SISSA 07/2021/FISI

Superconformal index of low-rank gauge
theories via the Bethe Ansatz

Francesco Benini1,2,3 and Giovanni Rizi1

1 SISSA, Via Bonomea 265, 34136 Trieste, Italy
2 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
3 ICTP, Strada Costiera 11, 34151 Trieste, Italy

We study the Bethe Ansatz formula for the superconformal index, in the case of
4d $\mathcal{N}=4$ super-Yang-Mills with gauge group $SU(N)$. We observe that
not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the
index, and thus formulate “reduced BAEs” such that all and only their
solutions contribute. We then propose, sharpening a conjecture of Arabi
Ardehali et al. [1], that there is a one-to-one correspondence between
branches of solutions to the reduced BAEs and vacua of the 4d
$\mathcal{N}=1^{*}$ theory. We test the proposal in the case of $SU(2)$ and
$SU(3)$. In the case of $SU(3)$, we confirm that there is a continuous family
of solutions, whose contribution to the index is non-vanishing.

##  1 Introduction and results

The AdS/CFT correspondence [2, 3, 4] provides us with a non-perturbative
definition of quantum gravity in anti-de-Sitter (AdS) space, in terms of an
ordinary quantum field theory (QFT) living on its conformal boundary. In
particular, ensembles of states in the boundary theory capture the physics of
black holes in the bulk [5]. This means that a counting of quantum microstates
in the boundary theory, performed by studying suitable partition functions in
the large $N$ limit, can reproduce and explain the Bekenstein-Hawking entropy
of black holes in AdS [6]. The last few years have seen a lot of activity in
this direction, in the context of supersymmetric black holes in which explicit
and precise computations can be performed, see for instance [7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

An interesting case is that of type IIB string theory (and its supergravity
low-energy limit) in AdS${}_{5}\times S^{5}$, whose physics is captured by the
four-dimensional $\mathcal{N}=4$ super-Yang-Mills (SYM) theory with gauge
group $SU(N)$ [2]. The microstates of 1/16 BPS black holes in AdS5 are
captured by the superconformal index [47, 48], a supersymmetric partition
function that can be computed exactly. The large $N$ limit can be studied with
a variety of methods [49, 50, 48, 21, 22, 36, 43]. From it, one can extract
the Bekenstein-Hawking entropy, but also find hints of new physics. For
instance, one can compute perturbative corrections expected to come from
higher-derivative terms [42], as well as non-perturbative corrections coming
from Euclidean complex saddles of the gravitational action [51, 52] (thus
highlighting the importance of Euclidean complex saddles [20]) and corrections
expected from D-branes.

A particularly effective way to compute the large $N$ limit is to use the
Bethe Ansatz (BA) approach [22], based on the Bethe Ansatz formulation of the
superconformal index [53, 54]. The standard integral formula for the index
[49, 50, 47, 48] can be recast as a sum over the solution set to certain
transcendental equations, dubbed Bethe Ansatz Equations (BAEs) because of
their similarity with the ones appearing in the context of integrable systems.
One particular solution to the BAEs reproduces the Bekenstein-Hawking entropy
of BPS black holes in AdS [22]. On the other hand, other solutions can be
associated with complex semiclassical saddles of the gravitational path-
integral [51, 52]. A comprehensive picture of all the contributions from BA
solutions and the physics they predict is an interesting open problem.

Even for finite $N$, the complete list of solutions to the BAEs is not known.
A subclass of exact solutions was found in [10, 55], and we will refer to them
as Hong-Liu (HL) solutions. These are discrete solutions, in one-to-one
correspondence with subgroups of $\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ of order
$N$, and are parametrized by three integers $\\{m,n,r\\}$ with $N=mn$ and
$r\in\mathbb{Z}_{n}$. On the other hand, a very interesting conjecture was put
forward in [1]: that there could be a sort of correspondence between some
solutions to the BAEs and the vacua of the 4d $\mathcal{N}=1^{*}$ theory —
which is $\mathcal{N}=4$ SYM deformed by $\mathcal{N}=1$ preserving mass terms
— compactified on $S^{1}$. The set of vacua of the $\mathcal{N}=1^{*}$ theory
on $\mathbb{R}^{3,1}$ [56] (that we will review below) include both discrete
massiva vacua, which are easily seen to be in one-to-one correspondence with
the HL solutions, as well as massless Coulomb vacua, which give rise to
continuous families of vacua upon compactification on $S^{1}$. Indeed, the
authors of [1] gave evidence for the existence of continuous families of
solutions to the BAEs. On the other hand, they also pointed out that there are
many more solutions to the BAEs (both discrete and continuous) that do not fit
into the pattern.

In this Letter we will make the conjectural correspondence sharper, by
studying in detail the low-rank cases of gauge groups $SU(2)$ and $SU(3)$.

Our first result (Section 2) is that, for generic $N$, most of the solutions
to the BAEs lead to contributions that cancel out. This means that we can
formulate a more restrictive set of equations, that we call “reduced Bethe
Ansatz Equations”, such that only solutions to the latter actually contribute.
We then conjecture that there is a one-to-one correspondence between branches
of solutions to the reduced BAEs and vacua of the $\mathcal{N}=1^{*}$ theory
on $\mathbb{R}^{3,1}$. In particular, isolated solutions to the reduced BAEs
are in correspondence with massive vacua of the $\mathcal{N}=1^{*}$ theory,
while complex $k$-dimensional manifolds of solutions are in correspondence
with massless Coulomb vacua with $k$ photons.

We test our conjecture in the cases of $SU(2)$ and $SU(3)$ gauge group. In the
case of $SU(2)$ (Section 3) we show that the reduced BAEs have only 3
solutions, which indeed are precisely the discrete HL solutions. We then show,
by expanding the index order by order in the operator dimension and computing
many terms, that those 3 solutions exactly reproduce the superconformal index
(as it follows from the integral formula).

In the case of $SU(3)$ (Section 4), we are able to analytically solve the
reduced BAEs and find all solutions. They comprise the 4 HL solutions, as well
as a continuous family which has complex dimension 1 and is made of a single
connected component. This is precisely in correspondence with the vacua of the
$\mathcal{N}=1^{*}$ theory. We also show that, in this case, the HL solutions
alone do not reproduce the index, and thus the contribution from the
continuous family must be non-trivial. We leave the interesting issue of
evaluating such a contribution for future work.

The method we use, in the $SU(3)$ case, to solve the reduced BAEs and to
exhibit the structure of discrete and continuous solutions sprouts from the
observation that the equations factorize into multiple hyperplanes and curves,
at whose intersections lie the solutions. We are hopeful that this method can
be generalized to higher $N$.

Note added: While this work was completed, its results had been announced in
[57], and we were in the process of writing up this Letter, we received the
paper [58] which has considerable overlap with our work.

##  2 Bethe Ansatz formula for the superconformal index

We study the superconformal index of 4d $\mathcal{N}=4$ SYM [47, 48], which
counts (with sign) operators in short representations (1/16 BPS) of the
superconformal algebra, preserving a complex supercharge $\mathcal{Q}$. Using
$\mathcal{N}=1$ notation, the field content of $\mathcal{N}=4$ SYM is given by
a vector multiplet and three chiral multiplets $X_{1},X_{2},X_{3}$, all in the
adjoint representation of the gauge group, with superpotential
$W=\operatorname{Tr}\bigl{(}X_{1}[X_{2},X_{3}]\bigr{)}$. The R-symmetry is
$SU(4)_{R}$, and we choose the Cartan generators $R_{1,2,3}$ such that each
one of them assigns charge 2 to one of the chiral multiplets and charge 0 to
the other two. States are further labeled by two angular momenta $J_{1,2}$
with semi-integer eigenvalues, generating rotations of two orthogonal planes
in $\mathbb{R}^{4}$, and we define the fermion number as $F=2J_{1}$. All
fields in the theory have integer charges under $R_{1,2,3}$, moreover

$F=2J_{1,2}=R_{1,2,3}\pmod{2}\;.$ (2.1)

The superconformal index is defined as

$\mathcal{I}(p,q,y_{1},y_{2})=\operatorname{Tr}\biggl{[}(-1)^{F}e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}}\,p^{J_{1}+\frac{1}{2}R_{3}}\,q^{J_{2}+\frac{1}{2}R_{3}}\,y_{1}^{\frac{1}{2}(R_{1}-R_{3})}\,y_{2}^{\frac{1}{2}(R_{2}-R_{3})}\biggr{]}\;.$
(2.2)

Here $p,q,y_{1},y_{2}$ are fugacities, and it is convenient to introduce
chemical potentials $\sigma,\tau,\Delta_{1},\Delta_{2}$ such that

$p=e^{2\pi i\sigma}\;,\qquad\qquad q=e^{2\pi i\tau}\;,\qquad\qquad
y_{a}=e^{2\pi i\Delta_{a}}\;.$ (2.3)

The index is well-defined for $|p|,|q|<1$. It is a single-valued function of
the fugacities, therefore periodic under integer shifts of the chemical
potentials, because all states have integer charges with respect to the
exponents in (2.2). Besides, it will be convenient to introduce an auxiliary
fugacity $y_{3}$ and chemical potential $\Delta_{3}$ such that111This will
restore the permutation symmetry acting on the index $a=1,2,3$ in $R_{a}$,
$y_{a}$ and $\Delta_{a}$. Such a symmetry is the Weyl group of the global
symmetry $SU(3)\subset SU(4)_{R}$ that commutes with $\mathcal{Q}$.

