††institutetext: Department of Mathematics, King’s College London,
The Strand, London WC2R 2LS, U.K.

# From multi-gravitons to Black holes: The role of complex saddles

Alejandro Cabo-Bizet<EMAIL_ADDRESS>
###### Abstract

By applying the Atiyah-Bott-Berline-Vergne equivariant integration formula
upon double dimensional integrals, we find a way to compute the matrix
integral representations of $4d$ $\mathcal{N}=1$ superconformal indices. The
final formula allows us to easily extract analytic results in the large-rank
expansion of certain theories. As an example, we compute the leading one-loop
corrections to the effective action of the known complex saddles in those
theories. For a superconformal index of $SU(N)$ $\mathcal{N}=4$ SYM, we use
the equivariant integration formula and the Picard-Lefschetz method to show
that at large enough values of $N$, only two, among the known complex saddles,
dominate the counting of  _large operators_ i.e. of operators with charges of
order $N^{2}$. Contributions from other known complex saddles are present, but
we show they are exponentially suppressed in that range of charges; the
smaller the charges the less suppressed they are, and eventually, to count
_small operators_ i.e. operators with charges smaller than $N^{\frac{2}{3}}$,
like multi-gravitons, they can not be neglected.

##  1 Introduction and summary

Following the general principles of string theory/gauge theory conjecture
Maldacena:1997re2 , and inspired by previous seminal results in the context of
string theory Strominger:1996sh ; a fresh perspective has emerged in the last
few years regarding the microscopic structure of supersymmetric $AdS_{d+1}$
black holes in $d=3$, starting with Benini:2015eyy , and $d=4$, starting with
Cabo-Bizet:2018ehj ; Choi:2018hmj ; Benini:2018ywd .

Regardless of which of the current approaches one chooses as tool to study the
asymptotic growth of BPS operators at large rank, via the use of matrix
integrals Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  Benini:2018ywd ;
Benini:2018mlo , there seems to be convincing evidence that complex eigenvalue
configurations play an important role in understanding the problem. Currently,
there are two approaches that work at large-$N$ and generic values of BPS
charges. They are consistent with each other, and in important ways, also with
the predictions coming from the gravitational side of the duality Cabo-
Bizet:2018ehj , they both include two complex eigenvalue configurations, whose
effective actions are entropy functionals dual to the Bekenstein-Hawking
entropy of the corresponding $AdS_{5}$ black holes Gutowski:2004yv ;
Cvetic:2004hs ; Cvetic:2004ny ; Kunduri:2005zg ; Kunduri:2006ek ; Silva:2006xv
; Hosseini:2017mds . These two approaches are $a)$ the Bethe Ansatz formula of
Benini:2018ywd ; Benini:2018mlo ; Closset:2017bse , and $b)$ the saddle-point
analysis of Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr . 111A very interesting
third perspective has been put forward recently Goldstein:2020yvj .  This
paper completes the approach of Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  by
promoting the saddle point approximation to an exact formula at finite values
of $N$. The problem that remains is that for the theories we have analysed so
far, it is hard to evaluate the formula at finite values of $N$. However, at
large $N$ the formula gives a natural expansion that matches exactly the
current expectations from the string theory side of the duality. 222An
incomplete subset of relevant references in both gravity and field theory
sides is Cassani:2019mms ; Kantor:2019lfo ; Suh:2018qyv ; Bobev:2019zmz
Benini:2020gjh ; Lezcano:2019pae ; Lanir:2019abx ArabiArdehali:2019orz
Murthy:2020rbd ; Agarwal:2020zwm Choi:2018vbz ; Honda:2019cio ;
ArabiArdehali:2019tdm ; Kim:2019yrz ; Cabo-Bizet:2019osg ; Amariti:2019mgp
Crichigno:2020ouj  Goldstein:2019gpz  Copetti:2020dil ; Agarwal:2020pol ;
Agarwal:2019crm , Larsen:2019oll ; Nian:2020qsk ; David:2020ems ; Melo:2020amq
; David:2020jhp ; Larsen:2020lhg . A pedagogical introduction to the topic can
be found in Zaffaroni:2019dhb .

There were two issues that Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  left open:

  1. 1.

A precise analysis of a deformation of the original contour passing through
all the complex saddles there studied (This would have justified the
assumption of considering all such configurations as competing ones in
determining the original integral).

  2. 2.

An exact formula that could be used to evaluate $1/N$ perturbative and non-
perturbative corrections.

This paper solves these two issues. The natural completion of the large-$N$
saddle point approach initiated in Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  is
the Atiyah-Bott-Beligne-Vergne (ABBV) equivariant integration formula
Duistermaat:1982vw ; 1983InMat..72..153D ; Witten:1982im ; berline1983 ;
Atiyah:1984px . 333We have followed as much as possible the presentation given
by Atiyah and Bott in Atiyah:1984px . But there are many other useful
references. An absolutely incomplete selection that we have found necessary to
follow at some points, being Duistermaat:1982vw ; 1983InMat..72..153D ;
Witten:1982im ; berline1983  and audin2012torus ; Cordes:1994fc ; Niemi:1994ej
; Blau:1995rs ; Pestun:2016qko ; JeffreyLectures ; Alekseev:2000fe ;
Cremonesi:2014dva . The formula does not require us to assume large $N$, but
to evaluate effective actions and one-loop determinants we will use the
large-$N$ expansion.

It is well known that the superconformal index can be written as a matrix
integral Romelsberger:2005eg ; Kinney:2005ej . We will see that the ABBV
formula can be used to compute a generic family of matrix integrals that
includes the one associated to the 4d superconformal index as a particular
case. For simplicity we will focus on unitary matrix integrals but the idea
can be explored in other cases as well. The answer is similar to the one of
$a)$ in that it can be expressed as a sum over configurations that solve a
Bethe Ansatz equation. 444That they compute the same quantity does not imply
that they are the same formula.

To prove the formula, we will make use of the ambiguity in the choice of
complex extension of the integrand of the original matrix integral, away from
the real contour of integration. In appendix B, we will find that these
extensions can be understood as regularizations of the divergent super-
determinants in the original 4d physical problem. The elliptic extension of
Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  is one of those. The (piecewise-)
meromorphic extension corresponds to another.

The second goal of this paper is to understand the competition of the many
large-$N$ complex configurations that are known to compete, at the moment: Is
there a finite/small number of saddles dominating the large $N$ expansion of
the microcanonical index? 555For $SU(N)$ $\mathcal{N}=4$ SYM the number of
known fixed points is finite and bounded by a polynomial function of $N$
Hong:2018viz ; ArabiArdehali:2019orz Cabo-Bizet:2019eaf . The observable we
will focus on, will be the logarithm of the number of operators – counted with
$(-1)^{F}$ – at a given value of a one-dimensional section in the space of
conserved BPS charges in $SU(N)$ $\mathcal{N}=4$ SYM. We will call it the
_microcanonical index_. The generating function of those integer numbers is
the superconformal index Romelsberger:2005eg ; Kinney:2005ej . 666Early
relevant work concerning the estimation of growth in the number of $1/16$-th
BPS operators in $\mathcal{N}=4$ SYM, can be found in Janik:2007pm ; Janik2008
; Berkooz:2006wc ; Chang:2013fba .

We will observe that for the counting of operators with charges of order
$N^{2}$ i.e. large operators, the competition among known saddles is dominated
by only two configurations, the $(1,0)$ and $(1,1)$. For large enough values
of $N$ the contributions from these two configurations become the same and the
absolute value of the microcanonical index matches the exponential of the
Bekenstein-Hawking entropy of the dual Gutowski-Reall black hole.
Contributions from other $(m,n)$ saddles are present but they are shown to be
exponentially suppressed for such counting problem. As one tries to count
operators of lesser charges, the contributions from the remaining $(m,n)$
configurations become less suppressed. In principle the formula could be used
even to count small operators of charges $\mathcal{O}(1)$ 777As shown in
Murthy:2020rbd ; Chang:2013fba ; Kinney:2005ej  these small operators are in
one-to-one correspond to multi-gravitons. , at leading order in large-$N$
approximation, but doing so with precision, would require the resummation of
contributions from all saddle configurations.

Before contributions from other $(m,n)$ saddles start being relevant,
something interesting happens for operators whose charges are not large
enough, but still finite in units of $N^{2}$. For those values of charges, the
ABBV formula predicts that the logarithm of the absolute value of the
microcanonical index must oscillate around the exponential of the Bekenstein-
Hawking entropy and that these oscillations are determined mainly by the
superposition of contributions coming from $(1,0)$ and $(1,1)$ saddles. For
smaller values of charges of order $0$ in units of $N^{2}$, more precisely of
order $N^{2/3}$ or smaller, the  _two-saddles approximation_ becomes invalid
and other $(m,n)$ configurations need to be considered.

The previous understanding is consistent with recent numerical results that
show that for relatively large (but still finite) values of $N$ ($N$ of order
$10$) Murthy:2020rbd ; Agarwal:2020zwm  indeed oscillations are present.
Moreover, the analytic interpolation of the numerical oscillations given in
reference Agarwal:2020zwm , is consistent with the theoretical understanding
summarised in the previous paragraph. Specifically, 888Up to a factor of $2$
that lies within the margin of systematic error that one makes after
discarding subleading contirbutions to the effective action at large $N$. such
interpolation corresponds to the profile of the previously mentioned two-
saddles approximation to the microcanonical index.

To summarise, we will focus on answering the following question: is there a
finite number of configurations $P$ dominating the large-$N$ counting problem?
Our answer will be:

  * •

For the counting of large enough operators, only two of them dominate the
large-$N$ expansion. Every other fixed point/saddle point/root known to us at
the moment is either non-contributing, or exponentially suppressed in that
expansion.

  * •

The presence of oscillations at relatively large (but still finite) values of
$N$ around the Bekenstein-Hawking entropy is a consequence of the sum over
saddles/fixed points/roots.

  * •

At leading order in large-$N$ expansion the oscillations become negligible and
the counting of large operators matches the exponential of the Bekenstein-
Hawking entropy.

Our goal with this paper is not to make precise numerical predictions at
subleading order in the $\frac{1}{N}$ expansion, but to understand the
competition and the relevant hierarchies of charges where the nature of the
competition among saddles changes. We will mainly focus on analyzing the
leading order in the $1/N$ expansion, but the ABBV formula does not require
assuming large $N$. To have analytic control at finite $N$, 999Or large $N$
and small enough scale of charges. though, a complete classification of the
set of fixed points is needed. That classification must be completed in a
case-by-case basis as the form of the conditions that determine the positions
of the fixed points $P$’s, depend on the representation content of the theory
one wishes to analyze. Unfortunately, this is an open problem even in the
simplest toy example that we study i.e. a simple ensemble of $1/16$-th BPS
operators in $SU(N)$ $\mathcal{N}=4$ SYM at large $N$. Thus, given our current
technical reach, we focus on analytically understanding the large-$N$
competition among known configurations, and use Picard-Lefschetz theory to
identify which of them contribute, and finally, among those, which one
dominates the large-$N$ counting. 101010The claim that large-$N$ counting of
large operators is dominated by two complex saddles, should not be
extrapolated to a statement about the large-$N$ asymptotics of the _grand
canonical index_ index $\mathcal{I}$. These are two different quantities. It
is possible that the canonical index could take relatively small values at
specific regions of the chemical potential $\tau$, meanwhile the underlying
Hilbert space could still have an exponentially large number of states at
large enough BPS charges, counted with the grading of $(-1)^{F}$.

In the remaining part of this introductory section we summarize the main two
results of the paper. In subsection 1.1 we introduce the superconformal index
and its relation to matrix integrals and elliptic hypergeometric integrals. In
subsection 1.3 we explain how the ABBV formula can be used to compute relevant
families of matrix integrals. In subsection 1.5 we comment on the relation to
the saddle point analysis of Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr  and the
Bethe Ansatz formula of Benini:2018ywd ; Benini:2018mlo ; Closset:2017bse . In
subsection 1.6 we explain how the leading large-$N$ behaviour of a specific
microcanonical index is controlled by the competition among complex saddles,
and how such competition depends on how large the operators to be counted are.
In subsection 1.7 we give interesting open questions that are left as future
work. Towards the end we summarize the content of the main sections of the
paper. The regularization of the 4d super-determinants is presented in
appendix B.

###  1.1 Superconformal indices, matrix integrals and Elliptic Hypergeometric
integrals: a lightening review

The Hamiltonian definition of the $\mathcal{N}=1$ superconformal index on
$S_{3}$ Romelsberger:2005eg ; Kinney:2005ej  is

$\mathcal{I}(p,q)\,=\,\,{\text{
Tr}}_{\mathcal{H}_{\text{phys}}}\,(-1)^{F}\text{e}^{-\beta\\{\mathcal{Q},\bar{\mathcal{Q}}\\}}p^{J_{1}+\frac{Q}{2}}\,q^{J_{2}+\frac{Q}{2}}\,t^{Q_{g}}\,.$
(1)

The $\mathcal{Q}$ and $\overline{\mathcal{Q}}$ are complex conjugated
supercharges. $F$ is the Fermionic number. $J_{1}$ and $J_{2}$ are the left
and right angular momenta on $S_{3}$. $Q$ is the $U(1)$ R-charge. The
$J_{1,2}\,+\,\frac{Q}{2}$ are the two combinations out of $J_{1,2}$ and $Q$
that commute with $\mathcal{Q}$ and $\overline{\mathcal{Q}}$. The $Q_{g}$
denotes a generic global charge. The $p\,=\,{\bf e}(\sigma)$, $q\,=\,{\bf
e}(\tau)$ and $t\,=\,{\bf e}(\Delta)$ are the rapidities associated to the
charges $J_{1,2}\,+\,\frac{Q}{2}$ and $Q_{g}$, respectively.

The series coefficients of (1) in an expansion around $p=q=0$ (and for $t=1$)
count the difference between bosonic and fermionic gauge invariant states in
the cohomology of $\mathcal{Q}$, with fixed values of energy $E$. In an abuse
of terminology, we will call the energy $E$ charge, as that is justified by
the BPS relation that all relevant BPS states obey

$E\,=\,2J\,+\,Q\,\quad\text{with}\quad 2J\,\equiv\,J_{1}\,+\,J_{2}\,.$ (2)

For a theory carrying a representation $R$ of a gauge group $G$, which we
assume to be a product of unitary groups

$\mathcal{I}\,=\,\int[dM]\,\exp\Bigl{(}\sum_{n=1}^{\infty}\,\frac{\mathcal{I}_{\text{sl}}(p^{n},q^{n})}{n}\,\chi_{R(G)}(M^{n})\Bigr{)}\,.$
(3)

The $\mathcal{I}_{sl}$ denotes the same index (1) but computed after
restricting the trace tr to run over gauge invariant operators that include
only single fundamental fields 111111For example, supposing the theory is
Lagrangian these are the fundamental fields in the free limit of such
Lagrangian. , and any number of derivative insertions, instead of over the
entire Hilbert space of the theory. $[dM]$ is the invariant measure of
integration over $G$ which is the product of Haar measures of each of the
simple factors in $M$. $\chi_{R(M)}$ is the character of the representation
$R$. If $\\{{\bf e}{(u)}\\}$ denotes the set of eigenvalues of $M$ then
$\chi_{R}(U)\,=\,\sum_{\rho}{\bf e}(\rho(u))$ where $\\{\rho\\}$ is the set of
weights of the representation $R$. From (3) one can recover the index of a
generic theory from the single-letter indices of chiral and vector multiplets

$\begin{split}\mathcal{I}^{\text{c}}_{\text{sl}}(t,p,q)\,=\,\frac{t-pq/t}{(1-p)(1-q)}\,,\quad\mathcal{I}^{\text{v}}_{\text{sl}}(p,q)\,=\,1\,-\,\frac{(1-pq)}{(1-p)(1-q)}\,.\end{split}$
(4)

For example, consider a $U(N)$ $\mathcal{N}=1$ SCFT composed by one vector
multiplet and three adjoint chiral multiplets. The vector and chiral
multiplets are assumed to be charged under the $U(1)_{R}$ superconformal
$R$-charge. The chiral multiplets are assumed to carry the fundamental
representation of the $SU(3)$ flavour symmetry group. In this case the
superconformal index is a unitary matrix integral

$\begin{split}\mathcal{I}&\,=\,\int[dU]\,\exp\Bigl{(}\sum_{n=1}^{\infty}\,\frac{\mathcal{I}_{sl}(t^{n}_{1},t^{n}_{2},t^{n}_{3},p^{n},q^{n})}{n}\,\text{Tr}U^{n}\text{Tr}U^{\dagger
n}\Bigr{)}\,,\\\
\mathcal{I}_{sl}&\,=\,1\,-\,\frac{(1\,-\,t_{1})(1\,-\,t_{2})(1\,-\,t_{3})}{(1-p)(1-q)}\,,\end{split}$
(5)

where the variables $t$, $p$ and $q$ are constrained by
$\frac{t_{1}t_{2}t_{3}}{pq}=1$ and initially one must assume
$0<|t_{1,2,3}|,\,|p|,|q|\,<1$ in order to guaranty this representation of
$\mathcal{I}$ to be well defined. The $\mathcal{I}$ can be analytically
continued in $t$’s, $p$ and $q$ to larger regions. Interestingly, it turns out
that $\mathcal{I}$ has singularities when $p=q$ becomes a root of unit. These
singularities are important in the large-$N$ counting of BPS operators. Their
neighborhood encodes the essential information needed to extract the
asymptotic form of such numbers.

