a ainstitutetext: Institute for Advanced Study & School of Physical Science
and Technology,
Soochow University, Suzhou 215006, P.R. China b binstitutetext: Mandelstam
Institute for Theoretical Physics, School of Physics, NITheCS, and DSI-NRF
Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS),
University of the Witwatersrand, Johannesburg 2050, South Africa

# Modularity in $d>2$ free conformal field theory

Yang Lei††*The authors are ordered purely alphabetically and should all be
viewed as the co-first authors.  b , Sam van Leuven<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We derive new closed form expressions for the partition functions of free
conformally-coupled scalars on $S^{2D-1}\times S^{1}$ which resum the exact
high-temperature expansion. The derivation relies on an identification of the
partition functions, analytically continued in chemical potentials and
temperature, with multiple elliptic Gamma functions. These functions satisfy
interesting modular properties, which we use to arrive at our expressions. We
describe a geometric interpretation of the modular properties of multiple
elliptic Gamma functions in the context of superconformal field theory. Based
on this, we suggest a geometric interpretation of the modular property in the
context of the free scalar CFT in even dimensions and comment on extensions to
odd dimensions and free fermions.

##  1 Introduction

In the seminal work Cardy:1986ie , Cardy showed that modular invariance
imposes powerful constraints on the operator spectrum of two-dimensional
conformal field theories. In particular, the operator growth at large
conformal dimension is completely determined by the conformal anomaly of the
theory. Many other applications followed. One early application was towards
the classification of rational conformal field theories Cappelli:1987xt  and
their fusion rules Verlinde:1988sn . In addition, through holography, there
have been important applications to quantum gravity, such as the
interpretation of black hole entropy Carlip:1995qv ; Strominger:1996sh ;
Strominger:1997eq  and the gravitational path integral Dijkgraaf:2000fq ;
Witten:2007kt ; Maloney:2007ud . Holography, in turn, inspired the modular
bootstrap program as formulated in Hellerman:2009bu  and the notion of a
“holographic CFT” Hartman:2014oaa  singles out interesting classes of modular
forms Belin:2016knb .

The connection between modular invariance and the operator content is
generally not expected for conformal field theories in higher dimensions.
Indeed, through the operator-state correspondence, the operator content is
encoded in the partition function on $S^{d-1}\times S^{1}$ with no apparent
modular symmetries for $d>2$.111Modular properties in higher dimensions do
occur for background manifolds involving tori, but such backgrounds are not
directly connected to the (local) operator content of the CFT
Shaghoulian:2015kta ; Shaghoulian:2015lcn ; Belin:2016yll ; Belin:2018jtf ;
Luo:2022tqy . Somewhat surprisingly, the partition function of a free
conformally coupled scalar in $d$ dimensions does admit some notion of
modularity, but it is not clear to what extent such properties extend to non-
trivial CFTs cardy1991operator . Quite likely, this is related to the
integrable properties of the free theory, reflected in its higher spin
symmetries. In the case of higher-dimensional strongly-coupled CFTs, other
successful methods have been developed to determine the asymptotic density of
states, for example through holography Carlip:1998wz ; Verlinde:2000wg ;
Kutasov:2000td ; Cai:2001jc ; Cai:2005kw ; Gibbons:2005vp ; Harlow:2021trr ;
Kang:2022orq , and more recently making use of the framework of thermal
effective field theory Benjamin:2023qsc ; Allameh:2024qqp ; Benjamin:2024kdg .
Existing geometric arguments for modular symmetry in higher dimensions seem to
involve a change of topology of the background Shaghoulian:2016gol ;
Horowitz:2017ifu .

For supersymmetric CFTs in higher dimensions, unconventional modular
properties have recently surfaced for which a geometric interpretation akin to
two dimensions is in fact possible. The relevant partition function is a
supersymmetric partition function on $S^{d-1}\times S^{1}$, known as the
superconformal index Romelsberger:2005eg ; Kinney:2005ej ; Bhattacharya:2008zy
. This object encodes aspects of the BPS operator content of a theory. More
precisely, it counts differences between the numbers of bosonic and fermionic
operators. Crucially, it is a renormalization group invariant, which allows
one to calculate this object for a rather general class of (strongly coupled)
SCFTs. The study of this object has led to Cardy-like constraints on the (BPS)
operator spectrum in terms of the conformal (and ’t Hooft) anomalies of 4d and
6d SCFTs DiPietro:2014bca ; ArabiArdehali:2015ybk ; DiPietro:2016ond , see
also Cabo-Bizet:2018ehj ; Benini:2018ywd ; Goldstein:2019gpz ;
ArabiArdehali:2019tdm ; Kim:2019yrz ; Cabo-Bizet:2019osg ; Goldstein:2020yvj ;
Cassani:2021fyv  for applications to AdS5 black hole entropy.

As it turns out, underlying these works there is indeed a notion of
modularity. A key building block of 4d superconformal indices, the elliptic
Gamma function, is known to fit into a generalized modular structure based on
$SL(3,\mathbb{Z})$ Felder_2000 . A geometric explanation of this structure was
pioneered in Gadde:2020bov  and extended to other dimensions as well. In
Jejjala:2022lrm , this idea was further explored in four dimensions. The key
idea relies on a factorization property of the index, which reflects the fact
that $S^{3}\times S^{1}$ can be split into two disk-times-torus ($D_{2}\times
T^{2}$) geometries with boundary $T^{3}$.222This follows from the genus $1$
Heegaard splitting of $S^{3}$. This property of the index is known as
holomorphic block factorization Nieri:2015yia , originally studied in the
context of 3d partition functions Beem:2012mb . Remarkably, when the 4d index
is expressed in factorized form, it has an unconventional type of modular
property with respect to a general $SL(2,\mathbb{Z})$ transformation acting
simultaneously on both tori in the $D_{2}\times T^{2}$ geometries. The
$SL(2,\mathbb{Z})$ action cannot be understood in terms of large
diffeomorphisms of the full background, as expected, and as such cannot be
understood in terms of ordinary modular forms. Instead, the modular action
reflects ambiguities in the splitting of a fixed $S^{3}\times S^{1}$ geometry
into $D_{2}\times T^{2}$ geometries. For this reason, the modular property was
referred to by _modular factorization_. See Jejjala:2022lrm  for more details.
There are strong indications that this basic idea extends to superconformal
indices in other dimensions, based on known modular properties of the so-
called multiple elliptic Gamma functions.333We present an overview in Section
3.1, including the relevant references.

In this work, we draw lessons from the supersymmetric story to shed new light
on the modular structure underlying free, non-supersymmetric CFTs. In
particular, we follow a recent observation by Nekrasov Nekrasov:2023xzm  that
the analytic continuation of parameters in a variety of physical quantities
can be naturally realized in supersymmetric quantum field theory. Inspired by
this, we first show that the multiple elliptic Gamma function
$G_{2D-1}(z|\underline{\tau})$ provides an analytic continuation in
temperature and angular potentials of the thermal partition function of a
conformally coupled free scalar on $S^{2D-1}\times S^{1}$. Using known modular
properties of $G_{2D-1}(z|\underline{\tau})$, we derive exact, closed form
expressions for the scalar partition functions in four and six dimensions in
terms of a modular transformed argument (i.e., $\beta\to 4\pi^{2}/\beta$). For
example, when all angular potentials are turned off, we find the following
expression in four dimensions:

$Z_{4}(\tau)=\exp\left[-\frac{2\pi
i}{120}B_{5}(\tau+1;\tau)\right]\frac{1}{\psi_{4}(3)}\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{\tau+k+1}{\tau})}{\psi_{4}(4-\frac{\tau-k-1}{\tau})}\,,$
(1)

where $\tau=i\frac{\beta}{2\pi}$, $Z_{4}(\tau)$ is the 4d partition function,
$B_{5}(z;\tau)$ is a degree five polynomial in $z$ and $\psi_{4}(z)$ can be
expressed in terms of the exponential of four polylogarithms $\text{Li}_{r\leq
4}(e^{-2\pi iz})$. This expression provides the exact high-temperature
asymptotics $\tau\to 0^{+i}$ of the partition function, including non-
perturbative corrections, as we will demonstrate. Our expression can be viewed
as a resummation of formulas recently derived in an appendix of
Benjamin:2023qsc . Apart from this, our formula exhibits an underlying,
unexpected modular structure. We further extend these results to include
angular potentials and to general modular transformations in order to study
the “root-of-unity” asymptotics $\tau\to-\frac{n}{m}+0^{+i}$ and find
consistency with the very recent work Benjamin:2024kdg . Finally, we speculate
that (1) provides an example of the generalization of modularity to higher
dimensions, by proposing a geometric interpretation of the formula inspired by
modular factorization.

The rest of the paper is organized as follows. In Section 2, we review modular
aspects of the free scalar partition functions in both $d=2$ and $d\geq 3$,
including Meinardus’ theorem. In Section 3, we deduce a relation between the
multiple elliptic Gamma functions and the free scalar partition function on
$S^{2D-1}\times S^{1}$. By applying modular properties of multiple elliptic
Gamma functions, we derive our expressions the partition functions and
demonstrate how they encode the exact high-temperature asymptotics. In Section
4, we extend this to general $SL(2,\mathbb{Z})$ transformations and the
corresponding root-of-unity asymptotics. Finally, in Section 5 we speculate on
a geometric interpretation of our expressions and list various future
directions.

##  2 Review of the free scalar partition function

The study of the asymptotics of $d=2$ CFT partition functions was initiated by
Cardy in Cardy:1986ie  and extended to $d\geq 3$ in cardy1991operator . In two
dimensions, one makes use of modular properties of the partition function. One
method to _derive_ the modular property, in the context of free CFTs, is based
on a reflection formula of the Riemann $\zeta$-function. This method can be
extended to higher dimensions and, even though a modular structure akin to 2d
does _not_ arise, there are still interesting traces of modularity and, in
particular, pertains to the high-temperature asymptotics of free CFT partition
functions. To contrast with our proposal in Section 3, in this section we
review this methodology, with a focus on the free conformally coupled scalar
on $S^{d-1}\times S^{1}$ for $d\geq 2$. Our discussion follows the more recent
work Melia:2020pzd , where these techniques were revisited and applied to the
asymptotic operator degeneracies in EFTs. We also briefly comment on the work
Benjamin:2023qsc , which calculates, in an appendix, the _exact_ high-
temperature asymptotics for the free scalar.

###  2.1 Mellin representation of the partition function

The partition function of a CFT quantized on $S^{d-1}$ is given by:

$Z(\beta)=\mathrm{Tr}_{\mathcal{H}(S^{d-1})}\,e^{-\beta(H-i\Omega_{i}J_{i})}\,,$
(2)

where $H$ is the Hamiltonian on $S^{d-1}$ and we have included charges for the
independent angular momenta $J_{i}$ on $S^{d-1}$. In the following, we will
often use the notation

$\tau=i\frac{\beta}{2\pi}\,,\qquad\theta_{i}=\frac{\beta}{2\pi}\Omega_{i}\,.$
(3)

In free field theories, one can write the full “multi-particle” partition
function $Z(\beta)$ in terms of a “single-particle” partition function
$Y(\beta)$ via plethystic exponentiation:

$Z(\beta)=\text{PE}[Y(\beta)]=\exp\left[\sum_{m=1}^{\infty}\frac{1}{m}Y(\beta^{m})\right]\,.$
(4)

To derive the $\beta\to 0$ asymptotics of $Z(\beta)$, it turns out to be
useful to represent it via a Mellin transformation of the so-called
Hamiltonian zeta function. The Hamiltonian $\zeta$-function is defined as
(see, e.g., Cardy:1986ie ; cardy1991operator ; Gibbons:2006ij ):

$D(d,s)=\text{Tr}(H^{-s})=\frac{1}{\Gamma(s)}\int_{0}^{\infty}Y(\beta)\beta^{s-1}d\beta\,.$
(5)

where $d$ denotes the spacetime dimension. The partition function can now be
written as the inverse Mellin transformation with in addition the insertion of
a Riemann zeta function $\zeta(s+1)$ kernel:

$\ln Z(\beta)=\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}\beta^{-s}\Gamma(s)\zeta(s+1)D(d,s)ds\,,$
(6)

where $\gamma$ is a positive number such that the integrand has no poles for
Re$(s)>\gamma$.