$\Delta_{1}+\Delta_{2}+\Delta_{3}-\sigma-\tau\in\mathbb{Z}\;.$ (2.4)

By standard arguments [59], the index only counts states annihilated by
$\mathcal{Q}$ and $\mathcal{Q}^{\dagger}$ and is therefore independent of
$\beta$.

When reading off the (weighted) multiplicities of BPS operators from the index
in a “low-temperature” expansion, it may be convenient to use the original
parametrization of [48] in terms of fugacities $(t,y,v_{1},v_{2})$. They are
related to our fugacities by

$p=t^{3}y\;,\qquad q=t^{3}/y\;,\qquad y_{1}=t^{2}v_{1}\;,\qquad
y_{2}=t^{2}v_{2}\;,\qquad y_{3}=t^{2}/v_{1}v_{2}\;,$ (2.5)

and recall that $y_{3}$ is an auxiliary variable defined by
$y_{1}y_{2}y_{3}=pq$. In terms of these fugacities, the index reads

$\mathcal{I}=\operatorname{Tr}_{\text{BPS}}(-1)^{F}\,t^{2(\Delta+J_{+})}\,y^{2J_{-}}\,v_{1}^{\mathfrak{q}_{1}}\,v_{2}^{\mathfrak{q}_{2}}\;.$
(2.6)

Here $J_{\pm}=(J_{1}\pm J_{2})/2$ are the Cartan generators of the
$SU(2)\times SU(2)$ Lorentz group, $\Delta$ is the dimension of the operators,
the trace is over BPS states — all and only states satisfying
$\Delta=2J_{+}+(R_{1}+R_{2}+R_{3})/2$ — while
$\mathfrak{q}_{1,2}=(R_{1,2}-R_{3})/2$ are the Cartan generators of the
$SU(3)\subset SU(4)_{R}$ that commutes with $\mathcal{Q}$.

#### Integral formulation.

The index is independent of the gauge coupling, and thus can be computed
exactly [49, 50, 47, 48] in terms of a certain contour integral. Specializing
to the case of gauge group $SU(N)$, the integral formula reads

$\mathcal{I}_{SU(N)}=\kappa_{N}\int_{\mathbb{T}^{N-1}}\mathcal{Z}(u;\Delta,\sigma,\tau)\,\prod_{j=1}^{N-1}\frac{dz_{j}}{2\pi
iz_{j}}\;,$ (2.7)

where $z_{j}=e^{2\pi iu_{j}}$, and $u_{j}$ with $j=1,\dots,N$ are gauge
holonomies along the Cartan generators. We regard $u_{j}$ with $j=1,\dots,N-1$
as the independent variables, while the last holonomy, $u_{N}$, is fixed by
the $SU(N)$ constraint

$\sum_{j=1}^{N}u_{j}=0\;.$ (2.8)

The prefactor is

$\kappa_{N}=\frac{1}{N!}\,\Bigl{(}\mathcal{I}_{U(1)}\Bigr{)}^{N-1}\;,$ (2.9)

written in terms of the index

$\mathcal{I}_{U(1)}(p,q,y_{1},y_{2})=(p;p)_{\infty}(q;q)_{\infty}\prod_{a=1}^{3}\widetilde{\Gamma}(\Delta_{a};\sigma,\tau)$
(2.10)

of the free $\mathcal{N}=4$ vector multiplet. Here $(z;q)_{\infty}$ is the
$q$-Pochhammer symbol while $\widetilde{\Gamma}(u;\sigma,\tau)$ is the
elliptic gamma function, both defined in Appendix A. The integrand is

$\mathcal{Z}(u;\Delta,\sigma,\tau)=\prod_{i\neq
j}^{N}\frac{\prod_{a=1}^{3}\widetilde{\Gamma}\bigl{(}u_{ij}+\Delta_{a};\sigma,\tau\bigr{)}}{\widetilde{\Gamma}\bigl{(}u_{ij};\sigma,\tau\bigr{)}}\;,$
(2.11)

where $u_{ij}=u_{i}-u_{j}$. This expression can be slightly simplified using
the identity

$\prod_{i\neq
j}^{N}\frac{1}{\widetilde{\Gamma}(u_{ij};\sigma,\tau)}=\prod_{i<j}^{N}\theta_{0}(u_{ij};\sigma)\>\theta_{0}(u_{ji};\tau)\;,$
(2.12)

written in terms of the elliptic function $\theta_{0}$ defined in Appendix A.
The contour of integration is along $N-1$ copies of the unit circle, as long
as the fugacities satisfy $|pq|<|y_{a}|<1$ for all $a=1,2,3$ (otherwise a
deformation of the contour is necessary in order to avoid crossing poles).

The superconformal index of the $\mathcal{N}=4$ SYM theory with gauge group
$U(N)$ is related to the one for gauge group $SU(N)$ by the simple relation

$\mathcal{I}_{U(N)}=\mathcal{I}_{U(1)}\,\mathcal{I}_{SU(N)}\;.$ (2.13)

#### Bethe Ansatz formulation.

In order to compute the large $N$ limit, it is convenient to consider the
alternative Bethe Ansatz formula [53, 54]. Such a formula applies whenever the
angular chemical potentials can be brought, with suitable integer shifts, to
have rational ratio,222Such a set is dense in the space of angular chemical
potentials. $\sigma/\tau\in\mathbb{Q}$. This means that $\sigma=a\omega$,
$\tau=b\omega$ for positive coprime integers $a,b$ and a chemical potential
$\omega$ with $\operatorname{\mathbb{I}m}\omega>0$. Our discussion of the
Bethe Ansatz equations will be general, however, for simplicity when computing
the index we will restrict to the case of two equal angular fugacities:

$p=q\qquad\Leftrightarrow\qquad\sigma=\tau\;.$ (2.14)

In this case the Bethe Ansatz formula simplifies to

$\mathcal{I}(q,y_{1},y_{2})=\kappa_{N}\sum_{\hat{u}\,\in\,\text{BAEs}}\mathcal{Z}(\hat{u};\Delta,\tau)\,H(\hat{u};\Delta,\tau)^{-1}\;.$
(2.15)

The sum is over the solution set $\\{\hat{u}\\}$ to a system of transcendental
equations, dubbed Bethe Ansatz Equations. Defining the $U(N)$ Bethe operators
as

$Q_{j}(u;\Delta,\tau)=e^{-2\pi
i\sum_{k=1}^{N}u_{jk}}\prod_{a=1}^{3}\prod_{k=1}^{N}\frac{\theta_{0}(u_{kj}+\Delta_{a};\tau)}{\theta_{0}(u_{jk}+\Delta_{a};\tau)}$
(2.16)

for $j=1,\dots,N$, the $SU(N)$ BA equations are given by

$1=\frac{Q_{i}}{Q_{N}}\qquad\qquad\text{for}\quad i=1,\dots,N-1\;.$ (2.17)

The unknowns are the “complexified $SU(N)$ holonomies” $u_{i}$ living on a
torus of modular parameter $\tau$, namely with identifications

$u_{i}\,\sim\,u_{i}+1\,\sim\,u_{i}+\tau\qquad\qquad\text{for}\quad
i=1,\dots,N-1\;,$ (2.18)

while $u_{N}$ is fixed by the constraint (2.8). The BAEs (2.17) are invariant
under such shifts. In fact, a stronger property holds: $Q_{i}$ are invariant
under shifts of the components of the antisymmetric tensor $u_{ij}$ by $1$ or
$\tau$, even relaxing the condition that $u_{ij}=u_{i}-u_{j}$. This will be
used later. It was proven in [54] that only the solutions that are not
invariant under any non-trivial element of the Weyl group of $SU(N)$ (namely,
only solutions with all $u_{i}$ different on the torus) actually contribute to
the sum in (2.15). In the general case $\sigma/\tau\in\mathbb{Q}$, the BA
formula takes a more complicated form, however the BAEs are the same as in
(2.17) with the only difference that $\tau$ is replaced by $\omega$. Therefore
the analysis we will make of the BAEs and their solutions will be valid in the
general case.

The prefactor $\kappa_{N}$ and the integrand $\mathcal{Z}$ in (2.15) are the
same as in the integral formula described before. On the other hand, $H$ is a
Jacobian defined as

$H=\det\left[\frac{1}{2\pi i}\,\frac{\partial\log(Q_{i}/Q_{N})}{\partial
u_{j}}\right]_{i,j=1,\dots,N-1}\;.$ (2.19)

Notice that $Q_{i}$, $\kappa_{N}$, $\mathcal{Z}$ and $H$ are all invariant
under integer shifts of $\tau$, $\Delta_{1}$ and $\Delta_{2}$, in accord with
the fact that the superconformal index (2.15) is a single-valued function of
the fugacities.