After integrating over angular variables, the integral (5) can be recast in
terms of integrals of products of elliptic functions Dolan:2008qi
Spiridonov:2010em

$\begin{split}\mathcal{I}&\,=\,\kappa\,\oint_{|\zeta|=1}\prod_{i=1}^{\text{rk}(G)}{\frac{d\zeta_{i}}{\zeta_{i}}}\,\left(\prod_{i\,<\,j=1}^{N}\,{\theta_{\text{ell}}(\frac{\zeta_{i}}{\zeta_{j}},p)}\,{\theta_{\text{ell}}(\frac{\zeta_{j}}{\zeta_{i}},q)}\right)\,\prod_{i,j=1\atop
i\neq
j}^{N}\,\,{\prod_{a=1}^{3}\Gamma_{\text{ell}}(\frac{\zeta_{i}}{\zeta_{j}}\,t_{a};\,p,\,q)}\,,\,\\\
\kappa&\equiv\frac{(p;p)^{\text{rk}(G)}(q;q)^{\text{rk}(G)}}{N!}\,\prod_{a=1}^{3}\Gamma_{\text{ell}}(t_{a};\,p,\,q)\,,\end{split}$
(6) 121212The definition of elliptic Gamma function $\Gamma_{\text{ell}}$ is
given in equation (161).

where for $G=U(N)$, $\text{rk}(G)\,=\,N$, and for $G=SU(N)$,
$\text{rk}(G)=N-1$. Moreover, for $G=~{}SU(N)$ the
$\xi^{-1}_{N}=\prod_{i=1}^{N-1}\xi_{i}$. 131313For generic theories similar
expressions exist. From this expression and after algebraic manipulations it
follows that
$\mathcal{I}_{SU(N)}\,=\,(p;p)^{-1}(q;q)^{-1}\,\mathcal{I}_{U(N)}\,.$ (7) In
particular, this implies that the answer at leading order in large $N$
expansion, is the same for both gauge groups. It is useful to note that the
vector multiplet contribution can also be expressed in terms of elliptic Gamma
functions by using the following identities

$\theta_{\text{ell}}(\zeta,p)\theta_{\text{ell}}(1/\zeta,q)\,=\,\Gamma_{\text{ell}}(\zeta\,pq;\,p,\,q)\Gamma_{\text{ell}}(1/\zeta\,pq;\,p,\,q)\,=\,\frac{1}{\Gamma_{\text{ell}}(\zeta;\,p,\,q)\Gamma_{\text{ell}}(1/\zeta;\,p,\,q)}\,.$
(8)

Sometimes it is convenient to express (6) in terms of the potentials
$\underline{u}$, $\sigma$, $\tau$ and $\Delta$’s

$\zeta_{\rho}\,=\,{\bf e}(\rho(u))\,,\,p\,=\,{\bf e}(\sigma)\,,\,q\,=\,{\bf
e}(\tau)\,,\,t_{a}\,=\,{\bf e}(\Delta_{(a)}+\frac{\tau+\sigma}{2})\,,$ (9)

where $\underline{u}=(u^{1},u^{2},\ldots)$ is an array that collects the $N$
Cartan angles. For the moment, we can consider the vector $\rho$ to be a
weight of the adjoint of either $U(N)$ or $SU(N)$. For the case of $SU(N)$ we
can consider $\rho$ to be a weight of $U(N)$ upon the imposition of the
constraint $u^{N}\,=\,-\sum^{N}_{i=1}u^{i}$.

The $\Delta$’s encode the dependence on $R$-charges, and chemical potentials
dual to global symmetries. They can be used to recover refined versions of the
superconformal index. 141414The point at which the flavour fugacities vanish
$\Delta_{g}=0$ corresponds to
$\Delta_{1}=\Delta_{2}=\Delta_{3}=-\frac{\tau\,+\,\sigma}{6}+\mathbb{Z}$. As
pointed out in Cabo-Bizet:2019eaf  for this value of the parameters there is a
pole in $A\times B$ and thus in principle one would need to use the ABBV
formula plus contributions coming from the non-vanishing residues. This
correction seems to be essential at finite $N$. However, at large-$N$ we argue
that in principle one can compute the integral (6) before imposing the
constraint (10) and then recover the constrained result after using
meromorphic flow in the $\Delta$’s. This is the approach we will follow in
this paper. In forthcoming work, we will comeback to analyze the details of
the contributions that come from residues.  In these variables the constraint
below (5) takes the form

$\Delta_{(1)}\,+\,\Delta_{(2)}\,+\,\Delta_{(3)}\,+\,\frac{\tau+\sigma}{2}\,\in\,\mathbb{Z}\,.$
(10)

As mentioned before, for a generic theory with gauge group $G$ of rank
$\text{rk}(G)$, the index can be written as elliptic hypergeometric integrals
as well Dolan:2008qi Spiridonov:2010em . The generic answer can be recast in
the form

$\begin{split}\mathcal{I}&\,=\,\int_{0}^{1}d\underline{u}\,e^{-\,S(u)}\,\equiv\,\kappa_{G}\int^{1}_{0}d\underline{u}\prod_{\alpha}\prod_{\rho_{\alpha}\,\neq\,0}\Gamma_{\text{e}}(z_{\alpha}(u)+\frac{\tau+\sigma}{2};\tau,\sigma)\,,\\\
\kappa_{G}&\equiv\frac{(p;p)^{\text{rk}(G)}(q;q)^{\text{rk}(G)}}{|\mathcal{W}|}\,\prod_{\alpha\atop\alpha\neq\text{vectors}}\prod_{\rho_{\alpha}=0}\Gamma_{\text{e}}(z_{\alpha}(0)+\frac{\tau+\sigma}{2};\,p,\,q)\,,\end{split}$
(11) 151515The definition of elliptic Gamma functions $\Gamma_{e}$ is given in
equation (161).

where the integration measure is defined as
$d{\underline{u}}\equiv\prod_{i=1}^{\text{rk}(G)}du_{i}$. The $|\mathcal{W}|$
is the dimension of the corresponding Weyl group. The $S(u)$ is a zero-
dimensional analog of QFT quantum effective action, and thus we will call it
effective action from now on. The label $\alpha$ runs over all the multiplets
in the theory. We define
$z_{\alpha}\,\equiv\,\rho_{\alpha}(u)\,+\,\Delta_{\alpha}\,$ where
$\rho_{\alpha}$ is the vector of charges, i.e a weight, that the multiplet
$\alpha$ carries (under the Cartan elements of $G$). The complex number
$\Delta_{\alpha}$ encodes both the dependence on the R-charges of $\alpha$ and
the dependence on the set of potentials associated to global charges $Q_{g}$,
as defined in (1). This parameter can be used to recover more refined versions
of the superconformal index that are obtained by turning on global symmetry
holonomies.

We find convenient to analyze the generic integrals like (11), without
imposing constraints like (10). Namely for each multiplet $\alpha$ we
associate a generic $\Delta_{\alpha}$ (even for the vector multiplets) and
only after the evaluation of the observable one wishes to study, the
$\Delta_{\alpha}$ can be fixed to specific values. For example, in the case of
(6) we can work by default with
$\Delta_{\alpha}\,=\,\\{\Delta_{(1)},\Delta_{(2)},\Delta_{(3)},\Delta_{(4)}\\}$
and in the very end, once a convergent result is obtained one can fix
$\Delta_{(3)}$ from condition (10).

###  1.2 Recovering the index from double periodic extensions: a first
incomplete but useful trial

Let $A\times B$ be a real torus with $A$ and $B$ cycles. Let
$e^{-S_{\lambda}}$ be a family of smooth complex functions in $A\times B$
labelled by a real parameter $\lambda\,$. Assume without loss of generality
that $\lambda$ flows from zero to one.

Figure 1: A first trial to the problem: One starts with integrals of a family
of exponentials of moment maps which are smooth on the torus $A\times B$.
These integrals are parameterized by a real number $\lambda$. As
$\lambda\,\to\,1^{-}$ the integrand develops a discontinuity at an $A$-period
and it is now well-defined and smooth in the cylinder (not in the torus). If
the limit integrand is holomorphic and the height of cylinder is one, the
integral over the cylinder collapses to an integral over an $A$-period. Every
$A$-period gives the same answer.

Suppose that as $\lambda\,\to\,1^{-}$ the $e^{-S_{\lambda}}$ develops a
discontinuity at an $A$-cycle of the torus. We can rephrase the previous
statement by saying that in the limit $\lambda\,\to\,1^{-}$ the domain on
which $S_{\lambda}$ was smooth, which used to be a torus $A\times B$ before
the limit, becomes a cylinder as sketched in Figure 1. If the limit of
$e^{-S_{\lambda}}$ is holomorphic, its integral over the cylinder collapses to
a middle-dimensional integral over any path homologous to an $A$-period. Under
such conditions, as it will be clear from the explanation we will give below,
the limit of the integral over the original torus equals the integral of the
limit function over any of the A-cycles. 161616This last statement follows
from the _Lebesgue’s dominated convergence theorem_ (DCT) Baxter1996 . We
assume integration to be defined in the Lebesgue’s sense. In summary, this
theorem states that the limit of a Lebesgue integral coincides with the
Lebesgue integral of the limit if the limit of the integrand is pointwise
convergent, and the integral over the same domain of the absolute value of the
limit is convergent. In particular, this implies that if the limit of the
integrand is bound and the domain of integration is compact, the limit can
always be commuted with the Lebesgue integral.

Now, think of the $A$-cycle in the previous paragraph as the Cartan torus of a
gauge group. Let the torus $A\times B$ be a double copy of $A$. Suppose there
exists a family of $e^{-S_{\lambda}}$ on $A\times B$ that flows to a
holomorphic function living in the cylinder obtained by slicing $A\times B$
across an $A$-cycle. If the limit of $e^{-S_{\lambda}}$ matches the integrand
of the superconformal index (or the integrand of any other integrals that one
wishes to compute) at one of these $A$-cycles, any of them, we conclude that
in such limit the integral over $A\times B$ converges to the superconformal
index.

A trivial example of how an integral over a slicing of $A\times B$ collapses
to a middle dimensional one over $A$ (assuming $\text{Im}(\Delta)>0$) is

$\int^{1}_{0}du_{1}\int^{1}_{0}du_{2}\,\Gamma_{\text{e}}(u_{1}+u_{2}\tau+\tau+\Delta;\tau,\tau)\,=\,\int^{1}_{0}du_{1}\,\Gamma_{\text{e}}(u_{1}+\tau+\Delta;\tau,\tau)\,.$
(12) 171717This does not mean that there is no dependence on $u_{2}$ in the
integrand of the left-hand side. On the contrary, such integrand does depend
on $u_{2}$ but still the equality holds in virtue of the Cauchy theorem.

From inspection of the position of poles of $\Gamma_{\text{e}}$ that are given
in appendix A, one reaches the conclusion that there are no poles in the
domain of integration of the left-hand side of (12). That implies that the
integral over $u_{2}$ is spurious. The next obvious example is the $n=2$ one
in the following family

$\begin{split}\int^{1}_{0}d\underline{u}_{1}\int^{1}_{0}\,d\underline{u}_{2}\,\prod_{i,j=1\atop
i\neq j}^{n}\Gamma_{\text{e}}(u_{1ij}+u_{2ij}\tau+\tau+\Delta;\tau,\tau)\\\
\,=\,\int^{1}_{0}d\underline{u}_{1}\,\prod_{i,j=1\atop i\neq
j}^{n}\Gamma_{\text{e}}(u_{1ij}+\tau+\Delta;\tau,\tau)\,,\end{split}$ (13)

where $u_{1,2ij}=u_{1,2}^{i}\,-\,u_{1,2}^{j}\,$. This can be generalized
further to integrals of the following kind (assuming
$\text{Im}(\Delta_{\alpha})>0$)

$\begin{split}\int^{1}_{0}d\underline{u}_{1}\int^{1}_{0}d\underline{u}_{2}\prod_{\alpha}&\prod_{\rho_{\alpha}\,\neq\,0}\Gamma_{\text{e}}(z_{\alpha}(u_{1}+\tau
u_{2})+\tau;\tau,\tau)\\\
&\,=\,\int^{1}_{0}d\underline{u}_{1}\prod_{\alpha}\prod_{\rho_{\alpha}\,\neq\,0}\Gamma_{\text{e}}(z_{\alpha}(u_{1})+\tau;\tau,\tau)\,,\end{split}$
(14)

with $\rho_{\alpha}$ being weights of a quiver gauge theory with matter in the
fundamental, adjoint and/or bi-fundamental of say $U(N)$ gauge nodes. The
analysis for $SU(N)$ nodes is more subtle, but one can always obtain it from
$U(N)$ by using the identity (7).

###  1.3 The index from Atiyah-Bott-Berline-Vergne localization formula

_What is the limitation of the previous approach?_

As we will explain in due time, integrals over $A\times B$ of the smooth
function $e^{-S_{\lambda}}$ can be computed by using the Atiyah-Bott-Berline-
Vergne formula. The expressions take the following form

$\int\,\frac{\omega^{n}}{n!}\,e^{-S_{\lambda}}\,=\,\sum_{P}\,\frac{e^{-S_{\mathcal{\lambda}}(P)}}{{E}(\nu_{P})}\,.$
(15)

Here $n=\text{rk}(G)$ and the volume form
$\int\frac{1}{n!}\omega^{n}\,=\,\int^{1}_{0}d\underline{u}_{1}\int^{1}_{0}d\underline{u}_{2}\,$,
is defined in terms of the two-form $\omega\,=\,\sum_{i=1}^{n}du^{i}_{1}\wedge
du^{i}_{2}$. The remaining objects will be properly defined in due time. Here
we just want to draw attention to a specific point.

The main claim of the previous subsection is summarized by the following
equality

$\mathcal{I}\,=\,\underset{\lambda\,\to\,1^{-}}{\lim}\,\int\,\frac{\omega^{n}}{n!}\,e^{-S_{\lambda}}\quad\Bigl{(}{\,\neq\,}\sum_{P}\,\underset{\lambda\,\to\,1^{-}}{\lim}\,\frac{e^{-S_{\mathcal{\lambda}}(P)}}{{E}(\nu_{P})}\Bigr{)}\,.$
(16)

However, as stated by the inequality in parenthesis, the limit
$\lambda\,\to\,1^{-}$ of the previous subsection does not commute with the sum
over $P$’s. The reason being that

$\underset{\lambda\,\to\,1^{-}}{\lim}{E}(\nu_{P})\,=\,0\,.$ (17)

This can be understood before doing computations. The quantity $E(\nu_{P})$,
which will be properly defined below, is proportional to the square root of
the determinant of the Hessian of $S_{\lambda}$, and inversely proportional to
the determinant of the matrix components of the two-form $\omega$. As in that
limit $S_{\lambda}$ becomes holomorphic, the Hessian develops anti-holomorphic
zero modes. Thus, as $\det\omega$ is constant, equality (17) follows through.

##### The integration measure

The previous obstacle can be solved by using an appropriate integration
measure, i.e by replacing the two form $\omega$ by a $\lambda$-dependent one
that we will denote as $\omega_{\lambda}$. The $\omega_{\lambda}$ and
$S_{\lambda}$ will be defined in such a way that the corresponding integral
will be independent of $\lambda$. Thus, for such definitions the limit
$\lambda\,\to\,1^{-}$ is well defined. Finally, from our choice of
$S_{\lambda}$ it follows that for specific values of chemical potentials the
integral along the real contour of integration is equal to the double
dimensional integral for any $0\leq\lambda<1$.

In particular, this implies that the ABBV equivariant integration formula can
be used to compute 4d unitary matrix integrals of the form (3) or equivalently
(11).

Let us go by steps. First we introduce the following measure of integration in
$A\times B$

$\int_{A\times
B}\frac{\omega^{n}_{\lambda}}{n!}\,\equiv\,\int_{0}^{1}d\underline{u}_{1}\int^{1}_{0}d\underline{u}_{2}\prod_{i=1}^{n}\mathcal{O}_{i,\lambda}\,.$
(18)

In this equation the closed two-form $\omega_{\lambda}$ is defined as

$\omega_{\lambda}\,=\,\sum_{i=1}^{n}du^{i}_{1}\wedge
du^{i}_{2}\,\mathcal{O}_{i,\lambda}\,.$ (19)

For $0\leq\lambda<1$

$\mathcal{O}_{i,\lambda}\,=\,\mathcal{O}_{\lambda}(u_{2i})\,\equiv\,1\,-\,\frac{\partial\\{u_{2i}\\}_{\lambda}}{\partial
u_{2i}}\,,$ (20)

is a smooth and strictly positive function of a single real variable $u_{2i}$
that is equal to $1$ at $\lambda\,=\,0$ i.e.

$\begin{split}\mathcal{O}_{\lambda}(u_{2})\,>\,0\,,\qquad\mathcal{O}_{\lambda}(u_{2}+1)\,=\,\mathcal{O}_{\lambda}(u_{2})\,,\qquad\mathcal{O}_{i,\lambda=0}\,=\,1\,.\end{split}$
(21)

These properties imply that $\omega_{\lambda}$ is simply a smooth and globally
well-defined passive diffeomorphism transformation of the two-form

$\omega\,=\,\omega_{\lambda=0}\,=\,\sum_{i=1}^{n}du^{i}_{1}\,\wedge\,du^{i}_{2}\,.$
(22)

The diffeomorphism is induced by the transformation
$u^{i}_{2}\,\mapsto\,u^{i}_{2}\,-\,\\{u^{i}_{2}\\}_{\lambda}$. The definition
and properties of the function $\\{\cdot\\}_{\lambda}$ will be given in the
main body of the paper, in equation (75). Here we just note that from that
definition it follows that

$\mathcal{O}_{\lambda}(u_{2})\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\sum_{m\in\mathbb{Z}}\,\delta(u_{2}\,+\,m)\,,$
(23)

as illustrated by the plots in figure 2. In fact, the specific details of the
function $\\{\cdot\\}_{\lambda}$ are irrelevant as long as we use a
$\\{\cdot\\}_{\lambda}$ that is periodic, smooth and that preserves the
conditions (21), (22) and (23). This is consistent with the results presented
in appendix B. There, it was shown that the choice of $\\{\cdot\\}_{\lambda}$
corresponds to a choice of regularisation of the divergent one-loop
determinants of the original 4d physical problem. Consequently different such
choices should not affect the outcome of physical observables. We have chosen
to work with the explicit form (75) for concreteness.

Figure 2:  The profile of $O_{\lambda}(u)$ that follows from the specific
choice of function $\\{\cdot\\}_{\lambda}$ that we have chosen to work with.
The plot shows three different values of $\lambda$. $O_{\lambda}(u)$ is
positive for any value of $0\leq\lambda<1$. $O_{\lambda}(u)\to 1$ for small
enough values of $\lambda$. $O_{\lambda}(u)$ tends to the periodic Dirac-delta
function centered at integer values of $u$ for $\lambda\,\to\,1^{-}$. In fact,
the specific details of the function $\\{\cdot\\}_{\lambda}$, the one that
defines the dependence on $\lambda$ in both the action $S_{\lambda}$, and the
two-form $\omega_{\lambda}$ (through $O_{\lambda}$), are irrelevant as long as
$\\{\cdot\\}_{\lambda}$ is periodic, smooth and it preserves the conditions
(21), (22) and (23).

From property (23) and (18) it follows that for any smooth action
$\mathcal{S}$ on $A\times B$

$\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-\mathcal{S}}\,\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\int_{u^{i}_{2}=0}\,\prod_{i=1}^{n}du^{i}_{1}\,e^{-\mathcal{S}}\,.$
(24)

If when restricted to the real contour of integration $u^{i}_{2}=0$ the action
$\mathcal{S}$ equals the effective action $S$ defined in equation (11) i.e.

$\mathcal{S}(u)\,=\,{S}(u)\qquad\text{at}\qquad u^{i}_{2}\,=\,0\,,$ (25)

then the desired integral e.g. the superconformal index, is recovered via the
following limit

$\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-\mathcal{S}}\,\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\mathcal{I}\,.$
(26)

##### The story is slightly more involved though

To guaranty smoothness of $\mathcal{S}$ at intermediate stages we will need to
introduce a cut-off $\Lambda$ BrownLevin  that eventually must be fixed to
$\Lambda\,=\,1^{-}$. The limit $\Lambda=1^{-}$ introduces discontinuities at
$u^{i}_{2}\,=\,0$. Thus, the correct version of (24) in that case is a product
of semi-sums of two single dimensional contour integrals i.e.

