,\alpha)$-dense in copies of $Z_{\ell^*}^{3,t^*}$, then there is some $S\in\cF_\ell^{r,t}(Y\cup Z)$ such that $S$ is $(r,\ell,t,C,\alpha)$-good for $(X,Y\cup Z)$.
For the proof of this proposition, we combine an application of the Counting Lemma and two uses of the pigeonhole principle. As a preparation for these steps we need to show that there exists a family $S^*\in\cF_{\ell^*}^{3,t^*}$ which is $(3,\ell^*,t^*,C^*,\alpha)$-good for $(X,Y)$ and ‘well-behaved’ in the following sense. For each of the sets $S^*_I\subset Y$ given by $S^*_I$ only a small positive fraction of the $(r-3)t$-element sets in $Z$ has a common neighbourhood in $S^*_I$ of less than $t$ vertices. To this end we shall use the following lemma.
Recall that for a set $T$ of vertices in a graph $G$, we write $$N(T)\colon = \bigcap_{x\in T} N(x)\,.$$
\[countST\] For all $r,t \in {\mathbb{N}}$ and $\mu,d > 0$, there exist $t^* =
t^*(r,t,\mu,d) \in {\mathbb{N}}$ and $\eps_0 = \eps_0(r,t,\mu,d) > 0$ such that for all $0<\eps<\eps_0$ the following holds.
Let $G$ be a graph, and suppose that $Y$ and $Z_1,\ldots,Z_{r-3}$ are disjoint subsets of $V(G)$ such that $(Y,Z_j)$ is $(\eps,d)$-regular for each $j \in
[r-3]$. Let $Z := Z_1 \cup \ldots \cup Z_{r-3}$, and define $$\cB(S) := \Big\{ T \in \binom{Z}{(r-3)t} \,:\, |N(T) \cap S | < t \Big\}$$ for each $S \subset Y$. Then we have $$\cS:=\Big\{ S \in \binom{Y}{t^*} \,:\, |\cB(S)| \ge \mu
|Z|^{(r-3)t} \Big\} \; \le \; \sqrt{\eps} |Y|^{t^*}\,.$$
Choose $t^*$ sufficiently large such that $$\label{eq:countST:t*}
\Pr\Big( {\textup{Bin}}\big( t^*, (d/2)^{(r-3)t} \big) < t \Big) \le
\mu \,,$$ where ${\textup{Bin}}(n,p)$ denotes a random variable with binomial distribution, and set $$\label{eq:countST:eps}
\eps_0:=\min\Big\{\Big(\frac d2 \Big)^{t^*}, \big(t^*\cdot 2^{t^*}(r-3)\big)^{-2}\Big\} \,.$$
In the first part of the proof we shall construct a family $\cS'$ of at least $\binom{|Y|}{t^*}-\sqrt\eps|Y|^{t^*}$ sets $S\in\binom{Y}{t^*}$. In the second part we will then show that $\cS'\subset
\binom{Y}{t^*}\setminus\cS$, which proves the lemma. For constructing the sets $S\in\cS'$ we proceed inductively and shall choose the vertices $v_1,\dots,v_{t^*}$ of $S$ one by one, in each step $k\in[t^*]$ avoiding a set $Y_k\subset Y$ of size at most $\eps 2^k(r-3)|Y|$. Clearly, by , this gives at least $\binom{|Y|}{t^*}-\sqrt\eps|Y|^{t^*}$ choices for $S$ as desired.
Indeed, suppose we have already chosen the vertices $v_1,\ldots,v_{k-1}$. In addition we have chosen for each $j\in[r-3]$ a partition $P_j^{k-1}$ of $Z_j$ with the following property (we shall make use of these partitions in part two of the proof): for each $I\subset\{v_1,\dots,v_{k-1}\}$ we have chosen a part $P_{j}^{k-1}(I)$ of size $(d-\eps)^{|I|}(1-d+\eps)^{k-1-|I|}|Z_j|$ such that $P_{j}^{k-1}(I)\subset N(I)$. Now we will explain how $v_k$ can be chosen together with partitions $P_j^{k}$ satisfying the above conditions. For this purpose consider the set $Y_k \subset Y$ of vertices $y$ such that for some $j\in[r-3]$ and some $I\subset\{v_1,\dots,v_{k-1}\}$ we have $$|N(y)\cap P_{j}^{k-1}(I)|< (d-\eps)^k|Z_j| \,,$$ where $P_j^{0}:=\{Z_j\}$ is the trivial partition of $Z_j$. The possible choices for $v_k$ now are the vertices in $Y\setminus Y_k$. The partitions $P_j^{k}$ with $j\in[r-3]$ are defined as follows. For each $I'\subset\{v_1,\dots,v_{k-1}\}$ we choose an arbitrary subset $P$ of $N(v_k)\cap P_j^{k-1}(I)$ with $|P|=(d-\eps)^k|Z_j|$, which is possible by the choice of $v_k$, and set $$P_j^{k}(I') := P_j^{k-1}(I') \setminus P
\quad\text{and}\quad
P_j^{k}\big(I'\cup\{z_k\}\big) := P \,.$$ Clearly, the partitions defined in this way satisfy that each part $P_j^{k}(I)$ is of size $(d-\eps)^{|I|}(1-d+\eps)^{k-|I|}|Z_j|$ and that $P_j^k(I)\subset N(I)$ as desired.
It remains to show that $|Y_k| \le \eps 2^k(r-3)|Y|$ as claimed above. If this is not true, then for some $j\in[r-3]$ and $I \subset [k-1]$, there exist $\eps |Y|$ vertices in $Y$ which have at most $(d-\eps)
|P_j^k(I)|$ neighbours in $P_j^{k-1}(I)$. Since $|P_j^{k-1}(I)| \ge
(d-\eps)^{k-1} |Z_j|\ge\eps |Z_j|$ by , this contradicts $(\eps,d)$-regularity of $(Y,Z_j)$.
We now turn to the second part of the proof: We claim that for every $S \in \cS'$ we have $|\cB(S)| < \mu |Z|^{(r-3)t}$. To see this, simply choose a random multiset $T \subset Z$ of size $(r-3)t$, and observe that $N(T) \cap S$ is given by the intersection of $(r-3)t$ sets $S_1,\ldots,S_{(r-3)t}
\subset S$ chosen (independently) according to the distribution $$\Pr\big( S_i = I \big)
=\frac{\big|\{z\in Z\,:\,I=N(z)\cap S\}\big|}{|Z|}
\qquad\text{for $I\subset S$}\,.$$ By construction we have $|P^{t^*}_j(I)| = (d -
\eps)^{|I|} (1-d+\eps)^{t^* - |I|} |Z_j| $ for every $j \in [r-3]$ and $I \subset S$. Hence $$\Pr(I \subset S_i)
\ge\frac{\big| \bigcup_{j=1}^{r-3} \bigcup_{I\subset I'\subset S}P^{t^*}_j(I')\big|}{|Z|}
= \sum_{I\subset I'\subset S}(d-\eps)^{|I'|} (1-d+\eps)^{t^* - |I'|}
=(d-\eps)^{|I|} \,.$$ This implies that for every $I\subset S$, we have $\Pr(I \subset S_i)\ge\Pr(I
\subset S'_i)$ for the random variable $S'_i$ with the following distribution: for every $u
\in S$ we take $u\in S'_i$ independently with probability $d - \eps$. We conclude that $$\begin{split}
\Pr\big(|S_1 \cap \ldots \cap S_{(r-3)t}| \ge t\big)
& = \Pr\big( I \subset S_1 \cap \dots \cap S_{(r-3)t} \quad\text{for
some $I\subset S$ with $|I|\ge t$} \big) \\
& \ge \Pr\big( I \subset S'_1 \cap \dots \cap S'_{(r-3)t} \quad\text{for
some $I\subset S$ with $|I|\ge t$} \big) \\
& = \Pr\big(|S'_1 \cap \ldots \cap S'_{(r-3)t}| \ge t\big)
= \Pr\Big( {\textup{Bin}}\big( t^*, (d - \eps)^{(r-3)t} \big) \ge t \Big) \\
&\ge 1-\mu \,,
\end{split}$$ where the last inequality follows from . This proves $|\cB(S)| < \mu |Z|^{(r-3)t}$ and hence finishes the proof of the lemma.
We shall now prove Proposition \[3-to-r-rich\].
We start by defining the constants. Given $r>3$, $\ell,t \in {\mathbb{N}}$ and $\gamma,d > 0$, we set $$\label{eq:3tor:setalphaell}
\mu:=\frac{\gamma^{(r-3)t}}{8\big((r-3)t\big)!(r-3)^{(r-3)t}}
\Big(\frac{d}{2}\Big)^{\binom{r-3}{2}t^2} \quad\text{and}\quad
\ell^*:=\frac{\ell}{2\mu}\,.$$ Let $t^*$ and $\eps_0$ be given by Lemma \[countST\] with input $r, t, \mu':=2^{-\ell^*}\mu, d$. Given $\alpha>0$ and $C$, we choose $$\label{eq:3tor:setepsCstar}
\eps_1:=\min\Big(\frac{\alpha^2}{2^{4\ell^*+1}},
\frac{d\gamma}{4(\gamma+1)(r-3)t},\eps_0\Big)\quad\text{and}\quad
C^*:=\frac{2^{\ell^*}C}{\alpha\mu}\,.$$
Now let $0<\eps<\eps_1$, let $G$ be a graph, and let $X$, $Y$ and $Z_1,\ldots,Z_{r-3}$ be disjoint subsets of $V(G)$ as described in the statement, so in particular, $(X,Y)$ is $(C^*,\alpha)$-dense in copies of $Z_{\ell^*}^{3,t^*}$. The goal is to show that there exists $S\in\cF_\ell^{r,t}(Y\cup Z)$ such that $S$ is $(r,\ell,t,C,\alpha)$-good for $(X,Y\cup Z)$.
Our first step is to show that there is a ‘well-behaved’ function $S^*\in\cF_{\ell^*}^{3,t^*}(Y)$.
\[clm:wellbvd\] There is a function $S^*\in\cF_{\ell^*}^{3,t^*}(Y)$ which is $(3,\ell^*,t^*,C^*,\alpha)$-good for $(X,Y)$ and has the property that for every $I\subset[\ell^*]$, the set $$\cB(S^*_I)=\Big\{T\in\binom{Z}{(r-3)t}\colon \big|N(T)\cap S^*_I\big|\le
t\Big\}$$ in $\binom{Z}{(r-3)t}$ has size at most $2^{-\ell^*}\mu |Z|^{(r-3)t}$.
By Lemma \[countST\] (with input $r,t,\mu'=2^{-\ell^*}\mu,d$), the total number of ‘bad’ $t^*$-subsets $S'$ of $Y$, i.e., those for which $\cB(S')\ge 2^{-\ell^*}\mu|Z|^{(r-3)t}$, is at most $\sqrt{\eps}|Y|^{t^*}$. Let $\cS$ be the set of functions $S^*$ in $\cF_{\ell^*}^{3,t^*}(Y)$ which do *not* have the property that for every $I\subset[\ell^*]$ we have $\cB(S^*_I)<2^{-\ell^*}\mu|Z|^{(r-3)t}$. We can obtain any function $S^*$ in $\cS$ by taking a set $I\subset [\ell^*]$ and one of the at most $\sqrt{\eps}|Y|^{t^*}$ ‘bad’ $t^*$-sets to be $S^*_I$, and choosing the $2^{\ell^*}-1$ remaining sets of $S^*$ in any way from $\binom{Y}{t^*}$. It follows that $$|\cS|\le
2^{\ell^*}\sqrt{\eps}|Y|^{t^*}|Y|^{(2^{\ell^*}-1)t^*}=2^{\ell^*}\sqrt{\eps}|Y|^{2^{\ell^*}t^*}\,.$$
Since $(X,Y)$ is $(C^*,\alpha)$-dense in copies of $Z_{\ell^*}^{3,t^*}$, there are at least $2^{-\ell^*}\alpha|Y|^{2^{\ell^*}t^*}$ functions in $\cF_{\ell^*}^{3,t^*}(Y)$ which are $(3,\ell^*,t^*,C^*,\alpha)$-good for $(X,Y)$. Since by we have $2^{-\ell^*}\alpha> 2^{\ell^*}\sqrt{\eps}$, at least one of these functions is not in $\cS$, as required.
For the remainder of the proof, $S^*$ will be a fixed function satisfying the conclusion of Claim \[clm:wellbvd\]. Since $S^*$ is $(3,\ell^*,t^*,C^*,\alpha)$-good for $(X,Y)$, there exist sets $$E^*_1,\ldots,E^*_{\ell^*} \subset E(X), \quad \text{with} \quad
{\overline}{d}(E^*_j) \ge 2^{-\ell^*} \alpha C^* \quad \text{ for each } \quad 1 \le j
\le \ell^*\,,$$ such that for every $e_{1} \in E^*_{1}, \ldots, e_{\ell^*} \in
E^*_{\ell^*}$, we have ${\mathbf{e}^{\ell^*}} \to S^*$.
Our next claim comprises two applications of the pigeonhole principle to find a copy of $K_{r-3}(t)$ in $Z$.
\[3-to-r:claim2\] There exists a copy $T$ of $K_{r-3}(t)$ with $t$ vertices in $Z_j$ for each $j \in [r-3]$, and a set $L \subset [\ell^*]$ of size $|L| = \ell$ such that:
1. \[3-to-r:a\] $|N(T) \cap S^*_I | \ge t$ for every $I \subset
[\ell^*]$,
2. \[3-to-r:b\] $N(T)$ contains at least $\mu
|E^*_j|$ edges of $E^*_j$, for each $j \in L$.
By assumption, for every $x \in X$ and $j \in [r-3]$ we have $$|N(x) \cap Z_j| \ge \left( \frac{1}{2} + \gamma \right)
|Z_j|\,,$$ and so each edge $e \in E^*_1 \cup
\ldots \cup E^*_{\ell^*}$ has at least $\gamma |Z_j|$ common neighbours in $Z_j$. By Fact \[prop:subpair\], the common neighbours of $e$ in $Z_i$ and $Z_j$ form an $(\eps/\gamma,d-\eps)$-regular pair for each $1\le i<j\le
r-3$. By we have $d-\eps-(r-3)t\eps/\gamma>d/2$. Hence, applying the Counting Lemma with $d$ replaced by $d-\eps$ and $\eps$ replaced by $\eps/\gamma$ to the graph $H=K_{r-3}(t)$, it follows that there are at least $$\begin{gathered}
\frac1{{\textup{Aut}}(H)}\Big(d-\eps-\frac\eps\gamma|H|\Big)^{e(H)}\Big(\frac{\gamma|Z|}{r-3}\Big)^{|H|}
\\ \ge\frac1{\big((r-3)t\big)!}\Big(\frac
d2\Big)^{\binom{r-3}{2}t^2}\Big(\frac{\gamma|Z|}{r-3}\Big)^{(r-3)t}
{ { {\overset{\mbox{\tiny{\eqref{eq:3tor:setalphaell}}}}{\ge}} } } 8\mu |Z|^{(r-3)t}
\end{gathered}$$ copies of $K_{r-3}(t)$ in $N(e) \cap Z$, each with $t$ vertices in each $Z_j$.
There are therefore, for each $j \in [\ell^*]$, at least $8\mu
|Z|^{(r-3)t} |E^*_j|$ pairs $(e,T)$, where $e \in E^*_j$ and $T$ is a copy of $K_{r-3}(t)$ as described, such that $T \subset N(e)$, or equivalently $e
\subset N(T)$. Since we have $$8\mu
|Z|^{(r-3)t} |E^*_j|=4 \mu |Z|^{(r-3)t}|E^*_j|+4 \mu
|E^*_j||Z|^{(r-3)t}\,,$$ by the pigeonhole principle, it follows that there are at least $4 \mu |Z|^{(r-3)t}$ copies of $K_{r-3}(t)$ in $Z$ each of which has at least $4 \mu |E^*_j|$ edges of $E^*_j$ in its common neighbourhood. Let us denote by $\cT_j$ the collection of such copies of $K_{r-3}(t)$. For a copy $T$ of $K_{r-3}(t)$, let $L(T) = \big\{ j : T
\in \cT_j \big\}$.
We claim that there is a set $\cT$ containing at least $2\mu
|Z|^{(r-3)t}$ copies $T$ of $K_{r-3}(t)$ in $Z$, each with $|L(T)|\ge\ell$. Indeed, this follows once again by the pigeonhole principle, since there are at least $$\ell^*\cdot 4 \mu |Z|^{(r-3)t} \,{ { {\overset{\mbox{\tiny{\eqref{eq:3tor:setalphaell}}}}{=}} } }\, \ell
|Z|^{(r-3)t}+\ell^*\cdot 2\mu|Z|^{(r-3)t}$$ pairs $(T,j)$ with $T \in \cT_j$.
Now, recall that $S^*$ satisfies the conclusion of Claim \[clm:wellbvd\], i.e., for each $I\subset[\ell^*]$, there are at most $2^{-\ell^*}\mu|Z|^{(r-3)t}$ sets $T \in \binom{Z}{(r-3)t}$ such that $|N(T) \cap S^*_I | \le t$. Since $|\cT|\ge 2\mu|Z|^{(r-3)t}$, there is a copy $T$ of $K_{r-3}(t)\in\cT$ such that for each $I\subset[\ell^*]$, we have $|N(T)\cap S^*_I|\ge t$. If we let $L$ be any subset of $L(T)$ of size $\ell$, then $T$ and $L$ satisfy the conclusions of the claim.
Let $T$ and $L$ be as given by Claim \[3-to-r:claim2\] and for each $j \in L$ let $E_j\subset X$ be a set of $\mu|E^*_j|$ edges of $E^*_j$ contained in $N(T)$ as promised by Claim \[3-to-r:claim2\]\[3-to-r:b\]. We construct a function $S \in \cF_\ell^{r,t}(Y)$ by choosing, for each $I \subset L$, a subset $S_I \subset S^*_I$ of size $t$ in $N(T)\cap Y$ (which is possible by Claim \[3-to-r:claim2\]\[3-to-r:a\]), and letting the sets $S_i$, $i\in[r-3]$, be the parts of $T$.
\[3-to-r:claim3\] $S$ is $(r,\ell,t,C,\alpha)$-good for $(X,Y \cup Z)$.
Recall that $|L| = \ell$, and assume without loss of generality that $L
= \{1,\ldots,\ell\}$. By the choice of $T$ and the definition of the sets $S_I$ with $I
\subset L$ and the sets $S_i$ with $i\in[r-3]$, we have that $S_i$ is completely adjacent to each $S_{i'}$ with $i\neq i'$, to each $S_I$, and to each edge $e\in \bigcup_{j\in L} E_j$. Since ${\mathbf{e}^{\ell^*}} \to S^*$ for each ${\mathbf{e}^{\ell^*}} \in E^*_1 \times \ldots \times E^*_{\ell^*}$, it follows that ${\mathbf{e}^{\ell}} \to S$ for each ${\mathbf{e}^{\ell}} \in E_1
\times \ldots \times E_{\ell}$. Finally, for each $j\in L$, since $|E_j|\ge\mu|E^*_j|$, we have $${\overline}{d}(E_j)\ge \mu{\overline}{d}(E^*_j)\ge \mu2^{-\ell^*}\alpha
C^*{ { {\overset{\mbox{\tiny{\eqref{eq:3tor:setepsCstar}}}}{=}} } } C\,,$$ as required.
Thus there exists a function $S \in \cF_\ell^{r,t}(Y)$ which is $(r,\ell,t,C,\alpha)$-good for $(X,Y \cup Z)$, as required.
It is possible to strengthen the conclusion of Proposition \[3-to-r-rich\]: under the same conditions, $(X,Y\cup
Z_1\cup\cdots\cup Z_{r-3})$ is $(C,\alpha')$-dense in copies of $Z_{\ell}^{r,t}$, for some $\alpha'=\alpha'(r,\ell,t,d,\gamma,\alpha)>0$. To see this, observe that the proofs of Claims \[clm:wellbvd\] and \[3-to-r:claim2\] both in fact yield a positive density of functions $S^*$ in $\cF_{\ell^*}^{3,t^*}(Y)$ and of copies $T$ of $K_{r-3}(t)$, respectively. From any such $S^*$ and $T$ can be obtained a function $S$ which is $(r,\ell,t,C,\alpha)$-good for $(X,Y \cup Z)$.
We can now deduce Theorem \[r-acyclic\].
The lower bound is given by Proposition \[lower\], so we are only required to prove the upper bound. Let $H$ be an $r$-near-acyclic graph, with $r \ge 4$, and let $\gamma > 0$. Because $H$ is $r$-near-acyclic, by Observation \[acy=Zyk\] there exist trees $T_1,\ldots,T_\ell$ and a number $t\in{\mathbb{N}}$ such that $H\subset
Z_\ell^{r,t}(T_1,\ldots,T_\ell)$. We now set constants as follows. First, we choose $d=\gamma$. Given $r$, $\ell$, $t$, $d$ and $\gamma$, Proposition \[3-to-r-rich\] returns integers $\ell^*$ and $t^*$. Now Proposition \[lem:VC\], with input $\ell^*,t^*$ and $d$, returns $\alpha>0$. Next, consistent with Lemma \[lem:denseembed\] we set $C:=2^{\ell+3}\alpha^{-1}\sum_{i=1}^\ell |T_i|$. Feeding $\alpha$ and $C$ into Proposition \[3-to-r-rich\] yields $\eps_1>0$ and $C^*$. Putting $C^*$ into Proposition \[lem:VC\] yields a constant $C'$. We choose $$\label{eq:r-acyclic:eps}
k_0:=2r/\gamma\quad\text{and}\quad \eps:=\min(\eps_1,\gamma)\,.$$ Finally, from the minimum degree form of the Szemerédi Regularity Lemma, with input $\eps$, $d$, $\delta=(\tfrac{r-3}{r-2}+3\gamma)$ and $k_0$, we obtain a constant $k_1$.
Let $G$ be an $H$-free graph on $n>k_1$ vertices, with $\delta(G) \ge \left( \frac{r-3}{r-2} + 3\gamma \right) n$. We shall prove that $\chi(G)\le 2\cdot2^{2k_1}C'$. First, applying the minimum degree form of the Szemerédi Regularity Lemma, we obtain a partition $V_0 \cup \ldots \cup V_k$ of $V(G)$, with reduced graph $R$, where $\delta(R) \ge \left( \frac{r-3}{r-2} + \gamma \right) k$. We form a second partition by setting $$\begin{aligned}
X(I_1,I_2) \,:=\, \bigg\{ v \in V(G) \,\colon\, i \in I_1 &
\Leftrightarrow & |N(v) \cap V_i| \ge \gamma |V_i|\\
\textup{ and} \quad i \in I_2 & \Leftrightarrow & |N(v) \cap V_i| \ge
\left( \frac{1}{2} + \gamma \right) |V_i| \bigg\}
\end{aligned}$$ for each pair of sets $I_2 \subset I_1 \subset [k]$. It obviously suffices to establish that for each $I_1$ and $I_2$ we have $\chi\big(G[X(I_1,I_2)]\big)\le 2C'$.
Hence let $I_2 \subset I_1 \subset [k]$ be fixed. Since $\chi\big(G[X(I_1,I_2)]\big)\le 2C'$ is obvious when $X(I_1,I_2)$ is empty, assume it is non-empty. Then the minimum degree condition on $G$ allows us to establish the following claim.
\[r-acyclic:claim\] There exist distinct clusters $Y,Y' \in I_1$ and $Z_1,Z'_1,\ldots,Z_{r-3},Z'_{r-3} \in I_2$ such that $(Y,Z_i),(Y',Z'_i),(Z_i,Z_j)$ and $(Z'_i,Z'_j)$ are $(\eps,d)$-regular for every pair $\{i,j\} \subset [r-3]$.
Let $x$ be any vertex in $X(I_1,I_2)$, and let $m=|V_1|=\cdots=|V_k|$. By the definition of $X(I_1,I_2)$, we have $|N(x) \cap V_i| \ge
\gamma m$ iff $i\in I_1$, and thus $$\big(\tfrac{r-3}{r-2}+3\gamma\big)n\le\delta(G)\le d(x)\le \eps
n+\big(k-|I_1|\big)\gamma m+|I_1|m\le (\eps+\gamma)n+|I_1|\tfrac{n}{k}\,.$$ Since by we have $\eps<\gamma$, we deduce $|I_1|\ge \big(\frac{r-3}{r-2}+\gamma\big)k$. Similarly, we have $|N(x) \cap V_i| \ge
(\frac12+\gamma)m$ iff $i\in I_2$ and therefore $$\big(\tfrac{r-3}{r-2}+3\gamma\big)n\le d(x)\le \eps
n+\big(k-|I_2|\big)\big(\tfrac{1}{2}+\gamma\big)m+|I_2|m\le
(\eps+ \tfrac12+\gamma) n+|I_2|\tfrac{n}{2k}\,,$$ from which we obtain $|I_2|\ge\big(\frac{r-4}{r-2}+\gamma\big)k$.
Since $\delta(R) \ge \big( \frac{r-3}{r-2} + \gamma \big) k$, each cluster in $R$ has at most $\big(\frac{1}{r-2}-\gamma\big)k$ non-neighbours. It follows that $$\delta\big( R[I_2] \big) \ge |I_2| - \tfrac{k}{r-2} + \gamma k \ge
\big( \tfrac{r-5}{r-4} + \gamma \big) |I_2|\,,$$ so by Turán’s theorem, $R[I_2]$ contains a copy of $K_{r-3}$. We let its clusters be $Z_1,\ldots,Z_{r-3}$. Since each $Z_i$ is non-adjacent to at most $\big(\frac{1}{r-2}-\gamma\big)k$ cluster in $I_1$, there is a cluster $Y$ in $I_1$ adjacent in $R$ to each $Z_i$ with $i\in[r-3]$. Since $k\ge k_0$, by we have $\gamma
k-(r-2)\ge\gamma k/2$ and therefore $$\delta\big( R[I_2\setminus\{Y,Z_1,\ldots,Z_{r-3}\}] \big) \ge |I_2| - \tfrac{k}{r-2} + \tfrac12\gamma k \ge
\big( \tfrac{r-5}{r-4} + \tfrac12\gamma \big) \big|I_2\setminus\{Y,Z_1,\ldots,Z_{r-3}\}\big|\,.$$ Thus we can again apply Turán’s theorem to $R[I_2\setminus\{Y,Z_1,\ldots,Z_{r-3}\}]$ to obtain a clique $Z'_1,\ldots,Z'_{r-3}$ in $I_2$, which has a common neighbour $Y'\in I_1\setminus\{Y,Z_1,\ldots,Z_{r-3}\}$, as required.
Let $Y,Y' \in I_1$ and $Z_1,Z'_1,\ldots,Z_{r-3},Z'_{r-3} \subset
I_2$ be the clusters given by Claim \[r-acyclic:claim\]. Let $X=X(I_1,I_2)\cap (Y'\cup
Z'_1\cup\cdots\cup Z'_{r-3})$, and $X'=X(I_1,I_2)\setminus X$. Observe that $X,Y,Z_1,\ldots,Z_{r-3}$ are pairwise disjoint (as are $X',Y',Z'_1,\ldots,Z'_{r-3}$). Our goal now is to show that $\chi\big(G[X]\big)\le C'$. Since an analogous argument gives $\chi\big(G[X']\big)\le C'$ and we have $X(I_1,I_2)=X\dcup X'$, this will imply $\chi\big(G[X(I_1,I_2)]\big)\le
2C'$, and thus complete the proof.
We apply Proposition \[lem:VC\], with input $\ell^*$, $t^*$, $d$ and $C^*$, to $(X,Y)$. Observe that, since $Y\in I_1$ and $X\subset
X(I_1,I_2)$, we have $|N(x)\cap
Y|\ge d|Y|$ for each $x\in X$. Recall that $\alpha$ and $C'$ were defined such that the conclusion of Proposition \[lem:VC\] is the following. Either $\chi\big(G[X]\big)\le C'$, or $(X,Y)$ is $(C^*,\alpha)$-rich in copies of $Z_{\ell^*}^{3,t^*}$. In the first case we are done, so we assume the latter. We will show that this contradicts our assumption that $G$ is $H$-free.
By Lemma \[q-induc\] the pair $(X,Y)$ is $(C^*,\alpha)$-dense in copies of $Z_{\ell^*}^{3,t^*}$. We now apply Proposition \[3-to-r-rich\], with input $r$, $\ell$, $t$, $d$, $\gamma$, $\alpha$, $C$, and $\eps$ to $X,Y,Z_1,\ldots,Z_{r-3}$. Observe that since $Z_1,\ldots,Z_{r-3}\in I_2$, we have $|N(x)\cap Z_i|\ge (\frac{1}{2}+\gamma)|Z_i|$ for each $x\in X$ and $i\in[r-3]$. Moreover, by Claim \[r-acyclic:claim\], any pair of $Y,Z_1,\ldots,Z_{r-3}$ is $(\eps,d)$-regular. Recall that $\ell^*$, $t^*$, $\eps_1$ and $C^*$ were defined such that the conclusion of Proposition \[3-to-r-rich\] is that there exists a function $S\in\cF_\ell^{r,t}(Y\cup Z_1\cup\cdots\cup Z_{r-3})$ which is $(r,\ell,t,C,\alpha)$-good for $(X,Y\cup Z_1\cup\cdots\cup Z_{r-3})$. Finally, we apply Lemma \[lem:denseembed\], with input $r,\ell,t,\alpha$ and $T_1,\ldots,T_\ell$, to $X$ and $Y\cup Z_1\cup\cdots\cup
Z_{r-3}$. By the definition of $C$, this lemma gives that $H\subset Z_\ell^{r,t}(T_1,\ldots,T_\ell)$ is contained in $G$, a contradiction.
Finally, we put the pieces together and complete the proof of Theorem \[mainthm\].
Let $H$ be a graph with $\chi(H) = r \ge 3$, and recall that $\cM(H)$ denotes the decomposition family of $H$. By Proposition \[noforest\], if $\cM(H)$ does not contain a forest then $\delta_\chi(H) = \frac{r-2}{r-1}$, and by Theorem \[thm:forest\], if $\cM(H)$ does contain a forest then $\delta_\chi(H) \le \frac{2r-5}{2r-3}$.
Now, by Theorem \[thm:borsuk\], if $H$ is not $r$-near-acyclic then $\delta_\chi(H) \ge \frac{2r-5}{2r-3}$, and by Theorem \[r-acyclic\], if $H$ is $r$-near-acyclic then $\delta_\chi(H) = \frac{r-3}{r-2}$. Thus $$\delta_\chi(H) \, \in \, \left\{ \frac{r-3}{r-2}, \, \frac{2r-5}{2r-3}, \,
\frac{r-2}{r-1} \right\},$$ where $\delta_\chi(H) \neq \frac{r-2}{r-1}$ if and only if $H$ has a forest in its decomposition family, and $\delta_\chi(H) =
\frac{r-3}{r-2}$ if and only if $H$ is $r$-near-acyclic, as required.
Open questions {#probsec}
==============
Although we have determined $\delta_\chi(H)$ for every graph $H$, there are still many important questions left unresolved. In this section we shall discuss some of these. We begin by conjecturing that the assumption on the minimum degree can be weakened to force the boundedness of the chromatic number, as Brandt and Thomassé [@BraTho] proved in the case of the triangle.
For every graph $H$ with $\delta_\chi(H) = \lambda(H)$, there exists a constant $C(H)$ such that the following holds. If $G$ is an $H$-free graph on $n$ vertices and $\delta(G) > \lambda(H)n$, then $\chi(G) \le C(H)$.
We mention that an analogous statement is not true for $H$ with $\delta_\chi(H)\in\{\theta(H),\pi(H)\}$ as a simple modification of our constructions for Propositions \[noforest\] and \[lower\] shows: we merely need to make the partite graphs used in these constructions slightly unbalanced and to guarantee that the Erdős graphs cover the whole partition class they are pasted into and have a sufficient minimum degree.
For graphs $H$ with $\delta_\chi(H) = 0$ one could still ask whether the minimum degree condition can be weakened to some function $f(n) =
o(n)$. The following well-known fact shows that this is not the case.
\[o(n)\] Let $H$ be a graph with $\chi(H) \ge 3$, and let $f(n) = o(n)$. For every $C$ and $n_1$, there exist $H$-free graphs $G$ on at least $n_1$ vertices with $\delta(G) \ge f\big(v(G)\big)$ and $\chi(G)\ge C$.
Given $H$, $f$, $C$ and $n_1$, let $G_0$ be a $(C,v(H)+1)$-Erdős graph. Without loss of generality, we may assume $\delta(G_0)\ge 1$. Let $n_0$ be such that $f(n)\le n/v(G_0)$ for each $n\ge
n_0$. Let $G$ be obtained from $G_0$ by blowing up each vertex to a set of size $\max(n_0,n_1)$. Then $G$ has at least $n_1$ vertices, and we have $\delta(G)\ge v(G)/v(G_0)\ge f\big(v(G)\big)$. Since $G_0$ contains no cycle on $v(H)$ or fewer vertices, $G$ contains no odd cycle with $v(H)$ or fewer vertices. In particular, every $v(H)$-vertex subgraph of $G$ is bipartite, and hence $G$ is $H$-free.
Proposition \[o(n)\] also implies that for graphs $H$ with $\delta_\chi(H)=0$ the upper bound on $\chi(G)$ for $H$-free graphs $G$ with $\delta(G)\ge\eps n$ increases as $\eps$ goes to zero. This suggests the following problem. Set $$\begin{aligned}
& \delta_\chi(H,k) \; := \; \inf \Big\{ d \,:\, \delta(G) \ge d |G|
\;\textup{ and }\; H \not\subset G \;\; \Rightarrow \;\; \chi(G) \le k\Big\}\,,\end{aligned}$$ or, equivalently, $$\begin{aligned}
& \chi_\delta(H,d) \; := \; \max \Big\{ \chi(G) \,:\, \delta(G) \ge d
|G| \;\textup{ and }\; H \not\subset G \Big\}\,,\end{aligned}$$ and call this the *chromatic profile* of $H$.
Determine the chromatic profile for every graph $H$.
As noted in the Introduction, we have, by the results of Andrásfai, Erdős and Sós [@AES], Brandt and Thomassé [@BraTho], Häggkvist [@Hagg] and Jin [@Jin], that $$\delta_\chi(K_3,2) = \frac{2}{5}, \quad
\delta_\chi(K_3,3) = \frac{10}{29} \quad \textup{ and } \quad \delta_\chi(K_3,k)
= \frac{1}{3} \quad \textup{for every $k \ge 4$.}$$ We remark that this problem was also asked by Erdős and Simonovits [@ES], who remarked that it seemed (in full generality) ‘too complicated’ to study; despite the progress made in recent years, we still expect it to be extremely difficult. Note that although our results give explicit upper bounds on $\chi_\delta(H,d)$ for every graph $H$, even in the case $\delta_\chi(H)=0$, where we do not use the Szemerédi Regularity Lemma, these bounds are very weak.
[Ł]{}uczak and Thomassé [@LT] suggested the following more general problem. Given a (without loss of generality monotone) family $\cF$ of graphs, we define $$\begin{aligned}
& \delta_\chi(\cF) \; := \; \inf \Big\{ \delta \,:\, \exists\, C = C(\cF,\delta)
\textup{ such that if } G\in\cF \textup{ is a graph on $n$ vertices } \\ &
\hspace{5cm} \textup{ with } \delta(G) \ge\delta n \textup{, then } \chi(G) \le
C\Big\}\,.\end{aligned}$$
What values can $\delta_\chi(\cF)$ take?
Our results settle this question completely when $\cF$ is defined by finitely many minimal forbidden subgraphs (in which case $\delta_\chi(\cF)$ is precisely the minimum of $\delta_\chi(H)$ over all minimal forbidden subgraphs $H$). For families $\cF$ defined by infinitely many forbidden subgraphs, however, this minimum provides only an upper bound on $\delta_\chi(\cF)$.
[Ł]{}uczak and Thomassé [@LT] suggested in particular to determine $\delta_\chi(\cB)$, where $\cB$ is the family of graphs $G$ such that for every vertex $v\in G$, the graph $G\big[N(v)\big]$ is bipartite (as a natural generalisation of the family of triangle-free graphs, in which every neighbourhood is an independent set). This family is indeed defined by infinitely many forbidden subgraphs: to be precise, by the odd wheels. [Ł]{}uczak and Thomassé gave a construction showing that $\delta_\chi(\cB)\ge\frac{1}{2}$, and conjectured that $\delta_\chi(\cB)=\frac{1}{2}$. Since the wheel $W_5$ (i.e.,the graph obtained from $C_5$ by adding a vertex adjacent to all its vertices) is a forbidden graph for $\cB$, and $\delta_\chi(W_5)=\frac{1}{2}$ by Theorem \[mainthm\], our results confirm that their conjecture is true.
One can generalise the concept of chromatic threshold to uniform hypergraphs. Recently, Balogh, Butterfield, Hu, Lenz and Mubayi [@BBHLM] extended the [Ł]{}uczak-Thomassé method to uniform hypergraphs $\cH$, and thereby proved that $\delta_\chi(\cH) = 0$ for a large family of such $\cH$. To quote from their paper, ‘Many open problems remain’.
Finally, we would like to introduce a new class of problems relating to the chromatic threshold. There has been a recent trend in Combinatorics towards proving ‘random analogues’ of extremal results in Graph Theory and Additive Number Theory (see, for example, the recent breakthroughs of Conlon and Gowers [@ConGow] and Schacht [@Schacht:KLR]). We propose the following variation on this theme: for each graph $H$ and every function $p = p(n) \in
[0,1]$, define $$\begin{aligned}
& \delta_\chi\big( H, p \big) \; := \; \inf \Big\{ d \,:\, \, \exists\,
C(H,d) \text{ such that for } G=G_{n,p}, \text{ asymptotically almost surely,}\\ & \hspace{4cm}
\textup{if } G' \subset G, \; \delta(G') \ge d pn \, \textup{ and } \,
H \not\subset G', \textup{ then } \chi(G') \le C(H,d) \Big\},\end{aligned}$$ where $G_{n,p}$ is the Erdős-Rényi random graph. Note that when $p(n) =
1$, we recover the definition of $\delta_\chi(H)$.
Determine $\delta_\chi(H,p)$ for every graph $H$, and every $p = p(n)$.
In a forthcoming paper [@CTRG] we intend to show that for every constant $p>0$ and every graph $H$, we have $\delta_\chi(H)=\delta_\chi(H,p)$. This is of course trivial in the case $\delta_\chi(H)=0$, when it follows from the results of this paper together with the well-known fact that for constant $p$, the minimum degree of $G_{n,p}$ is asymptotically almost surely at least $pn/2$. In the case $\delta_\chi(H)>0$, the result is not trivial: but much of the machinery developed in this paper can be used unchanged. The following construction shows that the result is best possible, in the sense that it fails to hold for $p= o(1)$.
\[thm:randomconst\] Let $r \ge 3$ and $C \in {\mathbb{N}}$, and let $H$ be a graph with $\chi(H) = r$ and $\delta_\chi(H) \ge \lambda(H) = \tfrac{2r-5}{2r-3}$. If $\eps > 0$ is sufficiently small, the following holds. If $n^{-\eps} < p < \eps^2$, then asymptotically almost surely the graph $G=G_{n,p}$ contains an $H$-free subgraph $G'$ with $\chi(G') \ge C$ and $\delta(G') \ge (1-\eps)\tfrac{r-2}{r-1}pn$.
Given $r$, $H$ and $C$, we let $F$ be a fixed $(C,v(H)+1)$-Erdős graph. We choose a sufficiently small $\eps>0$.
We now construct an $H$-free subgraph $G'$ of $G=G_{n,p}$ as follows. Let $V_1,\ldots,V_{r-1}$ be an arbitrary balanced partition of $[n]$. We fix a copy of $F$ within $G[V_1]$ (which exists asymptotically almost surely). Then we delete all edges within each part $V_i$ with $i\in[r-1]$, except those in the copy of $F$. Moreover, for each pair of vertices $u,v\in V(F)$, we delete the edges from $u$ and $v$ to the common neighbours of $u$ and $v$ in each of $V_2,\ldots,V_{r-1}$.
It follows that $\chi(G')\ge C$ and that $G'$ asymptotically almost surely has minimum degree $(1-\eps)\tfrac{r-2}{r-1}pn$. In addition it can easily be checked from our characterisation of graphs $H$ with $\delta_\chi(H)\ge\lambda(H)$ that $G'$ is also $H$-free.
Theorem \[thm:randomconst\] can be significantly strengthened, and we intend to do so in [@CTRG]. However, results of Kohayakawa, Rödl and Schacht [@KohRodlSchacht] show that we cannot increase the value $\frac{r-2}{r-1}$ in the minimum degree, i.e, $\delta_\chi(H,p) \le \pi(H)$. Thus, by Theorem \[thm:randomconst\], and in contrast to the $p=\Theta(1)$ case, if $\delta_\chi(H) \ge \lambda(H)$ and $n^{-o(1)}<p= o(1)$, then $\delta_\chi(H,p)=\frac{r-2}{r-1}=\pi(H)$.
|
---
abstract: |
The Large Hadron Collider will restart with higher energy and luminosity in 2015. This achievement opens the possibility of discovering new phenomena hardly described by the standard model, that is based on two neutral gauge bosons: the photon and the $Z$. This perspective imposes a deep and systematic study of models that predicts the existence of new neutral gauge bosons. One of these models is based on the gauge group $SU(3)_C \times SU(3)_L \times U(1)_N$ called the 3-3-1 model for short.
In this paper we perform a study with $Z^\prime$ predicted in two versions of the 3-3-1 model and compare the signature of this resonance in each model version. By considering the present and future LHC energy regimes, we obtain some distributions and the total cross section for the process $p + p \longrightarrow \ell^{+} + \ell^{-} + X$. Additionally, we derive lower bounds on $Z^\prime$ mass from the latest LHC results. Finally we analyze the LHC potential for discovering this neutral gauge boson at $14$ TeV center-of-mass energy.
author:
- 'Y. A. Coutinho[^1]'
- 'V. Salustino Guimarães'
- 'A. A. Nepomuceno'
title: 'Bounds on $Z^\prime$ from 3-3-1 model at the LHC energies'
---
Introduction
============
The search for new physics is one of the top priorities after a particle consistent with the Higgs boson has been found at the Large Hadron Collider (LHC) [@ATL1; @CMSQ]. Although this discovery can elucidate the mass-generation mechanism, it is still believed that the standard model (SM) is not the ultimate truth, and that physics beyond it must exist at the TeV scale. New phenomena are predicted in various alternative models and theoretical extensions from SM. The existence of a new neutral current, called $Z^\prime$, is a common feature of most of these models.
Among the models that have new physics content, the 3-3-1 model is the one that provides an elegant answer to one of the modern intriguing questions, the problem of fermion families in nature. The model is built so that anomalies cancel out when all families are summed over, so the family number must be a multiple of the color number.
The phenomenological consequences of the 3-3-1 model depend on its version. The different versions of this model are a consequence of the characteristics of the $SU(3)$ matrices. It is well known that two representations of the group generators can be simultaneously diagonalized. This makes the charge operator dependent on the ratio between $\lambda_3$ to $\lambda_8$ matrix representations leading to different model versions. There is a version with an extra neutral $Z^\prime$ and charged $V^\pm$ and $U^{\pm \pm}$ gauge bosons carrying double leptonic charge, called bileptons. Moreover, in this version the $Z^\prime$ width can be large, and it is usually called the minimal version of the model [@PIV; @FRA]. There are two versions of the model where there are no exotic charged quarks, one is called the right-handed neutrino version [@RHN1; @RHN2; @RHN3; @RHN4; @RHN5] and the other we call the Özer version [@OZ1; @OZ2]. For both, the $Z^\prime$ is a narrow resonance. As we will discuss in the next section, the properties of the new neutral boson depend on the model version, which is determined by the charge operator. Consequently, one needs to establish phenomenological criteria to disentangle these versions by analyzing the production cross section and some angular distributions that follow from each of them.
Several studies have been performed in order to derive bounds on the mass of new gauge bosons. These bounds come from either direct experimental searches or from phenomenological analysis using the available experimental data. In the universe of the 3-3-1 model, bounds on $M_{Z^\prime}$ were obtained from different analyses, such as the contribution from exotics to the oblique electroweak correction parameters ($S$, $T$ and $U$) [@LIU; @SAS; @OMR], corrections to the $Z$-pole observables for arbitrary values of $\beta$ [@OCH; @FRE; @GUT], the study of the energy region where perturbative treatment is still valid [@ALE], $Z^\prime$ and exotic boson mass contributions to the muon decay parameters [@NGL; @BEL], the decay $\mu \rightarrow 3 \ e $ [@SHE], and the contribution from neutral bosons to the flavor changing neutral current (FCNC) [@VAN; @DUM; @TAE; @LIUb; @LIUc; @JAI; @SHERb; @BEN; @DUM2; @CAB; @COG].
In the original work from F. Pisano and V. Pleitez [@PIV], a very restrictive bound on $Z^\prime$ mass was obtained ($M_{Z^\prime} > 40$ TeV) by considering the contribution from the $Z^\prime$ to the $K_S^0 - K_L^0$ mass difference. More recently, a work from V. Pleitez [*et al.*]{} [@ANA], based on additional contributions from a light scalar boson to FCNC, lowered the strong previous limit on the new neutral gauge boson for the minimal version of the 3-3-1 model. This new result allows the minimal 3-3-1 model predictions to be probed at LHC.
Direct experimental searches performed by DØ [@D0] and CDF [@CDF1; @CDF2] Collaborations derived bounds on $Z^\prime$ mass based on analyses with dielectron and dimuon final states at $\sqrt s = 1.96$ TeV. They established lower bounds for different models and had excluded a $Z^\prime$ with mass in the range from $963$ to $1030$ GeV.
Recently, the ATLAS and CMS Collaborations presented results on narrow resonances with dilepton final states ($e^+ e^-$ and $\mu^+ \mu^-$) [@ATL2; @ATL3; @CMS1; @CMS2] and excluded a sequential standard model $Z^\prime$ with mass smaller than $2.49$ TeV (ATLAS) and $2.59$ TeV (CMS). Although their data have been interpreted in terms of different scenarios for physics beyond the SM, no limits on the $Z^\prime$ from the 3-3-1 model was derived from the latest LHC results. The purpose of this article is to derive these unknown limits.
In this paper we consider the production and decay of the 3-3-1 $Z^\prime$ in the process $p + p \longrightarrow \ell^{+} + \ell^{-} + X$ ($\ell = e, \, \mu$) for different LHC energy regimes, when $Z^\prime$ is a narrow resonance as predicted in two versions of the model, namely, the right-handed neutrino model (RHN) [@RHN1; @RHN2; @RHN3; @RHN4; @RHN5] and the Özer model [@OZ1; @OZ2].
Studies using CDF results have excluded a $Z^\prime$ from the RHN model with mass below $920$ GeV [@MART]. For the Özer model, no limit on $Z^\prime$ mass has been derived so far. A previous study on $Z^\prime$ at the ILC energies was made by one of the authors, where it was possible to disentangle versions from the 3-3-1 model, considering the process $e^+ + e^- \longrightarrow \mu^+ + \mu^-$ and establishing from hadronic final states lower bounds on $M_{Z^\prime}$ with $95\%$ C. L. [@ELM]. The possibility to see signals from these models will considerably increased at the LHC running at $14$ TeV, a scenario that we also explore in this work.
This paper is organized as follows: in Sec. II we describe the right-handed neutrinos and Özer versions, highlighting the differences between them. In Sec. III we present the $Z^\prime$ width and the total cross section for the process investigated and for different $Z^\prime$ masses. In Secs. IV and V, we derive lower bounds on the $Z^\prime$ mass at $\sqrt s = 8$ and $\sqrt s = 14$ TeV and explore the LHC potential to find this new state at $14$ TeV. The conclusions are presented in Sec. VI.
Two versions of the 3-3-1 model
===============================
The 3-3-1 model has many attractive features: among them, it is free from anomalies considering the number of fermion families equal to the quantum number of color. The beginning is the electric charge operator that defines the version of the model, $$Q = T_3 - \beta \ T_8 + X I
\label {beta}$$ where the two generators $T_3$ and $ T_8$ satisfy the $SU(3)$ algebra, $I$ is the unit matrix, and finally, $X$ is the $U(1)$ charge.
Depending upon the $\beta$ value, the charge operator determines the arrangement of the fields for he minimal version $\beta = \sqrt 3$; $\beta = 1/ \sqrt 3$ leads to a model with right-handed neutrinos (RHN) and quarks with ordinary charges. Also another choice, $\beta = -1/\sqrt 3$ leads to a model without exotic charges.
We are interested in the two following versions: the right-handed neutrino version with $\beta = 1/ \sqrt 3$ called here version I [@RHN1; @RHN2; @RHN3; @RHN4], and the version with $\beta = -1/\sqrt 3$ [@OZ1; @OZ2], called version II.
Both versions present, besides the ordinary gauge bosons ($\gamma, \, Z, \, W^\pm $), neutral extra gauge bosons $Z^\prime$ and single charged bileptons $V^\pm$ and neutral one $X^0$, which carry a double lepton number. The heavy exotic quarks carry ordinary charges, $2/3$ for u-type and $-1/3$ for d-type.
Each lepton family is arranged in triplets; the first two elements are the charged and the neutral lepton and the third element is a conjugate of the charged lepton or neutral lepton, depending on the $\beta$ factor. In order to cancel anomalies, the quarks are arranged in triplets and antitriplets (one family must be different from the other two).
The Higgs structure to give mass to all particles is composed of three triplets ($\chi$, $\rho$, $\eta$), whose neutral fields develop nonzero vacuum expectation values, respectively, $v_{\chi}$, $v_{\rho}$, and $v_{\eta}$. To reproduce the SM phenomenology, a large scale is associated to the vacuum expectation value $v_\chi$, which gives mass to the exotic quarks and extra gauge bosons. Thus we have the conditions $v_\chi \gg v_\rho, v_\eta$, with $v_\rho^2 + v_\eta^2 = v_W^2= \left( 246 \right)^2$ GeV$^2$.
The general Lagrangian for the neutral current involving only the $Z^{\prime}$ contribution is $$\begin{aligned}
&&{\cal L}^{NC} =-\frac{g}{2 \cos\theta_W}\sum_{f} \Bigl[\bar
f\, \gamma^\mu\ (g^\prime_V + g^\prime_A \gamma^5)f \, { Z_\mu^\prime}\Bigr],\end{aligned}$$
where $f$ are leptons and quarks, the couplings $g^\prime_V$ and $g^\prime_A$ are shown in Tables \[tab2\] and \[tab1\] for RHN and Özer versions, $g$ is the $SU_L(3)$ coupling, and $\theta_W$ is the Weinberg angle.
[|c|c|c|]{}\
& $g^\prime_V$ & $g^\prime_A$\
$Z^{\prime} \bar \ell \ell $ & $\displaystyle{\frac{-1+4\sin^2\theta_W}{2\sqrt{3-4\sin^2\theta_W}}}$ & $\displaystyle{\frac{1}{2\sqrt{3-4\sin^2\theta_W}}}$\
$Z^{\prime} \bar u u$ & $\displaystyle{\frac{3-8\sin^2\theta_W}{{6\sqrt{3-4\sin^2\theta_W}}}}$ & $\displaystyle{-\frac{1}{2\sqrt{3-4\sin^2\theta_W}}}$\
$Z^{\prime} \bar d d$ & $\displaystyle{\frac{3- 2\sin^2\theta_W}{6\sqrt{3-4\sin^2\theta_W}}}$ & $\displaystyle{-\frac{{3-6\sin^2\theta_W}}{6\sqrt{3-4\sin^2\theta_W}}}$\
[|c|c|c|]{}\
& $g^\prime_V$ & $g^\prime_A$\
$Z^{\prime} \bar \ell \ell $ & $\displaystyle{-\frac{1+2\sin^2\theta_W}{2\sqrt{3-4\sin^2\theta_W}}}$ & $\displaystyle{\frac{1-2\sin^2\theta_W}{2\sqrt{3-4\sin^2\theta_W}}}$\
$Z^{\prime} \bar u u$ & $\displaystyle{-\frac{3+2\sin^2\theta_W}{6\sqrt{3-4\sin^2\theta_W}}}$ & $\displaystyle{\frac{1-2\sin^2\theta_W}{2\sqrt{3-4\sin^2\theta_W}}}$\
$Z^{\prime} \bar d d$ & $\displaystyle{\frac{-3+4\sin^2\theta_W}{6\sqrt{3-4\sin^2\theta_W}}}$ & $\displaystyle{\frac{1}{2\sqrt{3-4\sin^2\theta_W}}}$\
Numerical Implementation
========================
The two versions of 3-3-1 models discussed above were implemented in the COMPHEP package [@comp], which was used for cross-section calculation and event generation. The parton distribution functions (PDF) CTEQ6L were used and the QCD factorization scale was set as the dilepton invariant mass of the event. Concerning the particle parameters, we considered heavy quarks, heavy leptons, and bileptons masses to be $1$ TeV, and we took the $Z^\prime$ mass in the range from $500$ to $4000$ GeV.
In Fig. \[fig1\] we present the total $Z^\prime$ width as a function of its mass for the two versions studied here. As we can see, the resonance is narrow in both versions, varying from $2\%$ to $4\%$ of $M_{Z^\prime}$ in the mass range considered. At $M_{Z^\prime} = 2$ TeV the slope of the curve increases because, from this point, the decay of $Z^\prime$ into exotic quarks becomes kinematically allowed. In both versions, the new neutral gauge boson can also decay into exotic fermions with branching ratios of order of $2\%$.
Figure \[fig2\] shows the total cross section calculated at tree level for the process $p + p \longrightarrow \ell^{+} + \ell^{-} + X$ at $\sqrt{s} = 8$ TeV, where $\ell$ is either an electron or a muon. Figure \[fig3\] shows the same cross section calculated for $14$ TeV. Both versions foresee cross sections that can to be probed at the LHC. Version II is the most optimistic since the $Z^\prime$ coupling to leptons is stronger than in version I. Note that depending on $M_{Z^\prime}$, the cross sections increase by a factor of $10$ to $10^2$ at $14$ TeV in comparison with their value at $8$ TeV.
Exclusion limits at $\sqrt s=8$ TeV
===================================
The LHC experiments have performed many analyses searching for signals of new spin $1$ gauge bosons in different final states, but so far no deviation of SM has been found. These analyses are usually model dependent, where a set of benchmark model predictions are compared to data.
In the absence of any signal, ATLAS and CMS Collaborations have extended the $E_6$ superstring-inspired $Z^\prime$ exclusion mass to above $2$ TeV with $6$ fb$^{-1}$ and $4$ fb$^{-1}$ of collision data, respectively, at $\sqrt{s} = 8$ TeV [@ATL3; @CMS2]. In particular, the CMS Collaboration has combined the results from $7$ and $8$ TeV to set $95\%$ C. L. limits on the ratio $R_{\sigma}$ of the cross section times branching fraction for $Z^\prime$ to that of the SM,
$$R_{\sigma}= \frac{\sigma(p + p \longrightarrow Z^\prime \longrightarrow \ell^+ + \ell^-)}{\sigma(p + p \longrightarrow Z \longrightarrow
\ell^+ + \ell^-)}.$$
We use the CMS results to set lower limits on the $Z^\prime$ mass from 3-3-1 models. Following what was done by CMS, the $Z^\prime$ cross-sections for both versions are calculated in a range of $40\%$ about the $Z^\prime$ pole mass and the $Z$ cross-section is calculated in the interval $60$ GeV $< m_{\ell \ell} < 120$ GeV. The ratio $R_{\sigma}$ is evaluated for $Z^\prime$ masses in the range between $500$ and $3000$ GeV.
Figure \[fig4\] shows the CMS observed limits and the theoretical ratio $R_{\sigma}$ curve for both versions. The $Z^\prime$ lower mass limit is obtained from the point where the theoretical ratio curve crosses the observed limit. From the plot, we can conclude that the current data exclude with $95\%$ C. L. the version I new neutral gauge boson with mass below $2200$ GeV and the version II new resonance lighter than $2519$ GeV. This result does not change significantly if the value of exotic quark mass is changed.
Discovery potential and limits at $\sqrt{s}$ = 14 TeV
======================================================
After a shutdown in 2013 that is expected to take two years, the LHC will restart its operation at the design center-of-mass energy of $14$ TeV. Here we assume this scenario to investigate the LHC potential to find a $Z^\prime$ from the 3-3-1 model and determine the lower bounds on the $Z^\prime$ mass that can be set with this energy regime.
In order to determine the minimal integrated luminosity needed to claim a $Z^\prime$ discovery or to exclude it, the number of background and signal events expected in the processes $ p + p \longrightarrow e^{+} + e^{-} + X$ and $p + p \longrightarrow \mu^{+} + \mu^{-} + X$ are calculated. To make our results more realistic, we consider an overall efficiency of $66\%$ for the electron channel and $43\%$ for the muon channel, as determined by the ATLAS experiment [@ATL2]. These efficiencies take into account the geometrical acceptance of the detector ($\arrowvert \eta \arrowvert < 2.5$), cuts on lepton transverse momentum and lepton reconstruction and identification efficiencies.
The dominant and irreducible background taken into account in this paper is the Drell-Yan (DY) process. Although the $Z^\prime$ interfere with the $Z/ \gamma*$ process, the interference is minimal, and therefore we treat signal and background as independent. Other backgrounds include QCD jets and ttbar events, but at high masses these backgrounds can be heavily suppressed by isolation cuts and are not considered here.
Figures \[fig5\] and \[fig6\] show the invariant mass distributions for the DY and for a signal mass hypothesis of 3000 GeV for versions I and II, considering $100$ fb$^{-1}$ of data and the efficiencies mentioned above. Only the distribution for the electron channel is shown, but at the generator level the muon channel distributions looks the same. To determine the significance of a signal such as those shown in the plots, we estimate the number of signal and background events by calculating the cross sections within a window $[M_{Z^\prime} - 2 \Gamma_{Z^\prime}, M_{Z^\prime} + 2 \Gamma_{Z^\prime}]$ for both channels. This selection, represented by the two vertical lines in Figs. \[fig5\] and \[fig6\], suppresses considerably the background while maintaining high signal efficiency. We can also see in these plots the effect of $Z^\prime$-DY interference on the invariant mass, which is small in both models, and under the selected mass window around the $Z^\prime$ mass, it is highly suppressed.
The potential of the search to find a $ Z^\prime$ of a given mass is determined by the integrated luminosity needed to observe a signal with statistical significance of $5 \sigma$. The significance is obtained via the estimator [@cowan],
$$S = \sqrt{2((N_s+ N_b) \ln(1 + \frac{N_s}{N_b}) - N_s)}$$
where $N_s$ and $N_b$ are, respectively, the number of signal and backgrounds events expected in the mass window mentioned above.
Figures \[fig7\] and \[fig8\] show the amount of integrated luminosity required to have a 5$\sigma$ $Z^\prime$ discovery in the electron and muon channels for both versions. As we can see, a 3-3-1 $Z^\prime$ with mass just above the exclusion limit ($2519$ GeV) can be reached with an amount of data of order of $1$ fb$^{-1}$ to $10$ fb$^{-1}$, depending on the channel and model. This scenario can be achieved in the first year of LHC operation at $14$ TeV. For $M_{Z^\prime} \sim 4$ TeV in version II, the amount of data required to discover this new heavy state would be less than $100$ fb$^{-1}$, while for version I, at least $250$ fb$^{-1}$ of data would be needed to observe a boson with that mass.
If no resonance is found in the data, the current $Z^\prime$ limits can be considerably extended in the next years. Assuming the presence of only background in the data, we can calculate the expected limits on various $Z^\prime$ mass hypotheses considering different integrated luminosities. This is done by performing a single-bin likelihood analysis, using the estimated number of signal and background events and the algorithm described in [@junk]. It adopts a frequentist approach to compute the confidence level for exclusion of small signals by combining different searches. The electron and muon channels are combined to set $95\%$ C. L. exclusion on $\sigma \times Br (Z^\prime \longrightarrow \ell^+ + \ell^-)$, and these limits are translated to limits on $M_{Z^\prime}$.
Figure \[fig9\] shows the minimal integrated luminosity needed to exclude the new gauge boson as a function of $M_{Z^\prime}$. With $\sim 23$ fb$^{-1}$ of data, the version II $Z^\prime$ can be excluded up to masses of $4000$ GeV, but for version I, it would need at least $3$ times more luminosity to exclude a $Z^\prime$ with mass of $4000$ GeV. Note that for $M_{Z^\prime} \sim 3000$ GeV, less than $10$ fb$^{-1}$ of data is enough for exclusion. This is important to point out because, although we have not considered in this work the 3-3-1 version that has theoretical upper bounds on $Z^\prime$, our results suggest that such a version can be completely excluded in the very early stages of LHC running at $14$ TeV, since these upper bounds are usually below $3500$ GeV.
Conclusions
===========
New resonances are expected to manifest at LHC in the next years, and among them, the neutral heavy gauge boson $Z^\prime$ has a special role since it appears in different beyond-SM scenarios. In this paper we have presented a study involving the 3-3-1 model predictions, considering the process $p + p \longrightarrow \ell^{+} + \ell^{-} + X$. Lower limits on $Z^\prime$ mass from two versions of the 3-3-1 model were derived using the latest CMS published results. For the RHN model, a $Z^\prime$ with mass below $2200$ GeV is excluded. This limit is a considerable improvement of the bounds obtained with CDF results. On the other hand, we derived a first limit for the Özer version: a $Z^\prime$ lighter than $2519$ GeV is excluded.
Considering the LHC running at the design center-of-mass energy of $14$ TeV, we have shown that a new resonance with mass of $4000$ GeV can be reached at LHC with integrated luminosities of order of $100$ fb$^{-1}$. On the other hand, if no signal is found, the LHC can already exclude $M_{Z^\prime} = 4000$ GeV in the first year of operation at the high-energy regime. This is the first investigation of this kind performed for the 3-3-1 models considering the LHC upgraded energy. As the 3-3-1 model predicts a number of new particles, the observation of a $Z^\prime$ in combination with other exotic searches like bileptons and leptoquarks would provide a powerful way of discriminating between 3-3-1 versions and other BSM scenarios with new neutral heavy states.
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1.5cm
The authors would like to thank Prof. José de Sá Borges from UERJ (RJ - Brazil) for helpful suggestions.
Y. A. Coutinho thanks CNPq and FAPERJ, V. S. Guimarães thanks CAPES and A. A. Nepomuceno thanks FAPERJ for financial support.
[^1]: Now at CERN, Switzerland
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---
abstract: 'Modular structure is ubiquitous among real-world networks from related proteins to social groups. Here we analyze the modular organization of brain networks at a large-scale (voxel level) extracted from functional magnetic resonance imaging (fMRI) signals. By using a random walk-based method, we unveil the modularity of brain-webs, and show modules with a spatial distribution that matches anatomical structures with functional significance. The functional role of each node in the network is studied by analyzing its patterns of inter- and intra-modular connections. Results suggest that the modular architecture constitutes the structural basis for the coexistence of functional integration of distant and specialized brain areas during normal brain activities at rest.'
author:
- 'M. Valencia,$^{1}$ M. A. Pastor,$^{2}$ MA. Fern[á]{}ndez-Seara,$^{2}$ J. Artieda,$^{2}$ J. Martinerie,$^{1}$ and M. Chavez$^{1}$'
title: 'Modular organization as a basis for the functional integration/segregation in large-scale brain networks '
---
**There is a growing interest in studying the connectivity patterns extracted from brain signals during different mental states. Current studies suggest that brain architecture leads neural assemblies to be coordinated with an optimized wiring cost. Brain webs coordinate a mosaic of brain modules, carrying out specific functional tasks and integrated into a coherent process. We analyze the modular structure of brain networks extracted from fMRI signals in humans at rest. Using a random walk-based method we identify a non-random modular architecture of brain connectivity. This approach is fully data driven and relies on no a priori choice of a seed brain region or signal averaging in predefined brain areas. The analysis of intra- and inter-modules connections leads us to relate a node’s connectivity to a local information processing, or to the integration of distant anatomo/functional brain regions. We also find that the spatial distribution of the retrieved modules matches with brain areas associated with specific functions, assessing a functional significance to the modules. In our conclusions, we argue that a modular characterization of the functional brain webs constitutes an interesting model for the study of brain connectivity during different pathological or cognitive states.**
Introduction
============
From the brain over the Internet to social groups, complex networks are a prominent framework to describe collective behaviors in many areas [@Boccaletti2006]. Many of real-world networks exhibit topological features that can be captured neither by regular connectivity models as lattices, nor by random configurations [@Watts1998; @Barabasi1999]. Under this framework, recent studies of complex brain networks have attempted to characterize the connectivity patterns observed under functional brain states. Electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI) studies have consistently shown that human brain functional networks during different pathological and cognitive neurodynamical states display small world (SW) attributes [@Salvador2005a; @Eguiluz2005; @Achard2006; @Achard2007; @Bassett2006; @Reijneveld07]. SW networks are characterized by a small average distance between any two nodes while keeping a relatively highly clustered structure. Thus, SW architecture is an attractive model for brain connectivity because it leads distributed neural assemblies to be integrated into a coherent process with an optimized wiring cost [@LagoFernandez00; @buzsaki04].
Another property observed in many networks is the existence of a modular organization in the wiring structure. Examples range from RNA structures to biological organisms and social groups. A module is currently defined as a subset of units within a network such that connections between them are denser than connections with the rest of the network. It is generally acknowledged that modularity increases robustness, flexibility and stability of biological systems [@Barabasi2004; @Sole2008]. The widespread character of modular architecture in real-world networks suggests that a network’s function is strongly ruled by the organization of their structural subgroups.
Empirical studies have lead to the hypothesis that specialized neural populations are largely distributed and linked to form a web-like structure [@Varela2001]. The emergence of any unified brain process relies on the coordination of a scattered mosaic of modules, representing functional units, separable from -but related to- other modules. Characterizing the modular structure of the brain may be crucial to understand its organization during different pathological or cognitive states.
Previous studies over the mammalian and human brain networks have successfully used different methods to identify clusters of brain activities. Some classical approaches, such as those based on principal components analysis (PCA) and independent components analysis (ICA), make very strong statistical assumptions (orthogonality and statistical independence of the retrieved components, respectively) with no physiological justification [@Biswal1999; @McKeown2003]. Although a number of studies investigating the organization of anatomic and functional brain networks have shown very interesting properties of the macro-scale brain architecture [@Hagmann2008; @Chen2008], little is known about the network structure at a finer scale (at a voxel level). Current approaches are based on the use of a priori coarse parcellations of the cortex [@Salvador2005a; @Achard2006]; or on partial networks defined by a seed voxel [@Cordes2001]. Nevertheless, seed-based descriptions may fail to describe the global behavior of the brain, as they only consider the connectivity of the reference voxel. On the other hand, parcellation schemes reduce the analysis to a macro-scale fixed by an *a priori* definition of the brain areas. Further, a recent study shows that the topological organization of brain networks is affected by the different parcellation strategies applied [@Wang2008].
Here we focus on a completely data-driven framework to study the connectivity of brain networks extracted directly from functional magnetic resonance imaging (fMRI) signals at voxel resolution. A random walk-based algorithm is used to assess the modular organization of functional networks from healthy subjects in a resting-state condition. Results reveal that functional brain webs present a large-scale modular organization significatively different from that arising from random configurations. Further, the spatial distribution of some modules fits well with previously defined anatomo-functional brain areas, assessing a functional significance to the retrieved modules. Based on the patterns of inter- and intra-modular connectivities, we also study the roles played by different brain sites [@Guimera2005]. Results provide a characterization of the functional scaffold that underly the coordination of specialized brain systems during spontaneous brain behavior.
Data adquisition and preprocessing
==================================
BOLD fMRI data were acquired using a T2\*-weighted imaging sequence during a period of 10 minutes from 7 healthy right-handed subjects. The study was performed with written consent of the subjects and with the approval of local ethics committees. During the scan, all subjects were instructed to rest quietly, but alert, and keep their eyes closed. 500 volumes of gradient echoplanar imaging (EPI) data depicting BOLD contrast were acquired. In the acquisition, we used the following parameters: number of slices, $21$ (interleaved); slice thickness, $4$ mm; inter-slice gap, $1$ mm; matrix size, $64 \times 64$; flip angle, $90\,^{\circ}$; repetition time (TR), $1250$ ms; echo time, $30$ ms; in-plane resolution, $3 \times 3$ mm$^{2}$. Subsequently, a high resolution structural volume was acquired via a T1–weighted sequence (axial; matrix $192\times256\times160$; FOV $192\times256\times160$ mm$^{3}$; slice thickness; $1$ mm; in–plane voxel size, $1\times1$ mm$^{2}$; flip angle 15$\,^{\circ}$ ; TR, $1620$ ms, TI, $950$ ms; TE, 3.87 ms) to provide the anatomical reference for the functional scan.
All acquired brain volumes were corrected for motion and differences in slice acquisition times using the SPM5 (http://www.fil.ion.ucl.ac.uk) software package. After correction, fMRI datasets were coregistered to the anatomical dataset and normalized to the standard template MNI, enabling comparisons between subjects. Due to computational limitations, normalized and corrected functional scans were subsampled to a 4x4x4 mm resolution, yielding a total of 20898 voxels (nodes in the network). To eliminate low frequency noise (e.g. slow scanner drifts) and higher frequency artifacts from cardiac and respiratory oscillations, time-series were digitally filtered with a finite impulse response (FIR) filter with zero-phase distortion (bandwidth $0.01-0.1$ Hz) [@Cordes2001].
Estimation of functional connectivity
=====================================
A functional link between two time series $x_i(t)$ and $x_j(t)$ (normalized to zero mean and unit variance) was defined by means of the linear cross-correlation coefficient computed as $r_{ij} = \langle x_i(t)x_j(t)\rangle $, where $\langle\cdot\rangle$ denotes the temporal average. For the sake of simplicity, we only considered here correlations at lag zero. To determine the probability that correlation values are significantly higher than what is expected from independent time series, $r_{ij}(0)$ values (denoted $r_{ij}$) were firstly transformed by the Fisher’s Z transform $$Z_{ij} = 0.5\ln \left(\frac{1+r_{ij}}{1-r_{ij}} \right)$$ Under the hypothesis of independence, $Z_{ij}$ has a normal distribution with expected value 0 and variance $1/(df-3)$, where $df$ is the effective number of degrees of freedom [@Bartlett1946; @Bayley1946; @Jenkins1968]. If time series are formed of independent measurements, $df$ simply equals the sample size, $N$. Nevertheless, autocorrelated time series do not meet the assumption of independence required by the standard significance test, yielding a greater Type I error [@Bartlett1946; @Bayley1946; @Jenkins1968]. In presence of auto-correlated time series $df$ must be corrected by the following approximation: $$\frac{1}{df}\approx \frac{1}{N} + \frac{2}{N}\sum_\tau r_{ii}(\tau) r_{jj}(\tau),$$ where $r_{xx}(\tau)$ is the autocorrelation of signal $x$ at lag $\tau$. Other estimators of $df$, and statistical significance tests for auto-correlated time series can be found in [@Pyper1998]. To correct for multiple testing, the False Discovery Rate (FDR) method was applied to each matrix of $r_{ij}$ values [@Benjamini1995]. With this approach, the threshold of significance $r_{\text{th}}$ was set such that the expected fraction of false positives is restricted to $q \leq 0.001$.
In the construction of the networks, a functional connection between two brain sites was assumed as an undirected and unweighted edge ($A_{ij} = 1$ if $r_{ij} > r_{\text{th}}$; and zero otherwise). Although topological features can also be straightforwardly generalized to weighted networks, we obtained qualitative similar results (not reported here) for weighted networks with a functional connectivity strength between nodes given by $w_{ij} = r_{ij}$.
To characterize the topological properties of a network, a number of parameters have been described. Here we use three key parameters: mean degree $\langle K \rangle$, clustering index $C$ and global efficiency $E$ [@Watts1998; @Barabasi1999; @Boccaletti2006]. Briefly, the degree $k_{i}$ of node $i$ denotes the number of functional links incident with the node and the mean degree is obtained by averaging $k_{i}$ across all nodes of the network. The clustering index quantifies the local density of connections in a node’s neighborhood. For a node $i$, the clustering coefficient $c_{i}$ is calculated as the number of links between the node’s neighbors divided by all of their possible connections and $C$ is defined as the average of $c_{i}$ taken over all nodes of the network. The global efficiency $E$ provides a measure of the network’s capability for information transfer between nodes and is defined as the inverse of the harmonic mean of the shortest path length $L_{ij}$ between each pair of nodes.
Figure \[netDegreeCdf\] shows superimposed the degree distributions for the seven studied subjects. For each network, goodness-of-fit was compared here using Maximum Likelihood methods and the Kolmogorov-Smirnov statistic (KS) for four possible forms of degree distribution $p(k)$: a power law $p(k) \propto k^{-\gamma}$; an exponential $p(k) \propto \exp ^{-\lambda k}$; a truncated Pareto $p(k) \propto (\nu ^{\alpha+1} - \zeta^{\alpha+1})^{-1} k ^{\alpha}$; and an exponentially truncated power law $p(k) \propto k ^{\alpha-1} \exp(-k/k_c)$. The bestfitting were obtained for the truncated power law ($\text{KS} = 0.0421$ compared with $\text{KS} = 0.1028$, $0.2632$ and $0.3278$ for the exponential law, the truncated Pareto and the power law distribution, respectively). Estimated parameters for the truncated power law are $\alpha = 0.7688 \pm 0.1455$, $k_{c} = 410 \pm 351$.
$\text{S}_i$ S1 S2 S3 S4 S5 S6 S7
---------------------- -------- -------- -------- -------- -------- -------- --------
$\langle K\rangle$ 710.65 248.37 815.17 263.59 134.69 133.06 201.16
$C$ $^*$ 0.4954 0.3901 0.4865 0.3856 0.3541 0.3389 0.3638
$\overline{C}_{rnd}$ 0.0340 0.0119 0.0390 0.0126 0.0064 0.0064 0.0096
$E$ 0.3888 0.3569 0.4135 0.3447 0.3104 0.3132 0.3269
$\overline{E}_{rnd}$ 0.5170 0.4973 0.5195 0.5004 0.4337 0.4322 0.4810
: Parameters for real and randomized networks: $\langle K\rangle$, mean degree; $C$, clustering index; $E$, global efficiency; $\overline{\theta}_{rnd}$ denotes the average of parameter $\theta$ obtained from $10$ equivalent randomized networks. Single asterisks indicate that this parameter has a significance level of $p<10^{-4}$.[]{data-label="tableForNetsParams"}
Values of the topological parameters are summarized in Table \[tableForNetsParams\]. To asses the statistical significance of brain connectivity, we perform a benchmark comparison of the functional connectivity patterns. For this, the topological features of brain webs are compared with those obtained from equivalent random wirings. To create an ensemble of equivalent random networks we use the algorithm described in [@Boccaletti2006]. According to this procedure, each edge of the original network is randomly rewired avoiding self- and duplicate connections. The obtained randomized networks thus preserve the same mean degree as the original network, whereas the rest of the wiring structure is random. The significance of a given topological parameter $\theta$ was assessed by quantifying its statistical deviation from values obtained for the ensemble of randomized networks. Let $\mu$ and $\sigma$ be the mean and standard deviation of the parameter $\theta$ computed from such an ensemble. The significance is given by the ratio $\Sigma=(\theta - \mu)/\sigma$ whose p-value is given by the Chebyshev’s inequality (for *any* statistical distribution of $\theta $: $p(|\Sigma| \geqslant \zeta) \leqslant 1/\zeta^{2}$ where $\zeta$ is the chosen statistical threshold) [@Papoulis1991].
The sparse connectivity of functional brain networks was found to be significatively different from randomized wirings for all the subjects. Brain networks yielded larger clustering values ($p<10^{-4}$) than the equivalent random configurations, but similar efficiency values, indicating a connectivity with SW attributes.
Modular analysis of brain networks
==================================
A potential modularity of brain-webs is suggested by the fact that brain networks display a clustering index approximately one order of magnitude larger than that obtained from random configurations [@Ravasz2002]. Although the notion of module results very intuitive, in general it is difficult to define formally. It is currently accepted that a partition $\mathcal{P} = \{ \mathcal{C}_{1},\ldots,\mathcal{C}_{M} \}$ represents a good modular structure if the portion of edges inside each module $\mathcal{C}_{i}$ (intra–modular edges) is high compared to the portion of edges between them (inter–modular edges). The modularity $Q(\mathcal{P})$, for a given partition $\mathcal{P}$ of a network is formally defined as [@Newman2004]: $$Q(\mathcal{P})=\sum_{s=1}^{M}{\left[{\frac{l_{s}}{L}-\left(\frac{k_{s}}{2L}\right)}^{2}\right]},$$ where $M$ is the number of modules, $L$ is the total number of connections in the network, $l_{s}$ is the number of connections between vertices in module $s$, and $k_{s}$ is the sum of the degrees of the vertices in module $s$.
S1 S2 S3 S4 S5 S6 S7
---------------------- -------- -------- -------- -------- -------- -------- --------
$Q^{**}$ 0.4385 0.5814 0.4223 0.5538 0.5648 0.5749 0.5362
$\overline{Q}_{rnd}$ 0.0065 0.0169 0.0057 0.0160 0.0274 0.0279 0.0201
: Modularity for real ($Q$) and randomized networks. $\overline{Q}_{rnd}$ denotes the average obtained from $10$ equivalent randomized networks. Double asterisks denotes a significance level of $p<10^{-6}$.[]{data-label="tableForNetsModularity"}
To partition the functional networks in modules, we used a random walk-based algorithm [@Pons2006], because of its ability to manage very large networks, and its good performances in benchmark tests [@Pons2006; @Danon2005]. In a nutshell, a random walker on a connected graph tends to remain into densely connected subsets corresponding to modules. Let $P_{ij}= \frac{A_{ij}}{k_{i}}$ to be the transition probability from node $i$ to node $j$, where $A_{ij}$ denotes the adjacency matrix and $k_{i}$ is the degree of the i$^{\text{th}}$ node. This defines the transition matrix $(P^t)_{ij}$ for a random walk process of length $t$ (denoted here $P^t_{ij}$ for simplicity). The metric used to quantify the structural similarity between vertices is given by $$\rho_{ij} = \sqrt{\sum_{l=1}^{N}\frac{(P^{t}_{il}-P^{t}_{jl})^{2}}{k_{l}}}$$ Using matrix identities, the distance $\rho$ can be written as $\rho^{2}_{ij}=\sum^{n}_{\alpha=2}{\lambda^{2t}_{\alpha}{(v_{\alpha}(i)-v_{\alpha}(j))}^{2}}$; where $(\lambda_{\alpha})_{1 \leqslant \alpha \leqslant n}$ and $(v_{\alpha})_{1 \leqslant \alpha \leqslant n}$ are the $n$ eigenvalues and right eigenvectors of the matrix $P$, respectively [@Pons2006]. This relates the random walk algorithm to current methods using spectral properties of the graphs [@Newman2006; @Gfeller2007]. The current approach, however, needs not to explicitly compute the eigenvectors of the matrix; a computation that rapidly becomes intractable when the size of the graphs exceeds some thousands of vertices.
To find the modular structure, the algorithm starts with a partition in which each node in the network is the sole member of a module. Modules are then merged by an agglomerative approach based on a hierarchical clustering method. Following Ref. [@Pons2006], if two modules $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are merged into a new one $\mathcal{C}_{3} = \mathcal{C}_{1} \cup \mathcal{C}_{2}$, the transition matrix is updated as follows: $P^t_{\mathcal{C}_3 k}=\frac{|\mathcal{C}_1|P^t_{\mathcal{C}_1 k}+|\mathcal{C}_2|P^t_{\mathcal{C}_2 k}}{|\mathcal{C}_1|+|\mathcal{C}_2|}$, where $|\mathcal{C}_{i}|$ denotes the number of elements in module $\mathcal{C}_{i}$. The algorithm stops when all the nodes are grouped into a single component. At each step the algorithm evaluates the quality of partition $Q$. The partition that maximizes $Q$ is considered as the partition that better captures the modular structure of the network. In the calculation of $Q$, the algorithm excludes small isolated groups of connected vertices without any links to the main network. However, these isolated modules are considered here as part of the network for the calculation of the topological parameters.
As reported in Table \[tableForNetsModularity\], a modular structure is confirmed by the high values of $Q$ obtained for the optimal partition of the networks (a value of $Q \geq 0.3$ is in practice a good indicator of modularity in a network [@Clauset2004]). Further, values of modularity for all the subjects were statistically significant when compared with randomized wirings ($p<10^{-6}$). To assess the stability of the partition structure across subjects we used the Rand index $J$ [@Rand1971], which is a traditional criterion for comparison of different results provided by classifiers and clustering algorithms, including partitions with different numbers of classes or clusters. For two partitions $P$ and $P'$ the Rand index is defined as $J=\frac{a+d}{a+b+c+d}$; where $a$ is number of pairs of data objects belonging to the same class in $P$ and to the same class in $P'$, $b$ is number of pairs of data objects belonging to the same class in $P$ and to different classes in $P'$, $c$ is the number of pairs of data objects belonging to different classes in $P$ and to the same class in $P'$, and $d$ is number of pairs of data objects belonging to different classes in $P$ and to different classes in $P'$. Thus index $J$ yields a normalized value between $0$ (if the two partitions are randomly drawn) and $1$ (for identical partition structures). For our data, the values of $J$ indicate a moderate stability of the partition structure across all subjects ($J = 0.5148 $).
To assess a functionality to the different groups of the modular brain webs, we compared the spatial distribution of the recovered modules with a previously reported anatomical parcellation of the human brain [@Tzourio-Mazoyer2002]. For the sake of simplicity, we only consider here communities whose size was larger than 40 voxels (${\sim0.2\%}$ of the size of the whole network), which yields $N_C = 22$ modules.
Fig. \[commsDistribAndNet\] illustrates the spatial distribution of the modules retrieved from the averaged connectivity matrix computed over all subjects. Results show that the spatial distribution of recovered modules fits well some brain systems. Module 22 for instance, includes $75\%$ of the primary visual areas V1, while module 5 overlaps half of the ventral visual stream (brain areas V2 and V4), and visual areas of the V3 region (cuneus and precuneus) are included ($\sim40\%$) in the module 4. Module 20 includes most of the subcortical structures caudate and thalamus nuclei (covered at $70\%$ and $75\%$, respectively). The auditory system is included by module 12 that overlaps primary and secondary areas plus associative auditory cortex ($60-70\%$). Modules 11, 16 and 21 cover most ($40-70\%$) of the somatosensory and motor cortices; and language related areas are mainly included ($>60\%$) in module 10.
Importantly, some modules include distant brain locations that are functionally related, e.g. the language related areas (modules 10), the auditory system (module 12), or brain regions involved in high level visual processing tasks (module 5). This spatially distributed organization of modules rules out the possibility that modularity *simply* emerges as a consequence of vascular processes or local physiological activities independent of neuronal functions [@Logothetis2001; @Fox2007].
Modules assignment provides the basis for the classification of nodes according to their patterns of intra- and inter-modules connections, which conveys significant information about the importance of each node within the network [@Guimera2005].
The within-module degree $z$-score measures how well connected the node $i$ is to other nodes in the module, and is defined as: $$z_{i} = \frac{k_{i}-\overline{k}_{s_{i}}}{\sigma_{ks_{i}}}$$ where $k_{i}$ is the number of links of node $i$ to other nodes in its module $s_{i}$, $\overline{k}_{s_{i}}$ is the average of $k$ over all the nodes in $s_{i}$, and $\sigma_{k_{s_{i}}}$ is the standard deviation of $k$ in $s_{i}$. Thus node $i$ will display a large value of $z_{i}$ if it has a large number of intra-modular connections relative to other nodes in the same module, i.e. it measures how well connected a node is to other nodes in the module).
The extent a node $i$ connects to different modules is measured by the participation coefficient $pc_{i}$ defined as: $$pc_{i} = 1-\sum_{s=1}^{M} \left( \frac{k_{is}}{k_{i}}\right)^{2}$$ where $k_{is}$ is the number of links of node $i$ to nodes in module $s$, and $k_{i}$ is the degree of node $i$. The participation coefficient takes values of zero if a node has most of its connections exclusively with other nodes of its module. In contrast, $pc_{i} \sim 1$ if their links are distributed among different modules in the network.
The role (R$_{i}$) of a node in the network can be assessed by its within-module degree and its participation coefficient, which define how the node is positioned in its own module and with respect to other modules [@Guimera2005]. Figure \[rolesBrainNets\] shows the distribution of the roles obtained from all the analyzed networks over the $z-pc$ parameter space. Most of the nodes in the functional brain networks ($\sim 98\%$) can be classified as non-hubs (indicated by the gray area in Fig. \[rolesBrainNets\]-(b)), while only a minority of them are module hubs ($\sim2\%$). Non-hubs nodes were classified as ultra-peripheral (R1, $10.33\%$) having all their links within their own modules; peripherals (R2, $73.49\%$) with most links within their modules; or non hub-connectors (R3, $13.67\%$) with half of their links to other modules. This distribution of roles strongly contrasts with that obtained from random configurations (results not show) where most nodes have their links homogeneously distributed among all modules (R4 and R7).
The anatomical distribution of the parameters $z$ and $pc$ is depicted in Figure \[rolesBrainDistrib\]. Interestingly, this representation shows that the wiring structure of the brain has a non-homogeneous organization in terms of the $z-pc$ parameters distribution. Examples of the different behaviours that can be observed are: *i)* subcortical structures (indicated by the orange arrow) display relatively high values for both $z$-score and $pc$ parameters, indicating a dense inter- and intra-modular connectivity ; *ii)* nodes belonging to brain areas associated to the primary visual system (pointed by the red arrow) have a scatter connectivity, yielding low values for both $pc$ and $z$ parameters; *iii)* precuneus and cyngular gyrus areas (indicated by the yellow arrow) have a dense intra-modular connectivity (high values of $z$ ) but few links to other modules (low values of $pc$); *iv)* frontal areas and some visual regions related to associative functions (cian arrow) present more connections to other modules, which is reflected in their low values of $z$ and relatively high values of $pc$.
Conclusion
==========
In conclusion, here we address a fundamental problem in brain networks research: whether the spontaneous brain behavior relies on the coordination (integration) of a complex mosaic of functional brain modules (segregation). By using a random walk-based method we have identified a non-random modular structure of functional brain networks. In contrast to current approaches [@Cordes2001; @Salvador2005a; @Achard2006], our procedure requires neither of signal averaging in predefined brain areas, nor the definition of seed regions, nor subjective thresholds to assess the connectivities. To our knowledge, this work provides the first evidence of a modular architecture in functional human brain networks at a voxel level.
The modularity analysis of large-scale brain networks unveiled a modular structure in the functional connectivity. Although a one-to-one assignment of anatomical brain regions to each detected module is difficult to define, results reveal a strong correlation between the spatial distribution of the modules and some well-known functional systems of the brain, including some of the frequently reported circuits underlying the functional activity at rest [@Cordes2001]. It is worth to notice that, although the functional brain connectivity is strongly shaped by the underlying anatomical wiring (e.g. by the white matter pathways), future studied are needed to clearly examine the interplay between the structural substrate and the modular connectivity inferred from brain dynamics [@Honey09].
Our findings are in full agreement with previous studies about the structure of human brain networks. First, we have confirmed the degree distribution presents a power-law behavior over a wide range of scales, implying that there are a small number of regions with a large number of connections. We also found that brain connectivity shows a degree of clustering that is one order of magnitude higher than that of the equivalent random networks while keeping similar efficiency values, suggesting that spontaneous brain behavior involves an optimized (in a SW sense) functional integration of distant brain regions [@Salvador2005a; @Achard2006; @Eguiluz2005]. Further, the intrinsic non-random modular structure suggested by the high values of the clustering index of brain networks was confirmed by a high degree of modularity obtained for the ensemble of subjects.
Although the mechanisms by which modularity emerges in complex networks are not well understood, it is widely believed that the modular structure of complex networks plays a critical role in their functionality [@Variano2004; @Guimera2005]. Functional brain modules can be related to a local -segregate- information processing while inter-modular connections allows the integration of distant anatomo/functional brain regions [@Frackowiak1997]. On the other hand, the SW and scale-free characteristics of brain webs provide an optimal organization for the stability, robustnes, and transfer of information in the brain [@Achard2006; @Achard2007; @Bassett2006]. The modular structure constitutes therefore an attractive model for the brain organization as it supports the coexistence of a functional segregation of distant specialized areas and their integration during spontaneous brain activity [@Tononi1998; @Sporns2000]. Although the study of anatomical brain networks is a current subject of research, we suggest that a modular description might provide new insights into the understanding of human brain connectivity during pathological or cognitive states.
This work was supported by the EU-GABA contract no. 043309 (NEST) and CIMA-UTE projects.
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---
address:
- University of Toronto
- 'University of Wisconsin–Madison'
- University of Chicago
- Massachusetts Institute of Technology
author:
- 'J. Colliander'
- 'A. D. Ionescu'
- 'C. E. Kenig'
- 'G. Staffilani'
title: 'Weighted low-regularity solutions of the KP-I initial-value problem'
---
[^1]
[^2]
[^3]
[^4]
Introduction {#section1}
============
In this paper we consider the KP-I initial-value problem $$\label{eq-1}
\begin{cases}
\partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0;\\
u(0)=\phi,
\end{cases}$$ on $\mathbb{R}^2_{x,y}\times\mathbb{R}_t$. The dispersion function for this dispersive equation is for $(\xi,\mu)\in\mathbb{R}\setminus\{0\}\times\mathbb{R}$ $$\omega(\xi,\mu)=\xi^3+\mu^2/ \xi.$$ In [@CoKeSt] three of the authors studied with initial data $\phi$ in the space $E\cap P$ defined below. (See the introduction and the references of [@CoKeSt] for a discussion of , its relationship to the corresponding IVP for the KPII equation, and a discussion of the spaces $E$ and $P$ in connection with ). The main result in [@CoKeSt] is a weak form of local in time well-posedness (Theorem 1 in [@CoKeSt] ) for data which is small in $E\cap P$. Unfortunately, A. Ionescu discovered a counterexample to the main estimate used in [@CoKeSt] (Theorem 3 in [@CoKeSt]) to establish Theorem 1. The example exhibits a logarithmic divergence in the estimate, which shows that the proof of Theorem 1 in [@CoKeSt] is incorrect. The same applies to Theorem 2 in [@CoKeSt]. The counterexample is explained in subsection \[count\] below. Colliander, Kenig and Staffilani are very grateful to Ionescu for pointing out this mistake and for joining them in this work. Here we obtain a strengthening of Theorem 1 in [@CoKeSt] which yields the strong form of local in time well-posedness for small data in $E\cap P$. This is Theorem \[Main1\] below. The logarithmic divergence is avoided by introducing new resolution spaces, inspired by those used by Ionescu-Kenig ([@IK1; @IK3; @IK4]) in works on Benjamin-Ono equation and on the Schrödinger map problems. It seems very likely that using the tools developed here, a correct (and similarly strengthened) version of Theorem 2 in [@CoKeSt] could also be obtained. We have felt, however, that this would increase substantially the technicalities in an already very technical paper and we have therefore not pursued this issue.
We conclude by mentioning that our main theorem does not give local well-posedness in $E\cap P$ for large data; such a result would immediately yield global in time well-posedness.
The counterexample {#count}
------------------
We start this section with some notation and by recalling some spaces of functions introduced in [@CoKeSt]. We denote the Fourier transform of a function $f(x,y)$ as $$\label{FT}
\hat f(\xi,\mu)={\ensuremath{\mathcal{F}}}f(\xi,\mu)=\int_{{\ensuremath{\mathbb{R}}}^2}f(x,y)e^{i\langle(x,y)\cdot(\xi,\mu)\rangle}\, dx\, dy.$$ Now let $\chi_A$ denote a smooth characteristic function of the set $A$.
[**[Definition.]{}**]{} Let $\theta_{0}(s)=\chi_{[-1,1]}(s), \, \, \theta_{m}(s)=
\chi_{[2^{m-1},2^{m}]}(|s|), \, \, m\in \mathbb{N}$. For $({\xi},{\mu})\in
\mathbb{R}^{2}$ let $\chi_{1}({\xi},{\mu})=\chi_{\{|{\xi}|\geq {\frac{1}{2}}\frac{|\mu|}{|{\xi}|}\}}$, and $\chi_{2}({\xi},{\mu})=\chi_{\{|{\xi}|< {\frac{1}{2}}{\frac{|\mu|}{|\xi|}}\}}$. Let $\chi_0(s)=\chi_{\{|s|< 1\}}, \, \chi_j(s)=\chi_{\{2^{j-1}\leq |s|< 2^j\}}$ and $w(\xi,\mu)=
(1+|\xi|+|\mu|/|\xi|)$. We define the space $X_{s,b}$ through the norm $$\begin{aligned}
\label{xs}\|f\|_{X_{s,b}}&=&\sum_{j, m\geq 0}
2^{jb}\left(\int_{{\ensuremath{\mathbb{R}}}^{3}}
\chi_{j}(\tau-{\ensuremath{\omega}}(\xi,{\mu}))\chi_{1}(\xi,{\mu})\theta_{m}(\xi)
w^{2s}|\hat{f}|^{2}({\xi},{\mu},\tau)d{\xi}d{\mu}d\tau\right)^{{\frac{1}{2}}}\\
\nonumber&+&\sum_{j, m \geq 0}
2^{jb}\left(\int_{{\ensuremath{\mathbb{R}}}^{3}}
\chi_{j}(\tau-{\ensuremath{\omega}}(\xi,{\mu}))\chi_{2}(\xi,{\mu})\theta_{n}({\mu})
w^{2s}|\hat{f}|^{2}({\xi},{\mu},\tau)d{\xi}d{\mu}d\tau\right)^{{\frac{1}{2}}}.
\end{aligned}$$ We also define the space $$\label{yrs}Y_{s,r,b}=\{f : tf \in X_{s,b}, \mbox{ and } yf \in
X_{r,b}\},$$ and the spaces $$\label{zsb}Z_{s,b}=X_{s,b}\cap Y_{s,1-s,b}, \, \, \mbox{ and} \, \,
Z_{1-\epsilon}=Z_{1-\epsilon,{\frac{1}{2}}}.$$ \[spaces\] We recall here the statement of Theorem 3 in [@CoKeSt]:
\[Old1\] Assume $0 < \epsilon_{0}<\frac{1}{8}$. Then for any $\frac{1}{4}<\epsilon<1$, we have $$\begin{aligned}
\label{Xbilinear}
\|\partial_{x}(uv)\|_{X_{1-\epsilon_{0},-{\frac{1}{2}}}}&\leq& C
\|u\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}(\|v\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}+
\|v\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}^{1-\epsilon }
\|v\|_{Y_{1-\epsilon_{0},-\epsilon_{0}, {\frac{1}{2}}}}^{\epsilon })\\\nonumber
&+&C
\|v\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}(\|u\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}+
\|u\|_{X_{1-\epsilon_{0},{\frac{1}{2}}}}^{1-\epsilon }
\|u\|_{Y_{1-\epsilon_{0},-\epsilon_{0}, {\frac{1}{2}}}}^{\epsilon })\end{aligned}$$
This theorem unfortunately cannot hold since the following counterexample shows a logarithmic divergence. Let $\psi:\mathbb{R}\to[0,1]$ denote a smooth function supported in the interval $[-2,2]$ and equal to $1$ in the interval $[-1,1]$. Assume $N\gg 1$ is very large, $\omega(\xi,\mu)=\xi^3+\mu^2/\xi$, and define $$\label{w1}
\widehat{u}(\xi,\mu,\tau)=\psi(\xi-N)\psi((\mu-\sqrt{3}\xi^2)/\xi)\psi(\tau-\omega(\xi,\mu)),$$ and $$\label{w2}
\widehat{v}(\xi,\mu,\tau)=\psi(\xi-4)\psi((\mu+\sqrt{3}\xi^2)/\xi)\psi(\tau-\omega(\xi,\mu)).$$ Notice that in the definition $|\mu-\sqrt{3}N^2|\leq
CN$ and in the definition the variable $\xi$ is about $1$ (bounded away from $0$). The functions $\mu\to\psi((\mu-\sqrt{3}\xi^2)/\xi)$ in and $\mu\to\psi((\mu+\sqrt{3}\xi^2)/\xi)$ in are essentially the characteristic functions of the intervals $[\sqrt{3}N^2-N,\sqrt{3}N^2+N]$ and $[-16\sqrt{3}-1,-16\sqrt{3}+1]$ respectively. The precise formulas $\psi((\mu-\sqrt{3}\xi^2)/\xi)$ and $\psi((\mu+\sqrt{3}\xi^2)/\xi)$ are convenient for the nonlinear change of variables . Then $$\label{w3}
\begin{split}
&||u||_{X_{1-\epsilon_0,1/2}}\approx N^{1-\epsilon_0}N^{1/2},\,||u||_{Y_{1-\epsilon_0,-\epsilon_0,1/2}}
\approx N^{1-\epsilon_0}N^{1/2},\\
&||v||_{X_{1-\epsilon_0,1/2}}\approx 1,\,||v||_{Y_{1-\epsilon_0,-\epsilon_0,1/2}}\approx 1.
\end{split}$$ So the right-hand side in is $$\label{w4}
RHS\approx N^{1-\epsilon_0}N^{1/2}.$$ We look now at the left-hand side of : the function $\widehat{u}\ast\widehat{v}$ is supported in the set $\{(\xi,\mu,\tau):|\xi-N|\leq C\text{ and }|\mu-\sqrt{3}N^2|\leq CN\}$. So, $$||\partial_x(uv)||_{X_{1-\epsilon_0,-1/2}}\approx N\cdot N^{1-\epsilon_0}
\sum_{j\geq 0}2^{-j/2}||(\widehat{u}\ast\widehat{v})(\xi,\mu,\tau)
\chi_j(\tau-\omega(\xi,\mu))||_{L^2_{\xi,\mu,\tau}},
$$ where $\chi_j$ is the characteristic function of the set $\{s:|s|\in[2^{j-1},2^{j+1}]\}$. Using , it would follow from that $$\label{w5}
\sum_{j\geq 0}2^{-j/2}||(\widehat{u}\ast\widehat{v})(\xi,\mu,\tau)\chi_j(\tau-\omega(\xi,\mu))||_{L^2_{\xi,\mu,\tau}}\leq CN^{-1/2}.$$ We show now that if $100\leq 2^j\leq N^{1/10}$ then $$\label{w6}
||(\widehat{u}\ast\widehat{v})(\xi,\mu,\tau)\chi_j(\tau-\omega(\xi,\mu))||_{L^2_{\xi,\mu,\tau}}
\geq c2^{j/2}N^{-1/2}.$$ So the bound would fail by $\ln N$ since the sum in $j$ has $\approx \ln N$ terms. To prove , by duality, it suffices to prove that if $100\leq 2^j\leq N^{1/10}$ $$\label{w7}
\begin{split}
\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}&(\widehat{u}\ast\widehat{v})(\xi,\mu,\tau)\mathbf{1}_{[N-10,N+10]}(\xi)\\
&\mathbf{1}_{[\sqrt{3}N^2-100N,\sqrt{3}N^2+100N]}(\mu)\chi_j(\tau-\omega(\xi,\mu))\,d\xi d\mu d\tau\geq c2^{j},
\end{split}$$ where $\mathbf{1}_A$ denotes the characteristic function of the set $A$. We substitute the formulas and ; the left-hand side of becomes $$\label{w8}
\begin{split}
\int_{\mathbb{R}^6}&\psi(\xi_1-N)\psi((\mu_1-\sqrt{3}\xi_1^2)/\xi_1)\psi(\tau_1-\omega(\xi_1,\mu_1))\psi(\xi-\xi_1-4)\\
&\psi((\mu-\mu_1+\sqrt{3}(\xi-\xi_1)^2)/(\xi-\xi_1))\psi(\tau-\tau_1-\omega(\xi-\xi_1,\mu-\mu_1))\\
&\mathbf{1}_{[N-10,N+10]}(\xi)\mathbf{1}_{[\sqrt{3}N^2-100N,\sqrt{3}N^2+100N]}(\mu)\chi_j(\tau-\omega(\xi,\mu))\,d\xi_1 d\mu_1 d\tau_1d\xi d\mu d\tau.
\end{split}$$ In this expression we make the change of variables $$\begin{split}
&\xi_1=\xi_1,\,\mu_1=\mu_1,\,\xi=\xi_1+\xi_2,\,\mu=\mu_1+\mu_2\\
&\tau_1=\mu_1+\omega(\xi_1,\mu_1),\,\tau=\mu_2+\mu_1+\omega(\xi_1,\mu_1)+\omega(\xi_2,\mu_2).
\end{split}$$ Then we notice that
$$\begin{split}
&\psi(\xi_1-N)\psi(\xi_2-4)\mathbf{1}_{[N-10,N+10]}(\xi_1+\xi_2)=\psi(\xi_1-N)\psi(\xi_2-4);\\
&\psi((\mu_1-\sqrt{3}\xi_1^2)/\xi_1)\psi((\mu_2+\sqrt{3}\xi_2^2)/\xi_2)\mathbf{1}_{[\sqrt{3}N^2-100N,\sqrt{3}N^2+100N]}(\mu_1+\mu_2)\\
&=\psi((\mu_1-\sqrt{3}\xi_1^2)/\xi_1)\psi((\mu_2+\sqrt{3}\xi_2^2)/\xi_2)\text{ if }\xi_1\in[N-2,N+2]\text{ and }\xi_2\in[2,6].
\end{split}$$
Thus the expression in becomes $$\label{w10}
\begin{split}
\int_{\mathbb{R}^6}&\psi(\xi_1-N)\psi((\mu_1-\sqrt{3}\xi_1^2)/\xi_1)\psi(\mu_1)\psi(\xi_2-4)\psi((\mu_2+\sqrt{3}\xi_2^2)/\xi_2)\psi(\mu_2)\\
&\chi_j(\mu_1+\mu_2+\Omega(\xi_1,\mu_1,\xi_2,\mu_2))\,d\xi_1 d\mu_1 d\mu_1d\xi_2 d\mu_2 d\mu_2,
\end{split}$$ where $$\label{w20}
\begin{split}
\Omega(\xi_1,\mu_1,\xi_2,\mu_2)&=\omega(\xi_1,\mu_1)+\omega(\xi_2,\mu_2)-\omega(\xi_1+\xi_2,\mu_1+\mu_2)\\
&=-\frac{\xi_1\xi_2}{\xi_1+\xi_2}\Big[(\sqrt{3}\xi_1+\sqrt{3}\xi_2)^2-(\mu_1/\xi_1-\mu_2/\xi_2)^2\Big].
\end{split}$$ We make now the nonlinear change of variables $$\label{w11}
\mu_1=\sqrt{3}\xi_1^2+\beta_1\xi_1,\,\mu_2=-\sqrt{3}\xi_2^2+\beta_2\xi_2,$$ with $d\mu_1d\mu_2=\xi_1\xi_2d\beta_1d\beta_2\approx Nd\beta_1d\beta_2$. The expression in is bounded from below by $$\label{w15}
\begin{split}
(N/2)\int_{\mathbb{R}^6}&\psi(\xi_1-N)\psi(\beta_1)\psi(\mu_1)\psi(\xi_2-4)\psi(\beta_2)\psi(\mu_2)\\
&\chi_j(\mu_1+\mu_2+\widetilde{\Omega}(\xi_1,\beta_1,\xi_2,\beta_2))\,d\xi_1 d\beta_1 d\mu_1d\xi_2 d\beta_2 d\mu_2,
\end{split}$$ where, by , $$\label{w16}
\widetilde{\Omega}(\xi_1,\beta_1,\xi_2,\beta_2)=(\beta_1-\beta_2)\xi_1\xi_2\Big(2\sqrt{3}+\frac{\beta_1-\beta_2}{\xi_1+\xi_2}\Big).$$ It follows from that if $\xi_1\in[N-1/100,N+1/100]$,
$\xi_2\in[4-1/100,4+1/100]$,
$|\beta_1-\beta_2|\in[[1/(8\sqrt{3})-1/100]2^j/N,[1/(8\sqrt{3})+1/100]2^j/N]$,
$\mu_1,\mu_2\in[-2,2]$, and $2^j\in[100,N^{1/10}]$ then
$$\chi_j(\mu_1+\mu_2+\widetilde{\Omega}(\xi_1,\beta_1,\xi_2,\beta_2))=1.$$
Thus the only nontrivial restriction in the integral in is $$|\beta_1-\beta_2|\in[[1/(8\sqrt{3})-1/100]2^j/N,[1/(8\sqrt{3})+1/100]2^j/N],$$ which shows that this integral is bounded from below by $cN\cdot
2^j/N=c2^j$. This is the bound , which implies .
The main theorem
-----------------
In this section we introduce again the spaces of functions $E$ and $P$ already defined in [@CoKeSt] and state the main result that replaces Theorem 1 in [@CoKeSt]. We define the energy space $E$, $$\label{eq-2}
E=\{\phi:\mathbb{R}\times\mathbb{R}\to\mathbb{C}:\,\|\phi\|_{E}
:=\|\widehat{\phi}(\xi,\mu)\cdot (1+|\xi|+|\mu/ \xi| )\|_{L^2_{\xi,\mu}}<\infty\},$$ and the weighted space $P$, $$\label{eq-3}
P=\{\phi:\mathbb{R}\times\mathbb{R}\to\mathbb{C}:\,\|\phi\|_{P}
:=\| (y+i)\cdot \phi\|_{L^2}<\infty\}.$$ In Section \[section2\], see , we will define a Banach space $F\hookrightarrow C({\ensuremath{\mathbb{R}}}:E\cap P)$; let $$F_1=\{u\in C([-1,1]:E\cap P):\, \|u\|_{F_1}=\inf_{\widetilde{u}=u\text{ on }{\ensuremath{\mathbb{R}}}^2\times[-1,1]}\|\widetilde{u}\|_F<\infty \}.$$ For any Banach space $V$ and $r>0$ let $B(r,V)$ denote the open ball $\{v\in V:||v||_V<r\}$. Our main theorem concerns local well-posedness of the KP-I initial value problem for small data in $E\cap P$. \[section\]
\[Main1\] There are $\overline{r},\overline{R}\in(0,1]$, $\overline{r}\leq \overline{R}$, with the property that for any $\phi\in B({\overline{r}},E\cap P)$ there is a unique $u\in
B({\overline{R}},F_1)$ such that $$\begin{cases}
(\partial_t+\partial_x^3-\partial_x^{-1}\partial_y^2)u+\partial_x(u^2/2)=0
\text{ in }C((-1,1):H^{-2});\\
u(0)=\phi.
\end{cases}$$ In addition, the mapping $\phi\to u$ is Lipschitz continuous from $B(\overline{r},E\cap P)$ to $B({\overline{R}},F_1)$.
The rest of the paper is organized as follows: in section \[section2\] we define the main normed spaces $X_k$, $Y_k$, $V_k$, $W_k$, $F$, and $N$, and prove some of their basic properties. As explained in subsection \[count\], the use of standard $X^{s,b}$-type spaces seems to lead inevitably to logarithmic divergences in the modulation variable. To avoid these logarithmic divergences we work with high-frequency spaces that have two components: an $X^{s,b}$-type component measured in the frequency space (see the space $X_k$) and a normalized $L^1_yL^2_{x,t}$ component measured in the physical space (see the space $Y_k$). As in [@IK1], [@IK3], and [@IK4], for the physical space component we use a suitable normalization of the local smoothing space $L^1_yL^2_{x,t}$.
In section \[linear\] we prove two linear estimates. In section \[proofthm\] we prove Theorem \[Main1\], using a direct perturbative argument in the Banach space $F_1$, and assuming the dyadic bilinear estimates and . The remaining sections are concerned with the proofs of and : in sections \[prop\] and \[L2bi\] we prove preliminary linear estimates and an $L^2$ bilinear estimate. In sections \[bilinear2\], \[bilinear1\], and \[bilinear3\] we prove the dyadic bilinear estimate . In section \[bilinear4\] we prove the dyadic bilinear estimate .
The resolution spaces {#section2}
=====================
In this section we define the main normed spaces we will use in the rest of the paper, and prove some of their basic properties. Let $\mathbb{Z}_+=\mathbb{Z}\cap[0,\infty)$. For $k\in \mathbb{Z}$ let $I_k=\{\xi:|\xi|\in[2^{k-1},2^{k+1}]\}$, $\widetilde{I}_k=I_k$ if $k\geq 1$, $\widetilde{I}_k=[-2,2]$ if $k=0$, and $\widetilde{I}_k=\emptyset$ if $k\leq -1$. Let $\mu_0:\mathbb{R}\to[0,1]$ denote an even smooth function supported in $[-8/5,8/5]$ and equal to $1$ in $[-5/4,5/4]$. For $k\in\mathbb{Z}$ let $\chi_k(\xi)=\mu_0(\xi/2^k)-\mu_0(\xi/2^{k-1})$. Let $\mu_k=\chi_k$ for $k\in{\ensuremath{\mathbb{Z}}}\cap[1,\infty)$ and $\mu_k=0$ for $k\in {\ensuremath{\mathbb{Z}}}\cap(-\infty,-1]$. Let $$\chi_{[k_1,k_2]}=\sum_{k=k_1}^{k_2}\chi_k\text{ for any }k_1\leq
k_2\in\mathbb{Z}.$$ and, for $j\in{\ensuremath{\mathbb{Z}}}$, $$\mu_{\leq j}=\sum_{j'=-\infty}^j\mu_{j'}\text{ and }\mu_{\geq j}=\sum_{j'=j}^\infty \mu_{j'}.$$
For $(\xi,\mu)\in\mathbb{R}\setminus\{0\}\times\mathbb{R}$ let $$\label{omega}
\omega(\xi,\mu)=\xi^3+\mu^2/ \xi.$$ We define the relevant KP-I region[^5] $$\label{sp8}
R_{KP-I}=\{(\xi,\mu,\tau)\in\mathbb{R}^3:|\xi|\geq 1,\,|\mu|\in[
|\xi|^2/2^{20},2^{20}\cdot |\xi|^2],\,|\tau-\omega(\xi,\mu)|\leq
|\xi|\}.$$
For $k\in{\ensuremath{\mathbb{Z}}}$ let $k_+=\max(k,0)$. We define the normed spaces $X_k$, $$\label{sp7}
\begin{split}
X_{k}=\{f\in L^2(I_k&\times\mathbb{R}\times{\ensuremath{\mathbb{R}}}):\|f\|_{X_k}=\sum_{j=
0}^{2k_+-1}2^{j/2}\|\mu_j(\tau-\omega(\xi,\mu)) \cdot f\|_{L^2}\\
&+\Big[\sum_{j\geq 2k_+}2^{2j-2k_+}\|\mu_j(\tau-\omega(\xi,\mu)) \cdot f\|_{L^2}^2\Big]^{1/2}<\infty\}.
\end{split}$$ Notice that $$\label{sp7.7}
\|(\tau-\omega(\xi,\mu)+i)^{-1}\cdot \mu_{\geq J}(\tau-\omega(\xi,\mu))\cdot f\|_{X_k}\leq C(2^{-J/2}+2^{-2k_+/2})\cdot \|f\|_{L^2},$$ for any $f\in L^2({\ensuremath{\mathbb{R}}}^3)$ supported in $I_k\times{\ensuremath{\mathbb{R}}}\times{\ensuremath{\mathbb{R}}}$ and $J\in{\ensuremath{\mathbb{Z}}}_+$.
The spaces $X_{k}$ are not sufficient for a fixed-point argument, due to various logarithmic divergences. For $k\geq 100$ we also define the normed spaces $Y_k$, $$\label{sp2}
\begin{split}
Y_{k}=&\{f\in L^2(\mathbb{R}^3): f\text{ supported in }
R_{KP-I}\cap I_k\times \mathbb{R}\times\mathbb{R}\text{ and }\\
&\| f \|_{Y_{k}}=2^{-k/2}\|\mathcal{F}^{-1}
[(\tau-\omega(\xi,\mu)+i)\cdot f(\xi,\mu,\tau)]\|_{L^1_yL^2_{x,t}}<\infty \}.
\end{split}$$ For simplicity of notation, we define $Y_k=\{0\}$ for $k\leq 99$. Then we define the normed spaces $X_k+Y_k$, $k\in{\ensuremath{\mathbb{Z}}}$, $$\begin{split}
X_k+Y_{k}=&\{f\in L^2(\mathbb{R}^3): f\text{ supported in }I_k\times \mathbb{R}\times\mathbb{R}\text{ and }\\
&\| f \|_{X_k+Y_{k}}=\inf_{f=f_1+f_2}\|f_1\|_{X_k}+\|f_2\|_{Y_k}<\infty \}.
\end{split}$$
For $k\in\mathbb{Z}$ we define the normed spaces $V_k$ $$\label{sp1}
\begin{split}
V_{k}=\{f\in L^2&(\mathbb{R}^3): f\text{ supported in }
I_k\times \mathbb{R}\times\mathbb{R}\text{ and }\\
&\|f\|_{V_k}=\|f\cdot (1+2^k+i\mu/ 2^k)\|_{X_k+Y_k}<\infty\},
\end{split}$$ and the normed spaces $W_k$, $$\label{sp4}
\begin{split}
W_{k}=\{f\in L^2(\mathbb{R}^3)&: f\text{ supported in }
I_k\times \mathbb{R}\times\mathbb{R}\text{ and }\\
&\|f\|_{W_k}=\|(\partial_\mu+I)f\|_{X_k+Y_k}<\infty\}.
\end{split}$$ We define the (global) normed space $F=F(\mathbb{R}^3)$, $$\label{sp5}
\begin{split}
F= \{u\in L^2&(\mathbb{R}^3):\,u \text{ supported in }{\ensuremath{\mathbb{R}}}^2\times[-2,2]\text{ and }\\
&\|u\|_{F}^2=\sum_{k\in\mathbb{Z}}\|\chi_{k}(\xi)\cdot \mathcal{F}(u)\|_{V_k\cap W_k}^2<\infty\},
\end{split}$$ and the normed space $N=N({\ensuremath{\mathbb{R}}}^3)$, $$\label{sp6}
\begin{split}
N=&\{ u\in C({\ensuremath{\mathbb{R}}}:H^{-2}({\ensuremath{\mathbb{R}}}^3)):\\
&\|u\|_{N}^2
=\sum_{k\in\mathbb{Z}}\|\chi_{k}(\xi)(\tau-\omega(\xi,\mu)+i)^{-1}
\cdot \mathcal{F}(u)\|_{V_k\cap W_k}^2<\infty\}.
\end{split}$$
We start with a simple lemma concerning basic properties of our normed spaces.
\[section\]
\[Lemmaa1\] (a) If $m:\mathbb{R}\to\mathbb{C}$, $m':\mathbb{R}^2\to\mathbb{C}$, $k\in\mathbb{Z}$, and $f$ is supported in $I_k\times\mathbb{R}\times\mathbb{R}$ then $$\label{lb1}
\begin{cases}
&||m(\mu)\cdot f||_{X_k+Y_k}\leq C||\mathcal{F}^{-1}(m)
||_{L^1(\mathbb{R})}\cdot ||f||_{X_k+Y_k};\\
&||m(\mu)\cdot f||_{V_k\cap W_k}\leq C(||\mathcal{F}^{-1}(m)
||_{L^1(\mathbb{R})}+\|\partial_\mu m\|_{L^\infty(\mathbb{R})})\cdot ||f||_{V_k\cap W_k};\\
&||m'(\xi,\tau)\cdot f||_{X_k+Y_k}\leq
C||m'||_{L^\infty(\mathbb{R}^2)}||f||_{X_k+Y_k};\\
&||m'(\xi,\tau)\cdot f||_{V_k\cap W_k}\leq
C||m'||_{L^\infty(\mathbb{R}^2)}||f||_{V_k\cap W_k}.
\end{cases}$$
\(b) If $k\in\mathbb{Z}$, $j\geq 0$, and $f_k\in X_k+Y_k$ then $$\label{lb2}
||\mu_j(\tau-\omega(\xi,\mu))\cdot f_k||_{X_k}\leq C||f_k||_{X_k+Y_k}.$$ In particular, for any $J\in{\ensuremath{\mathbb{Z}}}_+$, $$\label{sp7.8}
\|\eta_{\geq J}(\tau-\omega(\xi,\mu))\cdot f_k\|_{L^2}\leq C2^{-J/2}(2^{(J-2k_+)/2}+1)^{-1}\cdot \|f_k\|_{X_k+Y_k},$$ and $$\label{lb9}
||f_k||_{X_k}\leq C(1+k_+)||f_k||_{X_k+Y_k}.$$
\(c) If $k\geq 0$, $j\in[0,k]\cap\mathbb{Z}$, and $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\in[2^{2k-100},2^{2k+100}]\},$$ then $$\label{lb7}
||\mathcal{F}^{-1}[\eta_{\leq j}(\tau-\omega(\xi,\mu))\cdot f]
||_{L^1_yL^2_{x,t}}\leq
C||\mathcal{F}^{-1}(f)||_{L^1_yL^2_{x,t}}.$$
Part (a) follows directly from the definitions.
For part (b), we may assume $k\geq 100$, $f_k\in Y_k$, so $f_k$ can be written as $$\label{me1}
\begin{split}
f_k(\xi,\mu,\tau)=2^{k/2}&\mathbf{1}_{I_k}(\xi)\chi_{[2k-30,2k+30]}(\mu)\eta_{\leq k+1}(\tau-\omega(\xi,\mu))\\
&\times (\tau-\omega(\xi,\mu)+i)^{-1}\cdot \int_{\mathbb{R}} e^{-iy\cdot\mu}g_k(y,\xi,\tau)\,dy,
\end{split}$$ with $$\label{me2}
\| f_k\|_{Y_k}=C\|g_k\|_{L^1_yL^2_{\xi,\tau}}.$$ The bound follows easily since $|\{\mu:|\tau-\omega(\xi,\mu)|\leq 2^{j+1}\}|\leq C2^{j-k}$ whenever $|\xi|\approx 2^k$, $|\mu|\approx 2^{2k}$, and $j\leq k+C$.
For part (c), using Plancherel theorem, it suffices to prove that $$\label{pr42}
\Big|\Big|\int_{\mathbb{R}}e^{iy\cdot \mu}\chi_{[k-1,k+1]}(\xi)\chi_{[2k-110,2k+110]}(\mu)\eta_{\leq j}(\tau-\omega(\xi,\mu))\,d\mu \Big|\Big|_{L^1_yL^\infty_{\xi,\tau}}\leq C.$$ In proving we may assume $k\geq 100$. Then the function in the left-hand side of is not zero only if $|\tau-\xi^3|\approx 2^{3k}$. Simple estimates using integration by parts show that $$\Big|\int_{\mathbb{R}}e^{iy\cdot \mu}\chi_{[k-1,k+1]}(\xi)\chi_{[2k-110,2k+110]}(\mu)\eta_{\leq j}(\tau-\omega(\xi,\mu))\,d\mu\Big|\leq C\frac{2^{j-k}}{1+(2^{j-k}y)^2}$$ if $|\tau-\xi^3|\approx 2^{3k}$, which suffices to prove .
We show now that $F\hookrightarrow C({\ensuremath{\mathbb{R}}}:E\cap P)$.
\[Lemmaa1\][Lemma]{}
\[Lemmaa2\] If $u\in F$ then $$\label{sd1}
\sup_{t\in{\ensuremath{\mathbb{R}}}} \|u(.,.,t)\|_{E\cap P}\leq C\|u\|_{F}.$$ Thus $F\hookrightarrow C({\ensuremath{\mathbb{R}}}:E\cap P)$.
Let $f_k=\chi_k(\xi)\cdot \mathcal{F}(u)$, $k\in{\ensuremath{\mathbb{Z}}}$. In view of the definition , it suffices to prove that for any $t\in {\ensuremath{\mathbb{R}}}$ and $k\in{\ensuremath{\mathbb{Z}}}$ $$\|\mathcal{F}^{-1}(f_k)(.,.,t)\|_{E\cap P}\leq C\|f_k\|_{V_k\cap W_k}.$$ In view of the last bound in , we may assume $t=0$. Thus it suffices to prove that if $k\in{\ensuremath{\mathbb{Z}}}$ and $f_k\in Z_k$ then $$\label{sd2}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^3}f_k(\xi,\mu,\tau)e^{ix\cdot \xi}e^{iy\cdot \mu}\,d\xi d\mu d\tau\Big|\Big|_{E\cap P}\leq C\|f_k\|_{V_k\cap W_k}.$$
We show first that $$\label{sd3}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^3}f_k(\xi,\mu,\tau)e^{ix\cdot \xi}e^{iy\cdot \mu}\,d\xi d\mu d\tau\Big|\Big|_{E}\leq C\|f_k\|_{V_k}.$$ Using the definition , it suffices to prove that $$\label{sd4}
\Big|\Big|(1+2^k+i\mu/ 2^k )\cdot \int_{{\ensuremath{\mathbb{R}}}}f_k(\xi,\mu,\tau)\,d\tau\Big|\Big|_{L^2_{\xi,\mu}}\leq C\|f_k\|_{V_k}.$$ Using the definition , it suffices to prove that $$\label{sd4.1}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}f_k(\xi,\mu,\tau)\,d\tau\Big|\Big|_{L^2_{\xi,\mu}}\leq C\|f_k\|_{X_k+Y_k}.$$ Assume first that $f_k\in X_k$ and write $f_k=\sum_{j\geq 0}f_k\cdot \eta_j(\tau-\omega(\xi,\mu))=\sum_{j\geq 0}f_{k,j}$. The left-hand side of is dominated by $$\begin{split}
&C\sum_{j\geq 0}\Big|\Big| \int_{{\ensuremath{\mathbb{R}}}}|f_{k,j}(\xi,\mu,\tau)|\,d\tau\Big|\Big|_{L^2_{\xi,\mu}}\leq C\sum_{j\geq 0}2^{j/2}\|f_{k,j}\|_{L^2_{\xi,\mu,\tau}}\leq C\|f_k\|_{X_k},
\end{split}$$ as desired. Asssume now that $f_k\in Y_k$ (so $k\geq 100$) and write $f_k$ as in . With $g_k$ as in and , the left-hand side of is dominated by $$\label{sd7}
\begin{split}
&C2^{k/2}\Big|\Big|\mathbf{1}_{I_k}(\xi)\chi_{[2k-30,2k+30]}(\mu)\int_{\mathbb{R}\times{\ensuremath{\mathbb{R}}}} e^{-i y\cdot \mu}\frac{\eta_{\leq k+1}(\tau-\omega(\xi,\mu))}{\tau-\omega(\xi,\mu)+i}\cdot g_k(y,\xi,\tau)\,dyd\tau\Big|\Big|_{L^2_{\xi,\mu}}.
\end{split}$$ We define the partial Hilbert transform operator $$\mathcal{L}_k(g)(y,\xi,\nu)=\int_{{\ensuremath{\mathbb{R}}}}g(y,\xi,\tau)\cdot \eta_{\leq k+1}(\tau-\nu)\cdot (\tau-\nu+i)^{-1}\,d\tau.$$ Using the Minkowski inequality, the expression in is dominated by $$C2^{k/2}\int_{{\ensuremath{\mathbb{R}}}}\Big|\Big|\mathbf{1}_{I_k}(\xi)\chi_{[2k-30,2k+30]}(\mu)\cdot \mathcal{L}_k(g_k)(y,\xi,\omega(\xi,\mu))\Big|\Big|_{L^2_{\xi,\mu}}\,dy.$$ A simple change of variables shows that this is dominated by $$C\int_{{\ensuremath{\mathbb{R}}}}\Big|\Big|\mathbf{1}_{I_k}(\xi)\mathcal{L}_k(g_k)(y,\xi,\nu)\Big|\Big|_{L^2_{\xi,\nu}}\,dy,$$ and the bound follows from and the estimate $\|\mathcal{L}_k(g)(y,\xi,\nu)\|_{L^2_{\xi,\nu}}\leq C\|g(y,\xi,\tau)\|_{L^2_{\xi,\tau}}$.
We show now that $$\label{sd10}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^3}f_k(\xi,\mu,\tau)e^{ix\cdot \xi}e^{iy\cdot \mu}\,d\xi d\mu d\tau\Big|\Big|_{P}\leq C\|f_k\|_{W_k}.$$ Using the definition and Plancherel theorem, it suffices to prove that $$\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}(\partial_\mu+I)f_k(\xi,\mu,\tau)\,d\tau\Big|\Big|_{L^2_{\xi,\mu}}\leq C\|f_k\|_{W_k},$$ which follows from . The bound follows from and .
Linear estimates {#linear}
================
In this section we prove two linear estimates. For $\phi \in L^2(\mathbb{R}^2)$ let $W\phi \in C(\mathbb{R}:L^2_{x,y})$ denote the solution of the free KP-I evolution given by $$\label{ni1}
W\phi(x,y,t)=C\int_{{\ensuremath{\mathbb{R}}}^2}e^{ix\cdot \xi}e^{iy\cdot \mu}e^{it\omega(\xi,\mu)}\widehat{\phi}(\xi,\mu)\,d\xi d\mu,$$ where $\omega(\xi,\mu)$ is defined in . Let $\psi=\widehat{\eta_0}\in\mathcal{S}(\mathbb{R})$.
\[section\]
\[Lemmab1\] If $\phi \in E\cap P$ then $$||\eta_0(t)\cdot W\phi||_{F}\leq C||\phi||_{E\cap P}.$$
A straightforward computation shows that $$\label{ni3}
\mathcal{F}[\eta_0(t)\cdot W\phi](\xi,\mu,\tau)=
\widehat{\phi}(\xi,\mu)\widehat{\eta_0}(\tau-\omega(\xi,\mu)).$$ Then, directly from the definitions, $$\begin{split}
|&|\eta_0(t)\cdot W\phi ||_{F}^2
\leq C\sum_{k\in\mathbb{Z}}||\chi_k(\xi)\cdot \widehat{\phi}(\xi,\mu)
\cdot \psi(\tau-\omega(\xi,\mu))||_{V_k\cap W_k}^2\\
&\leq
C\sum_{k\in\mathbb{Z}}||\chi_k(\xi)\widehat{\phi}(\xi,\mu)\cdot
(1+2^k+|\mu|/2^k)||_{L^2_{\xi,\mu}}^2+C\sum_{k\in\mathbb{Z}}||\chi_k(\xi)
(\partial_\mu\widehat{\phi})(\xi,\mu)||_{L^2_{\xi,\mu}}^2\\
&\leq C(||\phi||_{E}^2+||\phi ||_{P}^2),
\end{split}$$ as desired.
\[Lemmab1\][Proposition]{}
\[Lemmab3\] If $u\in N$ then $$\Big|\Big|\eta_0(t)\cdot \int_0^t[Wu(s)](t-s)\,ds\Big|\Big|_{F}\leq C||u||_{N}.$$
A direct computation shows that $$\label{ni2}
\begin{split}
\mathcal{F}&\Big[\eta_0(t)\cdot \int_0^t[Wu(s)](t-s)ds\Big](\xi,\mu,\tau)\\
&=C\int_\mathbb{R}\mathcal{F}(u)(\xi,\mu,\tau')\cdot \frac{\widehat{\eta_0}(\tau-\tau')-\widehat{\eta_0}(\tau-\omega(\xi,\mu))}{\tau'-\omega(\xi,\mu )}d\tau'.
\end{split}$$ For $k\in\mathbb{Z}$ let $$f_k(\xi,\mu,\tau')=\chi_k(\xi)(\tau'-\omega(\xi,\mu)+i)^{-1}\cdot \mathcal{F}(u)(\xi,\mu,\tau').$$ For $f_k\in V_k\cap W_k$ let $$\label{ar202}
T(f_k)(\xi,\mu,\tau)=\int_\mathbb{R}f_k(\xi,\mu,\tau')\frac{\psi(\tau-\tau')-
\psi(\tau-\omega(\xi,\mu))}{\tau'-\omega(\xi,\mu)}\cdot (\tau'-\omega(\xi,\mu)+i)\,d\tau'.$$ In view of the definitions, it suffices to prove that $$\label{ni5}
||T||_{V_k\cap W_k\to V_k\cap W_k}\leq C\text{ uniformly in }k\in\mathbb{Z}.$$
We prove first that $$\label{ni7}
||T(f_k)||_{X_k}\leq C\|f_k\|_{X_k}\text{ uniformly in }k\in {\ensuremath{\mathbb{Z}}}.$$ We observe the elementary bound $$\Big|\frac{\psi(\theta-\theta')-
\psi(\theta)}{\theta'}(\theta'+i)\Big|\leq C[(1+|\theta|)^{-4}+(1+|\theta-\theta'|)^{-4}],$$ for any $\theta,\theta'\in{\ensuremath{\mathbb{R}}}$. Thus, for , it suffices to prove that $$\label{ni7.3}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}|f_k(\xi,\mu,\tau')|\cdot [(1+|\tau-\tau'| )^{-4}+(1+|\tau-\omega(\xi,\mu)| )^{-4}]\,d\tau'\Big|\Big|_{X_k}\leq C\|f_k\|_{X_k},$$ for any $f\in X_k$. For this, we notice first that $$\Big|\Big|(1+|\tau-\omega(\xi,\mu)| )^{-4}\int_{{\ensuremath{\mathbb{R}}}}|f_k(\xi,\mu,\tau')|\,d\tau'\Big|\Big|_{X_k}\leq C\|f_k\|_{X_k},$$ using . In addition, for any $j\geq 0$, $$\begin{split}
\Big|\Big|\eta_j(\tau-\omega(\xi,\mu))&\cdot \int_{{\ensuremath{\mathbb{R}}}}|f_k(\xi,\mu,\tau')|\cdot (1+|\tau-\tau'| )^{-4}\,d\tau'\Big|\Big|_{L^2}\\
&\leq C\sum_{j'\in{\ensuremath{\mathbb{Z}}}}2^{-3|j-j'|}||\eta_{j'}(\tau'-\omega(\xi,\mu))\cdot f_k(\xi,\mu,\tau')||_{L^2}.
\end{split}$$ The bound follows from the definition .
We show now that $$\label{ni30}
||T(f_k)||_{X_k+Y_k}\leq C\|f_k\|_{Y_k}\text{ uniformly in }k\in {\ensuremath{\mathbb{Z}}}.$$ We may assume $k\geq 100$. Using and Lemma \[Lemmaa1\] (b), (c), we may also assume that $f_k\in Y_k$ is supported in the set $\{(\xi,\mu,\tau'):|\tau'-\omega(\xi,\mu)|\leq 2^{k-10}\}$. We write $$f_k(\xi,\mu,\tau')=\frac{\tau'-\omega(\xi,\mu)}{\tau'-\omega(\xi,\mu)+i}f_k(\xi,\mu,\tau')+\frac{i}{\tau'-\omega(\xi,\mu)+i}f_k(\xi,\mu,\tau').$$ Using Lemma \[Lemmaa1\] (b), $||i(\tau'-\omega(\xi,\mu)+i)^{-1}f_k(\xi,\mu, \tau')||_{X_k}\leq C||f_k||_{Y_k}$. In view of , it suffices to prove that $$\label{ni8}
\begin{split}
\Big|\Big|\int_\mathbb{R}f_k(\xi,\mu,\tau')&\psi(\tau-\tau')\,d\tau'\Big|\Big|_{X_k+Y_k}\\
&+\Big|\Big|\psi(\tau-\omega(\xi))\int_\mathbb{R}f_k(\xi,\mu,\tau')\,d\tau'\Big|\Big|_{X_k}\leq C||f_k||_{Y_k}.
\end{split}$$ The bound for the second term in the left-hand side of follows from . To bound the first term we write $$f_k(\xi,\mu,\tau')=f_k(\xi,\mu,\tau')\Big[\frac{\tau'-\omega(\xi,\mu)+i}{\tau-\omega(\xi,\mu)+i}+\frac{\tau-\tau'}{\tau-\omega(\xi,\mu)+i}\Big].$$ The first term in the left-hand side of is dominated by $$\label{ni9}
\begin{split}
&C\Big|\Big|\eta_{\leq k-5}(\tau-\omega(\xi,\mu))\int_\mathbb{R}f_k(\xi,\mu,\tau')\frac{\tau'-\omega(\xi,\mu)+i}{\tau-\omega(\xi,\mu)+i}\cdot \psi(\tau-\tau')\,d\tau'\Big|\Big|_{Y_k}\\
&+C\Big|\Big|\eta_{\leq k-5}(\tau-\omega(\xi,\mu))\int_\mathbb{R}f_k(\xi,\mu,\tau')\frac{\psi(\tau-\tau')\cdot (\tau-\tau')}{\tau-\omega(\xi,\mu)+i}\,d\tau'\Big|\Big|_{X_k}\\
&+C\Big|\Big|\eta_{\geq k-5}(\tau-\omega(\xi,\mu))\int_\mathbb{R}f_k(\xi,\mu,\tau')\psi(\tau-\tau')\,d\tau'\Big|\Big|_{X_k}.
\end{split}$$ For the first term in , we use Lemma \[Lemmaa1\] (c) to bound it by $$C2^{-k/2}||\mathcal{F}^{-1}(\psi)\cdot\mathcal{F}^{-1}[(\tau'-\omega(\xi)+i)f_k(\xi,\mu,\tau')]||_{L^1_yL^2_{x,t}}\leq C||f_k||_{Y_k},$$ as desired. To bound the second term, we observe that $$\Big|\frac{\psi(\tau-\tau')\cdot (\tau-\tau')}{\tau-\omega(\xi,\mu)+i}\Big|\leq C\frac{(1+|\tau-\tau'|)^{-4}}{1+|\tau'-\omega(\xi,\mu)|}.$$ Thus the second term in is bounded by $$\label{ni41}
\begin{split}
C&\Big|\Big|\int_\mathbb{R}\frac{|f_k(\xi,\mu,\tau')|}{1+|\tau'-\omega(\xi,\mu)|}\cdot (1+|\tau-\tau'|)^{-4}\,d\tau'\Big|\Big|_{X_k}\leq C\Big|\Big|\frac{f_k(\xi,\mu,\tau')}{1+|\tau'-\omega(\xi,\mu)|}\Big|\Big|_{X_k},
\end{split}$$ which is dominated by $C||f_k||_{Y_k}$ in view of Lemma \[Lemmaa1\] (b). To bound the third term in , recall that $f_k$ is supported in the set $\{(\xi,\mu,\tau'):|\tau'-\omega(\xi,\mu)|\leq 2^{k-10}\}$, thus $f_k(\xi,\mu,\tau')=f_k(\xi,\mu,\tau')\cdot \eta_{\leq k-10}(\tau'-\omega(\xi,\mu))$. In addition, it is easy to see that $$|\eta_{\geq k-5}(\tau-\omega(\xi,\mu))\cdot \eta_{\leq k-10}(\tau'-\omega(\xi,\mu))\cdot \psi(\tau-\tau')|\leq C\frac{(1+|\tau-\tau'|)^{-4}}{1+|\tau'-\omega(\xi,\mu)|},$$ so the third term in is also bounded as in .
Finally, we prove that $$\label{ni50}
||T(f_k)||_{W_k}\leq C\|f_k\|_{V_k\cap W_k}\text{ uniformly in }k\in {\ensuremath{\mathbb{Z}}}.$$ In view of the definition , the left-hand side of is dominated by $$\begin{split}
&C\|T[(\partial_\mu+I)f_k]\|_{X_k+Y_k}+C\Big|\Big|\psi'(\tau-\omega(\xi,\mu))\cdot (\mu/ \xi)\int_{{\ensuremath{\mathbb{R}}}}f_k(\xi,\mu,\tau')\,d\tau'\Big|\Big|_{X_k+Y_k}\\
&+C\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}f_k(\xi,\mu,\tau')\frac{d}{d\mu}\frac{\psi(\tau-\tau')-
\psi(\tau-\omega(\xi,\mu))}{\tau'-\omega(\xi,\mu)}\,d\tau'\Big|\Big|_{X_k+Y_k}.
\end{split}$$ The first term in the expression above is dominated by $C\|f_k\|_{W_k}$, in view of and . The second term is dominated by $C\|f_k\|_{V_k}$, in view of . Thus, for , it suffices to prove that $$\label{ni51}
\begin{split}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}f_k(\xi,\mu,\tau')\cdot \frac{d}{d\mu}\frac{\psi(\tau-\tau')-
\psi(\tau-\omega(\xi,\mu))}{\tau'-\omega(\xi,\mu)}\,d\tau'\Big|\Big|_{X_k}\leq C\|f_k\|_{V_k}.
\end{split}$$ By analyzing the cases $|\tau'-\omega(\xi,\mu)|\leq 1$ and $|\tau'-\omega(\xi,\mu)|\geq 1$, it is easy to see that $$\begin{split}
\Big|\frac{d}{d\mu}&\frac{\psi(\tau-\tau')-
\psi(\tau-\omega(\xi,\mu))}{\tau'-\omega(\xi,\mu)}\Big|\\
&\leq \frac{C|\mu/ \xi|}{1+|\tau'-\omega(\xi,\mu)|}\cdot[(1+|\tau-\tau'| )^{-4}+(1+|\tau-\omega(\xi,\mu)| )^{-4}].
\end{split}$$ In addition, using Lemma \[Lemmaa1\] (b), $\|f_k\cdot (1+|\tau-\omega(\xi,\mu)| )^{-1}\cdot |\mu/ \xi|\|_{X_k}\leq C\|f_k\|_{V_k}$. Thus, for , it suffices to prove that $$\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}}|f_k(\xi,\mu,\tau')|\cdot [(1+|\tau-\tau'| )^{-4}+(1+|\tau-\omega(\xi,\mu)| )^{-4}]\,d\tau'\Big|\Big|_{X_k}\leq C\|f_k\|_{X_k}.$$ This follows from .
The main bound follows from , , and .
The proof of Proposition \[Lemmab3\] above (in particular the bounds and ) shows that if $\varphi\in\mathcal{S}({\ensuremath{\mathbb{R}}})$ then $$\label{io1}
\begin{cases}
&\|\mathcal{F}[\varphi(t)\cdot \mathcal{F}^{-1}(f)]\|_{X_k}\leq C\|f\|_{X_k};\\
&\|\mathcal{F}[\varphi(t)\cdot \mathcal{F}^{-1}(f)]\|_{X_k+Y_k}\leq C\|f\|_{X_k+Y_k};\\
&\|\mathcal{F}[\varphi(t)\cdot \mathcal{F}^{-1}(f)]\|_{V_k\cap W_k}\leq C\|f\|_{V_k\cap W_k},
\end{cases}$$ for any $k\in {\ensuremath{\mathbb{Z}}}$ and $f\in L^2({\ensuremath{\mathbb{R}}}^3)$ supported in $I_k\times{\ensuremath{\mathbb{R}}}\times{\ensuremath{\mathbb{R}}}$.
Proof of Theorem \[Main1\] {#proofthm}
==========================
In this section we reduce Theorem \[Main1\] to proving the following two dyadic bilinear estimates: assume $k_i\in {\ensuremath{\mathbb{Z}}}$, $f_{k_i}\in V_{k_i}\cap W_{k_i}$, and $\mathcal{F}(f_{k_i})$ are supported in ${\ensuremath{\mathbb{R}}}^2\times [-2,2]$, $i=1,2$.
$\bullet$ If $k\in {\ensuremath{\mathbb{Z}}}$, $k_1\leq k-20$ and $|k_2-k|\leq 2$ then $$\label{BIG1}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}\\
&\leq C(2^{-|k_1|/8}+2^{-|k-k_1|/8})||f_{k_1}||_{V_{k_1}\cap W_{k_1}}||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
$\bullet$ If $|k_1-k_2|\leq 100$ then $$\label{BIG2}
\begin{split}
\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}^2\Big]^{1/2}\\
&\leq C||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
\[section\]
\[Lemmao1\] If $u,v\in F$ then $$||\partial_x(uv)||_{N}\leq C||u||_{F}\cdot \|v\|_{F}.$$
Let $f_k=\chi_k(\xi)\cdot \mathcal{F}(u)$ and $g_k=\chi_k(\xi)\cdot \mathcal{F}(v)$. Then $$\label{su1}
\begin{cases}
&\|u\|_F=\big[\sum_{k_1\in{\ensuremath{\mathbb{Z}}}}\|f_{k_1}\|_{V_{k_1}\cap W_{k_1}}^2\big]^{1/2};\\
&\|v\|_F=\big[\sum_{k_2\in{\ensuremath{\mathbb{Z}}}}\|g_{k_2}\|_{V_{k_2}\cap W_{k_2}}^2\big]^{1/2}.
\end{cases}$$ For $k\in{\ensuremath{\mathbb{Z}}}$ let $$\begin{cases}
&A=\{(k_1,k_2)\in{\ensuremath{\mathbb{Z}}}^2:\,|k_1-k_2|\leq 100\};\\
&A_1(k)=\{(k_1,k_2)\in{\ensuremath{\mathbb{Z}}}^2:\,|k_2-k|\leq 2\text{ and }k_1\leq k-20\};\\
&A_2(k)=\{(k_1,k_2)\in{\ensuremath{\mathbb{Z}}}^2:\,|k_1-k|\leq 2\text{ and }k_2\leq k-20\}.
\end{cases}$$ Clearly $$\chi_k(\xi)\cdot \mathcal{F}(\partial_x(uv))=C\chi_k(\xi)\xi\sum_{(k_1,k_2)\in A\cup A_1(k)\cup A_2(k)}(f_{k_1}\ast g_{k_2}).$$ Thus $$\label{su2}
\begin{split}
&||\partial_x(uv)||_{N}^2\leq C\sum_{k\in{\ensuremath{\mathbb{Z}}}}\Big(\sum_{(k_1,k_2)\in A}\big|\big|2^k\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
g_{k_2})\big|\big|_{V_k\cap W_k}\Big)^2\\
&+C\sum_{k\in{\ensuremath{\mathbb{Z}}}}\Big(\sum_{(k_1,k_2)\in A_1(k)}\big|\big|2^k\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
g_{k_2})\big|\big|_{V_k\cap W_k}\Big)^2\\
&+C\sum_{k\in{\ensuremath{\mathbb{Z}}}}\Big(\sum_{(k_1,k_2)\in A_2(k)}\big|\big|2^k\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
g_{k_2})\big|\big|_{V_k\cap W_k}\Big)^2.
\end{split}$$
Using , the first term in the right-hand side of is bounded by $$C\Big[\sum_{(k_1,k_2)\in A}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||g_{k_2}||_{V_{k_2}\cap W_{k_2}}\Big]^2\leq C\|u\|_F^2\cdot ||v||_{F}^2.$$ Using , the second term in the right-hand side of is bounded by $$C\sum_{k\in{\ensuremath{\mathbb{Z}}}}\Big(||u||_F\cdot \sum_{|k_2-k|\leq 2}||g_{k_2}||_{V_{k_2}\cap W_{k_2}}\Big)^2\leq C\|u\|_F^2\cdot ||v||_{F}^2.$$ The third term in the right-hand side of is similar, and the proposition follows.
If follows from Proposition \[Lemmab3\] and Proposition \[Lemmao1\] that $$\label{su4}
\Big|\Big|\eta_0(t)\cdot \int_0^t[W(\partial_x (uv))(s)](t-s)\,ds\Big|\Big|_{F}\leq C||u||_{F}\cdot ||v||_{F},$$ for any $u,v\in F$. It is easy to show that $F$ is a Banach space, and Theorem \[Main1\] follows from and Proposition \[Lemmab1\] by a standard fixed-point argument.
The rest of the paper is concerned with the proofs of the dyadic bilinear estimates and .
Preliminary estimates {#prop}
=====================
In this section we prove several localized $L^\infty_yL^2_{x,t}$ and $L^2_yL^\infty_{x,t}$ estimates and an $L^4$ Strichartz estimate. These bounds will be used in the bilinear estimates in Sections \[bilinear2\], \[bilinear1\], and \[bilinear3\]. We start with a representation formula for functions in $Y_k$, $k\geq 100$. Let $\mathbf{1}_+$ and $\mathbf{1}_-$ denote the characteristic functions of the intervals $[0,\infty)$ and $(-\infty,0]$ respectively.
\[section\]
\[Lemmav0\] If $k\geq 100$ and $f\in Y_k$, then $f$ can be written in the form $$\label{io10}
\begin{split}
f(\xi,\mu,\tau)&=2^{-k/2}\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\\
&\times \Big(\frac{\eta_0(M-\mu)}{M-\mu+i/2^k}+\frac{\eta_0(M+\mu)}{M+\mu+i/2^k}\Big)\cdot\int_{{\ensuremath{\mathbb{R}}}}e^{-iy\cdot \mu}g(y,\xi,\tau)\,dy+h,
\end{split}$$ where $M=M(\xi,\tau)=\sqrt{\xi\cdot (\tau-\xi^3)}$, $h$ is supported in the set $\{(\xi,\mu,\tau): \xi\in I_k,\,|\mu|\in[2^{2k-100},2^{2k+100}]\}$, and $$\label{io11}
\|h\|_{X_k}+\|g\|_{L^1_yL^2_{\xi,\tau}}\leq C\|f\|_{Y_k}.$$
We start from the identity . Since $|\xi|\in[2^{k-2},2^{k+2}]$, $|\mu|\in [2^{2k-35},2^{2k+35}]$, $|\tau-\xi^3-\mu^2/ \xi|\leq 2^{k+2}$, we have $\xi\cdot(\tau-\xi^3)\in [2^{4k-80},2^{4k+80}]$. So $M=M(\xi,\tau)=\sqrt{\xi\cdot (\tau-\xi^3)}$ is well-defined and $M\in[2^{2k-40},2^{2k+40}]$. For $\xi\in I_k$, an elementary computation shows that we can approximate $$\begin{split}
&\chi_{[2k-30,2k+30]}(\mu)\cdot \mathbf{1}_+(\mu)\cdot \frac{\eta_{\leq k+1}(\tau-\omega(\xi,\mu))}{\tau-\omega(\xi,\mu)+i}\\
&=\chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\cdot \frac{\xi}{2M}\cdot \frac{\eta_0(M-\mu)}{M-\mu+i/2^k}+E_+(\xi,\mu,\tau)
\end{split}$$ where, with $\beta=|\tau-\omega(\xi,\mu)|+1$, $$\label{io17}
|E_+(\xi,\mu,\tau)|\leq C\chi_{[2k-40,2k+40]}(\mu)\cdot\frac{\eta_{\leq k+100}(\beta)}{\beta }\cdot \Big(\frac{\beta}{2^k}+\frac{1}{\beta}\Big).$$ Similarly, we approximate $$\begin{split}
&\chi_{[2k-30,2k+30]}(\mu)\cdot \mathbf{1}_-(\mu)\cdot \frac{\eta_{\leq k+1}(\tau-\omega(\xi,\mu))}{\tau-\omega(\xi,\mu)+i}\\
&=\chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\cdot \frac{\xi}{2M}\cdot \frac{\eta_0(M+\mu)}{M+\mu+i/2^k}+E_-(\xi,\mu,\tau),
\end{split}$$ with $E_-$ satisfying the same bound . We substitute these formulas into and notice that the terms corresponding to $E_+$ and $E_-$ can be estimated in $X_k$ (as in the proof of Lemma \[Lemmaa1\] (b)). The bound follows from .
We prove now a localized $L^\infty_yL^2_{x,t}$ estimate.
\[Lemmav0\][Lemma]{}
\[Lemmav1\] Assume $k\geq 0$, $l\geq 2k-100$, and $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\in[2^{l-1},2^{l+1}]\}.$$
\(a) Then $$\label{am1}
||\mathcal{F}^{-1}(f)||_{L^\infty_yL^2_{x,t}}\leq C2^{-(l-k)/2}||f||_{X_k+Y_k}.$$
\(b) More generally, if $\varphi:\mathbb{R}\to[0,1]$ is a smooth function supported in the interval $[-2,2]$, $\epsilon\geq 2^{-k}$, and $$f^m_{\pm}(\xi,\mu,\tau)=f(\xi,\mu,\tau)\cdot \varphi((\mu/ \xi\pm \sqrt{3}\xi)/ \epsilon-m)\text{ for }m\in\mathbb{Z},$$ then $$\label{am100}
\Big[\sum_{m\in\mathbb{Z}}||\mathcal{F}^{-1}(f^m_{\pm})||_{L^\infty_yL^2_{x,t}}^2\Big]^{1/2}\leq C2^{-(l-k)/2}||f||_{X_k+Y_k}.$$
For part (a), assume first $f\in X_k$. Then (see [@CoKeSt p. 753]) $$\label{am2}
||\mathcal{F}^{-1}(f)||_{L^\infty_yL^2_{x,t}}\leq C2^{-(l-k)/2}||f||_{X_k},$$ as desired. Assume now that $f\in Y_k$, $k\geq 100$. We use the representation and the bound . In view of , and using Plancherel’s theorem, it suffices to prove that $$\label{pr41}
\Big |\int_{\mathbb{R}}e^{iy_0\cdot \mu }\cdot \frac{\eta_0(M\pm \mu)}{M\pm \mu+i/2^k}\,d\mu \Big|\leq C,$$ uniformly in $y_0$, $M\in [2^{2k-40},2^{2k+40}]$, and $\xi\in I_k$. This is a standard uniform estimate for the inverse Fourier transform of a Calderón–Zygmund kernel.
For part (b), if $f\in X_k$, then follows from by orthogonality. Assume now $f\in Y_k$, $k\geq 100$. We use , so we may assume $$\begin{split}
f^m_{\pm}(\xi,\mu,\tau)&=2^{-k/2}\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\cdot \varphi((\mu/ \xi\pm \sqrt{3}\xi)/ \epsilon-m) \\
&\times \Big(\frac{\eta_0(M-\mu)}{M-\mu+i/2^k}+\frac{\eta_0(M+\mu)}{M+\mu+i/2^k}\Big)\cdot\int_{R}e^{-iy\cdot \mu}g(y,\xi,\tau)\,dy.
\end{split}$$ By comparing the supports in $\mu$ of the functions and using the fact that $2^k\epsilon\geq 1$, we conclude that $f^m_{\pm}(\xi,\mu,\tau)\equiv 0$ unless $(\tau-\xi^3)/ \xi\in[C^{-1}2^{2k},C2^{2k}]$ and $$\Big|\frac{\sqrt{(\tau-\xi^3)/ \xi}\pm\sqrt{3}\xi}{\epsilon}-m\Big|\leq C_0\text{ or }\Big|\frac{-\sqrt{(\tau-\xi^3)/ \xi}\pm\sqrt{3}\xi}{\epsilon}-m\Big|\leq C_0.$$ We define $$\begin{split}
g^m_{\pm}&(y,\xi,\tau)=g(y,\xi,\tau)\\
&\times\Big[\eta_0\Big(\frac{\sqrt{(\tau-\xi^3)/ \xi}\pm\sqrt{3}\xi}{C_0\epsilon}-\frac{m}{C_0}\Big)+\eta_0\Big(\frac{-\sqrt{(\tau-\xi^3)/ \xi}\pm\sqrt{3}\xi}{C_0\epsilon}-\frac{m}{C_0}\Big)\Big].
\end{split}$$ In view of the support property above, we have $$\begin{split}
f^m_{\pm}(\xi,\mu,\tau)&=2^{-k/2}\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\cdot \varphi((\mu/ \xi\pm \sqrt{3}\xi)/ \epsilon-m) \\
&\times \Big(\frac{\eta_0(M-\mu)}{M-\mu+i/2^k}+\frac{\eta_0(M+\mu)}{M+\mu+i/2^k}\Big)\cdot\int_{R}e^{-iy\cdot \mu}g^m_{\pm}(y,\xi,\tau)\,dy.