#### Reduced Bethe Ansatz equations.

It turns out that not all solutions (faithfully acted upon by the Weyl group)
of (2.17) actually contribute to the sum in (2.15). In order to understand
this point, notice that $\prod_{i=1}^{N}Q_{i}=1$ identically. It follows that
the solutions to the BAEs (2.17) break into $N$ “sectors” parametrized by
$\lambda\in\mathbb{Z}_{N}$ and given by

$Q_{j}=e^{2\pi i\lambda/N}\qquad\qquad\text{for}\quad j=1,\dots,N\;,$ (2.20)

each sector corresponding to a different integer value of
$\lambda=0,1,\dots,N-1$ and containing a subset of the solutions. Now, the
Bethe operators have the property that they implement on $\mathcal{Z}$ the
shifts of complexified $SU(N)$ holonomies on the torus [53, 54]:

$\mathcal{Z}(u-\delta_{k}\tau)=\frac{Q_{k}(u)}{Q_{N}(u)}\,\mathcal{Z}(u)\qquad\qquad\text{for}\quad
k=1,\dots,N-1\;,$ (2.21)

where $u-\delta_{k}\tau=(u_{1},\dots,u_{k}-\tau,\dots,u_{N-1},u_{N}+\tau)$
denotes a shift of the $k$-th and $N$-th components of $u$, so as to preserve
the $SU(N)$ constraint. On the other hand, if we regard $\mathcal{Z}(u)$ as
function of $N$ independent holonomies $u_{i}$ of $U(N)$, we find

$\mathcal{Z}\bigl{(}u-\bar{\delta}_{k}\tau\bigr{)}=(-1)^{N-1}\,Q_{k}(u)\,\mathcal{Z}(u)\qquad\qquad\text{for}\quad
k=1,\dots,N\;,$ (2.22)

where $u-\bar{\delta}_{k}\tau=(u_{1},\dots,u_{k}-\tau,\dots,u_{N})$ denotes a
shift of the $k$-th component only.333The factor $(-1)^{N-1}$ did not appear
in [54] because that paper only dealt with semi-simple gauge groups. The BAEs
only depend on the differences $u_{ij}$ and, as already noted, are invariant
under shifts of $u_{ij}$ by multiples of $1$ and $\tau$. It follows that one
solution for $u_{ij}$ on the torus gives rise to multiple solutions for
$u_{i}$, of the form $u_{i}=u_{i}^{(0)}+(\alpha+\beta\tau)/N$ for
$i=1,\dots,N-1$ and $u_{N}=u_{N}^{(0)}+(1-N)(\alpha+\beta\tau)/N$, where
$\alpha,\beta=0,\dots,N-1$ (some of these solutions could be equivalent up to
the Weyl group action). The function $\mathcal{Z}$, to be evaluated in (2.15)
on those solutions, only depends on $u_{ij}$, thus the dependence on
$\alpha,\beta$ is the same as if $u_{N}$ were shifted by
$-(\alpha+\beta\tau)$. There is no dependence on $\alpha$, while the
dependence on $\beta$ is a phase $\bigl{(}(-1)^{N-1}Q_{N}(u)\bigr{)}^{\beta}$
described by (2.22). We conclude that in all but one sector of BA solutions,
the sum over $\beta$ leads to a cancelation because the sum of phases
vanishes. The exception is the sector in which $(-1)^{N-1}Q_{N}(u)=1$.

Therefore, in the Bethe Ansatz formula (2.15) we can restrict to BA solutions
solely in the sector $\lambda=\frac{N(N-1)}{2}\text{ mod }N$, namely to
solutions of

$Q_{i}=(-1)^{N-1}\qquad\qquad\text{for}\quad i=1,\dots,N$ (2.23)

with $Q_{i}$ given in (2.16), which moreover are faithfully acted upon by the
Weyl group. We call these the “reduced Bethe Ansatz Equations”. Recall that
the product of the $N$ equations is identically equal to 1, therefore one of
them could be removed from the set.

#### Hong-Liu (HL) solutions.

The full set of solutions to (2.23) is not known, however, a large set was
found in [10, 55] and we will refer to them as HL solutions. They are labelled
by three positive integers:

$\\{m,n,r\\}\qquad\text{such that}\qquad N=m\cdot n\;,\qquad
r\in\mathbb{Z}_{n}\;.$ (2.24)

The solutions are

$u_{j}\equiv
u_{\hat{\jmath}\hat{k}}=\bar{u}+\frac{\hat{\jmath}}{m}+\frac{\hat{k}}{n}\left(\tau+\frac{r}{m}\right)\;.$
(2.25)

Here we have decomposed the index $j=0,\dots,N-1$ into the indices
$\hat{\jmath}=0,\dots,m-1$ and $\hat{k}=0,\dots,n-1$. Moreover, $\bar{u}$ is a
constant chosen in such a way to solve the $SU(N)$ constraint. Since what
enters in all formulas are the differences $u_{ij}=u_{i}-u_{j}$, to each HL
solution is associated a multiplicity that we will discuss below. Notice that
the HL solutions are in one-to-one correspondence with subgroups of
$\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ of order $N$, that is with sublattices of
index $N$ in generic two-dimensional lattices. It is worth remarking that,
surprisingly enough, the HL solutions (2.25) do not depend on the flavor
chemical potentials $\Delta_{a}$, although the BAEs do.

The BAEs (2.23) are invariant under $SL(2,\mathbb{Z})$ modular transformations
of the torus, namely under the generators

$T:\begin{cases}\tau\mapsto\tau+1\\\ u\mapsto u\end{cases}\qquad
S:\begin{cases}\tau\mapsto-1/\tau\\\ u\mapsto u/\tau\end{cases}\qquad
C:\begin{cases}\tau\mapsto\tau\\\ u\mapsto-u\;.\end{cases}$ (2.26)

It follows that the HL solutions form orbits under $PSL(2,\mathbb{Z})$,
completely classified by the integer $d=\gcd(m,n,r)$. The action of
$PSL(2,\mathbb{Z})$ is given by

$T:\\{m,n,r\\}\mapsto\\{m,n,r+m\\}\;,\qquad
S:\\{m,n,r\\}\mapsto\left\\{\gcd(n,r)\,,\,\frac{m\,n}{\gcd(n,r)}\,,\,\frac{m(n-r)}{\gcd(n,r)}\right\\}\;.$
(2.27)

For given $N$, the total number of HL solutions is given by the divisor
function

$\sigma_{1}(N)=\sum_{k|N}k\;.$ (2.28)

For each integer $d$ such that $d^{2}$ divides $N$, there exist a separate
$PSL(2,\mathbb{Z})$ orbit of HL solutions generated by
$\bigl{\\{}d,\frac{N}{d},0\bigr{\\}}$, which is isomorphic to the orbit
generated by $\\{1,N/d^{2},0\\}$ in the case of $SU(N/d^{2})$. The number of
elements in such an orbit is given by $\psi\bigl{(}N/d^{2}\bigr{)}$, expressed
in terms of the Dedekind psi function444Notice that, consistently,
$\displaystyle\sum_{d^{2}|N}\psi\left(\frac{N}{d^{2}}\right)=\sigma_{1}(N)$.

$\psi(n)=n\prod_{p|n}\left(1+\frac{1}{p}\right)$ (2.29)

where the product is over all prime numbers that divide $n$.

The contributions to the BA formula (2.15) from HL solutions connected by the
action of $T$ have a simple relation. Indeed, $T$ acts on solutions to the
BAEs by $q\to e^{2\pi i}q$, and there is no extra effect of this shift on the
summand in (2.15). This implies that the action of $T$ on the contributions to
the BA formula is also given by $q\to e^{2\pi i}q$. In terms of the
alternative set of fugacities $(t,v_{1},v_{2})$ defined in (2.5) (we are
setting $y=1$ here), this is

$t\,\to\,e^{\frac{2\pi i}{3}}\,t\;,\qquad\qquad v_{a}\,\to\,e^{\frac{2\pi
i}{3}}\,v_{a}\;.$ (2.30)

Since the action of $T$ on HL solutions forms finite cyclic orbits, this can
pose constraints on the contributions from those solutions (although notice
that, as opposed to the full index, the separate contributions to (2.15) in
general are not single-valued functions of the fugacities). We will see this
in the examples in Section 3 and 4.

#### Multiplicities.

From each solution in terms of $u_{ij}$ on the torus, one gets up to $N^{2}$
solutions in terms of $u_{i}$, related by a shift of the “center of mass” of
the first $N-1$ components. Some of those, however, could be equivalent up to
the Weyl group action, and thus the exact multiplicity should be determined by
a case-by-case analysis. In the case of the HL solutions, one easily verifies
that each solution has multiplicity $N$, namely it gives rise to $N$
inequivalent solutions in terms of $u_{i}$ that cannot be identified by the
action of the Weyl group.

Besides, since we are only interested in BA solutions that are not fixed by
any non-trivial element of the Weyl group, there is an obvious multiplicity by
$N!$ related to the action of the Weyl group. Therefore, each HL solution has
total multiplicity $N\cdot N!$.

###  2.1 Correspondence with the $\boldsymbol{\mathcal{N}=1^{*}}$ theory

It has been pointed out in [1] that the Hong-Liu ones are not the only
solutions to the BAEs. Furthermore, by a combination of analytical and
numerical work, evidence was given that for $N\geq 3$ there exist continuous
families of solutions. Very interestingly, the authors of [1] put forward the
idea of a correspondence between a subset of the solutions to the BAEs and the
vacua of the $\mathcal{N}=1^{*}$ $SU(N)$ theory on $S^{1}$. This, in
particular, allowed them to predict the appearance of continuous branches of
BA solutions for various values of $N$. On the other hand, it was already
noticed in that work that, even for $N=2$, the BAEs admit solutions that do
not have a counterpart in the $\mathcal{N}=1^{*}$ theory,555The
compactification on $S^{1}$ poses another problem, if one wants to make the
correspondence precise. Indeed, certain gapped vacua of the
$\mathcal{N}=1^{*}$ theory on $\mathbb{R}^{3,1}$ give rise to multiple vacua
on $\mathbb{R}^{2,1}\times S^{1}$, due to the presence of a residual discrete
gauge symmetry [60, 61]. For instance, the $SU(2)$ theory has 3 vacua on
$\mathbb{R}^{3,1}$, but 4 vacua on $\mathbb{R}^{2,1}\times S^{1}$, while the
reduced BAEs have 3 isolated solutions. We are grateful to Jan Troost for
making us appreciate this point. therefore the problem of understanding in
which terms the correspondence could be correct and precise remained open.