$\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-\mathcal{S}}\,\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\prod_{i=1}^{n}\frac{\Bigl{(}\int_{C^{i}_{+}}\,+\,\int_{C^{i}_{-}}\Bigr{)}}{2}\,{e^{-\,\mathcal{S}(u)}}\,,$
(27)

where

$\int_{C^{i}_{+}}\,=\,\int^{1}_{0\,,\text{ at
}u^{i}_{2}\,=\,0^{+}}du^{i}_{1}\,,\qquad\int_{C^{i}_{-}}\,=\,\int^{1}_{0\,,\text{
at }u^{i}_{2}\,=\,1^{-}}du^{i}_{1}\,.$ (28)

(27) comes from the fact the the periodic delta function (23) picks up $1/2$
times the value of the corresponding integrand at $u^{i}_{2}=0^{+}$ plus $1/2$
times the value of the corresponding integrand at $u^{i}_{2}=1^{-}$.

At this point the global property (e.g. away from the $A$-period
$u^{i}_{2}\,=\,0$) of our choice of function $\mathcal{S}$ plays an essential
role. Thus, we choose the action mentioned in subsection 1.3,
$\mathcal{S}\,=\,S_{\lambda}\,$ where $S_{\lambda}$ will be defined in
equation (82). In the limit $\lambda,\,\Lambda\,\to\,1^{-}$

$\mathcal{S}\,\to\,S(u)\,,$ (29)

where $e^{-S(u)}$ is the meromorphic extension of the integrand of (11). In
that case due to the periodicity property $e^{-S(u)}\,=\,e^{-S(u\,+\,2\pi)}$
and Cauchy theorem, it follows that

$\int_{C^{i}_{-}}\,\blacksquare\,=\,\int_{C^{i}_{+}}\,\blacksquare\,+\,\widehat{\text{Res}}_{i}[\blacksquare]\,.$
(30)

Both linear operators, $\int_{C^{i}_{+}}$ and $\widehat{\text{Re}}_{i}\,$,
annihilate the dependence on a single complex variable
$u^{i}\,\equiv\,u^{i}_{1}\,+\,\tau\,u^{i}_{2}$ of the function they act upon
and give a new function of the remaining $u^{j\,\neq\,i}$ complex variables.
The integral fixes $u^{i}_{2}\,=\,0$ and integrates $u^{i}_{1}$ from $0$ to
$1$. The operator $\widehat{\text{Res}}_{i}$ when acting on a function
$f(u^{i})$ of the variable $u^{i}$’s, gives back a new function of the
remaining variables $u^{j\,\neq\,i}$ which equates to the sum over residues

$\sum_{\xi_{0}}\text{Res}\Bigl{[}\frac{f(u(\xi))}{\xi^{i}},\,\xi^{i}\,=\,\xi_{0}\Bigr{]},\qquad\xi^{i}\,\,=\,\,{\bf
e}{(u^{i})}\,.$ (31)

$\xi_{0}$ runs over all the simple poles of the function ${f(u^{i}(\xi^{i}))}$
located in the annular domain $|q|<|\xi^{i}|<1$. The corresponding residues
can be computed systematically, but we will not do that in here. Plugging (30)
into (27) and from the fact that $\mathcal{I}\,=\,\int_{C^{i}_{+}}\,e^{-S(u)}$
we obtain

$\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-{S}_{\lambda}}\,\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\mathcal{I}\,+\,\text{Residues}.$
(32)

By Residues we mean contributions that involve the application of at least one
of the operators $\widehat{\text{Res}}_{i}$, $i=1,\ldots,\text{rk}(G)$. These
contributions vanish if $e^{-S(u)}$ is a holomorphic function in the sliced
$A\times B$. We will assume there is always a chamber of $\Delta$’s, where
there is no contribution from simple poles. 181818This very same assumption
was used in reaching the Bethe Ansatz formula of Benini:2018mlo . The improved
integration formula includes contributions that involve the action of at least
one of the residue operators defined in equation (30). We will analyze that
point in forthcoming work. From now on we focus on the simplest variant of
(32)

$\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-\,S_{\lambda}(u)}\,\,\underset{\lambda\,\to\,1^{-}}{\longrightarrow}\,\mathcal{I}\,.$
(33)

A complete study of chambers of $\Delta$’s where residues do contribute is
left for future work.

###  1.4 The final formula

Let us wrap up and move forward. The relation (33) can be used to compute
unitary matrix integrals like (11) or (3) with the help of the Atiyah-Bott-
Berline-Vergne integration formula. The first step is to define the following
sequences of integrals

$\mathcal{I}_{\lambda}\equiv\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-S_{\lambda}}\,.$ (34)

As said earlier, the action $S_{\lambda}$ will be defined in equation (82).
The deformation parameter $\lambda$ ranges from $0$ to $1$, including $0$ but
not $1$! As we will explain in details in section 2 the sequence of integrals
(34) is computed exactly by the ABBV equivariant integration formula i.e.

$\mathcal{I}_{\lambda}\,=\,\sum_{P}\,\frac{e^{-S_{\mathcal{\lambda}}(P)}}{{E}(\nu_{P})}\,\,\equiv\,\,\sum_{P}\,\mathcal{I}_{\lambda}(P)\,,$
(35)

where the $P$’s are configurations of complex eigenvalues that are fixed
points of the moment maps defined by the real and imaginary parts of the
action $S_{\lambda}$. For $0\,\leq\,\lambda\,<\,1$ they are also saddle points
(or stationary phase points) of $S_{\lambda}$.

The factor in the denominator is an equivariant Euler class that equates to

${E}(\nu_{P})\,=\,\frac{1}{(2\pi{\rm i})^{\text{rk}(G)}}\det{\frac{\,\partial
B^{i}(P)}{\partial u_{1}^{j}}}\,.$ (36)

For the examples we will analyze, half of the fixed-point (saddle-point)
conditions can be written in terms of the objects $B^{i}(u)$ (See definition
(84)). They take the form of Bethe Ansatz equations $B^{i}(P)=1$. This object
does not seem to be the same as the Bethe operator of Benini:2018mlo , but it
is closely related. 191919A more detailed comparison will be left as future
work.

We will assume $G=\otimes SU(N)$ and choose $S_{\lambda}$ and
$\omega_{\lambda}$ in such a way that

$\frac{\text{d}\mathcal{I}_{\lambda}(P)}{\text{d}\lambda}\,=\,\Bigl{(}\frac{\partial}{\partial\lambda}\,+\,\frac{\partial
P}{\partial\lambda}\cdot\nabla_{P}\Bigr{)}\,\mathcal{I}_{\lambda}(P)\,=\,0\,.$
(37)

Here $\partial/\partial\lambda$ means derivative with respect to $\lambda$ at
fixed values of $P$.

For fixed points $P$ that are not continuously connected to other, the proof
will go in two steps. First, we will show that for any such $P$

$\partial_{\lambda}S_{\lambda}(P)\,=\,0\,,\qquad\partial_{\lambda}E(\nu_{P})\,=\,0\,.$
(38)

and thus $\frac{\partial\mathcal{I}_{\lambda}(P)}{\partial\lambda}\,=\,0$. The
first equation follows from the fact that $\partial_{\lambda}S_{\lambda}(u)$
is proportional to a linear combination of fixed point conditions and thus it
vanishes at any $P$ (not just for those with non-vanishing Hessian). The
second condition follows trivially from the fact
$\partial_{\lambda}B^{i}(u)\,=\,0$. Second, we will show that the position of
that type of fixed points $P$ can not depend on $\lambda$ in virtue of ABBV
theorem. That will complete the proof of (37) for such cases.

For a fixed point whose Hessian of the action vanishes, and thus it is
continuously connected to others, the variation in $\lambda$ generates motion
in the corresponding connected manifold of fixed points. As the ABBV formula
includes integration over such manifolds, it follows that even in that case
there is no dependence on $\lambda$.

The final conclusion is that in appropriate chambers of chemical potentials
$\dfrac{d\mathcal{I}_{\lambda}}{d\lambda}\,=\,0$ and thus in virtue of (33)

$\mathcal{I}\,=\,\mathcal{I}_{\lambda}\quad\text{ for }\quad
0\,\leq\,\lambda\,<\,1\,,$ (39)

where the superconformal index $\mathcal{I}$, was defined in (11). At last,
plugging (35) into (39) we obtain

$\mathcal{I}\,=\,\sum_{P}\,\frac{e^{-S_{\mathcal{\lambda}}(P)}}{E(\nu_{P})}\,.$
(40)

As just mentioned, the $P$’s can be part of connected manifolds and in that
case one needs to integrate over such components with the measure induced by
the inclusion map of the corresponding submanifold in $A\times B$ upon the
Liouville measure of $A\times B$. We will elaborate on this later on.

##### Comment about complex fixed points

We focus on quiver theories with $\nu$ $SU(N)$ gauge nodes and matter in the
adjoint and bi-fundamental representations. As $S_{\lambda}$ is smooth and
double periodic, there will always exist complex fixed points $P$ carrying
finite Abelian group structure. That follows from the analysis of Cabo-
Bizet:2019eaf ; Cabo-Bizet:2020nkr . For instance for $N$ prime, only
$\mathbb{Z}_{N}$ complex fixed points of such kind exist. They take the form

$u^{i\beta}\,=\,u^{i}\,=\,\Bigl{(}\frac{i}{N}\,-\,\frac{N+1}{2N}\Bigr{)}\,T\,,\qquad
i\,=\,1,\ldots,N-1\,.$ (41)

$T\,=\,m\tau\,+\,n$ and $m$, $n$ are co-prime integers s.t. $0\leq m<N$ and
$0\leq n<N$. $\beta=1,\ldots,\nu$ is the label of the $SU(N)$ gauge nodes. For
large-$N$ the $(m,n)=(1,0)$ and $(m,n)=(1,1)$ fixed points are the ones whose
on-shell action corresponds to the dual AdS5 BPS black holes. For fixed $N$
the number of solutions are finite. Note that these fixed points lie outside
the $A$-period $\underline{u}_{2}=0$ i.e they are not on the middle
dimensional contour of integration that defines the superconformal index.

###  1.5 Relation to other approaches and questions

##### Elliptic extension

Our initial motivation was to strengthen the basis of a large-$N$ saddle-point
approach of Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr . We have found that the
method put forward in those references is completed by the use of the ABBV
formula at $\lambda\,=\,0$. The saddle-point or stationary phase approximation
turns out to be exact in this case due to equivariant integration. In this way
we have shown that complex saddles do contribute to the original integral. On
the way, we have fixed minor ambiguities regarding the computation of
contributions coming from vector multiplets, and $\tau$-independent constant
phases in the effective action.

##### Bethe Ansatz formula

Possibly, formula (35) and the Bethe Ansatz formula of Benini:2018mlo ;
Closset:2017bse  are the same. 202020As previously said, the fact fact they
compute the same quantity does not mean that they are the same formula. There
are three observations we should make though:

  * •

The Bethe operator $B^{i}$ does not seem a priori to be the same one of
Benini:2018mlo ; Closset:2017bse . Finite $N$ checks are perhaps a good way to
start a more concrete comparison. We have computed the leading contribution of
$E(\nu_{P})$ at large-$N$ for any $(m,n)$ fixed point, and for the only case
at disposal for comparison, the $(1,0)$, the result matches an analogous semi-
analytic result obtained via the Bethe Ansatz formula GonzalezLezcano:2020yeb
. That is a large-$N$ check. Finite $N$ checks from both perspectives will be
addressed elsewhere.

  * •

The Bethe condition $B^{i}=1$ is only half of the saddle point conditions. If
all saddle point solutions carry Abelian group structure then the other half
is automatically satisfied. However, should other solutions of the Bethe root
conditions of Benini:2018mlo ; Closset:2017bse  not carrying Abelian group
structure exist, we do not know if they will also be fixed points of our
moment maps. This is an important issue that needs to be understood.

  * •

A complete classification of saddle points is missing. This is the main
obstacle to make checks at finite and generic values of $N$. 212121 It would
be interesting to explore if already-known integrability techniques could be
of help to complete such classification Nekrasov:2009uh ; Nekrasov:2009ui .

###  1.6 The large-$N$ expansion of the microcanonical index

In the second part of the paper, the ABBV formula will be combined with
Picard-Lefschetz method to study how complex eigenvalue configurations compete
at leading order in the large-$N$ expansion of the microcanonical index. So
far, that is the only case where numerical results have been reported in the
literature Murthy:2020rbd ; Agarwal:2020zwm . The analysis will focus on the
toy example of the case of $\mathcal{N}=4$ SYM with $\sigma=\tau$ and no
flavour potentials. More general cases $\sigma\neq\tau$ can be analyzed by
using the results in appendix B.

Before entering in details let us explain in simple terms the conclusions of
our study. As we will explain below, the perturbative parameter controlling
the competition among saddles will be the exponential of minus the Bekenstein-
Hawking entropy of the Gutowski-Reall black holes. To ease presentation let us
denote this parameter as $g\,\equiv\,e^{-\frac{A_{H}}{4}}$, where $A_{H}$ is a
function of the single independent charge carried by this BPS black hole, and
which from now on we identify with the very same charge of the dual operators
in the field theory side. For charges of order $N^{2}$, it turns out that
$g<<1$ for large enough values of $N$, and thus only the two saddle
contributions dominate the complete sum over $P$. The contributions from other
complex saddles to the microcanonical index are exponentially suppressed and
then can be safely truncated when counting large operators. For small enough
charges, the $g$ starts to approach one and indeed for charges of order
$N^{\frac{2}{3}}$, $\log g$ becomes of order one. When the charges become of
order one in the large-$N$ expansion ((.e for multi-gravitons) the $g\sim 1$
and then every known complex saddle contributes.

Next, we introduce some notation and explain our conclusions in more detail.
Further details are postponed until section 5.3.

We will find that at large $N$ the microcanonical index

$d(\mathfrak{q})\,=\,\text{tr}_{E}(-1)^{F}$ (42)

defined as the trace over $\frac{1}{16}$-th BPS operators with charge
$E=N^{2}\,\mathfrak{q}$ for $\mathfrak{q}$ finite, is defined by the
competition of only two fixed points $(1,0)$ and $(1,1)$ out of the many
$(m,n)$’s. We should mention that the associated grand canonical index is a
function of a single chemical potential $\tau$, namely

$\mathcal{I}(\tau)\,=\,\,\sum_{E\,=\,0}^{\infty}\,d(\mathfrak{q})\,q^{E}.$
(43)

This quantity can be obtained from the more general ensemble
$\mathcal{I}\,=\,\mathcal{I}(\Delta_{1},\Delta_{2},\Delta_{3},\sigma,\tau)$
(See (6)) after imposing the following linear constraint among chemical
potentials $\Delta_{1}+\Delta_{2}+\Delta_{3}+\frac{\sigma+\tau}{2}\,=\,p$,
$p\,\in\,\mathbb{Z}$. For instance $p$ can be fixed to an integer that we
denote as $n_{0}=\pm 1$ e.g. $p=n_{0}=\pm 1$. The ensemble to be compared
_versus_ the numerical results obtained in references Murthy:2020rbd ;
Agarwal:2020zwm  corresponds to fixing
$\Delta_{1}\,=\,\Delta_{2}\,=\,\Delta_{3}$, $\sigma\,=\,\tau$ and finally $p$
to a specific integer value. We will fix $p\,=\,n_{0}=-1$ and proceed.

The formula (35) predicts that the microcanonical index is a linear
combination of contributions from each fixed point $P$ i.e.

$d(\mathfrak{q})\,=\,\sum_{P}d_{P}(\mathfrak{q})\,.$ (44)

The single contributions $d_{P}$ are extracted out of the grand canonical
index by a contour integral in $\tau$-plane. We denote such integral as
$C_{\eta}$. For each $P$, this integral can be decomposed in an integer
combination of $6$ Lefschetz-thimbles in $\tau$-plane. To understand whether a
fixed point $P$ contributes or not to $d(\mathfrak{q})$, we need to compute
the intersection numbers of $C_{\eta}$ with its $6$ thimbels. In that way one
reaches a large-$N$ expansion of competing exponential terms the form

$\begin{split}d(\mathfrak{q})&\,=\,\sum_{0\,\leq\,m,\,n\,<\,N\atop\text{gcd}(m,n)\,=\,1}\,\eta_{(m,n)}\,e^{H_{(m,n)}(\mathfrak{q})\,+\,\dots}e^{{\rm
i}\,(I_{(m,n)}(\mathfrak{q})\,+\,\ldots)}\,+\,\ldots\,,\end{split}$ (45)

where the magnitude of the $(m,n)$-th exponential is controlled by the Morse
function

$H_{(m,n)}(\mathfrak{q})\,=\,\frac{1}{m}\,H_{(1,0)}(\mathfrak{q})\,=\,\frac{1}{4m}\,A_{H}\,\geq\,0\,,$
(46)

and $I_{(m,n)}$ are two real functions. $H_{(m,n)}$ is the would-be entropy of
the supposed-to-exist gravitational configuration dual to $(m,n)$ fixed
points. 222222As computed in section 4.3 of Cabo-Bizet:2019eaf . The $\ldots$
denote sub-leading contributions in the large-$N$ expansion at finite
$\mathfrak{q}$. It denotes also potential contributions from other large-$N$
fixed points which are unknown to us at the moment.

The integer intersection numbers $\eta_{(m,n)}$ take three possible values

$\eta_{(m,n)}\,=\,\pm 1\quad\text{or }\quad 0\,.$ (47)

For instance for $(m,n)=(1,0)$ or $(1,1),$ $\eta_{(m,n)}\,=\,1$. 232323The
$(0,1)$ does not contribute at leading order in large $N$, so effectively we
can take $\eta_{(0,1)}=0$. The sufficient condition for $(m,n)\neq(0,1)$ and
$\chi_{1}(m+n)\neq 0$ to contribute, i.e for $\eta_{m,n}\neq 0$, is that
$C_{\eta}$ must enclose its corresponding Cady-like point $\tau=-\frac{n}{m}$
Cabo-Bizet:2019eaf . For the chosen $C_{\eta}$ that condition is equivalent to

$-2<-\,\frac{n}{m}\,<\,1\,.$ (48)

In that case the intersection numbers will be non vanishing i.e.
$|\eta_{(m,n)}|=1$. The $(m,n)$ fixed points for which (48) does not hold,
have vanishing intersection number $\eta_{(m,n)}=0$, and thus they do not
contribute to the counting. That implies that  _replica_ 242424The grand
canonical index we study has periodicity $\tau\to\tau+3$. This implies that
the effective action of saddle $(m,n)$ is mapped to the effective action of
$(m,n+3m)$ by such transformation. We say then that they are _replica_ of each
other.  fixed points of $(1,0)$ e.g. like $(1,3)$, do not contribute to the
microcanonical index expansion defined by the contour $C_{\eta}$, as expected.

