#  Casimir Energy and Modularity in Higher-dimensional Conformal Field
Theories

###### Abstract

An important problem in Quantum Field Theory (QFT) is to understand the
structures of observables on spacetime manifolds of nontrivial topology. Such
observables arise naturally when studying physical systems at finite
temperature and/or finite volume and encode subtle properties of the
underlying microscopic theory that are often obscure on the flat spacetime.
Locality of the QFT implies that these observables can be constructed from
more basic building blocks by cutting-and-gluing along a spatial slice, where
a crucial ingredient is the Hilbert space on the spatial manifold. In
Conformal Field Theory (CFT), thanks to the operator-state correspondence, we
have a non-perturbative understanding of the Hilbert space on a spatial
sphere. However it remains a challenge to consider more general spatial
manifolds. Here we study CFTs in spacetime dimensions $d>2$ on the spatial
manifold $T^{2}\times{\mathbb{R}}^{d-3}$ which is one of the simplest
manifolds beyond the spherical topology. We focus on the ground state in this
Hilbert space and analyze universal properties of the ground state energy,
also commonly known as the Casimir energy, which is a nontrivial function of
the complex structure moduli $\tau$ of the torus. The Casimir energy is
subject to constraints from modular invariance on the torus which we spell out
using $PSL(2,{\mathbb{Z}})$ spectral theory. Moreover we derive a simple
universal formula for the Casimir energy in the thin torus limit using the
effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with
exponentially small corrections from worldline instantons. We illustrate our
formula with explicit examples from well-known CFTs including the critical
$O(N)$ model in $d=3$ and holographic CFTs in $d\geq 3$.

###### Contents

  1. 1 Introduction
  2. 2 General Structure of Casimir Energy Density
    1. 2.1 High Temperature Expansion from Effective Field Theory
    2. 2.2 Non-perturbative Corrections from Worldline Instantons
  3. 3 Modular Properties from $PSL(2,{\mathbb{Z}})$ Spectral Theory
    1. 3.1 Review of $PSL(2,{\mathbb{Z}})$ Spectral Theory
    2. 3.2 Spectral Decomposition of Casimir Energy Density
  4. 4 Casimir Energy Density in Free Scalar Theories
  5. 5 Casimir Energy Density in Interacting CFTs
    1. 5.1 Critical $O(N)$ Scalar CFT
      1. 5.1.1 Casimir Energy from Gap Equation
      2. 5.1.2 High Temperature Expansion and Worldline Instantons
    2. 5.2 Holographic CFTs with Einstein Gravity Duals
  6. 6 Discussions
  7. A Regularization of Lattice Sum on Torus

##  1 Introduction

Conformal Field Theories (CFTs) provide a unifying and powerful approach to a
large class of strongly coupled Quantum Field Theories (QFTs) that arise in
both high energy and condensed matter physics. Over the recent years, there
has been tremendous development in elucidating the structure of CFTs by the
conformal bootstrap method [1, 2, 3], which explores general constraints on
the CFT observables by basic principles such as unitarity and associativity of
the operator-product-expansion (OPE). In spacetime dimensions $d>2$, most of
the works focus on CFT observables defined on flat spacetime, namely the
spectrum of local operators and their OPE coefficients. These CFT data
determine completely the local correlation functions on the flat spacetime
${\mathcal{M}}_{d}={\mathbb{R}}^{d-1,1}$, which are central objects in QFT.
However there is much more to QFT than the flat spacetime. In particular,
local QFT is expected to be defined on general spacetime manifolds.