\end{split}$$ Using part (a) (in fact a slightly modified version of the bound ), $$||\mathcal{F}^{-1}(f^m_{\pm})||_{L^\infty_yL^2_{x,t}}\leq C2^{-k/2}||g^m_{\pm}||_{L^1_yL^2_{\xi,\tau}}.$$ Thus, the left-hand side of is dominated by $$C2^{-k/2}\Big[\sum_{m\in\mathbb{Z}}||g^m_{\pm}||_{L^1_yL^2_{\xi,\tau}}^2\Big]^{1/2}\leq C2^{-k/2}||g||_{L^1_yL^2_{\xi,\tau}},$$ which suffices in view of .
We prove now several localized maximal function estimates:
\[Lemmav0\][Lemma]{}
\[Lemmav2\] Assume $k,l,j\in\mathbb{Z}$, $k\leq 0$, $l\geq 0$, $j\geq 0$.
\(a) If $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\leq 2^l,\,|\tau-\omega(\xi,\mu)|\leq 2^j\},$$ then $$\label{max1}
||\mathcal{F}^{-1}(f)||_{L^2_yL^\infty_{x,t}}\leq C2^{j/2}\cdot 2^{(2l+k)/4}||(I-\partial_\tau^2)f||_{L^2}.$$
\(b) If $m\in\mathbb{R}$, $\epsilon\geq 2^{-l}$, and $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\leq 2^l,\,|\tau-\omega(\xi,\mu)|\leq 2^j,\,|\mu/
\xi\pm\sqrt{3}\xi-m|\leq \epsilon \},$$ then $$\label{max3}
||\mathcal{F}^{-1}(f)||_{L^2_yL^\infty_{x,t}}\leq C2^{j/2}\cdot (2^l\epsilon)^{1/2}\cdot 2^{k/2}||(I-\partial_\tau^2)f||_{L^2}.$$
For any $f:\mathbb{R}^3\to\mathbb{C}$ let $f^\#(\xi,\mu,\theta)=f(\xi,\mu,\theta+\omega(\xi,\mu))$. Then $$\label{timedecay}
\mathcal{F}^{-1}(f)(x,y,t)=C(t^2+1)^{-1}\int_{\mathbb{R}^3}[(I-\partial_\tau^2)f]^\#(\xi,\mu,\theta)e^{it\theta}e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu d\theta.$$ Thus, for , after noticing the time decay in , it suffices to prove that if $$g\text{ is supported in the set }\{(\xi,\mu):\xi\in I_k,\,|\mu|\leq 2^l\},$$ then $$\label{v22}
\Big|\Big|\int_{\mathbb{R}^2}g(\xi,\mu)e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^2_yL^\infty_{x,|t|\leq 1/2}}\leq C2^{(2l+k)/4}||g||_{L^2}.$$ A standard $TT^{\ast}$ argument (see, for example, [@KeZi p. 50]), shows that for it suffices to prove that $$\label{v23}
\Big|\Big|\int_{\mathbb{R}^2}\chi_{[k-1,k+1]}^2(\xi)\eta_0^2(\mu/2^l)e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^1_yL^\infty_{x,|t|\leq 1}}\leq C2^{(2l+k)/2}.$$ To prove we estimate the $\mu$-integral first. Simple integration by parts and van der Corput-type arguments show that if $y\in\mathbb{R}$, $|t|\leq 1$, $|\xi|\in [2^{k-2},2^{k+2}]$, and $k,l$ are as in the hypothesis then $$\Big|\int_{\mathbb{R}}\eta_0^2(\mu/2^l)e^{i(y\cdot\mu+t\cdot \mu^2/ \xi)}\,d\mu\Big|\leq C
\begin{cases}
2^{l-k}|y|^{-2}&\text{ if }|y|\geq 100\cdot 2^{l-k};\\
2^{l/2}|y|^{-1/2}&\text{ if }|y|\in[1,100\cdot 2^{l-k}];\\
2^l&\text{ if }|y|\leq 1.
\end{cases}$$ This leads to .
Similarly, for , it suffices to prove that if $$g\text{ is supported in the set }\{(\xi,\mu):\xi\in I_k,\,|\mu|\leq 2^l,\,|\mu/ \xi\pm\sqrt{3}\xi-m|\leq \epsilon\},$$ then $$\label{v26}
\Big|\Big|\int_{\mathbb{R}^2}g(\xi,\mu)e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^2_yL^\infty_{x,|t|\leq 1/2}}\leq C(2^{l}\epsilon)^{1/2}\cdot 2^{k/2}||g||_{L^2}.$$ In proving , by orthogonality, we may assume $\epsilon=2^{-l}$. We may also assume $|m|\leq C2^{l-k}$. As before, for , it suffices to prove that $$\label{v25}
\Big|\Big|\int_{\mathbb{R}^2}\chi_{[k-1,k+1]}^2(\xi)\eta_0^2(2^l(\mu/ \xi \pm\sqrt{3}\xi-m))e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^1_yL^\infty_{x,|t|\leq 1}}\leq C2^{k}.$$ The change of variables $\mu=\xi(\mp\sqrt{3}\xi+m+2^{-l}\beta)$, with $d\mu=2^{-l}\xi d\beta$, and integration by parts show that $$\Big|\int_{\mathbb{R}}\eta_0^2(2^l(\mu/ \xi \pm\sqrt{3}\xi-m))e^{i(y\cdot\mu+t\cdot\mu^2/ \xi)}d\mu\Big|\leq C2^{k-l}(1+2^{k-l}|y| )^{-2},$$ if $y\in\mathbb{R}$, $|m|\leq C2^{l-k}$, $|\xi|\in[2^{k-2},2^{k+2}]$, and $|t|\leq 1$. This leads to .
\[Lemmav0\][Lemma]{}
\[Lemmav3\] Assume $k,l,j\in\mathbb{Z}_+$.
\(a) If $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\leq 2^l,\,|\tau-\omega(\xi,\mu)|\leq 2^j\},$$ then, for any $\delta>0$, $$\label{max1.1}
||\mathcal{F}^{-1}(f)||_{L^2_yL^\infty_{x,t}}\leq C_\delta 2^{j/2}\cdot (2^k+2^{l-k})^{1/2+\delta }||(I-\partial_\tau^2)f||_{L^2}.$$
\(b) If $m\in\mathbb{R}$, $l\geq 2k$, $\epsilon\geq 2^{-l}$, and $f$ is supported in the set $$\{(\xi,\mu,\tau)\in\mathbb{R}^3:\xi\in I_k,\,|\mu|\leq 2^l,\,|\tau-\omega(\xi,\mu)|\leq 2^j,\,|\mu/
\xi\pm\sqrt{3}\xi-m|\leq \epsilon \},$$ then $$\label{max3.1}
||\mathcal{F}^{-1}(f)||_{L^2_yL^\infty_{x,t}}\leq C2^{j/2}\cdot (2^l\epsilon)^{1/2}||(I-\partial_\tau^2)f||_{L^2}.$$
As in the proof of Lemma \[Lemmav2\], for it suffices to show that if $$g\text{ is supported in the set }\{(\xi,\mu):\xi\in I_k,\,|\mu|\leq 2^l\},$$ then $$\Big|\Big|\int_{\mathbb{R}^2}g(\xi,\mu)e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^2_yL^\infty_{x,|t|\leq 1/2}}\leq C_\delta (2^k+2^{l-k})^{1/2+\delta }||g||_{L^2}.$$ This follows from [@KeZi Theorem 2.1 (b)].
Similarly, for , it suffices to prove that if $$g\text{ is supported in the set }\{(\xi,\mu):\xi\in I_k,\,|\mu|\leq 2^l,\,|\mu/ \xi\pm\sqrt{3}\xi-m|\leq \epsilon\},$$ then $$\label{v26.1}
\Big|\Big|\int_{\mathbb{R}^2}g(\xi,\mu)e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^2_yL^\infty_{x,|t|\leq 1/2}}\leq C(2^{l}\epsilon)^{1/2}||g||_{L^2}.$$ In proving , by orthogonality, we may assume $\epsilon=2^{-l}$. We may also assume $|m|\leq 2^{l-k+3}$. As before, for , it suffices to prove that $$\label{v25.1}
\Big|\Big|\int_{\mathbb{R}^2}\chi_{[k-1,k+1]}^2(\xi)\eta_0^2(2^l(\mu/ \xi \pm\sqrt{3}\xi-m))e^{i(x\cdot \xi+y\cdot\mu+t\cdot\omega(\xi,\mu))}\,d\xi d\mu\Big|\Big|_{L^1_yL^\infty_{x,|t|\leq 1}}\leq C.$$ We make the change of variables $\mu=\xi(\mp\sqrt{3}\xi+m+2^{-l}\beta)$, with $d\mu=2^{-l}\xi d\beta$. The estimate becomes $$\label{v27.1}
2^{-l}\Big|\Big|\int_{\mathbb{R}^2}\xi\cdot \chi_{[k-1,k+1]}^2(\xi)\eta_0^2(\beta)e^{i\Phi(x,y,t,\xi,\beta)}\,d\xi d\beta\Big|\Big|_{L^1_yL^\infty_{x,|t|\leq 1}}\leq C,$$ where $$\label{v28.1}
\Phi(x,y,t,\xi,\beta)=x\cdot \xi+y\cdot \xi(\mp\sqrt{3}\xi+m+2^{-l}\beta)+t\cdot \xi^3+t\cdot \xi(\mp\sqrt{3}\xi+m+2^{-l}\beta)^2.$$
It remains to prove . For $|y|\leq 2^{l-k+10}$ we notice that $|\partial_\xi^3\Phi(x,y,t,\xi,\beta)|\geq |t|$ and $|\partial_\xi^2\Phi(x,y,t,\xi,\beta)|\geq 2\sqrt{3}|y|-C2^{l-k}|t|$, provided that $|\xi|\approx 2^k$ and $|m|\leq2^{l-k+3}$. Thus, using van der Corput’s lemma for the integral in $\xi$, $$\label{v30.1}
2^{-l}\Big|\int_{\mathbb{R}^2}\xi\cdot \chi_{[k-1,k+1]}^2(\xi)\eta_0^2(\beta)e^{i\Phi(x,y,t,\xi,\beta)}\,d\xi d\beta\Big|\leq C2^{k-l}\cdot 2^{(l-k)/2}|y|^{-1/2}.$$ For $|y|\geq 2^{l-k+10}$ we integrate first by parts in $\beta$ (notice that $|\partial_\beta\Phi|\geq 2^{k-l-4}|y|$ and $|\partial^2_\beta\Phi|\leq C2^{k-2l}|$ if $|t|\leq 1$). Then we use van der Corput’s lemma for the integral in $\xi$ as before. The result is $$\label{v30.2}
2^{-l}\Big|\int_{\mathbb{R}^2}\xi\cdot \chi_{[k-1,k+1]}^2(\xi)\eta_0^2(\beta)e^{i\Phi(x,y,t,\xi,\beta)}\,d\xi d\beta\Big|\leq C2^{k-l}\cdot(2^{k-l}|y|)^{-1}\cdot |y|^{-1/2}.$$ The bound follows from and .
We conclude this section with an $L^4$ estimate.
\[Lemmav0\][Lemma]{}
\[Lemmav5\] If $k\in{\ensuremath{\mathbb{Z}}}$ and $f\in X_k+Y_k$ then $$\label{io2}
\|\mathcal{F}^{-1}(f)\|_{L^4_{x,y,t}}\leq C\|f\|_{X_k+Y_k}.$$
We use the scale-invariant Strichartz estimate of [@ArSa]: $$\label{io30}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^2}\phi(\xi,\mu)e^{ix\cdot\xi}e^{iy\cdot\mu}e^{it\cdot\omega(\xi,\mu)}\,d\xi d\mu\Big|\Big|_{L^4_{x,y,t}}\leq C\|\phi\|_{L^2},$$ for any $\phi\in L^2({\ensuremath{\mathbb{R}}}^2)$.
Assume first that $f\in X_k$. With $f^\#$ defined as in the proof of Lemma \[Lemmav2\], for $j\geq 0$ $$\label{re1}
\begin{split}
\Big|\Big|&\int_{{\ensuremath{\mathbb{R}}}^3}f(\xi,\mu,\tau)\cdot \eta_j(\tau-\omega(\xi,\mu))\cdot e^{ix\cdot\xi}e^{iy\cdot\mu}e^{it\cdot\tau}\,d\xi d\mu d\tau\Big|\Big|_{L^4_{x,y,t}}\\
&=\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^3}f^\#(\xi,\mu,\theta)\cdot \eta_j(\theta)e^{it\cdot\theta}\cdot e^{ix\cdot\xi}e^{iy\cdot\mu}e^{it\cdot\omega(\xi,\mu)}\,d\xi d\mu d\theta\Big|\Big|_{L^4_{x,y,t}}\\
&\leq C2^{j/2}||f^\#(\xi,\mu,\theta)\cdot \eta_j(\theta)||_{L^2},
\end{split}$$ which gives .
Assume now that $f\in Y_k$. We use the representation . With the notation in Lemma \[Lemmav0\], using and the bound for $f\in X_k$, it suffices to prove that $$\begin{split}
2^{-k/2}&\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^3}g(\xi,\tau)\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\\
&\times\frac{\eta_0(M\pm\mu)}{M\pm\mu+i/2^k}\cdot e^{ix\cdot\xi}e^{iy\cdot\mu}e^{it\cdot\tau}\,d\xi d\mu d\tau\Big|\Big|_{L^4_{x,y,t}}\leq C||g||_{L^2},
\end{split}$$ for any $g\in L^2({\ensuremath{\mathbb{R}}}^2)$. We take the integral in $\mu$ first; it remains to prove that $$\begin{split}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^2}g(\xi,\tau)&\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(M)\cdot \mathbf{1}_+(M)\\
&\times e^{ix\cdot\xi}e^{iy\cdot M}e^{it\cdot\tau}\,d\xi d\tau\Big|\Big|_{L^4_{x,y,t}}\leq C2^{k/2}||g||_{L^2}.
\end{split}$$ We make the change of variables $\tau=\xi^3+\nu^2/ \xi$, $\nu\in [C^{-1}2^{2k},C2^{2k}]$, $d\tau =2(\nu/ \xi)d\nu$. Clearly, $M(\xi,\tau)=\nu$. Thus, it suffices to prove that $$\begin{split}
\Big|\Big|\int_{{\ensuremath{\mathbb{R}}}^2}&g(\xi,\xi^3+\nu^2/ \xi)\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(\nu)\\
&\times \mathbf{1}_+(\nu)e^{ix\cdot\xi}e^{iy\cdot \nu}e^{it\cdot\omega(\xi,\nu)}\,d\xi d\nu\Big|\Big|_{L^4_{x,y,t}}\leq C2^{-k/2}||g||_{L^2}.
\end{split}$$ This follows from with $\phi(\xi,\nu)=g(\xi,\xi^3+\nu^2/ \xi)\mathbf{1}_{I_k}(\xi)\cdot \chi_{[2k-30,2k+30]}(\nu)\mathbf{1}_+(\nu)$.
An $L^2$ bilinear estimate {#L2bi}
==========================
In this section we prove an $L^2$ bilinear estimate. For $k\in{\ensuremath{\mathbb{Z}}}$ and $j\in {\ensuremath{\mathbb{Z}}}_+$ let $$D_{k,j}=\{(\xi,\mu,\tau):\,\xi\in I_k,\,\mu\in{\ensuremath{\mathbb{R}}},\,|\tau-\omega(\xi,\mu)|\leq 2^j\}.$$
\[section\]
\[Main9l\] Assume $k_1,k_2,k_3\in {\ensuremath{\mathbb{Z}}}$, $j_1,j_2,j_3\in{\ensuremath{\mathbb{Z}}}_+$, and $f_i:{\ensuremath{\mathbb{R}}}^3\to{\ensuremath{\mathbb{R}}}_+$ are $L^2$ functions supported in $D_{k_i,j_i}$, $i=1,2,3$. If $$\label{jj1.1}
\max(j_1,j_2,j_3)\leq k_1+k_2+k_3-20$$ then $$\label{jj1}
\int_{{\ensuremath{\mathbb{R}}}^3}(f_1\ast f_2)\cdot f_3\leq C2^{(j_1+j_2+j_3)/2}\cdot 2^{-(k_1+k_2+k_3)/2}\cdot \|f_1\|_{L^2}\|f_2\|_{L^2}\|f_3\|_{L^2}.$$
Before we proceed to the proof of this lemma we state a simple corollary that follows by duality. \[Main9l\][Corollary]{}
\[Main9\] Assume $k_1,k_2,k_3\in {\ensuremath{\mathbb{Z}}}$, $j_1,j_2,j_3\in{\ensuremath{\mathbb{Z}}}_+$, and $f_i:{\ensuremath{\mathbb{R}}}^3\to{\ensuremath{\mathbb{R}}}_+$ are $L^2$ functions supported in $D_{k_i,j_i}$, $i=1,2$. If $$\max(j_1,j_2,j_3)\leq k_1+k_2+k_3-20$$ then $$\|(f_1\ast f_2)\cdot \mathbf{1}_{D_{k_3,j_3}}\|_{L^2}\leq C2^{(j_1+j_2+j_3)/2}\cdot 2^{-(k_1+k_2+k_3)/2}\cdot \|f_1\|_{L^2}\|f_2\|_{L^2}.$$
Clearly, $$\int_{{\ensuremath{\mathbb{R}}}^3}(f_1\ast f_2)\cdot f_3=\int_{{\ensuremath{\mathbb{R}}}^3}(\widetilde{f}_1\ast f_3)\cdot f_2=\int_{{\ensuremath{\mathbb{R}}}^3}(\widetilde{f}_2\ast f_3)\cdot f_1,$$ where $\widetilde{f}_i(\xi,\mu,\tau)=f_i(-\xi,-\mu,-\tau)$, $i=1,2$. In view of the symmetry of we may assume $$\label{jj0}
j_3=\max(j_1,j_2,j_3).$$ As in the proof of Lemma \[Lemmav2\], we define $f_i^\#(\xi,\mu,\theta)=f_i(\xi,\mu,\theta+\omega(\xi,\mu))$, $i=1,2,3$, $\|f_i^\#\|_{L^2}=\|f_i\|_{L^2}$. We rewrite the left-hand side of in the form $$\label{jj3}
\begin{split}
&\int_{{\ensuremath{\mathbb{R}}}^6}f_1^\#(\xi_1,\mu_1,\theta_1)\cdot f_2^\#(\xi_2,\mu_2,\theta_2)\\
&\times f_3^\#(\xi_1+\xi_2,\mu_1+\mu_2,\theta_1+\theta_2+\Omega((\xi_1,\mu_1),(\xi_2,\mu_2)))\,d\xi_1d\xi_2d\mu_1d\mu_2d\theta_1d\theta_2,
\end{split}$$ where $$\label{jj2}
\begin{split}
\Omega((\xi_1,\mu_1),(\xi_2,\mu_2))&=-\omega(\xi_1+\xi_2,\mu_1+\mu_2)+\omega(\xi_1,\mu_1)+\omega(\xi_2,\mu_2)\\
&=\frac{-\xi_1\xi_2}{\xi_1+\xi_2}\Big[(\sqrt{3}\xi_1+\sqrt{3}\xi_2)^2-\Big(\frac{\mu_1}{\xi_1}-\frac{\mu_2}{\xi_2}\Big)^2\Big].
\end{split}$$ The functions $f_i^\#$ are supported in the sets $\{\xi,\mu,\theta):\,\xi\in I_{k_i},\,\mu\in{\ensuremath{\mathbb{R}}},\,|\theta|\leq 2^{j_i}\}$.
We will prove that if $g_i:{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}_+$ are $L^2$ functions supported in $I_{k_i}\times{\ensuremath{\mathbb{R}}}$, $i=1,2$, and $g:{\ensuremath{\mathbb{R}}}^3\to{\ensuremath{\mathbb{R}}}_+$ is an $L^2$ function supported in $I_k\times{\ensuremath{\mathbb{R}}}\times [-2^j,2^j]$, $j\leq k_1+k_2+k-15$, then $$\label{jj5}
\begin{split}
\int_{{\ensuremath{\mathbb{R}}}^4}g_1(\xi_1,\mu_1)\cdot &g_2(\xi_2,\mu_2)\cdot g(\xi_1+\xi_2,\mu_1+\mu_2,\Omega((\xi_1,\mu_1),(\xi_2,\mu_2)))\,d\xi_1d\xi_2d\mu_1d\mu_2\\
&\leq C2^{j/2}\cdot 2^{-(k_1+k_2+k)/2}\cdot\|g_1\|_{L^2}\|g_2\|_{L^2}\|g\|_{L^2}.
\end{split}$$ This suffices for , in view of and .
To prove , we observe[^6] first that we may assume that the integral in the left-hand side of is taken over the set $$\mathcal{R}_{++}=\{(\xi_1,\mu_1,\xi_2,\mu_2):\,\xi_1+\xi_2\geq 0\text{ and }\mu_1/ \xi_1-\mu_2/ \xi_2\geq 0\}.$$ Using the restriction $j\leq k_1+k_2+k-15$ and , we may assume also that the integral in the left-hand side of is taken over the set $$\widetilde{\mathcal{R}}_{++}=\{(\xi_1,\mu_1,\xi_2,\mu_2)\in\mathcal{R}_{++}:|\sqrt{3}(\xi_1+\xi_2)|-|\mu_1/ \xi_1-\mu_2/ \xi_2|\leq 2^{-10}|\xi_1+\xi_2|\}.$$ To summarize, it suffices to prove that $$\label{jj11}
\begin{split}
\int_{\widetilde{\mathcal{R}}_{++}}g_1(\xi_1,\mu_1)\cdot &g_2(\xi_2,\mu_2)\cdot g(\xi_1+\xi_2,\mu_1+\mu_2,\Omega((\xi_1,\mu_1),(\xi_2,\mu_2)))\,d\xi_1d\xi_2d\mu_1d\mu_2\\
&\leq C2^{j/2}\cdot 2^{-(k_1+k_2+k)/2}\cdot\|g_1\|_{L^2}\|g_2\|_{L^2}\|g\|_{L^2}.
\end{split}$$
We make the changes of variables $$\mu_1=\sqrt{3}\xi_1^2+\beta_1\xi_1\text{ and }\mu_2=-\sqrt{3}\xi_2^2+\beta_2\xi_2,$$ with $d\mu_1d\mu_2=\xi_1\xi_2\,d\beta_1d\beta_2$. The left-hand side of is bounded by $$\label{jj12}
\begin{split}
&C2^{k_1+k_2}\int_{S}g_1(\xi_1,\sqrt{3}\xi_1^2+\beta_1\xi_1)\cdot g_2(\xi_2,-\sqrt{3}\xi_2^2+\beta_2\xi_2)\\
&\times g(\xi_1+\xi_2,\sqrt{3}\xi_1^2-\sqrt{3}\xi_2^2+\beta_1\xi_1+\beta_2\xi_2,\widetilde{\Omega}((\xi_1,\beta_1),(\xi_2,\beta_2)))\,d\xi_1d\xi_2d\beta_1d\beta_2,
\end{split}$$ where $$\label{jj10}
S=\{(\xi_1,\beta_1,\xi_2,\beta_2):\xi_1+\xi_2\geq 0\text{ and }|\beta_1-\beta_2|\leq 2^{-10}(\xi_1+\xi_2)\},$$ and $$\label{jj15}
\widetilde{\Omega}((\xi_1,\beta_1),(\xi_2,\beta_2))=\xi_1\xi_2(\beta_1-\beta_2)\Big(2\sqrt{3}+\frac{\beta_1-\beta_2}{\xi_1+\xi_2}\Big).$$
We define the functions $h_i:{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}_+$ supported in $I_{k_i}\times{\ensuremath{\mathbb{R}}}$, $i=1,2$, $$\begin{cases}
&h_1(\xi_1,\beta_1)=2^{k_1/2}\cdot g_1(\xi_1,\sqrt{3}\xi_1^2+\beta_1\xi_1);\\
&h_2(\xi_2,\beta_2)=2^{k_2/2}\cdot g_2(\xi_2,-\sqrt{3}\xi_2^2+\beta_2\xi_2),
\end{cases}$$ with $\|h_i\|_{L^2}\approx\|g_i\|_{L^2}$. Thus, for it suffices to prove that $$\label{jj20}
\begin{split}
&2^{(k_1+k_2)/2}\int_{S}h_1(\xi_1,\beta_1)\cdot h_2(\xi_2,\beta_2)\\
&\times g(\xi_1+\xi_2,\sqrt{3}\xi_1^2-\sqrt{3}\xi_2^2+\beta_1\xi_1+\beta_2\xi_2,\widetilde{\Omega}((\xi_1,\beta_1),(\xi_2,\beta_2)))\,d\xi_1d\xi_2d\beta_1d\beta_2\\
&\leq C2^{j/2}\cdot 2^{-(k_1+k_2+k)/2}\cdot\|h_1\|_{L^2}\|h_2\|_{L^2}\|g\|_{L^2}.
\end{split}$$
To prove , we may assume without loss of generality that $$\label{jj11.1}
k_1\leq k_2.$$ We make the change of variables $\beta_1=\beta_2+\beta$. In view of , , and the restriction on the support of $g$, we may assume $|\beta|\leq 2^{j-k_1-k_2+4}$. Thus, the integral in the left-hand side of is equal to $$\label{jj21}
\begin{split}
&2^{(k_1+k_2)/2}\int_{\widetilde{S}}h_1(\xi_1,\beta+\beta_2)\cdot h_2(\xi_2,\beta_2)\cdot\mathbf{1}_{[-1,1]}( \beta/2^{j-k_1-k_2+4} )\\
&\times g(\xi_1+\xi_2,A(\xi_1,\xi_2,\beta)+\beta_2(\xi_1+\xi_2),B(\xi_1,\xi_2,\beta))\,d\xi_1d\xi_2d\beta d\beta_2,
\end{split}$$ where $\widetilde{S}=\{(\xi_1,\xi_2,\beta,\beta_2)\in{\ensuremath{\mathbb{R}}}^4:\xi_1+\xi_2\geq 0\text{ and }|\beta|\leq 2^{-10}(\xi_1+\xi_2)\}$, and $$\label{jj25}
\begin{cases}
&A(\xi_1,\xi_2,\beta)=\sqrt{3}\xi_1^2-\sqrt{3}\xi_2^2+\beta \xi_1;\\
&B(\xi_1,\xi_2,\beta)=\xi_1\xi_2\beta\cdot (2\sqrt{3}+\beta/(\xi_1+\xi_2)).
\end{cases}$$ Let $j'=j-k_1-k_2+4$ and decompose, for $i=1,2$, $$h_i(\xi',\beta')=\sum_{m\in{\ensuremath{\mathbb{Z}}}}h_i(\xi',\beta')\cdot \mathbf{1}_{[0,1)}(\beta'/2^{j'}-m)=\sum_{m\in{\ensuremath{\mathbb{Z}}}}h_i^m(\xi',\beta').$$ The expression in is dominated by to $$\label{jj26}
\begin{split}
&2^{(k_1+k_2)/2}\sum_{|m-m'|\leq 4}\int_{\widetilde{S}}h^m_1(\xi_1,\beta+\beta_2)\cdot h^{m'}_2(\xi_2,\beta_2)\\
&\times g(\xi_1+\xi_2,A(\xi_1,\xi_2,\beta)+\beta_2(\xi_1+\xi_2),B(\xi_1,\xi_2,\beta))\,d\xi_1d\xi_2d\beta d\beta_2.
\end{split}$$ Also, for $i=1,2$, $$\|h_i\|_{L^2}=\big[\sum_{m\in{\ensuremath{\mathbb{Z}}}}\|h_i^m\|_{L^2}^2\big].$$ Thus, to prove , we may assume $h_1=h_1^m$ and $h_2=h_2^{m'}$ for some fixed $m,m'\in{\ensuremath{\mathbb{Z}}}$ with $|m-m'|\leq 4$. To summarize, it suffices to prove that if $F_i:{\ensuremath{\mathbb{R}}}^2\to[0,\infty)$ are $L^2$ functions supported in $I_{k_i}\times{\ensuremath{\mathbb{R}}}$, $g$ is as before, and $m\in{\ensuremath{\mathbb{Z}}}$ then $$\label{jj30}
\begin{split}
&2^{(k_1+k_2)/2}\int_{\widetilde{S}}F_1(\xi_1,\beta+\beta_2)\cdot F_2(\xi_2,\beta_2)\cdot \mathbf{1}_{[m-1,m+1]}(\beta_2/2^{j'})\\
&\times g(\xi_1+\xi_2,A(\xi_1,\xi_2,\beta)+\beta_2(\xi_1+\xi_2),B(\xi_1,\xi_2,\beta))\,d\xi_1d\xi_2d\beta d\beta_2\\
&\leq C2^{j/2}\cdot 2^{-(k_1+k_2+k)/2}\cdot\|F_1\|_{L^2}\|F_2\|_{L^2}\|g\|_{L^2}.
\end{split}$$ To prove we use the Cauchy-Shwartz inequality in the variables $(\xi_1,\xi_2,\beta)$: with $$S'=\{(\xi_1,\xi_2,\beta)\in{\ensuremath{\mathbb{R}}}^3:\xi_i\in I_{k_i},\,\xi_1+\xi_2\geq 0,\,|\beta|\leq 2^{-10}(\xi_1+\xi_2)\},$$ the left-hand side of is dominated by $$\label{jj35}
\begin{split}
C2^{(k_1+k_2)/2}\int_{\ensuremath{\mathbb{R}}}\mathbf{1}_{[m-1,m+1]}(\beta_2/2^{j'})\cdot \Big(\int_{S'}|F_1(\xi_1,\beta+\beta_2)\cdot F_2(\xi_2,\beta_2)|^2\,d\xi_1d\xi_2d\beta\Big)^{1/2}\\
\times\Big(\int_{S'}|g(\xi_1+\xi_2,A(\xi_1,\xi_2,\beta)+\beta_2(\xi_1+\xi_2),B(\xi_1,\xi_2,\beta))|^2\,d\xi_1d\xi_2d\beta \Big)^{1/2}\,d\beta_2.
\end{split}$$ For , it is easy to see that it suffices to prove that $$\label{jj40}
\begin{split}
\Big(\int_{S'}|g(\xi_1+\xi_2,A(\xi_1,\xi_2,\beta)+\beta_2(\xi_1+\xi_2),B(\xi_1,\xi_2,\beta))|^2\,d\xi_1d\xi_2d\beta \Big)^{1/2}\\
\leq C2^{-(k_1+k_2+k)/2}||g||_{L^2}.
\end{split}$$ for any $\beta_2\in{\ensuremath{\mathbb{R}}}$. Indeed, assuming , we can bound the expression in by $$C2^{(k_1+k_2)/2}\int_{\ensuremath{\mathbb{R}}}\mathbf{1}_{[m-1,m+1]}(\beta_2/2^{j'})\cdot ||F_1||_{L^2}||F_2(.,\beta_2)||_{L^2_{\xi_2}}\cdot 2^{-(k_1+k_2+k)/2}||g||_{L^2}\,d\beta_2,$$ which suffices since $2^{j'/2}2^{(k_1+k_2)/2}\approx 2^{j/2}$.
Finally, to prove , we may assume first that $\beta_2=0$. We examine and make the change of variable $\beta=\sqrt{3}(\xi_1+\xi_2)\cdot \nu$. The left-hand side of is dominated by $$\label{jj41}
C\Big(2^{k}\int_{S''}|g(\xi_1+\xi_2,\sqrt{3}(\xi_1+\xi_2)(\xi_1-\xi_2+\nu\xi_1),3\xi_1\xi_2(\xi_1+\xi_2)\nu(2+\nu))|^2\,d\xi_1d\xi_2d\nu \Big)^{1/2},$$ where $S''=\{(\xi_1,\xi_2,\nu)\in{\ensuremath{\mathbb{R}}}^3:\xi_i\in I_{k_i},\,|\nu|\leq 2^{-10}\}$. We define the function $$h(\xi,x,y)=2^{2k}\cdot |g(\xi,\sqrt{3}\xi\cdot x,3\xi\cdot y)|^2,$$ so $||h|||_{L^1}\approx ||g||_{L^2}^2$. The expression in is dominated by $$C2^{-k/2}\Big(\int_{S''}|h(\xi_1+\xi_2,\xi_1-\xi_2+\nu\xi_1,\xi_1\xi_2\cdot \nu(2+\nu))|\,d\xi_1d\xi_2d\nu \Big)^{1/2}.$$ Therefore, it remains to prove that $$\int_{S''}|h(\xi_1+\xi_2,\xi_1-\xi_2+\nu\xi_1,\xi_1\xi_2\cdot \nu(2+\nu))|\,d\xi_1d\xi_2d\nu\leq C2^{-(k_1+k_2)}||h||_{L^1}$$ for any function $h\in L^1({\ensuremath{\mathbb{R}}}^3)$. This is clear since the absolute value of the determinant of the change of variables $(\xi_1,\xi_2,\nu)\to [\xi_1+\xi_2,\xi_1-\xi_2+\nu\xi_1,\xi_1\xi_2\cdot \nu(2+\nu)]$ is equal to $(2+\nu)|\xi_1|\cdot |\xi_2(2+\nu)+\xi_1\nu|\approx 2^{k_1+k_2}$, see and the definition of the set $S''$.
Dyadic bilinear estimates I {#bilinear2}
===========================
In this section we prove the bound for $k\geq 40$ and $k_1\in[0,k-20]$.
\[section\]
\[Lemmak1\] Assume $k\geq 40$, $k_2\in[k-2,k+2]$, $k_1\in[0,k-20]$, $f_{k_1}\in V_{k_1}\cap W_{k_1}$, $f_{k_2}\in V_{k_2}\cap W_{k_2}$, and $\mathcal{F}^{-1}(f_{k_1})(x,y,t)$ is supported in ${\ensuremath{\mathbb{R}}}^2\times[-2,2]$. Then $$\label{bt1}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}\\
&\leq
C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}\cap W_{k_1}}||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
Proposition \[Lemmak1\] follows from Lemma \[Lemmak2\], Lemma \[Lemmak3\], and Lemma \[Lemmak4\] below. We start by decomposing[^7] $$\begin{split}
f_{k_2}=f_{k_2,2k_2-10}+\sum_{l_2\geq 2k_2-9}f_{k_2,l_2}=f_{k_2}\cdot \eta_{\leq 2k_2-10}(\mu_2)+\sum_{l_2\geq 2k_2-9}f_{k_2}\cdot \eta_{l_2}(\mu_2).
\end{split}$$ and $$f_{k_1}=f_{k_1,2k_1}+\sum_{l_1\geq 2k_1+1} f_{k_1,l_1}=f_{k_1}\cdot \eta_{\leq 2k_1}(\mu_1)+\sum_{l_1\geq 2k_1+1}f_{k_1}\cdot\eta_{l_1}(\mu_1).$$
Finally for any $J\in{\ensuremath{\mathbb{Z}}}$ let $f_{k_i,l_i,J}=f_{k_i,l_i}\cdot \eta_{J}(\tau-\omega(\xi,\mu))$, $f_{k_i,l_i,\leq J}=f_{k_i,l_i}\cdot \eta_{\leq J}(\tau-\omega(\xi,\mu))$, and $f_{k_i,l_i,>J}=f_{k_i,l_i}\cdot \eta_{\geq J+1}(\tau-\omega(\xi,\mu))$, $i=1,2$.
\[Lemmak1\][Lemma]{}
\[Lemmak2\] With the notation in Proposition \[Lemmak1\], for any $l_2\in[2k_2-9,2k_2+9]$ $$\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{V_k\cap W_k}\\
&\leq C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}\cap W_{k_1}}||f_{k_2,l_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
In view of the definitions and Lemma \[Lemmaa1\] (b), it suffices to prove that $$\begin{split}
&\big|\big|\chi_k(\xi)\cdot(2^{2k}+i\mu)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&+2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast (\partial_\mu+I)f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\negmedspace(2^{k}||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}+\negmedspace||(\partial_\mu+I)f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}).
\end{split}$$ For this, it suffices to prove that $$\label{bt2.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot(2^{k}+&i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
In view of Lemma \[Lemmaa1\] (a) and (b), Lemma \[Lemmav3\] (a), , and the support assumption on $\mathcal{F}^{-1}(f_{k_1})$, $$\label{bj3.1}
\begin{split}
\|\mathcal{F}^{-1}(f_{k_1,l_1,>J})\|_{L^2_yL^\infty_{x,t}}&\leq C\sum_{j> J}2^{j/2}2^{(l_1-k_1)\cdot 3/5}||(I-\partial_{\tau_1}^2)f_{k_1,l_1,j}||_{L^2}\\
&\leq C2^{(l_1-k_1)\cdot 3/5}(1+2^{(J-2k_1)/2})^{-1}\|(I-\partial_{\tau_1}^2)f_{k_1,l_1}\|_{X_{k_1}}\\
&\leq C(k_1+1)2^{-(l_1-k_1)\cdot 2/5}(1+2^{(J-2k_1)/2})^{-1}\cdot \|f_{k_1}\|_{V_{k_1}}
\end{split}$$ for any $l_1\geq 2k_1$ and $J\in{\ensuremath{\mathbb{Z}}}\cap [-1,\infty)$.
We estimate first the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, $2k_1\leq l_1\leq k+k_1-10$. In this range we will show that $$\label{bj125.1}
\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{-(l_1-k_1)/8}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Let $$\label{j0}
J_0\text{ denote the smallest integer }\geq k-(l_1-k_1)/2-10.$$ Using , Lemma \[Lemmaa1\] (a), Lemma \[Lemmav1\] (a), and with $J=-1$, we estimate $$\label{bj10.1}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{k}\cdot 2^{-J_0/2}||f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2_{\xi,\mu,\tau}}\\
&\leq C2^{k}2^{-J_0/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{-(l_1-k_1)/8}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ We decompose $$\label{hh1}
\begin{split}
f_{k_2,l_2}=f^+_{k_2,l_2,\leq J_0}+f^-_{k_2,l_2,\leq J_0}+f_{k_2,l_2,> J_0}&=f_{k_2,l_2}\cdot \eta_{\leq J_0}(\tau_2-\omega(\xi_2,\mu_2))\mathbf{1}_+(\mu_2)\\
&+f_{k_2,l_2}\cdot \eta_{\leq J_0}(\tau_2-\omega(\xi_2,\mu_2))\mathbf{1}_-(\mu_2)\\
&+f_{k_2,l_2}\cdot \eta_{\geq J_0+1}(\tau_2-\omega(\xi_2,\mu_2))
\end{split}$$ Using , $$\label{hh2}
\|f_{k_2,l_2,>J_0}\|_{L^2}\leq C2^{-J_0/2}\|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.$$ Thus, using the definitions, Lemma \[Lemmaa1\] (a), (c), and we estimate $$\label{bj11.1}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2,>J_0})\big|\big|_{Y_k}\\
&\leq C2^{k/2}\cdot ||\mathcal{F}^{-1}(f_{k_1,l_1}\ast f_{k_2,l_2,>J_0})||_{L^1_yL^2_{x,t}}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2,>J_0})||_{L^2_yL^2_{x,t}}\\
&\leq C2^{-(l_1-k_1)/8}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ An estimate similar to , using gives $$\label{bj12.1}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>k+2k_1-10}\ast
f^\pm_{k_2,l_2,\leq J_0})\big|\big|_{Y_k}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
It remains to estimate $$2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq k+2k_1-10}\ast
f^\pm_{k_2,l_2,\leq J_0})\big|\big|_{X_k+Y_k}.$$ For $j_2\in{\ensuremath{\mathbb{Z}}}_+$ let $f^\pm_{k_2,l_2,j_2}=f_{k_2,l_2}\cdot \eta_{j_2}(\tau_2-\omega(\xi_2,\mu_2))\cdot \mathbf{1}_{\pm}(\mu_2)$. Using Corollary \[Main9\], Lemma \[Lemmaa1\] (b), and the definitions, we estimate $$\label{bj50.1}
\begin{split}
2^k&\sum_{j_1=J_0+1}^{k+2k_1-10}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f^\pm_{k_2,l_2,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{k}\sum_{j_1=J_0+1}^{k+2k_1-10}\sum_{j,j_2=0}^{J_0}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f^\pm_{k_2,l_2,j_2})\big|\big|_{L^2}\\
&\leq C2^{k}\sum_{j_1=J_0+1}^{k+2k_1-10}\sum_{j,j_2=0}^{J_0}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f^{\pm}_{k_2,l_2,j_2}\|_{L^2}\\
&\leq C2^{-k_1/2}\cdot k^3\cdot (2^{(J_0-2k_1)/2}+1)^{-1}\cdot 2^{-(l_1-k_1)}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
Finally, we prove that $$\label{bj40.1}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f^+_{k_2,l_2,\leq J_0})\big|\big|_{Y_k}\\
&\leq C2^{-(l_1-k_1)/8}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Recall that (see ) $$\label{bj13}
\begin{split}
&\Omega[(\xi_1,\mu_1),(\xi_2,\mu_2)]=-\omega(\xi_1+\xi_2,\mu_1+\mu_2)+\omega(\xi_1,\mu_1)+\omega(\xi_2,\mu_2)=-\frac{\xi_1\xi_2}{\xi_1+\xi_2}\\
&\times[(\sqrt{3}\xi_1-\mu_1/ \xi_1)+(\sqrt{3}\xi_2+\mu_2/ \xi_2)]\cdot [(\sqrt{3}\xi_1+\mu_1/ \xi_1)+(\sqrt{3}\xi_2-\mu_2/ \xi_2)].
\end{split}$$ Thus, for $\xi_2\in I_{k_2}$, $\mu_2\in[2^{2k-11},2^{2k+11}]$, $\xi_1\in I_{k_1}$, and $|\mu_1|\leq 2^{k-k_1/2-9}$ $$\label{bj20.1}
|\Omega[(\xi_1,\mu_1),(\xi_2,\mu_2)]|\geq 2^{k+k_1-4}|(\sqrt{3}\xi_1+\mu_1/ \xi_1)+(\sqrt{3}\xi_2-\mu_2/ \xi_2)|.$$
Let $\varphi:\mathbb{R}\to[0,1]$ denote a smooth function supported in $[-1,1]$ with the property that $$\sum_{m\in\mathbb{Z}}\varphi(s-m)\equiv 1.$$ Let $\epsilon=2^{-(l_1+k_1)/2}$. For $m\in\mathbb{Z}$ we define $$\label{bj131.1}
\begin{cases}
&f_{k_1,l_1,\leq J_0}^{+,m}(\xi_1,\mu_1,\tau_1)=f_{k_1,l_1,\leq J_0}(\xi_1,\mu_1,\tau_1)\cdot \varphi((\sqrt{3}\xi_1+\mu_1/ \xi_1)/ \epsilon-m);\\
&f_{k_2,l_2,\leq J_0}^{+,m}(\xi_2,\mu_2,\tau_2)=f_{k_2,l_2,\leq J_0}^+(\xi_2,\mu_2,\tau_2)\cdot \varphi((\sqrt{3}\xi_2-\mu_2/ \xi_2)/ \epsilon+m).
\end{cases}$$ The important observation is that, in view of and the definition of $J_0$, $$\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}^{+,m}\ast
f_{k_2,l_2,\leq J_0}^{+,m'})\equiv 0\text{ unless }|m-m'|\leq 4.$$ Thus, using the definitions and Lemma \[Lemmaa1\] (c), $$\label{bj30.1}
\begin{split}
&2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,l_2,\leq J_0}^+)\big|\big|_{Y_k}\\
&\leq \negmedspace \sum_{|m-m'|\leq 4}\negmedspace 2^k\big|\big|\chi_k(\xi)(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}^{+,m}\ast
f_{k_2,l_2,\leq J_0}^{+,m'})\big|\big|_{Y_k}\\
&\leq C\sum_{|m-m'|\leq 4}2^{k/2}\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}\cdot\|\mathcal{F}^{-1}(f_{k_2,l_2,\leq J_0}^{+,m'})\|_{L^\infty_yL^2_{x,t}}.
\end{split}$$ We use the elementary bound $$\label{bj120}
\|g\|_{L^1(\mathbb{R})}^2\leq C\|g\|_{L^2(\mathbb{R})}\cdot \|(y+i)\cdot g\|_{L^2(\mathbb{R})}$$ for any $g\in L^2({\ensuremath{\mathbb{R}}})$, Lemma \[Lemmav3\] (b), and the definitions to estimate $$\begin{split}
\|\mathcal{F}^{-1}&(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}\leq C2^{(l_1-k_1)/4}\Big(\sum_{j\leq J_0}2^{j/2}\|(I-\partial_{\tau_1}^2)f_{k_1,l_1,j}^{+,m}\|_{L^2}\Big)^{1/2}\\
&\times \Big(\sum_{j\leq J_0}2^{j/2}\|(I-\partial_{\tau_1}^2)(\partial_{\mu_1}+I)f_{k_1,l_1,j}^{+,m}\|_{L^2}\Big)^{1/2}\\
&\leq C2^{(l_1-k_1)/4}\|(1+|\tau_1-\omega(\xi_1,\mu_1)| )^{1/2+1/40}(I-\partial_{\tau_1}^2)f_{k_1,l_1,\leq J_0}^{+,m}\|_{L^2}^{1/2}\\
&\times \|(1+|\tau_1-\omega(\xi_1,\mu_1)| )^{1/2+1/40}(I-\partial_{\tau_1}^2)(\partial_{\mu_1}+I)f_{k_1,l_1,\leq J_0}^{+,m}\|_{L^2}^{1/2},
\end{split}$$ where $f_{k_1,l_1,j}^{+,m}=f_{k_1,l_1,j}\cdot \varphi((\sqrt{3}\xi_1+\mu_1/ \xi_1)/ \epsilon-m)$. Thus, using Lemma \[Lemmaa1\] (b), with $A=\|(1+|\tau_1-\omega(\xi_1,\mu_1)| )^{1/2+1/40}(I-\partial_{\tau_1}^2)f_{k_1,l_1}\|_{L^2}$ $$\begin{split}
&\Big[\sum_{m\in\mathbb{Z}}\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}^2\Big]^{1/2}\\
&\leq C2^{(l_1-k_1)/4}\cdot A^{1/2}\\
&\times \big[\|(1+|\tau_1-\omega(\xi_1,\mu_1)| )^{1/2+1/40}(I-\partial_{\tau_1}^2)(\partial_{\mu_1}+I)f_{k_1,l_1}\|_{L^2}+2^{l_1-k_1}\cdot A\big]^{1/2}\\
&\leq C2^{(l_1-k_1)/4}\cdot 2^{-(l_1-k_1)/2}\cdot (2^{k_1/20}(k_1+1))\|(I-\partial_{\tau_1}^2)f_{k_1,l_1}\|_{V_{k_1}}^{1/2}\\
&\times (2^{k_1/20}(k_1+1))\|(I-\partial_{\tau}^2)f_{k_1,l_1}\|_{V_{k_1}\cap W_{k_1}}^{1/2}.
\end{split}$$ We substitute this last bound into and, using Lemma \[Lemmav1\] (b), and $2k_1\leq l_1$, we conclude that the right-hand side of is dominated by $$\begin{split}
C2^{k/2}\Big[\sum_{m\in\mathbb{Z}}\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}^2\Big]^{1/2}\cdot \Big[\sum_{m\in\mathbb{Z}}\|\mathcal{F}^{-1}(f_{k_2,l_2,\leq J_0}^{+,m})\|_{L^\infty_yL^2_{x,t}}^2\Big]^{1/2}\\
\leq C2^{-(l_1-k_1)/8}\|f_{k_1}\|_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ This gives the bound . The bound follows from the bounds , , , , and .
We estimate now the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, $k+k_1-10\leq l_1\leq 2k_2+12$. In this range we will show that $$\label{bj125.2}
\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using , Lemma \[Lemmav1\] (a), and with $J=-1$, we estimate $$\label{bj10.2}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq k-4}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{k}\cdot 2^{-k/2}||f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2_{\xi,\mu,\tau}}\\
&\leq C2^{k}2^{-k/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using , Lemma \[Lemmaa1\] (a), (c), and we estimate $$\label{bj11.2}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq k-5}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2,>k-5})\big|\big|_{Y_k}\\
&\leq C2^{k/2}\cdot ||\mathcal{F}^{-1}(f_{k_1,l_1}\ast f_{k_2,l_2,>k-5})||_{L^1_yL^2_{x,t}}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2,>k-5})||_{L^2_yL^2_{x,t}}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ An estimate similar to gives $$\label{bj12.2}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq k-5}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>k+2k_1-10}\ast
f_{k_2,l_2,\leq k-5})\big|\big|_{Y_k}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Finally, we use Corollary \[Main9\] and Lemma \[Lemmaa1\] (b) to estimate $$\label{bj50.2}
\begin{split}
2^k&\sum_{j_1=0}^{k+2k_1-10}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq k-5}(\tau-\omega(\xi,\mu))(f_{k_1,l_1,j_1}\ast
f_{k_2,l_2,\leq k-5})\big|\big|_{X_k}\\
&\leq C2^{k}\sum_{j_1=0}^{k+2k_1-10}\sum_{j,j_2=0}^{k-5}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f_{k_2,l_2,j_2})\big|\big|_{L^2}\\
&\leq C2^{k}\sum_{j_1=0}^{k+2k_1-10}\sum_{j,j_2=0}^{k-5}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f_{k_2,l_2,j_2}\|_{L^2}\\
&\leq C2^{-k_1/2}\cdot k^3\cdot 2^{-(l_1-k_1)}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}\\
&\leq C2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ The bound follows from , , , and .
We estimate now the contribution of $\sum_{l_1\geq 2k_2+13}f_{k_1,l_1}\ast f_{k_2,l_2}$: using and Lemma \[Lemmav5\] $$\label{bj6.9}
\begin{split}
\big|\big|&\chi_k(\xi)\cdot (2^k+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq 2k_2+13}f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{-k}\big|\big|\chi_k(\xi)\cdot\mu \cdot (\sum_{l_1\geq 2k_2+13}f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{L^2}\\
&\leq C2^{-k}\big[\sum_{l_1\geq 2k_2+13}||2^{l_1}f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2}^2\big]^{1/2}\\
&\leq C2^{-k+k_1}\big[\sum_{ l_1\geq 2k_2+13}||2^{l_1-k_1}f_{k_1,l_1}||_{X_{k_1}+Y_{k_1}}^2\cdot ||f_{k_2,l_2}||_{X_k+Y_k}^2\big]^{1/2}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from , , and .
\[Lemmak1\][Lemma]{}
\[Lemmak3\] With the notation in Proposition \[Lemmak1\], $$\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,2k_2-10})\big|\big|_{V_k\cap W_k}\\
&\leq C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
As in Lemma \[Lemmak2\], it suffices to prove that $$\label{bo1.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot(2^{k}+&i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate first the contribution of $f_{k_1,l_1}\ast f_{k_2,2k_2-10}$, $l_1\in[2k_1,2k+10]$. Let $$\label{bo3.1}
J_0=2k+k_1-40.$$ Using , , and Lemma \[Lemmav5\], we estimate $$\label{rr1.1}
\begin{split}
2^{k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq 2k-39}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^k2^{-k}\big|\big|f_{k_1,l_1}\ast f_{k_2,2k_2-10}\big|\big|_{L^2}\\
&\leq C||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^4}\cdot ||\mathcal{F}^{-1}(f_{k_2,2k_2-10})||_{L^4}\\
&\leq C2^{k_1-l_1}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ We have the $L^\infty$ bound $$\label{nj1}
\begin{split}
||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^\infty}&\leq C\sum_{j\geq 0}2^{j/2}2^{k_1/2}2^{l_1/2}||f_{k_1,l_1,j}||_{L^2}\\
&\leq C(k_1+1)2^{3k_1/2}2^{-l_1/2}||f_{k_1}||_{V_{k_1}}.
\end{split}$$ Thus, using and , we estimate $$\label{rr2.1}
\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,2k_2-10,>J_0})\big|\big|_{X_k}\\
&\leq C2^k\big|\big|f_{k_1,l_1}\ast f_{k_2,2k_2-10,>J_0}\big|\big|_{L^2}\\
&\leq C2^k\|\mathcal{F}^{-1}(f_{k_1,l_1})\|_{L^\infty}\cdot \|f_{k_2,2k_2-10,>J_0}\|_{L^2}\\
&\leq C(k_1+1)2^{(k_1-l_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ As in the proof of Lemma \[Lemmav5\] (see ) we have $$\|\mathcal{F}^{-1}(f_{k_1,l_1,>J_0})\|_{L^4}\leq C\sum_{j\geq J_0+1}2^{j/2}\|f_{k_1,l_1,j}\|_{L^2}\leq C2^{-(2k-k_1)/2}2^{k_1-l_1}\|f_{k_1}\|_{V_{k_1}}.$$ Thus, using and and Lemma \[Lemmav5\], $$\label{rr1.2}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_0}\ast f_{k_2,2k_2-10,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^k\big|\big|f_{k_1,l_1,>J_0}\ast f_{k_2,2k_2-10,\leq J_0}\big|\big|_{L^2}\\
&\leq C2^k||\mathcal{F}^{-1}(f_{k_1,l_1,>J_0})||_{L^4}\cdot ||\mathcal{F}^{-1}(f_{k_2,2k_2-10,\leq J_0})||_{L^4}\\
&\leq C2^{3 k_1/2-l_1}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Finally, we observe that $$\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,2k_2-10,\leq J_0})\equiv 0,$$ unless $l_1\in[k+k_1-10,k+k_1+10]$, which is a consequence of the identity . Using Corollary \[Main9\], Lemma \[Lemmaa1\] (b), and the definitions, we estimate $$\label{rr3.1}
\begin{split}
2^k&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,2k_2-10,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{k}\sum_{j_1,j_2=0}^{J_0}\sum_{j=0}^{2k-40}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f_{k_2,2k_2-10,j_2})\big|\big|_{L^2}\\
&\leq C2^{k}\sum_{j,j_1,j_2=0}^{J_0}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f_{k_2,2k_2-10,j_2}\|_{L^2}\\
&\leq C2^{-k_1/2}\cdot k^3\cdot 2^{-(l_1-k_1)}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,2k_2-10}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Thus, using , , , and with $l_1\in[k+k_1-10,k+k_1+10]$, we have $$\label{bo4.1}
\begin{split}
\sum_{l_1=2k_1}^{2k+10}2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{-k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $\sum_{l_1\geq 2k+11}f_{k_1,l_1}\ast f_{k_2,2k_2-10}$: using and Lemma \[Lemmav5\], we estimate as in $$\label{rr9.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot &(2^k+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq 2k+11}f_{k_1,l_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,2k_2-10}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from and .
\[Lemmak1\][Lemma]{}
\[Lemmak4\] With the notation in Proposition \[Lemmak1\], for any $l_2\geq 2k_2+10$ $$\begin{split}
2^k\big|&\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{V_k\cap W_k}\\
&\leq C2^{-(l_2-2k_2)/4}(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
As in Lemma \[Lemmak2\], it suffices to prove that $$\label{go1.1}
\begin{split}
\big|\big|&\chi_k(\xi)\cdot(2^{l_2-k}+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{(l_2-2k_2)\cdot (3/4)}(2^{-k_1/8}+2^{-(k-k_1)/8})||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate first the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, for $$\label{l1}
l_1\in[2k_1,l_2+10]\setminus [l_2-k_2+k_1-10,l_2-k_2+k_1+10].$$ Let $$\label{go3.1}
J_0=l_2+k_1-40.$$ Using , , and Lemma \[Lemmav1\], we estimate $$\label{gr1.1}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}2^{-k}\big|\big|f_{k_1,l_1}\ast f_{k_2,l_2}\big|\big|_{L^2}\\
&\leq C2^{l_2-2k_2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{(l_2-2k_2)/2}2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using the $L^\infty$ bound , , and , we estimate $$\label{gr2.1}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2,>J_0})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\big|\big|f_{k_1,l_1}\ast f_{k_2,l_2,>J_0}\big|\big|_{L^2}\\
&\leq C2^{l_2-k}\|\mathcal{F}^{-1}(f_{k_1,l_1})\|_{L^\infty}\cdot \|f_{k_2,l_2,>J_0}\|_{L^2}\\
&\leq C(k_1+1)2^{(k_1-l_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using , , and Lemma \[Lemmav1\], $$\label{gr1.2}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_0}\ast f_{k_2,l_2,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\big|\big|f_{k_1,l_1,>J_0}\ast f_{k_2,l_2,\leq J_0}\big|\big|_{L^2}\\
&\leq C2^{l_2-k}||\mathcal{F}^{-1}(f_{k_1,l_1,>J_0})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2,\leq J_0})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Finally, we observe that for $l_1$ as in $$\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,l_2,\leq J_0})\equiv 0,$$ which is a consequence of the identity . Thus, for $l_1$ as in , $$\label{gr3.1}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{(l_2-2k_2)/2}2^{-(l_1-k_1)/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, for $$\label{l2}
l_1\in[l_2-k_2+k_1-10,l_2-k_2+k_1+10].$$ Let $$J_1=2k+k_1-40.$$ As in , , and , using also $2^{k_1-l_1}\approx 2^{k_2-l_2}$, we estimate $$\label{gr4.1}
\begin{split}
&2^{l_2-k}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq 2k-39}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&+2^{l_2-k}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2,>J_1})\big|\big|_{X_k}\\
&+2^{l_2-k}\big|\big|\chi_k(\xi)\cdot (\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_1}\ast f_{k_2,l_2,\leq J_1})\big|\big|_{X_k}\\
&\leq C 2^{(l_2-2k_2)/2}2^{-k/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ In addition, using Corollary \[Main9\], Lemma \[Lemmaa1\] (b), and the definitions, we estimate $$\label{gr4.2}
\begin{split}
2^{l_2-k}&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq 2k-40}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_1}\ast
f_{k_2,l_2,\leq J_1})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\sum_{j_1,j_2=0}^{J_1}\sum_{j=0}^{2k-40}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f_{k_2,l_2,j_2})\big|\big|_{L^2}\\
&\leq C2^{l_2-k}\sum_{j,j_1,j_2=0}^{J_1}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f_{k_2,l_2,j_2}\|_{L^2}\\
&\leq C2^{-k_1/2}\cdot k^3\cdot 2^{-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $\sum_{l_1\geq l_2+11}f_{k_1,l_1}\ast f_{k_2,l_2}$: using and Lemma \[Lemmav5\], we estimate as in $$\label{gr9.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot &(2^{l_2-k}+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq l_2+11}f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{-k}\big|\big|\mu\cdot (\sum_{l_1\geq l_2+11}f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{L^2}\\
&\leq C2^{k_1-k}\big[\sum_{l_1\geq l_2+11}\|2^{l_1-k_1}f_{k_1,l_1}\ast f_{k_2,l_2}\|_{L^2}^2\big]^{1/2}\\
&\leq C2^{k_1-k}\big[\sum_{l_1\geq l_2+11}\|2^{l_1-k_1}f_{k_1,l_1}\|_{X_{k_1}+Y_{k_1}}^2\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}^2\big]^{1/2}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from , , , and .
Dyadic bilinear estimates II {#bilinear1}
============================
In this section we prove the bound for $k\geq 40$ and $k_1\leq 0$.
\[section\]
\[Lemmac1\] Assume $k\geq 40$, $k_2\in[k-2,k+2]$, $k_1\leq0$, $f_{k_1}\in V_{k_1}\cap W_{k_1}$, $f_{k_2}\in V_{k_2}\cap W_{k_2}$, and $\mathcal{F}^{-1}(f_{k_1})$ is supported in ${\ensuremath{\mathbb{R}}}^2\times[-2,2]$. Then $$\label{bj1}
\begin{split}
2^k\big|\big|\chi_k&(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
Proposition \[Lemmac1\] follows from Lemma \[Lemmac2\], Lemma \[Lemmac3\], and Lemma \[Lemmac4\] below. We start by decomposing[^8]
$$\begin{split}
f_{k_2}=f_{k_2,2k_2-10}+\sum_{l_2\geq 2k_2-9}f_{k_2,l_2}=f_{k_2}\cdot \eta_{\leq 2k_2-10}(\mu_2)+\sum_{l_2\geq 2k_2-9}f_{k_2}\cdot \eta_{l_2}(\mu_2).
\end{split}$$
and $$f_{k_1}=f_{k_1,k_1}+\sum_{l_1\geq k_1+1} f_{k_1,l_1}=f_{k_1}\cdot \eta_{0}(\mu_1/2^{k_1})+\sum_{l_1\geq k_1+1}f_{k_1}\cdot\eta_{l_1}(\mu_1).$$ For any $J\in{\ensuremath{\mathbb{Z}}}$ let $f_{k_i,l_i,J}=f_{k_i,l_i}\cdot \eta_{J}(\tau-\omega(\xi,\mu))$, $f_{k_i,l_i,\leq J}=f_{k_i,l_i}\cdot \eta_{\leq J}(\tau-\omega(\xi,\mu))$, and $f_{k_i,l_i,>J}=f_{k_i,l_i}\cdot \eta_{\geq J+1}(\tau-\omega(\xi,\mu))$, $i=1,2$.
\[Lemmac1\][Lemma]{}
\[Lemmac2\] With the notation in Proposition \[Lemmac1\], for any $l_2\in[2k_2-9,2k_2+9]$ $$\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{V_k\cap W_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}||f_{k_2,l_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
In view of the definitions and Lemma \[Lemmaa1\] (b), it suffices to prove that $$\label{bj2}
\begin{split}
&\big|\big|\chi_k(\xi)\cdot(2^{2k}+i\mu)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&+2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast (\partial_\mu+I)f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot (2^{k}||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}+||(\partial_\mu+I)f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}).
\end{split}$$ For this, it suffices to prove that $$\label{bj2.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot(2^{k}+&i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
In view of Lemma \[Lemmaa1\] (a), Lemma \[Lemmav2\] (a), , and the support assumption on $\mathcal{F}^{-1}(f_{k_1})$, $$\label{bj3}
\begin{cases}
&\|\mathcal{F}^{-1}(\sum_{l_1=k_1}^0f_{k_1,l_1})\|_{L^2_yL^\infty_{x,t}}\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}};\\
&\|\mathcal{F}^{-1}(f_{k_1,l_1})\|_{L^2_yL^\infty_{x,t}}\leq C2^{(2l_1+k_1)/4}\cdot 2^{k_1-l_1}||f_{k_1}||_{V_{k_1}}\text{ for }l_1\geq 1.
\end{cases}$$ Also, using the elementary inequality , $$\label{bj4}
\|\mathcal{F}^{-1}(\sum_{l_1=k_1}^0f_{k_1,l_1})\|_{L^1_yL^\infty_{x,t}}\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}.$$
We start by estimating the contribution of $\sum_{l_1=k_1}^0f_{k_1,l_1}\ast f_{k_2,l_2}$. Using the definitions, Lemma \[Lemmaa1\] (a), (c), Lemma \[Lemmav1\] (a), , and , we estimate $$\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq k-10}(\tau-\omega(\xi,\mu))\cdot (\sum_{l_1=k_1}^0f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{Y_k}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(\sum_{l_1=k_1}^0f_{k_1,l_1}\ast f_{k_2,l_2})||_{L^1_yL^2_{x,t}}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(\sum_{l_1=k_1}^0f_{k_1,l_1})||_{L^1_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ In addition, using , $$\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq k-9}(\tau-\omega(\xi,\mu))\cdot (\sum_{l_1=k_1}^0f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{k}\cdot 2^{-k/2}||\sum_{l_1=k_1}^0f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2_{\xi,\mu,\tau}}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(\sum_{l_1=k_1}^0f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Thus $$\label{bj5}
\begin{split}
\big|\big|\chi_k(\xi)\cdot&(2^k+i\mu/ 2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1=k_1}^0f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, $1\leq l_1\leq k+2k_1-10$. In this range we will show that $$\label{bj125}
\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k+Y_k}\\
&\leq C2^{3k_1/4}2^{-l_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Let $$J_0\text{ denote the smallest integer }\geq k+k_1-l_1/2-10.$$ Using , Lemma \[Lemmaa1\] (a), Lemma \[Lemmav1\] (a), and we estimate $$\label{bj10}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{k}\cdot 2^{-J_0/2}||f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2_{\xi,\mu,\tau}}\\
&\leq C2^{k}2^{-J_0/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{3k_1/4}2^{-l_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ As in , we decompose $$\begin{split}
f_{k_2,l_2}=f^+_{k_2,l_2,\leq J_0}+f^-_{k_2,l_2,\leq J_0}+f_{k_2,l_2,> J_0}&=f_{k_2,l_2}\cdot \eta_{\leq J_0}(\tau_2-\omega(\xi_2,\mu_2))\mathbf{1}_+(\mu_2)\\
&+f_{k_2,l_2}\cdot \eta_{\leq J_0}(\tau_2-\omega(\xi_2,\mu_2))\mathbf{1}_-(\mu_2)\\
&+f_{k_2,l_2}\cdot \eta_{\geq J_0+1}(\tau_2-\omega(\xi_2,\mu_2)).
\end{split}$$ Using , $$\|f_{k_2,l_2,>J_0}\|_{L^2}\leq C2^{-J_0/2}\|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.$$ Thus, using the definitions, Lemma \[Lemmaa1\] (a), (c), and we estimate $$\label{bj11}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2,>J_0})\big|\big|_{Y_k}\\
&\leq C2^{k/2}\cdot ||\mathcal{F}^{-1}(f_{k_1,l_1}\ast f_{k_2,l_2,>J_0})||_{L^1_yL^2_{x,t}}\\
&\leq C2^{k/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2,>J_0})||_{L^2_yL^2_{x,t}}\\
&\leq C2^{3k_1/4}2^{-l_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Also, using Lemma \[Lemmaa1\](a), Lemma \[Lemmav2\] (a), and the definitions $$\|\mathcal{F}^{-1}(f_{k_1,l_1,>J_0})\|_{L^2_yL^\infty_{x,t}}\leq C2^{-J_0/2}2^{(2l_1+k_1)/4}\cdot 2^{k_1-l_1}||f_{k_1}||_{V_{k_1}}.$$ An estimate similar to then gives $$\label{bj12}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_0}\ast
f^\pm_{k_2,l_2,\leq J_0})\big|\big|_{Y_k}\\
&\leq C2^{3k_1/4}2^{-l_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
It remains to estimate $2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f^\pm_{k_2,l_2,\leq J_0})\big|\big|_{Y_k}$. We will use Lemma \[Lemmav1\] (b) and Lemma \[Lemmav2\] (b) to exploit some additional orthogonality. By symmetry, it suffices to prove that $$\label{bj40}
\begin{split}
2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f^+_{k_2,l_2,\leq J_0})\big|\big|_{Y_k}\\
&\leq C2^{k_1}2^{-l_1/4}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ For $(\xi_1,\mu_1),(\xi_2,\mu_2)\in\mathbb{R}^2$ recall that (see ) $$\begin{split}
&\Omega[(\xi_1,\mu_1),(\xi_2,\mu_2)]=-\omega(\xi_1+\xi_2,\mu_1+\mu_2)+\omega(\xi_1,\mu_1)+\omega(\xi_2,\mu_2)=-\frac{\xi_1\xi_2}{\xi_1+\xi_2}\\
&\times[(\sqrt{3}\xi_1-\mu_1/ \xi_1)+(\sqrt{3}\xi_2+\mu_2/ \xi_2)]\cdot [(\sqrt{3}\xi_1+\mu_1/ \xi_1)+(\sqrt{3}\xi_2-\mu_2/ \xi_2)].
\end{split}$$ Thus, for $\xi_2\in I_{k_2}$, $\mu_2\in[2^{2k-11},2^{2k+11}]$, $\xi_1\in I_{k_1}$, and $|\mu_1|\leq 2^{k+2k_1-9}$ $$\label{bj20}
|\Omega[(\xi_1,\mu_1),(\xi_2,\mu_2)]|\geq 2^{k+k_1-4}|(\sqrt{3}\xi_1+\mu_1/ \xi_1)+(\sqrt{3}\xi_2-\mu_2/ \xi_2)|.$$
Let $\varphi:\mathbb{R}\to[0,1]$ denote a smooth function supported in $[-1,1]$ with the property that $$\sum_{m\in\mathbb{Z}}\varphi(s-m)\equiv 1.$$ Let $\epsilon=2^{-l_1/2}$. For $m\in\mathbb{Z}$ we define $$\label{bj131}
\begin{cases}
&f_{k_1,l_1,\leq J_0}^{+,m}(\xi_1,\mu_1,\tau_1)=f_{k_1,l_1,\leq J_0}(\xi_1,\mu_1,\tau_1)\cdot \varphi((\sqrt{3}\xi_1+\mu_1/ \xi_1)/ \epsilon-m);\\
&f_{k_2,l_2,\leq J_0}^{+,m}(\xi_2,\mu_2,\tau_2)=f_{k_2,l_2,\leq J_0}^+(\xi_2,\mu_2,\tau_2)\cdot \varphi((\sqrt{3}\xi_2-\mu_2/ \xi_2)/ \epsilon+m).
\end{cases}$$ The important observation is that, in view of and the definition of $J_0$, $$\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}^{+,m}\ast
f_{k_2,l_2,\leq J_0}^{+,m'})\equiv 0\text{ unless }|m-m'|\leq 4.$$ Thus, using the definitions and Lemma \[Lemmaa1\] (c), $$\label{bj30}
\begin{split}
&2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,l_2,\leq J_0}^+)\big|\big|_{Y_k}\\
&\leq \negmedspace \sum_{|m-m'|\leq 4}\negmedspace 2^k\big|\big|\chi_k(\xi)(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}^{+,m}\ast
f_{k_2,l_2,\leq J_0}^{+,m'})\big|\big|_{Y_k}\\
&\leq C\sum_{|m-m'|\leq 4}2^{k/2}\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}\cdot\|\mathcal{F}^{-1}(f_{k_2,l_2,\leq J_0}^{+,m'})\|_{L^\infty_yL^2_{x,t}}.
\end{split}$$ Using the bound , Lemma \[Lemmav2\] (b), and the definitions $$\begin{split}
\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})&\|_{L^1_yL^\infty_{x,t}}\leq C2^{l_1/4}2^{k_1/2}\| (\tau_1-\omega(\xi_1,\mu_1)+i)\cdot (I-\partial_{\tau_1}^2)f_{k_1,l_1,\leq J_0}^{+,m}\|_{L^2}^{1/2}\\
&\times \| (\tau_1-\omega(\xi_1,\mu_1)+i)\cdot (I-\partial_{\tau_1}^2)(\partial_{\mu_1}+I)f_{k_1,l_1,\leq J_0}^{+,m}\|_{L^2}^{1/2}.
\end{split}$$ Thus $$\begin{split}
\Big[\sum_{m\in\mathbb{Z}}&\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}^2\Big]^{1/2}\\
&\leq C2^{l_1/4}2^{k_1/2}2^{-(l_1-k_1)/2}\|(I-\partial_{\tau}^2)f_{k_1,l_1}\|_{V_{k_1}}^{1/2}\cdot \|(I-\partial_{\tau}^2)f_{k_1,l_1}\|_{V_{k_1}\cap W_{k_1}}^{1/2}.
\end{split}$$ We substitute this last bound into and, using Lemma \[Lemmav1\] (b) and , we conclude that the right-hand side of is dominated by $$\begin{split}
C2^{k/2}\Big[\sum_{m\in\mathbb{Z}}\|\mathcal{F}^{-1}(f_{k_1,l_1,\leq J_0}^{+,m})\|_{L^1_yL^\infty_{x,t}}^2\Big]^{1/2}\cdot \Big[\sum_{m\in\mathbb{Z}}\|\mathcal{F}^{-1}(f_{k_2,l_2,\leq J_0}^{+,m})\|_{L^\infty_yL^2_{x,t}}^2\Big]^{1/2}\\
\leq C2^{-l_1/4}2^{k_1}\|f_{k_1}\|_{V_{k_1}\cap W_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ This gives the bound . The bound follows from the bounds , , , and .
We estimate now the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, $l_1\in[k+2k_1-10,3k]\cap [1,\infty)$: using , Lemma \[Lemmaa1\] (a), Lemma \[Lemmav1\] (a), and , $$\label{bj6}
\begin{split}
\big|\big|&\chi_k(\xi)\cdot(2^k+i\mu/ 2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C(2^k+2^{l_1-k})\cdot ||f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2_{\xi,\mu,\tau}}\\
&\leq C(2^{k}+2^{l_1-k})\cdot ||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{k_1/4}(2^{-(l_1-k-2k_1)/2}+2^{-(3k-l_1)/2})||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
Finally, we estimate the contribution of $\sum_{l_1\geq 3k}f_{k_1,l_1}\ast f_{k_2,l_2}$: using and Lemma \[Lemmav5\] $$\label{bj6.6}
\begin{split}
\big|\big|&\chi_k(\xi)\cdot (2^k+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq 3k}f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{-k}\big|\big|\chi_k(\xi)\cdot\mu \cdot (\sum_{l_1\geq 3k}f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{L^2}\\
&\leq C2^{-k}\big[\sum_{l_1\geq 3k}||2^{l_1}f_{k_1,l_1}\ast f_{k_2,l_2}||_{L^2}^2\big]^{1/2}\\
&\leq C\big[\sum_{l_1\geq 3k}||2^{l_1-k}f_{k_1,l_1}||_{X_{k_1}}^2\cdot ||f_{k_2,l_2}||_{X_k+Y_k}^2\big]^{1/2}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from , , , and .
\[Lemmac1\][Lemma]{}
\[Lemmac3\] With the notation in Proposition \[Lemmac1\], $$\begin{split}
2^k\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,2k_2-10})\big|\big|_{V_k\cap W_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
As in Lemma \[Lemmac2\], it suffices to prove that $$\label{bo1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot(2^{k}+&i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate first the contribution of $f_{k_1,l_1}\ast f_{k_2,2k_2-10}$, $l_1\in[k_1,2k+10]\cap{\ensuremath{\mathbb{Z}}}$. Using , for any $J\in{\ensuremath{\mathbb{Z}}}\cap[-1,\infty)$, $$\label{bo2}
\begin{split}
||\mathcal{F}^{-1}(f_{k_1,l_1,>J})]||_{L^\infty}&\leq C\sum_{j> J}2^{j/2}2^{k_1/2}2^{l_1/2}||f_{k_1,l_1,j}||_{L^2}\\
&\leq C2^{-J/2}2^{k_1/2}2^{l_1/2}||f_{k_1,l_1}||_{X_{k_1}}\\
&\leq C2^{-J/2}2^{3k_1/2}2^{-l_1/2}||f_{k_1}||_{V_{k_1}}.
\end{split}$$ Let $$\label{bo3}
J_0=2k+k_1-40.$$ Using , , and (with $J=-1$), we estimate $$\label{rr1}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^k2^{-J_0/2}\big|\big|f_{k_1,l_1}\ast f_{k_2,2k_2-10}\big|\big|_{L^2}\\
&\leq C2^k2^{-J_0/2}\cdot ||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^\infty}\cdot ||f_{k_2,2k_2-10}||_{L^2}\\
&\leq C2^{k_1-l_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Similarly, using , , and , we estimate $$\label{rr2}
\begin{split}
&2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast f_{k_2,2k_2-10,>J_0})\big|\big|_{X_k}\\
&+2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_0}\ast f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{k_1-l_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We observe now that $$\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,2k_2-10,\leq J_0})\equiv 0,$$ unless $l_1\in[k+k_1-10,k+k_1+10]\cap{\ensuremath{\mathbb{Z}}}$, which is a consequence of the identity . As in and , we estimate $$\label{rr5}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast f_{k_2,2k_2-10,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^k\big|\big|f_{k_1,l_1,\leq J_0}\ast f_{k_2,\leq 2k_2-10,\leq J_0}\big|\big|_{L^2}\\
&\leq C2^k2^{3k_1/2}2^{-l_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using Corollary \[Main9\], Lemma \[Lemmaa1\] (b), and the definitions, we estimate $$\label{rr3}
\begin{split}
2^k&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,2k_2-10,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{k}\sum_{j,j_1,j_2=0}^{J_0}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f_{k_2,2k_2-10,j_2})\big|\big|_{L^2}\\
&\leq C2^{k}\sum_{j,j_1,j_2=0}^{J_0}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f_{k_2,2k_2-10,j_2}\|_{L^2}\\
&\leq C2^{-k_1/2}\cdot k^3\cdot 2^{-(l_1-k_1)}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,2k_2-10}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$ It follows from and that $$\begin{split}
2^k&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,2k_2-10,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,2k_2-10}\|_{X_{k_2}+Y_{k_2}},
\end{split}$$ for $l_1\in[k+k_1-10,k+k_1+10]\cap{\ensuremath{\mathbb{Z}}}$. Thus, using also and , $$\label{bo4}
\begin{split}
\sum_{l_1=k_1}^{2k+10}2^k\big|\big|&\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,2k_2-10}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $\sum_{l_1\geq 2k+11}f_{k_1,l_1}\ast f_{k_2,2k_2-10}$: using and Lemma \[Lemmav5\], we estimate as in $$\label{rr9}
\begin{split}
\big|\big|\chi_k(\xi)\cdot &(2^k+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq 2k+11}f_{k_1,l_1}\ast
f_{k_2,2k_2-10})\big|\big|_{X_k}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,\leq 2k_2-10}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from and .
\[Lemmac1\][Lemma]{}
\[Lemmac4\] With the notation in Proposition \[Lemmac1\], for any $l_2\geq 2k_2+10$ $$\begin{split}
2^k\big|&\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{V_k\cap W_k}\\
&\leq C2^{-(l_2-2k_2)/4}2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
As in Lemma \[Lemmak4\], it suffices to prove that $$\label{fo1.1}
\begin{split}
\big|\big|&\chi_k(\xi)\cdot(2^{l_2-k}+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{(l_2-2k_2)\cdot (3/4)}2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using Lemma \[Lemmav2\] and the definitions, for any $J\in[-1,\infty)\cap{\ensuremath{\mathbb{Z}}}$, $k_1\leq 0$, and $l_1\geq k_1$, $$\label{fff1}
\begin{split}
\|\mathcal{F}^{-1}(f_{k_1,l_1,>J})\|_{L^2_yL^\infty_{x,t}}&\leq C2^{-J/2}2^{k_1/4}(2^{l_1/2}+1)\cdot 2^{k_1-l_1}||f_{k_1}||_{V_{k_1}}\\
&\leq C2^{-J/2}2^{k_1/4}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}.
\end{split}$$ Recall also the $L^\infty$ estimate $$\label{fff2}
\begin{split}
||\mathcal{F}^{-1}(f_{k_1,l_1,>J})]||_{L^\infty}\leq C2^{-J/2}2^{k_1}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}.
\end{split}$$
We estimate first the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$ for $$\label{f1}
l_1\in[k_1,l_2+10]\setminus [l_2-k_2+k_1-10,l_2-k_2+k_1+10].$$ Let $$\label{fo3.1}
J_0=l_2+k_1-40.$$ If $l_2-2k_2+k_1\geq 0$ then we use , , and Lemma \[Lemmav1\] to estimate $$\begin{split}
2^{l_2-k}&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}2^{-k}\big|\big|f_{k_1,l_1}\ast f_{k_2,l_2}\big|\big|_{L^2}\\
&\leq C2^{l_2-2k_2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^2_yL^\infty_{x,t}}\cdot ||\mathcal{F}^{-1}(f_{k_2,l_2})||_{L^\infty_yL^2_{x,t}}\\
&\leq C2^{(l_2-2k_2)/2}\cdot 2^{k_1/4}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ If $l_2-2k_2+k_1\leq 0$ then we use , , and to estimate $$\label{fr1.10}
\begin{split}
2^{l_2-k}&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}2^{-(l_2+k_1)/2}\big|\big|f_{k_1,l_1}\ast f_{k_2,l_2}\big|\big|_{L^2}\\
&\leq C2^{l_2-k_2}2^{-(l_2+k_1)/2}||\mathcal{F}^{-1}(f_{k_1,l_1})||_{L^\infty}\cdot ||f_{k_2,l_2}||_{L^2}\\
&\leq C2^{(l_2-2k_2)/2}\cdot 2^{k_1/2}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Thus $$\label{fr1.1}
\begin{split}
2^{l_2-k}&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_0+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{(l_2-2k_2)/2}\cdot 2^{k_1/4}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
Using the $L^\infty$ bound , , and , we estimate $$\label{fr2.1}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)\cdot&(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2,>J_0})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\big|\big|f_{k_1,l_1}\ast f_{k_2,l_2,>J_0}\big|\big|_{L^2}\\
&\leq C2^{l_2-k}\|\mathcal{F}^{-1}(f_{k_1,l_1})\|_{L^\infty}\cdot \|f_{k_2,l_2,>J_0}\|_{L^2}\\
&\leq C2^{(l_2-2k_2)/2}\cdot 2^{k_1/2}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Using , , and , $$\label{fr1.2}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_0}\ast f_{k_2,l_2,\leq J_0})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\big|\big|f_{k_1,l_1,>J_0}\ast f_{k_2,l_2,\leq J_0}\big|\big|_{L^2}\\
&\leq C2^{l_2-k}||\mathcal{F}^{-1}(f_{k_1,l_1,>J_0})||_{L^\infty}\cdot ||f_{k_2,l_2,\leq J_0}||_{L^2}\\
&\leq C2^{(l_2-2k_2)/2}\cdot 2^{k_1/2}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ Finally, we observe that for $l_1$ as in $$\eta_{\leq J_0}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_0}\ast
f_{k_2,l_2,\leq J_0})\equiv 0,$$ which is a consequence of the identity . Thus, for $l_1$ as in , $$\label{fr3.1}
\begin{split}
2^{l_2-k}\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{(l_2-2k_2)/2}2^{k_1/4}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$
We estimate now the contribution of $f_{k_1,l_1}\ast f_{k_2,l_2}$, for $$l_1\in[l_2-k_2+k_1-10,l_2-k_2+k_1+10].$$ Let $$J_1=2k+2k_1-40.$$ As in , , and , using also $2^{k_1-l_1}\approx 2^{k_2-l_2}$, we estimate $$\label{fr4.1}
\begin{split}
&2^{l_2-k}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\geq J_1+1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2})\big|\big|_{X_k}\\
&+2^{l_2-k}\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1}\ast f_{k_2,l_2,>J_1})\big|\big|_{X_k}\\
&+2^{l_2-k}\big|\big|\chi_k(\xi)\cdot (\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,>J_1}\ast f_{k_2,l_2,\leq J_1})\big|\big|_{X_k}\\
&\leq C 2^{l_2-k}(2^{J_1}+1)^{-1/2}2^{k_1}2^{-(l_1-k_1)/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}\\
&\leq C 2^{(l_2-2k_2)/2}2^{k_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2,l_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ In addition, using Corollary \[Main9\], Lemma \[Lemmaa1\] (b), and the definitions, we estimate $$\label{fr4.2}
\begin{split}
2^{l_2-k}&\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\eta_{\leq J_1}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,\leq J_1}\ast
f_{k_2,l_2,\leq J_1})\big|\big|_{X_k}\\
&\leq C2^{l_2-k}\sum_{j,j_1,j_2=0}^{J_1}2^{-j/2}\big|\big|\eta_{j}(\tau-\omega(\xi,\mu))\cdot (f_{k_1,l_1,j_1}\ast
f_{k_2,l_2,j_2})\big|\big|_{L^2}\\
&\leq C2^{l_2-k}\sum_{j,j_1,j_2=0}^{J_1}2^{-(2k+k_1)/2}\cdot 2^{j_1/2}\|f_{k_1,l_1,j_1}\|_{L^2}\cdot 2^{j_2/2}\|f_{k_2,l_2,j_2}\|_{L^2}\\
&\leq C2^{k_1/4}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}},
\end{split}$$ since we may assume that $k+k_1\geq 0$ (compare with the definition of $J_1$).
We estimate now the contribution of $\sum_{l_1\geq l_2+11}f_{k_1,l_1}\ast f_{k_2,l_2}$: using and Lemma \[Lemmav5\], we estimate as in $$\label{fr9.1}
\begin{split}
\big|\big|\chi_k(\xi)\cdot &(2^{l_2-k}+i\mu/2^k)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (\sum_{l_1\geq l_2+11}f_{k_1,l_1}\ast
f_{k_2,l_2})\big|\big|_{X_k}\\
&\leq C2^{k_1-k}||f_{k_1}||_{V_{k_1}}\cdot \|f_{k_2,l_2}\|_{X_{k_2}+Y_{k_2}}.
\end{split}$$
The main bound follows from , , , and .
Dyadic bilinear estimates III {#bilinear3}
=============================
In this section we prove the bound for $k\leq 40$.
\[section\]
\[Lemmaj1\] Assume $k\leq 40$, $k_2\in[k-2,k+2]$, $k_1\leq k-20$, $f_{k_1}\in V_{k_1}\cap W_{k_1}$, and $f_{k_2}\in V_{k_2}\cap W_{k_2}$. Then $$\label{ss1}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}\\
&\leq
C2^{k_1/2}||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
We show first that $$\label{ss2}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k}\leq
C2^{k_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}}.
\end{split}$$ Using and the definition , the left-hand side of is dominated by $$\label{ss3}
\begin{split}
&C||(1+|\mu|)\cdot \chi_k(\xi)\cdot (f_{k_1}\ast f_{k_2})\big|\big|_{L^2}\\
&\leq C||(|\mu_1f_{k_1}|)\ast |f_{k_2}|\,||_{L^2}+C||\,|f_{k_1}|\ast [(1+|\mu_2|)|f_{k_2}|]||_{L^2}.
\end{split}$$ We observe now that, for $i=1,2$ $$\label{ss4}
\begin{split}
||\mathcal{F}^{-1}(|f_{k_i}|)||_{L^\infty}&\leq C\sum_{l,j\geq 0}||f_{k_i}\cdot \eta_{l}(\mu)\cdot \eta_j(\tau-\omega(\xi,\mu))||_{L^1}\\
&\leq C\sum_{l,j\geq 0}2^{(k_i+l+j)/2}||f_{k_1}\cdot \eta_{l}(\mu)\cdot \eta_j(\tau-\omega(\xi,\mu))||_{L^2}\\
&\leq C2^{k_i/2}\sum_{j\geq 0}2^{j/2}||f_{k_1}\cdot (1+|\mu|)\cdot \eta_j(\tau-\omega(\xi,\mu))||_{L^2}\\
&\leq C2^{k_i/2}||f_{k_i}||_{V_{k_i}}.
\end{split}$$ Thus, using also , the right hand side of is bounded by $$\begin{split}
&C||\mu_1\cdot f_{k_1}||_{L^2}\cdot ||\mathcal{F}^{-1}(|f_{k_2}|)||_{L^\infty}+C||\mathcal{F}^{-1}(|f_{k_1}|)||_{L^\infty}||(1+|\mu_2|)f_{k_2}||_{L^2}\\
&\leq C2^{k_1}||(\mu_1/2^{k_1})\cdot f_{k_1}||_{X_{k_1}}||f_{k_2}||_{V_{k_2}}+C2^{k_1/2}||f_{k_1}||_{V_{k_1}}||(1+|\mu_2|)f_{k_2}||_{X_{k_2}}\\
&\leq C2^{k_1/2}||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}}.
\end{split}$$ This completes the proof of .
We show now that $$\label{ss8}
\begin{split}
2^k\big|\big|\chi_k(\xi)&\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{W_k}\leq
C2^{k_1/2}||f_{k_1}||_{V_{k_1}}||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$ In view of the definition , the left-hand side of is bounded by $$\label{ss9}
\begin{split}
&C2^k\big|\big|\chi_k(\xi)\cdot(\mu/\xi)(\tau-\omega(\xi,\mu)+i)^{-2}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{X_k}\\
&+C2^k\big|\big|\chi_k(\xi)\cdot(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
(\partial_\mu+I)f_{k_2})\big|\big|_{X_k}.
\end{split}$$ The first term in is dominated by the left-hand side of . We use , , and to estimate the second term in by $$\begin{split}
C2^k\big|\big|f_{k_1}\ast(\partial_\mu+I)f_{k_2}\big|\big|_{L^2}&\leq C||\mathcal{F}^{-1}(f_{k_1})||_{L^\infty}\cdot ||(\partial_\mu+I)f_{k_2}||_{L^2}\\
&\leq C2^{k_1/2}||f_{k_1}||_{X_{k_1}}\cdot ||(\partial_\mu+I)f_{k_2}||_{X_{k_2}},
\end{split}$$ which suffices for .
Dyadic bilinear estimates IV {#bilinear4}
============================
In this section we prove the bound .
\[section\]
\[Lemmal1\] Assume $k_1,k_2\in{\ensuremath{\mathbb{Z}}}$, $|k_1-k_2|\leq 100$, $f_{k_1}\in V_{k_1}\cap W_{k_1}$, and $f_{k_2}\in V_{k_2}\cap W_{k_2}$. Then $$\label{ll1}
\begin{split}
\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k\cap W_k}^2\Big]^{1/2}\\
&\leq C||f_{k_1}||_{V_{k_1}\cap W_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$
We show first that $$\label{ll2}
\begin{split}
\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{V_k}^2\Big]^{1/2}\leq C||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}}.
\end{split}$$ Using and the definition , the left-hand side of is dominated by $$\label{ll3}
\begin{split}
C\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}&\big|\big|(1+2^{2k}+|\mu|)\chi_k(\xi)\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{L^2}^2\Big]^{1/2}\\
&\leq C(2^{k_1+k_2}+1)||f_{k_1}\ast
f_{k_2}||_{L^2}+C||\mu\cdot (f_{k_1}\ast f_{k_2})||_{L^2}.
\end{split}$$ Using Lemma \[Lemmav5\], the first term in the right-hand side of is dominated by $$C(2^{k_1}+1)||\mathcal{F}^{-1}(f_{k_1})||_{L^4}\cdot (2^{k_2}+1)||\mathcal{F}^{-1}(f_{k_2})||_{L^4}\leq C||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}}.$$ The second term is bounded by $$\begin{split}
&C||\mathcal{F}^{-1}(\mu_1\cdot f_{k_1})||_{L^4}\cdot ||\mathcal{F}^{-1}(f_{k_2})||_{L^4}+C||\mathcal{F}^{-1}(f_{k_1})||_{L^4}\cdot ||\mathcal{F}^{-1}(\mu_2\cdot f_{k_2})||_{L^4}\\
&\leq C2^{k_1}||f_{k_1}||_{V_{k_1}}\cdot 2^{-k_2}||f_{k_2}||_{V_{k_2}}+C2^{-k_1}||f_{k_1}||_{V_{k_1}}\cdot 2^{k_2}||f_{k_2}||_{V_{k_2}}.
\end{split}$$ This completes the proof of .
We show now that $$\label{ll6}
\begin{split}
\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)&(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{W_k}^2\Big]^{1/2}\leq C||f_{k_1}||_{V_{k_1}}\cdot ||f_{k_2}||_{V_{k_2}\cap W_{k_2}}.
\end{split}$$ In view of the definition , the left-hand side of is bounded by $$\label{ll7}
\begin{split}
&C\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)\cdot (\mu/ \xi)(\tau-\omega(\xi,\mu)+i)^{-2}\cdot (f_{k_1}\ast
f_{k_2})\big|\big|_{X_k+Y_k}^2\Big]^{1/2}\\
&+C\Big[\sum_{k\in{\ensuremath{\mathbb{Z}}}}\big|\big|2^k\chi_k(\xi)(\tau-\omega(\xi,\mu)+i)^{-1}\cdot (f_{k_1}\ast
(\partial_\mu+I)f_{k_2})\big|\big|_{X_k+Y_k}^2\Big]^{1/2}.
\end{split}$$ The first term in is dominated by the left-hand side of , which suffices. Using and Lemma \[Lemmav5\], the second term in is bounded by $$\begin{split}
C2^{k_2}||f_{k_1}&\ast (\partial_\mu+I)f_{k_2}||_{L^2}\leq C||\mathcal{F}^{-1}(f_{k_1})||_{L^4}\cdot 2^{k_2}||\mathcal{F}^{-1}((\partial_\mu+I)f_{k_2})||_{L^4}\\
&\leq C||f_{k_1}||_{V_{k_1}}\cdot ||(\partial_\mu+I)f_{k_2}||_{X_{k_2}+Y_{k_2}}.
\end{split}$$ This completes the proof of .
The proposition follows from the estimates and .
[99]{}
M. Ben-Artzi and J.-C. Saut, Uniform decay estimates for a class of oscillatory integrals and applications, Differential Integral Equations [**[12]{}**]{} (1999), 137–145.
J. Colliander, C. E. Kenig, and G. Staffilani, Low regularity solutions for the Kadomtsev–Petviashvili I equation, Geom. Funct. Anal. [**[13]{}**]{} (2003), 737-794.
A. Ionescu and C Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, to appear in J. Amer. Math. Soc. (2007).
A. Ionescu and C Kenig, Low-regularity Schrödinger maps, to appear Diff. and Int. Eq. (2007)
A. Ionescu and C Kenig, Low-regularity Schrödinger maps II: global well-posedness in dimensions $d\geq 3$, to appear Comm. Math. Phys. (2007)
C Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. Inst. H. Poincaré Anal. Non. Linéaire [**[21]{}**]{} (2004), 827–838.
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[^1]: J.C. is supported in part by an NSERC Grant.
[^2]: A.D.I. is supported in part by an NSF grant and a Packard Fellowship.
[^3]: C.E.K. is supported in part by an NSF grant.
[^4]: G.S. is supported in part by an NSF grant.
[^5]: The main difficulties of the KP-I problem, including the counterexample of subsection \[count\], are caused by functions with Fourier support in this region.
[^6]: There are four identical integrals of this type.
[^7]: In the decomposition below we make an abuse of notation when we write that $f_{k_i,2k_i}=
\sum_{l_i< 2k_i+1} f_{k_i,l_i}$. One can see in the rest of the paper that this notation avoids some unnecessary technicalities. One example of its efficiency is in the fact that for any $l_i< 2k_i+1$ $$(1+|\xi|+|\mu/\xi|)| f_{k_i,l_i}|\sim (1+2^{k_i})| f_{k_i,l_i}|$$ and hence we can simply write $$(1+|\xi|+|\mu/\xi|)| f_{k_i,2k_i}|\sim (1+2^{k_i})| f_{k_i,2k_i}|.$$ Our notation also explains why in the proof of the lemmas below we will always assume that $l_1\geq 2k_1$.
[^8]: In the decomposition below we make an abuse of notation when we write that $f_{k_2,2k_2}=
\sum_{l_i< 2k_2+1} f_{k_2,l_2}$ and $f_{k_1,k_1}=
\sum_{l_1< k_1+1} f_{k_1,l_1}$ . One can see in the rest of the paper that this notation avoids some unnecessary technicalities. One example of its efficiency is in the fact that for any $k_1\leq 0, \, l_1< k_1+1$ $$(1+|\xi|+|\mu/\xi|)| f_{k_1,l_1}|\sim | f_{k_2,l_2}|$$ and hence we can simply write $$(1+|\xi|+|\mu/\xi|)| f_{k_1,k_1}|\sim | f_{k_1,k_1}|.$$ Our notation also explains why in the proof of the lemmas below we will always assume that $l_1\geq k_1$.
|
---
abstract: 'Data deduplication is able to effectively identify and eliminate redundant data and only maintain a single copy of files and chunks. Hence, it is widely used in cloud storage systems to save storage space and network bandwidth. However, the occurrence of deduplication can be easily identified by monitoring and analyzing network traffic, which leads to the risk of user privacy leakage. The attacker can carry out a very dangerous side channel attack, i.e., learn-the-remaining-information (LRI) attack, to reveal users’ privacy information by exploiting the side channel of network traffic in deduplication. Existing work addresses the LRI attack at the cost of the high bandwidth efficiency of deduplication. In order to address this problem, we propose a simple yet effective scheme, called randomized redundant chunk scheme (RRCS), to significantly mitigate the risk of the LRI attack while maintaining the high bandwidth efficiency of deduplication. The basic idea behind RRCS is to add randomized redundant chunks to mix up the real deduplication states of files used for the LRI attack, which effectively obfuscates the view of the attacker, who attempts to exploit the side channel of network traffic for the LRI attack. Our security analysis shows that RRCS could significantly mitigate the risk of the LRI attack. We implement the RRCS prototype and evaluate it by using three large-scale real-world datasets. Experimental results demonstrate the efficiency and efficacy of RRCS.'
author:
- |
[Pengfei Zuo, Yu Hua, Cong Wang[^^]{}, Wen Xia, Shunde Cao, Yukun Zhou, Yuanyuan Sun ]{}\
*Huazhong University of Science and Technology, Wuhan, China*\
[^^]{}*City University of Hong Kong, Hong Kong*\
bibliography:
- 'bibliography.bib'
title: 'Bandwidth-efficient Storage Services for Mitigating Side Channel Attack'
---
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---
author:
- 'Denis A. Leahy'
- Wenwu Tian
date: 'Received xx, 2006; accepted xx, 2006'
title: Radio Spectrum and Distance of the SNR HB9
---
Introduction
============
The production of high-energy particles in our Galaxy is closely related to shock acceleration in supernova remnants (SNR). The study of the radio spectra produced by high-energy electrons in SNRs allows one to learn about the electron energy spectrum. Due to its large diameter HB9 is one of the best candidates for a study of SNRs’ spectral index spatial variations. Variations have been observed to occur in some other large angular sized SNRs (Leahy $\&$ Tian 2005; Tian $\&$ Leahy 2005, Uyaniker et al. 2004, Alvarez et al. 2001). Spatial variations in the spectral index across HB9 has been previously noted by Leahy et al. (1998) and Leahy $\&$ Roger (1991), but a new set of higher resolution and sensitivity observations of the Canadian Galactic Plane Survey (CGPS) combined with new spectral index analysis methods, impel us to carry out a new spectral index study of HB9. A lack of direct estimates of the distance to HB9 in previous research (Leahy $\&$ Roger 1991, Lozinskaya 1981, Milne 1979) is corrected here by an analysis of the new HI observations of HB9.
Observations and Image Analysis
===============================
The continuum and HI emission data sets come from the CGPS, described in detail by Taylor et al. (2003). The data sets are mainly based on observations from the Synthesis Telescope (ST) of the Dominion Radio Astrophysical Observatory (DRAO). The spatial resolution of the continuum images of HB9 is 49$^{\prime}$$^{\prime}$ $\times$ 68$^{\prime}$$^{\prime}$ at 1420 MHz and 2.8$^{\prime}$$\times$3.9$^{\prime}$ at 408 MHz. The synthesized beam for the HI line images is 58$^{\prime}$$^{\prime}$$\times$ 80$^{\prime}$$^{\prime}$ and the radial velocity resolution is 1.32 km$/$s. The continuum images are noise limited with rms of $\sim$0.3 mJy$/$beam at 1420 MHz and $\sim$3 mJy$/$beam at 408 MHz. The DRAO ST observations are not sensitive to structures larger than an angular size scale of about 3.3$^{\circ}$ at 408 MHz and 56$^{\prime}$ at 1420 MHz. Thus the CGPS includes data from the 408 MHz all-sky survey of Haslam et al (1982), sensitive to structure greater than 51$^{\prime}$, and the Effelsberg 1.4 GHz Galactic plane survey of Reich et al. (1990, 1997), sensitive to structure greater than 9.4$^{\prime}$. The large scale HI data is from the single-antenna survey of the CGPS area (Higgs $\&$ Tapping 2000) with resolution of 36$^{\prime}$.
There are many compact sources (CS) overlapping the face of HB9. In order to better study the spectral index distribution, we compare two methods to reduce the effects of CS on the SNR’s spectral index. The first removes CS at both frequencies but is limited to the resolution of the 408 MHz image; the other removes CS only in the 1420 MHz image so is effective at higher spatial resolution. The first method was introduced by Tian and Leahy (2005) and the second was introduced by Leahy (2006). Both have been proved effective in removing CS contamination. We also analyze HI spectral line observations of HB9 and estimate HB9’s distance.
Results
=======
HB9 Flux Densities at 408 MHz and 1420 MHz
------------------------------------------
The first row of Fig. 1 shows the 408 MHz (left) and 1420 MHz (right) images of HB9 overlaid by boxes used for spectral index determination. The 408 MHz and 1420 MHz CS-subtracted images are shown in the second row of Fig. 1. 160 CS at 1420 MHz and 61 CS at 408 MHz are detected within HB9. The total integrated flux densities of these CS are 12.0 Jy at 408 MHz and 5.6 Jy at 1420 MHz. After subtracting the flux of CS, HB9’s integrated flux density is 117.8$\pm$5.3 Jy at 408 MHz and 65.9$\pm$3.4 Jy at 1420 MHz. This gives an integrated flux-density 408-1420 MHz spectral index $\alpha$=0.48$\pm$0.06 including CS, and 0.47$\pm$0.06 after subtracting CS. The results are consistent within errors.
HB9 T-T Plot Spectral Indices
-----------------------------
We have applied the T-T plot method to investigate spectral indices (e.g. Tian $\&$ Leahy 2005). The T-T plot method is first applied to the whole area covering HB9 (the single big box of Fig. 1). Fig. 2 gives the 408-1420 MHz T-T plots for three cases (from left to right): case 1- maps including CS ($\alpha_{auto}$=0.50$\pm$0.03); case 2- CS subtracted from both 408 MHz and 1420 MHz maps ($\alpha_{auto}$=0.48$\pm$0.03); case 3- maps with Gaussian fits to CS subtracted from the 1420 MHz map only ($\alpha_{manual}$=0.49). The detailed description of case 3 is given in Leahy (2006): the motivation is two-fold: to remove CS which are only clearly resolved at 1420 MHz and not at 408 MHz; and to be able to clearly separate points in the T-T plot which belong to CS from points which belong to diffuse SNR emission. The subscript auto refers to the case of an automated linear fit including all of the points, the subscript manual refers to the case of a manual fit done to the points excluding points due to CS. The points due to CS are clearly seen in the left plot of Fig. 2 if they have a significantly different slope than the points from the SNR. For the middle plot, CS are subtracted at both 408 and 1420 MHz. However, there are artifacts due to some CS subtracted at 1420 MHz but not at 408 MHz and vice versa. For the right plot, since CS are subtracted only at 1420 MHz, points due to CS show up as vertical lines of points, with 408 MHz flux but no 1420 MHz flux. This occurs even if a CS has a spectral index similar to the diffuse SNR emission.
To study spatial variations in HB9, we divide it into 52 regions (see the top plots of Fig. 1). The case 3 method removes fainter CS detected only at 1420 MHz (for discussion, see Leahy, 2006). We give the results in Table 1 and for comparison, the spectral index from case 1 including CS is also given. Table 1 shows that case 1 and case 3 results can be quite different. Fig. 3 shows T-T plots for region 21: for case 1 (left, $\alpha_{auto}=0.51\pm0.01$), case 2 (middle, $\alpha_{auto}=0.88\pm0.11$), and case 3 (right, $\alpha_{manual}=0.61$). The spectral index for case 1 is dominated by the single bright CS. Case 2 gives an erroneous (too large) value of spectral index. The reason is that in the presence of diffuse emission, the Gaussian fitting to the strongest CS produces very low flux at 408 MHz, resulting in a residual flux in the CS subtracted map at this frequency. The advantage of case 3 is more apparent here, since there is no overcrowding of the CS in the T-T plot, like there was in Fig. 2 (right). This allows easy identification of the CS, including the faint CS at TB(1420)=5.8K, A histogram of the T-T plot spectral index values from Table 1 is shown in Fig. 4. The dashed line shows the spectral indices obtained from automatic fits, including CS at both 408 MHz and 1420 MHz (case 1); the solid line shows the spectral indices obtained from manual fits for CS subtracted at 1420 MHz (case 3). The effect of CS on the spectral index distribution is clear: there is a second peak in the histogram introduced at 0.7 to 0.9 due to CS, whereas the SNR only emission (case 3) has a smooth distribution of indices between 0.4 and 0.8.
(200,300) (-34,460)
(230,460)
(-130,300)
(180,260)
(70,100) (-100,270)
(280,218)
(137,212)
(60,100) (-50,185)
(128,188)
(260,180)
(60,100) (0,0)
Region $\alpha_{case1}$ $\alpha_{case3}$ Region $\alpha_{case1}$ $\alpha_{case3}$
-------- ------------------ ------------------ -------- ------------------ ------------------
1 0.58$\pm$0.01 0.55 27 0.75$\pm$0.07 0.65
2 0.51$\pm$0.01 0.52 28 0.85$\pm$0.11 0.76
3 0.55$\pm$0.02 0.53 29 1.05$\pm$0.08 0.97
4 0.50$\pm$0.01 0.47 30 1.38$\pm$0.01 0.51
5 0.44$\pm$0.04 0.50 31 0.75$\pm$0.02 0.53
6 0.65$\pm$0.09 0.53 32 0.52$\pm$0.02 0.40
7 0.85$\pm$0.09 0.63 33 0.82$\pm$0.03 0.74
8 0.87$\pm$0.09 0.55 34 0.87$\pm$0.11 0.65
9 0.86$\pm$0.03 0.76 35 0.81$\pm$0.11 0.69
10 0.72$\pm$0.10 0.60 36 0.77$\pm$0.02 0.72
11 0.78$\pm$0.05 0.52 37 0.77$\pm$0.03 0.63
12 0.52$\pm$0.05 0.54 38 0.81$\pm$0.02 0.80
13 0.57$\pm$0.04 0.46 39 0.83$\pm$0.04 0.69
14 0.69$\pm$0.02 0.70 40 0.29$\pm$0.05 0.33
15 0.91$\pm$0.03 0.96 41 0.42$\pm$0.02 0.41
16 0.93$\pm$0.03 0.91 42 0.55$\pm$0.23 0.55
17 0.75$\pm$0.02 0.56 43 0.77$\pm$0.03 0.73
18 0.77$\pm$0.10 0.63 44 0.56$\pm$0.04 0.61
19 0.88$\pm$0.07 0.79 45 0.52$\pm$0.02 0.55
20 0.85$\pm$0.07 0.77 46 0.53$\pm$0.02 0.55
21 0.51$\pm$0.01 0.61 47 0.35$\pm$0.02 0.29
22 0.50$\pm$0.02 0.48 48 0.51$\pm$0.02 0.50
23 0.72$\pm$0.02 0.70 49 0.57$\pm$0.03 0.46
24 0.84$\pm$0.02 0.73 50 0.76$\pm$0.03 0.69
25 0.70$\pm$0.09 0.76 51 0.45$\pm$0.03 0.40
26 0.69$\pm$0.05 0.66 52 0.70$\pm$0.14 0.50
: HB9 408-1420 MHz Spectral Index for Regions 1 to 52
HI Emission
-----------
The CGPS data has 2 times better velocity resolution and better sensitivity than the older DRAO HI data (Leahy $\&$ Roger 1991). We find HI emission which is spatially associated with the boundary of HB9 and a deficit of emission associated with the interior in the velocity range -3 to -9 km/s, but not at other velocities. This suggests that this HI is likely to be physically associated with the SNR. Fig. 5 shows the average of the maps of HI emission in the 8 channels from -3 to -9 km/s, with a contour of continuum emission at 1420 MHz to show the boundary of HB9.
(80,60) (-10,390)
Discussion
==========
Distance of HB9, age, and possible association with pulsar 0458+46
------------------------------------------------------------------
For the velocity range of -3 to -9 km/s, using circular galactic rotation velocity V$_{R}$=220 km/s and solar distance R$_{0}$=8.5kpc from the Galactic center, yields a distance to HB9 of d=0.8$\pm$0.4 kpc. This distance is consistent within errors with previous estimates: greater than 1 kpc derived from measurements of radial velocity of H$\alpha$ filaments by Lozinskaya (1981); about 1.1 kpc from the X-ray properties of HB9 by Leahy (1987); and 1.3 - 1.8 kpc from surface-brightness-diameter relations by Milne (1979) and Caswell $\&$ Lerche (1979).
Since the angular size of HB9 is 130$^{\prime}$ by 120$^{\prime}$, the mean radius is 15 pc at 0.8 kpc. A Sedov model (Cox D., 1972) is applied, using the X-ray temperature and X-ray flux from Leahy and Aschenbach (1995). For a 15 pc radius, an age of $6600$ yrs and a ratio between explosion energy $E_{51}$(in $10^{51}$erg) and initial density $n_0$ (in $cm^{-3}$) $E_{51}$/$n_0$=5 are obtained.
As noted before by Leahy and Aschenbach (1995), based on the X-ray morphology and uniform temperature, it is likely that Sedov model does not apply in this case. The evaporative cooling model developed by White $\&$ Long (1991) provides a better explanation for the observed X-ray features. If White $\&$ Long’s (1991) model is applied for a distance of 0.8 kpc, a range of parameters is found which give a nearly flat temperature profile at 0.8 keV and an appropriately centrally brightened X-ray profile ($\tau=10$ and C=20 to C=50). Then the preshock intercloud density is 3-10$\times10^{-3}$cm$^{-3}$, the age is 4000-7,000 yr and the explosion energy is 0.15-0.3$\times10^{51}$erg.
The radio pulsar 0458+46 is only 23$^{\prime}$ from the center of HB9. It has a DM distance 1.8 kpc, spin-down age $1.8\times10^6$yr and transverse velocity 95.5 km/s (Manchester et al., 2005). The DM distance is expected to be larger than the distance derived from the HI data, due to the extra DM introduced by the extra electron density in the SNR shell. The spindown age is based on the assumption of a fast initial spin, so it probably overestimates the true age. The pulsar’s kinematic age based on its transverse velocity is $5.5\times10^4$yr, several times larger than HB9’s age from either Sedov or evaporative cloud models. Thus an association between the pulsar and HB9 is possible, but only if it was a somewhat off-center explosion: for instance, the kinematic age can be reduced to about 7,000 yr if the explosion center is $\sim3^\prime$ from the current pulsar position. Better X-ray observations of HB9 should determine the nature of SNR, and give an improved SNR age value.
HB9 Spectral Index
------------------
(70,100) (-10,-80)
(245,-95)
Based on T-T plots, Leahy $\&$ Roger (1991) obtained a mean T-T plot spectral index of 0.61 for HB9. The new T-T plot spectral index derived for the whole SNR HB9 ($\alpha$=0.48$\pm$0.03) is much smaller, but consistent with the integrated flux density-based spectral index ($\alpha$=0.47$\pm$0.06). Two main factors appear as responsible for this. First, there is a significant change (3-4 K difference) in SNR brightness temperature at 1420 MHz compared to earlier data set. Fig. 2 of Leahy $\&$ Roger (1991) shows a bright filament at $\sim$ 3.2 K compared to $\sim$ 6.5 K in the current 1420 MHz image, and no such change in the 408 MHz images. Second, the present observations have higher resolution and sensitivity, so more compact sources have been resolved and subtracted from the images of HB9.
In what follows, we discuss the spectral index variations with location within HB9. Since the integrated flux and T-T plot spectral indices for the whole of HB9 agree each other, the filamentary emission (measured by the T-T plot method) and the total emission (filamentary plus spatially smooth emission) should have the same spectral index. To analyze local variations we divided the 52 regions into 3 groups: (a) boxes that cover regions with strong filamentary emission (those labeled with numbers 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 19, 32, 33, 47, 50);(b) boxes overlapping weak filamentary emission (boxes 15, 16, 20, 26, 27, 30, 34, 35, 36, 37, 41, 42, 45, 51, 52); and (c) those with intermediate filamentary emission (the remainder). The means (and standard deviations) from the 3 groups were 0.58 (0.14), 0.64 (0.17) and 0.61 (0.14), respectively. Thus there is no clear difference in spectral index between regions including strong filaments and regions with weak filaments. We can therefore conclude that the spectral results confirm the scenario suggested by Leahy $\&$ Roger (1991), where the filaments are edge-on viewing of emitting sheets.
Larger differences in spectrum are apparent from the comparison of the SNR periphery with regions in the interior. When one looks at spectral index ($\alpha_{case3}$ in Table 1) vs. position, one finds that the flatter spectral indices ($<0.5$) are all associated with the SNR limb and all of the steeper values ($>0.7$) are associated with the interior. The mean and standard deviation of spectral indices for all limb regions are 0.49 and 0.11, whereas the values for the interior regions are 0.67 and 0.12, respectively. An alternate analysis of spectral index was carried out to verify this result. We made a spectral index map of HB9 using the running T-T plot method of Zhang et al. (1997): the method yields $T_B$ spectral index $\beta$, with $\beta=\alpha+2$. We used the CS subtracted images at 408 MHz and 1420 MHz to create the spectral index map shown in Fig. 6, left panel. The small white patches show the subregions where the fitting algorithm did not obtain a reliable value of spectral index. The spectral index map computed from images including CS is similar except it has a number of elliptical spots (one for each bright CS) at the locations of the CS and with shading corresponding to the CS spectral index. One can see from Fig. 6 (left) that the limb of HB9 has predominantly low spectral index and the interior has predominantly high spectral index. From the discussion here and in previous papers (e.g. Tian $\&$ Leahy (2005), Leahy (2006)), the reliability of the automatic fits used by the spectral map software is often poor, yet the map can give a general indication of spatial variations of spectral index.
A general discussion of mechanisms that govern the radio spectrum is given in Leahy & Roger (1998). Absorption processes and electron energy losses from ionization act more effectively at lower frequencies and result in spectral flattening. Synchrotron and inverse-Compton act at high energies and steepen the spectrum. If the electron energy spectrum is curved then spatial variations in spectral index can occur due to variations in magnetic field, which determines the observed synchrotron frequency for a given electron energy. The diffuse galactic electron spectrum has an electron energy index steepening from $\sim$2.0 below $\sim2$GeV to $\sim$2.6 above $\sim5$GeV, then to $\sim$3.5 above $\sim40$GeV, corresponding to synchrotron index increasing from $\alpha$=0.5 at low frequencies to $\alpha$=1.25 at high frequencies. For a fixed observing frequency, increasing magnetic field, B, means sampling lower electron energies, so that spectral index flattens with increasing B.
For HB9, the steeper index for interior regions, for which the line of sight goes through the whole SNR, could be interpreted as due to lower B for the interior compared to the rim; stronger synchrotron losses for the interior; or stronger absorption effects at the rim. For the Cygnus Loop (Leahy & Roger, 1998) low frequency observations confirm that the NE rim shows the effects of thermal plasma absorption. However for HB9, previous studies have not found significant weakening of the radio brightness at the rim at low frequencies. This leaves the first two causes to examine more closely. The radio morphology of HB9 is consistent with a low brightness interior surrounded by a brighter shell with filaments: the image at 1420 MHz in Figure 1 show this most clearly. The highest compression in an SNR should occur just inside the forward shock, where the optical filaments form. Thus one expects the highest magnetic fields and highest synchrotron emissivity in the high compression regions. This now explains the observed spectral index variations: higher magnetic field at larger radius within the SNR produces flatter radio spectrum for the outer regions. Thus a larger proportion of steep spectrum emission for lines-of-sight through the central body of the SNR results in an observed steeper spectral index for interior regions. The conclusion is consistent with the explanation given by Leahy et al. (1998) for spectral index variations at HB9, although that study did not do a comparison of limb and interior spectral index values.
The study of Leahy et al. (1998) used the old DRAO 408 MHz and 1420 MHz observations and combined those with existing 2695 MHz and 4850 MHz data from the Effelsberg telescope, observations at 151 MHz with the Cambridge Low-Frequency synthesis telescope and 232 MHz observations with the Beijing Astronomical Observatory Miyun telescope. The 151 and 232 MHz maps suffered from lack of low-order spacings, and the mean derived spectral indices for 151-232 MHz and 232-408 MHz indicate that the brightness scale for 232 MHz map was too large. To compare the effect on spectral index of the improved 408 and 1420 MHz data, we show the old 408-1420 MHz spectral index map in Fig. 6 (right panel) beside the spectral index map derived from the new data. The old map has an important artifact produced by the point source: the large dark area near RA 4H55M, Dec +46d20m is due to 4C46.09. The contribution removed in that analysis was apparently too small to take into account the high brightness of 4C46.09. The same artifact affects the spectral index map made using the new data with CS included, but not in the spectral index map (shown in the left panel of Fig. 6) with CS subtracted. The old and new spectral index maps are globally similar (e.g. the limb has a flatter spectral index than the interior, with the lowest index coming from the bright southeast filament). There are, however, significant differences in the details between the two spectral index maps. This might be attributed to the sensitivity of the automatic fitting routine to small changes: for example compare case 1 (automatic fit) and case 3 (manual fit) in Table 1. It is interesting to note that the frequency-averaged spectral index map, which should suffer least from these errors, bears similarity to the results presented here: steeper index from the central regions than the rim.
Conclusion
==========
We present new higher sensitivity and higher resolution images of the SNR HB9 at 408 MHz and 1420 MHz, and study its radio spectrum, corrected for compact source flux densities using new improved methods. The T-T plot spectral index for HB9 agrees with the integrated flux-density based 408-1420 MHz spectral index. A study of spatial variations shows no systematic difference in spectral index between weak and strong filaments, supporting the conclusion of Leahy $\&$ Roger (1991). We find steeper spectral index for interior regions than for rim regions. This can be explained by a standard curved interstellar electron energy spectrum combined with variable magnetic field. Due to lower compression, a lower magnetic field in the interior, compared to that near the outer shock, results in steeper spectrum emission from the interior. Thus a larger proportion of steep spectrum emission for lines-of-sight through the central body of the SNR results in an observed steeper spectral index for interior regions. Based on HI features associated with HB9, we obtain a distance of 0.8$\pm$0.4 kpc, and give updated Sedov and evaporating cloud model parameters for the SNR. We discuss the possible association between the pulsar and HB9, and conclude that more evidence is necessary to decide this possible association.
We acknowledge support from the Natural Sciences and Engineering Research Council of Canada. WWT thanks the NSFC for support. The DRAO is operated as a national facility by the National Research Council of Canada. The Canadian Galactic Plane Survey is a Canadian project with international partners.
Alvarez, H., Aparici, J. $\&$ Reich, P., 2001, A$\&$A, 372, 636 Caswell, J. L., Lerche, I., 1979, MNRAS, 187, 201 Cox, D., 1972, ApJ, 178, 159 Fürst, E., Reich, W., Reich, P. and Reif, K., 1990, A$\&$AS, 85, 691 Haslam, C.G.T., Salter, C.J., Stoffel, H. and Wilson, W.W., 1982, A$\&$AS, 47, 1 Higgs, L.A. and Tapping, K.F., 2000, AJ, 120, 2471 Leahy, D.A., 2006, ApJ, 647, 1125 Leahy, D.A. $\&$ Tian, W.W., 2005, A$\&$A, 440, 929 Leahy, D.A., Zhang, X.Z., Wu, X.J. and Lin, J.L., 1998, A$\&$A, 339, 601 Leahy, D.A., Aschenbach, B., 1995, A$\&$A, 293, 853 Leahy, D.A. $\&$ Roger, R.S., 1991, ApJ, 101, 1033 Leahy, D.A. $\&$ Roger, R.S., 1998, ApJ, 505, 784 Leahy, D.A., 1987, ApJ, 322, 917 Lozinskaya, T. A., 1981, SvAL, 7, 17 Manchester, R. N., Hobbs, G. B., Teoh, A. & Hobbs, M., 2005, AJ, 129, 1993 Milne, D. K., 1979, AuJPh, 32, 83 Reich, W., Reich, P. and Fürst, E., 1990, A$\&$AS, 83, 539 Reich, W., Reich, P. and Fürst, E., 1997, A$\&$AS, 126, 413 Taylor, A.R., Gibson, S.J., Peracaula, M. et al., 2003, AJ, 125, 3145 Tian, W.W. $\&$ Leahy, D.A., 2005, A$\&A$, 436, 187 Uyaniker, B., Reich, W., Yar, A. $\&$ Fürst, E., 2004, A$\&$A, 426, 909 White, R., Long, K., 1991, ApJ, 373, 543 Zhang X., Zheng, Y., Landecker, T., Higgs, L., 1997, A$\&$A, 324, 641
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abstract: 'It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another operator which satisfies a weak ergodic theorem. Similar results for other projection and reflection operators are also discussed.'
address: 'Computer Assisted Research Mathematics and its Applications (CARMA), The University of Newcastle, University Drive, Callaghan NSW 2308, Australia'
author:
- 'Jonathan M. Borwein'
- Ohad Giladi
title: 'Ergodic behaviour of a Douglas-Rachford operator away from the origin'
---
[^1]
Introduction
============
Background
----------
Given a set $A$ in a Hilbert space ${\mathbb H}$, denote by $P_A:{\mathbb H}\rightrightarrows {\mathbb H}$ the multi-valued projection operator, that is, $$\begin{aligned}
P_Ax = \Big\{y\in A~\Big|~ \|x-y\| = \inf_{z\in A}\|x-z\|\Big\},\end{aligned}$$ where here and in what follows, $\|\cdot\|$ denotes the Hilbert norm on ${\mathbb H}$. Also, if $I:{\mathbb H}\to {\mathbb H}$ is the identity operator, denote by $R_A:{\mathbb H}\rightrightarrows {\mathbb H}$ the reflection operator, which is given by $$\begin{aligned}
R_A = 2P_A - I.\end{aligned}$$ Given two sets $A,B\subseteq {\mathbb H}$, define the Doulgas-Rachford operator by $$\begin{aligned}
\label{def DR}
T_{A,B} = \frac {I+R_BR_A}{2}.\end{aligned}$$ Given $x\in {\mathbb H}$, let $\{x_n\}_{n=0}^{\infty}\subseteq {\mathbb H}$, be the sequence which is defined as follows, $$\begin{aligned}
\label{def iteration}
x_{n+1} = T_{A,B}x_n = T_{A,B}^nx_0, \quad x_0 = x.\end{aligned}$$ This sequence is also known as the *Douglas-Rachford iteration of $x$*. It was studied first in [@DR56] as an algorithm for finding an intersection point of two sets. Indeed, it is not hard to check that $$\begin{aligned}
\label{cond iff}
Tx = x \iff P_Ax \in A\cap B,\end{aligned}$$ and so any point $x\in A\cap B$ is a fixed point of $T_{A,B}$.
Analysing the Douglas-Rachford operator and the iteration sequence are well known questions with interesting applications. This question has been studied in a convex setting (that is, when both $A$ and $B$ are convex), as well as in a non-convex setting (when either $A$ or $B$ is not convex). See for example [@BCL; @02; @LM79] for the convex case and [@ERT07; @GE08] for the non-convex case.
In the case $A$ is convex, it is known that the projection operator $P_A$ is firmly non-expansive, that is, for every $x,y\in {\mathbb H}$, $$\begin{aligned}
\|P_Ax - P_Ay\|^2 + \|(I-P_A)x - (I-P_A)y\|^2 \le \|x-y\|^2.\end{aligned}$$ See for example [@GK90]\*[Thm. 12.2]{}. It then follows that the reflection operator $R_A$ is non-expansive, that is, for every $x,y\in {\mathbb H}$, $$\begin{aligned}
\|R_Ax-R_Ay\| \le \|x-y\|,\end{aligned}$$ and the Douglas-Rachford operator is firmly non-expansive. See for example [@GK90]\*[Thm. 12.1]{}. From the results of [@Opi67], it then follows that the Douglas-Rachford iteration is weakly convergent. In the case ${\mathbb H}$ is finite dimensional, the weak convergence implies strong (norm) convergence.
While the convex case is well understood, much less is known about the non-convex case. One of the simplest examples of a non-convex setting is the case of a sphere and a line. This case was studied in [@AB13; @BS11; @Ben15; @Gil16]. Let $$\begin{aligned}
\label{def sphere}
{\mathbb S}= \big\{x\in {\mathbb H}~|~\|x\|=1\big\},\end{aligned}$$ and for $\lambda \ge 0$, $$\begin{aligned}
\label{def line}
L_\lambda = \big\{t{\mathbf e}_1+\lambda {\mathbf e}_2\in {\mathbb H}~|~ t\in {\mathbb R}\big\},\end{aligned}$$ where here $\{{\mathbf e}_1,{\mathbf e}_2,\dots\}$ is an orthonormal basis of ${\mathbb H}$. It was shown in [@Ben15] that if $\lambda \in (0,1)$, then for every $x\in {\mathbb H}$ with $\langle x,{\mathbf e}_1\rangle \neq 0$, the Douglas-Rachford iteration converges in norm to one of the two intersection points of ${\mathbb S}$ and $L_\lambda$. Here and in what follows $\langle \cdot, \cdot \rangle$ denotes the inner product on ${\mathbb H}$. Global convergence for the case $\lambda =0$ was already proved in [@BS11]. The result in [@Ben15] improved previous results, which only gave local convergence. It was also shown in [@BS11], that if $\langle x,{\mathbf e}_1\rangle = 0 $ or if $\lambda \ge 1$, the Douglas-Rachford iteration is not convergent. Note that the case $\lambda \le 0$ is completely analogous. Other non-convex cases were considered in [@ABT16; @HL13; @Pha16].
An ergodic theorem for Lipschitz approximations of the Douglas-Rachford operator {#sec erg present}
--------------------------------------------------------------------------------
It follows from the results of [@Ben15], that the convergence of the Douglas-Rachford iteration is *uniform* on compact sets. See [@Gil16] for the exact argument (in [@Gil16] one considers a finite dimensional Hilbert space, but the case for an infinite dimensional space is similar). Define the following sets, $$\begin{aligned}
\label{def H}
{\mathbb H}_+ = \big\{x\in {\mathbb H}~|~\langle x,{\mathbf e}_1\rangle >0\big\}, \quad {\mathbb H}_- = \big\{x\in {\mathbb H}~|~\langle x,{\mathbf e}_1\rangle <0\big\}, \quad {\mathbb H}_0 = \big\{x\in {\mathbb H}~|~\langle x,{\mathbf e}_1\rangle =0\big\}.\end{aligned}$$ It is straightforward to show that if $T = T_{{\mathbb S}, L_\lambda}$, then $T({\mathbb H}_+)\subseteq {\mathbb H}_+$, $T({\mathbb H}_-)\subseteq {\mathbb H}_-$, $T({\mathbb H}_0) \subseteq {\mathbb H}_0$. In particular, it follows that if $K\subseteq {\mathbb H}_+$ or $K\subseteq {\mathbb H}_-$ is compact, then $$\begin{aligned}
\label{weak ergodic}
\sup_{x,y\in K}\|T^nx - T^ny\| \stackrel{n\to \infty}{\longrightarrow} 0.\end{aligned}$$ An estimate of the form is also known as a *weak ergodic theorem*. This type of theorems appears in the literature of population biology. See for example [@Coh79]. See also [@Nus90; @RZ03] for further discussion on weak ergodic theorems.
In this note, we are interested in an estimate of the form for the Douglas-Rachford operator in a more general setting where one of the sets is the unit sphere ${\mathbb S}$ and the other set is a convex set in ${\mathbb H}$, and the two sets have non-empty intersection (also known as the feasible case). This of course includes the case of the sphere and any affine subspace of ${\mathbb H}$. While we are unable to show an estimate of the form for the Douglas-Rachford operator itself, what we can show is that away from the origin, the Douglas-Rachford operator can be approximated by another operator that satisfies . The main result of this note reads as follows.
\[thm ergodic\] Assume that $C\subseteq {\mathbb H}$ is a convex set, let ${\mathbb S}$ be the unit sphere in ${\mathbb H}$ , and assume that ${\mathbb S}\cap C \neq \emptyset$. Let $T = T_{{\mathbb S},C}$, and let $x_0\in {\mathbb S}\cap C$. Assume also that $\alpha,{\beta},r \ge 0$ are such that ${\beta}\in [0,1)$, $r \ge \frac{2}{1-{\beta}}$, and $\alpha \le \frac 1{1-{\beta}}$. Then there exists $G:{\mathbb H}\to {\mathbb H}$ such that $$\begin{aligned}
\sup_{x\in B[x_0,r] \setminus B(0,1-{\beta})}\|Gx - Tx\| \le 2r\left(1-\alpha(1-{\beta})\right),\end{aligned}$$ and for all $n \in {\mathbb N}$, $$\begin{aligned}
\sup_{x,y \in B[x_0,r]}\|G^nx-G^ny\| \le 2r\alpha^n.\end{aligned}$$
In Theorem \[thm ergodic\] and in what follows, $B(x,r)$ denotes the open ball around $x$ with radius $r$ with respect to the norm $\|\cdot\|$, while $B[x,r]$ denotes the closed ball. If we consider $T=T_{C,{\mathbb S}}$ rather than $T_{{\mathbb S},C}$, Theorem \[thm ergodic\] does not necessarily hold. See Remark \[exchange sets lip\] and Remark \[exchange sets ergodic\] below.
The proof of Theorem \[thm ergodic\] is done in two steps. First, it is shown that away from the origin, the Douglas-Rachford operator satisfies a Lipschitz condition, and so using classical extension results, it can be extended to a Lipschitz map on all of ${\mathbb H}$. This is discussed in Section \[sec lip\]. By using further smoothing operations, it is shown that away from the origin, the Douglas-Rachford operator can be approximated by another operator which satisfies an estimate of the form . The proof of Theorem \[thm ergodic\] is presented in Section \[sec ergodic pf\].
In the special case where $C = L_\lambda$, as defined in , we have in fact a slightly stronger result, namely that we can construct $G$ such that ${\mathbb H}_+\cup {\mathbb H}_0$ (alternatively, ${\mathbb H}_-\cup {\mathbb H}_0$) is invariant under $G$. See Remark \[lip case line\] and Remark \[ergodic case line\] below.
Other projection and reflection operators
-----------------------------------------
Given two sets $A,B\subseteq {\mathbb H}$, the Douglas-Rachford operator is a special case of the following parametric family of operators. Given $s_1,s_2,s_3\in [0,1]$, define $$\begin{aligned}
\label{def family}
T_{A,B}^{s_1,s_2,s_3} = s_1I+(1-s_1)\left(s_2I+(1-s_2)R_B\right)\left(s_3I+(1-s_3)R_A\right).\end{aligned}$$ As before, $I$ denotes the identity operator and $R_A$, $R_B$, denote the reflection operators on $A$, $B$, respecitively. Note that the Douglas-Rachford operator defined in corresponds to the case $s_1=\frac 1 2$, $s_2=s_3=0$. See [@BST15] for a more detailed discussion on this family of operators. It is straightforward to show that the main result, Theorem \[thm ergodic\], holds in fact for this more general family . See Remark \[family lip\] and Remark \[family ergodic\] below.
\[thm family\] Assume that $C\subseteq {\mathbb H}$ is a convex set, let ${\mathbb S}$ be the unit sphere in ${\mathbb H}$ and assume that ${\mathbb S}\cap C \neq \emptyset$. Let $s_1,s_2,s_3\in [0,1]$, let $T = T^{s_1,s_2,s_3}_{{\mathbb S},C}$, and let $x_0\in {\mathbb S}\cap C$. Assume also that $\alpha,{\beta},r \ge 0$ are such that ${\beta}\in [0,1)$, and $r$ and $\alpha$ satisfy $$\begin{aligned}
r \ge \frac{2(1+{\beta}-2(s_1+(1-s_1)(s_2+s_3) + (1-s_1)(1-s_2)s_3){\beta})}{1-{\beta}},\end{aligned}$$ and $$\begin{aligned}
\alpha \le \frac{1+{\beta}-2(s_1+(1-s_1)(s_2+s_3) + (1-s_1)(1-s_2)s_3){\beta}}{1-{\beta}}.\end{aligned}$$ Then there exists $G:{\mathbb H}\to {\mathbb H}$ such that $$\begin{aligned}
\sup_{x\in B[x_0,r] \setminus B(0,1-{\beta})}\|Gx - Tx\| \le 2r\left(1-\frac{\alpha(1-{\beta})}{1+{\beta}-2(s_1+(1-s_1)(s_2+s_3) + (1-s_1)(1-s_2)s_3){\beta}}\right),\end{aligned}$$ and for all $n \in {\mathbb N}$, $$\begin{aligned}
\sup_{x,y \in B[x_0,r]}\|G^nx-G^ny\| \le 2r\alpha^n.\end{aligned}$$
Note that choosing $s_1 = \frac 1 2 $ and $s_2=s_3 =0$ in Theorem \[thm family\] gives Theorem \[thm ergodic\]. Another well known case is when $s_1=0$ and $s_2=s_3 = \frac 1 2$, in which case we obtain $$\begin{aligned}
T_{A,B}^{0,\frac 1 2 , \frac 1 2} = P_BP_A,\end{aligned}$$ also known as the Von-Neuman operator [@VN50]. Regarding the convergence of the iteration sequence $x_{n+1} = P_BP_Ax_n$, $x_0=x$, it was shown in [@VN50] that if $A$, $B$, are both subspaces in ${\mathbb H}$, then $x_n\stackrel{n\to \infty}{\longrightarrow} P_{A\cap B}x$ (norm convergence). It was later shown in [@BB93] that if $0\in \mathrm{int}(A-B)$ or $A-B$ is a closed subspace, then the iteration sequence converges linearly (that is, when the rate of convergence is $c\alpha^n$, where $c>0$ is a constant and $\alpha \in [0,1)$).
For the von Neumann operator, we have in fact a stronger result than Theorem \[thm ergodic\], which reads as follows.
\[thm VN\] Assume that $C\subseteq {\mathbb H}$ is a convex set, let ${\mathbb S}$ be the unit sphere in ${\mathbb H}$ , and assume that ${\mathbb S}\cap C \neq \emptyset$. Let $T = P_CP_{\mathbb S}$, and let $x_0 \in {\mathbb S}\cap C$. Also, assume that $\alpha,{\beta},r\ge 0$ are such that ${\beta}\in [0,1)$, $r \ge 2$, and $\alpha \le \frac 1 {1-{\beta}}$. Then there exists $G:{\mathbb H}\to {\mathbb H}$ such that $$\begin{aligned}
\sup_{x\in B[x_0,r]}\|Gx-Tx\| \le 2r\big(1-\alpha(1-{\beta})\big),\end{aligned}$$ and $$\begin{aligned}
\sup_{x,y\in B[x_0,r]}\|G^nx-G^ny\| \le 2r\alpha^n.\end{aligned}$$
Note that Theorem \[thm VN\] is slightly stronger than Theorem \[thm ergodic\] since we only require $r \ge 2$, rather than $r \ge \frac{1}{1-{\beta}}$. Similar to the case of Theorem \[thm ergodic\], we cannot change the order of the projections in Theorem \[thm VN\]. See Remark \[rem exchange VN\] below. Theorem \[thm VN\] is proved in Section \[sec family\].
Lipschitz behaviour of the Douglas-Rachford operator {#sec lip}
====================================================
Given two Banach spaces $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$, a set $D\subseteq X$, and a map $f: D\to Y$, define the Lipschitz constant of $f$ to be $$\begin{aligned}
\|f\|_{{\mathrm{lip}}} = \sup_{\substack{x,y \in D \\ x\neq y}}\frac{\|f(x)-f(y)\|_Y}{\|x-y\|_X}.\end{aligned}$$ A map $f: X\to Y$ is said to be Lipschitz if $\|f\|_{\mathrm{lip}}< \infty$. Note that if $C\subseteq {\mathbb H}$, then $T_{{\mathbb S},C}$ is not necessarily Lipschitz on ${\mathbb H}$, since $P_{\mathbb S}= x/\|x\|$, which is not Lipschitz. However, it is shown below that if $C\subseteq {\mathbb H}$ is convex, the Douglas-Rachford operator can be ‘smoothed’ in a neighbourhood of the origin such that the smoothed operator satisfies a Lipschitz condition.
\[thm lip approx\] Assume that $C\subseteq {\mathbb H}$ is a convex set, and let ${\mathbb S}$ be the unit sphere in ${\mathbb H}$ . Let $T = T_{{\mathbb S},C}$, and let ${\beta}\in [0,1)$. Then there exists $F:{\mathbb H}\to {\mathbb H}$ such that $$\begin{aligned}
F\big|_{{\mathbb H}\setminus B(0,1-{\beta})} = T,\end{aligned}$$ and $$\begin{aligned}
\|F\|_{\mathrm{lip}}\le \frac{1}{1-{\beta}}.\end{aligned}$$
We begin with the following proposition.
\[prop ref lip\] Assume that $x,y\in {\mathbb H}\setminus B(0,1-b)$. Then $$\begin{aligned}
\|R_{\mathbb S}x-R_{\mathbb S}y\| \le \frac{1+ {\beta}}{1-{\beta}}\|x-y\|.\end{aligned}$$
Recall that $$\begin{aligned}
R_{\mathbb S}x = 2P_{\mathbb S}x-x = \left(\frac 2 {\|x\|}-1\right)x.\end{aligned}$$ Hence, $$\begin{aligned}
\label{bound norm R}
\nonumber \|R_{\mathbb S}x-R_{\mathbb S}y\|^2 & = \|R_{\mathbb S}x\|^2 + \|R_{\mathbb S}y\|^2 - 2\langle R_{\mathbb S}x,R_{\mathbb S}y\rangle
\\
\nonumber & = (2-\|x\|)^2+(2-\|y\|)^2 - 2\left(\frac 2{\|x\|}-1\right)\left(\frac 2{\|y\|}-1\right)\langle x,y\rangle
\\
\nonumber & = 4-4\|x\|+\|x\|^2+4-4\|y\|+\|y\|^2-2\left(\frac 4{\|x\|\|y\|}-\frac 2{\|x\|}-\frac 2 {\|y\|}\right)\langle x,y\rangle -2\langle x,y\rangle
\\
\nonumber & = \|x\|^2+\|y\|^2-2\langle x,y\rangle + 8-4\|x\|-4\|y\| - \frac{2}{\|x\|\|y\|}\big(4 - 2\|x\| - 2\|y\|\big)\langle x,y\rangle
\\
& = \|x-y\|^2 + 4\big(2-\|x\|-\|y\|\big)\left(1-\frac{\langle x,y\rangle}{\|x\|\|y\|}\right)\end{aligned}$$ Now, since $\|x\|\|y\| \le \frac{\|x\|^2+\|y\|^2}{2}$ for all $x,y\in {\mathbb H}$, if $x,y \in {\mathbb H}\setminus B(0,1-{\beta})$, then $$\begin{aligned}
\|x\|\|y\| - \langle x,y \rangle & \le & \frac{\|x\|^2+\|y\|^2}{2} - \langle x,y \rangle
\\
& \stackrel{(*)}{\le} & \frac{\|x\|\|y\|}{(1-{\beta})^2}\left(\frac{\|x\|^2+\|y\|^2}{2} - \langle x,y \rangle\right)
\\
& = & \frac{\|x\|\|y\|}{2(1-{\beta})^2}\left(\|x\|^2+\|y\|^2 - 2\langle x,y \rangle\right)
\\
& = & \frac{\|x\|\|y\|}{2(1-{\beta})^2}\|x-y\|^2,\end{aligned}$$ where in ($*$) we used the fact that $$\begin{aligned}
\frac{\|x\|^2+\|y\|^2}{2} - \langle x,y \rangle \ge \|x\|\|y\| - \langle x,y\rangle \ge 0,\end{aligned}$$ and the fact that $\|x\| \ge 1-{\beta}$ and $\|y\| \ge 1-{\beta}$. Therefore, if $x,y\in {\mathbb H}\setminus B(0,1-{\beta})$, then $$\begin{aligned}
\label{bound one minus}
1-\frac{\langle x,y\rangle}{\|x\|\|y\|} \le \frac{1}{2(1-{\beta})^2}\|x-y\|^2.\end{aligned}$$ Plugging into , it follows that if $x,y\in {\mathbb H}\setminus B(0,1-{\beta})$, then $2-\|x\|-\|y\| \le 2{\beta}$, and so $$\begin{aligned}
\|R_{\mathbb S}x-R_{\mathbb S}y\|^2 & \le \left(1+ 4\big(2-\|x\|-\|y\|\big)\frac{1}{2(1-{\beta})^2}\right)\|x-y\|^2
\\ & \le \left(1+ \frac {4{\beta}} {(1-{\beta})^2}\right)\|x-y\|^2
\\ & = \frac{(1+{\beta})^2}{(1-{\beta})^2}\|x-y\|^2.\end{aligned}$$ Hence, $$\begin{aligned}
\|R_{\mathbb S}x-R_{\mathbb S}y\| \le \frac{1+{\beta}}{1-{\beta}}\|x-y\|,\end{aligned}$$ and this completes the proof.
Another tool which is needed in the proof of Theorem \[thm lip approx\] is the following theorem, known as Kirszbraun’s Theorem. See for example [@BL00; @GK90]. Given a set $D\subseteq {\mathbb H}$, let $\overline{\mathrm{conv}(D)}$ denote its closed convex hull, where the convex hull is given by $$\begin{aligned}
\mathrm{conv}(D) = \left\{\sum_{i=1}^n t_i x_i~\Big|~ x_i \in D, ~~t_i\ge 0, ~~1 \le i \le n, ~~ \sum_{i=1}^n t_i = 1, ~~ n \in {\mathbb N}\right\}.\end{aligned}$$
Kirsbraun’s theorem reads as follows.
\[thm Kirs\] Assume that $D_1,D_2 \subseteq {\mathbb H}$. Assume that $f:D_1 \to D_2$ is Lipschitz. Then there exists $F: {\mathbb H}\to \overline{\mathrm{conv}(D_2)}$ such that $F\big|_{D_1} = f$ and $\|F\|_{{\mathrm{lip}}} = \|f\|_{{\mathrm{lip}}}$.
We are now in a position to prove Theorem \[thm lip approx\]
Since $C$ is convex, it follows that $R_C$ is non-expansive. Let $x,y \in {\mathbb H}\setminus B(0,1-{\beta})$. Then $$\begin{aligned}
\|Tx-Ty\| & = & \left\|\left(\frac{I+R_CR_{\mathbb S}}{2}\right)x-\left(\frac{I+R_CR_{\mathbb S}}{2}\right)y\right\|
\\ &=& \left\|\frac{x-y}{2} + \frac{R_CR_{\mathbb S}x-R_CR_{\mathbb S}y}{2}\right\|
\\ & \le & \frac 1 2 \|x-y\| + \frac 1 2 \|R_CR_{\mathbb S}x-R_CR_{\mathbb S}y\|
\\ & \stackrel{(*)}{\le} & \frac 1 2 \|x-y\|+ \frac 1 2 \|R_{\mathbb S}x-R_{\mathbb S}y\|
\\ & \stackrel{(**)}{\le} & \frac 1 2 \|x-y\| + \frac {1+{\beta}}{2(1-{\beta})}\|x-y\|
\\ & = & \frac{\|x-y\|}{1-{\beta}},
$$ where in ($*$) we used the fact that $C$ is convex and thus $R_C$ is non-expansive, and in ($**$) we used Proposition \[prop ref lip\]. Applying Theorem \[thm Kirs\] to $T$ on the sets $ D_1 = {\mathbb H}\setminus B(0,1-{\beta})$ and $D_2 = {\mathbb H}$ completes the proof.
\[family lip\] Note that if $T = T^{s_1,s_2,s_3}_{{\mathbb S},C}$ is as defined in , then in particular, $$\begin{aligned}
T = (s_1+(1-s_1)(s_2+s_3))I + (1-s_1)s_2(1-s_3)R_{\mathbb S}+ (1-s_1)(1-s_2)s_3R_C+(1-s_1)s_2s_3R_CR_{\mathbb S}.\end{aligned}$$ Note also that $$\begin{aligned}
(s_1+(1-s_1)(s_2+s_3)) + (1-s_1)s_2(1-s_3) + (1-s_1)(1-s_2)s_3 + (1-s_1)s_2s_3 =1.\end{aligned}$$ Hence, if $C$ is convex, then since both $I$ and $R_C$ are non-expansive, using Proposition \[prop ref lip\], for every $x,y\in {\mathbb H}\setminus B(0,1-{\beta})$, $$\begin{aligned}
\label{lip const family}
\nonumber \|Tx-Ty\| & \le (s_1+(1-s_1)(s_2+s_3) + (1-s_1)(1-s_2)s_3)\|x-y\|
\\ \nonumber & \quad + ((1-s_1)s_2(1-s_3)+(1-s_1)s_2s_3)\frac{1+{\beta}}{1-{\beta}}\|x-y\|
\\ & = \frac{1+{\beta}-2(s_1+(1-s_1)(s_2+s_3) + (1-s_1)(1-s_2)s_3){\beta}}{1-{\beta}}\|x-y\|.\end{aligned}$$ Thus, repeating the proof of Theorem \[thm lip approx\], we obtain a similar result, but now the Lipschitz constant is the one given in . [$\diamond$]{}
\[exchange sets lip\] Even if $C\subseteq {\mathbb H}$ is convex, the map $x\mapsto R_{\mathbb S}R_Cx$ need not satisfy a Lipschitz condition, since $R_C$ might be arbitrarily close to $0$ (indeed, it might even not be defined). Thus, in general, Theorem \[thm lip approx\] does not hold for the operator $T=T_{C,{\mathbb S}}$. [$\diamond$]{}
\[lip case line\] In the case $C = L_\lambda$, as defined in , if $T = T_{{\mathbb S}, L_\lambda}$, then ${\mathbb H}_+$, ${\mathbb H}_-$, ${\mathbb H}_0$ as defined in are all invariant under $T$. Hence, by applying Theorem \[thm Kirs\] with $D_1 = ({\mathbb H}_+\cup {\mathbb H}_0)\setminus B(0,1-{\beta})$ (resp. $({\mathbb H}_-\cup {\mathbb H}_0)\setminus B(0,1-{\beta})$) and $D_2 = {\mathbb H}_+$ (resp. ${\mathbb H}_-$), it follows that in Theorem \[thm lip approx\] we can choose $F:{\mathbb H}_+\cup {\mathbb H}_0 \to {\mathbb H}_+\cup{\mathbb H}_0$ (resp. $F:{\mathbb H}_-\cup {\mathbb H}_0 \to {\mathbb H}_-\cup {\mathbb H}_0$). Note that we cannot choose $F:{\mathbb H}_+ \to {\mathbb H}_+$ or $F:{\mathbb H}_- \to {\mathbb H}_-$ as these are not closed sets. [$\diamond$]{}
Proof of Theorem \[thm ergodic\] {#sec ergodic pf}
================================
Given a set $D\subseteq {\mathbb H}$, define $$\begin{aligned}
{\mathrm{diam}}(D) = \sup_{x,y\in D}\|x-y\|.\end{aligned}$$
The next proposition shows that on a bounded convex set, we can ‘smooth’ Lipschitz maps, so that the smoothed map satisfies an estimate of the form . The smoothing operation is similar to the one which appeared in [@RZ03].
\[prop smooth\] Assume that $D\subseteq {\mathbb H}$ is bounded and convex, and let $F:D \to D$ be a Lipschitz map. Then for every $\alpha \le \|F\|_{\mathrm{lip}}$ there exists a map $G:D \to D$ such that $$\begin{aligned}
\|Gx-Fx\| \le \left(1-\frac \alpha{\|F\|_{\mathrm{lip}}}\right){\mathrm{diam}}(D),\end{aligned}$$ and for all $n\in {\mathbb N}$, $$\begin{aligned}
\sup_{x,y\in D}\|G^nx-G^ny\| \le \alpha^n{\mathrm{diam}}(D).\end{aligned}$$ In particular, if $\alpha \in [0,1)$, $$\begin{aligned}
\sup_{x,y\in D}\|G^nx-G^ny\| \stackrel{n\to \infty}{\longrightarrow} 0.\end{aligned}$$
Let $\theta \in D$ and $\gamma\in [0,1]$. Define $$\begin{aligned}
Gx = (1-\gamma)Fx + \gamma \theta.\end{aligned}$$ Then since $D$ is convex, it follows that $G(D)\subseteq D$, and $$\begin{aligned}
\sup_{x\in D}\|Gx-Fx\| = \sup_{x\in D}\gamma\|Fx-\theta\| \le \gamma\,{\mathrm{diam}}(D).\end{aligned}$$ Also, $$\begin{aligned}
\|G\|_{\mathrm{lip}}= (1-\gamma)\|F\|_{\mathrm{lip}}.\end{aligned}$$ Choosing $\gamma = 1-\frac \alpha{\|F\|_{\mathrm{lip}}} \in [0,1]$ and using the fact that $\|G^n\|_{\mathrm{lip}}\le \|G\|_{\mathrm{lip}}^n$ completes the proof.
We are now in a position to prove Theorem \[thm ergodic\].
Since $x_0\in {\mathbb S}\cap C$, we have $Tx_0 = x_0$, see . Let $F:{\mathbb H}\to {\mathbb H}$ be the map obtained from Theorem \[thm lip approx\]. Let $x\in B[x_0,r]$. If $x\notin B[0,1]$ then $R_{\mathbb S}= R_{\mathbb B}$, where $$\begin{aligned}
\mathbb B = \big\{x\in {\mathbb H}~|~ \|x\| \le 1\big\},\end{aligned}$$ which is convex. Thus, in this case, $R_C$, $R_{\mathbb S}$ and therefore $T$ are all non-expansive, and so $$\begin{aligned}
\|Fx-Fx_0\| =\|Tx-Tx_0\| \le \|x-x_0\| \le r.\end{aligned}$$ If $x\in B[0,1]$, then by Theorem \[thm lip approx\], $$\begin{aligned}
\|Fx - Fx_0\| \le \frac{\|x-x_0\|}{1-{\beta}} \le \frac{2}{1-{\beta}}.\end{aligned}$$ Therefore, if $r \ge \frac{2}{1-{\beta}}$, then $$\begin{aligned}
F(B[x_0,r]) \subseteq B[x_0,r].\end{aligned}$$ Now, ${\mathrm{diam}}(B[x_0,r]) = 2r$. Applying Proposition \[prop smooth\] to the function $F$ on the domain $D=B[x_0,r]$, it follows that for every $\alpha \le \frac1{1-{\beta}}$, there exists $G:{\mathbb H}\to {\mathbb H}$ which satisfies $G(B[x_0,r])\subseteq B[x_0,r]$, and such that $$\begin{aligned}
\sup_{x\in B[x_0,r]}\|Gx-Fx\| \le 2r\left(1- \alpha(1-{\beta})\right),\end{aligned}$$ and $$\begin{aligned}
\sup_{x,y\in B[x_0,r]}\|G^nx-G^ny\| \le 2r\alpha^n.\end{aligned}$$ Since $$\begin{aligned}
\sup_{x\in B[x_0,r]\setminus B(0,1-{\beta})}\|Gx-Tx\| \le \sup_{x\in B[x_0,r]}\|Gx-Fx\|,\end{aligned}$$ the proof is complete.