Consider for example the thermal partition function of a $d$-dimensional
conformally coupled free scalar cappelli1989stress ; cardy1991operator ;
Kutasov:2000td ; Beccaria:2014jxa ; Giombi:2014yra . The partition function
with all chemical potentials turned on is given by Melia:2020pzd ;
Nekrasov:2023xzm

$\displaystyle\begin{split}Z_{2D}(\tau,\theta_{1},...,\theta_{D})&=\exp\left[\sum_{l=1}^{\infty}\frac{q^{-l}-q^{l}}{l}\prod_{i=1}^{D}\frac{1}{q^{l}-2\cos(2\pi
l\theta_{i})+q^{-l}}\right]\,,\\\
Z_{2D+1}(\tau,\theta_{1},...,\theta_{D},\xi)&=\exp\left[\sum_{l=1}^{\infty}\frac{q^{-\frac{l}{2}}+q^{\frac{l}{2}}\xi^{l}}{l}\prod_{i=1}^{D}\frac{1}{q^{l}-2\cos(2\pi
l\theta_{i})+q^{-l}}\right]\,,\end{split}$ (7)

where $q=e^{2\pi i\tau}$ and $\theta_{i}$, defined in (3), and $\xi=\pm 1$
label $O(d)$ twisted boundary conditions. The partition function with
$\theta_{i}$ turned off is given by

$Z_{d}(\tau)=\exp\left[\sum_{l=1}^{\infty}\frac{1}{l}\frac{q^{\frac{d-2}{2}l}(1-q^{2l})}{(1-q^{l})^{d}}\right]\,.$
(8)

The single-particle partition function, i.e., the exponent, can be interpreted
as follows. In $d$ dimensions, the conformal dimension of the free scalar is
$(d-2)/2$. The factor $(1-q)^{d}$ in the denominator reflects the $d$
derivatives acting on a single-letter operator while the factor $1-q^{2}$ in
the numerator quotients by the equations of motion. Finally, the density of
states is calculated through an inverse Legendre transformation

$\rho(\Delta)=\int d\beta Z(\beta)e^{\beta\Delta}\,.$ (9)

Making use of the Mellin transformation, we now continue with the high-
temperature asymptotics of partition function and density of states in even
dimensions $d=2,4,6$. The method also applies to odd dimensions, but we will
only consider even dimensions in this work for reasons we come back to in
Section 5.

###  2.2 $d=2$

In two dimensions, the free scalar partition function can be written as

$\displaystyle\begin{split}Z_{2}(\tau,\bar{\tau})&=\exp\left[\sum_{l=1}^{\infty}\frac{1}{l}\left(\frac{(1-q^{l}\bar{q}^{l})}{(1-q^{l})(1-\bar{q}^{l})}-1\right)\right]=Z_{2}(\tau)\bar{Z}_{2}(\bar{\tau})\end{split}$
(10)

where we included a chemical potential for spatial rotations such that left-
and right-movers are distinguished by $q=e^{2\pi i\tau}$ and $\bar{q}=e^{-2\pi
i\bar{\tau}}$. Furthermore, we have not included the Casimir energy for
uniformity with the higher-dimensional examples. Finally the well-known
factorization into left- and right-movers is captured by

$Z_{2}(\tau)=\exp\left[\sum_{l=1}\frac{1}{l}\frac{q^{l}}{1-q^{l}}\right]=\prod_{l=1}^{\infty}\frac{1}{(1-q^{l})}$
(11)

The presence of $-1$ in the exponent of (10) is specific to two dimensions
Melia:2020pzd . It reflects the fact that 2d scalars are dimensionless and
therefore there is an infinite amount of operators with the same conformal
dimension obtained by decorating a given operator with a function $f(\phi)$.
The $-1$ cures the consequent divergence and can be viewed as modding out by
the equivalence relation between operators which only differ by such a
function $f(\phi)$.

Up to the ignored Casimir energies, $Z_{2}(\tau)$ is precisely the Dedekind
eta function. Using this connection, one deduces that $Z_{2}(\tau)$ satisfies
the following modular property

$Z_{2}\left(-\frac{1}{\tau}\right)=\frac{1}{\sqrt{\tau}}e^{-i\pi
P_{2}(\tau)}Z_{2}(\tau),\qquad
P_{2}(\tau)=\frac{1}{12}\left(\tau+\frac{1}{\tau}-3\right)\,,$ (12)

We now recall how Cardy Cardy:1986ie  rederived this property from the Mellin
representation of $Z_{2}(\tau)$:

$\ln Z_{2}(\tau)=\frac{1}{2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}ds(-2\pi
i\tau)^{-s}\zeta(s+1)\Gamma(s)\zeta(s)\,.$ (13)

In this case, the Hamiltonian zeta function is nothing but the Riemann zeta
function $D(2,s)=\zeta(s)$. The modular transformation $\tau\to-1/\tau$ is
translated to the $s\to-s$ transformation of Riemann zeta function. The
modular image $Z_{2}\left(-\frac{1}{\tau}\right)$ is related to its original
Mellin integral representation (13) by the famous reflection formula of
Riemann zeta function:

$\zeta(s)=\Gamma(1-s)2^{s}\pi^{s-1}\sin\left(\frac{\pi
s}{2}\right)\zeta(1-s)\,.$ (14)

Notice under the transformation $s\to-s$, the integration contour shifts
$\gamma$ to $-\gamma$. We can now calculate the difference between the
partition function at $\tau$ and $-1/\tau$ as a contour integral:

$\ln Z_{2}\left(-\frac{1}{\tau}\right)-\ln Z_{2}(\tau)=\frac{1}{2\pi
i}\oint_{C}ds(-2\pi i\tau)^{-s}\Gamma(s)\zeta(s)\zeta(1+s)\,.$ (15)

The integration contour $C$ is is made up of the lines
$[\gamma-i\infty,\gamma+i\infty]$, $[-\gamma-i\infty,-\gamma+i\infty]$ and
infinity. By collecting the residues at the simple poles at $s=0,\pm 1$, one
precisely reproduces the factor of automorphy (12). The asymptotics of the
full partition function $Z_{2}(\tau,\bar{\tau})$ for $\tau,\bar{\tau}\to 0$ is
given by

$Z_{2}(\tau,\bar{\tau})\sim\frac{1}{\sqrt{\tau\bar{\tau}}}\exp\left[\frac{\pi
i(\tau+\bar{\tau})}{12}+\frac{\pi
i}{12}\left(\frac{1}{\tau}+\frac{1}{\bar{\tau}}\right)-\frac{\pi
i}{2}\right]\,.$ (16)

The inverse Legendre transformation leads to the Cardy formula for the density
of states at large $\Delta\gg c$.

###  2.3 $d>2$

Following the procedure outlined above, we rewrite the partition function (8)
of a $d$-dimensional free scalar as cardy1991operator ; Kutasov:2000td ;
Dowker:2002ax ; Gibbons:2006ij :

$\ln Z_{d}(\tau)=\frac{1}{2\pi
i}\left(\oint_{C}+\int_{-\gamma-i\infty}^{-\gamma+i\infty}\right)(-2\pi
i\tau)^{-s}\Gamma(s)\zeta(s+1)D(d,s)ds\,,$ (17)

where the integral contour $C$ encloses all the simple poles of the integrand.
Let us assume that $D(d,s)$ a finite number of simple poles at $s=\alpha_{i}$
with residue $A_{i}$, which holds for free field theory in various dimensions
Melia:2020pzd . One then finds

$\displaystyle\begin{split}\ln
Z_{d}(\tau)&=\sum_{i}A_{i}\Gamma(\alpha_{i})\zeta(\alpha_{i}+1)(-2\pi
i\tau)^{-\alpha_{i}}-D(d,0)\ln(-2\pi i\tau)+D^{\prime}(d,0)\\\ &+\frac{1}{2\pi
i}\int_{-\gamma-i\infty}^{-\gamma+i\infty}ds(-2\pi
i\tau)^{-s}\Gamma(s)\zeta(s+1)D(d,s)\end{split}$ (18)

Here the first line contains the residues at the poles $s=\alpha_{i}$ and
$s=0$ respectively. In $d=2$, the second line would become $\ln
Z_{2}(-1/\tau)$ after $s\to-s$ and using the reflection formula. For $d>2$,
however, such a simplification does not occur. This reflects the fact that
higher-dimensional partition functions are not modular, at least not in the
same sense as in two dimensions.

Fortunately, for the $\tau\to 0$ asymptotics of the expression (18) is still
useful. To compute the asymptotics, it was observed in Melia:2020pzd  that the
“bosonic” Meinardus’ theorem can be used elizalde1994zeta ; andrews1998theory
; Melia:2020pzd . This theorem implies that the second line in (18) is
$\mathcal{O}(|\tau|^{C_{0}})$ when $\tau\to 0$ for some positive constant
$C_{0}<1$. To order $\mathcal{O}(\tau^{0})$, one is thus left with the first
line. This line is straightforwardly computed through explicit evaluation of
the Hamiltonian $\zeta$ function in (5) and its residues.

As a brief aside, it turns out that in $d=4$ the equation (18) can be
interpreted as a modular property if one takes a $\tau$ derivative on both
sides cardy1991operator ; Gibbons:2006ij . That is, consider the function
defined by

$E(\tau)=\frac{1}{2\pi i}\frac{\partial\ln Z(\tau)}{\partial\tau}\,.$ (19)

Taking the $\tau$ derivative on both sides of (18), one finds that the
equation can be interpreted as a modular property of $E(\tau)$:

$E(\tau)-\frac{B_{4}}{8}=\frac{1}{\tau^{4}}\left[E\left(-\frac{1}{\tau}\right)-\frac{B_{4}}{8}\right]$
(20)

Note that, up to a constant shift proportional to the fourth Bernoulli number
$B_{4}=-\frac{1}{30}$, $E(\tau)$ transform as a weight $4$ modular form. We
will come back to this observation in Section 3.3.1.

Using (18), it is straighforward to derive the asymptotics of the free scalar.
Explicitly, in $d=4,6$ dimensions one finds:

$\displaystyle\ln Z_{4}(\tau)$ $\displaystyle\sim$
$\displaystyle-\frac{i\pi}{360\tau^{3}}+\zeta^{\prime}(-2)+\mathcal{O}(\tau)$
(21) $\displaystyle\ln Z_{6}(\tau)$ $\displaystyle\sim$
$\displaystyle\frac{i\pi}{15120\tau^{5}}+\frac{i\pi}{4320\tau^{3}}+\frac{1}{12}\Big{[}\zeta^{\prime}(-4)-\zeta^{\prime}(-2)\Big{]}+\mathcal{O}(\tau)$
(22)

This method applies in general dimensions and can also be extended to free
fermions Melia:2020pzd . The exact, all-order asymptotics was recently derived
in an appendix of Benjamin:2023qsc  (based on the original work
cardy1991operator ), making use of a $\zeta$-function regularization of the
determinant of the $S^{d-1}\times S^{1}$ Laplacian elizalde1994zeta .

In the following, we will argue that, in the case of the free scalar, there is
in fact an interesting, albeit unconventional modular structure underlying the
above results, despite the dimension being greater than two. The relevant
modular property provides an exact, closed form expression for the free scalar
partition function which makes the high temperature asymptotics manifest.
Apart from recovering the above results, we also study implications for the
fully refined partition function and the root-of-unity asymptotics $\tau\to
0^{+i}-\frac{n}{m}$.

##  3 Exact asymptotics from generalized modularity

In this section, we will match the free scalar partition function in $D=2n$
with (a specialization of) certain multiple elliptic $\Gamma$ functions
nishizawa2001elliptic . These functions are generalizations of the
$q$-$\theta$ function and the elliptic $\Gamma$ function Felder_2000 , and
satisfy interesting modular properties kurokawa2003multiple ;
narukawa2004modular . They naturally arise in the context of supersymmetric
indices. As we will see, the superconformal index of a $2D$-dimensional
supersymmetric quantum field theory analytically continues the partition
function of the $D$-dimensional free scalar. This is part of a more general
observation in Nekrasov:2023xzm , namely that the analytic continuation of
parameters in a variety of physical quantities is naturally realized in
supersymmetric quantum field theory. After the identification, we show how the
modular properties of multiple elliptic Gamma functions lead to a new closed
form expression for the free scalar partition function which manifestly
encodes the exact high temperature asymptotics.

