$\frac{H_{(1,0)}}{N^{2}}\sim\frac{1}{2}\sqrt{3}\pi\mathfrak{q}^{2/3}-\frac{\pi\sqrt[3]{\mathfrak{q}}}{\sqrt{3}}+O\left(\sqrt[3]{\frac{1}{\mathfrak{q}}}\right)\,,\qquad\frac{I_{(1,0)}}{N^{2}}\sim\frac{1}{2}\pi\mathfrak{q}^{2/3}+\frac{\pi\sqrt[3]{\mathfrak{q}}}{3}+\frac{\pi}{6}+O\left(\sqrt[3]{\frac{1}{\mathfrak{q}}}\right)\,.$
(155) We should note that in terms of the charge $E$ of CFT operators,
$H_{(1,0)}\sim N^{\frac{2}{3}}E^{\frac{2}{3}}$.

If we discard sub-leading corrections coming from $E(\nu_{P})$, 444444These
were analyzed in subsection 4.1. and exponentially suppressed contributions
coming from other fixed points, we obtain

$d(\mathfrak{q})\,=\,e^{H_{(1)}(\mathfrak{q})\,+\,\ldots}\,\times
2\cos{\bigl{(}I_{(1,0)}(\mathfrak{q})\,+\,\ldots\bigr{)}}\,+\,\ldots\,.$ (156)

where we should stress that other fixed point configurations $(m,n)$ also
interfere to produce extra oscillations on top of (156). However, at finite
but relatively large values of $N$ and at finite $\mathfrak{q}$, these are
exponentially suppressed with respect to the contributions coming from the
pair $(1,0)$ and $(1,1)$, as one can note immediately from equation (153).
Moreover, at very first leading order in large-$N$ expansion all oscillations
attenuate and the microcanonical index matches the exponential of the
Bekenstein-Hawking entropy of the dual black holes (See figure 3)

$d(\mathfrak{q})\,\sim\,e^{H_{(1)}(\mathfrak{q})}\,=\,e^{\frac{A_{H}}{4}}\,.$
(157)

Here $A_{H}$ is the area of the horizon of the Gutowski-Reall black hole.
454545We have absorbed the $5d$ Newton constant $G_{5D}$, which is to be
identified as proportional to $\frac{1}{N^{2}}$, in the definition of $A_{H}$.

For small enough values of $\mathfrak{q}$ after the large-$N$ limit is taken,
the physical picture changes. In that case the relevant expansion is

$\frac{H_{(1,0)}}{N^{2}}\sim\pi\mathfrak{q}^{3/2}\,-\,\frac{21}{8}\pi\mathfrak{q}^{5/2}+O\left(\mathfrak{q}^{7/2}\right)\,,\qquad\frac{I_{(1,0)}}{N^{2}}\,\sim\,\frac{\pi}{2}\,+\,\pi\mathfrak{q}\,-\,\frac{3\pi\mathfrak{q}^{2}}{2}\,+\,5\pi\mathfrak{q}^{3}\,+\,O\left(\mathfrak{q}^{7/2}\right)\,.$
(158)

In terms of the charge $E$ of the CFT operators, $H_{(1,0)}\sim
N^{-1}E^{\frac{3}{2}}$. Roughly speaking, this implies that to count operators
with charges of the order $E\sim N^{\frac{2}{3}}$ or smaller, the competition
of other $(m,n)$ saddles can not be neglected. Moreover, other critical points
of the entropy functional $\mathcal{E}_{(m,n)}$ in (145), mainly the real one
(out of the triad of them), should also be analyzed in the regime of small
charges (at large $N$). 464646Interestingly, note that for small enough values
of $\mathfrak{q}$, at large $N$, the interference among $(1,0)$ and $(1,1)$ is
destructive i.e they tend to cancel each other.

Let us summarize in words what we have learnt. The parameter that controls the
magnitude of the exponential corrections coming from fixed points is
$\exp[-A_{H}/4]$ i.e. the inverse of the exponential of the entropy of the
Gutowski-Reall black hole. Generically, this parameter is small for large
enough $N$, meaning by this, values of $N$ such as the discrete sums over
eigenvalues are well approximated by the integrals. However, for the two-fixed
point approximation to work, at relatively large but still finite values of
$N$, operator charges need to be large enough in units of the specific values
$N$, in such a way that the perturbation parameter $\exp[-A_{H}/4]$ remains
small. That means that for small enough operators, the two-fixed point
approximation breaks down and one needs to start considering many more
configurations to approximate the correct micro-canonical index. This means
that when one approaches the transition from the black hole phase to the
multigraviton phase, the “two- fixed-point” approximation breaks down and
competition from all $(m,n)$ saddles must be considered. This observation
deserves deeper analysis and exploration. The ABBV and the Bethe ansatz
_formulae_ are tools that could be useful to make this analytic picture
precise at the numerical level (for instance at large $N$ but for small
operators).

## Acknowledgements

We thank Cyril Closset and Heeyeon Kim for a useful discussion regarding
regularization procedures. We are grateful to Davide Cassani and Dario
Martelli for useful comments and discussions. We are especially grateful to
Sameer Murthy for his careful reading of the manuscript, as well as for many
useful comments and discussions. This work is supported by the ERC
Consolidator Grant N. 681908, “Quantum black holes: A microscopic window into
the microstructure of gravity”.

##  Appendix A Definitions and identities

The quasi-elliptic function $\theta_{0}(z)=\theta_{0}(z;\tau)$ has the
following product representation

$\theta_{\text{ell}}(\zeta;q)\,=\,\theta_{0}(z;\tau)\,=\,(1-\zeta)\prod_{j=1}^{\infty}(1-q^{j}\zeta)\,(1-q^{j}\zeta^{-1})\,.$
(159)

The zeroes of $\theta_{0}$ are located at $z=j\tau+k$ where $j$ and $k$ run
over the integers. Representation (159) is convergent for $\tau_{2}>0$. It has
also the following quasi-periodicity properties

$\theta_{0}(z)\,=\,\theta_{0}(z+1)\,=\,-{\bf
e}(z)\,\theta_{0}(z+\tau)\,\quad\,\theta_{0}(\tau-z)\,=\,\theta_{0}(z)\,.$
(160)

The elliptic Gamma functions are defined out of the following product
representation

$\Gamma_{\text{ell}}(\zeta;p,q)\,=\,\Gamma_{\text{e}}(z;\sigma,\tau)\,=\,\prod_{j,\,k=0}^{\infty}\frac{1\,-\,\zeta^{-1}p^{j+1}q^{j+1}}{1\,-\,\zeta\,p^{j}\,q^{k}}\,.$
(161)

The poles and zeroes of $\Gamma_{\text{e}}$ are located at
$z=-j\sigma-k\tau+\ell$ and $z=(j+1)\sigma-(k+1)\tau+\ell$ respectively where
the indices $j$, $k$ run over the non-negative integers and $\ell$ runs over
the integers Felder2000 . It has also the following quasi-periodicity
properties

$\begin{split}\Gamma_{\text{e}}(z+\sigma;\sigma,\tau)&\,=\,\theta_{0}(z;\tau)\,\Gamma_{\text{e}}(z;\sigma,\tau)\,,\\\
\Gamma_{\text{e}}(z+\tau;\sigma,\tau)&\,=\,\theta_{0}(z;\sigma)\,\Gamma_{\text{e}}(z;\sigma,\tau)\,.\end{split}$
(162)

The first three Bernoulli polynomials are

$\displaystyle B_{1}(z)$ $\displaystyle\,\equiv\,$ $\displaystyle
z-\frac{1}{2}\,,$ (163) $\displaystyle B_{2}(z)$ $\displaystyle\,\equiv\,$
$\displaystyle z^{2}-z+\frac{1}{6}\,,$ (164) $\displaystyle B_{3}(z)$
$\displaystyle\,\equiv\,$ $\displaystyle
z^{3}-\frac{3\,z^{2}}{2}+\frac{z}{2}\,.$ (165)

Their Fourier series decomposition, for $k>1$ and $0\leq z<1$ is

$B_{k}(z)\,=\,-\frac{k!}{(2\pi{\rm i})^{k}}\,\sum_{j\neq 0}\frac{{\bf
e}(jz)}{j^{k}}\,.$ (166)

The $k=1$ is peculiar in the sense that (166) is almost fine, but fails to
hold at $z=0$. The equality (166) doesn’t hold for $k=1$ because at $z=0$ the
Fourier expansion vanishes meanwhile the Bernoulli polynomial
$B_{1}(\\{0\\})=-\frac{1}{2}$. Thus, (166) fails to hold at $z=0$. This will
have zero impact in generic computations, and thus, generically, we will only
highlight the difference when it will matter .

Clearly this also equals $B_{k}(\\{x\\})$ where $\\{x\\}\,\equiv\,x-\lfloor
x\rfloor$ is the fractional part of $x$. In particular we have for
$x\in\mathbb{R}$

$\displaystyle B_{0}(\\{x\\})$ $\displaystyle\,\equiv\,$ $\displaystyle
1-\sum_{j}\delta(x+j)\,,$ (167) $\displaystyle B_{1}(\\{x\\})$
$\displaystyle\,\equiv\,$ $\displaystyle-\frac{1}{2\pi{\rm i}}\sum_{j\neq
0}\frac{{\bf e}(jx)}{j}\,\quad\text{for}\,\quad x\notin\mathbb{Z}\,,$ (168)
$\displaystyle B_{2}(\\{x\\})$ $\displaystyle\,\equiv\,$
$\displaystyle\frac{1}{2\pi^{2}}\,\sum_{j\neq 0}\frac{{\bf e}(jx)}{j^{2}}\,,$
(169) $\displaystyle B_{3}(\\{x\\})$ $\displaystyle\,\equiv\,$
$\displaystyle-\frac{3\,{\rm i}}{4\pi^{3}}\,\sum_{j\neq 0}\frac{{\bf
e}(jx)}{j^{3}}\,.$ (170)

Equality (168) does not hold at $x\in\mathbb{Z}$. These are the positions of
the discontinuities of both he left and the right-hand objects. The value of
the function in the left-hand side at $x\in\mathbb{Z}$ coincides with the
limit from the right. The value of the function in the right-hand side,
matches the semisum of the left and right limits.

The generic polynomial $B_{k}$ for $k\,=\,1\,,\,2\,,\,\ldots$ obeys the
following three properties (the first and second also apply to $p=0$)

$\displaystyle\int_{0}^{1}dx\,B_{k}(\\{x\\})$ $\displaystyle\,=\,$
$\displaystyle 0\,,$ (171) $\displaystyle B_{k}(\\{x\\})$ $\displaystyle\,=\,$
$\displaystyle(k+1)\,\frac{d}{dx}B_{k+1}(\\{x\\})\,,$ (172) $\displaystyle
B_{k}(z+1)\,-\,B_{k}(z)$ $\displaystyle\,=\,$ $\displaystyle kz^{k-1}\,.$
(173)

At some points we will need to use the following smooth versions of the
periodic Bernoulli polynomials

$\displaystyle B_{1\Lambda}(\\{x\\})$ $\displaystyle\,\equiv\,$
$\displaystyle-\frac{1}{2\pi{\rm i}}\sum_{j\neq 0}\frac{\Lambda^{|j|}{\bf
e}(jx)}{j}\,=\,-\,\frac{1}{2\pi{\rm i}}\Bigl{(}\text{Li}_{1}(\Lambda{\bf
e}(x))\,-\,\text{Li}_{1}(\Lambda{\bf e}(-x))\Bigr{)}\,,$ (174) $\displaystyle
B_{2\Lambda}(\\{x\\})$ $\displaystyle\,\equiv\,$
$\displaystyle\frac{1}{2\pi^{2}}\,\sum_{j\neq 0}\frac{\Lambda^{|j|}{\bf
e}(jx)}{j^{2}}\,\,=\,\,\,\,\,\,\,\,\frac{1}{2\pi^{2}}\Bigl{(}\text{Li}_{2}(\Lambda{\bf
e}(x))\,+\,\text{Li}_{2}(\Lambda{\bf e}(-x))\Bigr{)}\,,$ (175) $\displaystyle
B_{3\Lambda}(\\{x\\})$ $\displaystyle\,\equiv\,$ $\displaystyle-\frac{3\,{\rm
i}}{4\pi^{3}}\,\sum_{j\neq 0}\frac{\Lambda^{|j|}{\bf
e}(jx)}{j^{3}}\,=\,-\,\frac{3\,{\rm
i}}{4\pi^{3}}\,\Bigl{(}\text{Li}_{3}(\Lambda{\bf
e}(x))\,-\,\text{Li}_{3}(\Lambda{\bf e}(-x))\Bigr{)}\,.$ (176)

For $x\in\mathbb{R}$ we will assume $0\leq\Lambda<1$. In cases where the
argument $x$ happens to be complex we will assume $0<\Lambda
e^{2\pi|\text{Im}x|}<1$ .