Note that by Proposition \[prop smooth\], the choice of $G$ in Theorem \[thm ergodic\] depends on $\alpha$ and on the centre point $x_0$. [$\diamond$]{}
\[family ergodic\] If we consider now the operator $T = T_{{\mathbb S},C}^{s_1,s_2,s_3}$ as defined in , then repeating the proof of Theorem \[thm ergodic\] but now using Remark \[family lip\], we obtain Theorem \[thm family\]. Note that the conditions on $\alpha$ and $r$ that we need are $r \ge 2\|F\|_{\mathrm{lip}}$ and $\alpha \le \|F\|_{{\mathrm{lip}}}$, where $F$ is the function obtained in Theorem \[thm lip approx\] (applied now to the operator $T$). These are exactly the conditions that appear in Theorem \[thm family\]. [$\diamond$]{}
\[exchange sets ergodic\] Since, by Remark \[exchange sets lip\], Theorem \[thm lip approx\] does not necessarily hold if we let $T = T_{C,{\mathbb S}}$, the same is true for Theorem \[thm ergodic\]. [$\diamond$]{}
\[ergodic case line\] In the case of the sphere and the line, $C = L_\lambda$, $\lambda \in [0,1]$ as defined in , it follows from Remark \[lip case line\] that we can choose $G:{\mathbb H}_+\cup{\mathbb H}_0 \to {\mathbb H}_+\cup {\mathbb H}_0$ such that $$\begin{aligned}
\sup_{\substack{ x\in B[x_0,r]\setminus B(0,1-{\beta}) \\ x\in {\mathbb H}_+\cup {\mathbb H}_0}}\|Gx - Tx\| \le 2r\left(1-\alpha(1-{\beta})\right),\end{aligned}$$ and for all $n\in {\mathbb N}$, $$\begin{aligned}
\sup_{\substack{x,y \in B[x_0,r] \\ x,y\in {\mathbb H}_+\cup {\mathbb H}_0}}\|G^nx-G^ny\| \le 2r\alpha^n.\end{aligned}$$ If we replace ${\mathbb H}_+$ by ${\mathbb H}_-$ we obtain a similar result. [$\diamond$]{}
Proof of Theorem \[thm VN\] {#sec family}
===========================
We begin with the following proposition, which shows that the projection operator on the sphere, $P_{\mathbb S}$, satisfies a Lipschitz condition away from the origin.
\[prop proj lip\] For every $x,y\in {\mathbb H}\setminus\{0\}$, $$\begin{aligned}
\left\|\frac x {\|x\|} - \frac y {\|y\|}\right\| \le \max\left\{\frac 1 {\|x\|}, \frac 1 {\|y\|}\right\}\|x-y\|.\end{aligned}$$ In particular, if ${\beta}\in[0,1)$, $x,y \in {\mathbb H}\setminus B(0,1-{\beta})$, and ${\mathbb S}$ is the unit sphere in ${\mathbb H}$ , $$\begin{aligned}
\|P_{\mathbb S}x-P_{\mathbb S}y\| \le \frac{\|x-y\|}{1-{\beta}}.\end{aligned}$$
Assume without loss of generality that $\|x\| \le \|y\|$. Then $$\begin{aligned}
&& \frac{1}{\|x\|^2}\|x-y\|^2 - \left\|\frac x{\|x\|}-\frac y {\|y\|}\right\|^2 = \frac{\|y\|^2}{\|x\|^2} -2\langle x,y\rangle \left(\frac 1{\|x\|^2}-\frac 1 {\|x\|\|y\|}\right)-1
\\
&& \quad\stackrel{(*)}{\ge}\frac{\|y\|^2}{\|x\|^2} -2\|x\|\|y\|\left(\frac 1{\|x\|^2}-\frac 1 {\|x\|\|y\|}\right)-1 = \frac{\|y\|^2}{\|x\|^2} - 2\frac{\|y\|}{\|x\|}+1 = \left(\frac{\|y\|}{\|x\|}-1\right)^2 \ge 0,\end{aligned}$$ where in ($*$) we used the fact that $\langle x,y \rangle \le \|x\|\|y\|$ and the fact that $\frac{1}{\|x\|^2}-\frac 1{\|x\|\|y\|} \ge 0$ (since $\|x\| \le \|y\|$). Thus, $$\begin{aligned}
\left\|\frac x{\|x\|}-\frac y {\|y\|}\right\| \le \frac{1}{\|x\|}\|x-y\|,\end{aligned}$$ which completes the proof of the first statement. The second statement follows as $P_{\mathbb S}x = x/\|x\|$ for all $x\in {\mathbb H}\setminus\{0\}$.
We are now in a position to prove Theorem \[thm VN\].
Note first that if $r \ge 2$, then since $x_0\in {\mathbb S}$, $B[0,1] \subseteq B[x_0,r]$. Therefore, $P_{\mathbb S}(B[x_0,r]\setminus\{0\}) = {\mathbb S}$. Now, since $C$ is convex, $P_C$ is non-expansive, and so for all $x\in B[x_0,r]\setminus \{0\}$, $$\begin{aligned}
\label{dist less 2}
\|P_CP_{\mathbb S}x - P_CP_{\mathbb S}x_0\| \le \|P_{\mathbb S}x - P_{\mathbb S}x_0\| \le 2 \le r.\end{aligned}$$ Therefore, $$\begin{aligned}
\label{inside ball}
P_CP_{\mathbb S}(B[x_0,r]\setminus \{0\}) \subseteq B[x_0,r]\end{aligned}$$ In particular, it follows that $$\begin{aligned}
P_CP_{\mathbb S}((1-{\beta}){\mathbb S}) \subseteq B[x_0,r],\end{aligned}$$ where $$\begin{aligned}
(1-{\beta}){\mathbb S}= \big\{x\in {\mathbb H}~|~ \|x\| = 1-{\beta}\big\}.\end{aligned}$$ Thus, by Theorem \[thm Kirs\], there exists $F:{\mathbb H}\to B[x_0,r]$ such that $\|F\|_{\mathrm{lip}}= \frac 1 {1-{\beta}}$ and $F\big|_{(1-{\beta}){\mathbb S}} = P_CP_{\mathbb S}$. Define, $F_1:{\mathbb H}\to {\mathbb H}$, $$\begin{aligned}
\label{def F1}
F_1x =\begin{dcases} Fx & x\in B[0,1-{\beta}] , \\ P_CP_{\mathbb S}x & x\in {\mathbb H}\setminus B(0,1-{\beta}). \end{dcases}\end{aligned}$$ If $x,y \in B[0,1-{\beta}]$ or $x,y\in {\mathbb H}\setminus B(0,1-{\beta})$ then since $\|F\|_{\mathrm{lip}}= \frac 1 {1-{\beta}}$ and by Proposition \[prop proj lip\], $$\begin{aligned}
\label{case both}
\|F_1x-F_1y\| \le \frac{\|x-y\|}{1-{\beta}}.\end{aligned}$$ If, without loss of generality, $x\in B[0,1-{\beta}]$ and $y\in {\mathbb H}\setminus B(0,1-{\beta})$, then there exists $t\in [0,1]$ such that $\|tx+(1-t)y\| = 1-{\beta}$. Thus, $$\begin{aligned}
\label{case one}
\nonumber \|F_1x-F_1y\| & \le & \|F_1x-F_1(tx+(1-t)y)\| + \|F_1(tx+(1-t)y) - F_1y\|
\\ \nonumber & \stackrel{\eqref{def F1}}{=} & \|Fx-F(tx+(1-t)y)\| + \|P_CP_{\mathbb S}(tx+(1-t)y) - P_CP_{\mathbb S}y\|
\\ \nonumber & \stackrel{(*)}{\le} & (1-t)\frac{\|x-y\|}{1-{\beta}}+ t\frac{\|x-y\|}{1-{\beta}}
\\ & = & \frac{\|x-y\|}{1-{\beta}},\end{aligned}$$ where in ($*$) we used the fact that $\|F\|_{\mathrm{lip}}= \frac 1{1-{\beta}}$ and Proposition \[prop proj lip\]. Combining and , it follows that $\|F_1\|_{\mathrm{lip}}= \frac 1 {1-{\beta}}$. Now, if $r\ge 2$, $$\begin{aligned}
F_1(B[0,1-{\beta}]) = F(B[0,1-{\beta}]) \stackrel{(*)}{\subseteq} B[x_0,r],\end{aligned}$$ and $$\begin{aligned}
F_1(B[x_0,r]\setminus B(0,1-{\beta})) = P_CP_{\mathbb S}(B[x_0,r]\setminus B(0,1-{\beta})) \stackrel{\eqref{inside ball}}{\subseteq} B[x_0,r],\end{aligned}$$ where in ($*$) we used the fact that $F({\mathbb H})\subseteq B[x_0,r]$. Altogether, $$\begin{aligned}
F_1(B[x_0,r]) \subseteq B[x_0,r],\end{aligned}$$ and $\|F_1\|_{\mathrm{lip}}= \frac 1 {1-{\beta}}$. Applying Proposition \[prop smooth\] to $F_1$ on the domain $B[x_0,r]$ completes the proof.
\[rem exchange VN\] Note that we cannot change the order of projections in Theorem \[thm VN\]. Indeed, it is possible that $P_Cx=0$ for some $x\in {\mathbb H}$, and then $P_{\mathbb S}P_Cx$ is not defined. Even if $\|P_Cx\| >0$, $\|P_Cy\| >0$, then by Proposition \[prop proj lip\], $$\begin{aligned}
\|P_{\mathbb S}P_C x - P_{\mathbb S}P_C y\| \le \max\left\{\frac 1{\|P_Cx\|}, \frac 1 {\|P_Cy\|}\right\}\|P_Cx - P_Cy\| \le \max\left\{\frac 1{\|P_Cx\|}, \frac 1 {\|P_Cy\|}\right\}\|x - y\|,\end{aligned}$$ but $\max\left\{\frac 1{\|P_Cx\|}, \frac 1 {\|P_Cy\|}\right\}$ can be very large. Thus, we do not obtain an estimate similar to . [$\diamond$]{}
Acknowledgements {#acknowledgements .unnumbered}
----------------
This note is a revised and much simplified version of a note whose original version can be found at <https://www.carma.newcastle.edu.au/jon/weak-ergodicity.pdf>. Note that the results in the original version apply only for the case of the sphere and a line in finite dimensional spaces, while here the results are more general. Sadly, the first named author passed away before this note was being revised. The second named author is grateful to Jon Borwein for many interesting conversations and for his warm friendship.
[99]{}
[^1]: This research was supported by ARC grant DP160101537
|
---
abstract: 'As is well-known, the compact groups ${\rm Spin}(7)$ and ${\rm SO}(7)$ both have a single conjugacy class of compact subgroups of exceptional type ${\bf G}_2$. We first show that if $\Gamma$ is a subgroup of ${\rm Spin}(7)$, and if each element of $\Gamma$ is conjugate to some element of ${\rm G}_2$, then $\Gamma$ itself is conjugate to a subgroup of ${\rm G}_2$. The analogous statement for ${\rm SO}(7)$ turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in ${\rm SO}(7)$ in a very specific way: ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$, ${\rm SL}_ 2({\mathbb{Z}}/3{\mathbb{Z}})$, ${\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$, as well as the nonabelian subgroups of ${\rm GO}_2({\mathbb{C}})$ with compact closure, similitude factors group $\{\pm1\}$, and which are not isomorphic to the dihedral group of order $8$. More generally, we consider the analogous problems in which the Euclidean space is replaced by a quadratic space of dimension $7$ over an arbitrary field.PS. This type of questions naturally arises in some formulation of a converse statement of Langlands’ global functoriality conjecture, to which the results above have thus some applications. Moreover, we give necessary and sufficient local conditions on a cuspidal algebraic regular automorphic representation of ${\mathrm{GL}}_7$ over a totally real number field so that its associated $\ell$-adic Galois representations can be conjugate into ${\rm G}_2(\overline{{\mathbb{Q}}_\ell})$.'
address: |
[Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS,\
Université Paris-Saclay, 91405 Orsay, France,\
[<gaetan.chenevier@math.cnrs.fr>]{}]{}
author:
- Gaëtan Chenevier
title: |
[Subgroups of ${\rm Spin}(7)$ or ${\rm SO}(7)$ with each element conjugate to some element of ${\rm G}_2$\
and applications to automorphic forms]{}
---
Introduction {#introduction .unnumbered}
============
For $n\geq 1$, let ${\rm Spin}(n)$ denote the Spin group of the standard Euclidean space ${\mathbb{R}}^n$ and ${\rm SO}(n)$ its special orthogonal group. As is well-known there is a unique conjugacy class of compact subgroups of ${\rm Spin}(7)$ (resp. ${\rm SO}(7)$) which are connected, semisimple, and whose root system is of type ${\bf G}_2$. We shall call such a subgroup a [*${\rm G}_2$-subgroup*]{}. PS. PS.
\[thmintroa\] Let $\Gamma \subset {\rm Spin}(7)$ be a subgroup such that each element of $\Gamma$ is contained in a ${\rm G}_2$-subgroup. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup.
This perhaps surprising result has a very simple proof. Indeed, let $W$ be a spin representation of ${\rm Spin}(7)$, an $8$-dimensional real vector space, and let $E$ be its standard $7$-dimensional real representation, which factors through ${\rm SO}(7)$. If $H$ is a ${\rm G}_2$-subgroup of ${\rm Spin}(7)$, then we have an isomorphism of ${\mathbb{R}}[H]$-modules $$W \simeq 1 \oplus E.$$ As a consequence, if $\Gamma$ is as in the statement of Theorem \[thmintroa\], we must have the equality $\det (t - \gamma_{|W}) = (t-1) \det(t- \gamma_{|E})$ for all $\gamma \in \Gamma$. But this implies that the (necessarily semisimple) ${\mathbb{R}}[\Gamma]$-modules $W$ and $1 \oplus E$ are isomorphic. In particular, $\Gamma$ fixes a nonzero vector in $W$. We conclude as the ${\rm G}_2$-subgroups of ${\rm Spin}(7)$ are well-known to be exactly the stabilizers of the nonzero elements of $W$. PS. PS.
Theorem \[thmintroa\] admits a generalization over an arbitrary field, that we prove in §\[pfspin7\]. Let $k$ be a field and $E$ a $7$-dimensional nondegenerate (or even “regular”, see §\[regqsp\]) quadratic space over $k$. If $C$ is an octonion $k$-algebra whose quadratic subspace of pure octonions is isometric to $E$, then the automorphism group of $C$ naturally embeds both in ${\rm Spin}(E)$ and in ${\rm SO}(E)$ (§\[g2emb\]). We call such a subgroup of ${\rm Spin}(E)$ or ${\rm SO}(E)$ a [*${\rm G}_2$-subgroup*]{}. These subgroups may not exist, but in any case they are all conjugate (by the action of ${\rm SO}(E)$, see Proposition\[carG2sbgp\]). When $k={\mathbb{R}}$ and $E$ is Euclidean they are exactly the subgroups introduced above. When $k$ is algebraically closed, they also do exist and coincide with the closed connected algebraic subgroups which are simple of type ${\bf G}_2$ (see Proposition \[algcarG2\]).
\[thmintrob\] Let $k$ be a field, $E$ the quadratic space of pure octonions of some octonion $k$-algebra, $W$ a spinor module for $E$, and $\Gamma$ a subgroup of ${\rm Spin}(E)$. Assume that the $k[\Gamma]$-module $W$ is semisimple. Then the following assertions are equivalent: PS.
- each element of $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm Spin}(E)$, PS. PS.
- for each $\gamma \in \Gamma$ we have $\det(1-\gamma_{|W})=0$, PS. PS.
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm Spin}(E)$. PS. PS.
This theorem is Corollary \[corthmbsecondev\] of Theorem \[thmbsecondev\], of which Theorem \[thmintroa\] is the special case $k={\mathbb{R}}$ (see Remark \[examplethmAR\]). Note that Theorem \[thmintroa\] also follows from the case $k={\mathbb{C}}$ of Theorem \[thmintrob\] by standard arguments from the theory of complexifications of compact Lie groups (see §\[pfthmc\]). PS. PS. The naive analogue of Theorem \[thmintroa\] to ${\rm SO}(7)$ turns out that to be false. Let us introduce three embeddings that will play some role in the correct statement. We fix a complex nondegenerate quadratic plane $P$, denote by ${\rm GO}(2)$ the unique maximal compact subgroup of the Lie group of orthogonal similitudes of $P$, and by ${\rm O}(2)^{\pm} \subset {\rm GO}(2)$ the subgroup[^1] of elements with similitude factor $\pm 1$. Recall that $E$ is the standard $7$-dimensional real representation of ${\rm SO}(7)$. Then:PS. PS.
— there is a group homomorphism $\alpha : {\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}\rightarrow {\rm SO}(7)$, unique up to conjugacy, such that the associated representation of ${\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$ on $E \otimes {\mathbb{C}}$ is the direct sum of its $7$ nontrivial characters;PS. PS.
— there is a morphism $\beta : {\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}) \rightarrow {\rm SO}(7)$, unique up to conjugacy, such that the representation of ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ on $E \otimes {\mathbb{C}}$ is the direct sum of its $4$ nontrivial irreducible representations of dimension $\leq 2$;PS. PS.
— there is a group morphism $\gamma : {\rm O}(2)^{\pm} \rightarrow {\rm SO}(7)$, unique up to conjugacy, such that the representation of ${\rm O}(2)^{\pm}$ on $E \otimes {\mathbb{C}}$ is isomorphic to the direct sum of $P$, $P^\ast$ and of the three order $2$ characters of ${\rm O}(2)^{\pm}$. PS. PS.
In each of these cases, if $\Gamma \subset {\rm SO}(7)$ denotes the image of the given morphism, we prove in §\[parexamples\] that each element of $\Gamma$ belongs to a ${\rm G}_2$-subgroup of ${\rm SO}(7)$. Let ${\rm D}_8$ denote the dihedral group of order $8$. PS. PS.
\[thmintroc\] Let $\Gamma$ be a subgroup of ${\rm SO}(7)$ such that each element of $\Gamma$ is contained in a ${\rm G}_2$-subgroup. Then exactly one of the following assertions holds: PS.
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(7)$,PS. PS.
- $\Gamma$ is conjugate in ${\rm SO}(7)$ to one of the groups $$\alpha({\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}), \, \, \,\,\beta({\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})),\, \, \,\,\beta({\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}}))\, \, \, \, {\rm or}\,\,\, \, \gamma(S),$$ where $S \subset {\rm O}(2)^{\pm}$ is a nonabelian subgroup, nonisomorphic to ${\rm D}_8$, and whose similitude factors group is $\{\pm 1\}$.
Let us discuss the main steps of the proof of this theorem, which turned out to be much harder than the one of Theorem \[thmintroa\]. Let $\Gamma$ be a subgroup of ${\rm SO}(7)$. We view $E$ as a (semisimple) representation of $\Gamma$. We first show in §\[basicid\] that each element of $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(7)$ if, and only if, we have a ${\mathbb{R}}[\Gamma]$-linear isomorphismPS. PS. PS. PS. It is an exercise to check that the images of $\alpha, \beta$ and $\gamma$ do satisfy ${\rm (\ast)}$. Note also that identity ${\rm (\ast)}$ implies that $\Gamma$ fixes a nonzero alternating trilinear form $f$ on $E$. In the special case $E \otimes {\mathbb{C}}$ is irreducible, the classification of trilinear forms given in [@cohen] shows that the stabilizer of $f$ is a ${\rm G}_2$-subgroup, and we are done. For reducible $E \otimes {\mathbb{C}}$, we proceed quite differently (and do not rely on any classification result). PS. PS.
Our starting point is the following necessary and sufficient condition for $\Gamma$ to belong to a ${\rm G}_2$-subgroup of ${\rm SO}(7)$. Let $\widetilde{\Gamma} \subset {\rm Spin}(7)$ denote the inverse image of $\Gamma$ under the natural surjective morphism ${\rm Spin}(7) \rightarrow {\rm SO}(7)$. We show that $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(7)$ if, and only if, the restriction to $\widetilde{\Gamma}$ of the spin representation $W$ of ${\rm Spin}(7)$ contains an order $2$ character (Corollary \[carbetaspin\]). Our strategy to prove Theorem \[thmintroc\] is then to analyze the simultaneous structure of the ${\mathbb{C}}[\widetilde{\Gamma}]$-modules $E \otimes {\mathbb{C}}$ and $W \otimes {\mathbb{C}}$, when the identity ${\rm (\ast)}$ holds. PS. PS.
We argue by descending induction on the maximal dimension of a $\Gamma$-stable isotropic subspace of the complex quadratic space $E$, an invariant that we call the [*Witt index*]{} of $\Gamma$ (see §\[defwindex\]). This invariant can only increase when we replace $\Gamma$ by a subgroup. Moreover, for a given Witt index we also argue according to the possible dimensions (and selfduality) of the irreducible summands of $E \otimes {\mathbb{C}}$. This forces us to study in details quite a number of special cases, which is somewhat unpleasant, but leads naturally to the discovery the counterexamples $\alpha,\beta$ and $\gamma$ above. Several exceptional properties of low dimensional spinor modules play a role along the proof. PS. PS.
Our arguments apply more generally to the special orthogonal group of a $7$-dimensional regular quadratic space over an arbitrary algebraically closed field $k$, that we assume for this introduction to have characteristic $\neq 2$. We define in §\[parexamples\] a natural analogue ${\rm O}_2^{\pm}(k)$ of the group ${\rm O}(2)^{\pm}$, as well as analogues of the morphisms $\alpha, \beta$ and $\gamma$ above (the morphism $\beta$ is actually only defined if the characteristic of $k$ is $\neq 3$). Our main theorem is then the following (see Theorem \[mainthmb\]).PS. PS.
\[thmintrod\] Let $k$ be an algebraically closed field, $E$ a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup. Assume that each element of $\Gamma$ has the same characteristic polynomial as some element of some ${\rm G}_2$-subgroup of ${\rm SO}(E)$, and that $\Gamma$ satisfies the “semisimplicity” assumption denoted by ${\rm (S)}$ in §\[parass\]. PS. Then the same conclusion as the one of Theorem \[thmintroc\] holds, with ${\rm SO}(7)$ replaced by ${\rm SO}(E)$, and ${\rm O}(2)^{\pm }$ replaced by ${\rm O}_2^{\pm}(k)$.
We refer to [*loc. cit.*]{} for a discussion of assumption ${\rm (S)}$. Let us simply say here that ${\rm (S)}$ holds if the $k[\Gamma]$-module $E$ is semisimple and if the characteristic of $k$ is either $0$ or $>13$. The special case $k={\mathbb{C}}$ of Theorem \[thmintrod\] is actually equivalent to Theorem \[thmintroc\] (see §\[pfthmc\]). PS. PS.
Let us end this group theoretic discussion by raising some natural questions. Let $G$ be a group and $H$ a subgroup of $G$. Denote by\[defcalp\] $\mathcal{P}(G,H)$ the following property of $(G,H)$: for all subgroup $\Gamma$ of $G$, if each element of $\Gamma$ is conjugate to an element of $H$, then $\Gamma$ is conjugate to a subgroup of $H$. We have explained that $\mathcal{P}({\rm Spin}(7),{\rm G}_2)$ holds and that $\mathcal{P}({\rm SO}(7),{\rm G}_2)$ does not.PS. PS.
[**Questions:**]{} [*Can we classify the couples $(G,H)$, with $G$ a compact connected Lie group and $H$ a closed connected subgroup, such that $\mathcal{P}(G,H)$ holds ? Are there “remarkable” couples of finite groups $(G,H)$ such that $\mathcal{P}(G,H)$ holds ?*]{}PS. PS.
For instance, one can show[^2] that for integers $a \geq b \geq 1$, the property $\mathcal{P}({\rm SO}(a+b),{\rm SO}(a) \times {\rm SO}(b))$ holds if, and only if, we have $b=1$ and $a \in \{1,3\}$. As another example, if $\got{S}_n$ denotes the symmetric group of $\{1,\dots,n\}$, then[^3] $\mathcal{P}(\got{S}_{n+1},\got{S}_n)$ holds if, and only if, we have $n\leq 3$.PS. PS. PS. PS. [Applications to automorphic and Galois representations]{} PS. PS.
Our original motivation for studying property $\mathcal{P}$ above is that it naturally arises in some formulation of a converse statement of Langlands’ functoriality conjecture. Before focusing on the specific case of our study, we first briefly give the general context, assuming some familiarity of the reader with Langlands philosophy [@langlandspb; @euler] and automorphic forms [@corvallis]. PS. PS.
Let $F$ be a number field, $G$ (resp. $H$) a connected semisimple linear algebraic group defined and split over $F$, $\widehat{G}$ (resp. $\widehat{H}$) a complex semisimple algebraic group dual to $G$ (resp. $H$) in the sense of Langlands, $\mathcal{G}$ (resp. $\mathcal{H}$) a maximal compact subgroup of $\widehat{G}$ (resp. $\widehat{H}$), $\rho : \mathcal{H} \rightarrow \mathcal{G}$ a continuous homomorphism, and $\pi$ a cuspidal tempered automorphic representation of $G({\mathbb{A}}_F)$. Assume that for all but finitely many places $v$ of $F$, the Satake parameter ${\rm c}(\pi_v)$ of $\pi_v$, viewed as a well-defined conjugacy class in $\mathcal{G}$, is the conjugacy class some element in $\rho(\mathcal{H})$. If $\mathcal{P}(\mathcal{G},\rho(\mathcal{H}))$ holds, then we may expect the existence of a cuspidal tempered automorphic representation $\pi'$ of $H({\mathbb{A}}_F)$ such that for almost all finite places $v$ of $F$, the $\mathcal{G}$-conjugacy class of $\rho({\rm c}(\pi'_v))$ coincides with ${\rm c}(\pi_v)$. PS. PS.
Indeed, here is a heuristical argument using the hypothetical Langlands group [@Lgl; @kott; @arthurconjlan]. \[lanintro\]Fix a finite set $S$ of places of $F$ such that for $v \notin S$ then $v$ is finite and $\pi_v$ is unramified, and let $\mathcal{L}_{F,S}$ be the Langlands group of $F$ unramified outside $S$. Following Kottwitz, this is a compact topological group, equipped for each $v \notin S$ with a distinguished conjugacy class ${\rm Frob}_v$. Langlands associates to $\pi$ some continuous morphism $\phi : \mathcal{L}_{F,S} \rightarrow \mathcal{G}$, whose image has a finite centralizer in $\mathcal{G}$, such that[^4] the $\mathcal{G}$-conjugacy class of $\phi({\rm Frob}_v)$ is ${\rm c}(\pi_v)$ for each $v \notin S$. The expected density (even equidistribution!) of $\cup_{v \notin S} {\rm Frob}_v$ in $\mathcal{L}_{F,S}$ implies that any element in $\phi(\mathcal{L}_{F,S})$ is conjugate to some element in $\rho(\mathcal{H})$, so that we have $\phi = \rho \circ \phi'$ for some continuous morphism $\phi' : \mathcal{L}_{F,S} \rightarrow \mathcal{H}$, by property $\mathcal{P}(\mathcal{G},\rho(\mathcal{H}))$. In turn, $\phi'$ is associated to some cuspidal tempered automorphic representation $\pi'$ of $H({\mathbb{A}}_F)$. PS. PS.
Of course, the properties $\mathcal{P}({\rm Spin}(7),{\rm G}_2)$ and $\mathcal{P}({\rm SO}(7),{\rm G}_2)$ studied before correspond to the special cases where $H$ is of type ${\bf G}_2$ and $G$ is either ${\rm PGSp}_6$ or ${\rm Sp}_6$. As a first application, Theorems \[thmintroa\] & \[thmintroc\] thus provide interesting conjectures in this context. We postpone the discussion of the case $G={\rm PGSp}_6$, about which more can be said, to the end of this introduction (see Theorem \[tig2pgsp6\]). Note that although $\mathcal{P}({\rm SO}(7),{\rm G}_2)$ does not hold, the argument above still shows that the existence of $\pi'$ is expected to hold in the case $G={\rm Sp}_6$, because none of the exceptions in the statement of Theorem \[thmintroc\] has a finite centralizer in $\mathcal{G}\simeq {\rm SO}(7)$ (their Witt index is nonzero). PS. PS.
When one knows how to associate to $\pi$ a compatible system of $\ell$-adic Galois representations, which requires at least some assumptions on the Archimedean components of $\pi$, the absolute Galois group of $F$ may be used as a substitute of the hypothetical Langlands group. This allows us to prove the following theorem (Corollary \[corcplx\]). We denote by ${\rm W}_{M}$ the Weil group of the local field $M$, ${\rm G}_2$ a fixed split semisimple group over ${\mathbb{Q}}$ of type ${\bf G}_2$, and if $k$ is an algebraically closed field of characteristic $0$ we fix an irreducible polynomial representation $\rho : {\rm G}_2(k) \rightarrow {\rm GL}_7(k)$. PS.
\[thmintrof\] Let $F$ be a totally real number field and $\pi$ a cuspidal automorphic representation of ${\rm GL}_7({\mathbb{A}}_F)$ such that $\pi_v$ is algebraic regular for each real place $v$ of $F$. The following properties are equivalent:
- for all but finitely many places $v$ of $F$, the Satake parameter of $\pi_v$ is the conjugacy class of an element in $\rho({\rm G}_2({\mathbb{C}}))$,PS. PS.
- for any finite place $v$ of $F$, there exists a continuous morphism $\phi_v : {\rm W}_{F_v} \times {\rm SU}(2) \rightarrow {\rm G}_2({\mathbb{C}})$, unique up to ${\rm G}_2({\mathbb{C}})$-conjugacy, such that $\rho \circ \phi_v$ is isomorphic to the Weil-Deligne representation attached to $\pi_v$ by the local Langlands correspondence [@harristaylor].PS.
We refer to §\[autgalrep\] for the unexplained terms of this statement. A key ingredient in the proof of this theorem is the following result, which is perhaps the main application of this paper (Corollary \[cor1thmgalois\]). PS. PS.
\[thmintroe\] Let $F$ be a totally real number field and $\pi$ a cuspidal automorphic representation of ${\mathrm{GL}}_7({\mathbb{A}}_F)$ satisfying assumption (i) of Theorem \[thmintrof\], and such that $\pi_v$ is algebraic regular for each real place $v$ of $F$. Let $E$ be a coefficient number field for $\pi$, $\ell$ a prime and $\lambda$ a place of $E$ above $\ell$. Then there exists a continuous semisimple morphism $$\widetilde{r}_{\pi,\lambda}: {\rm Gal}(\overline{F}/F) \longrightarrow {\rm G}_2(\overline{E_\lambda}),$$ unique up to ${\rm G}_2(\overline{E_\lambda})$-conjugacy, satisfying the following property: for each finite place $v$ of $F$ which is prime to $\ell$, and such that $\pi_v$ is unramified, the morphism $\widetilde{r}_{\pi,\lambda}$ is unramified at $v$ and we have the relation $\det (t - \rho(\widetilde{r}_{\pi,\lambda}({\rm Frob}_v))) = \det (t - {\rm c}(\pi_v))$, where ${\rm c}(\pi)$ is the Satake parameter of $\pi_v$ viewed as a semisimple conjugacy class in ${\rm GL}_7(E)$.
The proof of this theorem uses the existence and properties of the compatible system of $7$-dimensional $\ell$-adic Galois representations associated to $\pi$ [@harristaylor; @shin; @chharris; @bc] and Theorem \[thmintrod\]. We show that we are not in the exceptional cases of Theorem \[thmintrod\] using the knowledge of the Hodge-Tate numbers of those representations. The uniqueness assertion is a consequence of a result of Griess [@griess]. PS. PS. As promised earlier, we now go back to property ${\mathcal P}({\rm G}_2,{\rm Spin}(7))$. After the first version of this paper appeared on the arXiv, some discussions with Gan and Savin led to the conclusion that enough is known to prove unconditionally the conjecture mentioned above concerning the Langlands functorial lifting between ${\rm G}_2$ and ${\rm PGSp}_6$, using works of Arthur [@arthurbook], Ginzburg-Jiang [@gjiang] and Xu [@Xu]. We are grateful to them to let us include this result in this paper: see Theorem \[thmCGS\]. Recall that we may take $\widehat{{\rm G_2}}={\rm G}_2({\mathbb{C}})$ and $\widehat{{\rm PGSp}}_6={\rm Spin}_7({\mathbb{C}})$.PS. PS.
\[tig2pgsp6\] Let $F$ be a number field and $\pi$ a cuspidal automorphic representation of ${\rm PGSp}_6({\mathbb{A}}_F)$. Assume that $\pi$ is tempered, or more generally, that $\pi$ is [nearly generic]{} (see §\[localcarg2sp6\]). Then the following properties are equivalent:
- for all but finitely many places $v$ of $F$, the Satake parameter ${\rm c}(\pi_v)$ is the conjugacy class of an element of a ${\rm G}_2$-subgroup of ${\rm Spin}_7({\mathbb{C}})$,PS. PS.
- there exists a cuspidal automorphic representation $\pi'$ of ${\rm G}_2({\mathbb{A}}_F)$ such that for all but finitely many finite places $v$ of $F$, the image in ${\rm Spin}_7({\mathbb{C}})$ of the Satake parameter ${\rm c}(\pi'_v)$ is conjugate to ${\rm c}(\pi_v)$.PS. PS.
Preliminaries on quadratic spaces and spinors
=============================================
Regular quadratic spaces {#regqsp}
------------------------
Let $k$ be a field. A quadratic space over $k$ is a finite dimensional $k$-vector space $V$ equipped with a quadratic map ${\rm q} : V \rightarrow k$ [@bou §3 [no]{}.4]. By definition, the map $\beta_V(x,y):= {\rm q}(x+y)-{\rm q}(x)-{\rm q}(y)$, $V \times V \rightarrow k$, is a symmetric $k$-bilinear form, and we have ${\rm q}(\lambda v)=\lambda^2 {\rm q}(v)$ for all $v \in V$ and $\lambda \in k$. We refer to Bourbaki [@bou §3, §4] and Knus [@knuscampinas Ch. 1] for the basic notions concerning quadratic spaces (isometries, orthogonal sums, base change, etc...). PS. PS.
A quadratic space $V$ over $k$ will be called [*nondegenerate*]{} if the bilinear form $\beta_V$ is a perfect pairing. If the characteristic of $k$ is $2$, then $\beta_V$ is alternate, thus $\dim V$ has to be even if $V$ is nondegenerate. It will be convenient to say that a quadratic space $V$ over $k$ is [*regular*]{} if either $V$ is nondegenerate, or if we are in the following situation: $\dim V$ is odd, the characteristic of $k$ is $2$, the kernel of $\beta_V$ is one dimensional and ${\rm q}$ is not identically zero on it.[^5] PS. PS.
If $I$ is a finite dimensional $k$-vector space, we denote by $I^\ast$ its dual vector space and by ${\rm H}(I)$ the quadratic space $I \oplus I^\ast$ with ${\rm q}(x+\varphi)=\varphi(x)$ (“hyperbolic quadratic space over $I$”). This is a nondegenerate quadratic space. PS. PS.
The isometry group of a quadratic space $V$, denoted ${\rm O}(V)$, is the subgroup of elements $g \in {\rm GL}(V)$ such that ${\rm q} \circ g = g$. When $V$ is regular, there is a well-defined subgroup ${\rm SO}(V) \subset {\rm O}(V)$ of proper isometries: when $2 \in k^\times$ or $\dim V$ is odd, we set ${\rm SO}(V) = {\rm O}(V) \cap {\rm SL}(V)$, and in the remaining case ($\dim V$ even and $2 \not\in k^\times$) ${\rm SO}(V)$ is an index $2$ subgroup of ${\rm O}(V)$ defined as the kernel of the Dickson invariant of $V$ [@knuscampinas Ch. 6 p.59]. We also denote by ${\rm GO}(V) \subset {\rm GL}(V)$ the subgroup of orthogonal similitudes of the quadratic space $V$, and when $V$ is regular, by ${\rm GSO}(V)$ its subgroup of proper similitudes (see e.g. the end of §II.1 in [@cl]).PS. PS.
Clifford algebras, Spin groups, and their relatives {#parclag}
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Let $V$ be a quadratic space over $k$. Recall that the Clifford algebra ${\rm C}(V)$ of $V$ is a $k$-superalgebra [@bou §9] [@knuscampinas Ch. 4] [@deligne]. It is equipped with a canonical injective morphism $k \oplus V \rightarrow {\rm C}(V)$, that we shall always view as an inclusion. The [*Clifford group*]{} of $V$, denoted $\Gamma(V)$, is the supergroup[^6] of homogeneous elements in ${\rm C}(V)^\times$ normalizing $V$. The normal subgroup of even elements in $\Gamma(V)$ is the [*general spin group*]{} of $V$ and denoted ${\rm GSpin}(V)$. The superaction of $\Gamma(V)$ on $V$, given by $(\gamma,v) \mapsto (-1)^{{\rm p}(\gamma)} \gamma v \gamma^{-1}$, defines a group homomorphism $$\pi_V : \Gamma(V) \longrightarrow {\rm O}(V),$$ PS. PS. whose kernel is by definition the group of invertible homogeneous elements in the supercenter ${\rm Z}(V)$ of ${\rm C}(V)$. Following Chevalley, recall that ${\rm C}(V)$ is equipped with a unique anti-involution $x \mapsto x^{\rm t}$ fixing $V$ pointwise (hence preserving parity). It thus preserves $\Gamma(V)$ as well. If $\gamma \in \Gamma(V)$, the element $\nu(\gamma):=\gamma \gamma^{\rm t}$ belongs to ${\rm Z}(V)_0^\times$, which gives rise to a group homomorphism $$\nu_V : \Gamma(V) \rightarrow {\rm Z}(V)_0^\times.$$ The kernel of $\nu_V$ is the so-called [*pin*]{} supergroup of $V$ and denoted ${\rm Pin}(V)$. The subgroup of even elements in ${\rm Pin}(V)$ is the [*spin group*]{} of $V$, denoted ${\rm Spin}(V)$. We have $\nu_V (\lambda) = \lambda^2$ for all $\lambda \in k^\times$.PS. PS.
Assume from now on that $V$ is regular. If $\dim V$ is even (resp. odd) then ${\rm C}(V)$ (resp. ${\rm C}(V)_0$) is a central simple $k$-algebra [@bou §4] [@knuscampinas Ch. 4 Thm. 8]; in particular we have ${\rm Z}(V)_0=k$ in both cases. Applying the Skolem-Noether theorem, we can show that the homomorphism $\pi_V$ is surjective [@knuscampinas Prop. 6 Ch. 6] and induces an exact sequence $$\label{sexgspin} 1 \longrightarrow k^\times \longrightarrow {\rm GSpin}(V) \overset{\pi_V}{\longrightarrow} {\rm SO}(V) \longrightarrow 1.$$ Moreover, $\nu_V : {\rm GSpin}(V) \longrightarrow k^\times$ induces a homomorphism $\overline{\nu_V} : {\rm SO}(V) \rightarrow k^\times/{k^\times}^2$ called the [*spinor norm*]{}. We have thus an exact sequence $$\label{sexspin} 1 \longrightarrow \mu_2(k) \longrightarrow {\rm Spin}(V) \overset{\pi_V}{\longrightarrow} {\rm SO}(V) \overset{\overline{\nu_V}}{\longrightarrow} k^\times/{k^\times}^2,$$ where $\mu_2(k) \subset k^\times$ is the subgroup of square roots of $1$. PS. PS.
Let $U,V$ be quadratic spaces over $k$ with $U$ nondegenerate and $V$ regular. Then $W=U \bot V$ is regular and the natural $k$-superalgebra isomorphism ${\rm C}(U) \otimes^{\rm gr} {\rm C}(V) {\overset{\sim}{\rightarrow}}{\rm C}(U \bot V)$ induces a group homomorphism $$\label{rhoUV} \rho_{U; V} : {\rm GSpin}(U) \times {\rm GSpin}(V) \rightarrow {\rm GSpin}(U \bot V)$$ which satisfies $\nu_W \circ \rho_{U; V}(g,g') = \nu_{U}(g)\nu_{V}(g')$, $\pi_W \circ \rho_{U; V} = \pi_U \oplus \pi_{V}$ [@knusquadherm Ch. IV §(6.5)]. By , the image of $\pi_W \circ \rho_{U; V}$ is thus ${\rm SO}(U) \times {\rm SO}(V)$, and the kernel of $\rho_{U;V}$ is the subgroup of $k^\times \times k^\times$ whose elements $(\lambda,\lambda')$ satisfy $\lambda\lambda'=1$. PS. PS.
\[gpscheme\] [Most of the constructions of §\[regqsp\] and §\[parclag\] can be adapted in order to make sense over an arbitrary commutative ring $k$: see e.g. [@knusquadherm], [@cl Ch. I §1] and [@cf]. That being done, each group of the form ${\rm G}(V)$ associated above to a regular quadratic space $V$ over $k$ appears as the group of $k$-points of a natural corresponding affine group scheme of finite type over $k$. We shall not need this point of view in the sequel, but in a few places, and when the field $k$ is algebraically closed, it will be convenient to see ${\rm G}(V)$ as a (reduced) linear algebraic group over $k$ in the classical sense [@humphreys1]; it is indeed immediate from the definitions above that each ${\rm G}(V)$ does have a linear algebraic group structure. As is well-known, the algebraic groups ${\rm SO}(V)$, ${\rm Spin}(V)$ and ${\rm GSpin}(V)$ are connected and reductive if $\dim V > 1$ (see e.g. [@cf]). ]{}
[Notations:]{} Let $n\geq 0$ be an integer. We respectively denote by ${\rm O}_n(k)$, ${\rm SO}_n(k)$, ${\rm GO}_n(k)$, ${\rm GSO}_n(k)$, $\Gamma_n(k)$, ${\rm Spin}_n(k)$ and ${\rm GSpin}_n(k)$ the groups ${\rm O}(V)$, ${\rm SO}(V)$, ${\rm GO}(V)$, ${\rm GSO}(V)$, $\Gamma(V)$, ${\rm Spin}(V)$ and ${\rm GSpin}(V)$ with $$V = \left\{ \begin{array}{ll} {\rm H}(k^{r}) & \,\,{\rm \,if}\,\, n=2r, \\ {\rm H}(k^r) \bot k & \,\,{\rm \, if}\,\, n=2r+1.\end{array} \right.$$ In this latter case, $k$ is viewed as a quadratic space with ${\rm q}(x)=x^2$. In both cases, $V$ is regular. When $k$ is algebraically closed, it is the unique regular quadratic space of dimension $n$ over $k$ up to isometry. Moreover, when $k={\mathbb{R}}$, we shall also denote respectively by ${\rm O}(n)$, ${\rm SO}(n)$ and ${\rm Spin}(n)$ the corresponding groups with $V$ the standard Euclidean space ${\mathbb{R}}^n$ of dimension $n$. They are compact Lie groups in a natural way.
Spinor modules {#spinormodules}
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Let $V$ be a regular quadratic space over $k$. We say that the even Clifford algebra of $V$ is [*trivial*]{} if either $\dim V = 2r+1$ is odd and we have a $k$-algebra isomorphism ${\rm C}(V)_0 \simeq {\rm M}_{2^r}(k)$, or $\dim V = 2r$ is even and we have a $k$-algebra isomorphism ${\rm C}(V)_0 \simeq {\rm M}_{2^{r-1}}(k) \times {\rm M}_{2^{r-1}}(k)$. Assuming that $V$ has a trivial even Clifford algebra, we can then define a spinor module for $V$ as follows. Again, there are two cases:PS. PS.
Case (a). If $\dim V=2r$ is even, then we have a graded algebra isomorphism ${\rm C}(V) \simeq {\rm M}_{2^{r-1}|2^{r-1}}(k)$. A spinor module for $V$ is a supermodule $W$ for the $k$-superalgebra ${\rm C}(V)$ such that the associated graded morphism ${\rm C}(V) \rightarrow {\rm End}(W)$ is an isomorphism. Such a module $W$ is simple of dimension $2^{r-1}|2^{r-1}$, and its restriction to the even algebra ${\rm C}(V)_0 \simeq {\rm End}(W_0) \times {\rm End}(W_1)$ is the direct-sum of two nonisomorphic simple modules $W=W_0 \oplus W_1$ called half-spinor modules for $V$. The $k$-linear representations of the groups ${\rm GSpin}(V)$ and ${\rm Spin}(V)$ obtained by restriction of those modules $W$ and $W_i$ are respectively called spin and half-spin representations. PS. PS. Case (b). If $\dim V=2r+1$ is odd, a spinor module for $V$ is a simple module for the (matrix) $k$-algebra ${\rm C}(V)_0$. The restriction to ${\rm GSpin}(V)$ or ${\rm Spin}(V)$ of a spinor module for $V$ is by definition a spin representation. PS. PS.
Any hyperbolic quadratic space, or more generally any regular quadratic space containing a subspace of codimension $\leq 1$ which is hyperbolic, has a trivial even Clifford algebra. However, the converse does not hold: if $k={\mathbb{R}}$ and $V$ is positive definite, it is well-known that $V$ has a trivial even Clifford algebra if, and only if, we have $\dim V \equiv -1,0,1 \bmod 8$ [@deligne table p. 103]. PS. PS.
\[lemmeregularhyp\] Let $V$ be a nondegenerate quadratic space of even dimension, $e \in V$ an element such that ${\rm q}(e) \neq 0$, ${\rm O}(V)_e$ the stabilizer of $e$ in ${\rm O}(V)$, ${\rm SO}(V)_e={\rm SO}(V)\cap {\rm O}(V)_e$, and ${\rm GSpin}(V)_e$ the inverse image of ${\rm SO}(V)_e$ in ${\rm GSpin}(V)$ under the map $\pi_V$.
- The orthogonal $E$ of $e$ in $V$ is a regular quadratic subspace, and the natural map ${\rm O}(V)_e \rightarrow {\rm O}(E)$ induces a bijection $a: {\rm SO}(V)_e {\overset{\sim}{\rightarrow}}{\rm SO}(E)$.PS. PS.
- The natural map $b : {\rm C}(E) \rightarrow {\rm C}(V)$ is injective and induces a group isomorphism ${\rm GSpin}(E) \rightarrow {\rm GSpin}(V)_e$. Furthermore, we have the equality $\pi_V \circ b = a^{-1} \circ \pi_E$ of maps ${\rm GSpin}(E) \rightarrow {\rm SO}(V)$. PS. PS.
- If $V$ has a trivial even Clifford algebra, then so does $E$, and the restriction to $b: {\rm C}(E)_0 \rightarrow {\rm C}(V)_0$ of a half-spinor module for $V$ is a spinor module for $E$.
When $2$ is invertible in $k$, then $V = ke \bot E$ and the lemma is fairly standard. By lack of reference, we provide a proof which works in characteristic $2$ as well. We may and do choose a nondegenerate hyperplan $H \subset E$. Denote by $H^\bot$ the orthogonal of $H$ inside $V$. We have $\dim H^\bot = 2$, $V = H \oplus H^\bot$, $e \in H^\bot$, and $E = H \oplus ke$. PS. PS. The first statement of assertion (i) is immediate from the definitions. The second one is clear when $2 \in k^\times$. If $k$ has characteristic $2$, the surjectivity of the natural map ${\rm O}(V)_e \rightarrow {\rm O}(E)={\rm SO}(E)$ is a consequence of the Witt theorem (which is due to Arf in characteristic $2$ [@bou §4 [no]{}. 3 Thm. 1]). The kernel of this map is $(1 \times {\rm O}(H^\bot)) \cap {\rm O}(V)_e$, which is easily checked to be the group of order $2$ generated by the improper isometry $x \mapsto x - \frac{\beta(x,e)}{{\rm q}(e)} e$, which proves (i). PS. PS. The map $b$ of the statement is the one induced by applying the “Clifford functor” to the inclusion $E \rightarrow V$, it is injective by [@bou §9 [no]{}. 3 Cor. 3]. Let $\gamma \in {\rm GSpin}(E)$. We have $b(\gamma) \in {\rm C}(V)_0^\times$ and we claim that $b(\gamma) \in {\rm GSpin}(V)$. By surjectivity of $\pi_V$ in the sequence , we may choose $h \in {\rm GSpin}(V)_e$ such that $\pi_V(h) = a^{-1}\circ \pi_E(\gamma)$. If we can show that the even element $\lambda := b(\gamma)^{-1} h \in {\rm C}(V)_0^\times$ is a scalar, then (ii) follows at once. The relation $\pi_V(h) = a^{-1}\circ \pi_E(\gamma)$ asserts that $\lambda$ commutes with $E=H \oplus k e$. The natural isomorphism of graded algebras ${\rm C}(H) \otimes^{\rm gr} {\rm C}(H^\bot) {\overset{\sim}{\rightarrow}}{\rm C}(V)$ identifies the supercentralizer of $H$ in ${\rm C}(V)$ to ${\rm C}(H^\bot)$. It follows that $\lambda$ is identified with an element of the graded quaternion algebra ${\rm C}(H^\bot)$ which is even and commutes with some nonzero odd element $e \in H^\bot = {\rm C}(H^\bot)_1$ (we have $e^2 ={\rm q}(e) \neq 0$). But the commutator of such an element is $k[e]$, and $k[e] \cap {\rm C}(H^\bot)_0=k$, which proves $\lambda \in k^\ast$ and assertion (ii).PS. PS. The map $b$ induces a $k$-algebra morphism ${\rm C}(E)_0 \rightarrow {\rm C}(V)_0$. If $V$ has trivial even Clifford algebra, then ${\rm C}(V)_0$ is isomorphic to a direct product $A_1 \times A_2$ of two matrix $k$-algebras $A_i$, each of them being of rank $2^{\dim V-2}=2^{\dim E-1}$. On the other hand, as $E$ is regular the $k$-algebra ${\rm C}(E)_0$ is central and simple of rank $2^{\dim E-1}$ as well. It follows that each projection ${\rm C}(E)_0 \rightarrow A_i$ is an isomorphism, which proves (iii).
For later use, we end this paragraph by stating two classical results. PS.
\[cliffordhyp\] Let $I$ be a finite dimensional $k$-vector space, $V={\rm H(I)}$ the hyperbolic quadratic space over $I$ and $\rho_I : {\rm GL}(I) \rightarrow {\rm SO}({\rm H}(I))$ the natural homomorphism defined by $({\rho_I(g)}_{|I},{\rho_I(g)}_{|I^\ast})= (g,{}^{\rm t}\!g^{-1})$. Then the exterior superalgebra $\Lambda\, I$ is in a natural way a spinor module for $V$. Moreover, there is a natural homomorphism $$\widetilde{\rho}_I : {\rm GL}(I) \rightarrow {\rm GSpin}({\rm H}(I))$$ such that $\pi_V \circ \widetilde{\rho}_I = \rho_I$, $\nu_V \circ \widetilde{\rho}_I = \det$, and such that the restriction of the spin representation $\Lambda \,I$ to $\widetilde{\rho}_I$ is the canonical representation of ${\rm GL}(I)$ on $\Lambda \,I$.
The assertion on the exterior algebra is well-known, and proved [*e.g.*]{} in [@bou §9 [no]{} 4] or [@knusquadherm Ch. IV Prop. 2.1.1]. The other assertions are proved in [@knusquadherm Ch. IV §6.6].
\[decospin\] Let $U,V$ be regular quadratic spaces over $k$ with $\dim U$ even and $\dim V$ odd. If $U$ and $V$ have a trivial even Clifford algebra, then so does the regular quadratic space $U \bot W$. Moreover, the restriction to the canonical $k$-algebra morphism $${\rm C}(U)_0 \otimes {\rm C}(V)_0 \longrightarrow {\rm C}(U \bot V)_0$$ of a spinor module for $U \bot V$ is isomorphic to the (external) tensor product of a spinor module for $U$ and of a spinor module for $V$.
Set $R={\rm C}(U)_0$, $S={\rm C}(V)_0$ and $T={\rm C}(U \bot V)_0$. We have a natural injective $k$-algebra homomorphism $R \otimes S \rightarrow T$. Write $\dim U = 2a$ and $\dim V = 2b+1$. The we have $R \simeq {\rm M}_{2^{a-1}}(k) \times {\rm M}_{2^{a-1}}(k)$, $S \simeq {\rm M}_{2^b}(k)$, and $T$ is central simple of dimension $2^{2a+2b}$. As $T$ contains $R \otimes S$, which contains a $k$-algebra isomorphic to $k^{2^{a+b}}$, we have $T \simeq {\rm M}_{2^{a+b}}(k)$. We conclude using the following elementary fact: if $p$ and $q$ are integers, there is a unique ${\rm GL}_{p+q}(k)$-conjugacy class (resp. ${\rm GL}_{pq}(k)$-conjugacy class) of $k$-algebra morphisms ${\rm M}_p(k) \times {\rm M}_q(k) \rightarrow {\rm M}_{p+q}(k)$ (resp. ${\rm M}_p(k) \otimes {\rm M}_q(k) \rightarrow {\rm M}_{pq}(k)$).
The Witt index of a representation in ${\rm O}(V)$ {#defwindex}
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In this paragraph, we assume that $V$ is a nondegenerate quadratic space over the field $k$, that $\Gamma$ is a group and that $\rho : \Gamma \rightarrow {\rm O}(V)$ is a group homomorphism. PS. PS.
\[defindex\] The Witt index of $\rho$ is the maximal dimension of a subspace $I \subset V$ which is both totally isotropic [(]{}[i.e.]{} ${\rm q}(I)=0$[)]{} and stable by $\rho(\Gamma)$. When $\rho$ is the inclusion of a subgroup $\Gamma$ of ${\rm O}(V)$ we define the Witt index of $\Gamma$ as the one of $\rho$.
Denote by ${\rm Irr}(X)$ the set of isomorphism classes of the irreducible constituents of the $k[\Gamma]$-module $X$. As $V$ is isomorphic to $V^\ast$, the duality functor induces an action of ${\mathbb{Z}}/2{\mathbb{Z}}$ on ${\rm Irr}(V)$ (the “duality” action).
\[propwindex\] Assume $k$ is algebraically closed of characteristic $\neq 2$ and that $V$ is semisimple as a $k[\Gamma]$-module. Fix $\Phi \subset {\rm Irr}(V)$ a set of representatives for the duality action. There exists a decomposition $$\label{candeco} V \,=\, ( I \oplus J )\, \bot \, W$$ where $I,J$ and $W$ are $\Gamma$-stable subspaces of $V$ such that:
- $W$ is the orthogonal direct sum of nonisomorphic irreducible selfdual $k[\Gamma]$-modules, PS. PS.
- $I$ and $J$ are totally isotropic, and we have ${\rm Irr}(I) \subset \Phi$.PS. PS.
For any such decomposition, $\beta_V$ induces a $\Gamma$-equivariant isomorphism $I {\overset{\sim}{\rightarrow}}J^\ast$ and the dimension of $I$ is the Witt index of $\rho$. Morever, the isomorphism class of the $k[\Gamma]$-module $I$ does not depend on the choice of the decomposition , and the isomorphism class of the $k[\Gamma]$-module $W$ depends neither on the choice of nor on the choice of $\Phi$.
(This is presumably well-known so we use a small font.) Assume first that there is a $\Gamma$-stable totally isotropic subspace $I \subset V$ such that $I^\bot = I$ (so that $\dim I = \frac{\dim V}{2}$ is the Witt index of $\rho$). We want to show the existence of a decomposition satisfying (i) and (ii). As $V$ is semisimple we may find a $\Gamma$-stable subspace $F \subset V$ such that $V = I \oplus F$. The perfect pairing $\beta_V$ induces a $k[\Gamma]$-linear isomorphism $I {\overset{\sim}{\rightarrow}}F^\ast$, $x \mapsto (y \mapsto \beta_V(x,y))$. This shows that for $x \in F$, there exists a unique element $u(x) \in I$ such that $\beta_V(x,y) = \beta_V(u(x),y)$ for all $y \in F$. The map $u : F \rightarrow I$ is $k[\Gamma]$-linear. Observe that for all $x \in F$ we have $${\rm q}(x - \frac{u(x)}{2}) = {\rm q}(x) - \frac{1}{2}\,\beta_V(u(x),x) = \frac{1}{2} (\beta_V(x,x) - \beta_V(u(x),x)) = 0.$$ As a consequence, the subspace $J = \{x - \frac{u(x)}{2}, x \in F\}$ is totally isotropic, $\Gamma$-stable, and satisfies $I \oplus J = V$. This shows the existence of a decomposition satisfying (i) and (ii) except perhaps ${\rm Irr}(I) \subset \Phi$. As $V$ is semisimple, we may find a $k[\Gamma]$-module decomposition $I=I_0 \oplus I_1$ (resp. $J = J_0 \oplus J_1$) where $I_0$ (resp. $J_0)$ is the sum of the irreducible nonselfdual subrepresentations of $I$ (resp. $J$) whose isomorphism class do not (resp. do) belong to $\Phi$. By construction, any $k[\Gamma]$-linear morphism $J_0 \rightarrow I_1^\ast$ is zero, so we have $\beta_V(J_0,I_1)=0$, and $\beta_V$ induces an isomorphism $I_0^\ast \rightarrow J_0$. As a consequence, $I'=J_0 \oplus I_1$ is a $\Gamma$-stable totally isotropic subspace of $V$ of dimension $\frac{\dim V}{2}$, hence such that $I'^\bot = I'$, which satisfies furthermore ${\rm Irr}(I') \subset \Phi$. We conclude the proof by applying to $I'$ instead of $I$ the previous argument. PS. PS.
We now claim the existence of a decomposition satisfying (i), (ii) and such that $\dim I$ is the Witt index of $\rho$. Let $I$ be a $\Gamma$-stable totally isotropic subspace of $V$. By the semisimplicity assumption, we may choose a $\Gamma$-stable subspace $V' \subset I^\bot$ such that we have $I^\bot = I \oplus V'$. This forces $V'$ to be a nondegenerate subspace of $V$. By construction, we have $I^\bot \cap {V'}^\bot = I$. The previous paragraphs apply thus to the representation of $\Gamma$ on ${V'}^\bot$ and show the existence of a $\Gamma$-stable subspace $J$ such that ${V'}^\bot = I \oplus J$ satisfying (ii). Replacing $V$ by $V'$, we may thus assume that $V$ has no nonzero $\Gamma$-stable totally isotropic subspace. We have to show that $V$ is the direct sum of nonisomorphic irreducible selfdual $k[\Gamma]$-modules. PS. PS.
Let $U \subset V$ be any $\Gamma$-stable subspace. The isomorphism $\beta_V : V {\overset{\sim}{\rightarrow}}V^\ast$ induces a $\Gamma$-equivariant morphism $U \rightarrow U^\ast$ whose kernel is $U\cap U^\bot$. This later space is totally isotropic and $\Gamma$-stable, hence zero by assumption. It follows that each $\Gamma$-stable subspace of $V$ is nondegenerate, and in particular, selfdual as a $k[\Gamma]$-module. Assume that there are $\Gamma$-stable, orthogonal, subspaces $U,U' \subset V$ such that $U$ and $U'$ are isomorphic and irreducible as $k[\Gamma]$-modules. Choose a $k[\Gamma]$-linear isomorphism $f: U \rightarrow U'$. Both $\beta_V$ and $\beta_V \circ f$ define nondegenerate and $\Gamma$-invariant symmetric bilinear forms on $U$. As $U$ is irreducible and ${\rm char}\, k \neq 2$, there is a unique such form up to a scalar, so there is $\lambda \in k^\times$ such that we have ${\rm q}(u) = \lambda \,{\rm q}(f(u))$ for all $u \in U$. As $k$ is algebraically closed, there is $\mu \in k^\times$ such that $\lambda = -\mu^2$. It follows that $\{u + \mu f(u), u \in U\} \subset U \oplus U' \subset V$ is a $\Gamma$-stable totally isotropic subspace, a contradiction. This proves the claim. PS. PS.
We now prove the last assertions regarding an arbitrary decomposition satisfying (i) and (ii). Observe that the $k[\Gamma]$-module $W$ is isomorphic to the direct sum of the irreducible selfdual representations of $\Gamma$ which occur in $V$ with an odd multiplicity. In particular, its isomorphism class does not depend on the choice of or $\Phi$. As a consequence, the isomorphism class of $I \oplus I^\ast$ has the same property, and the one of $I$ is uniquely determined by $\Phi$. In particular, $\dim I$ does not depend either on the choice of . This concludes the proof, as we have shown the existence of a decomposition satisfying (i), (ii) and such that $\dim I$ is the Witt index of $\rho$.
\[corwindex\] Under the assumptions of Proposition \[propwindex\], the ${\rm O}(V)$-conjugacy class of $\rho$ only depends on the isomorphism class of the $k[\Gamma]$-module $V$. If furthermore $\dim V$ is odd, the same assertion also holds with ${\rm O}(V)$ [(]{}$ = \{\pm 1\} \times {\rm SO}(V)$[)]{} replaced by ${\rm SO}(V)$. PS. PS.
For $i=1,2$ let $V_i$ be a nondegenerate quadratic space over $k$ (algebraically closed of characteristic $\neq 2$) and $\rho_i : \Gamma \rightarrow {\rm O}(V_i)$ a group homomorphism. We assume that the $k[\Gamma]$-modules $V_i$ are semisimple and isomorphic. We have to show that there is an isometry $u : V_1 \rightarrow V_2$ such that $\rho_2 (\gamma) \circ u = u \circ \rho_1(\gamma)$ for all $\gamma \in \Gamma$. By Proposition \[propwindex\], we may assume that we have $V_i = {\rm H}(I_i) \bot W_i$, where $I_i,I_i^\ast$ and $W_i$ are preserved by $\rho_i(\Gamma)$ and the $W_i$ have Witt index $0$. By that proposition, we have $W_1 \simeq W_2$ and $I_1 \simeq I_2$ as $k[\Gamma]$-modules. Any $k[\Gamma]$-equivariant isomorphism $I_1 {\overset{\sim}{\rightarrow}}I_2$ induces a $\Gamma$-equivariant isometry ${\rm H}(I_1) {\overset{\sim}{\rightarrow}}{\rm H}(I_2)$. We may thus assume $V_i=W_i$ for each $i$. In this case, the isotypical components of the $V_i$ are nondegenerate, so we may even assume that each $V_i$ is irreducible. This implies that the $\Gamma$-invariants of $({\rm Sym}^2 V_i)^\ast$ have dimension $1$. It follows that if $u$ is any $k[\Gamma]$-linear isomorphism $V_1 {\overset{\sim}{\rightarrow}}V_2$, there is $\lambda \in k^\times$ such that $\lambda u$ is an isometry $V_1 {\overset{\sim}{\rightarrow}}V_2$.
\[windexgo\] [Assume more generally that $\rho$ is a morphism $\Gamma \rightarrow {\rm GO}(V)$, with similitude factor $\mu$. Then the considerations of this paragraph extend verbatim to this setting if one replaces the duality functor $X \mapsto X^\ast$ on $k[\Gamma]$-modules by the functor $X \mapsto X^\ast \otimes \mu$. ]{}
Octonion algebras and ${\rm G}_2$-subgroups
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The Clifford algebra of an octonion algebra {#octoalg}
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Let $C$ be an octonion $k$-algebra. Recall that this is a nondegenerate $8$-dimensional quadratic space over $k$ equipped with an element $e \in C$ and a $k$-bilinear map $C \times C \rightarrow C$, $(x,y) \mapsto xy$, such that for all $x,y \in C$ we have ${\rm q}(xy)={\rm q}(x){\rm q}(y)$ and $xe=ex=x$ [@blijspringer1][@springer Ch 1]. PS. PS.
In order to discuss some features of the spin representations of the group ${\rm Spin}(C)$, let us recall the description of the Clifford algebra of $C$ given in [@springer §3.6 p. 61] and [@kps Prop. 4.4 ]. Consider the $k$-supervector space $\mathcal{C}$ with $\mathcal{C}_0=\mathcal{C}_1=C$ (so that $\dim \mathcal{C} = 8|8$). If $x \in C$, denote by $\ell(x)$ the odd endomorphism of $\mathcal{C}$ sending $(a,b)$ to $(xb,\overline{x}a)$, where $x \mapsto \overline{x}:= x - \beta_C(x,e)e$ is the canonical involutive automorphism of $C$ [@blijspringer1]. Then we have $\ell(x) \circ \ell(x) = {\rm q}(x)\, {\rm Id}$, so the $k$-linear map $\ell : C \rightarrow {\rm End}(\mathcal{C})$ extends to a $k$-superalgebra morphism $$\label{isolc}
\ell_C : {\rm C}(C) \rightarrow {\rm End}(\mathcal{C}).$$ This is an isomorphism as the left-hand side is central simple and both $k$-algebras have the same dimension. PS. PS.
\[scholtrivclif\]If $C$ is an octonion $k$-algebra then $\ell_C$ is an isomorphism of ${\mathbb{Z}}/2{\mathbb{Z}}$-graded $k$-algebras. In particular, $C$ has a trivial even Clifford algebra, and $\mathcal{C}$ (resp. each copy of $C$ in it) is in a natural way a spinor module (resp. half-spinor modules) for $C$.
View $\mathcal{C}$ as a nondegenerate quadratic space for the quadratic form ${\rm q} \bot {\rm q}$ (orthogonal sum). Let $v,w \in C$ be two nonisotropic elements and consider their product $\gamma_{v,w}:=v \otimes w \in {\rm GSpin}(C)$. We have for all $a,b \in C$ the formula $$\label{formoctell} \ell_C(\gamma_{v,w})(a,b) = ( v(\overline{w}a), \overline{v}(wb)).$$ In particular, $\ell_C(\gamma_{v,w})$ is a similitude of $\mathcal{C}$ of factor ${\rm q}(v){\rm q}(w)=\nu_C(\gamma_{v,w})$. Recall that the Cartan-Dieudonné theorem [@dieudonne Prop. 8 & Prop. 14] asserts that the nonisotropic elements of $C$ generate $\Gamma(C)$, which implies that the elements of the form $\gamma_{v,w}$ generate ${\rm GSpin}(C)$. PS. PS.
For $i=0,1$, the space $\mathcal{C}_i=C$ viewed as a half-spinor module for $C$ defines a morphism $\rho_{C ; i} : {\rm GSpin}(C) \rightarrow {\rm GSO}(C)$ with similitude factor $\nu_C$.
Define a group homomorphism $$\tau_C : {\rm GSpin}(C) \rightarrow {\rm SO}(C) \times {\rm GSO}(C) \times {\rm GSO}(C)$$ by the formula $\tau_C(\gamma)=(\pi_C(\gamma),\rho_{C; 0}(\gamma),\nu_C(\gamma)^{-1}\rho_{C;1}(\gamma))$ (note the scalar factor $\nu_C(\gamma)^{-1}$ in the last component). Following [@springer], we say that a triple $(t_1,t_2,t_3) \in {\rm SO}(C) \times {\rm GSO}(C) \times {\rm GSO}(C)$ is [*related*]{} if we have $$t_1(xy)=t_2(x)t_3(y)\, \,\, \, \, \, \, \forall x,y \in C.$$As is shown in [@springer Thm. 3.2.1 (i) & (iii)], the subset ${\rm GRT}(C)$ of related triples is a subgroup of ${\rm SO}(C) \times {\rm GSO}(C) \times {\rm GSO}(C)$. The following proposition is a modest variant of [@springer Prop. 3.6.3].PS. PS.
\[groupeGRT\] Let $C$ be an octonion $k$-algebra. The morphism $\tau_C$ induces an isomorphism ${\rm GSpin}(C) {\overset{\sim}{\rightarrow}}{\rm GRT}(C)$. It restricts to an isomorphism between ${\rm Spin}(C)$ and the subgroup of all related triples in ${\rm SO}(C)^3$.
The kernel of $\tau_C$ is trivial as $\ker \ell_C = 0$. By [@springer Thm. 3.2.1] assertion (iii) and formula , we have $\tau_C(\gamma_{v,w}) \in {\rm GRT}(C)$, hence ${\rm Im} \tau_C \subset {\rm GRT}(C)$. The same theorem shows that the first projection ${\rm pr}_1 : {\rm GRT}(C) \rightarrow {\rm SO}(C)$, $(t_1,t_2,t_3) \mapsto t_1$, is surjective, and that $\ker \, {\rm pr}_1$ is the subgroup of elements of the form $(1,m_{\lambda},m_{\lambda^{-1}})$ where $\lambda \in k^\times$ and $m_{\lambda} : x \mapsto \lambda x$. But this kernel is exactly $\tau_C(k^\times)$ and the map ${\rm pr}_1 \circ \tau_C = \pi_C$ is surjective. This proves ${\rm Im}\, \tau_C = {\rm GRT}(C)$ and the last assertion follows.
The automorphism group of an octonion algebra {#g2emb}
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Let $C$ be an octonion $k$-algebra. We denote by ${\rm G}_{\tiny 2}(C)$ the automorphism group of the octonion $k$-algebra $C$. By [@springer Cor. 2.2.5], ${\rm G}_{\tiny 2}(C) \subset {\rm O}(C)$ is actually a subgroup of ${\rm SO}(C)$. We shall denote by $${\rm i}_C : {\rm G}_{\tiny 2}(C) \longrightarrow {\rm SO}(C)$$ this inclusion. Consider the “diagonal” group homomorphism $\iota_C : {\rm G}_{\tiny 2}(C) \rightarrow {\rm SO}(C) \times {\rm SO}( C)$ sending an automorphism $\phi$ of $C$ to $(a,b) \mapsto (\phi(a),\phi(b))$ (an even $k$-linear automorphism of $\mathcal{C}$). The isomorphism allows to define a map $\widetilde{{\rm i}}_C:=\ell_C^{-1} \circ \iota_C : {\rm G}_2(C) \rightarrow {\rm C}(C)_0^\times.$
\[iclift\] The map $\widetilde{{\rm i}}_C$ is a group homomorphism ${\rm G}_2(C) \rightarrow {\rm Spin}(C)$ such that $\pi_C \circ \widetilde{{\rm i}}_C = {\rm i}_C$.
Observe that for $x \in C$ and $\phi \in {\rm G}_{\tiny 2}(C)$, we have the relation $\iota_C(\phi) \ell_C(x) \iota_C(\phi)^{-1} = \ell_C(\phi(x))$. It follows that $\widetilde{{\rm i}}_C(\phi):=\ell_C^{-1} \circ \iota_C (\phi)$ is an element of ${\rm GSpin}(C)$ which satisfies $\pi_C(\widetilde{{\rm i}}_C(\phi))={\rm i}_C(\phi)$. We have $\nu_C(\widetilde{{\rm i}}_C(\phi))=1$ as the Chevalley anti-involution of ${\rm C}(C)$ corresponds via $\ell_C$ to the adjonction on ${\rm End}(C \oplus C)$ with respect to $\beta_{C \oplus C}$ by [@kps Prop. 4.4].
The subspace $P \subset C$ of [*pure octonions*]{} is defined as the orthogonal of the unit element $e$ for the bilinear form $\beta_C$. As ${\rm q}(e)=1 \neq 0$, we are in the context of Lemma \[lemmeregularhyp\]. In particular, $P$ is a regular quadratic subspace of $C$ of dimension $7$ and we have natural isomorphisms $a : {\rm SO}(C)_e {\overset{\sim}{\rightarrow}}{\rm SO}(P)$ and $b : {\rm GSpin}(P) {\overset{\sim}{\rightarrow}}{\rm GSpin}(C)_e$ satisfying $\pi_C \circ b = a^{-1} \circ \pi_P$. As ${\rm G}_2(C)$ preserves $e$ by definition, we get natural injective morphisms $${\rm j}_C:=a \circ {\rm i}_C : {\rm G}_2(C) \longrightarrow {\rm SO}(P) \, \, \, {\rm and}\, \, \, \widetilde{{\rm j}}_C= b^{-1} \circ \widetilde{{\rm i}}_C : {\rm G}_2(C) \longrightarrow {\rm Spin}(P)$$ satisfying $\pi_P \circ \widetilde{{\rm j}}_C = {\rm j}_C$. By Lemma \[lemmeregularhyp\] (iii), a half-spinor module for $C$ restricts via $b$ to a spinor module for $P$, and we shall denote by $$\rho_C := \rho_{C ; 0} \circ b : {\rm GSpin}(P) \longrightarrow {\rm GSO}(C)$$ the (canonical) associated spin representation of ${\rm GSpin}(P)$. PS. PS.
\[propgspin7\] Let $C$ be an octonion algebra over $k$, $P \subset C$ the $7$-dimensional quadratic space of pure octonions, $\widetilde{{\rm j}}_C : {\rm G}_2(C) \longrightarrow {\rm Spin}(P)$ be the canonical morphism, and $\rho_C : {\rm GSpin}(P) \longrightarrow {\rm GSO}(C)$ the canonical spin representation of ${\rm GSpin}(P)$. Then:
- $\rho_C \circ \widetilde{{\rm j}}_C$ coincides with the canonical morphism ${\rm i}_C : {\rm G}_2(C) \rightarrow {\rm SO}(C)$; PS. PS.
- the action of ${\rm GSpin}(P)$ on the set ${\rm Q}=\{v \in C, {\rm q}(v) \neq 0\}$, defined by $(\gamma,v) \mapsto \rho_C(\gamma)v$, is transitive;PS. PS.
- the stabilizer of the unit element $e \in {\rm Q}$ in ${\rm GSpin}(P)$ is ${\rm i}_C({\rm G}_2(C))$, and the stabilizer of $ke \subset C$ in ${\rm GSpin}(P)$ is $k^\times \times {\rm i}_C({\rm G}_2(C))$. PS. PS.
Assertion (a) is obvious from the definitions. To show (b), observe first that the action of the statement is well-defined as we have $\rho_C({\rm GSpin}(P)) \subset {\rm GSO}(C)$. In [@springer Lemma 3.4.2], the authors show that for any $v \in C$ such that ${\rm q}(v) \neq 0$, there is a related triple $(t_1,t_2,t_3) \in {\rm GRT}(C)$ such that $t_1 \in {\rm SO}(C)_e$ and $t_2(e)=v$. This is exactly assertion (b), by Proposition \[groupeGRT\]. Moreover, they also show in Proposition 3.4.1 [*loc. cit.*]{} that for an element $(t_1,t_2,t_3)$ in ${\rm GRT}(C)$ such that $t_1 \in {\rm SO}(C)_e$, we have $t_2(e) \subset k e$ if, and only if, we have $t_1 \in {\rm G}_2(C)$. It follows that the stabilizer of $e \in {\rm Q}$ in ${\rm GSpin}(P)$ is included in $k^\times \cdot \widetilde{{\rm j}}_C({\rm G}_2(C))$. But this stabilizer contains $\widetilde{{\rm j}}_C({\rm G}_2(C))$ by (a), and intersects $k^\times$ trivially by construction, which proves (c).
\[examplecayley\][Assume $k={\mathbb{R}}$ and $C$ is the Euclidean (Cayley-Graves’) octonion ${\mathbb{R}}$-algebra. Then ${\rm Spin}(P)$ is isomorphic to the compact Lie group ${\rm Spin}(7)$, the map ${\rm GSpin}(P) \rightarrow {\rm Spin}(P)$ is surjective, and ${\rm SO}(C)$ is isomorphic to ${\rm SO}(8)$. The spin representation ${\rm Spin}(P) \rightarrow {\rm SO}(C)$ defines a group homomorphism ${\rm Spin}(7)\rightarrow {\rm SO}(8)$. The proposition asserts that ${\rm Spin}(P)$ acts transitively on the Euclidean unit sphere $S$ ($ \simeq \mathbb{S}^7$) of $C$, and the stabilizers of this action form a conjugacy class of compact connected subgroups of ${\rm Spin}(P)$ of type ${\bf G}_2$. See [@adams Thm. 5.5] for another proof of this classical result. Let us add that any compact connected subgroup $H \subset {\rm Spin}(P)$ which is of type ${\rm G}_2$ fixes a point in $S$ (hence belongs to the aforementioned conjugacy class). Indeed, there is up to isomorphism a unique nontrivial irreducible complex representations of $H$ of dimension $\leq 8$, which is the $7$-dimensional representation defined by $\pi_P$ on $P \otimes {\mathbb{C}}$ (and thus defined over ${\mathbb{R}}$), so the natural representation of $H$ on $C$ contains the trivial representation. In the next paragraph, we prove more a general result along those lines, which applies to an arbitrary field $k$. ]{}
${\rm G}_2$-subgroups of ${\rm SO}(P)$ and ${\rm Spin}(P)$ with $\dim P=7$ {#defg2sb}
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Let $P$ be a regular quadratic space of dimension $7$ over a field $k$ and let $\Gamma$ be a subgroup of ${\rm SO}(P)$ [(]{}resp. ${\rm Spin}(P)$[)]{}. We say that $\Gamma$ is a ${\rm G_2}$-subgroup if there exists an octonion $k$-algebra $C$ and an isometric embedding of $P \hookrightarrow C$ onto the space of pure octonions of $C$, such that we have $\Gamma = {\rm j}_C({\rm G}_2(C))$ [(]{}resp. $\Gamma= \widetilde{{\rm j}}_C({\rm G_2}(C))$[)]{}.
Note that there may be no such subgroup in general, [*i.e.*]{} no octonion $k$-algebra with pure subspace isometric to $P$. They do exists, for instance, if $P$ is the orthogonal of an element of norm $1$ in ${\rm H}(k^4)$ (hence if $k$ is algebraically closed), as ${\rm H}(k^4)$ may be endowed with a structure of split octonion $k$-algebra. In other words, ${\rm SO}_7(k)$ and ${\rm Spin}_7(k)$ have ${\rm G}_2$-subgroups. PS. PS.
\[carG2sbgp\] Let $P$ be a regular quadratic space of dimension $7$ over $k$, the set of ${\rm G}_2$-subgroups of ${\rm SO}(P)$ [(]{}resp. ${\rm Spin}(P)$[)]{} form a single ${\rm SO}(P)$-conjugacy class [(]{}resp. ${\rm GSpin}(P)$-conjugacy class[)]{}.
For $i=1,2$, consider an octonion $k$-algebra $C_i$, with neutral element $e_i$, and an isometry $\mu_i : P {\overset{\sim}{\rightarrow}}e_i^\bot$. We show first that $C_i$ are isomorphic as octonion $k$-algebras. By [@blijspringer1 Assertion (2.3)], it is enough to show that they are isometric as quadratic spaces. This is obviously true if ${\rm char}\, k \,\neq 2$ as we have $C_i \simeq P \bot \langle 1\rangle $. If ${\rm char}\, k \, = 2$, and if $H$ is a nondegenerate hyperplane of $P$, then $C_i$ is isometric to the orthogonal sum of $H$ and of a $2$-dimensional nondegenerate quadratic space over $k$ which represents $1$. We conclude from the triviality of the Arf invariant of $C_i$, which in turn follows from the isomorphism . PS. PS. Unravelling the definitions, the ${\rm SO}(P)$ case of the statement is now obvious. In the ${\rm Spin}(P)$ case, it remains to show the following fact: let $\alpha \in {\rm SO}(P)$ and ${\rm C}(\alpha)$ be the associated even automorphism of ${\rm C}(P)$ defined by functoriality of the Clifford algebra construction, then ${\rm C}(\alpha)$ is an inner automorphism defined by some element in ${\rm GSpin}(P)$. But this follows from the Skolem-Noether theorem as: ${\rm C}(P)_0$ is a matrix $k$-algebra (Scholium \[scholtrivclif\]), ${\rm C}(\alpha)$ acts trivially on the center $Z$ of ${\rm C}(P)$ (as $\alpha$ belongs to ${\rm SO}(P)$), and the $k$-algebra ${\rm C}(P)$ is generated by ${\rm C}(P)_0$ and $Z$.
\[algcarG2\] Let $P$ be a regular quadratic space of rank $7$ over an algebraically closed field $k$, and let $\Gamma \subset {\rm SO}(P)$ [(]{}resp. $\Gamma \subset {\rm Spin}(P)$[)]{} be a subgroup. The following properties are equivalent:
- $\Gamma$ is a ${\rm G}_2$-subgroup,PS. PS.
- $\Gamma$ is a closed connected subgroup, which is a simple linear algebraic group of type ${\bf G}_2$.PS. PS.
[*Proof.* ]{} If $C$ is an octonion $k$-algebra, its automorphism group ${\rm G}_2(C)$ has a unique structure of linear algebraic group such that the morphism ${\rm i}_C : {\rm G}_2(C) \rightarrow {\rm SO}(C)$ is a closed immersion. By [@springer Thm. 2.3.5], it is connected, simple, of type ${\rm G}_2$ and the morphism ${\rm j}_C : {\rm G}_2(C) \rightarrow {\rm SO}(P)$ is a closed immersion as well. The morphism $\widetilde{{\rm j}}_C : {\rm G}_2(C) \rightarrow {\rm Spin}(P)$ defined above, which is a morphism of algebraic groups, satisfies $\pi_P \circ \widetilde{{\rm j}}_C = {\rm j}_C$, hence is a closed immersion as well. This proves (i) $\Rightarrow$ (ii).
Before proving (ii) $\Rightarrow$ (i) let us recall some properties of the (finite dimensional, algebraic) linear representations of reductive groups over a field $k$ of arbitrary characteristic, following [@humphreys1 Ch. XI] and [@humphreys2 Chap. 2 & 3] (see also [@rags]). Let $G$ be a semisimple algebraic group over $k$. If $\lambda$ is a dominant weight of $G$, the category of linear representations of $G$ equipped with an extremal vector of weight $\lambda$ has an initial object denoted $V(\lambda)$ and called the Weyl module of $\lambda$. The Weyl-module $V(\lambda)$ has a unique irreducible quotient $L(\lambda)$, and any irreducible representation of $G$ is isomorphic to a $V(\lambda)$ for a unique dominant weight $\lambda$. If $k$ has characteristic $0$, we have $V(\lambda)=L(\lambda)$, but not in general. Nevertheless, the dimension of $V(\lambda)$ is independent of the characteristic of $k$, and given by the classical “Weyl dimension formula”.
The following lemma is well-known in characteristic $0$, by lack of a reference we provide a proof in arbitrary characteristic.
\[lemmerepg2\] Let $k$ be an algebraically closed field and $G$ a connected linear algebraic group over $k$ of which is simple of type ${\bf G}_2$. Let $\omega_1$ [(]{}resp. $\omega_2$[)]{} be the fundamental weight of $G$ which is a short (resp. long) root. Let $V$ be an irreducible, nontrivial, linear representation of $G$ of dimension $\leq 11$. Then we have:
- [(]{}${\rm char}\, k \neq 2,3$[)]{} $\dim V=7$ and $V \simeq V(\omega_1) = L(\omega_1)$,
- [(]{}${\rm char}\, k = 3$[)]{} $\dim V=7$ and we have either $V \simeq V(\omega_1)=L(\omega_1)$ or $V \simeq L(\omega_2)$, and in the latter case $G\, \rightarrow {\rm GL}(V)$ is not a closed immersion.
- [(]{}${\rm char}\, k = 2$[)]{} $\dim V=6$ and $V \simeq L(\omega_1)$.
Moreover there is up to a scalar a unique quadratic form on the $7$-dimensional vector space $V(\omega_1)$ which is preserved by $G$ and such that the resulting quadratic space over $k$ is regular.
[*Proof.* ]{} Denote by $\alpha_1$ (resp. $\alpha_2$) the simple short (resp. long) root of $G$ and by $W$ its Weyl group. We have $\langle \omega_i, \alpha_j^\vee \rangle = \delta_{i,j}$, $\omega_1 = 2\alpha_1 + \alpha_2$ and $\omega_2 = 3 \alpha_1 +2 \alpha_2$. Let $X$ be the set of nonzero dominant weights $\mu$ of $G$ such that $|W \cdot \mu|<|W|=12$. Observe that we have $X= \coprod_{i=1,2} {\mathbb{Z}}_{>0} \omega_i$ and $|W \cdot \mu| =6$ for each $\mu \in X$. If $\mu$ is a weight of $V$, then so is $w \cdot \mu$ for all $w \in W$; as a consequence, if $\lambda$ denotes the dominant weight of $G$ such that $V \simeq V(\lambda)$, then the dominant weights of $V$ are in $\{\lambda,0\}$. If $k$ has characteristic zero, then as a general fact, a dominant weight $\mu$ occurs in $V(\lambda)$ if, and only if, $\mu \prec \lambda$, i.e. $\lambda-\mu$ is a sum of positive roots. As $0 \prec \omega_1 \prec \omega_2$, this implies $\lambda = \omega_1$. We have $\dim V(\omega_1)=7$.
Assume from now on that $k$ has prime characteristic $p$. The analysis above and Steinberg tensor product theorem show that $\lambda$ is $p$-restricted (as $\dim V(\lambda) < 6^2 =36$): we have $\langle \lambda, \alpha^\vee \rangle <p$ for each simple root $\alpha$. But then [@rags Chap. 2, Prop. 2.11] shows that $\lambda - n \alpha$ is a weight of $V(\lambda)$ for each simple root $\alpha$ and each integer $n$ such that $0 \leq n \leq \langle \lambda, \alpha^\vee \rangle$. The relations $2 \omega_1 = \omega_2 + \alpha_1$ and $2 \omega_2 = 3 \omega_1 + \alpha_2$ implies thus $\lambda = \omega_i$ with $i=1,2$. By [@springerweyl Cor. 4.3 & Table 1], we have $V(\omega_i)=L(\omega_i)$ if $p>3$. As $\dim V(\omega_2)=12$ (in fact, $V(\omega_2)$ is the adjoint representation of $G$) this excludes the case $i=2$ if ${\rm char} \,p\, >3$, hence prove assertion (i).
If $p=3$, the same reference shows $L(\omega_1)=V(\omega_1)$, $\dim L(\omega_2)=7$, that $V(\omega_2)$ is an extension of $L(\omega_2)$ by $L(\omega_1)$, and that $L(\omega_2)$ is isomorphic to the restriction of $L(\omega_1)$ via the inseparable isogeny $G \rightarrow G$. If $p=2$, then $V(\omega_2)$ (which is the adjoint representation of $G$) is actually irreducible, hence we must have $V \simeq L(\omega_1)$. This shows (ii) and (iii).
We now prove the last assertion. View $G$ as a ${\rm G}_2$-subgroup of ${\rm SO}(P)$, with $P$ regular of dimension $7$ over $k$. We have $P \simeq V(\omega_1)$ by [@springer §2.3]; these authors prove in particular that $P$ is irreducible, except in characteristic $2$ in which case the only invariant subspace is the kernel of $\beta_P$: see Thm. 2.3.3 [*loc. cit.*]{} The last statement follows in a standard way from this irreducibility property if ${\rm char} \,k\, \neq 2$.
Assume ${\rm char} \,k\, =\, 2$, set $P=V(\lambda_1)$, and let $q_1,q_2 : P \rightarrow k$ be two $G$-invariant quadratic forms on $P$ which both gives the vector space $P$ a structure of regular quadratic space, say $P_1$ and $P_2$. Then the unique $G$-invariant line $L$ in $P$ must be ${\rm ker} \,\beta_{P_1} \,=\, {\rm ker} \beta_{P_2}$. Replacing $q_2$ by some scalar multiple if necessary we may assume that $q_1$ and $q_2$ coincide on this line. Then $q_1-q_2$ factors through a $G$-invariant quadratic form on $P/L$. As $P/L$ is an irreducible $G$-module, the kernel of $\beta_{P_1}-\beta_{P_2}$ is either $L$ or $P$. Assume it is $L$. Then $P/L$ is a nondegenerate quadratic space for $q_1-q_2$. As $G$ is simply connected, we thus have an injective algebraic group morphism $G \rightarrow {\rm Spin}(P/L) \simeq {\rm Spin}_6(k)$. But the halfspinor modules of ${\rm H}(k^3)$ have dimension $4 <6$, hence are trivial as $G$-modules by (iii), a contradiction. It follows that $\beta_{P_1}=\beta_{P_2}$, i.e. $q_1-q_2$ is a Frobenius semilinear form. As $P$ has no $G$-invariant subspace of dimension $6$, and as $k$ is perfect, this must be the $0$ form, and we are done. $\Box$
We now prove $(ii) \Rightarrow (i)$ of Proposition \[algcarG2\]. Let $G$ be a simple algebraic group of type ${\bf G}_2$. Let $\rho : G \rightarrow {\rm SO}(P)$ be algebraic group homomorphism that is furthermore a closed immersion. If we can prove that the representation $P$ of $G$ is isomorphic to $V(\omega_1)$, then the last assertion of Lemma \[lemmerepg2\] shows that $\rho$ is unique up to conjugation, hence that $\rho(G)$ is a ${\rm G}_2$-subgroup by Proposition \[carG2sbgp\]. By Lemma \[lemmerepg2\] (i) and (ii), we are done if ${\rm char} \, k \neq 2$. If ${\rm char}\, k \, =2$, then Lemma \[lemmerepg2\] (iii) shows that $\omega_1$ is a highest weight of $P$, hence we have a nonzero morphism $f: V(\omega_1) \rightarrow P$. If $f$ is not an isomorphism, then $P$ has a subrepresentation $P'$ isomorphic to $L(\omega_1)$, and $P = P' \bot \ker \beta_P$. The space $P'$ is thus nondegenerate, but this is absurd as by last paragraph of the proof of Lemma \[lemmerepg2\], there is no nontrivial algebraic group homomorphism $G \rightarrow {\rm SO}_6(k)$.
Assume now that $G$ is a closed subgroup of ${\rm Spin}(P)$. Consider the central isogeny $\pi_P : {\rm Spin}(P) \rightarrow {\rm SO}(P)$. Then $G'=\pi_P(G)$ is a ${\rm G}_2$-subgroup by the previous paragraph. We have seen that the morphism $\pi_P : {\rm Spin}(P) \rightarrow {\rm SO}(P)$ admits a section $j$ over such a subgroup $G'$, so that we have a direct product $\pi_P^{-1}(G') = \mu_2(k) \times j(G')$. But we have an inclusion $G \subset \pi_P^{-1}(G')$, and the projection of $G$ on the finite $\mu_2(k)$ factor is necessarily trivial as $G$ is simple, so we have $G \subset j(G')$ and then $G=j(G')$. $\Box$
Polynomials of type ${\bf G}_2$
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\[defpg2\] Let $k$ be a field and $P(t) \in k[t]$. We say that the polynomial $P$ is [*of type $G_2$*]{} if there are elements $x,y$ in an algebraic closure of $k$ such that $$P(t) = (t-1)(t-x)(t-y)(t-xy)(t-x^{-1})(t-y^{-1})(t-x^{-1}y^{-1}).$$ In particular, $P$ is a monic polynomial and $t^7 P(1/t) = -P(t)$.
\[typeg2ss\] Let $P$ be a regular quadratic space of dimension $7$ over $k$ and $\gamma \in {\rm SO}(P)$. If $\gamma$ belongs to a ${\rm G}_2$-subgroup of ${\rm SO}(P)$, then the characteristic polynomial of $\gamma$ is of type ${\rm G}_2$. The converse holds if $k$ is algebraically closed and $\gamma$ is semisimple.
We may assume that $k$ is algebraically closed and, by Jordan decomposition, that $\gamma$ is semisimple. In this case, we have recalled in the proof of Proposition \[algcarG2\] that the linear representation $P$ of a ${\rm G}_2$-subgroup $G$ of ${\rm SO}(P)$ is isomorphic to the Weyl module $V(\omega_1)$. In particular, its weights are $0$ and the six short roots of $G$, which are $\pm \alpha_1, \pm (\alpha_2+\alpha_1), \pm (\alpha_2+2\alpha_1)$ in the notations of the proof [*loc. cit*]{}, which shows the first assertion. The last assertion is then an easy consequence of the fact that the conjugacy class of $\gamma$ in ${\rm SO}(P)$ is uniquely determined by $\det(t-\gamma)$ (a simple special case of Corollary \[corwindex\]).
\[rempotg2\][Let $P \in k[t]$ be a monic polynomial such that $t^7P(1/t)=-P(t)$. There is a unique monic polynomial $Q \in k[t]$ of degree $3$ such that $t^{-3}P(t) = (t-1)Q(t+t^{-1})$. Write $Q = t^3 - at^2 + bt - c$ with $a,b,c \in k$ (some simple universal integral polynomials in the coefficients of $P$). It is not difficult to check, [*e.g.*]{} using the equivalence between (i) and (iii) in Lemma \[calcpcar\] below, that $P$ is of type ${\bf G}_2$ if, and only if, we have the relation $a^2 = 2b+c+4$. ]{}
Proof of Theorems \[thmintroa\] & \[thmintrob\] {#pfspin7}
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The following lemma is well-known (at least in characteristic $0$).
\[pcarspin\] Let $k$ be an algebraically closed field, $V$ a regular quadratic space of odd dimension $2r+1$ over $k$, $(W,\rho)$ a spin representation of ${\rm GSpin}(V)$ and $\pi_V : {\rm GSpin}(V) \rightarrow {\rm SO}(V)$ the canonical map. For any element $\gamma \in {\rm GSpin}(V)$, there are elements $x_0, x_1,\dots,x_r \in k^\times$ such that:
- $\det(t - \rho(\gamma)) = \prod_{(\epsilon_i) \in \{-1, 1\}^r} (t - x_0 \prod_{i=1}^r x_i^{\epsilon_i})$,PS. PS.
- $\det(t - \pi_V(\gamma)) = (t-1) \prod_{i=1}^r (t-x_i^2)(t-x_i^{-2})$, PS. PS.
- $\nu_V(\gamma) = x_0^2$.
Both ${\rm GSpin}(V)$ and ${\rm SO}(V)$ are connected linear algebraic $k$-groups in a natural way [@springer p. 40], and both $\pi_V$ and $\rho$ are $k$-group homomorphisms. By considering a Jordan decomposition of $\gamma$, we may assume that $\gamma$ is semisimple, in which case so is $\pi_V(\gamma)$. We may thus assume that $V = {\rm H}(I) \bot k e$ and that $\pi_V(\gamma)$ preserves $I$, $I^\ast$ and fixes $e$. Choose $x_1,\dots,x_r \in k^\times$ whose squares are the eigenvalues of $\pi_V(\gamma)$ on $I$, then $\det(t - \pi_V(\gamma))$ is as in assertion (ii). The last exact sequence of §\[parclag\] shows that we may write $\gamma= \rho_{{\rm H}(I);ke}(\gamma' \times 1)$ with $\gamma' \in {\rm GSpin}({\rm H}(I))$, and Proposition \[decospin\] implies that $\det(t - \rho(\gamma))$ is the characteristic polynomial of $\gamma'$ in a spinor module for ${\rm H}(I)$. Proposition \[cliffordhyp\] shows that there is a unique element $\lambda \in k^\times$ such that we have $\gamma' = \lambda \cdot \widetilde{\rho}_I(\pi_{{\rm H}(I)}(\gamma'))$ and $\det(t-\rho(\gamma))=\det(t-\lambda \,\pi_V(\gamma) \, | \, \Lambda\, I)$. This concludes the proof with $x_0 = \lambda x_1\cdots x_r$.
Polynomials [*of type ${\bf G}_2$*]{} have been introduced in Definition \[defpg2\]. PS. PS.
\[calcpcar\] Let $k$ be a field, $V$ a regular quadratic space of dimension $7$ with trivial even Clifford algebra, $(W,\rho)$ a spin representation of ${\rm GSpin}(V)$, $\pi_V : {\rm GSpin}(V) \rightarrow {\rm SO}(V)$ the natural map, and $\gamma \in {\rm GSpin}(V)$. The following properties are equivalent: PS.
- the characteristic polynomial of $\pi_V(\gamma)$ is of type ${\bf G}_2$, PS. PS.
- $\rho(\gamma)$ has an eigenvalue whose square is $\nu_V(\gamma)$, PS. PS.
- $\nu_V(\gamma)$ is an eigenvalue of $\rho(\gamma^2)$, PS. PS.
- $\det(t - \nu_V(\gamma)^{-1} \rho(\gamma^2)) = (t-1) \det (t- \pi_V(\gamma^2))$. PS. PS.
Moreover, if $\rho(\gamma) \in {\mathrm{GL}}(W)$ has an eigenvalue $\lambda$ such that $\lambda^2 = \nu_V(\gamma)$, then we have $\det(t - \lambda^{-1} \rho(\gamma)) = (t-1) \det (t- \pi_V(\gamma))$.
Note that the equivalence (ii) $\Leftrightarrow$ (iii) and the implication (iv) $\Rightarrow$ (iii) are obvious. Let $K$ be an algebraic closure of $k$. By Lemma \[pcarspin\] applied to $\gamma$ viewed as an element of ${\rm GSpin}(V \otimes_k K)$, we may fix $x_0,x_1,x_2,x_3$ in $K^\times$ as in that lemma. PS. PS.
Assume assertion (i) of Lemma \[calcpcar\] holds. Up to replacing some of the $x_i$, with $i\geq 1$, by $x_i^{-1}$ if necessary, we may assume that we have $x_3^2 = x_2^2 x_1^2$. In particular, if we set $\epsilon = x_1x_2x_3^{-1}$, then $\epsilon=\pm 1$ and $\epsilon x_0$ is an eigenvalue of $\rho(\gamma)$, hence assertion (ii) above holds.PS. PS. Conversely, assume that $\epsilon \in \{\pm 1\}$ and that $\epsilon x_0$ is an eigenvalue of $\rho(\gamma)$. Again, up to replacing some of the $x_i$, with $i\geq 1$, by $x_i^{-1}$ if necessary, we may assume that we have $x_1x_2x_3^{-1}=\epsilon$. In particular, we have $x_3^2 = x_2^2x_1^2$ and assertion (i) holds. This shows (ii) $\Rightarrow$ (i). Moreover, the relation $\epsilon x_1x_2x_3^{-1} = 1$ implies $\epsilon x_1x_2x_3 = x_1^2x_2^2$, $\epsilon x_1x_2^{-1}x_3 = x_1^2$ and $\epsilon x_1x_2^{-1}x_3^{-1} = x_2^2$, hence the equality $$\det(t - x_0^{-1} \epsilon \rho(\gamma)) = (t-1) \det (t- \pi_V(\gamma)),$$ holds, which proves the last statement. This statement applied to $\gamma^2$ shows in turn (iii) $\Rightarrow$ (iv), which concludes the proof.
Let $V$ be a regular quadratic space of dimension $7$ over $k$ with trivial even Clifford algebra and $(W,\rho)$ be a spin representation of ${\rm Spin}(V)$. Let $\Gamma \subset {\rm Spin}(V)$ be a subgroup. Assume that: PS. PS.
- $W$ is semisimple as a $k[\Gamma]$-module, PS. PS.
- for each $\gamma \in \Gamma$, the element $\rho(\gamma)$ has the eigenvalue $1$, PS. PS.
then there is an element $w \in W-\{0\}$ such that $\rho(\gamma) w = w$ for all $\gamma \in W$.
Consider $E_1=W$ and $E_2=V \oplus k$ as $k$-linear representations of $\Gamma$. Assumption (ii) and the last assertion of Lemma \[pcarspin\] show that for each $\gamma \in \Gamma$, we have $\det ( t - \gamma\, |\, E_1) = \det (t - \gamma \,|\, E_2)$. The Brauer-Nesbitt theorem[^7] implies thus that $E_1$ and $E_2$ have isomorphic semisimplification. The theorem follows as $E_1$ is semisimple by assumption, and $E_2$ contains the trivial representation.
\[thmbsecondev\] Let $C$ be an octonion $k$-algebra, $P \subset C$ the quadratic space of pure octonions of $C$, $Q \subset C$ the subset of element $v \in C$ such that ${\rm q}(v) \neq 0$, and $\widetilde{{\rm j}}_C: {\rm G}_2(C) \rightarrow {\rm Spin}(P)$ the canonical morphism. Recall that the quadratic space $C$ itself may be endowed with a canonical structure of a spinor module for $P$ and denote by $\rho_C : {\rm GSpin}(P) \rightarrow {\rm GSO}(C)$ the associated spin representation. PS. PS.
Let $\Gamma \subset {\rm Spin}(P)$ be a subgroup such that $\rho_C(\Gamma)$ acts in a semisimple way on the $k$-vector space $C$. The following assertions are equivalent: PS. PS.
- for each $\gamma \in \Gamma$, there is $g \in {\rm GSpin}(P)$ such that $g \gamma g^{-1} \in \widetilde{{\rm j}}_C( {\rm G}_2(C) )$, PS. PS.
- for each $\gamma \in \Gamma$, there is $s \in Q$ such that $\rho_C(\gamma)s = s$, PS. PS.
- for each $\gamma \in \Gamma$, the element $\rho_C(\gamma) \in {\mathrm{GL}}(C)$ has the eigenvalue $1$, PS. PS.
- there is an element $g \in {\rm GSpin}(P)$ such that $g\, \Gamma\, g^{-1} \subset \widetilde{{\rm j}}_C( {\rm G}_2(C) )$, PS. PS.
- there is an element $s \in Q$ such that $\rho_C(\gamma)s=s$ for all $\gamma \in \Gamma$.
The equivalences $(a) \Leftrightarrow (b)$ and $(d) \Leftrightarrow (e)$ follow from Proposition \[propgspin7\], and the implications $(e) \Rightarrow (b) \Rightarrow (c)$ are obvious. Assume that assertion (c) holds. The previous theorem ensures that the trivial representation of $\Gamma$ occurs in the semisimple spinor module $C$. To conclude the proof it suffices to show that for any group $G$ and any nondegenerate quadratic space $U$, if we have a group homomorphism $G \rightarrow {\rm O}(U)$ whose underlying $k$-linear representation $U$ is semisimple and contains the trivial representation, then there exists a $G$-invariant vector in $U$ which is nonisotropic. But as $U$ is semisimple, we have $U \simeq U^G \oplus W$, where $-^G$ denotes $G$-invariants and $W$ is some $G$-stable subspace of $V$. As the trivial representation is selfdual and $W$ is semisimple, we have $(W^\ast)^G = W^G = 0$. It follows that $U^G$ is a nondegenerate subspace of $U$. We conclude as any nonzero nondegenerate quadratic space obviously contains some nonisotropic vector.
Given the definition of ${\rm G}_2$-subgroups in §\[defg2sb\], we get the following:
\[corthmbsecondev\] Let $P$ be a regular quadratic space of dimension $7$ over a field $k$ such that ${\rm Spin}(P)$ possesses ${\rm G}_2$-subgroups. Let $\Gamma \subset {\rm Spin}(P)$ be a subgroup acting in a semisimple way on a spinor module $W$ for $P$. The following assertions are equivalent: PS. PS.
- each element of $\Gamma$ belongs to some ${\rm G}_2$-subgroup of ${\rm Spin}(P)$,PS. PS.
- each element of $\Gamma$ has the eigenvalue $1$ in $W$, PS. PS.
- there is a ${\rm G}_2$-subgroup of ${\rm Spin}(P)$ containing $\Gamma$. PS. PS.
\[corcar2\] Let $k$ be a perfect field of characteristic $2$ and $P$ a regular quadratic space of dimension $7$ over $k$ such that ${\rm SO}(P)$ possesses ${\rm G}_2$-subgroups. Let $\Gamma \subset {\rm SO}(P)$ whose inverse image in ${\rm Spin}(P)$ acts in a semisimple way on a spinor module for $P$. The following assertions are equivalent: PS. PS.
- each element of $\Gamma$ belongs to some ${\rm G}_2$-subgroup of ${\rm SO}(P)$,PS. PS.
- for each $\gamma \in \Gamma$, then $\det(t-\pi_P(\gamma))$ is of type ${\bf G}_2$, PS. PS.
- there is a ${\rm G}_2$-subgroup of ${\rm SO}(P)$ containing $\Gamma$. PS. PS.
The map $\pi_P : {\rm Spin}(P) \rightarrow {\rm SO}(P)$ is surjective by the exact sequence . Moreover, an element $\gamma$ of ${\rm Spin}(P)$ possesses the eigenvalue $1=\pm 1$ in a spinor module for $P$ if, and only if, $\det(t-\pi_P(\gamma))$ is of type ${\bf G}_2$, by Lemma \[calcpcar\]. The corollary follows thus from Corollary \[corthmbsecondev\] applied to $\pi_P^{-1}(\Gamma)$.
\[corconnected\] Let $k$ be an algebraically closed field, $P$ be a regular quadratic space over $k$, and $\Gamma \subset {\rm SO}(P)$ a subgroup whose inverse image in ${\rm Spin}(P)$ acts in a semisimple way on a spinor module for $P$. Assume that:
- the characteristic polynomial of each element of $\Gamma$ is of type ${\bf G}_2$, PS. PS.
- $\Gamma$ is a Zariski closed and connected subgroup of ${\rm SO}(P)$. PS. PS.
Then there is a ${\rm G}_2$-subgroup of ${\rm SO}(P)$ containing $\Gamma$.
Let $G$ denote the connected component of the identity of the inverse image of $\Gamma$ in ${\rm Spin}(P)$. This is a connected linear algebraic subgroup of ${\rm Spin}(P)$ such that $\pi_P(G)=\Gamma$, as $\Gamma$ is irreducible as a variety. It is enough to show that $G$ is contained in a ${\rm G}_2$-subgroup of ${\rm Spin}(P)$. By Lemma \[calcpcar\], for each element $g \in G$, the element $g^2$ has the eigenvalue $1$, acting on a spinor module $W$ for $P$. But as $G$ is connected, any semisimple element of $G$ is a square, as tori are divisible as groups. As $G$ has a Zariski-dense subset consisting of semisimple elements, it follows that $\det(1-g | W)=0$ for all $g \in G$. We conclude by Corollary \[corthmbsecondev\].
In the following corollary, we denote by $-1 \in {\rm Spin}(E)$ the generator of the natural central subgroup $\mu_2(k)$. PS.
\[carbetaspin\] Let $k$ be an algebraically closed field, $P$ be a regular quadratic space over $k$, and $\Gamma \subset {\rm SO}(P)$ a subgroup whose inverse image $\widetilde{\Gamma}$ in ${\rm Spin}(P)$ acts in a semisimple way on a spinor module $W$ for $P$. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(P)$ if, and only if, there is a group homomorphism $\beta : \widetilde{\Gamma} \rightarrow k^\times$ which occurs in the restriction of $W$ to $\widetilde{\Gamma}$ and satisfies $\beta^2=1$.
Under the assumption on $k$, there exists ${\rm G}_2$-subgroups and we have $\pi_P({\rm Spin}(P))={\rm SO}(P)$. We already explained in the last paragraph of the proof of Proposition \[algcarG2\] that the inverse image in ${\rm Spin}(P)$ of a ${\rm G}_2$-subgroup of ${\rm SO}(P)$ has the form $\mu_2(k) \times H$ where $H$ is a ${\rm G}_2$-subgroup of ${\rm Spin}(P)$. Note that the generator $-1$ of $\mu_2(k)$ acts by $-{\rm id}$ on $W$. If $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(P)$, the existence of the character $\beta$ as in the statement follows [*e.g.*]{} from Proposition \[propgspin7\]. Conversely, if there is a character $\beta$ as in the statement, and if $\Gamma^1 \subset \widetilde{\Gamma}$ denotes the kernel of $\beta$, then we have $\widetilde{\Gamma} = \mu_2(k) \times \Gamma^1$ and Corollary \[corthmbsecondev\] applied to $\Gamma^1$ concludes the proof.
\[examplethmAR\] [Let $k={\mathbb{R}}$ as in Example \[examplecayley\]. Then any subgroup $\Gamma \subset {\rm Spin}(7)$ acts in a semisimple way on the Euclidean space $C$ (see e.g. the proof of Proposition \[critasss\] (i) below). Corollary \[corthmbsecondev\] thus asserts that if an arbitrary subgroup $\Gamma \subset {\rm Spin}(7)$ has the property that each element $\gamma \in \Gamma$ fixes a point of the unit sphere ${\mathbb S}^7$ (or equivalently, belongs to some ${\rm G}_2$-subgroup of ${\rm Spin}(7)$), then $\Gamma$ itself has a fixed point in this sphere (or equivalently, is included in some ${\rm G}_2$-subgroup of ${\rm Spin}(7)$). Moreover, if $\Gamma' \subset {\rm SO}(7)$ is a compact connected subgroup, and if the characteristic polynomials of the elements of $\Gamma'$ are all of type ${\bf G}_2$, then Corollary \[corconnected\] shows that $\Gamma'$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(7)$.]{}
Proof of Theorem \[thmintrod\] {#proofthmb}
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Basic identities and the irreducible case {#basicid}
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Let $k$ be a field and $\Gamma$ a group. We shall denote by ${\rm R}_k(\Gamma)$ the Grothendieck ring of finite dimensional $k$-linear representations of $\Gamma$. If $X$ is a finite dimensional $k$-linear representation of $\Gamma$, we denote by $[X]$ its class in ${\rm R}_k(\Gamma)$. The Adams endomorphism $\psi^2$ of the additive group of ${\rm R}_k(\Gamma)$ is defined by $\psi^2 [X] = [X \otimes X] - 2 [\Lambda^2 X] = [{\rm Sym}^2 X] - [\Lambda^2 X]$ for all $X$ as above. We denote by $1$ the trivial representation, as well as its class in ${\rm R}_k(\Gamma)$. PS. PS.
\[lemmesymlambda2\] Let $E$ be a $7$-dimensional regular quadratic space over a field $k$, with trivial even Clifford algebra, and let $W$ a spin representation of ${\rm GSpin}(E)$. The representation $W\otimes W \otimes \nu_{E}^{-1}$ of ${\rm GSpin}(E)$ factors via $\pi_E$ through a representation of ${\rm SO}(E)$ and we have $$[({\rm Sym}^2 \,W) \otimes \nu_E^{-1}] = [\Lambda^3 \,E] + 1 \hspace{.5 cm} {\rm and} \hspace{.5 cm} [(\Lambda^2\, W) \otimes \nu_E^{-1}] = [\Lambda^2 (1 \oplus E)]$$ in ${\rm R}_k({\rm SO}(E))$, as well as the following relation in ${\rm R}_k({\rm GSpin}(E))$: $$\psi^2 [W] \otimes \nu_E^{-1}- \psi^2 (1+[E]) = [\Lambda^3\, E] - [E] - [{\rm Sym}^2 \,E].$$ If furthermore ${\rm char}\, k \neq 2$ then we have $k[{\rm GSpin}(E)]$-module isomorphisms $({\rm Sym}^2 \,W) \otimes \nu_E^{-1} \simeq 1 \oplus \Lambda^3\, E$ and $(\Lambda^2 \,W) \otimes \nu_E^{-1} \simeq \Lambda^2 (1 \oplus E)$.
The second identity is a consequence of the first two ones and of the natural isomorphisms $\Lambda^2 (E \oplus k )\simeq \Lambda^2 E \oplus E$ and ${\rm Sym}^2 (E \oplus k) \simeq {\rm Sym}^2 E \oplus E$. To check the first two displayed equalities, it is enough to check that the characteristic polynomial of any element of ${\rm GSpin}(E)$ is the same on both representations in the two cases, by the Brauer-Nesbitt theorem. We omit the details, but this follows at once from the formulas given in Lemma \[pcarspin\] (see Remark \[decendc\] for another argument). PS. PS. To prove the last assertion, we may and do assume that $k$ is algebraically closed. Assume ${\rm char}\, k \neq 2$. It is well-known that the $k[{\rm SO}(E)]$-modules $\Lambda^i\, E$ are then irreducible for $0 \leq i \leq 3$ (see e.g. the tables in [@lubeck]). Recall from §\[octoalg\] that $W$ may be given a structure of a nondegenerate quadratic space over $k$ such that ${\rm GSpin}(E)$ acts by orthogonal similitudes of factor $\nu_{E}$. In particular, we have an isomorphism $W \simeq W^\ast\otimes \nu_{E}$, and both representations $({\rm Sym}^2 \,W)\otimes \nu_{E}^{-1}$ and $(\Lambda^2\, W)\otimes \nu_{E}^{-1}$ of are selfdual (for the first one it also uses $2 \in k^\ast$). As $\Lambda^i \, E$ is irreducible and selfdual for $0 \leq i \leq 3$, the (proven) first two displayed equalities of the statement imply $({\rm Sym}^2 \,W)\otimes \nu_{E}^{-1} \simeq 1 \oplus \Lambda^3 \, E$ and $(\Lambda^2 \,W)\otimes \nu_{E}^{-1} \simeq E \oplus \Lambda^2\,E$.
\[decendc\][Let $E$ be a regular quadratic space of odd dimension with trivial even Clifford algebra, and $W$ a spinor module for $E$. Then ${\rm GSpin}(E)$ acts by conjugation on ${\rm C}(E)_0 \simeq {\rm End}\, W$. This action factors through its quotient ${\rm SO}(E)$ and is induced by the obvious action of the latter on ${\rm C}(E)$ (defined by functoriality of the Clifford construction). This action preserves the canonical filtration of ${\rm C}(E)$ whose associated graded $k$-algebra is the exterior algebra $\Lambda \, E$. In particular, we obtain a more conceptual explanation of the identity $[{\rm End}\, W]= \sum_{0 \leq 2i < \dim E} [\Lambda^{2i}\, E]$ in ${\rm R}_k({\rm SO}(E))$. ]{}
\[propfondso7g2\] Let $E$ be a $7$-dimensional regular quadratic space over a field $k$, $K$ an algebraic closure of $k$, and $(W,\rho)$ a spin representation of ${\rm GSpin}(E \otimes_k K)$. Let $\Gamma \subset {\rm SO}(E)$ be a subgroup and set $\widetilde{\Gamma}:=\pi_E^{-1}(\Gamma) \subset {\rm GSpin}(E)$. The following properties are equivalent:PS.
- for all $\gamma \in \Gamma$, the polynomial $\det (t - \gamma \,|\, E)$ is of type ${\bf G}_2$, PS. PS.
- for all $\gamma \in \widetilde{\Gamma}$, $\rho(\gamma)$ has an eigenvalue whose square is $\nu(\gamma)$, PS. PS.
- the equality $\psi^2 [W] \otimes \nu_E^{-1} = 1 + \psi^2 [E \otimes_k K]$ holds in ${\rm R}_K(\widetilde{\Gamma})$,PS. PS.
- the equality $[\Lambda^3 E ] = [E] + [{\rm Sym}^2 E]$ holds in ${\rm R}_k(\Gamma)$. PS. PS.
In particular, if $\Lambda^3 E$ is semisimple as a representation of $\Gamma$, and if these properties hold, then $E$ admits a nonzero, $\Gamma$-invariant, alternating trilinear form.
The equivalences $(i) \Leftrightarrow (ii) \Leftrightarrow (iii)$ are exactly the equivalences $(i) \Leftrightarrow (ii) \Leftrightarrow (iv)$ of Lemma \[calcpcar\] (plus the Brauer-Nesbitt theorem). The equivalence between (iii) and (iv) follows from Lemma \[lemmesymlambda2\]. The last assertion comes from the fact that $({\rm Sym}^2 E)^\ast$ contains the nonzero $\Gamma$-invariant $\beta_E$ and from the relations in ${\rm R}_k(\Gamma)$ : $$[({\rm Sym}^2 E)^\ast]=[{\rm Sym}^2 E^\ast]=[{\rm Sym}^2\, E].$$
\[thmbirrcase\] Let $E$ be a $7$-dimensional regular quadratic space over an algebraically closed field $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup. Assume that the representation $E$ of $\Gamma$ is irreducible, with $\Lambda^3\, E$ and ${\rm Sym}^2\, E$ semisimple. If the characteristic polynomial of each element of $\Gamma$ is of type ${\bf G}_2$ then there is a unique ${\rm G}_2$-subgroup of ${\rm SO}(E)$ containing $\Gamma$.
Assume first that $E$ is an arbitrary $7$-dimensional vector space over $k$. The group ${\mathrm{GL}}(E)$ naturally acts on $\Lambda^3 E$; if $f \in \Lambda^3 E$, we set $H_f = \{g \in {\rm SL}(E), g \cdot f = f\}$, which is a closed subgroup of ${\rm SL}(E)$. By [@cohen Cor 2.3], if $H_f$ acts irreducibly on $E$ then, as an algebraic group, $H_f$ is simple of type ${\bf G}_2$. It follows from Lemma \[lemmerepg2\] that in this case, $k$ has characteristic $\neq 2$, $E$ is isomorphic to the representation $V(\omega_1)$ of $H_f$, and there is a structure of regular quadratic space on $E$ preserved by $H_f$. PS.
Let now $E$ and $\Gamma$ be as in the statement. By Proposition \[propfondso7g2\], there is a nonzero $\Gamma$-invariant element $f \in \Lambda^3 E$. In particular, $H_f \supset \Gamma$ acts irreducibly on $E$, hence the discussion above applies. We have thus ${\rm char} \,k\, \neq 2$ and there is a nondegenerate quadratic form on $E$ preserved by $H_f$. Such a form is preserved by $\Gamma$, which acts irreducibly on $E$, hence this form must be proportional to the quadratic form of $E$. In other words, we have $H_f \subset {\rm SO}(E)$ and $H_f$ is a ${\rm G}_2$-subgroup of ${\rm SO}(E)$ by Proposition \[algcarG2\]. PS.
If $H \subset {\rm SO}(E)$ is a ${\rm G}_2$-subgroup containing $\Gamma$, then we have $H = H_{f'}$ for some nonzero $f' \in \Lambda^3 E$ since $H$ is conjugate to $H_f$. It is thus enough to show that the form $f$ chosen above is unique up to a scalar. Under the assumptions we have $\Lambda^3 \,E\, \simeq \,E \,\oplus\, {\rm Sym}^2\, E$ as $k[\Gamma]$-modules by Proposition \[propfondso7g2\]. But as $E$ is irreducible and ${\rm char} \,k\, \neq 2$, the subspace of $\Gamma$-invariants is trivial in $E$ and one dimensional in ${\rm Sym}^2 \,E$.
Some (counter-)examples: ${\rm O}_2^{\pm}(k)$ and ${\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}})$ {#parexamples}
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As we shall see, the irreducibility assumption in Theorem \[thmbirrcase\] is actually necessary. We introduce now two important examples of subgroups $\Gamma$ of ${\rm SO}(E)$ (with $\dim E = 7$) such that each element of $\Gamma$ has a characteristic polynomial of type ${\bf G}_2$, but which will turn out not to be included in any ${\rm G}_2$-subgroup of ${\rm SO}(E)$.
[The group ${\rm O}_2^{\pm}(k)$]{}
Let $k$ be an algebraically closed field of characteristic $\neq 2$. Consider the similitude group ${\rm GO}_2(k)$ of the quadratic space $P:={\rm H}(k)$ which is hyperbolic of dimension $2$ over $k$. Denote by $\mu : {\rm GO}_2(k) \rightarrow k^\times$ the similitude factor (so ${\rm ker}\, \mu = {\rm O}_2(k)$) and set $\epsilon = \det : {\rm GO}_2(k) \rightarrow k^\times$. The character $\epsilon \mu^{-1}$ has order $2$ and its kernel is the subgroup ${\rm GSO}_2(k) \subset {\rm GO}_2(k)$ of proper similitudes, which is also the stabilizer of each of the two isotropic lines in $P$. Fix such a line and denote by $\chi : {\rm GSO}_2(k) \rightarrow k^\times$ the character defined by the action on this line; then ${\rm GSO}_2(k)$ acts by $\mu \chi^{-1}$ on the other line and the morphism $\chi \times \mu : {\rm GSO}_2(k) \rightarrow k^\times \times k^\times$ is an isomorphism. We set $${\rm O}_2^\pm (k) = \{ g \in {\rm GO}_2(k), \, \, \, \mu^2(g) = 1\}={\rm ker}\, \mu^2 = {\rm ker}\, \epsilon^2.$$ The orthogonal group ${\rm O}_2(k)$ is an index $2$ subgroup of ${\rm O}_2^{\pm}(k)$, and as $-1$ is a square in $k$ we have an exact sequence $$1 \longrightarrow \mu_2(k) \overset{\lambda \mapsto (\lambda,\lambda \,{\rm id}_P)}{\longrightarrow} \mu_4(k) \times {\rm O}_2(k) \longrightarrow {\rm O}_2^\pm(k) \longrightarrow 1$$ (with $\mu_n(k)=\{\lambda \in k^\times, \lambda^n=1\}$). The three characters $\mu, \epsilon, \mu\epsilon$ are distinct of order $2$ when restricted to ${\rm O}_2^\pm (k)$ and, as a $k[{\rm O}_2^\pm (k)]$-module, the space $P$ is irreducible and satisfies $P^\ast \simeq P \otimes \epsilon \simeq P \otimes \mu$. PS. PS.
\[o2pm\] Let $k$ be an algebraically closed field of characteristic $\neq 2$ and $E$ a nondegenerate quadratic space of dimension $7$ over $k$. There is a unique conjugacy class of injective group homomorphisms $$\rho : {\rm O}_2^\pm(k) \rightarrow {\rm SO}(E)$$ such that the $k[{\rm O}_2^\pm(k)]$-module $E$ is isomorphic to $P \oplus P^\ast \oplus \epsilon \oplus \mu \oplus \epsilon\mu$, where $P$, $\mu$ and $\epsilon$ are defined as above. For such a $\rho$, and any $\gamma \in {\rm O}_2^\pm(k)$, then $\det(t - \rho(\gamma))$ is of type ${\bf G}_2$.
The existence of such a morphism $\rho : {\rm O}_2^\pm(k) \rightarrow {\rm SO}(F)$ with $F = {\rm H}(P) \bot k^3$ and $k^3$ endowed with the standard quadratic form $(x_i) \mapsto \sum_i x_i^2$, is immediate from the equalities $\epsilon^2=\mu^2=1$. The existence part of the first assertion follows as we have $F \simeq E$ since $k$ is algebraically closed, as well as the uniqueness part (up to conjugation) by Corollary \[corwindex\]. It remains to show that for any $\gamma \in {\rm O}_2^\pm(k)$ then $\det(t - \rho(\gamma))$ is of type ${\bf G}_2$. Set $H = {\rm O}_2^\pm(k) \cap {\rm GSO}_2(k) = \ker \epsilon\mu$ and choose $\chi : H \rightarrow k^\ast$ as in the discussion above. We have an isomorphism $$E \simeq \chi \oplus \mu \chi^{-1} \oplus \chi^{-1} \oplus \mu^{-1} \chi \oplus \mu \oplus \mu \oplus 1$$ of $k[H]$-modules, which shows that $\det(t-\rho(\gamma))$ is of type ${\bf G}_2$ for all $\gamma \in H$ (recall $\mu^2=1$). Let $\gamma \in {\rm GO}_2(k)-{\rm GSO}_2(k)$, i.e. $\mu(\gamma) \neq \epsilon(\gamma)$. A simple computation shows that we have $\gamma^2 = \mu(\gamma) {\rm id}_P$ and $\det(t-\gamma)=t^2-\mu(\gamma)$. In particular, if we have $\gamma \in {\rm O}_2^\pm(k)$, then the element $s:=\mu(\gamma)$ is $\pm 1$, $\epsilon(\gamma)=-s$, and we have $$\det (t - \rho(\gamma)) = (t^2-s)^2(t-s)(t+s)(t+1)=(t^2\pm 1)^2(t+1)^2(t-1)$$ which is easily seen to be of type ${\bf G}_2$.
A consequence of our main result (Theorem \[mainthmb\]) will be that if $\rho$ is as in the statement of the proposition above, then $\rho({\rm O}_2^\pm(k))$ is not contained in any ${\rm G}_2$-subgroup of ${\rm SO}(E)$. We shall now give another argument for this fact. PS. PS.
\[lemmacentor2\] Let $\rho$ be as in Proposition \[o2pm\]. There is a unique conjugacy class of subgroups $\Gamma \subset {\rm O}_2^\pm(k)$ such that $\Gamma \simeq {\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$ and such that the $k[\Gamma]$-module $E$ is the direct sum of all nontrivial characters of $\Gamma$. These subgroups are exactly the centralizers of the order $2$ elements $s \in {\rm O}_2^\pm(k)$ such that $\mu(s)=1$ and $\epsilon(s)=-1$.
Indeed, this follows from an inspection of the centralizers of the order $2$ elements of ${\rm O}_2^\pm(k)$ that we leave as an exercise to the reader. PS. PS.
Let $\Gamma$ be a subgroup of ${\rm O}_2^\pm(k)$ isomorphic to ${\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$ and as in Lemma \[lemmacentor2\]. Then $\rho(\Gamma)$ [(]{}hence [*a fortiori*]{} $\rho({\rm O}_2^{\pm}(k))$[)]{} is not contained in any ${\rm G}_2$-subgroup of ${\rm SO}(E)$.
(see Proposition \[lemmaindex2\] for a rather different proof) Let $H$ be a ${\rm G}_2$-subgroup of ${\rm SO}(E)$ and $s \in H$ of order $2$ (and semisimple as ${\rm char}\, k \neq 2$). Then the centralizer $C$ of $s$ in $H$ is isomorphic to ${\rm SO}({\rm H}(k^2))$ and we have a decomposition $E \,= \,E_4\, \bot\, E_3$ where $E_4 = {\rm ker}(s+{\rm id})$ gives rise to the natural representation representation of $C$ of dimension $4$ (we omit the details here). In particular, we have $\det E_4 = 1$ as $k[C]$-module. On the other hand, assume that we have $s=\rho(s')$ with $s' \in {\rm O}_2^\pm(k)$ such that $\mu(s')=1$ and $\epsilon(s')=-1$, and that $\Gamma$ is the centralizer of $s'$. As $s'$ is not a square in ${\rm O}_2^\pm(k)$, the $4$ characters $\chi : \Gamma \rightarrow k^\times$ such that $\chi(s') = -1$ are of the form $c^i c'$, $0 \leq i \leq 3$, where $c$ has order $4$, $c'$ has order $2$, and $c(s') = 1 = - c'(s')$. In particular, their product is $c^2 \neq 1$, a contradiction.
The group ${\rm O}_2^\pm(k)$ will appear in practice using the following lemma. PS. PS.
\[crito2pm\] Let $\Gamma$ be a group, $k$ an algebraically closed field with ${\rm char}\, k\, \neq 2$, $Q$ an irreducible $k[\Gamma]$-module of dimension $2$, and $c : \Gamma \rightarrow k^\times$ a character with $c^2=1$ occurring in ${\rm Sym}^2 Q$. Then there is a unique $ {\rm O}_2(k)$-conjugacy class of morphisms $\Gamma \rightarrow {\rm O}_2^\pm(k)$ whose similitude factor is $c$ and such that the $k[\Gamma]$-module ${\rm H}(k)$ is isomorphic to $Q$.
As ${\rm char}\, k \neq 2$, we have natural isomorphisms ${\rm Hom}_k(Q^\ast,Q) \simeq Q \otimes Q \simeq {\rm Sym}^2 Q \oplus \det Q$; moreover, these $k[\Gamma]$-modules are semisimple as so is $Q$, by [@serre2]. By assumption, there is thus a nonzero $k[\Gamma]$-linear map ${\rm Sym}^2\, Q \rightarrow c$. Such a symmetric bilinear form on $Q$ is necessarily nondegenerate, as $Q$ is simple. It gives $Q$ a structure of nondegenerate quadratic space, necessarily isomorphic to ${\rm H}(k)$ as $k$ is algebraically closed. This proves the existence part of the lemma. Assume there are two $k[\Gamma]$-linear isomorphisms $u_i : Q {\overset{\sim}{\rightarrow}}{\rm H}(k)$, $i=1,2$, such that $\Gamma$ acts as similitude of factor $c$ on the nondegenerate quadratic form ${\rm q} \circ u_i$. By the first line of this proof, the $\Gamma$-invariants in ${\rm Hom}_k({\rm Sym}^2\,Q,c)$ have dimension $1$, so those two forms are proportional, i.e. $u_1 \circ u_2^{-1} \in {\rm GO}_2(k)$. We conclude as ${\rm GO}_2(k) = k^\times \cdot {\rm O}_2(k)$.
\[exempled8\] [Consider the dihedral group ${\rm D}_8$ of order $8$. Denote by $\mathcal{E}$ the set of characters ${\rm D}_8 \rightarrow k^\times$; we have $|\mathcal{E}|=4$. There is a unique $\eta_0 \in \mathcal{E}$ whose kernel is the unique subgroup of ${\rm D}_8$ of order $4$. We claim that there are exactly $3$ conjugacy classes of morphisms $r : {\rm D}_8 \rightarrow {\rm O}_2^\pm(k)$, distinguished by the character $\mu \circ r$, which can be any element of $\mathcal{E} -\{\eta_0\}$. Indeed, up to isomorphism, there is a unique faithful $2$-dimensional $k[{\rm D}_8]$-module $J$. It is irreducible and satisfies $\det J = \eta_0$. In particular, $J$ is selfdual and we have $J \simeq J \otimes \eta$ for all $\eta \in \mathcal{E}$, so we have ${\rm Sym}^2 J \simeq \bigoplus_{\eta \in \mathcal{E}-\{\eta_0\}} \eta$ and Lemma \[crito2pm\] proves the claim. Note however that the morphisms $\rho \circ r: {\rm D}_8 \rightarrow {\rm SO}(E)$, when $r$ varies as above, are all conjugate under ${\rm SO}(E)$ by Corollary \[corwindex\].]{}
[The group ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$]{}PS. PS.
The natural action of ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ on the set of lines in ${\mathbb{Z}}/3{\mathbb{Z}}\times {\mathbb{Z}}/3{\mathbb{Z}}$, numbered in an arbitrary way, defines a surjective morphism $${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}) \rightarrow \got{S}_4$$ with kernel $({\mathbb{Z}}/3{\mathbb{Z}})^\times$, which realizes ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ as a central extension of $\got{S}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$. This allows to view any $\got{S}_4$-module as a ${\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$-module by inflation.PS. PS.
Let $k$ be an algebraically closed field of characteristic $\neq 2,3$, so that $|{\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})| \in k^\times$. Up to isomorphism, the irreducible $k[\got{S}_4]$-modules are of the form $1, c, H, V$ and $V \otimes c$, where $c$ is the signature, $H$ is a $2$-dimensional representation inflated from a surjective homomorphism $\got{S}_4 \rightarrow \got{S}_3$ and $V$ is $3$-dimensional with determinant $1$. Moreover, still up to isomorphism, the irreducible $k[{\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})]$-modules with nontrivial central character are of the form $P$, $P^\ast$ and $P \otimes H$, where $P$ has dimension $2$. We have $\det H = \det P=c$, $H^\ast \simeq H \simeq H \otimes c$, $P^\ast \simeq P \otimes c$, $V^\ast \simeq V$, as well as the following identities: $$\label{idgl2f3}{\rm Sym}^2 H \simeq 1 \oplus H,\hspace{.4 cm} H \otimes V \simeq V \oplus V\otimes c \hspace{.4 cm} {\rm and} \hspace{.4 cm} P \otimes P^\ast \simeq 1 \oplus V.$$ PS.
\[rhogl23\]Let $k$ be an algebraically closed field of characteristic $\neq 2,3$ and $E$ a nondegenerate quadratic space of dimension $7$ over $k$. There is a unique conjugacy class of injective morphisms $\rho : {\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}}) \rightarrow {\rm SO}(E)$ such that the $k[{\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}})]$-module $E$ is isomorphic to $P \oplus P^\ast \oplus c \oplus H$, where $P$, $c$ and $H$ are as above. For such a $\rho$, and any $\gamma \in {\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$, then $\det(t - \rho(\gamma))$ is of type ${\bf G}_2$.
Recall that $c$ has order $2$ and that we have $\det H = c$ and $H^\ast \simeq H$, so there is a structure of nondegenerate quadratic space on $H$ such that $\got{S}_4$ acts as orthogonal isometries. The existence of $\rho$ (of Witt index $2$) follows, and its uniqueness up to conjugacy is a consequence of Corollary \[corwindex\]. To check the last assertion we check that assertion (iv) of Proposition \[propfondso7g2\] holds. Using the identities , it is tedious but straightforward to check that in ${\rm R}_k({\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}))$ both elements $[E] + [{\rm Sym}^2 E]$ and $[\Lambda^3 E]$ are equal to $3 + c + 3\,[H] + 2\,[P] +2\,[P^\ast] + [V] + 2 \,[V \otimes c] + 2\, [H \otimes P]$.
A consequence of our main Theorem \[mainthmb\] (or even Proposition \[lemmaindex3\]) will be that there is no ${\rm G}_2$-subgroup of ${\rm SO}(E)$ containing $\rho({\mathrm{SL}}_2({\mathbb{Z}}/3{\mathbb{Z}}))$ (hence $\rho({\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}}))$ [*a fortiori*]{}). PS. PS.
\[rhosl23\] [The group ${\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})$, which is the kernel of $c$, is the universal central extension of $\got{A}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$. Its abelianization is isomorphic to ${\mathbb{Z}}/3{\mathbb{Z}}$, and if $\eta : {\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}}) \longrightarrow k^\times$ is an order $3$ character, then we have $H_{|{\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})} \simeq \eta \oplus \eta^{-1}$. Moreover, the restriction of $P$ to ${\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ is, up to isomorphism, the unique two-dimensional irreducible $k[{\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})]$-module of determinant $1$. ]{}
Assumption [(S)]{} and statement of the main theorem {#parass}
----------------------------------------------------
\[defiasss\] Let $E$ be a regular quadratic space over an algebraically closed field $k$, $\Gamma$ a subgroup of ${\rm SO}(E)$, $\widetilde{\Gamma}$ the inverse image of $\Gamma$ under $\pi_E : {\rm Spin}(E) \rightarrow {\rm SO}(E)$, and $W$ a spinor module for $E$. We shall say that [*$\Gamma$ satisfies assumption [(S)]{}*]{} if $E$, ${\rm Sym}^2\,E$, $\Lambda^3 \, E$ and $W$ are semisimple $k[\widetilde{\Gamma}]$-modules.
This technical assumption [(S)]{} (where $S$ stands for [“semisimplicity”]{}) will be quite natural in our proof of Theorem \[mainthmb\]. It is of course satisfied if $\Gamma$ is finite with $2\,|\Gamma| \in k^\ast$. The next proposition shows that assumption [(S)]{} is not as strong as it may seem.PS. PS.
\[critasss\]Let $E$ be a nondegenerate quadratic space of dimension $7$ over an algebraically closed field $k$, and $\Gamma \subset {\rm SO}(E)$ a subgroup. Assume that at least one of the following assumptions holds:PS. PS.
- $k={\mathbb{C}}$ and the closure of $\Gamma$ in the Lie group ${\rm SO}(E)$ is compact, PS. PS.
- ${\rm char}\, k = 0$ or ${\rm char}\, k >13$, and the $k[\Gamma]$-module $E$ is semisimple, PS. PS.
- the $k[\Gamma]$-modules $E \otimes E$ and $\Lambda^3 E$ are semisimple.PS. PS.
Then $\Gamma$ satisfies assumption [(S)]{}.
Let $G \subset {\mathrm{GL}}_n({\mathbb{C}})$ be a subgroup with compact closure $\overline{G}$. A standard argument using a Haar measure of $\overline{G}$ shows that ${\mathbb{C}}^n$ possesses a Hermitian inner product which is $\overline{G}$-invariant, hence $G$-invariant as well. In particular the ${\mathbb{C}}[G]$-module ${\mathbb{C}}^n$ is semisimple. Under assumption (i), $\widetilde{\Gamma}$ has also a compact closure in the Lie group ${\rm SO}(E)$, which proves that $\Gamma$ satisfies assumption [(S)]{}. PS. PS.
Observe that the assumption on $E$ implies ${\rm char}\, k \neq 2$. By Lemma \[lemmesymlambda2\] we have $k[\widetilde{\Gamma}]$-linear isomorphisms $$\label{2isolemmsim} \Lambda^2 \, W \simeq E \oplus \Lambda^2 E \hspace{1 cm}{\rm and}\hspace{1 cm} {\rm Sym}^2 W \simeq 1 \oplus \Lambda^3 E.$$ We also have $W \otimes W \simeq {\rm Sym}^2\, W \oplus \Lambda^2\, W$ and $E \otimes E \simeq {\rm Sym}^2 E \oplus \Lambda^2 E$. By Serre [@serre2], the semisimplicity of $W \otimes W$ (resp $E \otimes E$) imply that of $W$ (resp. $E$). As a consequence, assumption (iii) implies that $\Gamma$ satisfies assumption [(S)]{}. PS. PS.
Assume that the $k[\Gamma]$-module $E$ is semisimple. If ${\rm char}\, k = 0$, a classical result of Chevalley ensures that $E^{\otimes n}$ is semisimple for each integer $n\geq 0$, so assumptions (iii) and [(S)]{} are satisfied. Set $p = {\rm char}\, k$ and assume $p>13$. As $p>2 \dim E -2$, $\Lambda^2 E$ and ${\rm Sym}^2$ are semisimple by the main result of Serre [@serre]. In particular, $\Lambda^2 \, W$ is also semisimple by the first isomorphism of . As $8 = \dim W \not \equiv 2,3 \bmod p$, another result of Serre [@serre2] ensures the semisimplicity of $W$. But by [@serre] again this implies that ${\rm Sym}^2\, W$ is semisimple as $p> 2 \dim W - 2$. We deduce that $\Lambda^3\, E$ is semisimple by the second isomorphism of . We have proved that assumption (iii) implies assumption [(S)]{}.
\[hypSex\][Let $\rho : {\rm O}_2^{\pm}(k) \longrightarrow {\rm SO}(E)$ be as in Proposition \[o2pm\] (in particular, we have ${\rm char}\, k \neq 2$) and let $\Gamma \subset {\rm O}_2^{\pm}(k)$ be a subgroup. We claim that $\rho(\Gamma)$ satisfies assumption [(S)]{}. Indeed, it is enough to show that $\rho(\Gamma')$ satisfies [(S)]{}, where $\Gamma'$ is the kernel of $\mu\epsilon$, as $|\Gamma/\Gamma'| \in k^\times$. But this kernel is a subgroup of ${\rm GSO}_2(k)$, which acts as a direct sum of two characters on $P$. It follows that $E^{\otimes n}$ is a direct sum of characters of $\Gamma'$ for all $n\geq 0$, and we are done by Proposition \[critasss\] (iii).]{}
We can now state our main theorem. PS. PS.
\[mainthmb\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over the algebraically closed field $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup. Assume that: PS. PS.
- $\Gamma$ satisfies assumption [(S)]{}, PS. PS.
- each element of $\Gamma$ has a characteristic polynomial of type ${\bf G}_2$,PS. PS.
Then exactly one of the following assertions holds:
- there is a ${\rm G}_2$-subgroup of ${\rm SO}(E)$ containing $\Gamma$,PS. PS.
- ${\rm char}\, k \neq 3$ and we have $\Gamma = \rho({\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}))$ or $\Gamma = \rho({\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}}))$, for some $\rho : {\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}) \longrightarrow {\rm SO}(E)$ as in Proposition \[rhogl23\],PS. PS.
- we have $\Gamma \simeq {\mathbb{Z}}/2{\mathbb{Z}}\times {\mathbb{Z}}/4{\mathbb{Z}}$ and the $k[\Gamma]$-module $E$ is the direct sum of the nontrivial characters of $\Gamma$,PS. PS.
- we have $\Gamma = \rho(\Gamma')$ where $\rho : {\rm O}_2^\pm(k) \longrightarrow {\rm SO}(E)$ is as in Proposition \[o2pm\], and $\Gamma' \subset {\rm O}_2^{\pm}(k)$ is a nonabelian subgroup, nonisomorphic to ${\rm D}_8$, and such that $\mu(\Gamma')=\{\pm1\}$.PS. PS.
In particular, we are in case [(a)]{} if the Witt index of $\Gamma$ is $\leq 1$.
Note that the assumption on $E$ in the statement above implies ${\rm char}\, k \neq 2$ (but when ${\rm char}\, k\,=2$ we can apply Corollary \[corcar2\] instead). Observe also that the “exceptional” cases (b), (c) and (d) do actually occur, as is shown by Propositions \[o2pm\] & \[rhogl23\], Lemma \[lemmacentor2\] and Remarks \[rhosl23\] & \[hypSex\]. PS. PS.
The end of this section is devoted to the proof of Theorem \[mainthmb\]. [*From now on and until the end of §\[proofthmb\], the letter $k$ will always denote an algebraically closed field of characteristic $\neq 2$*]{}. Note that under assumption [(S)]{}, Proposition \[propfondso7g2\] shows that assumption (ii) is equivalent to the existence of a $k[\Gamma]$-module isomorphism $$\label{eqfundg2} \Lambda^3 E \simeq E \oplus {\rm Sym}^2 E.$$ We shall study this identity and argue according to the Witt index of $\Gamma$, which is an element in $\{0,1,2,3\}$, in the decreasing order. One reason for that is that if $\Gamma' \subset \Gamma$ is a subgroup satisfying [(S)]{}, then $\Gamma'$ satisfies the assumptions of the theorem and its Witt index is at least the one of $\Gamma$. If $\Gamma$ satisfies (b) then its Witt index is $3$ if $\Gamma \simeq {\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ and $2$ if $\Gamma \simeq {\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$. If $\Gamma$ satisfies (c) its Witt index is $2$. If $\Gamma$ satisfies (d) then its Witt index is $3$ if $\epsilon(\Gamma')=1$ and $2$ otherwise. This explains the last assertion. PS.
The case of Witt index $3$ {#parcasewindex3}
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\[lemmaindex3\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup of Witt index $3$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. Then exactly one of the following assertions holds: PS.
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$, PS. PS.
- there is an order $2$ character $c$ and a $2$-dimensional irreducible representation $J$ of $\Gamma$ such that $[E]=1+2c+2[J]$ in ${\rm R}_k(\Gamma)$, $J \simeq J \otimes c$ and $\det J =1$,PS. PS. PS.
- ${\rm char}\, k \neq 3$, $\Gamma \simeq {\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})$ and $[E]=2[J]+1+c+c^{-1}$ where $J$ is “the” $2$-dimensional irreducible representation of $\Gamma$ with determinant $1$ and $c^{\pm 1}$ are the two order $3$ characters of $\Gamma$.
Let us first introduce a notation. If $X$ is a $k[\Gamma]$-module of finite dimension over $k$, we shall denote by ${\rm h}(X)$ the $k[\Gamma]$-module $X \oplus X^\ast$. PS. PS.
We may assume that $E = {\rm H}(I) \bot k $ with $\dim I = 3$ and that $\Gamma$ preserves the subspaces $I$ and $I^\ast$. In particular, we have a $k[\Gamma]$-linear isomorphism $$E \simeq {\rm h}(I) \oplus 1.$$ Let $\widetilde{\Gamma}$ denote the inverse image of $\Gamma$ under $\pi_E : {\rm Spin}(E) \rightarrow {\rm SO}(E)$. By Propositions \[cliffordhyp\] and \[decospin\], $\widetilde{\Gamma}$ is included in the natural subgroup $\rho_{{\rm H}(I);k}(\widetilde{\rho_I}({\rm GL}(I)) \times k^\times)$ of ${\rm GSpin}(E)$, and if $\alpha : \widetilde{\Gamma} \rightarrow k^\times$ denotes the projection on the $k^\times$-factor (which is also the center of ${\rm GSpin}(E)$), then we have $(\nu_E)_{|\widetilde{\Gamma}} = \alpha^2\, \det\,I=1$ and ${W}_{|\widetilde{\Gamma}} \simeq \,\alpha \,\otimes \,\Lambda\, I$. If we set $c = \Lambda^3 \, I = \det I$, we have thus $c \alpha^2 =1 $ and $\Lambda^2 I \simeq I^\ast \otimes c$, so we have a $k[\widetilde{\Gamma}]$-linear isomorphism $$\label{decowind3}W \simeq {\rm h}(\alpha \oplus \alpha \otimes I).$$ By assumption (S) and Corollary \[carbetaspin\], $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$ if, and only if, there is a character $\beta : \widetilde{\Gamma} \rightarrow k^\times$ occurring in $W_{|\widetilde{\Gamma}}$ and such that $\beta^2=1$. Note that $W_{|\widetilde{\Gamma}}$ contains $\alpha$ as well as the relation $\alpha^2 = c^{-1}$. PS. PS.
We have $\Lambda^2 I \simeq I^\ast \otimes c$, $\Lambda^2 I \otimes I^{\ast} \simeq {\rm Sym}^2 I^\ast \otimes c \oplus I$ and $E \simeq {\rm h}(I) \oplus 1$. A straightforward expansion shows that we have $k[\Gamma]$-linear isomorphisms $$\Lambda^3\, E \,\simeq \,{\rm h}(c \oplus ({\rm Sym}^2\, I^\ast) \otimes c \oplus I \oplus I^\ast \otimes c \, ) \oplus I \otimes I^\ast,$$ $${\rm Sym}^2\, E \,\simeq \,{\rm h}({\rm Sym}^2\, I \oplus I) \oplus I \otimes I^\ast \oplus 1.$$ In particular, $I$, $I \otimes I^\ast$ and ${\rm Sym}^2 I$ are semisimple by assumption ([S]{}). Equation is thus equivalent to $${\rm h}(x) \simeq {\rm h}(c^{-1} \otimes x) \, \, \, \,{\rm with}\, \,\, \, x = 1 \oplus I \oplus {\rm Sym}^2\, I.$$
In particular, the trivial representation $1$ of $\Gamma$ is a summand of $c^{-1}x$, i.e. the character $c$ is a summand of $x$. PS. PS.
[*Case 1 : the $k[\Gamma]$-module $I$ is irreducible*]{}. It follows that either $c=1$ or $c$ is a summand of ${\rm Sym}^2 I$. This second possibility implies that there is a nonzero $\Gamma$-equivariant map $I^\ast \rightarrow I \otimes c^{-1}$. As $I$ is irreducible, such a map is an isomorphism, and taking determinant gives the relation $c^{-1} = c c^{-3}=c^{-2}$. So we have $c=1$ in all cases, hence $\alpha^2=1$ and $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$. PS. PS.
[*Case 2 : $I \simeq \oplus_i \chi_i$ is a sum of $3$ characters $\chi_i : \Gamma \rightarrow k^\times$, $i=1,2,3$.*]{} We have $x \simeq 1 \oplus \bigoplus_i (\chi_i \oplus \chi_i^2 \oplus c \chi_i^{-1})$, hence also $c^{-1} x \simeq c^{-1} \oplus \bigoplus_i (c^{-1} \chi_i \oplus c^{-1}\chi_i^2 \oplus \chi_i^{-1})$. The relation ${\rm h}(x) \simeq {\rm h}(c^{-1}x)$ is thus equivalent to $${\rm h}(1\oplus \bigoplus_i \chi_i^2) \simeq {\rm h}(c^{-1}\oplus \bigoplus_{i} c^{-1}\chi_i^2).$$ As a consequence, $1$ is a summand of $c^{-1}\oplus \bigoplus_{i} c^{-1}\chi_i^2$, and we have either $c=1$ or $c=\chi_i^2$ for some $i$. We conclude the proof by choosing accordingly $\beta=\alpha$ or $\beta = \alpha \chi_i$, which occurs in $W$ and satisfies $\beta^2=1$. PS. PS.
[*Case 3 : $I \simeq \chi \oplus J$ with $J$ an irreducible representation of $\Gamma$ of dimension $2$.*]{} We have $\det J = c \chi^{-1}$. The isomorphism shows that $\Gamma$ is contained in a ${\rm G}_2$-subgroup if, and only if, we have $c=1$ or $c= \chi^2$. Assume $c \neq 1$ and $c \neq \chi^2$. We have $$x \simeq 1 \oplus \chi \oplus \chi^2 \oplus J \oplus \chi \otimes J \oplus {\rm Sym}^2 J,$$ so either $c=\chi$ or $c$ is a summand of ${\rm Sym}^2 J$. In this latter case, we have $J \simeq J^\ast \otimes c$ and by taking the determinant $c \chi^{-1} = c^2 c^{-1} \chi$, i.e. $\chi^2=1$. If follows that $1$ occurs twice in $x$, hence twice in $c^{-1}x$ as well. As ${\rm Hom}_{k[\Gamma]}(J,J^\ast \otimes c)$ has dimension $1$ by irreducibility of $J$, it follows that $c=\chi$ in all cases. In particular we have $\det J=1$ and $J^\ast \simeq J$. The equation ${\rm h}(x) \simeq {\rm h}(c^{-1}x)$ simplifies as $$\label{eqind3bis} {\rm h}(c^2) \oplus ({\rm Sym}^2 J)^{\oplus 2} \simeq {\rm h}(c) \otimes (1 \oplus {\rm Sym}^2 J).$$PS. PS.
[*Subcase [(a)]{}.*]{} Assume first that either $c$ or $c^{-1}$ occurs in ${\rm Sym}^2 J$. As $J$ is selfdual this implies $J \simeq J \otimes c$, hence $c$ has order $2$, and we are in case (ii). PS. PS. [*Subcase [(b)]{}.*]{} Assume now that $c^{\pm 1}$ does not occur in ${\rm Sym}^2 J$. As $c \neq 1$, equation shows $c=c^{-2}$, [*i.e.*]{} $c$ has order $3$, as well as a $k[\Gamma]$-linear isomorphism $$\label{eqind3bisbis} {\rm Sym}^2\, J \oplus \, {\rm Sym}^2\, J\,\simeq \, (c\oplus c^{-1}) \otimes {\rm Sym}^2 \, J.$$ We will eventually show that we are in case (iii). As $c$ has order $3$, observe that we have $3 \in k^\times$. PS. PS.
Let us show that ${\rm Sym}^2 \, J$ is irreducible. Otherwise, some character $\mu$ of $\Gamma$ occurs in ${\rm Sym}^2 \, J$ (which is $3$-dimensional and semisimple). The relation shows that each of the three distinct characters $\mu\, c^i$, $i \in \{-1,0,1\}$, occur in ${\rm Sym}^2 \, J$, so that we have ${\rm Sym}^2\, J \simeq \mu \oplus \mu c \oplus \mu c^{-1}$. As $J$ is selfdual and irreducible of determinant $1$, those characters must all have order $2$. But ${\rm Sym}^2\, J$ has determinant $(\det J)^3 = 1$, so we also have $\mu^3=1$: a contradiction. So ${\rm Sym}^2 \, J$ is irreducible and equation implies $${\rm Sym}^2 \, J \simeq c \otimes {\rm Sym}^2 \,J.$$
Consider the subgroup $\Gamma^0 := {\rm ker}\, c$, which is normal of index $3$ in $\Gamma$. Clifford theory[^8] shows that $({\rm Sym}^2\, J)_{|\Gamma^0}$ is a direct sum of $3$ distinct characters permuted transitively by the outer action of $\Gamma/\Gamma^0 \simeq {\mathbb{Z}}/3{\mathbb{Z}}$ on $\Gamma^0$. As $\Gamma^0$ is a normal subgroup of index $3$ in $\Gamma$, and as $\dim J = 2$, note that $J_{|\Gamma^0}$ is irreducible. It follows that each of the $3$ characters in $({\rm Sym}^2\, J)_{|\Gamma^0}$ has order $2$, the product of all of them being $1$. In particular, if $\Gamma^1$ denotes the kernel of the morphism $\Gamma \rightarrow {\rm GL}({\rm Sym}^2 J)$, then we have $\Gamma^1 \subset \Gamma^0$ and $\Gamma/\Gamma^1$ is an extension of $\Gamma/\Gamma^0 \simeq {\mathbb{Z}}/3{\mathbb{Z}}$ by $\Gamma^0/\Gamma^1 \simeq ({\mathbb{Z}}/2{\mathbb{Z}})^2$ such that the associated outer action of ${\mathbb{Z}}/3{\mathbb{Z}}$ on $({\mathbb{Z}}/2{\mathbb{Z}})^2$ permutes transitively the three nonzero elements. It follows that $\Gamma/\Gamma^1$ is isomorphic to $\got{A}_4$ (and that its irreducible $k$-linear representations are, up to isomorphism, ${\rm Sym}^2 J$, $1$, $c$ and $c^2$). PS. PS.
As $\Gamma^1$ acts trivially on ${\rm Sym}^2 \, J$, it acts by scalars of square $1$ in $J$, so $|\Gamma^1|\leq 2$. As $\got{A}_4$ has no irreducible $k$-linear representation of dimension $2$, we have $\Gamma^1 \neq 1$, [*i.e.*]{} $\Gamma^1 \simeq {\mathbb{Z}}/2{\mathbb{Z}}$. As $E \simeq J \oplus J \oplus 1 \oplus c \oplus c^{-1}$, one sees that $\Gamma^1$ is central in $\Gamma$. We know from Schur that there is a unique nontrivial central extension of $\got{A}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$, sometimes denoted $\widetilde{\got{A}_4}$, and which is isomorphic to ${\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}})$; moreover, such a group has exactly three $2$-dimensional irreducible $k$-linear representations, twists of each others, which are distinguished by their determinant (which can be any of the three characters of $\got{A}_4$). As a consequence, we have $\Gamma \simeq {\mathrm{SL}}_2({\mathbb{Z}}/3{\mathbb{Z}})$ and assertion (iii) holds.