Before starting, let us set up some notation following narukawa2004modular .
The chemical potentials $\tau_{j}$ and fugacities $q_{j}$ are related by
$q_{j}=e^{2\pi i\tau_{j}}$, where $0\leq j\leq r$. Then we denote

$\displaystyle\begin{split}\underline{q}=(q_{0},\cdots,q_{r}),&\qquad\underline{\tau}=(\tau_{0},\cdots,\tau_{r})\\\
\underline{q}^{-}(j)=(q_{0},\cdots,\check{q}_{j},\cdots,q_{r}),&\qquad\underline{\tau}^{-}(j)=(\tau_{0},\cdots,\check{\tau}_{j},\cdots,\tau_{r})\\\
\underline{q}[j]=(q_{0},\cdots,q_{j}^{-1},\cdots,q_{r}),&\qquad\underline{\tau}[j]=(\tau_{0},\cdots,-\tau_{j},\cdots,\tau_{r})\\\
\underline{q}^{-1}=(q_{0}^{-1},\cdots,q_{r}^{-1}),&\qquad-\underline{\tau}=(-\tau_{0},\cdots,-\tau_{r})\end{split}$
(23)

where $\check{q}_{j}$ means the $j$-th moduli is ruled out. We also define
$|\underline{\tau}|=\tau_{0}+...+\tau_{r}$. The generalized $q$-Pochhammer
symbol is denoted by

$(x;\underline{q})_{\infty}^{(r)}=\prod_{j_{0},..,j_{r}=0}^{\infty}(1-xq_{0}^{j_{0}}q_{1}^{j_{1}}...q_{r}^{j_{r}})$
(24)

where $x=e^{2\pi iz}$. The generalized $q$-Pochhammer symbols satisfy

$(x;\underline{q})_{\infty}^{(r)}=\frac{1}{(q_{j}^{-1}x;\underline{q}[j])^{(r)}_{\infty}},\qquad(q_{j}x;\underline{q})^{(r)}_{\infty}=\frac{(x;\underline{q})_{\infty}^{(r)}}{(x;\underline{q}^{-}(j))_{\infty}^{(r-1)}}$
(25)

where the first property extends (24) to
$\tau_{j}\in\mathbb{C}\setminus\mathbb{R}$. More generally:

$(x;\underline{q})_{\infty}^{(r)}=\left[\prod_{j_{0},...,j_{r}=0}^{\infty}(1-xq_{0}^{-j_{0}-1}...q_{k-1}^{-j_{k-1}-1}q_{k}^{j_{k}}...q_{r}^{j_{r}})\right]^{(-1)^{k}}$
(26)

where we take $|q_{0}|,...,|q_{k-1}|>1$ and $|q_{k}|,...,|q_{r}|<1$.

###  3.1 Modularity of the multiple elliptic Gamma function

The multiple elliptic Gamma function was first defined by Nishizawa in terms
of the generalized $q$-Pochhammers defined above nishizawa2001elliptic :444We
collect various properties in Appendix A.

$G_{r}(z|\underline{\tau})=(q_{0}...q_{r}x^{-1};\underline{q})^{(r)}_{\infty}\cdot\left[(x;\underline{q})_{\infty}^{(r)}\right]^{(-1)^{r}}$
(27)

By using plethystic exponentiation (see (4)), these functions can be viewed as
multi-particle partition functions whose single-particle partition function is
given by

$Y_{r}(z|\underline{\tau})=\frac{(-1)^{r+1}e^{2\pi iz}-e^{2\pi
i(|\underline{\tau}|-z)}}{\prod_{i=0}^{r}(1-e^{2\pi i\tau_{i}})}$ (28)

As mentioned above, the function $G_{r}(z|\underline{\tau})$ naturally arises
in the context of superconformal indices in dimension $D=2r+2$. In particular,
a single function $G_{r}$ represents the index of a free $D$-dimensional
matter multiplet of minimal supersymmetry, as we will detail below.

The function $G_{r}(z|\underline{\tau})$ satisfies an interesting type of
modular property narukawa2004modular

$\displaystyle\begin{split}\prod_{k=1}^{r+2}G_{r}\left(\frac{\zeta}{\omega_{k}}\Big{|}\frac{\omega_{1}}{\omega_{k}},...,\frac{\check{\omega}_{k}}{\omega_{k}},...,\frac{\omega_{r+2}}{\omega_{k}}\right)=\exp\left[-\frac{2\pi
i}{(r+2)!}B_{r+2,r+2}(\zeta|\underline{\omega})\right]\,.\end{split}$ (29)

Here, we use a homogeneous parametrization of the projective coordinates:

$(z|\underline{\tau})\equiv\left(\frac{\zeta}{\omega_{r+2}}\Big{|}\frac{\omega_{1}}{\omega_{r+2}},\ldots,\frac{\omega_{r+1}}{\omega_{r+2}}\right)\,.$
(30)

We note that the equation is defined when the arguments of the $G_{r}$
functions are in $\mathbb{C}\setminus\mathbb{R}$, as follows from the
definition (27). That is, $\frac{\omega_{j}}{\omega_{k}}\notin\mathbb{R}$ for
any $j,k$. Furthermore, $B_{r,n}$ is the multiple Bernoulli polynomial defined
by

$\frac{t^{r}e^{\zeta
t}}{\prod_{i=1}^{r}(e^{\omega_{i}t}-1)}=\sum_{n=0}^{\infty}B_{rn}(\zeta|\underline{\omega})\frac{t^{n}}{n!}\,.$
(31)

The relevant modular group for $G_{r}$ is $SL(r+2,\mathbb{Z})$. For example,
one may observe that the arguments of consecutive functions in (29) are
related by the $SL(r+2,\mathbb{Z})$ transformation which acts as a cyclic
permutation.555For $r$ even, the cyclic permutation matrix has one entry with
a $-$ sign so that the determinant condition is satisfied. This sign can be
made manifest in (29) through the extension property (25). The underlying
modular structure was further explored in Tizzano:2014roa ; Winding:2016wpw .

More recently, based on early work on the elliptic Gamma function Felder_2000
, it has been proposed in Gadde:2020bov  that the general modular theory of
$G_{r}$ can be phrased in terms of the group cohomology
$H^{r}(SL(r+2,\mathbb{Z}),N/M)$ with $N/M$ the space of meromorphic functions
of $(z|\underline{\tau})$ modulo holomorphic, nowhere vanishing functions.
There appears to be a beautiful geometric interpretation of this formalism, as
suggested in Gadde:2020bov  and studied in detail in Jejjala:2022lrm  in four
dimensions, which explains both the $SL(r+2,\mathbb{Z})$ modular group and the
number of functions, $r+2$, involved in the property. Let us briefly sketch
this interpretation for $r=0,1,2,3$:

  * •

$r=0$: $G_{0}(z;\tau)^{-1}\equiv\theta(z;\tau)^{-1}$ arises as the elliptic
genus of a 2d $\mathcal{N}=(0,2)$ chiral multiplet Kawai:1994np ;
Benini:2013nda . The group $SL(2,\mathbb{Z})$ reflects the large
diffeomorphisms of the $T^{2}$ geometric background and the modular property,
involving two functions, is standard.

  * •

$r=1$: $G_{1}(z|\tau_{0},\tau_{1})\equiv\Gamma(z;\tau_{0},\tau_{1})$ is the
superconformal index of a 4d $\mathcal{N}=1$ chiral multiplet Dolan:2008qi .
The group $SL(3,\mathbb{Z})$ arises by viewing $S^{3}\times S^{1}$ as a
$T^{2}$ fibration over the toric manifold $\mathbb{CP}^{1}$ or, equivalently,
as a $T^{3}$ fibered over an interval. Bisecting $\mathbb{CP}^{1}$, one
decomposes $S^{3}\times S^{1}$ as

$S^{3}\times S^{1}=D_{2}\times T^{2}\overset{g}{\sqcup}D_{2}\times T^{2}$ (32)

with $g\in SL(3,\mathbb{Z})$ a non-trivial identification of the boundary
$T^{3}$s which reflects the Hopf fibration of the full geometry. The index
respects this splitting, a phenomenon known as holomorphic block factorization
Nieri:2015yia . The three functions involved in the modular property can be
associated to these three geometries.666 One may wonder why the modular
property treats all functions on an equal footing, while the geometry clearly
does not. This is related to the fact that the one-loop determinants for a
chiral multiplet on $S^{3}\times S^{1}$ and $D_{2}\times T^{2}$ have the same
functional form Closset:2013sxa ; Longhi:2019hdh . This coincidence will be
addressed in an upcoming work vanLeuven:2024 .

  * •

$r=2$: $G_{2}(z|\tau_{0},\tau_{1},\tau_{2})^{-1}$ defines the 6d
superconformal index of an $\mathcal{N}=(0,1)$ hypermultiplet Lockhart:2012vp
; Imamura:2012efi ; Spiridonov:2012de . The group $SL(4,\mathbb{Z})$ arises by
viewing $S^{5}\times S^{1}$ as a $T^{2}$ fibration over the toric manifold
$\mathbb{CP}^{2}$ or, equivalently, as a $T^{4}$ fibered over a triangle.
Trisecting the triangle, one decomposes $S^{5}\times S^{1}$ as777See
Gukov:2017zao  for a nice exposition on how to trisect general four-manifolds,
including $\mathbb{CP}^{2}$.

$S^{5}\times S^{1}=(D_{2})^{2}\times
T^{2}\overset{g_{1}}{\sqcup}(D_{2})^{2}\times
T^{2}\overset{g_{2}}{\sqcup}(D_{2})^{2}\times T^{2}$ (33)

where we write $(D_{2})^{2}$ for the four-ball to make the toric structure
manifest and $g_{1,2}\in SL(4,\mathbb{Z})$ specify how the trisection is
glued. Again, the 6d index turns out to respect this splitting Lockhart:2012vp
; Qiu:2015rwp . The four functions involved in the modular property can be
associated to these four geometries, where we expect a similar comment as in
footnote 6 to apply.

  * •

$r=3$: it seems that $G_{3}(z|\tau_{0},\tau_{1},\tau_{2},\tau_{3})$ can be
interpreted as an index in 8 dimensions. Some evidence for this can be found
in Nekrasov:2017cih  and, through dimensional reduction to 7d and relations
between the multiple Sine function $S_{4}$ and the multiple elliptic Gamma
function $G_{3}$ (see Appendix A.1), in Minahan:2015jta ; Polydorou:2017jha ;
Polydorou:2020gxp ; Minahan:2022rsx . In this case, the group
$SL(5,\mathbb{Z})$ arises by viewing $S^{7}\times S^{1}$ as a $T^{2}$
fibration over the toric manifold $\mathbb{CP}^{3}$ or, equivalently, as a
$T^{5}$ fibered over a tetrahedron. Quadrisecting the tetrahedron, one
decomposes $S^{7}\times S^{1}$ as

$S^{7}\times S^{1}=(D_{2})^{3}\times
T^{2}\overset{g_{1}}{\sqcup}(D_{2})^{3}\times
T^{2}\overset{g_{2}}{\sqcup}(D_{2})^{3}\times
T^{2}\overset{g_{3}}{\sqcup}(D_{2})^{3}\times T^{2}$ (34)

where we write $(D_{2})^{3}$ for the six-ball to make the toric structure
manifest and $g_{1,2,3}\in SL(5,\mathbb{Z})$ specify how the quadrisection is
glued. We are not aware of references studying this, but it is natural to
conjecture that the 8d index respects this splitting and the five functions
involved in the modular property can be associated to these five geometries,
where we expect a similar comment as in footnote 6 to apply.

The key to the modular properties in all these cases is that upon splitting
the manifolds, the geometric factors always contain a $T^{2}$. The compact
manifolds for fixed complex structure can be split in a variety of ways,
labeled by elements in the $SL(2,\mathbb{Z})\subset SL(r+2,\mathbb{Z})$
subgroup corresponding to the large diffeomorphisms of these $T^{2}$s. This
was explored in detail in four dimensions Jejjala:2022lrm , where the
resulting modular structure was called modular factorization. We expect that a
very similar story extends to the other dimensions as well.

###  3.2 Identification with the free scalar partition function

Inspired by the observations of Nekrasov:2023xzm , we now note that the
partition function of the free scalar in _even_ dimensions $d=2D$ can be
written in terms of $G_{d-1}$ by comparing the plethystic expressions in (7)
and (28).888The scalar partition function in odd $d=2D+1$ dimensional
spacetime does not match with multiple elliptic Gamma functions. In
particular, the relative sign of the terms in the numerator of the plethystic
exponent in (7) does not match the number of factors in the denominator of the
multiple elliptic Gamma function $G_{2D}$. Nonetheless, the functions are
still rather closely related and we will make some further comments on odd
dimensions in Section 5. For example,

  * •

In $d=2$, the partition function (10) can be written in terms of the elliptic
Gamma function:

$Z_{2}(\tau,\bar{\tau})=\lim_{z\to 0}(1-e^{2\pi
iz})\Gamma(z,\tau,\bar{\tau})=Z_{2}(\tau)Z_{2}(\bar{\tau})$ (35)

This reflects the well-known fact that $\Gamma(z;\tau_{0},\tau_{1})$ has a
simple pole at $z=0$ with a factorized residue
$((q_{0};q_{0})_{\infty}(q_{1};q_{1})_{\infty})^{-1}$. This divergence only
occurs in $d=2$ and is related to the fact that free scalars in two dimensions
are dimensionless.