### Recalling the functions $\log P$, $\log Q$

The function $\log P$ is defined by the following double expansion

$\begin{split}\log P(z)&\,=\,\log P(z_{1},z_{2};\tau)\,=\,\log
P_{0}(z_{1},z_{2};\tau)\,+\,{\pi{\rm
i}\tau}B_{2}\bigl{(}\\{z_{2}\\}\bigr{)}+\pi{\rm i}\Psi_{P}(z)\,\\\
&\,=\,-\frac{{\rm i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\frac{{\bf e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,+\,{\pi{\rm
i}\tau}B_{2}\bigl{(}\\{z_{2}\\}\bigr{)}+\pi{\rm i}\Psi_{P}(z)\,,\end{split}$
(177)

where $\Psi_{P}(z)$ is an arbitrary $\tau$ independent double periodic real
function. The real part of $\log P$ is the Kronecker-Eisenstein series.
Without loss of generality, in the main body of the manuscript we fix
$\Psi_{P}\,=\,0$. This is equivalent to a trivial redefinition of the
regularization ambiguity $\Psi_{0}$ in (205).

The function $\log Q$ is defined by the following double expansion

$\begin{split}\log Q(z)&\,=\,\log Q(z_{1},z_{2};\tau)\,=\,\log
Q_{0}(z_{1},z_{2};\tau)\,+\,\frac{2\pi{\rm
i}\tau}{3}B_{3}\bigl{(}\\{z_{2}\\}\bigr{)}\,+\,\pi{\rm i}\,\Psi_{Q}(z)\,\\\
&\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\frac{{\bf e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,+\,\frac{2\pi{\rm
i}\tau}{3}B_{3}\bigl{(}\\{z_{2}\\}\bigr{)}\,+\,\pi{\rm
i}\,\Psi_{Q}(z)\,,\end{split}$ (178)

where $\Psi_{Q}(z)$ is an arbitrary $\tau$ independent double periodic real
function. The real part of $\log Q$ is the Bloch-Wigner elliptic dilogarithm.
Without loss of generality, in the main body of the manuscript we fix
$\Psi_{Q}\,=\,0$. This is equivalent to a trivial redefinition of the
regularization ambiguity $\Psi_{0}$ in (205).

##  Appendix B The supersymmetric partition function and the Index

In this appendix we revisit the problem of regularization of the divergent
super-determinants of the underlying four-dimensional field-theory problem
from a different perspective than usual. 474747The gravitational counterpart
of this perspective will not be studied here Assel:2014tba ; Genolini:2016ecx
; Papadimitriou:2017kzw ; An:2017ihs . This gives a first principles origin of
the sequence of actions $S_{\lambda}$ that was used to find a fixed point
expansion of the superconformal index.

Before entering in technical discussions, let us take a couple of paragraphs
to comment about regularization schemes in the context of supersymmetric
partition functions. The superconformal index and the supersymmetric partition
function that the gravitational side of the $AdS/CFT$ duality sets as the
appropriate observable to study Cabo-Bizet:2018ehj , can differ by a pre-
factor. We will show that there exist regularization schemes that, distinctly
to the schemes used in Cabo-Bizet:2018ehj ; Closset:2013sxa ; Assel:2014paa ;
Assel:2015nca , give the superconformal index out of the divergent 4d super-
determinants without the pre-factor. In that sense the schemes below will play
an analogous role to the one played by the background-subtraction method used
in the gravitational side of the duality, when evaluating the onshell action
Chen:2005zj  of the dual BPS black holes Cabo-Bizet:2018ehj ; Cassani:2019mms
.

The difference in between this regularization and the ones of Cabo-
Bizet:2018ehj ; Closset:2013sxa ; Assel:2014paa ; Assel:2015nca  484848The
method used in the first three references also differs from the one used in
the last one, by a finite quantity. is a finite contribution. It would be
interesting to better understand whether the counterterms accounting for such
finite differences can go beyond the hypothesis of the  _no-go_ arguments of
Assel:2014tba , or not; and in case the answer is yes, then how. In particular
the former scheme cancels the pre-factor, meanwhile the latter two do not. As
these schemes are used after supersymmetric paired contributions are
cancelled, then in principle the three of them could naively be thought to be
consistent with supersymmetry, and thus from this perspective, it is not clear
to us why one of them is supersymmetric and the others are not. This is a
puzzle to us. Perhaps an analysis along the lines of Kuzenko:2019vvi ;
Bzowski:2020tue  could help to resolve it.

After this small disgresion, let us summarize. The first outcome of this
appendix is to show that there exists a regularization, after supersymmetric
cancellations have been already taken into consideration, that cancels the
relative pre-factor between superconformal index and supersymmetric partition
function. 494949Note that this does not mean that the pre-factor lacks
physical meaning. Indeed, the pre-factor is known to be related to ’t Hooft
anomaly coefficients of the corresponding theory, as it will be recalled in
appendix C.1.  To claim that this scheme is consistent with supersymmetry a
more detailed analysis along the lines of Kuzenko:2019vvi ; Bzowski:2020tue
needs to be pursued. 505050It would be also interesting to compare with the
scheme presented in Closset:2019ucb . That lies beyond the scope of this
paper. The most important outcome of this section is that the freedom in
choice of that regularization scheme, includes the freedom in choice of
extension of the integrand (of the superconformal index) to the complex plane.
This gives a first principles derivation of the extensions $S_{\lambda}$ that
we used to find the fixed point expansion of the superconformal index.

For a while we will focus on the contribution of a single chiral multiplet. In
the following appendices we will reinstate the weights $\rho\,$ and the
multiplet index $\alpha$ and sum over them.

###  B.1 Two examples of regularization

We start by regularizing the one-loop determinants of the supersymmetric
partition functions on $S_{1}\times S_{3}$. Firstly, we illustrate two cases,
that we call the elliptic and meromorphic schemes. The elliptic scheme
corresponds to the double periodic extension of Cabo-Bizet:2019eaf ; Cabo-
Bizet:2020nkr . Both regularization schemes match the superconformal index at
the contour of integration $\underline{u}_{2}=0$.

### General strategy to regularize

The contribution of a $4d$ $\mathcal{N}=1$ chiral multiplet to the
supersymmetric partition function on $S_{1}\times S_{3}$ has been computed via
supersymmetric localization. The formal expression is Cabo-Bizet:2018ehj
Assel:2014paa ; Closset:2013sxa

$Z\,=\,e^{-S(z)}\,\equiv\,\prod_{k\,=\,-\infty}^{\infty}\,\prod_{m,\,n\,=\,0}^{\infty}\frac{z\,+\,k\,-\,(m\,+\,\frac{1}{2})\tau\,-\,(n\,+\,\frac{1}{2})\sigma}{z\,+\,k\,+\,(m+\frac{1}{2})\tau\,+\,(n\,+\,\frac{1}{2})\sigma}\,$
(179)

where $S$ is the divergent one-loop effective action. The complex parameters
$\sigma=\sigma_{1}\,+\,{\rm i}\sigma_{2}$ and $\tau=\tau_{1}\,+\,{\rm
i}\,\tau_{2}$ are in a beginning assumed to have positive imaginary part
$\sigma_{2}>0$ and $\tau_{2}>0\,$. These parameters can be thought of as a
complexification of the angular velocities of rotation around the two
independent rotational axis of the three-sphere. The three integer numbers
$k$, $m$ and $n$ label the modes associated to the non-contractible Euclidean
time cycle and two contractible cycles of the three-sphere. For the moment $z$
is just a complex variable that in due time will be identified with a linear
combination of the gauge variables $u$ and flavour and $R$-symmetry chemical
potentials. Momentarily, we will assume that $\sigma=\tau$. Later on in
appendix B.2 we will show that this analysis can be extended to infinitely
many other cases.

For $\sigma=\tau$ the potential in (179) can be written as the following
divergent series

$S\,=\,\sum_{(k,\,m)\,\in\,\mathbb{Z}^{2}}\,(m\,+\,1)\,\log\Bigl{(}z\,+\,k\,+\,\tau\,(m+1)\Bigr{)}\,.$
(180)

The series $S$ is formally related to

$\widehat{S}\,\equiv\,\sum_{(k,\,m)\,\in\,\mathbb{Z}^{2}}\,\,\log\,\Bigl{(}z\,+\,k\,+\,\tau\,(m+1)\Bigl{)}\,,$
(181)

by the following linear partial differential equation

$\partial_{z}\,S\,=\,\nabla_{\tau}\,\widehat{S}\,.$ (182)

The derivative

$\nabla_{\tau}\,\equiv\,\partial_{\tau}\,-\,e_{\tau}\,,\qquad\qquad
e_{\tau}\,\equiv\,\partial_{\tau}z\,\partial_{z}\,,$ (183)

is invariant under holomorphic re-parameterizations of $z$ as a function of
$\tau$. The choice of “frame” $e_{\tau}$ is part of the ambiguity in
regularization. Different such choices define different regularizations. Let
us choose a linear and holomorphic (in $\tau$) re-parameterization of the
complex variable $z$

$z\,=\,\chi\,+\,\tau\,\xi\,,$ (184)

with $\chi$ and $\xi$ being two auxiliary complex variables that are assumed
to be independent of $\tau$. Then, the differential equation (182) takes the
form

$\partial_{z}\,S\,=\,(\partial_{\tau}\,-\,\xi\,\partial_{z})\,\widehat{S}\,.$
(185)

To ease the work, we restrict (185) to the domain

$\xi\,\in\,\mathbb{R}\,,\qquad\chi\,\in\,\mathbb{C}\,.$ (186)

We recall that one can always write a generic complex variable $\mathfrak{X}$
such as $z,\,\chi\,$ or $\xi$, as a linear combination of $1$ and $\tau$. The
corresponding components are non-holomorphic functions of $\mathfrak{X}$ and
$\tau$

$\begin{split}\mathfrak{X}_{1}&\,=\,\frac{\overline{\mathfrak{X}}\,\tau\,-\,\mathfrak{X}\,\overline{\tau}}{\tau\,-\,\overline{\tau}}\,\qquad\mathfrak{X}_{2}\,=\,\frac{\overline{\mathfrak{X}}\,-\,\mathfrak{X}}{\tau\,-\,\overline{\tau}}\,.\end{split}$
(187)

We note that $\chi$ and $\xi$ are not the real components $z_{1}$ and $z_{2}$.
In fact $z_{1}\,=\,\chi_{1}$ and $z_{2}\,=\,\chi_{2}\,+\,\xi$, where
$\chi_{1}$ and $\chi_{2}$ are the components $1$ and $2$ of $\chi$.

So far $z$ in (184) is just a mere complex variable. We need to define its
relation to the physical variables i.e to the gauge variables $u$ and chemical
potentials $\Delta$’s . The relation will be

$u\,+\,\Delta\,\,\equiv\,z\,=\,\chi\,+\,\tau\,\xi.$ (188)

In particular we can always identify the components $1$ and $2$ of
$u\,+\,\Delta$ with the respective ones of $z\,=\,\chi\,+\,\tau\xi$. For
short, from now on we will use the notation $z(u)\,\equiv\,u\,+\,\Delta$.

The independent variable $\xi\in\mathbb{R}$ labels different regularization
schemes, i.e defines different regularization conditions, each of them,
associated to a differential equation (185) in the way that will be explained
below.

To regularize we will follow three steps :

  1. 1.

Regularize the real part of $S$ by demanding either double periodicity (in
subsection B), or meromorphy (in subsection B) in $u$. In both cases the
regular answer turns out to be a real-harmonic function of $\tau$

$(\partial^{2}_{\tau_{1}}\,+\,\partial^{2}_{\tau_{2}})\,\text{Re}S\,=\,0\,.$
(189)

  2. 2.

The regularized imaginary part of $S$ is constrained by requiring meromorphy
in $\tau$.

  3. 3.

There is a remaining $\tau$-meromorphic ambiguity that we denote as $R$. We
bound it to be, in both cases, of the form

$\begin{split}\log R\,\sim\,+{\rm i}\tau\sim-\infty\,\quad\text{for
}\quad\tau\to+\,{\rm i}\,\infty\,,\qquad\log
R\,\sim\,0\,\quad\text{for}\quad\tau\to+\,{\rm i}\,0\,.\end{split}$ (190)

515151As explained before, this condition can be relaxed. In principle, the
only constraint on $R$ is that its real part must be an arbitrary real
function of $u_{2}$ and $\tau$. This ambiguity has to be fixed by imposition
of a physical condition. In our case such condition is matching to the
superconformal index. As will be shown in the next section, the condition
(190) is only at half-way of the constraint that we must impose. To illustrate
connections to previous regularization schemes we find illustrative to leave
this ambiguity free for the moment.