Let us prove Theorem \[mainthmb\] when $\Gamma$ has Witt index $3$. Given Proposition \[lemmaindex3\] it is enough to show that: PS. PS. (1) we are in case (ii) of the proposition if, and only if, we are in case (d) of the statement of Theorem \[mainthmb\] with $\epsilon(\Gamma')=1$. PS. PS. (2) we are in case (iii) if, and only if, we are in case (b) of Theorem \[mainthmb\] and $\Gamma=\rho({\rm SL}_2({\mathbb{Z}}/3{\mathbb{Z}}))$.PS. PS.
Assertion (2) is an immediate consequence of Proposition \[rhogl23\] and Remark \[rhosl23\]. Let us check assertion (1). Observe first that a subgroup $\Gamma' \subset {\rm O}_2^{\pm}(k)$ acts irreducibly on $P={\rm H}(k)$ if, and only if, it is nonabelian, as $P$ is a $2$-dimensional semisimple $k[\Gamma']$-module by Remark \[hypSex\]. This shows that if we are in case (d) of the statement of Theorem \[mainthmb\] with $\epsilon(\Gamma')=1$, then we are in case (ii) of the statement of Proposition \[lemmaindex3\] (with $c$ playing the role of the character $\mu$). Assume conversely that we are in case (ii). As $c$ occurs in $J \otimes J^\ast$, the natural morphism $\Gamma \rightarrow {\rm GL}(J)$ is injective. Moreover, we have $c \neq 1$, $\det J = 1$ and $J \simeq J^\ast$, hence $c$ occurs in ${\rm Sym}^2 J$, and we conclude the proof by Lemma \[crito2pm\] and Proposition \[o2pm\]. Note that $\epsilon(\Gamma')=1$ implies $\Gamma' \not\simeq {\rm D}_8$ by the discussion in Example \[exempled8\].
The case of Witt index $2$
--------------------------
We shall now consider the Witt index $2$ case. If $\Gamma$ is a group and $P$ a finite dimensional $k$-linear representation of $\Gamma$, we shall denote by ${\rm ad}\, P\, \subset {\rm Hom}(P,P) \simeq P^\ast \otimes P$ the subspace of trace $0$ endomorphisms. If $\dim P =2$, we natural isomorphisms $P^\ast \otimes P \simeq 1 \oplus {\rm ad} \,P$ and ${\rm Sym}^2\, P\, \simeq \,\det P \otimes {\rm ad} \,P$ (recall ${\rm char}\, k \neq 2$).
\[lemmaindex2\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup of Witt index $2$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. Then exactly one of the following assertions holds: PS.
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$, PS. PS.
- $\Gamma \simeq {\mathbb{Z}}/4{\mathbb{Z}}\times {\mathbb{Z}}/2{\mathbb{Z}}$, and as a $k[\Gamma]$-module $E$ is isomorphic to the direct sum of the seven nontrivial characters of $\Gamma$,PS. PS.
- there is an irreducible, nonselfdual, $2$-dimensional representation $P$ of $\Gamma$, as well as two distinct order $2$ characters $c,\epsilon$ of $\Gamma$, such that $\det P = c$, $P \simeq P \otimes \epsilon$ and $[E] = [P] + [P^\ast] + c + \epsilon + c\epsilon$ in ${\rm R}_k(\Gamma)$. PS. PS.
- ${\rm char}\, k \neq 3$, $\Gamma \simeq {\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}})$, and as a $k[\Gamma]$-module $E$ is isomorphic to the direct sum of the three $2$-dimensional irreducible representations of $\Gamma$ and of its nontrivial (order $2$) character. PS. PS.
In case (i), we have $[E] = [P]+[P^\ast] + [{\rm ad}\, P]$ in ${\rm R}_k(\Gamma)$, where $P \subset E$ denotes any $2$-dimensional totally isotropic subspace stable by $\Gamma$, and $P$ is irreducible.
Let $P \subset E$ be a $\Gamma$-stable, two dimensional totally isotropic subspace of $E$. We may assume that we have $E = {\rm H}(P) \bot V$ and that $\Gamma$ preserves $P,P^\ast$ and $V$, with $V$ of Witt index $0$ and dimension $3$. In particular, we have a $k[\Gamma]$-linear isomorphism $E \simeq {\rm h}(P)\oplus V$ and $\det V = 1$. Set $c = \det P$. Equation on $\Gamma$ is easily seen to be equivalent to $$\label{eqind2} V \otimes (c \oplus c^{-1} \oplus {\rm ad} \, P) \simeq (1\oplus c \oplus c^{-1}) \otimes {\rm ad}\, P\oplus {\rm Sym}^2\, V,$$ both sides being semisimple $k[\Gamma]$-modules. Let $\widetilde{\Gamma}$ denote the inverse image of $\Gamma$ under $\pi_E : {\rm Spin}(E) \rightarrow {\rm SO}(E)$. The sequence and Proposition \[cliffordhyp\] show that the group $\widetilde{\Gamma}$ is contained in the subgroup $$\rho_{{\rm H}(P);V}(\widetilde{\rho_P}({\rm GL}(P)) \times {\rm GSpin}(V)) \simeq {\rm GL}(P) \times {\rm GSpin}(V)$$ of ${\rm GSpin}(E)$ such that $\nu_E = \det P \,\,\cdot \,\,\nu_V = 1$. Denote by $W_0$ a spinor module for $V$. It is well-known that the natural morphism ${\rm GSpin}(V) \rightarrow {\rm GL}(W_0)$ is an (exceptional) isomorphism, and that we have $\det W_0 = \nu_V$ and $V = \,{\rm ad} \,W_0$. If $W$ denotes a spinor module for $E$, it follows from Propositions \[cliffordhyp\] and \[decospin\] that we have a $k[\widetilde{\Gamma}]$-linear isomorphism $$\label{eqind2spin}
W \simeq \Lambda P \otimes W_0 \simeq (1 \oplus c \oplus P) \otimes W_0.$$ PS. PS. Observe that $W_0$ is irreducible as a representation of $\widetilde{\Gamma}$. Indeed, it is semisimple, being a direct summand of $W$. If $W_0$ is a sum of two characters of $\widetilde{\Gamma}$, then $V = \,{\rm ad} \,W_0$ contains the trivial character. But then we have a $\Gamma$-stable decomposition $V = k e \bot V'$ with $V'$ of dimension $2$ and determinant $1$. Such a $V'$ is reducible (as it is so as a representation of ${\rm SO}(V') \simeq k^\times$), of Witt index $0$, hence a sum of two order $2$ characters. But these characters are necessarily equal as $\det V'=1$, a contradiction. PS. PS.
We claim that the following properties are equivalent:\[pp1\]
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$. PS. PS.
- There exists a character $\beta : \widetilde{\Gamma} \rightarrow k^\times$ with $\beta^2=1$ such that $\beta$ is a summand of $W_{|\widetilde{\Gamma}}$, or which is the same, such that $W_0^\ast$ is isomorphic to $P \otimes \beta$,PS. PS.
- We have ${\rm ad}\, P \simeq V$ as $k[\Gamma]$-modules. PS. PS.
Indeed, given Corollary \[carbetaspin\], the irreducibility of $W_0$ and the isomorphism \[eqind2spin\], it only remains to explain (c) $\Rightarrow$ (b). Observe first that if $U$ is a two-dimensional $k$-vector space, viewed as a $k[{\rm GL}(U)]$-module, then the determinant on ${\rm End}\, U$ equip ${\rm ad} \,U$ with a structure of $3$-dimensional nondegenerate quadratic space over $k$, and the natural map ${\rm ad} : {\rm GL}(U)/k^\times \rightarrow {\rm SO}({\rm ad}\, U)$ is an isomorphism. Let $\rho_i : \Gamma \rightarrow {\mathrm{GL}}(U)$, $i=1,2$, be two representations such that ${\rm ad} \,\rho_i$ is semisimple and such that ${\rm ad} \rho_1 \simeq {\rm ad} \rho_2$. The last assertion of Proposition \[propwindex\] shows that there is some $g \in {\rm GL}(U)$ such that ${\rm ad}\, g \,\,{\rm ad} \rho_1\,\, {\rm ad} g^{-1} \,=\, {\rm ad} \rho_2$, i.e. $g \rho_1(\gamma) g^{-1} \rho_2(\gamma)^{-1} \in k^\ast$ for all $\gamma \in \Gamma$. There is thus a unique group homomorphism $\Gamma \rightarrow k^\ast$ such that $\rho_1 = \rho_2 \otimes \chi$. If $\det \rho_1 = \det \rho_2$ then we have $\chi^2 = 1$. These observations apply to the representations $\rho_i$ defined by $P$ and $W_0^\ast$, and prove the equivalence between (b) and (c). PS. PS.
[*Case 1 : $P$ is reducible*]{}. In particular, $\Gamma$ is not included in any ${\rm G}_2$-subgroup of ${\rm SO}(E)$ by the criterion (b) above. By assumption, there exists a character $\chi$ of $\Gamma$ such that $P \simeq \chi \oplus c \chi^{-1}$, and we thus have ${\rm ad}\, P \simeq 1 \oplus \chi^2 c^{-1} \oplus \chi^{-2} c$. Let ${\rm n}(V)$ be the number of $1$-dimensional summands of $V$. An inspection of equation shows the inequality $5 {\rm n}(V) \geq 9$, which implies ${\rm n}(V) \geq 2$, hence ${\rm n}(V)=3$ as $\dim V = 3$. So $V$ is a sum of $3$ characters, necessarily distinct of order $2$. In particular we have ${\rm Sym}^2 V \simeq 1 \oplus 1 \oplus 1 \oplus V$ and equation takes then the form: $$V \otimes (c \oplus c^{-1} \oplus \chi^2 c^{-1} \oplus \chi^{-2} c) \simeq (1\oplus c \oplus c^{-1}) \otimes (1 \oplus \chi^2 c^{-1} \oplus \chi^{-2} c) \oplus 1 \oplus 1 \oplus 1.$$ As $V$ has Witt index $0$, the character $c^{\pm 1}$ (resp. $\chi^2 c^{-1}$, $\chi^{-2}$) occurs at most once in $V$, and if it does we have $c^2= 1$ (resp. $\chi^2 c^{-1} = \chi^{-2}c$). As the trivial representation occurs at least $4$ times on the right-hand side of the equation above, it occurs exactly $4$ times on both sides. It follows that both $c$ and $\chi^2 c^{-1}$ occur in $V$, and that we have $\chi^2 \neq 1$ and $\chi^2 \neq c$. In particular, $c$ and $\chi^2 c^{-1}$ are distinct of order $2$, $\chi$ has order $4$, we have $V \simeq c \oplus \chi^2 \oplus \chi^2 c$ and $E \simeq \chi \oplus \chi^{-1} \oplus \chi\, c \oplus \chi^{-1}c \oplus \chi^2 \oplus c \oplus c \chi^2$. We are thus in case (ii) of the proposition. PS. PS.
[*Case 2 : $P$ is irreducible and ${\rm ad}\, P$ is not isomorphic to $V$.*]{} In particular, $\Gamma$ is not included in any ${\rm G}_2$-subgroup of ${\rm SO}(E)$. We will show that we are in case (iii) of the proposition if $V$ is a sum of $3$ characters, and in case (iv) otherwise. PS. PS.
As $P$ is irreducible, observe that for any character $\epsilon$ of $\Gamma$, the dimension of the space ${\rm Hom}_{k[\Gamma]} (\epsilon, P \otimes P^\ast ) \simeq {\rm Hom}_{k[\Gamma]}(P \otimes \epsilon,P)$ is at most $1$, and if it is nonzero we have $P \simeq P \otimes \epsilon$ and $\epsilon^2=1$. In particular, the semisimple representation ${\rm ad}\, P$ does not contain $1$. Moreover, both representations $V$ and ${\rm ad}\, P$ are semisimple, $3$-dimensional, multiplicity free and with trivial determinant. As a consequence, $\dim {\rm Hom}_{k[\Gamma]}(V,{\rm ad} \,P)$ is the number of irreducible summands of $V$ which occur in ${\rm ad}\, P$, and we have $\dim {\rm Hom}_{k[\Gamma]}(V,{\rm ad} \,P) \leq 1$ as $V$ and ${\rm ad}\, P$ are not isomorphic. PS. PS.
Let us show now that $c$ is a constituent of $V$. Observe that $1$ is a summand of ${\rm Sym}^2 V$, hence of the right-hand side of equation as well. Assume that $V \otimes c^{\pm 1}$ does not contain $1$. Then there is a nonzero $\Gamma$-equivariant morphism $V \rightarrow {\rm ad}\, P$. By assumption, this is not an isomorphism, so $V$ is reducible. It follows that ${\rm Sym}^2 V$ contains at least twice the trivial representation, and again by , that ${\rm Hom}_{k[\Gamma]}(V, {\rm ad}\, P)$ has dimension $\geq 2$, which is absurd by the previous paragraph. We conclude that $V$ does contain $c^{\pm 1}$, hence $c$ itself. We may thus write $$V \simeq c \oplus H,$$ and we have $c \neq 1$, $c^2 = 1$, $\det H = c$ and $H^\ast \simeq H \simeq H \otimes c$. Equation reduces then to $$\label{eqind2bis} 1 \oplus H \otimes ({\rm ad}\, P \oplus 1) \simeq {\rm ad}\, P \otimes (1 \oplus c) \oplus {\rm Sym}^2 H.$$
[*Subcase [(a)]{} : $H$ is reducible.*]{} Write $H = \chi_1 \oplus \chi_2$ for some distinct order $2$ characters $\chi_i$, with $\chi_1 \chi_2 =c$. Relation becomes $(\chi_1 \oplus \chi_2) \otimes ({\rm ad}\, P \oplus 1) \simeq ({\rm ad}\, P \oplus 1) \otimes (1 \oplus c)$. It follows that $1$ occurs in the left-hand side, so some $\chi_i$, call it $\epsilon$, occurs in ${\rm ad}\, P$. We have thus $V = c \oplus \epsilon \oplus \epsilon c$ and $P \simeq P \otimes \epsilon$. We have the following equivalences: ${\rm ad} \,P$ is not isomorphic to $V$, $c$ is not a constituent of ${\rm ad} \,P$, $P$ is not isomorphic to $P \otimes c$, $P$ is not selfdual (as we have $P^\ast \simeq P \otimes {\det P}^{-1} \simeq P \otimes c$). We are thus in case (iii) of the proposition. PS. PS.
[*Subcase [(b)]{} : $H$ is irreducible.*]{} Observe that if $H$ is a summand of ${\rm ad} \, P$, then ${\rm ad}\, P \simeq H \oplus \det H \simeq V$, which is not the case by assumption. The isomorphism $H \simeq H \otimes c$, and the relation $c^2=1$, show that $H$ is not a summand of ${\rm ad} \, P \otimes c$ either. Equation shows thus that $H$ is a summand of ${\rm Sym}^2 H$. This equation is then equivalent to the following system of isomorphisms: $$\label{eqind2bisbis} \left\{ \begin{array}{c} {\rm Sym}^2 H \simeq H \oplus 1 \\ \\ H \otimes {\rm ad}\, P \simeq (1\oplus c) \otimes {\rm ad}\, P \end{array} \right.$$ We claim that ${\rm ad}\, P$ is irreducible. Indeed, if we can write ${\rm ad}\, P = \epsilon \oplus T$ for some character $\epsilon$, we have already seen that $\epsilon$ has order $2$, the relation $\det T =\epsilon$, and the isomorphisms $T \simeq T^\ast \simeq T \otimes \epsilon$. By irreducibility of $H$, the second equation of shows that $T$ is irreducible and that $H \otimes \epsilon$ is isomorphic to $T$ or $T \otimes c$. But the isomorphisms $T \simeq T \otimes \epsilon$ and $H \simeq H \otimes c$ show $H \simeq T$ in both cases, which implies $V \simeq {\rm ad}\, P$ again: a contradiction. PS. PS.
Set $\Gamma^0 = {\rm ker}\, c$, it is an index $2$ subgroup of $\Gamma$. The relation $H \simeq H \otimes c$ and Clifford theory show that there is some character $\lambda : \Gamma^0 \rightarrow k^\times$ such that $H \simeq {\rm Ind}_{\Gamma^0}^\Gamma \lambda$. As $\det H = c$, we have $H_{|\Gamma^0} \simeq \lambda \oplus \lambda^{-1}$ and $\lambda^2 \neq 1$. As $H \simeq H^\ast$, we also have ${\rm Sym}^2 H \simeq 1 \oplus {\rm Ind}_{\Gamma^0}^\Gamma \lambda^2$. As a consequence, we have an isomorphism $H \simeq {\rm Ind}_{\Gamma^0}^\Gamma \lambda^2$. This forces either $\lambda = \lambda^2$ or $\lambda^{-1} = \lambda^2$. The first case is absurd as $\lambda \neq 1$, so $\lambda$ is an order $3$ character of $\Gamma^0$ (and ${\rm char}\, k \neq 3$). The image of $\Gamma \rightarrow {\rm GL}(H)$ is isomorphic to $\got{S}_3$, and its kernel is the subgroup $\Gamma^1:={\rm ker}\, \lambda \subset \Gamma^0$. PS. PS.
As $\Gamma^0$ has index $2$ in $\Gamma$, and ${\rm dim}\, {\rm ad}\, P = 3$, the irreducibility of ${\rm ad}\, P$ implies the irreducibility of $({\rm ad}\, P)_{|\Gamma^0}$, hence of $P_{|\Gamma^0}$ as well. As $P^\ast \simeq P \otimes c$, it follows that we have $[E]\,= \,2\, [P] + 1+ \lambda + \lambda^{-1}$ in ${\rm R}_k(\Gamma^0)$, with $\lambda$ of order $3$ and $P$ irreducible: we are thus in [*subcase (b)*]{} of [*Case 3.*]{} of the proof of Proposition \[lemmaindex3\] (for the triple $(\Gamma^0,P,\lambda)$ instead of $(\Gamma,J,c)$), and in particular in case (iii) of the statement of that proposition. It implies that $\Gamma^0$ is “the” nontrivial central extension of $\got{A}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$, and that is center ${\rm Z}(\Gamma^0)$ is the subgroup of order $2$ in ${\rm SO}(E)$ acting by $\pm {\rm id}$ on $P \oplus P^\ast$ and by ${\rm id}$ on $V$. This description shows that ${\rm Z}(\Gamma^0)$ lies in the center ${\rm Z}(\Gamma)$ of $\Gamma$, hence ${\rm Z}(\Gamma^0)={\rm Z}(\Gamma)$ ($\simeq {\mathbb{Z}}/2{\mathbb{Z}}$). PS. PS.
Let $\Gamma^2 \subset \Gamma$ be the kernel of the morphism $\Gamma \rightarrow {\rm GL}({\rm ad} P)$, we have ${\rm Z}(\Gamma) \subset \Gamma^2$. The second equation of shows $H_{|\Gamma^2} \simeq 1 \oplus c$, the subspace of $\Gamma^2$-invariants in $H$ is thus nonzero. But this subspace is stable by $\Gamma$ as $\Gamma^2$ is a normal subgroup, it follows that $\Gamma^2$ acts trivially in $H$, [*i.e.*]{} we have $\Gamma^2 \subset \Gamma^1$. The group $\Gamma^2$ acts thus by scalars of determinant $1$ in $P$, so we have $\Gamma^2 = {\rm Z}(\Gamma)$. Moreover, the finite group $\Gamma/\Gamma^2$ is isomorphic to a subgroup of ${\rm SO}({\rm ad} \,P)$; it contains $\Gamma^0/\Gamma^2$, which is a subgroup of index $2$ isomorphic to $\got{A}_4$ and acts irreducibly on ${\rm ad}\, P$, hence with a trivial centralizer in ${\rm SO}({\rm ad} \,P)$. As $\got{A}_4$ has an automorphism group isomorphic to $\got{S}_4$, this forces $\Gamma/\Gamma^2$ itself to be isomorphic to $\got{S}_4$. We have thus proved that $\Gamma$ is a central extension of $\got{S}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$ which does not split over $\got{A}_4$. PS. PS. We know from Schur that, up to isomorphism, there are exactly two central extensions $G$ of $\got{S}_4$ by ${\mathbb{Z}}/2{\mathbb{Z}}$ which do not split over $\got{A}_4$: we have either $G \simeq {\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}})$, or $G$ is isomorphic to the group usually denoted $\widetilde{\got{S}_4}$ (see e.g. [@serregalois §9.1.3]). In any case, character theory shows that such a $G$ has exactly $3$ isomorphism classes of irreducible $k$-linear representations which are nontrivial on the center ${\mathbb{Z}}/2{\mathbb{Z}}$: they have the form $J, J\otimes c$ and $J \otimes H$ where $\dim J = 2$, $c$ is the non-trivial (order $2$) character of $\got{S}_4$, and $H$ is the $2$-dimensional irreducible representation of $\got{S}_4$ (inflated from a surjective homomorphism $\got{S}_4 \rightarrow \got{S}_3$). The groups ${\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}})$ and $\widetilde{\got{S}_4}$ are distinguished as follows: we have $\det J = c$ in the first case (and thus $J^\ast \simeq J \otimes c$), and $\det J =1$ in the second (and then $J^\ast \simeq J$). As a consequence, we have $\Gamma \simeq {\mathrm{GL}}_2({\mathbb{Z}}/3{\mathbb{Z}})$ and assertion (iv) holds.
Let us prove Theorem \[mainthmb\] when $\Gamma$ has Witt index $2$. An immediate consequence of Proposition \[rhogl23\] is that we are in case (iv) of Proposition \[lemmaindex2\] if, and only if, we are in case (b) of the statement of Theorem \[mainthmb\] with $\Gamma=\rho({\rm GL}_2({\mathbb{Z}}/3{\mathbb{Z}}))$. It only remains to show that we are in case (iii) of the proposition if, and only if, we are in case (d) of Theorem \[mainthmb\] with $\epsilon(\Gamma')=\{\pm 1\}$. But this follows from a similar argument as the one given in the last paragraph of §\[parcasewindex3\], by Example \[exempled8\] and the following lemma. Note that the characters $c$ and $\epsilon$ here play respectively the role of the characters $\epsilon$ and $\epsilon\mu$ of subgroups of ${\rm O}_2^\pm(k)$.
Let $\Gamma \subset {\rm O}_2^\pm(k)$ be a subgroup on which both $\mu$ and $\epsilon$ are nontrivial. Then $\Gamma$ is isomorphic to ${\rm D}_8$ if, and only if, the $k[\Gamma]$-module ${\rm H}(k)$ is both irreducible and selfdual.
We have already seen in Example \[exempled8\] that if $\Gamma$ is isomorphic to ${\rm D}_8$ then ${\rm H}(k)$ is both irreducible and selfdual. Assume conversely that the $k[\Gamma]$-module $P={\rm H}(k)$ has these two properties. By irreducibility of $P$, $\epsilon\mu$ is nontrivial on $\Gamma$, and thus $\epsilon,\mu$ and $\epsilon\mu$ are of order $2$, hence distinct. As $P$ is selfdual we have $P^\ast \simeq P \otimes \chi$ for all $\chi$ in $\{1,\epsilon,\mu,\epsilon\mu\}$. In particular ${\rm Sym}^2\,P$ is isomorphic to $1 \oplus \mu \oplus \epsilon\mu$, which implies that $|\Gamma|$ divides $8$. As $P$ is irreducible $\Gamma$ is nonabelian, so we have $|\Gamma|=8$ and $P$ is the unique $2$-dimensional irreducible representation of $\Gamma$ up to isomorphism. But if $\Gamma$ was the “quaternion group” of order $8$ we would have $\det P = 1$. So $\Gamma$ is isomorphic to ${\rm D}_8$, and we are done.
The case of Witt index $1$
--------------------------
We shall now prove that there is no subgroup $\Gamma$ of ${\rm SO}(E)$ of Witt index $1$ which satisfies assumptions (i) and (ii) of Theorem \[mainthmb\]. We start with a lemma on representations of Witt index $0$. PS. PS.
If $V$ is a finite dimensional $k$-linear representation of $\Gamma$, we shall denote by $V^\Gamma$ the subspace $\{v \in V \, |\, \, \gamma v = v \, \, \,\,\forall \gamma \in \Gamma\}$ of $\Gamma$-invariants in $V$, and we set ${\rm t}(V) = \dim_k V^\Gamma$. If $U,V$ are finite dimensional $k$-linear representations of $\Gamma$, the natural isomorphism $U \otimes V {\overset{\sim}{\rightarrow}}{\rm Hom}(U^\ast,V)$ shows the equality $${\rm t}(U \otimes V) = \dim_k {\rm Hom}_{k[\Gamma]}(U^\ast,V),$$ that we shall freely use below. It implies that if both $U$ and $V$ are irreducible, then we have ${\rm t}(U \otimes V) \leq 1$. PS. PS.
\[lemmaind0\] Let $V$ be a nondegenerate quadratic space over $k$, $\Gamma$ a group, $\rho : \Gamma \longrightarrow {\rm O}(V)$ a semisimple representation of Witt index $0$, $r$ the number of irreducible summands of the $k[\Gamma]$-module $V$ and $c : \Gamma \rightarrow k^\times$ a group homomorphism. PS. PS.
- We have ${\rm t}(\Lambda^2 \, V)=0$ and ${\rm t}({\rm Sym}^2 V)=r$.PS. PS.
- We have ${\rm t}(\Lambda^2\, V \otimes c^{-1}) \leq r $ and ${\rm t}(\Lambda^2\, V \otimes c^{-1}) \leq \frac{\dim V}{2}$.PS. PS.
- If $\dim V$ is odd and ${\rm t}(\Lambda^2 \,V \otimes c^{-1}) \neq 0$ then we have $r\geq 2$. PS. PS.
- If $\dim V \leq 5$ then we have ${\rm t}(\Lambda^2 \,V \otimes c^{-1}) \leq 2$. PS. PS.
Assume furthermore $4 \leq \dim V \leq 5$ and ${\rm t}(\Lambda^2 \,V \otimes c^{-1})= 2$.
- We have $c^2 =1$ and a $k[\Gamma]$-module decomposition $$\label{casdeg2} V = U \oplus U' \oplus V'$$ with $\dim U = \dim U'=2$ and $\det U = \det U' = c$. PS. PS.
- Assume $r<\dim V$ and ${\rm t}(\Lambda^2 V \otimes \chi^{-1})\neq 0$ for some character $\chi : \Gamma \rightarrow k^\times$ with $\chi \neq c$. Then we have ${\rm t}(\Lambda^2 V \otimes \chi^{-1})=1$, and if $r=\dim V -2$ then we have $U' \simeq U \otimes \chi$ in . PS. PS.
We may write $V = \oplus_{i \in I} V_i$ where the $V_i$ are irreducible, selfdual and pairwise distinct, and with $r=|I|$. For each $i$ in $I$, the unique $\Gamma$-invariant line of ${\rm End}\, V_i \simeq V_i \otimes V_i \simeq {\rm Sym}^2 \,V_i \oplus \Lambda^2 \,V_i$ lies in ${\rm Sym}^2 \,V_i$, so we have ${\rm t}(\Lambda^2\, V_i) = 0$ and ${\rm t}({\rm Sym}^2 \,V_i)=1$. Moreover, if $i\in I$ we have $V_i \simeq V_i^\ast$ so ${\rm t}(V_i \otimes V_j) = 0$ if $i \neq j$. Choose a total ordering $<$ on $I$. We have a $k[\Gamma]$-linear isomorphism $$\Lambda^2\, V \simeq \left( \,\bigoplus_{i \in I} \Lambda^2 \,V_i\, \right) \oplus \left( \bigoplus_{i<j} V_i \otimes V_j \right),$$ as well as a similar isomorphism with each $\Lambda^2$ replaced by ${\rm Sym}^2$. This proves assertion (i). PS. PS.
Let $c : \Gamma \rightarrow k^\times$ be a character and consider the sets $${\rm I}(c) = \{ i \in I \, \,\, |\, \, \, {\rm t}(\Lambda^2 \, V_i \otimes c^{-1}) \neq 0\}$$ $${\rm J}(c) = \{ (i,j) \in I \times I\, \, \, |\, \, \, i<j, \,\,\,{\rm t}(V_i \otimes V_j \otimes c^{-1}) \neq 0\}.$$PS. PS.
Recall that $\Lambda^2 V_i$ injects into $V_i \otimes V_i$, and that ${\rm t}(V_i \otimes V_j) \leq 1$ for all $i,j$ by irreducibility of the $V_j$, so that we also have ${\rm t}(\Lambda^2 \, V_i \otimes c^{-1}) \leq 1$. In particular, we have the equality $$\label{formtij} {\rm t}(\Lambda^2 \,V \otimes c^{-1}) = |{\rm I}(c)|+|{\rm J}(c)|.$$ We now make a couple of simple observations. PS. PS.
– [(O1)]{} if $i \in {\rm I}(c)$ then $\dim V_i$ is even, and we have $\det V_i = c^{\frac{\dim V_i}{2}}$ and $c^{\dim V_i} = 1$. In particular, $\det V_i = c$ if $\dim V_i=2$.PS. PS.
Indeed, if $i \in {\rm I}(c)$ then there is a nonzero alternating pairing on $V_i$ such that $\Gamma$ acts by symplectic similitudes of factor $c$. Such a pairing is necessarily nondegenerate as $V_i$ is irreducible, so ${\rm (O1)}$ follows from general properties of similitude symplectic groups. As a consequence, if $\dim V$ is odd and $V$ is irreducible, then $\Lambda^2 V$ does not contain any character, which proves assertion (iii). PS. PS.
– [(O2)]{} If $(i,j) \in {\rm J}(c)$ then $\dim V_i = \dim V_j$ and $c^{2 \dim V_i} = 1$. PS. PS.
Indeed, if $(i,j) \in {\rm J}(c)$ then there is an isomorphism $V_i \simeq V_j \otimes c$.PS. PS.
– [(O3)]{} if $i \in {\rm I}(c)$ then there is no $j \in I$ such that $(i,j) \in {\rm J}(c)$ or $(j,i) \in {\rm J}(c)$ (since $V_i \simeq V_i \otimes c \simeq V_i \otimes c^{-1}$). PS. PS.
– [(O4)]{} if $(i,j) \in {\rm J}(c)$ then $(r,s) \in {\rm J}(c)$ implies $\{i,j\} \cap \{r,s\} = \emptyset$ or $\{i,j\} = \{r,s\}$ (observe that if $i,j \in I$ and $V_i \simeq V_j \otimes c$ then $V_j \simeq V_i \otimes c$). PS. PS.
It follows from those observations that we have the inequalities $$\begin{array}{ccc} \dim V & \geq & \sum_{i \in {\rm I}(c)} \dim V_i + 2\, \sum_{(i,j) \in {\rm J}(c)} \dim V_i \\ \\
& \geq & 2 \,|{\rm I}(c)| + 2\,|{\rm J}(c)| = 2\, {\rm t}(\Lambda^2 \, V \otimes c^{-1}), \,\,\,\,{\rm and}\end{array}$$PS. $$r \geq |{\rm I}(c)| + 2\,|{\rm J}(c)| = |{\rm J}(c)|+{\rm t}(\Lambda^2 \, V \otimes c^{-1}) \geq {\rm t}(\Lambda^2 \, V \otimes c^{-1}).$$ PS. PS.
This proves assertions (ii). Assertion (iv) is a special case of assertion (ii). PS. PS.
Assume now $4 \leq \dim V \leq 5$ and ${\rm t}(\Lambda^2 \, V \otimes c^{-1}) = 2$. We have $2 = |{\rm I}(c)|+|{\rm J}(c)|$ by . There are three cases:
- $|{\rm I}(c)|=2$ and $|{\rm J}(c)|=0$. Then [(O1)]{} shows $\dim V_i=2$ for $i \in {\rm I}(c)$, $c^2=1$, hence assertion (v) holds with $U$ and $U'$ irreducible of dimension $2$ and $r = \dim V -2$. PS. PS.
- $|{\rm J}(c)|=2$ and $|{\rm I}(c)|=0$. Then [(O2)]{} and [(O4)]{} show $\dim V_i=\dim V_j=1$ and $\det (V_i \oplus V_j) = V_i \otimes V_j = c$ if $(i,j) \in {\rm J}(c)$, $c^2=1$, and $V = V' \oplus \bigoplus_{(i,j) \in {\rm J}(c)} (V_i \oplus V_j)$. So assertion (v) holds with $U$ and $U'$ reducible of dimension $2$ and $r = \dim V$. PS. PS.
- $|{\rm I}(c)|=|{\rm J}(c)|=1$. Then [(O1)]{}, [(O2)]{} and [(O3)]{} show $\dim V_i=2$ for $i \in {\rm I}(c)$ as well as $\dim V_i = \dim V_j = 1$ and $\det (V_i \oplus V_j)=c$ if $(i,j) \in {\rm J}(c)$. So assertion (v) holds with either $U$ or $U'$ reducible (but not both) and $r = \dim V-1$.
Let now $\chi : \Gamma \rightarrow k^\times$ be a character different from $c$. We have ${\rm I}(\chi)=\emptyset$ by and [(O1)]{}, so ${\rm t}(\Lambda^2 \, V \otimes \chi^{-1})=|{\rm J}(\chi)|$. By [(O2)]{} and [(O4)]{}, we have $|{\rm J}(\chi)|\leq 2$ and if the equality holds then $V$ contains at least $4$ distinct characters, [*i.e*]{} $r=\dim V$. If we have $r=\dim V-2$, then $U$ and $U'$ are irreducible of determinant $\neq \chi$ in . Thus ${\rm J}(\chi) \neq \emptyset$ if, and only if, we have $U \simeq U' \otimes \chi$. This proves assertion (vi).
\[lemmaindex1\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over $k$. Then there is no subgroup $\Gamma \subset {\rm SO}(E)$ of Witt index $1$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\].
Let $\Gamma \subset {\rm SO}(E)$ be of Witt index $1$ and satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. There is a character $\chi : \Gamma \rightarrow k^\ast$, as well as a representation $\Gamma \rightarrow {\rm SO}(V)$ of dimension $5$ and Witt index $0$, such that $E \simeq \,\chi \oplus \chi^{-1} \,\oplus \, V $. As $V$ is selfdual and $\det V=1$, we have $\Lambda^3 V \simeq \Lambda^2 V$, and one easily checks that equation is equivalent to the following identity [$$\label{eqind1} \Lambda^2 V \, \otimes (1 \oplus \chi \oplus \chi^{-1}) \simeq 1 \oplus \chi \oplus \chi^{-1} \oplus \chi^2 \oplus \chi^{-2} \oplus {\rm Sym}^2 V \oplus V \otimes (\chi \oplus \chi^{-1}).$$]{} Denote by $r$ the number of irreducible constituents of $V$. By Lemma \[lemmaind0\] (i) and (iv), we have ${\rm t}({\rm Sym}^2\, V)=r$, ${\rm t}(\Lambda^2\,V)=0$ and ${\rm t}(\Lambda^2 V \otimes \chi^{\pm 1}) \leq 2$. As $V\otimes \chi$ and $\Lambda^2 V \otimes \chi$ are semisimple, since so are the representations occurring in equation by (S), and as $V$ is selfdual, we also have ${\rm t}(\Lambda^2 V \otimes \chi)={\rm t}(\Lambda^2 V \otimes \chi^{-1})$ and ${\rm t}(V \otimes \chi)={\rm t}(V \otimes \chi^{-1})$. We obtain $$2\, {\rm t}(\Lambda^2 V \otimes \chi) = 1 + r + 2\, ({\rm t}(\chi)+{\rm t}(\chi^2)+{\rm t}(V \otimes \chi)).$$ We deduce from this identity: ${\rm t}(\Lambda^2 V \otimes \chi) >0$, $r$ is odd and $1+r+2 \, {\rm t}(\chi^2) \leq2\, {\rm t}(\Lambda^2 V \otimes \chi) \leq 4$. But Lemma \[lemmaind0\] (iii) shows $r\geq 2$. The only possibility is thus ${\rm t}(\Lambda^2 V \otimes \chi)=2$, $r=3$ and $\chi^2 \neq 1$, which contradicts Lemma \[lemmaind0\] (v).
The case of Witt index $0$ {#parcasewindex0}
--------------------------
In order to prove Theorem \[mainthmb\], it only remains to prove the following Theorem.PS. PS.
\[thmindex0\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(E)$ a subgroup of Witt index $0$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$.
We shall divide the proof into several parts. PS. PS.
\[propso4\] For $i=3,4$, let $V_i$ be a nondegenerate quadratic space over $k$ of dimension $i$, set $V = V_3 \bot V_4$, and let $\Gamma \subset {\rm SO}(V)$ be a subgroup of Witt index $0$ such that $\Gamma \subset {\rm SO}(V_3) \times {\rm SO}(V_4)$. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(V)$ if, and only if, there is an injective $k[\Gamma]$-linear morphism $V_3 \rightarrow \Lambda^2 V_4$. PS. If the $k[\Gamma]$-module $V_3$ has an irreducible summand of dimension $\geq 2$ which injects into $\Lambda^2 V_4$, then these properties are satisfied.
Let $W$, $W_3$ and $W_4$ be spinor modules for $V$, $V_3$ and $V_4$ respectively, and write $W_4 = W_4^+ \oplus W_4^{-}$ as the direct sum of its two half-spinor modules $W_4^{\pm}$. Recall from the sequence that we have a natural homomorphism $\rho_{V_3;V_4} : {\rm GSpin}(V_3) \times {\rm GSpin}(V_4) \rightarrow {\rm GSpin}(V)$, whose image is the inverse image of ${\rm SO}(V_3) \times {\rm SO}(V_4)$ under the natural homomorphism $\pi_V : {\rm GSpin}(V) \rightarrow {\rm SO}(V)$. We may thus view $W$, $W_3$ and $W_4^{\pm}$ as $k[{\rm GSpin}(V_3) \times {\rm GSpin}(V_4)]$-modules. As such, there is an isomorphism $$\label{f0so4}W \simeq W_3 \otimes (W_4^+\oplus W_4^-)$$ by Proposition \[decospin\]. As is well-known, the image of the natural injective homomorphism ${\rm GSpin}(V_4) \rightarrow {\mathrm{GL}}(W_4^+) \times {\mathrm{GL}}(W_4^-)$ is the subgroup of elements $(g^+,g^-) \in{\mathrm{GL}}(W_4^+) \times {\mathrm{GL}}(W_4^-)$ such that $\det g^+=\det g^-$. Moreover, as a representation of ${\rm GSpin}(V_4)$ we have $$\label{f1so4} \det W_4^{\pm} = \nu_{V_4} \hspace{1 cm} {\rm and} \hspace{1cm} W_4^+ \otimes W_4^{-} \simeq V_4 \otimes \nu_{V_4}.$$ Besides, as already explained in the first paragraph of the proof of Proposition \[lemmaindex2\], the natural morphism ${\rm GSpin}(V_3) \rightarrow {\rm GL}(W_3)$ is an isomorphism, and as a representation of ${\rm GSpin}(V_3)$ we have $$\label{f2so4}\det W_3 = \nu_{V_3} \hspace{1 cm} {\rm and} \hspace{1cm} {\rm ad}\, W_3 \simeq V_3.$$PS.
Set $\widetilde{\Gamma} = \pi_V^{-1}(\Gamma) \cap {\rm Spin}(E)$. We may view $W, W_3, W_4^{\pm 1}, V, \nu_V, V_3, \nu_{V_3}$, $V_4, \nu_{V_4}$ as a representations of $\widetilde{\Gamma}$, and as such we have $\nu_{V_3}\nu_{V_4} = \nu_V =1$. We set $\nu = {\nu_{V_4}}_{|\widetilde{\Gamma}}$, we thus have $\det W_4^{\pm } = \nu$ and $\det W_3=\nu^{-1}$. PS. PS.
We claim that $W$, $W_3$ and $W_4^{\pm}$ are semisimple $k[\widetilde{\Gamma}]$-modules. We apply for this the main result of [@serre] as well as [@serre2 Thm. 2.4], using ${\rm char}\, k \neq 2$. Indeed, the semisimplicity of $V_3$ and $V_4$ implies that of $W_3 \otimes W_3$, $W_3$ and $W_4^{\pm}$ by , and [@serre2], which in turn implies the semisimplicity of $W$ by and [@serre].PS. PS.
Let us now check that $W_3$ and $W_4^{\pm}$ are simple $k[\widetilde{\Gamma}]$-modules. If $W_3$ is reducible, then we have $W_3 \simeq \chi \oplus \nu^{-1} \chi^{-1}$ for some character $\chi : \widetilde{\Gamma} \rightarrow k^\times$ as $W_3$ is semisimple. This implies $V_3 \simeq {\rm ad} \, W_3 \simeq 1 \oplus \chi^2 \nu \oplus \chi^{-2}\nu^{-1}$, a contradiction as $V_3$ has Witt index $0$ as a representation of $\Gamma$. Assume now that $W_4^+$ is reducible (the argument for $W_4^-$ will be similar), and write $W_4^+ \simeq \chi \oplus \chi^{-1} \nu$ as above. This implies $V_4 \simeq \chi \nu^{-1} \otimes W_4^- \oplus \chi^{-1} \otimes W_4^-$. Observe that the dual of $\chi \nu^{-1} \otimes W_4^-$ is $\chi^{-1} W_4^{-}$ as $\det W_4^- = \nu$. It follows that $V_4$ has Witt index $2$ as a representation of $\Gamma$, a contradiction. PS. PS.
By the same argument as the one given in the proof of Proposition \[lemmaindex2\], page , we obtain the equivalences between:
- $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(V)$, PS. PS.
- there exists a character $\beta : \widetilde{\Gamma} \rightarrow k^\times$ with $\beta^2=1$ such that $\beta$ is a summand of $W_{|\widetilde{\Gamma}}$, or which is the same, such that $W_3^\ast$ is isomorphic to $W_4^+ \otimes \beta$ or to $W_4^- \otimes \beta$,PS. PS.
- we have ${\rm ad}\, W_3 \simeq {\rm ad}\, W_4^+$ or ${\rm ad}\, W_3 \simeq {\rm ad}\, W_4^-$ as $k[\widetilde{\Gamma}]$-modules. PS. PS.
As we have $V_4 \simeq W_4^+ \otimes W_4^{-} \otimes \nu^{-1}$, we obtain a natural $k[\widetilde{\Gamma}]$-linear isomorphism $\Lambda^2 \, V_4 \simeq \Lambda^2 W_4^+ \otimes {\rm Sym}^2 W_4^- \otimes \nu^{-2} \oplus \Lambda^2 W_4^- \otimes {\rm Sym}^2 W_4^+ \otimes \nu^{-2}$, which also writes as $$\label{finpropso4} \Lambda^2 \, V_4 \simeq {\rm ad}\, W_4^- \oplus {\rm ad}\, W_4^+.$$ We are now ready to prove the proposition. Assume that $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(V)$. Assertion (c) above, relation and the isomorphism show that $V_3$ embeds in $\Lambda^2 \, V_4$. PS. PS.
Conversely, assume either that $V_3$ embeds in $\Lambda^2 \, V_4 \simeq {\rm ad}\, W_4^+ \oplus {\rm ad}\, W_4^-$, or that $V_3$ has an irreducible summand of dimension $\geq 2$ which injects into $\Lambda^2 V_4$. There exists in both cases a $k[\Gamma]$-submodule $H$ of $V_3$ of dimension $\geq 2$ such that $H$ embeds either in ${\rm ad}\, W_4^+$ or in ${\rm ad}\, W_4^-$. As both $V_3$ and ${\rm ad}\, W_4^+$ are semisimple of dimension $3$ and determinant $1$, this implies that $V_3$ is isomorphic either to ${\rm ad}\, W_4^+$ or to ${\rm ad}\, W_4^-$, thus (c) holds.
\[propso4cor\] Let $V$ be a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(V)$ a subgroup of Witt index $0$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. Assume furthermore that the $k[\Gamma]$-module $V$ has a summand of dimension $3$ (or $4$) and determinant $1$. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(V)$.
We have a $k[\Gamma]$-module decomposition $V \simeq V_3 \oplus V_4$ with $\dim V_i = i$ and $\det V_i=1$. As the $V_i$ are selfdual of determinant $1$ we easily checks that equation simplifies as $$1 \oplus \Lambda^2 \,V_4 \otimes V_3 \simeq V_3\oplus {\rm Sym}^2 \,V_3 \oplus {\rm Sym}^2 \,V_4.$$ Let $r_i$ be the number of irreducible summands of $V_i$. By Lemma \[lemmaind0\] (i), we have ${\rm t}(V_3) = {\rm t}(\Lambda^2 V_3) = 0$ and $${\rm t}(\Lambda^2 \,V_4 \otimes V_3) = r_3+r_4 -1.$$ In particular ${\rm t}(\Lambda^2 \,V_4 \otimes V_3)>0$, so there is a nonzero $k[\Gamma]$-linear morphism $V_3 \rightarrow \Lambda^2 \,V_4$. If $V_3$ is irreducible we are done by Proposition \[propso4\]. PS. PS.
Consider a character $\epsilon : \Gamma \rightarrow k^\ast$ and assume $V_3 \simeq \epsilon \oplus H$ with $H$ irreducible of dimension $2$ and determinant $\epsilon$. If $H$ occurs in $\Lambda^2 V_4$ then we are done again by the last assertion of Proposition \[propso4\]. We may thus assume ${\rm t}(\Lambda^2 \,V_4 \otimes H)=0$. We have then ${\rm t}(\Lambda^2 \,V_4 \otimes V_3) = {\rm t}(\Lambda^2 \,V_4 \otimes \epsilon) = 1+ r_4 \geq 2$. But thus is a contradiction as by Lemma \[lemmaind0\] (ii), we have ${\rm t}(\Lambda^2 V_4 \otimes \epsilon) \leq 2$, with $r_4\geq 2$ if the equality holds. PS. PS.
We may thus assume $V_3 = \oplus_{i=1}^3 \epsilon_i$ for some [*distinct*]{} characters $\epsilon_i$. Then: $$\sum_{i=1}^3 {\rm t}(\Lambda^2 \,V_4 \otimes \epsilon_i) = 2 + r_4 \geq 3.$$ If each term of the sum on the left is nonzero then $V_3$ embeds in $\Lambda^2\, V_4$ so we are done by Proposition \[propso4\], thus we may assume that one of them is $0$. If one term is $\geq 2$ then it is equal to $2$ and we have $r_4 \geq 2$ by Lemma \[lemmaind0\] (ii), so the two nonzero $ {\rm t}(\Lambda^2 \,V_4 \otimes \epsilon_i)$ are equal to $2$ and we have $r_4=2$, which contradicts Lemma \[lemmaind0\] (vi).
\[lemmaindex01\] Let $E$ be a $7$-dimensional nondegenerate quadratic space over $k$ and $\Gamma \subset {\rm SO}(E)$ of Witt index $0$ satisfying assumptions (i) and (ii) of the statement of Theorem \[mainthmb\]. Then the $k[\Gamma]$-module $E$ has no irreducible summand of dimension $5$ and at most $1$ irreducible summand of dimension $3$ and nontrivial determinant.
Assume first that we have a $\Gamma$-stable decomposition $E = P \bot V$ where $P$ has dimension $2$ and $V$ is irreducible of dimension $5$. The character $c=\det P$ satisfies $c^2=1$, and if we set $\Gamma^0 = {\rm ker}\, c$ then we have a natural morphism $\Gamma^0 \rightarrow {\rm SO}(V)$. In particular, there is a character $\chi : \Gamma^0 \rightarrow k^\times$ such that $P_{|\Gamma^0} \simeq \chi \oplus \chi^{-1}$. As $\Gamma^0$ has index $\leq 2$ in $\Gamma$, and as $\dim V = 5$ is odd, observe that $V_{|\Gamma^0}$ is irreducible. It follows that $\Gamma^0$ has Witt index $1$ and satisfies assumptions (i) and (ii) of Theorem \[mainthmb\]. This impossible by Proposition \[lemmaindex1\]. PS. PS.
Assume we have $E \simeq \epsilon \oplus V \oplus V'$ with $V,V'$ distinct, irreducible, selfdual, of dimension $3$. Set $c= \det V$ and $c'=\det V'$, we assume that $c$ and $c'$ are nontrivial, so the relation $cc'\epsilon=1$ implies $c \neq \epsilon$ and $c'\neq \epsilon$. We have $\Lambda^2 V \simeq V \otimes c$, $\Lambda^2 V' \simeq V' \otimes c'$, thus $$\Lambda^3 E \simeq c \oplus c' \oplus V \otimes V' \otimes (\epsilon \oplus c \oplus c') \oplus V \otimes c' \oplus V' \otimes c.$$ Let $\chi \in \{c,c',\epsilon\}$ be such that ${\rm t}(V \otimes V' \otimes \chi) \neq 0$. Then we have $\chi^2=1$, $V \simeq V' \otimes \chi$ and ${\rm t}(V \otimes V' \otimes \chi) = 1$. Taking determinant shows $c=c'\chi^3=c'\chi$, so $\chi = \epsilon$. This implies ${\rm t}(\Lambda^3\,E) = {\rm t}(V \otimes V' \otimes \epsilon) \leq 1$. On the other hand, equation and Lemma \[lemmaind0\] (i) imply ${\rm t}(\Lambda^3\,E) = {\rm t}(\epsilon) + 3$, a contradiction.
Theorem \[thmindex0\] holds when the $k[\Gamma]$-module $E$ contains at least two characters.
Assume we have $E \simeq \epsilon_1 \oplus \epsilon_2 \oplus V$ for some characters $\epsilon_i : \Gamma \rightarrow k^\times$. As $\dim V = 5$, $V^\ast \simeq V$ and $\det V = \epsilon_1 \epsilon_2$, we have $\Lambda^3 V \simeq \Lambda^2\, V \otimes \epsilon_1\epsilon_2$, hence an isomorphism $\Lambda^3 \,E \simeq \epsilon_1 \epsilon_2 \otimes V \oplus \Lambda^2\, V \otimes (\epsilon_1 \epsilon_2 \oplus \epsilon_1 \oplus \epsilon_2)$. If ${\rm t}(\epsilon_1 \epsilon_2 \otimes V ) \neq 0$, then $\epsilon_1\epsilon_2$ occurs in $V$, and $\epsilon_1 \oplus \epsilon_2 \oplus \epsilon_1 \epsilon_2$ is a $3$-dimensional summand of $E$ of determinant $1$, so we are done by Corollary \[propso4cor\]. We may thus assume ${\rm t}(V \otimes \epsilon_1 \epsilon_2)=0$, so that we have $${\rm t}(\Lambda^3 E) = {\rm t}(\Lambda^2\, V \otimes \epsilon_1 ) + {\rm t}(\Lambda^2\, V \otimes \epsilon_2 ) + {\rm t}(\Lambda^2\, V \otimes \epsilon_1 \epsilon_2 ),$$ and each term in the sum on the right is $\leq 2$ by Lemma \[lemmaind0\] (iv). On the other hand we have ${\rm t}(\Lambda^3 E) ={\rm t}(E) + 2+ r$ where $r$ is the number of irreducible summands of $V$ by Equation and Lemma \[lemmaind0\] (i). In particular, there exists a character $\chi$ of $\Gamma$ such that ${\rm t}(\Lambda^2\, V \otimes \chi) \neq 0$. This implies $r\geq 2$ by Lemma \[lemmaind0\] (iii), so there even exists a character $\chi$ of $\Gamma$ such that ${\rm t}(\Lambda^2\, V \otimes \chi) = 2$. But this implies that $\det V$, that is $\epsilon_1 \epsilon_2$, is a summand of $V$ by the same lemma assertion (v) (note $V' \simeq \det V$ in the notations [*loc. cit.*]{}), which contradicts ${\rm t}(V \otimes \epsilon_1 \epsilon_2)=0$.
Theorem \[thmindex0\] holds when the $k[\Gamma]$-module $E$ contains at least two irreducible summands of dimension $2$.
Assume we have $E \simeq P_1 \oplus P_2 \oplus V$ where each $P_i$ is irreducible of dimension $2$. Set $c_i = \det P_i$. As $V$ is selfdual of dimension $3$ and determinant $c_1 c_2$ we easily check the isomorphism $$\Lambda^3 E \simeq c_1c_2 \oplus P_1 \otimes c_2 \oplus P_2 \otimes c_2 \oplus V \otimes (c_1 \oplus c_2 \oplus P_1 \otimes c_1c_2 \oplus P_2 \otimes c_1c_2 \oplus P_1\otimes P_2).$$ By Corollary \[propso4cor\] we may assume that $E$ has no summand of dimension $3$ and determinant $1$. In particular, we have $c_1 c_2 \neq 1$ (consider $V$), ${\rm t}(V \otimes c_i) =0$ (consider $c_i \oplus P_i$) and ${\rm t}(V \otimes P_i \otimes c_1 c_2)=0$ otherwise we have $V \simeq P_i \otimes c_1c_2 \oplus c_j$ with $j\neq i$, and thus ${\rm t}(V \otimes c_j) \neq 0$. We obtain $${\rm t}(\Lambda^3 E) = {\rm t}(V \otimes P_1\otimes P_2).$$ On the other hand we have ${\rm t}(\Lambda^3 E) ={\rm t}(V) + 2+ r \geq 3$, where $r$ is the number of irreducible summands of $V$. As $P_1$ is irreducible of dimension $2$ and $\dim V \otimes P_2 = 6$, we have ${\rm t}(V \otimes P_1\otimes P_2) = \dim {\rm Hom}_{k[\Gamma]}(P_1,V\otimes P_2) \leq 3$. It follows that $r=1$, [*i.e.* ]{} $V$ is irreducible. As $\dim P_1 \otimes P_2 =4$, we thus have $3 \leq {\rm t}(V \otimes P_1\otimes P_2)= \dim {\rm Hom}_{k[\Gamma]}(V,P_1\otimes P_2) \leq 1$, a contradiction.
Theorem \[thmindex0\] holds when the $k[\Gamma]$-module $E$ contains exactly one irreducible summand of each dimension $1, 2$ and $4$.
Assume we have $E \simeq \epsilon \oplus P \oplus V$ where $P$ and $V$ are irreducible of respective dimension $2$ and $4$. Set $c = \det P$. It is an order $2$ character of $\Gamma$ as ${\rm SO}(P)$ acts reducibly on $P$. By Corollary \[propso4cor\] we may assume $c\epsilon\neq 1$. We easily check $\Lambda^3 E \simeq c\epsilon \oplus V \otimes (\epsilon c \oplus c \oplus P\otimes \epsilon ) \oplus \Lambda^2\, V \otimes (\epsilon \oplus P)$. By a standard argument (from now on) we thus get the equality $$\label{flemmpass} {\rm t}(\epsilon) + 3 = {\rm t}(\Lambda^2 V \otimes \epsilon) + {\rm t}(\Lambda^2 V \otimes P).$$ The number ${\rm t}(\Lambda^2 V \otimes P)$ is the multiplicity of $P$ as a submodule $\Lambda^2 V$. So we have ${\rm t}(\Lambda^2 V \otimes P) \leq \frac{\dim \Lambda^2 V}{\dim P} = 3$. As ${\rm t}(\Lambda^2 \,V )=0$ by Lemma \[lemmaind0\] (i), the identity implies $\epsilon \neq 1$. The irreducibility of $V$ implies ${\rm t}(\Lambda^2 V \otimes \epsilon)\leq 1$, so $P$ occurs at least twice in $\Lambda^2 V$. As $\det \Lambda^2 \,V = (\det V)^3 = \epsilon c$, and $\epsilon \neq 1$, we cannot have $\Lambda^2\, V \simeq P \oplus P \oplus P$. We have thus $\Lambda^2\, V \simeq P \oplus P \oplus \epsilon \oplus c$. PS. PS.
On the other hand the character $\epsilon$ is a summand of $E$, hence of $\Lambda^3\, E$ as well by equation . As $1$ does not occur in $\Lambda^2 V$, the only possibility is that $\epsilon$ is a summand of $\Lambda^2\, V \otimes P$. But we have $\Lambda^2\, V \otimes P \simeq P \otimes (P \oplus P) \oplus P \otimes (\epsilon \oplus c)$. As a consequence, $\epsilon$ occurs in $P \otimes P \simeq c \oplus {\rm Sym}^2\, P$, hence in ${\rm Sym}^2\, P$. As ${\rm Sym}^2\, P$ is semisimple of determinant $c^3=c$, we have proved ${\rm Sym}^2\, P \simeq 1\oplus \epsilon \oplus \epsilon c$. PS. PS.
Let $\Gamma^0$ be the kernel of the natural morphism $\Gamma \rightarrow {\rm O}(P)$. What we have just proved about ${\rm Sym}^2\, P$ implies that $\Gamma/\Gamma^0$ is a (nonabelian) finite group of order $8$ and $\epsilon(\Gamma^0)=1$. Moreover, $\Gamma^0$ acts trivially on $\Lambda^2 V$, hence by scalars of square $1$ on $V$, so $|\Gamma^0|\leq 2$ and $|\Gamma|\leq 16$. This contradicts the inequality $|\Gamma| \geq 1+1+ (\dim P)^2 + (\dim V)^2 = 22$.
Theorem \[thmindex0\] holds when the $k[\Gamma]$-module $E$ contains exactly one irreducible summand of each dimension $3$ and $4$.
Assume we have $E \simeq V_3 \oplus V_4$ with each $V_i$ irreducible of dimension $i$. Set $c=\det V_3 = \det V_4$. We have an isomorphism $\Lambda^3 \, E \simeq c \oplus V_4 \otimes c \oplus V_3 \otimes V_4 \otimes c \oplus V_3 \otimes \Lambda^2\, V_4$ and ${\rm t}(\Lambda^3 \, E)=2$. We thus get $$2 = {\rm t}(c) + {\rm t}(V_3 \otimes \Lambda^2\, V_4 ).$$ We have ${\rm t}(V_3 \otimes \Lambda^2\, V_4 )\leq \frac{\dim \Lambda^2 \, V_4}{\dim V_3}=2$. If $c \neq 1$ we obtain ${\rm t}(V_3 \otimes \Lambda^2\, V_4)=2$, hence $\Lambda^2\, V_4 \simeq V_3 \oplus V_3$. But this implies $\det \Lambda^2 \,V_4 = 1$, which contradicts $\det \Lambda^2 \,V_4 = (\det V_4)^3 = c^3 = c$. So we have $c = \det \, V_3 = 1$ and we conclude by Corollary \[propso4cor\].
Theorem \[thmindex0\] holds when the $k[\Gamma]$-module $E$ contains exactly one irreducible summand of each dimension $1$ and $6$.
Assume we have $E \simeq \epsilon \oplus V$ with $V$ irreducible of dimension $6$. Let us show first that $\epsilon$ has order $2$. If $\epsilon = 1$ then we have ${\rm t}(E \oplus {\rm Sym}^2 E)={\rm t}(\Lambda^3\, E) = 3$ and $\Lambda^3 E \simeq \Lambda^3 V \oplus \Lambda^2 V$. As ${\rm t}(\Lambda^2 V) =0 $ we obtain ${\rm t}(\Lambda^3 V) = 3$. As $V$ is selfdual, the $k[\Gamma]$-module $\Lambda^3 V$ embeds in $V \otimes \Lambda^2 V$. In particular we have ${\rm t}(V \otimes \Lambda^2\,V) \geq 3$. This contradicts the inequality $\dim \Lambda^2 V = 15 < 3 \dim V$. PS. PS.
Set $\Gamma^0 = {\rm ker }\, \epsilon$. As $\epsilon$ has order $2$ this is a subgroup of index $2$ in $\Gamma$ and $E_{|\Gamma^0}$ still satisfies assumption (i) and (ii) of Theorem \[mainthmb\]. The previous paragraph thus shows that $V_{|\Gamma^0}$ is reducible. By Clifford theory it is thus a direct sum of two nonisomorphic irreducible $3$ dimensional representations of $\Gamma^0$. Those representations are not selfdual by the last assertion of Lemma \[lemmaindex01\]. We have thus $V_{|\Gamma^0} \simeq W \oplus W^\ast$ for some irreducible, nonselfdual, representation $W$ of $\Gamma^0$. As a consequence, $\Gamma^0$ has Witt index $3$, and we are in [*Case 1.*]{} of the proof of Proposition \[lemmaindex3\], whose conclusion is $\det W = 1$. We conclude the proof by the following lemma.
Let $I$ be a finite dimensional $k$-vector space. Define ${\rm N}(I)$ as the subgroup of ${\rm O}({\rm H}(I))$ of elements $g$ such that the couple of subspaces $(g(I),g(I^\ast))$ is either $(I,I^\ast)$ or $(I^\ast,I)$. The action of ${\rm N}(I)$ on the set with two elements $\{I,I^\ast\}$ defines a group homomorphism ${\rm N}(I) \rightarrow {\mathbb{Z}}/2{\mathbb{Z}}$. The map $g \mapsto g_{|I}$ identifies the kernel of this morphism with the group ${\mathrm{GL}}(I)$, and the inverse image of $1 \in {\mathbb{Z}}/2{\mathbb{Z}}$ with the set of $k$-linear isomorphisms $I \rightarrow I^\ast$. If $\varphi$ is a symmetric isomorphism $I \rightarrow I^\ast$ and $g \in {\mathrm{GL}}(I)$ then $\varphi g^{-1} \varphi^{-1}$ is the adjoint of $g$ with respect to $\varphi$ and $\varphi$ has order $2$ in ${\rm N}(I)$. In particular we have ${\rm N}(I) \simeq {\rm GL}(I) \rtimes {\mathbb{Z}}/2{\mathbb{Z}}$. Set $E = {\rm H}(I) \bot k$. The morphism $g \mapsto g \bot \det g$ allows to see ${\rm O}({\rm H}(I))$ as a subgroup of ${\rm SO}(E)$. PS.
Let $I$ be a vector space of dimension $3$ over the algebraically closed field $k$ and $E = {\rm H}(I) \bot k$. Let ${\rm N}(I) \subset {\rm SO}(E)$ be as above and $\Gamma$ a subgroup of ${\rm N}(I)$ such that $\Gamma \cap {\rm {\mathrm{GL}}}(I) \subset {\rm SL}(I)$. Then $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(E)$.
We have defined above natural inclusions ${\mathrm{GL}}(I) \subset {\rm N}(I) \subset {\rm SO}(E)$. Let $H$ be the inverse image of ${\mathrm{GL}}(I)$ under the surjective morphism $\pi_E : {\rm GSpin}(E) \rightarrow {\rm SO}(E)$ and let $W$ be a spinor module for $E$. Recall from §\[parclag\] and Propositions \[cliffordhyp\] and \[decospin\] that there is a natural isomorphism $\epsilon : k^\times \times {\mathrm{GL}}(I) {\overset{\sim}{\rightarrow}}H$ such that the map ${\pi_E} \circ \epsilon : k^\times \times {\mathrm{GL}}(I) \rightarrow {\mathrm{GL}}(I)$ coincides with the projection onto the second factor and such that the restriction of $W$ to $H$ is isomorphic to the representation $\alpha \otimes \Lambda\,I$ where $\alpha : H \rightarrow k^\times$ is the character obtained as the composition of $\epsilon^{-1}$ and of the first projection $k^\times \times {\mathrm{GL}}(I) \rightarrow k^\times$. PS. PS.
Let $G$ be the inverse image of ${\rm N}(I)$ under $\pi_E$ and fix a symmetric isomorphism $\varphi : I \rightarrow I^\ast$. Then $H$ has index $2$ in $G$ and we may choose $\tau \in G-H$ such that the restriction to $I \subset E$ of the element $\pi_E(\tau) \in {\rm N}(I)$ coincides with $\varphi$. By the observations preceding the lemma there is a unique map $\psi : {\mathrm{GL}}(I) \rightarrow k^\times$ with $$\tau \,\epsilon(\lambda, g) \,\tau^{-1}\, = \,\epsilon(\lambda \psi(g),{}^{\rm t}\!g^{-1})$$ for all $\epsilon(\lambda,g) \in H$, where ${}^{\rm t}\!g$ is the adjoint of $g$ with respect to $\varphi$. This implies that $\psi$ is a group homomorphism. Applying $\nu_E$ we obtain the relation $(\det \, \cdot \,\psi^{-1})^2=1$, hence $\psi = \det $ as $k$ is algebraically closed. PS. PS.
The $k[H]$-module isomorphism $W \simeq \alpha \otimes \Lambda\, I$ shows that there are exactly $2$ lines in $W$ which are stable by $H$, let us call $P$ their direct sum. Then $P$ is stable by $G$ and we have a $k[H]$-linear isomorphism $P \simeq \alpha \oplus \alpha \det I$. The previous paragraph shows that the outer conjugation of $G$ on $H$ exchanges the (distinct) characters $\alpha$ and $\alpha \det I$. As $\dim I=3$, recall from §\[octoalg\] that $W$ may be equipped with a structure of nondegenerate quadratic space over $k$ such that ${\rm GSpin}(E)$ acts on $W$ as orthogonal similitudes of factor $\nu_E$. By definition, $P$ is necessarily nondegenerate, and as neither $\alpha$ nor $\alpha \det I$ has square $1$ on $H$ it follows that the two stable lines of $P$ are its isotropic lines, and that $\tau$ exchanges them. PS. PS.
So far the group $\Gamma$ did not play any role. Set now $\widetilde{\Gamma} = \pi_E^{-1}(\Gamma) \cap {\rm Spin}(E)$ and $\widetilde{\Gamma}^0 = H \cap \widetilde{\Gamma}$. We may and will view $\alpha$ and $I$ as representations of $\widetilde{\Gamma}$. We have a natural morphism $\rho : \widetilde{\Gamma} \rightarrow {\rm O}(P)$, with $P_{|\widetilde{\Gamma}^0} \simeq \alpha_{|\widetilde{\Gamma}} \oplus \alpha_{|\widetilde{\Gamma}} \det I $. But we also have the relation $\alpha^2 \det I = \nu_E = 1$ on $\widetilde{\Gamma}$, as well as $\det I = 1$ on $\widetilde{\Gamma}^0$ by assumption. As a consequence, $\widetilde{\Gamma}^0$ acts by $\pm {\rm id}$ on $P$. If $\widetilde{\Gamma}^0 \neq \widetilde{\Gamma}$, the previous paragraph shows $\rho(\widetilde{\Gamma}) \subsetneq {\rm SO}(P)$, thus the kernel of any orthogonal symmetry in $\rho(\widetilde{\Gamma})$ is a nondegenerate one-dimensional subspace of $P$ on which $\widetilde{\Gamma}$ acts via an order $2$ character. We conclude the proof by Proposition \[propgspin7\] (c).
This ends the proof of Theorem \[thmindex0\], hence of Theorem \[mainthmb\]. PS.
Proof of Theorem \[thmintroc\] {#pfthmc}
==============================
The existence, and uniqueness up to conjugacy, of the morphisms denoted by $\alpha, \beta, \gamma$ before the statement of Theorem \[thmintroc\] are consequences of §\[parexamples\] and assertion (b) of the following lemma applied to $K={\rm SO}(7)$ and $G = {\rm SO}(E \otimes {\mathbb{C}})$.
\[lemmecompactcomplex\] Let $K$ be a compact connected Lie group and $\iota : K \hookrightarrow G$ a complexification of $K$.
- If we have $x,y \in K$ and $g \in G$ with $g\,\iota(x)\,g^{-1}\,=\,\iota(y)$, and if $g\,=\,\iota(k)\,s$ is the polar [(]{}[*i.e.*]{} Cartan[)]{} decomposition of $g$ with respect to $\iota(K)$, then we have $k \,x\, k^{-1}\, =\, y$. PS.
- Let $\Gamma$ be a compact group and $\rho : \Gamma \rightarrow G$ a continuous group homomorphism. Then there exists a continuous group morphism $\rho' : \Gamma \rightarrow K$ such $\iota \circ \rho'$ is $G$-conjugate to $\rho$, and such a $\rho'$ is unique up to $K$-conjugacy.
(see e.g. [@serrelie p.1]) In order to prove (a), we rewrite the hypothesis as an equality $\iota(kx)\,(\iota(x)^{-1}\,s\,\iota(x)) \,=\, \iota(yk)\, s$ and conclude by the uniqueness of the polar decomposition. Let us prove (b). The group $\rho(\Gamma)$ is a compact subgroup of $G$ and a well-known result of Cartan asserts that the maximal compact subgroups of $G$ are the conjugate of $\iota(K)$. This shows the existence of $\rho'$, and its uniqueness up to $K$-conjugacy follows from (a).
Applying this lemma again both to the compact group ${\rm SO}(7)$ and its ${\rm G}_2$-subgroups we conclude that Theorem \[thmintroc\] of the introduction follows from the case $k={\mathbb{C}}$ of Theorem \[mainthmb\]. $\square$
Automorphic and Galois representations {#autgalrep}
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{#notationsaut}
Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers, and ${\mathbb{A}}_F$ its adèle ring. For each place $v$ of $F$ we denote by $F_v$ the associated completion of $F$ and $\mathcal{O}_v \subset F_v$ its ring of integers. PS. PS.
Let $G$ be a connected linear algebraic group defined and reductive over $\mathcal{O}_F$. We refer to [@boreljacquet] (see also [@cogdell §3] when $G={\rm GL}_n$) for the notion of [*cuspidal automorphic representation*]{} of $G({\mathbb{A}}_F)$. Such a representation $\pi$ has, for each place $v$ of $F$, a [*local component*]{} $\pi_v$ which is irreducible and well-defined up to isomorphism. When $v$ is a finite place, $\pi_v$ is a complex [*smooth representation*]{} of $G(F_v)$. When $v$ is an Archimedean place, $\pi_v$ is a complex $(\mathfrak{g}_v,K_v)$-[*module*]{} in the sense of Harish-Chandra, where $K_v$ is a fixed maximal compact subgroup of the Lie group $G(F_v)$ and $\mathfrak{g}_v$ is the Lie algebra of $G(F_v)$. PS. PS.
Let $v$ be a finite place of $F$. Recall that a complex, irreducible, smooth representation $\varpi$ of the locally profinite group $G(F_v)$ is called [*unramified*]{} if its underlying space has a nonzero vector invariant by the compact open subgroup $G(\mathcal{O}_v)$. According to Satake and Langlands, the set of isomorphism classes of unramified representations of $G(F_v)$ is in canonical bijection with the set of conjugacy classes of semisimple elements in $\widehat{G}$ [@euler; @borel]. Here, $\widehat{G}$ is a dual group for $G$ in the sense of Langlands, namely a reductive (connected) complex linear algebraic group whose based root datum is equipped with an isomorphism with the dual based root datum of $G$. In particular we may and shall take $\widehat{{\mathrm{GL}}_n}={\mathrm{GL}}_n({\mathbb{C}})$. We shall denote by ${\rm c}(\varpi) \subset \widehat{G}$ the semisimple conjugacy class associated to $\varpi$. The unramified representation $\varpi$ of $G(F_v)$ is said [*tempered*]{} if ${\rm c}(\varpi)$ is the conjugacy class of an element lying in a compact subgroup of $\widehat{G}$ (viewed as a Lie group). PS. PS.
If $\pi$ is a cuspidal automorphic representation of $G({\mathbb{A}}_F)$, then $\pi_v$ is unramified for all but finitely many finite places $v$ of $F$. Let ${\rm Ram}(\pi)$ denote the (finite) set of places $v$ of $F$ which are either Archimedean, or such that $v$ is finite and $\pi_v$ is not unramified. If $S$ is a finite set of places of $F$ containing ${\rm Ram}(\pi)$, we say that $\pi$ is $S$-tempered if $\pi_v$ is tempered for all $v \notin S$. We say that $\pi$ is tempered if it is $S$-tempered for some such $S$. Conjecturally, “$S$-tempered” implies “${\rm Ram}(\pi)$-tempered”. PS. PS.
Let $\pi$ be a cuspidal automorphic representation of $G({\mathbb{A}}_F)$, $S$ a finite set of places of $F$ containing ${\rm Ram}(\pi)$, and $\rho : \widehat{G} \rightarrow {\mathrm{GL}}_n({\mathbb{C}})$ a polynomial representation. Following Langlands [@euler] consider the Euler product $${\rm L}^S(s,\pi,\rho) = \prod_{v \notin S} \frac{1}{\det(1- q_v^{-s} \rho({\rm c}(\pi_v)))},$$ where $q_v$ denotes the cardinality of the residue field of $\mathcal{O}_v$. If $\pi$ is $S$-tempered, this product is well-defined and absolutely convergent in the half plane ${\rm Re}\, s >1$. In general, Langlands has shown that ${\rm L}^S(s,\pi,\rho)$ is absolutely convergent for ${\rm Re}\, s$ big enough. PS. PS.
Let $v$ be a real place of $F$, [*i.e.*]{} $F_v={\mathbb{R}}$. For any irreducible $(\mathfrak{g}_v,K_v)$-module $\varpi$, the center $\mathfrak{z}_v$ of the enveloping algebra of $\mathfrak{g}_v$ acts by scalars on $\varpi$. The resulting ${\mathbb{C}}$-algebra homomorphism $\mathfrak{z}_v \rightarrow {\mathbb{C}}$ is called the [*infinitesimal character*]{} of $\varpi$; it may be viewed according to Harish-Chandra and Langlands as a semisimple conjugacy class ${\rm c}(\varpi)$ in the complex Lie algebra $\widehat{\mathfrak{g}}$ of $\widehat{G}$ [@langlandspb; @euler; @borel]. When $G = {\mathrm{GL}}_n$, in which case we have $\widehat{\mathfrak{g}} = {\rm M}_n({\mathbb{C}})$, we shall say that $\varpi$ is [*algebraic*]{} if the eigenvalues of ${\rm c}(\varpi)$ are integers, which are then called the [*weights*]{} of $\varpi$. An algebraic $\varpi$ is [*regular*]{} if its weights are distinct. PS. PS.
{#algreg}
Assume $F$ is totally real and $n\geq 1$ is an odd integer. Let $\pi$ be a cuspidal automorphic representation of ${\rm GL}_n({\mathbb{A}}_F)$ such that $\pi_v$ is algebraic regular for each real place $v$ of $F$. Then there exists a number field $E \subset {\mathbb{C}}$, called [*a coefficient field for $\pi$*]{}, such that for each finite place $v$ of $F$ the representation $\pi_v$ is defined over $E$ [@clozel §3.5]. In particular, if $\pi_v$ is unramified then the characteristic polynomial $\det(t -{\rm c}(\pi_v))$ belongs to $E[t]$. Let $\ell$ be a prime number, $\lambda$ a place of $E$ above $\ell$, and $\overline{E_\lambda}$ an algebraic closure of $E_\lambda$. Assume furthermore that $\pi$ is [*selfdual*]{}, [*i.e.*]{} isomorphic to its contragredient $\pi^\vee$. Then by [@harristaylor; @shin; @chharris] we know that there exists a continuous semisimple representation $$\label{galoisrepeq} r_{\pi,\lambda} : {\rm Gal}(\overline{F}/F) \longrightarrow {\rm GL}_n(\overline{E_\lambda}),$$ unique up to isomorphism, satisfying the following property: for each finite place $v$ of $F$ which is prime to $\ell$, and such that $\pi_v$ is unramified, the Galois representation of $r_{\pi,\lambda}$ is unramified at $v$ (see e.g. [@serreabelian §2.1]) and the characteristic polynomial of a Frobenius element ${\rm Frob}_v$ at $v$ satisfies $\det (t -r_{\pi,\lambda}({\rm Frob}_v)) = \det ( t - {\rm c}(\pi_v))$. PS. PS.
{#thmsgalois}
We shall denote by ${\rm G}_2$ the automorphism group scheme of the standard split octonion algebra over ${\mathbb{Z}}$. This is a linear algebraic group, which is split and semisimple over ${\mathbb{Z}}$, and [*of type ${\bf G}_2$*]{}. Recall that for any algebraically closed field $k$ of characteristic $0$, there is up to isomorphism a unique $7$-dimensional irreducible $k$-linear algebraic representation of ${\rm G}_2(k)$ (see Lemma \[lemmerepg2\]). We shall denote by $\rho : {\rm G}_2(k) \rightarrow {\rm GL}_7(k)$ such a representation. PS. PS.
\[thmgalois\] Let $F$ be a totally real number field, $\pi$ a cuspidal automorphic representation of ${\rm GL}_7({\mathbb{A}}_F)$ such that $\pi_v$ is algebraic regular for each real place $v$ of $F$, and $E$ a coefficient field for $\pi$. Assume that $\pi$ satisfies the following condition: PS.
[(G)]{} [*For all but finitely many places $v$ of $F$ such that $\pi_v$ is unramified, then ${\rm c}(\pi_v)$ is the conjugacy class of an element in $\rho({\rm G}_2({\mathbb{C}}))$.*]{} PS. PS.
Then for any prime $\ell$ and any place $\lambda$ of $E$ above $\ell$, there is a morphism $$\widetilde{r}_{\pi,\lambda}: {\rm Gal}(\overline{F}/F) \longrightarrow {\rm G}_2(\overline{E_\lambda}),$$ unique up to ${\rm G}_2(\overline{E_\lambda})$-conjugacy, such that the representation $\rho \circ \widetilde{r}_{\pi,\lambda}$ is isomorphic to the representation $r_{\pi,\lambda}$ introduced in above.
By the Cebotarev theorem, an equivalent way to state the theorem, but without any reference to $r_{\pi,\lambda}$, is as follows.PS. PS.
\[cor1thmgalois\] Let $F$, $\pi$, $E$, $\ell$ and $\lambda$ be as in the statement of Theorem \[thmgalois\]. Then there exists a continuous semisimple morphism $$\widetilde{r}_{\pi,\lambda}: {\rm Gal}(\overline{F}/F) \longrightarrow {\rm G}_2(\overline{E_\lambda}),$$ unique up to ${\rm G}_2(\overline{E_\lambda})$-conjugacy, satisfying the following property: for each finite place $v$ of $F$ which is prime to $\ell$, and such that $\pi_v$ is unramified, the morphism $\widetilde{r}_{\pi,\lambda}$ is unramified at $v$ and we have $\det (t - \rho(\widetilde{r}_{\pi,\lambda}({\rm Frob}_v))) = \det (t - {\rm c}(\pi_v))$.
In this statement, the term [*semisimple*]{} means that the neutral component of the Zariski-closure of the image of $\widetilde{r}_{\pi,\lambda}$ is reductive. It is equivalent to ask that for some (resp. any) faithful polynomial representation $f$ of ${\rm G}_2$ over $\overline{E_\lambda}$ the representation $f \circ \widetilde{r}_{\pi,\lambda}$ is semisimple. PS. PS.
(of Theorem \[thmgalois\]) Assumption (G) on $\pi$ implies that for all but finitely many places $v$ of $F$ such that $\pi_v$ is unramified, the conjugacy class ${\rm c}(\pi_v) \subset {\rm GL}_7({\mathbb{C}})$ is equal to its inverse ${\rm c}(\pi_v^\vee)$. As $\pi^\vee$ is a cuspidal automorphic representation of ${\mathrm{GL}}_7({\mathbb{A}}_F)$, the strong multiplicity one theorem of Piatetski-Shapiro, Jacquet and Shalika shows that $\pi$ is selfdual, so that the discussion of §\[algreg\] applies to $\pi$. PS. PS. Consider a Galois representation $r_{\pi,\lambda}$ as in . By the main theorem of [@bc], there is a structure of non-degenerate quadratic space on the underlying space $V = \overline{E_\lambda}^7$ of $r_{\pi,\lambda}$ such that the image $\Gamma$ of $r_{\pi,\lambda}$ is a subgroup of ${\rm SO}(V)$. To conclude the existence of a morphism $\widetilde{r}_{\pi,\lambda}$ such that $\rho \circ \widetilde{r}_{\pi,\lambda}$ is isomorphic to $r_{\pi,\lambda}$ it is enough to show that $\Gamma$ is contained in a ${\rm G}_2$-subgroup of ${\rm SO}(V)$ (§\[defg2sb\]).PS. PS.
The semisimplicity of $r_{\pi,\lambda}$ implies that $\Gamma$ satisfies assumption (i) of Theorem \[mainthmb\] (i.e. assumption [(S)]{}). The assumption (G) on $\pi$, Cebotarev’s theorem and Remark \[rempotg2\] ensure that $\Gamma$ satisfies as well assumption (ii) of this theorem. So Theorem \[mainthmb\] applies, and it is enough to show that $\Gamma$ is not as in case (a), (b), (c) or (d) of its statement. By the concrete description of these specific cases, observe that it is enough to show that there is no open subgroup $\Gamma'$ of the profinite group $\Gamma$ such that the subspace of invariants $V^{\Gamma'} \subset V$ has dimension $\geq 3$ over $\overline{E_\lambda}$.PS. PS.
Let $w$ be a finite place of $F$ dividing $\ell$. The restriction to a decomposition group at $w$ of the representation $r_{\pi,\lambda}$, that we may view as a representation $r_w: {\rm Gal}(\overline{F_w}/F_w) \rightarrow {\mathrm{GL}}_7(\overline{E_\lambda})$, is known to be of Hodge-Tate type [@shin; @chharris]. More precisely, the Hodge-Tate weights of $r_w$, relative to any field embedding $F_w \longrightarrow \overline{E_\lambda}$, are the weights of the algebraic regular representation $\pi_v$, for a suitable real place $v$ of $F$ (depending on the choice of that field embedding). In particular, those weights are distinct integers and only one of them is zero (note that if $k$ is weight then so is $-k$ by selfduality of $r_{\pi,\lambda}$). As a general fact, the same properties trivially hold for the restriction of $r_w$ to ${\rm Gal}(\overline{F_w}/L)$ for any finite extension $F_w \subset L \subset \overline{F_w}$. In particular, if $L$ is any such extension, the multiplicity of the Hodge-Tate weight $0$ for ${r_w}_{|{\rm Gal}(\overline{F_w}/L)}$, relative to any field embedding $L \rightarrow
\overline{E_\lambda}$, is at most one. But this implies that the dimension of the space of invariants of $V$ under $r_w({\rm Gal}(\overline{F_w}/L))$ is at most one as well, and we are done. PS. PS.
It only remains to justify the uniqueness assertion. But this is a consequence of a result of Griess [@griess Thm. 1] (see also [@larsen Prop. 2.8]).
[If $r_{\pi,\lambda}$ is irreducible, which is conjectured to always be the case but is still an open problem at the moment, we can use in the proof above Theorem \[thmbirrcase\] instead of Theorem \[mainthmb\], whose proof is much easier. The known compatibilty of $r_{\pi,\lambda}$ with the local Langlands correspondence shows that this irreducibility assumption is automatically satisfied if, for some finite place $v$, the representation $\pi_v$ is square integrable (modulo the center). In the special case where $\pi_v$ is the Steinberg representation for some $v$, Theorem \[thmgalois\] even has a shorter proof, as already observed by Magaard and Savin in [@ms §7]. See also [@kls] for some quite specific special cases of Theorem \[thmgalois\]. ]{}
In the irreducible case, we can draw slightly further conclusions about the field of definition of $\widetilde{r}_{\pi,\lambda}$. PS. PS.
\[cor2thmgalois\] Let $F$, $\pi$, $E$, $\ell$ and $\lambda$ be as in the statement of Theorem \[thmgalois\]. Assume furthermore that $r_{\pi,\lambda}$ is irreducible. Then the conclusion of Corollary \[cor1thmgalois\] holds with ${\rm G}_2(\overline{E_\lambda})$ replaced [(]{}twice[)]{} by ${\rm G}_2(E_\lambda)$.
Choose $\widetilde{r}_{\pi,\lambda}$ as in Theorem \[thmgalois\] and denote by $C$ the centralizer of the image of $\widetilde{r}_{\pi,\lambda}$ in ${\rm G}_2(\overline{E_\lambda})$. The irreducibility of $r_{\pi,\lambda}$ and Schur’s lemma imply that $C$ is central in ${\rm G}_2(\overline{E_\lambda})$, hence the equality $C=\{1\}$. Set $k=E_\lambda$, $\overline{k}=\overline{E_\lambda}$, $\Gamma= {\rm Gal}(\overline{E_\lambda}/E_\lambda)$ and $G={\rm G}_2$. Then $\Gamma$ naturally acts on $G(\overline{k})$, with fixed points $G(k)$. Let $\sigma \in \Gamma$. Observe that we have $$\det(t - \rho \circ \sigma \circ \widetilde{r}_{\pi,\lambda}(g))=\sigma(\det(t-r_{\pi,\lambda}(g)))=\det(t-r_{\pi,\lambda}(g))=\det(t - \rho \circ \widetilde{r}_{\pi,\lambda}(g))$$ for all $g \in {\rm Gal}(\overline{F}/F)$. The uniqueness part of the statement of Theorem \[thmgalois\] implies the existence of an element $P_\sigma \in G(\overline{k})$ such that we have $\sigma \circ \widetilde{r}_{\pi,\lambda} = {\rm int}_{P^{-1}_\sigma} \circ \widetilde{r}_{\pi,\lambda}$ (where ${\rm int}_g$ denotes the inner automorphism $x \mapsto gxg^{-1}$ of $G(\overline{k})$). As $C=\{1\}$ the element $P_\sigma$ is uniquely determined by this relation. This implies that the map $\sigma \mapsto P_\sigma$ is a $1$-cocycle of $\Gamma$ acting on $G(\overline{k})$; it is continuous for the discrete topology on $G(\overline{k})$ as we have ${\rm Im}\, \widetilde{r}_{\pi,\lambda} \subset G(K)$ for some finite extension $k \subset K \subset \overline{k}$ by Baire’s theorem. The corresponding cohomology group classifies the isomorphism classes of octonion $k$-algebras. But since $k$ is a nonarchimedean local field, there is a unique such $k$-algebra up to isomorphism, as any $8$-dimensional non-degenerate quadratic space does represent $0$ [@blijspringer1 §2,§3]. This shows the existence of $P \in G(\overline{k})$ with $P_\sigma = P^{-1}\sigma(P)$ for all $\sigma \in \Gamma$, and then that ${\rm int}_{P} \circ \widetilde{r}_{\pi,\lambda}$ is fixed under $\Gamma$. This shows the existence assertion. The uniqueness follows from the uniqueness part of the statement of Theorem \[thmgalois\] and from $C=\{1\}$.
Our next application concerns the local components of a cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_7({\mathbb{A}}_F)$ satisfying assumption (G) of Theorem \[thmgalois\]. We refer to [@tate §4] for the various notions of Weil groups, Weil-Deligne groups and representations. Let $v$ be a finite place of $F$ and ${\rm W}_{F_v} \subset {\rm Gal}(\overline{F_v}/F_v)$ the Weil group of $F_v$. Recall that the local Langlands correspondence, proved by Harris and Taylor [@harristaylor], may be viewed as a natural bijection $\varpi \mapsto {\rm L}(\varpi)$ from the set of isomorphism classes of irreducible smooth complex representations of ${\rm GL}_n(F_v)$ and onto the set of isomorphism classes of continuous semisimple representations ${\rm W}_{F_v} \times {\rm SU}(2) \rightarrow {\mathrm{GL}}_n({\mathbb{C}})$. PS. PS.
\[corcplx\] Let $F$ be a totally real number field and $\pi$ a cuspidal automorphic representation of ${\rm GL}_7({\mathbb{A}}_F)$ such that $\pi_v$ is algebraic regular for each real place $v$ of $F$. The following properties are equivalent:
- $\pi$ satisfies property [(G)]{} of the statement of Theorem \[thmgalois\],PS. PS.
- for any finite place $v$ of $F$, there exists a continuous semisimple morphism $\phi_v : {\rm W}_{F_v} \times {\rm SU}(2) \rightarrow {\rm G}_2({\mathbb{C}})$, unique up to ${\rm G}_2({\mathbb{C}})$-conjugacy, such that $\rho \circ \phi_v$ is isomorphic to ${\rm L}(\pi_v)$.PS.
The nontrivial assertion is (i) $\Rightarrow$ (ii). Assume (i) holds. Let $v$ be a finite place of $F$, $E$ a coefficient field of $\pi$, $\ell$ a prime such that $v$ is not above $\ell$, $\lambda$ a place of $E$ above $\ell$, set $k=\overline{E_\lambda}$. Let $\widetilde{r}_{\pi,\lambda}$ be given by Theorem \[thmgalois\]. PS. PS.
Recall that if $G$ is a connected reductive group over $k$, there is a natural map from the set of $G(k)$-conjugacy classes of continuous morphisms $r : {\rm W}_{F_v} \rightarrow G(k)$, where $G(k)$ is equipped with the $\ell$-adic topology, onto the set of $G(k)$-conjugacy classes of semisimple morphisms $\varphi : {\rm W}_{F_v} \times {\rm SL}_2(k) \rightarrow G(k)$ which are $k$-algebraic on the second factor and trivial on $H \times 1$ for some open subgroup $H$ of (the inertia subgroup) of ${\rm W}_{F_v}$. The morphism $\varphi$ is a version of the so-called [*Frobenius-semisimple Weil-Deligne representation*]{} associated to $r$; its construction, which is standard at least in the $G={\mathrm{GL}}_n$ case, is an application of Grothendieck’s $\ell$-adic monodromy theorem and of the Jacobson-Morosov theorem (see [@tate §4] and [@grossgamma Prop. 2.2]). PS. PS.
The restriction of $\widetilde{r}_{\pi,\lambda}$ to a decomposition group at $v$ gives rise to a continuous morphism $\widetilde{r}_v: {\rm W}_{F_v} \rightarrow {\rm G}_2(k)$. If $\widetilde{\varphi}$ is an associated Frobenius-semisimple representation, then $\varphi = \rho \circ \widetilde{\varphi}$ is a Frobenius-semisimple representation associated to $\rho \circ r_v$. It is convenient to choose a field embedding $\sigma : k \longrightarrow {\mathbb{C}}$ whose restriction to $E$, which is naturally embedded in $E_\lambda$, is the given inclusion $E \subset {\mathbb{C}}$. The known compatibility of $r_{\pi,\lambda}$ with the local Langlands correspondence [@shin; @chharris] asserts that $\varphi \otimes_k {\mathbb{C}}$ (where the scalar extension is made through $\sigma$) is the complexification of ${\rm L}(\pi_v)$. We can thus take for $\phi_v$ the morphism $\widetilde{\varphi} \otimes_k {\mathbb{C}}$, restricted to ${\rm W}_{F_v} \times {\rm SU}(2)$. The uniqueness assertion is again a consequence of [@griess Thm. 1] .
\[remconjal1\] [A similar application of Theorem \[thmbirrcase\], but arguing with the hypothetical Langlands group instead of a Galois group as on page of the introduction, shows that properties (i) and (ii) of corollary \[corcplx\] should be equivalent without assuming $F$ is totally real or anything about the Archimedean places of $\pi$. Moreover, those assumptions should be equivalent to the existence of a cuspidal tempered automorphic representation $\pi'$ of ${\rm G}_2({\mathbb{A}}_F)$ such that for all but finitely many finite places $v$ of $F$, the conjugacy class of $\rho({\rm c}(\pi'_v))$ is ${\rm c}(\pi_v)$ (recall that we have ${\rm c}(\pi'_v) \subset \widehat{{\rm G}_2}={\rm G}_2({\mathbb{C}})$). ]{}
Examples {#exgalois}
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Examples of automorphic representations $\pi$ satisfying the assumptions of Theorem \[thmgalois\] can be constructed using Arthur’s recent results on automorphic representations of classical groups [@arthurbook]. PS. PS.
One method, developed by Gross and Savin [@gsduke], is to use an exceptional theta correspondence between anisotropic forms of ${\rm G}_2$ and ${\rm PGSp}_6$. This is also the point of view of Magaard and Savin in [@ms], who give some interesting examples. Another method, that we hope to pursue in the future, is to use triality for the algebraic group ${\rm PGSO}_8$ over $F$ (or for suitable forms of it). PS. PS.
Let us mention that in [@chrenard §8] the authors give an explicit conjectural formula for the number of cuspidal automorphic representations $\pi$ of ${\mathrm{GL}}_7({\mathbb{A}}_{\mathbb{Q}})$ satisfying assumption [(G)]{} and such that:
- $\pi_p$ is unramified for each prime $p$, PS. PS.
- $\pi_\infty$ is algebraic of weights $0, \pm u, \pm v, \pm (u+v)$ with $0<u<v$;PS. PS.
this is the number denoted ${\rm G}_2(2v,2u)$ [*loc. cit*]{}. For instance, by [@chrenard Table 11], there should be $3$ such representations with $u+v \leq 13$ (each one having ${\mathbb{Q}}$ as coefficient field), namely with $$(u,v)=(4,8), \,\,\,\,\,(3,10)\,\,\,\,\,{\rm and}\,\,\,\,\, (5,8).$$ Some informations about the Satake parameters of these representations $\pi$, such as ${\rm c}(\pi_2)$ and ${\rm trace}\, {\rm c}(\pi_p)$ for $p \leq 13$, have also been recently obtained by Megarbane [@megarbane Tables 8 & 9]. In particular, it should be possible to study the images of their associated Galois representations $\widetilde{r}_{\pi,\lambda} : {\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}}) \rightarrow {\rm G}_2(\overline{{\mathbb{Q}}_\ell})$ given by Theorem \[thmgalois\].PS. PS.
[Let $\pi$ be a cuspidal automorphic representation of ${\mathrm{GL}}_7({\mathbb{A}}_{\mathbb{Q}})$ satisfying assumption [(G)]{}, as well as (i) and (ii) above. The crystalline property of the restriction of $r_{\pi,\lambda}$ at a decomposition group at a prime $\ell$ with $\lambda\, |\, \ell$, as well as Minkowski’s theorem that any nontrivial number field has a ramified prime, imply that the Zariski-closure of the image of $r_{\pi,\lambda}$ is connected. In particular, in order to show the existence of $\widetilde{r}_{\pi,\lambda}$ in this case one may invoke (the much more elementary) Corollary \[corconnected\] instead of Theorem \[mainthmb\].]{}
A local characterization of Langlands’ functorial lifting between ${\rm G}_2$ and ${\rm PGSp}_6$ {#localcarg2sp6}
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In this last paragraph, we fix an injective algebraic group morphism $$\rho : \widehat{{\rm G}_2}={\rm G}_2({\mathbb{C}}) \longrightarrow \widehat{{\rm PGSp}_6} = {\rm Spin}_7({\mathbb{C}}).$$ Let $\pi$ be a cuspidal tempered automorphic representation of ${\rm PGSp}_6({\mathbb{A}}_F)$. As already explained on page of the introduction, if we assume the existence of the Langlands group of the number field $F$ then Theorem \[thmintroa\] shows the equivalence between the following properties:
- for all but finitely many places $v$ of $F$, the Satake parameter ${\rm c}(\pi_v)$ is the conjugacy class of an element of a ${\rm G}_2$-subgroup of ${\rm Spin}_7({\mathbb{C}})$,PS. PS.
- there exists a cuspidal automorphic representation $\pi'$ of ${\rm G}_2({\mathbb{A}}_F)$ such that for all but finitely many places $v$ of $F$, the conjugacy class of $\rho({\rm c}(\pi'_v))$ is ${\rm c}(\pi_v)$.PS. PS.
As $\pi$ was assumed to be tempered, note that if $\pi'$ is as in property (P2) then $\pi'$ is tempered as well (see §\[notationsaut\] for our convention on the meaning of the term [*tempered*]{} in this paper).PS. PS.
It turns out that the equivalence between these properties can be proved unconditionally. The following theorem grew out from a discussion with Wee Teck Gan and Gordan Savin, after the first version of this paper appeared on arXiv. PS. PS.
\[thmCGS\] Let $\pi$ be a cuspidal automorphic representation of ${\rm PGSp}_6({\mathbb{A}}_F)$. Assume that $\pi$ is tempered, or more generally that $\pi$ is [nearly generic]{}. Then $\pi$ satisfies [(P1)]{} if, and only if, it satisfies [(P2)]{}.
Let us first define the term [*nearly generic*]{} which occurs in this statement. If $\pi$ is an admissible irreducible representation of ${\rm GSp}_{2g}({\mathbb{A}}_F)$ we shall denote by ${\rm res}\, \pi$ its restriction to ${\rm Sp}_{2g}({\mathbb{A}}_F)$. It is an admissible semisimple representations of ${\rm Sp}_{2g}({\mathbb{A}}_F)$ : this follows from [@Xu Lemma 2.3.1] and the equality $\dim \pi_v^{{\rm Sp}_{2g}(\mathcal{O}_v)} =1$ when we have $\dim \pi_v^{{\rm GSp}_{2g}(\mathcal{O}_v)} =1$. Denote by $$\eta : \widehat{{\rm GSp}_{2g}} = {\rm GSpin}_{2g+1}({\mathbb{C}}) \longrightarrow {\rm SO}_{2g+1}({\mathbb{C}}) = \widehat{{\rm Sp}_{2g}}$$ the natural morphism. Let $\sigma$ be an irreducible constituent of ${\rm res}\, \pi$. If $\pi$ and $\sigma$ are unramified at the finite place $v$, which holds for almost all $v$, then the compatibility of the Satake isomorphism to isogenies (proved by Satake) shows the equality $$\label{satiso} \eta({\rm c}(\pi_v))={\rm c}(\sigma_v).$$ (see also [@Xupack Lemma 5.2] for another proof). If $\pi$ is cuspidal, then the restriction of functions along the inclusion ${\rm Sp}_{2g}({\mathbb{A}}_F) \rightarrow {\rm GSp}_{2g}({\mathbb{A}}_F)$ shows that a nonzero quotient of ${\rm res}\, \pi$ is a direct sum of cuspidal automorphic representations, by Proposition \[extcuspform\] (ii). By the equality , all the cuspidal constituents of ${\rm res}\, \pi$ belong to a same global packet of the form $\widetilde{\Pi}_{\psi}$ in the sense of Arthur [@arthurbook p. 45]. We shall say that $\pi$ is [*nearly generic*]{} if this packet (or $\psi$) is generic in the sense of [*loc. cit.*]{} p. 49. This condition is automatically satisfied if $\pi$ is tempered by the Jacquet-Shalika estimates. Note also that if $\pi$ is tempered then so is any constituent of ${\rm res}\, \pi$, by the equality and our definition of the term tempered in §\[notationsaut\]. The terminology [*nearly generic*]{} will be justified by Proposition \[propgengen\].
(of Theorem \[thmCGS\]) The implication (P2) $\Rightarrow$ (P1) is trivial. Assume now that (P1) holds. Denote by $({\rm spin},W)$ a spinor representation of ${\rm Spin}_7({\mathbb{C}})$, and $({\rm st},E)$ its standard representation. Let $S$ be a finite set of places of $F$ containing ${\rm Ram}(\pi)$, and such that for all $v \notin S$ then $\pi_v$ is unramified and ${\rm c}(\pi_v)$ is the conjugacy class of an element of a ${\rm G}_2$-subgroup of ${\rm Spin}_7({\mathbb{C}})$. Recall that we have a ${\mathbb{C}}[H]$-linear isomorphism $W \simeq 1 \oplus E$ over any ${\rm G}_2$-subgroup $H$ of ${\rm Spin}_7({\mathbb{C}})$. It follows that for ${\rm Re} \,s$ big enough we have a decomposition $${\rm L}^S(s,\pi,{\rm spin}) = \zeta_F^S(s) {\rm L}^S(s,\pi,{\rm st})$$ where $\zeta_F^S(s)={\rm L}^S(s,\pi,1)$ is the partial zeta function of $F$. PS. PS.
Let $\sigma$ be a cuspidal automorphic representation of ${\rm Sp}_6({\mathbb{A}}_F)$ which is a constituent of ${\rm res}\, \pi$. Up to enlarging $S$ if necessary we may assume that $\sigma_v$ is unramified for $v \notin S$. Formula shows ${\rm L}^S(s,\pi,{\rm st})={\rm L}^S(s,\sigma,{\rm st})$. As $\pi$ is nearly generic, the main global result of Arthur [@arthurbook Thm. 1.5.2] asserts that this ${\rm L}$-function is a finite product of Godement-Jacquet ${\rm L}$-functions of certain cuspidal unitary automorphic representations of the general linear groups over $F$. Such an ${\rm L}$-function has a meromorphic continuation to ${\mathbb{C}}$, and does not vanish at $s=1$ by a classical result of Jacquet and Shalika. It follows that ${\rm L}^S(s,\pi,{\rm spin})$ has a meromorphic continuation to ${\mathbb{C}}$ with a pole at $s=1$, coming from $\zeta_F^S(s)$. PS. PS. If $\pi$ is [*globally generic*]{} (see below), this last assertion and the main result of Ginzburg and Jiang [@gjiang Thm. 1.1] show that (P2) holds; the sought after automorphic representation of ${\rm G}_2$ is obtained there by an exceptional theta correspondence. In general, we reduce to this case by Proposition \[propgengen\] below.
Recall, following e.g. [@CKPS §1], that a cuspidal automorphic representation $\pi$ of a split reductive group $H$ is said [*globally generic*]{} if there exists an irreducible $H({\mathbb{A}}_F)$-submodule of the space of cuspforms on $H({\mathbb{A}}_F)$ which is isomorphic to $\pi$, and which is generic in the traditional sense (recalled in the appendix, before the statement of Proposition \[extcuspformgen\]).PS. PS.
\[propgengen\] Let $g\geq 1$ be an integer and $\pi$ a cuspidal automorphic representation of ${\rm GSp}_{2g}({\mathbb{A}}_F)$. Then $\pi$ is nearly generic if, and only if, there exists a globally generic cuspidal automorphic representation $\varpi$ of ${\rm GSp}_{2g}({\mathbb{A}}_F)$ with the same central character as $\pi$, and which is [*nearly equivalent*]{} to $\pi$, [i.e.]{} such that $\varpi_v \simeq \pi_v$ for all but finitely many places $v$ of $F$.
Set $G = {\rm Sp}_{2g}$ and $\widetilde{G}={\rm GSp}_{2g}$. Assume first that $\pi$ is nearly generic. We denote by $\widetilde{c}$ the central character of $\pi$ (a Hecke character of $F$). PS. PS.
Let $\sigma$ be a cuspidal automorphic constituent of ${\rm res} \,\,\pi$. Arthur’s theorem [@arthurbook Thm. 1.5.2] and the work of Cogdell, Kim, Piatetski-Shapiro and Shahidi [@CKPS Thm. 7.2] show that there is a globally generic cuspidal automorphic representation $\sigma'$ of $G({\mathbb{A}}_F)$ which is nearly equivalent to $\sigma$ (see also [@arthurbook Prop. 8.3.2]). We shall denote by $V$ a generic irreducible subspace of the space of cuspforms on $G({\mathbb{A}}_F)$ giving rise to $\sigma'$. PS. PS.
According to Arthur, $\sigma$ and $\sigma'$ are in a same (generic) global packet; in particular, they have the same central character (see the paragraph before [@Xupack Prop. 6.27]), namely the restriction $c$ of $\widetilde{c}$ to the center of $G({\mathbb{A}}_F)$. Denote by $\rho$ the natural restriction map $\mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}} \longrightarrow \mathcal{A}_{\rm cusp}(G)_c$ studied in Proposition \[extcuspform\]. By the assertion (iii) of this proposition, this map is surjective, so there is an irreducible admissible $\widetilde{G}({\mathbb{A}}_F)$-subrepresentation $\widetilde{V} \subset \mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$ whose image in $\mathcal{A}_{\rm cusp}(G)_c$ contains the space $V$ defined above. We denote by $\pi'$ the isomorphism class of the cuspidal automorphic $\widetilde{G}({\mathbb{A}}_F)$-representation $\widetilde{V}$. By Proposition \[extcuspformgen\], $\widetilde{V}$ is generic as so is $V$, so $\pi'$ is globally generic. PS. PS.
By construction, ${\rm res}\, \pi$ has a cuspidal automorphic constituent (namely $\sigma$) which is nearly equivalent to some cuspidal automorphic constituent of ${\rm res}\, \pi'$ (namely $\sigma'$). Let $\nu : \widetilde{G}({\mathbb{A}}_F) \rightarrow {\mathbb{A}}_F^\times$ denote the similitude factor. By Xu’s theorem [@Xupack Thm. 1.8], there exists a Hecke character $\omega$ of $F$ such that $\pi$ and $\pi' \otimes \omega \circ \nu$ belong to the same “global $L$-packet” that he defines. In particular, they are nearly equivalent and have the same central character (see [@Xupack Thm. 1.2]). The representation $\varpi := \pi' \otimes \omega \circ \nu$ of $\widetilde{G}({\mathbb{A}}_F)$ is cuspidal, globally generic, nearly equivalent to $\pi$, and with central character $c$, and we are done. PS. PS.
Conversely, up to replacing $\pi$ by $\varpi$ if necessary we may assume that $\pi$ is globally generic. In order to show that $\pi$ is nearly generic, it is enough to prove that ${\rm res}\,\pi$ has a cuspidal globally generic irreducible constituent, by [@CKPS Thm. 7.2]. But this follows from Proposition \[extcuspformgen\].
Some basic facts about restrictions and extensions of cusp forms {#appresext}
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The results of this appendix are probably well known to some specialists, however we have not been able to find their proofs in the litterature. Some of the arguments given below already appear in some form in Labesse-Langlands’ paper [@LL p. 49-52] and in [@Xulift Lemma 5.3]. Our main reference for what follows is Borel-Jacquet’s paper [@boreljacquet].
Let $F$ be a number field and $H$ a reductive linear algebraic group defined over $F$. We set $H_\infty=H(F \otimes_{\mathbb{Q}}{\mathbb{R}})$, the direct product of the Lie groups $H(F_v)$ for $v$ Archimedean, and denote by $H_f$ the locally profinite group $H({\mathbb{A}}_F^f)$ where ${\mathbb{A}}_F^f$ is the $F$-algebra of finite adèles of $F$: we have $H({\mathbb{A}}_F) = H_\infty \times H_f$. If $c$ is a character of the center of $H({\mathbb{A}}_F)$, we denote by $\mathcal{A}_{\rm cusp}(H)_c$ the space of cuspidal automorphic functions $H({\mathbb{A}}_F) \rightarrow {\mathbb{C}}$ with central character $c$ and with respect to a choice of maximal compact subgroup $K_\infty$ of $H_\infty$ (in the sense of [@boreljacquet §4.4]). This is an admissible, semisimple, $({\rm Lie}\, H_\infty,K_\infty) \times H_f$-module. For short, a $({\rm Lie}\, H_\infty,K_\infty) \times H_f$-module will be called an [*$H({\mathbb{A}}_F)$-[gk]{}-module*]{}.
\[extcuspform\] Let $\widetilde{G}$ be a reductive linear algebraic group, $T$ a torus, $\nu : \widetilde{G} \rightarrow T$ a surjective morphism, $\widetilde{Z}$ the center of $\widetilde{G}$, and $G$ the kernel of $\nu$. We assume that $\widetilde{G}$, $T$ and $\nu$ [(]{}hence $\widetilde{Z}$ and $G$[)]{} are defined over the number field $F$, and that $G$ is connected. Let $\widetilde{c}$ be an automorphic character of $\widetilde{Z}({\mathbb{A}}_F)$ and $c$ its restriction to the center of $G({\mathbb{A}}_F)$. Then the following assertions hold:
- the restriction map $\rho : \varphi \mapsto \varphi_{|G({\mathbb{A}}_F)}$ induces an equivariant morphism $\rho_{\rm cusp} : {\rm res}\, \mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}} \longrightarrow \mathcal{A}_{\rm cusp}(G)_c$ of $G({\mathbb{A}}_F)$-[gk]{}-modules,
- if $V$ is a nonzero $\widetilde{G}({\mathbb{A}}_F)$-[gk]{}-submodule of $\mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$ then we have $\rho_{\rm cusp}(V) \neq 0$,
- if furthermore $\widetilde{Z}$ is a split $F$-torus, then $\rho_{\rm cusp}$ is surjective.
[*Proof.* ]{} As $\nu$ is trivial on the derived subgroup of $\widetilde{G}$, we have $\nu(\widetilde{Z})=T$. We may thus find a torus $C \subset \widetilde{Z}$ defined over $F$ such that $\nu : C \rightarrow T$ is an isogeny. We shall denote by $$\iota : C \times G \rightarrow \widetilde{G}$$ the isogeny $(z,g) \mapsto zg$. For each place $v$ of $F$, note that $G(F_v)$ is a normal subgroup of $\widetilde{G}(F_v)$ with abelian quotient. Moreover, $\nu(C(F_v))$ is an open subgroup of finite index in $T(F_v)$, by the finiteness of the Galois cohomology group $H^1(F_v, C \cap G)$ (Tate). In particular, the image of the morphism $\iota_v : C(F_v) \times G(F_v) \rightarrow \widetilde{G}(F_v)$ induced by $\iota$ is an open subgroup of finite index in $\widetilde{G}(F_v)$. For each Archimedean place $v$, we choose a maximal compact subgroup $K_v$ of $G(F_v)$, as well as a maximal compact subgroup $\widetilde{K}_v$ of $\widetilde{G}(F_v)$ containing $K_v$. We have thus $K_v = G(F_v) \cap \widetilde{K}_v$. We define $K_\infty$ (resp. $\widetilde{K}_\infty$) as the product of the $K_v$ (resp. $\widetilde{K}_v$) for $v$ Archimedean. We may and do choose these maximal compact subgroups in order to define $\mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$ and $\mathcal{A}_{\rm cusp}(G)_c$. If we denote by $\mathfrak{z}(H)$ the ${\mathbb{C}}$-algebra defined as the center of the complex enveloping algebra ${\rm U}({\rm Lie}\, H)$ of the Lie group $H$, then for each Archimedean $v$ the local isomorphism $\iota_v$ defines a ${\mathbb{C}}$-algebra isomorphism $\mathfrak{z}(C(F_v)) \otimes \mathfrak{z}(G(F_v)) {\overset{\sim}{\rightarrow}}\mathfrak{z}(\widetilde{G}(F_v)) $. In particular, the natural inclusion ${\rm U}({\rm Lie}\, G(F_v)) \subset {\rm U}({\rm Lie}\, \widetilde{G}(F_v))$ induces an injection $\mathfrak{z}(G(F_v)) \subset \mathfrak{z}(\widetilde{G}(F_v))$.
Let us check assertion (i). Let $\varphi \in \mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$. There is a compact open subgroup $\widetilde{K}_f \subset \widetilde{G}_f$ such that $\varphi$ is $\widetilde{K}_f$-invariant on the right. Moreover, for any $g_f \in \widetilde{G}_f$ the function $g_\infty \mapsto \varphi(g_\infty \times g_f), \widetilde{G}_\infty \rightarrow {\mathbb{C}}$, is smooth, $\widetilde{K}_\infty$-finite on the right, $\mathfrak{z}(\widetilde{G}_\infty)$-finite, and slowly increasing (see [@boreljacquet §1.2]). Using a model over an open subset of ${\rm Spec}\,\mathcal{O}_F$ of the closed morphism $G \subset \widetilde{G}$, we see that $K_f= \widetilde{K}_f \cap G_f$ is a compact open subgroup of $G_f$. By the inclusions $K_f \subset \widetilde{K}_f$, $K_\infty \subset \widetilde{K}_\infty$, and $\mathfrak{z}(G_\infty) \subset \mathfrak{z}(\widetilde{G}_\infty)$ (justified above), it follows that $\rho(\varphi)$ is $K_f$-invariant and $K_\infty$-finite on the right, as well as $\mathfrak{z}(G_\infty)$-finite. It is also trivially slowly increasing (note [*e.g.*]{} that the restriction to $G_\infty$ of a [*norm*]{} on $\widetilde{G}_\infty$ is still a norm, in the sense of [@boreljacquet §1.2]). Of course, $\rho(\varphi)$ is $G(F)$-invariant on the left and has central character $c$. We have proved that $\rho(\varphi)$ is an automorphic form. It only remains to check that it is cuspidal. But for any parabolic subgroup $P$ of $G$ defined over $F$, there is a unique parabolic subgroup $\widetilde{P}$ of $\widetilde{G}$ such that $\widetilde{P} \cap G = P$, namely $\widetilde{P}=\widetilde{Z} \, P$. We conclude as the unipotent radical of $\widetilde{P}$ and $P$ coincide (in particular, they are included in $G$), and as $f$ is cuspidal. This shows that $\rho_{\rm cusp}$ is well defined. By construction, it is $({\rm Lie} \,G_\infty, K_\infty)\times G_f$-equivariant.
We now prove assertion (ii). Let $V$ be as in the statement and $\varphi \in V - \{0\}$. Choose $g = g_\infty g_f \in \widetilde{G}({\mathbb{A}}_F)$ such that $\varphi(g) \neq 0$. Up to replacing $\varphi$ by $g_f^{-1} \cdot \varphi \in V$ we may assume $g_f=1$. As the image $\iota_v$ is an open subgroup of $\widetilde{G}(F_v)$ for $v$ Archimedean, and as $\widetilde{K}_\infty$ meets every connected component of $\widetilde{G}_\infty$ by Cartan’s decomposition, we have: $$G_\infty \cdot \widetilde{Z}_\infty \cdot \widetilde{K}_\infty = \widetilde{G}_\infty.$$ It follows that up to replacing $\varphi$ by $h \cdot \varphi$ with $h \in \widetilde{Z}_\infty \cdot \widetilde{K}_\infty$, which preserves $V$, we may assume that we have $g \in G_\infty \subset G({\mathbb{A}}_F)$. In particular, we have $\rho(\varphi) \neq 0$.
Let us now check (the main) assertion (iii). Denote by $Z$ the center of $G$; we have $Z = \widetilde{Z} \cap G$ as $\widetilde{G}=\widetilde{Z}\,G$. Consider the subgroups $G({\mathbb{A}}_F) \subset H_1 \subset H_2 \subset \widetilde{G}({\mathbb{A}}_F)$ with $H_1=\widetilde{Z}({\mathbb{A}}_F) G({\mathbb{A}}_F)$ and $H_2=\widetilde{G}(F) H_2$. They are normal in $\widetilde{G}({\mathbb{A}}_F)$ (with abelian quotient), as so is $G({\mathbb{A}}_F)$. The assumption $\widetilde{c}_{|Z({\mathbb{A}}_F)}=c$ and the obvious inclusion $\widetilde{Z}({\mathbb{A}}_F)\cap G({\mathbb{A}}_F) \subset Z({\mathbb{A}}_F)$ show that there is a unique function $\varphi_1 : H_1 \rightarrow {\mathbb{C}}$ such that $\varphi_1(zg) = \widetilde{c}(z)\varphi(g)$ for all $(z,g) \in \widetilde{Z}({\mathbb{A}}_F) \times G({\mathbb{A}}_F)$.
We claim that there is a unique function $\varphi_2 : H_2 \rightarrow {\mathbb{C}}$ such that $\varphi_2(\gamma h)=\varphi_1(h)$ for all $(\gamma,h) \in \widetilde{G}(F) \times G({\mathbb{A}}_F)$. Indeed, such a function exists if, and only if, $\varphi_1$ is left-invariant under $\widetilde{G}(F) \cap H_1$. It is thus enough to show the inclusion $\widetilde{G}(F) \cap H_1 \subset \widetilde{Z}(F) G(F)$, or which is the same, the inclusion $\nu(\widetilde{G}(F) \cap H_1) \subset \nu( \widetilde{Z}(F))$. It suffices to prove $$\label{incltoprove} T(F) \cap \nu(\widetilde{Z}({\mathbb{A}}_F)) \subset \nu ( \widetilde{Z}(F)).$$
We have not used that $\widetilde{Z}$ is an $F$-split torus so far, but we shall do so now. As the morphism $\nu$ induces a surjective homomorphism $\widetilde{Z} \rightarrow T$ defined over $F$, it implies that $T$ is $F$-split. Moreover, all the subtori of $\widetilde{Z}$ are defined and split over $F$ as well. Let $Z^0$ be the neutral component of $Z=G \cap \widetilde{Z}$. Up to changing $C$ if necessary, we may thus assume that we have $\widetilde{Z} = C \times Z^0$. In particular, the following equalities hold $$\label{simplzc} \nu(\widetilde{Z}({\mathbb{A}}_F)) = \nu(C({\mathbb{A}}_F)), \, \, \, \widetilde{Z}({\mathbb{A}}_F)G({\mathbb{A}}_F) = C({\mathbb{A}}_F) G({\mathbb{A}}_F), \, \,\, \widetilde{Z}(F)G(F) = C(F)G(F).$$ By the theory of elementary divisors, we may also assume that we have $T=\mathbb{G}_m^r$ for some integer $r\geq 0$, as well as an isomorphism $\mu : \mathbb{G}_m^r {\overset{\sim}{\rightarrow}}C$ over $F$, and integers $m_1,\dots,m_r$, such that we have $\nu \circ \mu (z_1,\dots,z_r) = (z_1^{m_1},\dots,z_r^{m_r})$. But it is well-known that if $m \in {\mathbb{Z}}$ and $x \in F^\times$ is an $m$-th power in $F_v^\times$ for each $v$, then $x$ is an $m$-th power in $F^\times$. This shows , hence the existence of $\varphi_2$.
Let $S$ be a finite set of places of $F$ containing the Archimedean places. We denote by $\mathcal{O}_{F,S} \subset F$ the subring of $S$-integers. Up to enlarging $S$ if necessary, we may assume that $\widetilde{G}$ is an affine group schemes of finite type defined over $\mathcal{O}_{F,S}$, that $C \subset \widetilde{Z} \subset \widetilde{G}$ are closed subgroup schemes defined over $\mathcal{O}_{F,S}$, that $\nu : G \rightarrow \mathbb{G}_m$ is a group scheme homomorphism defined over $\mathcal{O}_{F,S}$, and that $\mu$ is a group scheme isomorphism $\mathbb{G}_m^r {\overset{\sim}{\rightarrow}}C$ over $\mathcal{O}_{F,S}$. We define again $G$ as the kernel of $\mu$; this is an affine group scheme of finite type over $\mathcal{O}_{F,S}$. By definition of the adelic topology, if we have a collection of open subsets $U_v \subset \widetilde{G}(F_v)$ for each $v \in S$, then $(\prod_{v \in S} U_v) \times (\prod_{v \notin S} \widetilde{G}(\mathcal{O}_v))$ is an open subset of $\widetilde{G}({\mathbb{A}}_F)$ (and a similar assertion holds with $\widetilde{G}$ replaced by $G$). For any $v \notin S$ we shall set $\widetilde{K}_v = \widetilde{G}(\mathcal{O}_v)$, and we shall also set $\widetilde{K}^S = \prod_{v \notin S} \widetilde{K}_v$. Up to enlarging $S$ if necessary, we may assume that $\varphi$ is $G(\mathcal{O}_v)$-invariant on the right, as well as $\widetilde{c}_v(C(\mathcal{O}_v))=1$, for all $v \notin S$. We may also assume $m_i \in \mathcal{O}_{F,S}^\times$ for $i=1,\dots,r$.
Define $M$ as the maximum of the $m_i$ for $ 1\leq i \leq r$. By Hermite’s theorem, there are only finitely many field extensions $F'/F$ of degree $\leq M$ and which are unramified everywhere. Up to enlarging $S$ if necessary, we may thus assume that the following extra property holds : if $F'/F$ is such an extension which is furthermore split at all places above $S$, then we have $F'= F$. In particular, for each $i=1,\dots, r$, if we have $x \in F^\times$ and $u \in {\mathbb{A}}_F^\times$ such that $u_v=1$ for $v \in S$, $u_v \in \mathcal{O}_v^\times$ for $v \notin S$, and such that $xu_v$ is an $m_i$-th power in $F_v^\times$ for each $v$, then $x$ is an $m_i$-th power in $F^\times$. This property shows $\widetilde{G}(F) \cap (C({\mathbb{A}}_F) G({\mathbb{A}}_F) \widetilde{K}^S) \subset C(F) G(F)$, hence $\widetilde{G}(F) \cap (\widetilde{Z}({\mathbb{A}}_F) G({\mathbb{A}}_F) \widetilde{K}^S) \subset \widetilde{Z}(F) G(F)$ by , which implies in turn the equality $$\label{eq1phi3} ( \widetilde{G}(F)\widetilde{Z}({\mathbb{A}}_F) G({\mathbb{A}}_F) ) \cap \widetilde{K}^S = (\widetilde{Z}({\mathbb{A}}_F) G({\mathbb{A}}_F)) \cap \widetilde{K}^S.$$ Observe moreover that for $v \notin S$ we have the inclusion $$\label{eq2phi3}(\widetilde{Z}(F_v) G(F_v)) \cap \widetilde{G}(\mathcal{O}_v) \subset C(\mathcal{O}_v) G(\mathcal{O}_v).$$ Indeed, we have $\widetilde{Z}(F_v) G(F_v) = C(F_v) G(F_v)$, and if $x \in \mathcal{O}_v^\times$ is an $m$-th power in $F_v^\times$, then $x$ is an $m$-th power in $\mathcal{O}_v^\times$.
Consider the subgroup $H_3 = H_2 \widetilde{K}^S \subset \widetilde{G}({\mathbb{A}}_F)$. The inclusions and show that there is a unique function $\varphi_3 : H_3 \rightarrow {\mathbb{C}}$ such that $\varphi_3(hk) = \varphi_2(h)$ for all $h \in H_2$ and $k \in \widetilde{K}^S$. Note that $H_3$ is open in $\widetilde{G}({\mathbb{A}}_F)$. Indeed, it contains the image of $\iota_v$ for each $v \in S$, which is open in $\widetilde{G}(F_v)$ by the first paragraph of the proof, as well as $\widetilde{K}^S$. Define now $\psi : \widetilde{G}({\mathbb{A}}_F) \rightarrow {\mathbb{C}}$ by $\psi_{|H_3} = \varphi_3$ and $\psi(h) = 0$ for $h \notin H_3$. This function has central character $c$, is $\widetilde{K}^S$-invariant on the right, $\widetilde{G}(F)$-invariant on the left, and satisfies $\rho(\psi)=f$. We claim that we have $\psi \in \mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$. One first observes that the $K_\infty$-finiteness of $\varphi$ implies the one of $\psi_{|H_3}$, hence the one of $\psi$ (use that $\psi$ vanishes outside $H_3$ and that $K_\infty$ is a subgroup of $H_3$). As the normal subgroup $K_\infty$ of $\widetilde{K}_\infty$ is easily seen to be of finite index, this implies the $\widetilde{K}_\infty$-finiteness of $\psi$. Let now $g \in \widetilde{G}({\mathbb{A}}_F)$. Observe that the map $\psi_g : C_\infty \times G_\infty \rightarrow {\mathbb{C}}$, $(z,h) \mapsto \psi(gzh)$, is either identically $0$ or a constant multiple of $\psi_{g'}$ with $g' \in G({\mathbb{A}}_F)$. For such a $g'$ we have $\psi_{g'}(z,h) =\widetilde{c}_\infty(z)\varphi(g'h)$. As $\iota_v$ is a smooth finite covering for each Archimedean $v$, this implies that $\psi$ is smooth, as so are $\varphi$ and $\widetilde{c}$. As $\varphi$ and $\widetilde{c}$ are slowly increasing, and as $C_\infty \times G_\infty$ has a finite index in $\widetilde{G}_\infty$, this also shows that $\psi$ is slowly increasing. We have proved that $\psi$ is an automorphic form.
It only remains to check that $\psi$ is a cuspform. Let $\widetilde{P}$ be a parabolic subgroup of $\widetilde{G}$ defined over $F$, and $N$ its unipotent radical. We have $N \subset G$, hence $N({\mathbb{A}}_F) \subset H_3$, so if $g \notin H_3$ we have $\psi(N({\mathbb{A}}_F)g) =0$. If $g =\gamma h z k$ with $\gamma \in \widetilde{G}(F)$, $h \in G({\mathbb{A}}_F)$, $z \in \widetilde{Z}({\mathbb{A}}_F)$ and $k \in \widetilde{K}^S$, then we have $$\int_{N(F) \backslash N({\mathbb{A}}_F)} \psi(ng) dn = \widetilde{c}(z) \int_{N(F) \backslash N({\mathbb{A}}_F)} \varphi (\gamma^{-1}n\gamma h) dn,$$ which vanishes as $\varphi$ is a cuspform and $\gamma^{-1} N \gamma$ is the unipotent radical of the $F$-parabolic subgroup of $\gamma^{-1} \widetilde{P} \gamma \cap G$ of $G$. $\Box$
Let $H$ be a linear reductive group defined and quasi-split over $F$. Recall that a [*Whittaker datum*]{} for $H$ is a quadruple $D=(B,T,U,\chi)$ where $B$ is a Borel subgroup of $H$ defined over $F$, $T$ is a maximal torus of $B$ defined over $F$, $U$ is the unipotent radical of $B$, and $\chi : U({\mathbb{A}}_F) \rightarrow {\mathbb{C}}^\times$ is a continuous character which is trivial on $U(F)$ and [*nondegenerate*]{} (see [*e.g*]{} [@KS p. 54]). When $H$ is split, it means [*e.g.*]{} that $\chi$ is nontrivial on the root subgroup $U_\alpha \subset U$ for each simple root $\alpha$ relative to $(B,T)$. Recall that an irreducible $H({\mathbb{A}}_F)$-[gk]{}-submodule $V$ of the space of cuspforms on $H({\mathbb{A}}_F)$ is said [*generic*]{} if there exists a Whittaker datum $D=(B,T,U,\chi)$, such that for all $\varphi \in V-\{0\}$ the function on $H({\mathbb{A}}_F)$ defined by $${\rm W}_\varphi^D(g)=\int_{N(F)\backslash N({\mathbb{A}}_F)} \varphi(ng) \chi(n) dn$$ is not identically zero. As the subspace of $\varphi \in V$ with ${\rm W}^{D}_\varphi =0$ is a ${\rm gk}$-submodule of $V$, this latter condition is equivalent to ask that we have ${\rm W}^{D}_\varphi \neq 0$ for [*some*]{} $\varphi \in V$, by irreducibility of $V$.
\[extcuspformgen\] Keep the assumptions of Proposition \[extcuspform\] and assume furthermore that $G$ is quasi-split over $F$. Let $\widetilde{V} \subset \mathcal{A}_{\rm cusp}(\widetilde{G})_{\widetilde{c}}$ be a nonzero irreducible $\widetilde{G}({\mathbb{A}}_F)$-[gk]{}-submodule. The following properties are equivalent :
- $\widetilde{V}$ is generic,
- $\rho_{\rm cusp}(\widetilde{V})$ contains a nonzero generic irreducible $G({\mathbb{A}}_F)$-[gk]{}-submodule.
[*Proof.* ]{} Assume first that (i) holds. Let $\widetilde{D}=(\widetilde{B},\widetilde{T},U,\chi)$ be a Whittaker datum such that for all $\varphi \in \widetilde{V}-\{0\}$ we have ${\rm W}_{\varphi}^{\widetilde{D}} \neq 0$. Then $D=(\widetilde{B}\cap G, \widetilde{T}\cap G, U, \chi)$ is a Whittaker datum for $G$. Choose a nonzero $\varphi \in \widetilde{V}$. Up to replacing $\varphi$ by some $\widetilde{K}_\infty \times \widetilde{G}({\mathbb{A}}_F^f)$-translate as in the proof of Proposition \[extcuspform\] (ii), we may assume that we have ${\rm W}^{\widetilde{D}}_\varphi \neq 0$ on $G({\mathbb{A}}_F)$. This implies that the cuspform $f := \rho(\varphi)=\varphi_{|G({\mathbb{A}}_F)}$ is nonzero and ${\rm W}_{f}^D \neq 0$. The $G({\mathbb{A}}_F)$-[gk]{}-submodule of $\rho_{\rm cusp}(\widetilde{V}) \subset \mathcal{A}_{\rm cusp}(G)_{c}$ generated by $f$ is semisimple, hence a finite direct sum of irreducible [gk]{}-modules $V_i$. Write $f = \sum_i f_i$ with $f_i \in V_i$. We have ${\rm W}_{f}^D = \sum_i {\rm W}_{f_i}^{D} \neq 0$, so there exists $i$ such that ${\rm W}_{f_i}^D \neq 0$, and $V_i$ is a generic irreducible constituent of $\rho_{\rm cusp}(\widetilde{V})$.
Assume now that $\rho_{\rm cusp}(\widetilde{V})$ contains some generic irreducible $G({\mathbb{A}}_F)$-submodule $V \neq 0$. Let $D=(B,T,U,\chi)$ be a Whittaker datum for $G$ such that ${\rm W}_\varphi^D$ is nonzero for all $\varphi \in V -\{0\}$. There exists a unique Whittaker datum $\widetilde{D}$ for $\widetilde{G}$ of the form $(\widetilde{B},\widetilde{T},U,\chi)$ with $\widetilde{B} \cap G = B$ and $\widetilde{T} \cap G = T$. As we have $0 \neq V \subset \rho(\widetilde{V})$, we may find some $\varphi \in \widetilde{V}$ such that the cuspform $f=\rho(\varphi)=\varphi_{|{\rm G}({\mathbb{A}}_F)}$ is nonzero and belongs to $V$. It is enough to show ${\rm W}^{\widetilde{D}}_\varphi \neq 0$. But for $g \in G({\mathbb{A}}_F)$ we have the identity ${\rm W}^{\widetilde{D}}_\varphi(g) = {\rm W}^{D}_{f}(g)$, and we are done. $\Box$
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[^1]: The group ${\rm O}(2)^{\pm}$ is isomorphic to the quotient of $\mu_4 \times {\rm O}(2)$ by the diagonal order $2$ subgroup, the similitude factor being the square of the factor $\mu_4$.
[^2]: Hints: consider subgroups $\Gamma$ of the form ${\rm SO}(a+b-c) \times {\rm SO}(c)$ with $c =0,1,3$ to show that $\mathcal{P}({\rm SO}(a+b),{\rm SO}(a) \times {\rm SO}(b))$ implies $b=1$ and $a \in \{1,3\}$. To check ${\mathcal P}({\rm SO}(4),{\rm SO}(3))$, observe that if $V$ is the restriction to ${\rm SO}(3)$ of the tautological $4$-dimensional real representation of ${\rm SO}(4)$, then we have an isomorphism $V \oplus V^\ast \simeq 1 \oplus 1 \oplus \Lambda^2\, V$ (a similar observation proves ${\mathcal P}({\rm SU}(4),{\rm SU}(3))$ as well).
[^3]: Hint: for $n=4,5$ consider for $\Gamma$ the subgroup of the alternating subgroup of $\got{S}_{n+1}$ preserving $\{n,n+1\}$ (we have $\Gamma \simeq \got{S}_{n-1}$).
[^4]: As pointed out by Langlands long ago, note that these properties of $\phi$ may however not determine it uniquely up to $\mathcal{G}$-conjugacy in general (see [*e.g.*]{} [@larsen]).
[^5]: We warn the reader that there does not seem to be any standard terminology for these classical notions. Several authors, such as Knus, use the term nonsingular for nondegenerate. Moreover, $V$ is $\frac{1}{2}-$regular is the sense of [@knuscampinas; @knusquadherm] if and only if $\dim V$ is odd and $V$ is regular in our sense.
[^6]: A supergroup is a group $G$ equipped with a group homomorphism ${\rm p} : G \rightarrow {\mathbb{Z}}/2{\mathbb{Z}}$.
[^7]: The Brauer-Nesbitt theorem is the following statement [@bou8 §20, [no]{} 6, Thm. 2 & Cor. 1]. Let $k$ be a field, $\Gamma$ a group, and $E_i$ a finite dimensional, semisimple, $k$-linear representation of $\Gamma$ for $i=1,2$. Assume we have $\det ( t - \gamma\, |\, E_1) = \det (t - \gamma \,|\, E_2)$ for each $\gamma \in \Gamma$. Then $E_1$ is isomorphic to $E_2$.
[^8]: By this we mean the following classical facts. Let $k$ be an algebraically closed field (of arbitrary characteristic), $\Gamma$ a group, $c : \Gamma \rightarrow k^\times$ a character of finite order $n$, and $V$ a finite dimensional irreducible $k$-linear representation of $\Gamma$ such that $V \simeq V \otimes c$. Then $n$ divides $\dim V$ (apply $\det$), and if $\Gamma^0$ denotes the kernel of $c$ then $V_{|\Gamma^0}$ is a direct sum of $n$ irreducible representations of $\Gamma^0$ of dimension $\frac{\dim V}{n}$, whose isomorphism classes are distinct and permuted transitively by the outer action of $\Gamma/\Gamma^0 \simeq {\mathbb{Z}}/n{\mathbb{Z}}$ on $\Gamma^0$. Moreover, if $W$ denotes any such representation we have $V \simeq {\rm Ind}_{\Gamma^0}^\Gamma W$.
|
---
abstract: 'We optimize the classical field approximation of the version described in J. Phys. B [**40**]{}, R1 (2007) for the oscillations of a Bose gas trapped in a harmonic potential at nonzero temperatures, as experimentally investigated by Jin [*et al.*]{} \[Phys. Rev. Lett. [**78**]{}, 764 (1997)\]. Similarly to experiment, the system response to external perturbations strongly depends on the initial temperature and on the symmetry of perturbation. While for lower temperatures the thermal cloud follows the condensed part, for higher temperatures the thermal atoms oscillate rather with their natural frequency, whereas the condensate exhibits a frequency shift toward the thermal cloud frequency ($m=0$ mode), or in the opposite direction ($m=2$ mode). In the latter case, for temperatures approaching critical, we find that the condensate begins to oscillate with the frequency of the thermal atoms, as in the $m=0$ mode. A broad range of frequencies of the perturbing potential is considered.'
author:
- 'Tomasz Karpiuk,$^1$ Miros[ł]{}aw Brewczyk,$^2$ Mariusz Gajda,$^{3,5}$ and Kazimierz Rzażewski$\,^{4,5}$'
title: 'Constructing classical field for a Bose-Einstein condensate in arbitrary trapping potential; quadrupole oscillations at nonzero temperatures'
---
Introduction
============
Experiments with atomic Bose-Einstein condensates driven by an external perturbation have allowed verification of the mean-field description of the condensed phase in a dynamical regime where the system responds by collective motion. Typically, a periodic perturbation (with a particular symmetry) of a trapping potential was used to excite the Bose gas [@Cornell-zero; @Ketterle-zero; @Cornell-osc; @Ketterle-osc], although other kinds of trap distortion leading, for instance, to scissors mode excitations [@Foot] or the transverse monopole mode excitation in an elongated condensate[@Dalibard] were also tried. The investigation of the low energy collective modes of the condensate in the zero-temperature limit [@Cornell-zero; @Ketterle-zero] has revealed that the mean-field description of that system (i.e., based on the Gross-Pitaevskii equation) works well [@theory-zero]. However, when the study of low-lying excitations was extended to include the measurement of frequencies and damping rates as a function of temperature [@Cornell-osc; @Ketterle-osc], it became clear that a new theory of an interacting Bose gas at nonzero temperatures is required.
The JILA experiment of Ref. [@Cornell-osc] showed two effects. First, it was found that two collective modes with different symmetries (quadrupole modes with angular momenta equal to $m=0$ and $m=2$) behave in qualitatively different ways. When the temperature increases, they exhibit a frequency shift in opposite directions. Moreover, for the $m=0$ mode a rather sudden upward shift is observed, suggesting the existence of a characteristic temperature which is approximately $0.65$ of the critical temperature for the corresponding ideal gas. Secondly, the damping of the collective oscillations turned out to be dramatically dependent on temperature, showing that the condensate modes are damped even faster than the noncondensed fraction while approaching the critical temperature. All these puzzling findings triggered a lot of theoretical work and after a few years resulted in the development of Zaremba-Nikuni-Griffin formalism [@ZNG; @Zaremba-osc] and the second-order quantum field theory [@Burnett-osc; @Morgan] for a Bose gas. Recently, another attempt to describe finite-temperature properties of low-lying collective modes was undertaken in Ref. [@Blakie-osc] within the approximation called the projected Gross-Pitaevskii equation.
The Zaremba-Nikuni-Griffin formalism [@Zaremba-osc] applied to the results of the JILA experiment gives relatively good agreement. Also the calculations based on the second-order quantum field theory [@Burnett-osc] show a good agreement with experimental data. However, already these two approaches differ when considering their fundamentals. For example, the first one neglects the phonon character of the low-lying energy modes, the anomalous average, as well as the Beliaev processes. Regarding the JILA experiment, for the $m=0$ mode, the Zaremba-Nikuni-Griffin method predicts an additional branch of condensate frequencies, so far not observed experimentally. No such branch is found within the second-order theory of Ref. [@Burnett-osc]. Furthermore, this approach predicts a single frequency of condensate response for a particular temperature, regardless of driving frequency, and hence the notion of a natural condensate frequency is meaningful. This strongly differs from what is reported in Jackson and Zaremba [@Zaremba-osc], where the condensate response depends on the driving frequency of the whole system (condensed and noncondensed parts). A comprehensive discussion of both approaches in the context of JILA experiment can be found in a recent review article [@Proukakis]. Another formalism, based on the projected Gross-Pitaevskii equation, used to model the JILA experiment [@Cornell-osc] produces good agreement with experimental data up to $0.65 T_c$, and for the $m=2$ mode at higher temperatures. However, it fails to predict the sudden upward frequency shift for the $m=0$ mode at this temperature [@Blakie-osc]. This failure is perhaps related to the way the cutoff parameter (which splits the space of modes into the highly occupied ones that are described in terms of the classical field, and the others that are sparsely occupied and require, in principle, a quantum treatment) is chosen. The details of the splitting procedure within the projected Gross-Pitaevskii equation approach are discussed in Ref. [@Advances].
In this paper we apply the classical field approximation in the version described in Ref. [@przeglad] to the case of a trapped interacting Bose gas at nonzero temperatures driven by an external perturbation as in the experiment of Ref. [@Cornell-osc]. The main purpose of this work is to check whether the classical field approximation is able to reproduce, at least qualitatively, the findings of JILA experiment. This is an especially important task because of the recently reported failure [@Blakie-osc] to explain the behavior of the $m=0$ mode within the projected Gross-Pitaevskii equation method, which is conceptually very close to our classical field approximation. On the other hand, although the other existing theories [@Zaremba-osc] and [@Burnett-osc] both lead to relatively good agreement with the experiment [@Cornell-osc], they somehow contradict each other conceptually as discussed in the previous paragraph. It would be nice to have an alternative view of the processes going on in the perturbed Bose gas. Finally, the approaches [@Zaremba-osc] and [@Burnett-osc] have some difficulties in describing the dynamics of the thermal cloud, especially in the $m=2$ mode. In fact, no predictions for thermal component frequencies in the $m=2$ mode are given in [@Zaremba-osc] or [@Burnett-osc]. Simultaneously, within the projected Gross-Pitaevskii equation method, at higher temperatures, the thermal cloud (in fact, for both the $m=0$ and $m=2$ modes) oscillates at frequencies much lower than in experiment.
The classical field approximation has already been applied in static and dynamical regimes for a uniform and harmonically trapped systems (for a review, see [@przeglad]). This approach was used to investigate the thermodynamics of an interacting gas [@rapid; @JPB] as well as dynamical processes like the photoassociation of molecules [@photo], the dissipative dynamics of a vortex [@vortex1], the superfluidity in ring-shaped traps [@super], or the thermalization in spinor condensates [@spinor]. The classical field approximation was also tested at a quantitative level when e.g. the Bogoliubov-Popov quasiparticle energy spectrum in a uniform Bose gas was obtained [@JPB] or when the process of splitting of doubly quantized vortices in dilute Bose-Einstein condensates [@vortex2] was studied.
The paper is organized as follows: In Sec. \[method\] we describe the classical field approximation for a trapped Bose gas with particular attention on how the equilibrium state is obtained and on the quality of the solution in comparison with the equilibrium states calculated within the self-consistent Hartree-Fock method. Sec. \[mode0\] discusses the results (response frequencies and damping rates) for the quadrupole $m=0$ collective mode for parameters as in JILA experiment [@Cornell-osc] while in Sec. \[mode2\] we do the same for $m=2$ excitation. Finally, we conclude in Sec. \[concl\].
Classical field approximation for a trapped Bose gas {#method}
====================================================
Formalism
---------
A good starting point to introduce the classical field approximation is the usual Heisenberg equation of motion for the bosonic field operator $\hat {\Psi }({\bf r},t)$ which annihilates an atom at point ${\bf r}$ and time $t$. The field operator $\hat {\Psi }({\bf r},t)$ fulfills standard commutation relations: $$\left[ {\hat {\Psi }({\rm {\bf r}},t),\hat {\Psi }^+({\rm {\bf r}'},t)}
\right]=\delta ({\rm {\bf r}}-{\rm {\bf r}'})
\label{comrel}$$ with other equal time commutation relations for $[\hat {\Psi},\hat {\Psi}]$ and $[\hat {\Psi}^+,\hat {\Psi}^+]$ being zeros. The equation of motion reads: $$\begin{aligned}
&&i\hbar \frac{\partial}{\partial t} \hat {\Psi }({\rm {\bf r}},t) =
\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{tr}({\rm {\bf r}},t) \right]
\hat {\Psi }({\rm {\bf r}},t) \nonumber \\
&&+ g\, \hat{\Psi }^+({\rm {\bf r}},t) \hat {\Psi }({\rm {\bf r}},t)
\hat {\Psi }({\rm {\bf r}},t) \,.
\label{Heisenberg}\end{aligned}$$ Here, we assume the time-dependent trapping potential $V_{tr}$ and the usual contact interaction for colliding atoms. The coupling constant $g=4\pi \hbar^2 a /m$ is expressed in terms of the s-wave scattering length $a$.
Next, we expand the field operator $\hat {\Psi }({\bf r},t)$ in the basis of one-particle wave functions $\psi_k({\bf r})$, where $k$ is a set of one-particle quantum numbers: $$\hat {\Psi }({\rm {\bf r}},t) = \sum_k \psi_k({\bf r}) \hat {a}_k(t)
\label{expansion}$$ Now, we assume that some of the modes used in the expansion (\[expansion\]) are macroscopically occupied and extend the original Bogoliubov idea [@Bogoliubov] by replacing all operators $\hat {a}_k(t)$ corresponding to these modes by c-numbers. When only macroscopically occupied modes are considered, the field operator $\hat {\Psi }({\bf r},t)$ is turned into the complex wave function $\Psi ({\bf r},t)$ and the expansion (\[expansion\]) takes the form: $$\Psi ({\rm {\bf r}},t) = \sum_{k=0}^{k_{max}} \psi_k({\bf r}) a_k(t) \,.
\label{expansion1}$$ The upper index in the summation tells us that, indeed, the wave function $\Psi ({\bf r},t)$ is expanded only over a finite number of states, i.e. those which are macroscopically occupied. We call the wave function $\Psi ({\bf r},t)$ the classical field. This is analogous to the way that intense electromagnetic waves can be treated. In spite of consisting of photons, an intense light beam is well characterized by the classical electric and magnetic fields. Since the experiments with dilute atomic gases are performed with millions of atoms it seems to be a plausible approximation to use the classical field to describe atoms in analogy with electric and magnetic fields for photons.
Obviously, the classical field obeys the following equation: $$\begin{aligned}
&&i\hbar \frac{\partial}{\partial t} {\Psi }({\rm {\bf r}},t) =
\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{tr}({\rm {\bf r}},t) \right]
{\Psi }({\rm {\bf r}},t) \nonumber \\
&&+ g\, {\Psi }^*({\rm {\bf r}},t) {\Psi }({\rm {\bf r}},t)
{\Psi }({\rm {\bf r}},t) \,.
\label{CFequation}\end{aligned}$$ We usually implement the cutoff parameter $k_{max}$ by solving the Eq. (\[CFequation\]) on a rectangular grid using the Fast Fourier Transform technique. The spatial grid step determines the maximal momentum per particle (and hence the energy) in the system whereas the use of the Fourier transform implies a projection in momentum space.
The equation (\[CFequation\]) looks like the usual Gross-Pitaevskii equation describing the Bose-Einstein condensate at zero temperature. However, here the interpretation of the complex wave function $\Psi ({\bf r},t)$ is different. It describes all the atoms in the system, both those in a condensate and in a thermal cloud. Therefore, the question appears how to split the classical field $\Psi ({\bf r},t)$ into the condensed and noncondensed fractions. For that we use the definition of a Bose-Einstein condensation proposed originally by Penrose and Onsager [@POdef]. According to this definition the condensate is assigned to the eigenvector corresponding to the dominant eigenvalue of a one-particle density matrix. So, we built the one-particle density matrix in the following way: $$\rho^{(1)}({\bf r},{\bf r}^{\,\prime};t) = \frac{1}{N}\, \Psi({\bf r},t)\,
\Psi^*({\bf r}^{\,\prime},t) \,,
\label{denmat}$$ where $N$ is the total number of particles. However, and here comes the surprise – since (\[denmat\]) is just the spectral decomposition of a one-particle density matrix, this would imply that the classical field $\Psi ({\bf r},t)$ is the condensate wave function and that all atoms are in the condensate. To split the classical field into the condensed and thermal fractions one first realizes that the high energy solutions of Eq. (\[CFequation\]) oscillate rapidly in time and space. On the other hand, the detection process is always performed with limited spatial and temporal resolution. Therefore, it becomes clear that the measurement process with its limited resolution involves a kind of averaging (coarse-graining) of Eq. (\[denmat\]). Again, an analogy with electromagnetic waves is in place. A partially coherent light beam exhibits complicated spatial and temporal behavior on short scales, in fact too complicated to be typically measured. What is important are then the correlation functions at long enough spatial and temporal separation, averaged over the smaller time intervals and space segments. Similarly here, the averaging over time and/or space of the one-particle density matrix (\[denmat\]) results in a partial loss of the information contained in the classical field [@rapid]. In other words, the mixed state emerges out of the pure one.
In a typical experiment, what is measured is the column density along some direction. Hence, we also implement the coarse-graining procedure in our numerics in this manner as: $$\bar{\rho}(x,y,x',y';t) = \frac{1}{N} \int dz \, \Psi(x,y,z,t) \, \Psi^*(x',y',z,t) \,.
\label{rhoave}$$
Solving the eigenvalue problem for a coarse-grained density matrix (\[rhoave\]) results in a decomposition: $$\bar{\rho}=\sum_k n_k \, \varphi_k(x,y,t) \, \varphi^*_k(x^{\,\prime},y^{\,\prime},t) \,,$$ where $n_k = N_k /N$ are the relative occupations of macroscopically occupied modes $\varphi_k$. Defining the dominant eigenvalue as $n_0$, the condensate wave function (normalized to $N_0/N$) can be written as: $$\psi_0(x,y,t) = \sqrt{\frac{N_0}{N}}\, \varphi_0(x,y,t) \,.$$ All the other modes contribute to the thermal density which is, therefore, given by: $$\rho_T(x,y,t) = \bar{\rho}(x,y,x,y;t) - |\psi_0(x,y,t)|^2 \,.$$
The appropriateness of the averaging (\[rhoave\]) for obtaining the large eigenvalues of a coarse-grained density matrix was verified by us in various ways. For example, we checked that for the classical field at equilibrium this procedure gives the same results as averaging over a long enough time. Prescription (\[rhoave\]) also works at zero temperature (when all atoms are expected to be in a condensate), having been successfully tested in this respect in the demanding case of a lattice of bent vortices (see Ref. [@mcv] for further details).
According to the description given above the splitting of the system into condensed and noncondensed components is a result of Bose statistics, interaction, and the measurement process. Unlike the alternative approaches [@ZNG; @Burnett-osc] we do not impose a two-component character of the system from the beginning. Also, our version of the classical field method is well suited to describe single realizations of the experiment since it corresponds to a microcanonical ensemble, as opposed to the competing approach [@CastinCFA], which deals with canonical ensembles.
Obtaining equilibrium states {#eqstates}
----------------------------
An initial classical field is generated from the ground state solution of Eq. (\[CFequation\]) by adding appropriate random disturbance. An inspection of that equation indicates that only the product of the coupling constant $g$ and the total number of atoms $N$ enters it. Therefore, we normalize the initial classical field $\Psi ({\bf r},t=0)$ to unity. The norm of $\Psi ({\bf r},t)$ is then one of the constants of motion of Eq. (\[CFequation\]). Another constant of motion is the total energy. Such an initial state is then propagated according to Eq. (\[CFequation\]) until the constituent energies (kinetic, trap, and interaction) cease to change systematically in time and exhibit only fluctuations. In this way the classical field at thermal equilibrium, corresponding to the particular values of $gN$ and $E_{tot}/N$, is obtained. An example is given in Fig. \[cfeq\], where we plot cuts of the total density, the condensate, and the thermal densities according to the prescription detailed in the previous section. Here, the equilibrium classical field describes a degenerate $^{87}$Rb Bose gas in a magnetic trap with frequencies $\omega_{\bot} \equiv \omega_{x,y} = 2\pi \times 129\,$Hz and $\omega_z = 2\pi \times 365\,$Hz as in the experiment of Ref. [@Cornell-osc]. Other parameters are: $gN=2911.9$ and $E_{tot}/N=21.2$ in units of $\hbar \omega_z (\hbar / m \omega_z)^{3/2}$ and $\hbar \omega_z$, respectively. They result in a condensate fraction $n_0=0.3$. A bimodal distribution (thick solid line) is clearly visible.
Now we have to solve the problem how to find the number of particles assigned to the classical field at equilibrium and how to determine the temperature of the system. This can be done in two ways. The first is given here, the second in Sec. \[HFmethod\]. To begin with, having the classical field at equilibrium one can project the field $\Psi ({\bf r},t)$ on the harmonic oscillator states obtaining in this way the relative populations of these states. Fig. \[relpop\] shows relative populations for $gN=1811.9$ and $E_{tot}/N=10.0$ as a function of the harmonic oscillator states’ energy (precisely, the time average over $274\,$ms is plotted). The maximal one-particle mode energy is determined by the momentum cutoff $p_{max}$ as $p_{max}^2/m$.
An important observation is made when one looks at the energy accumulated in harmonic oscillator states, see Fig. \[kTN\]. For higher energy states the product $n_i \, \varepsilon_i$ (where $n_i$ and $\varepsilon_i$ are the relative population and the harmonic oscillator state energy, respectively) becomes constant. On the other hand, for highly occupied modes (i.e., modes satisfying $\epsilon_i - \mu \lesssim k_B T$, where $\mu$ is the chemical potential) the quantum Bose-Einstein distribution reduces to the classical equipartition distribution. For the classical field studied here the equipartition extends all the way to the cutoff energy: $$N_i \, (\epsilon_i - \mu) = k_B T \,,$$ which can be written equivalently as $$n_i \, (\epsilon_i - \mu) = k_B T / N \,.
\label{qppop}$$ Therefore, since energy equipartition is established, the higher energy harmonic oscillator states become the quasiparticle modes. In Fig. \[kTN\] we determine the ratio $k_B T/N$ to be $4.62 \times 10^{-4}$. Having the ratio $k_B T/N$ we now find from Eq. (\[qppop\]) the relative populations of high energy quasiparticle modes. In particular, we get the relative population of the least occupied modes which belong to the classical field. In the example here, they are $2.99 \times 10^{-5}$ (based on formula (\[qppop\]) with $k_B T/N$ obtained above). In Ref. [@EW], arguments are given that the best match between this method and the ideal Bose gas occurs when the occupation of the least occupied mode is $N_{cut}=0.46$. These are based on the comparison between the probability distribution of the ideal Bose gas and its classical field counterpart. Assuming, then that the average number of atoms in this least-occupied mode is $0.46$ we can retrieve the total number of atoms in the system and its temperature separately. For the example here, these are $N=15342$ and $T=124.1\,$nK.