  * •

In $d=4$, we find a match with $G_{3}(z|\tau_{0},\tau_{1},\tau_{2},\tau_{3})$
through the identification:

$z=\tau,\quad\tau_{0}=\tau+\theta_{1},\quad\tau_{1}=\tau+\theta_{2},\quad\tau_{2}=\tau-\theta_{1},\quad\tau_{3}=\tau-\theta_{2}\,,$
(36)

where we recall our convention (3). That is, we can write the 4d partition
function as

$Z_{4}(\tau,\theta_{i})=G_{3}(\tau|\tau+\theta_{1},\tau+\theta_{2},\tau-\theta_{1},\tau-\theta_{2})\,.$
(37)

  * •

In general $d=2D$, the partition function (7) can be written as

$Z_{2D}(\tau,\theta_{i})=G_{2D-1}((D-1)\tau|\underline{\tau})$ (38)

where

$\tau_{2i-2}=\tau-\theta_{i},\quad\tau_{2i-1}=\tau+\theta_{i},\qquad
i=1,...,D$ (39)

In the following, we use this identification and the modular properties of
$G_{r}$ to present an exact, closed form expression for the partition function
which is a function of the $S$-transformed variables $-\frac{1}{\tau_{i}}$. We
leave speculation on a geometric interpretation to Section 5.

###  3.3 Unrefined partition function

In this section, we apply the modular property to calculate the asymptotics of
the unrefined scalar partition function. That is, we take $\theta_{i}=0$ so
that all chemical potentials are equal: $\tau_{i}=\tau$. We first set up
notation and collect the relevant formulas.

For convenience of notation we define the unrefined multiple Gamma function as

$G_{r}(z;\tau)=G_{r}(z|\tau,\tau,...,\tau)$ (40)

Note that to obtain the $d$-dimensional scalar partition function, we have to
further specialize $z$ as

$Z_{d=r+1}(\tau)\equiv G_{r}\left(\frac{r-1}{2}\tau;\tau\right)\,.$ (41)

We will sometimes refer to this specialization as the _trans-unrefined_ limit.
We also define the corresponding Bernoulli polynomial by

$B_{r}(z;\tau)\equiv B_{rr}(z|\tau,\tau,...,\tau,1)\,$ (42)

where we have simply set $\omega_{r+2}=1$ to avoid writing the homogeneous
variables.

When specializing the modular property (29) for $\tau_{i}=\tau$, some care is
required. Indeed, this specialization implies (setting $\omega_{r+2}=1$):

$\omega_{1}=\ldots=\omega_{r+1}=\tau\,.$ (43)

Comparing with (29), we immediately note that all but one of the $G_{r}$
functions involved obtain real arguments, but recall that
$G_{r}(z|\underline{\tau})$ is only defined when
$\tau_{i}\in\mathbb{C}\setminus\mathbb{R}$. However, it turns out that the
resulting divergences cancel among the $G_{r}$ functions, and a well-defined
non-trivial equation remains.999For the example of the elliptic Gamma
function, this was shown in detail in Theorem 5.2 of Felder_2000  (see also
narukawa2004modular ).

To avoid dealing with these divergences, we present an alternative form of the
modular property which is manifestly finite in the unrefined limit. To get
there, let us first recall Theorem 14 of Narukawa’s work narukawa2004modular .
This replaces the $G_{r}$ functions with an infinite product of normalized
multiple Sine functions $S_{r}(z|\underline{\tau})$:

$\displaystyle\begin{split}G_{r}(z|\underline{\tau})&=\exp\left[-\frac{2\pi
i}{(r+2)!}B_{r+2,r+2}(z|\underline{\tau},1)\right]\\\
&\times\prod_{k=0}^{\infty}\frac{S_{r+1}(z+k+1|\underline{\tau})^{(-1)^{r}}S_{r+1}(z-k|\underline{\tau})^{(-1)^{r}}}{\exp\\{\frac{i\pi}{(r+1)!}[B_{r+1,r+1}(z+k+1|\underline{\tau})-B_{r+1,r+1}(z-k|\underline{\tau})]\\}}\end{split}$
(44)

where $S_{r}(z|\underline{\tau})$ is defined in Appendix A.1. This equation is
defined for $\tau_{j}\in\mathbb{C}\setminus\mathbb{R}$ without any restriction
on the _ratios_ of $\tau_{j}$. Comparing with the conditions imposed below
(29), we see that the domain of analyticity is enlarged.101010This may be
viewed as a generalization of how Faddeev’s quantum dilogarithm extends a
ratio of two ordinary $q$-Pochhammer symbols, naively defined for
$\tau\in\mathbb{C}\setminus\mathbb{R}$, to
$\mathbb{C}\setminus\mathbb{R}_{\leq 0}$ (see Faddeev:1994fw  or, e.g., the
more recent in discussion in Dimofte:2009yn ).  To recover the original form
of the modular property from this expression, one uses the representation of
the multiple Sine function in terms of generalized $q$-Pochhammer symbols
(108) and the definition of $G_{r}$ in (27).

There is, however, a remaining subtlety with the expression (44): the infinite
products of numerator and denominator is not separately defined; only the
ratio is finite. Let us denote this ratio by
$\psi_{r}(z|\underline{\tau})$:111111See Appendix A.1 for more details.

$\psi_{r}(z|\underline{\tau})S_{r}(z|\underline{\tau})=\exp\left[(-1)^{r-1}\frac{\pi
i}{r!}B_{rr}(z|\underline{\tau})\right]\,.$ (45)

In terms of this function, the modular property becomes

$\displaystyle\begin{split}G_{r}(z|\underline{\tau})&=\exp\left[-\frac{2\pi
i}{(r+2)!}B_{r+2,r+2}(z|\underline{\tau},1)\right]\prod_{k=0}^{\infty}\frac{\psi_{r+1}(z+k+1|\underline{\tau})^{(-1)^{r+1}}}{\psi_{r+1}\left(|\underline{\tau}|-z+k|\underline{\tau}\right)}\end{split}$
(46)

where we used the reflection property121212This can be derived from (25) and
(110).

$\psi_{r}(z|\underline{\tau})\psi_{r}(|\underline{\tau}|-z|\underline{\tau})^{(-1)^{r}}=1\,.$
(47)

We can now safely take the unrefined limit $\tau_{i}=\tau$. We then use the
homogeneity property (106) of $S_{r}(z|\underline{\tau})$ and (45) to write

$\psi_{r}(z|\tau,...\tau)=\psi_{r}\left(\frac{z}{\tau}\Big{|}1,...,1\right)\equiv\psi_{r}\left(\frac{z}{\tau}\right)\,.$
(48)

This allows us to rewrite (46) in the unrefined limit as

$\displaystyle\begin{split}G(z;\tau)=\exp\left[-\frac{2\pi
i}{(r+2)!}B_{r+2,r+2}(z;\tau)\right]\prod_{k=0}^{\infty}\frac{\psi_{r+1}\left(\frac{z+k+1}{\tau}\right)^{(-1)^{r+1}}}{\psi_{r+1}\big{(}\frac{(r+1)\tau-z+k}{\tau}\big{)}}\,.\end{split}$
(49)

This is the modular property we will use in the remainder of this section. We
note that the specialization obscures the geometric interpretation of the
modular property mentioned in Section 3.1. It would be very interesting to
find an alternative expression which, in the unrefined limit, reflects the
underlying geometry more closely.

Finally, we will make use of integral representations of these functions. In
particular, $S_{r}(z)\equiv S_{r}(z|1,\ldots,1)$ has an integral
representation (see Appendix A.1)

$S_{r}(z)=\exp\left[-\pi\int_{z_{r}}^{z}\left(\begin{array}[]{c}t-1\\\
r-1\end{array}\right)\cot(\pi t)dt\right]\,.$ (50)

The $z_{r}$ indicate constants which are determined by $S_{r}(z_{r})=1$. From
here, we can work out the integral representation of
$\psi_{r}(z)\equiv\psi_{r}(z|1,...,1)$ up to a constant $a_{r}$

$\psi_{r}(z)=a_{r}\exp\left[2\pi i\int_{-i\infty}^{z}\frac{1}{e^{2\pi
it}-1}\left(\begin{array}[]{c}t-1\\\ r-1\end{array}\right)dt\right]\,.$ (51)

We can calculate $a_{r}=1$ making use of the explicit values of $z_{r}$, which
in turn can be determined from the values $S_{r}(1)$ 131313See also (115) for
the analytic expression for $S_{r}(1)$. kurokawa2003multiple . For example, we
have

$S_{2}(1)=S_{4}(2)=S_{6}(3)=1\,.$ (52)

For other values of $r$, such values can also be worked out
kurokawa2003multiple .

####  3.3.1 $d=4$

We now calculate the all-order asymptotics of the unrefined 4d free scalar
partition function, making use of the identification

$Z_{4}(\tau)=G_{3}(\tau;\tau)\,,$ (53)

and the modular property in the form (49):

$G_{3}(z;\tau)=\exp\left[-\frac{2\pi
i}{120}B_{5}(z+1;\tau)\right]\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{z+k+1}{\tau})}{\psi_{4}(4-\frac{z-k}{\tau})}\,.$
(54)

We can find a completely explicit expression by carrying out the integral
defining $\psi_{4}(z)$:

$\psi_{4}(z)=\exp\left[2\pi
i\int_{-i\infty}^{z}\frac{(t-1)(t-2)(t-3)}{6(e^{2\pi it}-1)}dt\right]$ (55)

which is defined for $\mathrm{Im}(z)<0$. By expanding the geometric series and
integrating terms, we find:

$\displaystyle\psi_{4}(z)$ $\displaystyle=$
$\displaystyle\exp\left[\frac{(z-1)(z-2)(z-3)}{6}\ln(1-e^{-2\pi
iz})+\frac{i(3z^{2}-12z+11)}{12\pi}\text{Li}_{2}(e^{-2\pi iz})\right.$ (56)
$\displaystyle+\left.\frac{z-2}{4\pi^{2}}\text{Li}_{3}(e^{-2\pi
iz})-\frac{i}{8\pi^{3}}\text{Li}_{4}(e^{-2\pi iz})\right]\,,$

where $\mathrm{Li}_{r}(x)$ is the $r^{\text{th}}$ polylogarithm. We now take
the trans-unrefined limit $z=\tau$ to find:

$G_{3}(\tau,\tau)=\exp\left[-\frac{2\pi
i}{120}B_{5}(\tau+1;\tau)\right]\frac{1}{\psi_{4}(3)}\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{\tau+k+1}{\tau})}{\psi_{4}(4-\frac{\tau-k-1}{\tau})}\,.$
(57)

The number $\psi_{4}(3)$ can be evaluated as:

$\psi_{4}(3)=\exp\left[\frac{19\pi i}{720}-\zeta^{\prime}(-2)\right]\,.$ (58)

Together with the explicit expression (56), the formula (57) provides a
beautiful rewriting of the 4d scalar partition function. In particular, this
function manifests the exact high temperature asymptotics $\tau\to 0^{+i}$. To
see this, we first note that $\lim_{z\to-i\infty}\psi_{4}(z)\to
1+\mathcal{O}(e^{-2\pi iz})$. Now, we have for Im$(\tau)>0$ that

$\displaystyle\begin{split}&\text{Im}\left(\frac{\tau+k+1}{\tau}\right)>0,\quad(k\geq
0);\qquad\text{Im}\left(4-\frac{\tau-k-1}{\tau}\right)>0,\quad(k\geq
0)\,.\end{split}$ (59)

Therefore, in the limit $\tau\to 0^{+i}$ the arguments of the $\psi_{4}(z)$ go
to $-i\infty$, and we can combine (57) and (58) to find:

$\displaystyle\begin{split}Z_{4}(\tau)&\sim\exp\left[-\frac{2\pi
i}{120}B_{5}(z+1;\tau)\right]\frac{1}{\psi_{4}(3)}\left(1+\mathcal{O}(e^{-\frac{2\pi
i}{\tau}})\right)\\\
&=\exp\left[-\frac{i\pi}{360\tau^{3}}+\zeta^{\prime}(-2)-\frac{i\pi\tau}{120}\right]\left(1+\mathcal{O}(e^{-\frac{2\pi
i}{\tau}})\right)\,.\end{split}$ (60)

This result is consistent with Meinardus’ theorem, as reviewed in Section 2.3,
but captures in addition the $\mathcal{O}(\tau)$ correction. More importantly,
our exact expression efficiently encodes all non-perturbative corrections as
well. To calculate those, we simply expand

$\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{\tau+k+1}{\tau})}{\psi_{4}(4-\frac{\tau-k-1}{\tau})}$
(61)

to any order $\mathcal{O}(e^{-\frac{2\pi in}{\tau}})$ for $n\geq 0$, making
use of the explicit expression for $\psi_{4}(z)$. In particular, we note that
the $k^{\text{th}}$ factor only contributes at order
$\mathcal{O}(e^{-\frac{2\pi i(k+1)}{\tau}})$. Equivalent results were recently
obtained in Appendix C of Benjamin:2023qsc , making use of the formalism
described in Section 2.3. Our result can be viewed as a resummation of their
exact asymptotics in terms of the special function $\psi_{4}(z)$.