Then, at large enough values of $N$ with fixed and finite
$\mathfrak{q}\,=\,\frac{E}{N^{2}}\,=\,\frac{\ell}{3N^{2}}$, and up to
exponentially suppressed contributions, the prediction coming from the ABBV
formula to the microcanonical index (45) is

$d(\mathfrak{q})\,=\,e^{\,H_{(1,0)}(\mathfrak{q})\,+\,\ldots}\,\times\,2\,\cos{\bigl{(}I_{(1,0)}(\mathfrak{q})\,+\,\ldots\bigr{)}}\,+\,\ldots\qquad\,.$
(49)

Here the $\ldots$ denote sub-leading corrections in large-$N$ expansion. Note
that in virtue of (46), the $(m,n)$ with $m>1$ are exponentially suppressed.
We will call (49) the  _two-saddles approximation_ to the microcanonical
index.

The Morse function $H_{(1,0)}$ coincides with the Bekenstein-Hawking entropy
of the dual AdS5 black hole Gutowski:2004ez . The cosine modulation

$\cos{\bigl{(}I_{(1,0)}(\mathfrak{q})\,+\,\ldots\bigr{)}}$ (50)

explains the oscillations of $d(\frac{\ell}{3N^{2}})$ as a function of $\ell$
at relatively large but still finite values of $N$ e.g. $N\sim 10$. These
oscillations have been recently observed in the $U(N)$ index 252525At leading
order in large $N$ approximation the $U(N)$ and $SU(N)$ indices are
equivalent. Differences come in at finite values of $N$. The two saddle
approximation can be used to approximate the microcanonical index of both,
$U(N)$ and $SU(N)$, finite values of $N$ e.g. $N\sim 10$. The approximation
should be even better for larger values of the charge $\ell\,$, or
equivalently for finite values of $\mathfrak{q}=\frac{\ell}{3N^{2}}$. This is
consistent with the outcome of the fit made in Agarwal:2020zwm  to their
numerical results Agarwal:2020zwm ; Murthy:2020rbd . The reason is that the
small parameter that controls the order of the corrections coming from other
fixed points is $e^{-H_{(1,0)}(\mathfrak{q})}$; and this parameter increases
when $\mathfrak{q}$ decreases, at fixed $N$. In particular it approaches
$e^{-\pi N^{2}\mathfrak{q}^{3/2}}\,\sim 1$ when $\mathfrak{q}\to 0$. Roughly
speaking, for values of charges $\ell$ (resp. $\mathfrak{q}$) smaller enough
than $N^{2/3}$ (resp. $\frac{1}{N^{\frac{4}{3}}}$) we need to start
considering contributions from other fixed points in order to increase
precision. Implementing this procedure to refine the counting lies beyond the
scope of this paper. Here we always assume that charges are large enough in
such a way that the two-fixed point approximation to the oscillations is a
good approximation (e.g. $\mathfrak{q}$ finite at large $N$).  via explicit
evaluation of the Fourier coefficients up to rank $N=10$ Murthy:2020rbd ;
Agarwal:2020zwm .

Here we claim that oscillations around the Bekenstein-Hawking curve, like the
example (50) are an interference effect produced by the superposition of
contributions of different complex fixed points of the matrix integral that
include the $(m,n)$ fixed points. However, for large enough value of $N$, but
not too large e.g. $N\sim 10$, the main contribution comes from the couple of
fixed points $(1,0)$ and $(1,1)$. This is, we claim that the oscillations are
a consequence of

  * •

the grand canonical index $\mathcal{I}(\tau)$ having an exact decomposition as
a sum (40) over fixed points $P$. 262626Note that $\mathcal{I}(\tau)$ depends
only on a single parameter, $\tau$.

In the region of charges where the two-fixed point approximation to the
oscillations can be trusted, we note that at large enough values of $N$ the
oscillatory pattern becomes negligible, and the curve $d(\mathfrak{q})$
approaches the profile of the Bekenstein-Hawking entropy of the Gutowski-Reall
solution Gutowski:2004ez ; Gutowski:2004yv  as shown in the plot in figure 3.

Figure 3:  In the figure in the top we have plotted values of the two-saddles
approximation to the graded number of states $d(\mathfrak{q})$ for $N=100$,
and the Bekenstein-Hawking entropy of the dual black holes. In this plot
$q\,=\,\mathfrak{q}$. In this scale, both plots are superposed and both are
the same. In this scale of charges and entropy, the two saddles oscillations
are negligible. For smaller values of $N$ oscillation become noticeable as
shown “experimentally” in Agarwal:2020zwm ( Murthy:2020rbd ) and as
analytically predicted by the two-saddles approximation. Something interesting
happens at the point $q=0$. This point encodes information about small
operators, i.e roughly speaking, operators with charges of order
$N^{\frac{2}{3}}$ or less. If we zoom in close enough into the point $q=0$,
oscillations start to enhance, first caused by the interference among
contributions coming from $(1,0)$ and $(1,1)$, and at some point contributions
from generic $(m,n)$ saddles start to become important and can not be
neglected. In principle, that flow evolution can be exactly probed (at least
the one driven by the competition among $(m,n)$ fixed points) with the help of
the ABBV formula, but implementing that with numerical precision lies beyond
the scope of the present paper and it is left for future work.

###  1.7 Final comments

Before entering in technical details, let us share few questions/problems that
we plan to study in the future:

  * 1.

To check the ABBV formula for $SU(N)$ $\mathcal{N}=4$ SYM against numerical
data for the lower rank $N=2,\,3,\ldots$.

  * 2.

As mentioned in the caption of figure 3, in principle, for $SU(N)$
$\mathcal{N}=4$ SYM, the flow evolution from large to small charges $E$,
always at large $N$, can be exactly probed (at least the approximation to the
exact flow driven by the competition among $(m,n)$ fixed points) with the help
of the ABBV formula. It would be extremely interesting to implement this with
numerical precision and to compare against numerical data.

  * 3.

To understand how this approach relates to the more orthodox large-$N$ methods
Brezin:1977sv ; Marino:2015yie . 272727A fresh revision of many related topics
have been recently given in Anninos:2020ccj . Is the continuum limit of the
fixed points $P$ related to large-$N$ cuts or not? Can one understand the
large-$N$ expansion of our approach from within the frame of loop-equations,
WKB expansions and/or resurgence? Answers to these questions could uncover
interesting observations. 282828Interesting approaches to the analysis of
unitary matrix models of the kind addressed in this paper, have been recently
put forward in GonzalezLezcano:2020yeb ; Copetti:2020dil . They suggest points
of contact between classical large-$N$ saddle point approaches and the one
presented in this paper.

  * 4.

Do these techniques give a hint on how to construct an explicit and tractable
representation of the Hilbert space of $\frac{1}{16}$-th BPS operators (in
$\mathcal{N}=4$ SYM language) in a large-$N$ expansion? The order zero problem
would be to try to understand the form of the large operators (in terms of
fundamental letters) that dominate the counting.

  * 5.

The exact solvability of this subsector of $SU(N)$ $\mathcal{N}=4$ SYM,
292929In relation with point $4.$, similar exact formulas, to the one here
presented, can be analogously found for other BPS observables. is indicative
of the existence of an analogous subsector in the string theory side of the
duality. It would be very interesting if such subsector could be explicitly
identified.

  * 6.

It would be very interesting to explore how this sector of large BPS operators
embeds and interact with non-BPS operators in the Hilbert space of weakly
coupled $\mathcal{N}=4$ SYM. This will allow us to probe how the complete
physical system behaves when perturbed away from zero temperature and
supersymmetry.

This paper is organized as follows. In section 2 we introduce the ABBV
equivariant integration formula. In section 3 contact is made with the
superconformal index. In section 4 we re-compute the effective actions of
$(m,n)$ fixed point and in the way fix previous ambiguities Cabo-Bizet:2019eaf
; Cabo-Bizet:2020nkr  associated to vector multiplet contributions. In the
second part of that section, in subsection 4.1, we compute the equivariant
Euler class of generic $(m,n)$ fixed points at leading order in large-$N$
expansion and compare against a known result in the simplest ensemble of
$SU(N)$ $\mathcal{N}=4$ SYM for the of $(1,0)$ fixed point. In section 5 the
ABBV formula and Picard-Lefschetz method are combined to analyze the
competition among fixed points. The focus is on the simplest ensemble of
$SU(N)$ $\mathcal{N}=4$ SYM that captures the growth of states of interest. In
appendix B we revisit the issue of regularization of the supersymmetric
partition function Cabo-Bizet:2018ehj  and show that there exists a scheme on
which the latter matches the superconformal index. Although we mainly work in
the case $\sigma\,=\,\tau$ that condition can be relaxed. In appendix B.2 we
show how the results obtained in this paper, can be easily generalized to
infinitely many other cases $\sigma\,\neq\,\tau$. In appendix D the deformed
action $S_{\lambda}$ that will be used to apply the equivariant localization
formula, is shown to correspond to a particular choice of regularization
scheme. Other supporting material is relegated to the remaining appendices.

##  2 Equivariant integration on a real torus

This section introduces the Atiyah-Bott-Berline-Vergne (ABBV) equivariant
integration formula. This formula can be used, for instance, to compute
integrals over symplectic manifolds. The relevant integrands can contain
exponentials of a linear combination of the so-called moment maps $f^{a}$.
These are real analytic functions whose gradients generate isometries and
preserve the symplectic structure of the manifold in question. The original
integral localises to lower dimensional ones over sub-manifolds that are left
invariant by the flows generated by the $f^{a}$’s. As pointed out by Atiyah
and Bott in Atiyah:1984px , one convenient feature of the formula is that the
linear combination of $f^{a}$’s can be complex. That will be advantageous to
our purposes.

###  2.1 The moment maps and the equivariant Euler class

Let $A$ be the Cartan torus of the gauge group $G$. Let
$\mathcal{M}\,=\,A\times B$ denote a real torus of rank
$2n\,\equiv\,2\text{rk}(G)$. The original Cartan torus is a middle dimensional
cycle $A$ in $A\times B$. $A\times B$ can be thought as a product of $n$ two-
tori with labels $i=1,\ldots,n$ i.e a double copy of a period $A$ with real
angles

$u^{i}_{1}\,\sim\,{u}^{i}_{1}\,+\,1\,,\qquad{u}^{i}_{2}\,\sim\,{u}^{i}_{2}\,+\,{1}\,.$
(51)

We find convenient to define the following symplectic 2-form on $A\times B$

$\omega_{\lambda}\,\equiv\,\sum_{i=1}^{n}du_{1}^{i}\,\wedge\,du_{2}^{i}\,\mathcal{O}_{i,\lambda}(u_{2}^{i})\,=\,\sum_{i=1}^{n}du^{i}\,\wedge\,dv^{i}\,\mathcal{O}_{i,\lambda}(v^{i})\,,$
(52)

with complex twisted differentials

$du\,\equiv\,du_{1}\,+\,\tau du_{2}\,,\qquad dv\,\equiv\,du_{2}\,.$ (53)

This 2-form allows to define integrals via the corresponding Liouville measure
$\frac{1}{n!}\,\omega_{\lambda}^{n}\,$,

$\int_{A\times
B}\,\frac{\omega_{\lambda}^{n}}{n!}\,e^{-\,\mathcal{S}}\,\equiv\,\int_{0}^{1}\,d\underline{u}_{1}\int^{1}_{0}\,d\underline{u}_{2}\,\prod_{i=1}^{n}\,\mathcal{O}_{i,\lambda}\,e^{-\mathcal{S}}\,.$
(54)

For any value of $\lambda$,
$\int^{1}_{0}du^{i}_{2}\,\mathcal{O}_{i\,\lambda}\,=\,1$.

Let $\mathcal{S}$ be an arbitrary linear combination of $d$ functions i.e.
$\mathcal{S}\equiv\sum_{a=1}^{d}{\rm i}t_{a}f^{a}\,$. The
$f^{a}:\mathcal{M}\mapsto\mathbb{R}$ with $a\,=\,1,\ldots,d\,$, are smooth and
real functions on $\mathcal{M}$. The functions $f^{a}$ are called moment maps
of a $d$-torus action on $\mathcal{M}$, if there exists a set of globally
defined $d$ vector fields $V^{a}$ on $\mathcal{M}$ that solve the following
set of equations

${\imath}(V^{a})\,\omega_{\lambda}\,=\,df^{a}\,.$ (55)

In that case, a particularization of the Atiyah-Bott-Beligne-Vergne (ABBV)
integration formula Atiyah:1984px  states that for $t^{a}\in\mathbb{C}$

$\begin{split}\int_{\mathcal{M}}\frac{\omega_{\lambda}^{n}}{n!}\,e^{-\,{\rm
i}\sum_{a}t_{a}\,f^{a}}\,=\,\sum_{P}\,\int_{P}\,\imath_{P}^{*}\Bigl{(}\frac{\omega_{\lambda}^{n}}{n!}\Bigr{)}\,\frac{e^{-\,{\rm
i}\sum_{a}t_{a}f^{a}(P)}}{E(\nu_{P})}\,.\end{split}$ (56)

Usually, the parameters $t^{a}$ are assumed to be pure imaginary numbers, and
thus the action in the exponent is a real function, a Morse function. That it
is possible to use complex $t^{a}$’s, was noticed by Atiyah and Bott in their
seminal work Atiyah:1984px . Here we build upon their suggestion (See footnote
32 below).

The $P$’s are connected sub-manifolds of $\mathcal{M}$ that solve the fixed-
point conditions

$\imath(V^{a})\,\omega_{\lambda}\,=\,df^{a}\,=\,0\,,\qquad
a\,=\,1,\ldots,d\,.$ (57)

The vector fields $V^{a}$ generate isometries of $\omega_{\lambda}$. That can
be shown by using Cartan’s formula upon the Lie derivative along $V^{a}$ of
$\omega_{\lambda}$, and the defining equation (55) i.e.

$\begin{split}\mathcal{L}_{V^{a}}\,\omega_{\lambda}&\,=\,i(V^{a})d\omega_{\lambda}\,+\,d(i(V^{a})\omega_{\lambda})\\\
&\,=\,d(i(V^{a})\,\omega_{\lambda})\,=\,d^{2}f^{a}\,=\,0\,.\end{split}$ (58)

The measure $\imath_{P}^{*}(\frac{\omega_{\lambda}^{n}}{n!})$ is the pullback
induced by the inclusion map $\imath_{P}:P\hookrightarrow\mathcal{M}$ over the
Liouville measure. Let us assume the $P$’s to be flat submanifolds of
$\mathcal{M}$. The $E(\nu_{P})$ is the equivariant Euler class of the normal
bundle $\nu_{P}$ to $P$ in $\mathcal{M}$. Let the real co-dimension of $P$ be
$2k$ then Niemi:1994ej ; Blau:1995rs

$E(\nu_{P})\,=\,\frac{1}{(2\pi)^{k}}\,\frac{\sqrt{\det\Bigl{(}\mathcal{H}_{\mu\nu}(p_{0})\Bigr{)}}}{\sqrt{\det\Bigl{(}(\imath_{P}^{*}\omega_{\lambda})_{\mu\nu}\Bigr{)}}}\,,\qquad
p_{0}\in P\,,$ (59)

where the greek indices $\mu$ and $\nu\,=\,1,\ldots,2k$ label the coordinates
normal to $P$ on $\mathcal{M}$ with $a=1,\ldots,k$. We note that for a complex
choice of equivariant parameters $t_{a}$’s the Hessian needs not to be real.
303030We obtained this expression by following the analysis in section 3 of
Niemi:1994ej  but implementing the new feature that we are working with
complex equivariant parameters and thus complex actions.

$(\imath_{P}^{*}\omega)_{\mu\nu}$ stands for the components of the two-form
$\imath_{P}^{*}\omega$. The Hessian matrix $\mathcal{H}$ is defined as

$\mathcal{H}\,=\,\left(\begin{array}[]{r@{}c|c@{}l}&\mbox{$\frac{\partial^{2}{\mathcal{S}}}{\partial
u^{a}\partial{u}^{b}}$}&\mbox{$\frac{\partial^{2}{\mathcal{S}}}{\partial
u^{a}\partial{v}^{b}}$}\\\
\hline\cr&\mbox{$\frac{\partial^{2}{\mathcal{S}}}{\partial
v^{a}\partial{u}^{b}}$}&\mbox{$\frac{\partial^{2}{\mathcal{S}}}{\partial
v^{a}\partial{v}^{b}}$}\end{array}\right)\,,\qquad a,b=1\,,\ldots,k\,,$ (60)

where the twisted derivatives are

$\partial_{\underline{u}}\,\equiv\,\partial_{\underline{u}_{1}}\,,\qquad\partial_{\underline{v}}\,\equiv\,\partial_{\underline{u}_{2}}-\tau\,\partial_{\underline{u}_{1}}\,.$
(61)

This twisted derivatives need not to be interpreted as coming from a complex
change of variables in $A\times B$. The coordinate will always be assumed to
be $u^{i}_{1}$ and $u^{i}_{2}$.

It is not rare to find continuous fixed sub-manifolds and not just isolated
fixed points $P$. For example, for $U(N)$ $\mathcal{N}=4$ SYM the fixed
manifolds are not points but two dimensional real surfaces. To avoid this
extra technical involvement only $SU(N)$ gauge groups will be analyzed.
313131The $U(N)$ case requires a minor redefinition of the closed $2$-from
$\omega_{\lambda}$ in (52) that we will not try to obtain in this paper.

###  2.2 The fixed point conditions

Let us focus on the case of a two-torus action i.e fix $d=2$. Fix the moment
maps $f^{a}$ to be the real and imaginary parts of the action $\mathcal{S}$
and the equivariant parameters accordingly i.e.

$f^{1}\,=\text{Re}\,\mathcal{S}\,,\qquad
f^{2}\,=\,\text{Im}\,\mathcal{S}\,,\qquad{\rm i}t_{1}=t_{2}=1\,.$ (62)

As announced before our choice of equivariant parameters is complex. 323232As
pointed out in Atiyah:1984px  in page 15 in the paragraph below the comment
numbered as 2). For the case of a 2-torus action, the component
$u_{a}\in\mathbb{C}$ in the $l$-array mentioned in that paragraph (in this
case $l=2$), corresponds to the ${\rm i}t_{a}$ in here.

From this choice of moment maps and equivariant parameters $t_{a}$ in (62), it
is immediate to see that the fixed point conditions (57) are the saddle-point
condition of the corresponding action $\mathcal{S}$ i.e.
$\partial_{\underline{u}_{1}}\mathcal{S}=\partial_{\underline{u}_{2}}\mathcal{S}=0$.
Or equivalently

$\partial_{\underline{u}}\,\mathcal{S}\,=\,0\,,\qquad\partial_{\underline{v}}\,\mathcal{S}\,=\,0\,,$
(63)

when written in terms of the twisted derivatives $\partial_{\underline{u}}$
and $\partial_{\underline{v}}$. This form of the equations will simplify the
analysis later on.

###  2.3 Examples of fixed points: quiver gauge theories

Let $\mathcal{S}$ be a smooth and double periodic extension to the double
torus $A\times B$, of the effective action $S(u)$ in (11).