We call $R$ the  _remainder function_. The function $\log R$ determines the
large-$\tau$ asymptotic behaviour of the regularized version of the action
$S(z)$.

There are two technical issues that we need to highlight before moving on

  * •

The infinities to regularize, come from a zero Fourier mode $(m,n)=0$ in the
double Fourier expansions below. The Poisson summation dual of the divergent
expressions will make that clear. The first step will be to excise such mode.
After that, finite results are obtained, but there is still a finite ambiguity
that needs to be constrained by the properties of the physical observable that
one wishes to compute.

  * •

Some of the regular series below do not converge fast enough. This means they
converge to functions which are non smooth. To render such limit functions
smooth at intermediate steps, further “regularization” is required. 525252In
order to apply ABBV integration formula it is safer to use smooth functions.
To solve that issue we insert an intermediate real cut-off
$0\,\leq\,\Lambda\,<\,1$. We will implicitly assume $\Lambda\,=\,1^{-}$ at
every step. See appendix F.

##### An elliptic scheme

We start by working out two particular schemes. The first one we call the
_elliptic_ scheme. It corresponds to the following choice of frame

$\xi\,=\,\Delta_{2}\,,\qquad\Delta_{2}\,\in\,\mathbb{R}\,,$ (191)

where $\Delta_{2}$ is the second real component of the complex chemical
potential $\Delta\,=\,\Delta_{1}\,+\,\Delta_{2}\tau$.

The divergent series $S$ and $\widehat{S}$ are analytic in the independent
complex variables $z$ and $\tau$. To avoid clumsiness in our presentation it
is convenient to introduce the following two real variables $x$ and $y$ to
denote the real and imaginary parts of $z$ i.e $z=\,x\,+\,{\rm i}y\,$. Thus,
from (182) it follows that 535353For a holomorphic function $h(z)$ of $z$ it
follows that
$\text{Re}\,\partial_{z}h(z)=\partial_{\text{Re}\,z}\text{Re}\,h\,.$

$\partial_{x}\,\text{Re}S\,=\,\,(-\,\xi\,\partial_{x}+\partial_{\tau_{1}})\,\text{Re}\widehat{S}\,.$
(192)

The series $\widehat{S}$ is an average over the lattice
$\mathbb{Z}\tau+\mathbb{Z}$. Thus, at formal level we can recast it as a
double Fourier expansion with the use of Poisson summation formula

$\sum_{(m,\,n)\,\in\,\mathbb{Z}^{2}}\,f(x+k,\,y+m)\,=\,\,\sum_{(m,\,n)\,\in\,\mathbb{Z}^{2}}\widehat{f}(m,n)\,{\bf
e}(xm+yn)\,.$

We define the Fourier transform of $f(x,y)$ as
$\widehat{f}(m,n)\,\equiv\,\int_{0}^{1}dxdy\,{\bf e}(-xm-yn)f(x,y)\,.$ A
computation shows that

$\text{Re}\widehat{S}\,=\,-\sum_{(k,\,m)\,\in\,\mathbb{Z}^{2}}\,\frac{\tau_{2}}{2\pi}\,\frac{{\bf
e}\bigl{(}nz_{2}\,-\,mz_{1}\bigr{)}}{|m\tau\,+\,n|^{2}}\,.$ (193)

This series is divergent due to the presence of the zero mode $(m,n)=0$.
Indeed, this is the origin of the divergences in the original series $S$. From
now on we excise the zero mode $(m,n)=(0,0)$ and proceed with the new
convergent expression of $\widehat{S}$ which can be presented in the form

$\text{Re}\widehat{S}\,\equiv\,\log|P(z_{1},z_{2};\tau)|\,,$ (194)

where

$\log|P(z_{1},z_{2};\tau)|\equiv-\frac{\tau_{2}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop(m,n)\neq(0,0)}{\sum}}\;\frac{{\bf
e}(nz_{2}-mz_{1})}{|m\tau+n|^{2}}\,,$ (195)

is a well known real harmonic and real analytic modular invariant function of
$\tau$.

If we plug (194) in the right-hand side of (192) we can integrate to obtain a
regular expression for Re$S$. We obtain two contributions. One contribution
comes from the antiderivative with respect to $x$ of
$\partial_{x}\text{Re}\widehat{S}$, which equals $-\xi$ times $\widehat{S}$,
up to an arbitrary function of $z_{2}$ and $\tau$,

$-\xi\int\,dx\,\partial_{x}\,\text{Re}\widehat{S}\,=\,\,\,-\,\xi\,\log|P(z_{1},z_{2};\tau)|\,+\,\ldots\,,$
(196)

and a second comes from the antiderivative with respect to $x$ of the
derivative of (194) with respect to $\tau_{1}$, which equals

$\int\,dx\,\partial_{\tau_{1}}\text{Re}\widehat{S}\,=\,\,\,\,\log|Q(z_{1},z_{2};\tau)|\,+\,\ldots\,,$
(197)

where

$\begin{split}\log|Q(z_{1},z_{2};\tau)|&\equiv\frac{{\rm
i}\,\tau_{2}}{2\pi^{2}}\,{\underset{m,n\in\mathbb{Z}\atop(m,n)\neq(0,0)}{\sum}}\frac{\left(m\,\tau_{1}\,+\,n\right)\,{\bf
e}\bigl{(}nz_{2}-mz_{1}\bigr{)}}{|m\tau+n|^{4}}\,.\end{split}$ (198)

This double periodic function is closely related to the Bloch-Wigner elliptic
dilogarithm ZagierOnBloch , as was pointed out in Cabo-Bizet:2019eaf , see
section $3.1$ in that reference. This function is a real harmonic function of
$\tau$ but it is not modular covariant.

Collecting partial results we obtain the regularized expression for the real
part of the action after evaluating $\xi=\Delta_{2}$

$\text{Re}S_{\text{ell}}\,=\,\log|Q(z_{1},z_{2};\tau)|\,-\,\Delta_{2}\,\log|P(z_{1},z_{2};\tau)|\,+\,\log|R(z_{2};\tau)|\,,$
(199)

where $\log|R|$ depends on $z_{2}$ and $\tau$. If we assume $\log|R|$ to be
the real part of a function that is meromorphic in $\tau$, then we must assume
it to be a real harmonic function of $\tau\,$.

Holomorphy in $\tau$ constraints the imaginary part of the action
$\text{Im}S\,$. As $\log|P|$ and $\log|Q|$ are the real parts of (177) and
(178) we obtain

$S_{\text{ell}}(z(u))\,=\,\log Q(z(u))\,-\,\Delta_{2}\,\log P(z(u))\,+\,\log
R(z(u);\tau)\,.$ (200)

The asymptotic conditions (190) constraint $\log R$ to be of the form

$\log R\,=\,2\pi{\rm i}\tau E(z_{2})\,\,+\,\pi{\rm i}\Phi(z_{1},z_{2})\,,$
(201)

where $\Phi$ and $E$ are real functions. Note that the term proportional to
$E$ dictates the asymptotic behaviour of $S_{\text{ell}}$ at
$\tau\,\to\,+\,{\rm i}\,\infty$. Note that when the function $E$ is a
constant, we can interpret it as an insertion of a background charge (or
energy). That is because $2\pi{\rm i}\tau$ is the chemical potential dual to
the BPS charge $E$.

##### A meromorphic scheme

In this subsection we present a second scheme. It corresponds to the following
choice

$\xi\,=\,z_{2}\,=\,u_{2}\,+\,\Delta_{2}\,,\qquad z_{2}\,\in\,\mathbb{R}\,,$
(202)

where $u_{2}$ and $\Delta_{2}$ are one of the components of $u$ and $\Delta$
in their linear decomposition in terms of $1$ and $\tau$,
$u\,=\,u_{1}\,+\,\tau u_{2}$ and $\Delta\,=\,\Delta_{1}\,+\,\tau\Delta_{2}\,$.

The first two steps are identical to the ones of the previous section. Compute
the anti-derivative with respect to $x$ of the term coming from
$\xi\,\partial_{x}$ and obtain again (196). Then, compute the antiderivative
with respect to $x$ of the term coming from $\partial_{\tau_{1}}$ which is
again (197). Collecting partial results, and finally substituting
$\xi\,=\,z_{2}=u_{2}+\Delta_{2}$, we obtain

$\begin{split}\text{Re}S&\,=\,\text{Re}S_{\text{ell}}\,-\,u_{2}\log|P(z(u))|\\\
&\,=\,\log|Q(z(u))|\,-\,(u_{2}+\Delta_{2})\,\log|P(z(u))|\,+\,\log|R(z_{2}(u))|\,,\end{split}$
(203)

where $\log|R|$ is (before imposing boundary conditions (190)) an arbitrary
function of $z_{2}$ and harmonic in $\tau$. Finally, by demanding meromorphy
in $\tau$ we obtain

$\begin{split}S&\,=\,S_{\text{ell}}\,-\,u_{2}\log P(z(u))\\\ &\,=\,\log
Q(z(u))\,-\,z_{2}(u)\,\log P(z(u))\,+\,\log R(z_{2}(u))\,,\end{split}$ (204)

where again the asymptotic conditions (190) constraint $\log R$ to be of the
form

$\log R\,=\,2\pi{\rm i}\tau E(z_{2})\,+\,\pi{\rm i}\Phi(z_{1},z_{2}),$ (205)

where $\Phi$ and $E$ are real functions. Note that this term dictates the
asymptotic behaviour of $S$ at $\tau\to+{\rm i}\infty\,$. In the next section
we will show that there exists a choice of $R$ such that for $u\in\mathbb{C}$

$\,e^{-S(z(u))}\,=\,\Gamma_{\text{e}}(z(u)+\tau;\tau,\tau)\,.$ (206)

In these two subsections, we have assumed $\sigma=\tau$ but that condition can
be relaxed and the same regularization process applies (See appendix B.2).

###  B.2 Generalization to the case $\sigma\,\neq\,\tau$

In this appendix we show that the same regularization used for the cases
$\sigma\,=\,\tau$ in section B can be adapted to more general cases with
$\sigma\neq\tau$. This implies that everything we have said in this paper
relative to the case $\sigma=\tau$ can be straightforwardly generalized to the
more general case studied in this subsection.

Let $\sigma\,=\,a\omega$ and $\tau\,=\,b\omega$ with $a$ and $b$ positive
integers. For simplicity we will assume $a$ and $b$ to be primes, but the same
result applies to the generic case. Let us define
$\widetilde{z}\,=\,z\,+\,\frac{\tau\,+\,\sigma}{2}$ and analyze the logarithm
of (179)

$\sum_{k\in\mathbb{Z}}\sum_{m,n\,=\,0}^{\infty}\log\Bigl{(}\widetilde{z}\,+\,k\,+\,\Bigl{(}\frac{m}{b}+\frac{n}{a}\Bigr{)}ab\omega\Bigr{)}\,-\,\log\Bigl{(}\widetilde{z}\,+\,k-\Bigl{(}\frac{m+1}{b}+\frac{n+1}{a}\Bigr{)}ab\omega\Bigr{)}\,.$
(207)

Here we have conveniently written
$ma\omega+nb\omega\,=\,\Bigl{(}\frac{m}{b}+\frac{n}{a}\Bigr{)}ab\omega$. For
positive integers $m$ and $n$ it is always possible to rewrite

$\frac{m}{b}\,=\,\frac{r_{m}}{b}\,+\,\widetilde{m}\,,\qquad\frac{n}{a}\,=\,\frac{s_{n}}{a}\,+\,\widetilde{n}\,,$
(208)

with integers $\widetilde{m}$, $\widetilde{n}$, $r_{m}$ and $s_{n}$ ranging
over

$\widetilde{m}\,\geq\,0\,,\quad\widetilde{n}\,\geq\,0\,,\quad
r_{m}\,=\,0,\,\ldots\,,b-1\,,\quad s_{n}\,=\,1,\ldots,a-1\,.$ (209)

It is straightforward to check that (207) takes the form

$\sum_{r=0}^{a-1}\sum_{s=0}^{b-1}\sum_{k\in\mathbb{Z}}\sum_{\widetilde{m},\,\widetilde{n}\,=\,0}^{\infty}\log\Bigl{(}\widetilde{z}_{r,s}\,+\,k\,+\,\bigl{(}\widetilde{m}\,+\,\widetilde{n}\bigr{)}ab\omega\Bigr{)}\,-\,\log\Bigl{(}\widetilde{z}_{r,s}\,+\,k-\bigl{(}\widetilde{m}\,+\,\widetilde{n}\,+\,2\bigr{)}ab\omega\Bigr{)}\,,$
(210)

where

$\widetilde{z}_{r,s}\,=\,\widetilde{z}\,+\,ra\omega\,+\,sb\omega\,.$ (211)

In reaching this expansion we have used the following redefinition in the
argument of the second logarithm

$\widetilde{m}_{\text{after}}\,=\,b\,-\,1\,-\,\widetilde{m}_{\text{before}}\,,\qquad\widetilde{n}_{\text{after}}\,=\,a\,-\,1\,-\,\widetilde{n}_{\text{before}}\,.$
(212)

As the summand of (210) only depends on the combination
$\widetilde{m}\,+\,\widetilde{n}$ we can reorganize the sum as follows, by
collecting degeneracies

$\sum_{r=0}^{a-1}\,\sum_{s=0}^{b-1}\,\sum_{(k,m)\in\mathbb{Z}^{2}}\,(m\,+\,1)\,\log\Bigl{(}z^{a,b}_{\,r,s}\,+\,k\,+\,(m+1)\,ab\omega\Bigr{)}\,.$
(213)

In this expression

$z^{ab}_{rs}\,\equiv\,\widetilde{z}_{r,s}\,-\,ab\omega\,=\,z\,+\,\Bigl{(}r+\frac{1}{2}\Bigr{)}a\omega\,+\,\Bigl{(}s+\frac{1}{2}\Bigr{)}b\omega\,-\,ab\omega\,.$
(214)

Note that (213) is a combination of terms like the one we regularized before
in subsection B, see equation (180). Thus, assuming (213) as starting point,
one can regularize as explained in the previous subsections, but for generic
values of $\tau$ and $\sigma$, with a common complex divisor $\omega$.