The modular property (57) can be viewed as the integrated version of the more
conventional modular property (20) of $E(\tau)$, the $\tau$-derivative of
$Z_{4}(\tau)$, as derived a long time ago by Cardy cardy1991operator  (see
also Gibbons:2006ij ). In particular, we can check that the $\tau$-derivative
of our Bernoulli polynomial precisely reproduces their modular anomaly:

$\partial_{\tau}\left[-\frac{2\pi i}{120}B_{5}(\tau+1;\tau)\right]=2\pi
i\frac{B_{4}}{8}\left(1-\frac{1}{\tau^{4}}\right)=E(\tau)-\frac{1}{\tau^{4}}E\left(-\frac{1}{\tau}\right)\,.$
(62)

An enlightening way to understand this modular property is to write

$E(\tau)=\sum_{n}\frac{d_{n}E_{n}}{e^{-2\pi i\tau E_{n}}-1}$

and recall Ramanujan’s formula Dowker:2002ax ; Gibbons:2006ij

$(\pi i\tau)^{p}\sum_{n}\frac{n^{2p-1}}{e^{2\pi in\tau}-1}-\left(\frac{\pi
i}{\tau}\right)^{p}\sum_{n}\frac{n^{2p-1}}{e^{-\frac{2\pi
in}{\tau}}-1}=\left[(\pi i\tau)^{p}-\left(\frac{\pi
i}{\tau}\right)^{p}\right]\frac{B_{2p}}{4p}\,.$ (63)

The fact that the summand is close to the integrand in the integral
representation of $\psi_{r}(z)$ explains why the derivative of partition
function $\ln Z$ behaves as a modular form.

The asymptotic growth of the density of states can be computed by inverse
Legendre transformation followed by saddle point approximation, resulting in

$\rho(\Delta)\sim\frac{1}{2\sqrt{2}\sqrt[8]{15}\left(\Delta+\frac{1}{240}\right)^{\frac{5}{8}}}\exp\left[\frac{4\pi}{3\sqrt[4]{15}}\left(\Delta+\frac{1}{240}\right)^{\frac{3}{4}}+\zeta^{\prime}(-2)\right]\,.$
(64)

The difference between asymptotic growth (64) and the prediction by Meinardus’
theorem Melia:2020pzd  is the shift by $1/240$ in the conformal dimension.
This reminds us of the Casimir energy $-c/24$ in two dimensional conformal
field theory, which similarly appears as a shift in the conformal dimension in
Cardy’s formula Cardy:1986ie . To see this analogy more explicitly, we can
consider $c$ free scalars. In this case, the asymptotics simply become

$Z(\tau)\equiv\exp\left[-\frac{ci\pi}{360\tau^{3}}+c\zeta^{\prime}(-2)-\frac{i\pi
c\tau}{120}\right]\,.$ (65)

The density of states, in turn, reads:

$\rho\sim\frac{c^{\frac{1}{8}}}{2\sqrt{2}\sqrt[8]{15}\left(\Delta+\frac{c}{240}\right)^{\frac{5}{8}}}\exp\left[\frac{4\pi}{3\sqrt[4]{15}}c^{\frac{1}{4}}\left(\Delta+\frac{c}{240}\right)^{\frac{3}{4}}+c\zeta^{\prime}(-2)\right]\,.$
(66)

####  3.3.2 6d free scalar

The free scalar theory in six dimensions can be analyzed similarly. In
particular, the 6d partition function with angular potentials turned off can
be written in terms of $G_{5}(z|\underline{\tau})$:

$Z_{6}(\tau)=\exp\left[\sum_{l=1}^{\infty}\frac{1}{l}\frac{q^{2l}(1-q^{2l})}{(1-q^{l})^{6}}\right]\equiv
G_{5}(2\tau;\tau)\,.$ (67)

Specializing the modular property in the form (46) for $\tau_{i}=\tau$ we find

$\displaystyle\begin{split}G_{5}(z;\tau)&=\exp\left[-\frac{2\pi
i}{7!}B_{7}(z+1,\tau)\right]\prod_{k=0}^{\infty}\frac{\psi_{6}(\frac{z+k+1}{\tau})}{\psi_{6}(6-\frac{z-k}{\tau})}\,.\end{split}$
(68)

From (51) we have the integral representation of $\psi_{6}(z)$:

$\psi_{6}(z)=\exp\left[2\pi i\int_{-i\infty}^{z}\frac{1}{e^{2\pi
it}-1}\left(\begin{array}[]{c}t-1\\\ 5\end{array}\right)dt\right]\,.$ (69)

We now take $z=2\tau$ and calculate the $\tau\to 0^{+i}$ asymptotics in
complete analogy with the 4d case. The asymptotics up to non-perturbative
corrections is given by

$Z_{6}(\tau)\sim\exp\left[-\frac{2\pi
i}{7!}B_{77}(2\tau+1,\tau)\right]\frac{1}{\psi_{6}(4)}\left(1+\mathcal{O}(e^{-\frac{2\pi
i}{\tau}})\right)\,.$ (70)

From the integral representation, $\psi_{6}(4)$ is evaluates to

$\psi_{6}(4)=\frac{120689i\pi}{60480}-\frac{1}{12}\Big{[}\zeta^{\prime}(-4)-\zeta^{\prime}(-2)\Big{]}\,.$
(71)

Therefore, the asymptotics of the 6d scalar partition function is

$\ln
Z_{6}(\tau)\sim\frac{i\pi}{15120\tau^{5}}+\frac{i\pi}{4320\tau^{3}}+\frac{1}{12}\Big{[}\zeta^{\prime}(-4)-\zeta^{\prime}(-2)\Big{]}+\frac{31i\pi\tau}{30240}+\mathcal{O}(e^{-\frac{2\pi
i}{\tau}})\,.$ (72)

This result is again consistent with Meinardus’ theorem (22) but also produces
the $\mathcal{O}(\tau)$ correction. Furthermore, the non-perturbative
corrections are also easily derived from our exact expression (68). In
particular, one first integrates (69) to obtain an expression in terms of
poly-logarithms similar to the 4d case, and then expand

$\displaystyle\begin{split}\prod_{k=0}^{\infty}\frac{\psi_{6}(\frac{2\tau+k+1}{\tau})}{\psi_{6}(6-\frac{2\tau-k-1}{\tau})}\,.\end{split}$
(73)

Again, this provides an exact, closed-form expression for the all order
asymptotics, which resums the exact asymptotics obtained in Benjamin:2023qsc .

###  3.4 Refined partition function

We now extend our analysis to the 4d partition function including angular
potentials. This was identified with $G_{3}(z|\underline{\tau})$ in (37),
which we recall for convenience:

$Z_{4}(\tau,\theta_{i})=G_{3}(\tau|\tau+\theta_{1},\tau+\theta_{2},\tau-\theta_{1},\tau-\theta_{2})\,.$
(74)

We write $\theta_{i}=\tau\Omega_{i}$ with $|\Omega_{i}|<1$ and find it
convenient to again use the modular property in the form (46). Explicitly, the
modular property reads

$\displaystyle\begin{split}&G_{3}\Big{(}\tau\Big{|}\tau(1+\Omega_{1}),\tau(1+\Omega_{2}),\tau(1-\Omega_{1}),\tau(1-\Omega_{2})\Big{)}\\\
=&\exp\left[\frac{2\pi
i}{120}B_{55}(\tau\big{|}\tau(1+\Omega_{1}),\tau(1+\Omega_{2}),\tau(1-\Omega_{1}),\tau(1-\Omega_{2}),-1)\right]\\\
&\times\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{\tau+k+1}{\tau}|1+\Omega_{1},1+\Omega_{2},1-\Omega_{1},1-\Omega_{2})}{\psi_{4}(3+\frac{k}{\tau}|1+\Omega_{1},1+\Omega_{2},1-\Omega_{1},1-\Omega_{2})}\,.\end{split}$
(75)

Similar to the unrefined case, the $k=0$ term in the denominator contributes
at $\mathcal{O}(\tau^{0})$ while all other terms in the product contribute
only at non-perturbative order $\mathcal{O}(e^{-\frac{2\pi i}{\tau}})$. We
notice, however, that in this case we require an integral representation of
$\psi_{4}(z|\underline{\tau})$, which is considerably more complicated than
the one for $\psi_{4}(z)$ narukawa2004modular . An alternative definition of
$\psi_{4}(z|\underline{\tau})$, in terms of generalized $q$-Pochhammer symbols
(110), does not apply since the arguments in this specialization are real. For
this reason, it is more difficult, although in principle possible, to
calculate the non-perturbative corrections order by order in this case as
compared with the unrefined case.

We can still easily calculate the $\tau\to 0$ asymptotics up to non-
perturbative order. For convenience, we define the $k=0$ term as

$\psi_{4}(3|1+\Omega_{1},1+\Omega_{2},1-\Omega_{1},1-\Omega_{2})=\exp[2\pi
i\Psi(\Omega_{1},\Omega_{2})]$ (76)

for which we only have an implicit definition in terms of the integral
representation of $\psi_{4}(z|\underline{\tau})$ narukawa2004modular . The
asymptotics is then calculated as

$\displaystyle\begin{split}\ln Z_{4}(\tau,\tau\Omega_{i})&=\frac{\pi
i}{360\tau^{3}(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})}+\frac{\pi
i(\Omega_{1}^{2}+\Omega_{2}^{2})}{72(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})\tau}-2\pi
i\Psi(\Omega_{1},\Omega_{2})\\\ &+\pi
i\frac{3(\Omega_{1}^{4}+\Omega_{2}^{4})+5\Omega_{1}^{2}\Omega_{2}^{2}-17(\Omega_{1}^{2}+\Omega_{2}^{2})-19}{720(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})}\\\
&+\pi
i\frac{3(\Omega_{1}^{4}+\Omega_{2}^{4})+5\Omega_{1}^{2}\Omega_{2}^{2}+3(\Omega_{1}^{2}+\Omega_{2}^{2})-3}{360(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})}\tau+\mathcal{O}\left(e^{-\frac{2\pi
i}{\tau}}\right)\end{split}$ (77)

where the rest of terms are non-perturbative in the $\tau\to 0$ limit.

The asymptotics enables us to compute the asymptotic degeneracy of states for
given energy _and_ angular momenta. This was also studied in Benjamin:2023qsc
, but we include the analysis for completeness.

#### Small angular momenta

Let’s first consider the regime where the angular momenta are small compared
to the energy but still large enough to do a saddle point approximation.

We assume $\Delta,J_{1},J_{2}\gg 1$ but subject to

$1\ll J_{1},J_{2}\ll\Delta\,.$ (78)

In this case, the chemical potentials related to angular momenta can be
assumed to be much smaller than $1$. We can therefore consider the first term
in (77) and expand in terms of $\Omega_{i}$:

$Q_{5}(\Omega_{1},\Omega_{2})\sim\frac{1}{720\tau^{3}(1-\Omega_{1}^{2})(1-\Omega_{2}^{2})}=\frac{1}{720\tau^{3}}(1+\Omega_{1}^{2}+\Omega_{2}^{2})+\mathcal{O}(\Omega^{4})$
(79)

To calculate the density of state, we extremize:

$F_{s}(\tau,\Omega_{1},\Omega_{2})=-\frac{\pi
i}{360\tau^{3}}(1+\Omega_{1}^{2}+\Omega_{2}^{2})-2\pi i\tau\Delta-2\pi
i\tau\Omega_{1}J_{1}-2\pi i\tau\Omega_{2}J_{2}$ (80)

This results in the asymptotic density of states in the regime of interest:

$\ln\rho(\Delta,J_{1},J_{2})=\frac{2^{\frac{9}{4}}\pi\Delta^{\frac{3}{4}}}{\sqrt{15}}\left[\left(\frac{x}{1-\sqrt{1-15x}}\right)^{\frac{3}{4}}+\frac{1}{5}\left(\frac{1-\sqrt{1-15x}}{x}\right)^{\frac{1}{4}}\right]$
(81)

where $x=\frac{J_{1}^{2}+J_{2}^{2}}{E^{2}}$. We can see when $x\to 0$, the
(64) is reproduced.