Any such $\mathcal{S}$, with the spectrum of gauge charges $\rho$ of the
adjoint and/or bi-fundamental of $SU(N)$ nodes, contains a simple set of
stationary configurations that carries a finite Abelian group structure of
rank $N$. That was shown in Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr . See
section $6$ of that reference for a proof. 333333For finite $N$ there are a
finite number of these fixed points. For instance for solutions with
$\mathbb{Z}_{N}$ structure, which are those labelled by a couple of integers
$(m,n)$ with gcd$(m,n)=1$, and rank $N=2$ there are only three solutions
$(1,0)$, $(1,1)$ and $(0,1)$. For simplicity in this paper we will always
assume $N$ to be prime. In that case, among solutions carrying finite Abelian
group representations, only those carrying $\mathbb{Z}_{N}$ representation
exist. It is possible that there are other kind of solutions that do not carry
these previously mentioned finite Abelian group structure.

Let us assume the action $\mathcal{S}$ to be of the form

$\mathcal{S}(u)\,=\,\sum_{\alpha}\sum_{\rho_{\alpha}\,\neq\,0}\mathcal{S}_{\alpha}(\rho(u))\,-\,\log\kappa_{G}\,,$
(64)

where for $z\,=\,z_{1}\,+\,\tau z_{2}$ and the single-particle action
$\mathcal{S}_{\alpha}(\rho(u))$ is a smooth and double periodic complex
function of the real components $z_{1}$ and $z_{2}$ of
$z_{\alpha}(u)=\rho_{\alpha}(u)+\Delta_{\alpha}$, and $\rho_{\alpha}$ are
weights with respect to the total gauge group
$G={\oplus}_{\beta}\,SU_{\beta}(N)$. The index ${\beta}=1,\ldots,\nu$ labels
the gauges nodes. $\rho_{\alpha}$ is the array of $\nu(N-1)$ charges under the
action Cartan generators of $G$. We will always assume multiplets in the the
adjoint, or the bi-fundamental representations. The variable $u$ stands for
the array of $\nu(N-1)$ complexified Cartan angles i.e the complexified Cartan
angles of all gauge nodes are collected in a single array of dimension
$\nu\times(N-1)$ and denoted by the letter $u$ in this equation.

For any smooth and double periodic single particle action
$\mathcal{S}_{\alpha}$ the solutions to (63) include configurations

$u^{i\beta}\,=\,u^{i}\,=\,\Bigl{(}\frac{i}{N}\,-\,\frac{N+1}{2N}\Bigr{)}\,T\,,\qquad
i\,=\,1,\ldots,N-1\,.$ (65)

To simplify presentation we can insert an extra auxiliary $N$-th gauge
variable in some intermediate equations

$u^{N}\,=\,\Bigl{(}1\,-\,\frac{N+1}{2N}\Bigr{)}\,T\,.$ (66)

In the previous equations

$T\,=\,m\tau+n\,,\qquad~{}0\leq m\leq N-1\,,\qquad~{}0\leq n\leq N-1\,,\qquad
gcd(m,n)=1\,.$ (67)

As the ansatz (65) is the same for every node $\beta$, effectively, we can
consider all weights $\rho_{\alpha}$ to be weights of the Adjoint of the
diagonal $SU(N)$ subgroup of $G$. Thus, for any such quiver gauge theory, the
effective action of ansatz (65) can be written in the convenient form

$\nu\,\sum_{\alpha}\sum_{\rho\,\in\,\text{Adj}(SU(N))\atop\rho\,\neq\,0}\mathcal{S}_{\alpha}\Bigl{(}\rho(u)\Bigr{)}\,\,=\,\,\nu\,\sum_{\alpha}\sum_{i,\,j\,=\,1\atop
i\,\neq\,j}^{N}\mathcal{S}_{\alpha}\Bigl{(}\frac{i-j}{N}\,T\Bigr{)}\,,$ (68)

where all the effective weights are in the adjoint of the diagonal
$SU(N)\subset G$ and the index $\alpha$ runs over every multiplet in the
quiver theory.

In the large-$N$ limit the set of fixed points (65) becomes a continuum i.e.
$\frac{i}{N}\to x\in[0,1)$ and $\sum_{i}\to N\int^{1}_{0}dx$. The effective
action in that limit is computed by the Fourier averages along the period
$T=m\tau+n$

$\begin{split}\mathcal{S}_{\text{eff}}(m,n)&\,=\,\,\sum_{\alpha}\mathcal{S}_{\text{eff}\,\alpha}(m,n)\,\\\
&\,\equiv\,\nu
N^{2}\,\sum_{\alpha}\int_{0}^{1}\int_{0}^{1}dxdy\,\mathcal{S}_{\alpha}\bigl{(}(x-y)T\bigr{)}\\\
&\,=\,\nu
N^{2}\,\sum_{\alpha}\int_{0}^{1}dx\,\mathcal{S}_{\alpha}(xT)\,.\end{split}$
(69)

The index $\alpha$ runs over every multiplet in the theory. Note that we have
not specified the form of the action $\mathcal{S}_{\alpha}$ yet.

##  3 The superconformal index

As mentioned in section 1.3 the first step to obtain a formula for the
superconformal index $\mathcal{I}$ is to construct the action $S_{\lambda}$
that together with the measure $\omega_{\lambda}$ defines the
$\lambda$-independent quantity $\mathcal{I}_{\lambda}$ introduced in equation
(34). The real and imaginary parts of the action $S_{\lambda}$ must be smooth
moment maps in $A\times B$ and thus, we can use the ABBV equivariant
integration formula to compute $\mathcal{I}_{\lambda}$ at any value of
$\lambda$ in between zero and one, including $0$ and excluding $1$. If by
construction $\mathcal{I}_{\lambda}$ reduces to $\mathcal{I}$ in the limit
$\lambda\,\to\,1^{-}$, and if the former happens to be independent of
$\lambda$, then the ABBV formula for $\mathcal{I}_{\lambda}$ is guarantied to
compute $\mathcal{I}$. 343434 In specific regions of chemical potentials for
which there are no poles emerging in the limit $\lambda\to 1^{-}$(and
$\Lambda=1^{-}$) of the integrand of $\mathcal{I}_{\lambda}$ inside the
fundamental domain $A\times B$. In case there are poles, the formula needs to
be ammended as explained in section 1.3.

To simplify the presentation, at first we will focus on a single multiplet
$\alpha$ with gauge charge vector $\rho_{\alpha}$. Later on a theory will be
assembled.

For $0\,\leq\,\lambda\,<\,1$ we define

$e^{-S_{\lambda\alpha}}\,=\,\Gamma_{\lambda}(z_{\alpha})\,\equiv\,\frac{{R}_{\Lambda}^{-1}(z_{\alpha})}{Q_{0\Lambda}(z_{\alpha})}\,\Bigl{(}P_{0\Lambda}(z_{\alpha})\Bigr{)}^{\,z_{2\alpha}(\\{u_{2}\\}_{\lambda})}\,.$
(70)

Here
$z_{\alpha}\,=\,z_{\alpha}(u)\,=\,\rho_{\alpha}(u)\,+\,\Delta_{\alpha}\,$, and
$\Delta\,=\,\Delta_{\alpha 1}\,+\,\tau\Delta_{\alpha 2}$ with real
$\Delta_{\alpha 1}$ and $\Delta_{\alpha 2}$. From now on we assume
$-1\,<\,\Delta_{2\alpha}\,<\,1\,$.

The double periodic functions $P_{0\Lambda}$ and $Q_{0\Lambda}$ are defined
via the logarithms

$\log P_{0\Lambda}(z)\,=\,\log P_{0\Lambda}(z_{1},z_{2})\,\equiv-\frac{{\rm
i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\Lambda^{|m|\,+\,|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,,$ (71)

and

$\log Q_{0\Lambda}(z)\,=\,\log
Q_{0\Lambda}(z_{1},z_{2})\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop
m\neq 0}{\sum}}\;\Lambda^{|m|\,+\,|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,.$ (72)

Here $0\,\leq\,\Lambda\,<\,1$. The role of $\Lambda$ is explained in appendix
F.

The pre-factor $R_{\Lambda}^{-1}$ has two contributions. One is a power of
$q={\bf e}(\tau)$ and the other a pure $\tau$-independent phase

$R_{\Lambda}^{-1}(z)\,=\,R_{\Lambda}^{-1}(z_{1},z_{2})\,=\,q^{-E_{\Lambda}}\,\,{\bf
e}\bigl{(}-\frac{\Phi_{\Lambda}}{2}\bigr{)}\,.$ (73)

Explicit definitions of $E_{\Lambda}$ and $\Phi_{\Lambda}$ are given in
appendix D equation (238). But for concrete purposes we will only be
interested in their form at $\Lambda\,=\,1^{-}$.

From (70) we obtain the following expression for the action of a generic
chiral multiplet

$\begin{split}S_{\lambda\alpha}&\,=\,\,\log
Q_{0\Lambda}(z_{\alpha}(u))\,-\,z_{2\alpha}(\\{u_{2}\\}_{\lambda})\,\log
P_{0\Lambda}(z_{\alpha}(u))\\\ &\qquad\qquad+\,2\pi{\rm i}\tau
E_{\Lambda}(z_{2\alpha}(u))\,+\,\pi{\rm
i}\Phi_{\Lambda}(z_{\alpha}(u))\,.\end{split}$ (74)

We define the bracket function of a real variable $x$ as

$\\{x\\}_{\lambda}\,\equiv\,B_{1\lambda}(\\{x\\})+\,\frac{1}{2}\,.$ (75)

It is a smooth version of the usual $\\{x\\}\,\equiv\,x\,-\,\lfloor x\rfloor$.
$\\{x\\}$ is almost equal to $\lim_{\lambda\,\to\,1^{-}}\\{x\\}_{\lambda}\,$.
The difference is located at integer values of $x$. For such $x$’s

$\lim_{\lambda\,\to\,1^{-}}\\{x\\}_{\lambda}\,\,=\,\frac{\\{x^{+}\\}\,+\,\\{x^{-}\\}}{2}\,.$
(76)

We define the bracket of the vector $\underline{x}$ as the vector of the
brackets of the components $\\{\underline{x}^{i}\\}$. The function
$B_{1\lambda}$ is defined in (174).

The phase function $\Phi_{\Lambda}(z(u))$ can be any smooth function that in
the limit $\lambda\,\to\,1^{-}$ equates to a specific discrete periodization
of the phase $\Phi$ in identity (292). As mentioned before, one such an
example is given in the second line of equation (238). However, for concrete
purposes there is no unique such choice i.e one can choose any real
$\Phi_{\Lambda}(z(u))$ s.t. in the limit $\Lambda\to 1^{-}$ the exponential
${\bf e}\Bigl{(}\frac{\Phi_{\Lambda}}{2}\Bigr{)}$ goes to

$\begin{split}\frac{{\bf
e}\Bigl{(}\frac{1}{2}\,B_{1}(\rho(u_{1})+\Delta_{1}-K)\Bigl{(}B_{2}(\lfloor\rho(\\{u_{2}\\})+\Delta_{2}\rfloor+1)\Bigr{)}\Bigr{)}}{\,{\bf
e}\Bigl{(}\frac{1}{2}\,B_{1}(\rho(u_{1})+\Delta_{1}-K)\Bigl{(}B_{2}(\lfloor\Delta_{2}\rfloor+1)\Bigr{)}\Bigr{)}}\,.\end{split}\,$
(77)

For large-$N$ evaluation, we will take the weights $\rho$ to be the ones of
the adjoint representation of $U(N)$ which are labelled by two integers
$i,j=1,\ldots N$ and are s.t. $\rho(u)\,\equiv\,u_{ij}\,=\,u^{i}-u^{j}$.
Although we are interested in $SU(N)$, in order to obtain closed expressions
for the contribution coming from the phase function $\Phi_{\Lambda}$, it will
be technically convenient to effectively work with $U(N)$ instead of $SU(N)$.
In virtue of identity (7), the final contribution is bound to be the same at
leading order in large $N$ expansion. Thus, to compute the large-$N$
contribution to the effective action of an $(m,n)$ fixed point out of equation
(77) we will consider

$\rho(u_{1})\,=\,u^{i}_{1}\,-\,u^{j}_{1}\,,\qquad\rho(\\{u_{2}\\})\,=\,u_{2\\{i\\}\\{j\\}}\,\equiv\,\\{u^{i}_{2}\\}-\\{u^{j}_{2}\\}\,.$
(78)

The object $K\in\mathbb{Z}$ is a choice of branch. It is an arbitrary piece-
wise constant and integer function, which guaranty the exponent to be double
periodic. When computing large-$N$ objects by using the continuum limit,
different definitions of $K$ could seem to lead to different contributions to
the phase of the effective actions of $(m,n)$ fixed points. However, coming
from the discrete finite $N$ case, one can check that the difference between
any two such choices is always an integer multiple of $2\pi{\rm i}$, as it is
bound to be the case. This implies that in order to compute values of the on-
shell actions, we can even choose a branch $K$ that breaks double periodicity,
and that could even depend on specific saddles. This is because the value of
the imaginary part of the effective action is only determined up to an integer
multiple of $2\pi{\rm i}$, thus we can always compute the double periodic
value of the exponential by relaxing double periodicity of the exponent up to
$2\pi{\rm i}$ identifications. In due time 353535This is a trick and it will
only be used in the very end, to evaluate effective actions of specific fixed
points. we will use this feature for technical convenience.

Before moving on, let us summarize three relevant properties of
$e^{-S_{\lambda}(u)}$ that follow from definition (70)

  * a.

At finite $0\leq\lambda<1$ and $\Lambda$, it is smooth and double periodic,
but non-meromorphic in $\underline{u}$.

  * b.

In the limits $\lambda=1^{-}$ and $\Lambda=1^{-}$, it becomes locally
meromorphic i.e.

$\Gamma_{\lambda}(z_{\alpha}(u))\,=\,\Gamma_{\text{e}}\bigl{(}z_{1\alpha}\,+\,\tau
z_{2\alpha}(\\{u_{2}\\})\,+\,\tau;\,\tau,\,\tau\bigr{)}\,.$ (79)

  * c.

For any $0\leq\lambda<1$ and $\Lambda\,=\,1^{-}$, it matches the elliptic
gamma function in the limit to the $A$-cycle $u^{i}_{2}\,=\,0^{+}$ i.e at
$\Lambda=1^{-}$

$\Gamma_{\lambda}(z_{1\alpha}(u)\,+\,\tau
z_{2\alpha}(0^{+}))\,=\,\Gamma_{\text{e}}\bigl{(}z_{1\alpha}(u)\,+\,\tau\Delta_{2\alpha}\,+\,\tau;\,\tau,\,\tau\bigr{)}\,.$
(80)

Property a. b. and c. were the initial assumptions needed in order to reach
(33).

###  3.1 Half of the fixed point conditions: A Bethe Ansatz form

In this subsection we show that for the choice of action $S_{\lambda}$ to be
defined below, half of the fixed point conditions can be written as a Bethe
Ansatz condition.

Now that we have defined $S_{\lambda\alpha}$ for a multiplet $\alpha$, we can
further refine the definition of $\mathcal{I}_{\lambda}$ initially given in
equation (34) for the case of a generic quiver theory

$\begin{split}\mathcal{I}_{\lambda}&\,=\,\int_{A\times
B}\,\frac{\omega^{n}_{\lambda}}{n!}\,e^{-\,S_{\lambda}(u)}\,\\\
&\equiv\,\kappa_{G}\int_{0}^{1}d\underline{u}_{1}\int_{0}^{1}d\underline{u}_{2}\,\prod_{i=1}^{n}\,\mathcal{O}_{i,\lambda}\,\prod_{\alpha}\prod_{\rho_{\alpha}\,\neq\,0}\Gamma_{\lambda}(z_{1\alpha}(u)\,+\,\tau\,z_{2\alpha}(u))\,.\end{split}$
(81)

The action of the complete theory is

$S_{\lambda}(u)\,\equiv\,\sum_{\alpha}\sum_{\rho_{\alpha}}\,S_{\lambda\alpha}(u)\,,$
(82)

where $z_{1\alpha}\,=\,\rho_{\alpha}(u_{1})\,+\,\Delta_{1\alpha}$ and
$z_{2\alpha}\,=\,\rho_{\alpha}(u_{2})\,+\,\Delta_{2\alpha}$. The function
$\Gamma_{\lambda}(z_{\alpha})=e^{-S_{\lambda\alpha}}$ was defined in (70). It
is a combination of $P_{0\Lambda}$, $Q_{0\Lambda}$ and Polylogs.

The main goal of this subsection is to complement the explanations given in
subsection 1.4. To do so let us highlight the explicit form of half of the
fixed point conditions. Once $\Lambda=1^{-}$, such half takes the form

$\begin{split}\partial_{v^{i}}\,\mathcal{S_{\lambda}}&\,=\,\sum_{\alpha}\sum_{\rho_{\alpha}}\rho^{i}\,\mathcal{O}_{i,\lambda}\log
P_{0\Lambda=1^{-}}(z_{\alpha}(u))\,+\,\text{c.t.}\,\\\
&\,=\,\mathcal{O}_{i,\lambda}\,\log B^{i}\,+\,\text{c.t.}\,=\,0\,.\end{split}$
(83)

Recall that
$\partial_{v^{i}}\,\equiv\,\partial_{u^{i}_{2}}\,-\,\tau\partial_{u^{i}_{1}}\,$.
In reaching (83) we have used the definition of $S_{\lambda\alpha}$ (70), the
equations in appendix F.3, and also (239). The c.t. stands for contact terms
contributions. The $B^{i}$ stands for

$\begin{split}B^{i}&\,\equiv\,\prod_{\alpha}\,\prod_{\rho_{\alpha}}\,P^{\rho_{\alpha}^{i}}_{0\Lambda=1^{-}}(z_{\alpha}(u))\,.\end{split}$
(84)

The vector multiplets do not contribute to the definition of $B^{i}$ and can
be ignored. That can be seen even before taking the limit $\Lambda=1^{-}$. It
follows from property $\log P_{0\Lambda}(z)\,=\,\log P_{0\Lambda}(-z)$, and
from the fact that vector multiplets carry the weights of the adjoint
representation which is a real representation. For the same reason, the same
happens for the Bethe Ansatz equations of Closset:2017bse ; Benini:2018mlo .

From the positivity property (21) i.e. $\mathcal{O}_{i,\lambda}>0$ and the
condition $\partial_{v^{i}}S_{\lambda}=0$, it follows that the fixed points
$P$ must necessarily solve a Bethe Ansatz like condition

$\log B^{i}(P)\,=\,0\,\implies\,B^{i}(P)\,=\,1\,.$ (85)

Moreover, they must not intersect the positions of the contact terms c.t..
363636In case the latter do not vanish identically after summing over the
corresponding matter content, which is the case for the examples we have
studied. Note that (85) is only half of the fixed point conditions.