In this Letter we present a sharper version of the conjecture: we propose that
there is a one-to-one correspondence between branches of solutions to the
reduced Bethe Ansatz equations, and vacua of the $\mathcal{N}=1^{*}$ $SU(N)$
theory on $\mathbb{R}^{3,1}$. In particular, isolated solutions to the reduced
BAEs are in correspondence with massive vacua of the $\mathcal{N}=1^{*}$
theory, while complex $k$-dimensional manifolds of solutions are in
correspondence with massless Coulomb vacua with $k$ photons. If the conjecture
is correct, it implies that the HL solutions are the only discrete solutions
to the reduced BAEs. It also implies a precise characterization of the number
of continuous families and their dimensionality, for each value of $N$. In
Sections 3 and 4 we provide some evidence of the conjecture.

Let us review the $\mathcal{N}=1^{*}$ $SU(N)$ theory, which is obtained from
$\mathcal{N}=4$ SYM by an $\mathcal{N}=1$ preserving mass deformation

$W_{\text{def}}=\sum_{a=1}^{3}\frac{m_{a}}{2}\,X_{a}^{2}\;.$ (2.31)

Its vacua have been analyzed in [56] (see also [62, 61]). F- and D-term
equations reduce to the classification of isomorphism classes of homomorphisms
$\rho:\mathfrak{su}(2)\to\mathfrak{su}(N)$ modulo $SL(N,\mathbb{C})$, which
correspond to partitions of $N$. Since the chiral superfields $X_{a}$ all have
bare masses, possible massless fields come from the gauge sector. At the
classical level, the $SU(N)$ gauge group is broken to the subgroup $H$ that
commutes with the image of $\rho$. Quantum mechanically, all simple $SU(n)$
factors of $H$ confine leading to a number $n$ of massive sectors; on the
other hand, if $H$ contains $U(1)$ factors, then one obtains a massless
Coulomb phase.

The net result can be summarized as follows. Consider all integer partitions
of $N$. For each partition, let $n_{j}$ be the number of times the integer $j$
appears in it, and so label the partition by a string

$(n_{1},\dots,n_{j})\qquad\text{such that}\qquad\sum_{j=1}^{N}j\,n_{j}=N\;.$
(2.32)

Then the subgroup $H$ is given by666More precisely, $H$ can contain extra
discrete factors [61]. The latter would affect the number of vacua upon
compactification on $S^{1}$.

$H=\Biggl{[}\;\prod_{j\text{ s.t. }n_{j}\neq
0}^{N}U(n_{j})\,\Biggr{]}/U(1)\;.$ (2.33)

Thus, such a partition contributes to the moduli space with a certain number
of vacua that can be either gapped or contain massless photons:

  * •

$\text{number of photons}=(\text{number of non-zero $n_{j}$'s})-1$ (these are
isolated gapped vacua if the number is zero, or Coulomb branch vacua if it is
bigger than zero);

  * •

number of vacua = product of non-zero $n_{j}$’s.

It turns out [56] that the massive vacua are in one-to-one correspondence with
subgroups of order $N$ of $\mathbb{Z}_{N}\times\mathbb{Z}_{N}$, and thus in
correspondence with the HL solutions to the BAEs. On the other hand, we put
each Coulomb vacuum in correspondence with a continuous family of solutions to
the BAEs, with complex dimension equal to the number of photons.

##  3 The case of $\boldsymbol{SU(2)}$

Let us analyze in detail the case of gauge group $SU(2)$. Defining $u\equiv
u_{21}=u_{2}-u_{1}$, the refined Bethe Ansatz equation (2.23) can be rewritten
as

$\frac{\theta_{1}(\Delta_{1}+u)\,\theta_{1}(\Delta_{2}+u)\,\theta_{1}(-\Delta_{1}-\Delta_{2}+u)}{\theta_{1}(\Delta_{1}-u)\,\theta_{1}(\Delta_{2}-u)\,\theta_{1}(-\Delta_{1}-\Delta_{2}-u)}=-1\;,$
(3.1)

in terms of the standard Jacobi theta function $\theta_{1}$ (see Appendix A).
For the sake of clarity, here and in the following we leave the dependence of
the Jacobi theta functions $\theta_{r}$ on $\tau$ implicit. Let us define the
function

$f(u;\Delta,\tau)=\theta_{1}(u)\,\theta_{1}(\Delta_{1}+u)\,\theta_{1}(\Delta_{2}+u)\,\theta_{1}(-\Delta_{1}-\Delta_{2}+u)\;.$
(3.2)

Then, multiplying and dividing (3.1) by $\theta_{1}(u)$ and using that
$\theta_{1}$ is an odd function of $u$ (while $\theta_{2,3,4}$ are even
functions of $u$), we can rewrite the reduced BAE as

$\frac{f(u;\Delta,\tau)-f(-u;\Delta,\tau)}{f(-u;\Delta,\tau)}=0\;.$ (3.3)

Using standard identities among the theta functions, see for instance [63], we
obtain

$f(u;\Delta,\tau)=c_{1}\,\theta_{1}(2u)+c_{2}\,\theta_{2}(2u)+c_{3}\,\theta_{3}(2u)+c_{4}\,\theta_{4}(2u)$
(3.4)

where the coefficients are given by

$\displaystyle c_{1}$
$\displaystyle=-\frac{1}{2}\,\theta_{1}(\Delta_{1})\,\theta_{1}(\Delta_{2})\,\theta_{1}(\Delta_{1}+\Delta_{2})$
(3.5) $\displaystyle c_{2}$
$\displaystyle=-\frac{1}{2}\,\theta_{2}(\Delta_{1})\,\theta_{2}(\Delta_{2})\,\theta_{2}(\Delta_{1}+\Delta_{2})$
$\displaystyle c_{3}$
$\displaystyle=+\frac{1}{2}\,\theta_{3}(\Delta_{1})\,\theta_{3}(\Delta_{2})\,\theta_{3}(\Delta_{1}+\Delta_{2})$
$\displaystyle c_{4}$
$\displaystyle=-\frac{1}{2}\,\theta_{4}(\Delta_{1})\,\theta_{4}(\Delta_{2})\,\theta_{4}(\Delta_{1}+\Delta_{2})\;.$

From the parity properties of the theta functions, the reduced BAE becomes

$\frac{c_{1}(\Delta,\tau)\,\theta_{1}(2u)}{f(-u;\Delta,\tau)}=0\;.$ (3.6)

For generic values of $\Delta_{a}$ and $\tau$, the coefficient $c_{1}$ is non-
zero and thus the full set of solutions is given by $u=(m+n\tau)/2$ for
integers $m,n$ that are not both even.777When $m,n$ are both even, the zero of
$\theta_{1}$ in the numerator cancels with the zero of $f$ in the denominator.
On the torus there are 3 solutions:888Notice also that $u=0$ would lead to
configurations $(u_{1},u_{2})$ that are fixed by the non-trivial element of
the Weyl group, and thus would have to be excluded even if it was a solution.

$u=\frac{1}{2}\,,\;\frac{\tau}{2}\,,\;\frac{\tau+1}{2}\;.$ (3.7)

These are precisely the 3 Hong-Liu solutions $\\{2,1,0\\}$, $\\{1,2,0\\}$,
$\\{1,2,1\\}$, respectively, as it follows from (2.25). Therefore, for $N=2$,
the HL solutions exhaust the full set of solutions to the reduced BAEs, in
agreement with the proposed correspondence with vacua of the
$\mathcal{N}=1^{*}$ theory.

#### Evaluation of the index.

According to the BA formula (in the special case $\sigma=\tau$ for
simplicity), we have the identity

$\mathcal{I}_{SU(2)}(q,y_{1},y_{2})=\mathcal{I}_{SU(2)}^{\\{2,1,0\\}}+\mathcal{I}_{SU(2)}^{\\{1,2,0\\}}+\mathcal{I}_{SU(2)}^{\\{1,2,1\\}}\;.$
(3.8)

On the left-hand side of this identity, $\mathcal{I}_{SU(2)}$ is full index
computed with the integral formula (2.7). On the right-hand side, instead,
each of the three terms $\mathcal{I}_{SU(2)}^{\\{m,n,r\\}}$ is the
contribution to the Bethe Ansatz formula (2.15) from one of the HL solutions:

$\mathcal{I}_{SU(N)}^{\\{m,n,r\\}}=N\\!\cdot\\!N!\,\kappa_{N}\,\mathcal{Z}(\hat{u};\Delta,\tau)\,H(\hat{u};\Delta,\tau)^{-1}\,\Big{|}_{\hat{u}\in\\{m,n,r\\}}\;,$
(3.9)

where $\hat{u}_{j}$ are as in the HL solution $\\{m,n,r\\}$ given in (2.25),
while the factor $N\cdot N!$ comes from the multiplicity. We have verified
(3.8) numerically for many values of the fugacities. This confirms that, for
$N=2$, the contributions from the three HL solutions exactly reproduce the
superconformal index.