#### Fast rotating regime

We also consider the fast rotating limit for which $\Omega_{i}\to-1$. The
limit $\Omega\to 1$ is similar due to the symmetry in (75). In this case, we
parametrize $\Omega_{i}=-1-\omega_{i}$. The entropy functional becomes

$F_{f}(\tau,\omega_{1},\omega_{2})=-\frac{\pi
i}{1440\omega_{1}\omega_{2}\tau^{3}}-2\pi i(\Delta-J_{1}-J_{2})+2\pi
i\tau\omega_{1}J_{1}+2\pi i\tau\omega_{2}J_{2}$ (82)

This functional is extremized by

$\omega_{1}=-\frac{\Delta-J_{1}-J_{2}}{J_{1}},\qquad\omega_{2}=-\frac{\Delta-
J_{1}-J_{2}}{J_{2}}\,,$ (83)

and the asymptotic density of states in this limit is computed to be

$\ln\rho(\Delta,J_{1},J_{2})=\frac{2\sqrt{2}}{\sqrt{3}\sqrt[4]{5}}\pi\sqrt[4]{(\Delta-
J_{1}-J_{2})J_{1}J_{2}}$ (84)

As we expect $\omega_{i}$ to be small, then $\Delta$ should be close to the
unitarity bound:

$\Delta-J_{1}-J_{2}\geq 0,\quad J_{i}>0\,.$ (85)

##  4 Root-of-unity asymptotics

In Section 3, we derived explicit and exact expressions of free scalar
partition functions in terms of the modular transformed variable
$-\frac{1}{\tau}$. To derive these expressions, we made use of the
identification of the partition function with the multiple elliptic Gamma
functions $G_{r}(z|\underline{\tau})$ and their modular properties.

In this section, we demonstrate how this connection can also be used to derive
expressions in terms of a general modular transformed variable
$\frac{k\tau+l}{m\tau+n}$. The resulting expressions can then be used to
calculate the “root-of-unity” asymptotics of the partition function, where
$\tau\to-\frac{n}{m}+0^{+i}$. Such limits are well-studied in the context of
2d CFT and play a prominent role in holography Dijkgraaf:2000fq ;
Maldacena:1998bw . The very recent work Benjamin:2024kdg  studies these limits
in higher dimensional CFTs.141414The root-of-unity asymptotics has also been
of interest in the context of the 4d superconformal index, in particular in
connection to AdS5 black holes Cabo-Bizet:2019eaf ; ArabiArdehali:2021nsx ;
Jejjala:2021hlt ; Cabo-Bizet:2021plf . As in Section 3, we will see that our
method produces explicit, closed form expressions, which make use of the
underlying modular structure, and make the asymptotics manifest.

We start with a multiplication formula for $G_{3}(z|\underline{\tau})$:

$\displaystyle\begin{split}&G_{3}(z|\tau_{0},\tau_{1},\tau_{2},\tau_{3})=\\\
&\prod_{a_{0}=0}^{m-1}\prod_{a_{1}=0}^{m-1}\prod_{a_{2}=0}^{m-1}\prod_{a_{3}=0}^{m-1}G_{3}\left(z+\sum_{i=0}^{3}a_{i}\tau_{i}\Big{|}m\tau_{0}+n_{0},m\tau_{1}+n_{1},m\tau_{2}+n_{2},m\tau_{3}+n_{3}\right)\end{split}$
(86)

This formula is derived in Appendix B. We will only be interested in the
unrefined limit with $\tau_{i}=\tau$. In this case, the multiplication formula
can be rewritten as

$G_{3}(z;\tau)=\prod_{c=0}^{4m-4}G_{3}(z+c\tau+N_{c};m\tau+n)^{d(c)}$ (87)

where $d(c)$ is an integer depending on $c$. Its explicit form is parametrized
in terms of four functions $d_{i}(c)$ which are explicitly listed in Appendix
B. We plot an example for $m=7$ in Figure 1.

Figure 1: The piece-wise function $d_{A}(c)$ for $m=7$ as $0\leq c\leq 4m-4$.
From left to right, the different colors represent difference pieces. There
are four disconnected pieces, labelled as I, II, III, IV. Notice $d(c)$ are
only defined on integer points.

We now obtain a formula in terms of the modular transformed variable by simply
applying (54), the modular property of $G_{3}(z;\tau)$, to each function on
the right hand side of (87). More specifically, each factor in (87) is
replaced by:

$\displaystyle\begin{split}G_{3}(z+c\tau+N_{c};m\tau+n)&=\exp\left[-\frac{2\pi
i}{120}B_{5}\left(z+c\tau+N_{c}+1;m\tau+n\right)\right]\\\
&\times\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{z+c\tau+N_{c}+k+1}{m\tau+n})}{\psi_{4}(4-\frac{z+c\tau+N_{c}-k}{m\tau+n})}\end{split}$
(88)

The resulting formula provides us with an exact, albeit unwieldy expression
for $G_{3}(z;\tau)$ in terms of a general modular transformed variable. In
particular, upon setting $z=\tau$, the formula encodes the all-order
asymptotics of the unrefined 4d free scalar partition function in the limit
$\tau\to-\frac{n}{m}+0^{+i}$. Notice that the dependence on $N_{c}$ on the
right hand side should cancel since the left hand side is periodic under $z\to
z+1$. We use this independence below to set $N_{c}$ such that the limit
$\tau\to-\frac{n}{m}$ is most easily taken.

Before proceeding with the asymptotics, let us make some comments. Compared to
the analysis for $\tau\to-\frac{1}{\tau}$ in Section 3.3, in particular
formula (54), the number of factors in the resulting expression for
$G_{3}(z;\tau)$ is much larger and in particular depends on $m$, the
$SL(2,\mathbb{Z})$ parameter. In principle, this does not pose a fundamental
problem for, say, the asymptotic analysis. However, in practice, the
expression becomes rather unwieldy to work with. Furthermore, this formula
obscures the already unconventional modular property in that the number of
functions involved seems to depend on the $SL(2,\mathbb{Z})$ parameter. We
strongly suspect that, in fact, there exists a much simpler formula where all
factors collapse into a single term.151515See also Felder_2008 ;
Tizzano:2014roa ; Winding:2016wpw . This suspicion is based on a similar
phenomenon occurring for the elliptic Gamma function. Very briefly, the
derivation of the general modular property of the elliptic Gamma function in
Jejjala:2022lrm  (Appendix D) involves the same strategy as above, but
includes one further step. Namely, just as above, one first uses the
appropriate multiplication formula and applies the analogue of (88) to each
factor. However, in the final step, the multiplication formula is used again
in the opposite direction to collapse the number of factors on the right hand
side. We leave this step in the present case to future work.

Let us now continue with the root-of-unity asymptotics. Similar to Section
3.3, there are two types of contributions. In terms of $\ln G_{3}(\tau;\tau)$,
there are power law contributions coming from the Bernoulli polynomial and
there is a constant contribution and non-perturbative contributions coming
from the product over $\psi_{4}$ functions. To take the limit, we now show
that there is a unique choice of $N_{c}$ such that the $\psi_{4}$ functions do
not diverge in the $\tau\to-\frac{n}{m}$ limit. This choice turns out to be:

$N_{c}=\bigg{\lfloor}\frac{(c+1)n}{m}\bigg{\rfloor}$ (89)

where $\lfloor\cdot\rfloor$ is the floor function. To see this, just like in
Section 3.3, we check that the arguments of all $\psi_{4}$ functions approach
$-i\infty$ in the limit, or in exceptional cases, are constant. Setting
$z=\tau$, the arguments of the $\psi_{4}$ functions in the numerator involve

$\tau+c\tau+N_{c}+k+1=(c+1)\frac{(m\tau+n)-n}{m}+\bigg{\lfloor}\frac{(c+1)n}{m}\bigg{\rfloor}+k+1\,.$
(90)

This is always positive for $\tau\to-\frac{n}{m}+0^{+i}$ so that

$\text{Im}\left(\frac{z+c\tau+N_{c}+k+1}{m\tau+n}\right)\to-\infty$ (91)

and the functions $\psi_{4}\left(\frac{z+c\tau+N_{c}+k+1}{m\tau+n}\right)\to
1+\mathcal{O}(e^{-\frac{2\pi i}{m\tau+n}})$. For the $\psi_{4}$ functions in
the denominator, we similarly find

$4-\frac{\tau+c\tau+N_{c}-k}{m\tau+n}\to
4+\frac{k+\frac{(c+1)n}{m}-\lfloor\frac{(c+1)n}{m}\rfloor}{m\tau+n}-\frac{c+1}{m}\,.$
(92)

Again, the imaginary part generically goes to $-i\infty$ except when
$\lfloor\frac{(c+1)n}{m}\rfloor$ is an integer and $k=0$. These exceptions
correspond to three values of $c$:

$c=m-1,\quad 2m-1,\quad 3m-1\,.$ (93)

In these cases, the $k=0$ terms in the respective factors in (87) contribute a
constant listed in the following table:

$c$ value |  $\psi_{4}$ value | Degeneracy
---|---|---
$m-1$ | $\psi_{4}(3)$ | $d_{\text{II}}(m-1)$
$2m-1$ | $\psi_{4}(2)$ | $d_{\text{III}}(2m-1)$
$3m-1$ | $\psi_{4}(1)$ | $d_{\text{IV}}(3m-1)$
Table 1: Constant order contributions to the asymptotics of (88).

Similar to the case of the elliptic Gamma function Felder_2008 ;
Jejjala:2021hlt ; Jejjala:2022lrm , the sum of all $B_{5}$ Bernoulli
polynomials, coming from the replacement of (88) in (87), for the choice of
$N_{c}$ in (89) combine into a simple expression. We will denote this sum by
$Q_{5}^{(m,n)}(z;\tau)$ and find that it is equal to:

$Q_{5}^{(m,n)}(z;\tau)=\frac{1}{m}B_{5}(mz+n+1;m\tau+n)+C_{5}(m,n)$ (94)

where we refer to Appendix B for more details. We note that this holds even
for $z\neq\tau$. We have not determined a simple expression for the constant
$C_{5}(m,n)$, but we note here that in the case of the elliptic Gamma function
a similar constant can be expressed in terms of interesting number-theoretic
quantities such as the Dedekind sum $s(n,m)$ and generalizations
Jejjala:2022lrm . An explicit formula for this constant is written in (132).

Using the expression (94) and the constant contributions from Table 1, we find
the leading order asymptotics for $\tau\to-\frac{n}{m}$ to be

$\displaystyle\begin{split}G_{3}(\tau;\tau)&\sim\exp\left[-2\pi
iQ_{5}^{(m,n)}(\tau;\tau)\right]\exp\left[m^{2}\zeta^{\prime}(-2)+\frac{i\pi(m^{2}-20)m}{720}\left(1+\mathcal{O}(e^{-\frac{2\pi
i}{m\tau+n}})\right)\right]\end{split}$ (95)

The real constant term in (95) generalizes the prediction by Meinardus’
theorem. We also note that the $\zeta^{\prime}(-2)$ term has appeared before
and was called the subleading topological entropy in Asorey:2012vp . We thus
find the root-of-unity asymptotics of the 4d free scalar partition function to
be

$\displaystyle\begin{split}\ln
Z_{4}(\tau)\sim&\frac{-i\pi}{360m(m\tau+n)^{3}}+m^{2}\zeta^{\prime}(-2)-2\pi
iC_{5}(m,n)\\\
&+\frac{i\pi(m^{2}-1)(m^{2}-19)}{720m}-\frac{i\pi(m\tau+n)}{120m}\,.\end{split}$
(96)

Pulling out $m$ in the denominator of the first terms, we obtain $1/m^{4}$
scaling at leading order consistent with Benjamin:2024kdg . From this
equation, to leading order, we can evaluate the asymptotic density of states
to be

$\rho(\Delta)\sim\exp\left(\frac{4\pi}{3m\sqrt[4]{15}}\Delta^{\frac{3}{4}}\right)\,.$
(97)

The $1/m$ factor in the exponent indicates a $\mathbb{Z}_{m}$ quotient, which
features explicitly in Benjamin:2024kdg . The presence of such a factor was
also noted in the context of superconformal indices at roots of unity Cabo-
Bizet:2019eaf ; ArabiArdehali:2021nsx ; Jejjala:2021hlt ; Aharony:2021zkr ;
Cabo-Bizet:2021plf .

##  5 Discussion and future work

In this paper, we identified the multiple elliptic Gamma functions $G_{r}$ as
analytic continuations in the temperature and chemical potentials of the
thermal partition function of conformally coupled free scalars in _even_
dimensions $d=r+1$ on $S^{r}\times S^{1}$. The natural habitat of these
functions, however, is in the context of superconformal indices in dimension
$d=2r+2$, i.e., supersymmetric partition functions on $S^{2r+1}\times S^{1}$.
The multiple elliptic Gamma functions satisfy interesting modular properties,
which have a beautiful geometric interpretation in terms of splitting
properties of $S^{2r+1}\times S^{1}$ geometric backgrounds, as summarized in
Section 3.1. The torus consisting of the time circle and the Hopf fiber of the
odd-dimensional spheres plays a crucial role. As we have seen in Section 3.2,
the specialization required to match the multiple elliptic Gamma functions
with the free scalar obscures the geometric interpretation of the modular
property. Nonetheless, one still finds an equation which expresses the free
scalar partition function $Z_{d}(\tau)$ in terms of functions of a modular
transformed variable $-\frac{1}{\tau}$, or more generally
$\frac{k\tau+l}{m\tau+n}$.