The explicit form of these c.t. contributions can be obtained from identity
(273). From the sufficient condition

$\forall_{i\,,\,j\,,\,\alpha}\,u^{i}\,\,-u^{j}\,\neq\,\Delta_{\alpha}\,\implies\,\text{c.t.}\,=\,0\,,\,\qquad
u^{N}\,=\,-\sum^{N-1}_{i=1}\,u^{i}\,.$ (86)

where $i\,,\,j\,=\,1\,,\,\ldots\,N\,$, it follows that c.t.$\,=\,0$ is solved
by demanding the constraint to the left of the implication symbol upon any
solution of (85). For the theories we analyze, and the solutions of (85) that
we are aware of (See subsection 2.3), that condition can be always satisfied.
373737For instance by restricting to certain domains of the
$\Delta_{\alpha}$’s and/or by assuming $N$ to be prime. From now on we assume
such choices and thus ignore the contact terms.

Before moving on, let us complete the proof of the first equation (38),
$\partial_{\lambda}S_{\lambda}(P)\,=\,0$. Recall that $\partial_{\lambda}$ in
that equation means at fixed $P$. This equation follows trivially from (85) as
the only contribution of $S_{\lambda}(P)$ that depends explicitly on $\lambda$
is a linear combination of the logarithm of (84) which by definition vanishes
at any fixed point $P$.

###  3.2 The final form of the equivariant Euler class

In this subsection we refine the definition of the equivariant Euler class
associated to the action $S_{\lambda}$ and the symplectic form
$\omega_{\lambda}$.

For simplicity we focus on the generic case in which $P$ is a point. That
means that $k=n$. From the definition of $B^{i}$ given in (84) and identity
(273), it follows that for $\Lambda=1^{-}$ and after avoiding contact terms

$\frac{\partial^{2}S_{\lambda}}{\partial{v^{i}}\partial{v^{j}}}\,=\,0\,.$ (87)

Recall that
$\partial_{\underline{u}}\,\equiv\,\partial_{\underline{u}_{1}}\,,\,\partial_{\underline{v}}\,\equiv\,\partial_{\underline{u}_{2}}-\tau\,\partial_{\underline{u}_{1}}\,$.
Thus from the definition of the Hessian $\mathcal{H}_{\mu\nu}$ of the action
$S_{\lambda}$ (60) and (87) it follows that only the off-diagonal blocks
contribute to the corresponding determinant and thus

$\sqrt{\det\Bigl{(}\mathcal{H}_{\mu\nu}(p_{0})\Bigr{)}}\,=\,\pm\,{\rm
i}^{n}\,\det\Bigl{(}\frac{\partial^{2}B^{i}(u)}{\partial
u^{j}}\Bigr{)}\,\times\,\prod_{i=1}^{n}\mathcal{O}_{i,\lambda}\,.$ (88)

$\sqrt{\det\Bigl{(}(\imath_{P}^{*}\omega_{\lambda})_{\mu\nu}\Bigr{)}}\,=\,\sqrt{\det\Bigl{(}(\omega_{\lambda})_{\mu\nu}\Bigr{)}}\,=\,\pm\,\prod_{i=1}^{n}\mathcal{O}_{i,\lambda}\,.$
(89)

From these equations and the definition of $E(\nu_{P})$ in (59), it follows
that the equivariant Euler class equates to

${E}(\nu_{P})\,=\,\frac{1}{(2\pi{\rm i})^{n}}\det{\frac{\,\partial
B^{i}(P)}{\partial u_{1}^{j}}}\,.$ (90)

In this definition we have fixed an overall constant ambiguity by comparing
against other results in the literature. Note that this expression does not
depend on $\lambda$, thus it implies the second equation in (38).

###  3.3 The final integration formula: independence of $\lambda$

In this subsection we show that the integral $\mathcal{I}_{\lambda}$ is
independent of $\lambda$.

Again focus on $\Lambda=1^{-}$. Assume that $p_{\lambda}$ is a fixed point
whose position depends on $\lambda$, i.e.

$p_{\lambda}\,=\,(\underline{u}_{\lambda},\underline{v}_{\lambda})\,,\qquad
p_{\lambda+\delta\lambda}\,=\,(u_{\lambda}+\delta\lambda\,{u}^{(1)},v_{\lambda}+\delta\lambda\,{v}^{(1)})\,.$
(91)

It then follows that

$B^{i}(p_{\lambda})\,=\,B^{i}(p_{\lambda+\delta\lambda})\,=\,1\,.$ (92)

The linear variation of this equation gives

$\sum_{j=1}^{n}\,u^{(1)j}\,\frac{\partial
B^{i}(p_{\lambda})}{\partial{u^{j}}}+\,v^{(1)j}\,\frac{\partial
B^{i}(p_{\lambda})}{\partial{v^{j}}}\,=\,\sum_{j=1}^{n}\,u^{(1)j}\,\frac{\partial
B^{i}(p_{\lambda})}{\partial{u^{j}}}\,=\,0\,,$ (93)

The first equality follows from (273) and from the fact that only solutions of
(92) which do not collide with contact terms are fixed points. The second
equality is telling us that the matrix $\frac{\partial
B^{i}(P_{\lambda})}{\partial u_{1}^{j}}$ has a zero eigenvalue and thus

$\det{\frac{\,\partial B^{i}(p_{\lambda})}{\partial
u_{1}^{j}}}\,=\,0\,=\,{E}(\nu_{p_{\lambda}})\,.$ (94)

In virtue of ABBV theorem, this can not be true for a fixed point
$p_{\lambda}$ that is not smoothly connected with others. Thus we conclude
that for such class, $p_{\lambda}$ can not depend on $\lambda$ i.e.

$\frac{dp_{\lambda}}{d\lambda}\,=\,0\,.$ (95)

When $p_{\lambda}$ is part of a connected set of fixed points $P$, Atiyah-Bott
localization theorem tells us that the matrix $\frac{\,\partial
B^{i}(p_{\lambda})}{\partial u_{1}^{j}}$ can indeed have zero eigenvalues and
the corresponding eigenvectors should be tangent vectors to the connected
manifold $P$ at $p_{\lambda}$. The second equality in (93) is telling us that
the variation induced by $\lambda$ is constrained to move along the connected
manifold of fixed points $P$. But as the Atiyah-Bott prescription includes a
covariant integration over such manifold, it then follows that even when
$p_{\lambda}$ belongs to a continuum set of fixed points $P$, the associated
contribution to the equivariant integral is independent of $\lambda$. Of
course it would be great to check this in a concrete fixed point, but we do
not have an example of this type of solution yet.

In summary, for $\Lambda\,=\,1^{-}$ the integral (81) does not depend on
$\lambda$, and in specific chambers in the space of potentials
$\Delta^{\prime}s$, it is by definition equal to the limit
$\underset{\lambda\,\to\,1^{-}}{\lim}\mathcal{I}_{\lambda}$ (at
$\Lambda=1^{-}$). If we restrict attention to those specific regions in the
domain of potentials $\Delta$’s then

$\mathcal{I}\,\,=\,\mathcal{I}_{\lambda}\,=\,\sum_{P}\,\frac{e^{-S_{\lambda}(P)}}{\,E(\nu_{P})}\,.$
(96)

$E{(\nu_{P})}$ was defined in (90). We should point out that generic fixed
points $P$ need to obey the saddle conditions of $S_{\lambda}$. The Bethe
Ansatz condition (85) is only half of the latter. $(m,n)$ fixed points satisfy
both conditions trivially, but there could be other kind of solutions not
carrying Abelian group structure.

##  4 The leading contributions in large-$N$ expansion

In this subsection we revisit the evaluation of the large-$N$ effective action

$S_{\text{eff}}(p)\,=\,S_{\lambda}(p)\,,$ (97)

when $p$ is an $(m,n)$ fixed point. In Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr
the answer for (97) was fixed up to a $\tau$ independent phase. Here we
constraint such ambiguity.

In the large-$N$ limit, we can introduce the $N$-th auxiliary variable $u^{N}$
that at $(m,n)$ saddles is defined as (66) and work with the weights of $U(N)$
instead of with the ones of $SU(N)$. After substituting $z$ by
$z_{\alpha}(u)=u_{ij}\,+\,\Delta_{\alpha}$ with the ansatz

$u_{ij}=\frac{i-j}{N}\,(m\tau\,+\,n)\,,$ (98)

the discrete variables become continuum and the sums become integrals i.e.

$\frac{i}{N}\rightarrow x\in(0,1]\,,\qquad\sum_{i=1}^{N}\rightarrow
N\int_{0}^{1}dx\,.$ (99)

In the large $N$ for $m\neq 0$ one obtains

$\begin{split}\int_{0}^{1}\int_{0}^{1}dxdy\,\log
Q_{0\Lambda}\bigl{(}(x-y)(m\tau+n)+\Delta_{\alpha}\bigr{)}&\,=\,\frac{\pi{\rm
i}\,B_{3\Lambda}(\\{m\Delta_{\alpha 1}\,-\,n\Delta_{\alpha
2}\\})}{3m(m\tau\,+\,n)^{2}}\,,\\\ \int_{0}^{1}\int_{0}^{1}dxdy\,\log
P_{0\Lambda}\bigl{(}(x-y)(m\tau+n)+\Delta_{\alpha}\bigr{)}&\,=\,-\frac{\pi{\rm
i}\,B_{2\Lambda}(\\{m\Delta_{\alpha 1}\,-\,n\Delta_{\alpha
2}\\})}{m(m\tau\,+\,n)}\,,\\\ \int_{0}^{1}\int_{0}^{1}dxdy\,2\pi{\rm
i}\tau\,E_{\Lambda}\bigl{(}(x-y)(m\tau+n)+\Delta_{\alpha}\bigr{)}&\,=\,\frac{\pi{\rm
i}\tau}{6}\,\bigl{(}2\Delta_{\alpha 2}^{3}\,-\,\Delta_{\alpha
2}\bigr{)}\,+\,\ldots\,.\end{split}$ (100)

The Polylogs in (100) become periodic Bernoulli polynomials at
$\Lambda\,=\,1^{-}$. The $\ldots$ stands for corrections that go to zero at
$\Lambda=1^{-}$.

From the limit (77) of $\Phi_{\Lambda}$ and after making a specific choice of
branch $K$ that we will detail below, we obtain a definite answer for the
phase

$\lim_{\Lambda\to 1^{-}}\sum_{i,j\,=\,0}^{N-1}\,\pi{\rm
i}\Phi_{\Lambda}\Bigl{(}\frac{i-j}{N}\,(m\tau+n)+\Delta_{\alpha}\Bigr{)}\,=\,\pi{\rm
i}N^{2}\Phi(\Delta_{a})\,.$ (101)

Let us go step by step. For convenience we work in continuum variables $x$ and
$y$ instead of discrete ones $i$ and $j$. To ease notation, let us drop the
index $\alpha$ for a moment. We recall that
$\Delta\,=\,\Delta_{1}\,+\,\tau\,\Delta_{2}\,$. For $0\,<\,\Delta_{2}\,<\,1$
we can choose (for $m>0$)

$K\Bigl{(}(x\,-\,y)\,(m\tau\,+\,n)\,+\,\Delta\Bigr{)}\,=\,2m\lfloor
m(\Delta_{1}\,-\,1)\,-\,n(\Delta_{2}\,-\,1)\rfloor\,+\,m(m\,+\,1)\,$ (102)

for
$1\,-\,\Delta_{2}\,<\,(mx\,-\,my)\,<\,{1-\sqrt{1\,-\,\frac{1}{2m}}\,\Delta_{2}}$
and zero otherwise. Below it will become clear why we have made this choice of
$K$. A similar $K$ can be defined for the case $-1<\Delta_{2}<0$.

For $-1\leq\Delta_{2}<1$ we can take the definition of $\Phi_{\Lambda}$ in the
exponent in the second line of (77) and obtain

$\begin{split}\pi{\rm i}N^{2}\Phi(\Delta)&\,=\,\pi{\rm
i}N^{2}\Phi(\Delta_{1},\Delta_{2})\,\\\ &\equiv\,\pi{\rm
i}N^{2}\,\int_{0}^{1}\int_{0}^{1}dxdy\,\bigl{(}B_{1}(nx\,-\,ny\,+\,\Delta_{1})\,-\,K\bigr{)}\times\\\
&\qquad\qquad\hfill\Bigl{(}B_{2}(\lfloor\\{mx\\}-\\{my\\}\,+\,\Delta_{2}\rfloor+1)\,-\,B_{2}(\lfloor\Delta_{2}\rfloor+1)\Bigr{)}\\\
&\,=\,\pi{\rm
i}N^{2}\,\Bigl{(}\frac{n}{3m}\,\Delta^{3}_{2}\,+\,\frac{1}{m}\,\Delta^{2}_{2}\,B_{1}(\\{m\Delta_{1}\,-\,n\Delta_{2}\\})\Bigr{)}\,.\end{split}$
(103)

The choice of $K$ mentioned before was chosen in order to reach the result in
the third line. That will be convenient later on.

Next, we show an useful identity. Introducing the $N$-th auxiliary variable
$u^{N}\,\equiv\,u^{N}_{1}\,+\,\tau u^{N}_{2}\,=\,-\sum_{i=1}^{N-1}u^{i}$, and
defining its bracket as
$\\{u^{N}_{2}\\}_{\lambda}\,\equiv\,-\sum_{i\,=\,1}^{N-1}\,\\{u^{i}_{2}\\}_{\lambda}\,$
it follows that

$\begin{split}\sum_{\rho\in\text{Adj}(SU(N))}\rho(\\{u_{2}\\}_{\lambda})\log
P_{0\Lambda}(z_{a}(u))&\,=\,\sum_{i,j=1}^{N}\,(\\{u^{i}_{2}\\}_{\lambda}-\\{u^{j}_{2}\\}_{\lambda})\log
P_{0\Lambda}(z_{a}(u))\\\
&\,=\,\sum_{i,j=1}^{N}\,(\\{\frac{i}{N}m\\}_{\lambda}-\\{\frac{j}{N}m\\}_{\lambda})\log
P_{0\Lambda}(\frac{(i-j)}{N}T+\Delta_{\alpha})\\\
&\,=\,\sum_{i,j=1}^{N}\,\\{\frac{i}{N}m\\}_{\lambda}\,\Bigl{(}\log
P_{0\Lambda}(\frac{(i-j)}{N}T+\Delta_{\alpha})\,-\,i\leftrightarrow
j\Bigr{)}\,\\\
&=\sum_{i=1}^{N}\,\\{\frac{i}{N}m\\}_{\lambda}\,\sum_{j=1}^{N}\Bigl{(}\log
P_{0\Lambda}(\frac{(i-j)}{N}T+\Delta_{\alpha})\,-\,i\leftrightarrow
j\Bigr{)}\,\\\ &\,=\,\sum_{i=1}^{N}\,\\{\frac{i}{N}m\\}_{\lambda}\,\times
0\,=\,0\,.\end{split}$ (104)

In the last step we have used that

$\sum_{j=1}^{N}\Bigl{(}\log
P_{0\Lambda}(\frac{(i-j)}{N}T+\Delta_{\alpha})\,-\,i\leftrightarrow
j\Bigr{)}\,$ (105)

is a combination of the following vanishing quantities

$\sum_{j=1}^{N}\sin{2\pi p\frac{(i-j)}{N}}\,.$ (106)

The integer numbers $p$’s depend on $m$ and $n$. One way to prove this
analytically, is just to split the $sin$ in the difference of exponentials,
and then use the geometric sum
$\sum_{i=1}^{N}X^{i}\,=\,\frac{X(1-X^{N})}{1-X}$ for $X={\bf
e}(\frac{p}{N})\neq 1$, then as $X^{N}=1$ the sum cancels. When $X=1$ every
term in the sum (106) vanishes. Note that identity (104) holds for any finite
$N$, $\lambda$ and $\Lambda$.

For every chiral multiplet, after using (69) we can obtain a closed expression
for the deformed effective action of a chiral multiplet in terms of Polylogs:

$\begin{split}\frac{S_{\lambda\alpha}(p)}{N^{2}}\,\equiv\,\frac{{S}_{\text{eff}\alpha}(m,n)}{N^{2}}&\,=\,\frac{\pi{\rm
i}\,B_{3\Lambda}(\\{m\Delta_{\alpha 1}\,-\,n\Delta_{\alpha 2}\\})}{3mT^{2}}\\\
&\,+\,\Delta_{\alpha 2}\,\frac{\pi{\rm i}\,B_{2\Lambda}(\\{m\Delta_{\alpha
1}\,-\,n\Delta_{\alpha 2}\\})}{mT}\,\\\ &\,+\,\frac{\pi{\rm
i}\tau}{6}\,\,\bigl{(}2\Delta_{\alpha 2}^{3}\,-\,\Delta_{\alpha
2}\bigr{)}\,+\,\pi{\rm i}\Phi_{\Lambda}\,.\end{split}$ (107)

Essentially, this effective action reduces to the one computed in Cabo-
Bizet:2020nkr  at $\Lambda=1^{-}$. This time the phase $\Phi$ is fixed up to
the previously mentioned ambiguity in the choice of central term $K$. See
(103).

After taking the limits, we can use property Cabo-Bizet:2020nkr

$B_{3}(x+y)\,=\,B_{3}(x)\,+\,3B_{2}(x)y\,+\,3B_{1}(x)y^{2}\,+\,y^{3}\,,\qquad
x,\,y\,\in\,\mathbb{C}\,,$ (108)

to obtain the following form of the effective action Cabo-Bizet:2020nkr

$\begin{split}{S}_{\text{eff}\lambda\alpha}(m,n)&\,=\,\frac{\pi{\rm
i}N^{2}}{3mT^{2}}\,[\Delta_{\alpha}]^{m}_{T}\,\big{(}[\Delta_{\alpha}]^{m}_{T}+\frac{1}{2}\big{)}\big{(}[\Delta_{\alpha}]^{m}_{T}+1\big{)}\\\
&\,-\,\frac{\pi{\rm i}\tau}{6}\,N^{2}\,\Delta_{\alpha 2}\,+\,\pi{\rm
i}N^{2}(C+\Phi)\,+\,\ldots.\end{split}$

For $m\Delta_{\alpha 1}\,-\,n\Delta_{\alpha 2}\notin\mathbb{Z}$, the symbol
bracket is the same as the one defined in Cabo-Bizet:2020nkr  i.e.