To gain a more interesting physical understanding, we can perform a “low-
temperature” expansion of both sides of (3.8), corresponding to an expansion
of the index into BPS operators, order by order in the operator dimension and
starting from the identity. In order to do that, it is convenient to use the
set of fugacities $(t,y,v_{1},v_{2})$ in (2.6). The restriction to $p=q$
corresponds to $y=1$. On the other hand, the dependence on $v_{1},v_{2}$
organizes into characters of $SU(3)$ that we indicate by $\chi_{\text{dim}}$.
For instance

$\displaystyle\chi_{\mathbf{1}}$ $\displaystyle=1$ (3.10)
$\displaystyle\chi_{\mathbf{3}}$
$\displaystyle=v_{1}+v_{2}+v_{1}^{-1}v_{2}^{-1}$
$\displaystyle\chi_{\mathbf{6}}$
$\displaystyle=v_{1}^{2}+v_{2}^{2}+v_{1}v_{2}+v_{1}^{-1}+v_{2}^{-1}+v_{1}^{-2}v_{2}^{-2}\;,$

and so on. With computer assistance, we perform a Taylor or Laurent expansion
of both sides of (3.8) around $t=0$.

Expanding the integral expression, left-hand side of (3.8), we obtain the
following. Switching off also the flavor fugacities ($v_{1}=v_{2}=1$) we
find999A much longer list of coefficients can be found in [39] for gauge group
$U(2)$, and then the coefficients for $SU(2)$ can be derived using (2.13).
Switching off flavor fugacities, one finds
$\mathcal{I}_{U(1)}=1+3t^{2}-2t^{3}+3t^{4}+6t^{7}-6t^{8}+12t^{10}-18t^{11}+27t^{12}-12t^{13}-27t^{14}+\mathcal{O}(t^{15})\;.$
(3.11)

$\mathcal{I}_{SU(2)}=1+6\,t^{4}-6\,t^{5}-7\,t^{6}+18\,t^{7}+6\,t^{8}-36\,t^{9}+6\,t^{10}+84\,t^{11}-80\,t^{12}-132\,t^{13}+309\,t^{14}+\mathcal{O}(t^{15}).$
(3.12)

Notice that the expansion only contains integer powers of $t$, and the
coefficients are well-defined because the number of operators with bounded
dimension is finite. Including the flavor fugacities we find

$\displaystyle\mathcal{I}_{SU(2)}$
$\displaystyle=1+\chi_{\mathbf{6}}t^{4}-2\chi_{\mathbf{3}}t^{5}+\bigl{(}1-\chi_{\mathbf{8}}\bigr{)}t^{6}+2\bigl{(}\chi_{\mathbf{\bar{3}}}+\chi_{\mathbf{6}}\bigr{)}t^{7}+\bigl{(}\chi_{\mathbf{15}}-3\chi_{\mathbf{3}}\bigr{)}t^{8}$
(3.13)
$\displaystyle\qquad{}-2\bigl{(}\chi_{\mathbf{10}}+\chi_{\mathbf{8}}\bigr{)}t^{9}+\bigl{(}3\chi_{\mathbf{6}}+4\chi_{\mathbf{\bar{3}}}-\chi_{\mathbf{24}}\bigr{)}t^{10}+\mathcal{O}(t^{11})\;.$

In terms of Dynkin labels: $\mathbf{10}=[3,0]$, $\mathbf{15}=[4,0]$,
$\mathbf{15^{\prime}}=[2,1]$, $\mathbf{21}=[5,0]$ and $\mathbf{24}=[3,1]$.

Similarly, we expand the BA contributions (3.9) for each one of the three HL
solutions. Suppressing flavor fugacities, we find

$\displaystyle\mathcal{I}_{SU(2)}^{\\{2,1,0\\}}$
$\displaystyle=-2-12t-42t^{2}-124t^{3}-348t^{4}-900t^{5}-2142t^{6}-4860t^{7}-10644t^{8}+\mathcal{O}(t^{9})$
(3.14) $\displaystyle\mathcal{I}_{SU(2)}^{\\{1,2,0\\}}$
$\displaystyle=-\frac{1}{4}t^{-\frac{3}{2}}-\frac{3}{4}t^{-\frac{1}{2}}+\frac{3}{2}-3t^{\frac{1}{2}}+6t-\frac{43}{4}t^{\frac{3}{2}}+21t^{2}-\frac{153}{4}t^{\frac{5}{2}}+62t^{3}-105t^{\frac{7}{2}}$
$\displaystyle\quad\;+177t^{4}-\frac{1131}{4}t^{\frac{9}{2}}+447t^{5}-\frac{2775}{4}t^{\frac{11}{2}}+\frac{2135}{2}t^{6}-1635t^{\frac{13}{2}}+2439t^{7}+\mathcal{O}\bigl{(}t^{\frac{15}{2}}\bigr{)}$
$\displaystyle\mathcal{I}_{SU(2)}^{\\{1,2,1\\}}$
$\displaystyle=+\frac{1}{4}t^{-\frac{3}{2}}+\frac{3}{4}t^{-\frac{1}{2}}+\frac{3}{2}+3t^{\frac{1}{2}}+6t+\frac{43}{4}t^{\frac{3}{2}}+21t^{2}+\frac{153}{4}t^{\frac{5}{2}}+62t^{3}+105t^{\frac{7}{2}}$
$\displaystyle\quad\;+177t^{4}+\frac{1131}{4}t^{\frac{9}{2}}+447t^{5}+\frac{2775}{4}t^{\frac{11}{2}}+\frac{2135}{2}t^{6}+1635t^{\frac{13}{2}}+2439t^{7}+\mathcal{O}\bigl{(}t^{\frac{15}{2}}\bigr{)}\;.$

Interestingly, the contributions from the two $\\{1,2,r\\}$ solutions contain
many unwanted features: fractional coefficients, and fractional as well as
negative powers of $t$. However the unwanted powers of $t$ cancel out in the
sum, and the coefficients sum up to integers. We have verified that the sum of
the three contributions above exactly matches the expansion (3.12) of the
integral expression, up to order $\mathcal{O}(t^{40})$. Including the flavor
fugacities we find

$\displaystyle\mathcal{I}_{SU(2)}^{\\{2,1,0\\}}$
$\displaystyle=-2-4\chi_{\mathbf{\bar{3}}}t-2\bigl{(}3\chi_{\mathbf{3}}+2\chi_{\mathbf{\bar{6}}}\bigr{)}t^{2}-4\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}+5\bigr{)}t^{3}$
(3.15)
$\displaystyle\qquad-4\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+2\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+2\chi_{\mathbf{6}}+10\chi_{\mathbf{\bar{3}}}\bigr{)}t^{4}+\mathcal{O}(t^{5})$
$\displaystyle\mathcal{I}_{SU(2)}^{\\{1,2,0\\}}$
$\displaystyle=-\frac{1}{4}t^{-\frac{3}{2}}-\frac{1}{4}\chi_{\mathbf{\bar{3}}}t^{-\frac{1}{2}}+\frac{3}{2}-\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{6}}}+2\chi_{\mathbf{3}}\bigr{)}t^{\frac{1}{2}}+2\chi_{\mathbf{\bar{3}}}t-\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}+17\bigr{)}t^{\frac{3}{2}}$
$\displaystyle\;\;+\bigl{(}2\chi_{\mathbf{\bar{6}}}+3\chi_{\mathbf{3}}\bigr{)}t^{2}-\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+2\chi_{\mathbf{15^{\prime}}}+3\chi_{\mathbf{6}}+30\chi_{\mathbf{\bar{3}}}\bigr{)}t^{\frac{5}{2}}+2\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}+5\bigr{)}t^{3}$
$\displaystyle\;\;-\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{1}}}+2\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{4}}}+3\chi_{\mathbf{15^{\prime}}}+31\chi_{\mathbf{\bar{6}}}+40\chi_{\mathbf{3}}\bigr{)}t^{\frac{7}{2}}+\bigl{(}2\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+4\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+\tfrac{9}{2}\chi_{\mathbf{6}}+20\chi_{\mathbf{\bar{3}}}\bigr{)}t^{4}+\mathcal{O}\bigl{(}t^{\frac{9}{2}}\bigr{)}$
$\displaystyle\mathcal{I}_{SU(2)}^{\\{1,2,1\\}}$
$\displaystyle=+\frac{1}{4}t^{-\frac{3}{2}}+\frac{1}{4}\chi_{\mathbf{\bar{3}}}t^{-\frac{1}{2}}+\frac{3}{2}+\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{6}}}+2\chi_{\mathbf{3}}\bigr{)}t^{\frac{1}{2}}+2\chi_{\mathbf{\bar{3}}}t+\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}+17\bigr{)}t^{\frac{3}{2}}$
$\displaystyle\;\;+\bigl{(}2\chi_{\mathbf{\bar{6}}}+3\chi_{\mathbf{3}}\bigr{)}t^{2}+\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+2\chi_{\mathbf{15^{\prime}}}+3\chi_{\mathbf{6}}+30\chi_{\mathbf{\bar{3}}}\bigr{)}t^{\frac{5}{2}}+2\bigl{(}\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}+5\bigr{)}t^{3}$
$\displaystyle\;\;+\frac{1}{4}\bigl{(}\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{1}}}+2\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{4}}}+3\chi_{\mathbf{15^{\prime}}}+31\chi_{\mathbf{\bar{6}}}+40\chi_{\mathbf{3}}\bigr{)}t^{\frac{7}{2}}+\bigl{(}2\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+4\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+\tfrac{9}{2}\chi_{\mathbf{6}}+20\chi_{\mathbf{\bar{3}}}\bigr{)}t^{4}+\mathcal{O}\bigl{(}t^{\frac{9}{2}}\bigr{)}$

Notice that the action of $T$ on these expressions is correctly given by
(2.30). In particular, $\mathcal{I}_{SU(2)}^{\\{2,1,0\\}}$ is invariant, while
$\mathcal{I}_{SU(2)}^{\\{1,2,1\\}}$ is obtained from
$\mathcal{I}_{SU(2)}^{\\{1,2,0\\}}$ (and viceversa). In both cases, the fact
that $T$ has finite cyclic orbits and that only integer or half-integer powers
of $t$ appear, implies that the triality of the $SU(3)$ characters is
correlated with the power of $t$. We have verified that the sum of the three
contributions above reproduces the expansion (3.13) of the integral
expression, up to order $\mathcal{O}(t^{10})$.