Considerations of QFT on a spacetime manifold of nontrivial topology open the
door to a profusion of new observables. For example the correlation functions
of local operators (even the partition function itself) may now depend
nontrivially on topology and finer geometric data of the manifold. Such
dependence detects global topological features of QFT that are crucial to
distinguish theories are otherwise identical at the level of local correlation
functions [4, 5]. Of course QFT on a general spacetime manifold contains far
more information than such topological data. The challenge is to develop
methods to extract QFT observables on nontrivial manifolds, understand their
general properties, and uncover underlying structures. On the surface this
seems an intractable task simply because of the sheer complexity of the
geometry of the manifolds. Nonetheless, the locality principle dictates that
QFT observables must obey consistency conditions known as cutting-and-gluing
axioms [6, 7], which implies that the study of QFT on general manifolds can be
reduced to more basic building blocks and their gluing. It also points to
intricate relations among seemingly distinct observables on different
manifolds. Unsurprisingly, such questions are most well-formulated for CFTs.
Indeed the conformal bootstrap equations for local correlation functions
amount to the cutting-and-gluing consistencies on the flat spacetime. Here we
decompose a CFT observable (e.g. a correlation function of local operators)
into two pieces by cutting along a sphere $S^{d-1}$, and the basic building
blocks consist of the local operator spectrum which specifies the Hilbert
space of states on $S^{d-1}$ via the state-operator correspondence and the OPE
coefficients that determine the wavefunction associated to each piece from the
decomposition. To pursue this axiomatic approach for CFTs on $d>2$ spacetime
manifolds which in general do not admit an $S^{d-1}$ spatial slice, we will
need to understand the Hilbert space on non-spherical spatial manifold
${\mathcal{M}}_{d-1}$, which is a main motivation for this work.

In this paper, we initiate a systematic study of $d>2$ CFTs on spatial
manifold ${\mathcal{M}}_{d-1}=T^{2}\times{\mathbb{R}}^{d-3}$, which is one of
the simplest non-spherical manifolds with nontrivial topology and nontrivial
geometric moduli (coming from the $T^{2}$ factor). We write the metric on
$T^{2}$ as

$\displaystyle ds^{2}_{T^{2}}=L_{1}^{2}|dt+\tau
dx|^{2}\,,\quad\tau={L_{2}\over L_{1}}e^{i\theta}\,,$ (1.1)

where $L_{1}$ and $L_{2}$ are the lengths of two cycles with a tilt angle
$\theta\in(0,\pi)$ and $t,x\in[0,1)$ are the periodic coordinates. In other
words, the area of $T^{2}$ is $A=L_{1}L_{2}\sin\theta$ and
$\tau=\tau_{1}+i\tau_{2}$ denotes the usual shape moduli (complex structure
moduli) of the torus which take values on the upper half-plane ${\mathbb{H}}$
(i.e. $\tau_{2}>0$).

The immediate task is to investigate the Hilbert space
${\mathcal{H}}_{T^{2}\times{\mathbb{R}}^{d-3}}$ on this spatial manifold, for
which little is known since there is no correspondence with local CFT
operators (see comments on the $d=3$ case in [8]). Here we make progress by
focusing on the ground state of
${\mathcal{H}}_{T^{2}\times{\mathbb{R}}^{d-3}}$ in an ambient spacetime
${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-3,1}$ and exploring general
properties of the ground state energy, also known as the Casimir energy,
measured with respect to the Hamiltonian

$\displaystyle
H_{T^{2}\times{\mathbb{R}}^{d-3}}=\int_{T^{2}\times{\mathbb{R}}^{d-3}}T_{00}\,.$
(1.2)

The Casimir energy is extensive in the noncompact ${\mathbb{R}}^{d-3}$
directions, whose energy density we denote by $E_{\rm vac}$ takes the
following form

$\displaystyle E_{\rm vac}=-{{\mathcal{E}}(\tau)\over A^{d-2\over 2}}\,,$
(1.3)

where the dependence on the torus area $A$ is fixed by dimensional analysis
and the dimensionless coefficient ${\mathcal{E}}(\tau)$ will be the central
object in this paper that contains nontrivial dependence of the shape moduli
$\tau$ of the torus. We will refer to ${\mathcal{E}}(\tau)$ as the
(dimensionless) Casimir energy density. We emphasize that $E_{\rm vac}$ has a
counterterm ambiguity that is fully-extensive on
$T^{2}\times{\mathbb{R}}^{d-3}$ and shifts $E_{\rm vac}\to E_{\rm
vac}+A\Lambda^{d}$ where $\Lambda$ is a UV cutoff scale. This comes from the
cosmological constant counterterm on $T^{2}\times{\mathbb{R}}^{d-3,1}$. There
are no other counterterm ambiguities since $T^{2}$ is flat. Consequently,
${\mathcal{E}}(\tau)$ is unambiguous and well-defined.