A key question is now: does the geometric interpretation of the modular
property in the SCFT living in $d=2r+2$ imply a geometric interpretation of
the (admittedly obscure) modular property of the free scalar in $d=r+1$? Our
tentative answer is: yes! Let us sketch the idea, leaving a detailed analysis
to future work. First note that the variables $\underline{\tau}$ of the
multiple elliptic Gamma function can be understood as complex structure moduli
of the $S^{2r+1}\times S^{1}$ geometry.161616 See, e.g., Closset:2013vra ;
Assel:2014paa , where the chemical potentials of the 4d superconformal index
are matched with the complex structure moduli of a Hopf surface. A similar
story applies to other dimensions as well. The specialization of these
parameters, required to match the free scalar on $S^{r}\times S^{1}$, appears
to be a degenerating limit of the $S^{2r+1}\times S^{1}$ geometry. It seems
possible that this degeneration precisely reduces the $d=(2r+2)$-dimensional
geometry to a $d=(r+1)$-dimensional geometry.

As an example, consider the connection between the
$G_{3}(z;\underline{\tau})$, interpreted as the index on $S^{7}\times S^{1}$,
and the free scalar partition function on $S^{3}\times S^{1}$. A natural way
in which $S^{3}\times S^{1}$ could arise from a degeneration of $S^{7}\times
S^{1}$ is as follows. View $S^{7}$ as an $S^{3}\times S^{3}$ fibration over an
interval. Now consider a squashing limit in which the Hopf fiber of both
$S^{3}$ becomes much larger than their base $S^{2}$. In this case, $S^{7}$
degenerates to $S^{3}$. In particular, the quadrisection of $S^{7}\times
S^{1}$, as discussed in Section 3.1, becomes the bisection of $S^{3}\times
S^{1}$. Now, whereas the full modular property of $G_{3}(z;\underline{\tau})$
has a natural interpretation in terms of the quadrisection $S^{7}\times
S^{1}$, we conjecture that the specialization can be interpreted in terms of
bisection $S^{3}\times S^{1}$. More specifically, we propose that the right
hand side of the modular property:

$Z_{4}(\tau)=\exp\left[-\frac{2\pi
i}{120}B_{5}(\tau+1;\tau)\right]\frac{1}{\psi_{4}(3)}\prod_{k=0}^{\infty}\frac{\psi_{4}(\frac{\tau+k+1}{\tau})}{\psi_{4}(4-\frac{\tau-k-1}{\tau})}\,,$
(98)

could indeed be interpreted in terms of the splitting of $S^{3}\times S^{1}$
into two $D_{2}\times T^{2}$ geometries and the associated concept of modular
factorization Jejjala:2022lrm .171717In work in progress, we present a
Hamiltonian picture of modular factorization in the context of 4d
supersymmetric field theories. The emerging picture seems to suggest that this
idea could be extended to (at least) free non-supersymmetric CFTs
vanLeuven:2024 . To actually identify which terms on the right correspond to
which $D_{2}\times T^{2}$ partition function, we see two ways forward. Either,
one should independently calculate the $D_{2}\times T^{2}$ partition functions
and identify an appropriate product of two of such partition functions with
(98). Alternatively, one may attempt to derive an alternative version of (1)
which does not obscure the underlying geometry. For example, starting off with
the full modular property, one could try to take the degenerating limit while
carefully keeping track of the indivdual $G_{3}$ functions, which have a clear
geometric origin. If this indeed works out, we expect that this interpretation
could apply to other free theories and dimensions as well.

As future directions, we would like to study: (i) Scalars in odd dimension;
(ii) Fermions in various dimensions; (iii) Scalars in AdS, dS spaces
Gibbons:2006ij . The difficulty of constructing a modular property of the
scalar partition function in odd dimensions ($d=3$) was already observed in
cardy1991operator . Technically, this is due to the Hamiltonian zeta function
$D(d,s)$ being a Hurwitz $\zeta$-function instead of a Riemann
$\zeta$-function. An anti-periodic boundary condition along the time circle
can ameliorate the issues cardy1991operator ; Oshima:1992yy . From our
perspective, both free fermions in all dimensions and the free scalars in odd
dimensions share the same obstacle: their partition functions are not of the
form of the multiple elliptic Gamma function. However, the basic building
blocks of their partition functions are generalized $q$-Pochhammer symbols,
which also appear in the definition of the multiple elliptic Gamma functions.
As such, these theories are not that far removed from our general framework
either. The original $q$-Pochhammer symbol $(x;q)_{\infty}$ has recently been
understood as potentially the simplest example of a (holomorphic) quantum
Jacobi form Garoufalidis:2022wij . It would be very interesting to generalize
this notion to generalized $q$-Pochhammer symbols, and study the implications
for the larger class of free CFTs as listed above along the lines of our work.

A final interesting direction is to revisit the mysterious $SL(2,\mathbb{Z})$
duality between the partition functions on two topologically distinct
manifolds Shaghoulian:2016gol :

$\lim_{k,p\to\infty}Z\left[S_{2\pi/k}^{1}\times
S^{3}/\mathbb{Z}_{p}\right]=\lim_{k,p\to\infty}Z\left[S_{2\pi/p}^{1}\times
S^{3}/\mathbb{Z}_{k}\right],\qquad k,p\in\mathbb{Z}^{+}$ (99)

This duality was checked for free field theory, supersymmetric theories and
holographic CFTs. It would be interesting if our perspective could shed some
light on this proposal, perhaps also in the context of the recent work on
twisted boundary conditions on the time circle Benjamin:2024kdg .

## Acknowledgement

YL thanks 13th Joburg Workshop on String Theory and University of
Witwatersrand for hospitality where this work was initiated. YL would also
like to thank the organizers of the “Frontier forum on Black holes and
wormholes” 2024 in Thousands island lake county, the ICTP-AP UCAS for
hospitality where this work is completed. Y.L. is supported by a Project
Funded by the Priority Academic Program Development of Jiangsu Higher
Education Institutions (PAPD) and by National Natural Science Foundation of
China No.12305081. SvL is in part supported by the DSI-NRF Centre of
Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa,
grant #2022-59-Phy-Indices. SvL would also like to thank the Tata Institute of
Fundamental Research for hospitality, where part of this work was done.

##  Appendix A Special functions

###  A.1 Multiple Sine function

The _Barnes multiple zeta function_ was studied by barnes1904theory  as the
generalization of Hurwitz zeta function 181818This is also called multiple
zeta functions in some literatures, but the terminology “multiple zeta
function” is usually used as the following generalization of Riemann zeta
function  $\zeta(s,t)=\sum_{m,n\geq 0}^{\prime}\frac{1}{m^{s}n^{t}}$ (100) We
therefore emphasize (101) as the Barnes multiple zeta function in order to
distinguish from (100). :

$\zeta_{r}(s,z,\underline{\omega})=\sum_{\vec{n}\geq
0}\frac{1}{(\vec{n}\cdot\underline{\omega}+z)^{s}}$ (101)

The Barnes-MZ function (101) can be analytically continued to the complex
plane $s\in\mathbb{C}$. The related multiple Gamma function is then defined as

$\Gamma_{r}(z|\underline{\omega})=\exp\left[-\frac{\partial}{\partial
s}\zeta_{r}(s,z,\underline{\omega})\Big{|}_{s=0}\right]$ (102)

Then the multiple Sine functions $S_{r}(z|\underline{\omega})$ are defined as

$S_{r}(z|\underline{\omega})=\Gamma_{r}(z|\underline{\omega})^{-1}\Gamma_{r}(|\underline{\omega}|-z|\underline{\omega})^{(-1)^{r}}$
(103)

The refined multiple sine functions have a few useful properties
kurokawa2003multiple .

  * •

Shift property

$S_{r}(z+\omega_{i}|\underline{\omega})=S_{r}(z|\underline{\omega})S_{r-1}(z|\underline{\omega}^{-}(j))^{-1}$
(104)

  * •

Multiplication formula

$S_{r}(z|\underline{\omega})=\prod_{k_{i}=0}^{m-1}S_{r}\left(z+\underline{k}\cdot\underline{\omega}|\,m\underline{\omega}\right)$
(105)

  * •

Homogeneity: for $c\in\mathbb{C}-\\{0\\}$

$S_{r}(cz|c\underline{\omega})=S_{r}(z|\underline{\omega})$ (106)

  * •

Reflection

$S_{r}(z|\underline{\omega})S_{r}(|\underline{\omega}|-z|\underline{\omega})^{(-1)^{r}}=1$
(107)

The explicit expression of multiple sine function given by the generalized
$q$-Pochhammer symbol is given in narukawa2004modular , which is

$\displaystyle\begin{split}S_{r}(z|\underline{\tau})&=\exp\left[(-1)^{r}\frac{\pi
i}{r!}B_{rr}(z|\underline{\tau})\right]\prod_{k=1}^{r}(x_{k};\underline{q_{k}})_{\infty}^{(r-2)}\\\
&=\exp\left[(-1)^{r-1}\frac{\pi
i}{r!}B_{rr}(z|\underline{\tau})\right]\prod_{k=1}^{r}(x_{k}^{-1};\underline{q_{k}^{-1}})_{\infty}^{(r-2)}\end{split}$
(108)

given Im$(\tau_{i}/\tau_{j})\neq 0$ and

$\displaystyle\begin{split}&x_{k}=e^{2\pi i\frac{z}{\tau_{k}}},\quad
q_{jk}=e^{2\pi
i\frac{\tau_{j}}{\tau_{k}}}\,,\quad\underline{q_{k}}=(q_{1k},...,\check{q}_{kk},...,q_{rk})\end{split}$
(109)

It is sometimes convenient to introduce $\psi_{r}(z)$ functions for (108) such
that the phase part of multiple Sine function is extracted (45):

$\psi_{r}(z|\underline{\tau})=\left[\prod_{k=1}^{r}(x_{k}^{-1};\underline{q_{k}^{-1}})^{(r-2)}_{\infty}\right]^{-1}$
(110)

The two lines of (108) are defined for $|q_{j}|<1$ and $|q_{j}|>1$
respectively. Due to the homogeneity condition, we can further simplify the
multiple sine function by setting one of the elliptic moduli to $1$. For
example, the $S_{2}(z|\tau,1)$ is simply

$S_{2}(z|\tau,1)=\exp\left[\frac{\pi
i}{2}B_{22}(z|\tau,1)\right]\prod_{j=0}^{\infty}\frac{1-e^{2\pi
i(z+j\tau)}}{1-e^{2\pi i(\frac{z}{\tau}-\frac{j+1}{\tau})}}\,.$ (111)

This structure is very useful in studying the holomorphic factorization
properties of partition functions in three dimensional superconformal field
theory Nieri:2015yia .

The multiple sine functions defined in (103) are called _normalized multiple
sine function_ in the literature kurokawa2003multiple . There is another type
of multiple sine function called _primitive multiple sine function_
kurokawa2003multiple , defined as

$\displaystyle\begin{split}\mathcal{S}_{r}(z)=\exp\left(\frac{z^{r-1}}{r-1}\right)\prod_{n=-\infty,\neq
0}^{+\infty}P_{r}\left(\frac{z}{n}\right)^{n^{r-1}}\end{split}$ (112)

where

$P_{r}(z)=(1-z)\exp\left(z+\frac{z^{2}}{2}+...+\frac{z^{r}}{r}\right)$ (113)

The paper kurokawa2003multiple  showed the $S_{r}(z)$ and $\mathcal{S}_{r}(z)$
are closely related. The relations is actually very transparent in the
following representations

$\displaystyle\begin{split}\frac{S_{r}^{\prime}}{S_{r}}(z)&=(-1)^{r-1}\left(\begin{array}[]{c}z-1\\\
r-1\end{array}\right)\pi\cot(\pi z)\\\
\frac{\mathcal{S}_{r}^{\prime}}{\mathcal{S}_{r}}(z)&=z^{r-1}\pi\cot(\pi
z)\end{split}$ (114)

Given the differential form of multiple sine function (114), the functions are
completely fixed by an initial condition at special values. The Lemma 3.2 of
kurokawa2003multiple  claims

$\displaystyle\begin{split}S_{r}(1)&=\exp\left(-2\sum_{2\leq k\leq
r-1,\text{even}}a(r,k)\zeta^{\prime}(-k)\right)\end{split}$ (115)

where $a(r,k)$ are defined as

$C_{X+r-2}^{r-1}=\sum_{k=1}^{r-1}a(r,k)X^{k}$ (116)

For concrete examples, we can use (115) to determine a few special values such
as

$S_{4}(2)=1,\quad S_{2}(1)=1,\quad S_{4}(1)=e^{-\zeta^{\prime}(-2)}$ (117)

These can then determine following integral representations of multiple sine
functions

$\displaystyle\begin{split}&S_{2}(z)=\exp\left[-\pi\int_{1}^{z}(t-1)\cot(\pi
t)dt\right]\\\
&S_{4}(z)=\exp\left[-\pi\int_{2}^{z}\frac{(t-1)(t-2)(t-3)}{6}\cot(\pi
t)dt\right]\end{split}$ (118)

In order to derive the $S_{2}(z)$ from the refine expression (111), we should
carefully take the limit $\tau\to 1$. It turns out we can apply the Theorem
5.2 of Felder_2000  as follows.