##  4 The case of $\boldsymbol{SU(3)}$

We consider now the case of gauge group $SU(3)$. We define $u\equiv u_{21}$
and $v\equiv u_{31}$. The Weyl group $S_{3}$ of the $SU(3)$ gauge group
permutes the triplet $(u_{1},u_{2},u_{3})$ while it acts in the following way
on $(u,v)$:

$\displaystyle s_{12}:(u,v)$ $\displaystyle\mapsto(-u,v-u)\qquad\qquad$
$\displaystyle s_{R}:(u,v)$ $\displaystyle\mapsto(v-u,-u)$ (4.1)
$\displaystyle s_{13}:(u,v)$ $\displaystyle\mapsto(u-v,-v)\qquad\qquad$
$\displaystyle s_{L}:(u,v)$ $\displaystyle\mapsto(-v,u-v)$ $\displaystyle
s_{23}:(u,v)$ $\displaystyle\mapsto(v,u)\;.$

The three reduced BAEs (2.23) are mapped one into the others by the action of
the Weyl group, therefore let us discuss how to manipulate the first of them.
Introducing the function $f$ defined in (3.2), we can recast that equation in
the form101010Notice that the first reduced BAE concerns the antisymmetric
part of the product of two $f$’s. This generalizes to any $N$: the solutions
to the first reduced BAE can always be written as the zeros of the
antisymmetric part of the product of $N-1$ functions $f$ (while the other ones
are obtained by the action of the Weyl group).

$0=Q_{1}(u,v;\Delta,\tau)-1=\frac{f(u;\Delta,\tau)\,f(v;\Delta,\tau)-f(-u;\Delta,\tau)\,f(-v;\Delta,\tau)}{f(-u;\Delta,\tau)\,f(-v;\Delta,\tau)}\;.$
(4.2)

Using the identity (3.4), the numerator becomes

$\text{num}=\biggl{[}c_{1}\,\theta_{1}(2u)\sum_{k=2,3,4}\\!c_{k}\,\theta_{k}(2v)\biggr{]}+\biggl{[}c_{1}\,\theta_{1}(2v)\sum_{k=2,3,4}\\!c_{k}\,\theta_{k}(2u)\biggr{]}\;.$
(4.3)

Collecting terms with the same coefficient $c_{i}$ and using the identities
(A.6), we rewrite it as

num
$\displaystyle=\frac{c_{1}\,\theta_{1}(u+v)}{\theta_{2}(0)\,\theta_{3}(0)\theta_{4}(0)}\Biggl{[}[3]\,\theta_{3}(u+v)\,\theta_{2}(u-v)\,\theta_{4}(u-v)$
(4.4) $\displaystyle\hskip
56.9055pt{}-[4]\,\theta_{4}(u+v)\,\theta_{2}(u-v)\,\theta_{3}(u-v)-[2]\,\theta_{2}(u+v)\,\theta_{3}(u-v)\,\theta_{4}(u-v)\Biggr{]}\;,$

where we defined
$[r]=\theta_{r}(0)\,\theta_{r}(\Delta_{1})\,\theta_{r}(\Delta_{2})\,\theta_{r}(\Delta_{1}+\Delta_{2})$.
They obey $[3]=[4]+[2]$ [63], then using (A.6) again we obtain

num
$\displaystyle=\frac{2\,c_{1}\,\theta_{1}(u+v)\,\theta_{1}(u)\,\theta_{1}(v)}{\theta_{2}(0)^{2}\,\theta_{3}(0)^{2}\,\theta_{4}(0)^{2}}\times{}$
(4.5) $\displaystyle\hskip
56.9055pt{}\times\biggl{[}[2]\,\theta_{4}(0)\,\theta_{4}(u)\,\theta_{4}(v)\,\theta_{4}(u-v)-[4]\,\theta_{2}(0)\,\theta_{2}(u)\,\theta_{2}(v)\,\theta_{2}(u-v)\biggr{]}\;.$

This allows us to rewrite the first reduced BAE as

$0=Q_{1}-1=\frac{2\,c_{1}\,\theta_{1}(u+v)\,h(u,v;\Delta,\tau)}{\theta_{2}(0)\,\theta_{3}(0)^{2}\,\theta_{4}(0)\prod_{\Delta\in\\{\Delta_{1},\Delta_{2},-\Delta_{1}-\Delta_{2}\\}}\theta_{1}(\Delta-u)\,\theta_{1}(\Delta-v)}$
(4.6)

where we defined the function

$h(u,v;\Delta,\tau)=\theta_{2}(\Delta_{1})\,\theta_{2}(\Delta_{2})\,\theta_{2}(\Delta_{1}+\Delta_{2})\,\theta_{4}(u)\,\theta_{4}(v)\,\theta_{4}(u-v)-\bigl{(}2\leftrightarrow
4\bigr{)}\;.$ (4.7)

Crucially, this function is invariant under the gauge Weyl group. The second
reduced BAE can be obtained by acting with the gauge Weyl group on the first
one. Since the denominator of (4.6) does not have poles, the reduced BAEs can
be brought to the factorized form

$\left\\{\begin{aligned} 0&=\theta_{1}(u+v)\,h(u,v;\Delta,\tau)\\\
0&=\theta_{1}(2u-v)\,h(u,v;\Delta,\tau)\;.\end{aligned}\right.$ (4.8)

From here we clearly see the structure of the solutions.

First, there are discrete solutions that follow from solving

$0=\theta_{1}(u+v)=\theta_{1}(2u-v)\;.$ (4.9)

These equations represent hyperplanes inside $T^{2}$, and the discrete
solutions are at the intersections of two hyperplanes. There are four
solutions on $T^{2}$, up to identifying those that are related by the Weyl
group action and dropping those that are fixed by some non-trivial element of
the Weyl group (i.e., solutions in which either $u$ or $v$ vanish):

$(u,v)=\left(\frac{1}{3},\frac{2}{3}\right),\;\left(\frac{\tau}{3},\frac{2\tau}{3}\right),\;\left(\frac{\tau+1}{3},\frac{2\tau+2}{3}\right),\;\left(\frac{\tau+2}{3},\frac{2\tau+1}{3}\right).$
(4.10)

These are precisely the 4 HL solutions $\\{3,1,0\\}$, $\\{1,3,0\\}$,
$\\{1,3,1\\}$, $\\{1,3,2\\}$, respectively, according to (2.25).

Second, there is a continuous family of solutions obtained by solving the
single equation

$0=h(u,v;\Delta,\tau)\;.$ (4.11)

We will analyze some properties of these solutions below, but we already see
that the continuous family has complex dimension 1. The solutions to (4.11),
as opposed to the HL solutions, depend on the flavor fugacities $\Delta_{a}$.
There are however six special points on the curve,111111These points have been
studied in [64] in the context of the $\mathcal{N}=1^{*}$ theory on $S^{1}$.
In the context of the Bethe Ansatz equations, they have been noticed in [1].
all related by the action of the $S_{3}$ Weyl group, that do not depend on
$\Delta_{a}$:

$(u,v)=\left(\frac{1}{2},\frac{\tau}{2}\right)\;.$ (4.12)

On these points, the action of $PSL(2,\mathbb{Z})$ reduces to the action of
$S_{3}$. These are smooth points, which do not seem to have any other special
property.

#### Evaluation of the index.

The Bethe Ansatz formula (2.15) was derived in [53, 54] assuming that all
solutions to the BAEs are isolated. It is then clear that, in the presence of
continuous families of solutions as it happens in this case, the formula has
to be modified somehow in order to correctly evaluate the contribution of
those families. We could hope that, for some reason, continuous families do
not contribute. Let us test this possibility, by comparing the result of the
integral formula (2.7) with that of the BA formula (2.15), in which the sum is
restricted to isolated solutions to the BAEs (here the HL solutions), in the
low-temperature limit.