We note the following immediate properties of ${\mathcal{E}}(\tau)$. Firstly,
${\mathcal{E}}(\tau)$ is a real function of the complex moduli $\tau$ since
$H_{T^{2}\times{\mathbb{R}}^{d-2}}$ is Hermitian. Secondly, for bosonic CFTs,
${\mathcal{E}}(\tau)$ is modular invariant under $PSL(2,{\mathbb{Z}})$
transformations of $\tau$,

$\displaystyle{\mathcal{E}}(\tau)={\mathcal{E}}({\gamma}\tau)\,,\quad{\gamma}\tau={a\tau+b\over
c\tau+d}\,,$ (1.4)

with $a,b,c,d\in{\mathbb{Z}}$ and $ad-bc=1$. This follows from the
orientation-preserving large diffeomorphisms on $T^{2}$ and is also a general
property of the Hamiltonian $H_{T^{2}\times{\mathbb{R}}^{d-2}}$ and its entire
energy spectrum.111For fermionic CFTs, to define the theory on $T^{2}$, a
choice of spin structure must be specified. This breaks the
$PSL(2,{\mathbb{Z}})$ modular invariance to congruence subgroups. We study the
toroidal Casimir energy density of fermionic CFTs in a subsequent publication
[9]. Thirdly, if the CFT is parity-invariant then

$\displaystyle{\mathcal{E}}(\tau)={\mathcal{E}}(-\bar{\tau})\,,$ (1.5)

which is a consequence of the orientation-reversing large diffeomorphism
$(x_{1},x_{2})\to(x_{1},-x_{2})$ on $T^{2}$.

To derive further universal constraints on ${\mathcal{E}}(\tau)$, we consider
the Euclidean CFT on $T^{2}\times{\mathbb{R}}^{d-2}$ (see previous works [10,
11]). We first take one of ${\mathbb{R}}^{d-2}$ directions to be the Euclidean
time, then the partition function is clearly determined by the Casimir energy.
Alternatively, choosing the cycle $S^{1}_{\beta}$ on $T^{2}$ of length
${\beta}=L_{1}$ to be the Euclidean time, the partition function becomes a
trace over the Hilbert space on the base $S^{1}\times{\mathbb{R}}^{d-2}$. This
second perspective enables us to infer the structure of ${\mathcal{E}}(\tau)$
in the limit $\tau\to i\infty$ where $L_{1}$ is small compared to $L_{2}$
using the effective field theory (EFT) of the CFT from Kaluza-Klein (KK)
reduction on $S^{1}_{\beta}$. A main result is the following universal
behavior for the Casimir energy density,222Despite the funny looking powers in
$\tau_{2}$ (which come from the rescaling (1.3)), the physical interpretations
of the perturbative terms are quite simple: $c_{1}$ controls the extensive
dependence of the CFT Casimir energy on the large circle of size $L_{2}$ while
$c_{2}$ controls the sub-extensive dependence on $L_{2}$.

$\displaystyle\lim_{\tau\to i\infty}{\mathcal{E}}(\tau)=c_{1}\tau_{2}^{d\over
2}+c_{2}\tau_{2}^{1-{d\over 2}}+{\mathcal{O}}\left(\tau_{2}^{\alpha}e^{-2\pi
nM\tau_{2}}e^{2\pi inQ\tau_{1}}\right)\,,$ (1.6)

where $n>0$ and $Q$ are integers and $c_{1},c_{2},{\alpha},M$ are theory-
dependent constants. Namely the cusp behavior of ${\mathcal{E}}(\tau)$ is
controlled by two perturbative terms in $\tau_{2}$ and a tower of non-
perturbative (exponentially small) terms which may also come with perturbative
corrections. As we explain in Section 2.1, the constants $c_{1},c_{2}$ are
captured by the EFT from integrating out massive KK modes and the absence of
other perturbative terms follows from a simple EFT argument. On the other
hand, the non-perturbative terms in (1.6) come from worldline instantons from
massive KK particles with (rescaled) mass $M$ and KK charge $Q$ going
$n$-times along the $S^{1}$ on the base of the KK reduction.