Since no data on the population of the cutoff mode are available for a weakly interacting Bose gas considered here we compare in Table \[tab0\] the temperature of the system and the total number of atoms for various values of $N_{cut}$.
$N_{cut}$ $T$\[nK\] $N$
----------- ----------- -------
0.44 119.5 14778
0.46 124.1 15342
0.48 130.4 16121
0.50 135.8 16793
0.52 141.2 17465
: Temperature and the total number of atoms obtained by projecting the classical field on the harmonic oscillator states of the example described in Sec. \[eqstates\] for different values of cutoff parameter.[]{data-label="tab0"}
Comparison with the self-consistent Hartree-Fock model {#HFmethod}
------------------------------------------------------
Another approach to obtain the number of atoms and the temperature of the system described by the classical field at equilibrium is based on the self-consistent Hartree-Fock method [@Pethick]. Since this method works well for a Bose gas at equilibrium as verified experimentally in [@Aspect], a comparison between this model and the classical field approximation should be instructive. The Hartree-Fock description is defined by the following set of equations [@Pethick]: $$\begin{aligned}
&&n_c({\bf r}) = \frac{1}{g}\left[ \mu - V_{tr}({\bf r}) - 2\, g\, n_{th}({\bf r}) \right]
\label{SCHF1} \\
&&f({\bf r},{\bf p}) = \left( e^{ [{\bf p}^2/2m + V_{eff}({\bf r}) - \mu ] / k_B T} -1 \right)^{-1}
\label{SCHF2} \\
&&n_{th}({\bf r}) = \frac{1}{\lambda_T^3} \;\;
g_{3/2}\left( e^{\left[\mu - V_{eff}({\bf r}) \right] /k_B T } \right )
\label{SCHF3} \\
&&V_{eff}({\bf r}) = V_{tr}({\bf r}) + 2\, g\, n_c({\bf r}) + 2\, g\, n_{th}({\bf r})
\\
&&\mu = g\, n_c(0) + 2\, g\, n_{th}(0) \,,\end{aligned}$$ where $$\lambda_T = \left( \frac{2\pi\hbar^2}{mk_B T} \right)^{1/2}$$ is the thermal de Broglie wavelength and the $g_{3/2}(z)$ function is given by the expansion: $$g_{3/2}(z) = \sum_{n=1}^\infty \frac{z^n}{n^{3/2}} \,.$$
The basic variables in this approach are the condensate density, $n_c({\bf r})$, and the distribution function in phase space for thermal atoms, $f({\bf r},{\bf p})$. They are calculated according to the Eqs. (\[SCHF1\]) and (\[SCHF2\]). The thermal density, $n_{th}({\bf r})$, is just an integral of the distribution function $f({\bf r},{\bf p})$ over momenta and can be found analytically (Eq. (\[SCHF3\])). Since both the effective potential, $V_{eff}({\bf r})$, and the chemical potential, $\mu$, appearing on the right-hand sides of Eqs. (\[SCHF1\]), (\[SCHF2\]), and (\[SCHF3\]), depend on the condensate and thermal densities the set of Eqs. (\[SCHF1\]) and (\[SCHF2\]) is well suited to be solved iteratively. For that, however, we have to first choose the temperature of the system and then keep fixed the total number of atoms (which is $N=\int d{\bf r}\, (n_c({\bf r}) + n_{th}({\bf r}))$) during the iterations. Having the densities of condensed and noncondensed fractions one can easily calculate two important parameters: the condensate fraction and the total energy per particle. Now the strategy is as follows: find the input parameters $(N,T)$ in such a way that the condensate fraction and the total energy per atom calculated within the Hartree-Fock method match the values calculated from the classical field at equilibrium. This procedure allows one to determine the number of atoms and the temperature assigned to the classical field separately.
These parameters for the example discussed in the context of Figs. \[relpop\] and \[kTN\] are found to be $N=17306$ and $T=128.7\,$nK. These values are very close to what was obtained in the previous section with the cutoff occupation $N_{cut}=0.46$, and differ by $3.7\%$ in the temperature and $12.8\%$ in the total number of atoms. The agreement is good even though the average occupation of the highest energy modes (the last modes considered in the classical field approximation as being macroscopically occupied) was taken the same as for an ideal gas. Unfortunately, there is no data available for the average occupation of the cutoff region modes for the weakly interacting gas considered here. Changing slightly this cutoff occupation number, the agreement between both approaches to the system parameters can be made even better (for example, for $N_{cut}=0.48$ the difference is $1.3\%$ and $7.4\%$ in the temperature and the total number of atoms, respectively). In Fig. \[SCHF-CFA\] the total Hartree-Fock and classical field densities are plotted together for parameters as in Figs. \[relpop\] and \[kTN\] showing a good agreement. The classical field density, however, exhibits all the fluctuations which are not present in the Hartree-Fock model.
In Table \[tab1\] we again compare the parameters of the system (total number of atoms and temperature) for the methods described in this and in the previous sections. However, this time the comparison is for several different equilibrium states. All the states being compared are used later in the simulations related to JILA experiment [@Cornell-osc] (see Secs. \[mode0\] and \[mode2\]).
$ $ $CFA$ $HF$
----------- ------- -------
$T$\[nK\] 79.8 91.8
$N$ 6751 11300
$T$\[nK\] 124.1 128.7
$N$ 15342 17306
$T$\[nK\] 142.1 153.9
$N$ 16347 18699
$T$\[nK\] 154.4 168.8
$N$ 19313 21509
$T$\[nK\] 177.8 190.4
$N$ 26984 27875
$T$\[nK\] 195.3 204.8
$N$ 27770 31222
$T$\[nK\] 264.1 240.9
$N$ 61876 43572
: Comparison (for several final equilibrium states as per Table \[tab2\]) between the results (the temperature and the total number of atoms) obtained by projecting the classical field on the harmonic oscillator states as described in Sec. \[eqstates\] ($CFA$ column) and by utilizing the self-consistent Hartree-Fock method according to Sec. \[HFmethod\] ($HF$ column). For $CFA$ data $N_{cut}=0.46$. []{data-label="tab1"}
Obtaining equilibrium states on demand {#ondemand}
--------------------------------------
In Secs. \[eqstates\] and \[HFmethod\] we detailed the methods for retrieving the total number of atoms and the temperature assigned to the classical field at equilibrium. Since the product $gN$ is an initial parameter it means that the coupling constant $g$ (and consequently the scattering length $a$) is known only after the classical field is thermalized. Unfortunately, it usually differs from the value that was used to calculate an initial value of $gN$. Therefore, an approach for obtaining the classical field at equilibrium corresponding to given values of the total number of atoms and the temperature is required.
The method we have developed is rather simple although demanding from a numerical point of view. Let’s assume that we need the classical field which at equilibrium describes the system with given parameters $N$ and $T$. We start from a solution obtained from the self-consistent Hartree-Fock method corresponding to the chosen parameters. Then we build an initial classical field as follows: $$\Psi ({\rm {\bf r}},t=0) = \sqrt{n_c({\bf r}) } + \sqrt{n_{th}({\bf r})}\, e^{i \varphi({\bf r}) }
\label{psi}$$ and randomize the phase $\varphi({\bf r})$ and the density $n_{th}({\bf r})$ in such a way that the total energy per atom in the classical field equals the corresponding energy in the Hartree-Fock model. The presence of the phase factor in the second term in (\[psi\]) is necessary. Without this, the classical field suffers from a lack of kinetic energy in comparison with the Hartree-Fock method, where it is calculated from the distribution function $f({\bf r},{\bf p})$: $$\begin{aligned}
&&E_{kin} = \frac{1}{h^3} \int d{\bf r}\, d{\bf p}\, \frac{{\bf p}^2}{2m}
f({\bf r},{\bf p}) \nonumber \\
&&= \frac{3 k_B T}{2 \lambda_T^3}
\int d{\bf r}\, g_{5/2}\left( e^{\left[\mu - V_{eff}({\bf r}) \right] /kT } \right )
\,.
\label{kinene}\end{aligned}$$
Now we evolve the classical field according to Eq. (\[CFequation\]) and let the field thermalize. During the thermalisation, the total energy per atom is a constant of motion, but the condensate fraction usually changes. However, there will be a particular value of the spatial step of the grid for which the condensate fraction does not change in time. Then, since the total energy per atom is a constant of motion the values of parameters $n_0$ and $E_{tot}/N$ at the end of thermalization process are the same as at the beginning. Consequently, the number of particles $N$ and the temperature $T$ must be the same as chosen at the beginning as well. Although, the procedure just described is numerically time consuming (since it requires several trials to obtain the proper value of the spatial step), it is much more efficient than attempts to match final $T$ and $N$ simultaneously, which require search in two-dimensional parameter space. Here, the energy matching is computationally fast because it requires no thermalisation, while the final $T,N$ matching is done with only one free parameter, the spatial lattice spacing.
Results for the m=0 mode {#mode0}
========================
Our numerical procedure involves the following steps: First, we find the classical field (as described in \[ondemand\]) corresponding to the Bose gas at equilibrium confined in a harmonic trap with frequencies $\omega_{\bot} \equiv \omega_{x,y} = 2\pi \times 129\,$Hz and $\omega_z = 2\pi \times 365\,$Hz. According to the experiment [@Cornell-osc] the total number of atoms was of the order of ten to a few tens of thousands and the initial temperature was ranging up to the critical temperature. The number of condensed atoms remained on a level of several thousand. The numerical values of $N$, $N_0$, $T$, and the reduced temperature $T^{\prime} \equiv T/T_c$ (with $T_c$ being the transition temperature for a harmonically confined ideal gas) for the states used in the simulations are shown in Table \[tab2\].
$N$ $N_0$ $T$\[nK\] $T^{\prime}$
------- ------- ----------- --------------
11300 8558 92.8 0.497
17306 10349 128.7 0.605
18700 7905 153.9 0.705
21509 7768 168.8 0.738
27875 8496 190.4 0.763
31222 7753 204.8 0.791
43572 7111 240.9 0.832
: Numerical values of $N$, $N_0$, $T$, and $T^{\prime}$ used in the simulations.[]{data-label="tab2"}
Next, the Bose gas is disturbed by a sinusoidal perturbation to the trapping potential. Since the classical field describes both the condensed and noncondensed atoms, the disturbance of the classical field means that both components are simultaneously perturbed. To excite the $m=0$ and the $m=2$ quadrupole modes, the perturbation of the trapping potential takes the form: $$\begin{aligned}
\delta V_{tr}({\rm {\bf r}},t) = A(t)\, [\omega_x^2 x^2 \sin(\omega_d\, t + \phi) +
\omega_y^2 y^2 \sin(\omega_d\, t)] \,, \nonumber \\
\label{pertur}\end{aligned}$$ where $\omega_d$ is the driving frequency and $\phi$ is a phase shift between the $x$ and $y$ direction perturbations. The choice of the phase shift $\phi$ determines the symmetry of the excited collective mode - for $\phi =0$ ($\phi =\pi$) the $m=0$ ($m=2$) mode is excited. The perturbation, as in the experiment, lasts for $14\,$ms and the amplitude $A(t)\, (=0.05)$ takes small value to avoid any nonlinear effects.
After the perturbation is turned off, the classical field is oscillating in time. Is it possible that the condensed and noncondensed components (extracted from the single classical field) exhibit oscillations with different frequencies? To answer these question we split the classical field into the condensate and the thermal cloud in the way described in Sec. \[formalism\] and calculate the widths of both components from the formula: $$\begin{aligned}
w_{c,th} = \int dx\, dy\, (x^2 + y^2)\, n_{c,th}(x,y) \,.
\label{width}\end{aligned}$$ Results are shown in Fig. \[oscillations\] for the $m=0$ mode. Solid symbols (upper frame) represent the condensate widths, whereas the open symbols (middle frame) stand for thermal cloud widths. Solid lines in both frames are fits by an exponentially damped sine waves: $$\begin{aligned}
A_{c,th}\; \exp(-\gamma_{c,th} t)\, \sin(\omega_{c,th} t + \vartheta_{c,th}) + B_{c,th}
\label{fit}\end{aligned}$$ to numerical data. As in the experiment, fits are performed based on five initial oscillations. The lower frame shows, indeed, that condensate and thermal components oscillate with different frequencies. Moreover, these oscillations are phase shifted and the condensate oscillates slower than the thermal cloud.
In Figs. \[m0fr\] and \[m0dr\] we summarize our results for the $m=0$ mode. In Fig. \[m0fr\] we plot the frequencies of the condensate and thermal fractions response to the external perturbation for various temperatures together with the experimental data of Ref. [@Cornell-osc]. Black solid and open symbols represent data for a condensate (upper frame) and a thermal cloud (lower frame) for various driving frequencies according to the legend attached to the figure. Gray symbols with error bars are the experimental results. Up to temperature $T \approx 0.6 T_c$ both components oscillate with the same frequency, which is the natural condensate frequency for the $m=0$ collective mode. At approximately $0.65 T_c$ a rather sudden upward shift in condensate frequency is observed in the experiment. Our numerics shows that at this temperature the dynamics of the thermal cloud changes. The thermal component starts to oscillate with a higher frequency approaching eventually $2 \omega_{\bot}$ which is the oscillation frequency of a thermal gas alone. For higher temperatures the thermal fraction becomes dominant and finally thermal atoms change the dynamics of the condensate in such a way that condensed atoms oscillate along with the thermal ones. Unfortunately, the influence of the thermal cloud on the condensate is apparently too weak and consequently the condensate starts to oscillate with higher frequencies only for higher temperatures.
Our results also show that the notion of natural condensate frequency breaks down when the thermal cloud is present. The condensate response depends on the dynamics of the thermal cloud and, in fact, the possible frequencies for the condensate oscillation lie in an interval which gets wider for higher temperatures. In particular, no two branches of frequencies are visible as reported in Ref. [@Zaremba-osc]. In Fig. \[m0dr\] we compare the numerical and experimental damping rates for the oscillations of the condensed and noncondensed components. There is some discrepancy for higher temperatures where the thermal component seems to be damped too strongly in comparison with the experimental data. Perhaps this increases the temperature at which the frequency of the condensate fraction shifts up.
Results for the m=2 mode {#mode2}
========================
We now do the same analysis for the $m=2$ mode. The system is disturbed by changing the trapping potential according to (\[pertur\]) with the phase shift $\phi =\pi$. Such a perturbation excites the quadrupole $m=2$ oscillations. The classical field is split into the condensed and thermal parts and the condensate and thermal widths are calculated using formulas exhibiting the $m=2$ mode’s symmetry: $$\begin{aligned}
w_{c,th} = \int dx\, dy\, (x^2 - y^2)\, n_{c,th}(x,y) \,.
\label{width2}\end{aligned}$$ As before, these data are fitted by exponentially damped sine waves.
Frequencies and damping rates for the condensate and the thermal cloud are plotted in Figs. \[m2fr\] and \[m2dr\]. Again, the system responds in a way that depends on the driving frequency. A comment on how the authors of the experiment [@Cornell-osc] choose the driving frequency is in place here. They say that the driving frequency is set to match the frequency of the excitation being studied. This could make sense for $m=0$ mode where supposedly the condensed and noncondensed fractions always oscillate in phase. Supposedly, because in [@Cornell-osc] we see only two data points (both having frequency approximately $2 \omega_{\bot}$) showing the behavior of the thermal cloud in the $m=0$ mode. However, for the $m=2$ mode the frequencies at which the condensate and thermal cloud oscillate are different, and the meaning of the driving frequency as the frequency of the excitation being studied is not clear. Therefore, in Fig. \[m2fr\] we show our data for various driving frequencies. Good agreement with experiment is found for appropriate driving frequencies up to temperatures $\approx 0.8 T_c$. For temperatures approaching the critical one, however, we observe the same effect as for the $m=0$ mode. The thermal cloud becomes dominant and the frequencies at which the condensate oscillates become higher. Fig. \[m2fr\] shows that already for a temperature $T = 0.83 T_c$, with the driving frequency $\omega_d = 1.75 \omega_{\bot}$, the condensate oscillates with the frequency of the thermal cloud. Such a behavior was not observed experimentally. On the other hand, it is an expected behavior, since the condensed fraction becomes smaller and smaller while the critical temperature is approached, and the dynamics should be dominated by the thermal cloud. In the case of the thermal cloud, our model predicts frequencies in agreement with the experiment (other theoretical studies of the JILA experiment either do not show results for the thermal atoms for the $m=2$ mode, or predict frequencies that are very different than observed experimentally).
In Fig. \[m2dr\] we compare the numerical and experimental damping rates for the oscillations of the condensed and noncondensed parts of the system. As for the $m=0$ mode, there is some discrepancy for higher temperatures where the thermal cloud is damped too strongly, whereas the condensate is damped too weakly.
Conclusions {#concl}
===========
In conclusion, we have presented in detail the construction of the classical field describing the desired number of atoms, confined in any trapping potential, at a prescribed temperature. We have studied the oscillations of the Bose-Einstein condensate in the presence of a thermal cloud. As in the experiment, we find the temperature dependent condensate frequency shift for both the $m=0$ and $m=2$ collective oscillation modes. For the $m=0$ mode, the thermal atoms pull the condensate fraction along, and above some characteristic temperature $(\approx 0.8 T_c)$ the condensate tends to oscillate with the frequency of the thermal part (approximately equal to $2 \omega_{\bot}$). Unfortunately, in the present version of the classical field approximation, the value of this characteristic temperature turns out to be about $20 \%$ higher than the one observed in the experiment. For the $m=2$ mode, on the other hand, the frequency at which the condensate oscillates first decreases with an increase of temperature (the thermal cloud oscillating with its natural frequency equal to $2 \omega_{\bot}$ damps the condensate motion) but when the temperature gets closer to the critical temperature the condensate starts to oscillate with higher frequencies, approaching the frequency of the thermal cloud.
We are grateful to Piotr Deuar for his critical reading of the manuscript and his valuable comments. The authors acknowledge support by Polish Government research funds for 2009-2011. Some of the results have been obtained using computers at the Interdisciplinary Centre for Mathematical and Computational Modeling of Warsaw University. The Centre for Quantum Technologies is a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore.
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---
abstract: 'This paper extends some recent results on the controllability/observability of networked systems to a system in which the system matrices of each subsystem are described by a linear fractional transformation (LFT). A connection has been established between the controllability/observability of a networked system and that of a descriptor system. Using the Kronecker canonical form of a matrix pencil, a rank based condition is established in which the associated matrix affinely depends on a matrix formed by the parameters of each subsystem and the subsystem connection matrix (SCM). One of the attractive properties of this condition is that in obtaining the associated matrices, all the involved numerical computations are performed on each subsystem independently, which makes the condition verification scalable for a networked system formed by a large number of subsystems. In addition, the explicit expression of the condition associated matrix on subsystem parameters and subsystem connections may be helpful in system topology design and parameter selections. As a byproduct, this investigation completely removes the full normal rank condition required in the previous works.'
author:
- 'Tong Zhou[^1][^2]'
title: '**Topology and Subsystem Parameter Based Verification for the Controllability/Observability of a Networked Dynamic System**'
---
controllability, descriptor system, first principle parameter, LFT, networked system, observability
Introduction
============
In system designs, it is essential to at first construct a plant that is possible to achieve a good performance. When a networked system is to be designed, this problem is related to both subsystem parameter selections and the design of subsystem connections. To achieve this objective, it appears preferable to establish an explicit relation between system achievable performances and its subsystem parameters and connections. On the other hand, both controllability and observability are essential requirements for a system to work properly, noting that they are closely related to a number of important system properties. For example, the existence of an optimal control, possibilities of stabilizing a plant and/or locating its poles to a desirable area, convergence of a state estimation procedure like the extensively utilized Kalman filter, etc., are closely related to the controllability and/or observability of the plant at hand [@Kailath80; @sbkkmpr11; @zdg96; @zyl18].
Controllability and observability are well developed concepts in system analysis and synthesis, and various criteria have been established, such as the PBH test, controllability/observability matrix, etc. [@ksh00; @zdg96; @zyl18]. It is now extensively known that for many systems, both controllability and observability are generic system property, which means that rather than a numerical value of the system matrices, it is the connections among the system states, as well as the connections from an input to the system states (the connections from the system states to an output), that determine the controllability (observability) of a system [@dcv03; @zyl18]. When a networked system is under investigation, various new theoretical issues arise, which include influences from subsystem dynamics, subsystem connections, etc., to the controllability/observability of the whole system. Another challenging issue is computational costs and numerical stability [@ccv12; @cpa17; @zyl18].
In this paper, we reinvestigate the controllability/observability verification problem for networked systems in which the system matrices of each subsystem are described by a linear fractional transformation (LFT), which has been studied in [@zz19]. It has been observed that this verification problem can be converted to the verification of the controllability/observability of a particular descriptor system. Using the Kronecker canonical form of a matrix pencil, a rank based condition is established in which the associated matrix affinely depends on a matrix formed by the parameters of each subsystem and the subsystem connection matrix (SCM). This condition keeps the attractive properties of the verification procedure reported in [@zz19] that in obtaining the associated matrices, the involved numerical computations are performed on each subsystem independently, which means that the associated condition verification is also scalable for a networked system formed by a large number of subsystems. In deriving these results, there are neither any restrictions on a subsystem first principle parameter, nor any restriction on an element of the SCM. Note that in [@zz19], it is required that the matrix that is constructed from all subsystem first principle parameters and the SCM, must have a diagonal parametrization, which is a great restriction on the applicability of the obtained results to a practical problem. This assumption removal appears to be the most significant advances of this paper. In addition, the explicit expression of the matrix on subsystem parameters and subsystem connections may be helpful in system topology design and parameter selections. As a byproduct, this investigation also completely removes the full normal rank condition required in [@zhou15; @zyl18].
The outline of this paper is as follows. At first, in Section II, a state space model like representation is given for a networked dynamic system, together with some preliminary results. Controllability and observability of a networked dynamic system are investigated respectively in Sections III and IV. Finally, some concluding remarks are given in Section V in which some further issues are discussed. Two appendices are included to give proofs of some technical results.
The following notation and symbols are adopted. ${\rm\bf det} \left(\cdot\right)$ represents the determinant of a square matrix, ${\rm\bf null} \left(\cdot\right)$ and ${\rm\bf span} \left(\cdot\right)$ the null space of a matrix and the space spanned by the columns of a matrix. ${\rm\bf
diag}\!\{X_{i}|_{i=1}^{L}\}$ denotes a block diagonal matrix with its $i$-th diagonal block being $X_{i}$, while ${\rm\bf
col}\!\{X_{i}|_{i=1}^{L}\}$ the vector/matrix stacked by $X_{i}|_{i=1}^{L}$ with its $i$-th row block vector/matrix being $X_{i}$. For a $m\times n$ dimensional matrix $A$, $A(1:k)$ stands for the matrix consisting of its first $k$ columns with a $k$ satisfying $1\leq k\leq n$, while $A(1\!\sim\!k)$ the matrix consisting of its first $k$ rows with a $k$ satisfying $1\leq k\leq m$. $0_{m}$ and $0_{m\times n}$ represent respectively the $m$ dimensional zero column vector and the $m\times n$ dimensional zero matrix. The subscript is usually omitted if it does not lead to confusions. The superscript $T$ and $H$ are used to denote respectively the transpose and the conjugate transpose of a matrix/vector.
System Description and Some Preliminaries
=========================================
In actual engineering problems, the subsystems of an NDS may have distinguished input-output relations. A possible approach to describe the dynamics of a general linear time invariant (LTI) NDS is to model each of its subsystems as that in [@zhou15; @zyl18]. To express the dependence of the system matrices of a subsystem on its first principle parameters, the following model is used in this paper to describe the dynamics of the $i$-th subsystem ${\bf{\Sigma}}_i$ of an NDS ${\bf{{\bf \Sigma}}}$ composing of $N$ subsystems, which is also utilized in [@zz19] for a continuous time NDS. $$\hspace*{-0.25cm}\begin{array}{l}
\left[\!\! {\begin{array}{*{20}{c}}
{{{x}}(t+1,i)}\\
{{z}(t,i)}\\
{{y}(t,i)}
\end{array}}\!\! \right] \!=\! \left\{\! {\left[\!\! {\begin{array}{*{20}{c}}
{A_{\rm\bf xx}^{[0]}(i)} & {A_{\rm\bf xv}^{[0]}(i)} & {B_{\rm\bf x}^{[0]}(i)}\\
{A_{\rm\bf zx}^{[0]}(i)} & {A_{\rm\bf zv}^{[0]}(i)} & {B_{\rm\bf z}^{[0]}(i)}\\
{C_{\rm\bf x}^{[0]}(i)} & {C_{\rm\bf v}^{[0]}(i)} & {D^{[0]}(i)}
\end{array}}\!\! \right] \!\!+\!\! \left[ \!\!{\begin{array}{*{20}{l}}
{E_1{(i)}}\\
{E_2{(i)}}\\
{E_3{(i)}}
\end{array}} \!\!\right]\!\!\times } \right.\\
\left. {\left. {\begin{array}{*{20}{l}}
\\
\\
\end{array}} \right.{P(i)}{{\left[I - {G{(i)}}{P{(i)}}\right]}^{ - 1}}\left[\!\! {\begin{array}{*{20}{c}}
{F_1{(i)}} & {F_2{(i)}} & {F_3{(i)}}
\end{array}}\!\! \right]}\! \right\}\!\!\left[\!\! {\begin{array}{*{20}{c}}
{{x}(t,i)}\\
{{v}(t,i)}\\
{{u}(t,i)}
\end{array}}\!\! \right]
\end{array}
\label{eqn:1}$$ Here, $t$ represents the temporal variable, $x(t,i)$ its state vector, $u(t,i)$ and $y(t,i)$ respectively its external input and output vectors, $v(t,i)$ and $z(t,i)$ respectively its internal input and output vectors which denote signals received from other subsystems and signals transmitted to other subsystems. All the parameters of this subsystem are included in the matrix $P(i)$, which may be the masses, spring and damper coefficients of a mechanical system, concentrations and reaction ratios of a chemical/biological process, resistors, inductor and capacitor coefficients of an electronic/electrical system, etc. These parameters are usually called the first principle parameter as they can be selected or adjusted in designing an actual system. The matrix $G{(i)}$, together with the matrices $E_j{(i)}$ and the matrices $F_j{(i)}$ with $j=1,2,3$, are known matrices reflecting how these first principle parameters affect the system matrices of this subsystem. These matrices, together with the matrices ${A_{\rm\bf *\#}^{[0]}(i)}$, ${B_{\rm\bf *\#}^{[0]}(i)}$, ${C_{\rm\bf *}^{[0]}(i)}$ and ${D^{[0]}(i)}$ with ${\rm\bf *,\#}={\rm\bf x}$, ${\rm\bf u}$, ${\rm\bf v}$, ${\rm\bf y}$ or ${\rm\bf z}$, are in general known and can not be selected or adjusted in system designs, as they reflect the physical, chemical or electrical principles governing the dynamics of this subsystem, such as the Kirchhoff¡¯s current law, Netwon’s mechanics, etc.
In the above description, the matrix $P(i)$ consists of fixed zero elements and elements which are from the set consisting of all the first principle parameters of the subsystem ${\bf{\Sigma}}_i$, $i=1,2,\cdots,N$. In some situations, it may be more convenient to use a simple function of some first principle parameters, such as the reciprocal of a first principle parameter, the product of several first principle parameters, etc. These transformations do not affect results of this paper, provided that the corresponding global transformation is a bijective mapping. To avoid an awkward presentation, these elements are called pseudo first principle parameters (PFPP) in this paper, and are usually assumed to be algebraically independent of each other.
Compared with the subsystem model adopted in [@zhou15; @zyl18], it is clear that each of its system matrices in the above model, that is, ${A_{\rm\bf *\#}(i)}$, ${B_{\rm\bf *\#}(i)}$, ${C_{\rm\bf *}(i)}$ and ${D(i)}$ with ${\rm\bf *,\#}={\rm\bf x}$, ${\rm\bf u}$, ${\rm\bf v}$, ${\rm\bf y}$ or ${\rm\bf z}$, is a matrix valued function of the parameter matrix $P(i)$. This reflects the fact that in an actual system, elements of its system matrices are usually not independent of each other, and some of them can not be changed in system designs. It can therefore be declared that this model is closer to the input-output relations of a dynamic plant. To have a concise presentation, the dependence of a system matrix of the subsystem ${\bf{\Sigma}}_i$ on its parameter matrix $P(i)$ is usually not explicitly expressed, except under a situation in which this omission may lead to confusions.
Obviously, the aforementioned model is also applicable to situations in which we are only interested in the influences from part of the subsystem first principle parameters on the performances of the whole NDS. This can be simply done through fixing all other first principle parameters to a particular numerical value.
Define vectors $v(t)$ and $z(t)$ respectively as $v(t)={{\rm{{\bf{{\rm {col}}}}}}}\{v(t,i)|_{i=1}^N\}$, $z(t)={{\rm{{\bf{{\rm {col}}}}}}\{z(t,i)|_{i=1}^N\}}$. It is assumed in this paper that the interactions among subsystems of a NDS are described by $$v(t)=\Phi z(t)
\label{eqn:2}$$ The matrix $\Phi$ is called the subsystem connection matrix (SCM), which describes influences between different subsystems of a NDS. A graph can be assigned to a NDS when each subsystem is regarded as a node and each nonzero element in the SCM $\Phi$ is regarded as an edge. This graph is usually referred as the structure or topology of the associated NDS.
The following assumptions are adopted throughout this paper.
- The dimensions of the vectors $u(t,i)$, $v(t,i)$, $x(t,i)$, $y(t,i)$ and $z(t,i)$ are respectively $m_{{\rm\bf u}i}$, $m_{{\rm\bf v}i}$, $m_{{\rm\bf x}i}$, $m_{{\rm\bf y}i}$ and $m_{{\rm\bf z}i}$.
- Each subsystem ${\bf{\Sigma}}_i$, $i=1,2,\cdots,N$, is well posed.
- The whole NDS ${\bf{\Sigma}}$ is well posed.
Note that the first assumption is only for indicating the size of the involved vectors. On the other hand, well-posedness is an essential requirement for a system to work properly. It appears safe to declare that all the above three assumptions must be satisfied for a practical system. Therefore, the adopted assumptions seem not very restrictive in actual applications.
Using these symbols, define integers $M_{{\rm\bf x}i}$, $M_{{\rm\bf
v}i}$, $M_{\rm\bf x}$ and $M_{\rm\bf v}$ as $M_{\rm\bf
x}={\sum_{k=1}^{N} m_{{\rm\bf x}k}}$, $ M_{\rm\bf v}={\sum_{k=1}^{N}
m_{{\rm\bf v}k}}$, and $M_{{\rm\bf x}i}=M_{{\rm\bf v}i}=0$ when $i=1$, $M_{{\rm\bf x}i}={\sum_{k=1}^{i-1} m_{{\rm\bf x}k}}$, $M_{{\rm\bf v}i}={\sum_{k=1}^{i-1} m_{{\rm\bf v}k}}$ when $2\leq
i\leq N$. Obviously, the SCM $\Phi$ is a $M_{\rm\bf v}\times M_{\rm\bf z}$ dimensional real matrix. In addition, if we partition this matrix according to the dimensions of the vectors $v(t,i)|_{i=1}^{N}$ and $z(t,i)|_{i=1}^{N}$ and denote its $i$-th row $j$-th column block by $\Phi_{ij}$, then $\Phi_{ij}$ is a $m_{{\rm\bf v}i}\times m_{{\rm\bf z}j}$ dimensional real matrix, which reflects direct influence from the subsystem ${\bf{\Sigma}}_j$ to the subsystem ${\bf{\Sigma}}_i$, $i,j=1,2,\cdots,N$.
The following results on a matrix pencil are required in deriving a computationally checkable necessary and sufficient condition for the aforementioned NDS, which can be found in many references, for example, [@bv88; @it17].
For two arbitrary $m\times n$ dimensional real matrices $G$ and $H$, a first degree matrix valued polynomial $\Psi(\lambda)=\lambda G+H$ is called a matrix pencil. When $m=n$ and ${\rm\bf det}(\Psi(\lambda))\not\equiv 0$, this matrix pencil is called regular. A regular matrix pencil is called strictly regular if both the associated matrix $G$ and the associated matrix $H$ are invertible. A matrix pencil $\bar{\Psi}(\lambda)$ is said to be strictly equivalent to the matrix pencil $\bar{\Psi}(\lambda)$, if there exist two invertible real matrices $U$ and $V$ satisfying $\Psi(\lambda)=U\bar{\Psi}(\lambda)V$.
Given a positive integer $m$, two $m\times m$ matrix pencils $K_{m}(\lambda)$ and $N_{m}(\lambda)$, a $m\times (m+1)$ matrix pencil $L_{m}(\lambda)$, as well as a $(m+1)\times m$ matrix pencil $J_{m}(\lambda)$, are defined as follows. $$\begin{aligned}
& &
\hspace*{-1.0cm} K_{m}(\lambda)\!=\!\lambda I_{m}\!+\!\left[\!\!\begin{array}{cc}
0 & I_{m-1} \\ 0 & 0 \end{array}\!\!\!\right]\!,\hspace{0.1cm}
N_{m}(\lambda)\!=\!I_{m}\!+\!\left[\!\!\begin{array}{cc}
0 & \lambda I_{m-1} \\ 0 & 0 \end{array}\!\!\!\right] \label{eqn:4} \\
& &
\hspace*{-1.0cm} L_{m}(\lambda)=\left[\begin{array}{cc}
K_{m}(\lambda) & \left[\begin{array}{c} 0 \\ 1 \end{array}\right] \end{array}\right],\hspace{0.15cm}
J_{m}(\lambda)= \left[\begin{array}{c}
K_{m}^{T}(\lambda) \\ \left[0 \;\;\;\;\; 1\right] \end{array}\right] \label{eqn:5}\end{aligned}$$
Obviously, $J_{m}(\lambda)=L_{m}^{T}(\lambda)$. For a clear presentation, however, it appears better to introduce this matrix pencil. Moreover, when $m=0$, $L_{m}(\lambda)$ is a $0\times 1$ zero matrix, while $J_{m}(\lambda)$ is a $1\times 0$ zero matrix.
It is well known that any matrix pencil is strictly equivalent to a block diagonal form with its diagonal blocks being in the form of the matrix pencils $K_{m}(\lambda)$, $N_{m}(\lambda)$, $L_{m}(\lambda)$ and $J_{m}(\lambda)$, which is extensively called the Kronecker canonical form. More precisely, we have the following results [@bv88; @Gantmacher59; @it17].
For any two $m\times n$ dimensional real matrices $G$ and $H$, there exist some unique nonnegative integers $\mu$, $a$, $b$, $c$ and $d$, some unique positive integers $\xi_{j}|_{j=1}^{a}$ and $\eta_{j}|_{j=1}^{b}$, some unique nonnegative integers $\kappa_{j}|_{j=1}^{c}$ and $\rho_{j}|_{j=1}^{d}$, as well as a strictly regular $\mu\times \mu$ dimensional matrix pencil $H_{\mu}(\lambda)$, such that the matrix pencil $\Psi(\lambda)=\lambda G+H$ is strictly equivalent to a block diagonal form $\bar{\Psi}(\lambda)$ with $$\begin{aligned}
\bar{\Psi}(\lambda)\!\!&=&\!\!{\rm\bf diag}\!\left\{\!H_{\mu}(\lambda),\;K_{\xi_{j}}(\lambda)|_{j=1}^{a},\; L_{\kappa_{j}}(\lambda)|_{j=1}^{c}, \right.\nonumber \\
& & \hspace*{2.5cm}\left. N_{\eta_{j}}(\lambda)|_{j=1}^{b},\; J_{\rho_{j}}(\lambda)\!|_{j=1}^{d}\!\right\}
\label{eqn:6}\end{aligned}$$ \[lemma:2\]
Observability of an NDS
=======================
Note that parallel, cascade and feedback connections of LFTs can still be expressed as a LFT [@zdg96]. On the other hand, [@zhou15] has already made it clear that the system matrices of the whole NDS can be represented as a LFT of its SCM, provided that all the subsystems are connected by their internal inputs/outputs. These make it possible to rewrite the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) in a form which is completely the same as that of [@zhou15], in which all the pseudo first principle parameters of each subsystem, as well as the subsystem connection matrix, are expressed with a single matrix. This has also been performed in [@zz19].
More precisely, for each Subsystem ${\bf{\Sigma}}_i$, $i=1,2,\cdots,N$, introduce the following two auxiliary internal input and output vectors $v^{[a]}(t,i)$ and $z^{[a]}(t,i)$, $$\begin{aligned}
& & \hspace*{-1cm} z^{[a]}(t,i) = [{F_1{(i)}}\ {F_2{(i)}}\ {F_3{(i)}}]\!\left[\!\!\begin{array}{c} x(t,i) \\ v(t,i) \\ u(t,i) \end{array}\!\!\right] \!+\! {G{(i)}}v^{[a]}(t,i)
\label{eqn:7} \\
& & \hspace*{-1cm} v^{[a]}(t,i) = P(i)z^{[a]}(t,i)
\label{eqn:8}\end{aligned}$$ Let $\bar{z}(t,i)={\rm\bf col}\{z(t,i),z^{[a]}(t,i)\}$, $\bar{v}(t,i) ={\rm\bf col}\{v(t,i),v^{[a]}(t,i)\}$. Denote the dimensions of $\bar{v}(t,i)$ and $\bar{z}(t,i)$ respectively by $m_{{\rm\bf\bar{v}}i}$ and $m_{{\rm\bf\bar{z}}i}$. Then it can be straightforwardly shown that, under the assumption that this subsystem is well-posed, which is equivalent to that the matrix $I-G{(i)} P{(i)}$ is invertible, the input-output relation of Subsystem ${\bf{\Sigma}}_i$ can be equivalently expressed by (\[eqn:8\]) and the following equation $$\left[ \begin{matrix}
{{ x}(t+1,i)}\\
{{\bar{z}}(t,i)}\\
{{y}(t,i)}
\end{matrix} \right] = \left[ \begin{matrix}
{A_{\rm\bf xx}{(i)}} & {A_{\rm\bf xv}{(i)}} & {B_{\rm\bf x}{(i)}}\\
{A_{\rm\bf zx}{(i)}} & {A_{\rm\bf zv}{(i)}} & {B_{\rm\bf z}{(i)}}\\
{C_{\rm\bf x}{(i)}} & {C_{\rm\bf v}{(i)}} & {D{(i)}}
\end{matrix} \right]\left[ \begin{matrix}
{x(t,i)}\\
{\bar{v}(t,i)}\\
{u(t,i)}
\end{matrix} \right]
\label{eqn:11}$$ in which $$\begin{aligned}
& & \hspace*{-0.8cm} A_{\rm\bf xx}{(i)}=A_{\rm\bf xx}^{[0]}{(i)},\hspace{0.1cm} A_{\rm\bf xv}{(i)} = \left[ \begin{matrix}
{A_{\rm\bf xv}^{[0]}{(i)}}& {E_1{(i)}}
\end{matrix} \right] \\
& & \hspace*{-0.8cm} A_{\rm\bf zx}{(i)}{\rm{ = }}\left[ \begin{matrix}
{A_{\rm\bf zx}^{[0]}{(i)}} \\
{F_1{(i)}}
\end{matrix} \right],\hspace{0.1cm} A_{\rm\bf zv}{(i)} = \left[ \begin{matrix}
{A_{\rm\bf zv}^{[0]}{(i)}} & {E_2{(i)}}\\
{F_2{(i)}} & {{G^{(i)}}}
\end{matrix} \right] \\
& & \hspace*{-0.8cm} B_{\rm\bf x}^{(i)}=B_{\rm\bf x}^{[0]}{(i)}, \hspace{0.25cm} B_{\rm\bf z}{(i)} = \left[ \begin{matrix}
{B_{\rm\bf z}^{[0]}{(i)}}\\
{F_3{(i)}}
\end{matrix} \right] \\
& & \hspace*{-0.8cm} C_{\rm\bf x}(i) = C_{\rm\bf x}^{[0]}{(i)}, \hspace{0.25cm} C_{\rm\bf v}(i) = \left[\begin{matrix}
{C_{\rm\bf v}^{[0]}{(i)}}&{E_3{(i)}}
\end{matrix} \right]\end{aligned}$$ and $D{(i)} = D^{[0]}{(i)}$. Define vectors $\bar{v}(t)$ and $\bar{z}(t)$ respectively as $$\begin{aligned}
& & \bar{v}(t) = {\rm\bf col}\left\{ \bar{v}(t,i)|_{i=1}^N\right\} \\
& & \bar{z}(t) = {\rm\bf col}\left\{ \bar{z}(t,i)|_{i=1}^N\right\}\end{aligned}$$ Moreover, define a matrix $\bar{\Phi}$ as $$\bar{\Phi} = \begin{pmat}[{.|.||}]
{{\Phi _{11}}}&{}&{{\Phi _{12}}}&{}&{\cdots}&{{\Phi _{1N}}}&{}\cr
{}&{ P{(1)}}&{}&0& \cdots &{}&0\cr\-
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \cr\-
{{\Phi _{N1}}}&{}&{{\Phi _{N2}}}&{}&{\cdots}&{{\Phi _{NN}}}&{}\cr
{}&0&{}&0&{\cdots}&{}&{ P{(N)}}\cr
\end{pmat}.
\label{eqn:10}$$ Then Equations (\[eqn:2\]) and (\[eqn:8\]) can be compactly expressed as $$\bar{v}(t) = \bar{\Phi} \bar{z}(t)
\label{eqn:9}$$ Here, $\Phi_{ij}$, $i,j=1,2,\cdots,N$, is the $i$-th row block $j$-th column block submatrix of the SCM $\Phi$ when it is partitioned consistently with the dimensions of the system internal input and output vectors.
To emphasize similarities in system analysis and synthesis between the matrix $\bar{\Phi}$ and the matrix $\Phi$, as well as to distinguish the matrix $\bar{\Phi}$ from the matrix $\Phi$ in their engineering significance, etc., it is called the augmented subsystem connection matrix in the remaining of this paper.
Equations (\[eqn:11\]) and (\[eqn:9\]) give an equivalent description for the input-output relations of the NDS ${\bf\Sigma}$, which has completely the same form as that for the NDS investigated in [@zhou15]. This equivalent form is benefited from the invariance properties of LFTs, and makes results of [@zhou15] applicable to the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]).
From these observations, the following results have been established which can be directly obtained from [@zhou15] for the observability of the NDS described by Equations (\[eqn:1\]) and (\[eqn:2\]).
Assume that the networked dynamic system ${\rm\bf\Sigma}$, as well as all of its subsystems ${\bf{\Sigma}}_i|_{i=1}^{N}$ , are well-posed. Then, this NDS is observable if and only if for every complex scalar $\lambda$, the following matrix pencil $M(\lambda)$ is of full column rank, $$M(\lambda)=\left[\begin{array}{cc}
\lambda I_{M_{\rm\bf x}}-A_{\rm\bf xx} & -A_{\rm\bf xv} \\
-C_{\rm\bf x} & -C_{\rm\bf v} \\
-\bar{\Phi} A_{\rm\bf zx} & I_{M_{\rm\bf z}}-\bar{\Phi} A_{\rm\bf zv} \end{array}\right]
\label{eqn:3}$$ \[lemma:1\]
Here, $A_{\rm\bf
*\#}\!\!=\!\!{\rm\bf diag}\!\left\{\!A_{\rm\bf
*\#}(i)|_{i=1}^{N}\!\right\}$, $C_{\rm\bf *}\!\!=\!\!{\rm\bf diag}\!\!\left\{\!C_{\rm\bf
*}(i)|_{i=1}^{N}\!\right\}$, in which ${\rm\bf *,\#}={\rm\bf x}$, ${\rm\bf v}$, or ${\rm\bf z}$.
From these results, the following conclusions are derived, which establish some relations between the observability of the NDS investigated in this paper and that of a descriptor system. Their proof is deferred to Appendix A.
Assume that for each $i=1,2,\cdots,N$, $${\rm\bf Null}\left([C_{\rm\bf x}(i)\;\;C_{\rm\bf v}(i)]\right)
={\rm\bf Span}\left(\left[\begin{array}{c} N_{\rm\bf cx}(i) \\ N_{\rm\bf cv}(i) \end{array}\right]\right)
\label{eqn:12}$$ in which the matrices $N_{\rm\bf cx}(i)$ and $N_{\rm\bf cv}(i)$ have respectively $m_{{\rm\bf x}i}$ rows and $m_{\bar{\rm\bf v}i}$ rows. Define matrices $N_{\rm\bf cx}$ and $N_{\rm\bf cv}$ respectively as $$N_{\rm\bf cx}={\rm\bf diag}\!\left\{\!N_{\rm\bf cx}(i)|_{i=1}^{N}\!\right\}, \hspace{0.5cm}
N_{\rm\bf cv}={\rm\bf diag}\!\left\{\!N_{\rm\bf cv}(i)|_{i=1}^{N}\!\right\}
\label{eqn:13}$$ Then, the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is observable if and only if for every complex scalar $\lambda$, the following matrix pencil $\Psi(\lambda)$ is of full column rank. $$\Psi(\lambda)=\lambda\left[\begin{array}{c}
N_{\rm\bf cx} \\ 0 \end{array}\right]+
\left[\begin{array}{c}
-A_{\rm\bf xx}N_{\rm\bf cx}-A_{\rm\bf xv}N_{\rm\bf cv} \\
N_{\rm\bf cv}-\bar{\Phi} \left(A_{\rm\bf zx}N_{\rm\bf cx}+A_{\rm\bf zv}N_{\rm\bf cv}\right) \end{array}\right]
\label{eqn:3-a}$$ \[theorem:1\]
Note that when ${\rm\bf det} \left(\lambda N_{\rm\bf cx} -\left[ A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \right)\not\equiv 0$, the condition that the aforemention matrix pencil $\Psi(\lambda)$ is of full column rank at every complex $\lambda$ is necessary and sufficient for the observability of the following descriptor system, $$\begin{aligned}
& & N_{\rm\bf cx}x(t+1)= \left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right]x(t) \\
& & y(t)=\left[ N_{\rm\bf cv}-\bar{\Phi} \left(A_{\rm\bf zx}N_{\rm\bf cx}+A_{\rm\bf zv}N_{\rm\bf cv}\right) \right] x(t)\end{aligned}$$ It is clear that results on the observability of descriptor systems can be directly applied to that of the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]), while the former has been extensively studied and various conclusions have been established [@Dai89; @Duan10].
A direct application of these results, however, may not efficiently use the block diagonal structure of the associated matrices, which may introduces some unnecessary computational costs that is not quite attractive for the analysis and synthesis of a large scale NDS. On the other hand, there is in general no guarantee that the matrix pencil $\lambda N_{\rm\bf cx} -\left[ A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right]$ is regular. As a matter of fact, this matrix pencil may sometimes even not be square.
To derive a computationally attractive condition for the observability of the NDS described by Equations (\[eqn:1\]) and (\[eqn:2\]), the Kronecker canonical form for a matrix pencil, which is given in the previous section, appears helpful.
From Lemma \[lemma:2\], it can be declared that for every $i=1,2,\cdots,N$, there exist an invertible real matrix $U(i)$, an invertible real matrix $V(i)$, as well as a matrix pencil $\Xi(\lambda,i)$, such that $$\lambda N_{\rm\bf cx}(i)-\left[A_{\rm\bf xx}(i)N_{\rm\bf cx}(i)+A_{\rm\bf xv}(i)N_{\rm\bf cv}(i)\right]=U(i)\Xi(\lambda,i)V(i)
\label{eqn:14}$$ in which $$\begin{aligned}
{\Xi}(\lambda,i)\!\! & =& \!\!{\rm\bf diag}\!\left\{\!H_{\mu(i)}(\lambda),\;K_{\xi_{j}(i)}(\lambda)|_{j=1}^{a(i)},\; L_{\kappa_{j}(i)}(\lambda)|_{j=1}^{c(i)}, \; \right.\nonumber\\
& & \hspace*{1.5cm} \left. N_{\eta_{j}(i)}(\lambda)|_{j=1}^{b(i)},\; J_{\rho_{j}(i)}(\lambda)\!|_{j=1}^{d(i)}\!\right\}
\label{eqn:15}\end{aligned}$$ Here, $\mu(i)$, $a(i)$, $b(i)$, $c(i)$ and $d(i)$ are some nonnegative integers, $\xi_{j}(i)|_{j=1}^{a(i)}$ and $\eta_{j}(i)|_{j=1}^{b(i)}$ are some positive integers, $\kappa_{j}(i)|_{j=1}^{c(i)}$ and $\rho_{j}(i)|_{j=1}^{d(i)}$ are some nonnegative integers. Moreover, $H_{\mu(i)}(\lambda)$ is a strictly regular $\mu(i)\times \mu(i)$ dimensional matrix pencil. In addition, all these numbers are uniquely determined by the matrices $A_{\rm\bf xx}(i)$, $A_{\rm\bf xv}(i)$, $N_{\rm\bf cx}(i)$ and $N_{\rm\bf cv}(i)$.
In the decomposition of Equation (\[eqn:14\]), the calculations are performed for each subsystem individually. On the other hand, there are extensive studies on expressing a matrix pencil with the Kronecker canonical form and various computationally attractive algorithms have already been established [@bv88; @Gantmacher59; @Kailath80]. It can therefore be declared that computations involved in the aforementioned decomposition are in general possible, while the total computational complexity increases linearly with the increment of the subsystem number $N$.
Rank Deficiency and Null Space of Some Matrix Pencils
-----------------------------------------------------
To apply the decomposition in Equation (\[eqn:14\]) to the verification of the observability of the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]), it appears necessary to clarify the values of the complex variable $\lambda$ at which the matrix pencil $\Xi(\lambda)$ is rank deficient, as well as the associated null space. For this purpose, we investigate in this subsection some associated properties of the matrix pencils $H_{\mu(i)}(\lambda)$, $K_{\xi_{j}(i)}(\lambda)$, $L_{\kappa_{j}(i)}(\lambda)$, $N_{\eta_{j}(i)}(\lambda)$ and $J_{\rho_{j}(i)}(\lambda)$, in which $1\leq \xi_{j}(i)\leq a(i)$, $1\leq \kappa_{j}(i)\leq c(i)$, $1\leq \eta_{j}(i)\leq b(i)$, $1\leq \rho_{j}(i)\leq d(i)$, and $1\leq i\leq N$. In particular, the values of the complex variable $\lambda$ are given at which these matrix pencils are rank deficient, as well as a matrix whose column vectors are independent of each other and span the associated null space.
Note that $H_{\mu(i)}(\lambda)$ is strictly regular. This means that there are two real and nonsingular $\mu(i)\times \mu(i)$ dimensional matrices $X_{i}$ and $Y_{i}$, such that $$H_{\mu(i)}(\lambda)=\lambda X_{i}+Y_{i}$$ Hence, $$\begin{aligned}
& & H_{\mu(i)}(\lambda)\alpha =X_{i}\left[\lambda I+X_{i}^{-1}Y_{i}\right]\alpha \\
& &
{\rm\bf det}\left[H_{\mu(i)}(\lambda)\right]={\rm\bf det}\left[X_{i}\right]\times {\rm\bf det}\left[\lambda I+X_{i}^{-1}Y_{i}\right]\end{aligned}$$ in which $\alpha$ is an arbitrary vector with a consistent dimension. It can therefore be declared that ${\rm\bf det}\left[H_{\mu(i)}(\lambda)\right]=0$ if and only if ${\rm\bf det}\left[\lambda I+X_{i}^{-1}Y_{i}\right]=0$. On the other hand, note that $${\rm\bf det}\left[X_{i}^{-1}Y_{i}\right]=\frac{{\rm\bf det}\left[Y_{i}\right]}{{\rm\bf det}\left[X_{i}\right]}$$ We therefore have that the matrix $X_{i}^{-1}Y_{i}$ is always nonsingular. Moreover, $H_{\mu(i)}(\lambda)\alpha=0$ if and only if $\left[\lambda I+X_{i}^{-1}Y_{i}\right]\alpha=0$.
These observations mean that the matrix pencil $H_{\mu(i)}(\lambda)$ is rank deficient only at an eigenvalue of the matrix $X_{i}^{-1}Y_{i}$. As $X_{i}^{-1}Y_{i}$ is a nonsingular matrix, all of its eigenvalues are different from zero. Moreover, at an eigenvalue of the nonsingular matrix $X_{i}^{-1}Y_{i}$, say $\lambda_{0}$, the null space of $H_{\mu(i)}(\lambda_{0})$ is spanned by the associated independent eigenvectors of the matrix $X_{i}^{-1}Y_{i}$. The latter is now related to a standard problem in numerical computations and can be easily calculated in general.
From the definitions of the matrix pencils $N_{\eta_{j}(i)}(\lambda)$ and $J_{\rho_{j}(i)}(\lambda)$, which are given in Equations (\[eqn:4\]) and (\[eqn:5\]), it is straightforward to prove that the matrix pencil $N_{\eta_{j}(i)}(\lambda)$ is always of full rank, while the matrix pencil $J_{\rho_{j}(i)}(\lambda)$ is always of full column rank. As a matter of fact, we have that $${\rm\bf det}\left[N_{\eta_{j}(i)}(\lambda)\right]\equiv 1$$ Moreover, if there is a vector $\alpha=[\alpha_{1}\; \alpha_{2}\; \cdots\; \alpha_{\rho_{j}(i)}]^{T}$, such that $J_{\rho_{j}(i)}(\lambda)\alpha=0$ is satisfied at a particular value of the complex variable $\lambda$, say, $\lambda_{0}$. Then $$\begin{aligned}
& & \lambda_{0}\alpha_{1}=0 \\
& & \alpha_{i}+\lambda_{0}\alpha_{i+1}=0,\hspace{0.25cm} i=1,2,\cdots,\rho_{j}(i)-1 \\
& & \alpha_{\rho_{j}(i)}=0\end{aligned}$$ which lead to $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{\rho_{j}(i)}=0$. That is, $\alpha=0$. Hence, the matrix $J_{\rho_{j}(i)}(\lambda)$ is of full column rank, no matter what value is taken by the complex variable $\lambda$.
On the other hand, from the definition of the matrix pencil $K_{\xi_{j}(i)}(\lambda)$, it can be proved that $K_{\xi_{j}(i)}(\lambda)$ is singular only at $\lambda=0$. Moreover, $${\rm\bf Null}\left\{ K_{\xi_{j}(i)}(0)\right\}={\rm\bf Span}\left\{\left[\begin{array}{c} 1 \\ 0 \end{array}\right]\right\}
\label{eqn:18}$$ More precisely, assume that there is a vector $\alpha=[\alpha_{1}\; \alpha_{2}\; \cdots\; \alpha_{\xi_{j}(i)}]^{T}$, such that $K_{\xi_{j}(i)}(\lambda)\alpha=0$ is satisfied at a particular value of the complex variable $\lambda$. Denote it by $\lambda_{0}$. Then $$\begin{aligned}
& & \lambda_{0}\alpha_{i}+\alpha_{i+1}=0,\hspace{0.25cm} i=1,2,\cdots,\xi_{j}(i)-1
\label{eqn:17}\\
& & \lambda_{0}\alpha_{\xi_{j}(i)}=0\end{aligned}$$
If $\lambda_{0}\neq 0$, then the last equation means that $\alpha_{\xi_{j}(i)}=0$, which further leads to that $\alpha_{\xi_{j}(i)-1}=\alpha_{{\xi_{j}(i)}-2}=\cdots=\alpha_{1}=0$. Hence, the matrix $K_{\xi_{j}(i)}(\lambda)$ is of full rank when $\lambda\neq 0$. On the other hand, if $\lambda_{0}= 0$, then Equation (\[eqn:17\]) implies that $\alpha_{\xi_{j}(i)}=\alpha_{{\xi_{j}(i)}-1}=\cdots=\alpha_{2}=0$, while $\alpha_{1}$ can be an arbitrary complex number. This proves Equation (\[eqn:18\]).
Furthermore, from the definition of the matrix pencil $L_{\kappa_{j}(i)}(\lambda)$, it can be proved that this matrix pencil is singular at an arbitrary complex $\lambda$. Moreover, $${\rm\bf Null}\left\{ L_{\kappa_{j}(i)}(\lambda)\right\}=
{\rm\bf Span}\left\{\left[\begin{array}{c} 1 \\ -\lambda \\ \vdots \\
(-\lambda)^{\kappa_{j}(i)} \end{array}\right]\right\}
\label{eqn:19}$$ As a matter of fact, assume that there is a vector $\alpha=[\alpha_{1}\; \alpha_{2}\; \cdots\; \alpha_{\kappa_{j}(i)}]^{T}$, such that $L_{\kappa_{j}(i)}(\lambda)\alpha=0$ is satisfied at a particular value of the complex variable $\lambda$. Denote this $\lambda$ by $\lambda_{0}$. Then $$\lambda_{0}\alpha_{i}+\alpha_{i+1}=0,\hspace{0.25cm} i=1,2,\cdots,\kappa_{j}(i)
\label{eqn:20}$$
If $\lambda_{0}\neq 0$, then Equation (\[eqn:20\]) means that $\alpha_{i+1}=-\lambda_{0}\alpha_{i}$, $i=1,2,\cdots,\kappa_{j}(i)$. On the other hand, if $\lambda_{0}= 0$, this equation implies that $\alpha_{\kappa_{j}(i)}=\alpha_{{\kappa_{j}(i)}-1}=\cdots=\alpha_{2}=0$, while $\alpha_{1}$ can take any complex value. These prove Equation (\[eqn:19\]).
In summary, the matrix pencils in $\Xi(\lambda,i)$ is either of full column rank or has a null space with its bases computable. Moreover, the block diagonal structure of the matrix pencils $\Xi(\lambda,i)|_{i=1}^{N}$ and the matrix pencil $\Xi(\lambda)$ imply that these properties hold also for them.
Observability Verification for the NDS
---------------------------------------
Using the observations in the previous subsection, as well as the expressions of Equation (\[eqn:15\]), the following condition is obtained for the observability of the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]). Its derivations are given in Appendix B.
For each $i=1,2,\cdots,N$, let $m(i)$ denote $\mu(i)+\sum_{j=1}^{a(i)}\xi_{j}(i)+\sum_{j=1}^{c(i)}\kappa_{j}(i)$. Moreover, let $V_{i}^{-1}(1:m(i))$ represent the matrix constructed from the first $m(i)$ columns of the inverse of the matrix $V(i)$. Denote the following matrix pencil [$$\left[\!\!\!\!\!\!\!\!\!\!\begin{array}{c}
\hspace*{0.3cm}{\rm\bf diag}\!\left\{\!
{\rm\bf diag}\!\left\{\!H_{\mu(i)}(\lambda),\;K_{\xi_{j}(i)}(\lambda)|_{j=1}^{a(i)},\; L_{\kappa_{j}(i)}(\lambda)|_{j=1}^{c(i)}\right\}|_{i=1}^{N}\!\!\right\} \\
{\rm\bf diag}\!\left\{\! N_{\rm\bf cv}(i)V_{i}^{-1}(1:m(i))|_{i=1}^{N}\!\right\}- \bar{\Phi} \left({\rm\bf diag}\!\left\{\!A_{\rm\bf zx}(i)N_{\rm\bf cx}(i)\times \right.\right.\\
\hspace*{0.5cm} \left.\left.V_{i}^{-1}(1:m(i))|_{i=1}^{N}\!\right\}
+{\rm\bf diag}\!\left\{\! A_{\rm\bf zv}(i) N_{\rm\bf cv}(i)V_{i}^{-1}(1:m(i))\!\right\}\!\right)\!\!|_{i=1}^{N} \end{array}\!\!\!\!\!\right]
\label{eqn:16}$$]{} by $\bar{\Psi}(\lambda)$. Then, the networked dynamic system of Equations (\[eqn:1\]) and (\[eqn:2\]) is observable, if and only if the matrix pencil $\bar{\Psi}(\lambda)$ is of full column rank at each value of the complex variable $\lambda$. \[theorem:2\]
From the proof of Theorem \[theorem:2\], it is clear that if $\mu(i)=a(i)=c(i)=0$ in each $\Xi(\lambda,i)$ with $i\in\{1,2,\cdots,N\}$, which is essentially a condition required for each subsystem individually, then the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is always observable, no matter how the subsystems are connected and the parameters of a subsystem are selected. It is interesting to see whether or not this condition can be satisfied by a practical system.
For simplicity, let $\bar{\Xi}(\lambda,i)$ denote the matrix pencil $${\rm\bf diag}\!\left\{\!H_{\mu(i)}(\lambda),\;K_{\xi_{j}(i)}(\lambda)|_{j=1}^{a(i)},\; L_{\kappa_{j}(i)}(\lambda)|_{j=1}^{c(i)}\right\}$$ in which $i=1,2,\cdots,N$. Moreover, let $\bar{\Xi}(\lambda)$ denote the matrix pencil $${\rm\bf diag}\!\left\{\!
{\rm\bf diag}\!\left\{\!H_{\mu(i)}(\lambda),\;K_{\xi_{j}(i)}(\lambda)|_{j=1}^{a(i)},\; L_{\kappa_{j}(i)}(\lambda)|_{j=1}^{c(i)}\right\}|_{i=1}^{N}\!\!\right\}$$ Clearly, $\bar{\Xi}(\lambda)={\rm\bf diag}\!\left\{\!\bar{\Xi}(\lambda,i)|_{i=1}^{N}\!\!\right\}$. Moreover, for each $i\in\{1,2,\cdots,N\}$, let $\Lambda(i)$ denote the set consisting of the values of the complex variable $\lambda$ at which $\bar{\Xi}(\lambda,i)$ is not of full column rank. From the discussions of Subsection III.A, this set is the whole complex plane if $c(i)\neq 0$. On the other hand, if $c(i)=0$, then, this set is simply formed by zero and all the complex values that lead to a singular $H_{\mu(i)}(\lambda)$. For a $\lambda_{0}\in\Lambda(i)$, let $N(\lambda_{0},i)$ denote a matrix whose columns are independent of each other and span the null space of $\bar{\Xi}(\lambda_{0},i)$. Obviously, this matrix is also block diagonal with each of its blocks equal to a matrix constructed from base vectors of the null space of $H_{\mu(i)}(\lambda_{0})$ or $K_{\xi_{j}(i)}(\lambda_{0})$ with $j\in \{1,2,\cdots,a(i)\}$ or $L_{\kappa_{j}(i)}(\lambda_{0})$ with $j\in\{1,2,\cdots,c(i)\}$.
Furthermore, let $\Lambda$ denote the set consisting of the values of the complex variable $\lambda$ at which $\bar{\Xi}(\lambda)$ is not of full column rank, and $N(\lambda_{0})$ a matrix whose columns are independent of each other and span the null space of $\bar{\Xi}(\lambda_{0})$ with $\lambda_{0}$ belonging to $\Lambda$. From the definitions of $\bar{\Xi}(\lambda)$ and $\bar{\Xi}(\lambda,i)|_{i=1}^{N}$, it is obvious that $$\Lambda=\bigcup_{i=1}^{N}\Lambda(i),\hspace{0.25cm}
N(\lambda_{0})={\rm\bf diag}\!\left\{N(\lambda_{0},i)|_{i=1}^{N}\!\!\right\}
\label{eqn:31}$$
The following results are well known in matrix analysis [@Gantmacher59; @hj91].
Partition a matrix $M$ as $M = \left[\begin{array}{c} M_{1} \\ M_{2}\end{array}\right]$. Let $M_{1}^{\perp}$ denote a matrix consisting of column vectors that are independent of each other and span the null space of the submatrix $M_{1}$. Then, the matrix $M$ is of full column rank, if and only if $M_{2}M_{1}^{\perp}$ is of full column rank. \[lemma:3\]
The following results can be immediately established from Theorem \[theorem:2\] and Lemma \[lemma:2\].
For a prescribed complex $\lambda_{0}$, denote the following two matrices $$\begin{aligned}
& &\hspace*{-0.7cm} {\rm\bf diag}\!\left\{\! N_{\rm\bf cv}(i)V_{i}^{-1}(1:m(i))|_{i=1}^{N}\!\right\}N(\lambda_{0}) \\
& &\hspace*{-0.7cm} \left\{\!{\rm\bf diag}\!\left[ A_{\rm\bf zx}(i)N_{\rm\bf cx}(i)\!+\!A_{\rm\bf zv}(i) N_{\rm\bf cv}(i)\right]V_{i}^{-1}\!(1\!:\!m(i))|_{i\!=\!1}^{N}\!\right\}\!\!N(\lambda_{0})\end{aligned}$$ respectively by $X(\lambda_{0})$ and $Y(\lambda_{0})$. Then the networked dynamic system of Equations (\[eqn:1\]) and (\[eqn:2\]) is observable, if and only if for each $\lambda_{0}\in \Lambda$, the matrix $$X(\lambda_{0})-\bar{\Phi} Y(\lambda_{0})
\label{eqn:32}$$ is of full column rank. \[theorem:3\]
The proof is omitted due to its obviousness.
The above theorem makes it clear that the existence of a matrix pencil in a form of $L_{*}(\lambda)$ in the matrix pencil $\bar{\Xi}(\lambda)$ may make the observability condition difficult to be satisfied by a NDS, as it makes the set $\Lambda$ equal to the whole complex plane and requires that the matrix of Equation (\[eqn:32\]) is of full column rank at each complex $\lambda_{0}$. It is interesting to see possibilities to avoid occurrence of this type of matrix pencils in subsystem constructions for a NDS.
It is worthwhile to mention that in both the definition of the matrix $X(\lambda_{0})$ and the definition of the matrix $Y(\lambda_{0})$, all the involved matrices have a consistent block diagonal structure. This means that these two matrices are also block diagonal, and the computational costs for obtaining them increase only linearly with the increment of the subsystem number $N$. This is a quite attractive property in dealing with a large scale NDS which consists of numerous subsystems.
Moreover, the augmented SCM $\bar{\Phi}$, which is defined by Equation (\[eqn:10\]), clearly has a sparse structure. This means that results about sparse computations, which have been extensively and well studied in fields like numerical analysis, can be applied to the verification of the condition in Theorem \[theorem:3\]. It is interesting to see possibilities of developing more numerically efficient methods for this condition verification, using the particular sparse structure of the augmented SCM $\bar{\Phi}$ and the consistent block diagonal structure of both the matrix $X(\lambda_{0})$ and the matrix $Y(\lambda_{0})$.
The matrix of Equation (\[eqn:32\]) has completely the same form as that of our previous work reported in [@zhou15; @zyl18]. In the derivations of these results, however, except the well-posedness assumptions, there are not any other requirements on a subsystem of the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]). That is, the full normal column rank condition on each subsystem, which is required in [@zhou15; @zyl18] to get the associated transmission zeros of each subsystem, is completely removed.
Compared with [@zz19], the results of the above theorem are in a pure algebraic form. In system analysis and synthesis, they may not be as illustrative as the results of [@zz19] which are given in a graphic form. It is interesting to see whether or not a graphic form can be obtained from Theorem \[theorem:3\] on the observability of a NDS. On the other hand, in the derivations of this theorem, except well posedness of each subsystem and the whole system which is also asked in [@zz19] and is an essential requirement for a system to work properly, there are not any other constraints on either a subsystem or the whole system of the NDS. This appears to be a significant progress, as the augmented subsystem connection matrix, that is, the matrix $\bar{\Phi}$, is required to have a diagonal parametrization in [@zz19], which may not be easily satisfied by a practical system and significantly restricts applicability of the corresponding results.
Controllability of an NDS
=========================
Recall that controllability of a linear time invariant system is equal to observability of its dual system, and this is also true for a networked dynamic system [@zhou15; @zyl18]. This means that the results of Section III can be directly applied to controllability analysis for the same NDS. As a matter of fact, using the duality between controllability and observability of a system, as well as the equivalence representation of the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]), which is given by Equations (\[eqn:11\]) and (\[eqn:9\]), it can be declared that the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is controllable, if and only if the following matrix pencil is of full row rank for each complex $\lambda$, $$\left[\begin{array}{ccc}
\lambda I_{M_{\rm\bf x}} - A_{\rm\bf xx} & B_{\rm\bf x} & -A_{\rm\bf xv}\bar{\Phi} \\
-A_{\rm\bf zx} & B_{\rm\bf z} & I_{M_{\rm\bf z}} - A_{\rm\bf zv}\bar{\Phi}
\end{array}\right]$$ in which $B_{\rm\bf x}\!\!=\!\!{\rm\bf diag}\!\left\{\!B_{\rm\bf x}(i)|_{i=1}^{N}\!\right\}$, $B_{\rm\bf z}\!\!=\!\!{\rm\bf diag}\!\left\{\!B_{\rm\bf z}(i)|_{i=1}^{N}\!\right\}$, and the other matrices have the same definitions as those of Lemma \[lemma:1\].
Using the Kronecker canonical form of a matrix pencil, as well as a basis for the left null space of the matrix $\left[\!\!\!\begin{array}{c} B_{\rm\bf x}^{T} \;\; B_{\rm\bf z}^{T} \end{array}\!\!\!\right]^{T}$, similar algebraic manipulations lead to a necessary and sufficient condition for the controllability of the NDS, which has a similar form as that of Theorem \[theorem:3\].
The details are omitted due to their obviousness.
The results of Sections III and IV are also applicable to the analysis of the controllability/observability of a continuous time NDS.
Through similar arguments as those of this paper and [@zhou15; @zyl18], corresponding results can also be obtained for the controllability and observability of a networked dynamic system, in which the dynamics of each subsystem is described by a descriptor system with its system matrices depending on some parameters in the way of a linear fractional transformation. This type of subsystems are also well encountered in practical applications [@it17; @Dai89; @Duan10].
Concluding Remarks
==================
In this paper, we revisit controllability/observability of networked dynamic systems in which the system matrices of each subsystem are described by a linear fractional transformation of its (pseudo) first principle parameters. An explicit connection has been established between the controllability/observability of a networked dynamic system and that of a descriptor system. Using the Kronecker canonical form of a matrix pencil, a rank based condition is established in which the associated matrix affinely depends on a matrix formed by the parameters of each subsystem and the subsystem connection matrix. This matrix form completely agrees with that of [@zhou15; @zyl18], but in its derivations, the full normal rank condition asked there is no longer required. On the other hand, this matrix keeps the attractive property that in obtaining the involved matrices, the associated numerical computations are performed on each subsystem independently, which makes the condition verification scalable for a networked dynamic system formed by a large number of subsystems. In addition, except well-posedness of each subsystem and the whole system, there are not any other restrictions on either a subsystem or the subsystem connections.
Further efforts include finding engineering significant explanations for the obtained results, as well as extending the obtained results to structural controllability/observability of a networked dynamic system.
Proof of Theorem 1
==================
Let $\alpha_{{\rm\bf x}}(i)$ be an arbitrary $m_{{\rm\bf x}i}$ dimensional real vector, while $\alpha_{{\rm\bf v}}(i)$ be an arbitrary $m_{{\rm\bf v}i}$ dimensional real vector, $i=1,2,\cdots,N$. Denote ${\rm\bf col}\!\left\{\alpha_{\rm\bf x}(i)|_{i=1}^{N},\;\alpha_{\rm\bf v}(i)|_{i=1}^{N}\right\}$ by $\alpha$. From the block diagonal structure of the matrix $C_{\rm\bf x}$ and $C_{\rm\bf v}$, it is immediate that $$\left[ C_{\rm\bf x}\;\; C_{\rm\bf v} \right]\alpha
=\left[\begin{array}{c}
C_{\rm\bf x}(1)\alpha_{{\rm\bf x}}(1)+ C_{\rm\bf v}(1)\alpha_{{\rm\bf v}}(1) \\
C_{\rm\bf x}(2)\alpha_{{\rm\bf x}}(2)+ C_{\rm\bf v}(2)\alpha_{{\rm\bf v}}(2) \\
\vdots \\
C_{\rm\bf x}(N)\alpha_{{\rm\bf x}}(N)+ C_{\rm\bf v}(N)\alpha_{{\rm\bf v}}(N) \end{array}\right]$$ It can therefore be declared that $\left[C_{\rm\bf x}\;\; C_{\rm\bf v}\right]\alpha=0$ if and only if $C_{\rm\bf x}(i)\alpha_{{\rm\bf x}}(i)+ C_{\rm\bf v}(i)\alpha_{{\rm\bf v}}(i)=0$ for each $i=1,2,\cdots,N$. The latter is equivalent to that $$\left[\begin{array}{c}
\alpha_{{\rm\bf x}}(i) \\
\alpha_{{\rm\bf v}}(i) \end{array}\right] \in
{\rm\bf Null}\left([C_{\rm\bf x}(i)\;\;C_{\rm\bf v}(i)]\right)
={\rm\bf Span}\left(\left[\begin{array}{c} N_{\rm\bf cx}(i) \\ N_{\rm\bf cv}(i) \end{array}\right]\right)
\label{eqn:21}$$ which can be further expressed as that there is a vector $\xi(i)$, such that $$\left[\begin{array}{c}
\alpha_{{\rm\bf x}}(i) \\
\alpha_{{\rm\bf v}}(i) \end{array}\right]=
\left[\begin{array}{c} N_{\rm\bf cx}(i) \\ N_{\rm\bf cv}(i) \end{array}\right]\xi(i)
\label{eqn:22}$$ Hence $$\begin{aligned}
\alpha&=& \left[\begin{array}{c} {\rm\bf col}\left\{N_{\rm\bf cx}(i)\xi(i)|_{i=1}^{N}\right\} \\
{\rm\bf col}\left\{ N_{\rm\bf cv}(i)\xi(i)|_{i=1}^{N}\right\} \end{array}\right]\nonumber\\
&=&
\left[\begin{array}{c} {\rm\bf diag}\left\{N_{\rm\bf cx}(i)|_{i=1}^{N}\right\} \\
{\rm\bf diag}\left\{ N_{\rm\bf cv}(i)|_{i=1}^{N}\right\} \end{array}\right]
{\rm\bf col}\left\{\xi(i)|_{i=1}^{N} \right\} \nonumber\\
&=&
\left[\begin{array}{c} N_{\rm\bf cx} \\
N_{\rm\bf cv} \end{array}\right]\xi
\label{eqn:23}\end{aligned}$$ in which $\xi={\rm\bf col}\left\{\xi(i)|_{i=1}^{N} \right\}$.
On the other hand, it can be straightforwardly shown that the matrix $\left[\!\!\!\begin{array}{c} N_{\rm\bf cx}^{T} \;\;
N_{\rm\bf cv}^{T} \end{array}\!\!\!\right]^{T}$ is of full column rank. It can therefore be declared that $${\rm\bf Null}\left([C_{\rm\bf x} \;\;C_{\rm\bf v}]\right)
={\rm\bf Span}\left(\left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\right)
\label{eqn:24}$$
Assume that the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is observable. Then according to Lemma \[lemma:1\], the matrix pencil $M(\lambda)$ is of full column rank at each complex $\lambda$. This is equivalent to that for every nonzero vector $\alpha$ with a consistent dimension, $M(\lambda)\alpha \neq 0$.
Now, let $\alpha$ belong to the null space of $[C_{\rm\bf x} \;\;C_{\rm\bf v}]$. Then according to Equation (\[eqn:24\]), there is a nonzero vector $\xi$, such that $$\alpha
=\left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\xi
\label{eqn:25}$$ Hence $$\begin{aligned}
M(\lambda)\alpha
&=&\left[\begin{array}{cc}
\lambda I_{M_{\rm\bf x}}-A_{\rm\bf xx} & -A_{\rm\bf xv} \\
-C_{\rm\bf x} & -C_{\rm\bf v} \\
-\bar{\Phi} A_{\rm\bf zx} & I_{M_{\rm\bf z}}-\bar{\Phi} A_{\rm\bf zv} \end{array}\right] \left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\xi \nonumber\\
&=&\left[\begin{array}{cc}
\lambda N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \\
0 \\
N_{\rm\bf cv}-\bar{\Phi} \left[A_{\rm\bf zx}N_{\rm\bf cx}+ A_{\rm\bf zv}N_{\rm\bf cv}\right] \end{array}\right] \xi \nonumber\\
\neq 0
\label{eqn:26}\end{aligned}$$ Therefore, $$\left[\begin{array}{cc}
\lambda N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \\
N_{\rm\bf cv}-\bar{\Phi} \left[A_{\rm\bf zx}N_{\rm\bf cx}+ A_{\rm\bf zv}N_{\rm\bf cv}\right] \end{array}\right] \xi \neq 0
\label{eqn:26-a}$$
As the vector $\xi$ can take an arbitrary nonzero value, this means that the matrix pencil $\left[\begin{array}{cc}
\lambda N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \\
N_{\rm\bf cv}-\bar{\Phi} \left[A_{\rm\bf zx}N_{\rm\bf cx}+ A_{\rm\bf zv}N_{\rm\bf cv}\right] \end{array}\right]$ is always of full column rank.
On the contrary, assume that the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is observable, but the matrix pencil $\left[\begin{array}{cc}
\lambda N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \\
N_{\rm\bf cv}-\bar{\Phi} \left[A_{\rm\bf zx}N_{\rm\bf cx}+ A_{\rm\bf zv}N_{\rm\bf cv}\right] \end{array}\right]$ is not of full column rank at a particular $\lambda$. Denote it by $\lambda_{0}$. Then there exists a nonzero vector $\xi$ such that $$\left[\begin{array}{cc}
\lambda_{0} N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right] \\
N_{\rm\bf cv}-\bar{\Phi} \left[A_{\rm\bf zx}N_{\rm\bf cx}+ A_{\rm\bf zv}N_{\rm\bf cv}\right] \end{array}\right] \xi
= 0
\label{eqn:27}$$ which is equivalent to $$\left[\begin{array}{cc}
\lambda_{0} I_{M_{\rm\bf x}}-A_{\rm\bf xx} & -A_{\rm\bf xv} \\
-\bar{\Phi} A_{\rm\bf zx} & I_{M_{\rm\bf z}}-\bar{\Phi} A_{\rm\bf zv} \end{array}\right] \left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\xi=0
\label{eqn:28}$$
Combing Equation (\[eqn:28\]) with Equation (\[eqn:24\]), we have that $$\left[\begin{array}{cc}
\lambda_{0} I_{M_{\rm\bf x}}-A_{\rm\bf xx} & -A_{\rm\bf xv} \\
-C_{\rm\bf x} & -C_{\rm\bf v} \\
-\bar{\Phi} A_{\rm\bf zx} & I_{M_{\rm\bf z}}-\bar{\Phi} A_{\rm\bf zv} \end{array}\right] \left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\xi=0
\label{eqn:29}$$
Note that $\xi\neq 0$ by assumption. On the other hand, $\left[\!\!\!\begin{array}{c} N_{\rm\bf cx}^{T} \;\; N_{\rm\bf cv}^{T} \end{array}\!\!\!\right]^{T}$ is of full column rank. These mean that $$\left[\begin{array}{c} N_{\rm\bf cx} \\ N_{\rm\bf cv} \end{array}\right]\xi=0$$ It can therefore be decalred from Equation (\[eqn:29\]) that the NDS of Equations (\[eqn:1\]) and (\[eqn:2\]) is not observable, which is a contradiction to the assumption about the observability of the NDS described by Equations (\[eqn:1\]) and (\[eqn:2\]). Therefore, the existence of the aforementioned $\lambda_{0}$ is impossible.
This completes the proof. $\Diamond$
Proof of Theorem 2
==================
Define matrices $U$ and $V$, as well as a matrix pencil $\Xi(\lambda)$ as $$\begin{aligned}
& & U={\rm\bf diag}\left\{U(i)|_{i=1}^{N}\right\},\hspace{0.25cm}
V={\rm\bf diag}\left\{V(i)|_{i=1}^{N}\right\} \\
& & \Xi(\lambda)={\rm\bf diag}\left\{\Xi(\lambda,i)|_{i=1}^{N}\right\}\end{aligned}$$ Then both $U$ and $V$ are invertible, while $\Xi(\lambda)$ is block diagonal and consists only of strictly regular matrix pencils and the matrix pencils in the forms of $K_{*}(\lambda)$ and/or $N_{*}(\lambda)$ and/or $L_{*}(\lambda)$ and/or $J_{*}(\lambda)$. Recall that matrices $A_{\rm\bf xx}$, $A_{\rm\bf xv}$, $N_{\rm\bf cx}$ and $N_{\rm\bf cv}$ are block diagonal and the dimensions of their diagonal blocks are consistent with each other. It is not difficult to see from Equation (\[eqn:14\]) and the above definitions that $$\lambda N_{\rm\bf cx}-\left[A_{\rm\bf xx}N_{\rm\bf cx}+A_{\rm\bf xv}N_{\rm\bf cv}\right]=U\Xi(\lambda)V$$
Based on this expression and Equation (\[eqn:3-a\]), we have that $$\begin{aligned}
\Psi(\lambda)\!\!\!\!\!&=&\!\!\!\!\!\!\!
\left[\begin{array}{c}
U\Xi(\lambda)V \\
N_{\rm\bf cv}-\bar{\Phi} \left(A_{\rm\bf zx}N_{\rm\bf cx}+A_{\rm\bf zv}N_{\rm\bf cv}\right) \end{array}\right] \nonumber\\
&=&\!\!\!\!\!\!\!
\left[\!\!\begin{array}{cc}
U & 0 \\ 0 & I \end{array}\!\!\right]\!\!
\times \nonumber\\
& & \hspace*{-0.4cm}\left[\!\!\!\begin{array}{c}
\Xi(\lambda) \\
N_{\rm\bf cv}V^{-1}\!-\!\bar{\Phi} \left(A_{\rm\bf zx}N_{\rm\bf cx}\!\!+\!A_{\rm\bf zv}N_{\rm\bf cv}\right)V^{-1} \end{array}\!\!\!\right]\!\!V
\label{eqn:30}\end{aligned}$$
Define a matrix pencil $\tilde{\Psi}(\lambda)$ as $$\tilde{\Psi}(\lambda)=\left[\begin{array}{c}
\Xi(\lambda) \\
N_{\rm\bf cv}V^{-1}\!-\!\bar{\Phi} \left(A_{\rm\bf zx}N_{\rm\bf cx}V^{-1}+A_{\rm\bf zv}N_{\rm\bf cv}V^{-1}\right) \end{array}\right]$$ As both the matrix $U$ and the matrix $V$ are invertible, it is obvious from Equation (\[eqn:30\]) that the matrix pencil $\Psi(\lambda)$ is of full column rank at every complex $\lambda$, if and only if the matrix pencil $\tilde{\Psi}(\lambda)$ holds this property.
Recall that a matrix pencil in the form $N_{*}(\lambda)$ or $J_{*}(\lambda)$ is always of full column rank. This characteristic of these matrix pencils in the matrix pencil $\Xi(\lambda)$, as well as its block diagonal structure, imply that the matrix pencil $\tilde{\Psi}(\lambda)$ is always of full column rank, if and only if the matrix pencil $\bar{\Psi}(\lambda)$ meets this requirement, which is defined in Equation (\[eqn:16\]).
This completes the proof. $\Diamond$
[20]{}
[^1]: This work was supported in part by the NNSFC under Grant 61733008 and 61573209.
[^2]: Tong Zhou is with the Department of Automation, Tsinghua University, Beijing, 100084, P. R. China [(email: [tzhou@mail.tsinghua.edu.cn]{}).]{}
|
---
abstract: 'We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah-Segal.'
address:
- 'Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, GERMANY'
- 'Departamento de matemáticas, Universidad de Sonora, Blvd Luís Encinas y Rosales S/N, Colonia Centro. Edificio 3K-1. C.P. 83000. Hermosillo, Sonora, MÉXICO '
- 'Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstrasse 62, 48149 Münster, GERMANY'
- ' Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bogotá, COLOMBIA'
author:
- 'Noé Bárcenas, Jesús Espinoza, Michael Joachim and Bernardo Uribe'
bibliography:
- 'TwistedK.bib'
title: 'Universal twist in Equivariant K-theory for proper and discrete actions'
---
[^1]
Introduction
============
Topological K-theory is a generalized cohomology theory [@Atiyah-book] that in the case of compact spaces can be represented by isomorphism classes of vector bundles. A remarkable theorem of Atiyah [@Atiyah-Fredholm] and Jänich [@Janich] tells us that $K^0(X) \cong
\pi_0(Maps(X,{\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})_{\rm{norm}})$, namely that the K-theory groups of a compact space $X$ can be alternatively obtained as the homotopy classes of maps from the space $X$ to the space ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ of Fredholm operators on a fixed separable Hilbert space ${\mathcal{H}}$, whenever ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ is endowed with the norm topology. Note that the space of maps from $X$ to ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ can also be defined as the space of sections of the trivial bundle ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}}) \times X \to X$; this simple remark leads the way to consider spaces of sections of non trivial bundles over $X$ with fibers ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$, and by doing so we reach one of the definitions of the twisted K-theory groups [@AtiyahSegal]. Let us see how this works: the structural group of a bundle with fiber ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ will be the group ${{\mathcal U}}({\mathcal{H}})$ of unitary operators on the Hilbert space endowed with the norm topology acting on ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ by conjugation. As the conjugation by complex numbers of norm one is trivial, the action of ${{\mathcal U}}({\mathcal{H}})$ factors through the group of projective unitary operators $P{{\mathcal U}}({\mathcal{H}}):={{\mathcal U}}({\mathcal{H}})/S^1$. Therefore any principal $P{{\mathcal U}}({\mathcal{H}})$-bundle, or in other words, any projective unitary bundle $P{{\mathcal U}}({\mathcal{H}}) \to P \to X$, provides the essential information in order to define a bundle over $X$ with fibers ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ by taking the associated bundle ${\ensuremath{{\mathrm{Fred}}}}(P):= P \times_{P{{\mathcal U}}({\mathcal{H}})}
{\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$. With these bundles at hand, the twisted K-theory groups of $X$ twisted by $P$ are defined as $$K^{-i}(X;P) := \pi_i(\Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P)))$$ where $\Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P))$ denotes the space of sections of the bundle ${\ensuremath{{\mathrm{Fred}}}}(P)$.
The equivariant version of the construction presented above turns out to be very subtle. Not only there are issues with the topology of the spaces ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ and ${{\mathcal U}}({\mathcal{H}})$, as it was noted and resolved in [@AtiyahSegal], but moreover there is not a Hilbert space endowed with a group action that will serve as a universal equivariant Hilbert space for projective representations. This last fact makes the classification of twists for equivariant K-theory a more elaborate task, that we have pursued in this paper in a systematic way, and whose results are the main point of this publication.
The paper is divided in three chapters. Chapter \[section PU(H)\] is devoted to understanding the equivariant projective unitary bundles over a point, which are classified by the moduli space of homomorphisms $f: G \to
P{{\mathcal U}}({\mathcal{H}})$ from a compact Lie group to the group of projective unitary operators, such that the induced action of the group $\widetilde{G}:=f^*{{\mathcal U}}({\mathcal{H}})$ on ${\mathcal{H}}$, makes ${\mathcal{H}}$ into a representation of $\widetilde{G}$ on which all its irreducible representations where $S^1$ acts by multiplication, appear infinitely number of times. It turns out that the equivariant homotopy groups $\pi_i$ of this moduli space are isomorphic to the cohomology groups $H^{3-i}(BG, {\ensuremath{{\mathbb Z}}})$ for $i \geq 0$. We note that some of the results of this chapter were already in [@AtiyahSegal], but we believe that the main construction of this chapter, which is the homotopy quotient $$EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}})),$$ where $Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}}))$ is the space of homomorphisms on which all representations of $\widetilde{G}$ appear, has not been studied in the literature before.
In Chapter \[section Equivariant stable bundles\] we proceed to define the projective unitary stable and equivariant bundles over $G$ spaces, and devote the rest of the chapter to the classification of the equivariant projective unitary stable bundles over the orbit types $G/K$ for $K$ compact subgroup of $G$. We show that the universal projective unitary bundle over the orbit type $G/K$ can be constructed from the moduli space of stable homomorphisms from $K$ to $P{{\mathcal U}}({\mathcal{H}})$, and we furthermore show that this model serves as the universal moduli space for $K$-equivariant projective unitary stable bundles over trivial $K$-spaces. The gluing of these local universal bundles built out from stable homomorphisms turns out very subtle and it is not at all clear how to do it. Therefore we propose larger models for the universal bundles of the orbit types, models that are built out of functors instead of homomorphisms; the advantage of these larger models is that they are constructed in such a way that the gluing becomes very clear, but the disadvantage is that we have to restrict our attention to discrete groups since these larger models may not have the desired properties in the general case.
In Chapter \[chapter universal bundle\] we restrict our attention to the case on which $G$ is discrete and it acts properly. We devote the whole chapter to the construction of the universal projective unitary stable and equivariant bundle; this is the main result of this paper and can be found in Theorem \[theorem the universal bundle\]. Using stable functors from the groupoid $G \ltimes G/K$ to the group $P{{\mathcal U}}({\mathcal{H}})$, we construct a category that we denoted $\widetilde{D}_{G/K}$ endowed with a free right $P{{\mathcal U}}({\mathcal{H}})$ action and a left $N(K)/K$ action, whose geometrical realization is the universal equivariant projective unitary stable bundle over the orbit type $G/K$. Then we use the fact that the categories $\widetilde{D}_{G/K}$ were defined from functors from $G \ltimes G/K$ in order to explicitely define the gluing of these universal spaces, and therefore obtaining the universal projective unitary stable and equivariant bundle. We finish this chapter by calculating the homotopy type of this universal space, and in particular we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group..
Finally, in Appendix \[appendix Twisted equivariant K-theory\] we define the twisted equivariant K-theory groups for proper actions using the projective unitary stable and equivariant bundles defined in Chapter \[section Equivariant stable bundles\]. We show that this twisted equivariant K-theory satisfies the axioms of a generalized cohomology theory, that satisfies Bott periodicity, that is endowed with an [*[induction structure]{}*]{} as in [@Lueck1 Section 1] and that restricted to orbit types $G/K$ recovers the Grothendieck group $R_{S^1}(\widetilde{G})$ of representations of $\widetilde{G}$ on which $S^1$ acts by multiplication. We finish the paper by relating our definition of twisted equivariant K-theory groups to the definition of Dwyer in [@Dwyer] which is given through projective representations characterized by discrete torsion. We added this appendix in order to show that the twisted equivariant K-theory of proper $G$-actions defined through twisted Fredholm bundles satisfies all the axioms of a generalized equivariant cohomology theory.
Notation: {#Notation}
---------
$G$ will be the global group and $K \subset
G$ will be a compact subgroup of $G$; $BG$ will be the classifying space of $G$-principal bundles and it will be defined as $EG/G$ where $EG$ is the universal $G$-principal bundle. Letters in calligraphic style will denote groupoids or categories; and for a topological category ${\mathcal{C}}$ we will denote by $| {\mathcal{C}}|$ the geometric realization of the category ${\mathcal{C}}_{\bf N}$, that is, the associated category unravelled over the ordered set ${\bf N}$ of natural numbers in the following way: ${\mathcal{C}}_{\bf N}$ is the subcategory of ${\bf N} \times {\mathcal{C}}$ obtained by deleting all morphisms of the form $(n,c) \to (n,c')$ except for the identity morphisms. Having defined the geometrical realization in this way we have that $|G|$ is a the classifying space for principal $G$-bundles, and the map $|G\times G|\to |G|$ is a principal $G$-bundle where $G\times G$ denotes the product category of the set $G$; for details see [@Segal page 107].
Acknowledgments
---------------
The fourth author acknowledges and thanks the financial support of the Alexander Von Humboldt Foundation. Part of this work was carried out while the fourth author held a “Humboldt Research Fellowship for experienced researchers".
Properties of the group $P{{\mathcal U}}({\mathcal{H}})$ of projective unitary operators {#section PU(H)}
========================================================================================
Topology of $P{{\mathcal U}}({\mathcal{H}})$
--------------------------------------------
Let ${\mathcal{H}}$ be a separable Hilbert space and $${{\mathcal U}}({\mathcal{H}}):= \{ U : {\mathcal{H}}\to {\mathcal{H}}| U\circ U^*= U^*\circ U = \mbox{Id} \}$$ the group of unitary operators acting on ${\mathcal{H}}$. Let ${\rm End}({\mathcal{H}})$ denote the space of endomorphisms of the Hilbert space and endow ${\rm End}({\mathcal{H}})_{c.o.}$ with the compact open topology. Consider the inclusion $$\begin{aligned}
{{\mathcal U}}({\mathcal{H}}) &\to {\rm End}({\mathcal{H}})_{c.o.} \times {\rm End}({\mathcal{H}})_{c.o.}\\
U &\mapsto (U,U^{-1})\end{aligned}$$ and induce on ${{\mathcal U}}({\mathcal{H}})$ the subspace topology. Denote the space of unitary operators with this induced topology by ${{\mathcal U}}({\mathcal{H}})_{c.o.}$ and note that this is different from the usual compact open topology on ${{\mathcal U}}({\mathcal{H}})$.
It was pointed out in [@AtiyahSegal Appendix 1] that the group ${{\mathcal U}}({\mathcal{H}})_{c.o.}$ fails to be a topological group, since the composition of operators is only continuous on compact subspaces. Knowing this, we can endow ${{\mathcal U}}({\mathcal{H}})$ with the compactly generated topology induced by the topology of ${{\mathcal U}}({\mathcal{H}})_{c.o.}$ and in this way the composition of operators becomes continuous; denote by ${{\mathcal U}}({\mathcal{H}})_{c.g.}$ the group of unitary operators with the compactly generated topology induced by ${{\mathcal U}}({\mathcal{H}})_{c.o.}$
Since in [@AtiyahSegal Prop. A2.1] it was contructed a homotopy $h: {{\mathcal U}}({\mathcal{H}})_{c.o.} \times[0,1] \to {{\mathcal U}}({\mathcal{H}})_{c.o.}$ such that $h(U,1)=U$ and $h(U,0)=constant$, with the property that $h$ was continuous on compact sets, then the same map $h: {{\mathcal U}}({\mathcal{H}})_{c.g.} \times[0,1] \to {{\mathcal U}}({\mathcal{H}})_{c.g.}$ becomes continuous and therefore it proves the contractibility of the space ${{\mathcal U}}({\mathcal{H}})_{c.g.}$. Summarizing
\[Proposition contractibility of U(H)\] Let ${{\mathcal U}}({\mathcal{H}})_{c.g.}$ denote the group of unitary operators ${{\mathcal U}}({\mathcal{H}})$ endowed with the compactly generated topology of ${{\mathcal U}}({\mathcal{H}})_{c.o.}$. Then ${{\mathcal U}}({\mathcal{H}})_{c.g.}$ becomes a topological group and moreover it is contractible.
Let $P{{\mathcal U}}({\mathcal{H}})$ be the projectivization of ${{\mathcal U}}({\mathcal{H}})$ and endow it with the quotient topology of the quotient ${{\mathcal U}}({\mathcal{H}})_{c.g.}/S^1$. The topological group $P {{\mathcal U}}({\mathcal{H}})$ will be called the [*group of projective unitary operators*]{} and it fits into the short exact sequence of topological groups $$1 {\longrightarrow}S^1 {\longrightarrow}{{\mathcal U}}({\mathcal{H}})_{c.g.} {\longrightarrow}P{{\mathcal U}}({\mathcal{H}}) {\longrightarrow}1.$$
Since we know that ${{\mathcal U}}({\mathcal{H}})_{c.g.}$ is contractible, then we have that the homotopy type of $P {{\mathcal U}}({\mathcal{H}})$ is the one of an Eilenberg-Maclane space $K({\ensuremath{{\mathbb Z}}},2)$ and therefore the underlying space of $P {{\mathcal U}}({\mathcal{H}})$ is a universal space for complex line bundles.
Central $S^1$ extensions
------------------------
The group structure on $P {{\mathcal U}}({\mathcal{H}})$ also permits to define $S^1$ central extensions of a group $G$ out of continuous homomorphisms from $G$ to $P {{\mathcal U}}({\mathcal{H}})$ in the following way.
Let $G$ be a Lie group and $Ext(G, S^1)$ be the isomorphism classes of $S^1$ central extensions of $G$ $$1 \to S^1 \to
\widetilde{G} \to G \to 1$$ where $\widetilde{G}$ also has the structure of an $S^1$-principal bundle over $G$.
Let us define the map $\Psi$ from the space of continuous homomorphisms from $G$ to $P{{\mathcal U}}({\mathcal{H}})$ endowed with the compact open topology, to the set of isomorphism classes of $S^1$ central extensions of $G$ $$\begin{aligned}
\Psi : Hom(G, P {{\mathcal U}}({\mathcal{H}}))& \to & Ext(G, S^1)\\
a : G \to P {{\mathcal U}}({\mathcal{H}}) & \mapsto & \widetilde{G} : = a^* {{\mathcal U}}({\mathcal{H}}),\end{aligned}$$ where $\widetilde{G}$ and $\widetilde{a}$ denote respectively the Lie group and the continuous homomorphism defined by the pullback square $$\xymatrix{
\widetilde{G} \ar[r]^{\widetilde{a}} \ar[d] & {{\mathcal U}}({\mathcal{H}}) \ar[d] \\
G \ar[r]^a & P{{\mathcal U}}({\mathcal{H}}).}$$
\[lemma Psi surjective\] For $G$ a compact Lie group, the map $$\Psi : Hom(G, P {{\mathcal U}}({\mathcal{H}}))
\to Ext(G, S^1)$$ is surjective.
Let $\widetilde{G}$ be a $S^1$-central extension of $G$ and consider the Hilbert space $L^2(\widetilde{G}) \otimes L^2([0,1])$ which is tensor product of square Hilbert space of integrable functions on $\widetilde{G}$ and the Hilbert spaces of square integrable functions on the unit interval. By Peter-Weyl’s theorem, $L^2(\widetilde{G}) \otimes L^2([0,1])$ contains all irreducible representations of $\widetilde{G}$ infinitely number of times. Now consider $V_c(\widetilde{G})$ to be the subspace of $L^2(\widetilde{G}) \otimes L^2([0,1])$ of elements on which the central $S^1$ of $\widetilde{G}$ acts by multiplication by scalars. Take ${\mathcal{H}}:= V_c(\widetilde{G})$ and define the homomorphism $$\widetilde{a} : \widetilde{G} \to {{\mathcal U}}({\mathcal{H}}), \ \ \ \ \ \ \widetilde{a}(g) v : = gv$$ induced by the left action of $\widetilde{G}$ on $V_c(\widetilde{G})$; note that this homomorphism $\widetilde{a}$ is continuous because we have endowed ${{\mathcal U}}({\mathcal{H}})$ with the compactly generated compact open topology. Taking the projectivization $$a : G \to P{{\mathcal U}}({\mathcal{H}})$$ of the map $\widetilde{a}$ we get the desired homomorphism such that $$a^* {{\mathcal U}}({\mathcal{H}}) \cong
\widetilde{G}.$$
Clearly conjugate homomorphisms define isomorphic central extensions, therefore we get an induced map $$Hom(G, P {{\mathcal U}}({\mathcal{H}})) / P{{\mathcal U}}({\mathcal{H}}) \to Ext(G, S^1).$$
Now, take any two continuous homomorphisms $a,b : G \to P
{{\mathcal U}}({\mathcal{H}})$ such that their induced central extensions are isomorphic, i.e. $\Psi(a) \cong \Psi(b)$. The maps $a$ and $b$ are $P {{\mathcal U}}({\mathcal{H}})$-conjugate whenever the induced homomorphisms $\widetilde{a} , \widetilde{b} : \widetilde{G} \to {{\mathcal U}}({\mathcal{H}})$ are conjugate. The two maps $\widetilde{a}$ and $\widetilde{b}$ are conjugate whenever the induced $\widetilde{G}$ representations on ${\mathcal{H}}$ can be written as the direct sum of the same number of irreducible representations for each irreducible representation in $V_c(\widetilde{G})$. Therefore we would like to focus our attention to the $\widetilde{G}$ actions on ${\mathcal{H}}$ such that all irreducible representations in $V_c(\widetilde{G})$ appear infinitely number of times.
\[definition stable\] A continuous homomorphism $a : G \to P{{\mathcal U}}({\mathcal{H}})$ is called [ **stable**]{} if the unitary representation ${\mathcal{H}}$ induced by the homomorphism $\widetilde{a}: \widetilde{G}= a^*{{\mathcal U}}({\mathcal{H}}) \to
{{\mathcal U}}({\mathcal{H}})$ contains each of the irreducible representations of $\widetilde{G}$ that appear in $V_c(\widetilde{G})$ infinitely number of times. We denote $$Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}})) \subset Hom(G,P{{\mathcal U}}({\mathcal{H}}))$$ the subspace of continuous stable homomorphisms.
\[proposition pi zero of stable hom\] The map $\Psi$ induces a bijection of sets $$Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}}))/P{{\mathcal U}}({\mathcal{H}}) \stackrel{\cong}{{\longrightarrow}} Ext(G,S^1).$$ between the set of isomorphism classes of continuous stable homomorphisms and the set isomorphism classes of $S^1$-central extensions of $G$.
Let $\widetilde{G}$ be a $S^1$-central extension of $G$. In Lemma \[lemma Psi surjective\] we showed that if we take ${\mathcal{H}}:=V_c(\widetilde{G})$ the subspace of $L^2(\widetilde{G})
\otimes L^2([0,1])$ where $S^1$ acts by multiplication, then the projectivization $a: G \to P{{\mathcal U}}({\mathcal{H}})$ of the induced homomorphism $\widetilde{a}: \widetilde{G} \to {{\mathcal U}}({\mathcal{H}})$ is a continuous stable homomorphism and it produces the desired central extension $a^*{{\mathcal U}}({\mathcal{H}}) \cong \widetilde{G}$; this shows surjectivity.
Let us suppose now that we have two continuous stable homomorphisms $a,b$ such that $a^*{{\mathcal U}}({\mathcal{H}}) \cong \widetilde{G}
\cong b^*{{\mathcal U}}({\mathcal{H}})$. The Hilbert space ${\mathcal{H}}$ becomes a $\widetilde{G}$ representation with respect to the map $\widetilde{a}: \widetilde{G} \to {{\mathcal U}}({\mathcal{H}})$ and therefore there is a (non-canonical) $\widetilde{G}$-equivariant isomorphism $f_a
:{\mathcal{H}}\stackrel{\cong}{\to} V_c(\widetilde{G})$ that can be taken to be unitary. We get the same result for the map $b$ and we get another $\widetilde{G}$-equivariant isomorphism $f_b :{\mathcal{H}}\stackrel{\cong}{\to} V_c(\widetilde{G})$.
The $\widetilde{G}$-equivariant isomorphism $f_b^{-1} \circ f_a:
{\mathcal{H}}\to {\mathcal{H}}$ makes the following diagram commute $$\xymatrix{
{\mathcal{H}}\ar[rr]^{f_b^{-1} \circ f_a} \ar[d]_{\widetilde{a}(g)} & & {\mathcal{H}}\ar[d]^{\widetilde{b}(g)}\\
{\mathcal{H}}\ar[rr]^{f_b^{-1} \circ f_a} & &{\mathcal{H}}}$$ for all $g \in \widetilde{G}$. Therefore the homomorphisms $\widetilde{a}$ and $\widetilde{b}$ are conjugate; the injectivity follows.
Groupoid of stable homomorphisms {#subsection homotopy quotient}
--------------------------------
Let us now consider the action groupoid $$[Hom_{st}(G,
P{{\mathcal U}}({\mathcal{H}}))/P{{\mathcal U}}({\mathcal{H}})]$$ associated to the action of the group $P{{\mathcal U}}({\mathcal{H}})$ on the continuous stable homomorphisms by conjugation; this groupoid can also by understood as the groupoid whose objects are functors from the category defined by $G$ to the category defined by $P{{\mathcal U}}({\mathcal{H}})$, and whose morphisms are natural transformations.
We would like to understand the homotopy type of the classifying space $$B[Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}}))/P{{\mathcal U}}({\mathcal{H}})]$$ which can be modelled by the homotopy quotient $$EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})}
Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}})) .$$
By Proposition \[proposition pi zero of stable hom\] we know that its connected components are parametrized by the $S^1$-central extensions of $G$, $$\pi_0(EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})}
Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}}))) \cong Ext(G,S^1).$$ If we choose the connected component of the classifying space determined by a stable map $a: G \to P{{\mathcal U}}({\mathcal{H}})$, its higher homotopy groups are determined by the homotopy groups of $$EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})_a} \{a\} \simeq B
(P{{\mathcal U}}({\mathcal{H}})_a)$$ where $$P{{\mathcal U}}({\mathcal{H}})_a =\{ b \in P{{\mathcal U}}({\mathcal{H}}) | \forall g \in G, \ b^{-1}\, a(g)\, b=a(g) \}$$ is the subgroup of $P{{\mathcal U}}({\mathcal{H}})$ that stabilizes $a$.
Note that if we call $\widetilde{G}_a$ the $S^1$ central extension $a^* {{\mathcal U}}({\mathcal{H}})$ defined by $a$, then the induced homomorphism $\widetilde{a} : \widetilde{G}_a \to {{\mathcal U}}({\mathcal{H}})$ defines an action on the Hilbert space ${\mathcal{H}}$ making it isomorphic to the Hilbert space $V_c(\widetilde{G}_a)$ defined in Lemma \[lemma Psi surjective\]. We can now take the action of $\widetilde{G}_a$ on ${{\mathcal U}}({\mathcal{H}})$ by conjugation through the map $\widetilde{a}$; this induces an action of the group $G$ on $P{{\mathcal U}}({\mathcal{H}})$ by conjugation through the map $a$. Let us denote the stabilizer of the action $G$ on $P{{\mathcal U}}({\mathcal{H}})$ by conjugation through the map $a$ by $G_a$.
If we call the fixed points of the action of $G_a$ on $P{{\mathcal U}}({\mathcal{H}})$ by $$P{{\mathcal U}}({\mathcal{H}})^{G_a}:= \{ b \in P{{\mathcal U}}({\mathcal{H}}) |\forall g \in G, \ a(g)^{-1} \, b \,
a(g) = b \}$$ then we have that $$P{{\mathcal U}}({\mathcal{H}})^{G_a} =P{{\mathcal U}}({\mathcal{H}})_a.$$ This means that the fixed point set of the action of $G_a$ is the same as the stabilizer of the homomorphism $a$.
Now take a projective operator $F \in P{{\mathcal U}}({\mathcal{H}})^{G_a}$ that commutes with the $G_a$ action. Take $\widetilde{F} \in {{\mathcal U}}({\mathcal{H}})$ a lift of $F$ and note that $\widetilde{F}$ commutes with the $\widetilde{G}_a$ action up to some phase, i.e. for all $\widetilde{g} \in \widetilde{G}_a$ we have that $$\widetilde{a}( \widetilde{g})^{-1} \, \widetilde{F} \, \widetilde{a}(
\widetilde{g}) = \widetilde{F} \, \sigma_F( \widetilde{g})$$ for some map $\sigma_F: \widetilde{G}_a \to S^1$. Because the action of $\widetilde{G}_a$ is by conjugation, we have that the map $\sigma_F$ is a homomorphism of groups, that it does not depend on the choice of lift for $F$, and moreover that $\sigma_F$ is trivial when restricted to $S^1= kernel(\pi: \widetilde{G}_a \to G_a).$ We have then,
For any $F \in P{{\mathcal U}}({\mathcal{H}})^{G_a}$ there exists a homomorphism $\sigma_F \in Hom(G_a, S^1)$ such that for all lifts $\widetilde{F} \in {{\mathcal U}}({\mathcal{H}})$ of $F$ and all $ \widetilde{g} \in
\widetilde{G}_a$, we have that $$\widetilde{a}( \widetilde{g})^{-1} \, \widetilde{F} \, \widetilde{a}(
\widetilde{g}) = \widetilde{F}\, \sigma_F(g)$$ where $g$ is the image of $\widetilde{g}$ on $G_a$ under the natural map $\widetilde{G}_a \to G_a$.
If we take two operators $F_1, F_2 \in P{{\mathcal U}}({\mathcal{H}})^{G_a}$ with their respective lifts $\widetilde{F}_1, \widetilde{F}_2$ and the induced homomorphisms $\sigma_{F_1}, \sigma_{F_2} \in
Hom(G_a,S^1)$, we have that the composition $\widetilde{F}_1
\widetilde{F}_2$ is a lift for the composition ${F}_1 {F}_2$. Therefore we have that the induced homomorphism $\sigma_{{F}_1
{F}_2}$ for the composition ${F}_1 {F}_2$ is equal to the product of the homomorphisms $\sigma_{F_1}$ and $ \sigma_{F_2}$, i.e. $$\sigma_{{F}_1\, {F}_2} = \sigma_{F_1} \sigma_{F_2}.$$ Note that the product of homomorphisms endows the set $Hom(G_a,
S^1)$ with a natural group structure. We have then
\[Lemma surjective sigma\] The map $$\sigma: P{{\mathcal U}}({\mathcal{H}})^{G_a} \to Hom(G_a,S^1), \ \ \ \ \ F
\mapsto \sigma_F$$ is a surjective homomorphism of groups.
We have already shown that $\sigma$ is a homomorphism of groups, let us show that is surjective. Take $\rho \in Hom(G_a,S^1)$ and consider ${\ensuremath{{\mathbb C}}}(\rho)$ the linear $\widetilde{G}_a$ representation defined by multiplication of scalars, i.e. for $\lambda \in {\ensuremath{{\mathbb C}}}(\rho)$, $\widetilde{g} \cdot \lambda =
\rho(g) \lambda$.
Now, if $V$ is an irreducible representation of $\widetilde{G}_a$ where the kernel of $\widetilde{G}_a \to G_a$ acts by multiplication of scalars, the vector space $V \otimes
{\ensuremath{{\mathbb C}}}(\rho)$ becomes a $\widetilde{G}_a$ representation by the diagonal action, and moreover it is irreducible and the kernel $\widetilde{G}_a \to G_a$ also acts by multiplication of scalars. On the other hand, any irreducible representation of $\widetilde{G}_a$, where the kernel of $\widetilde{G}_a \to G_a$ acts by multiplication of scalars, is isomorphic to a representation of the form $W \otimes {\ensuremath{{\mathbb C}}}(\rho)$ for a suitable irreducible representation $W$ of $\widetilde{G}_a$ where the kernel of $\widetilde{G}_a \to G_a$ acts by multiplication of scalars. Therefore we have that by tensoring the space $V_c(\widetilde{G}_a)$ with ${\ensuremath{{\mathbb C}}}(\rho)$ we get a $\widetilde{G}_a$-equivariant isomorphism $$\eta:
V_c(\widetilde{G}_a) \otimes {\ensuremath{{\mathbb C}}}(\rho) \to
V_c(\widetilde{G}_a)$$ where $V_c(\widetilde{G}_a)$ is the Hilbert space defined in Lemma \[lemma Psi surjective\].
Let us consider the map $$\gamma : V_c(\widetilde{G}_a) \to V_c(\widetilde{G}_a) \otimes
{\ensuremath{{\mathbb C}}}(\rho), \ \ \ \ \ \gamma(v)= v \otimes 1$$ and note that $$\widetilde{g} \gamma ( \widetilde{g}^{-1} v) = \widetilde{g}
\left( \gamma (\widetilde{g}^{-1} v) \otimes 1 \right) = v \otimes
\rho(g) = \rho(g) \gamma(v).$$
Choosing an explicit $\widetilde{G}_a$-equivariant isomorphism $f_a : {\mathcal{H}}\to V_c(\widetilde{G}_a)$ such as the one defined in Proposition \[proposition pi zero of stable hom\], we can define the operator in ${{\mathcal U}}({\mathcal{H}})$ $$\widetilde{F} : = f_a^{-1} \circ \eta \circ \gamma \circ f_a$$ which by definition has the property that for all $\widetilde{g} \in \widetilde{G}_a$ $$\label{equation equivariant lift up to a phase} \widetilde{a}(
\widetilde{g})^{-1} \, \widetilde{F} \, \widetilde{a}(
\widetilde{g}) = \widetilde{F} \, \sigma_F(g).$$
If $F$ is the projectivization of $\widetilde{F}$, then we have that equation implies that $F \in P {{\mathcal U}}({\mathcal{H}})^{G_a}$, and moreover that $\sigma_F
= \rho$. This shows that the homomorphism $\sigma$ is surjective.
With the help of the surjective map $\sigma$ let us try to understand in a more explicit way the group $P{{\mathcal U}}({\mathcal{H}})^{G_a}$. Take $\rho \in Hom(G_a,S^1)$ and define the space $${{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)} := \{ \widetilde{F} \in
{{\mathcal U}}({\mathcal{H}}) | \forall \widetilde{g} \in \widetilde{G}_a, \
\widetilde{a}(\widetilde{g})^{-1} \, \widetilde{F} \,
\widetilde{a}( \widetilde{g}) = \widetilde{F} \, \sigma_F(g)\}.$$ From the proof of Lemma \[Lemma surjective sigma\] we get that the inverse image of $\rho$ under $\sigma$ is isomorphic to the projectivization of the space ${{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}$, i.e. $$\sigma^{-1}(\rho) = P \left( {{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}\right),$$ and moreover we have that any operator $\widetilde{F} \in
{{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}$ induces a homeomorphism of spaces $$\beta_{\widetilde{F}} : {{\mathcal U}}({\mathcal{H}})^{\widetilde{G}_a}
\stackrel{\cong}{{\longrightarrow}} {{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}, \ \ \ W
\mapsto F \circ W$$ by composing the operators.
\[theorem homotopy groups of PU(H)G\] Let $a \in Hom_{st}(G, P{{\mathcal U}}({\mathcal{H}}))$ be a continuous stable homomorphism, and denote with $G_a$ the action on $P{{\mathcal U}}({\mathcal{H}})$ by conjugation defined by the map $a$. Then there is a natural isomorphism of groups $$\pi_0\left(P{{\mathcal U}}({\mathcal{H}})^{G_a} \right) \cong Hom(G;S^1),$$ and each connected component of $P{{\mathcal U}}({\mathcal{H}})^{G_a}$ has the homotopy type of a $K({\ensuremath{{\mathbb Z}}},2)$.
We have seen from the arguments above that $$P{{\mathcal U}}({\mathcal{H}})^{G_a} \cong \bigsqcup_{\rho \in Hom(G,S^1)} P \left(
{{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}\right)$$ and that as spaces we have homeomorphisms $$P \left( {{\mathcal U}}({\mathcal{H}})^{\widetilde{G}_a}\right) \cong P \left(
{{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}\right).$$
Following the same argument as in [@SegalEqContractibility Proof of Proposition 1], we have that $${{\mathcal U}}({\mathcal{H}})^{\widetilde{G}_a} \cong \prod_{\alpha} {{\mathcal U}}({\mathcal{H}}_\alpha)^{\widetilde{G}_a}$$ where ${\mathcal{H}}= \oplus_\alpha {\mathcal{H}}_\alpha$ is a decomposition of ${\mathcal{H}}$ into isotypical $\widetilde{G}_a$-representations, and $${\mathcal{H}}_\alpha \cong M_\alpha \otimes {\mathcal{H}}_{\alpha,0}$$ where $M_\alpha$ is a simple $\widetilde{G}_a$-representation and ${\mathcal{H}}_{\alpha,0}$ is a separable Hilbert space. Since $${{\mathcal U}}({\mathcal{H}}_\alpha)^{\widetilde{G}_a} \cong {{\mathcal U}}({\mathcal{H}}_{\alpha,0})$$ and each of the spaces $ {{\mathcal U}}({\mathcal{H}}_{\alpha,0})$ is contractible by Proposition \[Proposition contractibility of U(H)\], then we can conclude that $ {{\mathcal U}}({\mathcal{H}})^{\widetilde{G}_a}$ is contractible and therefore we have that $$\pi_i\left(P\left ({{\mathcal U}}({\mathcal{H}})^{\widetilde{G}_a}\right) \right) = \left\{
\begin{array}{cl}
{\ensuremath{{\mathbb Z}}}& i =0,2\\
0 & \mbox{otherwise.}
\end{array} \right.$$ Therefore the spaces $P \left
({{\mathcal U}}({\mathcal{H}})^{(\widetilde{G}_a,\rho)}\right)$ have the homotopy type of a $K({\ensuremath{{\mathbb Z}}},2)$, and moreover, the connected components of $P{{\mathcal U}}({\mathcal{H}})^{G_a}$ are parameterized by the elements in $Hom(G_a,S^1)$; hence we have an isomorphism of groups $$\pi_0\left(P{{\mathcal U}}({\mathcal{H}})^{G_a} \right) \cong Hom(G;S^1).$$
We can now bundle up all previous results in the following theorem
\[theorem homotopy groups BPU(H)\] Let $G$ be a compact Lie group and $Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}}))/P{{\mathcal U}}({\mathcal{H}})$ the category whose space of objects consist of continuous stable homomorphisms from $G$ to the projective unitary group endowed with the compact open topology, and whose morphisms are natural transformations. Then the connected components of the homotopy quotient are parameterized by the $S^1$-central extensions of $G$, $$\pi_0\left(EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})) \right) = Ext_c(G,S^1),$$ and the higher homotopy groups of any connected component are $$\pi_i\left(EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}}))\right)=\left\{
\begin{array}{cl}
Hom(G,S^1) & i=1\\
{\ensuremath{{\mathbb Z}}}& i=3\\
0 & \mbox{otherwise}.
\end{array}
\right.$$
From the analysis done at the beginning of Section \[subsection homotopy quotient\] we know that the connected components of the homotopy quotient $$EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})}
Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}}))$$ are parameterized by $Ext_c(G,S^1)$. Also from Section \[subsection homotopy quotient\], we know that if we take any stable homomorphism $a:G \to P{{\mathcal U}}({\mathcal{H}})$, the connected component defined by $a$ is homotopy equivalent to $$EP{{\mathcal U}}({\mathcal{H}})
\times_{P{{\mathcal U}}({\mathcal{H}})_a} \{a\} \cong B (P{{\mathcal U}}({\mathcal{H}})_a ) = B(
P{{\mathcal U}}({\mathcal{H}})^{G_a} ).$$
By Theorem \[theorem homotopy groups of PU(H)G\] we know that $\pi_0(P{{\mathcal U}}({\mathcal{H}})^{G_a})=Hom(G,S^1)$ is an isomorphism of groups, and therefore we have that $$\pi_1\left(B( P{{\mathcal U}}({\mathcal{H}})^{G_a} )\right) = Hom(G,S^1).$$ Also from Theorem \[theorem homotopy groups of PU(H)G\] we know that $\pi_2(P{{\mathcal U}}({\mathcal{H}})^{G_a})= {\ensuremath{{\mathbb Z}}}$ and that the other homotopy groups are trivial. Therefore $$\pi_3\left(B( P{{\mathcal U}}({\mathcal{H}})^{G_a} )\right)= {\ensuremath{{\mathbb Z}}}$$ and $$\pi_i\left(B( P{{\mathcal U}}({\mathcal{H}})^{G_a} )\right)= 0$$ for $i=2$ and $i>3$.
The homotopy groups described previously are precisely the cohomology groups for the classifying space $BG$ in certain degree, let us see:
\[corollary equivalence of homotopy type for G\] For $G$ a compact Lie group there are isomorphisms $$\pi_i\left(EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})) \right) =
H^{3-i}(BG, {\ensuremath{{\mathbb Z}}}).$$
The action of $G$ on $EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}}))$ given by equation (\[left K action\]) defines a projective unitary bundle on the homotopy quotients $$\xymatrix{
P{{\mathcal U}}({\mathcal{H}}) \ar[r] & (EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})))
\times_G EG \ar[d]& \\
& EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})) \times BG &
}$$ classified by a map from the base to $BP{{\mathcal U}}({\mathcal{H}})$, which induces an adjoint map $$\begin{aligned}
\label{induced map to BG}
EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})) \to Maps(BG,
BP{{\mathcal U}}({\mathcal{H}})). \end{aligned}$$
At the level of homotopy groups we know that $$\pi_i(Maps(BG, BP{{\mathcal U}}({\mathcal{H}})) \cong H^{3-i}(BG, {\ensuremath{{\mathbb Z}}})$$ and therefore we get the desired map $$\pi_i\left(EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(G,P{{\mathcal U}}({\mathcal{H}})) \right)
\to H^{3-i}(BG, {\ensuremath{{\mathbb Z}}}).$$
The space $BG$ is connected and therefore $H^0(BG, {\ensuremath{{\mathbb Z}}})=
{\ensuremath{{\mathbb Z}}}$, agreeing with the third homotopy group of the homotopy quotient.
The group $G$ being compact implies that $\pi_0(G)$ is finite. Hence $\pi_1(BG)$ is also finite and we have that $H^1(BG,{\ensuremath{{\mathbb Z}}})=0$; this shows the isomorphism for $i=2$.
In [@LashofMaySegal Prop. 4] it is proven that the natural map $$Hom(G,A) \to [BG, BA]$$ is an isomorphism whenever $G$ and $A$ are compact Lie groups and $A$ is abelian. Therefore $$Hom(G, S^1) \stackrel{\cong}{\to} [BG,BS^1] = H^2(BG,
{\ensuremath{{\mathbb Z}}});$$ this shows the isomorphism for $i=1$.
Finally, in [@AtiyahSegal Prop. 6.3] Atiyah and Segal prove that $H^3(BG, {\ensuremath{{\mathbb Z}}}) \cong Ext_c(G,S^1)$. This, together with the fact that the connected component of $[(a, f)]$ in the left hand side of (\[induced map to BG\]), maps to the connected component of $Bf: BG \to BP{{\mathcal U}}({\mathcal{H}})$ imply that the isomorphism holds for $i=0$.
Equivariant stable projective unitary bundles and their classification {#section Equivariant stable bundles}
======================================================================
This chapter is devoted to set-up the framework for generalizing the twistings of K-theory obtained by principal $P{{\mathcal U}}({\mathcal{H}})$-bundles to the equivariant case. We start the chapter by giving the definition of an [*equivariant stable projective unitary bundle*]{} and the rest of the chapter is devoted to understand how the local objects look like, namely the equivariant stable projective unitary bundles over the spaces $G/K$, and then we proceed to show how they can be classified.
Let us begin by clarifying which type of spaces we will be working with.
Preliminaries
-------------
###
Throughout this chapter $G$ will be a Lie group and $X$ will be a proper $G-ANR$; let us recall what all this means:
A $G$-space $X$ is [*proper*]{} if the map $$G \times X \to X \times X \ \ \ \ \ (g,x) \mapsto (gx,x)$$ is a proper map of topological spaces (preimages of compact sets are compact) and the action map $G \times X \to X, \ (g,x) \mapsto gx$ is a closed map.
A $G$ space $X$ is a $G-ANR$ ($G$-equivariant absolute neighborhood retract [@JamesSegal]) if $X$ is a separable and metrizable $G$-space such that whenever $B$ is a normal $G$-space and $A$ is an invariant closed subspace of $B$, any $G$ map $A \to
X$ can be extended over an invariant neighborhood of $A$ in $B$. In the case that $G$ is compact Lie acting smoothly on a compact manifold $M$, then it can be shown that $M$ is always a $G-ANR$.
Note that any proper $G-ANR$ space can be covered with a [*good $G$-cover*]{}, namely a countable $G$-cover such that any non trivial intersection of finite $G$-open sets of the cover is equivariantly homotopicaly equivalent to a $G$-set of the form $G/H$ with $H$ a compact subgroup of $G$.
When studying the homotopy theory of $G$-spaces is important to understand the homotopy theory of the system of fixed points of the action. This is done with the category of canonical orbits, let us recall its definition and some of its properties [@DavisLueck Section 7].
### System of fixed points {#subsection system of
fix points}
The category of canonical orbits for proper $G$-actions denoted by ${\mathcal{O}}_G^P$ is a topological category with discrete object space $${\rm{Obj}}({\mathcal{O}}_G^P) = \{G/H \colon H \ \mbox{is a compact
subgroup of} \ G \}$$ and whose morphisms consist of $G$-maps $${\rm Mor}_{{\mathcal{O}}^P_G} (G/H,G/K) = {\rm Maps}(G/H,G/K)^G$$ with a topology such that the natural bijection $${\rm Mor}_{{\mathcal{O}}^P_G} (G/H,G/K) \cong (G/K)^H$$ be a homeomorphism.
An [*${\mathcal{O}}_G^P$-space*]{} shall be a contravariant functor from ${\mathcal{O}}_G^P$ to the category of topological spaces, and these functors will form the objects of a topological category whose morphisms will consist of natural transformations.
The [*fixed point set system*]{} of $X$, denoted by $\Phi X$, is the ${\mathcal{O}}_G^P$-space defined by: $$\Phi X (G/H) := {\rm{Maps}}(G/H, X)^G= X^H$$ and if $\theta: G/H \to G/K$ corresponds to $gK \in (G/K)^H$ then $$\Phi X (\theta)(x) := gx \in X^H$$ whenever $x \in X^K$. The functor $\Phi$ becomes a functor from the category of proper $G$-spaces to the category of ${\mathcal{O}}_G^P$-spaces.
If ${{\mathcal X}}$ is a contravariant functor from ${\mathcal{O}}_G^P$ to spaces and ${\mathcal{Y}}$ is a covariant functor from ${\mathcal{O}}_G^P$ to spaces one can define the space $${{\mathcal X}}\times_{{\mathcal{O}}_G^P}{\mathcal{Y}}:= \bigsqcup_{c \in {\rm{Obj}}({\mathcal{O}}_G^P)} {{\mathcal X}}(c) \times {\mathcal{Y}}(c) / \sim$$ where $\sim$ is the equivalence relation generated by $({{\mathcal X}}(\phi)(x),y) \sim (x, {\mathcal{Y}}(\phi)(y))$ for all morphisms $\phi: c \to d$ in ${\mathcal{O}}_G^P$ and points $x \in {{\mathcal X}}(d)$ and $y \in {\mathcal{Y}}(c)$.
A model for the homotopical version of the previous construction is defined as follows: consider the category ${{\mathcal F}}({{\mathcal X}}, {\mathcal{Y}})$ whose objects are $${\rm{Obj}}({{\mathcal F}}({{\mathcal X}}, {\mathcal{Y}}))=\bigsqcup_{c \in {\rm{Obj}}({\mathcal{O}}_G^P)} {{\mathcal X}}(c) \times {\mathcal{Y}}(c)$$ and whose morphisms consist of all triples $(x, \phi,y)$ where $\phi: c \to d$ is a morphism in ${\mathcal{O}}_G^P$ and $x \in {{\mathcal X}}(d)$ and $y \in {\mathcal{Y}}(c)$, with ${\mathsf{source}}(x, \phi,y)= ({{\mathcal X}}(\phi)(x),y)$ and ${\mathsf{target}}(x, \phi,y)= (x, {\mathcal{Y}}(\phi)(y))$. Define the space $${{\mathcal X}}\times_{h{\mathcal{O}}_G^P}{\mathcal{Y}}:= |{{\mathcal F}}({{\mathcal X}}, {\mathcal{Y}})|$$ as the geometric realization of the category ${{\mathcal F}}({{\mathcal X}}, {\mathcal{Y}})$.
Now, if we consider the covariant functor $\nabla$ from ${\mathcal{O}}_G^P$ to spaces that to each orbit type $G/H$ it assigns the set $G/H$, then we can consider the functors $$\begin{aligned}
\_ \times_{{\mathcal{O}}_G^P}\nabla : {\mathcal{O}}_G^P-{\rm spaces} & \to & {\rm proper} \ G-{\rm spaces} \\
{{\mathcal X}}& \mapsto & {{\mathcal X}}\times_{{\mathcal{O}}_G^P}\nabla\\
\_ \times_{h{\mathcal{O}}_G^P}\nabla : {\mathcal{O}}_G^P-{\rm spaces} & \to & {\rm proper} \ G-{\rm spaces} \\
{{\mathcal X}}& \mapsto & {{\mathcal X}}\times_{h{\mathcal{O}}_G^P}\nabla.\end{aligned}$$
Since the orbit types $G/H$ are endowed with a natural left $G$ action, then the spaces ${{\mathcal X}}\times_{{\mathcal{O}}_G^P}\nabla$ and ${{\mathcal X}}\times_{h{\mathcal{O}}_G^P}\nabla$ become $G$-spaces by endowing them with the left $G$ action included by the $G$ action on the objects and morphisms of $\nabla$.
We shall quote the following result:
\[theorem Davis-Lueck\] For $G$ a discrete group [@DavisLueck lemma 7.2] the functors $\Phi$ and $ \_ \times_{{\mathcal{O}}_G^P}\nabla$ are adjoint, i.e. for a ${\mathcal{O}}_G^P$ space ${{\mathcal X}}$ and a proper $G$-space $Y$ there is a natural homeomorphism $${\rm{Maps}}({{\mathcal X}}\times_{{\mathcal{O}}_G^P}\nabla,Y)^G \stackrel{\cong}{{\longrightarrow}} {\rm{Hom}}_{{\mathcal{O}}_G^P} ({{\mathcal X}}, \Phi Y),$$ and moreover, the adjoint of the identity map on $\Phi Y$ under the above adjunction, is a natural $G$-homeomorphism $$(\Phi Y) \times_{{\mathcal{O}}_G^P}\nabla \stackrel{\cong}{{\longrightarrow}} Y.$$
Note furthermore that there is a natural $G$-homotopy equivalence $$(\Phi Y) \times_{h{\mathcal{O}}_G^P}\nabla \stackrel{\simeq}{{\longrightarrow}} (\Phi Y) \times_{{\mathcal{O}}_G^P}\nabla,$$ and a ${\mathcal{O}}_G^P$-homotopy equivalence $${{\mathcal X}}\to \Phi({{\mathcal X}}\times_{h{\mathcal{O}}_G^P}\nabla).$$
Similar statements in the case that $G$ is a compact Lie group can be found in [@Elmendorf].
Equivariant stable projective unitary bundles
---------------------------------------------
A first guess may suggest that the appropriate equivariant twistings for K-theory from a point of view of index theory are $G$-equivariant projective unitary bundles. These bundles may be used to define associated bundles whose fibers are $Fred({\mathcal{H}})$ the space of Fredholm operators on ${\mathcal{H}}$, but a closer look at them leads us to think that we need to put further conditions on how the group $G$ acts on the fibers; this local condition is that the local isotropy group should act on the fibers by stable homomorphisms. The precise definition is as follows:
\[def projective unitary G-equivariant stable bundle\] A [*projective unitary $G$-equivariant stable bundle*]{} over $X$ is a principal $P{{\mathcal U}}({\mathcal{H}})$-bundle $$P{{\mathcal U}}({\mathcal{H}}) {\longrightarrow}P {\longrightarrow}X$$ where $P{{\mathcal U}}({\mathcal{H}})$ acts on the right, endowed with a left $G$ action lifting the action on $X$ such that:
- the left $G$-action commutes with the right $P{{\mathcal U}}({\mathcal{H}})$ action, and
- for all $x \in X$ there exists a $G$-neighborhood $V$ of $x$ and a $G_x$-contractible slice $U$ of $x$ with $V$ equivariantly homeomorphic to $ U \times_{G_x} G$ with the action $$G_x \times (U \times G) \to U \times G, \ \ \ \
k \cdot(u,g)= (ku, g k^{-1}),$$ together with a local trivialization $$P|_V \cong (P{{\mathcal U}}({\mathcal{H}}) \times U) \times_{G_x} G$$ where the action of the isotropy group is: $$\begin{aligned}
G_x \times \left( (P{{\mathcal U}}({\mathcal{H}}) \times U) \times G \right)& \to & (P{{\mathcal U}}({\mathcal{H}}) \times U) \times G
\\
\ \left(k , ((F,y),g)\right)& \mapsto & ((f_x(k)F, ky), g k^{-1})\end{aligned}$$ with $f_x : G_x \to P{{\mathcal U}}({\mathcal{H}})$ a fixed stable homomorphism (see Definition \[definition stable\]).
Two projective unitary $G$-equivariant stable bundles $P', P$ over $X$ will be isomorphic if there is a $G$-equivariant homeomorphism $P' \to P$ of principal $P{{\mathcal U}}({\mathcal{H}})$ bundles. The isomorphism classes of projective unitary $G$-equivariant stable bundles over $X$ will be denoted by $${\ensuremath{{\mathrm{Bun}}}}^G_{st}(X, P{{\mathcal U}}({\mathcal{H}})).$$
###
The results of Chapter \[section PU(H)\] will provide us with the first examples of projective unitary equivariant stable bundles. These examples will be the building blocks of the construction of the universal projective unitary $G$-equivariant stable bundle as well as their classification; this will be done in Chapter \[chapter universal bundle\] for the case of discrete and proper actions.
Let $K$ be a compact Lie group and consider the projective unitary bundle $$\label{projective unitary bundle K
compact}
\xymatrix{
P{{\mathcal U}}({\mathcal{H}}) \ar[r] & EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \ar[d] \\
& EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) }$$ where the base is the space whose homotopy groups are calculated in Theorem \[theorem homotopy groups BPU(H)\].
The right-$P{{\mathcal U}}({\mathcal{H}})$ action on the total space of the bundle in (\[projective unitary bundle K compact\]) is defined as: $$\begin{aligned}
\nonumber EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}}) & \to &
EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))\\
((a,f),F) & \mapsto & (aF, F^{-1} f F) \label{right PU(H) action}\end{aligned}$$ where $F^{-1} f F$ denotes the homomorphism conjugate of $f$ by $F$.
The left $K$-action on the total space of (\[projective unitary bundle K compact\]) is defined as: $$\begin{aligned}
\nonumber K \times EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) & \to
&
EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))\\
(g,(a,f)) & \mapsto & (a f(g), f(g)^{-1} f f(g)), \label{left K
action}\end{aligned}$$ where a simple calculation shows that the $K$-action is indeed a left action.
It follows that the total space of the bundle in (\[projective unitary bundle K compact\]) has a left $K$-action inducing a trivial $K$ action on the base. We claim that
The projective unitary bundle $$\xymatrix{
P{{\mathcal U}}({\mathcal{H}}) \ar[r] & EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \ar[d] \\
& EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) }$$ is a projective unitary $K$-equivariant stable bundle.
Let us first prove that the left $K$-action commutes with the right $P{{\mathcal U}}({\mathcal{H}})$ action; on the one side we have $$\begin{aligned}
\left( g(a,f) \right) F& =& \left( a f(g), f(g)^{-1}f f(g))
\right) F \\
& = & \left(a f(g) F, F^{-1} f(g)^{-1} f f(g) F \right)\end{aligned}$$ and on the other $$\begin{aligned}
g\left( (a,f)F \right) & =& \left( a F, F^{-1}f F)
\right) F \\
& = & \left(a F F^{-1} f(g) F, F^{-1} f(g)^{-1}F F^{-1} f F F^{-1}
f(g) F\right) \\
& = & \left(a f(g) F, F^{-1} f(g)^{-1} f f(g) F \right);\end{aligned}$$ which shows that the actions commute.
Now, for the second condition let us take any point $x \in
EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))$ and let us choose a contractible neighborhood $V$ of $x$. Because the restricted bundle $$\left( EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K,
P{{\mathcal U}}({\mathcal{H}}))\right)|_V$$ is trivializable we may find a section $$\begin{aligned}
\label{diagram section to EPxHom}\xymatrix{
& EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \ar[d] \\
V \ar[ur]^\alpha \ar@{->}[r] & EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})}
Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) }\end{aligned}$$ which can be decomposed as $$\alpha(v) = \left(\lambda(v), \eta(v) \right)$$ with $\lambda: V
\to EP{{\mathcal U}}({\mathcal{H}})$ and $\eta: V \to Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))$.
Let us denote $f := \eta(x)$ and let us consider the connected component $W_f \subset Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))$ containing $f$. In view of Proposition \[proposition pi zero of stable hom\] we know that the group $P{{\mathcal U}}({\mathcal{H}})$ acts transitively on $W_f$ and therefore we have a non canonical homeomorphism $$P{{\mathcal U}}({\mathcal{H}})/P{{\mathcal U}}({\mathcal{H}})_f \stackrel{\cong}{{\longrightarrow}} W_f, \ \ \ \ \ [F] \mapsto F^{-1}f F$$ where $$P{{\mathcal U}}({\mathcal{H}})_f= \{F \in P{{\mathcal U}}({\mathcal{H}}) | F^{-1}fF = f \}$$ is the isotropy group of $f$ and acts on $P{{\mathcal U}}({\mathcal{H}})$ by conjugation on the right.
Because the bundle $$P{{\mathcal U}}({\mathcal{H}})_f {\longrightarrow}P{{\mathcal U}}({\mathcal{H}}) {\longrightarrow}P{{\mathcal U}}({\mathcal{H}})/P{{\mathcal U}}({\mathcal{H}})_f$$ is a principal bundle and $V$ is a contractible set, we may find a lift $\sigma: V \to P{{\mathcal U}}({\mathcal{H}})$ of the map $\eta: V \to W_f$ making the diagram commutative $$\xymatrix{
& & P{{\mathcal U}}({\mathcal{H}}) \ar[d] \\
V \ar[r]_\eta \ar[urr]^\sigma & W_f \ar[r]^(.3)\cong &
P{{\mathcal U}}({\mathcal{H}})/P{{\mathcal U}}({\mathcal{H}})_f }$$ Hence the map $\sigma$ satisfies the equation $$\eta(v) = \sigma(v)^{-1} f \sigma(v)$$ for $v \in V$.
Let us now consider a different section $$\alpha' : V \to
EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}}))$$ of the diagram (\[diagram section to EPxHom\]) defined by the action of $\sigma$ on $\alpha$, namely: $$\alpha'(v) := \alpha(v) \cdot \sigma(v)^{-1} =
\left(\lambda(v)\sigma(v)^{-1}, \sigma(v) \eta(v) \sigma(v)^{-1}
\right) = (\lambda'(v), f)$$ where we denoted $\lambda'(v):=\lambda(v) \sigma(v)^{-1}$.
With the section $\alpha'$ at hand we can define a local trivialization as follows $$\begin{aligned}
\nonumber V \times P{{\mathcal U}}({\mathcal{H}}) & \stackrel{\phi}{{\longrightarrow}} & \left(
EP{{\mathcal U}}({\mathcal{H}})
\times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})) \right)|_V \\
(v,F) & \mapsto & \alpha'(v) \cdot F = \left( \lambda'(v) F,
F^{-1} f F \right). \label{local trivialization}\end{aligned}$$ Transporting the $K$-action to the left-hand side of (\[local trivialization\]) we have that for $g \in K$ and $(v,F) \in V
\times P{{\mathcal U}}({\mathcal{H}})$, $$\begin{aligned}
g\cdot (v, F) & := & \phi^{-1} \left( g ( \phi(v, F))\right)\\
&=& \phi^{-1} \left( g( \lambda'(v)F, F^{-1}fF) \right)\\
&=& \phi^{-1}\left( \lambda'(v)f(g)F, F^{-1}f(g)^{-1} f f(g) F
\right) \\
&=& \left( v,f(g)F\right),\end{aligned}$$ which implies that the $K$ action on $P{{\mathcal U}}({\mathcal{H}})$ is by multiplication on the left by the fixed stable homomorphism $f$.
This finishes the proof.
Local objects
-------------
Following the notation of [@tomDieck; @Murayama] we say that:
For $K$ a compact subgroup of $G$, a projective unitary stable $G$-equivariant bundle over $G/K$ is a [*[local object]{}*]{}.
We would like to classify the local objects. Let $P \to G/K$ be a local object and consider the restriction $P|_{[K]}$ of $P$ to the point $[K] \in G/K$. The canonical map $$P|_{[K]} \times_K G \to
P, \ \ \ \ \ [(x,g)] \mapsto gx$$ produces a $G$-equivariant isomorphism of projective unitary bundles, and by Definition \[def projective unitary G-equivariant stable bundle\] we know that $P|_{[K]}$ trivializes via a map $\phi: P{{\mathcal U}}({\mathcal{H}}) \cong
P|_{[K]}$ where the left $K$ action on $P|_{[K]}$ induces a $K$ action on $P{{\mathcal U}}({\mathcal{H}})$ via a stable homomorphism $f \in Hom_{st}(K,
P{{\mathcal U}}({\mathcal{H}}))$.
We can take the $K$ action on $P{{\mathcal U}}({\mathcal{H}}) \times G$ given by $k
\cdot (F,g) := (f(k)F, gk^{-1})$ and we obtain an isomorphism $$P{{\mathcal U}}({\mathcal{H}}) \times_K G \stackrel{\cong}{\to} P, \ \ \ \ \ [(F,g)] \mapsto g\phi(F)$$ of projective unitary stable $G$-equivariant bundles.
We can conclude that
The isomorphism classes of local objects over $G/K$ are in one to one correspondence with conjugacy classes of stable homomorphisms from $K$ to $P{{\mathcal U}}({\mathcal{H}})$, which in view of Proposition \[proposition pi zero of stable hom\] is isomorphic to the set of isomorphism classes of $S^1$ central extensions of $K$.
We have classified the projective unitary stable $G$-equivariant bundles over the $G$-space $Y \times_K G$ when $Y$ is a point. In the next section we will generalize the classification whenever $Y$ is a trivial $K$-space.
Universal projective unitary stable $K$-equivariant bundle over trivial $K$-spaces {#section universal for trivial K spaces}
----------------------------------------------------------------------------------
We will show that the universal projective unitary stable $K$-equivariant is bundle precisely the bundle $$\begin{aligned}
\label{universal K bundle}
\xymatrix{
P{{\mathcal U}}({\mathcal{H}}) \ar[r] & EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})) \ar[d] \\
&EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})).}\end{aligned}$$ To carry out the proof we will use two groupoids ${{\mathcal D}}_K$ and ${\mathcal{C}}_K$ and a functor between the two ${{\mathcal D}}_K \to {\mathcal{C}}_K$ such that the geometric realization of the functor $| {{\mathcal D}}_K| \to |{\mathcal{C}}_K|$ is homotopy equivalent to the bundle in (\[universal K bundle\]). The groupoid ${\mathcal{C}}_K$ will be the action groupoid $${\mathcal{C}}_K:=Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \rtimes P{{\mathcal U}}({\mathcal{H}})$$ whose objects are the stable homomorphisms and whose space of morphisms is $$Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}}),$$ together with the structural maps $${\mathsf{source}}(f,F)=f, \ \ \ {\mathsf{target}}(f,F)=F^{-1}fF, \ \ \ {\mathsf{inverse}}(f,F)=(F^{-1}fF, F^{-1}),$$ $${\mathsf{comp}}((f,F),(F^{-1}fF,G))=(f,FG) \ \ \ {\rm{and}} \ \ {\mathsf{identity}}(f) = (f,1).$$
The groupoid ${{\mathcal D}}_K$ will consist of $${\rm{Mor}}({{\mathcal D}}_K) := Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}}) \times P{{\mathcal U}}({\mathcal{H}})$$ $${\rm{Obj}}({{\mathcal D}}_K):=Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}})$$ together with structural maps $${\mathsf{source}}(f,F,G)=(f,F), \ \ \ {\mathsf{target}}(f,F,G)=(GF^{-1}fFG^{-1},G),$$ $${\mathsf{inverse}}(f,F,G)=(GF^{-1}fFG^{-1}, G,F),$$ $${\mathsf{comp}}((f,F,G),(GF^{-1}fFG^{-1},G,L))=(f,F,L)$$ $${\rm{and}} \ \ {\mathsf{identity}}(f,F) = (f,F,1).$$
The map $\gamma:{\rm{Mor}}({{\mathcal D}}_K) \to {\rm{Mor}}({\mathcal{C}}_K)$, $(f,F,G)
\mapsto (f,FG^{-1})$ induces a functor $\gamma:{{\mathcal D}}_K \to {\mathcal{C}}_K$, and therefore we have a map $|\gamma|:|{{\mathcal D}}_K| \to |{\mathcal{C}}_K|$ at the level of their geometric realizations. The results of [@Segal Section 3] show that $|{{\mathcal D}}_K|
\stackrel{|\gamma|}{\to} |{\mathcal{C}}_K|$ is indeed projective unitary bundle $$P{{\mathcal U}}({\mathcal{H}}) \to |{{\mathcal D}}_K| \to |{\mathcal{C}}_K|$$ and also that it is homotopy equivalent to the bundle in (\[universal K bundle\]); recall from the notation, section \[Notation\], that in this paper the geometrical realization of a category ${{\mathcal W}}$ is defined as the geometrical realization of the category ${{\mathcal W}}_{\bf N}$.
Endowing the groupoid ${{\mathcal D}}_K$ with the following left action of the group $K$ $$K \times {{\mathcal D}}_K \to {{\mathcal D}}_K, \ \ \ \ (g,(f,F,G)) \mapsto (f,f(g)F, GF^{-1}f(g)F),$$ a simple calculation shows that the induced action on ${\mathcal{C}}_K$ is trivial. Hence we have
The bundle $P{{\mathcal U}}({\mathcal{H}}) \to |{{\mathcal D}}_K| \stackrel{|\gamma|}{\to}
|{\mathcal{C}}_K|$ is a projective unitary stable $K$-equivariant bundle over the trivial $K$ space $|{\mathcal{C}}_K|$.
Now let us do the main construction of this section
\[theorem classifying map for trivial K spaces\] Let $P{{\mathcal U}}({\mathcal{H}}) \to Q \to Y$ be a projective unitary stable $K$-equivariant bundle over the trivial $K$-space $Y$, then there is a map $\alpha:Y \to |{\mathcal{C}}_K|$ such that $\alpha^*|{{\mathcal D}}_K| \cong
Q$ as projective unitary stable $K$-equivariant bundles.
Choose a cover $\{U_i\}_{i \in I}$ of $Y$ where $Q$ is trivialized, i.e. $$\phi_i : Q|_{U_i} \stackrel{\cong}{\to} U_i \times P{{\mathcal U}}({\mathcal{H}})$$ and the $K$ action on the right hand side comes form a stable homomorphism $f_i : K \to P{{\mathcal U}}({\mathcal{H}})$ satisfying the equation $\phi_i(g \cdot q) = (x, f_i(g)F)$ where $\phi(q) = (x,F)$ and $g
\in K$. Define the transition functions $\rho_{ji} : U_i \cap U_j
\to P{{\mathcal U}}({\mathcal{H}})$ through the equations $$\left( \phi_j|_{U_i \cap
U_j} \circ \left( \phi_i|_{U_i \cap U_j} \right)^{-1} \right)(x,F)
= (x, \rho_{ji}(x)F),$$ and note that we have the compatibility conditions $$\begin{aligned}
\label{compatibility of functions}
\rho_{kj} \circ \rho_{ji} = \rho_{ki}, \ \ \ \ \ \ \rho_{ij}\circ \rho_{ji}= \rho_{ii}=1, \ \ {\rm{and}} \ \ \nonumber
\rho_{ji}(x)^{-1} f_i(g) \rho_{ij}(x) = f_j(g) \end{aligned}$$ for all $x \in U_i \cap U_j$ and $g \in K$.
Now define the categories ${\mathcal{Y}}, {{\mathcal Q}}$ associated to the open cover [@Segal] whose objects and morphism are respectively $${\rm{Obj}}({\mathcal{Y}}) := \{(y,i) \in Y \times I | y \in U_i \} \cong \bigsqcup_{i \in I} U_i$$ $${\rm{Mor}}({\mathcal{Y}}): = \{(y,i,j) \in Y \times I \times I | y \in U_i \cap U_j \} \cong \bigsqcup_{(i,j) \in I^2} U_i \cap U_i$$ $${\rm{Obj}}({{\mathcal Q}}) := \{(y,i,F) \in Y \times I \times P{{\mathcal U}}({\mathcal{H}})| y \in U_i \} \cong \bigsqcup_{i \in I} U_i \times P{{\mathcal U}}({\mathcal{H}})$$ $${\rm{Mor}}({{\mathcal Q}}): = \{(y,i,j,F) \in Y \times I \times I \times P{{\mathcal U}}({\mathcal{H}}) | y \in U_i \cap U_j \} \cong \bigsqcup_{(i,j) \in I^2} U_i \cap U_i\times P{{\mathcal U}}({\mathcal{H}})$$ and whose structural maps for ${\mathcal{Y}}$ are $${\mathsf{source}}(y,i,j)=(y,i), \ \ \ {\mathsf{target}}(y,i,j)=(y,j), \ \ \ {\mathsf{inverse}}(y,i,j)=(y,j,i),$$ $${\mathsf{comp}}((y,i,j),(y,j,k))=(y,i,k) \ \ \ {\rm{and}} \ \ {\mathsf{identity}}(y,i) = (y,i,i),$$ and for ${{\mathcal Q}}$ are $${\mathsf{source}}(y,i,j,F)=(y,i,F), \ \ \ {\mathsf{target}}(y,i,j,F)=(y,j,\rho_{ji}(y)F),$$ $${\mathsf{inverse}}(y,i,j,F)=(y,j,i,\rho_{ji}(y)F), \ \ {\mathsf{identity}}(y,i,F) = (y,i,i,F)$$ $${\rm{and}} \ \ \ {\mathsf{comp}}((y,i,j,F),(y,j,k,\rho_{ji}(y)F))=(y,i,k,F).$$
The forgetful functor $\beta:{{\mathcal Q}}\to {\mathcal{Y}}$, $(y,i,j,F) \mapsto (y,i,j)$ induces a map at the level of the geometric realizations that makes the map $|{{\mathcal Q}}| \stackrel{\beta}{\to} |{\mathcal{Y}}|$ into a projective unitary bundle.
Now define the functors $\Phi: {{\mathcal Q}}\to {{\mathcal D}}_K$ and $\phi: {\mathcal{Y}}\to
{\mathcal{C}}_K$ that at the level of morphisms are respectively $$\Phi(y,i,j,F) := (f_i, F, \rho_{ji}(y)F), \ \ \ \phi(y,i,j) := (f_i, \rho_{ij}(y))$$ (the equations in (\[compatibility of functions\]) imply that $\Phi$ and $\phi$ are indeed functors), which at the level of morphisms make the following diagram commute $$\xymatrix{
{{\mathcal Q}}\ar[d]^\beta \ar[r]^\Phi & {{\mathcal D}}_K \ar[d]^\gamma & (y,i,j,F) \ar@{|->}[d]^\beta \ar@{|->}[r]^\Phi & (f_i,F,\rho_{ji}(y)F) \ar@{|->}[d]^\gamma \\
{\mathcal{Y}}\ar[r]^\phi & {\mathcal{C}}_K & (y,i,j) \ar@{|->}[r]^\phi & (f_i,
\rho_{ij}(y)). }$$
Endowing the category ${{\mathcal Q}}$ with the left $K$-action $$K \times {{\mathcal Q}}\to {{\mathcal Q}}, \ \ (g,(y,i,j,F)) \mapsto (y,i,j,f_i(g)F)$$ we see that the functor $\Phi$ is $K$-equivariant. Therefore we get the at the level of the geometric realizations $$\xymatrix{P{{\mathcal U}}({\mathcal{H}}) \ar[d] & P{{\mathcal U}}({\mathcal{H}}) \ar[d] \\
|{{\mathcal Q}}| \ar[d]^{|\beta|} \ar[r]^{|\Phi|} & |{{\mathcal D}}_K| \ar[d]^{|\gamma|} \\
|{\mathcal{Y}}| \ar[r]^{|\phi|} & |{\mathcal{C}}_K| . }$$ we obtain a map of projective unitary stable $K$-equivariant bundles.
Following the procedure defined in [@Segal Section 4], from a partition of unity subordinated to the open cover $\{U_i\}_{\{i \in I\}}$ one can construct maps $\Theta, \theta$ $$\xymatrix{
Q \ar[r]^{\Theta} \ar[d] & |{{\mathcal Q}}| \ar[d]^{|\beta|} \\
Y \ar[r]^{\theta} & |{\mathcal{Y}}| .