##  Appendix C The real functions $E$ and $\Phi$

Here we fix $R$ in such a way that the finite result coincides with the
elliptic Gamma function $\Gamma_{\text{e}}(z(u)+\tau;\tau,\tau)$ in the entire
plane $(u_{1},u_{2})$ with $u\,=\,u_{1}\,+\,\tau u_{2}$. The details are given
in appendix F.2.

We choose the same scheme as before but modifying $z(u)$ in (188) by
substituting $u$ by a linear combination of $N$ complexified Cartan angles
$\underline{u}$, $\rho(u)$. $\rho$ is a weight. This gives again (204) with
$u$ substituted by $\rho(u)$. Next we must fix $R$ by demanding that for all
$u^{i}\in\mathbb{C}$

$\begin{split}\Gamma_{\text{e}}(z(u)+\tau;\tau,\tau)\,=\,{R}^{-1}\,\,\frac{\,\,P_{0}(z_{1},z_{2})^{z_{2}}}{Q_{0}(z_{1},z_{2})}\,.\\\
\end{split}$ (215)

This equation is the exponential version of (204), after assuming (206), with
the substitution $u\,\mapsto\,\rho(u)$. This is the completion of the equation
(2.28) of Cabo-Bizet:2020nkr Zudilin . There, this identity was defined up to
a $\tau$-independent imaginary constant. The derivation of the remainder $R$
that fixes this pure phase ambiguity, will be postponed until appendix F.2.
Here we will just quote the result.

We define the functions $P_{0}$ and $Q_{0}$ via their logarithms

$\log P_{0}\,=\,\log P_{0}(z_{1},z_{2})\,\equiv-\frac{{\rm
i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq 0}{\sum}}\;\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,,$ (216)

and

$\log Q_{0}\,=\,\log
Q_{0}(z_{1},z_{2})\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop
m\neq 0}{\sum}}\;\frac{{\bf e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,.$ (217)

They are related, via a redefinition of $R$, to the functions $\log P$ and
$\log Q$ (See appendix A). In the left-hand side of (215) we define

$z(u)\,=\,\rho(u)\,+\,\Delta\,.$ (218)

In the right-hand side, the variables $z_{1}$ and $z_{2}$ are related to the
$2$rk$(G)$ variables
$\underline{u}\,=\,\underline{u}_{1}\,+\,\tau\underline{u}_{2}$ and
$\underline{v}$ as follows

$\begin{split}\,z_{1}(u,v)&\,=\,z(u)\,-\,\rho(v)\,\tau\,-\,\Delta_{2}\,\tau\,,\\\
\,z_{2}(v)&\,=\,\rho(v)\,+\,\Delta_{2}\,.\end{split}$ (219)

In the right-hand side of (215) the auxiliary variable $v$ is fixed in terms
of $u$ i.e.

$\underline{v}\,=\,\underline{u}_{2}\,.$ (220)

Note that $z_{1}$ is a real function of the real variables $\underline{u}_{1}$
and not of $\underline{u}_{2}\,$. The $R^{-1}$ then takes the following form

$\begin{split}R^{-1}\,&\,=\,R^{-1}(u,v)\,\,\equiv\,q^{-E}\,\,{\bf
e}\bigl{(}-\frac{\Phi}{2}\bigr{)}\,,\end{split}$ (221)

where $E$ is related to the following cubic polynomial in two variables

$\mathcal{A}(u,z_{2}(v))\,=\,\frac{B_{3}(z_{2}(v))}{3}\,-\,\bigl{(}\rho(u_{2})+\Delta_{2}\bigr{)}\,\frac{B_{2}(z_{2}(v))}{2}$
(222)

by the following relation

$E\,\equiv\,-\mathcal{A}(u,z_{2}(v))\,+\,\mathcal{A}(u,\\{z_{2}(v)\\})\,.$
(223)

The polynomial $\mathcal{A}$ is related to the contribution of the given
multiplet to the cubic polynomial of anomaly coefficients. If we define

$\mathcal{A}(u,z_{2}(u))\,\,=\,\,-\,\frac{1}{6}\,\Bigl{(}z_{2}(u)^{3}\,-\,\frac{z_{2}(u)}{2}\Bigr{)}\,,$
(224)

then the cubic polynomial of anomaly coefficients for flavour and R-charges
can be defined as

$\mathcal{A}_{anom}\,=\,\sum_{\alpha}\sum_{\rho_{\alpha}}\mathcal{A}(0,z_{2\alpha}(0))\,,$
(225)

where $z_{\alpha}\,=\,\rho_{\alpha}(u)+\Delta_{\alpha}$. More details on this
can be found in section 2.4 of Cabo-Bizet:2020nkr . Finally, the pure phase
$e^{\pi{\rm i}\Phi}$ takes the form

$\begin{split}{\bf e}\bigl{(}-\frac{\Phi}{2}\bigr{)}&\,\equiv\,\frac{{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{z_{1}(u)\\})\,B_{2}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}{{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{z_{1}(u)\\})\,B_{2}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}\,,\\\ &\equiv\frac{{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(z_{1}(u))\,B_{2}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}{{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(z_{1}(u))\,B_{2}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}\,.\end{split}$ (226)

In going to the second line we have used the $2\pi{\rm i}\mathbb{Z}\,$-shift
ambiguity of the exponent. In the main body of the paper we reinstated this
ambiguity in the form of a piecewise integer function that we denoted with the
letter $K$ (See equation (77)).

###  C.1 The universal contribution at large-$N$ of $R$: The susy Casimir
energy

In a gauge anomaly-free theory, the $u_{2}$-dependence cancels out after
summing over multiplets. In those theories, the non-vanishing contribution of
$E$ to the supersymmetric partition function comes from the following two
terms in (223). From the first, one obtains

$-\,\mathcal{A}_{anom}\,.$ (227)

The other contribution comes from the sum over matter multiplets of the term

$\mathcal{A}(u,\\{z_{2}(u_{2})\\})\,=\,\frac{B_{3}(\\{\rho(u_{2})+\Delta_{2}\\})}{3}\,-\,\bigl{(}\rho(u_{2})+\Delta_{2}\bigr{)}\,\frac{B_{2}(\\{\rho(u_{2})+\Delta_{2}\\})}{2}\,.$
(228)

The contribution (227), is the cubic polynomial of flavour and R-symmetry
anomaly coefficients. This is a universal contribution to the large-$N$
effective action of complex saddles, that is linear in $\tau$ Cabo-
Bizet:2020nkr . As pointed out in Cabo-Bizet:2020nkr , this contribution
matches the susy Casimir energy of Assel:2015nca .

For the complex fixed-points $P$ that we know contribute,

$\mathcal{A}(u,\\{z_{2}(u_{2})\\})\Big{|}_{u\,=\,P}\quad\underset{N\,\to\,\infty}{\longrightarrow}\quad
0\,.$ (229)

Let us explain how this happens. For $(m,n)$ fixed points with $m>0$, the
$\rho(u_{2})$ is always of the form

$\frac{i-j}{K},\qquad i,j=1,\ldots,K\in\mathbb{Z}\,,$ (230)

where $K$ is a divisor of $N$ such that $K\to\infty$ as $N\to\infty$. In that
limit the sum over $i$ and $j$ can be reduced to an integral. (229) follows
from the identities

$\int^{1}_{0}dxB_{k}(x)\,=\,0\,,\qquad\int^{1}_{0}dx\,x\,B_{k}(x)\,=\,0\,.$
(231)

Thus, for the meromorphic scheme

$E(P)\quad\underset{N\,\to\,\infty}{\longrightarrow}\quad-\mathcal{A}_{anom}\,.$
(232)

This contribution is the same for every $P$. For the $(1,0)$ this contribution
is not present.

##  Appendix D Another choice of regularization: $\Gamma_{\lambda}$

As we have mentioned before. The regularization ambiguity that we have in the
form of a choice of frame $e_{\tau}$ and remainder $R$, includes other schemes
that are not the elliptic and meromorphic ones. 545454 Note that part of the
degeneracy corresponds to reparametrizations of $S_{1}$ upon $S_{1}$ that
preserve one point. The point being a representation of the real contour of
integration. In this subsection we define a family of schemes that corresponds
to the deformed action $S_{\lambda}$ of section 3.

The choice of frame in this case is

$\xi\,=\,\rho(\\{u_{2}\\}_{\lambda})\,+\,\Delta_{2}\,,\qquad\rho(\\{u_{2}\\}_{\lambda})\,+\,\Delta_{2}\in\,\mathbb{R}\,.$
(233)

By $\\{\underline{u_{2}}\\}_{\lambda}$ we mean the vector with components
$\\{u^{i}_{2}\\}_{\lambda}\,$; and for $x\in\mathbb{R}$ and $0\leq\lambda<1$
was defined in (75).

Here we comment on three relevant properties of this function:

  * a.

The $\\{x\\}_{\lambda}$ is a smooth function for $0\leq\lambda<1$, and it is
periodic $\\{x+1\\}_{\lambda}=\\{x\\}_{\lambda}$.

  * b.

When $x\to 0^{+}$ the $\\{x^{i}\\}_{\lambda}\to 0\,$.

  * c.

When $\lambda\to 0^{+}$ the $\\{x\\}_{\lambda}\to 0\,$.

Repeating the same regularization steps used in appendix B.1, and introducing
the cut-off $\Lambda$ in the final double series expansions we obtain

$e^{-S_{\lambda}}\,=\,{\Gamma_{\lambda}}(z(u))\,\equiv\,{R}_{\Lambda}^{-1}\,\,\frac{\,\,P_{{0}\Lambda}(z_{1},z_{2})^{z_{2}(\\{u_{2}\\}_{\lambda})}}{Q_{0\Lambda}(z_{1},z_{2})}\,.$
(234)

As it will be explained in great detail in appendix F, to ensure continuity
and smoothness of their derivatives, we will need to introduce a cut-off
$\Lambda$ in the double series representation of the logarithms of $P_{0}$,
$Q_{0}$, and $R^{-1}$ i.e for $0\leq\lambda<1\,$, we define

$\begin{split}\log P_{0\Lambda}&\,=\,-\frac{{\rm
i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\Lambda^{|m|\,+\,|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,.\end{split}$ (235) $\begin{split}\log
Q_{0\Lambda}(z;\tau)&\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop
m\neq 0}{\sum}}\;\Lambda^{|m|\,+\,|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,.\end{split}$ (236)

Moreover, we choose

$\begin{split}R^{-1}_{\Lambda}&=q^{-E_{\Lambda}}\,\,{\bf
e}{\left(-\frac{\Phi_{\Lambda}}{2}\right)}\,,\end{split}$ (237)

where

$\begin{split}q^{-E_{\Lambda}}&\,\equiv\,q^{\frac{B_{3}(\lfloor
z_{2}(\\{u_{2}\\}_{\Lambda})\rfloor_{\Lambda}+1)\,-\,B_{3}(\lfloor
z_{2}(0^{+})\rfloor+1)}{3}}\,\times\,\\\ &\qquad\qquad\hfill\times\,{\bf
e}\left(-\,\tau\,(\\{u_{2}\\}_{\Lambda}+\Delta_{2})\,\int^{\\{u_{2}\\}_{\Lambda}\,+\,\Delta_{2}}_{\Delta_{2}}d\xi\,\xi\,\,\Bigl{(}\sum_{m\in\mathbb{Z}}\,{\Lambda}^{|m|}\,{\bf
e}(m\,\xi)\Bigr{)}\right)\,.\\\ {\bf
e}{\left(-\frac{\Phi_{\Lambda}}{2}\right)}&\,\equiv\,{\bf
e}\left(-\,B_{1\Lambda}(z_{1}(u)))\,\,\int^{\\{u_{2}\\}_{\Lambda}\,+\,\Delta_{2}}_{\Delta_{2}}d\xi\,\xi\,\,\Bigl{(}\sum_{m\in\mathbb{Z}}\,{\Lambda}^{|m|}\,{\bf
e}(m\,\xi)\Bigr{)}\right)\,.\end{split}$ (238)

Here by $\lfloor\cdot\rfloor_{\Lambda}$ denotes a generic smooth version of
the floor function. We note that for $0\leq\Lambda<1\,$, the right-hand side
of equation (237) is not holomorphic in $u$ when $v=u_{2}$, but that is not a
problem. Basic algebraic manipulations upon (238) imply that

$\underset{\Lambda\,\to\,1}{\lim}\,\partial_{v^{i}}(2\,Q_{\Lambda}\,+\,\Phi_{\Lambda})\,=\,0\,+\,\text{c.t.}\,.$
(239)

Here
$\partial_{v^{i}}\,=\,\partial_{u^{i}_{2}}\,-\,\tau\,\partial_{u^{i}_{1}}$ .
This relation was used in reaching (83) in the main body of the paper.