Moreover we will argue that the leading perturbative coefficient $c_{1}\geq 0$
and is strictly positive unless the theory is topological, thus providing a
measure of local degrees of freedom in the $d$-dimensional CFT. Similarly, the
second perturbative coefficient $c_{2}\geq 0$ is a measure of the gapless
degrees of freedom in the $d-1$-dimensional EFT from KK reduction, and
$c_{2}=0$ if the EFT is gapped.

The perturbative coefficients $c_{1},c_{2}$ in (1.6) are closely related to
CFT observables at finite temperature (in $d$ and $d-1$ dimensions
respectively). Such observables are of great interest in both condensed matter
and high energy literature because they encode finite temperature properties
of quantum critical points [12, 13, 14] and via the AdS/CFT correspondence
provide a CFT description of black holes [15, 16, 17, 18]. The relevant
spacetime manifold here is
${\mathcal{M}}_{d}=S^{1}_{\beta}\times{\mathbb{R}}^{d-1}$ for the
$d$-dimensional CFT, and the temperature is related to the inverse size of the
circle $T={1\over{\beta}}$.333The thermal background
${\mathcal{M}}_{d}=S^{1}_{\beta}\times{\mathbb{R}}^{d-1}$ is one of the
simplest manifolds that is not conformally flat. Nonetheless all known
observables on ${\mathcal{M}}_{d}=S^{1}_{\beta}\times{\mathbb{R}}^{d-1}$ are
still determined by the flat space CFT data [19]. The basic observables here
are the one-point functions of conformal primaries which determine higher-
point functions by OPE. In contrast to the flat spacetime, the one-point
function on the thermal background is in general non-vanishing and determined
by the residual conformal symmetry up to an overall constant [19]. A
distinguished case is the one-point function of the stress energy tensor

$\displaystyle\langle
T_{tt}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}=(1-d)\langle
T_{ii}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}={(d-1)b_{T}\over
d{\beta}^{d}}\,,\quad\langle
T_{ti}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}=0\,,$ (1.7)

where $t$ labels the $S^{1}_{\beta}$ direction and $i$ labels the spatial
${\mathbb{R}}^{d-1}$ directions. The same coefficient $b_{T}$ determines the
free energy density

$\displaystyle f({\beta})={1\over{\beta}V_{d-1}}\log
Z_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}={b_{T}\over d{\beta}^{d}}\,,$ (1.8)

where $V_{d-1}$ regulates the infinite spatial volume. This follows from (1.7)
and the relation

$\displaystyle\langle
T_{tt}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}={1\over
V_{d-1}}\partial_{\beta}\log Z_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}\,.$
(1.9)

The thermal background can be thought of a limit of
${\mathcal{M}}_{d}=S^{1}_{\beta}\times S^{d-1}_{R}$ with
${R\over{\beta}}\to\infty$. The latter comes naturally from the radial
quantization of the CFT on the sphere $S^{d-1}_{R}$ of radius $R$. The
corresponding partition function counts states in the Hilbert space
${\mathcal{H}}_{S^{d-1}}$ graded by their energy measured by the radial
Hamiltonian (equivalently local operators graded by their scaling dimensions
$\Delta$),