}$$ such that $\Theta$ is a map of projective unitary stable $K$-equivariant bundles, and moreover such that $\Theta$ and $\theta$ are homotopy equivalences.
We have then constructed a map of projective unitary stable $K$-equivariant bundles $$\xymatrix{
Q \ar[r]^{|\Phi| \circ \Theta} \ar[d] & |{{\mathcal D}}_K| \ar[d]^{|\gamma|} \\
Y \ar[r]^{|\phi| \circ \theta} & |{\mathcal{C}}_K| . }$$ which by taking $\alpha:=|\phi| \circ \theta$ implies the proposition.
Let $Y$ be a trivial $K$-space, then $${\ensuremath{{\mathrm{Bun}}}}_{st}^K(Y, P{{\mathcal U}}({\mathcal{H}}))
=[Y , EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}}))].$$
Standard arguments in the classification of principal bundles (see [@Segal Section 3], [@Milnor1; @Milnor2]), together with Theorem \[theorem classifying map for trivial K spaces\] implies that $${\ensuremath{{\mathrm{Bun}}}}_{st}^K(Y, P{{\mathcal U}}({\mathcal{H}})) =[Y , |{\mathcal{C}}_K|].$$
Now, using Theorem \[theorem classifying map for trivial K spaces\] once again for the bundle in (\[universal K bundle\]) we get a map of projective unitary stable $K$-equivariant bundles $$\xymatrix{
EP{{\mathcal U}}({\mathcal{H}}) \times Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})) \ar[r] \ar[d] & |{{\mathcal D}}_K| \ar[d]^{|\gamma|} \\
EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})) \ar[r] &
|{\mathcal{C}}_K| . }$$ which at the level of the horizontal maps are homotopy equivalences [@Segal]. Therefore we could take the bundle on the left hand side of the previous diagram as our universal projective unitary stable $K$-equivariant bundle for trivial $K$ spaces. This finishes the proof.
Gluing of local objects
-----------------------
We have seen in Section \[subsection system of fix points\] that there is a standard procedure in order to construct a $G$-space whose $K$-fixed point set has a prescribed homotopy type. For the purpose of this section we would need to construct a contravariant functor from the category ${\mathcal{O}}_G^P$ of canonical orbits to the category of topological spaces, which at each orbit type $G/K$ specifies a space which classifies the isomorphism classes of projective unitary $N(K)$ equivariant bundles over trivial $K$-spaces endowed with a $N(K)/K$ actions
The first choice that comes to one’s mind is to make the assignment $$G/K \mapsto EP{{\mathcal U}}({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})}
Hom_{st}(K,P{{\mathcal U}}({\mathcal{H}})),$$ but unfortunately this assignment fails to provide a contravariant functor from ${\mathcal{O}}_G^P$ to topological spaces. One way to fix the problem is to choose spaces with the same homotopy type for each $G/K$, with the extra property that the functoriality of the category ${\mathcal{O}}_G^P$ is encoded in the selection. Let us see how this selection can be carried out.
Consider the action groupoid $G \ltimes G/K$ associated to the left action of $G$ on $G/K$, i.e. the space of objects is $G/K$ and the space of morphisms is $G \times G/K$ with ${\mathsf{source}}(g,h[K])
= h[K] $ and ${\mathsf{target}}(g,h[K]) = gh[K]$.
Note that the functor $$K \to G\ltimes G/K, \ \ \ k \mapsto (k, [K])$$ from the category defined by the group $K$ to $G \ltimes G/K$ is an equivalence of categories (Morita equivalence of groupoids as in [@Moerdijk-Madison]).
One would expect that this equivalence of categories would induce another equivalence of categories obtained by the restriction functor, from the category $$Funct_{st}(G\ltimes G/K, P{{\mathcal U}}({\mathcal{H}}))\rtimes Maps(G/K, P{{\mathcal U}}({\mathcal{H}}))$$ whose objects are stable functors from $G\ltimes G/K$ to $P{{\mathcal U}}({\mathcal{H}})$ $$\begin{aligned}
Funct_{st}(G\ltimes G/K,
P{{\mathcal U}}({\mathcal{H}})) := &\\
\{F \in Funct(G\ltimes G/K, & P{{\mathcal U}}({\mathcal{H}})) \colon F|_K \in Hom_{st}(K,
P{{\mathcal U}}({\mathcal{H}}))\},\end{aligned}$$ and whose morphisms are defined by natural transformations, to the category $$Hom_{st}(K, P{{\mathcal U}}({\mathcal{H}})) \rtimes P{{\mathcal U}}({\mathcal{H}}).$$ Unfortunately again, this is not true in general for topological groups (see Section \[remark equivalence of categories\]), but it is the case when the group $G$ is discrete (see Proposition \[proposition equivalence of categories\] ), let us see why.
Consider the projective unitary $G$-equivariant bundle $$P \stackrel{p}{\to} G/K$$ and let us suppose that there is a trivialization of $P$ $$\phi :
P \stackrel{\cong}{\to} P{{\mathcal U}}({\mathcal{H}}) \times G/K$$ as a unitary projective bundle. Note that such trivialization always exists in the case that the group $G$ is discrete.
Associated to the trivialization $\phi$ define the map $\psi: G
\times G/K \to P{{\mathcal U}}({\mathcal{H}})$ by $$\begin{aligned}
\label{functor psi}\psi(g,k[H]):= \pi_1(\phi(gx))
\pi_1(\phi(x))^{-1}\end{aligned}$$ where $\pi_1$ is the projection on the first coordinate, and $x$ is any point in $P$ such that $p(x)=k[K]$.
The map $\psi$ is a functor from $G\ltimes G/K$ to $P{{\mathcal U}}({\mathcal{H}})$.
The map $\psi$ is well defined; because if we take $y =xF$ with $F
\in P{{\mathcal U}}({\mathcal{H}})$, since $\phi$ is a map of principal bundles, we have that $\phi(xF)= \phi(x)F$, and therefore $$\pi_1(\phi(gy)) \pi_1(\phi(y))^{-1}=\pi_1(\phi(gx))F \left(\pi_1(\phi(x))F
\right)^{-1}=\pi_1(\phi(gx)) \pi_1(\phi(x))^{-1}.$$
By definition we have that $\psi(1,k[H])=1$; and the composition follows from $$\begin{aligned}
\psi(h,gk[K]) \psi(g,k[K]) & = & \pi_1(\phi(hgx))
\pi_1(\phi(gx))^{-1} \pi_1(\phi(gx)) \pi_1(\phi(x))^{-1}\\
&= & \pi_1(\phi(hgx)) \pi_1(\phi(x))^{-1} \\
&=& \psi(hg,k[K]).\end{aligned}$$
If we have two different trivializations $\phi_1,\phi_2: P \cong
P{{\mathcal U}}({\mathcal{H}}) \times G/K$, they are related via a gauge transformation $\sigma: G/K \to P{{\mathcal U}}({\mathcal{H}})$ by the equation $$\sigma(p(x))= \pi_1(\phi_1(x)) \pi_1(\phi_2(x))^{-1}.$$
Therefore the associated functors $\psi_1$ and $\psi_2$ are related by the natural transformation $\sigma$ in the following way $$\sigma(gk[K])^{-1} \psi_1(g,k[K]) \sigma(k[K]) =
\psi_2(g,k[K])$$ which can be written as $\sigma^{-1} \psi_1
\sigma= \psi_2$.
If we denote the category $$\widetilde{{\mathcal{C}}}_{G/K}: =
Funct_{st}(G\ltimes G/K, P{{\mathcal U}}({\mathcal{H}}))\rtimes Maps(G/K, P{{\mathcal U}}({\mathcal{H}}))$$ and recalling the category ${\mathcal{C}}_K$ from \[section universal for trivial K spaces\] we can conclude
\[proposition equivalence of categories\] Let us suppose that for any local object $P
\to G/K$ there exists a trivialization $P \cong P{{\mathcal U}}({\mathcal{H}}) \times
G/K$ as principal bundles, then the restriction functor $R:\widetilde{{\mathcal{C}}}_{G/K} \to {\mathcal{C}}_K$ which to any functor $\psi$ assigns the homomorphism $R(\psi)(k):= \psi(k,[K])$, is an equivalence of categories.
By definition of $\widetilde{{\mathcal{C}}}_{G/K}$, any stable functor restricts to a stable homomorphism. Let us see that the functor is essentially surjective.
For $f : K \to P{{\mathcal U}}({\mathcal{H}})$ a stable homomorphism, consider $$P= P{{\mathcal U}}({\mathcal{H}}) \times_f G := P{{\mathcal U}}({\mathcal{H}}) \times_K G$$ where $K$ acts on $P{{\mathcal U}}({\mathcal{H}})$ via the homomorphism $f$. By hypothesis $P$ is trivializable via $\phi
: P \stackrel{\cong}{\to} P{{\mathcal U}}({\mathcal{H}}) \times G/K$ and therefore the associated functor $\psi : G \ltimes G/K \to P{{\mathcal U}}({\mathcal{H}})$ is defined by the formula (\[functor psi\]). The restriction $R(\psi)$ is a homomorphism which is conjugate to $f$, and therefore the functor $R$ is essentially surjective.
The maps on morphisms $$Hom_{\widetilde{{\mathcal{C}}}_{G/K}} (\psi_1, \psi_2) \to Hom_{{{\mathcal{C}}}_{K}} (R(\psi_1),
R(\psi_2))$$ are bijective, as any natural transformation is defined by its value in $[K]$: $$\sigma(g[K]) = \psi_1(g,[K]) \sigma([K]) \psi_2(g,[K])^{-1}.$$ Therefore the restriction functor is moreover full and faithful, hence it gives an equivalence of categories.
\[corollary local objects trivial imply equivalence of categories\] If all local objects $P \to G/K$ trivialize as $P{{\mathcal U}}({\mathcal{H}})$-principal bundles, then the restriction functor induces a homotopy equivalence $$R: |\widetilde{{\mathcal{C}}}_{G/K}| \stackrel{\simeq}{\to} |{\mathcal{C}}_K|.$$
### Triviality as principal bundles of the local objects {#remark equivalence of categories}
The triviality of the local objects as principal $P{{\mathcal U}}({\mathcal{H}})$-bundles may be characterized in topological terms in the following form.
Let $P \to G/K$ be a projective unitary stable $G$-equivariant bundle over $G/K$ for $K$ compact. We have seen that $$P \cong P|_{[K]} \times_K G \cong P{{\mathcal U}}({\mathcal{H}}) \times_f G$$ as projective unitary stable $G$-equivariant bundles where $K$ acts on $P{{\mathcal U}}({\mathcal{H}})$ through the stable homomorphism $f: K \to
P{{\mathcal U}}({\mathcal{H}})$.
We have already seen that the homotopy class of the map $Bf: BK
\to BP{{\mathcal U}}({\mathcal{H}})$ determines the isomorphism class of the bundle defined by $f$, and that the isomorphism class of its associated bundle over $G/K$ is determined by the homotopy class of a map $F
: EG \times_G G/K \to BP{{\mathcal U}}({\mathcal{H}})$ defined using a homotopy equivalence $EG \times_G G/K \stackrel{\simeq}{\to} BK$. In this framework we can see that the bundle $P$ is trivial after forgetting the $G$ action, if the composition of the maps $$G/K \to EG \times_G G/K \to BP{{\mathcal U}}({\mathcal{H}})$$ is homotopy trivial; or in other words that the homomorphism in third cohomology groups $$\begin{aligned}
\label{homomorphism in
third cohomology} H^3_G(G/K, {\ensuremath{{\mathbb Z}}}) \to H^3(G/K,
{\ensuremath{{\mathbb Z}}})\end{aligned}$$ is trivial.
In the case that the group $G$ is discrete, the orbit types are also discrete and therefore any bundle over $G/K$ is trivializable. When the group $G$ is a Lie group there is no reason to expect that the homomorphism of (\[homomorphism in third cohomology\]) is trivial in general. On the contrary, applying the Eilenberg-Moore spectral sequence to the fibration $G/K \to BK \to BG$ one gets the isomorphism [@Wolf] $$H^*(G/K, {\ensuremath{{\mathbb Z}}}) \cong {\mathrm{Tor}}_{C^*(BG, {\ensuremath{{\mathbb Z}}})}( {\ensuremath{{\mathbb Z}}},
C^*(BK, {\ensuremath{{\mathbb Z}}})),$$ that in principle would make one guess that there are several cases on which the homomorphism in (\[homomorphism in third cohomology\]) is not trivial.
Take for example the case of $G=SU(3)$ and $K=SO(3)$ whose quotient space is $X_{-1}=SU(3)/SO(3)$ which is known as the Wu manifold. In this case $H_2(X_{-1}, {\ensuremath{{\mathbb Z}}}) = {\ensuremath{{\mathbb Z}}}/2$ (see [@Barden]) which implies that $H^3(X_{-1}, {\ensuremath{{\mathbb Z}}}) =
{\ensuremath{{\mathbb Z}}}/2$. Applying the Serre spectral sequence to the fibration $G/K \to EG \times_K G \to BG $ we get that the only non trivial term of the second page at total degree three is $$E_2^{0,3}=H^0(BSU(3), H^3(X_{-1}, {\ensuremath{{\mathbb Z}}}))= {\ensuremath{{\mathbb Z}}}/2,$$ which moreover survives to the infinity page $E_\infty^{0,3} =
{\ensuremath{{\mathbb Z}}}/2$. This fact implies that the homomorphism in (\[homomorphism in third cohomology\]) is an isomorphism $$H^3(BSO(3), {\ensuremath{{\mathbb Z}}}) \stackrel{\cong}{\to} H^3(SU(3)/SO(3),
{\ensuremath{{\mathbb Z}}})= {\ensuremath{{\mathbb Z}}}/2.$$
From the previous discussion we see that we can improve Proposition \[proposition equivalence of categories\] in the following way.
\[proposition restriction functor R is equivalence of categories\] The restriction functor $R
:\widetilde{{\mathcal{C}}}_{G/K} \to {\mathcal{C}}_K$ is an equivalence of categories, if and only if the following homomorphism is trivial $$H^3_G(G/K, {\ensuremath{{\mathbb Z}}}) \to
H^3(G/K,{\ensuremath{{\mathbb Z}}}).$$
We have seen that the local triviality of the local objects is equivalent to the triviality of the homomorphism, and moreover that it implies that the functor $R$ is an equivalence of categories.
Now, if the functor $R$ is essentially surjective, then any stable homomorphism $f: K \to P{{\mathcal U}}({\mathcal{H}})$ is the restriction of a stable functor $\psi: G \ltimes G/K \to P{{\mathcal U}}({\mathcal{H}})$ , and therefore the bundle $P{{\mathcal U}}({\mathcal{H}}) \times_f G$ can be trivialized with the information of the functor $\psi$.
The functor $R :\widetilde{{\mathcal{C}}}_{SU(3)/SO(3)} \to {\mathcal{C}}_{SO(3)}$ is not an equivalence.
Once one has classified all projective unitary stable $G$-equivariant bundles $P \to G/K$ over the orbit types $G/K$, one may proceed to glue all these bundles via the information provided by their gauge groups. When the bundle $P \to G/K$ is trivializable as a $P{{\mathcal U}}({\mathcal{H}})$-bundle, we have shown that the category $\widetilde{{\mathcal{C}}}_{G/K}$ contains all the relevant information to perform the gluing, and since we would like to construct the universal equivariant projective unitary stable bundle using the categories $\widetilde{{\mathcal{C}}}_{G/K}$, we restric to the case on which $G$ is discrete and the groups $K$ are finite. The general case on which $G$ is not discrete will not be done in this paper since it would take us away from the main focus of this work which is the study of the category of functors from $G \ltimes G/K $ to $P{{\mathcal U}}({\mathcal{H}})$.
Universal equivariant projective unitary stable bundle for proper and discrete actions {#chapter universal bundle}
======================================================================================
In this chapter we construct a universal model for the equivariant projective unitary stable bundles for proper actions of a discrete group $G$. Hence throughout this chapter the group $G$ is discrete and ${\mathcal{O}}_P^G$ has for objects the sets $G/K$ for all $K$ finite subgroups of $G$.
\[section universal bundle\]
A model for the universal equivariant projective unitary bundle for the orbit type $G/K$
----------------------------------------------------------------------------------------
We will proceed in a similar way as in Section \[section universal for trivial K spaces\] by constructing a category $\widetilde{{{\mathcal D}}}_{G/K}$ together with a functor $\widetilde\gamma: \widetilde{{{\mathcal D}}}_{G/K} \to \widetilde{{\mathcal{C}}}_{G/K}$, whose geometric realization provides a concrete description of the pullback of the $P{{\mathcal U}}({\mathcal{H}})$-bundle $|\gamma|:|{{\mathcal D}}_K| {\to} |{\mathcal{C}}_K|$ along the restriction map $R: |\widetilde{{\mathcal{C}}}_{G/K}| \to |{\mathcal{C}}_K|$.
Let $\widetilde{{{\mathcal D}}}_{G/K}$ be the category whose morphisms ${\rm{Mor}}({\widetilde{{{\mathcal D}}}_{G/K}})$ are $$Funct_{st}(G\ltimes G/K,
P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}})\times Maps(G/K,
P{{\mathcal U}}({\mathcal{H}})),$$ whose objects are $$Funct_{st}(G\ltimes
G/K, P{{\mathcal U}}({\mathcal{H}})) \times P{{\mathcal U}}({\mathcal{H}})$$ and whose structural maps are defined by $$\begin{aligned}
{\mathsf{source}}(\psi, F, \sigma)& = &(\psi, F)\\
{\mathsf{target}}(\psi, F, \sigma)& =&
(\sigma F^{-1} \psi F \sigma^{-1}, \sigma([K]))\end{aligned}$$ $$\begin{aligned}
{\mathsf{comp}}((\psi, F, \sigma),(\sigma F^{-1}
\psi F \sigma^{-1}, \sigma([K]), \delta)) =
(\psi, F, & \delta \sigma([K])^{-1} \sigma ).\end{aligned}$$
Let us consider the functor that forgets the $P{{\mathcal U}}({\mathcal{H}})$ coordinate $$\widetilde{\gamma} : \widetilde{{{\mathcal D}}}_{G/K} \to \widetilde{{\mathcal{C}}}_{G/K}$$ that at the level of morphisms and objects is defined by $$\widetilde{\gamma}(\psi, F, \sigma):= (\psi,F \sigma^{-1}), \
\ \widetilde{\gamma}(\psi, F):= \psi.$$
\[proposition bundle for G/K\] The bundle $$P{{\mathcal U}}({\mathcal{H}}) \to |\widetilde{{{\mathcal D}}}_{G/K}|
\stackrel{|\widetilde{\gamma}|}{\to} |\widetilde{{\mathcal{C}}}_{G/K}|$$ is a projective unitary stable $N(K)$-equivariant bundle.
Since $\widetilde{{\mathcal{C}}}: G/K \mapsto \widetilde{{\mathcal{C}}}_{G/K}$ is a functor from the category of canonical orbits ${\mathcal{O}}_G^P$ to the category of small categories we have an induced left action of ${\rm{Mor}}_{{\mathcal{O}}_G^P}(G/K,G/K) = N(K)/K$ on $\widetilde{{\mathcal{C}}}_{G/K}$ as well as on its geometric realization. On the objects of $\widetilde{{\mathcal{C}}}_{G/K}$ the left $N(K)/K$-action is explicitly given by $$n \cdot \psi= \psi^n,\text{ with } \psi^n(g,k[K]) = \psi(g,kn[K]),$$ while on morphisms it is given by $n \cdot (\psi, \sigma) = (\psi^n, \sigma^n)$, where $\sigma^n= \sigma \circ r_n$ and $r_n: G/K \to G/K$ is the $G$-equivariant map with $r_n(gK)=gnK$. A straightforward calculation shows that with the previous action, the group $N(K)/K$ acts on the category $\widetilde{{\mathcal{C}}}_{G/K}$.
Let us now endow the category $\widetilde{{{\mathcal D}}}_{G/K}$ with a right $P{{\mathcal U}}({\mathcal{H}})$ action and with a left $N(K)$ action, commuting one with each other.
The left $N(K)$ action $N(K) \times \widetilde{{{\mathcal D}}}_{G/K} \to
\widetilde{{{\mathcal D}}}_{G/K}$ at the level of morphisms it is defined by $$n \cdot (\psi, F, \sigma) := (\psi^n, \psi(n,[K])F,
\sigma^n F^{-1} \psi(n,[K]) F)$$ and at the level of objects by $$n \cdot (\psi, F) := (\psi^n, \psi(n,[K]) F);$$ and it is a simple calculation to show that indeed it is an action (which follows from the explicit calculation done in ) and that it commutes with the composition of the category.
The right $P{{\mathcal U}}({\mathcal{H}})$ action $\widetilde{{{\mathcal D}}}_{G/K} \times
P{{\mathcal U}}({\mathcal{H}}) \to \widetilde{{{\mathcal D}}}_{G/K}$ can be defined at the level of morphisms $$(\psi, F, \sigma) \cdot L := (\psi, FL,\sigma L)$$ and it is a straightforward calculation to show that it is indeed an action, that commutes with the composition of the category, and that it commutes with the $N(K)$ action.
By definition of the functor $\widetilde{\gamma}$ the induced $P{{\mathcal U}}({\mathcal{H}})$ action on $\widetilde{{\mathcal{C}}}_{G/K}$ is trivial, and the induced $N(K)$ action on $\widetilde{{\mathcal{C}}}_{G/K}$ coincides with the $N(K)/K$-action on $\widetilde{{\mathcal{C}}}_{G/K} $ referred to at the beginning of the proof.
The stability of the action of the subgroup $K$ on the fibers is satisfied by construction. Therefore we have that the bundle $$P{{\mathcal U}}({\mathcal{H}}) \to |\widetilde{{{\mathcal D}}}_{G/K}| \stackrel{|\widetilde{\gamma}|}{\to}
|\widetilde{{\mathcal{C}}}_{G/K}|$$ is a unitary stable $N(K)$-equivariant bundle.
A model for the universal equivariant projective unitary stable bundle
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The individual spaces $|\widetilde{{{\mathcal D}}}_{G/K}|$ assemble like the restrictions of a projective unitary stable $G$-equivariant bundle to the fixed point sets of the base. We will glue them together accordingly in order to obtain a universal projective unitary stable $G$-equivariant bundle.
Let $\widetilde{\mathcal{O}}_G^P$ denote the following category. Its objects are copies of $G$, one for each canonical proper orbit $G/K$, and we will write $G_{G/H}$ for the copy of $G$ which belongs to a proper orbit $G/H$. The set of morphisms from $G_{G/H}$ to $G_{G/K}$ is given by $${\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G_{G/H},G_{G/K}) = \{m \in G\ |\ m^{-1}Hm \subset K\}.$$ A morphism $m\in {\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G/H,G/K)$ should be thought of as a $G$-equivariant map from $G$ to $G$ given through right multiplication by $m$, hence it covers the induced map $r_m: G/H \to G/K, gH \mapsto gmK$. Consequently the composition $n \circ m$ of two morphisms $m\in {\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G/H,G/K)$ and $n\in {\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G/K,G/L)$ is defined as the right multiplication by $m \cdot n$.
There is a canonical functor $\pi: \widetilde{\mathcal{O}}_G^P \to {\mathcal{O}}_G^P$, which maps the object $G_{G/K}$ to $G/K$, and which maps a morphism $m\in {\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G_{G/H},G_{G/K})$ to $r_m: G/H \to G/K$, i.e. for every morphism $m$ the functor $\pi$ decodes the commutative diagram $$\xymatrix{G_{G/H} \ar[r]^{m} \ar[d] & G_{G/K} \ar[d]
\\ G/H \ar[r]^{r_m} & G/K.}$$
The categories $\widetilde{{{\mathcal D}}}_{G/K}$ provide a contravariant functor $\widetilde{{{\mathcal D}}}$ from ${\widetilde{\mathcal{O}}_G^P}$ to the category of small categories which on the level of objects is given by $$\widetilde{{{\mathcal D}}}(G_{G/K}) = \widetilde{{{\mathcal D}}}_{G/K},$$ while for a morphism $m\in{\rm Mor}_{\widetilde{\mathcal{O}}_G^P}(G_{G/H},G_{G/K})$ the corresponding map $$\widetilde{{{\mathcal D}}}(m): \widetilde{{{\mathcal D}}}_{G/K} \to \widetilde{{{\mathcal D}}}_{G/H}$$ is given at the level of morphisms by $$\widetilde{{{\mathcal D}}}(m) (\psi, F, \sigma) = (\psi^m, \psi(m,[K])F, \sigma^m F^{-1}\psi(m,[K])F )$$ and at the level of objects by $$\widetilde{{{\mathcal D}}}(m) (\psi, F) = (\psi^m, \psi(m,[K])F)$$ where $\psi^m$ and $\sigma^m$ are defined by the formulas $$\begin{aligned}
\psi^m(g,h[H]) & := & \psi(g,hm[K])\\
\sigma^m(h[H]) & := & \sigma(hm[K]).\end{aligned}$$
Note that $\widetilde{{{\mathcal D}}}$ respects composition in $\widetilde{\mathcal{O}}_G^P$ since for two composable morphisms $m:G_{G/H}
\to G_{G/K}$ and $n: G_{G/K} \to G_{G/L}$ and a morphism $(\psi,F,\sigma)$ in $\widetilde{{\mathcal D}}_{G/L}$ we have that $$\begin{aligned}
\nonumber \widetilde{{{\mathcal D}}}(m) \circ \widetilde{{{\mathcal D}}}(n) & (\psi, F, \sigma)\\
\nonumber = & \widetilde{{{\mathcal D}}}(m) (\psi^n, \psi(n,[L])F, \sigma^n F^{-1}\psi(n,[L])F)\\
\nonumber = & \left((\psi^n)^m, \psi^n(m,[K])\psi(n,[L])F, \right. \\
\label{explicit calculation} & \left. (\sigma^n)^m
F^{-1}\psi(n,[L])F
(\psi(n,[L])F)^{-1}\psi^n(m,[K])\psi(n,[L])F\right)\\
\nonumber = & \left(\psi^{m \cdot n}, \psi(m, n[L]) \psi(n,[L])
F, \sigma^{m\cdot n}F^{-1}\psi(m,n[L])\psi(n,[L])F\right)\\
\nonumber = & \left(\psi^{m \cdot n}, \psi(m \cdot n,[L]) F,
\sigma^{m\cdot n}F^{-1}\psi(m \cdot n,[L]) F\right)\\
= & \widetilde{{{\mathcal D}}}(n \circ m)
(\psi, F, \sigma). \nonumber\end{aligned}$$ Note also that there is the antihomomorphism $${Mor}_{\widetilde{\mathcal{O}}_G^P}(G_{G/K},G_{G/K}) \to N(K),\ \ \ \ \ \ n \circ m \mapsto m \cdot n,$$ and that the contravariant action of ${\rm{Mor}}_{\widetilde{\mathcal{O}}_G^P}(G_{G/K},G_{G/K})$ on $\widetilde{{{\mathcal D}}}_{G/K}$ coincides with the left $N(K)$ action introduced in the proof of Proposition \[proposition bundle for G/K\].
The restriction functors $R':\widetilde{{\mathcal D}}_{G/K} \to {{\mathcal D}}_K$ and $R:\widetilde{\mathcal{C}}_{G/K} \to {\mathcal{C}}_K$ induced by the inclusion $K
\ltimes K/K \hookrightarrow G \ltimes G/K$ are compatible with the functors $\widetilde\gamma$ and $\gamma$ respectively, so that we have a commutative diagram $$\xymatrix{|\widetilde{{\mathcal D}}_{G/K}| \ar[r]^{R'} \ar[d] & |{{\mathcal D}}_K|\ar[d]
\\ |\widetilde{\mathcal{C}}_{G/K}| \ar[r]^{R} & |{\mathcal{C}}_K|.}$$ The diagram is a map of $K$-equivariant $P{{\mathcal U}}({\mathcal{H}})$-bundles, hence in particular the bundle $|\widetilde\gamma|: |\widetilde{{\mathcal D}}_{G/K}| \to |\widetilde{\mathcal{C}}_{G/K}|$ is a pullback of the projective unitary stable $K$-equivariant bundle $|\gamma|: |{{\mathcal D}}_K| \to |{\mathcal{C}}_K|$.
Moreover, since the restriction functor $R:\widetilde{\mathcal{C}}_{G/K} \to
{\mathcal{C}}_K$ is an equivalence of categories (see Proposition \[proposition equivalence of categories\]), the induced map $R:|\widetilde{\mathcal{C}}_{G/K}| \to
|{\mathcal{C}}_K|$ is a homotopy equivalence and therefore the bundle $|\widetilde{{\mathcal D}}_{G/K}|\to |\widetilde{\mathcal{C}}_{G/K}|$ is another universal bundle for projective unitary $K$-equivariant bundles of $K$-trivial spaces. The main difference between the two bundles $|\widetilde{{\mathcal D}}_{G/K}|$ and $|\widetilde{{\mathcal D}}_K|$ is that on the first we have a canonical way to extend the $K$ action to the group $N(K)$, meanwhile on the second we do not know whether the action could be extended in a canonical way. It is precisely this extension property which later on will allow us to glue together the various $|\widetilde{{\mathcal D}}_{G/K}|$ to obtain $G$-space.
The various functors $\widetilde\gamma: \widetilde{{\mathcal D}}_{G/K} \to \widetilde{\mathcal{C}}_{G/K}$ provide a natural transformation $\widetilde\gamma$ from $\widetilde{{\mathcal D}}$ to $\widetilde{\mathcal{C}}\circ \pi$. Now for constructing the universal bundle consider the covariant functors which at the level of objects are given by $$\begin{aligned}
\widetilde\nabla & : \widetilde{\mathcal{O}}_G^P \to Spaces,\ G_{G/K} \mapsto G\\
\nabla & : {\mathcal{O}}_G^P \to Spaces, \ G/K \mapsto G/K\end{aligned}$$ and at the level of morphisms are the $G$-equivariant maps defined by multiplication on the right.
Assemble the covariant functors $\widetilde{\nabla} $ and $\nabla$ with the contravariant functors $|\widetilde{{{\mathcal D}}}|$ and $|\widetilde{{\mathcal{C}}}|$ respectively in order to get spaces that we denote $${{\mathcal E}}:= |\widetilde{{{\mathcal D}}}| \times_{h\widetilde{\mathcal{O}}_G^P} \widetilde{\nabla} \ \ \ {\rm{and}}
\ \ \ \ {{\mathcal B}}:=|\widetilde{{\mathcal{C}}}| \times_{h{\mathcal{O}}_G^P} \nabla$$ where we have that $|\widetilde{{{\mathcal D}}}|(G_{G/K}):=
|\widetilde{{{\mathcal D}}}_{G/K}|$ and $|\widetilde{{\mathcal{C}}}|({G/K}):=
|\widetilde{{\mathcal{C}}}_{G/K}|$.
The natural transformation $\widetilde\gamma: \widetilde{{\mathcal D}}\to \widetilde{\mathcal{C}}\circ \pi$ provides a map $$|\widetilde{{{\mathcal D}}}| \times_{h\widetilde{\mathcal{O}}_G^P} \widetilde{\nabla}
\longrightarrow |\widetilde{{\mathcal{C}}}| \times_{h{\mathcal{O}}_G^P} \nabla$$ which makes $$P{{\mathcal U}}({\mathcal{H}}) \to {{\mathcal E}}\to {{\mathcal B}}$$ into a projective unitary stable $G$-equivariant bundle.
\[theorem the universal bundle\] The bundle $$P{{\mathcal U}}({\mathcal{H}}) \to {{\mathcal E}}\to {{\mathcal B}}$$ is a universal projective unitary stable $G$-equivariant bundle for proper $G$-actions.
The bundles $$P{{\mathcal U}}({\mathcal{H}}) \to |\widetilde{{{\mathcal D}}}_{G/K}| \times_{N(K)}
G \to |\widetilde{{\mathcal{C}}}_{G/K}| \times_{N(K)/K} G/K$$ are projective unitary stable $G$-equivariant bundles, and therefore the bundle ${{\mathcal E}}\to {{\mathcal B}}$ is also a projective unitary stable $G$-equivariant bundle since we defined the spaces ${{\mathcal E}}$ and ${{\mathcal B}}$ via a homotopy colimit.
Let us check the universality: take a projective unitary stable $G$-equivariant bundle $P \to X$ over the proper $G$-space $X$ and take any point $x \in X$. By Definition \[def projective unitary G-equivariant stable bundle\] we know that there exists a $G$-invariant neighborhood $V$ of $x$ and a $G_x$-contractible slice $U$ such that $$\alpha: P|_{V} \stackrel{\cong}{\to} (P{{\mathcal U}}({\mathcal{H}}) \times U) \times_{G_{x}} G$$ where $G_x$ acts on $P{{\mathcal U}}({\mathcal{H}})$ via a stable homomorphism $f$. Contracting $U$ to the point $x$, and trivializing the bundle in the bottom with the map $\phi$ we get the diagram of projective unitary stable bundles $$\xymatrix{
P|_{V}\ar[r]^(0.3){\alpha}_(0.3)\cong & (P{{\mathcal U}}({\mathcal{H}}) \times U)
\times_{G_{x}} G
\ar[d]^p &\\
& P{{\mathcal U}}({\mathcal{H}}) \times_{f} G \ar[r]^\phi & P{{\mathcal U}}({\mathcal{H}}) \times G/G_x. }$$
Define the map $$P|_{V} \to Obj(\widetilde{{{\mathcal D}}}_{G/G_{x}}), \ \ \ z \mapsto (\psi,
\pi_1 (\phi (p (\alpha(z)))))$$ where $\psi : G \ltimes G/K \to
P{{\mathcal U}}({\mathcal{H}})$ is the functor induced by $\phi$ constructed in , and note that this map is a $G$-equivariant map of projective unitary stable bundles.
Changes on the trivialization $\phi$ are parameterized by the gauge group ${Maps}(G/G_x, P{{\mathcal U}}({\mathcal{H}}))$ and they are controlled by the morphisms of the category $\widetilde{{{\mathcal D}}}_{G/G_{x}}$, and changes on the base point $x$ are parameterized by the morphisms in $\widetilde{{\mathcal{O}}}_G^P$. The composition of a change of base point together with a change of trivialization induce a path in the space ${{\mathcal E}}$. Therefore the standard argument of the classification of principal bundles permits us to conclude that by choosing a [*[good $G$ cover]{}*]{} of $X$, we can find a $G$-equivariant map of projective unitary stable bundles which makes the diagram into a pullback diagram $$\xymatrix{P \ar[r] \ar[d] & {{\mathcal E}}\ar[d]
\\ X \ar[r] & {{\mathcal B}}.}$$
We have that the space ${{\mathcal B}}$ is the base for the universal projective unitary stable $G$-equivariant bundle and therefore we have that the isomorphism classes of projective unitary stable bundles over a proper $G$-space are in 1-1 correspondence with homotopy classes of maps to the space ${{\mathcal B}}$.
The homotopy type of the classifying space of equivariant projective unitary stable bundles
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Let us do a first attempt in trying to obtain a homotopy equivalent space which relates to equivariant cohomology as in Corollary \[corollary equivalence of homotopy type for G\]. Taking the Borel construction of the bundle ${{\mathcal E}}\to {{\mathcal B}}$ we obtain a projective unitary bundle $$P{{\mathcal U}}({\mathcal{H}}) \to {{\mathcal E}}\times_G EG \to {{\mathcal B}}\times_G EG$$ whose classifying map to $BP{{\mathcal U}}({\mathcal{H}})$ induces an adjoint map $$\Psi:{{\mathcal B}}\to Maps(EG, BP{{\mathcal U}}({\mathcal{H}}))$$ which is $G$-equivariant. This map restricted to the fixed point set of a finite group $K$ $$\begin{aligned}
\label{homotopy equivalence of B with Maps for K finite}
\Psi^K: {{\mathcal B}}^K \stackrel{\simeq}{\to} Maps(EG, BP{{\mathcal U}}({\mathcal{H}}))^K \cong Maps (BK,BP{{\mathcal U}}({\mathcal{H}}))\end{aligned}$$ is a homotopy equivalence; this follows from Corollary \[corollary equivalence of homotopy type for G\] and the homotopy equivalences $$|\widetilde{{\mathcal{C}}}_{G/K}| \simeq |\widetilde{{\mathcal{C}}}_{G/K}| \times_{N(K)/K} E(N(K)/K)
\stackrel{\simeq}{\to} {{\mathcal B}}^K$$ .
When the subgroup $K$ is not finite (the orbit type $G/K$ is not an object in ${\mathcal{O}}_G^P$) the fixed point set ${{\mathcal B}}^K$ is empty, meanwhile $Maps(BK,BP{{\mathcal U}}({\mathcal{H}}))$ is far from being empty. Therefore the map $\Psi$ is not in general a $G$-homotopy equivalence, but in the case that $G$ is finite we have just shown that $\Psi$ induces a $G$-homotopy equivalence.
\[theorem homotopy equivalence for finite group\] The $G$-equivariant map $$\Psi : {{\mathcal B}}\to Maps(EG, BP{{\mathcal U}}({\mathcal{H}})),$$ which is the adjoint of the classifying map of the projective unitary bundle ${{\mathcal E}}\times_G EG \to {{\mathcal B}}\times_G EG$, is a $G$-homotopy equivalence in the case that $G$ is finite.
Now let us see the case on which $G$ might not be finite.
\[definition category MM\] Let $M : {\mathcal{O}}_G^P \to Spaces$ be the contravariant functor that is defined on objects by $$M(G/K):=Maps(G/K \times_G EG, BP{{\mathcal U}}({\mathcal{H}}))$$ and that to any morphism $\alpha \in {\rm{Mor}}_{{\mathcal{O}}_G^P}(G/K,
G/H)$ the induced morphism $M(\alpha): M(G/H) \to M(G/K)$ is obtained by the composition with the induced map $G/K \times_G EG
\to G/H \times_G EG$.
Define the space ${\mathcal{M}}:= M
\times_{h{\mathcal{O}}_G^P} \nabla$.
By the classification of principal bundles, there exist horizontal maps making the following diagram commutative $$\xymatrix{
\left(|\widetilde{{{\mathcal D}}}_{G/K}| \times_{N(K)} G\right)\times_G EG \ar[r] \ar[d] &
EP{{\mathcal U}}({\mathcal{H}})
\ar[d] \\
\left(|\widetilde{{\mathcal{C}}}_{G/K}| \times_{N(K)/K} G/K\right) \times_G EG \ar[r] &
BP{{\mathcal U}}({\mathcal{H}}).}$$ By taking the adjoint map of the bottom horizontal map, we get a $N(K)/K$ equivariant map $$|\widetilde{{\mathcal{C}}}_{G/K}| \to Maps(G/K \times_G EG, BP{{\mathcal U}}({\mathcal{H}}))$$ for all orbit types; these maps can be assembled into a natural transformation of functors between the functors $|\widetilde{{\mathcal{C}}}|$ and $M$ and therefore we get a $G$-equivariant map $$\Psi: {{\mathcal B}}\to {\mathcal{M}}.$$
\[proposition BB = MM\] The map $\Psi: {{\mathcal B}}\to {\mathcal{M}}$ is a $G$-equivariant homotopy equivalence.
Let $K$ be a finite subgroup of $G$, and let us consider restriction of $\Psi$ to the fixed point set of the group $K$. We obtain the following diagram $$\xymatrix{ {{\mathcal B}}^K \ar[r]^{\Phi^K} & {\mathcal{M}}^K \\
|\widetilde{{\mathcal{C}}}_{G/K}| \ar[u]_\simeq \ar[d]^\simeq \ar[r] &
Maps(G/K \times_G
EG, BP{{\mathcal U}}({\mathcal{H}})) \ar[u]_\simeq \ar[d]^\simeq\\
|{\mathcal{C}}_K| \ar[r]^(0.3){\simeq} & Maps(BK,BP{{\mathcal U}}({\mathcal{H}})) }$$ where the top vertical arrows are homotopy equivalences induced by the inclusion of the spaces associated to the orbit type $G/K$, the bottom vertical arrows are homotopy equivalences induced by restriction (see Corollary \[corollary local objects trivial imply equivalence of categories\]), and the bottom horizontal arrow is a homotopy equivalence as it was shown in Corollary \[corollary equivalence of homotopy type for G\].
Therefore $\Psi$ induces a homotopy equivalence for all finite subgroups of $G$ which by Theorem 7.4 of [@DavisLueck] implies that $\Psi$ is a $G$-equivariant homotopy equivalence.
Note that from the construction of the functor $M$ and from Corollary \[corollary equivalence of homotopy type for G\] we see that ${\mathcal{M}}$ is a classifying space for the degree 3 cohomology of finite groups of $G$, namely, for all orbit types $G/K$ for $K$ finite, we have the isomorphism in homotopy groups $$\pi_i((Maps(G/K,{\mathcal{M}})^G) \cong H^{3-i}(BK,{\ensuremath{{\mathbb Z}}}).$$
Classification of projective unitary stable equivariant bundles por proper and discrete actions
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We have seen in that the map $\Psi : {{\mathcal B}}\to Maps(EG, BP{{\mathcal U}}({\mathcal{H}}))$ is a homotopy equivalence once restricted to the fixed point set of a finite subgroup $K$ of $G$. This fact is precisely what is needed in order to prove the following theorem.
For $X$ a proper $G$-space, the map $\Psi$ induces a bijective map between the isomorphism classes of projective unitary stable $G$-equivariant bundles over $X$ and the elements of the third $G$-equivariant cohomology group of $X$, i.e. $$\widetilde{\Psi}: {\ensuremath{{\mathrm{Bun}}}}_{st}^G(X,P{{\mathcal U}}({\mathcal{H}})) \stackrel{\cong}{{\longrightarrow}} H^3(X \times_G EG; {\ensuremath{{\mathbb Z}}}).$$ Therefere, the isomorphism classes of projective unitary stable $G$-equivariant bundles over $X$ are classified by the elements in $H^3(X \times_G EG, {\ensuremath{{\mathbb Z}}})$.
Consider the following isomorphisms $$\begin{aligned}
{\ensuremath{{\mathrm{Bun}}}}_{st}^G(X,P{{\mathcal U}}({\mathcal{H}}))& \cong &
\pi_0(Maps(X,{{\mathcal B}})^G)\\
&\cong &\pi_0(Maps(X,Maps(EG,BP{{\mathcal U}}({\mathcal{H}})))^G) \\
& \cong & \pi_0(Maps(X\times_G EG, BP{{\mathcal U}}({\mathcal{H}}))) \\
& \cong & H^3(X \times_G EG, {\ensuremath{{\mathbb Z}}}).
\end{aligned}$$ where the isomorphism of the first line follows from Theorem \[theorem the universal bundle\], the isomorphism of the third line follows from the compact open topology, and the isomorphism in the fourth line follows from the fact that $BP{{\mathcal U}}({\mathcal{H}})$ is an Eilenberg-Maclane $K({\ensuremath{{\mathbb Z}}},3)$ space. We are left with the isomorphism of the second line.
In view of the results of section \[subsection system of fix points\], let us consider the map $${\rm Hom}_{{\mathcal{O}}_G^P}(\Phi X, \Phi {{\mathcal B}}) \to {\rm Hom}_{{\mathcal{O}}_G^P}(\Phi X, \Phi Maps(EG,BP{{\mathcal U}}({\mathcal{H}}))$$ induced by the map $\Psi$. At the orbit type $G/K$ with $K$ finite, we get that the map $${\rm Maps}(X^K , {{\mathcal B}}^K) \stackrel{\simeq}{\to} {\rm Maps}(X^K , Maps(EG,BP{{\mathcal U}}({\mathcal{H}})^K), \ \ f \mapsto \Psi^K \circ f$$ induces a homotopy equivalence since we know that the map $\Psi^K$ defined in induces a homotopy equivalence. Therefore we can conclude that the map $${\rm Hom}_{{\mathcal{O}}_G^P}(\Phi X, \Phi {{\mathcal B}}) \stackrel{\simeq}{{\longrightarrow}} {\rm Hom}_{{\mathcal{O}}_G^P}(\Phi X, \Phi Maps(EG,BP{{\mathcal U}}({\mathcal{H}}))$$ is a homotopy equivalence, and this implies that the map $$Maps(X,{{\mathcal B}})^G) \stackrel{\simeq}{{\longrightarrow}} Maps(X,Maps(EG,BP{{\mathcal U}}({\mathcal{H}})))^G$$ is also a homotopy equivalence. This shows the isomorphism of the second line. Hence we obtain the desired isomorphism $${\ensuremath{{\mathrm{Bun}}}}_{st}^G(X,P{{\mathcal U}}({\mathcal{H}})) \cong H^3(X \times_G EG; {\ensuremath{{\mathbb Z}}}).$$
Twisted equivariant K-theory for proper actions, definition and properties {#appendix Twisted equivariant K-theory}
==========================================================================
In this appendix we show that the twisted equivariant K-theory defined as the homotopy groups of bundles of Fredholm operators satisfies the axioms of an equivariant generalized cohomology theory. We first give the definition of the twisted equivariant K-theory for proper actions of Lie groups twisted by equivariant projective unitary bundles, and then we show that these groups satisfy the axioms of a generalized equivariant cohomology theory in the sense of [@Lueck1], namely that this generalized cohomology theory is endowed with induction and restriction structures.
Representing space for K-theory
-------------------------------
To define the twisted equivariant K-theory we need to use the appropriate representing space of K-theory defined by Fredholm operators, with the extra property that the group $P{{\mathcal U}}({\mathcal{H}})$ endowed with the compactly generated compact open topology acts continuously on it by conjugation.
The space ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$ of Fredholm operators on the Hilbert space ${\mathcal{H}}$ endowed with the norm topology is a representing space for K-theory, but the group $P{{\mathcal U}}({\mathcal{H}})$ with the compactly generated compact open topology does not act continuously on ${\ensuremath{{\mathrm{Fred}}}}({\mathcal{H}})$. Atiyah and Segal in [@AtiyahSegal] construct an alternative space ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ representing K-theory by Fredholm operators and on which $P{{\mathcal U}}({\mathcal{H}})$ acts continuously. Let us recall the definition of ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$.
[@AtiyahSegal Chapter 3] Let ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ consist of pairs $(A,B)$ of bounded operators on ${\mathcal{H}}$ such that $AB -1$ and $BA -1$ are compact operators. Endow ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ with the topology induced by the embedding $$\begin{aligned}
{\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}}) & \to & {\mathsf{B}}({\mathcal{H}}) \times {\mathsf{B}}({\mathcal{H}})
\times {\mathsf{K}}({\mathcal{H}})
\times {\mathsf{K}}({\mathcal{H}}) \\
(A,B) & \mapsto & (A,B,AB-1, BA-1)\end{aligned}$$ where ${\mathsf{B}}({\mathcal{H}})$ is the bounded operators on ${\mathcal{H}}$ with the compact open topology and ${\mathsf{K}}({\mathcal{H}})$ is the compact operators with the norm topology.
Proposition 3.1 of [@AtiyahSegal] shows that ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ is a representing space for K-theory and moreover that ${{\mathcal U}}({\mathcal{H}})_{c.o.}$ with the compact open topology acts continuously on ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ by conjugation. It is a simple exercise in topology to show that the continuity of the ${{\mathcal U}}({\mathcal{H}})_{c.o.}$ action implies the continuity of the ${{\mathcal U}}({\mathcal{H}})_{c.g.}$ action; therefore we can conclude that the group $P{{\mathcal U}}({\mathcal{H}})$ acts continuously on ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ by conjugation.
Let us choose the identity operator $({\rm{Id}},{\rm{Id}})$ as the base point in ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$.
Definition of Twisted Equivariant K-theory for proper actions
-------------------------------------------------------------
Let $X$ be a proper $G$ space and $P \to X$ a projective unitary stable $G$-equivariant bundle over $X$. Recall that the space of Fredholm operators defined above is endowed with a continuous right action of the group $P{{\mathcal U}}({\mathcal{H}})$ by conjugation, therefore we could take the associated bundle over $X$ $${\ensuremath{{\mathrm{Fred}}}}(P) := P \times_{P{{\mathcal U}}({\mathcal{H}})} {\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$$ with fibres ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$ with the induced $G$ action given by $$g \cdot [(\lambda, (A,B))] := [(g \lambda, (A,B))]$$ for $g$ in $G$, $\lambda$ in $P$ and $(A,B)$ in ${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$.
Denote by $$\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P))$$ the space of sections of the bundle ${\ensuremath{{\mathrm{Fred}}}}(P) \to X$ and choose as base point in this space the section which chooses the base point on each fiber. This section exists because the $P{{\mathcal U}}({\mathcal{H}})$ action on $({\rm{Id}},{\rm{Id}})$ is trivial, and therefore $$X \cong P/P{{\mathcal U}}({\mathcal{H}}) \cong P
\times_{P{{\mathcal U}}({\mathcal{H}})} \{({\rm{Id}},{\rm{Id}}) \} \subset {\ensuremath{{\mathrm{Fred}}}}(P);$$ let us denote this [*[identity section]{}*]{} by $s$.
The group $G$ acts on $\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P))$ in the natural way, namely for $g \in G$ and $\sigma$ a section $(g \cdot \sigma)(x) := g \sigma( g^{-1}x)$ and therefore the fixed point set $$\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P))^G$$ is precisely the space of $G$-equivariant sections. Note that the base point in $\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P))$ is fixed by $G$ because the identity operators commute with all operators, and therefore the space of $G$-equivariant sections has also a base point.
\[definition K-theory of X,P\] Let $X$ be a connected $G$-space and $P$ a projective unitary stable $G$-equivariant bundle over $X$. The [*[Twisted $G$-equivariant K-theory]{}*]{} groups of $X$ twisted by $P$ are defined as $$K^{-n}_G(X;P) := \pi_n \left( \Gamma(X;{\ensuremath{{\mathrm{Fred}}}}(P))^G, s \right)$$ where the base point $s$ is the identity section.
For an inclusion $j: A \to X$ of $G$-spaces we have a restriction map on invariant sections $$\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P))^G \to \Gamma(A; {\ensuremath{{\mathrm{Fred}}}}(P|_A))^G$$ which induces homomorphisms at the level of the homotopy groups, hence homomorphisms in the twisted equivariant K-theories $$j^* : K^*_G(X;P) \to K^*_G(A;P|_A).$$
The relative K-theory groups for the pair $(X,A)$ twisted by the bundle $P$ over $X$, whenever $X$ is connected, will be defined as the homotopy groups of the homotopy fiber of the restriction map $$j^*:\Gamma(X; {\ensuremath{{\mathrm{Fred}}}}(P)) \to \Gamma(A; {\ensuremath{{\mathrm{Fred}}}}(P|_A)).$$
The homotopy fiber of the restriction map can be written in terms of the mapping cylinder of the inclusion $j:A \to X$ $${\rm{Cyl}}(X,A) = (X \sqcup A \times [0,1]) / a \sim (a,0) \ \ \forall a \in A,$$ as the space of relative sections $$\Gamma({\rm{Cyl(X,A)}},A\times\{1\};{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))$$ where the relative sections for an inclusion $B \to Y$ are defined as $$\Gamma(Y,B;{\ensuremath{{\mathrm{Fred}}}}(Q)) := \{ \sigma \in \Gamma(Y; {\ensuremath{{\mathrm{Fred}}}}(Q)) \colon \sigma|_B = s \}.$$
\[definition relative twisted k-theory groups\] The [*[Relative Twisted Equivariant K-theory groups]{}*]{} of the triple $(X,A;P)$ are the groups $$K^{-n}_G(X,A;P) := \pi_n \left(\Gamma({\rm{Cyl(X,A)}},A\times \{1\};{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))^G
\right).$$
If the space $X$ is a disjoint union of $G$-spaces $X=\bigsqcup_\alpha
X_\alpha$, we define the Twisted Equivariant K-theory groups as $$K^{-n}_G(X, A;P) := \prod_\alpha K^{-n}_G(X_\alpha, A \cap X_\alpha;P|_{X_\alpha}).$$
Properties of the Twisted Equivariant K-theory groups
-----------------------------------------------------
### Functoriality
Let us consider the category ${\it
prop}\; G-\mathcal{CW}^{2}_{\rm twist}$ whose objects are triples $(X,A;P)$ consisting of an inclusion $A \to X$ of proper $G$-CW spaces, together with a projective unitary stable $G$-equivariant bundle $P \to X$, and whose morphisms $f : (Y,B;Q) \to (X,A; P)$ consist of $G$ equivariant maps $f : Q \to P$ of principal $P{{\mathcal U}}({\mathcal{H}})$-bundles, inducing an equivariant map $\overline{f}:(Y,B) \to (X,A)$ on the bases such that the diagram $$\xymatrix{ Q \ar[r]^f \ar[d] & P \ar[d] \\
Y \ar[r]^{\overline{f}} & X }$$ is a pullback diagram
Then the map $f$ induces a pullback diagram at the level of the associated bundles $$\xymatrix{ {\ensuremath{{\mathrm{Fred}}}}(Q) \ar[r]^f \ar[d] & {\ensuremath{{\mathrm{Fred}}}}(P) \ar[d] \\
Y \ar[r]^{\overline{f}} & X }$$ which implies that any section on ${\ensuremath{{\mathrm{Fred}}}}(P)$ defines a unique section on ${\ensuremath{{\mathrm{Fred}}}}(Q)$. Thus the map $f$ induces an equivariant map at the level of sections $$f^\#: \Gamma(X;{\ensuremath{{\mathrm{Fred}}}}(P)) \to \Gamma(Y;{\ensuremath{{\mathrm{Fred}}}}(Q))$$ and therefore $f$ induces a homomorphism at the level of the homotopy groups of the $G$-invariant sections $$f^*: \pi_n \left( \Gamma(X;{\ensuremath{{\mathrm{Fred}}}}(P))^G \right) \to \pi_n \left( \Gamma(Y;{\ensuremath{{\mathrm{Fred}}}}(Q))^G \right)$$ which produces the desired homomorphism in Twisted K-theory groups $$f^*: K^{-n}_G(X;P) \to K^{-n}_G(Y; Q).$$
The relative case follows the same principle as we have the following pullback diagram $$\xymatrix{ {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(Q,Q|_B)) \ar[r]^f \ar[d] & {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)) \ar[d] \\
{\rm{Cyl}}(Y,B) \ar[r] & {\rm{Cyl}}(X,A)}$$ which induces an equivariant map in relative sections $$\Gamma({\rm{Cyl(X,A)}},A;{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))
\to \Gamma({\rm{Cyl(Y,B)}},B;{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(Q,Q|_B)))$$ inducing the desired homomorphism in twisted equivariant K-theory groups $$f^*: K^{-n}_G(X,A;P) \to K^{-n}_G(Y,B; Q).$$
This implies that the twisted equivariant K-theory groups provide a functor from ${\it prop}\; G-\mathcal{CW}^{2}_{\rm twist}$ to graded abelian groups $$K_G^{*}: {\it prop}\; G-\mathcal{CW}^{2}_{\rm
twist} \to \mbox{Graded abelian groups}.$$
### Twisted Equivariant K-theory as a generalized cohomology theory
The twisted equivariant K-theory groups satisfy the axioms of a generalized cohomology theory.
- [**[Homotopy axiom:]{}**]{} Two morphisms $f_0, f_1 :(Y,B;Q) \to
(X,A;
P)$ in ${\it prop}\; G-\mathcal{CW}^{2}_{\rm
twist}$ are homotopy equivalent if there exists a left $G$-equivariant and right $P{{\mathcal U}}({\mathcal{H}})$ equivariant homotopy $F: Q
\times I \to P$ such that $F(\_,0) = f_0$ and $F(\_,1) = f_1$, and moreover that the diagram $$\xymatrix{ Q\times I \ar[r]^F \ar[d] & P \ar[d] \\
Y \times I \ar[r]^{\overline{F}} & X }$$ is a pullback square.
The fact that the square above is a pullback square implies that there is an induced map on relative sections $$\begin{aligned}
\Gamma({\rm{Cyl(X,A)}},A;{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A))) \to & \\
\Gamma({\rm{Cyl(Y,B)}} \times I, &B \times
I; {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(Q,Q|_B)\times I)) \end{aligned}$$ whose adjoint can be seen as the desired homotopy $$\Gamma(X,A; {\ensuremath{{\mathrm{Fred}}}}(P))\times I \stackrel{F^\#}{\to} \Gamma({\rm{Cyl(Y,B)}} , B
; {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(Q,Q|_B)))$$ between the induced maps $f_0^\#$ and $f_1^{\#}$.
Therefore the homomorphisms $$f_0^*,f_1^*:K^{-n}_G(X,A;P) \to
K^{-n}_G(Y,B;Q)$$ are equal.
- [**[Additivity axiom:]{}**]{} Follows by Definition \[definition relative twisted k-theory groups\].
- [**[Excision axiom:]{}**]{} Let $Z\subset X$ be an open, $G$-invariant subset such that the closure of $Z$ is contained in the interior of $A$. Then the restriction map induced by $(X-Z, A-Z)\to (X,A)$ induces a homeomorphism of spaces of relative sections $$\begin{aligned}
\Gamma({\rm{Cyl}}(X,A),A \times I;{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))
\stackrel{\cong}{\to} & \nonumber \\
\Gamma({\rm{Cyl}}(X-Z,A-Z),(A-Z) \times
I; & {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P|_{X-Z},P|_{X-A})))
\label{homeomorphism for relative sections}
\end{aligned}$$ because any section $\sigma$ of the bundle ${\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P|_{X-Z},P|_{X-A}))$ which restricts to the base point in $(A-Z) \times I$, can be uniquely extended to a section $\overline{\sigma}$ in ${\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A))$ by defining $\overline{\sigma}_{A \times I}:=s$ and $\overline{\sigma}|_{{\rm{Cyl}}(P|_{X-Z},P|_{X-A})}:=\sigma$.
Now, since the inclusion of $A$ into $X$ is a $G$-cofibration the inclusion of pairs of spaces $$({\rm{Cyl}}(X,A),A
\times\{1\}) \to ({\rm{Cyl}}(X,A),A \times I)$$ is a $G$-homotopy equivalence, and therefore the induced map on the spaces of relative sections is also a $G$-homotopy equivalence, $$\begin{aligned}
\Gamma({\rm{Cyl}}(X,A),A \times I;{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))
\stackrel{\simeq}{\to} & \\
\Gamma({\rm{Cyl}} (X,A),A \times \{1\}; &
{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A))).\end{aligned}$$
The previous argument can also be carried out for the pair $(X-Z,
A-Z)$ yielding a $G$-homotopy equivalence $$\begin{aligned}
\Gamma({\rm{Cyl}}(X-Z,A-Z),(A-Z) \times
I; {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P|_{X-Z},P|_{X-A})))
\stackrel{\simeq}{\to} & \\
\Gamma({\rm{Cyl}}(X-Z,A-Z),(A-Z) \times
\{1\}; {\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P|_{X-Z}, P|_{X-A} & ))).\end{aligned}$$ Therefore, the homotopies outlined above, together with the homeomorphism of (\[homeomorphism for relative sections\]) implies that there is an isomorphism of relative groups $$K_{G}^{n}(X,A;P) \cong K_{G}^{n}(X-Z, A-Z; P|_{X-Z}).$$
- [**[Long Exact Sequence axiom for pairs:]{}**]{} We have defined the relative twisted equivariant K-theory groups as the homotopy groups of the relative sections on the mapping cylinder $$K^{-n}_G(X,A;P) := \pi_n \left(\Gamma({\rm{Cyl(X,A)}},A\times \{1\};{\ensuremath{{\mathrm{Fred}}}}({\rm{Cyl}}(P,P|_A)))^G
\right).$$
The relative sections of the mapping cylinder is weakly homotopicaly equivalent to the homotopy fiber of the restriction map $$\Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P)) \to \Gamma(A; {\ensuremath{{\mathrm{Fred}}}}(P|_A)).$$ Therefore the long exact sequence on homotopy groups induces the long exact sequence for the twisted equivariant K-theory groups $$\to K_G^n(X,A;P) \to K_G^n(X;P) \to K_G^n(A,P|_A) \to K^{n+1}(X,A;P) \to.$$
### Bott periodicity
The existence of a homotopy equivalence $${\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}}) \stackrel{\simeq}{\to} \Omega^2 {\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})$$ which is ${{\mathcal U}}({\mathcal{H}})$-equivariant (see Section 2 in [@Karoubi]), yields isomorphisms $$K^{-n}_G(X,A;P) \stackrel{\cong}{\to} K^{-n-2}_G(X,A;P)$$ which makes the twisted equivariant K-theory groups into a ${\ensuremath{{\mathbb Z}}}$-graded 2-periodic cohomology if we define the positive twisted equivariant K-theory groups by the information on $K^0$ and $K^{-1}$; namely, for $p>0$ we define $$K_G^p(X,A;P) = \left\{
\begin{array}{ccl}
K_G^0(X,A;P) & {\rm {if}} & p \ \mbox{ is even}\\
K_G^{-1}(X,A;P) & {\rm {if}} & p \ \mbox{ is odd}\\
\end{array}\right.$$
### Twisted Equivariant K-theory over $G/K$
Let $P \to G/K$ be a projective unitary stable $G$-equivariant bundle over $G/K$ for $K$ finite subgroup of $G$, and recall that $$P \cong P{{\mathcal U}}({\mathcal{H}}) \times_K G$$ as equivariant bundles where $K$ acts on $P{{\mathcal U}}({\mathcal{H}})$ by the stable homomorphism $f:K \to P{{\mathcal U}}({\mathcal{H}})$.
Therefore we have the index map $$\pi_0\left( \Gamma(G/K;{\ensuremath{{\mathrm{Fred}}}}(P) \right) = \pi_0 \left( {\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}})^K
\right) \stackrel{\cong}{\to} R_{S^{1}}(\widetilde{K})$$ where the second is obtained by the index map and $R_{S^{1}}(\widetilde{K})$ denotes the Grothendieck group of the semi-group $\widetilde{K}=f^*{{\mathcal U}}({\mathcal{H}})$ of isomorphism classes representations on which $S^1= Ker(\widetilde{K} \to K)$ acts by multiplication. The index map is an isomorphism. It surjective because the $K$ action on $P{{\mathcal U}}({\mathcal{H}})$ is stable, and the injectivity follows from the $G$-equivariant contractibility of ${{\mathcal U}}({\mathcal{H}})$.
We can conclude that the twisted equivariant K-theory groups for the orbit type $G/K$ twisted by $P \to G/K$ are: $$K_G^0(G/K;P) \cong R_{S^{1}}(\widetilde{K}), \ \ \ \
K^{-1}_G(G/K;P)=0.$$
### Induction structure
The twisted equivariant K-theory groups for proper actions can be endowed with an [*[Induction structure]{}*]{} as it is defined in Section 1 of [@Lueck1]. Let $\alpha: H \to G$ be a group homomorphism and $X$ be a $H$-proper space such that $ker(\alpha)$ acts freely on $X$. Let us denote by $X
\times_\alpha G$ the quotient space $(X \times G)/H$ where the action is defined by $$H \times (X \times G ) \to X \times G \ \ \
\ \ h(x,l) \mapsto (hx,l\alpha(h)^{-1}),$$ and endow the space $X
\times_\alpha G$ with the left $G$ action defined by $g[x,l]:=
[x,gl]$. Then we must show that there exists natural graded isomorphisms $${\rm{ind}}_\alpha: K^*_H(X,A;P) \stackrel{\cong}{\to} K_G^*(X
\times_\alpha G, A\times_\alpha G ;P \times_\alpha G)$$ which are functorial with respect to homomorphisms of groups $\beta:G \to K$ on which $ker(\beta)$ acts trivially, that induce isomorphisms at the level of the long exact sequences of the pairs $(X,A)$ and $(X
\times_\alpha G,A \times_\alpha G)$, and moreover that are compatible with respect to conjugation; these three conditions will follow from the following lemmas.
\[lemma induction subgroup\] Let $H \subset G$, $X$ be a proper $H$-space and $P$ a projective unitary stable $H$-equivariant bundle over $X$. Then spaces of invariant sections $$\Gamma(X,{\ensuremath{{\mathrm{Fred}}}}(P))^H \cong \Gamma(X \times_H G, {\ensuremath{{\mathrm{Fred}}}}(P
\times_H G))^G$$ are homeomorphic, and therefore we have an isomorphism $${\rm{ind}}_H^G : K^*_H(X,P) \stackrel{\cong}{\to} K_G^*(X
\times_H G, {\ensuremath{{\mathrm{Fred}}}}(P \times_H G)).$$
Since the left $H$-action commutes with the right $P{{\mathcal U}}({\mathcal{H}})$ action on $P$ we have that $$\begin{aligned}
{\ensuremath{{\mathrm{Fred}}}}(P)\times_H G & =
& \left({\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} P \right)\times_H G \\&= &
{\ensuremath{{\mathrm{Fred}}}}'({\mathcal{H}}) \times_{P{{\mathcal U}}({\mathcal{H}})} \left( P \times_H G \right) \\ & =
& {\ensuremath{{\mathrm{Fred}}}}(P \times_H G).\end{aligned}$$
The $G$-invariant sections in ${\ensuremath{{\mathrm{Fred}}}}(P) \times_H G$ are determined uniquely by the $H$ invariant sections of ${\ensuremath{{\mathrm{Fred}}}}(P) \times_H H$ and therefore the restriction map induces a homeomorphism $$R:\Gamma(X \times_H G, {\ensuremath{{\mathrm{Fred}}}}(P)
\times_H G)^G \stackrel{\cong}{\to} \Gamma(X\times_H H,{\ensuremath{{\mathrm{Fred}}}}(P)
\times_H H)^H.$$
Now, the canonical map $X \to X \times_H H$, $x \mapsto [(x,1)]$ is an $H$-equivariant homomorphism and it induces an homeomorphism $$\phi:\Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P))^H \stackrel{\cong}{\to} \Gamma(X\times_H H,{\ensuremath{{\mathrm{Fred}}}}(P) \times_H
H)^H.$$
On the level of homotopy groups the homeomorphism $$\phi^{-1} \circ R: \Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P))^H \stackrel{\cong}{\to}
\Gamma(X \times_H G, {\ensuremath{{\mathrm{Fred}}}}(P)
\times_H G)^G$$ induces the desired isomorphism $${\rm{ind}}_H^G : K^*_H(X,P) \stackrel{\cong}{\to} K_G^*(X
\times_H G,P \times_H G).$$
\[lemma induction normal free action\] Let $X$ be a proper $H$-space together and $P$ a projective unitary stable $H$-equivariant bundle over $X$. Let $N \subset H$ be a normal subgroup of $H$ acting freely on $X$. Then there is a canonical homeomorphism between the spaces of invariant sections $$\Gamma(X,{\ensuremath{{\mathrm{Fred}}}}(P))^H \cong \Gamma(X/N , {\ensuremath{{\mathrm{Fred}}}}(P/N))^{H/N}$$ which induces an isomorphism $${\rm{inv}}_H^N: K^*_H(X,P) \stackrel{\cong}{\to} K_{H/N}^*(X/N
,P/N).$$
We have the following homeomorphisms $$\begin{aligned}
\Gamma(X,{\ensuremath{{\mathrm{Fred}}}}(P))^H & = & \left( \Gamma(X,{\ensuremath{{\mathrm{Fred}}}}(P))^N
\right)^{H/N} \\
& \cong & \Gamma(X/N,{\ensuremath{{\mathrm{Fred}}}}(P)/N)^{H/N} \\
& \cong & \Gamma(X/N,{\ensuremath{{\mathrm{Fred}}}}(P/N))^{H/N}\end{aligned}$$ where the homeomorphism from the first line to the second follows from the fact that $N$ acts freely on $X$, and the homeomorphism from the second line to the third follows from the facts that $N$ acts freely on $P$ and that the $H$ action commutes with the $P{{\mathcal U}}({\mathcal{H}})$-action.
The composition of the homeomorphisms above induces the desired isomorphism in the twisted K-theory groups $${\rm{inv}}_H^N: K^*_H(X,P) \stackrel{\cong}{\to} K_{H/N}^*(X/N
,P/N).$$
For a homomorphism $\alpha : H \to G$ such that $N := ker(\alpha)$ acts freely on the $H$-proper space $X$ we have the following diagram of homomorphisms $$\xymatrix{ H \ar[rr]^\alpha \ar[rd]^p & & G \\
& H/N \ar@{^{(}->}[ru]^{\overline{\alpha}}}$$ which induce the following homeomorphisms $$\xymatrix{ \Gamma(X, {\ensuremath{{\mathrm{Fred}}}}(P))^H \ar[d]^\cong\\
\Gamma(X/N,{\ensuremath{{\mathrm{Fred}}}}(P/N))^{H/N} \ar[d]^\cong \\
\Gamma(X/N \times_{H/N} G, {\ensuremath{{\mathrm{Fred}}}}(P/N \times_{H/N} G))^G \ar[d]^\cong \\
\Gamma(X \times_{\alpha} G, {\ensuremath{{\mathrm{Fred}}}}(P \times_{\alpha} G))^G}$$ where the first comes from Lemma \[lemma induction normal free action\], the second from Lemma \[lemma induction subgroup\] and the third from the canonical $G$-equivariant homeomorphism $$X \times_\alpha G \stackrel{\cong}{\to} X/N \times_{H/N} G, \ \
[(x,g)] \mapsto [([x],g)].$$
The compositions of the homeomorphisms described above gives us the desired isomorphism $${\rm{ind}}_\alpha : = {\rm{ind}}_{H/N}^G \circ {\rm{inv}}^N_H :
K^*_H(X;P) \stackrel{\cong}{\to} K_G^*(X\times_\alpha G,
P\times_\alpha G).$$
Since the induction structure comes from explicit homeomorphisms at the level of invariant sections, we claim that the twisted equivariant K-theory groups possess an induction structure as it is defined in Section 1 of [@Lueck1]. We will not reproduce the proofs here.
From Lemma \[lemma induction normal free action\] we also obtain the following relation between the twisted equivariant K-theory groups and non-equivariant twisted K-theory groups.
Let $X$ be a free $G$ space and let $P$ be a projective unitary stable $G$-equivariant bundle over $X$. Then there a canonical isomorphism $$K_G^*(X;P) \stackrel{\cong}{\to} K^*(X/G;P/G)$$ between the $P$-twisted equivariant K-theory groups of $X$ and the $P/G$-twisted K-theory groups of $X/G$.
### Discrete torsion twistings
In this last section we would like to describe the relation between the twisted equivariant K-theory groups defined in this paper and the twisted equivariant K-theory groups defined via projective representations characterized by discrete torsion as it is defined in [@Dwyer].
For any cohomology class $\beta \in H^3(BG, {\ensuremath{{\mathbb Z}}})$ denote by $F_\beta: EG \to BP{{\mathcal U}}({\mathcal{H}})$ a $G$-invariant map whose homotopy class represents $\beta$.
If $X$ is a proper $G$-space we consider the map $$X^K \to Maps(EG \times G/K, BP{{\mathcal U}}({\mathcal{H}}))^G, \ \ x \mapsto
F_\beta \circ \pi_1$$ at the level of fixed point sets for all $K$ compact subgroups of $G$.
This map can be assembled into a $G$-equivariant map $$\Psi_\beta: X \to {\mathcal{M}}$$ where ${\mathcal{M}}$ is the space defined in Definition \[definition category MM\]. By Proposition \[proposition BB = MM\] we know that the map $\Psi_\beta$ induces a map $\overline{\Psi}_\beta: X \to {{\mathcal B}}$ and therefore $$(\overline{\Psi}_\beta)^* {{\mathcal E}}\to X$$ is a projective unitary $G$-equivariant stable bundle.
We claim that the $\beta$-twisted $G$-equivariant K-theory groups of $X$ defined in [@Dwyer] are isomorphic to the twisted $G$-equivariant K-theory groups $$K^*_G(X,(\overline{\Psi}_\beta)^* {{\mathcal E}}).$$
The proof will be postponed to a forthcoming publication where the appropriate tools for proving such fact, and many others, will be developed.
[^1]: The first author was partially supported by the grant SFB 878 “Groups, geometry and actions" and the Hausdorff Center for Mathematics. The first and fourth author were partially supported with funds from the Leibniz prize of Prof. Dr. Wolfgang Lück. Part of this work was done while the fourth author was financially supported by the Alexander Von Humboldt Foundation.
|
---
abstract: 'A numerical equivalence class of $k$-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We develop several geometric criteria that distinguish big classes from boundary classes. In particular, we construct for arbitrary cycle classes an analogue of the volume function for divisors.'
address: |
Department of Mathematics, Rice University\
Houston, TX 77005
author:
- Brian Lehmann
bibliography:
- 'mobility.bib'
nocite: '[@*]'
title: Geometric characterizations of big cycles
---
[^1]
Introduction
============
Let $X$ be an integral projective variety over an algebraically closed field. We will let $N_{k}(X)$ denote the vector space of numerical classes of $k$-cycles on $X$ with $\mathbb{R}$-coefficients. The pseudo-effective cone $\operatorname{\overline{Eff}}_{k}(X) \subset N_{k}(X)$ is defined to be the closure of the cone generated by all effective $k$-cycles. Classes that lie in the interior of the cone – known as big classes – are expected to exhibit special geometric properties. Our goal is to give geometric characterizations of big cycles similar to the well-known criteria for codimension $1$ cycles and dimension $1$ cycles.
An important tool for understanding big divisor classes is the volume function. The volume of a Cartier divisor $L$ is the asymptotic rate of growth of dimensions of sections of $L$. More precisely, if $X$ has dimension $n$, $$\operatorname{vol}(L) := \limsup_{m \to \infty} \frac{\dim H^{0}(X,\mathcal{O}_{X}(mL))}{m^{n}/n!}.$$ It turns out that the volume is an invariant of the numerical class of $L$ and satisfies many advantageous geometric properties. On a smooth variety $X$, divisors with positive volume are precisely the divisors with big numerical class.
One can interpret the volume of a divisor $L$ as an asymptotic measurement of the number of general points contained in members of $|mL|$ as $m$ increases. [@delv11] suggests studying a similar notion for arbitrary cycles. Let $N_{k}(X)_{\mathbb{Z}} \subset N_{k}(X)$ and $N_{k}(X)_{\mathbb{Q}} \subset N_{k}(X)$ denote the subsets generated by cycles with $\mathbb{Z}$-coefficients and $\mathbb{Q}$-coefficients respectively. Given a class $\alpha \in N_{k}(X)_{\mathbb{Z}}$, one easily verifies that there is a constant $C$ such that a cycle with class $m\alpha$ can pass through at most $Cm^{\frac{n}{n-k}}$ general points of $X$. The mobility function identifies the best possible constant $C$. (See Definition \[mobdefn\] for a more precise formulation.)
Let $X$ be an integral projective variety of dimension $n$ and suppose $\alpha \in N_{k}(X)_{\mathbb{Z}}$ for $0 \leq k < n$. The mobility of $\alpha$ is $$\operatorname{mob}(\alpha) = \limsup_{m \to \infty} \frac{\max \left\{ b \in \mathbb{Z}_{\geq 0} \, \left| \, \begin{array}{c} \textrm{Any }b\textrm{ general points of } X \textrm{ are} \\ \textrm{contained in a cycle of class } m\alpha \end{array} \right. \right\}}{m^{\frac{n}{n-k}}/n!}$$
If $X$ is a smooth variety and $L$ is a Cartier divisor then $\operatorname{mob}([L]) = \operatorname{vol}(L)$ as shown in Example \[divisormob\].
Let $\ell$ denote the class of a line on $\mathbb{P}^{3}$. The mobility of $\ell$ is determined by an enumerative question: what is the minimal degree of a curve in $\mathbb{P}^{3}$ going through $b$ general points?
It turns out that the answer to this question is not known (even asymptotically as the degree increases). [@perrin87] conjectures that the “optimal” curves are complete intersections of two divisors of equal degree, which would imply that $\operatorname{mob}(\ell)=1$. We discuss this interesting question in more depth in Section \[p3mobility\].
We define the rational mobility of a class $\alpha \in N_{k}(X)_{\mathbb{Z}}$ in a similar way by counting the number of general points lying on cycles in a fixed rational equivalence class inside of $\alpha$ (see Definition \[mobdefn\]). Rational mobility is interesting even for $0$-cycles.
Let $A_{0}(X)$ denote the set of rational equivalence classes of $0$-cycles on $X$. Recall that $A_{0}(X)$ is said to be representable if the addition map $X^{(r)} \to A_{0}(X)_{\deg(r)}$ is surjective for some $r>0$. In Section \[ratmob0cycles\] we show for varieties over $\mathbb{C}$ that $A_{0}(X)$ is representable if and only if the rational mobility of the class of a point is the maximal possible value $(\dim X)!$.
The mobility function shares many of the important properties of the volume function for divisors. In particular $\operatorname{mob}$ is a homogeneous function, so the definition extends naturally to every element of $N_{k}(X)_{\mathbb{Q}}$. Our first main theorem shows that bigness is characterized by positive mobility, confirming [@delv11 Conjecture 6.5].
Let $X$ be an integral projective variety and suppose $\alpha \in N_{k}(X)_{\mathbb{Q}}$ for $0 \leq k < \dim X$. Then $\alpha$ is big if and only if $\operatorname{mob}(\alpha) > 0$. In fact, $\operatorname{mob}$ extends to a continuous function on $N_{k}(X)$.
One might expect that a subvariety with “positive” normal bundle will have a big numerical class. However, [@voisin10 Example 2.4] shows that even a subvariety with an ample normal bundle need not be big, indicating the need for a different approach.
In contrast to the situation for divisors, it is possible for a subvariety $V$ to have big numerical class even if no multiple of $V$ moves in an algebraic family. For example, [@fl82] constructs a surface $S$ with ample normal bundle in a fourfold $X$ such that no multiple of $S$ moves in $X$. [@peternell09 Example 4.10] and a calculation of Fulger verify that $[S] \in N_{2}(X)$ is big.
Theorem A has analogues in the setting of other equivalence relations on cycles. The main step in the proof of Theorem A is to show that if $\operatorname{mob}(\alpha)>0$ then $\alpha$ is big; the proof does not use any special feature of $N_{k}(X)$ besides the ability to intersect against Cartier divisors. To prove the converse implication, one needs to work with an equivalence relation whose classes form a finitely generated group.
For example, suppose $X$ is an integral projective variety over $\mathbb{C}$. The statement of Theorem A holds for the subspace $N'_{k}(X) \subset H_{2k}(X,\mathbb{R})$ spanned by classes of cycles and for the homological analogue of the mobility function. (The other theorems below can be extended in a similar way.)
There is another natural generalization of the volume of a divisor. We can loosely interpret $\dim H^{0}(X,\mathcal{O}_{X}(L))$ as measuring “how much” the cycle $L$ can vary. For an arbitrary class $\alpha \in N_{k}(X)_{\mathbb{Z}}$, the analogous notion is the dimension of the components of the Chow variety which parametrize cycles of class $\alpha$. Theorem \[familydim\] shows that as $m$ increases the dimension of the components representing $m\alpha$ is bounded above by $Cm^{k+1}$ for some constant $C$.
Let $X$ be an integral projective variety and suppose $\alpha \in N_{k}(X)_{\mathbb{Z}}$ for $0 \leq k < \dim X$. The variation of $\alpha$ is $$\operatorname{var}(\alpha) = \limsup_{m \to \infty} \frac{\max \left\{ \dim W \, \left| \, \begin{array}{c} W \subset \operatorname{Chow}(X) \textrm{ parametrizes} \\ \textrm{cycles of class }m\alpha \end{array} \right. \right\}}{m^{k+1}/(k+1)!}$$
It turns out that $\operatorname{var}$ is homogeneous and so extends naturally to classes in $N_{k}(X)_{\mathbb{Q}}$.
\[curvesinpn\] [@eh92] computes the dimension of the Chow variety of curves on $\mathbb{P}^{n}$. Let $\ell$ denote the class of a line in $\mathbb{P}^{n}$. [@eh92 Theorem 3] shows that for sufficiently high degrees $d$ the maximal dimension of a component of $\operatorname{Chow}(\mathbb{P}^{n})$ with class $d\ell$ is $$\frac{d^{2} + 3d}{2} + 3(n-2)$$ so that $\operatorname{var}(\ell) = 1$. The corresponding component of the Chow variety parametrizes planar curves of degree $d$.
The previous example is typical: components of $\operatorname{Chow}(X)$ with large dimension tend to parametrize cycles that are as “degenerate” as possible. Thus variation usually does not accurately reflect the positivity of a class on $X$. Indeed, it is possible for non-big classes to have positive variation.
\[p3blowup\] Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a closed point. Let $E$ denote the exceptional divisor and let $\alpha$ be the class of a line in $E$. For any positive integer $m$, every effective cycle of class $m\alpha$ is contained in $E$. Thus, the variation coincides with the variation of the line class on $\mathbb{P}^{2}$, showing that that $\operatorname{var}(\alpha)=1$. Note however that $\alpha$ is not big.
More generally, the pushforward of a big class on a subvariety $V$ of $X$ will have positive variation. Our second main theorem shows that this is essentially the only way to construct classes of positive variation. In other words, variation is a measure of positivity along subvarieties.
Let $X$ be an integral projective variety and suppose $\alpha \in N_{k}(X)_{\mathbb{Q}}$ for $0 \leq k < \dim X$. Then $\operatorname{var}(\alpha) > 0$ if and only if there is a morphism $f: Y \to X$ from an integral projective variety of dimension $k+1$ that is generically finite onto its image and a big class $\beta \in N_{k}(Y)_{\mathbb{Q}}$ such that some multiple of $\alpha - f_{*}\beta$ is represented by an effective cycle.
The proof of Theorem B is modeled on a different geometric characterization of bigness for dimension $1$ cycles. [@8authors Theorem 2.4] shows that a class $\alpha \in N_{1}(X)$ is big if and only if any two points of $X$ can be connected via a chain of effective cycles with numerical classes proportional to $\alpha$. For cycles of higher dimension, the correct analogue is to consider chains of effective cycles that intersect in codimension $1$ along “positive” subvarieties. The following definition encodes a strong version of this property.
Let $X$ be an integral projective variety of dimension $n$. Suppose that $W$ is an integral variety and $U \subset W \times X$ is a family of effective $k$-cycles. Denote the projection maps by $p: U \to W$ and $s: U \to X$. We say that the family is strongly big-connecting if $s: U \to X$ is dominant and there is a big effective Cartier divisor $B$ on $X$ such that every $p$-horizontal component of $s^{*}B$ is contracted by $s$ to a subvariety of $X$ of dimension at most $k-1$.
The strongly big-connecting property forces our cycles to intersect in codimension $1$ along a fixed “positive” $(k-1)$-dimensional subvariety (controlled by the divisor $B$). It also implies that two general points in $X$ can be connected by a length-two chain of members of the family. A typical example is given by fixing a $(n-k+1)$-dimensional subspace $V \subset H^{0}(X,\mathcal{O}_{X}(A))$ for a very ample divisor $A$ and taking complete intersections of $(n-k)$ general elements of $V$. This family is strongly big-connecting for any divisor $A \in |V|$: the intersection of $A$ with a general element of our family is contained in the base locus of $V$ which has dimension $k-1$.
The study of connecting chains leads to the following characterization of bigness for cycle classes. The special case when $\alpha$ is a divisor class is one of the key steps in the proof of Theorem B.
Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)$ for $0 \leq k < \dim X$. Then $\alpha$ is big if and only if there is some strongly big-connecting family of $k$-cycles of class $\beta$ and a positive constant $C>0$ such that $\alpha - C\beta$ is pseudo-effective.
A slightly different perspective on the geometry of big cycles has been proposed by Voisin. [@voisin10] conjectures that bigness can be characterized using the tangency behavior of cycles representing $\alpha$ and shows that this conjecture has interesting ties to the Hodge theory of complete intersections in projective space. Theorem C can be viewed as a step in this direction.
Organization
------------
Section \[prelimsection\] reviews background material on cycles. Section \[cyclefamilysection\] describes several geometric constructions for families of cycles. In Section \[divisorsection\] we bound the dimension of components of $\operatorname{Chow}(X)$ parametrizing divisors. Section \[variationsection\] introduces the variation function and proves Theorem B and Theorem C. Section \[mobilitysection\] defines mobility and proves Theorem A. Finally, Section \[mobilityexamplesection\] discusses some examples of the mobility.
Acknowledgements
----------------
My sincere thanks to A.M. Fulger for numerous discussions and for his suggestions. I am grateful to B. Bhatt for sharing his work concerning the content of Section \[divisorsection\]. Thanks to C. Voisin for recommending a number of improvements on an earlier draft. Thanks also to B. Hassett and R. Lazarsfeld for helpful conversations and to Z. Zhu and X. Zhao for reading a draft of the paper.
Preliminaries {#prelimsection}
=============
Throughout we work over a fixed algebraically closed field $K$. A variety will mean a quasiprojective scheme of finite type over $K$ (which may be reducible and non-reduced).
\[dimensionlemma\] Let $X$ be an irreducible variety. Suppose that $f: X \dashrightarrow Y$ is a rational map to a variety $Y$ and $g: X \dashrightarrow Z$ is a rational map to a variety $Z$. Let $F$ be a general fiber of $f$ over a closed point of $Y$ (so in particular $F$ is not contained in the locus where $g$ is not defined). Then $\dim(\overline{g(X)}) \leq \dim(\overline{g(F)}) + \dim Y$.
Let $U$ be an open subset of $X$ where both $f$ and $g$ are defined and let $h: U \to Y \times Z$ be the induced map. Since $F$ is a general fiber, $h(F \cap U)$ is dense in its closure in $Y \times Z$. By considering the first projection, we see that $\dim(\overline{h(U)}) \leq \dim(\overline{g(F \cap U)}) + \dim Y$.
We will often use the following special case of [@gr71 Théorème 5.2.2].
Let $f: X \to S$ be a projective morphism of varieties such that some component of $X$ dominates $S$. There is a birational morphism $\pi: S' \to S$ such that the morphism $f': X' \to S'$ is flat, where $X' \subset X \times_{S} S'$ is the closed subscheme defined by the ideal of sections whose support does not dominate $S'$.
Cycles
------
Suppose that $X$ is a projective variety. A $k$-cycle on $X$ is a finite formal sum $\sum a_{i} V_{i}$ where the $a_{i}$ are integers and each $V_{i}$ is an integral closed subvariety of $X$ of dimension $k$. The support of the cycle is the union of the $V_{i}$ (with the reduced structure). The cycle is said to be effective if each $a_{i} \geq 0$. For a $k$-dimensional closed subscheme $V$ of $X$, the fundamental cycle of $V$ is $\sum m_{i}V_{i}$ where the $V_{i}$ are the $k$-dimensional components of the reduced scheme underlying $V$ and the $m_{i}$ are the lengths of the corresponding Artinian local rings $O_{V,V_{i}}$.
The group of $k$-cycles is denoted $Z_{k}(X)$ and the group of $k$-cycles up to rational equivalence is denoted $A_{k}(X)$. We will follow the conventions of [@fulton84] in the use of various intersection products on $A_{k}(X)$.
[@fulton84 Chapter 19] defines a $k$-cycle on $X$ to be numerically trivial if its rational equivalence class has vanishing intersection with every weighted homogeneous degree-$k$ polynomial in Chern classes of vector bundles on $X$. Two cycles are numerically equivalent if their difference is numerically trivial. We let $N_{k}(X)_{\mathbb{Z}}$ denote the abelian group of numerical equivalence classes of $k$-cycles on $X$. By [@fulton84 Example 19.1.4] $N_{k}(X)_{\mathbb{Z}}$ is a finitely generated free abelian group.
We also define $$\begin{aligned}
N_{k}(X)_{\mathbb{Q}} & := N_{k}(X)_{\mathbb{Z}} \otimes \mathbb{Q} \\
N_{k}(X) & := N_{k}(X)_{\mathbb{Z}} \otimes \mathbb{R}\end{aligned}$$
Thus $N_{k}(X)$ is a finitely generated $\mathbb{R}$-vector space and there are natural injections $N_{k}(X)_{\mathbb{Z}} \hookrightarrow N_{k}(X)_{\mathbb{Q}} \hookrightarrow N_{k}(X)$. We denote the dual group of $N_{k}(X)_{\mathbb{Z}}$ by $N^{k}(X)_{\mathbb{Z}}$ and the dual vector spaces of $N_{k}(X)_{\mathbb{Q}}$ and $N_{k}(X)$ by $N^{k}(X)_{\mathbb{Q}}$ and $N^{k}(X)$ respectively.
[@fulton84] defines the Chern class of a vector bundle $c_{i}(E)$ as an operation $A_{k}(X) \to A_{k-i}(X)$. It follows formally from the definition that Chern classes descend to maps $N_{k}(X) \to N_{k-i}(X)$.
\[dimconv\] When we discuss $k$-cycles on an integral projective variety $X$, we will always implicitly assume that $0 \leq k < \dim X$. This allows us to focus on the interesting range of behaviors without repeating hypotheses.
For a cycle $Z$ on $X$, we let $[Z]$ denote the numerical class of $Z$, which can be naturally thought of as an element in $N_{k}(X)_{\mathbb{Z}}$, $N_{k}(X)_{\mathbb{Q}}$, or $N_{k}(X)$. If $\alpha$ is the class of an effective cycle $Z$, we say that $\alpha$ is an effective class.
Let $X$ be a projective variety. The pseudo-effective cone $\operatorname{\overline{Eff}}_{k}(X) \subset N_{k}(X)$ is the closure of the cone generated by all classes of effective $k$-cycles. The big cone is the interior of the pseudo-effective cone. The cone in $N^{k}(X)$ dual to the pseudo-effective cone is known as the nef cone and denoted $\operatorname{Nef}^{k}(X)$.
We say that $\alpha \in N_{k}(X)$ is pseudo-effective (resp. big) if it lies in the pseudo-effective cone (resp. big cone), and $\beta \in N^{k}(X)$ is nef if it lies in the nef cone. For $\alpha,\alpha' \in N_{k}(X)$ we write $\alpha \preceq \alpha'$ when $\alpha' - \alpha$ is pseudo-effective.
Since $X$ is projective, $\operatorname{\overline{Eff}}_{k}(X)$ is a full-dimensional salient cone. For any morphism of projective varieties $f: X \to Y$, there is a pushforward map $f_{*}: N_{k}(X) \to N_{k}(Y)$. It is clear that $f_{*}(\operatorname{\overline{Eff}}_{k}(X)) \subset \operatorname{\overline{Eff}}_{k}(Y)$. There is also a formal dual $f^{*}: N^{k}(Y) \to N^{k}(X)$ that preserves nefness.
Let $B$ be a Cartier divisor on an equidimensional projective variety $X$ of dimension $n$. We say that $B$ is big if for each reduced component $X_{i}$ of $X$ we have that $h^{0}(X_{i},mB|_{X_{i}}) > \lfloor Cm^{n} \rfloor$ for some positive constant $C$. This implies that the corresponding Weil divisor has big class. With this definition bigness of a Cartier divisor is preserved by generically-finite pullback.
The following lemmas record some basic properties of pseudo-effective cycles.
\[surjpushforward\] Let $f: X \to Y$ be a surjective morphism of projective varieties. For any effective cycle $Z$ on $Y$, there is an effective cycle $V$ on $X$ such that $f_{*}V = cZ$ for some $c>0$. In particular $f_{*}$ takes big classes to big classes.
One can not conclude immediately from Lemma \[surjpushforward\] that the induced morphism $f_{*}: \operatorname{\overline{Eff}}_{k}(X) \to \operatorname{\overline{Eff}}_{k}(Y)$ is surjective, because the image of a closed cone under a linear map need not be closed. Nevertheless this statement is true; see [@fl13 Corollary 3.20].
It suffices to prove the first statement when $Z$ is an irreducible subvariety of $Y$. Let $T$ be any integral subvariety of $X$ that is mapped surjectively onto $Z$ by $f$. By cutting $T$ down by very ample divisors, we may find a subvariety $V$ of $X$ that has the same dimension as $Z$ such that $f|_{V}$ surjects onto $Z$. Then $f_{*}V$ is a positive multiple of $Z$.
To see the final statement, fix a big effective class $\alpha \in N_{k}(Y)_{\mathbb{Z}}$. Choose an effective class $\beta \in N_{k}(X)$ such that $f_{*}\beta = \alpha$. Then for any big class $\gamma \in N_{k}(X)$, we have that $\gamma - c \beta$ is pseudo-effective for some $c>0$; thus $f_{*}\gamma \succeq c\alpha$.
\[ampleintisbig\] Let $X$ be a projective variety of dimension $n$ and let $A$ be an ample Cartier divisor on $X$. Then $[A^{n-k}] \in N_{k}(X)$ is big.
Let $Z$ be a big effective $k$-cycle on $X$. Then $V = \operatorname{Supp}(Z)$ is also big, so it suffices to prove that $j[A^{n-k}] - [V]$ is an effective class for some integer $j>0$. Choose a sufficiently large integer $m$ such that $\mathcal{O}_{X}(mA) \otimes \mathcal{I}_{V}$ is globally generated. Let $V'$ be the intersection of $(n-k)$ general divisors in $|mA|$ containing $V$; then $V'-V$ is an effective $k$-cycle. Set $j = m^{n-k}$.
\[intlem\] Let $X$ be a projective variety.
1. If $A$ is a nef Cartier divisor then $\cdot A: N_{k}(X) \to N_{k-1}(X)$ takes $\operatorname{\overline{Eff}}_{k}(X)$ into $\operatorname{\overline{Eff}}_{k-1}(X)$.
2. If $\alpha \in N_{k}(X)$ is a big class and $A$ is an ample Cartier divisor then $\alpha \cdot A \in N_{k-1}(X)$ is also big.
3. Let $D$ be the support of an effective big $j$-cycle $Z$ with injection $i: D \to X$. If $\alpha \in N_{k}(D)$ is big (for $0 \leq k \leq j$) then $i_{*}\alpha \in N_{k}(X)$ is big.
By continuity and homogeneity it suffices to prove (1) when $A$ is very ample. Let $Z$ be an integral $k$-dimensional subvariety; for sufficiently general elements $H \in |A|$, the cycle underlying $H|_{Z}$ is an effective cycle of class $[Z] \cdot A$, proving (1). To see (2), write $\alpha = \alpha' + cA^{n-k}$ for some pseudo-effective class $\alpha'$ and some small $c>0$. Applying (1), it suffices to note that $A^{n-k+1}$ is a big class by Lemma \[ampleintisbig\]. Similarly, to show (3) fix an ample Cartier divisor $A$ on $X$ and consider the class $\beta := (i^{*}A)^{j-k}$ in $N_{k}(D)$. Choose $m$ large enough so that $m[D] \succeq [Z]$. By the projection formula $i_{*}(m\beta) \succeq [A^{j-k} \cdot Z]$, so that $i_{*}\beta$ is big on $X$ by (2). Writing $\alpha = \alpha' + c\beta$ for a sufficiently small $c$, the claim follows from the fact that $i_{*}$ preserves pseudo-effectiveness.
Analytic lemmas
---------------
We are interested in invariants constructed as asymptotic limits of functions on $N_{k}(X)_{\mathbb{Z}}$. The following lemmas will allow us to conclude several important properties of these functions directly from some easily verified conditions.
\[lazlemma\] Let $f: \mathbb{N} \to \mathbb{R}_{\geq 0}$ be a function. Suppose that for any $r,s \in \mathbb{N}$ with $f(r) > 0$ we have that $f(r+s) \geq f(s)$. Then for any $k \in \mathbb{R}_{>0}$ the function $g: \mathbb{N} \to \mathbb{R} \cup \{ \infty \}$ defined by $$g(r) := \limsup_{m \to \infty} \frac{f(mr)}{m^{k}}$$ satisfies $g(cr) = c^{k}g(r)$ for any $c,r \in \mathbb{N}$.
Although [@lazarsfeld04 Lemma 2.2.38] only explicitly address the volume function, the essential content of the proof is the more general statement above.
\[easyconelem\] Let $V$ be a finite dimensional $\mathbb{Q}$-vector space and let $C \subset V$ be a salient full-dimensional closed convex cone. Suppose that $f: V \to \mathbb{R}_{\geq 0}$ is a function satisfying
1. $f(e) > 0$ for any $e \in C^{int}$,
2. there is some constant $c > 0$ so that $f(me) = m^{c}f(e)$ for any $m \in \mathbb{Q}_{>0}$ and $e \in C$, and
3. for every $v \in C$ and $e \in C^{int}$ we have $f(v+e) \geq f(v)$.
Then $f$ is locally uniformly continuous on $C^{int}$.
Endow $V$ with the Euclidean metric for some fixed basis. Let $T \subset C^{int}$ be any bounded set such that $$\inf_{p \in T, q \not \in C} \Vert p - q \Vert > 0.$$ We show that $f$ is uniformly continuous on $T$. Let $\mathcal{T}$ be the cone over $T$. There is some constant $\xi > 0$ such that if $v \in \mathcal{T}$ satisfies $\Vert v \Vert = \mu$ then the open ball $B_{\xi \mu}(v)$ satisfies $B_{\xi \mu}(v) \subset C^{int}$.
Let $M = \sup_{w \in T}f(w)$; since there is some element $x \in C^{int}$ such that $T \subset x-C^{int}$, we see that $M$ is a positive real number.
Fix $\epsilon > 0$ and let $v \in T$. Note that the set $\left(1-\frac{\epsilon}{M}\right)^{1/c}v + C^{int}$ contains the open ball $B_{r_{v}}(v)$, where $$r_{v} = \xi \left( 1 - \left(1 - \frac{\epsilon}{M} \right)^{1/c} \right) \Vert v \Vert$$ Every $e \in B_{r_{v}}(v)$ satisfies $f(e) \geq f(v) - \epsilon$. Similarly, the set $\left(1+\frac{\epsilon}{M}\right)^{1/c}v + (-C^{int})$ contains the open ball $B_{s_{v}}(v)$ where $$s_{v} = \xi \left(\left(1 + \frac{\epsilon}{M} \right)^{1/c} - 1 \right) \Vert v \Vert$$ Every $e \in B_{s_{v}}(v)$ satisfies $f(e) \leq f(v) + \epsilon$.
As we vary $v \in T$, the length $\Vert v \Vert$ has a positive lower bound (since by assumption $T$ avoids a sufficiently small neighborhood of the origin). Thus, there is some $\delta > 0$ such that $\delta < \inf_{v \in T}\min\{s_{v},r_{v}\}$. Then $|f(v') - f(v)| \leq \epsilon$ for every $v$ and $v'$ in $T$ satisfying $\Vert v' - v \Vert<\delta$, showing uniform continuity on $T$. By varying $T$, we obtain local uniform continuity on $C^{int}$.
Families of cycles {#cyclefamilysection}
==================
Although there are several different notions of a family of cycles in the literature, the theory we will develop is somewhat insensitive to the precise choices. It will be most convenient to use a simple geometric definition.
\[familydef\] Let $X$ be a projective variety. A family of $k$-cycles on $X$ consists of an integral variety $W$, a reduced closed subscheme $U \subset W \times X$, and an integer $a_{i}$ for each component $U_{i}$ of $U$, such that for each component $U_{i}$ of $U$ the first projection map $p: U_{i} \to W$ is flat dominant of relative dimension $k$. If each $a_{i} \geq 0$ we say that we have a family of effective cycles. We say that $\sum a_{i}U_{i}$ is the cycle underlying the family.
In this situation $p: U \to W$ will denote the first projection map and $s: U \to X$ will denote the second projection map unless otherwise specified. We will usually denote a family of $k$-cycles using the notation $p: U \to W$, with the rest of the data implicit.
For a closed point $w \in W$, the base change $w \times_{W} U_{i}$ is a subscheme of $X$ of pure dimension $k$ and thus defines a fundamental $k$-cycle $Z_{i}$ on $X$. The cycle-theoretic fiber of $p: U \to W$ over $w$ is defined to be the cycle $\sum a_{i}Z_{i}$ on $X$. We will also call these cycles the members of the family $p$.
Let $X$ be a projective variety. We say that a family of $k$-cycles $p: U \to W$ on $X$ is a rational family if every cycle-theoretic fiber lies in the same rational equivalence class.
Definition \[familydef\] has a number of deficiencies. For example, many intuitive constructions of families of cycles fail to meet the criteria: the map $\mathbb{A}^{2} \times \mathbb{A}^{2} \to \mathrm{Sym}^{2}\mathbb{A}^{2}$ is not flat over a characteristic $2$ field as pointed out in [@kollar96]. Since we are primarily interested in the “generic” behavior of families of cycles, these shortcomings are not important for us. On the other hand, the geometric flexibility of Definition \[familydef\] will be very useful.
The following constructions show how to construct families of cycles from subsets $U \subset W \times X$.
\[cycletofamilyconstr\] Let $X$ be a projective variety and let $W$ be an integral variety. Suppose that $Z = \sum a_{i}V_{i}$ is a $(k+\dim W)$-cycle on $W \times X$ such that the first projection maps each $V_{i}$ dominantly onto $W$. Let $W^{0} \subset W$ be the (non-empty) open locus over which every projection $p: V_{i} \to W$ is flat and let $U \subset \operatorname{Supp}(Z)$ denote the preimage of $W^{0}$. Then the map $p: U \to W^{0}$ defines a family of cycles where we assign the coefficient $a_{i}$ to the component $V_{i} \cap U$ of $U$.
\[equidimsubschemeconstr\] Suppose that $Y$ is a reduced variety and that $X$ is a projective variety. Let $\tilde{U} \subset Y \times X$ be a closed subscheme such that the fibers of the projection $p: \tilde{U} \to Y$ are equidimensional of dimension $k$. There is a natural way to construct a finite collection of families of effective cycles associated to the subscheme $\tilde{U}$.
Consider the image $p(\tilde{U})$ (with its reduced induced structure). Let $\{ \tilde{W}_{j} \}$ denote the irreducible components of $p(\tilde{U})$. For each there is a non-empty open subset $W_{j} \subset \tilde{W}_{j}$ such that the restriction of $p$ to each component of $p^{-1}(W_{j})_{red}$ is flat. Since furthermore $p$ has equidimensional fibers, we obtain a family of effective $k$-cycles $p_{j}: U_{j} \to W_{j}$ where $U_{j} = p^{-1}(W_{j})_{red}$ and we assign coefficients so that the cycle underlying the family $p_{j}$ coincides with the fundamental cycle of $p^{-1}(W_{j})$. We can then replace $\tilde{U}$ by the closed subscheme obtained by taking the base change to $p(\tilde{U}) - \cup_{j} W_{j}$ and repeat. The end result is a collection of families $p_{i}: U_{i} \to W_{i}$ parametrizing the cycles contained in $\tilde{U}$.
If $p(\tilde{U})$ is irreducible and we are interested only in the generic behavior of the cycles in $\tilde{U}$, we can stop after the first step to obtain a single family of cycles.
Chow varieties and the Chow map
-------------------------------
Fix a projective variety $X$ and an ample divisor $H$ on $X$. For any reduced scheme $Z$ over the ground field, [@kollar96 Chapter I.3] introduces a more refined definition of a family of $k$-cycles of $X$ of $H$-degree $d$ over $Z$. Kollár then constructs a semi-normal projective variety $\operatorname{Chow}_{k,d,H}(X)$ that parametrizes families of effective $k$-cycles of $H$-degree $d$. $\operatorname{Chow}(X)$ denotes the disjoint union over all $k$ and $d$ of $\operatorname{Chow}_{k,d,H}(X)$ for some fixed ample divisor $H$; it does not depend on the choice of $H$.
The precise way in which $\operatorname{Chow}(X)$ parametrizes cycles is somewhat subtle in characteristic $p$. For a discussion of the Chow functor and universal families, see [@kollar96]. We will need the following properties of $\operatorname{Chow}(X)$:
- Any family of cycles in the sense of Definition \[familydef\] naturally yields a family of cycles in the refined sense of [@kollar96 I.3.11 Definition] by applying [@kollar96 I.3.14 Lemma] with the identity map (see also [@kollar96 I.3.15 Corollary]).
- For any weakly normal integral variety $W$ and any (refined) family of effective cycles $p: U \to W$, there is an induced morphism $\operatorname{ch}_{p}: W \to \operatorname{Chow}(X)$ by [@kollar96 I.4.8-I.4.10]. (We will denote this map simply by $\operatorname{ch}$ when the family $p$ is clear from the context.)
For any family of effective cycles $p: U \to W$ the base change to the normal locus $W^{0} \subset W$ is still a family of cycles (where we assign the same coefficients). Thus there is an induced rational map $\operatorname{ch}_{p}: W \dashrightarrow \operatorname{Chow}(X)$ that is a morphism on the normal locus of $W$.
The following crucial lemma encapsulates the set-theoretical nature of the Chow functors constructed in [@kollar96 Chapter I.3].
\[injectivechowlemma\] Let $X$ be a projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$ over a weakly normal $W$. A curve $C \subset W$ is contracted by $\operatorname{ch}: W \to \operatorname{Chow}(X)$ if and only if every cycle-theoretic fiber over $C$ has the same support.
We will freely use the notation of [@kollar96] in the verification.
First suppose that every cycle-theoretic fiber of $p$ over $C$ has the same support. Since $\operatorname{Chow}_{k,d,H}(X)$ is constructed by taking a semi-normalization (which is set-theoretically bijective), we may instead consider the induced map to $\operatorname{Chow}'_{k,d,H}(X)$. This map factors through the map $\operatorname{ch}$ for a projective space containing an embedding of $X$; therefore it suffices to consider the case when $X = \mathbb{P}$. Then the construction following [@kollar96 Ch. I Eq. (3.23.1.5)] shows that the Cartier divisors on $(\mathbb{P}^{\vee})^{k+1}$ parametrized by the image of $C$ in $\mathbb{H}$ must all have the same support. But this implies they are equal.
Conversely, suppose that $C$ is contracted by $\operatorname{ch}$. As discussed in [@kollar96 I.3.27.3], $C$ is also contracted by the morphism to $\mathrm{Hilb}((\mathbb{P}^{\vee})^{k+1})$. Again comparing with [@kollar96 Ch. I Eq. (3.23.1.5)], we see that the support of each of the cycles parametrized by $C$ is the same.
It will often be helpful to replace a family $p: U \to W$ by a slightly modified version.
\[goodfamilymodification\] Let $X$ be a projective variety and let $p: U \to W$ be a family of effective cycles on $X$. Then there is a normal projective variety $W'$ that is birational to $W$ and a family of cycles $p': U' \to W'$ such that $\operatorname{ch}(W') = \overline{\operatorname{ch}(W)}$.
Let $\tilde{W}$ be any projective closure of $W$ and let $\tilde{U}$ be the closure of $U$ in $\tilde{W} \times X$. Let $\phi: W' \to \tilde{W}$ be the normalization of a simultaneous flattening of the morphisms $\tilde{p}: \tilde{U}_{i} \to \tilde{W}$ for the components $\tilde{U}_{i}$ of $\tilde{U}$. Let $U'$ denote the reduced subscheme of $W' \times X$ defined by the components of $\tilde{U} \times_{\tilde{W}} W'$ that dominate $W'$. Since the components of $U'$ are in bijection with the components of $U$, we can assign to each component of $U'$ the coefficient of the corresponding component of $U$. Then $p': U' \to W'$ is a family of effective $k$ cycles. Since the $p'$ and $p$ agree over an open normal subset of the base, the closure of the images under the map $\operatorname{ch}$ agree.
It is also important to know whether a rational family $p: U \to W$ can be extended to a *rational* family over a projective closure of $W$ (although we will not need such statements below). The arguments of [@samuel56 Theorem 3] show that the subset of $\operatorname{Chow}(X)$ parametrizing cycles in a fixed rational equivalence class is a countable union of closed subvarieties. Thus we can extend families in this way when working over an uncountable algebraically closed field $K$.
Chow dimension of families
--------------------------
Let $p: U \to W$ be a family of effective $k$-cycles on a projective variety $X$. Then all the cycle-theoretic fibers of $p$ are algebraically equivalent. Indeed, for any two closed points of $W$, let $C$ be the normalization of a curve through those two points; since the base change of $U$ to $C$ is a union of flat families of effective cycles, we see that the corresponding cycle-theoretic fibers are algebraically equivalent.
Let $p: U \to W$ be a family of effective $k$-cycles on a projective variety $X$. We say that $p$ represents $\alpha \in N_{k}(X)_{\mathbb{Z}}$ if the cycle-theoretic fibers of our family have class $\alpha$.
Let $X$ be a projective variety and let $p: U \to W$ be an effective family of $k$-cycles. We define the Chow dimension of $p$ to be $$\operatorname{chdim}_{X}(p) := \dim(\overline{\mathrm{Im} \, \operatorname{ch}\!: W \dashrightarrow \operatorname{Chow}(X)})$$ If $\alpha \in N_{k}(X)_{\mathbb{Z}}$, we define $$\operatorname{chdim}_{X}(\alpha) = \max \{ \operatorname{chdim}(p) | p: U \to W \textrm{ represents } \alpha \}.$$ We will usually omit the subscript when it is clear from the context.
When our ground field $K$ has characteristic $0$, [@kollar96] constructs a universal family over any component of $\operatorname{Chow}(X)$. Using Construction \[cycletofamilyconstr\] this can be turned into a family of effective cycles in the sense of Definition \[familydef\]. Thus $$\operatorname{chdim}(\alpha) = \max \{ \dim(Y)\, | \, Y \textrm{ is a component of } \operatorname{Chow}(X) \textrm{ representing }\alpha \}.$$ Even when $K$ has characteristic $p$, for any component of $\operatorname{Chow}(X)$ [@kollar96 I.4.14 Theorem] constructs a family of cycles whose chow map $\operatorname{ch}$ is dominant, so that we still have the same interpretation.
We can also consider the analogous construction for rational families.
Let $X$ be a projective variety. For $\alpha \in N_{k}(X)_{\mathbb{Z}}$, we define $$\operatorname{rchdim}(\alpha) = \max \{ \operatorname{chdim}(p) | p: U \to W \textrm{ is a rational family representing } \alpha \}.$$ Similarly, for $\tau \in A_{k}(X)$ we define $$\operatorname{rchdim}(\tau) = \max \{ \operatorname{chdim}(p) | p: U \to W \textrm{ is a rational family of class } \tau \}.$$
Geometry of families
--------------------
Let $X$ be a projective variety and let $p: U \to W$ be a family of effective cycles on $X$.
- We say that $p$ is a reduced family if every coefficient $a_{i}$ is $1$. Any family of effective cycles yields a reduced family by simply changing the coefficients.
- We say that $p$ is an irreducible family if $U$ only has one component. For any component $U_{i}$ of $U$, we have an associated irreducible family $p_{i}: U_{i} \to W$ (with coefficient $a_{i}$).
\[genreduceddimest\] Let $X$ be a projective variety and let $p: U \to W$ be an effective family of $k$-cycles. Let $p': U' \to W$ denote the reduced family for $p$. Then $\operatorname{chdim}(p') = \operatorname{chdim}(p)$.
Note that $p: U \to W$ and $p': U' \to W$ agree set-theoretically. The statement then follows immediately from Lemma \[injectivechowlemma\].
\[componentsdimest\] Let $X$ be a projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$. For each component $U_{i}$ of $U$, let $p_{i} := p|_{U_{i}}: U_{i} \to W$ denote the irreducible family induced by $U_{i}$. Then $\operatorname{chdim}(p) \leq \sum_{i} \operatorname{chdim}(p_{i})$.
By Lemma \[injectivechowlemma\], the map $\operatorname{ch}_{p}: W \dashrightarrow \operatorname{Chow}(X)$ factors rationally through the map $\prod_{i} \operatorname{ch}_{p_{i}}: W \dashrightarrow \prod_{i} \operatorname{Chow}(X)$.
We will also need several geometric constructions.
\[flatpullbackfamilies\] Let $g: Y \to X$ be a flat morphism of projective varieties of relative dimension $d$. Suppose that $p: U \to W$ is a family of effective $k$-cycles on $X$ with underlying cycle $V$. The flat pullback cycle $g^{*}V$ on $W \times Y$ is effective and has relative dimension $(d+k)$ over $W$. We define the flat pullback family $g^{*}p: U' \to W^{0}$ of effective $(d+k)$-cycles on $Y$ over an open subset $W^{0} \subset W$ by applying Construction \[cycletofamilyconstr\] to $g^{*}V$.
\[pushforwardoffamilies\] Let $f: X \to Y$ be a morphism of projective varieties. Suppose that $p: U \to W$ is a family of effective $k$-cycles on $X$ with underlying cycle $V$. Consider the cycle pushforward $f_{*}V$ on $W \times Y$ and assume $f_{*}V \neq 0$. Construction \[cycletofamilyconstr\] yields a family of $k$-cycles $f_{*}p: \tilde{U} \to W^{0}$ over an open subset of $W$. We call $f_{*}p$ the pushforward family. Note that this operation is compatible with the pushforward on cycle-theoretic fibers over $W^{0}$ by [@kollar96 I.3.2 Proposition].
\[restrictionfamilies\] Let $X$ be a projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$. Let $W' \subset W$ be an integral subvariety. For each component $U_{i}$ of $U$, the restriction $U_{i} \times_{W} W'$ is flat over $W'$ of relative dimension $k$. Consider the cycle $V$ on $W' \times X$ defined as the sum $V = \sum_{i} a_{i}V_{i}$ where $V_{i}$ is the fundamental cycle of $U_{i}$ restricted to $W'$. We define the restriction of the family $p$ to $W'$ over an open subset $W'^{0} \subset W'$ by applying Construction \[cycletofamilyconstr\] to $V$. Note that this operation leaves the cycle-theoretic fibers unchanged over $W'^{0}$. Note also that if $W' \subset W$ is open, then we may take $W^{0} = W'$ and the family $p$ is simply the base-change to $W^{0}$.
\[familysumconstr\] Let $X$ be a projective variety and let $p: U \to W$ and $q: S \to T$ be two families of effective $k$-cycles on $X$. We construct the family sum of $p$ and $q$ over an open subset of $W \times T$ as follows. Let $V_{p}$ and $V_{q}$ denote the underlying cycles for $p$ and $q$ on $W \times X$ and $T \times X$ respectively. The family sum of $p$ and $q$ is the family defined by applying Construction \[cycletofamilyconstr\] to the sum of the flat pullbacks of $V_{p}$ and $V_{q}$ to $W \times T \times X$.
Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$. Suppose that $\phi: X \dashrightarrow Y$ is a birational map. We define the strict transform family of effective $k$-cycles on $Y$ as follows.
First, modify $U$ by removing all irreducible components whose image in $X$ is contained in the locus where $\phi$ is not an isomorphism. Then define the cycle $U'$ on $W \times Y$ by taking the strict transform of the remaining components of $U$. We define the strict transform family by applying Construction \[cycletofamilyconstr\] to $U'$ over $W$.
\[intersectionfamilyconstr\] Let $X$ be a projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$. Let $D$ be an effective Cartier divisor on $X$. If every cycle in our family has a component contained in $\operatorname{Supp}(D)$, we say that the intersection family of $p$ and $D$ is empty.
Otherwise, let $s: U \to X$ denote the projection map. By assumption the effective Cartier divisor $s^{*}D$ does not contain any component of $U$, so we may take a cycle-theoretic intersection of $s^{*}D$ with the cycle underlying the family $p$ to obtain a $(k-1+\dim W)$-cycle $V$ on $W \times \operatorname{Supp}(D)$. We then apply Construction \[cycletofamilyconstr\] to obtain a family of cycles on $\operatorname{Supp}(D)$ over an open subset of $W$. We can also consider the intersection as a family of cycles on $X$ by pushing forward and we denote this family by $p \cdot D$.
Finally, suppose that we have a linear series $|L|$. We define the intersection of $|L|$ with a family $p: U \to W$ as follows. Consider the flat pullback family $q: U' \to W^{0}$ on $\mathbb{P}(|L|) \times X$. Then intersect the family $q$ against the pullback of the universal divisor on $\mathbb{P}(|L|) \times X$ to obtain a family of cycles on $\mathbb{P}(|L|) \times X$. The underlying cycle has dimension $k-1+\dim W + \dim(\mathbb{P}(|L|))$; by using Construction \[cycletofamilyconstr\], we can convert this cycle to a family of effective $(k-1)$-cycles on $X$ over an open subset of $W \times \mathbb{P}(|L|)$.
We conclude this section with a brief analysis of how these constructions affect the Chow dimension.
\[genfinitepushlem\] Let $f: X \to Y$ be a morphism of projective varieties. Let $p: U \to W$ be a family of effective $k$-cycles on $X$ such that for every component $U_{i}$ of $U$ the image $\overline{s(U_{i})}$ is not contracted to a variety of smaller dimension by $f$. Then $\operatorname{chdim}(p) = \operatorname{chdim}(f_{*}p)$.
Let $T$ be an integral curve through a general point of $W$ that is not contracted by $\operatorname{ch}_{p}$ and set $S = p^{-1}(T)$. Then $\dim(\overline{s(S)}) = \dim(f(\overline{s(S)}))$. Thus the cycle-theoretic fibers parametrized by $T$ do not pushforward to the same cycle on $Y$. We conclude by Lemma \[injectivechowlemma\] that that $T$ is not contracted by $\operatorname{ch}_{f_{*}p}$.
\[chdimfamilysumlem\] Let $X$ be a projective variety and let $p: U \to W$ and $q: S \to T$ be two families of effective $k$-cycles on $X$. Then $\operatorname{chdim}(p+q) = \operatorname{chdim}(p) + \operatorname{chdim}(q)$.
A curve through a general point of $(W \times T)^{0}$ is contracted by $\operatorname{ch}_{p+q}$ if and only if its projection to $W$ and to $T$ are contracted by $\operatorname{ch}_{p}$ and $\operatorname{ch}_{q}$ respectively. We conclude by Lemma \[injectivechowlemma\].
Families of divisors {#divisorsection}
====================
In this section we analyze the Chow dimension of families of effective divisors. The goal is to find bounds on the Chow dimension that depend only on the numerical class of the divisor. The key result is Proposition \[basicvarestimate\], which is the base case of inductive arguments used in the following sections.
A Lefschetz hyperplane-type theorem for Weil divisors
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Let $X$ be an integral projective variety. Fix a closed point $p$ in the smooth locus of $X$; then $X$ admits an Albanese mapping $\operatorname{alb}: X \dashrightarrow \operatorname{Alb}(X)$ sending $p$ to $0$ characterized by the properties:
- the image of $\operatorname{alb}$ generates $\operatorname{Alb}(X)$, and
- the map $\operatorname{alb}$ is universal among rational maps from $X$ to abelian varieties that map $p$ to $0$.
We will let $P(X)$ denote the abelian variety dual to $\operatorname{Alb}(X)$. Note that $P(X)$ is a birational invariant of $X$.
Suppose that $A$ is a very ample Cartier divisor on $X$. A general $H \in |A|$ is an integral variety by the Bertini theorems. After choosing a basepoint $p \in H$ that is also in the smooth locus of $X$, by the universal property of the Albanese we obtain an induced morphism $\operatorname{Alb}(H) \to \operatorname{Alb}(X)$. We let $r: P(X) \to P(H)$ denote the dual map. Note that the restriction map is unchanged if we replace the inclusion $i: H \to X$ by a birationally equivalent map (that is compatible with the basepoint).
When $X$ is normal, [@bgs11] shows that the $K$-points of $P(X)$ parametrize the rational equivalence classes of Weil divisors on $X$ that are algebraically equivalent to $0$. More precisely, let $X_{sm} \subset X$ be the open smooth locus of $X$ and fix a closed point $p \in X_{sm}$ (corresponding to the base point of the Albanese map). Define the functor $P_{X}^{0}$ which assigns to a reduced $K$-variety $T$ the collection of invertible sheaves $\mathcal{L}$ on $X_{sm} \times T$ (up to isomorphism) such that $\mathcal{L}|_{\{p\} \times T}$ is trivial and $\mathcal{L}|_{X_{sm} \times \{t\}}$ is algebraically equivalent to zero for every $K$-point $t$ of $T$. [@bgs11 Proposition 3.2] shows that $P(X)$ represents the functor $P_{X}^{0}$ on the category of reduced $K$-varieties.
### Relationship with $\operatorname{Pic}^{0}$
For an integral projective variety $X$, let $X^{\nu}$ denote the normalization of $X$ and let $X^{\nu}_{sm} \subset X^{\nu}$ denote the smooth locus. Let $\operatorname{Pic}^{0}(X)$ denote the connected component of the identity of $\operatorname{Pic}(X)$. Then $\operatorname{Pic}^{0}(X)_{red}$ represents the functor $Pic_{X}^{0}$ on the category of reduced $K$-varieties.
We define a morphism $q: \operatorname{Pic}^{0}(X)_{red} \to P(X)$ via the following natural transformation on the corresponding functors. Let $T$ be a reduced variety. For an invertible sheaf $\mathcal{L}$ on $X \times T$ representing an element of $Pic_{X}^{0}(T)$, we pull-back $\mathcal{L}$ to $X^{\nu} \times T$ and restrict to $X_{sm}^{\nu} \times T$. After twisting by the pullback of a line bundle on $T$ to ensure triviality over the basepoint, we obtain an element of $P_{X}^{0}(T)$. Since this natural transformation is injective on the set-theoretic level, $q$ is a monomorphism in the category of reduced $K$-varieties; in particular it is set-theoretically injective.
\[divisorrestriction\] Let $X$ be an integral projective variety. Let $A$ be a very ample Cartier divisor on $X$ and let $H$ be a general element of $|A|$. Then the kernel of the restriction map $r: P(X) \to P(H)$ is supported on a finite number of points of $P(X)$. In particular, $r$ is finite flat onto its image.
Let $X^{\nu}$ denote the normalization of $X$ and let $X_{sm}^{\nu}$ denote the smooth locus of $X^{\nu}$. We will first need a special case of a result of [@benoist12] (obtained by applying [@benoist12 Lemma 6] when $U$ is the smooth locus of $X^{\nu}$).
\[benoistlem\] Let $X^{\nu}$ be a projective integral normal variety over an algebraically closed field. Then there exists a birational map $\phi: Y \to X^{\nu}$ from a projective normal variety $Y$ that is an isomorphism over the smooth locus of $X^{\nu}$ such that every rational equivalence class of divisors corresponding to a $K$-point of $P(Y)$ is Cartier as a class on $Y$.
Let $Y_{sm} \subset Y$ be the smooth locus of $Y$ and let $\psi: Y \to X$ denote the composition $\nu \circ \phi$. Suppose $T$ is a normal reduced variety and $\mathcal{L}$ is an invertible sheaf on $Y_{sm} \times T$ parametrizing algebraically-trivial line bundles. $\mathcal{L}$ extends to a reflexive rank one sheaf $\mathcal{D}$ on the (normal) variety $Y \times T$. Furthermore, [@bgs11 Lemme 2.6] shows that $\mathcal{D}$ is an invertible sheaf on $Y \times T$. Thus we see that on the category of normal reduced $K$-varieties the defining functors for $\operatorname{Pic}^{0}(Y)_{red}$ and $P(Y)$ are naturally isomorphic, so that $r: \operatorname{Pic}^{0}(Y)_{red} \to P(Y)$ induces an isomorphism of normalizations. Since both $\operatorname{Pic}^{0}(Y)_{red}$ and $P(Y)$ are smooth, $r$ is in fact an isomorphism.
Let $H$ be a general element of $|A|$ on $X$ so that $H$ is integral. We may also suppose that $H$ does not contain any exceptional center for the birational morphism $\psi: Y \to X$, so that the pullback $H' := \psi^{*}H$ is integral as well. Consider the diagram $$\begin{CD}
\operatorname{Pic}^{0}(Y)_{red} @>>> P(Y) \\
@VrVV @VVrV \\
\operatorname{Pic}^{0}(H')_{red} @>>> P(H')
\end{CD}$$ This diagram commutes; on the level of functors, both maps $\operatorname{Pic}^{0}(Y)_{red} \to P(H')$ are defined by the pull-back of invertible sheaves. The morphism on the top is an isomorphism and the morphism on the bottom is set-theoretically injective. Since $H'$ is big and basepoint free, the kernel of the morphism on the left is supported on a finite number of points by the following theorem. (In the statement $\operatorname{Pic}^{\tau}$ denotes the torsion components of $\operatorname{Pic}$.)
Let $Y$ be a normal integral projective variety of dimension $\geq 2$ over a field $K$ and let $D$ be a big semiample divisor on $X$. Then the restriction map $\operatorname{Pic}^{\tau}(Y) \to \operatorname{Pic}^{\tau}(D)$ is
- injective if $\operatorname{char}(K) = 0$.
- has finite and $p^{\infty}$-torsion kernel if $\operatorname{char}(K) = p > 0$.
By identifying $P(X) = P(Y)$ and $P(H') = P(H)$, we obtain the desired statement.
Restrictions to very ample divisors
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Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective divisors on $X$. Fix a general closed point $w \in W$. This data yields a rational map $v_{p,w}: W \dashrightarrow P(X)$ as follows. Let $\nu: X^{\nu} \to X$ be a normalization of $X$. Let $p': U' \to W^{0}$ denote the strict transform family of $p$ to $X^{\nu}$. After shrinking $W^{0}$ we may suppose that it is smooth. Let $w \in W^{0}$ be a closed point and let $L$ denote the corresponding divisor. The effective cycle underlying the family $p'$ defines a reflexive rank $1$ sheaf $\mathcal{D}$ on the normal variety $W^{0} \times X^{\nu}$. The restriction $\mathcal{D}$ to the smooth locus $W^{0} \times X^{\nu}_{sm}$ is invertible. Furthermore, the divisor $L$ restricts to define an invertible sheaf $\mathcal{L}$ on $X^{\nu}_{sm}$. There is an induced morphism $v: W^{0} \to P(X^{\nu}) = P(X)$ given by twisting $\mathcal{D}$ by the pull-back of $\mathcal{L}^{-1}$ and using the functorial definition of $P(X^{\nu})$.
\[chdimandpx\] Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective divisors on $X$. Fix a general closed point $w$ in $W$ and let $F$ be a component (with the reduced structure) of a general fiber of $v_{p,w}: W \dashrightarrow P(X)$. Then $\operatorname{chdim}(p) = \dim(\overline{v_{p,w}(W)}) + \operatorname{chdim}(p|_{F})$.
Let $p'$ denote the strict transform family on $X^{\nu}$ as above. Let $W^{0} \subset W$ denote the open locus where the maps $\operatorname{ch}_{p}$, $\operatorname{ch}_{p'}$, and $v_{p,w}$ are all defined. Note that the only divisors removed by the construction of $p'$ are contained in $\nu$-exceptional centers and are thus fixed components of $p$. By Lemma \[injectivechowlemma\] a curve $C \subset W^{0}$ through a general point is contracted by $\operatorname{ch}_{p}$ if and only if it is contracted by $\operatorname{ch}_{p'}$. But if $C$ is contracted by $\operatorname{ch}_{p'}$ it must also be contracted by $v_{p,w}$, and the lemma follows.
We next prove two lemmas in preparation for Proposition \[restricttoample\].
\[stricttransformdivisors\] Let $\phi: Y \to X$ be a birational map of integral projective varieties. Suppose that $p: U \to W$ is a family of effective divisors on $X$ and that some cycle in our family intersects the locus where $\phi$ is an isomorphism.
1. The strict transform family $p': U' \to W'$ on $Y$ has $\operatorname{chdim}(p') = \operatorname{chdim}(p)$.
2. Suppose that $D$ is an effective Cartier divisor on $X$ that does not contain any $\phi$-exceptional center or any component of $\overline{s(U)}$. Then $\operatorname{chdim}_{D}(p \cdot D) = \operatorname{chdim}_{\phi^{*}D}(p' \cdot \phi^{*}D)$.
\(1) Recall that the strict transform family is constructed by throwing away any components of $U$ with image contained in a $\phi$-exceptional center and then taking the strict transform of a general element of the remaining components of $U$. In particular, two general cycles in the family $p$ have the same support if and only if the corresponding cycles in $p'$ have the same support. We conclude by Lemma \[injectivechowlemma\].
\(2) Note that over an open subset of $W$ the pushforward family of $p' \cdot \phi^{*}D$ to $X$ only differs from $p \cdot D$ by a constant cycle (namely, the intersection of $D$ with the divisor removed in constructing $p'$). Furthermore pushing forward a family of effective divisors over a generically finite morphism does not change the Chow dimension: since the only components of the family that can be contracted are those representing a constant cycle, this follows from Lemma \[genfinitepushlem\]. Combining these two facts yields the statement.
\[ratfamdivrest\] Let $X$ be a normal integral projective variety and let $p: U \to W$ be a rational family of effective divisors on $X$ representing the class $\alpha$. Suppose that $\pi: X \to Y$ is a birational morphism and $A$ is a very ample Cartier divisor on $Y$ such that $\alpha - [\pi^{*}A]$ is not pseudo-effective. Then for a general element $H \in |A|$ we have $\operatorname{chdim}_{X}(p) = \operatorname{chdim}_{\pi^{*}H}(p \cdot \pi^{*}H)$.
Let $\mathcal{D}$ be the reflexive rank one sheaf on $X$ defined by the cycle-theoretic fibers of $p$. Let $T \subset \mathbb{P}(H^{0}(X,\mathcal{D}))$ denote the closure of the locus defined by the cycle-theoretic fibers of $p$. Over an open subset of $T$ we obtain a family of divisors and we replace $p$ by this family; note that this change does not affect the Chow dimension $\dim T$.
By [@takagi07 Proposition 1.1-1.4] there is a birational morphism $\phi: X' \to X$ from a normal projective variety $X'$ and a line bundle $\mathcal{L}$ on $X'$ such that $\phi_{*}\mathcal{L} = \mathcal{D}$ and there is an equality $H^{0}(X',\mathcal{L}) = H^{0}(X,\mathcal{D})$. Thus our (modified) family $p$ yields a new family $p'$ on $X'$ defined over an open subset of the same subvariety $T$ of $\mathbb{P}(H^{0}(X,\mathcal{L}))$. Note that $p'$ pushes forward to $p$ (over an open subset of $T$). In particular, if $\beta$ is the class of $p'$ then $\beta - [\phi^{*}\pi^{*}A]$ is still not pseudo-effective. Furthermore Lemma \[stricttransformdivisors\] (2) shows that the Chow dimension of $p' \cdot \phi^{*}\pi^{*}H$ coincides with $p \cdot \pi^{*}H$. In sum, after a birational modification we may suppose that $\mathcal{D}$ is a line bundle.
A general element $H \in |A|$ does not contain any $\pi$-exceptional center. We may also assume that $H$ does not contain the $\pi$-image of any component of $\overline{s(U)}$. Thus the pullback $\pi^{*}H$ is an integral projective variety such that the intersection family $p \cdot \pi^{*}H$ is defined. Consider the exact sequence $$0 \to \mathcal{O}_{X}(-\pi^{*}A) \otimes \mathcal{D} \to \mathcal{D} \to \mathcal{O}_{\pi^{*}H} \otimes \mathcal{D} \to 0.$$ By assumption $H^{0}(X,\mathcal{O}_{X}(-\pi^{*}A) \otimes \mathcal{D}) = 0$ since the corresponding numerical class is not pseudo-effective. Thus there is an injection $H^{0}(X,\mathcal{D}) \to H^{0}(\pi^{*}H,\mathcal{D}|_{\pi^{*}H})$.
Let $\psi: H' \to \pi^{*}H$ be a normalization. There is an injective morphism $H^{0}(\pi^{*}H,\mathcal{D}|_{\pi^{*}H}) \to H^{0}(H',\psi^{*}(\mathcal{D}|_{\pi^{*}H}))$. Precomposing with the injection above, we can identify $T$ as a subvariety of $\mathbb{P}(H^{0}(H',\psi^{*}(\mathcal{D}|_{\pi^{*}H})))$. The corresponding family of divisors on $H'$ has Chow dimension equal to $\dim T$ and its pushforward to $\pi^{*}H$ agrees with $p \cdot \pi^{*}H$ over an open subset of $T$. Since pushing forward this family does not change the Chow dimension by Lemma \[genfinitepushlem\], we have proved the result.
\[restricttoample\] Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective divisors on $X$ representing $\alpha$. Suppose that $A$ is a very ample Cartier divisor on $X$ such that $\alpha - [A]$ is not pseudo-effective. Then for a general element $H \in |A|$ we have $$\operatorname{chdim}_{X}(p) = \operatorname{chdim}_{H}(p \cdot H).$$
We return to the setting of the proof of Theorem \[divisorrestriction\] keeping the notation there. Let $p': U' \to W^{0}$ denote the strict transform family of $p$ to $Y$. Let $\beta$ denote the class of $p'$ and note that $\beta - [\psi^{*}A]$ is not pseudo-effective.
Let $H \in |A|$ be a general element so that $H' = \psi^{*}H$ is an integral projective variety and $p' \cdot H'$ is defined. By shrinking $W^{0}$ we may suppose that $p' \cdot H'$ is defined over $W^{0}$. Fix a general closed point $w$ of $W^{0}$ yielding a map $v_{p',w}: W^{0} \dashrightarrow P(X)$. We first show that the map $v_{p' \cdot H', w}: W^{0} \dashrightarrow P(H')$ coincides over an open subset of $W^{0}$ with the composition $r \circ v_{p',w}$. This is a consequence of the key property of $Y$: every rational equivalence class of divisors parametrized by $P(Y)$ is Cartier on $Y$. Thus if $H^{\nu}$ is the normalization of $H'$ the pullbacks of the rational classes parametrized by $v_{p',w}(W^{0}) \subset P(Y)$ to $H^{\nu}$ are the same as the rational classes on $H^{\nu}$ given in the construction of $v_{p' \cdot H', w}$.
Lemma \[chdimandpx\] shows that $$\operatorname{chdim}(p' \cdot H') = \dim(\overline{v_{p' \cdot H',w}(W)}) + \operatorname{chdim}(p' \cdot H'|_{F})$$ where $F$ is a component of a general fiber of $v_{p' \cdot H',w}$. Since $r$ is finite flat onto its image by Theorem \[divisorrestriction\] the argument above shows that $$\dim(\overline{v_{p' \cdot H',w}(W)}) = \dim(\overline{v_{p',w}(W)}).$$ Similarly, since $r$ is finite flat onto its image a component $F$ of a general fiber of $v_{p' \cdot H', w}$ can be identified with a component of a general fiber of $v_{p',w}$. Lemma \[ratfamdivrest\] shows that $\operatorname{chdim}(p' \cdot H'|_{F}) = \operatorname{chdim}(p'|_{F})$. We conclude that $\operatorname{chdim}_{H'}(p' \cdot H') = \operatorname{chdim}_{Y}(p')$.
By Lemma \[stricttransformdivisors\] (1) this implies $\operatorname{chdim}_{H'}(p' \cdot H') = \operatorname{chdim}_{X}(p)$. By Lemma \[stricttransformdivisors\] (2) the Chow dimension of $p' \cdot H'$ agrees with the Chow dimension of $p \cdot H$, giving the desired statement.
\[restricttoamplecor\] Let $X$ be an integral projective variety of dimension $n$ and let $\alpha \in N_{n-1}(X)$. Suppose that $A$ is a very ample Cartier divisor on $X$ such that $\alpha - [A]$ is not pseudo-effective. Let $H \in |A|$ be a general element. Then $$\operatorname{chdim}_{X}(\alpha) \leq \operatorname{chdim}_{H}(\alpha \cdot H).$$
Let $p: U \to W$ be a family of effective divisors representing $\alpha$ with maximal Chow dimension. Apply Proposition \[restricttoample\] to $p$.
Dimensions of families of divisors
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We next develop bounds several dimension estimates that govern the behavior of families of divisors. The goal is to construct bounds that depend only on intersection numbers.
\[basicvarestimate\]
1. Let $X$ be an equidimensional projective variety of dimension $n$ and let $\alpha \in N_{n-1}(X)_{\mathbb{Z}}$. Let $A$ be a Cartier divisor that is the pullback of a very ample divisor by a generically finite map. Then $$\operatorname{chdim}(\alpha) < \left( \begin{array}{c} \alpha \cdot A^{n-1} + n \\ n \end{array} \right)$$
2. Let $X$ be an integral projective variety of dimension $n$ and let $\alpha \in N_{n-1}(X)_{\mathbb{Z}}$. Let $H$ be a very ample divisor such that $\alpha - [H]$ is not pseudo-effective and let $A$ be a Cartier divisor that is the pullback of a very ample divisor by a generically finite map. Then $$\operatorname{chdim}(\alpha) < \left( \begin{array}{c} \alpha \cdot A^{n-2} \cdot H + n-1 \\ n-1 \end{array} \right).$$
\(1) Let $\pi: X \to \mathbb{P}^{n}$ be a generically finite morphism defined by a general subspace of $|A|$. Suppose that $p: U \to W$ is a family of effective divisors representing $\alpha$. Note that any component of $\overline{s(U)}$ that is contracted by $\pi$ must map to a fixed divisor in $X$. By Lemma \[genfinitepushlem\] we see that $\operatorname{chdim}(p) = \operatorname{chdim}(\pi_{*}p)$. But $\pi_{*}p$ is a family of divisors on $\mathbb{P}^{n}$ of degree $\alpha \cdot A^{n-1}$, yielding the result.
\(2) Replace $H$ by a general element in its linear series. By Corollary \[restricttoamplecor\] we have $\operatorname{chdim}_{X}(\alpha) \leq \operatorname{chdim}_{H}(\alpha \cdot H)$ and we conclude by (1).
For later reference we restate these bounds in a simpler (but weaker) form.
\[oldversion\] Let $X$ be an integral projective variety of dimension $n$ and let $p: U \to W$ be a family of effective $(n-1)$-cycles on $X$ representing $\alpha \in N_{k}(X)_{\mathbb{Z}}$.
1. Suppose that $A$ is a very ample Cartier divisor on $X$ and $s$ is a positive integer such that $\alpha \cdot A^{n-1} \leq sA^{n}$. Then $$\operatorname{chdim}(p) < (n+1)s^{n-1}\alpha \cdot A^{n-1}.$$
2. Let $A$ and $H$ be very ample divisors and let $s$ be a positive integer such that $\alpha - [H]$ is not pseudo-effective and $\alpha \cdot A^{n-2} \cdot H \leq s A^{n-1} \cdot H$. Then $$\operatorname{chdim}(p) < n s^{n-2} \alpha \cdot A^{n-2} \cdot H.$$
An easy inductive argument using difference functions shows that for a positive integer $d$ and non-negative integer $n$ we have $$\left( \begin{array}{c} d+ n \\ n \end{array} \right) \leq (n+1)d^{n}.$$
Finally, we prove a similar statement for sections of line bundles.
\[basicsectionestimate\] Let $X$ be an equidimensional projective variety of dimension $n$ and let $A$ be a very ample Cartier divisor on $X$. Then $$h^{0}(X,\mathcal{O}_{X}(A)) \leq (n+1)A^{n}.$$
Computing the cohomology of the exact sequence $$0 \to \mathcal{O}_{X}(-A) \to \mathcal{O}_{X} \to \mathcal{O}_{A} \to 0$$ shows by induction on dimension that $h^{0}(X,\mathcal{O}_{X}) \leq A^{n}$. Computing the cohomology of the exact sequence $$0 \to \mathcal{O}_{X} \to \mathcal{O}_{X}(A) \to \mathcal{O}_{A}(A) \to 0$$ gives the desired statement by induction on dimension.
The variation function {#variationsection}
======================
The variation of a class $\alpha \in N_{k}(X)_{\mathbb{Z}}$ measures the rate of growth of the dimensions of components of $\operatorname{Chow}(X)$ that represent $m\alpha$ as $m$ increases. The main theorem in this section is Theorem \[variationandbigness\] which shows that variation is in some sense a measure of bigness along subvarieties of $X$.
Dimensions of families of cycles
--------------------------------
Before defining the variation, we need to find bounds for the dimension of components of $\operatorname{Chow}(X)$. The following theorem incorporates a suggestion of Voisin who pointed out that the coefficient could be improved by considering a generically finite map to projective space.
\[familydim\] Let $X$ be an equidimensional projective variety of dimension $n$ and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Suppose that $A$ is a very ample divisor on $X$ and set $d = \alpha \cdot A^{k}$. Then we have $$\operatorname{chdim}(\alpha) < (n-k)\left( \begin{array}{c} d + k + 1 \\ k+1 \end{array} \right).$$
Let $p: U \to W$ be a family of effective cycles representing $\alpha$. Since the desired upper bound is superadditive in $d$, by Lemma \[componentsdimest\] we may prove the bound for each irreducible component of $U$ separately. By Lemma \[genreduceddimest\] we may furthermore suppose that $p$ is a reduced family.
Since $p$ is an irreducible family, the image $\overline{s(U)}$ is irreducible, and $\operatorname{chdim}(p)$ is the same whether we consider it as a family of cycles on $X$ or on $\overline{s(U)}$ (with its reduced induced structure) by Lemma \[injectivechowlemma\]. In sum, we have reduced to the case where $X$ is integral and $p$ is a irreducible reduced family such that the map $s: U \to X$ is dominant.
Let $\pi: X \to \mathbb{P}^{n}$ be a generically finite morphism defined by a generic subspace of $|A|$. Lemma \[genfinitepushlem\] shows that $\operatorname{chdim}(p) = \operatorname{chdim}(\pi_{*}p)$. But $\pi_{*}p$ is a family of effective cycles on $\mathbb{P}^{n}$ of degree $d$, so it suffices to consider this case.
Suppose that $q: R \to T$ is a dominant irreducible family of effective $k$-cycles on $\mathbb{P}^{n}$ with degree $d$. We prove the desired bound on $\operatorname{chdim}(q)$ by induction on the codimension $n-k$. The base case $n-k=1$ is clear.
Let $f: Y \to \mathbb{P}^{n}$ be the blow-up of a general closed point and $g: Y \to \mathbb{P}^{n-1}$ the resolution of the projection away from the point. Since $q$ is a dominant irreducible family on $\mathbb{P}^{n}$, Lemma \[genfinitepushlem\] shows that $\operatorname{chdim}(q) = \operatorname{chdim}(q')$ where $q'$ is the strict transform family on $Y$. Let $F$ be a component of a general fiber of $\operatorname{ch}_{g_{*}q'}: T \dashrightarrow \operatorname{Chow}(\mathbb{P}^{n-1})$. Note that the cycles parametrized by $F$ all map to the same $k$-dimensional subvariety $V \subset \mathbb{P}^{n-1}$. Then the restriction $q'|_{F}$ of $q'$ to (an open subset of) $F$ parametrizes a family of $k$-cycles on the variety $g^{-1}(V)$ of pure dimension $k+1$. Altogether $$\begin{aligned}
\operatorname{chdim}(q') & \leq \operatorname{chdim}(g_{*}q') + \operatorname{chdim}(q'|_{F}) \textrm{ by Lemma \ref{dimensionlemma}} \\
& < (n-k-1)\left( \begin{array}{c} d + k+1 \\ k+1 \end{array} \right) + \operatorname{chdim}(q'|_{F}) \textrm{ by induction} \\
& < (n-k-1)\left( \begin{array}{c} d + k+1 \\ k+1 \end{array} \right) + \left( \begin{array}{c} d + k+1 \\ k+1 \end{array} \right) \textrm{ by Proposition \ref{basicvarestimate}}.\end{aligned}$$
Theorem \[familydim\] shows that for any class $\alpha \in N_{k}(X)_{\mathbb{Z}}$, there is some positive constant $C$ such that $\operatorname{chdim}(m\alpha) < Cm^{k+1}$. Furthermore, this order of growth can always be achieved by some class on $X$ as in the following example.
\[bigvariationexample\] Let $H_{1},\ldots,H_{n-k}$ be general very ample divisors on $X$ and set $\alpha = H_{1} \cdot \ldots \cdot H_{n-k}$. Let $V$ denote the scheme-theoretic intersection $H_{1} \cap \ldots \cap H_{n-k-1}$. The linear series $|m(H_{n-k}|_{V})|$ defines a rational family of $k$-cycles representing $m\alpha$. Thus $$\operatorname{rchdim}(m\alpha) \geq (H_{1} \cdot \ldots \cdot H_{n-k-1} \cdot H_{n-k}^{k+1})\frac{m^{k+1}}{(k+1)!} + O(m^{k})$$
Definitions
-----------
Theorem \[familydim\] and Example \[bigvariationexample\] suggest that one should compare the growth rate of $\operatorname{chdim}(m\alpha)$ against $m^{k+1}$. The choice of the coefficient $(k+1)!$ is justified by Example \[cyclesinpn\].
\[variationdefn\] Let $X$ be an integral projective variety. For any $\alpha \in N_{k}(X)_{\mathbb{Z}}$, we define the variation of $\alpha$ to be $$\operatorname{var}(\alpha) := \limsup_{m \to \infty} \frac{\operatorname{chdim}(m\alpha)}{m^{k+1}/(k+1)!}.$$ We define the rational variation of $\alpha$ in the analogous way using $\operatorname{rchdim}(\alpha)$ in place of $\operatorname{chdim}(\alpha)$.
[@nollet97 Corollary 1.6] shows that there are infinitely many components of $\operatorname{Hilb}(\mathbb{P}^{3})$ parametrizing subschemes whose underlying cycle is a double line. Furthermore the dimensions of these components is unbounded. Thus there is no analogue of Theorem \[familydim\] for components of the Hilbert scheme with a fixed underlying cycle class.
Any attempt to formulate an analogue of the variation using the Hilbert scheme will need to either consider only special components of $\operatorname{Hilb}(X)$ or account for all the terms in the Hilbert polynomial.
\[variationofdivisors\] Let $X$ be a normal integral projective variety of dimension $n$ and suppose that $X$ admits a resolution $\phi: Y \to X$ such that the kernel of $\phi_{*}: N_{n-1}(Y) \to N_{n-1}(X)$ is spanned by $\phi$-exceptional divisors. (For example $X$ could be smooth or normal $\mathbb{Q}$-factorial over $\mathbb{C}$.) Then for any Cartier divisor $D$ on $X$ we have $\operatorname{ratvar}([D]) = \operatorname{var}([D]) = \operatorname{vol}(D)$.
To verify this, note that since $\operatorname{chdim}$ is preserved by passing to strict transform families of divisors we have $$\operatorname{chdim}(m[D]) = \max \left\{ \operatorname{chdim}(\beta) \left| \begin{array}{l} \beta \textrm{ is a class on } Y \\ \textrm{with } \phi_{*}\beta = m[D] \end{array} \right. \right\}.$$ Let $L$ denote any divisor in the class $\beta$ attaining this maximum value. We may write $L \equiv m\phi^{*}D + E$ where $E$ is some $\phi$-exceptional divisor. Since increasing the coefficients in $E$ can only increase $\operatorname{chdim}$, we may assume that $E$ is effective. But then $$\begin{aligned}
h^{0}(Y,\mathcal{O}_{Y}(L)) = h^{0}(Y,\mathcal{O}_{Y}(L - E))\end{aligned}$$ by the negativity of contraction lemma (see for example [@nakayama04 III.5.7 Proposition]). Thus $$\begin{aligned}
h^{0}(X,\mathcal{O}_{X}&(mD))-1 \leq \operatorname{rchdim}(m[D]) \leq \operatorname{chdim}(m[D]) \\
& \leq \dim \operatorname{Pic}^{0}(Y) + \max_{D' \equiv m\phi^{*}D} h^{0}(Y,\mathcal{O}_{Y}(D')) - 1.\end{aligned}$$ While the rightmost term may be greater than $h^{0}(X,\mathcal{O}_{X}(mD))-1$, the difference is bounded by a polynomial of degree $n-1$ in $m$ (see the proof of [@lazarsfeld04 Proposition 2.2.43]). Thus $\operatorname{ratvar}([D])$ and $\operatorname{var}([D])$ agree with the volume.
\[0cyclevol\] Suppose that $X$ is an integral projective variety of dimension $n$. There is an isomorphism $\deg \!: N_{0}(X)_{\mathbb{Z}} \to \mathbb{Z}$. For $\alpha \in N_{0}(X)_{\mathbb{Z}}$ of positive degree $\operatorname{chdim}(\alpha) = n\deg(\alpha)$ so that $$\operatorname{var}(\alpha) = n\deg(\alpha)$$
The behavior of $\operatorname{ratvar}(\alpha)$ is somewhat more subtle. For simplicity suppose that $\alpha$ is the positive generator of $N_{0}(X)_{\mathbb{Z}}$. A result of [@roitman72] shows that there are non-negative integers $d(X), j(X)$ such that for sufficiently large $m$ $$\operatorname{rchdim}(m\alpha) \geq m(n - d(X)) - j(X).$$ This gives a lower bound on $\operatorname{ratvar}$. However, since $\operatorname{ratvar}$ calculates the maximal variation (and not the “minimum variation” as in [@roitman72]), it may happen that $\operatorname{ratvar}(\alpha) > n-d(X)$. At the very least we know that $\operatorname{ratvar}(\alpha)$ is always positive. In fact, by considering families of points lying on a fixed curve $C \subset X$ we see that $\operatorname{ratvar}(\alpha) \geq 1$.
\[cyclesinpn\] Let $h \in N_{k}(\mathbb{P}^{n})$ denote the class of a $k$-dimensional hyperplane. By analogy with the case of curves, [@eh92] predicts that for sufficiently large $d$ the maximal dimension of a component of $\operatorname{Chow}(\mathbb{P}^{n})$ representing $dh$ parametrizes degree $d$ hypersurfaces in a $(k+1)$-dimensional hyperplane. The expected value of $\operatorname{chdim}(dh)$ is thus $$\left( \begin{array}{c} d+k+1 \\ k+1 \end{array} \right) - 1 + (k+2)(n-k-1)$$ which would yield $\operatorname{var}(h)=1$. Note that this conjecture would yield a better bound in Theorem \[familydim\].
The components of $\operatorname{Chow}(X)$ of maximal dimension tend to parametrize degenerate subvarieties. To measure the bigness of a class, one should instead only consider the dimensions of components of $\operatorname{Chow}(X)$ that are “general” in some sense. This intuition is captured by the mobility function defined in Section \[mobilitysection\]; however, it would be interesting to see a formulation using the Chow variety directly. For curves on $\mathbb{P}^{3}$, [@perrin87] conjectures that calculating the dimensions of components of $\operatorname{Chow}(X)$ parametrizing “general” curves of degree $d$ – in the sense that the corresponding cycles are not contained in any hypersurface of degree $<d^{1/2}$ – will yield the value of the mobility function.
Basic properties
----------------
We next verify some of the basic properties of the variation.
\[rescalingvariation\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Then for any positive integer $c$ we have $\operatorname{var}(c\alpha) = c^{k+1}\operatorname{var}(\alpha)$ (and similarly for $\operatorname{ratvar}$).
If $\operatorname{chdim}(\alpha) > 0$ then $\alpha$ is represented by an effective cycle $Z$. Thus $\operatorname{chdim}(\alpha + \beta) \geq \operatorname{chdim}(\beta)$ for any class $\beta$: if $p$ is a family of effective cycles of class $\beta$ then we can add the constant cycle $Z$ to $p$ (using the family sum construction) to obtain a family representing $\alpha + \beta$ with the same Chow dimension. We conclude by Lemma \[lazlemma\].
Lemma \[rescalingvariation\] allows us to extend the definition of variation to any $\mathbb{Q}$-class by homogeneity. Thus we obtain a function $$\operatorname{var}: N_{k}(X)_{\mathbb{Q}} \to \mathbb{R}_{\geq 0}.$$
\[variationincreases\] Let $X$ be an integral projective variety. Suppose that $\alpha, \beta \in N_{k}(X)_{\mathbb{Q}}$ are classes such that some positive multiple of each is represented by an effective cycle. Then $\operatorname{var}(\alpha + \beta) \geq \operatorname{var}(\alpha) + \operatorname{var}(\beta)$ (and similarly for $\operatorname{ratvar}$).