$[\Delta_{\alpha}]^{T}_{m}\,\equiv\,T\Delta_{\alpha 2}\,+\,\\{m\Delta_{\alpha
1}\,-\,n\Delta_{\alpha 2}\\}\,-\,1\,,$ (109)

for $T=m\tau+n\,$. 383838To simplify the presentation, we have changed the
definition of $\Delta$ relative to the one we used in Cabo-Bizet:2020nkr . The
translation is $\Delta_{there}=\Delta_{here}+\tau$. However, in virtue of
property (76), for $\gamma\equiv m\Delta_{\alpha 1}\,-\,n\Delta_{\alpha
2}\,\in\,\mathbb{Z}$ that definition needs to be slightly modified i.e for
$\gamma\in\mathbb{Z}$

$\\{\gamma\\}\,\equiv\,\frac{\\{\gamma^{+}\\}+\\{\gamma^{-}\\}}{2}\,.$ (110)

This follows from the fact that the corresponding discontinuities arise from
limits of smooth and double-periodic Fourier expansions where uniform
convergence is lost, and thus Dirichlet theorem applies (See around F below
equation (269) for more information on this theorem). The constant

$C\,\equiv\,-\,\frac{n}{3m}\,\Delta^{3}_{\alpha_{2}}\,\,-\,\frac{1}{m}\,\Delta^{2}_{\alpha
2}\,B_{1}(\\{\gamma\\})\,=\,-\Phi\,,$ (111)

where the $\Phi$ was defined in (103). Thus we reach

${S}_{\text{eff}\lambda\alpha}(m,n)\,=\,\frac{\pi{\rm
i}N^{2}}{3mT^{2}}\,[\Delta_{\alpha}]^{m}_{T}\,\big{(}[\Delta_{\alpha}]^{m}_{T}+\frac{1}{2}\big{)}\big{(}[\Delta_{\alpha}]^{m}_{T}+1\big{)}\,-\,\frac{\pi{\rm
i}\tau}{6}\,N^{2}\,\Delta_{\alpha 2}\,$ (112)

but with the modification (110).

##### Contribution from a vector multiplet

The result for a vector multiplet can be recovered from the limit
$\Delta_{2\alpha}\to 1^{-}$ and $\Delta_{1\alpha}\to 0^{\pm}$ of (112), by
using the modified definition of (109). In this case the effective action is

${S}_{\text{eff}\lambda\text{v}}(m,n)\,=\,\frac{\pi{\rm
i}N^{2}}{6mT}\,+\,\frac{\pi{\rm i}\tau N^{2}}{3}\,+\,\frac{\pi{\rm
i}N^{2}n}{3m}\,.$ (113)

Thus, having the previous detail in mind, we can write, for a complete gauge-
anomaly free theory

$\begin{split}{S}_{\text{eff}}(m,n)\,\equiv\,{S}_{\text{eff}lambda}(m,n)&\,=\,\sum_{\alpha}\,\frac{\pi{\rm
i}N^{2}}{3mT^{2}}\,[\Delta_{\alpha}]^{m}_{T}\,\big{(}[\Delta_{\alpha}]^{m}_{T}+\frac{1}{2}\big{)}\big{(}[\Delta_{\alpha}]^{m}_{T}+1\big{)}\,-\,\frac{\pi{\rm
i}\tau}{6}\,N^{2}\,\Delta_{\alpha 2}\,\\\ &\,=\,\sum_{\alpha}\,\frac{\pi{\rm
i}N^{2}}{3mT^{2}}\,[\Delta_{\alpha}]^{m}_{T}\,\big{(}[\Delta_{\alpha}]^{m}_{T}+\frac{1}{2}\big{)}\big{(}[\Delta_{\alpha}]^{m}_{T}+1\big{)}\,.\end{split}$
(114)

In going to the second line we have used that

$\sum_{\alpha}\Delta_{2\alpha}\,=\,\text{Tr}{R}\,=\,0\,+\,\mathcal{O}\Bigl{(}\frac{1}{N}\Bigr{)},$
(115)

where Tr${R}$ is the sum of superconformal R-charges of the fundamental
fermionic fields of the corresponding theory. As noticed in Cabo-Bizet:2020nkr
, this quantity vanishes at large $N$ due to ABJ anomaly cancellation. Please
refer to subsection 4.1 of Cabo-Bizet:2020nkr  around equations (4.12) and
(4.21) for a detailed explanation.

##### The simplest ensemble of $\mathcal{N}=4$ SYM

In this case we have one vector and three chiral multiplets with
$\Delta_{\alpha}\to\frac{1}{3}(-\tau-n_{0})$ where we can choose $n_{0}=1$ or
$n_{0}=-1$. 393939 As it follows from setting up $N=4$ SYM on the conformal
boundary conditions dictated by the dual BPS black hole solution Cabo-
Bizet:2018ehj . We then proceed to complete the effective action computation
of Cabo-Bizet:2019eaf

$S_{\text{eff}}(m,n;\tau)\,=\,\frac{1}{m}\,\frac{\pi{\rm
i}}{27}\,N^{2}\,\frac{\bigl{(}2T+\chi_{1}(-n_{0}m\,+\,n)\bigr{)}^{3}}{T^{2}}+N^{2}\pi{\rm
i}\,\varphi(m,n)\,.$ (116)

For our choice of branch $K$

$\varphi(m,n)\,=\,-\frac{\chi_{1}(-n_{0}m\,+\,n)}{2m}\,.$ (117)

This answer is unique up to an addition of $\tau$-independent real numbers
that after multiplication by $N^{2}$ become an even integer. From now on we
will fix $n_{0}=-1$.

###  4.1 The equivariant Euler class at leading order in the large-$N$
expansion

In this subsection we study the equivariant Euler class $E(\nu_{P})$. The goal
is to compute its leading behaviour in large-$N$ expansion.

For the $S_{\lambda}(u)$ of a theory with a single $SU(N)$ node and fields in
the adjoint, let us say for $\mathcal{N}=4$ SYM (if $P$ is isolated),

$\begin{split}E(\nu_{P})&\,=\,\frac{1}{(2\pi{\rm
i})^{N-1}}\,\underset{i,j}{\det}\,\frac{\partial}{\partial
u^{j}}\,B^{i}(P)\,\\\ &\,=\,\frac{1}{(2\pi{\rm
i})^{N-1}}\,\underset{i,j}{\det}\,\frac{\partial}{\partial
u_{1}^{j}}\,B^{i}(P)\,\\\ &\,=\,\frac{1}{(2\pi{\rm
i})^{N-1}}\,\underset{i,j}{\det}\,\frac{\partial}{\partial u_{1}^{j}}\,\log
B^{i}(P)\,.\end{split}$ (118)

In going to the second line we have used (61). In going to the third line we
have used $B^{i}(P)=1$ .

Basic algebraic manipulations show that the matrix elements can be arranged as
follows

$\partial_{u^{j}_{1}}\log
B^{i}(u)\,=\,\sum_{\alpha}\,\Bigl{(}\sum_{k\,=\,1\atop k\neq
i}^{N-1}\,\mathcal{B}_{\alpha}(u_{jk})\,\delta_{ij}\,+\,\sum_{k\,=\,1\,\atop\,k\,\neq\,i\,\wedge\,k\,\neq\,j}^{N-1}\mathcal{B}_{\alpha}(u_{Nk})\Bigr{)}\,+\,\ldots\,.$
(119)

For $x=x_{1}+\tau x_{2}$, with $x_{1}$ and $x_{2}$ being real, one obtains

$\begin{split}\mathcal{B}_{\alpha}(x)&\,\equiv\,\frac{\partial}{\partial
x_{1}}\Bigl{(}\log P_{0\Lambda}(x+\Delta_{\alpha})\,-\,\log
P_{0\Lambda}(-x+\Delta_{\alpha})\Bigr{)}\,\\\ &\,=\,\\!-\,2{\rm
i}{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\Lambda^{|m|\,+\,|n|}\,\frac{\sin\Bigl{(}2\pi(n\Delta_{\alpha
2}-m\Delta_{\alpha
1})\Bigr{)}}{(m\tau+n)}\,\cos\Bigl{(}2\pi(nx_{2}-mx_{1})\Bigr{)}\,.\end{split}$
(120)

The $\ldots$ in (119) stand for terms with single elements, namely without sum
over $N-1$. These terms are suppressed in the large-$N$ expansion. We write
them for completeness

$\ldots\,=\,\,-\,\mathcal{B}_{\alpha}(u_{ij})\,|\epsilon_{ij}|\,+\,2\,\bigl{(}\mathcal{B}_{\alpha}(u_{Ni})+\mathcal{B}_{\alpha}(u_{Nj})\bigr{)}|\epsilon_{ij}|\,+\,4\,\mathcal{B}_{\alpha}(u_{Ni})\,\delta_{ij}\,,$
(121)

where $1\leq i,\,j\leq N-1$. At finite $N$ these terms are relevant. For
instance at $N=2$ the only non-vanishing contribution comes from the last term
in (121).

At large $N$ the $(N-1)\times(N-1)$ matrix (119) takes the form

$D^{N-1}\times\begin{pmatrix}2&1&\dots&1\\\ \vdots&&2&\vdots\\\
1&\ldots&1&2\end{pmatrix}\,,$ (122)

with $2$’s as diagonal elements and $1$’s otherwise. The matrix that
multiplies $D^{N-1}$ has determinant $N-1$ and so the final form of
$E(\nu_{P})$ has the following form at leading order in large-$N$ expansion

$(N-1)\,D^{N-1}\,\sim\,N\,D^{N-1}\,.$ (123)

Let us show these statements and determine $D$. The first contribution in
(119) is

$\sum_{k\,=\,1\atop k\neq j}^{N-1}\,\mathcal{B}_{\alpha}(u_{jk})\,.$

At large $N$, $\frac{j}{N}\to w\in[0,1)$ and $\frac{k}{N}\to x\in[0,1)$ and
the sums become integrals. In that limit (4.1) becomes

$\begin{split}N\,\int^{1}_{0}dx\mathcal{B}_{\alpha}((w-x)T)\,=\,N\,\int^{1}_{0}dx\mathcal{B}_{\alpha}((x-w)T)\,=\,N\,\int^{1}_{0}dx\mathcal{B}_{\alpha}(xT)\,.\end{split}$
(124)

After fixing $\Lambda\,=\,1^{-}$, the last term in (124) becomes

$N\,\int^{1}_{0}dx\mathcal{B}_{\alpha}(xT)\,=\,\frac{4\pi{\rm
i}N}{T}\,\Bigl{(}-T\,\Delta_{\alpha
2}\,+\,[\Delta_{\alpha}]^{T}_{m}\,+\,\frac{1}{2}\Bigr{)}\,.$ (125)

There is still a second contribution to analyze from (119). Evaluating it at
fixed points, the sum becomes an integral at leading order in large-$N$
expansion

$\sum_{k\,=\,1\,\atop\,k\neq i\,\wedge\,k\neq
j}^{N-1}\,\mathcal{B}_{\alpha}(u_{Nk})\longrightarrow
N\,\int^{1}_{0}dx\mathcal{B}_{\alpha}(xT)\,.$ (126)

This integral equals the contribution from the first term in (119). Thus, we
conclude that at leading order in large-$N$ expansion, (119) takes the form
(122), with

$D\,=\,\frac{4\pi{\rm i}N}{T}\,\sum_{\alpha}\Bigl{(}-T\,\Delta_{\alpha
2}\,+\,[\Delta_{\alpha}]^{T}_{m}\,+\,\frac{1}{2}\Bigr{)}\,.$ (127)

The final result for $(m,n)$ solutions is

$\underset{i,j}{\det}\,{\partial_{u_{j}}B^{i}}(P)\,=\,N^{N}\,\Bigl{(}\frac{2}{T}\,\sum_{\alpha}\Bigl{(}-T\,\Delta_{\alpha
2}\,+\,[\Delta_{\alpha}]^{T}_{m}\,+\,\frac{1}{2}\Bigr{)}\Bigr{)}^{N-1}\,+\,\text{sub-
leading}\,.$ (128)

In the limit $\Lambda\to 1^{-}$ the effective action $S_{\lambda}$ (112)
develops discontinuities at walls defined by the condition
$m\Delta_{1}\,-\,n\Delta_{2}\in\mathbb{Z}\,$. Dirichlet theorem fixes the
value of $S_{\text{eff}\alpha}(m,n)$ and its derivatives
$\partial^{\ell}\,S_{\text{eff}\alpha}(m,n)$, at such discontinuities to

$\frac{(\partial^{\ell}\,S_{\text{eff}\alpha}(m,n))^{L}\,+\,(\partial^{\ell}\,S_{\text{eff}\alpha}(m,n))^{R}}{2}\,,$
(129)

where $L$ and $R$ denote the limit values from the left and right sides of the
corresponding discontinuity of $\partial^{\ell}\,S_{\text{eff}\alpha}(m,n)\,$.

To compare with other literature, let us analyze the previously introduced
case of $\mathcal{N}=4$ SYM. Focus on the saddle/fixed point $(1,0)$ i.e on
$T=\tau$, then

$\sum_{\alpha}\,\Bigl{(}-T\,\Delta_{\alpha
2}\,+\,[\Delta_{\alpha}]^{\tau}_{1}\,+\,\frac{1}{2}\Bigr{)}\,=\,-\,\frac{1}{2}\,.$
(130)

We recall that the function bracket was defined in (109). For a generic gauge
anomaly-free theory $\sum_{\alpha}\Delta_{\alpha 2}\,=\,\text{Tr}R$ where
${R}$ denotes the superconformal R-charge of the fermionic fields in a given
multiplet and Tr denotes sum over all multiplets. This quantity vanishes at
large $N$ in virtue of ABJ anomaly cancellation Cabo-Bizet:2020nkr . For
$\mathcal{N}=4$ SYM this quantity vanishes at any value of $N$.

Finally, we use (130) in the logarithm of (128) and obtain at leading order

$\log E(\nu_{(1,0)})\,=\,N\,\log N\,-\,(N-1)\log(\tau)\,+\,\text{sub-
leading}\,.$ (131)

This answer matches a recent result obtained for the fixed point $(1,0)$ with
a combination of a small $\tau$ expansion and numerical extrapolation, via the
use of the Bethe Ansatz formula GonzalezLezcano:2020yeb .

##  5 Towards an analytic approach to large-$N$ counting

In this section we combine the ABBV formula with Picard-Lefschetz method to
study how complex eigenvalue configurations compete at leading order in the
large-$N$ expansion of the microcanonical index. We will use as toy example
the case of $\mathcal{N}=4$ SYM with $\sigma=\tau$ and no flavour potentials.
The generalization to the cases $\sigma\neq\tau$ is explained in appendix B.2.
The outline of this section was given in the introduction section 1. Here we
will focus on the details.

###  5.1 The contour of integration

We focus on the simplest ensemble of $SU(N)$ $\mathcal{N}=4$ SYM that captures
the graded counting of $\frac{1}{16}$-th BPS operators. That is the family of
integrals (6) with $t_{(1)}=t_{(2)}=t_{(3)}=tq$ and $p=q$ and the constraint
(10) $t^{3}=q^{-1}\,$. This is a function of a single chemical potential.

For finite $N$, the integral $\mathcal{I}$ (6) can be computed order by order
in a Taylor expansion around $p=q=0$. The coefficients $a_{n}$ in such
expansion can be computed as a contour integral over the Cartan torus
Murthy:2020rbd ; Agarwal:2020zwm  or via decomposition in characters of
$SU(N)$ Murthy:2020rbd ; Dolan:2007rq . As the R-charges are quantized in
multiples of $\frac{1}{3}$, the variable with the correct monodromy properties
is $q^{1/3}$. The Fourier coefficients can then be extracted from the
$q^{\frac{1}{3}}$-series, via the Laurent integral

$a(\ell)\,=\,\oint_{C}\,{dq^{\frac{1}{3}}}\,\frac{\mathcal{I}(q^{\frac{1}{3}})\,}{\,q^{\frac{\ell+1}{3}}\,}\,,\qquad
j\in\mathbb{Z}\,.$ (132)

The contour $C$ is a closed contour surrounded by $|q|=1$. In terms of $\tau$
the contour $C$ is a segment that for convenience we take to run in between
the vertical lines $\tau_{1}=-2$ and $\tau_{1}=1$. As
$\mathcal{I}(\tau)\,=\,\mathcal{I}(\tau+3)$ we can add to $C$ two vertical
contours running in opposite sense from (resp. to) $\tau_{2}=-\eta$ to (resp.
from) the left (resp. right) extremum of $C$. The positive number $\eta$ is
assumed to be as large as wished, see Figure 4. On the contrary to $C$,
$C_{\eta}$ has a natural decomposition in terms of Lefschetz thimbles. We will
explain this later on. As explained in appendix F.1 the vertical pieces of
$C_{\eta}$ must be slightly deformed (while preserving their mutual
cancellation) in such a way they cross the real axis across two irrational
values of $\tau$ at distance $3$ of each other.

Plugging the ABBV formula (35) into (132), changing integration variables from
$q^{\frac{1}{3}}\mapsto\tau$, using the fact that the integral of
$\mathcal{I}(\tau)$ over $C$ is equal to the integral over $C_{\eta}$, and
finally commuting the sum over $P$’s with the integral over $C_{\eta}$ 404040
If we consider only $(m,n)$ saddles, as we are doing, then the number of terms
in the sum over fixed points, is less than $N^{2}$. Thus for large but finite
$N$, if the integrals of the summands is finite, then the integral can be
commuted with the sum. one obtains the microcanonical index as a sum over
fixed points $P$

$\begin{split}d(\mathfrak{q})\,=\,\sum_{P}\,d_{P}\,,\end{split}$ (133)

where $d(\mathfrak{q})\equiv a(\ell+1)$ and
$\mathfrak{q}\,=\,\frac{{E}}{N^{2}}\equiv\frac{\ell+1}{3N^{2}}>0$ and the
contributions of single fixed point $P$, $d_{P}$, are defined as

${d}_{P}\,\equiv\,\int_{C_{\eta}}d\tau\,\frac{e^{-S_{\lambda}(P)\,-\,2\pi{\rm
i}\tau
E}}{E(\nu_{P})}\,=\,\int_{C_{\eta}}d\tau\,e^{-S_{\lambda}(P)\,-\,2\pi{\rm
i}\tau{E}\,+\,\ldots}\,.$ (134)

The $\ldots$ denote sub-leading contributions in the large-$N$ expansion. The
goal of this section is to study the leading in $N$ asymptotics of the full
sum by comparing the individual contributions of the $d_{P}$.

Figure 4:  The contour $C_{\eta}$ in $q^{\frac{1}{3}}$ and $\tau$-planes. The
original contour in $q$-plane is a circle inside the unit disk $|q|<1$. The
grey dashed circle (segment) represents $|q|=1$ ($\text{Im}(\tau)=0$). The
integration along the two vertical contours in $\tau$-plane cancel each other
due to the periodicity properties of the index. Due to reasons explained in
F.1 they must cross the real axis across irrational values. Thus, the integral
along $C_{\eta}$ equals the integral along the original contour $C$.

###  5.2 Picard-Lefschetz method

In this section we use Picard-Lefschetz method to analyze the integrals at
large $N\,$. Our approach follows the presentation of Witten:2010cx . The
large-$N$ limit of integrals $d_{P}$ is determined by the critical points of

$\mathcal{E}_{P}(\tau)\,\equiv\,-\,S(P)\,-\,2\pi{\rm i}\tau{E}\,.$ (135)

This function is analytic in the region

$\mathcal{X}_{P}\,\equiv\,\\{\tau\,:\,\tau\in\mathbb{C}-\tau_{P}\\}\,,$ (136)

and the point $\tau_{P}$ is an essential singularity. The behaviour of
$\mathcal{E}_{P}$ in a limit to $\tau_{P}$ depends on how the limit is taken.
For the example that we are considering, $\tau_{P}$ is always a rational
number.