$\displaystyle\begin{split}\ln S_{2}&=\frac{\pi
i}{2}B_{22}(z|\tau,1)+\sum_{j=0}^{\infty}\ln(1-e^{2\pi
i(z+j\tau)})-\sum_{j=0}^{\infty}\ln(1-e^{2\pi
i(\frac{z}{\tau}-\frac{j+1}{\tau})})\\\ &=\frac{\pi
i}{2}B_{22}(z|\tau,1)-\sum_{l=1}^{\infty}\frac{1}{l}\frac{e^{2\pi
ilz}}{1-e^{2\pi i\tau l}}-\sum_{l=1}^{\infty}\frac{1}{l}\frac{e^{2\pi
il\frac{z-1}{\tau}}}{1-e^{-2\pi il/\tau}}\end{split}$ (119)

By taking $\tau=1+\epsilon$ and expand the expression to the order
$\epsilon^{(0)}$, we can see the divergent piece in the limit $\epsilon\to 0$
will be cancelled and the finite piece will be non-trivial. The result will
exactly match with (118) after explicit integration.

###  A.2 Higher rank of elliptic Gamma

The multiple elliptic Gamma function is defined in the work by Nishizawa
nishizawa2001elliptic  as the generalization to the elliptic Gamma functions
studied in Felder_2000 . Given the the generalized q-Pochhammer symbol
$(x;\underline{q})_{\infty}^{(r)}=\prod_{j_{0},..,j_{r}=0}^{\infty}(1-xq_{0}^{j_{0}}q_{1}^{j_{1}}...q_{r}^{j_{r}})$,
the definition of multiple elliptic Gamma function is

$G_{r}(z|\underline{\tau})=(q_{0}...q_{r}x^{-1};\underline{q})^{(r)}_{\infty}\cdot\left[(x;\underline{q})_{\infty}^{(r)}\right]^{(-1)^{r}}$
(120)

The known elliptic Gamma function $\Gamma(z,\tau,\sigma)$ studied in
Felder_2000  is identically the $G_{1}(z|\tau,\sigma)$, while the $q$-$\theta$
function $\theta(z,\tau)$ is precisely the $G_{0}(z|\tau)$. Thus for
$r\in\mathbb{Z}$, the $G_{r}(z|\underline{\omega})$ construct a family of
multiple elliptic Gamma functions. These functions share many similar
properties as elliptic Gamma functions. Just as the elliptic Gamma function
satisfying the following shift properties

$\Gamma(z+\sigma;\tau,\sigma)=\theta(z;\tau)\Gamma(z;\tau,\sigma),\quad\Gamma(z+\tau;\tau,\sigma)=\theta(z;\sigma)\Gamma(z;\tau,\sigma)$
(121)

the multiple elliptic Gamma functions will satisfy following properties
narukawa2004modular :

$\displaystyle\begin{split}&G_{r}(z+1|\underline{\tau})=G_{r}(z|\underline{\tau}),\qquad
G_{r}(z|\underline{\tau})G(z-\tau_{j}\big{|}\underline{\tau}[j])=1\\\
&G_{r}(-z|-\underline{\tau})G(z|\underline{\tau})=1,\quad
G_{r}(z+\tau_{j}|\underline{\tau})=G_{r}(z|\underline{\tau})G_{r-1}(z|\underline{\tau}^{-}(j))\end{split}$
(122)

##  Appendix B Multiplication formula

The multiplication formula is a useful tool in relating elliptic Gamma
functions with moduli $\tau$ to those with moduli $m\tau$
felder2002multiplication ; Felder_2000 . It is also applied in Jejjala:2022lrm
to understand the holomorphic factorization of superconformal indices in the
lens geometry $L(p,q)$. The multiplication formula of multiple elliptic Gamma
function is based on the multiplication formula of generalized $q$-Pochammer
symbols. Namely, one can decompose the $j_{i}$ according to the equivalence
class modulo a given integer $m$. Then

$(x;\underline{q})_{\infty}^{(r)}=\prod_{a_{i}=0}^{m-1}\prod_{j_{i}=0}^{\infty}(1-xq_{0}^{mj_{0}+a_{0}}...q_{r}^{mj_{r}+a_{r}})=\prod_{a_{i}=0}^{m-1}(xq_{0}^{a_{0}}...q_{r}^{a_{r}},\underline{q^{m}})_{\infty}^{(r)}$
(123)

where $i=0,...,r$. The $G_{r}(z|\underline{\tau})$ can be shown to satisfy

$\displaystyle\begin{split}G_{r}(z|\underline{\tau})=\prod_{\underline{a}=0}^{m-1}G_{r}(z+\underline{a}\cdot\underline{\tau}|m\underline{\tau}+\underline{n})\end{split}$
(124)

where
$\underline{a}\cdot\underline{\tau}=\sum_{i=0}^{r}a_{i}\tau_{i}\,,m\underline{\tau}+\underline{n}=(m\tau_{0}+n_{0}\,,...,\,m\tau_{r}+n_{r})$.
Similar formula for elliptic Gamma function $\Gamma(z,\tau,\sigma)$ was
displayed in felder2002multiplication .

In this work, we are interested in unrefinement limit of the multiplication
formula, by taking $\tau_{i}=\tau\,,n_{i}=n$ for $G_{3}(z|\underline{\tau})$,
as we studied in section 4. The multiplication formula reduces to

$G_{3}(z\,;\tau)=\prod_{\underline{a}=0}^{m-1}G_{3}(z+\left(a_{0}+a_{1}+a_{2}+a_{3}\right)\tau\,;m\tau+n)$
(125)

We then need to count for given integer $c$ subject to the condition

$c=a_{0}+a_{1}+a_{2}+a_{3},\qquad 0\leq a_{i}\leq m-1$ (126)

what is the degeneracy of $c$ for different values, considering the factors
are independent of $a_{i}$. The useful fact is that two variable summation can
be decomposed as

$\sum_{r=0}^{m-1}\sum_{s=0}^{m-1}=\sum_{c=0}^{m-2}\sum_{s=0}^{c}+\sum_{c=m-1}^{2m-2}\sum_{s=c+1-m}^{m-1}=\sum_{c=0}^{m-2}(c+1)+\sum_{c=m-1}^{2m-2}(2m-1-c)$
(127)

We can use this formula to split the summation over $a_{i}$ into two pairwise
groups.

$\displaystyle\begin{split}&\sum_{a_{0}=0}^{m-1}\sum_{a_{1}=0}^{m-1}\sum_{a_{2}=0}^{m-1}\sum_{a_{3}=0}^{m-1}=\left(\sum_{a=0}^{m-2}\sum_{a_{1}=0}^{a}+\sum_{a=m-1}^{2m-2}\sum_{a_{1}=a+1-m}^{m-1}\right)\left(\sum_{b=0}^{m-2}\sum_{a_{3}=0}^{b}+\sum_{b=m-1}^{2m-2}\sum_{a_{3}=b+1-m}^{m-1}\right)\end{split}$
(128)

After summing over $a_{1}$ and $a_{3}$ we can then define $c=a+b$. Then we can
see four different regions of summation of $c$ will appear automatically

$\displaystyle\begin{split}&\text{I}:\quad 0\leq c\leq m-2\\\ &\text{II}:\quad
m-1\leq c\leq 2m-2\\\ &\text{III}:\quad 2m-1\leq c\leq 3m-3\\\
&\text{IV}:\quad 3m-2\leq c\leq 4m-4\end{split}$ (129)

and degeneracy function $d(c)$ in each piece-wise domain will be

$\displaystyle\begin{split}&d_{\text{I}}(c)=\frac{(c+1)(c+2)(c+3)}{6},\\\
&d_{\text{II}}(c)=\frac{2m}{3}(m^{2}+11)+2c(4+c)m-2(2+c)m^{2}-\frac{(c+1)(c+2)(c+3)}{2},\\\
&d_{\text{III}}(c)=\frac{(c+1)(c+2)(c+3)}{2}-\frac{4}{3}(11+12c+3c^{2})m+10(2+c)m^{2}-\frac{22m^{3}}{3}\\\
&d_{\text{IV}}(c)=\frac{(1+c-4m)(2+c-4m)(3+c-4m)}{6}\,.\end{split}$ (130)

For consistency checks, we can find

$\sum_{c=0}^{m-2}d_{\text{I}}(c)+\sum_{c=m-1}^{2m-2}d_{\text{II}}(c)+\sum_{c=2m-1}^{3m-3}d_{\text{III}}(c)+\sum_{c=3m-2}^{4m-4}d_{\text{IV}}(c)\equiv
m^{4}$ (131)

We can then replace each triple elliptic Gamma function on RHS of (125) by the
corresponding phase polynomial $B_{5}$. They will add up to give the phase
polynomial of $G_{3}(z,\tau)$ as $\tau\to-n/m$:

$\displaystyle\begin{split}Q_{5}^{(m,n)}(z;\tau)&=\sum_{c=0}^{m-2}d_{\text{I}}(c)\,B_{5}\left(z+c\tau+\Big{\lfloor}\frac{(c+1)n}{m}\Big{\rfloor}+1;m\tau+n\right)\\\
&+\sum_{c=m-1}^{2m-2}d_{\text{II}}(c)\,B_{5}\left(z+c\tau+\Big{\lfloor}\frac{(c+1)n}{m}\Big{\rfloor}+1\,;m\tau+n\right)\\\
&+\sum_{c=2m-1}^{3m-3}d_{\text{III}}(c)\,B_{5}\left(z+c\tau+\Big{\lfloor}\frac{(c+1)n}{m}\Big{\rfloor}+1\,;m\tau+n\right)\\\
&+\sum_{c=3m-2}^{4m-4}d_{\text{IV}}(c)\,B_{5}\left(z+c\tau+\Big{\lfloor}\frac{(c+1)n}{m}\Big{\rfloor}+1\,;m\tau+n\right)\\\
&=\frac{1}{m}B_{5}(mz+n+1;m\tau+n)+C_{5}(m,n)\end{split}$ (132)

We can then use (130) to compute the extra constants in (132). Simply we can
extract the constant term $K(m,c)$ in the $Q_{5}(z+c\tau+N_{c},m\tau+n)$ and
the ones in $\frac{1}{m}Q_{5}(mz+n,m\tau+n)$. The computation directly gives

$K(m,c)=\frac{30c^{4}-240c^{3}m+660c^{2}m^{2}-720cm^{3}+251m^{4}}{12m^{5}}\left(m-2cn+2m\Big{\lfloor}\frac{(c+1)n}{m}\Big{\rfloor}\right)$
(133)

Then $C_{5}(m,n)$ in the phase (132) is precisely

$\displaystyle\begin{split}C_{5}(m,n)&=\sum_{c=0}^{m-2}K(m,c)d_{\text{I}}(c)+\sum_{c=m-1}^{2m-2}K(m,c)d_{\text{II}}(c)\\\
&+\sum_{c=2m-1}^{3m-3}K(m,c)d_{\text{III}}(c)+\sum_{c=3m-2}^{4m-4}K(m,c)d_{\text{IV}}(c)-\frac{251(1+2n)}{12m}\end{split}$
(134)

where the last rational function comes from the constant term in
$\frac{1}{m}Q_{5}(mz+n;m\tau+n)$.

The summand in (134) includes various structures

$\mathcal{S}_{k,1}^{(i)}(n,m)=\sum_{c=1}^{m-1}c^{k}\left\lfloor\frac{(c-i)n}{m}\right\rfloor,\qquad
k=4,5,6,7$ (135)

Some functions of these structures can be analytically computed by following
the algorithm developed by Carlitz carlitz1975some , for $i=0$. For example,
we can confirm by numerical checks

$\displaystyle\begin{split}\mathcal{S}_{1,1}^{(0)}&=\frac{(m-1)(4mn-3m-2n)}{12}-ms(n,m)\\\
\mathcal{S}_{2,1}^{(0)}&=\frac{m(m-1)(3mn-2m-3n+1)}{12}-m^{2}s(n,m)\end{split}$
(136)

where $s(n,m)$ is the Dedekind sum. However, it is not clear how to give an
analytic formula for general $i$. In (134), there will be constants computed
from the summand $\mathcal{S}_{k,1}^{(i)}(n,m)$ for $i=0,1,2,3$. The analytic
form of these are not clear to our knowledge. But we can always do numerical
checks for each given coprime pair $(m,n)$.

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