Finally, properties $a.$, $b.$ and $c$ imply three relevant properties of the
deformed action $S_{\lambda}$

  * •

a. implies that the $S_{\lambda}$ is smooth in the torus $A\times B$.

  * •

b. implies that the value of $S_{\lambda}$ at the real cycle of integration
$u^{i}_{2}=v^{i}=0^{+}$ is independent of $\lambda\,$.

  * •

c. implies that in limit $\lambda\to 0^{+}$ the $S_{\lambda}\to
S_{\text{ell}}\,$. Where $S_{\text{ell}}$ is essentially the elliptic
deformation of Cabo-Bizet:2019eaf ; Cabo-Bizet:2020nkr . 555555Up to a term
linear in $\tau$ that does not contribute to the on-shell effective actions of
$(m,n)$ fixed points and that can be reinstated by a trivial redefinition of
the remainder function $R$.

##  Appendix E Cauchy-Riemann conditions

In this section we will prove equations (269) and (280), in that order, with
the use of Cauchy-Riemann relations. Let $z\in\mathbb{C}$, we define its real
and imaginary parts as

$\text{Re}z\,=\,x\equiv
z_{1}+\tau_{1}z_{2}\,,\quad\text{Im}(z)\,=\,y\,=\,\tau_{2}\,z_{2}\,.$ (240)

Let $\log G\,=\,U+{\rm i}V$ be a holomorphic function of $z$. Then, its real
and imaginary parts $U$ and $V$, obey the Cauchy-Riemann relations

$\frac{\partial U}{\partial x}\,-\,\frac{\partial V}{\partial
y}\,=\,0\,,\quad\frac{\partial U}{\partial y}\,+\,\frac{\partial V}{\partial
x}\,=\,0\,.$ (241)

In terms of the variables $z_{1}$ and $z_{2}$ these relations read

$\frac{\partial U}{\partial z_{1}}\,+\,\frac{\tau_{1}}{\tau_{2}}\frac{\partial
V}{\partial z_{1}}\,-\,\frac{1}{\tau_{2}}\frac{\partial V}{\partial
z_{2}}\,=\,0\,,\quad\frac{1}{\tau_{2}}\frac{\partial U}{\partial
z_{2}}\,-\,\frac{\tau_{1}}{\tau_{2}}\frac{\partial U}{\partial
z_{2}}\,+\,\frac{\partial V}{\partial z_{1}}\,=\,0\,.$ (242)

Let us define the real function $U$ as

$\begin{split}U\,=\,\text{Re}\log G\,&\equiv\,-\,\text{Re}\log
Q_{0}(z_{1},z_{2};\tau)\,+\,z_{2}\,\text{Re}\log
P_{0}(z_{1},z_{2};\tau)\,.\end{split}$ (243)

For $z_{2}\notin\mathbb{Z}$ the harmonic dual to $U$ can be checked to be, up
to a $\tau$ dependent constant

$\begin{split}V\,=\,&-\sum_{(m,n)\,\in\,\mathbb{Z}^{2}\atop
m\,\neq\,0}\frac{2\pi
z_{2}\text{Re}(m\tau+n)\,\cos\left(2\pi\left(mz_{1}-nz_{2}\right)\right)}{4\pi^{2}m|m\tau+n|^{2}}\\\
&\,-\,\sum_{(m,n)\,\in\,\mathbb{Z}^{2}\atop
m\,\neq\,0}\,\frac{\Bigl{(}\text{Re}^{2}\bigl{(}m\tau+n\bigr{)}-\text{Im}^{2}\bigl{(}m\tau+n\bigr{)}\Bigr{)}\sin\left(2\pi\left(mz_{1}-nz_{2}\right)\right)}{4\pi^{2}m|m\tau+n|^{4}}\\\
&\,=\,-\,\text{Im}\log Q_{0}(z_{1},z_{2};\tau)\,+\,z_{2}\,\text{Im}\log
P_{0}(z_{1},z_{2};\tau)\,.\end{split}$ (244)

Thus, the function

$G(z;\tau)\equiv\frac{P_{0}(z_{1},z_{2};\tau)^{z_{2}}}{Q_{0}(z_{1},z_{2};\tau)}\,$
(245)

is meromorphic for $z_{2}\notin\mathbb{Z}$. Moreover, in Cabo-Bizet:2019eaf
(See below equation (3.20) in that reference) we have shown that

$\Gamma_{\text{e}}(z_{1}+\tau;\tau,\tau)\,=\,\frac{1}{Q_{0}(z_{1},0^{+};\tau)}\,.$
(246)

Let us assume that $0<z_{2}<1$. In this region we can take the limit $z_{2}\to
0^{+}$. From equations (245) and (246) it follows that in such limit

$G(z_{1}+0^{+}\tau;\tau)\,=\,\Gamma_{\text{e}}(z_{1}+0^{+}\tau+\tau;\tau,\tau)\,.$
(247)

From this initial condition and uniqueness of solutions to the Cauchy-Riemann
equations it follows that for $z\in\mathbb{C}$ and $0<z_{2}<1$

$G(z;\tau)\,=\,\Gamma_{\text{e}}(z\,+\,\tau;\tau,\tau)\,=\,\frac{P_{0}(z_{1},z_{2};\tau)^{z_{2}}}{Q_{0}(z_{1},z_{2};\tau)}\,.$
(248)

Next we note that

$\frac{\Gamma_{\text{e}}(z_{1}+1^{-}\tau+\tau;\tau,\tau)}{\Gamma_{\text{e}}(z_{1}+0^{+}\tau+\tau;\tau,\tau)}\,=\,\theta_{0}(z_{1}+1^{-}\tau;\tau)\,=\,P_{0}(z_{1},1^{-};\tau)\,.$
(249)

A computation shows that the real and imaginary parts of
$P_{0}(z_{1},z_{2};\tau)$ obey the Cauchy-Riemann relations (242) for
$z_{2}\notin\mathbb{Z}$. Then, from the initial condition in the last equality
in (249) and uniqueness theorem, it follows that for $0<z_{2}<1$

$\theta_{0}(z_{1}+z_{2}\tau;\tau)\,=\,P_{0}(z_{1},z_{2};\tau)\,.$ (250)

From (248) and (250) it follows that for $z_{2}\notin\mathbb{Z}$

$\Gamma_{\text{e}}(z_{1}\,+\,\\{z_{2}\\}\tau\,+\,\tau;\tau,\tau)\,=\,\frac{P_{0}(z_{1},z_{2};\tau)^{\\{z_{2}\\}}}{Q_{0}(z_{1},z_{2};\tau)}\,,$
(251)

and

$\theta_{0}(z_{1}+\\{z_{2}\\}\tau;\tau)\,=\,P_{0}(z_{1},z_{2};\tau)\,.$ (252)

Using quasi-periodicity properties of $\Gamma_{\text{e}}$ and $\theta_{0}$ one
can find other identities of these kind like for instance

$\Gamma_{\text{e}}(z_{1}+\tau(-\\{-z_{2}\\}+1);\tau,\tau)\,=\,\frac{\,\,P_{0}(z;\tau)^{-\\{-z_{2}\\}}}{Q_{0}(z;\tau)}\,.$
(253)

###  E.1 Cancelling discontinuities and contact terms

The property (173) is particularly useful to generate contact terms that need
to be cancelled in order to relate the double periodic and meromorphic
functions. From this identity it is easy to note that if one composes the
function $B_{p}$ with the ceiling function evaluated at real number $x$ the
new composed function is piecewise constant with jumps (from left to right) of
length $(-1)^{p+1}m^{p}$ at every integer value $m$ of $x$. This new function
can be used to generate primitive functions $B_{p}(x)$ of contact terms of the
form

$\sum_{m}\,(-1)^{p}\,m^{p}\delta(x\,+\,m)\,=\,\frac{1}{p+1}\,\partial_{x}B_{p+1}(\lfloor
x\rfloor\,+\,1)\,,$ (254)

where the primitive is defined as

$B^{\circ}_{p+1}(\lfloor
x\rfloor\,+\,1)\,\equiv\,\left\\{\begin{array}[]{lllr}B_{p+1}(\lfloor
x\rfloor\,+\,1)&&\text{for}&\qquad x\notin\mathbb{Z}\\\
\frac{B_{p+1}(m\,+\,1)\,+\,B_{p+1}(m)}{2}&&\text{for}&\qquad
x\,=\,m\in\mathbb{Z}\end{array}\right.\,.$ (255)

The case $x=m\in\mathbb{Z}$ will be of relevance in due time, but not for the
technical manipulations that will be presented in this appendix. Thus, from
now on in this appendix we will ignore the supra-index $\circ$ i.e. we will
ignore the difference between $B^{\circ}_{p}\,$ and $B_{p}$. Only in due time
(in the main part of the manuscript) when remarking the difference between the
two will turn out to be of relevance to reach partial conclusions, we will
reinstate the symbol $\circ$.

In the main body of the paper relation (254) turns out to be relevant at
various instances. One example being the following. Suppose we want to compute
the primitive of

$\begin{split}B_{1}(u_{1})\sum_{m}\delta(v\,+\,m)&\,=\,(B_{1}(u)\,-\,v\tau)\sum_{m}\delta(v\,+\,m)\\\
&\,=\,B_{1}(u)\sum_{m}\delta(v\,+\,m)\,+\,\tau\sum_{m}\,m\,\delta(v\,+\,m)\,,\end{split}$
(256)

with respect to the variable $v$ at fixed $u$ and evaluate the result at the
section $u_{2}=v$. Then, from property (254) it follows that the answer is

$B_{1}(u)B_{1}(\lfloor v\rfloor+1)\,-\,\frac{\tau}{2}B_{2}(\lfloor
v\rfloor+1)\,.$ (257)

A second relevant example is the following one. Suppose we want to compute the
primitive of

$\begin{split}v\,B_{1}(u_{1})\sum_{m}\delta(v\,+\,m)&\,=\,v\,(B_{1}(u)\,-\,v\tau)\sum_{m}\delta(v\,+\,m)\\\
&\,=\,-\,B_{1}(u)\sum_{m}m\delta(v\,+\,m)\,-\,\tau\sum_{m}\,m^{2}\,\delta(v\,+\,m)\,,\end{split}$
(258)

with respect to the variable $v$ at fixed $u$ and evaluate the result on the
section $u_{2}=v$. From property (254) it follows that the answer is

$\frac{1}{2}B_{1}(u)B_{2}(\lfloor v\rfloor+1)\,-\,\frac{\tau}{3}B_{3}(\lfloor
v\rfloor+1)\,.$ (259)

##### A more subtle case

These two previous examples are closely related to another two examples that
turn out to be relevant, from a technical viewpoint, to ones for our scope.
The first example is computing the primitive of

$B_{1}(\\{u_{1}\\})\sum_{m}\delta(v\,+\,m)\,=\,B_{1}(\\{u-v\tau\\})\sum_{m}\delta(v\,+\,m)\Big{|}_{u_{2}=v}\,,$
(260)

with respect to the variable $v$ at fixed $u$ and evaluating the result at the
section $u_{2}=v$. We first must define what we mean by

$\\{u-v\tau\\}\,,$ (261)

for $u_{2}\neq v$ when the argument of the bracket is not real. In that case
we define for $x\in\mathbb{C}$

$\\{x\\}\equiv x-\lfloor P_{1}[x]\rfloor\,,\qquad
P_{1}[x]\,\,\equiv\frac{\overline{\tau}x-\tau\overline{x}}{\overline{\tau}-\tau}\,\in\,\mathbb{R}\,,$
(262)

where $P_{1}$ is the projector operator i.e. $P_{1}=P_{1}^{2}$, that projects
$x=x_{1}+x_{2}\tau$ to $x_{1}$.