Note that we may check the inequality after rescaling $\alpha$ and $\beta$ by the same positive integer $c$. Thus we may suppose that every multiple of $\alpha$ and $\beta$ is represented by an effective cycle.
Suppose that $p: U \to W$ is a family representing $m\alpha$ and $q: S \to T$ is a family representing $m\beta$. Then the family sum $p+q$ represents $m(\alpha+\beta)$. Lemma \[chdimfamilysumlem\] shows that $$\operatorname{chdim}(p+q) = \operatorname{chdim}(p) + \operatorname{chdim}(q)$$ and the desired inequality follows.
By Example \[bigvariationexample\], we find:
\[bigvarcor\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Q}}$ be a big class. Then $\operatorname{var}(\alpha) > 0$.
As a consequence, we see that $\operatorname{var}$ is a continuous function on the big cone.
Let $X$ be an integral projective variety. The function $\operatorname{var}: N_{k}(X)_{\mathbb{Q}} \to \mathbb{R}_{\geq 0}$ is locally uniformly continuous on the interior of $\operatorname{\overline{Eff}}_{k}(X)_{\mathbb{Q}}$ (and similarly for $\operatorname{ratvar}$).
$\operatorname{var}$ verifies conditions (1)-(3) of Lemma \[easyconelem\].
The behavior of the variation along the pseudo-effective boundary is more subtle. Probably the most one can hope for is:
Let $X$ be an integral projective variety. Is the function $\operatorname{var}: \operatorname{\overline{Eff}}_{k}(X)_{\mathbb{Q}} \to \mathbb{R}_{\geq 0}$ upper semi-continuous?
By analogy with the volume, the variation should satisfy some form of concavity on the big cone. The following conjecture is a weak statement in this direction. It is easy to show that the conjecture would imply the upper semi-continuity of $\operatorname{var}$ as a function on $\operatorname{\overline{Eff}}_{k}(X)_{\mathbb{Q}}$.
\[volconvexconj\] Let $X$ be an integral projective variety. Suppose that $\alpha,\beta,\gamma \in N_{k}(X)_{\mathbb{Z}}$ are classes with $\alpha$ pseudo-effective and $\beta$ and $\gamma$ big. Then $$\operatorname{chdim}(\alpha + \beta) - \operatorname{chdim}(\alpha) \leq \operatorname{chdim}(\alpha + \beta + \gamma) - \operatorname{chdim}(\alpha + \gamma).$$
Finally we note that variation behaves well with respect to inclusions of subvarieties.
\[variationinclusionlem\] Let $X$ be an integral projective variety and $i: W \to X$ an integral closed subvariety. For any class $\beta \in N_{k}(W)_{\mathbb{Q}}$ we have $\operatorname{var}(\beta) \leq \operatorname{var}(i_{*}\beta)$ (and similarly for $\operatorname{ratvar}$).
Let $p$ be a family of effective cycles on $W$ and consider the push-forward family $q$ on $X$. Recall that for a general cycle-theoretic fiber $Z$ of $p$ the corresponding cycle in the push-forward family is just $i_{*}Z$; thus Lemma \[injectivechowlemma\] shows that $\operatorname{chdim}(p) = \operatorname{chdim}(q)$ and the result follows.
Variation, connecting chains, and bigness {#connectionsandbignesssection}
-----------------------------------------
Example \[p3blowup\] shows that a class may have positive variation even when it is not big. This class is constructed by pushing forward a big class on a subvariety. In this section we show that every class with positive variation arises in this way.
The main step in the proof is to develop a criterion for bigness of a class using connecting chains of cycles. This criterion is modeled on [@8authors Theorem 2.4] which describes big curve classes via connecting chains. The correct analogue in higher dimensions should require that the cycles in our chain intersect “positively” in some sense. The next theorem shows that such a statement holds under a very strong positivity condition.
\[stronglyampleconnectingdefn\] Let $X$ be an integral projective variety of dimension $n$ and let $p: U \to W$ be a family of effective $k$-cycles. We say that $p$ is strongly big-connecting if $s: U \to X$ is dominant and there is a big effective divisor $B$ on $X$ such that every $p$-horizontal component of $s^{*}B$ is contracted to a subvariety of $X$ of dimension at most $k-1$.
The following lemma is in preparation for Theorem \[stronglyampleisbig\].
\[stramphelp\] Let $p: U \to W$ be a flat map of projective varieties of relative dimension $k$ with $W$ integral. Let $A$ be a big effective Cartier divisor on $U$. There is a big effective $k$-cycle $Z$ on $U$ such that every component of $\operatorname{Supp}(Z)$ is contained either in a fiber of $p$ or in a $p$-horizontal component of $A$.
Set $n=\dim U$. The proof is by induction on the codimension $n-k$. For the base case $n-k=1$ we can simply take $Z=A$.
For the general case, note that by Lemma \[intlem\] (3) there is a big effective cycle $V$ on $U$ contained in $\operatorname{Supp}(A)$. Let $D$ denote the support of the $p$-vertical components of $A$ and let $V'$ denote the part of $V$ whose support is not contained in any $p$-horizontal component of $A$.
Choose a very ample divisor $H$ on $W$ sufficiently positive so that $p^{*}H - D$ is linearly equivalent to an effective divisor. For $H$ general, and hence integral, we can apply the induction hypothesis to $p: p^{*}H \to H$ and $A|_{p^{*}H}$ to obtain a big effective $k$-cycle $Z'$ on $p^{*}H$ satisfying the support condition. In particular, for an ample divisor $\tilde{H}$ on $U$ there is some $c$ sufficiently large so that $c[Z'] \succeq p^{*}H \cdot \tilde{H}^{n-k-1}$. This shows that some multiple of $[Z']$ will also dominate any effective cycle supported in $D$, so for some $c'$ we have $c'[Z'] \succeq [V']$. Set $Z = c'Z' + (V-V')$.
\[stronglyampleisbig\] Let $X$ be an integral projective variety. A class $\alpha \in N_{k}(X)$ is big if and only if there is some strongly big-connecting family of effective $k$-cycles $p: U \to W$ with class $\beta$ and a positive constant $c$ such that $c\alpha - \beta$ is pseudo-effective.
We first prove the forward implication. It suffices to construct an example of a strongly big-connecting family of effective $k$-cycles on $X$. Fix a very ample divisor $A$ on $X$ and an $(n-k+1)$-dimensional subspace $V \subset |A|$. Consider the family of effective $k$-cycles $p: U \to \mathbb{P}(V^{\vee})$ defined by taking intersections of $(n-k)$ general elements of $V$. Choose an element $A \in |V|$. A general cycle in our family $p$ only intersects $A$ along the base locus of $V$; thus any $p$-horizontal component of $A$ must be mapped under $s$ to the base-locus of $V$ which has dimension $k-1$. So $p$ is a strongly big-connecting family for the divisor $A$.
Conversely, it suffices to show that a strongly big-connecting family $p$ has big class $\beta$. We may restrict our family $p$ to (an open subset of) a general complete intersection of very ample divisors on $W$ to ensure that $\dim U = \dim X$ without changing the strongly big-connecting property. We may then modify $p$ as in Lemma \[goodfamilymodification\] to make $U$ and $W$ projective without changing the strongly big-connecting property.
Then $s^{*}A$ is a big effective Cartier divisor on $U$. Lemma \[stramphelp\] shows that there is a big effective $k$-cycle $Z$ on $U$ whose support is contained in fibers of $p$ and in $p$-horizontal components of $s^{*}A$. The former components are dominated by the cycle-theoretic fibers of $p$; the latter push forward to $0$. Thus there is some constant $d$ such that $d\beta - {s}_{*}[Z]$ is a pseudo-effective class. Since $s_{*}[Z]$ is a big class on $X$ by Lemma \[surjpushforward\], we find that $\beta$ is also a big class.
Suppose that $X$ is a smooth variety and $D$ is an irreducible divisor. This proposition is similar to the fact that $D$ is big if $D|_{D}$ is ample (see [@voisin10 Lemma 2.3]).
\[variationandbigness\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Q}}$. Then the following conditions are equivalent:
1. $\operatorname{ratvar}(\alpha)>0$.
2. $\operatorname{var}(\alpha) > 0$.
3. There is an integral $(k+1)$-dimensional projective variety $Y$, a big class $\beta \in N_{k}(Y)_{\mathbb{Q}}$, and a morphism $f: Y \to X$ that is generically finite onto its image such that some multiple of $\alpha - f_{*}\beta$ is represented by an effective cycle.
The proof shows that Theorem \[variationandbigness\] is also true if the morphism $f$ in (3) is instead required to be a closed immersion.
The case when $k=0$ is explained in Example \[0cyclevol\], so we may assume $k \geq 1$.
$(1) \implies (2)$ is obvious.
We next show $(2) \implies (3)$. Suppose that $\operatorname{var}(\alpha) > 0$. We may rescale $\alpha$ so that $\alpha \in N_{k}(X)_{\mathbb{Z}}$ and every positive multiple of $\alpha$ is represented by an effective cycle. Fix a very ample Cartier divisor $A$, and choose some positive integer $m$ sufficiently large so that $$\operatorname{chdim}(m\alpha) > (n-k+1)\left( \begin{array}{c} m\alpha \cdot A^{k} + k \\ k \end{array} \right).$$ Let $p: U \to W$ denote a family of effective $k$-cycles that has maximal Chow dimension among all the families representing $m\alpha$. Denote the projection map to $X$ by $s: U \to X$. By replacing $A$ by a linearly equivalent divisor, we may suppose that $A$ that does not contain any component of $s(U)$.
Let $q: R \to W^{0}$ denote the intersection family of $p$ with $A$ as in Construction \[intersectionfamilyconstr\] (where $W^{0}$ is an appropriately chosen open set of $W$). The family $q$ has class $\beta := m\alpha \cdot A \in N_{k-1}(X)$. By Theorem \[familydim\], we have $$\operatorname{chdim}(\beta) \leq (n-k+1)\left( \begin{array}{c} m\alpha \cdot A^{k} + k \\ k \end{array} \right) < \operatorname{chdim}(p).$$ Thus there is a curve $T \subset W^{0}$ through a general point of $W$ that is contracted by $\operatorname{ch}_{q}$ but not by $\operatorname{ch}_{p}$. Let $q_{T}: R_{T} \to T$ denote the restriction of the family to (an open subset of) $T$. Using Lemma \[goodfamilymodification\] we may extend the family $q_{T}$ to a projective closure of $T$; from now on we let $q_{T}$ denote this family over a projective base.
By Lemma \[injectivechowlemma\] there is some irreducible component $R'$ of $R_{T}$ whose $s$-image in $X$ has dimension $k+1$. Set $Y = s(R')$ with the reduced induced structure and $f: Y \to X$ the corresponding closed immersion. By construction every $q_{T}$-horizontal component of $s^{*}A$ on $R'$ has $s$-image of dimension at most $k-1$. Thus $q_{T}': R' \to T$ defines a strongly big-connecting family of divisors on $Y$ with respect to $A$. If we set $\beta'$ to be the class on $Y$ of the family $q_{T}'$, then $\beta'$ is a big class on $Y$ by Theorem \[stronglyampleisbig\]. Furthermore $m\alpha - f_{*}\beta'$ is the class of an effective cycle. Setting $\beta = \frac{1}{m}\beta'$ finishes the second implication.
To show $(3) \implies (1)$, suppose that there is a generically finite morphism $f: Y \to X$ from an integral $(k+1)$-dimensional projective variety $Y$ and a big class $\beta \in N_{k}(Y)_{\mathbb{Q}}$ so that some multiple of $\alpha - f_{*}\beta$ is represented by an effective cycle. Let $V = f(Y)$ with the induced reduced structure, so we have morphisms $f': Y \to V$ and $i: V \subset X$. By Lemma \[surjpushforward\] $f'_{*}\beta$ is big on $V$ so that $\operatorname{ratvar}(f'_{*}\beta) > 0$. By Lemma \[variationinclusionlem\] $\operatorname{ratvar}(i_{*}f'_{*}\beta) > 0$. Thus $\operatorname{ratvar}(\alpha) > 0$ as well.
The mobility function {#mobilitysection}
=====================
As suggested by [@delv11], we will define the mobility of a class $\alpha \in N_{k}(X)_{\mathbb{Z}}$ to be the asymptotic growth rate of the number of general points contained in cycles representing multiples of $\alpha$. We prove that big classes are precisely those with positive mobility, confirming [@delv11 Conjecture 6.5]. Recall that by Convention \[dimconv\] we only consider $k$-cycles for $0 \leq k < \dim X$.
Mobility count
--------------
The mobility count of a family of effective cycles can be thought of informally as a count of how many general points of $X$ are contained in members of the family. Although we are mainly interested in families of cycles, it will be helpful to set up a more general framework.
\[mcdefn\] Let $X$ be an integral projective variety and let $W$ be a reduced variety. Suppose that $U\subset W \times X$ is a subscheme and let $p: U \to W$ and $s: U \to X$ denote the projection maps. The mobility count $\operatorname{mc}(p)$ of the morphism $p$ is the maximum non-negative integer $b$ such that the map $$U \times_{W} U \times_{W} \ldots \times_{W} U \xrightarrow{s \times s \times \ldots \times s} X \times X \times \ldots \times X$$ is dominant, where we have $b$ terms in the product on each side. (If the map is dominant for every positive integer $b$, we set $\operatorname{mc}(p) = \infty$.)
For $\alpha \in N_{k}(X)_{\mathbb{Z}}$, the mobility count of $\alpha$, denoted $\operatorname{mc}(\alpha)$, is defined to be the largest mobility count of any family of effective cycles representing $\alpha$. We define the rational mobility count $\operatorname{rmc}(\alpha)$ in the analogous way by restricting our attention to rational families.
\[mcandvar\] Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective $k$-cycles on $X$. Then $\operatorname{mc}(p) \leq (\dim W)/(\dim X - k)$. Indeed, if the map of Definition \[mcdefn\] is dominant then dimension considerations show that $\operatorname{mc}(p)k + \dim W \geq \operatorname{mc}(p) \dim X$.
\[amplemc\] Let $X$ be an integral projective variety and let $A$ be a very ample divisor such that $h^{i}(X,\mathcal{O}_{X}(A)) = 0$ for every $i>0$. For any positive integer $s$, one can construct by induction a collection of $s$ distinct reduced closed points $P_{s} \subset X$ with $$h^{1}(X,I_{P_{s}}(A)) = \max \{ 0 , s - h^{0}(X,\mathcal{O}_{X}(A)) \}.$$ Furthermore this is the maximal value of $h^{1}(X,I_{P}(A))$ for any collection of $s$ distinct closed points $P$.
Let $p: U \to W$ be the family of hyperplanes in $|A|$. Then $\operatorname{rmc}(p) = h^{0}(X,\mathcal{O}_{X}(A)) - 1$, since this is the largest number of points for which we are guaranteed to have $h^{0}(X,I_{p}(A)) > 0$.
\[fibercontainedmc\] Let $X$ be an integral projective variety. Let $W$ be a reduced variety and let $p: U \to W$ denote a closed subscheme of $W \times X$. Suppose that $T$ is another reduced variety and $q: S \to T$ is a closed subscheme of $T \times X$ such that every fiber of $p$ over a closed point of $W$ is contained in a fiber of $q$ over some closed point of $T$ (as subsets of $X$). Then $\operatorname{mc}(p) \leq \operatorname{mc}(q)$.
The conditions imply that for any $b>0$, the $s^{b}$-image of any fiber of $p^{b}: U^{\times_{W} b} \to W$ over a closed point of $W$ is set-theoretically contained in the image of a fiber of $q^{b}: S^{\times_{T} b} \to T$ over a closed point of $T$ (as subsets of $X^{\times b}$). The statement follows.
\[opensubsetsforcycles\] Let $X$ be an integral projective variety.
1. Let $W$ be an integral variety and let $U \subset W \times X$ be a closed subscheme such that $p: U \to W$ is flat. For an open subvariety $W^{0} \subset W$ let $p^{0}: U^{0} \to W^{0}$ be the base change to $W^{0}$. Then $\operatorname{mc}(p) = \operatorname{mc}(p^{0})$.
2. Let $p: U \to W$ be a family of effective cycles on $X$. For an open subvariety $W^{0} \subset W$ let $p^{0}: U^{0} \to W^{0}$ be the restriction family. Then $\operatorname{mc}(p) = \operatorname{mc}(p^{0})$.
3. Let $W$ be a normal integral variety and let $U \subset W \times X$ be a closed subscheme such that:
- Every fiber of the first projection map $p: U \to W$ has the same dimension.
- Every component of $U$ dominates $W$ under $p$.
Let $W^{0} \subset W$ be an open subset and $p^{0}: U^{0} \to W^{0}$ be the preimage of $W^{0}$. Then $\operatorname{mc}(p) = \operatorname{mc}(p^{0})$.
Proposition \[opensubsetsforcycles\] indicates that the mobility count is insensitive to the choice of definition of a family of effective cycles.
\(1) The map $p^{b}: U^{\times_{W} b} \to W$ is proper flat, so that every component of $U^{\times_{W} b}$ dominates $W$. Then $(U^{0})^{\times_{W^{0}} b}$ is dense in $U^{\times_{W} b}$ for any $b$. Thus $\operatorname{mc}(p^{0}) = \operatorname{mc}(p)$.
\(2) Let $\{ U_{i} \}$ denote the irreducible components of $U$. Every irreducible component of $U^{\times_{W} b}$ is contained in a product of the $U_{i}$ over $W$. Since each $p|_{U_{i}}: U_{i} \to W$ is flat, we can apply the same argument as in (1).
\(3) The inequality $\operatorname{mc}(p) \geq \operatorname{mc}(p^{0})$ is clear. To show the converse inequality, we may suppose that $U$ is reduced. We may also shrink $W^{0}$ and assume that $p^{0}$ is flat.
Let $p': U' \to W'$ be a flattening of $p$ via the birational morphism $\phi: W' \to W$. We may ensure that $\phi$ is an isomorphism over $W^{0}$. Choose a closed point $w \in W$ and let $T \subset W'$ be the set-theoretic preimage. Since $W$ is normal $T$ is connected.
Choose a closed point $w' \in T$. By construction the fiber $U'_{w'}$ is set theoretically contained in $U_{w}$ (as subsets of $X$). Since they have the same dimension, $U'_{w'}$ is a union of components of $U_{w}$. Since $p'$ is flat over $T$ and $T$ is connected, in fact $U'_{w'}$ and $U_{w}$ have the same number of components and thus are set-theoretically equal. Applying Lemma \[fibercontainedmc\] and part (1) we see that $\operatorname{mc}(p) \leq \operatorname{mc}(p') = \operatorname{mc}(p^{0})$.
We can now describe how the mobility count changes under certain geometric constructions of families of cycles.
\[ignorecomponents\] Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective $k$-cycles. Suppose that $U$ has a component $U_{i}$ whose image in $X$ is contained in a proper subvariety. Then $\operatorname{mc}(p) = \operatorname{mc}(p')$ where $p'$ is the family defined by removing $U_{i}$ from $U$.
This is immediate from the definition.
\[stricttransformmc\] Let $\psi: X \dashrightarrow Y$ be a birational morphism of integral projective varieties. Let $p: U \to W$ be a family of effective $k$-cycles on $X$ and let $p'$ denote the strict transform family on $Y$. Then $\operatorname{mc}(p) = \operatorname{mc}(p')$.
By Lemma \[ignorecomponents\] we may assume that every component of $U$ dominates $X$. Using Proposition \[opensubsetsforcycles\] (2), we may replace $p$ by the restricted family $p^{0}: U^{0} \to W^{0}$, where $W^{0}$ is the locus of definition of the strict transform family $p': U' \to W^{0}$. The statement is then clear using the fact that the morphisms $(U^{0})^{\times_{W^{0}}b} \to X^{\times b}$ and $(U')^{\times_{W^{0}}b} \to Y^{\times b}$ are birationally equivalent for every $b$.
\[familysummc\] Let $X$ be an integral projective variety. Suppose that $W$ and $T$ are reduced varieties and that $p_{1}: U \to W$ and $p_{2}: S \to T$ are closed subschemes of $W \times X$ and $T \times X$ respectively. Let $q: V \to W \times T$ denote the subscheme $$U \times T \cup W \times S \subset W \times T \times X.$$ Then $\operatorname{mc}(q) = \operatorname{mc}(p_{1}) + \operatorname{mc}(p_{2})$.
In particular, if $p_{1}$ and $p_{2}$ are families of effective $k$-cycles, then the mobility count of the family sum is the sum of the mobility counts.
Set $b_{1} = \operatorname{mc}(p_{1})$ and $b_{2} = \operatorname{mc}(p_{2})$. There is a dominant projection map $$\left( U^{\times_{W} b_{1}} \times T \right) \times_{W \times T} \left( W \times S^{\times_{T} b_{2}} \right) \to X^{\times (b_{1} + b_{2})}.$$ Since the domain is naturally a subscheme of $V^{\times_{W \times T} b_{1} + b_{2}}$, we obtain $\operatorname{mc}(q) \geq \operatorname{mc}(p_{1}) + \operatorname{mc}(p_{2})$.
Conversely, any irreducible component of $V^{\times_{W \times T} c}$ is (up to reordering the terms) a subscheme of $$\left( U^{\times_{W} c_{1}} \times T \right) \times_{W \times T} \left( W \times S^{\times_{T} c_{2}} \right)$$ for some non-negative integers $c_{1}$ and $c_{2}$ with $c = c_{1} + c_{2}$ where the map to $X^{\times b}$ is component-wise. This yields the reverse inequality.
To extend the lemma to the family sum, first replace $p_{1}$ and $p_{2}$ by their restrictions to the normal locus of $W$ and $T$ respectively; this does not change the mobility count by Proposition \[opensubsetsforcycles\] (2). Then by Proposition \[opensubsetsforcycles\] (3) the mobility count of the family sum is the same as the mobility count of the subscheme $U \times T \cup W \times S$ as defined in Construction \[familysumconstr\].
\[mobilitydecompositioncor\] Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective $k$-cycles. Let $p_{i}: U_{i} \to W$ denote the irreducible components of $U$. Then $\operatorname{mc}(p) \leq \sum_{i} \operatorname{mc}(p_{i})$.
Let $q: S \to T$ denote the family sum of the $p_{i}$. By Lemma \[fibercontainedmc\] we have $\operatorname{mc}(p) \leq \operatorname{mc}(q)$; by Lemma \[familysummc\] $\operatorname{mc}(q) = \sum_{i} \operatorname{mc}(p_{i})$.
The mobility function {#the-mobility-function}
---------------------
As indicated by [@delv11], we have:
\[mcbound\] Let $X$ be an integral projective variety of dimension $n$ and let $\alpha \in N_{k}(X)$. Fix a very ample divisor $A$ and choose a positive constant $c<1$ so that $h^{0}(X,mA) \geq \lfloor c m^{n} \rfloor$ for every positive integer $m$. Then any family $p: U \to W$ representing $\alpha$ has $$\operatorname{mc}(p) \leq (n+1)2^{n} \left(\frac{2(k+1)}{c} \right)^{\frac{n}{n-k}} (\alpha \cdot A^{k})^{\frac{n}{n-k}}A^{n}.$$ In particular, there is some constant $C$ so that $\operatorname{mc}(m\alpha) \leq Cm^{\frac{n}{n-k}}$.
We will develop a similar bound that does not depend on the constant $c$ in Theorem \[mobprecisebound\].
By Lemma \[basicsectionestimate\], the support $Z$ of any effective cycle representing $\alpha$ satisfies $$h^{0}(X,\mathcal{I}_{Z}(dA)) \geq \lfloor c d^{n} \rfloor - (k+1)d^{k}(\alpha \cdot A^{k})$$ for any positive integer $d$. Thus an effective cycle representing $\alpha$ is set-theoretically contained in an element of $|\lceil d \rceil A|$ as soon as $d$ is sufficiently large to make the right hand side greater than $1$, and in particular, for $$d = \left( \frac{2(k+1)}{c} \right)^{\frac{1}{n-k}} (\alpha \cdot A^{k})^{\frac{1}{n-k}}.$$ Let $q: \tilde{U} \to \mathbb{P}(| \lceil d \rceil A|)$ denote the family of divisors defined by the linear series. By Lemma \[fibercontainedmc\] we have $\operatorname{mc}(p) \leq \operatorname{mc}(q)$. Since $c<1$, $d \geq 1$ so that $\lceil d \rceil < 2d$. Applying Lemma \[basicsectionestimate\] again, Example \[amplemc\] indicates that $$\operatorname{mc}(p) \leq h^{0}(X,\lceil d \rceil A) -1 < (n+1)2^{n} \left(\frac{2(k+1)}{c} \right)^{\frac{n}{n-k}} (\alpha \cdot A^{k})^{\frac{n}{n-k}}A^{n}.$$
Furthermore, the growth rate of $Cm^{\frac{n}{n-k}}$ is always achieved by some big class as demonstrated by the next example.
\[bigmobpos\] Let $X$ be an integral projective variety and let $A$ be a very ample divisor such that $h^{i}(X,\mathcal{O}_{X}(mA)) = 0$ for every $i>0, m >0$ and $h^{0}(X,\mathcal{O}_{X}(mA))>0$. Choose a positive constant $c$ such that we have $h^{0}(X,\mathcal{O}_{X}(mA)) > c m^{n} $ for every positive integer $m$.
Let $P \subset X$ be any collection of $k$ reduced closed points on $X$. Example \[amplemc\] shows that the ideal sheaf $I_{P}$ is $m$-regular as soon as $c(m-n)^{n}>k$. In particular, for large $m$ we can find a complete intersection of $n$ elements of $|mA|$ that has dimension $0$ and contains any $\lfloor c(m-n)^{n}\rfloor$ closed points of $X$. Setting $\alpha = A^{n-k}$, it is then clear that $$\operatorname{rmc}(m^{n-k} \alpha) \geq \lfloor c(m-n)^{n} \rfloor.$$
Proposition \[mcbound\] and Example \[bigmobpos\] indicate that we should make the following definition.
\[mobdefn\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. The mobility of $\alpha$ is $$\operatorname{mob}_{X}(\alpha) = \limsup_{m \to \infty} \frac{\operatorname{mc}(m\alpha)}{m^{n/n-k}/n!}.$$ We will omit the subscript $X$ when the ambient variety is clear from the context. We define the rational mobility $\operatorname{ratmob}(\alpha)$ in an analogous way using $\operatorname{rmc}$.
The coefficient $n!$ is justified by Section \[p3mobility\]. We verify in Example \[divisormob\] that the mobility agrees with the volume function for Cartier divisors on a smooth integral projective variety.
\[mobzerocycles\] Let $X$ be an integral projective variety of dimension $n$ and let $\alpha \in N_{0}(X)$ be the class of a point. Then $\operatorname{mc}(m\alpha) = m$, so that the mobility of the point class is $n!$. Rational mobility is more interesting; we will analyze it in more detail in Section \[ratmob0cycles\].
Basic properties
----------------
We now turn to the basic properties of the mobility function.
\[rescalingmob\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Fix a positive integer $a$. Then $\operatorname{mob}(a\alpha) = a^{\frac{n}{n-k}}\operatorname{mob}(\alpha)$ (and similarly for $\operatorname{ratmob}$).
If $\operatorname{mc}(\alpha) > 0$ then $\alpha$ is represented by a family of effective cycles. Thus $\operatorname{mc}(\alpha + \beta) \geq \operatorname{mc}(\beta)$ for any class $\beta$ by the additivity of mobility count under family sums as in Lemma \[familysummc\]. Then apply Lemma \[lazlemma\].
Lemma \[rescalingmob\] allows us to extend the definition of mobility to any $\mathbb{Q}$-class by homogeneity, obtaining a function $\operatorname{mob}: N_{k}(X)_{\mathbb{Q}} \to \mathbb{R}_{\geq 0}$.
\[divisormob\] Let $X$ be a normal integral projective variety of dimension $n$ and suppose that $X$ admits a resolution $\phi: Y \to X$ such that the kernel of $\phi_{*}: N_{n-1}(Y) \to N_{n-1}(X)$ is spanned by $\phi$-exceptional divisors. (For example $X$ could be smooth or normal $\mathbb{Q}$-factorial over $\mathbb{C}$.) For any Cartier divisor $L$ on $X$ we have $$\operatorname{ratmob}([L]) = \operatorname{mob}([L]) = \operatorname{vol}(L).$$ To prove this, note first that by Example \[mcandvar\] we have $$\operatorname{rmc}(m[L]) \leq \operatorname{mc}(m[L]) \leq \operatorname{chdim}(m[L])$$ so that by Example \[variationofdivisors\] $\operatorname{vol}(L) = \operatorname{var}([L]) \geq \operatorname{mob}([L]) \geq \operatorname{ratmob}([L])$.
We first show the converse inequality when $L$ is ample. After rescaling, we may suppose that $L$ is very ample and that $h^{i}(X,\mathcal{O}_{X}(mL)) = 0$ for every $i>0,m>0$. Example \[amplemc\] indicates that $h^{0}(X,\mathcal{O}_{X}(mL)) - 1 \leq \operatorname{rmc}(m[L])$, showing that $\operatorname{ratmob}(L) = \operatorname{vol}(L)$ in this case. By homogeneity, we also obtain equality for any ample $\mathbb{Q}$-divisor $A$.
Suppose now that $L$ is big. Let $\phi: Y \to X$ be an $\epsilon$-Fujita approximation for $L$ (constructed by [@takagi07] in arbitrary characteristic), so that there is an ample $\mathbb{Q}$-divisor $A$ and an effective $\mathbb{Q}$-divisor $E$ on $Y$ with $\phi^{*}L \equiv A + E$ and $\operatorname{vol}(A) > \operatorname{vol}(L) - \epsilon$. Then $$\begin{aligned}
\operatorname{vol}(L) - \epsilon & < \operatorname{vol}(A) = \operatorname{ratmob}([A]) \\
& \leq \operatorname{ratmob}(f_{*}[A]) \leq \operatorname{ratmob}([L]).\end{aligned}$$ Since $\epsilon > 0$ is arbitrary, we obtain the desired equalities.
\[mobadditive\] Let $X$ be an integral projective variety. Suppose that $\alpha, \beta \in N_{k}(X)_{\mathbb{Q}}$ are classes such that some positive multiple of each is represented by an effective cycle. Then $\operatorname{mob}(\alpha + \beta) \geq \operatorname{mob}(\alpha) + \operatorname{mob}(\beta)$ (and similarly for $\operatorname{ratmob}$).
We may verify the inequality after rescaling $\alpha$ and $\beta$ by the same positive constant $c$. Thus we may suppose that every multiple of each is represented by an effective class. Using the additivity of mobility counts under family sums as in Lemma \[familysummc\], we see that $$\operatorname{mc}(m(\alpha + \beta)) \geq \operatorname{mc}(m\alpha) + \operatorname{mc}(m\beta)$$ and the conclusion follows.
By Example \[bigmobpos\], we find:
\[bigposcor\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Q}}$ be a big class. Then $\operatorname{mob}(\alpha) > 0$ (and similarly for $\operatorname{ratmob}$).
\[mobcontbigcone\] Let $X$ be an integral projective variety. The function $\operatorname{mob}: N_{k}(X)_{\mathbb{Q}} \to \mathbb{R}_{\geq 0}$ is locally uniformly continuous on the interior of $\operatorname{\overline{Eff}}_{k}(X)_{\mathbb{Q}}$ (and similarly for $\operatorname{ratmob}$).
Theorem \[mobcontinuous\] extends this result to prove that $\operatorname{mob}$ is continuous on all of $N_{k}(X)$.
The conditions (1)-(3) in Lemma \[easyconelem\] are verified by Lemma \[rescalingmob\], Lemma \[mobadditive\], and Corollary \[bigposcor\].
The mobility should also have good concavity properties. Here is a strong conjecture in this direction:
\[mobconvexconj\] Let $X$ be an integral projective variety. Then $\operatorname{mob}$ is a log-concave function on $\operatorname{\overline{Eff}}_{k}(X)$: for any classes $\alpha, \beta \in \operatorname{\overline{Eff}}_{k}(X)$ we have $$\operatorname{mob}(\alpha + \beta)^{\frac{n-k}{n}} \geq \operatorname{mob}(\alpha)^{\frac{n-k}{n}} + \operatorname{mob}(\beta)^{\frac{n-k}{n}}$$
Mobility and bigness
--------------------
We now show that big cycles are precisely those with positive mobility:
\[mobilityandbigness\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Q}}$. The following statements are equivalent:
1. $\alpha$ is big.
2. $\operatorname{ratmob}(\alpha) > 0$.
3. $\operatorname{mob}(\alpha) > 0$.
The implication (1) $\implies$ (2) follows from Corollary \[bigposcor\] and (2) $\implies$ (3) is obvious. The implication (3) $\implies$ (1) is a consequence of the more precise statement in Corollary \[iitakacor\].
Let $X$ be a normal integral projective variety with $\mathbb{Q}$-factorial singularities over $\mathbb{C}$. By Example \[divisormob\], Theorem \[mobilityandbigness\] is equivalent to the usual characterization of big divisors using the volume function.
\[mobilityofcurves\] Let $\alpha$ be a curve class on a complex variety $X$. [@8authors Theorem 2.4] shows that if two general points of $X$ can be connected by an effective curve whose class is proportional to $\alpha$, then $\alpha$ is big. Theorem \[mobilityandbigness\] is a somewhat weaker statement in this situation.
For positive integers $n$ and for $0 \leq k < n$, define $\epsilon_{n,k}$ inductively by setting $\epsilon_{n,n-1} = 1$ and $$\epsilon_{n,k} = \frac{ \frac{n-k-1}{n-k}\epsilon_{n-1,k} }{\frac{n-1}{n-k-1} - \epsilon_{n-1,k}}.$$ For positive integers $n$ and for $0 \leq k < n$, define $\tau_{n,k}$ inductively by setting $\tau_{n,n-1} = 1$ and $$\tau_{n,k} = \min \left\{ \frac{n-k-1}{n-1} \tau_{n-1,k}, \frac{ \frac{n-k-1}{n-k}\tau_{n-1,k} }{\frac{n-1}{n-k-1} - \tau_{n-1,k}} \right\}.$$ It is easy to verify that $0 < \tau_{n,k} \leq \epsilon_{n,k} \leq \frac{1}{n-k}$ and that the last inequality is strict as soon as $n-k>1$.
\[mobprecisebound\] Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Let $A$ be a very ample divisor and let $s$ be a positive integer such that $\alpha \cdot A^{k} \leq sA^{n}$. Then
1. $$\operatorname{mc}(\alpha) \leq 2^{kn+3n} (k+1) s^{\frac{n}{n-k}}A^{n}.$$
2. Suppose furthermore that $\alpha - [A]^{n-k}$ is not pseudo-effective. Then $$\operatorname{mc}(\alpha) \leq 2^{kn+3n} (k+1) s^{\frac{n}{n-k} - \epsilon_{n,k}} A^{n}.$$
3. Suppose that $t$ is a positive integer such that $t \leq s$ and $\alpha - t[A]^{n-k}$ is not pseudo-effective. Then $$\operatorname{mc}(\alpha) \leq 2^{kn+3n} (k+1) s^{\frac{n}{n-k} - \tau_{n,k}} t^{\tau_{n,k}} A^{n}.$$
We prove (1) by induction on the dimension $n$ of $X$. The induction step may reduce the codimension $n-k$ of our cycle class by at most $1$. Thus, for the base case it suffices to consider when $k=0$ or when $n$ is arbitrary and $n-k = 1$. These cases are proved by Example \[mobzerocycles\] and Corollary \[oldversion\] (1) respectively.
Let $p: U \to W$ be a family of effective $k$-cycles representing $\alpha$. By Proposition \[opensubsetsforcycles\] we may modify $p$ by Lemma \[goodfamilymodification\] to assume $W$ is projective without changing the mobility count of $p$. Choose a general divisor $H$ in the very ample linear series $|\lceil s^{\frac{1}{n-k}} \rceil A|$ on $X$ such that $H$ is integral and does not contain the image of any component of $U$. We can associate several families of subschemes to $p$ and to $H$.
- Consider the base change $U \times_{X} H$. We can view this as a subscheme $U_{H} \subset W \times H$ with projection map $\pi: U_{H} \to W$.
- We can intersect the family $p$ with the divisor $H$ to obtain a family of effective $(k-1)$ cycles $q: S \to T$ on $H$ as in Construction \[intersectionfamilyconstr\]. Note that $T$ is an open subset of $W$; we may shrink $T$ so that it is normal. By Lemma \[goodfamilymodification\] we may extend $q$ over a projective closure of $T$ which we continue to denote by $q: S \to T$.
- Let $V \subset U_{H}$ be the reduced closed subset consisting of points whose local fiber dimension for $\pi$ attains the maximal possible value $k$. The map $\pi|_{V}: V \to W$ has equidimensional fibers of dimension $k$. Thus we can associate to $\pi|_{V}$ a collection of families $p_{i}: V_{i} \to W_{i}$ as in Construction \[equidimsubschemeconstr\]. We will think of these as families of effective $k$-cycles on $H$.
It will be useful to combine $q$ and the $p_{i}$ as follows. Let $\widetilde{W}$ denote the disjoint union of the irreducible varieties $T \times W_{i}$ as we vary over all $i$. This yields the subscheme $S \times \left( \cup_{i} W_{i} \right) \cup T \times \left( \cup_{i} V_{i} \right)$ of $\widetilde{W} \times X$. We denote the first projection map by $\tilde{p}$.
Using the universal property, one can see that $$(U \times_{X} H)^{\times_{W} b} \cong (U^{\times_{W} b}) \times_{X^{\times b}} H^{\times b}.$$ Since the base change of a surjective map is surjective, we see that $\operatorname{mc}_{H}(\pi) \geq \operatorname{mc}_{X}(p)$. Furthermore, by Krull’s principal ideal theorem every component of a fiber of $\pi$ over a closed point of $W$ has dimension $k$ or $k-1$. In particular, any member of the family $\pi$ is set theoretically contained in a member of the family $\tilde{p}$. Applying Lemmas \[familysummc\] and \[fibercontainedmc\], we obtain $$\operatorname{mc}_{X}(p) \leq \operatorname{mc}_{H}(\pi) \leq \operatorname{mc}_{H}(\tilde{p}) = \operatorname{mc}_{H}(q) + \sup_{i} \operatorname{mc}_{H}(p_{i}).$$ We will use induction to bound the two terms on the right, giving us our overall bound for $\operatorname{mc}_{X}(p)$.
The family $q$ of effective $(k-1)$-cycles on $H$ has class $\alpha \cdot H$. Note that $(\alpha \cdot H) \cdot A|_{H}^{k-1} \leq sA|_{H}^{n-1}$. By induction on the dimension of the ambient variety, $$\begin{aligned}
\operatorname{mc}_{H}(q) & \leq 2^{(k-1)(n-1) + 3(n-1)} k s^{\frac{n-1}{n-k}}(A|_{H}^{n-1}) \\
& \leq 2^{(k-1)(n-1) + 3(n-1)} k s^{\frac{n-1}{n-k}} (2s^{\frac{1}{n-k}} A^{n}) \\
& \leq 2^{(k-1)(n-1) + 3(n-1)+1} (k+1) s^{\frac{n}{n-k}} A^{n}.\end{aligned}$$
Next consider a family $p_{i}$ of effective $k$-cycles on $H$. Let $\alpha_{i}$ denote the corresponding class. Let $j: H \to X$ be the inclusion; by construction, it is clear that $\alpha - j_{*}\alpha_{i}$ is the class of an effective cycle. In particular $$\alpha_{i} \cdot A|_{H}^{k} \leq \alpha \cdot A^{k} \leq \lceil s^{\frac{n-k-1}{n-k}} \rceil A|_{H}^{n-1}.$$ By induction on the dimension of the ambient variety, $$\begin{aligned}
\operatorname{mc}_{H}(p_{i}) & \leq 2^{(n-1)k + 3(n-1)} (k+1) \lceil s^{\frac{n-k-1}{n-k}} \rceil^{\frac{n-1}{n-k-1}}(A|_{H}^{n-1}) \\
& \leq 2^{(n-1)k + 3(n-1)} (k+1) 2^{\frac{n-1}{n-k-1}} s^{\frac{n-1}{n-k}}(2s^{\frac{1}{n-k}}A^{n}) \\
& \leq 2^{(n-1)k + 3(n-1) + k + 2} (k+1) s^{\frac{n}{n-k}} A^{n}.\end{aligned}$$ By adding these contributions, we see that $$\operatorname{mc}_{X}(p) \leq 2^{kn+3n} (k+1) s^{\frac{n}{n-k}}A^{n}.$$
\(2) is proved in a similar way. The argument is by induction on the codimension $n-k$ of $\alpha$. The base case – when $n$ is arbitrary and $n-k=1$ – is a consequence of Corollary \[oldversion\] (2) (applied with $H=A$).
Let $p: U \to W$ be a family of effective $k$-cycles representing $\alpha$. Set $c := \frac{1}{n-k} - \epsilon_{n,k}$. Let $H$ be an integral element of $|\lceil s^{c} \rceil A|$ that does not contain the image of any component of $U$. We construct the families $q: S \to T$ and $p_{i}: V_{i} \to W_{i}$ just as in (1). The same argument shows that $$\operatorname{mc}_{X}(p) \leq \operatorname{mc}_{H}(q) + \sup_{i} \operatorname{mc}_{H}(p_{i}).$$
The family $q$ of effective $(k-1)$-cycles on $H$ has class $\alpha \cdot H$. Note that $(\alpha \cdot H) \cdot A|_{H}^{k-1} \leq sA|_{H}^{n-1}$. By (1), we have $$\begin{aligned}
\operatorname{mc}_{H}(q) & \leq 2^{(k-1)(n-1)+3(n-1)} ks^{\frac{n-1}{n-k}}(A|_{H}^{n-1}) \\
& \leq 2^{(k-1)(n-1)+3(n-1)}k s^{\frac{n-1}{n-k}} (2s^{c} A^{n}) \\
& \leq 2^{(k-1)(n-1)+3(n-1) + 1} (k+1) s^{\frac{n}{n-k} - \epsilon_{n,k}} A^{n}.\end{aligned}$$
Next consider the family $p_{i}$ of effective $k$-cycles on $H$. Let $\alpha_{i}$ denote the class of the family $p_{i}$ on $H$. Let $j: H \to X$ be the inclusion; by construction, it is clear that $\alpha - j_{*}\alpha_{i}$ is the class of an effective cycle. In particular $$\begin{aligned}
\alpha_{i} \cdot A|_{H}^{k} \leq \alpha \cdot A^{k} \leq \left\lceil s^{1-c} \right\rceil A|_{H}^{n-1}.\end{aligned}$$ Note furthermore that $\alpha_{i} - [A|_{H}]^{n-1-k}$ is not pseudo-effective; otherwise it would push forward to a pseudo-effective class on $X$, contradicting the fact that $\alpha - [A]^{n-k}$ is not pseudo-effective. By induction on the codimension of the cycle, $$\begin{aligned}
\operatorname{mc}_{H}(p_{i}) & \leq 2^{k(n-1)+3(n-1)} (k+1) \lceil s^{1-c} \rceil^{\frac{n-1}{n-k-1} - \epsilon_{n-1,k}}(A|_{H}^{n-1}) \\
& \leq 2^{k(n-1)+3(n-1)} (k+1) 2^{\frac{n-1}{n-k-1}} s^{\frac{(1-c)(n-1)}{n-k-1} - (1-c)\epsilon_{n-1,k}}(2s^{c}A^{n}) \\
& \leq 2^{k(n-1)+3(n-1) + k + 2} (k+1) s^{\frac{n}{n-k} - \epsilon_{n,k}} A^{n}.\end{aligned}$$ Adding the two contributions proves the statement as before.
The proof of (3) is also very similar. The argument is by induction on the codimension $n-k$ of $\alpha$. The base case – when $n$ is arbitrary and $n-k=1$ – is a consequence of Corollary \[oldversion\] (2) (applied with $H=tA$).
Let $p: U \to W$ be a family of effective $k$-cycles representing $\alpha$. Set $c := \frac{1}{n-k} - \tau_{n,k}$. Let $H$ be an integral element of $|\lceil s^{c} t^{\tau_{n,k}} \rceil A|$ that does not contain the image of any component of $U$. We construct the families $q: S \to T$ and $p_{i}: V_{i} \to W_{i}$ just as in (1). The same argument shows that $$\operatorname{mc}_{X}(p) \leq \operatorname{mc}_{H}(q) + \sup_{i} \operatorname{mc}_{H}(p_{i}).$$
The family $q$ of effective $(k-1)$-cycles on $H$ has class $\alpha \cdot H$. Note that $(\alpha \cdot H) \cdot A|_{H}^{k-1} \leq sA|_{H}^{n-1}$. By (1), we have $$\begin{aligned}
\operatorname{mc}_{H}(q) & \leq 2^{(k-1)(n-1)+3(n-1)} ks^{\frac{n-1}{n-k}}(A|_{H}^{n-1}) \\
& \leq 2^{(k-1)(n-1)+3(n-1)}k s^{\frac{n-1}{n-k}} (2s^{c}t^{\tau_{n,k}} A^{n}) \\
& \leq 2^{(k-1)(n-1)+3(n-1) + 1} (k+1) s^{\frac{n}{n-k} - \tau_{n,k}} t^{\tau_{n,k}} A^{n}.\end{aligned}$$
Next consider the family $p_{i}$ of effective $k$-cycles on $H$. Let $\alpha_{i}$ denote the class of the family $p_{i}$ on $H$. Let $j: H \to X$ be the inclusion; by construction, it is clear that $\alpha - j_{*}\alpha_{i}$ is the class of an effective cycle. In particular $$\begin{aligned}
\alpha_{i} \cdot A|_{H}^{k} \leq \alpha \cdot A^{k} & \leq \left\lceil s^{1-c} t^{-\tau_{n,k}} \right\rceil A|_{H}^{n-1}.\end{aligned}$$ Also, we have that $$\begin{aligned}
\alpha_{i} - \lceil t^{1-\tau_{n,k}}s^{-c} \rceil [A|_{H}]^{n-k-1}\end{aligned}$$ is not pseudo-effective, since the difference between $\alpha - t[A]^{n-k}$ and the push forward of this class to $X$ is pseudo-effective. Finally, note that $\lceil s^{1-c}t^{-\tau_{n,k}} \rceil \geq \lceil t^{1-\tau_{n,k}}s^{-c} \rceil$ so that we may apply (3) inductively to the family $p_{i}$ with the constants $s' = \lceil s^{1-c}t^{-\tau_{n,k}} \rceil$ and $t' = \lceil t^{1-\tau_{n,k}}s^{-c} \rceil$.
There are two cases to consider. First suppose that $t^{1-\tau_{n,k}}s^{-c} \geq 1$. Then by induction on the codimension of the cycle, $$\begin{aligned}
\operatorname{mc}_{H}(p_{i}) & \leq 2^{k(n-1)+3(n-1)} (k+1) \lceil s^{1-c} t^{-\tau_{n,k}} \rceil^{\frac{n-1}{n-k-1} - \tau_{n-1,k}} \\
& \qquad \qquad \lceil t^{1-\tau_{n,k}} s^{-c} \rceil^{\tau_{n-1,k}} (A|_{H}^{n-1}) \\
& \leq 2^{k(n-1)+3(n-1)} (k+1) 2^{\frac{n-1}{n-k-1}} s^{\frac{(1-c)(n-1)}{n-k-1} - \tau_{n-1,k}} \\
& \qquad \qquad t^{\tau_{n-1,k} - \frac{n-1}{n-k-1} \tau_{n,k}}(2s^{c}t^{\tau_{n,k}}A^{n}) \\
& \leq 2^{k(n-1)+3(n-1) + k + 2} (k+1) s^{\frac{n}{n-k} - \tau_{n,k} + \left(\frac{n-1}{n-k-1}\tau_{n,k} - \tau_{n-1,k} \right)} \\
& \qquad \qquad t^{\tau_{n,k} + \left(\tau_{n-1,k} - \frac{n-1}{n-k-1}\tau_{n,k}\right)} A^{n}.\end{aligned}$$ Since $\tau_{n-1,k} \geq \frac{n-1}{n-k-1}\tau_{n,k}$, the part of the exponents in parentheses is negative for $s$ and positive for $t$. By assumption $s \geq t$, so $$\begin{aligned}
\operatorname{mc}_{H}(p_{i}) \leq 2^{kn+3(n-1) + 2} (k+1) s^{\frac{n}{n-k} - \tau_{n,k}} t^{\tau_{n,k}} A^{n}\end{aligned}$$ Next suppose that $t^{1-\tau_{n,k}}s^{-c} < 1$. Then by (2) we find $$\begin{aligned}
\operatorname{mc}_{H}(p_{i}) & \leq 2^{k(n-1)+3(n-1)} (k+1) \lceil s^{1-c}t^{-\tau_{n,k}} \rceil^{\frac{n-1}{n-k-1} - \epsilon_{n-1,k}} (A|_{H}^{n-1}) \\
& \leq 2^{k(n-1)+3(n-1)} (k+1) \lceil s^{1-c} \rceil^{\frac{n-1}{n-k-1} - \tau_{n-1,k}} (A|_{H}^{n-1}) \\
& \leq 2^{k(n-1)+3(n-1)} (k+1) 2^{\frac{n-1}{n-k-1}} s^{\frac{(1-c)(n-1)}{n-k-1} - (1-c)\tau_{n-1,k}} (2s^{c}t^{\tau_{n,k}}A^{n}) \\
& \leq 2^{kn + 3(n-1)+2}(k+1) s^{\frac{n}{n-k} - \tau_{n,k}} t^{\tau_{n,k}} A^{n}.\end{aligned}$$ This upper bound for the two cases is the same; by adding it to the upper bound for $\operatorname{mc}_{H}(q)$ we obtain the desired upper bound for $\operatorname{mc}(p)$.
We can apply Theorem \[mobprecisebound\] (2) to any class $\alpha$ in $\partial \operatorname{\overline{Eff}}_{k}(X) \cap N_{k}(X)_{\mathbb{Z}}$ to obtain the following corollary.
\[iitakacor\] Let $X$ be an integral projective variety and suppose that $\alpha \in N_{k}(X)_{\mathbb{Z}}$ is not big. Let $A$ be a very ample divisor and let $s$ be a positive integer such that $\alpha \cdot A^{k} \leq sA^{n}$. Then $$\operatorname{mc}(\alpha) \leq 2^{kn + 3n}(k+1)s^{\frac{n}{n-k} - \epsilon_{n,k}} A^{n}.$$
The exponent $\frac{n}{n-k} - \epsilon_{n,k}$ in Corollary \[iitakacor\] is not optimal in general. For example, [@8authors Theorem 2.4] shows that for a curve class $\alpha$ that is not big there is a positive constant $C$ such that $\operatorname{mc}(m\alpha) < Cm$.
Let $f: X \to Z$ be a surjective morphism from a smooth integral projective variety of dimension $n$ to a smooth integral projective variety of dimension $k$ for some $1 < k < n$. Fix ample divisors $A$ on $X$ and $H$ on $Z$ and define $\alpha = [A]^{n-k-1} \cdot [f^{*}H]$. $\alpha$ is not big since $\alpha \cdot [f^{*}H]^{k} = 0$. By taking the complete intersection of $(n-k-1)$ elements of $H^{0}(X,\mathcal{O}_{X}(\lfloor m^{\frac{k}{(n-k)(k+1)}} \rfloor A))$ with an element of $H^{0}(X,\mathcal{O}_{X}(\lfloor m^{\frac{n}{(n-k)(k+1)}} \rfloor f^{*}H))$ we see $$\operatorname{mc}(m\alpha) \geq Cm^{\frac{nk}{(n-k)(k+1)}}$$ for some positive constant $C$. Rewriting $$\frac{nk}{(n-k)(k+1)} = \frac{n}{n-k} - \frac{n}{(n-k)(k+1)}$$ shows that the optimal value of $\epsilon_{n,k}$ is at most $\frac{n}{(n-k)(k+1)}$.
Continuity of mobility
----------------------
Theorem \[mobprecisebound\] also allows us to prove the continuity of the mobility function.
\[mobcontinuous\] Let $X$ be an integral projective variety. Then the mobility function $\operatorname{mob}: N_{k}(X)_{\mathbb{Q}} \to \mathbb{R}$ can be extended to a continuous function on $N_{k}(X)$.
Note that $\operatorname{mob}$ can be extended to a continuous function on the interior of $\operatorname{\overline{Eff}}_{k}(X)$ by Theorem \[mobcontbigcone\]. Furthermore $\operatorname{mob}$ is identically $0$ on every element in $N_{k}(X)_{\mathbb{Q}}$ not contained in $\operatorname{\overline{Eff}}_{k}(X)$. Thus it suffices to show that $\operatorname{mob}$ approaches $0$ for classes approaching the boundary of $\operatorname{\overline{Eff}}_{k}(X)$.
Let $\alpha$ be a point on the boundary of $\operatorname{\overline{Eff}}_{k}(X)$. Fix $\mu > 0$; we show that there exists a neighborhood $U$ of $\alpha$ such that $\operatorname{mob}(\beta) < \mu$ for any class $\beta \in U \cap N_{k}(X)_{\mathbb{Q}}$.
Fix a very ample divisor $A$ and a positive integer $s$ such that $\alpha \cdot A^{k} \leq \frac{s}{2}A^{n}$. Choose $\delta$ sufficiently small so that $$n! 2^{kn+3n+1} (k+1) s^{\frac{n}{n-k}} A^{n} \delta^{\tau_{n,k}} < \mu.$$ Let $U$ be a sufficiently small neighborhood of $\alpha$ so that:
- $\beta \cdot A^{k} \leq sA^{n}$ for every $\beta \in U$, and
- $\beta - \delta s [A]^{n-k}$ is not pseudo-effective for every $\beta \in U$.
Suppose now that $\beta \in U \cap N_{k}(X)_{\mathbb{Q}}$ and that $m$ is any positive integer such that $m\beta \in N_{k}(X)_{\mathbb{Z}}$. Then:
- $m\beta \cdot A^{k} \leq smA^{n}$ and
- $m\beta - \lceil \delta ms \rceil [A]^{n-k}$ is not pseudo-effective.
Theorem \[mobprecisebound\] shows that $$\operatorname{mc}(m\beta) \leq 2^{kn+3n} (k+1) (ms)^{\frac{n}{n-k} - \tau_{n,k}} (\lceil \delta ms \rceil)^{\tau_{n,k}} A^{n}.$$ When $m$ is sufficiently large, $\lceil \delta ms \rceil \leq 2 \delta ms$, so we obtain for such $m$ $$\operatorname{mc}(m\beta) \leq 2^{kn+3n+1} (k+1) m^{\frac{n}{n-k}} s^{\frac{n}{n-k}} A^{n} \delta^{\tau_{n,k}} < \frac{\mu}{n!} m^{\frac{n}{n-k}}$$ showing that $\operatorname{mob}(\beta)<\mu$ as desired.
Alternative definitions of mobility
-----------------------------------
The mobility seems difficult to calculate explicitly. In this section we discuss two alternatives which may be easier to compute. However, they seem to be less flexible from a theoretical perspective.
### Smooth mobility
Let $X$ be an integral projective variety and let $p: U \to W$ be a family of effective cycles on $X$. We say that $p$ is a mostly smooth family if every component of a general cycle-theoretic fiber of $p$ is smooth.
To obtain a good theory of mostly smooth families, it is important that mostly smoothness is compatible with the geometric constructions outlined in Section \[cyclefamilysection\]. In particular, mostly smoothness is compatible with
- closure of families,
- family sums,
- intersections against general very ample divisors, and
- pushforwards from subvarieties.
Let $X$ be an integral projective variety and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Define $\operatorname{mc}_{sm}(\alpha)$ to be the maximum mobility count of any mostly smooth family of effective cycles representing a class $\beta$ such that $\alpha - \beta$ is an effective class. Define $$\operatorname{mob}_{sm}(\alpha) = \limsup_{m \to \infty} \frac{\operatorname{mc}_{sm}(m\alpha)}{m^{n/n-k}/n!}.$$
Since mostly smoothness is compatible with family sums, just as for the mobility function one verifies that $\operatorname{mob}_{sm}$ is homogenous and for $\alpha,\beta \in N_{k}(X)_{\mathbb{Q}}$ we have $\operatorname{mob}_{sm}(\alpha + \beta) \geq \operatorname{mob}_{sm}(\alpha) + \operatorname{mob}_{sm}(\beta)$. Furthermore, if $X$ is a smooth integral variety, the Bertini theorems imply that by taking complete intersections of very ample divisors we obtain mostly smooth families. Thus $\operatorname{mob}_{sm}$ is positive on any big class on a smooth variety.
By Lemma \[easyconelem\], $\operatorname{mob}_{sm}$ extends to a continuous function on the interior of $\operatorname{\overline{Eff}}_{k}(X)$. Since $\operatorname{mob}_{sm} \leq \operatorname{mob}$, Theorem \[mobcontinuous\] then implies:
Let $X$ be an integral projective variety. Then $\operatorname{mob}_{sm}$ extends to a homogeneous continuous function on $N_{k}(X)$. If $X$ is smooth, then $\operatorname{mob}_{sm}$ is positive on the big cone.
### ACM mobility
Let $X$ be an integral projective variety with a fixed embedding into some projective space $\mathbb{P}^{r}$. $X$ is said to be arithmetically Cohen-Macaulay for this embedding (or simply ACM) if its homogeneous coordinate ring is Cohen-Macaulay.
Let $X$ be an integral projective variety with a fixed embedding into projective space and let $p: U \to W$ be a family of effective $k$-cycles on $X$. We say that $p$ is a mostly ACM family if for a general cycle-theoretic fiber $Z$ of $p$ there is a finite collection of positive integers $a_{j}$ and of ACM subschemes $Y_{j}$ (with respect to the fixed embedding) of pure dimension $k$ with fundamental cycles $Z_{j}$ such that $Z = \sum a_{j}Z_{j}$.
Note that mostly ACMness is compatible with
- closure of families,
- family sums,
- intersections against multiples of the hyperplane class on the ambient projective space, and
- pushforwards from subvarieties (with the compatible embedding).
Let $X$ be an integral projective variety with a fixed embedding into projective space and let $\alpha \in N_{k}(X)_{\mathbb{Z}}$. Define $\operatorname{mc}_{ACM}(\alpha)$ to be the maximum mobility count of any mostly ACM family of effective cycles representing a class $\beta$ such that $\alpha - \beta$ is an effective class. Define $$\operatorname{mob}_{ACM}(\alpha) = \limsup_{m \to \infty} \frac{\operatorname{mc}_{ACM}(m\alpha)}{m^{n/n-k}/n!}.$$
If $X$ is ACM with respect to its fixed embedding in $\mathbb{P}^{r}$, then families of complete intersections of $X$ with hyperplanes in $\mathbb{P}^{r}$ yield mostly ACM families of effective cycles. Thus on an ACM variety $X$ any big class has positive $\operatorname{mob}_{ACM}$. Since $\operatorname{mob}_{ACM}$ is compatible with family sums, arguing as before for $\operatorname{mob}_{sm}$ we see that
Let $X$ be a integral projective variety with a fixed embedding in projective space. Then $\operatorname{mob}_{ACM}$ extends to a homogeneous continuous function on $N_{k}(X)$. If $X$ is ACM, then $\operatorname{mob}_{ACM}$ is positive on the big cone.
Examples of mobility {#mobilityexamplesection}
====================
The mobility seems difficult to calculate explicitly. By analogy with the volume, one wonders whether the mobility is related to intersection numbers for “sufficiently positive” classes (just as the volume of an ample divisor is a self-intersection product). In particular, we ask:
\[intersectiontheoreticques\] Let $X$ be an integral projective variety and let $H$ be an ample Cartier divisor. For $0<k<n$, is $$\operatorname{mob}(H^{n-k}) = \operatorname{vol}(H)?$$ More generally, if $L$ is a big Cartier divisor, and $\alpha = \langle L^{n-k} \rangle$ for some $0<k<n$, where $\langle - \rangle$ denotes the positive product of [@bdpp04], is $$\operatorname{mob}(\alpha) = \operatorname{vol}(L)?$$
An affirmative answer would imply that the “optimal cycles” with respect to the mobility count for $H^{n-k}$ are complete intersections of $(n-k)$ general elements of $|dH|$
\[intersectiontheoreticrmk\] Note that the statements in Question \[intersectiontheoreticques\] do not hold for point classes: for an ample divisor $H$ we have $\operatorname{vol}(H) = H^{n}$ but $\operatorname{mob}(H^{n}) = n!H^{n}$.
In the remainder of this section we discuss two examples in detail. We will work over the base field $\mathbb{C}$ to cohere with the cited references.
Curves on $\mathbb{P}^{3}$ {#p3mobility}
--------------------------
Let $\ell$ denote the class of a line on $\mathbb{P}^{3}$ over $\mathbb{C}$. The mobility of $\ell$ is determined by the following enumerative question: what is the minimal degree of a curve in $\mathbb{P}^{3}$ going through $b$ very general points? The answer is unknown (even in the asymptotic sense).
[@perrin87] conjectures that the “optimal” curves are the complete intersections of two hypersurfaces of degree $d$. Indeed, among all curves not contained in a hypersurface of degree $(d-1)$, [@gp77] shows that these complete intersections have the largest possible arithmetic genus, and thus conjecturally the corresponding Hilbert scheme has the largest possible dimension.
Complete intersections of two hypersurfaces of degree $d$ have degree $d^{2}$ and pass through $\approx \frac{1}{6}d^{3}$ general points. Letting $d$ go to infinity, we find the lower bound $$1 \leq \operatorname{mob}(\ell)$$ and conjecturally equality holds.
Let $\ell$ be the class of a line on $\mathbb{P}^{3}$. Then:
1. $1 \leq \operatorname{mob}(\ell) < 3.54$.
2. $1 \leq \operatorname{mob}_{sm}(\ell) \leq 3$. ([@perrin87 Proposition 6.29])
### Mobility
To prove the first bound on the mobility, we simply repeat the argument of Theorem \[mobprecisebound\] with more careful constructions of families and better estimates.
Fix a degree $d$. Let $s = \left \lceil \sqrt{\frac{9-\sqrt{69}}{2}d} \right \rceil$ and let $S$ be a Noether-Lefschetz general hypersurface of degree $s$. Then every curve on $S$ is the restriction of a hypersurface on $\mathbb{P}^{3}$. In particular, $\operatorname{Pic}(S) \cong \mathbb{Z}$, and if $H$ denotes the hyperplane class on $\mathbb{P}^{3}$ then the mobility count of $cH|_{S}$ is $$\left( \begin{array}{c} c+3 \\ 3 \end{array} \right) - \left( \begin{array}{c} c-s+3 \\ 3 \end{array} \right)$$ (where we use the convention that the rightmost term is $0$ when $c<s$). Let $p: U \to W$ be a family of degree $d$ curves on $\mathbb{P}^{3}$. Consider the base change $p': U \times_{\mathbb{P}^{3}} S \to W$. Every component of a fiber of $p'$ has dimension $1$ or $0$. We can stratify $W$ by locally closed subsets $W_{i}$ based on the degree $d'$ of the components of the fibers of dimension $1$; the components of dimension $0$ then have degree $(d-d')s$. Applying Construction \[equidimsubschemeconstr\] to construct families of cycles as in the proof of Theorem \[mobprecisebound\], Lemma \[fibercontainedmc\] implies that $$\begin{aligned}
\operatorname{mc}(d\ell) & \leq \max_{0 \leq d' \leq d} \operatorname{mc}_{S}((d-d')\ell \cdot S) + \operatorname{mc}_{S}\left(\left\lceil \frac{d'}{s} \right\rceil H|_{S}\right) \\
& \leq \max_{0 \leq d' \leq d} (d-d')s + \left( \begin{array}{c} \lceil \frac{d'}{s} \rceil+3 \\ 3 \end{array} \right) - \left( \begin{array}{c} \lceil \frac{d'}{s} \rceil-s+3 \\ 3 \end{array} \right).\end{aligned}$$ A straightforward computation shows that for $\lceil d'/s \rceil \geq s$ the maximum value is achieved when $d' = d$ and for $\lceil d'/s \rceil < s$ the maximum value is achieved when $d'=0$. In either case, the asymptotic value of the computation above is $$\operatorname{mc}(d\ell) \leq \sqrt{\frac{9-\sqrt{69}}{2}} d^{3/2} + O(d)$$ yielding the desired bound.
### Smooth mobility
The bound $\operatorname{mob}_{sm}(\ell) \leq 3$ is proved by [@perrin87 Proposition 6.29]. The argument below is more-or-less the same as that of Perrin. Suppose that $p: U \to W$ is a mostly smooth family of degree $d$ curves on $\mathbb{P}^{3}$. We show that $$\operatorname{mc}(p) \leq \frac{1}{2}d^{3/2} + O(d)$$ which immediately yields $\operatorname{mob}_{sm}(\ell) \leq 3$. By Lemmas \[familysummc\] and \[opensubsetsforcycles\], it suffices to prove this bound when $p$ is generically irreducible and reduced. By replacing $p$ by the flattening of a Stein factorization, we may suppose that a general fiber is an integral smooth curve (since we are working over a field of characteristic $0$).
Set $s = \lfloor \sqrt{d} \rfloor$. First suppose that a general cycle-theoretic fiber of $p$ is contained in a surface of degree $<s$. Then Lemma \[fibercontainedmc\] shows that $\operatorname{mc}(p) \leq \operatorname{mc}(|sH|) \approx \frac{1}{6}d^{3/2}$.
Otherwise, a general cycle-theoretic fiber $C$ of $p$ is a smooth curve not contained on a surface of degree $<s$. Let $g$ denote the genus of $C$. The exact sequence $$0 \to T_{C} \to T_{\mathbb{P}^{3}}|_{C} \to N_{C/\mathbb{P}^{3}} \to 0$$ shows that $h^{0}(C,N_{C/\mathbb{P}^{3}}) \leq h^{0}(C,T_{\mathbb{P}^{3}}|_{C}) - \chi(T_{C})$. An elementary argument shows that $T_{\mathbb{P}^{3}}|_{C}$ admits a filtration by line bundles of positive degree; in particular $h^{0}(C,T_{\mathbb{P}^{3}}|_{C}) \leq \deg(T_{\mathbb{P}^{3}}|_{C}) = 4d$. Thus $$h^{0}(C,N_{C/\mathbb{P}^{3}}) \leq 4d + g - 1.$$ [@gp77 Théorème 3.1] shows that curves not lying on a surface of degree $<s$ have arithmetic genus at most $s^{3}-2s^{2}+1$. Thus, $$h^{0}(C,N_{C/\mathbb{P}^{3}}) \leq d^{3/2} + 2d.$$ This gives an upper bound on the dimension of the corresponding component of the Hilbert scheme, hence also the corresponding component of the Chow scheme. Since general points impose at least $2$ conditions on curves, we see that in this case $$\operatorname{mc}(p) \leq \frac{1}{2}d^{3/2} + d.$$
Rational mobility of points {#ratmob0cycles}
---------------------------
In this section we relate rational mobility with the theory of rational equivalence of $0$-cycles. In order to cohere with the cited references, we work only with normal integral varieties $X$ over $\mathbb{C}$ (although the results easily extend to a more general setting). Recall that $A_{0}(X)$ denotes the group of rational-equivalence classes of $0$-cycles on $X$. We will denote the $r$th symmetric power of $X$ by $X^{(r)}$; by [@kollar96 I.3.22 Exercise] this is the component of $\operatorname{Chow}(X)$ parametrizing $0$-cycles of degree $r$.
The universal family of $0$-cycles of degree $r$ (in the sense of Definition \[familydef\]) is not $u: X^{\times r} \to X^{(r)}$ but a flattening of this map. However, note that the rational mobility computations are the same whether we work with $u$ or a flattening by Lemma \[opensubsetsforcycles\]. For simplicity we will work with $u$ and $X^{(r)}$ despite the slight incongruity with Definition \[familydef\].
We start by recalling the results of [@roitman72] concerning $A_{0}(X)$. Consider the map $\gamma_{m,n}: X^{(m)} \times X^{(n)} \to A_{0}(X)$ sending $(p,q) \mapsto p-q$. [@roitman72 Lemma 1] shows that the fibers of $\gamma_{m,n}$ are countable unions of closed subvarieties.
A subset $V \subset A_{0}(X)$ is said to be irreducible closed if it is the $\gamma_{m,n}$-image of an irreducible closed subset $Y$ of $X^{(m)} \times X^{(n)}$ for some $m$ and $n$. The dimension of such a subset $V$ is defined to be the dimension of $Y$ minus the minimal dimension of a component of a fiber of $\gamma_{m,n}|_{Y}$. [@roitman72 Lemma 9] shows that the dimension is independent of the choice of $Y$, $m$, and $n$.
\[containmentdimineq\] Let $V,W \subset A_{0}(X)$ be irreducible closed subsets with $V \subsetneq W$. Then $\dim(V) < \dim(W)$.
Let $Z \subset X^{(m)} \times X^{(n)}$ be an irreducible closed subset whose $\gamma_{m,n}$-image is $W$. Let $Y \subset Z$ denote the preimage of $V$; [@roitman72 Lemma 5] shows that $Y$ is a countable union of closed subsets. By [@roitman72 Lemma 6] some component $Y' \subset Y$ dominates $V$. But then $\dim(Y') < \dim(Z)$, proving the statement.
We are mainly interested in when $A_{0}(X)$ is an irreducible closed set. This is equivalent to the following notion:
$A_{0}(X)$ is said to be representable if there is a positive integer $r$ such that the addition map $a_{r}: X^{(r)} \to A_{0}(X)_{\deg r}$ is surjective.
We now relate these notions to the rational mobility of $0$-cycles on $X$.
\[0cyclepositiveratmob\] Let $X$ be an integral projective variety of dimension $n$ and let $\alpha$ be the class of a point in $N_{0}(X)$. Let $A$ be a very ample divisor on $X$; for sufficiently large $m$ we have $h^{0}(X,\mathcal{O}_{X}(mA)) \approx \frac{1}{n!}A^{n}$. By taking complete intersections of $n$ elements of $|mA|$, we see that $\operatorname{ratmob}(\alpha) \geq 1$.
\[representabilityandratmob\] Let $X$ be a normal integral projective variety and let $\alpha$ denote the class of a point in $N_{0}(X)$. Then the following are equivalent:
1. $A_{0}(X)$ is representable.
2. $\operatorname{ratmob}(\alpha) = n!$.
3. $\operatorname{ratmob}(\alpha) > n!/2$.
\(1) $\implies$ (2). Suppose that $A_{0}(X)$ is representable. There is some positive integer $r$ such that the addition map $X^{(r)} \to A_{0}(X)_{\deg r}$ is surjective. Fix $m>0$ and choose some class $\tau \in A_{0}(X)_{\deg(m+r)}$. For any effective $0$-cycle $Z$ of degree $m$, there is an effective $0$-cycle $T_{Z}$ of degree $r$ such that $T_{Z} + Z \subset \tau$. As $Z \in X^{(m)}$ varies, the effective cycles $Z + T_{Z}$ are rationally equivalent, showing that $\operatorname{rmc}((m+r)\alpha) \geq m$ and $\operatorname{ratmob}(\alpha) = n!$.
\(3) $\implies$ (1). Suppose that $A_{0}(X)$ is not representable. Fix a closed point $p_{0} \in X$. Note that non-representability implies that $a_{m}(X^{(m)}) + p_{0} \subsetneq a_{m+1}(X^{(m+1)})$ for every $m$: if we had equality for some $m$, we would also have equality for every larger $m$ and the map $a_{m}$ would be surjective. By Lemma \[containmentdimineq\], $\dim(a_{m}(X^{(m)}))$ strictly increases in $m$.
Suppose that $\operatorname{rmc}(m\alpha) = b$. This implies that there is some rational equivalence class $\tau$ of degree $m$ so that for any $p \in X^{(b)}$, there is an element $q \in X^{(m-b)}$ such that $p+q \in \tau$. In particular, the subset $\tau - X^{(b)} \subset A_{0}(X)_{\deg m-b}$ is contained in $a_{m-b}(X^{(m-b)})$. By Lemma \[containmentdimineq\], $\dim(a_{b}(X^{(b)})) \leq \dim(a_{m-b}(X^{(m-b)}))$. But since these dimensions are strictly increasing in $m$ we must have $m \geq 2b$. Thus we see that $\operatorname{ratmob}(\alpha) \leq n!/2$, proving the statement.
Let $X$ be a smooth surface and let $\alpha$ be the class of a point. By combining Example \[0cyclepositiveratmob\] with Proposition \[representabilityandratmob\] we see that there are two possibilites:
- $A_{0}(X)$ is representable and $\operatorname{ratmob}(\alpha) = 2$.
- $A_{0}(X)$ is not representable and $\operatorname{ratmob}(\alpha) = 1$.
[^1]: The author is supported by NSF Award 1004363.
|
---
abstract: 'We characterize Salem numbers which have some power arising as dynamical degree of an automorphism on a complex (projective) 2-Torus, K3 or Enriques surface.'
address: 'Insitut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany'
author:
- Simon Brandhorst
bibliography:
- 'literature.bib'
date: 'August, 2017'
title: On the stable dynamical spectrum of complex surfaces
---
Introduction
============
To a bimeromorphic transformation $F: X \dashrightarrow X$ of a Kähler surface one can associate its *dynamical degree* $$\lambda(F)=\limsup_{n \rightarrow \infty} (||(F^n)^*||)^{1/n},$$ where $F^*$ denotes the action on $H^2(X,{\mathbb{Z}})$ and $||\cdot||$ is any norm on $\operatorname{End}(H^2(X,{\mathbb{Z}}))$. The dynamical degree is a bimeromorphic invariant of $(X,F)$ which measures the dynamical complexity of $F$. In the projective case, it describes the asymptotic degree growth of defining equations for $F$. We call $F$ algebraically stable if $(F^n)^*=(F^*)^n$. Passing to a bimeromorphic model of $(X,F)$ this can be achieved for surfaces [@diller_favre:dynamics_surface]. Hence, in this case the dynamical degree is an algebraic integer - the spectral radius of $F^*$. Its logarithm $\log \lambda(F)$ is an upper bound for the topological entropy $h(F)$. If moreover $F: X \rightarrow X$ is an automorphism, then they agree and $\lambda(F)$ is a Salem number, that is, an algebraic integer $\lambda>1$ which is Galois conjugate to $1/\lambda$ and all whose other conjugates lie on the unit circle. Conversely, if $\lambda(F)$ is a Salem number of degree at least $4$, then $F$ is conjugate to an automorphism on some birational model of $X$ [@cantat:dynamical_degrees Thm A]. If the dynamical degree of an automorphism $F$ of a surface $X$ is $\lambda(F)>1$, then $X$ is a blow up of the projective plane in at least $10$ points, or a blow up of a 2-Torus, a K3 or an Enriques surface [@cantat:classification]. Let $\mathcal{K}$ be a class of surfaces. We define the dynamical spectrum of surfaces of type $\mathcal{K}$ as $$\Lambda(\mathcal{K},{\mathbb{C}})= \bigcup \{\lambda(F) | F \in \operatorname{Bir}(X),\; X/{\mathbb{C}}\mbox{ is a } \mathcal{K} \mbox{ surface}\},$$ and its counterpart for projective surfaces $$\Lambda^{proj}(\mathcal{K},{\mathbb{C}})= \bigcup \{\lambda(F) | F \in \operatorname{Bir}(X),\; X/{\mathbb{C}}\mbox{ is a projective } \mathcal{K} \mbox{ surface}\}.$$ If $X$ is a surface of type $\mathcal{K} \in \{$ 2-Torus, K3, Enriques $\}$, then its canonical divisor is nef and hence $\operatorname{Bir}(X)=\operatorname{Aut}(X)$. The Kummer construction and the fact that the universal cover of an Enriques surface is a K3 surface give the inclusions $$\Lambda(\mbox{2-Tori},{\mathbb{C}}), \Lambda(\mbox{Enriques},{\mathbb{C}})\subseteq \Lambda(K3,{\mathbb{C}}).$$ For rational surfaces the contribution to the dynamical spectrum coming from automorphisms is described in terms of Weyl groups in [@uehara:rational_automorphism_entropy]. However, for a concrete Salem number it seems to be hard to decide whether it is a spectral radius of an element of a certain Weyl group. For genuinely birational maps on rational surfaces, dynamical degrees may also be Pisot numbers and less seems to be known [@cantat:dynamical_degrees]. The dynamical spectrum of complex 2-Tori (respectively Abelian surfaces) is completely described in [@reschke:tori_salem; @reschke:abelian_salem]. In a very recent preprint [@lenny:equivariant_witt Cor. 1.1] Bayer-Fluckiger and Taelman completely characterize which Salem numbers of degree $22$ arize as dynamical degrees of (non-projective) K3 surfaces.
However, in lower degrees or projective K3 surfaces as well as Enriques surfaces our picture is much less complete. In each even degree $d$ there is a minimal Salem number $\lambda_d$. Conjecturally, the smallest Salem number is Lehmer’s number $\lambda_{10}\approx 1.17628$. In [@mcmullen:minimum] McMullen gives a strategy to decide whether a single given Salem number $\lambda$ is the dynamical degree of an automorphism of a complex projective K3 surface. This strategy is then applied in [@mcmullen:minimum; @brandhorst_alonso:minimal_salem] to show that the minimal Salem numbers $$\lambda_d \in \Lambda^{proj}( K3,{\mathbb{C}}) \quad \Leftrightarrow \quad 14,16\neq d \leq 18.$$ Using the strategy in [@mcmullen:minimum] and the improved positivity test from [@brandhorst_alonso:minimal_salem] one easily obtains $$\lambda_{14}^9,\lambda_{16}^7,\lambda_{20}^{11} \in \Lambda^{proj}( K3,{\mathbb{C}}).$$ In the non-projective realm we get $$\lambda_{14},\lambda_{16},\lambda_{20},\lambda_{22} \in \Lambda( K3,{\mathbb{C}}).$$
Given the dynamical context of the question, it is natural to ask for stable realizations of dynamical degrees instead, that is, whether there exists some power of a given Salem number which arises as a dynamical degree.
\[thm:main2\] Let $\lambda$ be a Salem number of degree $d\leq 20$. Then there is an $n \in {\mathbb{N}}$, a projective K3 surface $X$ and an automorphism $F: X \rightarrow X$ with dynamical degree $\lambda(F)=\lambda^n$.
Considering the same question for complex tori and Enriques surfaces one finds that the answer depends only on the Betti and Hodge numbers. We can derive Theorem \[thm:main2\] from the more general Theorem \[thm:main\] below.
\[thm:main\] Let $\lambda$ be a Salem number with minimal polynomial $s(x)\in {\mathbb{Z}}[x]$ of degree $d$. Fix a class of surfaces $\mathcal{K} \in \{$2-Torus, K3, Enriques$\}$ and denote by $b_2(\mathcal{K})\in\{6,22,10\}$ the second Betti number of a surface in this class. Then there exists an $n \in {\mathbb{N}}$ with $\lambda^n \in \Lambda(\mathcal{K},{\mathbb{C}})$ if and only if
1. $d< b_2(\mathcal{K})$ or
2. $d=b_2(\mathcal{K})$ and $-s(1)s(-1) \in \left({\mathbb{Q}}^\times\right)^2$.
If additionally $d\leq h^{1,1}(\mathcal{K})$, then we can find $n'\in {\mathbb{N}}$ with $\lambda^{n'} \in \Lambda^{proj}(\mathcal{K},{\mathbb{C}})$.
The proof proceeds as follows. All Tori (resp. K3/Enriques) are diffeomorphic. Hence, the isometry class of the lattice $H^2(X,{\mathbb{Z}})$ is independent of which Torus (resp. K3/Enriques) $X$ we have chosen. It is abstractly isomorphic to some fixed lattice $L$. Given an isometry $f \in O(L)$ it is possible, using some Torelli theorem, to decide whether $f$ is in the image of the natural representation $\operatorname{Aut}(X) \rightarrow O(H^2(X,{\mathbb{Z}}))\cong O(L)$. For this to be the case $f$ has to preserve some extra linear data such as a Hodge structure or has trivial mod $2$ reduction. The most intricate case is that of *projective* K3 surfaces. There $f$ has to preserve a chamber of the positive cone - corresponding to the ample cone. Since it is usually infinite sided, it is notoriously difficult to control. For a given concrete $f$ it is now algorithmically possible to decide whether this cone is preserved [@mcmullen:minimum; @brandhorst_alonso:minimal_salem]. However, the algorithm can only deal with a single isometry at a time. In Proposition \[prop:preserves\_chamber\], we give a sufficient condition for a chamber to be preserved. We expect that it will be useful to study the stable dynamical spectrum of supersingular K3 surfaces (as in [@brandhorst_alonso:minimal_salem]) and IHSM manifolds (as in [@amerik:automorphisms_ihsm]) as well.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank Víctor Gonzalez-Alonso, Curtis T. McMullen and Matthias Schütt for comments and discussions on an early version of this paper. The financial support of the research training group GRK 1463 ”Analysis, Geometry and String Theory” is gratefully acknowledged.
Preliminaries
=============
In this section we review the necessary material from [@nikulin:quadratic_forms; @mcmullen:minimum] concerning the theory of lattices, their isometries, discriminant forms, gluings and twists.
Lattices
--------
A *lattice* is a finitely generated free abelian group $L$ equipped with a non-degenerate integer valued bilinear form $$\langle \cdot , \cdot \rangle \colon L \times L \rightarrow {\mathbb{Z}}.$$ It is called *even* if $\langle x , x \rangle \in 2{\mathbb{Z}}$ for all $x \in L$. For brevity we sometimes write $x.y$ for $\langle x,y\rangle$ and $x^2$ for $x.x$ where $x,y \in L$. The *dual lattice* $L^\vee$ of $L$ is given by $$L^\vee = \{x \in L \otimes {\mathbb{Q}}\mid \langle x , L \rangle \subseteq {\mathbb{Z}}\}.$$ Let $(e_i)$ be any ${\mathbb{Z}}$-basis of $L$, then the *determinant* of $L$ is defined as the determinant of the Gram matrix $( e_i. e_j)_{ij}$. We call $L$ *unimodular* if it is of determinant $\pm 1$. An *isometry* $M\rightarrow L$ of lattices is a homomorphism of ${\mathbb{Z}}$-modules preserving the bilinear forms. The orthogonal group $O(L)$ consists of the self isometries of the lattice $L$. The *signature* (pair) of a lattice is denoted by $(s_+,s_-)$ where $s_+$ (respectively $s_-$) is the number of positive (respectively negative) eigenvalues of the Gram matrix. A lattice is called *indefinite* if both $s_+$ and $s_-$ are non-zero and *hyperbolic* if it is indefinite and $s_+=1$. We denote by $U$, resp. $E_8$, the even unimodular lattices of signature $(1,1)$, respectively $(0,8)$. Indefinite, even unimodular lattices are classified up to isometry by their signature pair.
The *discriminant group* $D_L=L^\vee/L$ has cardinality $|\det L |$. If $L$ is an even lattice, then its discriminant group carries the *discriminant form*, given by $$q_L \colon D_L \rightarrow {\mathbb{Q}}/2{\mathbb{Z}}\quad \langle x , x\rangle \mod 2{\mathbb{Z}}.$$ If $M\subseteq L$ are lattices of the same rank, then we call $L$ an *overlattice* of $M$. Even overlattices $L$ of a lattice $M$ correspond bijectively to isotropic subgroups $L/M=H\subseteq D_M$, i.e., with $q_M|H=0$. For a prime number $p$ we denote by ${\mathbb{Z}}_p$ the p-adic integers and by ${\mathbb{Q}}_p$ the p-adic numbers. The discriminant form (and group) has an orthogonal decomposition into its $p$-primary parts $(q_L)_p$ $$q_L = \bigoplus_p \left((q_L)_p \colon (D_L)_p \rightarrow {\mathbb{Q}}_p/2{\mathbb{Z}}_p\right)$$ where $(q_L)_p$ is the discriminant form of $L\otimes {\mathbb{Z}}_p$ (defined analogously).
Embeddings and gluing
---------------------
An embedding of lattices $M \rightarrow L$ is called *primitive* if $L/M$ is torsion free. Let $M\rightarrow L$ be a primitive embedding into a unimodular lattice $L$ and $N=M^\perp$ the orthogonal complement. It is primitive as well. We get an isomorphism $ \phi \colon D_M \rightarrow D_N$, with $q_N(\phi(x))=-q_M(x)$, called glue map. Conversely, given such a glue map, its graph $$\Gamma =\{x + \phi(x) \in D_M \oplus D_N \mid x \in D_M\}$$ is isotropic with respect to the discriminant quadratic form $q_{M\oplus N}$. Hence, it defines a unimodular overlattice $L$ via $L/\left(M\oplus N \right)=\Gamma$. Let $f \in O(M)$, $g\in O(N)$ be isometries. Then $f\oplus g \in O(M\oplus N)$ extends to the overlattice $L$ if and only if $\phi \circ \bar f = \bar g \circ \phi$ where $\bar f \in O(q_M)$ and $\bar g \in O(q_N)$ are the induced actions.
Twists
------
A pair $(L,f)$ of a lattice $L$ and an isometry $f$ of $L$ with characteristic polynomial $s(x)\in \mathbb{Z}[x]$ is called a $s(x)$-lattice. Given a $s(x)$-lattice $(L,f)$ and $a \in \mathbb{Z}[f+f^{-1}]$, we obtain a new symmetric bilinear form on $L$ by setting $$\left\langle g_1 , g_2 \right \rangle_a=\left\langle ag_1 , g_2 \right \rangle.$$ The lattice $L$ equipped with this new product is called the *twist* of $L$ by $a$ and is denoted by $(L(a),f)$. Note that the twist of an even lattice stays even. Twisting may change the signature and determinant of a lattice. However, we will only twist by a square $t^2 \in {\mathbb{Z}}[f+f^{-1}]$. Then $L(t^2)$ is isomorphic, via $x\mapsto tx$, to the sublattice $tL$ of $L$. In particular, the signature of $L(t^2)$ and $L$ coincide. Of particular interest is the case when the characteristic polynomial $s(x)$ of $f$ is irreducible. Then $K={\mathbb{Q}}[f]$ is a degree two extension of the field $k={\mathbb{Q}}[f+f^{-1}]$.
\[lem:twist\_split\] Let $(L,f)$ be a $s(x)$-lattice with $s(x)$ irreducible. Suppose that $t \in {\mathbb{Z}}[f+f^{-1}]$ is a prime of $k$ split in $K$ of norm $p$ not dividing $2 \cdot \det L \cdot \operatorname{discr}p(x)$. Then the twisted p(x)-lattice $L(t^n)$ has determinant $\pm p^{2n}$ and discriminant quadratic form isomorphic to $$q_{L(t^n)}\cong \frac{1}{p^n} \left( \begin{matrix}
0 & 1\\
1 & 0
\end{matrix}
\right).$$
Since the statement is local in $p$, we can tensor with ${\mathbb{Z}}_p$. The factorization of $t$ in $K$ corresponds to the factorization of $s(x)$ in ${\mathbb{Z}}_p[x]$. Since $t$ is split, the corresponding factorization is $t=(f-a)(f-a^{-1})u(f)$ for some $a \in {\mathbb{Z}}_p$ and $u[f]\in \left({\mathbb{Z}}_p[f]\right)^\times$. Hence, we can split off the combined eigenspace $E=\ker (f+f^{-1}-a-a^{-1})$ of $f$ for $a$ and $a^{-1}$. We get $L(t^n)=E(t^n) \oplus E^\perp(t^n)$. Since $t|E^\perp$ is invertible, we get that $\det E^\perp (t^n)=\det E^\perp$ is unimodular. Hence, up to units $\det E=\det L(t^n) =\det t^{n}=N^K_{\mathbb{Q}}(t^n)=p^{2n}$. Then, in an eigenbasis the Gram Matrix of $E(t^n)$ is given by $\left( \begin{matrix}
0 & p^n\\
p^n & 0
\end{matrix}
\right)$. The matrix representing the discriminant form is now obtained by inverting the Gram matrix. Since $\det u(f)$ is a unit, $E^\perp(t^n)$ is unimodular.
Positivity
----------
Let $L$ be an even lattice. A *root* of $L$ is $r \in L$ with $r^2 = -2$. We denote the set of roots by $\Delta_L$. If $L$ is hyperbolic, we set $$V_L=\{x \in L \mid x^2 >0, r.x \neq 0 \; \forall r \in \Delta_L\}$$ which is an open set. If $L$ is negative definite, we define $$V_L=\{x \in L \mid x^2 <0, r.x \neq 0 \; \forall r \in \Delta_L\}.$$ In both cases the connected components of $V_L$ are called the *chambers* of $V_L$. An isometry $f\in O(L)$ is called *positive* if it preserves a chamber. We denote by $O^+(L)$ the subgroup stabilizing each connected component of the light cone $V^0=\{x \in L \otimes {\mathbb{R}}\mid x\neq 0, \; x^2=0\}$. A dual perspective on positivity is that of obstructing roots. An *obstructive root* for $f$ is $r \in \Delta_L$ such that there is no $h \in L$ with $h^\perp$ negative definite and $ h . f^i(r) >0$ for all $i \in {\mathbb{Z}}$. We call $r$ a *cyclic root* for $f$ if $$r + f(r) + f^2(r) + \dots + f^i(r)=0$$ for some $i>0$. Cyclic roots are obstructing. If $L$ is negative definite, then every obstructing root is cyclic. We have the following
[@mcmullen:minimum 2.1]\[thm:obstructing\_roots\] A map $f\in O^+(L)$ is positive if and only if it has no obstructing roots. The set of obstructing roots, modulo the action of $f$, is finite.
Let $L$ be hyperbolic and $f\in O(L)$ with spectral radius a Salem number $\lambda$. Denote by $\gamma$ the real plane spanned by the eigenspaces for $\lambda$ and $\lambda^{-1}$. Then the obstructing roots for $f$ are the cyclic roots together with the roots $r$ such that $r^\perp \cap \gamma$ is positive definite. To see this, note that the closure of any $f$-invariant chamber contains the intersection of $\gamma$ with the positive cone.
Surfaces and their automorphisms
================================
Let $X$ be a either a $2$-Torus, a K3 surface or an Enriques surface. Its second singular cohomology group modulo torsion $H^2(X,{\mathbb{Z}})/tors$ equipped with the cup product is a unimodular lattice isomorphic to $$H^2(X,{\mathbb{Z}})/tors \cong \begin{cases}
3U & \mbox{for } $X$ \mbox{ a 2-Torus} \\
U \oplus E_8 & \mbox{for } $X$ \mbox{ an Enriques surface} \\
3U \oplus 2 E_8 &\mbox{for } $X$ \mbox{ a K3 surface.}
\end{cases}$$ It admits a Hodge decomposition $$H^2(X,\mathbb{Z})\otimes \mathbb{C} \cong H^2(X,\mathbb{C})=H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X)$$ where $H^{i,j}(X)\cong H^j(X,\Omega_X^i)$, $H^{i,j}(X)=\overline{H^{j,i}(X)}$ and $H^{1,1}(X)=(H^{2,0}(X) \oplus H^{0,2}(X))^\perp$ is hyperbolic. By Lefschetz’ Theorem on $(1,1)$ classes we can recover the numerical divisor classes from the Hodge structure as $$\operatorname{Num}(X)= H^{1,1}(X)\cap H^2(X,\mathbb{Z})/tors.$$ We note that a (compact) Kähler surface $X$ is projective if and only if there is a divisor of positive square, i.e. $\operatorname{Num}(X)$ has signature $(1,\operatorname{rk}\operatorname{Num}(X)-1)$. The transcendental lattice is defined as the smallest primitive sublattice $T\subseteq H^2(X,\mathbb{Z})$ whose complexification contains $H^{2,0} \subseteq T\otimes \mathbb{C}$. Since in our case $h^{2,0}\in \{0,1\}$, the Hodge structure on $T$ is irreducible.
Let $F\in \operatorname{Aut}(X)$ be an automorphism with dynamical degree $\lambda>1$. We call the minimal polynomial $s(x)\in {\mathbb{Z}}[x]$ of $\lambda$ a *Salem polynomial*. Then $F^*$ is semisimple with characteristic polynomial $s(x)c(x)$ where $c(x)$ is a product of cyclotomic polynomials [@mcmullen:siegel_disk Thm. 3.2]. Set $S=\ker s(F^*)\subseteq H^2(X,{\mathbb{Z}})/tors$. By irreducibility of $T(X)$, the minimal polynomial of $F^*|T(X)$ must be irreducible in ${\mathbb{Q}}[x]$ too. Hence, either $T(X)= S$ or $S\subseteq \operatorname{Num}(X)$. Since the real eigenvectors for $\lambda$ and $\lambda^{-1}$ span a hyperbolic plane, the signature of $S$ is either $(1,\deg s(x)-1)$ if $S\subseteq \operatorname{Num}(X)$ or in case $S=T(X)$ it is $(3,\deg s(x)-3)$. In the first case $X$ is projective and in the second not.\
In the following three Lemmas we collect criteria for an isometry of the cohomology lattice to come from an automorphism of a surfaces.
[@barth:automorphism_enriques]\[lem:torelli\_enriques\] The automorphism group of a very general Enriques surface $X$ is the $2$-congruence subgroup given by the kernel of $$O^+(H^2(X,{\mathbb{Z}}))\rightarrow O(H^2(X,{\mathbb{Z}})\otimes {\mathbb{F}}_2).$$ It is of finite index in the orthogonal group of $H^2(X,{\mathbb{Z}})/tors\cong U\oplus E_8$.
\[lem:torelli\_tori\] Let $s(x)$ be a Salem polynomial of degree $d$ and $f \in O(3U)$ an isometry with characteristic polynomial $s(x)(x-1)^{6-d}$ which acts trivially on $3U \otimes {\mathbb{F}}_2$. Then one can find a complex $2$-torus $T$ with $3U=H^2(T,{\mathbb{Z}})$ and $F \in \operatorname{Aut}(T)$ such that $F^* =f$. The $2$-torus $T$ is projective if and only if $S=\ker s(f)$ has signature $(1,d-1)$.
Let $L$ be a free ${\mathbb{Z}}$ module of rank $4$. An orientation on $L$ is an isomorphism $\det\colon \bigwedge^4 L \xrightarrow{\sim} {\mathbb{Z}}$. It gives rise to an even unimodular lattice which we identify with $3U$. We can choose an eigenvector $\eta \in 3U \otimes {\mathbb{C}}$ of $f \otimes {\mathbb{C}}$ such that $\eta^2=0$ and $\eta.\bar \eta>0$. Since ${\mathbb{C}}\eta$ is isotropic, it is in the image of the Plücker embedding $Gr(2,L\otimes {\mathbb{C}}) \rightarrow \bigwedge^2 L \cong 3U$. Hence, we get a complex $2$-plane $S$ with $\bigwedge^2S={\mathbb{C}}\eta$ and $S \oplus \bar S=L\otimes {\mathbb{C}}$. This defines a weight one Hodge structure on $L$, i.e., a complex $2$-torus $T$. We can view $f$ as an isometry of $H^2(T,{\mathbb{Z}})=\bigwedge^2 L\cong 3U$ which, by construction, preserves the Hodge structure on $H^2(T,{\mathbb{Z}})$. Since $f\otimes {\mathbb{F}}_2$ is the identity, we can apply, [@BHPV:compact_complex_surfaces V (3.2)] to get an automorphism $F$ of $T$ with $F^*=\pm f$. However, both $F^*$ and $f$ stabilize each connected component of the positive cone of $H^{1,1}(T)$. Hence, they are equal.
\[lem:torelli\_k3\] Let $f \in O(3U\oplus 2E_8)$ be an isometry with characteristic polynomial $s(x)(x-1)^{22-d}$.Then one can find a K3 surface $X$, $F \in \operatorname{Aut}(X)$ and an isometry $\phi:3U\oplus 2E_8 \rightarrow H^2(X,{\mathbb{Z}})$ such that $F^* =\phi \circ f \circ \phi^{-1}$ if and only if
1. $S=\ker s(f)$ has signature $(3,d-3)$ or
2. $S$ has signature $(1,d-1)$ and $f|S$ is positive.
In case (2) $X$ is projective and in case (1) not.
The lemma follows once we check the conditions of [@mcmullen:minimum 6.1]. If the signature of $S$ is $(3,d-3)$, then we take as period an eigenvector $\eta\in S\otimes {\mathbb{C}}$ of $f$ with $\eta.\bar{\eta}>0$. Since $f$ is the identity on $S^\perp$ there are no cyclic roots, and $f|S^\perp$ is positive. If the signature of $S$ is $(1,d-1)$, then we take as period a very general line in $S^\perp \otimes {\mathbb{C}}$.
Proof of Theorem \[thm:main\]
=============================
In order to prove the main Theorem \[thm:main\], we need to produce isometries of certain lattices with given spectral radius. In general this can be difficult. Hence, we simplify the problem by asking for *rational* isometries first. Indeed, here the answer is known as is displayed by the following Lemma \[lem:iso\_QQ\]. We postpone its proof till the end of this paper.
\[lem:iso\_QQ\] Let $L \in \{3U, U\oplus E_8, 3U\oplus 2E_8\}$ and $s(x)$ be a Salem polynomial of degree $d$. Then there exists a rational isometry $f\in O(L\otimes {\mathbb{Q}})$ with characteristic polynomial $\det (xId-f)=s(x)(x-1)^{\operatorname{rk}L-d}$ if and only if either
1. $d\leq \operatorname{rk}L-2$ or
2. $d=\operatorname{rk}L$ and $-s(1)s(-1)$ is a square.
In case $(1)$ we can find $f$ such that $\ker s(f)$ is hyperbolic. If the signature of $L$ is $(3,\operatorname{rk}L-3)$, then we can find $f$ such that $\ker s(f)$ has signature $(3,d-3)$.
Typically, a rational isometry $f\in O(L\otimes {\mathbb{Q}})$ does not preserve $L$. Since we are only considering the *stable* dynamical spectrum, we may replace $f$ by some power $f^n$.
\[lem:iso\_ZZ\] Let $L$ be a lattice and $f\in O(L\otimes {\mathbb{Q}})$ a rational isometry with $$\det(xId-f) \in {\mathbb{Z}}[x].$$ Then one can find $n\in {\mathbb{N}}$ such that $f^n\in O(L)$.
Since the characteristic polynomial of $f$ is integral, the ${\mathbb{Z}}$-module ${\mathbb{Z}}[f]L$ is finitely generated and of the same rank as $L$. Consequently, for the index $k=[{\mathbb{Z}}[f]L:L]$ we get the chain of inclusions $$k {\mathbb{Z}}[f]L \subseteq L \subseteq {\mathbb{Z}}[f]L.$$ Conclude by taking $n\in {\mathbb{N}}$ such that $f^n$ acts as the identity on the finite quotient ${\mathbb{Z}}[f]L/k{\mathbb{Z}}[f]L$.
We now have all the ingredients for the
A combination of Lemmas \[lem:iso\_QQ\] and \[lem:iso\_ZZ\] provides us with isometries of $L \in \{3U, U\oplus E_8\}$ with spectral radius some power of the desired Salem number. After raising the isometries to some sufficiently divisible power, we can assume that they satisfy the conditions of Lemmas \[lem:torelli\_enriques\] and \[lem:torelli\_tori\].
The same argument completes the proof of the main theorem for non-projective K3 surfaces. It remains for us to control the positivity of the isometries to prove the result for projective K3 surfaces as well.
\[prop:preserves\_chamber\] Let $f \in O(N)$ be an isometry of a hyperbolic lattice $N$ with characteristic polynomial a Salem polynomial $s(x)$. If $$|\det N|> 4 \operatorname{discr}s(x),$$ then $f$ preserves a chamber of the positive cone.
Let $L$ and $f$ be as in the proposition and denote by $$\pi: S\otimes {\mathbb{R}}\rightarrow \ker(f+f^{-1}-\lambda-\lambda^{-1})$$ the orthogonal projection where $\lambda>1$ is a root of $s(x)$. Suppose that $f$ does not preserve a chamber. Then, by Theorem \[thm:obstructing\_roots\], there is an obstructive root $r\in L$. This means that $r^2=-2$ and $r^\perp$ crosses the geodesic $\gamma$ of $f$, i.e. $\pi(r)^2<0$. Since $s(x)$ is irreducible over ${\mathbb{Q}}$, ${\mathbb{Z}}[f]r$ is a sublattice of full rank, and hence $$|\det N|\leq |\det {\mathbb{Z}}[f]r|.$$ The basic idea at this point is that the obstructing roots modulo the action of $f$ lie in some compact fundamental domain in $N\otimes {\mathbb{R}}$ depending only on $s(x)$. Then we can maximize $|\det {\mathbb{Z}}[f]r|$ over all $r$ in this fundamental domain. We extend the bilinear form to a ${\mathbb{C}}$-linear form on $N\otimes {\mathbb{C}}$ and compute the determinant in an eigenbasis of $f$. We can find $u_1,u_2 \in N \otimes {\mathbb{R}}$ and $v_i \in N \otimes {\mathbb{C}}$ such that $$f(u_1)=\lambda u_1, \;\; f(u_2)=1/\lambda u_2, \;\;f(v_i)=\alpha_iv_i, \;\;i\in \{1,\dots, k\}$$ where $\lambda>1, 1/\lambda$, $\alpha_i,\overline{\alpha}_i$ are the complex roots of $s(x)$ and $\deg s(x)=2k+2$. After rescaling, we may assume that $\langle u_1,u_2 \rangle=1$ and $\langle v_i,\overline{v_i} \rangle=-1$ for $i \in \{1, \dots, k\}$. Now, write $$r=x_1u_1+x_2u_2+\sum_{i=1}^k (y_i v_i + \overline{y_iv_i})$$ for $x_1,x_2 \in {\mathbb{R}},y_i \in {\mathbb{C}}$. The Van-der-Monde determinant yields that $$|\det \langle f^i(r),f^j(r)\rangle|= |x_1x_2|^2 \prod_{i=1}^k|y_i|^4 \operatorname{discr}s.$$ Since $r$ is obstructing, $x_1x_2=\langle x_1u_1,x_2u_2\rangle \in [-2,0)$ and in these coordinates $r^2=x_1x_2 - \sum_{i=1}^k |y_i|^2=-2$, i.e. the coordinates of $r$ lie in the set $$K=\left\{(x,y) \in {\mathbb{R}}^{2}\times {\mathbb{C}}^{k} : x_1x_2 \in [-2,0), \; x_1x_2 - \sum_{i=1}^k |y_i|^2=-2 \right\}.$$ Then, assuming $k\neq 0$, $$\begin{aligned}
|\det N| &\leq &|\det \langle f^i(r),f^j(r)\rangle|\\
&\leq& \sup \left\{|x_1x_2| \prod_{i=1}^k|y_i|^2 : (x,y) \in K\right\}^2 \operatorname{discr}s\\
&=& \sup_{c \in (0,2]} c^2 \cdot \sup \left\{\prod_{i=1}^k |y_i|^2 : \sum_i |y_i|^2=2-c\right\}^2\operatorname{discr}s\\
&=& \left[\sup_{c \in (0,2]} c(2-c)^{k} \sup \left\{\prod_{i=1}^k |y_i|^2 : \sum_i |y_i|^2=1\right\}\right]^2\operatorname{discr}s\\
&=&\left[\frac{2}{1+k}\left(\frac{2}{k}\right)^k\left(1-\frac{1}{1+k}\right)^k\right]^2\operatorname{discr}s\\
&\leq & \operatorname{discr}s \end{aligned}$$ In line (5) we have used that $$1/k^k=\max\left\{\prod_{i=1}^k y_i^2 \mid (y_1,\dots,y_k) \in {\mathbb{R}}^k, \sum_{i=1}^{k} y_i^2=1\right\}.$$ Here the maxima lie at $|y_i|=1/\sqrt{k}$, $i \in \{1,\dots,k\}$. For the case of $k=0$ one obtains $4\operatorname{discr}s$.
For any natural number $n$ let $s_n(x)$ denote the minimal polynomial of $\lambda^n$, and set $L_{K3}=3U\oplus 2 E_8$. By Lemmas \[lem:iso\_QQ\] and \[lem:iso\_ZZ\], for some $n \in {\mathbb{N}}$ we get an isometry $f\in O(L_{K3})$ with characteristic polynomial $s_n(x)(x-1)^{22-d}$ and such that $S=\ker s_n(f)$ is hyperbolic.
What remains is to modify $S=\ker s_n(f)$ in order to increase the determinant of $S$ such that $f|S$ preserves a chamber by Proposition \[prop:preserves\_chamber\]. Set $R=S^\perp=\ker(f-id)$. The primitive extension $S\oplus R\hookrightarrow L_{K3}$, provides us with an isomorphism of discriminant quadratic forms $q_S \cong q_R(-1)$. By Chebotarev’s density theorem there are infinitely many prime ideals $\mathfrak{p}<{\mathcal{O}}_k$ of degree one, split in $K/k$ such that $$p \equiv 1 \mod (8\det R)$$ where $(p)={\mathbb{Z}}\cap \mathfrak{p}$. Choose one such $\mathfrak{p}$ with $p>\operatorname{discr}s_n(x)$. We can find $l\in {\mathbb{N}}$ and $t \in {\mathcal{O}}_k$ with $\mathfrak{p}^l=t{\mathcal{O}}_k$, and then $$|\det S(t^{2})|=|\det S|p^{2l}>\operatorname{discr}s_n(x).$$ For primes $p'\neq p$, $S(t^{2})\otimes {\mathbb{Z}}_{p'} \rightarrow S\otimes {\mathbb{Z}}_{p'}$, $x\mapsto tx$ is an isometry. Hence, $$(q_{S(t^{2})})_{p'}\cong (q_{S})_{p'} \cong (q_{R})_{p'}(-1)\cong (q_{R(p^{l})})_{p'}(-1).$$ By Lemma \[lem:twist\_split\] $$(q_{S(t^{2})})_{p}\cong p^{-2l} \left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right)\cong p^{-2l}\left(\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right).$$ Further, a direct computation reveals $$(q_{R(p^{2l})})_{p}(-1)\cong p^{-2l} \left(\begin{matrix} 1 & 0 \\ 0 & \det R \end{matrix}\right).$$ The two forms are isomorphic if and only if both $-1$ and $\det R$ are of the same square class in ${\mathbb{Z}}_p^\times/\left({\mathbb{Z}}_p^\times\right)^2$. This is computed by the Legendre symbols $\left(\frac{-1}{p}\right)$ and $\left(\frac{\det R}{p}\right)$. Since $p\equiv 1 \mod (8\det R)$, we have $\left(\frac{-1}{p}\right)=1$ and $\left(\frac{\det R}{p}\right)=\left(\frac{p}{\det R}\right)=1$. Piecing together the isometries for the different primes, we have constructed a glue map $q_{S(t^2)}\cong q_{R(p^{2t})}(-1)$. Its graph provides us with an overlattice of $S(t^2)\oplus R(p^{2t})$ isomorphic to $L_{K3}$. In other words, we have a primitive embedding of $S(t^{2})$ into $L_{K3}$ with orthogonal complement $R(p^{2l})$. By construction, $f|S(t^{2})$ preserves a chamber, and after replacing $f$ by a sufficiently divisible power $f^k$, we may glue it to the identity on $R(p^{2t})$ to obtain an isometry $f^k|_{S(t^{2})} \oplus id_{R(p^{2t})}$ which extends to $L$. We can apply Lemma \[lem:torelli\_k3\] (2) to get a projective K3 surface $X$ and $F \in \operatorname{Aut}(X)$ with dynamical degree some power of $\lambda$.
Proof of Lemma \[lem:iso\_QQ\]
------------------------------
Except for the signature condition, Lemma \[lem:iso\_QQ\] is a special case of [@bayer-fluckiger:rationa_isometries Cor. 9.2, Prop. 11.9]. In what follows, we inspect the original proof. For the readers convenience, we recall some of the notation involved. We need only the following special case: $k={\mathbb{Q}}$ and $\Sigma_k$ is the set of its places. Then $q$ is the quadratic form on $L\otimes {\mathbb{Q}}$. An isometry $t\in O(q)$ induces the structure of a self-dual torsion ${\mathbb{Q}}[x]$-module on $L_{\mathbb{Q}}$ via $p(x).v=p(t)v$ for $v\in L_{\mathbb{Q}}$ and $p(x) \in {\mathbb{Q}}[x]$. We set $$M_0=\left({\mathbb{Q}}[x]/(x-1)\right)^{22-r}, \quad M_1={\mathbb{Q}}[x]/s(x), \quad \mbox{ and } \quad M=M_0\oplus M_1.$$ Then $\mathcal{C}_{M,q}$ is the set of all collections of forms $C=\{q_i^\nu \}$ for $i \in I_0=\{0,1\}$ and $\nu \in \Sigma_k$, such that $q_i^\nu$ has an isometry with module $M_i\otimes {\mathbb{Q}}_\nu$ and $q_0^\nu \oplus q_1^\nu$ is isomorphic to the localization $q^\nu=q\otimes {\mathbb{Q}}_\nu$ of $q$ at $\nu$. For $i\in I_0$, we set $$T_i(C)=\{ \nu \in \Sigma_k \mid w(q_i^\nu)=1 \}$$ where $w(q_i^\nu)$ is the Hasse invariant of $q_i^\nu$. Let $\mathcal{F}_{M,q}$ be the subset of $\mathcal{C}_{M,q}$ such that for all $i \in I_0$, $T_i(C)$ is a finite set.
The main step involved is
[@bayer-fluckiger:rationa_isometries Thm. 10.8]\[thm:existence\_rat\_module\] Let $M$ be a self-dual torsion $k[x]$-module which is finite dimensional as a $k$-vector space. Suppose that the quadratic form $q$ over the global field $k$ has an isometry with module $M$ over $k_\nu$ for all places $\nu$ of $k$. Then $q$ has an isometry with module $M$ if and only if there exists a collection $C=\{q_i^\nu \} \in \mathcal{F}_{M,q}$ such that for all $i \in I_0$, the cardinality of $T_i(C)$ is even. In this case $q_i^\nu=q_i\otimes \kappa_\nu$.
By [@bayer-fluckiger:rationa_isometries Prop. 11.9], we find $f\in O(q)$ with module $M$. Hence its characteristic polynomial has the desired form $$\det(xId-f)=s(x)(x-1)^{22-r}.$$ If $q_1=q|S_{\mathbb{Q}}$ has signature $(1,r-1)$, we are done, else $q_1$ has signature $(3,r-3)$. By Theorem \[thm:existence\_rat\_module\], this provides us with a collection $C=\{q_i^\nu\} \in \mathcal{F}_{M,q}$ with $q_i^\nu=q_i\otimes \kappa_\nu$ such that $|T_i(C)|$ is even for $i\in \{0,1\}$. We set $$d_1\equiv\det q_1 \equiv s(1)s(-1) \mod {\mathbb{Q}}^{\times 2}$$ and $$d_0\equiv\det q_0\equiv d_1 \det q =-s(1)s(-1) \mod {\mathbb{Q}}^{\times2}.$$ Denote by $\Omega(M_i,d_i)$ the set of finite places $\nu$ of ${\mathbb{Q}}$ such that for any $\epsilon \in \{0,1\}$ there is a quadratic space $Q$ over ${\mathbb{Q}}_\nu$ with determinant $d_i$, Hasse invariant $w(Q)=\epsilon$ and which has an isometry with module $M_i$. By [@bayer-fluckiger:rationa_isometries Prop. 11.9] every finite place is in $\Omega( M_0,d_0)$. Hence $$\Omega_{0,1}=\Omega(M_0,d_0) \cap \Omega(M_1,d_1)=\Omega(M_1,d_1),$$ which is non-empty by [@bayer-fluckiger:rationa_isometries Lem. 9.4, 9.6] and Chebotarev’s density theorem for the degree two extension ${\mathbb{Q}}[\lambda]$ of ${\mathbb{Q}}[\lambda+\lambda^{-1}]$. Choose a place $p\in \Omega_{0,1}$. We can define a new collection $\tilde{C}=\{\tilde{q}_i^\nu\}$ by $\tilde{q}_i^\nu=q_i^\nu$ for $i\in \{0,1\}$ and $\nu \neq p, \infty$, For $\tilde{q}_i^\infty$, we take forms of signature $(1,d-1)$, respectively $(2,\operatorname{rk}L-2-d)$. At $p$ we just switch the Hasse invariants of $q_i^p$. By [@bayer-fluckiger:rationa_isometries Lem. 9.6], $\tilde{C} \in \mathcal{C}_{M,q}$, and moreover $\tilde{C}\in \mathcal{F}_{M,q}$. Since the Hasse invariants have changed at two places, $|T_i(\tilde{C})|$ is still even. Finally, $\tilde{C}$ meets the conditions of Theorem \[thm:existence\_rat\_module\] and the claim follows.
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---
abstract: 'We discuss how to compute certified enclosures for the eigenvalues of benchmark linear magnetohydrodynamics (MHD) operators in the plane slab and cylindrical pinch configurations. For the plane slab, our method relies upon the formulation of an eigenvalue problem associated to the Schur complement, leading to highly accurate upper bounds for the eigenvalue. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigourously justified. Therefore in this case we rely on a specialized technique based on a method proposed by Zimmermann and Mertins. In turns this technique is also applicable for finding accurate complementary bounds in the case of the plane slab. We establish convergence rates for both approaches.'
address:
- 'Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK'
- 'Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK'
author:
- Lyonell Boulton
- Michael Strauss
title: Eigenvalue enclosures and convergence for the linearized MHD operator
---
Introduction
============
Let $\Omega\subset {{\mathbb R}}^3$. The linearized ideal MHD equation $$\rho \partial_t^2 \xi(t) + K\xi(t)=0 \qquad \xi(0)=\xi_0 \qquad \partial_t \xi(0) =v_0$$ for displacement vector $\xi:\Omega\longrightarrow {{\mathbb R}}^3$ and force operator $$K\xi={\operatorname{grad}}[\gamma P({\operatorname{div}}\xi)+({\operatorname{grad}}P)\cdot \xi]+
\frac{1}{\mu} [{\mathrm{B}}\times{\operatorname{curl}}({\operatorname{curl}}(\mathbf{B}\times \xi))-({\operatorname{curl}}\mathbf{B})\times {\operatorname{curl}}(\mathbf{B}\times \xi)]$$ arises in applications from plasma confinement in thermonuclear fusion. The constants $\mu$ and $\gamma$ here denote the magnetic permeability and heat ratio. The smooth function $\rho$ is the density, $P>0$ is the pressure and $\mathbf{B}$ the divergence-free magnetic field of the given equilibrium, satisfying $\mu {\operatorname{grad}}P=({\operatorname{curl}}\mathbf{B})\times\mathbf{B}$.
In the study of this equation a fundamental role is played by the eigenvalue problem associated to $K$. The appropriate Hilbert space setting ensures that $K$ has a self-adjoint realization. A considerable amount of research has been devoted to the formulation of a rigorous operator theoretic framework for $K$ and to the structure of the spectrum, [@1989Lifschitz]. Particular attention has been payed to the plane slab (plasma layer) and the cylindrical (plasma pinch) configurations [@1987Kako; @1991Kakoetal; @1991Raikov; @atla] where $K$ is reduced to a block ordinary differential operator matrix. A systematic description (analytical or numerical) of properties of the eigensolutions turns out to be difficult even for these, the simplest configurations. This is due to the presence of regions of essential spectrum near the bottom end of the spectrum.
The plane slab configuration has been the subject of thorough analytical investigation and has become a benchmark model for the top dominant class of block operator matrix, see [@2007Tretter] and references therein. Precise eigenvalue asymptotics can be found in this case by means of the WKB method [@1989Lifschitz §7.5] or by means of specialised variational principles, see [@2007Tretter Theorem 3.1.4] and references therein. The cylindrical configuration is more involved due to the presence of singularities in the coefficients of the differential expression; however, eigenvalue asymptotics are known in this case, [@1986Raikov].
Specialized variational approaches are extremely useful for examining analytic asymptotics for the eigenvalues in the case of the plasma layer configuration. Unfortunately, as they usually involve a triple variation formulation, it is arguable whether they are well suited for direct numerical implementations.
If a sequence of subspaces is guaranteed not to produce spectral pollution, then the standard Galerkin method can be used. A prescribed recipe for avoiding spurious modes when these subspaces are generated by the finite element method dates back to [@1977Rappaz; @1991Kakoetal]. In this classical approach convergence is guaranteed, however, it is never clear whether a computed eigenvalue is on the left or on the right of the exact eigenvalue. In this respect the method is not certified.
A technique for finding certified enclosures for the eigenvalues of $K$ in the case of the plane slab configuration was considered in [@2011Strauss] based on the method proposed in [@1998Davies; @shar; @LS2004]. The approach was based on computing the so-called second order spectrum of $K$ for given finite dimensional subspaces generated by the spectral basis. In the present paper we consider a further computational strategy which improves upon this technique in terms of accuracy. Our main approach is to combine two complementary Galerkin-type methods for computing eigenvalue enclosures which, by construction, never produce spectral pollution.
For the plane slab, our method relies on the formulation of an eigenvalue problem associated to the Schur complement, this leads to highly accurate upper bounds. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigorously justified. Therefore in this case we rely on a specialized technique based on a method proposed by Zimmermann and Mertins [@zim] as described by Davies and Plum in [@dapl Section 6]. This approach is intimately related to classical methods, see [@1949Kato; @1963Lehmann; @1985Goerisch]. We also apply this technique to the Schur complement and find accurate complementary lower bounds for the plane slab.
In Section 2 we give a mathematical formulation of the MHD operators under investigation, and some of their spectral properties. In Section 3 we examine the approximation technique due to Zimmermann and Mertins. We present a formulation of this technique in terms of the Galerkin method which establishes both approximation and, importantly, the convergence of the method. Our main results are contained in Section 4. We present a highly efficient method for obtaining upper bounds for eigenvalues above the essential spectrum of top-dominant block operator matrices, an example of which is the matrix $K$ associated to the plasma layer configuration. We show in Theorem \[super\] that the convergence rate for this approach is the same as that achieved by the Galerkin method when applicable (below the essential spectrum). Our method is therefore extremely efficient. We also combine this approach with the Zimmermann and Mertins technique to obtain complementary lower bounds for the eigenvalues. In Theorem \[enclose\] we use our results from Section 3 to obtain convergence rates for these lower bounds. In Sections 5 and 6 we apply our results to the plasma and cylindrical configurations, respectively.
One-dimensional MHD operators
=============================
The reduction process for the force operator, the precise constraints on the equilibrium quantities and the boundary conditions on $\Omega$, which yield the one-dimensional boundary value problems associated to the plane slab and cylindrical configurations, are described in detail in [@1989Lifschitz §7.2 and §8.2], respectively. Through this reduction $K$ becomes similar to a superposition of operators which are self-adjoint extensions of block matrix differential operators of the form $$\label{matrix}
M_0 = \begin{pmatrix} A & B \\ B^* & D \end{pmatrix}$$ acting on $L^2$-spaces of a one-dimensional component.
For the plasma layer, the components of $M_0$ are explicitly given by $$\label{slab}
\left\{ \begin{aligned}
A&= -\rho_0^{-1}\partial_x \rho_0(v_a^2 + v_s^2)\partial_x + k^2v_a ^2 ,
\\ B&= \begin{pmatrix}(-i\rho_0^{-1}\partial_x\rho_0(v_a^2 + v_s^2)+ig)k_{\perp},
(-i\rho_0^{-1}\partial_x \rho_0v_s^2+ig)k_{\parallel} \end{pmatrix},\\
D&=\begin{pmatrix} k^2v_a^2 + k_{\perp}^2v_s^2
& k_{\perp}k_{\parallel}v_s^2 \\ k_{\perp}k_{\parallel}v_s^2 & k_{\parallel}^2v_s^2
\end{pmatrix},
\end{aligned} \right.$$ where $$\begin{aligned}
{{\rm Dom}}(A) &= H^{2}((0,1);\rho_0 {\mathrm{d}}x)\cap H_{0}^{1}((0,1);\rho_0 {\mathrm{d}}x) \\
{{\rm Dom}}(B) &= [H^{1}((0,1);\rho_0 {\mathrm{d}}x)]^2 \\
{{\rm Dom}}(D) &= [L^2((0,1);\rho_0\,{\mathrm{d}}x)]^2.\end{aligned}$$ We assume that the Alfvén speed $v_a$, the sound speed $v_s$, and the coordinates of the wave vector $k_\perp$ and $k_\parallel$, are bounded differentiable functions. We also assume that $\rho_0$ and $v_s$ are bounded away from $0$. Following standard notation in the literature $k^2(x)=k_\perp^2(x)+k_\parallel^2(x)$ and $g$ is the gravitation constant. In this case [@2007Tretter Proposition 3.1.2] $$\label{dom_self}
M_0:[{{\rm Dom}}(A)\cap{{\rm Dom}}(B^*)]\times {{\rm Dom}}(B) \longrightarrow\left[L^2((0,1);\rho_0\,{\mathrm{d}}x)\right]^3$$ is essentially self-adjoint. Denote by $M_{\slb}$ the closure of $M_0$. Then $M_{\slb}$ is bounded from below, and the essential spectrum is given by the range of the Alfvén frequency $v_a^2k_{\parallel}$ and the mean frequency $v_a^2v_s^2k_{\parallel}/(v_a^2 + v_s^2)$, see [@1989Lifschitz §7.6]. The discrete spectrum always accumulates at $+\infty$. We show in Example \[constant\_coeff\] that endpoints of the essential spectrum can also be points of accumulation.
\[constant\_coeff\] Let $k_{\|}=\rho=v_a=v_s=g=1$ and $k_\perp=0$. Then ${{\rm Spec}}_{\mathrm{ess}}(M_\slb)=\{1/2,1\}$ and ${{\rm Spec}}_{\mathrm{dis}}(M_\slb)=\{\lambda_k^\pm\}_{k=1}^\infty$ where $$\begin{aligned}
\lambda_k^+&=1+k^2\pi^2+\sqrt{1+k^2\pi^2+k^4\pi^4} =\mathcal{O}(k^2) & \text{and} \\
\lambda_k^-&= 1+k^2\pi^2-\sqrt{1+k^2\pi^2+k^4\pi^4}\to\frac12 \qquad &\text{as } k\to\infty.
\end{aligned}$$ Both $\lambda_k^\pm$ are positive and increasing in $k$. Also $$\begin{gathered}
\operatorname{Range} \int_{(-\infty,3/4]} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mathrm{d}}E_\lambda ={{\rm span}}\{ \phi^-_{k}\} , \qquad
\operatorname{Range} \int_{(3/4,3/2]} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\mathrm{d}}E_\lambda ={{\rm span}}\{ \phi^1_{k}\} \\
\text{and} \qquad \operatorname{Range} \int_{(3/2,\infty)} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mathrm{d}}E_\lambda ={{\rm span}}\{ \phi^+_{k}\},
\end{gathered}$$ where $$\label{eigenfunctions}
\begin{gathered}
\phi^-_{k}(x) = \alpha^-_{k}
\begin{pmatrix} \sin k\pi x \\ 0 \\ \frac{i}{1-\lambda^-_k}(k\pi \cos k\pi x +\sin k\pi x) \end{pmatrix}, \quad
\phi^1_{k}(x) = \frac{1}{\sqrt{2}}
\begin{pmatrix} 0\\ \sin k\pi x \\ 0 \end{pmatrix} \\
\text{and} \qquad \phi^+_{k}(x) = \alpha^+_{k}
\begin{pmatrix} \sin k\pi x \\ 0 \\ \frac{i}{1-\lambda^+_k}(k\pi \cos k\pi x +\sin k\pi x) \end{pmatrix},
\end{gathered}$$ the constants $\alpha^\pm_k$ chosen so that $\{\phi_k^\pm\}_{k=1}^\infty$ are normalized. Note that $\{\phi_k^-,\phi_k,\phi_k^+\}$ is an orthonormal basis of $[L^2(0,1)]^3$.
\[non-constant\_coeff\] Let $\rho_0 = 1$, $k_{\perp} = 1$ , $k_{\parallel} = 1$, $g=1$, $v_a(x)=\sqrt{7/8 - x/2}$, and $v_s(x) = \sqrt{1/8 + x/2}$. The essential spectrum of $M_\slb$ is given by $${{\rm Ran}}(v_a^2k_{\parallel})\cup{{\rm Ran}}(v_a^2v_s^2k_{\perp}/(v_a^2 + v_s^2)) = [7/64,1/4]\cup[3/8,7/8].$$ Below we will use the fact that $d=\max{{\rm Spec}}(D) = 1+\sqrt{17/32}<\pi^2$.
The plasma pinch configuration yields a differential operator $M_0$ with singular coefficients, $$\label{cylinder} \left\{
\begin{aligned}
A&= -\partial_r (b^2+\gamma P) \partial_r^\ast + r\left(
\frac{b^2\sin^2 \phi }{r^2}\right)'+b^2k^2_\phi
\\ B&= \begin{pmatrix}(-i\partial_r (b^2+\gamma P) m_\phi +2i b^2k\frac{\sin \phi}{r},
-i\partial_r \gamma P k_\phi \end{pmatrix}\\
D&= \begin{pmatrix} m_\phi^2 (b^2 + \gamma P)+b^2 k^2_\phi
& m_\phi k_\phi \gamma P \\ m_\phi k_\phi \gamma P & k_\phi^2 \gamma P
\end{pmatrix}
\end{aligned} \right.$$ acting on $[L^2((0,R_0);r\, {\mathrm{d}}r)]^3$. Here $\partial_r^\ast=\frac1r \partial_r r$, $$\begin{aligned}
\mathbf{B}&=(0,b(r) \sin(\phi(r)),b(r)\cos \phi(r)), \\
P'(r)&=-b(r)b'(r)=\frac{1}{r} b(r)^2\sin^2 \phi(r),
\end{aligned}$$ $b(r)$ and $\phi(r)$ are smooth functions with $b'(0)=\phi(0)=0$, $$k_\phi=k \cos \phi + \frac{m}{r} \sin \phi \qquad
\text{and} \qquad m_\phi=\frac{m}{r} \cos \phi -k \sin \phi.$$ The indices $R_0 k$ and $m$ are integer numbers corresponding to the Fourier mode decomposition of $K$.
In order to define rigourously the domain of $M_0$ for this configuration, a further change of variables $r=e^{s}$ is usually implemented, [@1987Kako]. Under this change of variables, $M_0$ becomes similar to an operator acting on $[L^2((-\infty, \log R_0);{\mathrm{d}}x)]^3$ which is essentially self-adjoint in the space of rapidly decreasing functions at $-\infty$ vanishing at $ \log R_0$, [@1987Kako Theorem 2.3]. Operator $M_0$ is essentially self-adjoint in the pre-image of this space under the similarity transformation. We denote by $M_{\mathrm{c}}$ the unique self-adjoint extension of $M_0$ in the latter domain.
The original formulation is numerically more stable for the treatment of the eigenvalues via a projection method. The finite element space generated by Hermite elements of order $3$, $4$ and $5$, subject to Dirichlet boundary conditions at $0$ and $R_0$, considered below are $C^1$-conforming and hence are all contained in ${{\rm Dom}}(M_{{\mathrm{c}}})$ for the benchmark equilibrium quantities considered in our examples.
The essential spectrum of $M_{{\mathrm{c}}}$ consists of an Alfvén band determined by the range of $b^2 k_\phi^2$, and a slow magnetosonic band determined by the range of $(b^2 k_\phi^2 \gamma P)/(b^2+\gamma P)$, [@1987Kako Theorem 3.5]. As in the previous configuration, these bands are located near the bottom of the spectrum and $+\infty$ is always an accumulation point if the discrete spectrum.
\[example\_cylinder\] Let $P\equiv 0$, $b\equiv 1$, $\phi\equiv 0$, and $k=m=1$. Then ${{\rm Spec}}_{\mathrm{ess}}(M)=\{0,1\}$ (the point $0$ is the slow magnetosonic spectrum and the point $1$ is the Alfvén spectrum). On the other hand ${{\rm Spec}}_{\mathrm{dis}}(M)=\{E^2+1:J'_1(E)=0\}$ where $J_v(x)$ is the Bessel function of index $v$.
Pollution-free bounds for eigenvalues {#zimmerman}
=====================================
The essential spectrum of both operators $M_{\slb}$ and $M_{{\mathrm{c}}}$ is non-negative. Therefore unstable spectrum can only occur in the discrete spectrum. The eigenvalues below the bottom of the essential spectrum can be computed using the standard Galerkin method. Hence, the stability of the configuration can be determined by means of the Rayleigh-Ritz variational principle.
By contrast, computing the eigenvalues above the essential spectrum is problematic due to the possibility of variational collapse. The technique described in this section avoids spectral pollution and can be implemented on the finite element method. It gives certified enclosures up to machine precision for eigenvalues above the essential spectrum. In Section \[supersection\] we argue that this technique should be applied, not only to $M_{\slb}$, but also to its Schur complement. In order to keep a neat notation, we formulate the general procedure for a generic semi-bounded self-adjoint operator $T$ acting on a Hilbert space $\mathcal{H}$.
Basic notation
--------------
Let the dense subspace ${{\rm Dom}}(T)\subset \mathcal{H}$ be the domain of $T$. For $\mu=\min{{\rm Spec}}(T)$, let ${\frak{t}}$ be the close bilinear form induced by the non-negative operator $T-\mu$ with domain ${{\rm Dom}}({\frak{t}})={{\rm Dom}}(| T|^{\frac{1}{2}})$. We denote the inner products and norms that render ${{\rm Dom}}(T)$ and ${{\rm Dom}}({\frak{t}})$ with a Hilbert space structure, respectively by $\langle u,v\rangle_{T} = \langle Tu,Tv\rangle + \langle u,v\rangle$, $\langle u,v\rangle_{{\frak{t}}} = {\frak{t}}(u,v)+\langle u,v\rangle$, $\|\cdot \|_{T}$ and $\|\cdot\|_{{\frak{t}}}$.
Let ${\mathcal{ E}}$ be a subspace of ${{\rm Dom}}(T)$. For another subspace $\mathcal{L}$, we denote $$\begin{aligned}
\delta({\mathcal{ E}},{\mathcal{L}}) &= \sup_{\phi\in{\mathcal{ E}},~\|\phi\|=1}{{\rm dist}}[\phi,{\mathcal{L}}]
& \text{if } {\mathcal{L}}\subset \mathcal{H} \\
\delta_{{\frak{t}}}({\mathcal{ E}},{\mathcal{L}}) &= \sup_{\phi\in{\mathcal{ E}},~\|\phi\|_{{\frak{t}}}=1}{{\rm dist}}_{{\frak{t}}}[\phi,{\mathcal{L}}] & \text{if } {\mathcal{L}}\subset {{\rm Dom}}({\frak{t}})\\
\delta_T({\mathcal{ E}},{\mathcal{L}}) &= \sup_{\phi\in{\mathcal{ E}},~\|\phi\|_T=1}{{\rm dist}}_T[\phi,{\mathcal{L}}]
& \text{if } {\mathcal{L}}\subset {{\rm Dom}}(T).\end{aligned}$$ Here and elsewhere ${{\rm dist}}_\bullet [\phi,{\mathcal{L}}]$ refers to the Haussdorff distance in the norm $\|\cdot\|_\bullet$ between $\{\phi\}$ and ${\mathcal{L}}$.
Below we establish spectral approximation results by following the classical framework of [@chat]. These results will be formulated in a general context for sequences of subspaces ${\mathcal{L}}_n \subset {\mathcal{ H}}$ which are dense as $n\to \infty$ in the following precise senses. We will say $$\begin{aligned}
({\mathcal{L}}_n)\in \Lambda \equiv \Lambda(I) & \quad \iff \quad {{\rm dist}}[u,{\mathcal{L}}_n]\to 0 & \forall u\in{\mathcal{ H}}\\
({\mathcal{L}}_n)\in\Lambda({\frak{t}}) & \quad \iff \quad {{\rm dist}}_{{\frak{t}}}[u,{\mathcal{L}}_n]\to 0 & \forall u\in {{\rm Dom}}({\frak{t}})\\
({\mathcal{L}}_n)\in\Lambda(T) & \quad \iff \quad {{\rm dist}}_{T}[u,{\mathcal{L}}_n]\to 0 & \forall u\in {{\rm Dom}}(T).\end{aligned}$$
Let ${\mathcal{L}}\subset {{\rm Dom}}({\frak{t}})$. Below we denote by ${{\rm Spec}}(T,{\mathcal{L}})$ the spectrum of the classical weak Galerkin problem: $$\label{galerkin}
\exists~u\in{\mathcal{L}}\backslash\{0\} \text{ and }\mu\in\mathbb{R}\quad\text{such that}\quad {\frak{t}}(u,v) = \mu\langle u,v\rangle\quad\textrm{for all}\quad v\in{\mathcal{L}}.$$
Assume that an interval $(a,b)\subset {{\mathbb R}}$ is such that $\operatorname{Tr}[\int_{(-\infty,a)} {\mathrm{d}}E_\lambda]=\operatorname{Tr}[\int_{(b,\infty)} {\mathrm{d}}E_\lambda]=\infty$. Then for general $({\mathcal{L}}_n)\in \Lambda(T)$, the set $$(a,b) \cap \bigcap_{n=1}^{\infty} \overline{\bigcup _{k\geq n} {{\rm Spec}}(T,{\mathcal{L}}_k)}$$ could be a much larger set than $(a,b)\cap {{\rm Spec}}(T)$. This phenomenon is usually called spectral pollution. See [@1977Rappaz] for further details on this in case $T=M_{\slb}$ and ${\mathcal{L}}_n$ chosen as finite element spaces.
The following classical convergence result will play a fundamental role below, see [@chat Theorem 6.11]. Assume that $\|T\|<\infty$. Let $({\mathcal{L}}_n)\in\Lambda$. For any isolated eigenvalue $\lambda\in {{\rm Spec}}(T) \setminus {{\rm conv}}[{{\rm Spec}}_{\mathrm{ess}}(T)]$, $$\label{galerkin2}
\delta(\ker(T-\lambda),{\mathcal{L}}_n)=\varepsilon_n\to 0 \quad \Rightarrow \quad{{\rm dist}}[\lambda,{{\rm Spec}}(T,{\mathcal{L}}_n)] = \mathcal{O}(\varepsilon_n^2).$$
If $T$ is only bounded from below, the condition ${\mathcal{L}}_n\subset{{\rm Dom}}({\frak{t}})$ and $({\mathcal{L}}_n)\in\Lambda$, is typically not sufficient to ensure approximation. By applying the spectral mapping theorem it can be shown that still holds true for a $\lambda<\min {{\rm Spec}}_{\mathrm{ess}}(T)$ whenever $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$ and $\delta$ is replaced by $\delta_{\frak{t}}$, see for example the trick applied in the proof of [@2010bs Corollary 3.6]. This type of convergence is often called superconvergence.
The Zimmermann-Mertins method {#zimmermethod}
-----------------------------
The following method for computing eigenvalue enclosures originated from [@zim] and is closely related to the classical Lehmann method. Below we show that it may be described in simple terms by means of mapping theorems at the level of reducing spaces for the resolvent. As we will see subsequently, convergence estimates will follow easily from .
This method turns out to be efficient for computing eigenvalue enclosures for $M_{\slb}$ and $M_{{\mathrm{c}}}$ in their original matrix formulation. In Section \[supersection\] we will discuss a further technique which allows improvement in accuracy for $M_{\slb}$ and depends on re-writing the eigenvalue problem in terms of the Schur complement.
According to [@dapl Theorem 11] the present approach is equivalent with optimal constant to another method formulated in [@1998Davies]. The latter is closely related to the so-called second order relative spectrum [@shar] which was applied to $M_\slb$ in [@2011Strauss]. We should stress that the latter is probably best for obtaining preliminary information about the spectrum, [@2010bs; @2011bStrauss]. This *a priori* information includes a reliable guess on the interval $[a,b]$ below.
Let $a<b$ be such that $(a,b)\cap {{\rm Spec}}(T)\not= \varnothing$. Assume that a finite-dimensional subspace ${\mathcal{L}}\subset {{\rm Dom}}(T)$ is such that $$\label{zimcon}
\min_{u\in {\mathcal{L}}}\frac{\langle Tu,u \rangle}{\|u\|^2}<b \quad\textrm{and}\quad
\max_{u\in {\mathcal{L}}}\frac{\langle Tu,u \rangle}{\|u\|^2}>a.$$ Note that this condition is certainly satisfied by ${\mathcal{L}}={\mathcal{L}}_n$ for $n$ sufficiently large, if $({\mathcal{L}}_n)\in \Lambda({\frak{t}})$. We define two inverse residuals associated to the interval $(a,b)$: $$\label{zimmer}
\tau^+=\max_{u\in {\mathcal{L}}} \frac{\langle (T-a)u,u \rangle}{\langle (T-a) u,(T-a) u \rangle}\quad
\text{and} \quad\tau^-=\min_{u\in {\mathcal{L}}} \frac{\langle (T-b)u,u \rangle}{\langle (T-b) u,(T-b) u \rangle}.$$ By virtue of , we have $\tau_+>0$ and $\tau_-<0$.
\[non-pollu\] Suppose that $[a,b]\cap {{\rm Spec}}(T)= \{\lambda\}$ and $a<\lambda<b$. For any subspace ${\mathcal{L}}\subset {{\rm Dom}}(T)$ satisfying the inverse residuals are such that $$\label{zimbound}
b + \frac{1}{\tau^-}\le\lambda\le a+\frac{1}{\tau^+}.$$
We prove the first inequality, the second my be proved similarly. Let $\hat{{\mathcal{L}}} = (T-b){\mathcal{L}}$, then $$\begin{aligned}
(\lambda - b)^{-1}&= \min[{{\rm Spec}}(T-b)^{-1}] = \min_{v\in {{\rm Dom}}(T)}\frac{\langle(T-b)^{-1}v,v \rangle}{\|v\|^2} \\ & \leq \min_{v\in\hat{{\mathcal{L}}}}\frac{\langle(T-b)^{-1}v,v \rangle}{\|v\|^2}
= \min_{u\in {\mathcal{L}}} \frac{\langle (T-b)u,u \rangle}{\langle (T-b) u,(T-b) u \rangle}=\tau^-.\end{aligned}$$
Note that $$\tau^-=\min[{{\rm Spec}}((T-b)^{-1},\hat{\mathcal{L}})]\quad\textrm{and}\quad\tau^+=\max[{{\rm Spec}}((T-a)^{-1},(T-a){\mathcal{L}})].$$ This observation turns out to be useful when studying convergence of the enclosure as ${\mathcal{L}}$ increases in dimension. Below $\tau^{\pm}_n=\tau^{\pm}$ for ${\mathcal{L}}_n={\mathcal{L}}$.
\[daplcon\] Let $\lambda\in{{\rm Spec}}_{{\mathrm{dis}}}(T)$ and assume that $[a,b]\cap{{\rm Spec}}(T) = \{\lambda\}$ with $a<\lambda<b$. Let ${\frak{B}}$ be an orthonormal basis of $\ker(T-\lambda)$, and $({\mathcal{L}}_n)\in\Lambda(T)$ be such that ${{\rm dist}}_T[{\frak{B}},{\mathcal{L}}_n] = \varepsilon_n\to 0$. For all sufficiently large $n\in\mathbb{N}$ $$\label{zimconv}
b + \frac{1}{\tau_n^-}\le\lambda\le a+\frac{1}{\tau_n^+}\quad\text{and}\quad\left(a+\frac{1}{\tau_n^+}\right) - \left(b + \frac{1}{\tau_n^-}\right) = \mathcal{O}(\varepsilon_n^2).$$
The condition is satisfied for all sufficiently large $n\in\mathbb{N}$, so the left hand side of follows from Lemma \[non-pollu\].
Let $\hat{{\mathcal{L}}}_n=(T-b){\mathcal{L}}_n$. Since $({\mathcal{L}}_n)\in\Lambda(T)$, it follows that $(\hat{{\mathcal{L}}}_n)\in \Lambda$. Let ${\frak{B}}= \{\phi_1,\dots,\phi_{k}\}$. Then there exist vectors $u_{n,j}\in{\mathcal{L}}_n$, such that $\|(T-b)(\phi_j - u_{n,j})\|\le (1+| b|)\varepsilon_n$ for each $1\le j\le k$. Set $\hat{u}_{n,j} = (T-b)u_{n,j}\in\hat{{\mathcal{L}}}_n$, then for any normalised $\phi\in\ker(T-\lambda)$ we have $$\begin{aligned}
\Big\|\phi - \sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{\lambda-b}\hat{u}_{n,j}\Big\| &=
\Big\|\sum_{j=1}^k\langle\phi_j,\phi\rangle\phi_j -
\sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{\lambda-b}\hat{u}_{n,j}\Big\|\\
&=\Big\|(T-b)\sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{\lambda-b}(\phi_j - u_{n,j})\Big\|\\
&\le \frac{k(1+| b|)\varepsilon_n}{b-\lambda}.\end{aligned}$$ Thus $\delta(\ker(T-\lambda),\hat{{\mathcal{L}}}_n)\le k(1+| b|)\varepsilon_n/(b-\lambda)$. Hence, applying to the operator $(T-b)^{-1}$ and eigenvalue $(\lambda - b)^{-1}$ yields $|\tau_n^- - (\lambda - b)^{-1}| = \mathcal{O}(\varepsilon_n^2)$, and therefore $b+\frac{1}{\tau_n^-}-\lambda=\mathcal{O}(\varepsilon_n^2)$. Similarly we have $a+\frac{1}{\tau_n^+}-\lambda=\mathcal{O}(\varepsilon_n^2)$, and the right hand side of follows.
By means of an example we now show that $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$ and $({\mathcal{L}}_n)\subset{{\rm Dom}}(T)$, is not generally sufficient to ensure a decrease in the size of the enclosure as $n\to \infty$. The crucial point here is that $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$ does not ensure that ${{\rm dist}}_T({\frak{B}},{\mathcal{L}}_n)$ decrease to $0$.
\[ex3\] Let $T=M_\slb$ be as in Example \[constant\_coeff\]. For $j\in {{\mathbb N}}$, let $\phi_{3k-2}=\phi^-_k$, $\phi_{3k-1}=\phi^1_k$ and $\phi_{3k}=\phi^+_k$, where the right hand sides are given by . Let $\lambda_{3k-2}=\lambda^-_k$, $\lambda_{3k-1}=1$ and $\lambda_{3k}=\lambda_k^+$. Then $T\phi_j=\lambda_j \phi_j$ and $\{\phi_j\}$ is an orthonormal basis of $\mathcal{H}=[L^2(0,1)]^3$.
For $n>2$ consider the subspaces ${\mathcal{L}}_n = {{\rm span}}\{\alpha_n \phi_1+\varepsilon_n\phi_{3n},\phi_2,\dots,\phi_{3n-1}\}$ where $\varepsilon_n=\frac{1}{\lambda_n^+}$ and $\alpha_n=\sqrt{1-\varepsilon_n^{2}}$. Then ${\mathcal{L}}_n\subset{{\rm Dom}}(T)$ for every $n\in\mathbb{N}$. We show that $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$. Let $u\in{{\rm Dom}}({\frak{t}})$ and $\gamma_j=\langle u,\phi_j\rangle$. Then $$\quad\| u\|_{{\frak{t}}} = \sqrt{\sum_{j=1}^\infty (1+\lambda_j )|\gamma_j|^2}<\infty.$$ Let $u_n=\gamma_1(\alpha_n\phi_1+\varepsilon_n\phi_{3n}) + \sum_{j=2}^{3n-1}\gamma_j\phi_j$. Then $$\begin{aligned}
\| u - u_n\|_{{\frak{t}}}^2 &= \Big\|\gamma_1(\alpha_n-1) \phi_1 +
(\gamma_1\varepsilon_n-\gamma_{3n})\phi_{3n} - \sum_{j=3n+1}^\infty\gamma_j\phi_j\Big\|_{{\frak{t}}}^2\\
&=(1+\lambda_1^-)|\gamma_1|^2|\alpha_n-1|^2 + (1+\lambda_n^+)|\gamma_1\varepsilon_n-\gamma_{3n}|^2 + \sum_{j=3n+1}^\infty (1+\lambda_j)|\gamma_j|^2\\
&\le(1+\lambda_1^-)|\gamma_1|^2|\alpha_n-1|^2 + \frac{(1+\lambda_n^+)|\gamma_1|^2}{(\lambda_n^+)^2} + \sum_{j=3n}^\infty (1+\lambda_j)|\gamma_j|^2\longrightarrow 0,\end{aligned}$$ therefore $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$.
This ensures that ${{\rm dist}}_{{\frak{t}}}(\phi_1,{\mathcal{L}}_n)\to 0$. On the other hand, a straightforward calculation shows that ${{\rm dist}}_T(\phi_1,{\mathcal{L}}_n)\to 1+\sqrt{3}$ as $n\to \infty$. Choose $a=0$ and $b=\frac{\lambda_1^-+\lambda_2^-}{2}$ so that $[a,b]\cap{{\rm Spec}}(T)=\{\lambda_1^-\}$. For $n$ large enough, $$\begin{gathered}
\frac{\langle T(\alpha_n\phi_1+\varepsilon_n\phi_{3n}),\alpha_n\phi_1+\varepsilon_n\phi_{3n}\rangle}
{\|\alpha_n\phi_1+\varepsilon_n\phi_{3n}\|^2} - b=\alpha_n^2 \lambda_1^-+\frac{1}{\lambda_n^+}-
\left( \frac{\lambda_1^-+\lambda_2^-}{2} \right) < 0\quad\textrm{and} \\
\frac{\langle T\phi_{n-1},\phi_{n-1}\rangle}
{\|\phi_{n-1}\|^2} - a=\lambda_n^+ > 0.
\end{gathered}$$ Thus is satisfied and we may obtain upper and lower bounds on the eigenvalue $\lambda_1^-$ from the left side of .
Let us now prove that the length of the enclosure in does not decrease. Indeed $$\begin{gathered}
\tau_n^+=\frac{\alpha_n^2 \lambda_1^-+ (\lambda_n^+)^{-1}}{\alpha_n^2(\lambda_1^-)^2+1}\to
\frac{\lambda_1^-}{(\lambda_1^-)^2+1}<(\lambda_1^-)^{-1} \\
\tau_n^-=\frac{\alpha_n^2(\lambda_1^- -b)+\varepsilon_n^2(\lambda_n^+-b)}{\alpha_n^2(\lambda_1^- -b)^2+\varepsilon_n^2(\lambda_n^+-b)^2} \to \frac{(\lambda_1^- -b)}{(\lambda_1^- -b)^2+1}<
(\lambda_1^- -b)^{-1}
\end{gathered}$$ as $n\to \infty$.
It is easily verified that $$\delta_T(\ker(T-\lambda),{\mathcal{L}}_n)=\mathcal{O}(\varepsilon_n)\qquad \Rightarrow
\qquad\delta_{{\frak{t}}}(\ker(T-\lambda),{\mathcal{L}}_n)=\mathcal{O}(\varepsilon_n).$$ As the following example shows, $\delta_T(\ker(T-\lambda),{\mathcal{L}}_n)$ and $\delta_{{\frak{t}}}(\ker(T-\lambda),{\mathcal{L}}_n)$ can converge at the same rate, but the latter may be faster. Therefore, there is a potential loss in convergence of the method when compared with the standard Galerkin method in the case where the latter is applicable. This loss of convergence is compensated by the fact that the enclosures found are certified and free from spectral pollution.
\[ex1\] Let $T$ and $\phi_n$ be as in Example \[ex3\]. Let $\varepsilon_n=(\lambda_n^+)^{-2}$ and $\alpha_n = \sqrt{1-\varepsilon_n^{2}}$. If we consider ${\mathcal{L}}_n = {{\rm span}}\{\alpha_n\phi_1+\varepsilon_n\phi_{3n-1},\phi_2,\dots,\phi_{3n-2}\}$, then $({\mathcal{L}}_n)\in\Lambda(T)$, and both $\delta_{{\frak{t}}}(\ker(T-1),{\mathcal{L}}_n)$ and $\delta_{T}(\ker(T-1),{\mathcal{L}}_n)$ are $\mathcal{O}(n^{-2})$. If we consider ${\mathcal{L}}_n = {{\rm span}}\{\alpha_n\phi_1+\varepsilon_n\phi_{3n},\phi_2,\dots,\phi_{3n-1}\}$, then $({\mathcal{L}}_n)\in\Lambda(T)$ once again but now $\delta_{{\frak{t}}}(\ker(T-\lambda_1^-),{\mathcal{L}}_n)= \mathcal{O}(n^{-3})$ and $\delta_{T}(\ker(T-\lambda_1^-),{\mathcal{L}}_n)=\mathcal{O}(n^{-2})$.
Operator matrices and eigenvalue enclosures {#supersection}
===========================================
The linearized MHD operator associated to the plasma layer configuration falls into the class of top dominant block matrices. We show that enclosures for the eigenvalues of $M_\slb$ which lie above the essential spectrum can be obtained from enclosures for the eigenvalues of its Schur complement. Denoted by $S(\mu)$, the latter is a $\mu$-dependant holomorphic family of semi-bounded operators. Upper bounds for its eigenvalues can be found from a direct application of the Galerkin method. We show in Section \[evub\] that these upper bounds are superconvergent as the dimension of ${\mathcal{L}}_n$ increases, hence they turn out to be asymptotically sharper than the upper bounds found from the method of Section \[zimmerman\] applied directly to $M_\slb$. In Section \[evlb\], on the other hand, we show how to find lower bounds for the eigenvalues of $M_\slb$ from corresponding lower bounds on the eigenvalues of $S(\mu)$. The latter are found from the left side of with $T=S(\mu)$ for a particular choice of $\mu$.
Basic notation
--------------
The results established below apply to any block operator matrix $M_0$ as in which is top dominant in the following precise sense, see [@2007Tretter].
1. \[top\_1\] $A$ and $D$ are self-adjoint operators acting on Hilbert spaces ${\mathcal{ H}}_1$ and ${\mathcal{ H}}_2$, respectively.
2. \[top\_2\] $A$ is bounded from below, $D$ is bounded from above, $B$ is closed and densely defined on a domain of ${\mathcal{ H}}_2$ with values in ${\mathcal{ H}}_1$.
3. \[top\_3\] ${{\rm Dom}}(| A|^{\frac{1}{2}})\subset{{\rm Dom}}(B^*)$, ${{\rm Dom}}(B)\subset{{\rm Dom}}(D)$ and ${{\rm Dom}}(B)$ is a core for $D$.
Without further mention, we assume that the entries of $M_0$ are subject to these conditions. They are satisfied by the plane slab configuration MHD operator, however, for $m\not=0$ the ansatz \[top\_3\] does not hold in general for the cylindrical pinch configuration.
The first condition in \[top\_3\] and the semi-boundedness of $A$, together, imply the existence of constants $\alpha,\beta\ge 0$ such that $$\label{domcons}
\| B^*u\|^2 \le \alpha\frak{a}[u] + \beta\| u\|^2\quad\textrm{for all}\quad u\in{{\rm Dom}}(| A|^{\frac{1}{2}})={{\rm Dom}}(\frak{a})$$ where $\frak{a}$ is the closure of the quadratic form associated to $A$. These two constraints also imply that $(A-\nu)^{-1}B$ is a bounded operator, so ${{\rm Dom}}(\overline{(A-\nu)^{-1}B})=\mathcal{H}_2$, for an arbitrary $\nu<\min{{\rm Spec}}(A)$. The self-adjoint closure of $M_0$, which we denote here by $M$, is explicitly given by $$\begin{aligned}
{{\rm Dom}}(M)&=\bigg\{\left(
\begin{array}{c}
x\\
y
\end{array} \right):y\in{{\rm Dom}}(D),~x + \overline{(A-\nu )^{-1}B}y\in{{\rm Dom}}(A)\bigg\}\label{top1}\\
M\left(
\begin{array}{c}
x\\
y
\end{array} \right) &= \left(
\begin{array}{c}
A(x + \overline{(A-\nu )^{-1}B}y) - \nu\overline{(A-\nu )^{-1}B}y\\
B^*x + Dy
\end{array} \right)\label{top2},\end{aligned}$$ see [@math0 Section 4.2] and references therein.