We need to study real one-dimensional curves in the space $\mathcal{X}_{P}$,
say $\tau\,=\,\tau(s)\in\mathcal{X}_{P}\,$, with worldline parameter
$s\in(0,\infty)$. They must end or start at a non-degenerate critical
$\tau_{c}$ of $\mathcal{E}_{P}$ i.e. they must satisfy the asymptotic initial
condition

$\tau_{c}^{+}\,\equiv\,\tau(0^{+})\,=\,\tau_{c}\quad\text{or}\quad\tau^{-}_{c}\,=\,\tau(\infty)\,=\,\tau_{c}\,.$
(137)

Moreover, they should preserve the imaginary part of $\mathcal{E}_{P}$ along
the trajectory, and thus are determined by the differential equation

$\partial_{s}\,I_{P}(\tau(s))\,=\,0\,,\qquad
I_{P}(\tau)\,\equiv\,\text{Im}\mathcal{E}_{P}(\tau)\,,$ (138)

subject to one of the initial conditions in (137). If the Morse function

$H_{P}(\tau)\,=\,\text{Re}\mathcal{E}_{P}(\tau)$ (139)

is strictly decreasing along the flow i.e.

$\partial_{s}H_{P}(\tau(s))<0\,,$ (140)

then such flow $\mathcal{J}$ is called a Lefschetz thimble and it satisfies
the first of the conditions in equation (137). A more refined way of imposing
condition (140) upon solutions to (138) is instead, to demand the flow to be a
solution of the so-called  _downward flow equations_ Witten:2010cx  with the
initial condition (137). In the end, every thimble $\mathcal{J}$ obtained
directly from  _downward flow equations_ can be obtained from restricting the
space of solutions of (138) to the ones satisfying (140), and  _viceversa_.

Instead of solving the flow equations, it is convenient to solve the algebraic
relations in between imaginary and real part of the complex curves $\tau(s)$
that arise from demanding the imaginary part of the entropy function,
$I_{P}(\tau(s))$, to coincide with the imaginary part of $I_{P}(\tau_{c}))$ of
the fixed point $\tau_{c}$ of interest. The problem then reduces to that of
solving algebraic equations. Out of those solutions one then needs to
understand which ones are ascent or descent paths. Below we explain how that
is done in this particular example.

The next step is to decompose the contour $C_{\eta}$ in Figure 4 into an
integral combination of Lefschetz thimbles $\mathcal{J}_{P}$

$C_{\eta}\,=\,\sum_{\mathcal{J}_{P}}\,n_{\langle
C_{\eta},\mathcal{J}_{P}\rangle}\,\mathcal{J}_{P}\,,\qquad
n_{\mathcal{J}_{P}}\in\mathbb{Z}\,.$ (141)

That this decomposition exists can be understood in two ways. One homological,
and the other uses meromorphy of $\mathcal{E}_{P}$. In the example we study,
the latter turns out to be convenient to obtain the decomposition in thimbles
(141). Let us briefly comment on why the former turns out to be more involved
in this case. This requires to recall the concept of relative homology. Again,
we follow the presentation of Witten:2010cx .

The thimbles $\mathcal{J}_{P}$ are a basis of the relative homology group
$H_{1}(\mathcal{X}_{P},\mathcal{X}_{P_{-T}},\mathbb{Z})$. This is the group of
one-real dimensional paths that originate and end at the given connected piece
of “boundary” of $\mathcal{X}_{P}$ (the space defined in (136)). One-cycles
that can be contracted to a point without crossing any obstruction are
identified as trivial. We will see an instance of this in the next subsection.
By one-cycles one means the following. Select a very large positive real
number $T$, and excise from $\mathcal{X}_{P}$ all the regions at which the
Morse function $H_{P}<-T$. Then, identify each connected subpart of such
region as a “point” and define the new punctured domain as
$\mathcal{X}_{P_{-T}}$. The one-cycles are the one-curves in $\mathcal{X}_{P}$
that start and end at the “same” puncture of $\mathcal{X}_{P_{-T}}$. Those
cycles can be expanded as an integral combination of Lefschetz-thimbles
$\mathcal{J}_{P}$. Usually the integers are computed as intersection numbers
between the corresponding one-cycle and the dual ascent paths
$\mathcal{K}_{P}$. 414141Ascent paths are one-dimensional curves that solve
the flow equation (138). They are also asymptotically close to a critical
point, but this time they obey the opposite condition to (140) i.e they flow
towards the critical point. Equivalently, they solve the so-called  _upward
flow equations_ Witten:2010cx .  However, that method needs to be refined in
the presence of Stokes’ lines i.e when there exists a flow running among two
different critical points. In that case a thimble can be an ascent path as
well, and one needs to slightly deform $\mathcal{E}_{P}$ to solve this issue
(See Witten:2010cx  for a detailed discussion on this, and many other related
issues).

The present case “sits at” a Stokes line i.e. there is a thimble
$\mathcal{J}_{P}$ that it is also ascent path $\mathcal{K}_{P}$. That happens
for every $P$. The reason is that for every $P$ there are two critical points
of $\mathcal{E}_{P}$ with the same imaginary part $I_{P}$. Using meromorphy
this will turn out not to be an issue to compute the intersection numbers.

Using the contour decomposition (141) upon the integrals $d_{P}$ defined in
(134) one obtains

$d_{P}\,=\,\sum_{\mathcal{J}_{P}}n_{\langle
C_{\eta},\mathcal{J}_{P}\rangle}\,\int_{\mathcal{J}_{P}}\,d\tau\,e^{\mathcal{E}_{P}(\tau)\,+\,\ldots\,}\,.$
(142)

We will see that for many $P$’s the intersection numbers
$n_{\langle\ldots\rangle}$ vanish. But many others could contribute. The
$\ldots$ denote sub-leading contributions in the large-$N$ expansion.

###  5.3 The competition among $(m,n)$ fixed points

Let $N$ be a large integer, that we assume to be prime. 424242This implies the
$(m,n)$ configurations here-analyzed to be the only fixed points carrying
finite Abelian group structure of rank $N$ Cabo-Bizet:2019eaf ; Cabo-
Bizet:2020nkr .  The fixed points $P$ are the $(m,n)$’s configurations (65).
The on-shell action of the $(m,n)$ fixed point with $0\leq m,n<N$ integer co-
primes and $(m,n)\neq(0,0)$, was computed in Cabo-Bizet:2019eaf

$S(m,n)\equiv S_{\text{eff}}(m,n;\tau)\,=\,\frac{1}{m}\,\frac{\pi{\rm
i}}{27}\,N^{2}\,\frac{\bigl{(}2T+\chi_{1}(m+n)\bigr{)}^{3}}{T^{2}}+N^{2}\pi{\rm
i}\,\varphi(m,n)\,.$ (143)

In this formula $T=m\tau+n$ and $\chi_{1}(\ell)=\\{-1,0,1\\}$ if
$\ell=\\{-1,0,1\\}$ mod $3$ respectively. From now on we focus on the cases
$\chi_{1}\,\neq\,0$ i.e. $m\,+\,n\,\neq\,0$ mod $3$. Incorporating the cases
$\chi_{1}\,=\,0$ will not affect the conclusions that will be presented below.
It will be useful to note the following conjugation relation

$S_{\text{eff}}(1,0;\tau_{R})\,=\,S_{\text{eff}}(1,1;\tau)^{*}\,$ (144)

where $\tau_{R}\,=\,-\,1\,-\,\tau_{1}\,+\,{\rm i}\,\tau_{2}\,$. For a given
$(m,n)$ we must extremise the functional (135)

$\mathcal{E}_{(m,n)}(\tau)\,=\,-S_{\text{eff}}(m,n;\tau)-2\pi{\rm
i}\,\tau{E}\,.$ (145)

The extremization of this functional was analyzed in some detail in Cabo-
Bizet:2019eaf . Here we will complete the analysis by following the Picard-
Lefschetz method.

For any $(m,n)$ there are three critical points in $\mathcal{X}_{(m,n)}$. Let
us denote them as

$\tau^{*}_{+}\,,\qquad\tau^{*}_{-}\,,\qquad\tau^{*}_{0}\,.$ (146)

We have not written down their explicit dependence on $(m,n)$, but they do
depend on $m$ and $n$ Cabo-Bizet:2019eaf . For the critical points
$\tau^{*}_{\pm}$ the Morse function $H_{(m,n)}$ has the same absolute value
and opposite sign: say plus for $\tau^{*}_{+}$ and minus for $\tau^{*}_{-}$.
For the critical point $\tau^{*}_{0}$

$H_{(m,n)}(\tau^{*}_{0})\,=\,0\,.$ (147)

The critical points $\tau^{*}_{\pm}$ have equal phase function $I_{(m,n)}$.
Indeed for all $(m,n)$ there will always be a descent/ascent flow from
$\tau^{*}_{+}$ to $\tau^{*}_{-}\,$.

Let us start by studying the thimbles associated to the fixed point $(1,0)$.
For the critical point $\tau^{*}_{+}$ there are two thimbles, one flows to
$\tau_{2}=-\infty$ and the other flows to $\tau\,=\,0$. The point $\tau\,=\,0$
is the essential singularity of $\mathcal{E}_{(1,0)}$, namely the difference
between $\mathcal{X}_{(1,0)}$ and $\mathbb{C}$, as defined previously in
equation (136). For a limit that approaches $\tau=0$ from the first and third
quadrants, $H_{(1,0)}$ goes to $-\infty$. For a limit that approaches $\tau=0$
from the second and fourth quadrants, $H_{(1,0)}$ goes to $+\infty$. This
means that Lefschetz thimbles of $(1,0)$ that flow to $\tau=0$, must approach
it from the first and third quadrants, meanwhile ascent paths emanating from
$\tau=0$ must do it through the second and/or fourth quadrants. Indeed this is
what happens.

For $(1,1)$ the pattern is analogous, but this time the essential singularity
is located at $\tau\,=\,-1$ and now the first (second) and third (fourth)
quadrants take the roles of the second (first) and the fourth (third) had in
the previous case, respectively. For a generic $(m,n)$ fixed point, the story
is also the same. Now the essential singularity is located at
$\tau=-\frac{n}{m}$ and the two opposite quadrants of that singularity through
which the thimbles can enter, are determined by the sign of $\chi_{1}(m+n)$.
If $\chi_{1}(m+n)=+1$ then thimbles should approach it through the first and
third quadrants, otherwise through the second and fourth. The other region,
apart from the essential singularity $\tau=-\frac{n}{m}$, to which the
thimbles can flow towards is the asymptotic region $\tau\,=\,-{\rm i}\infty$.
Namely in this region $H_{(m,n)}$ goes to $-\infty$. This is the same region
where our integration cycle starts and ends at.

Figure 5:  The plots of the analytic solutions (of the relevant algebraic
equations) obtained for the Lefschetz thimbles and ascent paths associated to
the $(1,1)$ (left) and $(1,0)$ (right) fixed points. The solution for generic
$(m,n)$ has the same form as one of the two plots above. Each critical point
of the entropy functional $\mathcal{E}_{(m,n)}$ in (145), has two thimbles and
two ascent paths associated. In the figure we have only indicated the
direction of the flows corresponding to the thimbles that appear (have non-
zero intersection number) in the decomposition of the original integration
contour $C_{\eta}$ which we have denoted as $I$, $II$, $III$, $IV$, $V$. The
different $H$’s in the plots denote the values of the Morse function at the
closest critical point and also at $\tau_{2}\approx\pm\infty$ where $H=\pm T$,
respectively, with $T$ a very large positive number.

To know whether the intersection numbers $n_{\langle\ldots\rangle}$ that
determine the contribution of fixed points $(m,n)$ to the integral along
$C_{\eta}$ vanish or not, we need to know whether the cycle $C_{\eta}$ is
trivial in the relative homology
$H_{1}(\mathcal{X}_{(m,n)},\mathcal{X}_{(m,n)_{-T}};\mathbb{Z})\,$. That means
to check whether the $C_{\eta}$ encloses the essential singularity
$\tau=-\frac{n}{m}$. If the cycle $C_{\eta}$ does not enclose
$\tau=-\frac{n}{m}$ the integral $\mathcal{I}_{(m,n)}$ vanishes. It vanishes
because the integrand is meromorphic and the contour can be contracted to a
point at $\tau=-{\rm i}\infty$ (where the integrand vanishes). If on the
contrary the cycle $C_{\eta}$ does enclose $\tau=-\frac{n}{m}$ then the fixed
point $(m,n)$ does contribute and the precise intersection numbers can be
computed easily.

Given our choice of contour $C_{\eta}$, only fixed points for which

$-2<-\,\frac{n}{m}\,<\,1\,,$ (148)

contribute to the entropy counting formula (133). Take an $(m,n)$ that obeys
this condition. The symmetry transformation $\tau\mapsto\tau+3$ maps
$(m,n)\mapsto(m,n+3m)$. Naively, these two saddles should contribute with the
same exponential weight, namely $\mathcal{E}_{m,n}\,=\,\mathcal{E}_{m,n+3m}$,
however $(m,n+3m)$ does not satisfy (148) and thus its intersection numbers
vanish. Thus, these  _replica_ saddles do not contribute to the large-$N$
microcanonical index defined by the contour integral $C_{\eta}$.

In figure 4 we have plotted the analytic solutions of all the three thimbles
and three ascent paths, together with the corresponding critical points, for
fixed points $(1,0)$ and $(1,1)$. In both cases the contour $C_{\eta}$ can be
explicitly checked to have intersection number equal to either $1$, $-1$ or
$0$ associated to thimbles $\mathcal{J}_{(1,0)}$ and $\mathcal{J}_{(1,1)}$
i.e.

$\begin{split}C_{\eta}&\,\,=\,\,-\,\mathcal{J}_{(1,0){I}}\,-\,\mathcal{J}_{(1,0){II}}\,+\,\mathcal{J}_{(1,0){III}}\,-\,\mathcal{J}_{(1,0){IV}}\,+\,\mathcal{J}_{(1,0){V}}\,,\\\
&\,\,=\,\,\quad\,\mathcal{J}_{(1,1){I}}\,+\,\mathcal{J}_{(1,1){II}}\,-\,\mathcal{J}_{(1,1){III}}\,+\,\mathcal{J}_{(1,1){IV}}\,-\,\mathcal{J}_{(1,1){V}}\,.\end{split}$
(149)

This decomposition can be obtained by simply comparing the contour $C_{\eta}$
with the thimbles in figure 5.

Figure 6:  For large enough $N$ we can focus on combinations of subregions of
the dominating thimbles that include $\tau^{*}_{+}$. This is the plot of two
such combinations, one of them associated to $(1,1)$ (to the left), the other
to $(1,0)$ (to the right). These two combinations transform into each other
under the $\mathbb{Z}_{2}$ operation $\tau\mapsto\tau^{R}$. The arrows denote
the sense in which the original contour $C_{\eta}$ flows. For convenience we
define $\widetilde{\mathcal{J}}_{(1,0)}$ and $\widetilde{\mathcal{J}}_{(1,1)}$
to flow in the same sense $C_{\eta}$ does.

Integrals along these contours can be computed exactly with all the data we
have. However, in this paper we are interested in the leading approximation at
large $N$ so it will be enough to identify the contribution of the leading
critical point. That is always the $\tau^{*}_{+}$, thus, for instance, to
analyze the contribution of the fixed points $(1,0)$ and $(1,1)$ we can focus
on the following two portions of thimble

$\begin{split}C_{\eta}\,\sim\widetilde{\mathcal{J}}_{(1,0)}\,+\,\ldots\,,\\\
C_{\eta}\,\sim\widetilde{\mathcal{J}}_{(1,1)}\,+\,\ldots\,.\end{split}$ (150)

A computation shows that $\widetilde{\mathcal{J}}_{(1,0)}$ and
$\widetilde{\mathcal{J}}_{(1,1)}$ transform into each other under the
$\mathbb{Z}_{2}$ operation

$(\tau_{1},\tau_{2}(\tau_{1}))\mapsto(\tau^{R}_{1},\tau_{2}(\tau^{R}_{1}))\,.$
(151)

Moreover, from property (144), definition (145) and the fact
${E}\in\mathbb{R}$, it follows that

$\int_{\widetilde{\mathcal{J}}_{(1,1)}}d\tau_{1}\,e^{\mathcal{E}_{(1,1)}(\tau(\tau_{1}))}\,=\,\int_{\widetilde{\mathcal{J}}_{(1,0)}}d\tau_{1}\,e^{\bigl{(}\mathcal{E}_{(1,0)}(\tau(\tau_{1}))\bigr{)}^{*}}\,.$
(152)

This relation implies that contributions to the large-$N$ microcanonical index
that come from $(1,0)$ and $(1,1)$, are complex conjugated to each other. We
expect that this property will be preserved by sub-leading corrections in the
large-$N$ expansion as well. That is because by direct evaluation one can
check that the Fourier coefficients $a_{n}$ are always real integers.

As explained before, generic $(m,n)$ fixed points have analogous thimble
structure. Thus, we must compare the contribution coming from thimbles
associated to generic $(m,n)$ fixed points that obey the condition (148). For
those $(m,n)$, the intersection number $n_{\langle C_{\eta},\ldots\rangle}\,$
with the  _leading_ thimble will always have absolute value one. Thus, the
contribution of those fixed points is determined by the value of the Morse
function at $\tau^{*}_{+}$ which is

$\begin{split}H_{(m)}(\mathfrak{q})\,=\,H_{(m,n)}(\tau^{*}_{+})&\,\equiv\,\frac{N^{2}}{m}\,\frac{\pi\sqrt{(2T^{*}+\,1)^{3}\,(1\,-\,6T^{*})}}{24\sqrt{3}T^{*2}}\,,\\\
T^{*}&\,\equiv\,-\frac{54\,\mathfrak{q}\,+\,\sqrt[3]{2}\,D^{2}\,+\,8}{2\,2^{2/3}\,(27\,\mathfrak{q}\,+\,4)\,D}\,,\\\
D&\,\equiv\,\sqrt[3]{(27\mathfrak{q}+4)^{2}+3\sqrt{3}\sqrt{\mathfrak{q}(27\mathfrak{q}+4)^{3}}}\,.\end{split}$
(153)

The Morse function is inversely proportional to $m$ and it does not depend on
$n$. At a given level $m$ there are only a finite number of $n$’s with
$|\chi(m+n)|=1$ in the intersection domain (148). For $m=1$ there are two,
$n=0$ and $n=1$. These are the two dominating contributions at large $N$, and
for charges of order $N^{2}$. As explained above, both have the same leading
exponential contribution in absolute value, but their phases are complex
conjugated to each other and are given by the imaginary part

$I_{(1,0)}\,=\,-\,I_{(1,1)}\,=\,-\,\frac{\pi\,N^{2}\,(2T^{*}+1)(10T^{*}-1)}{72T^{*2}}\,+\,\frac{\pi
N^{2}}{2}.$ (154) 434343For large $\mathfrak{q}$ after large-$N$