The expansion of the integral formula (2.7), setting the flavor fugacities
$v_{1}=v_{2}=1$, gives

$\mathcal{I}_{SU(3)}=1+6\,t^{4}-6\,t^{5}+3\,t^{6}+6\,t^{7}-16\,t^{9}+27\,t^{10}+18\,t^{11}-87\,t^{12}+96\,t^{13}+54\,t^{14}+\mathcal{O}(t^{15}),$
(4.13)

while switching on the $SU(3)$ fugacities it gives

$\displaystyle\mathcal{I}_{SU(3)}$
$\displaystyle=1+\chi_{\mathbf{6}}t^{4}-2\chi_{\mathbf{3}}t^{5}+\bigl{(}\chi_{\mathbf{10}}-\chi_{\mathbf{8}}+1\bigr{)}t^{6}+2\chi_{\mathbf{\bar{3}}}t^{7}+\bigl{(}\chi_{\mathbf{15}}-\chi_{\mathbf{15^{\prime}}}+\chi_{\mathbf{\bar{6}}}-2\chi_{\mathbf{3}}\bigr{)}t^{8}$
(4.14)
$\displaystyle\quad-2\chi_{\mathbf{8}}t^{9}+\bigl{(}\chi_{\mathbf{21}}-\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+\chi_{\mathbf{6}}+5\chi_{\mathbf{\bar{3}}}\bigr{)}t^{10}+\bigl{(}2\chi_{\mathbf{15^{\prime}}}-2\chi_{\mathbf{15}}+4\chi_{\mathbf{\bar{6}}}-2\chi_{\mathbf{3}}\bigr{)}t^{11}+\mathcal{O}(t^{12})\;.$

The expansion of the contributions (3.9) of the HL solutions to the BA formula
gives:

$\displaystyle\mathcal{I}_{SU(3)}^{\\{3,1,0\\}}$
$\displaystyle=3+6\chi_{\mathbf{\bar{3}}}t+3\bigl{(}5\chi_{\mathbf{\bar{6}}}-2\chi_{\mathbf{3}}\bigr{)}t^{2}+6\bigl{(}3\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}+2\chi_{\mathbf{8}}-6\bigr{)}t^{3}$
$\displaystyle\;\;+3\bigl{(}10\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}-\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+24\chi_{\mathbf{6}}-13\chi_{\mathbf{\bar{3}}}\bigr{)}t^{4}+6\bigl{(}5\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{1}}}+6\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{4}}}+8\chi_{\mathbf{15^{\prime}}}-22\chi_{\mathbf{\bar{6}}}+30\chi_{\mathbf{3}}\bigr{)}t^{5}$
$\displaystyle\;\;+3\bigl{(}15\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{8}}}-\chi_{\mathbf{\bar{3}\\!\\!\bar{\phantom{1}}\\!\bar{5}}}+16\chi_{\mathbf{\bar{2}\\!\\!\bar{\phantom{1}}\\!\bar{4}}}+71(\chi_{\mathbf{27}}+\chi_{\mathbf{24}})-99\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{0}}}-69\chi_{\mathbf{10}}-85\chi_{\mathbf{8}}+164\bigr{)}t^{6}+\mathcal{O}(t^{7})$
$\displaystyle\mathcal{I}_{SU(3)}^{\\{1,3,0\\}}$
$\displaystyle=\frac{\chi_{\mathbf{3}}-\chi_{\mathbf{\bar{3}}}}{9\bigl{(}\chi_{\mathbf{6}}-6\chi_{\mathbf{3}}+\chi_{\mathbf{\bar{3}}}+9\bigr{)}}\,t^{-4}$
$\displaystyle\;\;+\frac{2\bigl{(}\chi_{\mathbf{15^{\prime}}}-\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+6\chi_{\mathbf{8}}-7\chi_{\mathbf{6}}+\chi_{\mathbf{\bar{6}}}+11\chi_{\mathbf{3}}-17\chi_{\mathbf{\bar{3}}}+6\bigr{)}}{9\bigl{(}\chi_{\mathbf{10}}+2\chi_{\mathbf{8}}-9\chi_{\mathbf{6}}-9\chi_{\mathbf{\bar{3}}}+27\chi_{\mathbf{3}}-26\bigr{)}}\,t^{-3}+\mathcal{O}(t^{-2})$
(4.15) $\displaystyle\mathcal{I}_{SU(3)}^{\\{1,3,1\\}}$
$\displaystyle=\frac{\chi_{\mathbf{\bar{3}}}+e^{\pi
i/3}\chi_{\mathbf{3}}}{9\bigl{(}6\chi_{\mathbf{3}}+e^{-\pi
i/3}(\chi_{\mathbf{6}}+\chi_{\mathbf{\bar{3}}})+9e^{\pi i/3}\bigr{)}}\,t^{-4}$
$\displaystyle\;\;-\frac{2\bigl{(}\chi_{\mathbf{15^{\prime}}}+\chi_{\mathbf{\bar{6}}}+11\chi_{\mathbf{3}}-e^{\pi
i/3}(6\chi_{\mathbf{8}}+6)+e^{-\pi
i/3}(\chi_{\mathbf{\bar{1}\\!\\!\bar{\phantom{1}}\\!\bar{5}^{\prime}}}+7\chi_{\mathbf{6}}+17\chi_{\mathbf{\bar{3}}})\bigr{)}}{9\bigl{(}e^{\pi
i/3}(\chi_{\mathbf{10}}+2\chi_{\mathbf{8}}-26)-9e^{-\pi
i/3}(\chi_{\mathbf{6}}+\chi_{\mathbf{\bar{3}}})-27\chi_{\mathbf{3}}\bigr{)}}\,t^{-3}+\mathcal{O}(t^{-2})$

while $\mathcal{I}_{SU(3)}^{\\{1,3,2\\}}$ is obtained from
$\mathcal{I}_{SU(3)}^{\\{1,3,1\\}}$ by taking the complex conjugate of all
coefficients. The action of $T$ on these expressions is given by (2.30).
Notice that $\mathcal{I}_{SU(3)}^{\\{1,3,r\\}}$ does not have a well-defined
limit as $v_{1,2}\to e^{-\frac{2\pi i}{3}r}$, for $r=0,1,2$.

One immediately sees that the sum of the contributions from the 4 HL solutions
does not reproduce the index. In particular, there remain negative powers of
$t$, fractional coefficients and characters in the denominators. This implies
that the contribution from the continuous family of solutions is necessary in
order to correctly reproduce the index. It is an open issue how to compute the
contribution from the continuous family, to which we hope to return in future
work.

###  4.1 The continuous family of solutions

Let us study in more detail the curve $h(u,v;\Delta,\tau)=0$, for generic
values of the flavor chemical potentials $\Delta_{a}$.121212In the case of the
$SU(3)$ $\mathcal{N}=1^{*}$ theory on $S^{1}$, the continuous family of vacua
has been studied in [65]. We rewrite the equation as

$\frac{\theta_{4}(u)\,\theta_{4}(v)\,\theta_{4}(u-v)}{\theta_{2}(u)\,\theta_{2}(v)\,\theta_{2}(u-v)}=\frac{\theta_{4}(\Delta_{1})\,\theta_{4}(\Delta_{2})\,\theta_{4}(\Delta_{1}+\Delta_{2})}{\theta_{2}(\Delta_{1})\,\theta_{2}(\Delta_{2})\,\theta_{2}(\Delta_{1}+\Delta_{2})}\;.$
(4.16)

If we regard the left-hand side as a function of $u$ (for fixed $v$), we see
that it is doubly-periodic with periods $1$ and $\tau$, with two simple poles
at $u=\frac{1}{2}$ and $u=v+\frac{1}{2}$, and two simple zeros at
$u=\frac{\tau}{2}$ and $u=v+\frac{\tau}{2}$, i.e., it is an elliptic function
of order 2. This means that it gives two values $u_{\pm}(v)$ of $u$ for each
value of $v$, and they satisfy
$u_{+}+u_{-}=v\pmod{\mathbb{Z}+\mathbb{Z}\tau}$. In other words we have a
branched double cover of the torus.

The equation can be formally solved in terms of Jacobi elliptic functions (see
for instance [66]). Define the new variables $z=2K(m)\,u$, $\omega=2K(m)\,v$
and $z_{i}=2K(m)\,\Delta_{i}$, where $K(m)$ is the complete elliptic integral
of the first kind, and $m$ is the parameter (see Appendix A). Then the curve
can be rewritten as

$\operatorname{cn}(z)\operatorname{cn}(\omega)\operatorname{cn}(z-\omega)=\operatorname{cn}(z_{1})\operatorname{cn}(z_{2})\operatorname{cn}(z_{1}+z_{2})\;.$
(4.17)

We use the Jacobi elliptic functions $\operatorname{cn}(z,m)$,
$\operatorname{sn}(z,m)$, $\operatorname{dn}(z,m)$ and keep the dependence on
the parameter $m$ implicit. Then, employing addition formulas, we can isolate
the dependence on $z$:

$\operatorname{sn}^{2}\left(z-\frac{\omega}{2}\right)=\frac{\operatorname{cn}(z_{1})\operatorname{cn}(z_{2})\operatorname{cn}(z_{1}+z_{2})-\operatorname{cn}^{2}(\omega/2)\operatorname{cn}(\omega)}{m\operatorname{sn}^{2}(\omega/2)\operatorname{cn}(z_{1})\operatorname{cn}(z_{2})\operatorname{cn}(z_{1}+z_{2})-\operatorname{dn}^{2}(\omega/2)\operatorname{cn}(\omega)}\;.$
(4.18)

The two values for $u$ can finally be written in terms of the inverse function
$\operatorname{arcsn}$:

$u_{\pm}(v)=\frac{v}{2}\pm\frac{1}{2K(m)}\operatorname{arcsn}\left(\sqrt{\frac{\operatorname{cn}(z_{1})\operatorname{cn}(z_{2})\operatorname{cn}(z_{1}+z_{2})-\operatorname{cn}^{2}(\omega/2)\operatorname{cn}(\omega)}{m\operatorname{sn}^{2}(\omega/2)\operatorname{cn}(z_{1})\operatorname{cn}(z_{2})\operatorname{cn}(z_{1}+z_{2})-\operatorname{dn}^{2}(\omega/2)\operatorname{cn}(\omega)}}\,\right)\;.$
(4.19)

The two equations $u=u_{\pm}(v)$ describe the curve as a double cover of the
torus. The two sheets of the covering meet at all points where
$u_{+}(v)=u_{-}(v)=v/2$. We could ask whether these are smooth points where
the covering is branched, or rather are self-intersection points of the curve
or intersection points of two connected components the curve is composed of.
We can see that these must be smooth branch points with a simple argument. The
points lie on the hyperplane $\theta_{1}(2u-v)=0$ on which $Q_{2}=1$. However,
we could have equivalently described the curve as $v=v_{\pm}(u)$. If those
points were intersection points, and so if they had an intrinsic geometric
characterization, they would also solve $v_{+}(u)=v_{-}(u)=u/2$ and so would
also lie on the hyperplane $\theta_{1}(u-2v)=0$ on which $Q_{3}=1$. However
the intersections of the two hyperplanes (which intersect also the hyperplane
$\theta_{1}(u+v)=0$) are precisely the HL solutions, and one can easily check
that they do not lie on the curve (4.11). This contradiction implies that
there are no special intersection points on the curve, only smooth branch
points. In particular this implies that the curve is composed of a single
connected component.