From definition (262) it follows that $\\{u-v\tau\\}\,=\,\\{u\\}\,-\,v\tau$,
and the primitive of (260) can be written as

$\frac{1}{2\pi{\rm i}}\log
R_{P}(u,v)\,\equiv\,B_{1}(\\{u\\})\bigl{(}B_{1}(\lfloor
v\rfloor+1)\,+\,c_{1}\bigr{)}\,-\,\frac{\tau}{2}\bigl{(}B_{2}(\lfloor
v\rfloor+1)\,+\,c_{2}\bigr{)}\,.$ (263)

The $c_{1}=c_{1}(u)$ and $c_{2}=c_{2}(u)$ are $v$-independent quantities that
we can fix at convenience. For any such choice we get

$\left(\partial_{v}\Bigl{(}\frac{1}{2\pi{\rm i}}\log
R_{P}(u,v))\Bigr{)}\right)\Big{|}_{u_{2}=v}\,=\,B_{1}(\\{u_{1}\\})\sum_{m}\delta(v\,+\,m)\,.$
(264)

Suppose we want to compute the primitive of

$\begin{split}v\,B_{1}(\\{u_{1}\\})\sum_{m}\delta(v\,+\,m)\,,\end{split}$
(265)

with respect to the variable $v$ at fixed $u$, and evaluate the result at the
section $u_{2}=v$. Then, the answer is

$\frac{1}{2\pi{\rm i}}\log
R_{vP}(u,v)\,\equiv\,\frac{1}{2}B_{1}(\\{u\\})\bigl{(}B_{2}(\lfloor
v\rfloor+1)\,+\,c_{1}\bigr{)}\,-\,\frac{\tau}{3}\bigl{(}B_{3}(\lfloor
v\rfloor+1)\,+\,c_{2}\bigl{)}\,.$ (266)

The $c_{1}=c_{1}(u)$ and $c_{2}=c_{2}(u)$ are $v$-independent quantities that
we can fix at convenience. Namely

$\left(\partial_{v}\Bigl{(}\frac{1}{2\pi{\rm i}}\log
R_{vP}(u,v))\Bigr{)}\right)\Big{|}_{u_{2}=v}\,=\,vB_{1}(\\{u_{1}\\})\sum_{m}\delta(v\,+\,m)\,.$
(267)

##  Appendix F Double periodic and meromorphic functions

In this section we study the relation between the double periodic functions
$P_{0}$ (essentially the $P$ that was introduced in the elliptic
regularization scheme) and $Q_{0}$ (essentially the $Q$ that was introduced in
the elliptic regularization scheme) and $\theta_{0}$ and
$\Gamma_{\text{e}}\,$. 565656We shall call these _double periodic_ functions
and sometimes (non-holomorphic)  _elliptic_ functions ZagierOnBloch .

The precise definition of $P_{0}$ and $Q_{0}$ will be given below. The
$\log|P_{0}|$ is closely related to the real-analytic Kronecker-Eisenstein
series (See equation (177)). The function $\log|Q_{0}|$ is closely related to
the Bloch-Wigner elliptic dilogarithm (See equation (178)). Understanding the
relations with $\theta_{0}$ and $\Gamma_{\text{e}}$, is useful to understand
how to construct a family of functions on the torus $A\times B$ that flows to
the meromorphic extension of the integrand of (11) (truncated at a portion of
a cylinder). The functions $\log Q_{0}$ and $\log P_{0}$ can be used as
building blocks of such family of double periodic extensions. Interestingly,
the relation of $\theta_{0}$ and $\Gamma_{\text{e}}$ to $P_{0}$ and $Q_{0}$
allows us to use the latter two to understand how the former two behave in a
vicinity of the real axis $\text{Im}\tau=0$ and further extend them to the
lower-half plane. Existence of this extension is important in order to be able
to define the contour of integration in $\tau$-plane that could be used to
extract the microcanonical index out of the single exponential blocks of the
ABBV formula for the grand canonical one, see section 5.

#### The properties of $P_{0}$

The $P_{0}$ is a double-periodic extension of the Jacobi theta function
$\theta_{0}$. Equivalently, it can be defined via the following logarithmic
branch

$\log P_{0}(z)\,=\,\log P_{0}(z_{1},z_{2})\,\equiv-\frac{{\rm
i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq 0}{\sum}}\;\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,.$ (268)

This function is discontinuous at $z_{2}\in\mathbb{Z}$. The discontinuity is
clear from the following identity, which holds for $z_{2}\notin\mathbb{Z}$

$P_{0}(z)\,=\,\theta_{0}(z_{1}+\\{z_{2}\\}\tau;\tau)\,.$ (269)

The curly bracket means $\\{z_{2}\\}\equiv z_{2}-\lfloor z_{2}\rfloor$.

The identity (269) fails to hold at $z_{2}\in\mathbb{Z}$ but it holds in the
limits from the left and the right to $z_{2}\in\mathbb{Z}$. However, that is
not an issue for the purpose of computing integrals. That the (269) fails to
hold at $z_{2}\in\mathbb{Z}$ has a simple explanation: The
$\theta_{0}(z_{1}+z_{2}\tau;\tau)$ is a continuous function of $z_{2}$. After
composition with $\\{z_{2}\\}$ the limits from the right at
$z_{2}\in\mathbb{Z}$ coincide with the value of the function at the point due
to continuity of $\theta_{0}$ and continuity from the right of the function
$\\{\cdot\\}$.

On the other hand the function $P_{0}$ is not continuous neither from the
right nor from the left at $z_{2}\in\mathbb{Z}$. The series (268) diverges at
$(z_{1},z_{2})\in\mathbb{Z}^{2}$. Its real part diverges to $-\infty$ at such
positions. These are the positions of the zeroes of $\theta_{0}$. For any
other value of $z^{\prime}s$ this series converges. However, if
$z_{2}\in\mathbb{Z}$ and $z_{1}\notin\mathbb{Z}$ this series does not converge
uniformly. In that case Dirichlet theorem states that the series converges to
a function that is discontinuous at $z_{2}\in\mathbb{Z}$, and the value of the
function at the discontinuity equals the semi-sum of the limits from both
sides of the discontinuity. This is, the function $P_{0}$ is not continuous at
$z_{2}\in\mathbb{Z}$, neither from the right nor from the left, and its value
at the discontinuity differs from that of
$\theta_{0}(z_{1}+\\{z_{2}\\};\tau)$.

The discontinuity at $z_{2}\in\mathbb{Z}$ from the left and the right of the
double series representation (268) of $\log P_{0}$, originates from the fact
that the Fourier coefficients of these series do not decrease fast enough for
large values of $m$ and $n$. We solve this with the insertion of a cut-off
$\Lambda$. We will not make it explicit at every step but we will work, unless
we explicitly say otherwise, with the $\Lambda$-smoothened Levin ; BrownLevin
double Fourier expansions

$\begin{split}\log P_{0\Lambda}&\,=\,-\frac{{\rm
i}}{2\pi}\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\Lambda_{1}^{|m|}\,\Lambda_{2}^{|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)}\,.\end{split}$ (270)

The real cut-offs $0<\Lambda_{1}<1$ and $0<\Lambda_{2}<1$, must be taken to
$\Lambda_{1}=1^{-}$ and $\Lambda_{2}=1^{-}$ at the very end. The smoothened
version $\log P_{0\Lambda}$ and its derivatives are absolutely and thus
uniformly convergent for every real $z_{1}$ and $z_{2}$. Thus, $\log
P_{0\Lambda}$ is smooth i.e. even divergences are also resolved by the
insertion of $\Lambda$. There is a price to pay though as the piecewise
meromorphy of $\log P_{0}$ gets spoiled in an infinitesimal band of width of
order $1-\Lambda$ centered at $z_{2}\in\mathbb{Z}$. We give an analytic
description of this in terms of the cut-offs in appendix F.3. Next we will do
it with distributions

### Recovering $\theta_{0}$ from $P_{0}$

The right-hand side of (269) can be triviallly de-periodised by using quasi-
periodic properties of $\theta_{0}$ (See appendix F). Here we would like to
prove that, starting from the double Fourier expansion of $P_{0}$ and upon the
requirement of meromorphy. Quasi-periodicity will be an outcome. It is also a
convenient introduction for the slightly less trivial case of elliptic Gamma
functions that will be presented next.

Let us introduce the following new auxiliary complex variables $u=u_{1}+\tau
u_{2}\,$, $v=u_{2}$ and the derivatives

$\partial_{u}\,=\,\partial_{u_{1}}\,,\qquad\partial_{v}\,=\,\partial_{u_{2}}\,-\,\tau\,\partial_{u_{1}}\,.$
(271)

A function of $u$ and $v$ is a holomorphic function of $u$ if its derivative
with respect to $v$ vanishes.

After acting on $\log P_{0}$, with $\partial_{v}$, using the Fourier expansion
of $B_{1}(\\{z\\})$ given in (168), and the Poisson summation identity

$\sum_{m\in\mathbb{Z}}{\bf e}(mz)\,=\,\sum_{m\in\mathbb{Z}}\delta(z+m)\,,$
(272)

we obtain the following identity

$\begin{split}\partial_{v}\,\log P_{0}&\,=\,\,2\pi{\rm
i}\,B_{1}(\\{u_{1}\\})\,\Bigl{(}\sum_{m\in\mathbb{Z}}\,\delta(v\,+\,m)\Bigr{)}\,.\end{split}$
(273)

where the derivative $\partial_{v}$ is taken at fixed $u\,\in\,\mathbb{C}$ and
$u_{2}\,=\,v$. In this expression $u_{1}\,\in\,\mathbb{R}$ stands for the
combination $u\,-\,v\,\tau$ with $u\in\mathbb{C}$ and $u_{2}\,=\,v$. Equation
(273) is telling us that $P_{0}$ is meromorphic in $u$ for every
$u_{2}\,\notin\,\mathbb{Z}$. It is also telling us that the lack of meromorphy
corresponds to a discontinuity located at $u_{2}\,\in\,\mathbb{Z}$.

This discontinuity can be cancelled with the help of the function $R_{P}$ that
we define in (263). Namely, the combination $P_{0}\,R^{-1}_{P}$ is meromorphic
in $u$. Moreover, from the relation (269) and uniqueness of solutions to
Cauchy-Riemann relations, it follows that for complex $u$

$\begin{split}\theta_{0}(u;\tau)\,=\,R^{-1}_{\theta_{0}}\,P_{0}(u-v\tau,v)\,,\qquad
v=u_{2}\,,\\\ \end{split}$ (274)

where

$R^{-1}_{\theta_{0}}\,\equiv\,\frac{R^{-1}_{P}(u,v)}{R^{-1}_{P}(u,0^{+})}\,=\,{\bf
e}\Bigl{(}\frac{1}{2}\lfloor v\rfloor(1-2\\{u\\}+(\lfloor
v\rfloor+1)\tau)\Bigr{)}\,.$ (275)

Note that $\\{u\\}$ can be safely substituted by $u$ when evaluating the
exponential. Using the bracket corresponds to a $v$-dependent choice of branch
for the corresponding logarithm.

Relation (274) can proven as follows: in virtue of (269) the function

$\frac{P_{0}(u-v\tau,v)R^{-1}_{P}(u,v)}{R^{-1}_{P}(u,0^{+})}$ (276)

equals the function $\theta_{0}(u)$ at $v=0^{+}$ and $u\in\mathbb{R}$.
Finally, Cauchy-Riemann relations in the complex extension of the variable
$u$, with the component $u_{2}$ identified with $v$, imply equality (284) for
all $u\in\mathbb{C}$ with $v=u_{2}$. A much simpler way to prove it, by using
quasi-periodicity properties of $\theta_{0}$ can be found in appendix F.

### Properties of $Q_{0}$

The $P_{0}$ and $Q_{0}$ determine a double-periodic extension of the elliptic
Gamma function $\Gamma_{e}$ (as will be recalled below). We can define $Q_{0}$
via the following logarithmic branch

$\log Q_{0}(z)\,=\,\log
Q_{0}(z_{1},z_{2})\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop
m\neq 0}{\sum}}\;\frac{{\bf e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,.$ (277)

This function is continuous, and it has no divergences for $u_{1}$,
$u_{2}\,\in\,\mathbb{R}$. Its derivatives have discontinuities at
$u_{2}\,\in\,\mathbb{Z}$. Indeed, after acting with
$\partial_{v}\,=\,\partial_{u_{2}}\,-\,\tau\partial_{u_{1}}$ on the right-hand
side of (277) one obtains

$\partial_{v}\log Q_{0}\,=\,\log P_{0}\,.$ (278)

Then, the previous analysis for $\log P_{0}$ applies. We should stress that
$P_{0}$ and $Q_{0}$ are always thought to be functions of $u_{1}$ and $u_{2}$.
The meaning of $\partial_{v}$ in (278) is that of considering the complex
linear combination of $\partial_{u_{1}}$ and $\partial_{u_{2}}$ as stated in
equation (271).