Set $d=\max{{\rm Spec}}(D)$ and $U=\{z\in\mathbb{C}:{{\rm Re}\;}z >d\}$. For $\mu\in U$ consider the following family of forms $$\label{schurform}
\frak{s}(\mu)[x,y] = \frak{a}[x,y] - \mu\langle x,y\rangle - \langle(D-\mu)^{-1}B^*x,B^*y\rangle$$ with common domain ${{\rm Dom}}(\frak{s})={{\rm Dom}}(\frak{s}(\mu))={{\rm Dom}}(\frak{a})$. Then $\frak{s}(\mu)$ is a holomorphic family of type (a), see [@math Proposition 2.2]. Associated to these forms is a holomorphic family of type (B) sectorial operators $S(\mu)$: $$\begin{aligned}
{{\rm Dom}}(S(\mu)) &= \{x\in{{\rm Dom}}(\frak{a}): x - \overline{(A-\nu)^{-1}B}(D-\mu)^{-1}B^*x\in{{\rm Dom}}(A)\}\label{schurdom}\\
S(\mu)x &= (A-\nu)(x - \overline{(A-\nu)^{-1}B}(D-\mu)^{-1}B^*x) + (\nu-\mu)x\label{schuract};\end{aligned}$$ see [@math0 Proposition 4.4]. Here, as above, $\nu<\min{{\rm Spec}}(A)$ is fixed, but can be chosen arbitrarilly. We note that for any $x\in{{\rm Dom}}(\frak{s})$ and $\mu\in U\cap\mathbb{R}$, $$\label{derivative}
\frac{\partial \frak{s}(\mu)[x]}{\partial \mu} = -\| x\|^2 - \|(D-\mu)^{-1}B^*x\|^2.$$ The families $\frak{s}(\cdot)$ and $S(\cdot)$ are called the Schur form and the Schur complement associated to $M_0$, respectively.
The form $\frak{s}(\mu)$ is symmetric and semi-bounded whenever $\mu\in {{\mathbb R}}\cap U$. The corresponding operator $S(\mu)$ is therefore self-adjoint and bounded from below. We set the spectra of the Schur complement as $$\begin{aligned}
{{\rm Spec}}(S)&=\{\mu\in U:0\in{{\rm Spec}}(S(\mu))\}, \\
{{\rm Spec}}_{\mathrm{dis}}(S)&=\{\mu\in U:0\in{{\rm Spec}}_{\mathrm{dis}}(S(\mu))\} \qquad and \\
{{\rm Spec}}_{\mathrm{ess}}(S)&=\{\mu\in U:0\in{{\rm Spec}}_{\mathrm{ess}}(S(\mu))\}.\end{aligned}$$
Upper bounds via Schur complement {#evub}
---------------------------------
We denote $$\lambda_{\mathrm{e}}=\inf\{{{\rm Spec}}_{{\mathrm{ess}}}(M)\cap(d,\infty)\}$$ and $\lambda_1\le\lambda_2\le\cdots$ the repeated eigenvalues of $M$ which lie in the interval $(d,\lambda_{\mathrm{e}})$.
The spectra of $S$ and $M$ coincide on $(d,\lambda_{\mathrm{e}})$ and ${{\rm Spec}}_{{\mathrm{ess}}}(S)\cap(d,\lambda_{\mathrm{e}}) = \varnothing$. Moreover $\dim{{\rm Ker}}(S(\lambda_j)) = \dim{{\rm Ker}}(M-\lambda_j)$.
For the first and third assertions, see [@math0 Proposition 4.4]. For the second assertion we proceed by contradiction.
Suppose there exists $\lambda\in (d,\lambda_{\mathrm{e}})$ such that $0\in {{\rm Spec}}_{\mathrm{ess}}(S(\lambda))$. Since $S(\lambda)=S(\lambda)^*$, there is a singular Weyl sequence $x_n\in {{\rm Dom}}(S(\lambda))$ such that $\|x_n\|=1$, $x_n{\rightharpoonup}0$ and $\|S(\lambda)x_n\|\to 0$. Let $$\begin{aligned}
y_n=\begin{pmatrix}
x_n\\
-(D-\lambda_j)^{-1}B^*x_n
\end{pmatrix}.\end{aligned}$$ Then $y_n\in{{\rm Dom}}(M)$ and $\| y_n\|\ge 1$. A direct calculation shows that $$(M-\lambda) y_{n}=\begin{pmatrix} S(\lambda)x_n \\ 0 \end{pmatrix} \to 0.$$
We prove that $y_n$ has a subsequence $y_{n(k)}{\rightharpoonup}0$, which in turn is a contradiction because $\lambda\not \in{{\rm Spec}}_{\mathrm{ess}}(M)$. Let $\mathcal{D}=(D-\lambda){{\rm Dom}}(B)$. By virtue of the second and third ansatz in \[top\_3\], $\mathcal{D}$ is a dense subspace of $\mathcal{H}_2$. Moreover, $$\label{weak_dense}
\langle (D-\lambda)^{-1}B^* x_n,y \rangle \to 0 \qquad \text{for all } y\in \mathcal{D}.$$ According to , $$\begin{aligned}
\|B^* x_n \|^2 & \leq \alpha \frak{a}[x_n]+\beta \leq \alpha( \frak{a}[x_n] -\lambda) +\alpha\lambda +\beta \\
&\leq \alpha \frak{s}(\lambda)[x_n] +\alpha\lambda +\beta = \alpha\langle S(\lambda)x_n,x_n\rangle +\alpha\lambda +\beta.\end{aligned}$$ As the right hand side of this identity is uniformly bounded for all $n$, there exists a subsequence $x_{n(k)}$ and $z\in \mathcal{H}_2$ such that $(D-\lambda)^{-1} B^* x_{n(k)}{\rightharpoonup}z$. Since implies that $z=0$, the subsequence $y_{n(k)}$ is as needed.
For $\mu>d$, we denote the spectral subspace of $S(\mu)$ corresponding to an interval $J$ by $${\mathcal{ E}}_{J}(S(\mu))={\mathrm{Range}}\int_{J} {\mathrm{d}}E_\lambda.$$ Here we abuse the notation and write ${\mathrm{d}}E_\lambda$ for the spectral measure associated to the self-adjoint operator $S(\mu)$ also. Let the dimension of ${\mathcal{ E}}_{(-\infty,0)}(S(\mu))$ be $$\kappa(\mu) ={{\rm tr}}\int_{(-\infty,0)} {\mathrm{d}}E_\lambda.$$ Throughout this section we assume that $\kappa(\mu)<\infty$ for some $\mu>d$. By [@math0 Theorem 4.5], this assumption and \[top\_1\]-\[top\_3\] imply the existence of $\gamma>d$ such that $(d,\gamma]\cap {{\rm Spec}}(M)=\varnothing$. We write $\kappa:=\kappa(\gamma)<\infty$ and note that $\kappa$ is independent of the particular choice of $\gamma\in (d,\min\{{{\rm Spec}}(M)\cap(d,\infty)\})$.
Let $l_1(m) = \min\{j\in\mathbb{N}:\lambda_j = \lambda_m\}$ and $l_2(m) = \max\{j\in\mathbb{N}:\lambda_j = \lambda_m\}$. Then $\kappa(\cdot)$ is constant on intervals contained in $(d,\lambda_{\mathrm{e}})\setminus {{\rm Spec}}(M)$ and $$\begin{aligned}
&\dim{\mathcal{ E}}_{(-\infty,0)}(S(\lambda_m)) = \kappa + l_1(m) - 1,\label{dims1}\\
&\dim{\mathcal{ E}}_{(-\infty,0]}(S(\lambda_m)) = \kappa + l_2(m)\label{dims0};\end{aligned}$$ see [@math0 Section 2] for further details.
\[ex\_case\_plasma\] In the case of the plasma layer configuration, $S(\mu)$ is a family of Sturm-Liouville operators. It is readily seen from the results of [@1989Lifschitz §7.5] that $\lambda_{\mathrm{e}}=\infty$, $\kappa<\infty$ and ${{\rm Spec}}(M)\cap(d,\lambda_{\mathrm{e}})$ consists of a sequence of simple eigenvalues which accumulate at $+\infty$.
We now describe the theoretical framework and basic procedure for approximating a fixed eigenvalue $\lambda_m$. Denote by $E_1(\mu)\le\dots\le E_{\kappa+m}(\mu)$ the first $\kappa+m$ eigenvalues of $S(\mu)$ repeated according to their multiplicity. Let $${\mathcal{L}}={{\rm span}}\{u_1,\dots,u_n\}\subset {{\rm Dom}}(\frak{s}) \quad\textrm{where}\quad \langle u_i, u_j\rangle =\delta_{ij},$$ be an $n$-dimensional subspace where $n\ge \kappa+m$. Consider the family of matrices $S_{{\mathcal{L}}}(\mu)\in {{\mathbb C}}^{n\times n}$ whose entries are given by $$S_{{\mathcal{L}}}(\mu)_{i,j} = \frak{s}(\mu)[u_j,u_i] \qquad i,j=1,\ldots,n.$$ Denote by $E_1({\mathcal{L}},\mu)\le\dots\le E_{\kappa+m}({\mathcal{L}},\mu)$ the first $\kappa+m$ eigenvalues of $S_{{\mathcal{L}}}(\mu)$ repeated according to their multiplicity.
\[upbdslem\] Let $\mu\in(d,\infty)$ be such that $E_{\kappa+m}({\mathcal{L}},\mu) \le 0$, then $\lambda_m\le\mu$.
We suppose that $\lambda_m>\mu$. The Rayleigh-Ritz variational principle ensures that $E_{\kappa+m}(\mu) \leq E_{\kappa + m}({\mathcal{L}},\mu) \le 0$, and therefore $$\label{dims2}
\dim{\mathcal{ E}}_{(-\infty,0]}(S(\mu))\ge\kappa + m.$$ If $\mu=\lambda_j$ for some some $1\le j<l_1(m)$, then from we have $\dim{\mathcal{ E}}_{(-\infty,0]}(S(\mu))
= \kappa + l_2(j) < \kappa + l_1(m)$, which contradicts . Suppose now that $\mu\not \in{{\rm Spec}}(M)$. Then from we have $\frak{s}(\mu)\ge\frak{s}(\lambda_m)$ from which it follows that $\dim{\mathcal{ E}}_{(-\infty,0)}(S(\mu))\le\dim{\mathcal{ E}}_{(-\infty,0)}(S(\lambda_m))$. From we then deduce that $\dim{\mathcal{ E}}_{(-\infty,0)}(S(\mu))<\kappa + m$, which contradicts .
An upper bound for $\lambda_m$ may be obtained by applying the Galerkin method to the Schur complement, then finding a $\mu\in(d,\infty)$ such that $S(\mu)$ has at least $\kappa+m$ non-positive eigenvalues via a root finding algorithm. We now turn our attention to the convergence properties of this approach. For this we employ assuming $T=S(\lambda_m)$ and denote ${\mathcal{ E}}=\ker(T)$.
\[super\] Let $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$ be such that $\delta_{{\frak{t}}}({\mathcal{ E}},{\mathcal{L}}_n)=\varepsilon_n\to 0$ as $n\to\infty$. Let $\mu_n^+\in\mathbb{R}$ be such that $E_{\kappa+m}({\mathcal{L}}_n,\mu_n^+) = 0$. Then $\mu_n^+>\lambda_m$ and $\mu_n^+-\lambda_m = \mathcal{O}(\varepsilon_n^2)$.
From it follows that $S(\lambda_m)$ has $\kappa + l_1(m)-1$ negative eigenvalues counting multiplicity. Therefore, the density condition $({\mathcal{L}}_n)\in\Lambda({\frak{t}})$ implies that for all sufficiently large $n\in\mathbb{N}$ there are precisely $\kappa+l_1(m) - 1$ elements from ${{\rm Spec}}(T,{\mathcal{L}}_n)$ which are negative. Since $\delta_{{\frak{t}}}({\mathcal{ E}},{\mathcal{L}}_n)=\varepsilon_n\to 0$, there are precisely $l_2(m)-l_1(m)+1$ ($=\dim{\mathcal{ E}}$) elements from ${{\rm Spec}}(T,{\mathcal{L}}_n)$ which are non-negative and of the order $\mathcal{O}(\varepsilon_n^2)$. The result now follows from and .
Lower bounds via Schur complement {#evlb}
---------------------------------
In the previous section we found upper bounds for an eigenvalue $\lambda_m\in(d,\lambda_{\mathrm{e}})$. We now turn our attention to finding complementary lower bounds for this eigenvalue via the method described in Section \[zimmermethod\].
\[1\] For $\mu\in(d,\lambda_{\mathrm{e}})$ let $0<\varepsilon<\mu-d$. If $[-\varepsilon,0]\cap{{\rm Spec}}(S(\mu))\ne\varnothing$, then $[\mu-\varepsilon,\mu]\cap{{\rm Spec}}(M)\ne\varnothing$.
Let $\delta=\max\{{{\rm Spec}}(S(\mu))\cap[-\varepsilon,0]\}$ and assume that $[\mu-\varepsilon,\mu]\cap{{\rm Spec}}(M)=\varnothing$. According to [@math0 Lemma 2.6], $\kappa(\mu) = \kappa(\mu-\varepsilon)$. By virtue of and the Rayleigh-Ritz variational principle, $$0>\inf_{{\genfrac{}{}{0ex}{}{V\subset{{\rm Dom}}(\frak{s})}{\dim V=\kappa(\mu)}}}\sup_{{\genfrac{}{}{0ex}{}{u\in V}{u\ne 0}}} \frac{\frak{s}(\mu-\varepsilon)[u]}{\| u\|^2}
\ge\varepsilon + \inf_{{\genfrac{}{}{0ex}{}{V\subset{{\rm Dom}}(\frak{s})}{\dim V=\kappa(\mu)}}}\sup_{{\genfrac{}{}{0ex}{}{u\in V}{u\ne 0}}} \frac{\frak{s}(\mu)[u]}{\| u\|^2}
=\varepsilon + \delta,$$ where the right hand side is non-negative. The result follows from the contradiction.
By applying Lemma \[upbdslem\], we can find $\mu\in\mathbb{R}$ such that $\lambda_m\le\mu$. If $S(\mu)$ has $\kappa + l_2(m)$ non-positive eigenvalues and $b>0$ is such that $(0,b]\cap {{\rm Spec}}(S(\mu))=\varnothing$, then we employ the Zimmermann-Mertins method with $T=S(\mu)$ to obtain a lower bound on the first non-positive eigenvalue. Combined with Lemma \[1\], this yields a lower bound for $\lambda_m$. We now find the rate of convergence of this lower bound.
\[lemres1\] Let $b>0$ be such that $(0,b]\cap{{\rm Spec}}(S(\lambda_m))=\varnothing$. Let $\mu_n$ be a sequence of real numbers such that $0\le\mu_n-\lambda_m=\varepsilon_n\to 0$ as $n\to \infty$. For all $n$ sufficiently large $(0,b]\cap{{\rm Spec}}(S(\mu_n))=\varnothing$. Moreover, $$\label{reso_Schur}
\|(S(\lambda_m)-b)^{-1} - (S(\mu_n)-b)^{-1}\| = \mathcal{O}(\varepsilon_n).$$
We first show that $b\not \in {{\rm Spec}}(S(\mu_n))$ for all sufficiently large $n\in\mathbb{N}$, and that holds true. Let $x\in{{\rm Dom}}(\frak{s})$ and $\alpha_n=\mu_n-\lambda_m$, then $$\begin{aligned}
\frak{s}(\lambda_m)[x] &= \frak{a}[x] - \lambda_m\| x\|^2 - \langle(D-\lambda_m)^{-1}B^*x,B^*x\rangle\\
&=\frak{s}(\mu_n)[x] + \alpha_n\| x\|^2 + \langle[(D-\mu_n)^{-1} - (D-\lambda_m)^{-1}]B^*x,B^*x\rangle\\
&=\frak{s}(\mu_n)[x] + \alpha_n\| x\|^2+
\alpha_n\langle(D-\mu_n)^{-1}(D-\lambda_m)^{-1}B^*x,B^*x\rangle .\end{aligned}$$ Set $\hat{\frak{s}} = \frak{s}(\mu_n)-\frak{s}(\lambda_m)$. Note that $\frak{a}[x]\le\frak{s}(\lambda)[x]+\lambda\|x\|^2$. By virtue of , we have $$\begin{aligned}
|\hat{\frak{s}}[x]|&= \alpha_n\| x\|^2+
\alpha_n\langle(D-\mu_n)^{-1}(D-\lambda_m)^{-1}B^*x,B^*x\rangle\\
&\le \alpha_n\| x\|^2+
\alpha_n\|(D-\mu_n)^{-1}(D-\lambda_m)^{-1}\|\| B^*x\|^2\\
&\le \alpha_n\| x\|^2+
\alpha_n\|(D-\mu_n)^{-1}(D-\lambda_m)^{-1}\|(\alpha\frak{a}[x]+\beta\| x\|^2)\\
&\leq \alpha_n(a_n\| x\|^2+b_n\frak{s}(\lambda_m)[x]).\end{aligned}$$ where $$\begin{aligned}
a_n&= 1+\beta\|(D-\mu_n)^{-1}(D-\lambda_m)^{-1}\| + \alpha\lambda_m \to 1+\beta(\lambda_m-d)^{-2}+\alpha \lambda_m
\\
b_n&=\alpha\| (D-\mu_n)^{-1}(D-\lambda_m)^{-1}\|\to \alpha(\lambda_m-d)^{-2}\quad\textrm{as}\quad n\to\infty.\end{aligned}$$ Set $c_1 = \max\{\|(S(\lambda_m) - b)^{-1}\|,\| S(\lambda_m)(S(\lambda_m) - b)^{-1}\|\}$, and let $n\in\mathbb{N}$ be sufficiently large to ensure that $\alpha_n(a_n+b_n)<c_1^{-1}$. By virtue of [@katopert Theorem VI-3.9], we obtain $b\not\in{{\rm Spec}}(S(\mu_n))$ and $$\label{normres}
\|(S(\lambda_m)-b)^{-1} - (S(\mu_n)-b)^{-1}\|\le
\frac{4\alpha_n(a_n+b_n)c_1^2}
{(1-\alpha_n(a_n+b_n)c_1)^2}$$ which immediately implies .
It remains to show that $(0,b]\cap{{\rm Spec}}(S(\mu_n))=\varnothing$ for all sufficiently large $n\in\mathbb{N}$. By virtue of [@katopert Theorem VII-4.2], there exists a constant $\nu<\min{{\rm Spec}}(S(\mu_n))$ for all sufficiently large $n\in\mathbb{N}$. Let $\Gamma$ be a circle with center $(b + \nu)/2$ and radius $(b-\nu)/2$, and set $$c_2 = \max_{z\in\Gamma}\Big\{\|(S(\lambda_m) - z)^{-1}\|,\| S(\lambda_m)(S(\lambda_m) - z)^{-1}\|\Big\}.$$ Then $\alpha_n(a_n+b_n)<c_2^{-1}$ for all sufficiently large $n\in\mathbb{N}$. Applying the same argument as above, we obtain $$\max_{z\in\Gamma}\Big\{\|(S(\lambda_m)-z)^{-1} - (S(\mu_n)-z)^{-1}\|\Big\}\le
\frac{4\alpha_n(a_n+b_n)c_2^2}
{(1-\alpha_n(a_n+b_n)c_2)^2}.$$ The right hand side of converges to zero as $n\to\infty$. Thus, the spectral subspaces of $S(\lambda_m)$ and $S(\mu_n)$ corresponding to the eigenvalues below $b$ have the same dimension for all sufficently large $n\in\mathbb{N}$. The desired conclusion follows from and the Rayleigh-Ritz variational principle.
According to Theorem \[super\], if $({\mathcal{L}}_n)\in\Lambda(\frak{s}(\lambda_m))$ and $\delta_{\frak{s}(\lambda_m)}({\mathcal{ E}},{\mathcal{L}}_n)=\mathcal{O}({\varepsilon}_n)$, then we obtain a sequence of upper bounds $$\label{upper_seq}
\mu_n^+\searrow\lambda_m \qquad \text{ satisfying } \qquad
\mu_n^+-\lambda_m = \mathcal{O}({\varepsilon}_n^2).$$ If we now pick $b$ satisfying the hypothesis of Lemma \[lemres1\], by virtue of we find a lower bound for the smallest in modulus non-positive eigenvalue of $T=S(\mu_n^+)$. That is $$b + \frac{1}{\tau_n^-} \le \max \left\{{{\rm Spec}}(S(\mu_n^+))\cap(-\infty,0]\right\}.$$ Lemma \[1\] ensures corresponding lower bounds $$\label{lower_seq}
\mu_n^- = \mu_n^+ + b+1/\tau_n^-\le\lambda_m.$$ In the theorem below we find bounds on the speed of convergence $\mu_n^-\to \lambda_m$.
Before proceeding further, we note that $S(\mu)$ and $B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*$ for the plane slab configuration are sectorial Sturm-Liouville operators for all $\mu\in U$. Both families of operators are closed in the domain $$\label{domain_schur}
\mathcal{D}=H^{2}((0,1);{\mathrm{d}}x)\cap H_{0}^{1}((0,1);{\mathrm{d}}x),$$ which coincides with and is independent of $\mu$. Moreover, they are both holomorphic families of type (A), see [@katopert Example VII-2.12].
\[enclose\] Suppose that the entries of $M_0$ satisfy \[top\_1\] - \[top\_3\]. Assume that $S(\mu)$ is a holomorphic family of type (A) with ${{\rm Dom}}(S(\mu))=\mathcal{D}$ independent of $\mu$. Assume additionally that $B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*$ is closed on $\mathcal{D}$. Let $({\mathcal{L}}_n)\in\Lambda(S(\lambda_m))$ with $\delta_{S(\lambda_m)}({\mathcal{ E}},{\mathcal{L}}_n)=\varepsilon_n$. If $\mu_n^-$ is constructed as in , then $\mu_n^-\leq \lambda_m$ and $\lambda_m -\mu_n^- = \mathcal{O}(\varepsilon_n^2)$ as $n\to \infty$.
Let $\mu\in(d,\lambda_{\mathrm{e}})$, $x\in\mathcal{D}$ and $\nu<\min{{\rm Spec}}(A)$. According to we have $$\begin{aligned}
S(\mu)x - S(\lambda_m)x &= (A-\nu)(x - \overline{(A-\nu)^{-1}B}(D-\mu)^{-1}B^*x) + (\nu-\mu)x\\
&\quad - (A-\nu)(x - \overline{(A-\nu)^{-1}B}(D-\lambda_m)^{-1}B^*x) - (\nu-\lambda_m)x\\
&= (A-\nu)(\overline{(A-\nu)^{-1}B}[(D-\lambda_m)^{-1} - (D-\mu)^{-1}]B^*x)\\
&\quad + (\lambda_m-\mu)x\\
&= (\lambda_m-\mu)(A-\nu)(\overline{(A-\nu)^{-1}B}(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*x)\\
&\quad + (\lambda_m-\mu)x\\
&= (\lambda_m-\mu)B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*x+ (\lambda_m-\mu)x.\end{aligned}$$ We consider the closed operator $B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*$. Since $$\begin{aligned}
B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*x = \frac{S(\mu)x - S(\lambda_m)x}{\lambda_m - \mu} - x\quad\textrm{for}\quad\mu\ne\lambda_m,\end{aligned}$$ and $S(\mu)$ is holomorphic of type (A), then $\Vert B(D-\lambda_m)^{-1}(D-\mu)^{-1}Bx\Vert$ is uniformly bounded in a neighbourhood of $\mu$. Moreover, for any $y\in{{\rm Dom}}(B^*)$ the function $$\langle B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*x,y\rangle =
\langle (D-\mu)^{-1}B^*x,(D-\lambda_m)^{-1}B^*y\rangle$$ is analytic for $\mu\in U$. It follows that $B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*$ is a holomorphic family of type (A).
Let $J\subset(d,\lambda_{\mathrm{e}})$ be any compact interval containing a neighbourhood of $\lambda_m$. By virtue of [@katopert Section VII.2.1], there always exists a constant $c_3>1$ such that $$\label{M}
\frac{\|x\| + \| B(D-\lambda_m)^{-1}(D-\mu)^{-1}B^*x\|}{\|x\| + \| B(D-\lambda_m)^{-2}B^*x\|} \le c_3\quad\textrm{for all}\quad \mu\in J.$$ Since the operators $B(D-\lambda_m)^{-2}B^*$ and $S(\lambda_m)$ have the same domain $\mathcal{D}$, there exist constants $\tilde{\alpha},\tilde{\beta}\ge 0$ such that $$\begin{aligned}
\| B(D-\lambda_m)^{-2}B^*x\|&\le \tilde{\alpha}\| S(\lambda_m)x\| +
\tilde{\beta}\|x\|\quad\textrm{for all}\quad x\in\mathcal{D}\label{consts}.\end{aligned}$$ As $\mu_n^+-\lambda=\mathcal{O}(\varepsilon_n^2)$, for some $N\in\mathbb{N}$ large enough $\mu_n^+\in J$ whenever $n\ge N$. Combining with , gives $$\begin{aligned}
\| B(D-\lambda_m)^{-1}(D-\mu_n^+)^{-1}B^*x\|&\le (c_3-1)\|x\| + c_3\| B(D-\lambda_m)^{-2}B^*x\|\\
&\le (c_3-1)\|x\| + \tilde{\alpha}c_3\|S(\lambda_m)x\| + \tilde{\beta}c_3\|x\|\\
&\le c_4(\| S(\lambda_m)x\| + \|x\|),\end{aligned}$$ where $c_4\ge 0$ is independent of $n\ge N$. Thus $$\label{mess1}
\| S(\mu_n^+)x - S(\lambda_m)x\| \le (\mu_n^+-\lambda_m)(c_4+1)(\| S(\lambda_m)x\|+\|x\|)\quad\textrm{for all}\quad n\ge N.$$
Let $\hat{{\mathcal{L}}}_n = (S(\mu_n^+)-b){\mathcal{L}}_n$. We show that $(\hat{{\mathcal{L}}}_n)\in\Lambda$. Let $v\in\mathcal{H}$. There exists $u\in\mathcal{D}$ such that $(S(\lambda_m)-b)u = v$. Since $({\mathcal{L}}_n)\in\Lambda(S(\lambda_m))$ we have a sequence $u_n\in{\mathcal{L}}_n$ satisfying $(S(\lambda_m)-b)u_n \to v$. As $b\not \in {{\rm Spec}}(S(\lambda_m))$, the sequences $\| u_n\|$ and $\| S(\lambda_m)u_n\|$ are uniformly bounded. Hence, it follows from that $\|S(\mu_n^+)u_n-S(\lambda_m)u_n\|\to 0$. Since $$v - (S(\mu_n^+)-b)u_n + (S(\mu_n^+)-S(\lambda_m))u_n = v - (S(\lambda_m)-b)u_n \to 0,$$ clearly also $(S(\mu_n^+)-b)u_n\to v$. Thus $\hat{{\mathcal{L}}}_n\in\Lambda$.
We now show that $\delta({\mathcal{ E}},\hat{{\mathcal{L}}}_n)=\mathcal{O}(\varepsilon_n)$. Let $\phi_1,\dots,\phi_{k}$ be an orthonormal basis for ${\mathcal{ E}}$. There exist vectors $u_{n,j}\in{\mathcal{L}}_n$, such that $\|S(\lambda_m)(\phi_j - u_{n,j})\|\le \varepsilon_n$ and $\|\phi_j - u_{n,j}\|\le\varepsilon_n$ for each $1\le j\le k$. We set $\hat{u}_{n,j} = (S(\mu_n^+)-b)u_{n,j}\in\hat{{\mathcal{L}}}_n$. Using we have for any normalised $\phi\in{\mathcal{ E}}$ $$\begin{aligned}
\Big\|\phi + \sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{b}\hat{u}_{n,j}\Big\| &=
\Big\|\sum_{j=1}^k\langle\phi_j,\phi\rangle\phi_j +
\sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{b} (S(\mu_n^+)-b)u_{n,j}\Big\|\\
&\le\Big\|\sum_{j=1}^k\langle\phi_j,\phi\rangle(\phi_j -u_{n,j})\Big\| +
\Big\| \sum_{j=1}^k\frac{\langle\phi_j,\phi\rangle}{b}S(\mu_n^+)u_{n,j}\Big\|\\
&\le k\varepsilon_n + \frac{1}{b}\sum_{j=1}^k\left(\|S(\mu_n^+)u_{n,j} - S(\lambda_m)u_{n,j}\| + \|S(\lambda_m)u_{n,j}\|\right)\\
&\le 2k\varepsilon_n + \frac{1}{b}\sum_{j=1}^k (\mu_n^+-\lambda_m)(c_4+1)(\| S(\lambda_m)u_{n,j}\|+\|u_{n,j}\|)\\
&\le 2k\varepsilon_n + \frac{1}{b}\sum_{j=1}^k (\mu_n^+-\lambda_m)(c_4+1)(\varepsilon_n+\|u_{n,j}\|).\end{aligned}$$ Therefore $\delta({\mathcal{ E}},\hat{{\mathcal{L}}}_n)=\mathcal{O}(\varepsilon_n)$.
We complete the proof of the theorem as follows. By applying to the operator $T=(S(\lambda_m)-b)^{-1}$ and eigenvalue $(-b)^{-1}=\min\{{{\rm Spec}}((S(\lambda_m) - b)^{-1})\}$, we obtain $$\label{gal1}
\min\{{{\rm Spec}}((S(\lambda_m)-b)^{-1},\hat{{\mathcal{L}}}_n)\} + b^{-1} = \mathcal{O}(\varepsilon_n^2).$$ Using Lemma \[lemres1\] and $0\le\mu_n^+-\lambda_m=\mathcal{O}(\varepsilon_n^2)$, we have $$\label{gal2}
\|(S(\lambda_m)-b)^{-1} - (S(\mu_n^+)-b)^{-1}\|=\mathcal{O}(\varepsilon_n^2).$$ From , and the Rayleigh-Ritz variational principle, it becomes clear that $$\label{gal3}
\tau_n^- = \min\{{{\rm Spec}}((S(\mu_n^+)-b)^{-1},\hat{{\mathcal{L}}}_n)\}\quad\textrm{satisfies}\quad\tau_n^- + b^{-1} =
\mathcal{O}(\varepsilon_n^2).$$ Moreover, $b + 1/\tau_n^-$ is precisely the lower bound on the smallest in modulus non-positive eigenvalue of $S(\mu_n^+)$ which is obtained from the Zimmermann-Mertins method. Then $\mu_n^- = \mu_n^+ + b+1/\tau_n^-\le\lambda_m$ follows from Lemma \[1\], and $\lambda_m -\mu_n^- = \mathcal{O}(\varepsilon_n^2)$ follows from .
![Log-log graph. Vertical axis: $\frac{\mu_n^+-\mu_n^-}{\mu_n^-}\times 100$. Horizontal axis: eigenvalue index $m$. We depict the relative size of the enclosure in the calculation of the first 50 eigenvalues of $M_{\slb}$ of Example \[non-constant\_coeff\]. The subspace ${\mathcal{L}}_n$ is chosen to be: ${\mathcal{L}}_{67}^{\sin}$ and ${\mathcal{L}}(h,1,r)$ for Hermite elements of order $r=3$, 4 and 5 on an uniform mesh with $h$ chosen so the dimension of the spaces is approximately $10\times 67$. We have chosen $\tau_{\mathrm{i}}=10^{-14}$, $\tau_{\mathrm{s}}=10^{-12}$, $\tau_{\mathrm{b}}=\mathcal{O}(10^{-5})$ in the case of the sine basis, and $\tau_{\mathrm{i}}=10^{-10}$, $\tau_{\mathrm{s}}=10^{-6}$ and $\tau_{\mathrm{b}}=\mathcal{O}(10^{-3})$ in the case of the finite element method. \[fig2\]](many_eigenvalues_different_hermite_same_dof){height="8cm"}
![Log-log graph. Vertical axis: $|\lambda^{{\eqref{constant_coeff}}}_{1,2,3}-\mu^{\pm}_{n,h}|$. Horizontal axis: maximum element size $h$. The subspace ${\mathcal{L}}_n$ is chosen to be: ${\mathcal{L}}(h,1,3)$ for decreasing values of $h$. For these calculations $\tau_{\mathrm{i}}=10^{-10}$, $\tau_{\mathrm{s}}=10^{-12}$, $\tau_{\mathrm{b}}=\mathcal{O}(10^{-3})$.\[fig4\]](Order_bounds_constant_coeff_2){height="8cm"}
Numerical examples: plane slab configuration {#numerical_slab}
============================================
An optimal strategy in terms of convergence for calculating enclosures for the configuration can be established from the approach in Section \[supersection\]. We now illustrate the practical applicability of this strategy by performing various numerical experiments on benchmark models. Our equilibrium quantities will be chosen from examples \[constant\_coeff\] and \[non-constant\_coeff\].
For a fixed $\mu\in(d,\infty)$, the eigenvalues of $S(\mu)$ are simple in both examples. The corresponding eigenvectors are in $C^{\infty}(0,1)$ and they satisfy Dirichlet boundary conditions at the endpoints of the interval. We have $${{\rm Spec}}(M_{\slb})\cap(d,\infty)=\{{\lambda}_1<{\lambda}_2<\ldots\}$$ where each eigenvalue is simple and ${\lambda}_j\to\infty$. Below we distinguish the model used by denoting these eigenvalues by ${\lambda}^{{\eqref{constant_coeff}}}_m$ and ${\lambda}^{{\eqref{non-constant_coeff}}}_m$ respectively. For all $u\in{{\rm Dom}}(\frak{s})=H^1_0((0,1);{\mathrm{d}}x)$, $$\label{Schur_comp_pd}
\begin{aligned}
\frak{s}(\mu)[u] & \ge \frak{a}[u] - \mu\| u\|^2 \\
& \ge \pi^2\| u\|^2 + \langle(7/4 - x)u,u\rangle-\mu\| u\|^2 \\
& \ge (\pi^2 -\mu)\| u\|^2.
\end{aligned}$$ Thus $S(\mu)$ is positive definite for $\mu\in(d,\pi^2)$. Hence upper bounds $\mu_n^+$ for $\lambda_m$ can be found from Theorem \[super\] with $\kappa = 0$.
In practice we find $\mu_n^+$ as follows. For a fixed ${\mathcal{L}}_n$ we compute a few eigenvalues of $S(\mu)$ for $\mu$ in an uniform partition with $p$ points of a suitable interval $(a,b)$ containing only $\lambda_m$. We then approximate $\mu_n^+$ via one iteration of Newton’s method. Below, the integrations involved in the assembling of the matrix problems are set to a tolerance of the order $\mathcal{O}(\tau_{\mathrm{i}})$, the eigenvalue solver is set to a tolerance of order $\mathcal{O}(\tau_{\mathrm{s}})$ and $\tau_{\mathrm{b}}=(b-a)/p$. These are different for the different experiments. By virtue of , the root finding step is accurate to $\mathcal{O}(\tau_{\mathrm{b}}^2)$.
To find complementary lower bounds from Theorem \[enclose\], we require $b>0$ such that $(0,b]\cap{{\rm Spec}}(S(\mu_n^+))=\varnothing$ for all sufficiently large $n\in\mathbb{N}$. From it follows that $\frak{s}(\mu_n^+)[u] \ge \frak{a}[u] - \mu_n^+\| u\|^2$ for all $u\in{{\rm Dom}}(\frak{s})$ (in both Example \[constant\_coeff\] and Example \[non-constant\_coeff\]). By the minmax principle, the $(m+1)$-th eigenvalue of $S(\mu_n^+)+\mu_n^+$ lies above the $(m+1)$-th eigenvalue of $A$. In fact $\lambda_m<(m+1)^2\pi^2<\lambda_{m+1}$ and we may choose $b\in(0,(m+1)^2\pi^2 - \mu_n^+)$. Integration and eigenvalue solver tolerances are set as for the upper bounds.
We consider two canonical basis to generate ${\mathcal{L}}_n\subset \mathcal{D}$, see . A first natural choice is the sine basis, $${\mathcal{L}}_n^{\sin}={{\rm span}}\{u_1,\dots,u_n\}\qquad
\text{where} \qquad u_n = \sqrt{2}\sin(n\pi x).$$ Standard arguments show that ${\mathcal{L}}_n\in\Lambda(S(\lambda_m))$ and $$\delta_{S(\lambda_m)}(\ker S(\lambda_m),{\mathcal{L}}_n)=\mathcal{O}(n^{-r})$$ where $r$ can be chosen arbitrarilly large. Applying Theorem \[super\] and Theorem \[enclose\] we obtain $$\mu_n^+\searrow\lambda_m,\quad\mu_n^-\nearrow\lambda_m,\quad\textrm{and}\quad
\mu_n^+-\mu_n^-=\mathcal{O}(n^{-r}).$$ This means that the enclosures should converge to zero super-polynomially fast for the family of subspaces ${\mathcal{L}}_n^{\sin}$. See Table \[table1\] and Figure \[fig2\]. All calculations involving this basis were coded in Matlab.
n $\lambda^{{\eqref{non-constant_coeff}}}_1$ $\lambda^{{\eqref{non-constant_coeff}}}_2$ $\lambda^{{\eqref{non-constant_coeff}}}_3$ $\lambda^{{\eqref{non-constant_coeff}}}_4$ $\lambda^{{\eqref{non-constant_coeff}}}_5$
---- -------------------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- --------------------------------------------
5 $12.350^{47799}_{38099}$ $41.9106^{5300}_{3750}$ $91.2474^{7057}_{6613}$ $160.3305^{8480}_{7817}$ $ 249.155^{19069}_{07400}$
10 $12.3504^{7592}_{2524}$ $41.9106^{5224}_{4418}$ $91.2474^{7031}_{6778}$ $160.33058^{264}_{158}$ $249.155079^{76}_{13}$
20 $12.3504^{7563}_{4946} $ $41.9106^{5214}_{4796}$ $91.2474^{7026}_{6895}$ $160.330582^{62}_{04}$ $249.155079^{73}_{43}$
40 $12.3504^{7559}_{6228}$ $41.91065^{213}_{001}$ $91.2474^{7026}_{6958}$ $160.330582^{61}_{31}$ $249.155079^{73}_{57}$
: Approximation of the first five eigenvalues of $M_{\slb}$ for Example \[non-constant\_coeff\] and test spaces chosen as ${\mathcal{L}}_n^{\mathrm{s}}$. For these calculations $\tau_{\mathrm{i}}=10^{-14}$, $\tau_{\mathrm{s}}=10^{-12}$, $\tau_{\mathrm{b}}=\mathcal{O}(10^{-5})$. \[table1\]
![Log-log graph. Vertical axis: $\frac{\mu_n^+-\mu_n^-}{\mu_n^-}\times 100$. Horizontal axis: eigenvalue index $m$. We depict the relative size of the enclosure in the calculation of the first 50 eigenvalues of $M_{\slb}$ of Example \[non-constant\_coeff\]. The subspace ${\mathcal{L}}_n$ is chosen to be: ${\mathcal{L}}(h,1,5)$ for decreasing values of $h$. For these calculations $\tau_{\mathrm{i}}=10^{-10}$, $\tau_{\mathrm{s}}=10^{-6}$, $\tau_{\mathrm{b}}=\mathcal{O}(10^{-3})$. \[fig3\]](many_eigenvalues_hermite5_different_h){height="8cm"}
Another natural basis is obtained by applying the finite element method. Let $\Xi$ be an equidistant partition of $[0,1]$ into $n$ sub-intervals $I_l=[x_{l-1},x_l]$ of length $h=1/n=x_l-x_{l-1}$. Consider the subspaces $$\label{fe_subspaces}
{\mathcal{L}}(h,k,r)=V_h(k,r,\Xi)=\{v\in C^k(0,1): v\upharpoonright _{I_l}\in P_r(I_l),
1\leq l\leq n, v(0)=0=v(1) \}.$$ The ${\mathcal{L}}(h,k,r)$ are the finite element spaces generated by $C^k$-conforming elements of order $r$ subject to Dirichlet boundary conditions at $0$ and $1$. Then $$\|v-v_h\|_{W^{p,2}(0,1)}\leq c\|v\|_{H^{p+1}_\Xi(0,1)} h^{r+1-p}$$ for $v_h\in {\mathcal{L}}(h,k,r)$ the finite element interpolant of $v\in C^k\cap H^{r+1}_\Xi(0,1)$. For fixed $k,r\ge 1$, let $\mu^+_{m,h}$ and $\mu^-_{m,h}$ be the upper and lower bounds for $\lambda_m$ given by Theorem \[super\] and Theorem \[enclose\], respectively. Then $$\label{order_upper_bound}
\mu^+_{m,h} -\lambda_m = \mathcal{O}(h^{2r}) \quad\textrm{and}\quad\mu^-_{m,h}-\lambda_m=\mathcal{O} (h^{2(r-1)}).$$ All calculations involving this basis were coded in Comsol.
Figure \[fig4\] shows that the orders of convergence found in are optimal in the case $r=3$ for Hermite elements and $m=1,2,3$. In order to compare the quality of the upper and lower bounds, we have chosen Example \[constant\_coeff\] and calculated the value of $\lambda^{{\eqref{constant_coeff}}}_m$ with the exact formula in machine precision. Observe that the upper bounds are all roughly 4 orders of magnitude more accurate than the lower bounds. This is certainly expected from the fact that the calculation of the upper bound involves the solution of a second order problem, whereas that of the lower bound involves the eigenproblem with $T=S(\mu^+_n)$ which is of fourth order. Here we have purposely chosen large values of $h$, so the calculation of the bounds for $\lambda_2$ and $\lambda_3$ is not particularly accurate.
The aim of the experiment performed in Figure \[fig2\] is to compare accuracies in the computation of the bounds by picking ${\mathcal{L}}_n$ of roughly the same dimension, but generated by different bases. For this we have fixed ${\mathcal{L}}_n$ of a given dimension and compute the size of the enclosure $(\mu^-_n,\mu^+_n)$ relative to the size of the lower bound $\mu^-_n$. We consider Example \[non-constant\_coeff\]. We have chosen $\dim {\mathcal{L}}_n=67$ for the sine basis and $\dim {\mathcal{L}}_n\approx 670$ for the finite element bases (remember that the sine basis is exponentially accurate).
The accuracy deteriorates (even in relative terms) as the eigenvalue counting number $m$ increases. For the same dimension of ${\mathcal{L}}_n$, accuracy increases as the order of the polynomial $r$ increases. In this figure, the enclosures found for $\lambda_m$ for $m<5$ ($r=3$), $m<10$ ($r=4$) and $m<20$ ($r=5$) should not be trusted and it is just included for illustration purposes. This locking effect is consistent with the fact that the calculation of the enclosures can never be more accurate than a factor of $\max\{\tau_{\mathrm{i}},\tau_{\mathrm{s}},\tau_{\mathrm{b}}^2\}$.
We can examine this phenomenon in more detail from Figure \[fig3\] and the blue line in Figure \[fig4\]. As the dimension of the test subspace decreases, for each individual eigenvalue, the residual starts decreasing and eventually hits the accuracy threshold. From Figure \[fig4\] it should be noted that the lower bound hits the threshold earlier than the upper bound, however this threshold for the lower bound is three to four orders of magnitude larger that that of the lower bound.
Numerical examples: cylindrical pinch configuration
===================================================
The approach considered in Section \[supersection\] cannot be implemented on the cylindrical pinch configuration for $m\not=0$ as the block operator matrix does not satisfy condition \[top\_3\]. We now report on a set of numerical experiments performed on the benchmark model in Example \[example\_cylinder\], by directly applying the method described in Section \[zimmerman\] to $T=M$.
In this case we have chosen ${\mathcal{L}}_n={\mathcal{L}}(h,1,r)\times {\mathcal{L}}(h,1,r)$ where ${\mathcal{L}}(h,1,r)$ is defined by and is generated by Hermite elements. The Dirichlet boundary condition inposed at both ends of the interval $[0,1]$ ensures that ${\mathcal{L}}_n \in {{\rm Dom}}(M)$. In Table \[tab1\] we show computation of the first three eigenvalues above ${{\rm Spec}}_{\mathrm{ess}}(M)$. Similar calculations can be found in [@1991Kakoetal Table 1]. Note that in the latter, for $N=32$ the approximated eigenvalue appears to be below $\lambda_1$ whereas for $N=64$ it appears to be above $\lambda_1$. This phenomenon is not present in the method described in Section \[zimmerman\] as it always provide a certified enclosure for the eigenvalue.
$j$ exact $\lambda_m$ $(a,b)$ enclosure d.o.f.
----- ------------------- ------------ ------------------------- --------
$1$ $4.38995771667$ $(3,20)$ $4.3_{895445}^{903962}$ $5004$
$2$ $29.4242820473$ $(20,60)$ $29.42_{3873}^{4656}$ $5720$
$3$ $73.8686971063$ $(60,100)$ $73.86_{8030}^{9378}$ $8004$
: Enclosures for the first three eigenvalues in ${{\rm Spec}}_{\mathrm{dis}}(M)$ above the essential spectrum for Example by direct application of the method of Section \[zimmerman\]. For these calculations we have chosen $r=3$ and $\tau_{\mathrm{i}}=\tau_{\mathrm{s}}=10^{-6}$. \[tab1\]
The eigenfunctions of $M$ associated to $\lambda_m$ possess a singularity at the origin, so neither the upper nor the lower bounds obey an estimate analogous to that of . On the left of Figure \[fig\_cyl\_1\] we show a log-log plot of the size of the enclosure against maximum element size for $r=3$ and $r=5$. The graph clearly indicates that the order of decrease of the enclosure does not seem to decrease with the order of the polynomial. On the right of Figure \[fig\_cyl\_1\] we show the absolute residuals for lower and upper bounds separately. Both graphs indicate that $$|\lambda_m-\mu_{m,h}^\pm|=O(h^{1}) \qquad \text{as }h\to 0$$ equally for $r=3$ and $r=5$.
![Log-log graphs. Vertical axis: $\mu_{m,h}^+-\mu_{m,h}^-$ (left) and $|\lambda_m-\mu^\pm_{m,h}|$ (right). Horizontal axis: Maximum element size $h$. For the subspaces ${\mathcal{L}}_n$ we choose $r=3,5$ (left) and $r=3$ (right). Note that the order of decrease of all the residuals is roughly $O(h^1)$ for both polynomial orders (left) and both bounds (right). For these calculations $\tau_{\mathrm{i}}=\tau_{\mathrm{s}}=10^{-10}$. \[fig\_cyl\_1\]](Order_const_coeff_cylindrical "fig:"){height="6cm"} ![Log-log graphs. Vertical axis: $\mu_{m,h}^+-\mu_{m,h}^-$ (left) and $|\lambda_m-\mu^\pm_{m,h}|$ (right). Horizontal axis: Maximum element size $h$. For the subspaces ${\mathcal{L}}_n$ we choose $r=3,5$ (left) and $r=3$ (right). Note that the order of decrease of all the residuals is roughly $O(h^1)$ for both polynomial orders (left) and both bounds (right). For these calculations $\tau_{\mathrm{i}}=\tau_{\mathrm{s}}=10^{-10}$. \[fig\_cyl\_1\]](absolute_res_cylinder "fig:"){height="6cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
This research was funded by EPSRC grant number 113242.
[99]{} , The essential spectrum of some matrix operators. Math. Nachr. 167 (1994) 5–20. , Lower and Upper Bounds for Sloshing Frequencies. International Series of Numerical Mathematics, Vol. 157 (2008) 13–22. , On the convergence of second-order spectra and multiplicity. Proc. R. Soc. A 467 (2011) 264–275. , Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1 (1998) 42–74. , Spectral Approximation of Linear Operators. Academic Press (1983). , Spectral pollution. IMA J. Numer. Anal. 24 (2004) 417–438. , Variational principles for eigenvalues of self-adjoint operator functions, Inter. Equ. Oper. Theory 49 (2004) 287–321. , Eigenwertschranken fur Eigenwertaufgaben mit partiellen Differentialgleinschungen. Z. Angew. Math. Mech. 65 (1985) 129–135. , Essential spectrum of linearized MHD operator in cylindrical region. J. Appl. Maths. Phys. ZAMP. 38 (1987) 433–449. , Spectral approximation for the linearized MHD operator in cylindrical region. Japan J. Indust. Appl. Math. 8 (1991) 221–244. , On the upper and lower bounds of eigenvalues. J. Phys. Soc. Japan 4 (1949) 334–339. , Perturbation theory for linear operators, Springer-Verlag (1966). , Spectral pollution and second order relative spectra for self-adjoint operators, IMA J. Numer. Anal. 24 (2004) 393–416. , Variational principles and eigenvalue estimates for unbounded block operator matrices and applications, J. Comp. and App. Math. 171 (2004) 311–334. , Variational principles for unbounded operator functions and applications, preprint (2011). , Optimale Eigenwerteinschliessungen. Numer. Math. 5 (1963) 246–272. , *Magnetohydrodynamics and Spectral Theory*. Kluwer Academic Publisher (1989). , Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lust operator. Proc. AMS. 130 (2001) 1699-1710. , Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwendungen 14 (1995) 327–345. , The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry. C.R. Acad. Bulg. Sci. 39 (1986) 17–20. , The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry. Arch. Rational Mech. Anal. 116 (1991) 161–198. , Approximation of the spectrum of a non-compact operator given by the magnetohydrodynamic stability of a plasma. Numer. Math. 28 (1977), 15–24. , Geometry of higher order relative spectra and projection methods. J. Operator Theory 44 (2000) 43–62. , *Quadratic Projection Methods for Approximating the Spectrum of Self-Adjoint Operators* IMA J. Numer. Anal. 31 (2011) 40–60. , *The Second Order Spectrum and Optimal Convergence* preprint. , *Spectral Theory Of Block Operator Matrices And Applications*. Imperial College Press (2007).
|
---
abstract: |
We compute explicitly several abstract metrics for RNA secondary structures defined by Reidys and Stadler.
**Keywords:** RNA secondary structure, metric, symmetric group.
author:
- |
F. Rosselló\
[Departament de Matemàtiques i Informàtica,]{}\
[Institut Universitari d’Investigació en Ciències de la Salut (IUNICS),]{}\
[Universitat de les Illes Balears,]{}\
[07122 Palma de Mallorca (Spain)]{}\
[*E-mail:* `cesc.rossello@uib.es`]{}
title: 'On Reidys and Stadler’s metrics for RNA secondary structures[^1]'
---
Introduction
============
As it is well known, an RNA molecule can be viewed as a chain of (ribo)nucleotides with a definite orientation. Each of these nucleotides is characterized by (and in practice identified with) the base attached to it, which can be adenine (A), cytosine (C), guanine (G), or uracil (U). Thus, an RNA molecule with $N$ nucleotides can be mathematically described as a word of length $N$ over the alphabet $\{A,C,G,U\}$, called the *primary structure* of the molecule.
In the cell and *in vitro* each RNA molecule folds into a three-dimensional structure, which determines its biochemical function. This structure is held together by weak interactions called *hydrogen bonds* between pairs of non-consecutive bases: actually, a hydrogen bond can only form between bases that are several positions apart in the chain, but we shall not take this restriction into account here. Most of these bonds form between *Watson-Crick complementary bases*, i.e., between $A$ and $U$ and between $C$ and $G$, but a significant amount of bonds also form between other pairs of bases [@beyondWC]. The *secondary structure* of an RNA molecule is a simplified model of this three-dimensional structure, consisting of an undirected graph with nodes its bases and arcs its base pairs or *contacts*; the *length* of a secondary structure is the number of its nodes. A restriction is added to the definition of secondary structure: a base can only pair with at most one base. This restriction is called the *unique bonds condition*.
An important problem in molecular biology is the comparison of these RNA secondary structures, because it is assumed that a preserved three-dimensional structure corresponds to a preserved function. Moreover, the comparison of RNA secondary structures of a fixed length is used in the prediction of RNA secondary structures to reduce the output of alternate structures when suboptimal solutions, and not only optimal, are considered [@Zuk §IX]. In a seminal paper on the algebraic representation of biomolecular structures [@RS96], C. Reidys and P. F. Stadler introduced three abstract metrics on the set of RNA secondary structures of a fixed length based on their algebraic models and independent of any notion of graph edition, and they discussed their biophysical relevance. They ended that paper by asking, among other questions, whether there exists any relation between the metrics for RNA secondary structures they had defined. In this paper we answer this question by explicitly computing these metrics. In a subsequent paper [@LR] we plan to generalize these metrics to contact structures without unique bonds, as for instance protein structures.
Main results
============
From now on, let $[n]$ denote the set $\{1,\ldots,n\}$, for every positive integer $n$.
An *RNA secondary structure* of length $n$ is an undirected graph without multiple edges or self-loops $\Gamma=([n],Q)$, for some $n\geq
1$, whose arcs $\{j,k\}\in Q$, called *contacts*, satisfy the following two conditions:
i\) For every $j\in [n]$, $\{j,j+1\}\notin Q$.
ii\) For every $j\in [n]$, if $\{j,k\},\{j,l\}\in Q$, then $k=l$.
Condition (i) translates the impossibility of a contact between two consecutive bases, while condition (ii) translates the [unique bonds condition]{}. We should point out that this definition of RNA secondary structure is not the usual one, as the latter forbids the existence of (*pseudo*)*knots*: pairs of contacts $\{i,j\}$ and $\{k,l\}$ such that $i<k<j<l$. This rather unnatural condition is usually required in order to enable the use of dynamic programming methods to predict RNA secondary structures [@Zuk], but real secondary structures can contain knots and thus we shall not impose this restriction here. Therefore, our RNA secondary structures correspond to what in the literature on secondary structure modelling has been called *contact structures with unique bonds* [@RS96; @SS99] or *1-diagrams* [@HS99].
We shall denote from now on a contact $\{j,k\}$ by $j\!\cdot\! k$ or $k\!\cdot\! j$, without distinction. A node is said to be *isolated* in an RNA secondary structure when it is not involved in any contact.
Let ${\mathbb{S}_{n}}$ stand for the set of all RNA secondary structures of length $n$ and let ${\mathcal{S}_{n}}$ be the symmetric group of permutations of $[n]$.
For every $\Gamma=([n],Q)\in {\mathbb{S}_{n}}$, say with $Q=\{i_{1}\!\cdot\!j_{1},\ldots,i_{k}\!\cdot \!j_{k}\}$, let $$\pi(\Gamma)=\prod_{t=1}^k (i_{t},j_{t})\in {\mathcal{S}_{n}},$$ where $(i,j)$ denotes the transposition in ${\mathcal{S}_{n}}$ defined by $i\leftrightarrow j$.
Reidys and Stadler proved in [@RS96] that the mapping $\pi:{\mathbb{S}_{n}}\to
{\mathcal{S}_{n}}$ is injective and that $\pi(\Gamma)$ is an involution for every $\Gamma\in {\mathbb{S}_{n}}$. This representation of RNA secondary structures as involutions is then used by these authors to define the following metric, called the *involution metric*.
\[inv-dist\] The mapping $d_{inv}:{\mathbb{S}_{n}}\times {\mathbb{S}_{n}}\to {\mathbb{R}}$ sending every $(\Gamma_{1},\Gamma_{2})\in {\mathbb{S}_{n}}^2$ to the least number $d_{inv}(\Gamma_{1},\Gamma_{2}) $ of transpositions necessary to represent the permutation $\pi(\Gamma_{1})\pi(\Gamma_{2})$, is a metric.
The following proposition computes explicitly this metric. In it, and henceforth, $A\Delta B$ denotes the symmetric difference $(A\cup B)-(A\cap B)$ of the sets $A$ and $B$, and $|A|$ stands for the cardinal of the finite set $A$.
For every $\Gamma_{1}=([n],Q_{1}),\Gamma_{2}=([n],Q_{2})\in {\mathbb{S}_{n}}$, $$d_{inv}(\Gamma_{1},\Gamma_{2})=|Q_{1}\Delta Q_{2}|-2\Omega,$$ where $\Omega$ is the number of cyclic orbits of length greater than 2 induced by the action on $[n]$ of the subgroup $\langle
\pi(\Gamma_{1}),\pi(\Gamma_{2})\rangle$ of ${\mathcal{S}_{n}}$.
Let $\Gamma_{1}=([n],Q_{1})$ and $\Gamma_{2}=([n],Q_{2})$ be two RNA secondary structures of length $n$. To simplify the language, we shall refer to the orbits induced by the action of $\langle
\pi(\Gamma_{1}),\pi(\Gamma_{2})\rangle$ on $[n]$ simply by *orbits*. Notice that we can understand such an orbit as a subset $\{i_{1},i_{2},\ldots,i_{m}\}$ of $[n]$, $m\geq 1$, such that $$i_{1}\!\cdot\! i_{2},i_{2}\!\cdot\! i_{3},\ldots,
i_{m-1}\!\cdot\!i_{m} \in Q_{1}\cup Q_{2}$$ and maximal with this property, i.e., such that any other contact in $Q_{1}\cup Q_{2}$ involving $i_{1}$ or $i_{m}$ can only be $i_{1}\!\cdot \! i_{m}$. The unique bonds condition (or, in group-theoretical terms, the fact that the transpositions defining each $\pi(\Gamma_{i})$ are pairwise disjoint) implies that if $\{i_{1},i_{2},\ldots,i_{m}\}$ is an orbit, then either $$i_{1}\!\cdot\!
i_{2},i_{3}\!\cdot\!
i_{4},\ldots,\in
Q_{1}\mbox{ and }
i_{2}\!\cdot\!i_{3},i_{4}\!\cdot\!i_{5},\ldots\in
Q_{2}$$ or $$i_{1}\!\cdot\!
i_{2},i_{3}\!\cdot\!
i_{4},\ldots,\in
Q_{2}\mbox{ and }
i_{2}\!\cdot\!i_{3},i_{4}\!\cdot\!i_{5},\ldots\in
Q_{1}.$$ Such an orbit is *cyclic* if $m=2$ and $i_{1}\!\cdot\! i_{2}\in
Q_{1}\cap Q_{2}$, or $m\geq 3$ and $i_{1}\!\cdot\! i_{m}\in Q_{1}\cup
Q_{2}$, and an orbit is *linear* in all other cases. The fact that $\pi(\Gamma_{1}),\pi(\Gamma_{2})$ are both involutions implies that the cardinal of cyclic orbits is always even: roughly speaking, if $i_{1}\!\cdot\! i_{2}\in Q_{1}$ in a cyclic orbit, then $i_{1}\!\cdot\! i_{m}\in Q_{2}$ and hence $i_{m-1}\!\cdot\! i_{m}\in
Q_{1}$.
If two transpositions appearing in the product $\pi(\Gamma_{1})\pi(\Gamma_{2})$ are not disjoint, then the indexes involved in them belong to the same orbit. Moreover, two disjoint transpositions always commute. This allows us to reorganize the transpositions in the product $\pi(\Gamma_{1})\pi(\Gamma_{2})$, assembling them into subproducts corresponding to orbits. More specifically, if for every orbit $O$ and for every $i=1,2$ we let $$\pi(O,\Gamma_{i})=\prod_{\scriptstyle k\cdot l\in Q_{i}\atop \scriptstyle k,l\in O}
(k,l),$$ then $$\pi(\Gamma_{1})\pi(\Gamma_{2})=\prod_{O\in\{\mathrm{orbits}\}}
\pi(O,\Gamma_{1})\pi(O,\Gamma_{2}).$$ Since the orbits are pairwise disjoint, this finally shows that the least number of transpositions which $\pi(\Gamma_{1})\pi(\Gamma_{2})$ decomposes into is equal to the sum of the least numbers of transpositions which $\pi(O,\Gamma_{1})\pi(O,\Gamma_{2})$ decompose into, for every orbit $O$. It remains to compute this last number for each type of orbit $O$.
If $O$ is a linear orbit of length $m=1$, then $\pi(O,\Gamma_{1})\pi(O,\Gamma_{2})=\mathrm{Id}$, and it corresponds to a node that is isolated both in $\Gamma_{1}$ and in $\Gamma_{2}$.
Let now $O=\{i_{1},\ldots,i_{m}\}$ be a linear orbit of length $m\geq
2$. Consider first the case when $i_{1}\!\cdot\!
i_{2},i_{3}\!\cdot\! i_{4},\ldots,i_{m-1}\!\cdot\!i_{m}\in Q_{1}$ and $i_{2}\!\cdot\!i_{3},i_{4}\!\cdot\!i_{5},\ldots\in Q_{2}$; in particular, $m$ is even. Then $$\begin{array}{rl}
\pi(O,\Gamma_{1})\pi(O,\Gamma_{2}) & = (i_{1},i_{2})(i_{3},i_{4})\cdots
(i_{m-1},i_{m})(i_{2},i_{3})\cdots (i_{m-2},i_{m-1})\\ & =
(i_{2},i_{4},\ldots,i_{m},i_{m-1},i_{m-3},\ldots,i_{3},i_{1}),
\end{array}$$ a cycle of length $m$ that decomposes into the product of $m-1$ transpositions (and it is the least number of transpositions required to represent it), which is exactly the number of contacts of $Q_{1}\cup Q_{2}$ involved in this orbit.
A similar argument shows that in all other cases for a linear orbit $O$, the permutation $\pi(O,\Gamma_{1})\pi(O,\Gamma_{2})$ is equal to a cycle of length the number of elements of the orbit, and thus the least number of transpositions this product decomposes into is equal to the number of contacts of $Q_{1}\cup Q_{2}$ involved in this orbit $O$, all of them belonging to $Q_{1}\Delta Q_{2}$.
If $O$ is a cyclic orbit of length $m=2$, say $O=\{i_{1},i_{2}\}$, then $\pi(O,\Gamma_{1})\pi(O,\Gamma_{2})=(i_{1},i_{2})(i_{1},i_{2})=
\mathrm{Id}$. Notice that cyclic orbits of length 2 correspond to contacts in $Q_{1}\cap Q_{2}$.
Finally, assume that $O$ is a cyclic orbit of length $m\geq 3$, say $O=\{i_{1},\ldots,i_{m}\}$ with $i_{1}\!\cdot\! i_{2},i_{3}\!\cdot\!
i_{4},\ldots,i_{m-1}\!\cdot\!i_{m}\in Q_{1}$ and $i_{2}\!\cdot\!i_{3},\ldots,i_{m-2}\!\cdot\! i_{m-1},i_{m}\!\cdot\!
i_{1}\in Q_{2}$; remember that $m$ is in this case even. Then $$\begin{array}{rl}
\pi(O,\Gamma_{1})\pi(O,\Gamma_{2}) & = (i_{1},i_{2})(i_{3},i_{4})\cdots
(i_{m-1},i_{m})(i_{2},i_{3})\cdots (i_{m-2},i_{m-1})(i_{m},i_{1})\\ & =
(i_{2},i_{4},\ldots,i_{m})(i_{m-1},i_{m-3},\ldots,i_{3},i_{1}),
\end{array}$$ the product of two disjoint cycles of length $m/2$. Since each cycle requires $m/2-1$ transpositions, the least number of transpositions the permutation $\pi(O,\Gamma_{1})\pi(O,\Gamma_{2})$ decomposes into is equal to $m-2$, the number of contacts of $Q_{1}\cup Q_{2}$ involved in this orbit $O$ (all of them belonging again to $Q_{1}\Delta Q_{2}$) minus 2.
To sum up, and if we call $\Omega$ the number of cyclic orbits of length greater than 2, $$\begin{array}{rl}
d_{inv}(\Gamma_{1},\Gamma_{2}) & =|\{\mbox{contacts involved in linear orbits}\}|\\
& \qquad +|\{\mbox{contacts involved in cyclic orbits
of length greater than 2}\}| -2\Omega \\
& =|Q_{1}\Delta Q_{2}|-2\Omega,
\end{array}$$ as we claimed.
The number and structure of the orbits induced by the action of $\langle
\pi(\Gamma_{1}),\pi(\Gamma_{2})\rangle$ on $[n]$ are related to the probability of transition from the neutral network of $\Gamma_{1}$ (the set of sequences that fold into it) to that of $\Gamma_{2}$: see [@RS96 §3] and the references cited therein.
Let now $\mbox{Sub}({\mathcal{S}_{n}})$ be the set of subgroups of ${\mathcal{S}_{n}}$.
For every $\Gamma=([n],Q)\in {\mathbb{S}_{n}}$, say with $Q=\{i_{1}\!\cdot\!j_{1},\ldots,i_{k}\!\cdot \!j_{k}\}$, let $$T(\Gamma)=\{(i_{1},j_{1}),\ldots,(i_{k},j_{k})\}$$ be the set of the transpositions corresponding to the contacts in $Q$ and let $G(\Gamma)=\langle T(\Gamma)\rangle$ be the subgroup of ${\mathcal{S}_{n}}$ generated by this set of transpositions.
Reidys and Stadler also proved in [@RS96] that the mapping $G:{\mathbb{S}_{n}}\to \mbox{Sub}({\mathcal{S}_{n}})$ is injective, and then they used this representation of RNA secondary structures as permutation subgroups to define the following *subgroup metric*.
The mapping $d_{sgr}: {\mathbb{S}_{n}}\times {\mathbb{S}_{n}}\to {\mathbb{R}}$ defined by $$d_{sgr}(\Gamma_{1},\Gamma_{2})= \ln\left(\frac{\displaystyle |G(\Gamma_{1})\cdot
G(\Gamma_{2})|}{\displaystyle |G(\Gamma_{1})\cap G(\Gamma_{2})|}\right)$$ is a metric.
Next proposition shows that this metric simply measures, up to a constant factor, the cardinal of the symmetric difference of the sets of contacts.
For every $\Gamma_{1}=([n],Q_{1}),\Gamma_{2}=([n],Q_{2})\in {\mathbb{S}_{n}}$, $$d_{sgr}(\Gamma_{1},\Gamma_{2})= (\ln 2)|Q_{1}\Delta Q_{2}|.$$
Since the transpositions generating a group $G(\Gamma)$, with $\Gamma\in {\mathbb{S}_{n}}$, are pairwise disjoint, there is a bijection between $G(\Gamma)$ and the powerset $\mathcal{P}(T(\Gamma))$: each element of $G(\Gamma)$ is the product of a subset of $T(\Gamma)$ in a unique way. Hence, $|G(\Gamma_{1})|=2^{|Q_{1}|}$ and $|G(\Gamma_{2})|=2^{|Q_{2}|}$.
On the other hand, by the uniqueness of the decomposition of a permutation into a product of disjoint cycles, a permutation belongs to $G(\Gamma_{1})\cap G(\Gamma_{2})$ if and only if it is a product of transpositions belonging to both $G(\Gamma_{1})$ and $G(\Gamma_{2})$. Therefore, $$G(\Gamma_{1})\cap G(\Gamma_{2})=\langle T(\Gamma_{1})\cap
T(\Gamma_{2})\rangle=
\langle (i,j)\mid i\!\cdot \! j\in Q_{1}\cap Q_{2}\rangle,$$ and then, arguing as in the previous paragraph, we see that $|G(\Gamma_{1})\cap G(\Gamma_{2})|=2^{|Q_{1}\cap Q_{2}|}$.
Now, it is well known that $$|G(\Gamma_{1})\cdot G(\Gamma_{2})|=\frac{|G(\Gamma_{1})|\cdot
|G(\Gamma_{2})|}{|G(\Gamma_{1})\cap G(\Gamma_{2})|},$$ and hence $$d_{sgr}(\Gamma_{1},\Gamma_{2})=
\ln \left(\frac{|G(\Gamma_{1})|\cdot
|G(\Gamma_{2})|}{|G(\Gamma_{1})\cap G(\Gamma_{2})|^2}\right)=
\ln 2^{|Q_{1}|+|Q_{2}|-2|Q_{1}\cap Q_{2}|}=\ln 2^{|Q_{1}\Delta
Q_{2}|},$$ as we claimed.
Notice in particular that, should Reidys and Stadler had defined their subgroup metric as $\log_{2}(|G(\Gamma_{1})\cdot G(\Gamma_{2})|/|G(\Gamma_{1})\cap
G(\Gamma_{2})|)$, it would coincide with $|Q_{1}\Delta Q_{2}|$.
The third metric on ${\mathbb{S}_{n}}$ proposed by Reidys and Stadler is actually a general way of defining metrics, rather than a single one, and it uses Magarshak and coworkers’ algebraic representation of RNA secondary structures [@KMM93; @Mag93; @MB92], recently extended in [@CMR] to cope with contacts other than Watson-Crick complementary base pairs. These authors represent an RNA secondary structure $\Gamma=([n],Q)$ as an $n\times n$ complex symmetric matrix $S_{\Gamma}=(s_{i,j})_{i,j=1,\ldots,n}$ where $$s_{i,j}=\left\{
\begin{array}{rl}
-1 & \mbox{ if $i\neq j$ and $i\!\cdot\! j\in Q$}\\
1 & \mbox{ if $i=j$ and $i\!\cdot\! l\notin Q$ for every $l$}\\
0 & \mbox{ otherwise}
\end{array}
\right.$$ Since $S_{\Gamma}^{-1}=S_{\Gamma}$ for every $\Gamma\in {\mathbb{S}_{n}}$, one can define for any $\Gamma_{1},\Gamma_{2}\in {\mathbb{S}_{n}}$ the *transfer matrix* $T_{\Gamma_{1},\Gamma_{2}}=S_{\Gamma_{2}}\circ
S_{\Gamma_{1}}$. Then, Reidys and Stadler propose to measure the difference between two RNA secondary structures by defining a metric through $$(\Gamma_{1},\Gamma_{2})\mapsto \|T_{\Gamma_{1},\Gamma_{2}}\|,$$ where $\|\cdot \|$ stands for some *length function* on the group $GL(n,{\mathbb{C}})$ of $n\times n$ invertible complex matrices [@RS96 Def. 9, Lem. 6] (actually, Reidys and Stadler propose to use a matrix norm $\|\cdot \|$, but it is probably a misprint, as it would not yield a metric). A simple and well-known length function on $GL(n,{\mathbb{C}})$ is $$\|A\|=\mathrm{rank}(A-\mathrm{Id}),$$ which allows to define a metric on ${\mathbb{S}_{n}}$ $$d_{mag}(\Gamma_{1},\Gamma_{2})=\mathrm{rank}(T_{\Gamma_{1},\Gamma_{2}}-\mathrm{Id}).$$ This metric turns out to be equal to the involution metric $d_{inv}$ defined above.
For every $\Gamma_{1},\Gamma_{2}\in {\mathbb{S}_{n}}$, $d_{mag}(\Gamma_{1},\Gamma_{2})=d_{inv}(\Gamma_{1},\Gamma_{2})$.
The proof of this proposition is similar to (and simpler than) the proof of [@CMR Thm. 17], which establishes essentially this equality for the generalized algebraic representation of RNA secondary structures in the sense of Magarshak introduced in that paper, and therefore we omit it.
[99]{}
J. Casasnovas, J. Miró, F. Rosselló, On the algebraic representation of RNA secondary structures with G.U pairs, to appear in *Journal of Mathematical Biology* (DOI: 10.1007/s00285-002-0188-0).
C. Haslinger, P. F. Stadler, RNA structures with pseudo-knots: Graph-theoretical, combinatorial, and statistical properties, *Bulletin of Mathematical Biology* **61** (1999), 437–467.
A. Kister, Y. Magarshak, J. Malinsky, The theoretical analysis of the process of RNA molecule self-assembly, *BioSystems* **30** (1993), 31–48.
M. Llabrés, F. Rosselló, A new abstract metric for arbitrary contact structures, in preparation; a preliminary version will be presented at the *First Joint AMS-RSME Conference* (Sevilla 2003).
Y. Magarshak, Quaternion representation of RNA sequences and tertiary structures, *BioSystems* **30** (1993) 21–29.
Y. Magarshak, C. J. Benham, An algebraic representation of RNA secondary structures, *J. of Biomolecular Structures &Dynamics* **10** (1992) 465–488.
C. Reidys, P. F. Stadler, Bio-molecular shapes and algebraic structures, *Computers & Chemistry* **20** (1996), 85–94.
P. Schuster, P. F. Stadler, Discrete models of biopolymers, to appear in *Handbook of Computational Chemistry* (M.J.C. Crabbe, M. Drew and A. Konopka, eds.), Marcel Dekker (in press); see also Univ. Wien TBI Preprint No. pks-99-012 (1999).
E. Westhof, V. Fritsch, RNA folding: Beyond Watson-Crick pairs, *Structure with Folding & Design* **8** (2000) R55–R65.
M. Zuker, The use of dynamic programming algorithms in RNA secondary structure prediction, In *Mathematical methods for DNA sequences* (M. Waterman, ed.), CRC Press (1989), 159–184.
[^1]: This work has been partially supported by the Spanish DGES, grant BFM2000-1113-C02-01.
|
---
abstract: 'Discs that contain dead zones are subject to the Gravo-Magneto (GM) instability that arises when the turbulence shifts from gravitational to magnetic. We have previously described this instability through a local analysis at some radius in the disc in terms of a limit cycle. A disc may be locally unstable over a radial interval. In this paper, we consider how the local instability model can describe global disc outbursts. The outburst is triggered near the middle of the range of locally unstable radii. The sudden increase in turbulence within high surface density material causes a snow plough of density that propagates both inwards and outwards. All radii inside of the trigger radius become unstable, as well as locally unstable radii outside the trigger radius. In addition, a locally stable region outside of the trigger radius may also become unstable as the gravitational instability is enhanced by the snow plough. For the circumstellar disc model we consider, we find that a quarter of the disc mass is accreted on to the central object during the outburst. The radius out to which the disc is globally unstable is twice that for which it is locally unstable.'
author:
- |
Rebecca G. Martin$^1$[^1] and Stephen H. Lubow$^2$\
$^1$NASA Sagan Fellow, JILA, University of Colorado, Boulder, CO 80309, USA\
$^2$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\
title: 'Propagation of the Gravo-Magneto Disc Instability'
---
\[firstpage\]
accretion, accretion disks –- planets and satellites: formation –- protoplanetary disks –- stars: pre-main sequence
Introduction
============
Accretion discs are ubiquitous in the Universe, they form on all scales from planetary, to stellar, to galactic. They transport angular momentum outwards allowing material to spiral inwards through the disc [e.g. @pringle81]. If the disc is fully ionised, the magneto rotational instability (MRI) drives turbulence and thus angular momentum transport [@balbus91]. However, below a critical level of ionisation, a dead zone may form at the mid-plane where the MRI does not operate [@gammie96; @gammie98; @turner08]. The inner parts of the disc are hot enough to be thermally ionised. However, further out, only the surface layers of the disc may be ionised by external sources such as cosmic rays or X-rays from the central star [@glassgold04]. The restricted flow through the disc causes a build up of material in the dead zone. If there is sufficient accretion infall, some turbulence in the dead zone may be driven by self-gravity when the layer becomes massive enough [@paczynski78; @lodato04]. However, if it does not transport angular momentum fast enough, the gravo-magneto (GM) disc instability can result. This occurs when the temperature in the dead zone reaches the critical required for the MRI to be triggered and an accretion outburst ensues [@armitage01; @zhu09; @zhu10a; @zhu10b; @martin11].
The GM disc instability can be explained as transitions between steady state disc solutions plotted on a state diagram showing the accretion rate through the disc against the surface density for a fixed radius [@martin11]. There are three steady state disc solutions, two fully MRI turbulent solutions (one that is thermally ionised and one that is fully ionised by external sources such as cosmic rays or X-rays) and the GM solution. The GM solution consists of a self gravitating, MRI inactive mid-plane region with MRI active surface layers such that the flow through the disc is at the steady rate. For a given radius, there may be a range of accretion rates for which no steady solution exists. If the accretion rate on to the disc lies in this range then the disc will be locally GM unstable. The disc transitions between the steady thermal MRI and GM solutions cause repeating outbursts over time. This is similar to the “S-curve” used to explain dwarf nova outbursts [@bath82; @faulkner83].
For a range of radii in the disc, there exists a local limit cycle. In this work, we extend the model of [@martin11] to consider how the local limit cycles at different radii are coordinated and thus how the global disc evolution operates. We examine the radius at which the outburst is triggered and how far the outburst propagates through the disc. We find that the radii are connected through a snow plough effect where the dense material is pushed both inwards and outwards during the outburst. Previous works have considered the propagation of the outburst [e.g. @armitage01; @zhu10b], but not within the framework of the limit cycle.
The GM disc instability may occur in accretion discs on many scales. It is thought to describe FU Orionis outbursts in discs around young stellar objects [e.g. @armitage01; @zhu10b; @martin12a; @martin12b; @stramatellos12]. It is also likely to occur in circumplanetary discs that form around massive planets as they form within the circumstellar disc [@martin11b; @lubow12a; @lubow12b]. In this work we restrict our analysis to that of a circumstellar disc, but note that the underlying mechanism is the same for all scales.
Local Disc Instability
======================
The layered disc model we apply is based on [@armitage01] and further developed in [@zhu10b] and [@martin11]. Following [@martin11], we first solve the steady state layered accretion disc equation $$\dot M =3 \pi ( \nu_{\rm m} \Sigma_{\rm m} + \nu_{\rm g} \Sigma_{\rm g})$$ with the steady energy equation $$\sigma T^4= \frac{9}{8} \Omega^2(\tau_{\rm m} \nu_{\rm m} \Sigma_{\rm m} + \tau \nu_{\rm g} \Sigma_{\rm g}),$$ where $\dot M$ is the steady accretion rate through the disc, $T$ is the midplane temperature, $\Sigma_{\rm g}$ is the surface density in the dead zone layer, $\Sigma_{\rm m}$ is the MRI active surface density, the total surface density is $\Sigma=\Sigma_{\rm
m}+\Sigma_{\rm g}$, $\Omega$ is the Keplerian angular velocity, $\tau_{\rm m}$ is the optical depth of the magnetic layer and $\tau$ is the optical depth of the whole disc. The viscosity due to the MRI is $$\nu_{\rm m}=\alpha_{\rm m}\frac{c_{\rm m}^2}{\Omega},$$ where $c_{\rm m}$ is the sound speed in the active surface layers and $\alpha_{\rm m}$ is the [@shakura73] $\alpha$ parameter in the layer. The disc is self-gravitating if the [@toomre64] parameter is less than the critical, $Q<Q_{\rm crit}=2$, and in this case $\nu_{\rm g}$ depends on $Q$. The viscosity due to self-gravity is $$\nu_{\rm g}=\alpha_{\rm g}\frac{c_{\rm g}^2}{\Omega},$$ where $c_{\rm g}$ is the sound speed in the mid-plane layer and $$\alpha_{\rm g}=\alpha_{\rm m} \left[\left(\frac{Q_{\rm crit}}{Q}\right)^2-1\right]$$ for $Q<Q_{\rm crit}$ and zero otherwise [@lin87; @lin90]. We find that the exact form of $\alpha_{\rm g}$ does not affect the disc evolution provided it is a steeply declining function of $Q$ [see also @zhu10b and the Discussion section].
In this work we consider a constant infall accretion rate, $\dot
M_{\rm infall}$, on to the disc. For a steady disc solution, the infall accretion rate is equal to the accretion rate through the disc, $\dot M$, at all radii. However, if at some radius there is no steady solution, then the disc is unstable to the GM disc instability.
As described in the Introduction, there are three different steady solutions, two MRI active solutions and the GM solution. The fully MRI active solutions are found by setting $\nu_{\rm g}=0$. They exist in two regions of the disc, first in the inner parts where the mid-plane temperature is sufficiently high for thermal ionisation ($T>T_{\rm
crit}$, where $T_{\rm crit}$ is the temperature above which the MRI operates) and secondly in the outer parts where the surface density is sufficiently small that cosmic rays or X-rays penetrate the whole disc ($\Sigma<\Sigma_{\rm crit}$, where $\Sigma_{\rm crit}$ is the critical surface density that is ionised through external sources). The third steady solution is the GM solution, where there is gravitational turbulence near the mid-plane and MRI turbulence in disc surface layers. This solution occurs where the mid-plane temperature is too low for the MRI ($T<T_{\rm crit}$) and the surface density is greater than the critical that can be ionised by external sources ($\Sigma>\Sigma_{\rm crit}$), and large enough for the disc to be self-gravitating. The steady state solutions at a given radius cover a range of accretion rates. In general, there are cases where there is a gap in accretion rates for steady state solutions. Accretion rates that fall in that gap at some radius are locally unstable [@martin11]. For a fixed accretion rate on to the disc, there is generally a range of radii where that rate falls within the local gaps and thus a range of radii where the disc can be locally unstable.
In this work we take an example of a circumstellar disc, but note that the same mechanisms apply to circumplanetary discs on a smaller scale. We choose a star of mass $M=1\,\rm M_\odot$ and a disc with a [@shakura73] viscosity parameter $\alpha_{\rm m}=0.01$, $T_{\rm
crit}=800\,\rm K$ and $\Sigma_{\rm crit}=200\,\rm
g\,cm^{-2}$. Fig. \[data\] shows the state diagrams of the steady solutions at various radii in the disc. The surface temperature is shown against the surface density for the steady state solutions at a fixed radius. The surface temperature in the disc is related to an effective, steady accretion rate, $\dot M_{\rm s}$, with $$T_e =\left(\frac{3\dot M_{\rm s}\Omega^2}{8 \pi \sigma}\right)^\frac{1}{4}.$$
All radii shown, except $R=0.1\,\rm au$, show the three steady branches. There are two fully MRI active solutions, the upper branch with $T>T_{\rm crit}$ and the lower branch with $\Sigma<\Sigma_{\rm
crit}$. The third branch is the self gravitating GM solution. The steady state solutions are shown in the solid thick lines and the dead zone branch (that is not a steady branch) is shown as the dashed line. The branches are labeled in the state diagram at $R=5\,\rm au$. The shaded region is where there is no steady state solution and the disc is locally unstable. If the imposed disc accretion rate (infall accretion rate onto the disc), represented as the dotted line, lies within the shaded region, then the disc is locally unstable at that radius. The state diagram at a radius of $R=0.1\,\rm au$ does not have an unstable region because a fully MRI active disc solution exists at all accretion rates because either $\Sigma<\Sigma_{\rm
crit}$ or $T>T_{\rm crit}$.
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Innermost Unstable Radius
-------------------------
For a disc with an unstable region, the innermost radius that is locally unstable occurs where the thermally ionised fully MRI active solution has a mid-plane temperature equal to the critical temperature. The surface density is larger than the critical that is ionised by external sources, if the unstable region exists.
This location can been seen in the state diagrams to occur at a radius where the accretion rate through the disc is such that it lies on the lower end of the upper MRI branch (similar to the middle left plot in Fig. \[data\]). That is, the radius where the dotted line intersects the lower end of the upper, solid sloped line. The upper MRI branch begins where the mid-plane temperature is equal to the critical, $T=T_{\rm crit}$. We scale the variables to $\dot M'=\dot M
/(10^{-6}\,\rm M_\odot\, yr^{-1})$, $M'=M/{\rm M_\odot}$, $\alpha'=\alpha/0.01$ and $T_{\rm crit}'=T_{\rm crit}/{800\,\rm
K}$. We find steady state fully turbulent solutions (with $\Sigma_{\rm m}=\Sigma$ and $\nu_{\rm g}=0$). The radius at which the fully turbulent solution has the critical mid-plane temperature is $$R= 1.87 \, \frac{ \dot M'^{4/9} M'^{1/3}}{\alpha'^{2/9} T_{\rm crit}'^{14/15}}
\,\rm au.$$ This is the innermost radius that is locally unstable to the GM instability.
Outermost Unstable Radius
-------------------------
The outermost radius for which no steady solution exists (and thus the disc is locally unstable to the GM instability) occurs where the GM solution has a mid-plane temperature equal to the critical. In the state diagram this is where the accretion rate through the disc lies at the upper end of the GM branch (similar to the lower left plot in Fig. \[data\]). This is where the self gravitating, GM solution reaches the critical mid-plane temperature. We again solve the steady state equations, but now include the self gravitating term. However, the equations must be solved numerically. For an accretion rate of $\dot M_{\rm infall}=10^{-6}\,\rm M_\odot\, yr^{-1}$, we find the radius to be $R=4.51\,\rm au$. Thus, the locally unstable region for this disc extends from $R=1.87\,\rm au$ to $R=4.51\,\rm au$. Outside of this region there is a steady GM disc solution, and inside there is a fully MRI solution. We note that the innermost and outmost unstable radii in the disc do not necessarily correspond to the inner and outer edges of a dead zone in a time-dependent disc.
Outer Transition to MRI Branch
------------------------------
The outer parts of the disc are fully MRI active where the surface density is smaller than the critical, $\Sigma<\Sigma_{\rm crit}$. This occurs at a radius $$R=29.15\, \frac{\dot M'^{11/9} M'^{1/3}}{\alpha'^{16/9}} \,\rm au.$$ Hence, in summary, in the region $R<1.87\,\rm au$ the disc has a fully MRI active solution, $1.87<R/{\rm au}<4.51$ the disc has no steady solution (and is locally unstable to the GM instability). In $4.51<R/{\rm au}<29.2$ there is a steady GM solution. In $R>29.2\,\rm
au$ the disc is fully MRI active. In Figs. \[st\] and 3 we label these regions as a function of radius in the disc and shade the region for which there is no steady solution.
{width="7.5cm"} {width="7.5cm"}
Global Disc Instability
=======================
{width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"}
We solve the time-dependent accretion disc equations that consist of a surface density evolution equation [@pringle81] and an energy equation [@pringle86; @cannizzo93]. The equations are described in more detail in [@martin11] (see their equations 1 and 3). We take a grid of 200 points equally spaced in $\log R$ that extends from $R_{\rm in}=5\,\rm R_\odot$ to $R_{\rm out}=40\,\rm au$ [e.g. @armitage01; @martin07]. At the inner boundary we have a zero torque boundary condition to allow the inward flow of gas on to the star. At the outer boundary, we choose a zero radial velocity condition to prevent material from leaving the disc there. There is a constant infall accretion rate on to the disc of $\dot M_{\rm
infall}=10^{-6}\,\rm M_\odot\,yr^{-1}$ at a radius of $35\,\rm
au$. As shown in the previous section, for this accretion rate the disc is locally unstable for a range of radii.
The disc evolves through several outbursts before the limit cycle repeats. Thus, the initial conditions of the disc have no effect on the limit cycles shown. Fig. \[data\] shows the local state diagrams, for various radii, of the surface temperature against the surface density. The thick lines show the analytic steady state solutions (as described in the previous section) and the thin lines show the numerical time-dependent evolution. The dashed line shows the steady infall accretion rate.
The crosses mark the position at the time when the outburst is triggered at that radius (i.e. where $T>T_{\rm crit}$), and the stars mark the position when the outburst is first triggered globally. For the $R=3\,\rm au$ case, the two positions coincide with the end of the GM branch and hence also with a temperature of $T=T_{\rm
crit}$. However, at other radii, the position when the outburst is triggered locally does not coincide with the position where it is triggered globally. Once the outburst is triggered somewhere in the disc, it propagates very quickly and adjacent radii undergo their limit cycles similarly fast. At $R=0.1\,\rm au$, the mid-plane temperature of the disc does not drop below the critical during the cycle, hence there is no cross shown. Similarly, at $R=10\,\rm au$, the mid-plane temperature during the cycle does not exceed the critical temperature and so there is no cross shown.
The outburst propagates inwards and hence all radii inside of the trigger radius show outburst behaviour. The total mass of the disc before the outburst is $0.45\,\rm M_\odot$ of which $0.34\,\rm
M_\odot$ is within the dead zone and $0.024\,\rm M_\odot$ is not self-gravitating. During the outburst, a mass of $0.1\,\rm M_\odot$ is accreted on to the star. This amount of material is initially distributed up to a radius of $2.9\,\rm au$ (that roughly coincides with the trigger radius). The outburst also propagates outwards. The dead zone has a high surface density that suddenly becomes more turbulent when the MRI is triggered, causing material to move outwards. For example, at $R=5\,\rm au$, the disc has a steady state GM solution. However, this solution is not found because of the outward propagation. The disc is depleted after an outburst, leaving a less massive and cooler disc than the steady state would predict. The disc moves towards the steady state but the outburst is first triggered at an accretion rate lower than the steady rate. Even further out, where the steady solution is lower down the GM branch, the outward propagation is not sufficient to trigger an outburst and the disc remains close to the GM branch and then moves downwards after the outburst. The outermost radius for which the disc moves up to the upper MRI branch during the outburst is around $R=9\,\rm au$, twice the radius of outer edge of the locally GM unstable region.
Fig. \[st\] shows the disc structure leading up to an outburst. The branch labels in the figure (Upper MRI etc.) refer to where steady state solutions exist. For the time-dependent evolution being plotted, these labels may not reflect the instantaneous structure at each radius. The shaded region shows where there is no steady state disc solution. The innermost parts of the disc are fully MRI turbulent. However, beyond radius of $R=0.41\,\rm au$, the mid-plane temperature of the disc drops below the critical value. At this radius, $R=0.41\,\rm au$, there is a sharp increase in the surface density and this marks the inner edge of the dead zone. The narrow density peak is due to a local pileup of gas. It is the result of the outward spread of turbulent gas from the active, inner (inside 0.41 AU) region into the nonturbulent dead zone where it cannot spread outward any further. The narrowness is a consequence of the sharp transition criteria we apply between active and dead zones. If we applied a smoother transition criteria, the peak would be spread out over a larger range of radii. The dead zone covers some of the region where there exists a steady state MRI disc solution (as labeled Upper MRI). This overlap exists because the outer parts of the flow near $R=1\,\rm au$ are limited to the accretion rate of the active surface layer. There is not sufficient flow in this region to reach the steady solution at the infall accretion rate.
The local peak in the surface density within the unstable region marks the radius in the disc at which the dead zone becomes self gravitating. This local peak moves inwards in time as material builds up. There is a similar local peak in the temperature profile that moves inwards slightly behind the surface density peak in time. Once this peak reaches the critical temperature required for the MRI, the outburst is triggered (as in the last time shown in Fig. \[st\]). The outburst is first triggered at a radius of $R=2.6\,\rm au$.
It is the outer parts of the dead zone that become self gravitating first (as shown in Fig. \[st\]). Hence, in the limit cycles in Fig. \[data\], the larger the radius, the sooner the disc reaches the GM branch. However, the speed at which the disc moves up this branch, decreases with radius. As a function of radius, there is then opposite behavior in the time that the disc reaches this branch and the speed at which it travels up the branch. Hence, the outburst is triggered somewhere in the middle of the locally unstable region.
Fig. \[mr\] shows the radius dependent accretion rate through the disc at various times during the outburst cycle. The upper portion of each plot represents the inward flow and the lower portion the outward flow. The top left plot shows the accretion rate just before an outburst, that corresponds to the upper lines for the surface density and temperature shown in Fig. \[st\]. There are three main parts to the disc, the inner part that is fully MRI active, the middle part where there is a dead zone and the large outer part that is self gravitating. In the very outermost parts of the disc, the accretion drops off because there is a zero flow outer boundary condition. This only affects the parts of the disc outside of the radius where material is injected, at $35\,\rm au$.
The top right hand plot shows the accretion rate shortly after the outburst is triggered. There are both inward and outward propagating waves moving away from the trigger point at $2.6\,\rm au$. This is the snow plough effect, material moves both inwards and outwards away from the trigger radius. It is this snow plough that causes the various radii to be coordinated. The middle left plot shows the inner wave has propagated all the way in and the disc is now in outburst phase with a high accretion rate on to the star. The outward propagating waves continue to move outwards. In the middle right hand plot, the dead zone has begun to re-form. This initially occurs at a radius of $8.1\,\rm au$. The dead zone formation propagates both inwards and outwards as seen in the lower left plot. As the dead zone forms, the viscous torque there drops. Thus, material just interior to the dead zone initially moves outwards because the outward viscous torque is larger than the inward torque. This is the cause of the narrow peak in the surface density at the inner edge of the dead zone (as described in Fig. 2). Eventually the inner parts of the disc move back to the quiescent phase as the dead zone blocks the flow, as seen in the bottom right hand plot. This marks the end of the outburst, although there is still some ongoing outward propagation in the outer regions. The material in the disc builds up again as shown in Fig. \[st\] until the cycle repeats.
Discussion
==========
There are a number of simplifications in our model that should be investigated in future work. Some of these were discussed in more detail in [@martin11] and [@lubow12a]. For example, we have assumed that the critical surface density that is ionised by external sources in constant with radius. However, a more realistic way to find the extent of the dead zone may be with a critical magnetic Reynolds number [e.g. @fleming00; @matsumura03]. The critical value however remains uncertain [e.g. @martin12b]. The ionisation may be further suppressed by effects such as ambipolar diffusion and the presence of dust and polycyclic aromatic hydrocarbons [e.g. @bai09; @bai11; @perez11; @simon12; @dzyurkevich13]. No matter how the dead zone is determined, the outburst mechanism works in the same way. The shape of the limit cycle may be slightly different [see @martin12a], but the principles remain the same.
There remains some uncertainty in the prescriptions for the viscosities in the disc. Observations of the fully MRI turbulent discs in dwarf novae and X-ray binaries suggest $\alpha \approx
0,1-0.4$ [@king07]. However, in discs around T Tauri stars, such a high $\alpha$ would cause the outer discs to expand too quickly [@hartmann98]. Moreover, theoretical work suggests a smaller $\alpha \approx 0.01$ [e.g. @fromang07]. We have taken $\alpha_{\rm m}=0.01$ in this work, but note that the qualitative evolution of the disc would be similar had we taken a larger value for $\alpha_{\rm m}$. There is also some uncertainty in the value of $\alpha_{\rm g}$ in a self gravitating disc. But its exact value has little influence of the disc structure and evolution, as noted by [@zhu10a]. Within the self gravitating region the Toomre parameter in our model is nearly constant, $Q\approx Q_{\rm crit}$. In our prescription the value of $\alpha_{\rm
g}$ is not fixed and depends on the Toomre parameter, $Q$. In steady state, implicitly it varies with the local cooling timescale, $t_{\rm
cool}$, such that $\alpha_{\rm g}\propto 1/(t_{\rm cool}\Omega)$ [see also @gammie01].
In a real disc the infall accretion rate is not likely to be constant in time, but rather decrease. The locally unstable region would decrease in time until the accretion rate is so low that a fully MRI active solution is found. For the parameters chosen in this work, this would be for accretion rates $\dot M_{\rm infall} \lesssim
10^{-8}\,\rm M_\odot\, yr^{-1}$.
The existence of dead zones in protoplanetary discs may have important implications for planet formation. Planets that survive after the disc has dispersed [e.g. by photoevaportation @hollenbach94; @alexander06] likely formed after the final GM disc outburst. If planets form between outbursts, they likely accrete on to the star during the outburst. As the infall accretion rate drops over time, the outbursts cease but a dead zone may remain. This dead zone provides a quiescent region suitable for planet formation. The presence of a dead zone also affects the migration of planets through the disc. Planets embedded in low viscosity discs can undergo very slow and sometimes chaotic migration [@li09; @yu10]. The nature of gap opening and migration of planets through a layered disc should be investigated in future work.
Conclusions
===========
The GM disc instability can be explained as transitions between steady state solutions in a state diagram that plots the surface temperature against the surface density at a fixed radius. We have examined local cycles at different radii. There is a range of radii for which a given disc is locally GM unstable. For the parameters chosen in this work ($\Sigma_{\rm crit}=200\,\rm g\,cm^{-2}$, $T_{\rm crit}=800\,\rm K$, $\dot M_{\rm infall}=10^{-6}\,\rm M_\odot\, yr^{-1}$ and $\alpha=0.01$), we find the unstable range to be $1.87<R/{\rm au}<4.51$.
The outburst is first triggered in the middle of the range of locally unstable radii. For the parameters we chose, the trigger radius is at $2.6\,\rm au$. The resulting snow plough of material that propagates both inwards and outwards links the different radii in the disc. All radii inside of the trigger radius become unstable and the mass in this region moves inwards to be accreted during the outburst. The snow plough effect pushes material outwards through the disc to radii which would otherwise lie along the steady GM branch and be locally stable. The higher density causes these outer radii to also become unstable. The outburst propagates to a radius of around $9\,\rm au$, about double that of the outermost locally unstable radius. Even further out than this, there are increases in density but these are not sufficient for a change in the state.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for useful comments. RGM’s support was provided in part under contract with the California Institute of Technology (Caltech) funded by NASA through the Sagan Fellowship Program. SHL acknowledges support from NASA grant NNX11AK61G.
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\[lastpage\]
[^1]: E-mail: rebecca.martin@jila.colorado.edu
|
---
abstract: |
With the growing reliance on the ubiquitous availability of IT systems and services, these systems become more global, scaled, and complex to operate. To maintain business viability, IT service providers must put in place reliable and cost efficient operations support. Artificial Intelligence for IT Operations (AIOps) is a promising technology for alleviating operational complexity of IT systems and services. AIOps platforms utilize big data, machine learning and other advanced analytics technologies to enhance IT operations with proactive actionable dynamic insight.
In this paper we share our experience applying the AIOps approach to a production cloud object storage service to get actionable insights into system’s behavior and health. We describe a real-life production cloud scale service and its operational data, present the AIOps platform we have created, and show how it has helped us resolving operational pain points.
author:
- |
Anna Levin, Shelly Garion, Elliot K. Kolodner,\
Dean H. Lorenz, Katherine Barabash\
IBM Research – Haifa\
`{lanna,shelly,kolodner,dean,kathy}@il.ibm.com`
- |
Mike Kugler, Niall McShane\
\
IBM Cloud and Cognitive Software\
`Mike.Kugler@ibm.com, nmcshane@us.ibm.com`
bibliography:
- 'bibtex.bib'
title: AIOps for a Cloud Object Storage Service
---
****
Anna Levin, Shelly Garion, Elliot K. Kolodner, Dean H. Lorenz, Katherine Barabash IBM Research – Haifa lanna,shelly,kolodner,dean,kathy@il.ibm.com
Mike Kugler, Niall McShane IBM Cloud and Cognitive Software Mike.Kugler@ibm.com, nmcshane@us.ibm.com
Published in: 2019 IEEE International Congress on Big Data (BigDataCongress)
$\copyright$2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. DOI: 10.1109/BigDataCongress.2019.00036
This work has been partially supported by the SODALITE project, grant agreement 825480, funded by the EU Horizon 2020 Programm.
|
[**On a family of hypergeometric Sobolev orthogonal polynomials on the unit circle.**]{}
[**S.M. Zagorodnyuk**]{}
Introduction.
=============
The theories of orthogonal polynomials on the real line and on the unit circle have many similarities as well as considerable differences [@cit_50000_Gabor_Szego], [@cit_5000_Ismail], [@cit_48000_Simon_1], [@cit_48000_Simon_2]. For a long time they have been developed side by side by efforts of numerous mathematicians. The theory of Sobolev orthogonal polynomials is a much more *terra incognita* [@cit_5150_M_X], [@cit_49000_Sri_Ranga]. In this theory one can also see that some ideas come from the real line to the unit circle. Examples of such ideas are adding of Dirac deltas to the classical inner products and considering of coherent pairs of measures (see, e.g., [@cit_5000_Castillo], [@cit_5500_GMPC_coherent_pairs] and references therein). In the present paper we shall follow the same line: we shall develop the ideas from [@cit_80000_Zagorodnyuk_JAT_2020] to get some new hypergeometric polynomials and study their properties.
Let $\mu$ be a (non-negative) measure on $\mathfrak{B}(\mathbb{T})$ with an infinite support, $\mu(\mathbb{T})=1$. Denote by $p_n$ orthogonal polynomials on $\mathbb{T}$ with respect to $\mu$ ($\deg p_n = n$, but the positivity of leading coefficients is not assumed): $$\label{f1_10}
\int_{\mathbb{T}} p_n(z) \overline{p_m(z)} d\mu = A_n \delta_{n,m},\qquad A_n>0,\ n,m\in\mathbb{Z}_+.$$ Fix an arbitrary positive integer $\rho$. Consider the following differential equation: $$\label{f1_20}
( e^{-x} y(x) )^{ (\rho) } = e^{-x} p_n(x),\qquad n\in\mathbb{Z}_+.$$ Expanding the derivative by the Leibniz formula and canceling $e^{-x}$ we get $$\label{f1_25}
\sum_{k=0}^\rho (-1)^{\rho -k} \left( \begin{array}{cc} \rho \\
k \end{array} \right)
y^{ (k) }(x)
= p_n(x),\qquad n\in\mathbb{Z}_+.$$
**Condition A.** *Suppose that for each $n\in\mathbb{Z}_+$, there exists a $n$-th degree polynomial solution $y=y_n(x)$ of (\[f1\_25\])*.
If Condition A is satisfied, then $y_n$ are Sobolev orthogonal polynomials on $\mathbb{T}$: $$\label{f1_30}
\int_{\mathbb{T}} \left( y_n(z), y_n'(z),..., y_n^{(\rho)}(z) \right) M \overline{
\left( \begin{array}{cccc} y_m(z) \\
y_m'(z) \\
\vdots \\
y_m^{(\rho)}(z) \end{array} \right)
}
d\mu = A_n \delta_{n,m},\qquad n,m\in\mathbb{Z}_+,$$ where $$\label{f1_35}
M =
\left(
(-1)^{l+j}
\left( \begin{array}{cc} \rho \\
l \end{array} \right)
\left( \begin{array}{cc} \rho \\
j \end{array} \right)
\right)_{l,j=0}^\rho.$$
In this paper we shall only consider the following case: $p_n(x) = x^n$, and $\mu$ being the normalized arc length measure on $\mathbb{T}$. Let us briefly describe the content of the paper. Equating coefficients of the same powers on the both sides of equation (\[f1\_25\]) one obtains a linear system of equations for the coefficients of an unknown polynomial $y(x)$ (the same idea was used in [@cit_5_Azad]). However, for large values of $\rho$ it is not easy to get a convenient expression for solutions, without huge determinants or recurrences. In this case equation (\[f1\_20\]) turned out to be useful. It gives a possibility to express $y_n(x)$ for $\rho=1$ in terms of the incomplete gamma function. A step-by-step analysis for $\rho = 1,2,...,$ allows to obtain an explicit representation of $y_n(x)$. Explicit representations, differential equations and orthogonality relations for $y_n$ will be given by Theorem 1. Using Fasenmeier’s method ([@cit_5150_Rainville]) for the reversed polynomials $y_n^*(x) = x^n y_n\left( \frac{1}{x} \right)$, we shall derive recurrence relations for $y_n(x)$ (Theorem 2) as well.
[**Notations.** ]{} As usual, we denote by $\mathbb{R}, \mathbb{C}, \mathbb{N}, \mathbb{Z}, \mathbb{Z}_+$, the sets of real numbers, complex numbers, positive integers, integers and non-negative integers, respectively. Set $\mathbb{T} := \{ z\in\mathbb{C}:\ |z|=1 \}$. By $\mathfrak{B}(\mathbb{T})$ we mean the set of all Borel subsets of $\mathbb{T}$. By $\mathbb{P}$ we denote the set of all polynomials with complex coefficients. For a complex number $c$ we denote $(c)_0 = 1$, $(c)_1=c$, $(c)_k = c(c+1)...(c+k-1)$, $k\in\mathbb{N}$ (*the shifted factorial or Pochhammer symbol*). The generalized hypergeometric function is denoted by $${}_m F_n(a_1,...,a_m; b_1,...,b_n;x) = \sum_{k=0}^\infty \frac{(a_1)_k ... (a_m)_k}{(b_1)_k ... (b_n)_k} \frac{x^k}{k!},$$ where $m,n\in\mathbb{N}$, $a_j,b_l\in\mathbb{C}$.
Some Sobolev orthogonal polynomials on $\mathbb{T}$.
====================================================
As it was stated in the Introduction, in what follows we shall consider the following case: $p_n(x) = x^n$, and $\mu=\mu_0$ being the (probability) normalized arc length measure on $\mathbb{T}$. Rewrite equations (\[f1\_20\]),(\[f1\_25\]) for this case: $$\label{f2_20}
( e^{-x} y_n(x) )^{ (\rho) } = e^{-x} x^n,\qquad n\in\mathbb{Z}_+;$$ $$\label{f2_25}
\sum_{k=0}^\rho (-1)^{\rho -k} \left( \begin{array}{cc} \rho \\
k \end{array} \right)
y_n^{ (k) }(x)
= x^n,\qquad n\in\mathbb{Z}_+.$$ We start with the case $\rho=1$. In this case equation (\[f2\_25\]) has the following form: $$\label{f2_27}
y_n'(x) - y_n(x) = x^n,\qquad n\in\mathbb{Z}_+.$$ Fix an arbitrary $n\in\mathbb{Z}_+$. We shall seek for a solution of the required form: $$\label{f2_29}
y_n(x) = \sum_{k=0}^n \mu_{n,k} x^k,\qquad \mu_{n,k}\in\mathbb{C}.$$ Substitute for $y_n$ into (\[f2\_27\]) to get $$\sum_{k=0}^{n-1}
\{
(k+1) \mu_{n,k+1} - \mu_{n,k}
\}
x^k
-
\mu_{n,n} x^n = x^n.$$ Comparing the coefficients of the same powers on the both sides we obtain that $$\label{f2_31}
\mu_{n,n} = -1,\quad \mu_{n,k} = (k+1) \mu_{n,k+1},\quad k=n-1,n-2,...,0.$$ It can be verified by the induction argument that $$\mu_{n,j} = - (n)_{n-j} = - \frac{n!}{j!},\qquad j=0,1,...,n.$$ Thus, $$\label{f2_35}
y_n(x) = - n! \sum_{k=0}^n \frac{ x^k }{ k! },$$ is a solution of (\[f2\_27\]). In the case $\rho > 1$, it is not easy to solve the corresponding recurrence relation for the coefficients and we shall proceed in another way.
Observe that $$\label{f2_37}
y_n(x) = - e^x \Gamma(n+1,x),$$ where $$\Gamma(\alpha,x) = \int_x^\infty e^{-t} t^{\alpha - 1} dt,\qquad \alpha > 0,$$ is *the complementary incomplete gamma function* ([@cit_3_Andrews_book]). In fact, integrating (\[f2\_20\]) (with $\rho=1$) from $a$ to $b$ ($a,b\in\mathbb{R}$) we get $$e^{-b} y_n(b) - e^{-a} y_n(a) = \int_a^b e^{-x} x^n dx.$$ Passing to the limit as $b\rightarrow +\infty$ we get $$\label{f2_38}
y_n(a) = - e^a \int_a^\infty e^{-x} x^n dx,$$ and relation (\[f2\_37\]) follows.
Suppose that we have constructed a polynomial solution (of the required form) $y_n(\rho;x) = y_n(x)$ of equation (\[f2\_20\]) for some positive integer $\rho$. Let us show how to get a polynomial solution $y_n(\rho+1;x)$ of equation (\[f2\_20\]) with $\rho+1$. Notice that we do not state the uniqueness of such solutions for $\rho>2$. We shall need the following auxiliary equation: $$\label{f2_42}
( e^{-x} y_n(\rho+1;x) )' = e^{-x} y_n(\rho;x),\qquad n\in\mathbb{Z}_+,$$ with an unknown $y_n(\rho+1;x)$. Equation (\[f2\_42\]) has a unique $n$-th degree polynomial solution. This can be verified comparing the coefficients of polynomials, in the same manner as for equation (\[f2\_27\]). It is not easy to solve the corresponding recurrence relation in this case, but the existence and the uniqueness of a $n$-th degree polynomial solution is obvious.
Integrating relation (\[f2\_42\]) from $t$ to $b$ we get $$e^{-b} y_n(\rho+1;b) - e^{-t} y_n(\rho+1;t) = \int_t^b e^{-x} y_n(\rho;x) dx.$$ Passing to the limit as $b\rightarrow +\infty$ we get $$\label{f2_44}
y_n(\rho+1;t) = - e^t \int_t^\infty e^{-x} y_n(\rho;x) dx.$$ By (\[f2\_20\]),(\[f2\_42\]) we may write: $$e^{-x} x^n = ( e^{-x} y_n(\rho;x) )^{ (\rho) } = ( e^{-x} y_n(\rho+1;x) )^{ (\rho+1) }.$$ Therefore $y_n(\rho+1;\cdot)$ given by (\[f2\_44\]) is a required polynomial solution of (\[f2\_20\]) for $\rho+1$.
Equation (\[f2\_44\]) shows how to construct polynomial solutions step by step for $\rho = 1,2,...$. However, we are interested to get an explicit representation for every $y_n(\rho;x)$. Let $$\label{f2_46}
y_n(\rho;x) = \sum_{j=0}^n d_j(\rho) \frac{x^j}{j!},\qquad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N},$$ with some unknown complex numbers $d_j(\rho)$. By (\[f2\_44\]),(\[f2\_38\]),(\[f2\_35\]) we may write $$y_n(\rho+1;t) = - e^t \sum_{j=0}^n d_j(\rho) \frac{1}{j!} \int_t^\infty e^{-x} x^j dx =
\sum_{j=0}^n d_j(\rho) \frac{1}{j!} y^j(1;t) =$$ $$\label{f2_50}
= -\sum_{j=0}^n \sum_{k=0}^j d_j(\rho) \frac{x^k}{k!},\qquad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N}.$$ Changing the order of summation in (\[f2\_50\]) we write: $$y_n(\rho+1;t) = -\sum_{k=0}^n \sum_{j=k}^n d_j(\rho) \frac{x^k}{k!}.$$ Therefore $$\label{f2_52}
d_k(\rho+1) = -\sum_{j=k}^n d_j(\rho),\qquad j=0,1,...,n;\ \rho\in\mathbb{N}.$$ Relation (\[f2\_52\]) can be written in a matrix form for the vectors of coefficients $\vec d(\rho) := (d_0(\rho),...,d_n(\rho))^T$, and a $(n+1)\times(n+1)$ upper-diagonal Toeplitz matrix $T$, having all nonzero elements equal to $1$: $$\label{f2_54}
\vec d(\rho+1) = - T \vec d(\rho),\qquad \rho\in\mathbb{N}.$$ Therefore $$\label{f2_56}
\vec d(\rho) = (-1)^\rho T^\rho (0,...,0,1)^T,\qquad \rho\in\mathbb{N}.$$ Applying the Riesz calculus for evaluating $T^\rho$, one obtains the following solution: $$\label{f2_58}
d_k(\rho) = (-1)^\rho \left( \begin{array}{cc} n-k+\rho-1 \\
n-k \end{array} \right),\qquad k=0,1,...,n;\ n\in\mathbb{Z}_+,\ \rho\in\mathbb{N}.$$ We shall omit the details of calculating the resolvent $(T-\lambda E)^{-1}$. We only notice that it was convenient to subtract the subsequent rows when solving the linear system of equations $(T-\lambda E) f = (0,...,0,1)^T$. It can be directly verified that the resulting expression (\[f2\_58\]) obeys (\[f2\_52\]), by using the Pascal identity and the induction argument.
Thus, we have obtained the following representation for $y_n$: $$\label{f2_60}
y_n(\rho;x) = (-1)^\rho \sum_{j=0}^n \left( \begin{array}{cc} n-k+\rho-1 \\
n-k \end{array} \right)
\frac{x^j}{j!},\qquad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N}.$$
\[t2\_1\] Let $y_n(\rho;x)$ be polynomials given by relation (\[f2\_60\]) ($\rho\in\mathbb{N}$, $n\in\mathbb{Z}_+$). They have the following properties:
- Polynomials $y_n(\rho;x)$ admit the following representation: $$\label{f2_62}
y_n(\rho;x) = \frac{ (-1)^\rho }{n!} x^n {}_2 F_0 \left(-n,\rho;-;-\frac{1}{x}\right),\quad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N};\
x\in\mathbb{C}\backslash\{ 0 \}.$$
- Polynomials $y(x) = y_n(\rho;x)$ satisfy the following differential equation: $$\label{f2_64}
x
\sum_{k=0}^\rho (-1)^{\rho -k} \left( \begin{array}{cc} \rho \\
k \end{array} \right)
y^{ (k+1) }(x)
-
n
\sum_{k=0}^\rho (-1)^{\rho -k} \left( \begin{array}{cc} \rho \\
k \end{array} \right)
y^{ (k) }(x)
= 0.$$
- Polynomials $y(x) = y_n(\rho;x)$ obey the following differential equation: $$\label{f2_65}
x y''(x) - (x+\rho-1) y'(x) - n
\left[
y'(x) - y(x)
\right]
= 0.$$
- Polynomials $y_n(x) = y_n(\rho;x)$ are Sobolev orthogonal polynomials on $\mathbb{T}$: $$\label{f2_67}
\int_{\mathbb{T}} \left( y_n(z), y_n'(z),..., y_n^{(\rho)}(z) \right) M \overline{
\left( \begin{array}{cccc} y_m(z) \\
y_m'(z) \\
\vdots \\
y_m^{(\rho)}(z) \end{array} \right)
}
d\mu_0 = \delta_{n,m},\qquad n,m\in\mathbb{Z}_+,$$ where $M$ is given by (\[f1\_35\]).
**Proof.** $(a)$: It is readily checked that the reversed polynomial for $y_n$ is given by $$y_n^*(\rho;x) =
\frac{ (-1)^\rho }{n!} x^n {}_2 F_0 \left(-n,\rho;-;-x\right),\qquad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N},$$ and relation (\[f2\_62\]) follows.
$(b)$: Substitute for $x^n$ from (\[f2\_25\]) into the following identity: $$x (x^n)' = n x^n.$$
$(c)$: Hypergeometric polynomials $$\label{f2_68}
u = u_n(z) := F_0 \left(-n,\rho;-;z\right),\qquad n\in\mathbb{Z}_+,\ \rho\in\mathbb{N},$$ satisfy the following differential equation: $$\label{f2_70}
z(-n+\theta)(\rho+\theta) u - \theta u = 0,$$ where $\theta = z\frac{d}{dz}$. The differential equation for the generalized hypergeometric function ${}_p F_q$ is usually written when $p,q\geq 1$. However, the arguments in [@cit_5150_Rainville p. 75] can be applied in the case $q=0$ as well. Then for $z\not=0$ we may write $$\label{f2_72}
z^2 u''(z) +(\rho+1)z u'(z)- n(zu'(z)+\rho u(z)) - u'(z) = 0.$$ Observe that $$\label{f2_73}
u_n(z) = \frac{n!}{(-1)^{n+\rho}} z^n y_n(\rho;-\frac{1}{z}).$$
Calculating the derivatives $u_n',u_n''$ and inserting them into relation (\[f2\_72\]), after some algebraic simplifications, we get relation (\[f2\_65\]).
$(d)$: This follows from our motivation and relation (\[f1\_30\]) in the Introduction. $\Box$
In order to obtain a recurrence relation for polynomials $y_n(\rho;x)$ we shall apply Fasenmeier’s method ([@cit_5150_Rainville]) to hypergeometric polynomials $u_n(z)$ from (\[f2\_68\]). In the following considerations, we shall admit for $\rho$ not only positive integer values but $\rho>0$ as well. We shall express $u_n,u_{n-1},u_{n-2},zu_n(z),zu_{n-1}(z)$, using $u_{n+1}(z)$. Choose and fix an arbitrary integer $n$ greater or equal to $2$. We may write $$u_{n+1}(z) = \sum_{k=0}^\infty (-n-1)_k (\rho)_k \frac{z^k}{k!} = \sum_{k=0}^\infty \varepsilon_{n+1}(k),$$ where $\varepsilon_{n+1}(k) = \varepsilon_{n+1}(z;\rho;k) := (-n-1)_k (\rho)_k \frac{z^k}{k!}$. Using $$(-n)_k = (-n-1)_k \frac{ (n+1-k) }{ (n+1) },\qquad k\in\mathbb{Z}_+,$$ $$(-n+1)_k = (-n-1)_k \frac{ (n+1-k)(n-k) }{ (n+1)n },\qquad k\in\mathbb{Z}_+,$$ and similar relations we obtain that $$\label{f2_74}
u_n(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{ (n+1-k) }{ (n+1) },$$ $$\label{f2_76}
u_{n-1}(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{ (n+1-k)(n-k) }{ (n+1)n },$$ $$\label{f2_78}
u_{n-2}(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{ (n+1-k)(n-k)(n-1-k) }{ (n+1)n(n-1) },$$ $$\label{f2_80}
z u_n(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{ (-k) }{ (n+1)(\rho+k-1) },\quad \rho\not=1,$$ $$\label{f2_82}
z u_{n-1}(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{ (-k)(n+1-k) }{ n(n+1)(\rho+k-1) },\quad \rho\not=1.$$ We now assume that $\rho\not=1$. Consider the following expression $R_n(z)$: $$R_n(z) := \varphi_1 u_{n-1}(z) + \varphi_2 u_{n}(z) + \varphi_3 u_{n+1}(z) +
\varphi_4 z u_{n}(z) +$$ $$\label{f2_84}
+ \varphi_5 z u_{n-1}(z) + \varphi_6 u_{n-2}(z),\qquad \varphi_k\in\mathbb{C}.$$ We intend to choose parameters $\varphi_k$ (depending on the chosen $n$) in such a way that $R_n(z)= 0$, $\forall z\in\mathbb{C}$ . Substitute above expressions for $u_{n-2},u_{n-1},u_n,zu_n,zu_{n-1}$ into (\[f2\_84\]) to get $$R_n(z) = \sum_{k=0}^\infty \varepsilon_{n+1}(k) \frac{1}{ (n-1)n(n+1)(\rho+k-1) } I_{n,k},$$ where $$I_{n,k} =
\varphi_1 (n-k)(n+1-k) (n-1)(\rho+k-1) +
\varphi_2 (n+1-k) (n-1)n(\rho+k-1) +$$ $$+ \varphi_3 (n-1)n(n+1)(\rho+k-1) + \varphi_4 (-1) k (n-1)n + \varphi_5 (n+1-k) (-1)k (n-1) +$$ $$\label{f2_86}
+ \varphi_6 (n+1-k)(n-k)(n-1-k) (\rho+k-1).$$ Observe that $I_{n,k}$ is a polynomial of degree $\leq 4$. Therefore we may check that $I_{n,k}=0$ for some distinct five values of $k$ to get $R_n(z)\equiv 0$. This is a crucial point in Fasenmeier’s method.
We choose $k=-\rho+1; n+1; n; n-1; 0$. After some obvious simplifications we get the following five equations: $$\label{f2_88}
\varphi_5 = -\frac{ \varphi_4 }{ n+\rho },$$ $$\label{f2_90}
\varphi_3 = \frac{ \varphi_4 }{ n+\rho },$$ $$\label{f2_92}
\varphi_2 (\rho+n-1) + \varphi_3 (n+1)(\rho+n-1) + \varphi_4 (-1)n - \varphi_5 = 0,$$ $$\varphi_1 2(\rho+n-2) + \varphi_2 2n (\rho+n-2) + \varphi_3 n(n+1) (\rho+n-2) -$$ $$\label{f2_94}
- \varphi_4 (n-1)n - \varphi_5 2(n-1) = 0,$$ $$\label{f2_96}
\varphi_1 + \varphi_2 + \varphi_3 + \varphi_6 = 0.$$ Set $\varphi_4 = n+\rho$. Then $$\varphi_5 = -1,\quad \varphi_3 = 1.$$ By (\[f2\_92\]) we get $$\varphi_2 = \frac{ n-\rho }{ n+\rho-1 }.$$ By (\[f2\_94\]) we obtain that $$\varphi_1 = -\frac{ n(n+1) }{ 2 } -\frac{ (n-\rho)n }{ n+\rho-1 } -
\frac{ n-1 }{ n+\rho-2 } + \frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) }.$$ Finally, by (\[f2\_96\]) we conclude that $$\varphi_6 = \frac{ n(n+1) }{ 2 } + \frac{ (n-\rho)(n-1) }{ n+\rho-1 } +
\frac{ n-1 }{ n+\rho-2 } -\frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) } - 1.$$ Consequently, polynomials $u_n(z)$ satisfy the following relation: $$\left(
-\frac{ n(n+1) }{ 2 } -\frac{ (n-\rho)n }{ n+\rho-1 } -
\frac{ n-1 }{ n+\rho-2 } + \frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) }
\right)
u_{n-1}(z) +$$ $$+ \frac{ n-\rho }{ n+\rho-1 }
u_{n}(z) + u_{n+1}(z) +
(n+\rho)
z u_{n}(z) - z u_{n-1}(z) +$$ $$+
\left(
\frac{ n(n+1) }{ 2 } + \frac{ (n-\rho)(n-1) }{ n+\rho-1 } +
\frac{ n-1 }{ n+\rho-2 } -\frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) } - 1
\right) *$$ $$\label{f2_98}
* u_{n-2}(z),\qquad n=2,3,...; \rho>0,\rho\not=1.$$
\[t2\_2\] Let $y_n(x) = y_n(\rho;x)$ be polynomials from relation (\[f2\_60\]) with $\rho\in\mathbb{N}\backslash\{ 1\}$. They satisfy the following recurrence relation: $$\left(
-\frac{ n(n+1) }{ 2 } -\frac{ (n-\rho)n }{ n+\rho-1 } -
\frac{ n-1 }{ n+\rho-2 } + \frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) }
\right)
x^2 *$$ $$* (n-1) y_{n-1}(x) +
\frac{ n-\rho }{ n+\rho-1 }
x (n-1)n y_n(x) +
(n-1)n(n+1) y_{n+1}(x) +$$ $$+
\left(
\frac{ n(n+1) }{ 2 } + \frac{ (n-\rho)(n-1) }{ n+\rho-1 } +
\frac{ n-1 }{ n+\rho-2 } -\frac{ (n+\rho) (n-1)n }{ 2(n+\rho-2) } - 1
\right) x^3 *$$ $$* y_{n-2}(x) -
(n+\rho) (n-1)n y_n(x) +
x (n-1) y_{n-1}(x) = 0,$$ $$\label{f2_100}
n=2,3,....$$
**Proof.** Use relations (\[f2\_73\]) and (\[f2\_98\]). $\Box$
Observe that $y_1(\rho;x) = (-1)^\rho (x+\rho)$. Thus $y_1(1;x)$ has its root on the unit circle, while the roots of $y_1(\rho;x)$, for $\rho>1$, are outside $\mathbb{T}$. Consequently, polynomials $y_n(\rho;x)$ are not orthogonal on the unit circle with respect to a scalar measure.
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[ **On a family of hypergeometric Sobolev orthogonal polynomials on the unit circle.**]{}
[**S.M. Zagorodnyuk**]{}
In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$. Differential equations of orders $\rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.
V. N. Karazin Kharkiv National University School of Mathematics and Computer Sciences Department of Higher Mathematics and Informatics Svobody Square 4, 61022, Kharkiv, Ukraine
Sergey.M.Zagorodnyuk@gmail.com; Sergey.M.Zagorodnyuk@univer.kharkov.ua
|
---
abstract: 'We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large $N$ matrix model. Using a related function $\Xi(z)$ we develop an analog between this function and the Airy function $Ai(z)$ of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $\Xi(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we develop a Kontsevich matrix model which describes $n$ FZZT branes. The Kontsevich model associated with the $\Xi(z)$ function is given by a superposition of Liouville type matrix models that have been used to describe matrix model instantons.'
author:
- |
Michael McGuigan\
Brookhaven National Laboratory\
Upton NY 11973\
mcguigan@bnl.gov
title: ' Riemann Hypothesis, Matrix/Gravity Correspondence and FZZT Brane Partition Functions'
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Introduction
============
It is an old idea that if a Hermitian operator can be found which has eigenvalues of the form $\lambda_n = -i(\rho_n - \frac{1}{2})$ where $\rho_n$ are the nontrivial zeros of the zeta function then the Riemann hypothesis would be true. This would follow because the eigenvalues of a Hermitian operator are real. No such operator has yet been found however.
In the Heisenberg matrix formulation of quantum mechanics [@Heisenberg] one represents observables with infinite matrices which are Hermitian. The eigenvalues of the matrix are what are measured in an experiment and hence are real. The infinite matrix can be constructed by forming an $N \times N$ matrix and taking the large $N$ limit. No such large $N$ matrix whose eigenvalues are related to the Riemann zeros has been found.
One can also consider theories called matrix models in which the dynamical variables are such large $N$ matrices [@Wigner][@Dyson:1972tm][@Martinec:2004td]. There is a remarkable correspondence between such matrix theories and continuous theories describing a quantum theory of world sheet gravity and low dimensional string theory. In such a correspondence invariants of the matrix theory are related to geometric observables in the world-sheet gravity usually through an integral transform [@Martinec:2004td]. This integral transform takes one from a variable in the expansion of the characteristic polynomial of the large $N$ Hermitian matrix description to a Liouville variable describing the size of the string or of a of a macroscopic loop of 2d gravity in the continuum description. The correspondence arises because Feynman graphs in the matrix theory description can yield discrete representation of surfaces which become continuous as one takes $N$ to infinity [@'tHooft:1973jz].
In this paper we interpret the Riemann zeta function as being related to a particular observable in the matrix/gravity correspondence namely the FZZT brane partition function of a matrix model and interpret it’s master matrix as the Riemann operator. The potential for this matrix model is more complicated than most of the ones considered in the literature. Nevertheless the techniques of simpler matrix theories can be applied to this case as well. This paper is organized as follows. In section 2 we discuss the master matrix approach to matrix models. We discuss some of the conceptual advantages of the approach as well as the difficulties. In section 3 we discuss introduce the FZZT brane partition function from the matrix model point of view. In section 4 we determine the Kontsevich integrand associated with the Riemann zeta function and develop an analogy between the Riemann zeta function and the Airy function which is the FZZT partition function of the $(2,1)$ minimal matrix model. In section 5 we discuss how to approximate the matrix model associated with Riemann zeta function using the generalized $(p,1)$ matrix model for large $p$ whose FZZT partition function is a generalized Airy integral. In section 6 we review the main conclusions of the paper.
Master matrix
=============
If one can find a special infinite Hermitian matrix $M_0$ such that: $$\Xi (z ) = \det (M_0 - z I)$$ where $$\Xi (z ) = \zeta (i z + \frac{1}{2})\Gamma (\frac{{z }}{2} + \frac{1}{4})\pi ^{ - 1/4} \pi ^{ - iz /2} (
- \frac{{z ^2 }}{2} - \frac{1}{8})$$ then the Riemann hypothesis would be true. This is because this function can be written in product form as: $$\Xi (z ) = \frac{1}{2}\prod\limits_n {(1 - \frac{{iz + 1/2}}{{\rho _n }}} )$$ The eigenvalues of the Hermitian matrix $M_0$ are denoted by $\lambda_n$ and are related to the Riemann zeros via $\rho_n = i\lambda_n + 1/2$. Then the product becomes: $$\Xi (z ) = \frac{1}{2}\prod\limits_n {(1 - \frac{{iz + 1/2}}{{i\lambda _n + 1/2}}} ) = \frac{1}{2}\prod\limits_n {\frac{{\lambda _n - z }}{{\lambda _n - i/2}}}$$ This vanishes at the values $\lambda_n$ just as the formal determinant expression. The $\lambda_n$ are real if the matrix $M_0$ is Hermitian and thus the Riemann Hypothesis would be true. Unfortunately just as for the Riemann operator referred to above no such infinite matrix $M_0$ has ever been constructed.
The difficulty in constructing $M_0$ is somewhat similar to the difficulty in constructing a master field or master matrix in large $N$ field theory of matrix theory [@Gopakumar:1994iq][@Gopakumar:1995bk]. A master field or master matrix is a special large $N$ matrix such that statistical averages of an observable can be computed by simply evaluating the observable on the the special large $N$ matrix. The reason that a master matrix exists is because at large $N$ expectation values factorize as: $$\begin{array}{l}
\left\langle {O_1 O_2 } \right\rangle = \left\langle {O_1 } \right\rangle \left\langle {O_2 } \right\rangle + O(1/N^2 ) \\
\left\langle {(O - \left\langle O \right\rangle )^2 } \right\rangle = \left\langle {O^2 } \right\rangle - \left\langle O \right\rangle ^2 = O(1/N^2 ) \\
\end{array}$$ where: $$\left\langle {O } \right\rangle = \int DM O(M) e^{-V(M)}$$ and $V(M)$ is a matrix potential. Thus variances vanish so the observable’s value is localized on a particular matrix as $N \to \infty$ just as particle trajectories are localized on classical solutions as $\hbar$ goes to zero. Once such a master field is found the above observables are simply given by: $$\left\langle {O } \right\rangle = O(M_0)$$ There are several such observables in matrix theory. We discuss some of these in the next section.
For a general matrix model with potential $V(M)$ the master matrix can be written [@Gopakumar:1994iq][@Gopakumar:1995bk]: $$M_0 = S^{ - 1} TS = S^{ - 1} (a + \sum\limits_{n = 0}^\infty {t_n a^{ + n} )S}$$ where the similarity transformation $S$ is defined so that $M_0$ is Hermitian and the operators $a,a^+$ obey $[a,a^+] = I$. One can expand the master matrix as a function of the Hermitian operator $\hat x = a + a^{+}$ as: $$M_0 (\hat x) = g_1 \hat x + g_2 \hat x^2 + \ldots$$ One can also define an associated complex function: $$M_0 (y) = \frac{1}{y} + \sum\limits_{n = 0}^\infty {t_n y^n }$$ as well as a conjugate matrix $P_0$ that satisfies: $$[P_0 ,M_0 ] = I$$ The Master matrix can be determined from the equation [@Gopakumar:1994iq][@Gopakumar:1995bk]: $$(V'(M_0 (\hat x)) + 2P_0)\left| 0 \right\rangle = 0$$ Here $\left| 0 \right\rangle$ is the vacuum state annihilated by $a$. The master matrix is closely connected with the resolvent $R(z)$ and eigenvalue density $\rho(x)$ through: $$R(z) = Tr(\frac{1}{{z - M_0 }}) = \int {dx\frac{{\rho (x)}}{{z - x}}} = - \oint\limits_C {\frac{{dw}}{{2\pi i}}} \log (z - M_0 (w))$$ The associated function $M_0(y)$ obeys the relation: $$R(M_0 (y)) = M_0 (R(y)) = y$$ The function $yM_0(y)$ is the generating functional of connected Green functions for the generalized matrix model. While the concept of the master matrix is appealing, to construct the master matrix explicitly is equivalent to finding all the connected Green functions which amounts to solving the theory. This can be done for the potential $V(M) = Tr(M^2)$ but for the general matrix model is quite difficult. In the next three sections we turn to other methods of dealing with the generalized matrix model which are somewhat more tractable and apply them to the interpretation of the zeta function.
FZZT brane
==========
One observable of matrix models is the exponentiated macroscopic loop or FZZT brane partition function [@Fateev:2000ik][@Teschner:2000md][@Giusto:2004mt][@Ellwood:2005nt]. This is given by: $$B(z ) = \det (M - z I)$$ This is the characteristic polynomial associated with the matrix $M$. It’s argument $z$ can be complex. In the context of the Riemann zeta function $\zeta(s)$ the variable is related to the usual argument of the zeta function by $s = iz + \frac{1}{2}$. Another observable is the macroscopic loop which is the transform of the Wheeler-DeWitt wave function defined on the gravity side of the correspondence [@Klebanov:2003wg]. $$W(z ) = - Tr\log (M - z I) = \lim _{\varepsilon \to 0}
(\int\limits_\varepsilon ^\infty {\frac{{d\ell }}{\ell }} Tr(e^{\ell (-z I + M)} ) + \log \varepsilon )$$ where $\epsilon$ is a UV cutoff.
The resolvent observable mentioned above is defined by: $$R(z ) = \frac{{\partial W(z )}}{{\partial z }} = Tr(\frac{1}{{M - z I}})$$ Finally one has the inverse determinant observable defined in [@Klebanov:2003wg].
If a special master matrix $M_0$ can be found then expectation values such as $$\left\langle {B(z )} \right\rangle = \left\langle {\left. {\det (M - z I)} \right\rangle } \right. = \int {DM\det (M - z I)e^{ - V(M)} } = \det (M_0 - z I)$$ reduce to evaluating the observable at $M_0$. In the context of the $\Xi(z)$ function the desired relation is of the form: $$\Xi (z ) = \det (M_0 - z I) = \left\langle {B(z )} \right\rangle = \left\langle {\left. {\det (M - z I)} \right\rangle } \right. = \int {DM\det (M - z I)e^{ - V(M)} }$$
Some matrix potentials that have been considered are $$V(M) = Tr(M^2)$$ which describes 2d topological gravity or the (2,1) minimal string theory [@Witten:1989ig] [@Maldacena:2004sn] [@Kutasov:2004fg]. A quartic potential: $$V(M) = Tr( - M^2 + gM^4 )$$ is used to describe minimal superstring string theory [@Seiberg:2004ei][@Seiberg:2003nm][@Fukuma:2006ny][@Johnson:2003hy]. A more complicated matrix potential is $$V(M) = - Tr(M + \log (I - M)) = \sum\limits_{m = 2}^\infty {\frac{1}{m}Tr(M^m
)}$$ which defines the Penner matrix model [@Penner][@Distler:1990pg][@Distler:1990mt][@Imbimbo:1995ns][@Matsuo:2005nw] and is used to compute the Euler characteristic of the moduli space of Riemann surfaces.
Another matrix model that has been introduced is the Liouville matrix model [@Mukhi:2003sz][@Imbimbo:1995yv] with potential given by: $$V(M)= Tr (\alpha M + \mu e^M)$$ with cosmological constant $\mu$ so that: $$e^{-V(M)} = e^{-\alpha Tr M}e^{-\mu Tr e^M}$$ In this paper we will encounter the matrix potential determined by: $$e^{-U(M)} = \sum\limits_{q = 1}^\infty {(q^4 \pi ^2 e^{2TrM} - \frac{3}{2}} q^2 \pi e^{TrM})e^{ - q^2 \pi Tr( e^M) }$$ The partition function for this matrix model can be seen as a superposition of partition functions of Liouville matrix models with cosmological constants of the form. $$\mu = q^2\pi$$ for integer $q$. The origin of this particular matrix model and it’s relation to the zeta function will be discussed in the next section.
Kontsevich integrand
====================
To see how the matrix potential (3.1) arises it is helpful to consider how the coefficients of the characteristic polynomial observable $B(z)$ can be determined by expanding as a series in $z$. If the function $\Xi(z)$ is interpreted as a characteristic polynomial then one can obtain these coefficients from the expansion: $$\Xi (z ) = \sum\limits_{n = 0}^\infty {a_{2n} \frac{{( - 1)^n }}{{\left(
{2n} \right)!}}} z ^{2n}$$ where $$a_{2n} = 4\int\limits_1^\infty {d\ell (\ell ^{ - 1/4} f(\ell )(\frac{1}{2}\log \ell )^{2n} } )$$ and $$f(\ell ) = \sum\limits_{q = 1}^\infty {(q^4 \pi ^2 \ell } - \frac{3}{2}q^2 \pi )\ell ^{1/2} e^{ - q^2 \pi \ell }$$ Inserting the coefficients $a_{2n}$ into $\Xi(z)$ and summing over $n$ we can represent $\Xi(z)$ as an integral transform: $$\Xi (z ) = 4\int\limits_1^\infty {\frac{{d\ell }}{\ell }} \ell ^{(iz + 1/2)/2} \sum\limits_{q = 1}^\infty {(q^4 \pi ^2 \ell ^2 - \frac{3}{2}} q^2 \pi \ell )e^{ - q^2 \pi \ell } = 4\int\limits_1^\infty {\frac{{d\ell }}{\ell }}
\ell ^{(iz + 1/2)/2} \ell ^{1/2} f(\ell )$$ Defining the variable $\phi$ by $\ell = e^\phi$ we have: $$\Xi[z] = \int {d\phi e^{iz\phi} } \sum\limits_{k = 1}^\infty {(\pi ^2 k^4 }
e^{2\phi } - \frac{3}{2}\pi k^2 e^{\phi } )e^{ - \pi k^2 e^\phi }$$ which is a well known integral expression for the function $\Xi(z)$.
For the simple potential $V(M) = Tr(M^2)$ the exponentiated macroscopic loop observable (FZZT brane) can be computed. It is given by the Airy function [@Kutasov:2004fg]: $$Ai(z) = \int DM det(M-zI) e^{-Tr(M^2)}= \int d\phi e^{iz\phi + i\phi^3\frac{1}{3}}$$ Because this function is associated with an Hermitian matrix model it’s zeros are real. This is the analog of the Riemann hypothesis for $V(M) = Tr(M^2)$. The similarity between the integral representations of (4.1) and (4.2) suggest an analogy between the Airy and zeta functions.
To illustrate a comparison between the Airy function and the zeta function consider figures 1 and 2. The zeros disappear as one moves off the critical line which corresponds to $z$ real in both cases. This suggests that the zeta function corresponds to a Hermitian matrix model. Table 1 illustrates the comparison on both sides of the correspondence. The question mark indicates the (substantial) missing information involved in a matrix/gravity approach to the Riemann hypothesis.
Qualitative differences exist between between the functions $Ai(z)$ and $\Xi(z)$. The $Ai(z)$ function is exponentially decaying to the positive $z$ axis. This is a result of Stokes phenomena where an exponentially growing form of the Airy function is completely absent in the right $z$ axis. For the $\Xi(z)$ function one does not see exponentially decaying function in the positive $z$ axis. Instead one has identical behavior in the positive and negative $z$ axis. One way to see the difference is to use a Riemann-Hilbert Problem approach to both functions [@Its][@Konig][@Brezin][@Kitaev][@Strahov][@Gangardt]. In the case of the Airy function this leads to the differential equation [@Its]: $$Ai''(z) = zAi(z)$$ whereas in the case of the $\Xi(z)$ function one does not obtain a differential equation but a discrete equation [@Its] : $$\Xi (z) = \Xi ( - z)$$ Indeed it is known that the zeta function does not obey a finite order differential equation so this may be a possible explanation for the qualitative difference between the two functions. It would be interesting to explore further the differences between the two functions using the Riemann-Hilbert approach of and their interpretations as FZZT brane partition functions. .
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Observable General Airy Zeta
---------------------- ---------------- ----------------- ---------------------------------------------------------------------------------------------------------
Master Matrix $M_0$ $a + a^+$ ?
Potential $V(M)$ $Tr(M^2)$ $
\mathop {\lim }\limits_{p \to \infty } Tr(V_p (M) + \sum\limits_{k = 1}^{p - 2} {s_k V_k (M))}$
FZZT Brane $B(z)$ $Ai(z)$ $\zeta( iz +1/2)$
Macroscopic Loop $W(z)$ $\log{Ai(z)}$ $\log{\zeta(iz+1/2)}$
Kontsevich Integrand $e^{-U(\phi)}$ $e^{i\phi^3/3}$ $\sum\limits_{q = 1}^\infty(q^4 \pi ^2 e^{2\phi} - \frac{3}{2} q^2 \pi e^\phi ) \exp(-\pi q^2 e^\phi)$
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: Analogy between the Airy function and the Riemann zeta function. The quantities $V_k(M)$ and $s_k$ defined by a generalized $(p,1)$ matrix model in the following section. []{data-label="tab1"}
The integral representation of the Airy function has a matrix integral generalization. The matrix potential is defined from: $$e^{ - U(\Phi )} = e^{i\frac{1}{3}Tr(\Phi ^3 )}$$ The matrix generalized Airy function is given by: $$Ai(Z) = \int d\Phi e^{iTr(Z\Phi )}e^{-U(\Phi)}$$ In the above $\Phi$ and $Z$ are $n \times n$ matrices. The interpretation of this matrix integral is that it describes $n$ FZZT branes. The matrix $\Phi$ in the Kontsevich integrand is an effective degree of freedom describing open strings stretched between $n$ FZZT branes [@Kontsevich:1992ti][@Kharchev:1992dj][@Aganagic:2003qj][@Gaiotto:2003yb].
One can try to interpret the integrand of the $\Xi(z)$ function in a similar manner. In that case the analog of the potential defined by: $$e^{ - U(\Phi )} = \sum\limits_{k = 1}^\infty {(\pi ^2 k^4 } e^{2Tr\Phi } - \frac{3}{2}\pi k^2 e^{Tr\Phi } )e^{ - \pi k^2 Tre^\Phi }$$ and the analog of the the matrix integral describing $n$ FZZT branes is: $$\Xi[Z] = \int {D\Phi e^{iTr(Z\Phi )} } \sum\limits_{k = 1}^\infty {(\pi ^2 k^4 }
e^{2Tr\Phi } - \frac{3}{2}\pi k^2 e^{Tr\Phi } )e^{ - \pi k^2 Tre^\Phi }$$ This is the origin of the matrix model given by (3.1). As discussed in section 2 this can treated as a sum of Liouville type matrix models.
Relation to generalized $(p,1)$ matrix models
=============================================
The Airy function is the FZZT partition function for the $(2,1)$ minimal matrix model. In [@Hashimoto:2005bf] the FZZT partition function was given for the generalized $(p,1)$ minimal matrix model with parameters $s_k$. This theory has a characteristic polynomial or FZZT partition function given by: $$B(z) = \frac{1}{{2\pi }}\int {d\phi e^{iz\phi - \frac{1}{{p + 1}}(i\phi )^{p + 1} + \sum\nolimits_{k = 1}^{p - 2} {s_k \frac{1}{{k + 1}}(i\phi )^{k + 1} } } }$$ Unlike the $(2,1)$ matrix model the definition of the generalized $(p,1)$ matrix model requires a two matrix integral of the form [@Hashimoto:2005bf][@Daul:1993bg][@Kazakov:2004du][@Alexandrov:2003qk][@Kharchev:1991cu][@Alexandrov:2006sb][@Mironov:2005qn]: $$Z_{(p,1)} (g) = \int {DMDAe^{ - \frac{1}{g}(V(M + I) - AM)} }$$ Comparison with the integral representation of the $\Xi(z)$ function shows that a generalized matrix model for large $p$ can be constructed as an approximation. This can be compared with the formulas from the previous section to compute the corresponding coefficients $s_k$. One writes: $$\log \left( {\sum\limits_{k = 1}^\infty {(\pi ^2 k^4 } e^{2\phi } - \frac{3}{2}\pi k^2 e^\phi )e^{ - \pi k^2 e^\phi } } \right) = - \frac{1}{{p + 1}}(i\phi )^{p + 1} + \sum\nolimits_{k = 1}^{p - 2} {s_k \frac{1}{{k + 1}}(i\phi )^{k +
1} }$$ In the above formula the function on the left is expanded to order $p+1$ in the variable $\phi$. We denote this terminated expansion by $\Xi_p(z)$. Another way to compute the coefficients $s_k$is to differentiate the left hand side and set: $$s_k = \frac{{i^{ - (k + 1)} }}{{k!}}\left. {\partial _\phi ^k \log \left( {\sum\limits_{k = 1}^\infty {(\pi ^2 k^4 } e^{2\phi } - \frac{3}{2}\pi k^2 e^\phi
)e^{ - \pi k^2 e^\phi } } \right)} \right|_{\phi = 0}$$ From the integral representation one has: $$\begin{array}{l}
Q\Xi _p (z) = z\Xi _p (z) \\
P\Xi _p (z) = - \partial _z \Xi _p (z) \\
\end{array}$$ where: $$Q = (P^p + \sum\limits_{k = 0}^{p - 1} {s_k P^k } )$$ Inserting this operator into the above equation one has the generalization of the Airy equation given by: $$(P^p + \sum\limits_{k = 0}^{p - 1} {s_k P^k } )\Xi _p (z) = z\Xi _p (z)$$ To recover the equation for the full $\Xi(z)$ function one has to take $p$ to infinity which agrees with the fact that the zeta function does not obey a finite order differential equation.
Note that $z$ and $\phi$ are in some sense canonically conjugate [@Hashimoto:2005bf]. Denote the Fourier transform of the $\Xi(z)$ function as $\tilde \Xi(p)$ then: $$\Xi (z) = \int {d\phi e^{i\phi z} \tilde \Xi } (\phi )$$ The generalized Airy equation them becomes in Fourier space: $$(\phi ^p + \sum\limits_{k = 0}^{p - 1} {s_k \phi ^k } )\tilde \Xi _p (\phi ) =
Q\tilde \Xi _p (\phi )$$ This can be written: $$(U'(\phi ) - Q)\tilde \Xi (\phi ) = 0$$ where: $$e^{ - U(\phi )} = \sum\limits_{k = 1}^\infty {(\pi ^2 k^4 } e^{2\phi } - \frac{3}{2}\pi k^2 e^\phi )e^{ - \pi k^2 e^\phi }$$ Equation (5.2) is very similar to the equation for the master matrix (2.1). Indeed if we set: $$\begin{array}{l}
\phi = M_0 (y) \\
z = P_0 (y) \\
\end{array}$$ we see that $y$ can be thought of as coordinates of a parametrization of the Riemann surface $M_{p,1}$ which is determined from the $\phi$ and $z$ constraint $U'(\phi ) - z = 0$. If we make these variables into operators through: $$\begin{array}{l}
\hat \phi = \hat M_0 (a,a^ + ) \\
\hat z = \hat P_0 (a,a^ + ) \\
\end{array}$$ this classical surface is turned into a quantum Riemann surface similar to those studied using noncommunative geometry [@Connes:1994yd].
Once one has obtained the coefficients $s_k$ one can define matrix potential associated with a finite $N$ theory as [@Hashimoto:2005bf]: $$V(M) = \mathop {\lim }\limits_{p \to \infty } Tr(V_p (M) + \sum\limits_{k = 1}^{p - 2} {s_k V_k (M))}$$ where: $$V_k (M) = \sum\limits_{j = 1}^p {\frac{1}{j}(M^j } - I)$$ This is the matrix potential of Table 1 in the previous section.
A set of orthogonal polynomials with this matrix potential through the integral equation: $$B_n (z) = \frac{{n!}}{{2\pi i}}\oint {e^{ - \mathop {\lim }\limits_{p \to \infty } (V_p (y + 1) + \sum\limits_{k = 1}^{p - 2} {s_k V_k (y + 1))} + 2zy} \frac{1}{{y^{n + 1} }}} dy$$ Or equivalently though the generating function definition: $$e^{ - \mathop {\lim }\limits_{p \to \infty } (V_p (y + 1) + \sum\limits_{k =
1}^{p - 2} {s_k V_k (y + 1))} + 2zy} = \sum\limits_{n = 0}^\infty {B_n (z)\frac{{y^n }}{{n!}}}$$ These are the generalizations of the integral and generating function definitions of the Hermite polynomials associated with the $(2,1)$ minimal model.
We note that some other entire functions can be treated in a similar manner. For example the reciprocal factorial function $\frac{1}{\Pi(z)} = \frac{1}{\Gamma(z+1)}$ has a product representation: $$\frac{1}{\Pi (z)} = e^{\gamma z} \prod\limits_{n = 1}^\infty {(1 + \frac{z}{n}} )e^{ - z/n}$$ and integral representation: $$\frac{1}{\Pi (z)} = \int {d\phi e^{iz\phi } e^{ - e^\phi } }$$ The identity $$\frac{1}{\Pi (z - 1)} = z\frac{1}{{\Pi (z)}}$$ implies the equation $$e^{ - \partial _z } \frac{1}{\Pi (z)} = z\frac{1}{{\Pi (z)}}$$ or: $$(e^P - z)\frac{1}{\Pi (z)} = 0$$ This is the analog of equations (5.1) for the $\Xi(z)$ function. The product representation shows that the zeros of the inverse factorial function are of the form $\lambda _n = - 1, - 2, - 3, \ldots$. The inverse factorial function is similar to the zeta function in that it does not obey a finite order differential equation. It is similar to the Airy function in that it has all it’s zeros on the negative real axis. It differs from both the Airy and zeta function in that it’s zeros are of a simple form namely the negative integers. The integral representation of the reciprocal factorial function seems related to the Liouville matrix model with Kontsevich integrand $e^{-U(\phi)} = e^{-e^\phi}$ and generalized $(p,1)$ matrix model with $s_k= \frac{1}{k!}$. The matrix integral representation of the Gamma function in terms of the Liouville matrix model has been discussed in [@Mukhi:2003sz].
Most of our analysis has centered on the matrix side of the matrix/gravity correspondence. The gravity side is related through an integral transform. For example the macroscopic loop observable associated with the Riemann zeta function is given by: $$\log \zeta (iz + 1/2) = \int\limits_0^\infty {\ell ^{ - iz - 1/2} W(\ell )d\ell }$$ In terms of the $\lambda_n$ this observable takes the form [@Edwards]: $$W(\ell ) = \frac{1}{{\log \ell }} - \sum\limits_n {\frac{{2\cos (\lambda _n \log \ell )}}{{\ell ^{1/2} \log \ell }}} - \frac{1}{{\ell (\ell ^2 - 1)\log \ell }}$$ The indefinite integral of this Wheeler-DeWitt wave function is connected to the prime numbers $p$ through: $$\int\limits_2^\ell {W(\ell ')d\ell '} = \frac{1}{2}(\sum\limits_{p^n < x}
{\frac{1}{n}} + \sum\limits_{p^n \le \ell} {\frac{1}{n}} )$$ The FZZT brane partition function can also be represented by prime numbers as: $$\log \zeta (iz + 1/2) = \sum\limits_p {\sum\limits_n {\frac{1}{n}p^{ - n(iz + 1/2)} } }$$ Both of the above formulas follow from the Euler product formula of the zeta function. Much of the physical intuition about the meaning of the FZZT brane and the Wheeler-DeWitt wave function occurs on the gravity side of the correspondence. Thus the connection of number theory and gravity in this context is quite intriguing.
Finally to approach the generalized Riemann hypothesis using the\
matrix/gravity correspondence one can replace the Kontsevich integrand $e^{-U(\phi)}$ with a modular function. Indeed such modular functions already arise in the matrix/gravity $CFT_2/AdS_3$ correspondence between two dimensional conformal field theory and three dimensional gravity with negative cosmological constant [@Witten:2007kt].
Conclusion
==========
In this paper we have examined the Riemann zeta function as a FZZT brane partition function involved in matrix models. The FZZT description gives rise to the physical interpretation of the Riemann hypothesis, that the $\Xi(z)$ is an entire function and has zeros on the critical line with $z$ on the real axis (this corresponds to the $Re(s) = \frac{1}{2}$). The zeros are interpreted as eigenvalues of the master matrix. The macroscopic loop observable and resolvent also have physical interpretations in terms of the matrix model. In the gauge gravity correspondence the macroscopic loop is identified with the Wheeler-DeWitt wave function of the 2d world sheet gravity. The variable $z$ is identified with the boundary cosmological constant in the 2d gravity. The matrix gravity correspondence is the mapping between the matrix quantities and the 2d gravity computations. In a string theory context these in turn describe target space time processes.
The Kontsevich integrand was identified using the Fourier transform of the $\Xi(z)$ function. Replacing the $z$ variable by $n\times n$ matrix $Z$ and the Kontsevich integrand by a matrix integrand one obtains representation of a matrix model describing $n$ FZZT branes. The Kontsevich integrand is given by a superposition Liouville matrix models that have been used to represent instanton matrix models for the $c=1$ string.
Some long standing issues are indicated in Table 1. To identify and interpret the Master matrix associated with the Riemann zeta function.
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