More directly, we can compute the gradient of $h$. Exploiting the relations
(A.6) and (A.7), with some algebra we obtain that on the curve
$h(u,v;\Delta,\tau)=0$ it holds

$\partial_{u}h=-\frac{\theta_{1}^{\prime}(0)\,\theta_{1}(2u-v)}{\theta_{1}(u)\,\theta_{1}(u-v)}\,h(0,v;\Delta,\tau)\;,\qquad\partial_{v}h=\frac{\theta_{1}^{\prime}(0)\,\theta_{1}(2v-u)}{\theta_{1}(u)\,\theta_{1}(u-v)}\,h(u,0;\Delta,\tau)\;.$
(4.20)

There are no points where $h=\partial_{u}h=\partial_{v}h=0$, therefore all
points on the curve are smooth.

## Acknowledgements

We thank Ofer Aharony, Arash Arabi Ardehali and Jan Troost for useful
discussions and correspondence, and in particular Paolo Milan for
collaboration in the early stage of this work. We also thank the organizers
and participants of the SCGP seminar series on “Supersymmetric black holes,
holography and microstate counting”, where part of these results were
announced, for interesting discussions. F.B. is partially supported by the
ERC-COG grant NP-QFT No. 864583, by the MIUR-SIR grant RBSI1471GJ, by the
MIUR-PRIN contract 2015 MP2CX4, as well as by the INFN “Iniziativa Specifica
ST&FI”.

##  Appendix A Special functions

We use fugacities and chemical potentials related by

$z=e^{2\pi iu}\;,\qquad\qquad p=e^{2\pi i\sigma}\;,\qquad\qquad q=e^{2\pi
i\tau}\;,$ (A.1)

with $|p|,|q|<1$. The $q$-Pochhammer symbol is defined as

$(z;q)_{\infty}=\prod_{j=0}^{\infty}(1-zq^{j})\;.$ (A.2)

The elliptic theta function $\theta_{0}$ is defined as

$\theta_{0}(u;\tau)=(z;q)_{\infty}\,(q/z;q)_{\infty}\;.$ (A.3)

The elliptic gamma function is defined as

$\widetilde{\Gamma}(u;\sigma,\tau)=\prod_{m,n=0}^{\infty}\frac{1-p^{m+1}q^{n+1}/z}{1-p^{m}q^{n}z}\;.$
(A.4)

We use the following definitions for the Jacobi theta functions:

$\displaystyle\theta_{1}(u;\tau)$ $\displaystyle=2\,e^{\pi i\tau/4}\sin(\pi
u)\prod_{n=1}^{\infty}(1-q^{n})(1-q^{n}z)(1-q^{n}/z)=i\,e^{\pi i\tau/4-\pi
iu}\,(q;q)_{\infty}\,\theta_{0}(u;\tau)$ $\displaystyle\theta_{2}(u;\tau)$
$\displaystyle=2\,e^{\pi i\tau/4}\cos(\pi
u)\prod_{n=1}^{\infty}(1-q^{n})(1+q^{n}z)(1+q^{n}/z)=e^{\pi i\tau/4-\pi
iu}\,(q;q)_{\infty}\,\theta_{0}\bigl{(}u+\tfrac{1}{2};\tau\bigr{)}$
$\displaystyle\theta_{3}(u;\tau)$
$\displaystyle=\prod_{n=1}^{\infty}(1-q^{n})\bigl{(}1+q^{n-\frac{1}{2}}z\bigr{)}\bigl{(}1+q^{n-\frac{1}{2}}/z\bigr{)}=(q;q)_{\infty}\,\theta_{0}\bigl{(}u+\tfrac{1}{2}+\tfrac{\tau}{2};\tau\bigr{)}$
$\displaystyle\theta_{4}(u;\tau)$
$\displaystyle=\prod_{n=1}^{\infty}(1-q^{n})\bigl{(}1-q^{n-\frac{1}{2}}z\bigr{)}\bigl{(}1-q^{n-\frac{1}{2}}/z\bigr{)}=(q;q)_{\infty}\,\theta_{0}\bigl{(}u+\tfrac{\tau}{2};\tau)\;.$
(A.5)

Notice that $\theta_{1}$ is odd while $\theta_{2,3,4}$ are even under
$u\to-u$. One set of identities we use is

$\displaystyle\theta_{1}(u+v)\,\theta_{2}(u-v)\,\theta_{3}(0)\,\theta_{4}(0)$
$\displaystyle=\theta_{1}(u)\,\theta_{2}(u)\,\theta_{3}(v)\,\theta_{4}(v)+\theta_{3}(u)\,\theta_{4}(u)\theta_{1}(v)\,\theta_{2}(v)$
(A.6)
$\displaystyle\theta_{1}(u+v)\,\theta_{3}(u-v)\,\theta_{2}(0)\,\theta_{4}(0)$
$\displaystyle=\theta_{1}(u)\,\theta_{3}(u)\,\theta_{2}(v)\,\theta_{4}(v)+\theta_{2}(u)\,\theta_{4}(u)\theta_{1}(v)\,\theta_{3}(v)$
$\displaystyle\theta_{1}(u+v)\,\theta_{4}(u-v)\,\theta_{2}(0)\,\theta_{3}(0)$
$\displaystyle=\theta_{1}(u)\,\theta_{4}(u)\,\theta_{2}(v)\,\theta_{3}(v)+\theta_{2}(u)\,\theta_{3}(u)\theta_{1}(v)\,\theta_{4}(v)$
$\displaystyle\theta_{2}(u+v)\,\theta_{3}(u-v)\,\theta_{2}(0)\,\theta_{3}(0)$
$\displaystyle=\theta_{2}(u)\,\theta_{3}(u)\,\theta_{2}(v)\,\theta_{3}(v)-\theta_{1}(u)\,\theta_{4}(u)\theta_{1}(v)\,\theta_{4}(v)$
$\displaystyle\theta_{3}(u+v)\,\theta_{4}(u-v)\,\theta_{3}(0)\,\theta_{4}(0)$
$\displaystyle=\theta_{3}(u)\,\theta_{4}(u)\,\theta_{3}(v)\,\theta_{4}(v)-\theta_{1}(u)\,\theta_{2}(u)\theta_{1}(v)\,\theta_{2}(v)\;,$

see for instance [63]. Here $\theta_{r}(u)\equiv\theta_{r}(u;\tau)$. We also
use

$\partial_{u}\left(\frac{\theta_{1}(u)}{\theta_{4}(u)}\right)=\pi\,\frac{\theta_{4}(0)^{2}\,\theta_{2}(u)\,\theta_{3}(u)}{\theta_{4}(u)^{2}}$
(A.7)

and a similar relation with $2$ and $4$ exchanged, as well as
$\theta_{1}^{\prime}(0)=\pi\,\theta_{2}(0)\,\theta_{3}(0)\,\theta_{4}(0)$.

The Jacobi elliptic functions are related to the Jacobi theta functions by

$\displaystyle\operatorname{cn}\bigl{(}2K(m)u,\,m\bigr{)}$
$\displaystyle=\biggl{(}\frac{1-m}{m}\biggr{)}^{1/4}\,\frac{\theta_{2}(u;\tau)}{\theta_{4}(u;\tau)}$
(A.8) $\displaystyle\operatorname{sn}\bigl{(}2K(m)u,\,m\bigr{)}$
$\displaystyle=\biggl{(}\frac{1}{m}\biggr{)}^{1/4}\;\frac{\theta_{1}(u;\tau)}{\theta_{4}(u;\tau)}$
$\displaystyle\operatorname{dn}\bigl{(}2K(m)u,\,m\bigr{)}$
$\displaystyle=(1-m)^{1/4}\,\frac{\theta_{3}(u;\tau)}{\theta_{4}(u;\tau)}\;.$

The parameter $m$ is related to the modulus $\tau$ by

$m(\tau)=\frac{\theta_{2}(0;\tau)^{4}}{\theta_{3}(0;\tau)^{4}}\;.$ (A.9)

The function $K(m)$ is the complete elliptic integral of the first kind, and
is related to $\tau$ by

$\tau=i\,\frac{K(1-m)}{K(m)}\qquad\text{or}\qquad
2K(m)=\frac{\theta_{3}(0;\tau)^{2}}{\pi}\;.$ (A.10)

We use the addition formula

$\operatorname{cn}(x+y,m)\operatorname{cn}(x-y,m)=\frac{\operatorname{cn}^{2}(y,m)-\operatorname{sn}^{2}(x,m)\operatorname{dn}^{2}(y,m)}{1-m\operatorname{sn}^{2}(x,m)\operatorname{sn}^{2}(y,m)}\;.$
(A.11)

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