Although we will not make it explicit at every step, we will always work,
unless we indicate explicitly otherwise, with the smooth functions

$\begin{split}\log
Q_{0\Lambda}(z;\tau)&\,\equiv\,-\frac{1}{4\pi^{2}}\\!{\underset{m,n\in\mathbb{Z}\atop
m\neq 0}{\sum}}\;\Lambda_{1}^{|m|}\,\Lambda_{2}^{|n|}\,\frac{{\bf
e}(nz_{2}-mz_{1})}{m(m\tau+n)^{2}}\,.\end{split}$ (279)

The real cut-offs $\Lambda_{1}$ and $\Lambda_{2}$ are the same as before. The
smooth version $\log Q_{0\Lambda}$ and its derivatives are absolutely and thus
uniformly convergent for every real $z_{1}$ and $z_{2}$. This is, the function
$\log Q_{0\Lambda}$ is smooth. In the very end the cut-offs must be taken to
one.

### Towards fixing the remainder function in (204)

In appendix E we have shown the following identity which holds for
$z_{2}\notin\mathbb{Z}$

$\frac{\,\,P_{0}(z;\tau)^{\\{z_{2}\\}}}{Q_{0}(z;\tau)}\,=\,\Gamma_{\text{e}}(z_{1}+\tau(\\{z_{2}\\}+1);\tau,\tau)\,.$
(280)

Here $\\{z_{2}\\}\equiv z_{2}-\lfloor z_{2}\rfloor$ and

$P_{0}^{\\{z_{2}\\}}\,\equiv\,\exp(\\{z_{2}\\}\,\log P_{0})\,.$ (281)

The function $Q_{0}$ is understood to be the exponential of $\log Q_{0}$. This
identity is saying that the combination in the left-hand side is piecewise
meromorphic in $z$. Here, we construct a combination of $P_{0}$ and $Q_{0}$
that is meromorphic. This is, let us de-periodise relation (280). In appendix
F we do so by using quasi-periodic properties of $\theta_{0}$ and
$\Gamma_{\text{e}}$. Here, we would like to do it starting from a deformation
of the natural “de-periodization” of the left-hand side of (280), upon
demanding meromorphy. Quasi-periodicity will be an outcome.

We start by noticing the following identity which holds for $v\in\mathbb{R}$

$\partial_{v}\log\Bigl{(}\frac{\,\,P_{0}^{\,v\,}}{Q_{0}}\Bigr{)}\,=\,\,2\pi{\rm
i}\,v\,B_{1}(\\{u_{1}\\})\,\Bigl{(}\sum_{m\in\mathbb{Z}}\,\delta(v\,+\,m)\Bigr{)}\,.$
(282)

We define

$P_{0}^{\,v\,}\equiv\,\exp(v\,\log P_{0})\,.$ (283)

Equation (282) follows from the definitions of $P^{v}_{0}$, $Q_{0}$ and (273).
As before, the derivative $\partial_{v}$ is taken at fixed $u\in\mathbb{C}$
and $u_{2}=v$. In this expression $u_{1}\in\mathbb{R}$ stands for the
combination $u\,-\,v\tau$ with $u\in\mathbb{C}$ and $u_{2}=v$. Equation (273)
is telling us that $\frac{P_{0}^{v}}{Q_{0}}$ is holomorphic in $u$ for every
$u_{2}\notin\mathbb{Z}\,$. It is also telling us that the lack of holomorphy
corresponds to a discontinuity located at $u_{2}\in\mathbb{Z}\,$.

This discontinuity can be cancelled with the help of the function $R_{vP}$
that we define in (266). Namely, the combination
$\frac{P^{v}_{0}}{Q_{0}}\,R^{-1}_{vP}$ is holomorphic in $u\,$. 575757….
whenever the expression in terms of double Fourier expansions converges. If
such expression diverges at some point, then that point is guarantied to be a
pole. Moreover, from the relation (269) and uniqueness of solutions to Cauchy-
Riemann relations, it follows that for $u\in\mathbb{C}$

$\begin{split}\Gamma_{\text{e}}(u+\tau;\tau,\tau)\,=\,R_{\Gamma_{\text{e}}}^{-1}\,\,\frac{P_{0}(u-v\tau,v)^{v}}{Q_{0}(u-v\tau,v)}\,,\qquad
v=u_{2}\,.\end{split}$ (284)

The remainder function (205) in this case is fixed to be

$\begin{split}R^{-1}_{\Gamma_{\text{e}}}\,\equiv\,\frac{R^{-1}_{vP}(u,v)}{R^{-1}_{vP}(u,0^{+})}&\,=\,\frac{q^{\frac{B_{3}(\lfloor
v\rfloor+1)}{3}}\,{\bf e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{u\\})B_{2}(\lfloor
v\rfloor+1)\Bigr{)}}{q^{\frac{B_{3}(\lfloor 0^{+}\rfloor+1)}{3}}\,{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{u\\})B_{2}(\lfloor
0^{+}\rfloor+1)\Bigr{)}}\,\\\ &\,=\,q^{\frac{B_{3}(\lfloor
v\rfloor+1)}{3}}\,{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{u\\})\bigl{(}B_{2}(\lfloor
v\rfloor+1)\,-\,\frac{1}{6}\bigr{)}\Bigr{)}\,\\\ &\,=\,q^{\frac{B_{3}(\lfloor
v\rfloor+1)}{3}}\,{\bf e}\Bigl{(}-\,\frac{1}{2}B_{1}(u)\bigl{(}B_{2}(\lfloor
v\rfloor+1)\,-\,\frac{1}{6}\bigr{)}\Bigr{)}\,,\end{split}$ (285)

where $R^{-1}_{vP}$ was defined in (266).

Note that if $u_{1}\notin\mathbb{Z}$, then $\\{u\\}$ in the second line can be
safely substituted by $u$ in the third, when evaluating the exponential.
585858Definition of the $\\{u\\}$ when $u$ is complex has been given in
appendix E.1 equation (262). Using the bracket corresponds to picking up a
$u$-dependent choice of branch for the corresponding logarithm.

Relation (284) is proven as follows: In virtue of (280)

$\,\frac{R^{-1}_{vP}(u,v)}{R^{-1}_{vP}(u,0^{+})}\,\frac{P_{0}(u-v\tau,v)^{v}}{Q_{0}(u-v\tau,v)}$
(286)

equals the function $\Gamma_{\text{e}}(u+\tau;\tau,\tau)$ at $v=0^{+}$ and
$u\in\mathbb{R}$. Finally, Cauchy-Riemann relations in the complex extension
of the variable $u$, with the component $u_{2}$ identified with $v$, implies
equality (284) for all $u\in\mathbb{C}$ with $v=u_{2}$.

##### Comments on $R_{\Gamma_{\text{e}}}$

Note that if we change the definition of (285) by multiplying it by a phase
${\bf e}(K)$ where we assume $K$ to be an arbitrary piecewise-constant
function taking values in the integers, then identity (284) also applies to
the new definition of $R_{\Gamma_{\text{e}}}$. Obviously, this is because
${\bf e}(K)\,=\,1$. As we have said, by using this ambiguity, which is a
choice of branch, one can remove the bracket from $\\{u\\}$ in the exponent of
(285).

###  F.1 Holomorphic extension of $\theta_{0}$ and $\Gamma_{\text{e}}$ to the
lower-half of $\tau$-plane

We pause and comment about a subtle but important result that is implicit in
the previously proven identities. Notice that the product representation (161)
is absolutely convergent for $\text{Im}(\tau)>0$. This does not mean that the
elliptic gamma functions $\Gamma_{\text{e}}$ can not be continued to
$\text{Im}(\tau)=0$. In fact $\Gamma_{\text{e}}$ is well-defined at infinitely
many points in the real axis $\text{Im}(\tau)=0$. That was shown in subsection
3.5 of Felder2000 . Specifically they showed that (at least) for irrational
values of $\tau$ the function $\Gamma_{\text{e}}(z;\tau+{\rm i}\epsilon)$ is
continuous as the infinitesimal real $\epsilon$ crosses $0$ from the negatives
to the positives and  _viceversa_. This result is consistent with Identity
(284).

Moreover, the right-hand side of (284) can be understood as the definition of
$\Gamma_{\text{e}}$ in the lower-half plane $\text{Im}(\tau)<0$ i.e. as its
analytic continuation to the lower half-plane. From (284) it follows that
rational $\tau$’s are singularities of $\Gamma_{\text{e}}$. This means that
holomorphic flows to the lower-half $\tau$-plane are possible when they go
across irrational values of $\tau$. That claim is consistent with the results
of Felder2000 .

Everything we have just said for (284) relative to $\Gamma_{\text{e}}$,
applies also to the representation (274) of $\theta_{0}$.

The existence of these holomorphic flows from the upper to the lower half
$\tau$-plane is a highly non-trivial property that proves to be very hard to
show existence of, in many cases. Identities (274) and (284) demonstrate that
such extensions exist for the quasi-elliptic functions $\theta_{0}$ and
$\Gamma_{\text{e}}$. Their existence will be important to the analysis
regarding large-$N$ counting of operators that will be presented in section 5.
It will allow us to define the contour of integration $C_{\eta}$ in section 5.

From these identities one can see that the index $\mathcal{I}$ will have
essential singularities at rational values of $\tau$. Take for instance
$\tau=0$. The asymptotic behaviour of the index at such point depends on how
one approaches it. This was previously pointed out in Cabo-Bizet:2019osg .
More generally, for rational values $\tau=-\frac{n}{m}$ an analogous
conclusion holds, as pointed out in Cabo-Bizet:2020nkr . We encountered this
issue in section 5. There it was further argued that these essential
singularities are of paramount importance for operator counting.

Moreover, the relation of $\theta_{0}$ and $\Gamma_{\text{e}}$ to $P_{0}$ and
$Q_{0}$ allows us to use the latter two to understand how the former two
behave in a vicinity of the real axis $\text{Im}(\tau)=0$ and further extend
them to the lower-half plane.

### Relation between $Q_{0}$ and $\Gamma_{\text{e}}$ from quasi-periodicity

In this appendix we prove (284) with the use of quasi-periodicity properties
of $\Gamma_{\text{e}}$. We start from the logarithm of relation (251)

$\begin{split}\log
G(z_{1}+\tau\\{z_{2}\\};\tau)&\,=\,\log\Gamma_{\text{e}}(z_{1}+\tau
z_{2}+\tau;\tau,\tau)\,=\,-\log Q_{0}(z;\tau)+\\{z_{2}\\}\log
P_{0}(z;\tau)\,\\\ &\,=\,-\log
Q_{0}(z;\tau)+\\{z_{2}\\}\log\theta_{0}(z_{1}+\tau\\{z_{2}\\};\tau)\,.\end{split}$
(287)

Next we prove a couple of identities that will help in “de-periodising” (287).
Assume without loss of generality that $\lfloor z_{2}\rfloor>0$ then

$\begin{split}\log\theta_{0}(z_{1}+\tau\\{z_{2}\\};\tau)&\,=\,\log\theta_{0}(z;\tau)+2\pi{\rm
i}\sum_{i=1}^{\lfloor z_{2}\rfloor}(z-i\tau-\frac{n}{2})\\\
&\,=\,\log\theta_{0}(z;\tau)+2\pi{\rm i}\lfloor
z_{2}\rfloor(z-\frac{n}{2})-2\pi{\rm i}\tau\frac{\lfloor z_{2}\rfloor(\lfloor
z_{2}\rfloor+1)}{2}\,.\end{split}$ (288)

The final answer for the case $\lfloor z_{2}\rfloor<0$ is the same. In this
equation $n\in\mathbb{Z}$ is an arbitrary quantity that corresponds to a
choice of branch. Note that $n$ can be a discontinuous function of $z$. Note
that identity (288) is a version of (275) in terms of logarithms after
identifying $z$ and $u$ and $z_{2}$ with $v$. The second identity is

$\begin{split}\log G(z_{1}+\tau\\{z_{2}\\};\tau)&\,=\,\log
G(z;\tau)-\sum_{i=0}^{\lfloor z_{2}\rfloor-1}\log\theta_{0}(z-i\tau;\tau)\\\
&\,=\,\log G(z;\tau)-\sum_{i=0}^{\lfloor
z_{2}\rfloor-1}\bigl{(}\log\theta_{0}(z;\tau)+2\pi{\rm
i}\sum_{j=1}^{i}(z-j\tau-\frac{n}{2})\bigr{)}\\\ &\,=\,\log G(z;\tau)-\lfloor
z_{2}\rfloor\log\theta_{0}(z;\tau)-2\pi{\rm i}\frac{\lfloor
z_{2}\rfloor(\lfloor z_{2}\rfloor-1)}{2}(z-\frac{n}{2})\\\
&\qquad\,+\frac{2\pi{\rm i}\tau}{6}\lfloor z_{2}\rfloor(\lfloor
z_{2}\rfloor^{2}-1)\,.\end{split}$ (289)

Finally, from (287), (288) and (289) it follows that

$\log\Gamma_{\text{e}}(z_{1}+\tau z_{2}+\tau;\tau,\tau)\,=\,-\log
Q_{0}(z_{1},z_{2};\tau)\,+\,z_{2}\log P_{0}(z_{1},z_{2};\tau)\,+\,\log
R_{\Gamma_{\text{e}}}^{-1}\,,$ (290)

where

$\log R_{\Gamma_{\text{e}}}^{-1}\,=\,\,\frac{2\pi{\rm i}\tau}{3}B_{3}(\lfloor
z_{2}\rfloor+1)\,-\,\pi{\rm
i}\,\bigl{(}z\,-\,\frac{n}{2}\bigr{)}\Bigl{(}B_{2}(\lfloor
z_{2}\rfloor+1)\,-\,\frac{1}{6}\Bigr{)}.$ (291)

After the following identifications $z\,=\,u\,=\,u_{1}+\tau v$, one can check
that the exponential of (291) maps to (285). We note that the two equations
were computed in different ways. Equation (285) was computed via
“differential” methods, meanwhile (291) was computed via quasi-periodicity
properties. They both represent the same object, which as  _a priori_
expected, can be recast as a remainder function of the regularization
procedure introduced in section B.1.

###  F.2 Placing the discontinuities at $u=0$ for every $\rho$

This corresponds to a proper choice of integration constants $c_{1}(u)$ and
$c_{2}(u)$, piece-wise continuous functions of $u$, to be more precise, in the
solution to equation (267) given in (266). Let us start by assuming
$-1<\Delta_{2}<1\,$. A choice of remainder function $R$ that solves equation
(215) is

$\begin{split}R^{-1}\,&\,=\,R^{-1}(u,v)\,\equiv\,\,\frac{R^{-1}_{vP}(z(u),z_{2}(v))}{R^{-1}_{vP}(z(u),z_{2}(0^{+}))}\\\
\ &\,=\,\frac{q^{\frac{B_{3}(\lfloor z_{2}(v)\rfloor+1)}{3}}\,{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{z(u)\\})B_{2}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}{q^{\frac{B_{3}(\lfloor
z_{2}(0^{+})\rfloor+1)}{3}}\,{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(\\{z(u)\\})B_{2}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}\,,\\\ &\,=\,\frac{q^{\frac{B_{3}(\lfloor
z_{2}(v)\rfloor+1)}{3}}\,{\bf e}\Bigl{(}-\,\frac{1}{2}B_{1}(z(u))B_{2}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}{q^{\frac{B_{3}(\lfloor
z_{2}(0^{+})\rfloor+1)}{3}}\,{\bf
e}\Bigl{(}-\,\frac{1}{2}B_{1}(z(u))B_{2}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}\,,\end{split}$ (292)

where $R_{vP}^{-1}$ was defined in (266). In going from the second to the
third line we have used the $2\pi{\rm i}\mathbb{Z}\,$-shift ambiguity of the
exponent. Moreover, as explained in appendix E.1 below (255), when the
argument of the function $floor$ hits integer values, we must use the
definitions of $B^{\circ}_{2}$ and $B^{\circ}_{3}$ that were given in (255),
instead of the Bernoulli polynomials $B_{2}$ and $B_{3}\,$. This is what
Dirichlet theorem indicates to do. The bracket function of the complex
variable $z(u)$ is defined by replacing the real component $z_{1}$ by
$\\{z_{1}\\}$ and leaving $z_{2}$ unchanged (See definition (262)). The term
in the denominator of (292) is independent of $v$. It corresponds to a choice
of integration constant that guarranties $R^{-1}\to 1$ to vanish as $v^{i}\to
0^{+}$.

For $-1<\Delta_{2}<1$ this choice of $R^{-1}$ corresponds to the choice
$c_{1}=-\frac{1}{6}$ and $c_{2}\,=\,0$ in (266).

To prove that the choice (292) solves (215) we proceed as follows. Start by
noticing that for $v\in\mathbb{R}$

$\partial_{v^{i}}\log\Bigl{(}\frac{\,\,P_{0}^{\,z_{2}(v)\,}}{Q_{0}}\Bigr{)}\,=\,\,2\pi{\rm
i}\,\rho^{i}\,z_{2}(v)\,B_{1}(\\{z(u)\\})\,\Bigl{(}\sum_{m\in\mathbb{Z}}\,\delta(z_{2}\,+\,m)\Bigr{)}\,.$
(293)

This derivative is taken at fixed $u^{i}\,$. The Dirac-delta term comes from a
discontinuity of the function upon which $\partial_{v^{i}}$ is acting. Such
discontinuity is cancelled by adding a multiplying pre-factor $R^{-1}$, the
one defined in (292), to the discontinuous function
${\,\,P_{0}^{\,z_{2}(v)\,}}/{Q_{0}}$. This was explained in appendix E.1.

Finally, in virtue of identity (280), it follows that upon substitution of
(292) in (215), the latter holds at least for $v=0^{+}$ for
$0\leq\Delta_{2}<1$. This together with uniqueness of solutions to Cauchy-
Riemann relations, imply that (215) holds for any $u\in\mathbb{C}$ and
$v=u_{2}$. More precisely, in virtue of (280) and for $0\leq\Delta_{2}<1$ the
function

$\,\frac{R^{-1}_{vP}(z(u),z_{2}(v))}{R^{-1}_{vP}(z(u),z_{2}(0^{+}))}\,\frac{P_{0}(z_{1}(u,v),z_{2}(v))^{z_{2}(v)}}{Q_{0}(z_{1}(u,v),z_{2}(v))}$
(294)

equals $\Gamma_{\text{e}}(\rho(u)+\Delta+\tau;\tau,\tau)$ at $v=0^{+}$ and
$u\in\mathbb{R}$. At last, Cauchy-Riemann relations in the complex extension
of the variable $u$, with the component $u_{2}$ identified with $v$, imply
equality (215) to hold upon using the definition of $R$ given in (292), for
all $u\in\mathbb{C}$ and $v=u_{2}\,$.

Figure 7:  The plots of the real (left) and imaginary (right) parts of the
left- and right-hand sides of (215) for gauge group $U(1)$ and $\rho(u)=u$, as
a function of $0<u_{1}<1$ and $0<u_{2}<1$, and for $\tau=1+{\rm i}/3$,
$\Delta_{1}=0.4325$, $\Delta_{2}=0.54731$. The blue (resp. green) points are
obtained out of the left (resp. right)-hand side of (215). We have used the
product representation of $\Gamma_{\text{e}}(z;\tau,\tau)$ for the left hand
side. Both product and double Fourier representations were truncated at level
$20$. More precisely, the product representation has been truncated at the
term $q^{20}$.

If $\Delta_{2}$ is not in between $0$ and $1$ we can not identify the limit
$v=0^{+}$ of (294) with the left-hand side of (280) (for $u\in\mathbb{R}$). In
that case the proof of (215) needs to be slightly modified by using an
alternative to (280) that we wrote in (253). After identifying the limit
$v=0^{+}$ of (294) with the left-hand side of that equation (for
$u\in\mathbb{R}$) it follows that (215) holds also when $\Delta_{2}$ ranges in
between $-1$ and $0$ i.e. $-1<\Delta_{2}\leq 0\,$.

##### A non Abelian formula for $\theta_{0}$ in terms of $P_{0}$

From completeness we present a non Abelian formula that is analogous to (215).

This one relates the $\theta_{0}$ and $P_{0}$ when evaluated at non Abelian
arguments $z(u)=\rho(u)+\Delta$, for any value of $u^{i}$’s and
$0\leq\Delta_{2}<1$. The formula is

$\begin{split}\theta_{0}(z(u);\tau)\,=\,{R}_{\theta_{0}}^{-1}\,P_{0}(z_{1},z_{2})\,,\qquad
v=u_{2}\,,\\\ \end{split}$ (295)

where $z_{1}$ and $z_{2}$ are the ones defined in (219) and

$\begin{split}{R}_{\theta_{0}}^{-1}&\,=\,{R}^{-1}_{\theta_{0}}(u,v)\,\equiv\,\frac{R^{-1}_{P}(z(u),z_{2}(v))}{R^{-1}_{P}(z(u),z(0^{+}))}\,\\\
&\,=\,\frac{q^{\frac{B_{2}(\lfloor z_{2}(v)\rfloor\,+\,1)}{2}}\,{\bf
e}{\Bigl{(}-B_{1}(\\{z(u)\\})B_{1}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}}{q^{\frac{B_{2}(\lfloor
z_{2}(0^{+})\rfloor\,+\,1)}{2}}\,{\bf
e}{\Bigl{(}-B_{1}(\\{z(u)\\})B_{1}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}}\,\\\ &\,=\,\frac{q^{\frac{B_{2}(\lfloor
z_{2}(v)\rfloor\,+\,1)}{2}}\,{\bf e}{\Bigl{(}-B_{1}(z(u))B_{1}(\lfloor
z_{2}(v)\rfloor+1)\Bigr{)}}}{q^{\frac{B_{2}(\lfloor
z_{2}(0^{+})\rfloor\,+\,1)}{2}}\,{\bf e}{\Bigl{(}-B_{1}(z(u))B_{1}(\lfloor
z_{2}(0^{+})\rfloor+1)\Bigr{)}}}\,.\end{split}$ (296)

In going from the second to the third line we have used the $2\pi{\rm
i}\mathbb{Z}\,$-shift ambiguity of the exponent. This formula can be proven by
following analogous steps to the ones followed in reaching (215). It can also
be checked to follow from (269) and the use of quasi-periodic properties of
$\theta_{0}\,$.

###  F.3 Analysis in the presence of cut-off

In this subsection we arrive to the relations (273) and (278) with the use of
cut-offs $0<\Lambda_{1,2}<1$. 595959We are abusing notation in the sense that
now we consider two independent cut-offs $\Lambda_{1}$ and $\Lambda_{2}$
instead of one. The single cut-off mentioned in the main body of this paper
corresponds to $\Lambda_{1}=\Lambda_{2}=\Lambda$.

The second of these relations follows trivially from (270) and (279)

$\begin{split}\partial_{v}\log Q_{0\Lambda}\,=\,\log
P_{0\Lambda}\,.\end{split}$ (297)

The first one follows from

$\begin{split}\,\partial_{v}\log
P_{0\Lambda}&\,=\,\\!{\underset{m,n\in\mathbb{Z}\atop m\neq
0}{\sum}}\;\Lambda_{1}^{|m|}\,\Lambda_{2}^{|n|}\,\frac{{\bf
e}(nu_{2}-mu_{1})}{m}\,\\\ &\,=\,2\pi{\rm
i}\,B_{1\Lambda}(\\{u_{1}\\})\sum_{n}\Lambda^{|n|}_{2}{\bf e}(nu_{2})\,\\\
&\,=\,2\pi{\rm
i}\,\,B_{1\Lambda}(\\{u_{1}\\})\,\frac{1\,-\,\Lambda_{2}^{2}}{1\,+\,\Lambda_{2}^{2}\,-\,2\,\Lambda_{2}\cos(2\pi
u_{2})}\,.\\\ \end{split}$ (298)

Note that the right-hand side is a pure phase, and it is $\tau$-independent at
fixed $u_{1}$ and $u_{2}$. Note also that upon taking the limit
$\Lambda_{2}\to 1$ (298) vanishes everywhere except for at the points
$z_{2}\in\mathbb{Z}$ where the result goes to infinity. Moreover, a
computation shows that if one integrates the pre-factor in the right-hand side
of (298) along the segment $-\epsilon<v<\epsilon$, with $\epsilon$ being an
infinitesimal positive real number, and extracts the leading contribution to
such result after expanding it around $\Lambda=1$, the final result turns out
to be independent of $\epsilon$ and it matches the analogous integral of the
right-hand side of (273). These facts together with periodicity under the
shifts $u_{2}\mapsto u_{2}+1$, imply that in the limit $\Lambda\to 1$ the
right-hand side of (298) approaches the Dirac-delta term in the right-hand
side of (273).

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