$\displaystyle Z_{S^{1}_{\beta}\times S^{d-1}_{R}}\equiv{\rm
Tr\,}_{{\mathcal{H}}_{S^{d-1}}}e^{-{{\beta}\over
R}\Delta}\quad\xrightarrow[]{{R\over{\beta}}\to\infty}\quad
Z_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}=\exp\left(-{{\rm
vol}(S_{R}^{d-1})\over d{\beta}^{d-1}}b_{T}\right)\,,$ (1.10)

which is related to the thermal partition function on
$S^{1}_{\beta}\times{\mathbb{R}}^{d-1}$ in the ${R\over{\beta}}\to\infty$
limit where the heavy operator contributions are no longer suppressed.
Therefore $b_{T}$ measures the asymptotic density of high dimension local
operators, akin to the role of the conformal central charge $c_{2d}$ in $d=2$
CFT,444We emphasize that in $d>2$, the coefficient $b_{T}$ is in general a
nontrivial function on the conformal manifold of the CFT. and we see that the
finite temperature one-point function $\langle
T_{\mu\nu}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}$ is determined by
the flat space operator data.555Similarly, the one-point function of a general
local operator
$\langle{\mathcal{O}}\rangle_{S^{1}_{\beta}\times{\mathbb{R}}^{d-1}}$ is
related to the ${R\over{\beta}}\to\infty$ limit of
$\langle{\mathcal{O}}(x)\rangle_{S^{1}_{\beta}\times S^{d-1}_{R}}$,
$\displaystyle\langle{\mathcal{O}}(x)\rangle_{S^{1}_{\beta}\times
S^{d-1}_{R}}=\sum_{\phi}e^{-{{\beta}\over
R}\Delta_{\phi}}\langle\phi|{\mathcal{O}}(x)|\phi\rangle\,,$ (1.11) which
depends on the OPE coefficients with general CFT operators $\phi$.

Coming back to the coefficients $c_{1}$ and $c_{2}$ in our universal formula
for the Casimir energy (1.6), as we explain in Section 2.1, $c_{1}$ is
determined by the thermal one-point function of the CFT stress energy
tensor,666In the notation of [19], $c_{1}^{\rm here}=-f^{\rm there}$.

$\displaystyle c_{1}=-{b_{T}\over d}\,.$ (1.12)

This is natural because the ${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-2}$
background we consider is a generalization of the thermal background by
including another compact direction which becomes large in the limit of (1.6).

If the $d$-dimensional CFT upon reduction on $S^{1}$ contains a gapless sector
described by a $(d-1)$-dimensional CFT with stress energy tensor
$\widehat{T}$, $c_{2}$ is determined by the corresponding thermal one-point
function in $d-1$ dimensions,

$\displaystyle c_{2}=-{b_{\widehat{T}}\over d-1}\,,$ (1.13)

which no longer has an obvious interpretation in terms of flat space operator
data in $d$-dimensions.

The rest of the paper is organized as follows. In Section 2, we carry out the
effective field theory analysis to derive the universal behavior of the
Casimir energy density ${\mathcal{E}}(\tau)$ in the thin torus limit. In
Section 3, we spell out the modular properties of ${\mathcal{E}}(\tau)$ using
$PSL(2,{\mathbb{Z}})$ spectral theory. We then illustrate our formula (1.6)
with concrete CFT examples, starting from the free scalar theory in $d>2$ in
Section 4 where the Casimir energy ${\mathcal{E}}(\tau)$ is given by familiar
real analytic Eisenstein series. For interacting CFTs, ${\mathcal{E}}(\tau)$
is much harder to compute exactly. In Section 5.1, we study the $d=3$ critical
$O(N)$ CFT in the large $N$ limit on $T^{2}\times\mathbb{R}$ extending
previous works. We derive new formulae for ${\mathcal{E}}(\tau)$ in the $O(N)$
CFT in terms of a generalized Eisenstein series and determine the behavior in
the thin torus limit which matches onto our formula (1.6) in an interesting
way. In large $N$ CFTs with a semi-classical holographic dual, the ground
state on $T^{2}\times{\mathbb{R}}^{d-3,1}$ is dual to certain AdS soliton in
the bulk which depends on the moduli $\tau$ of the boundary $T^{2}$. The
Casimir energy density ${\mathcal{E}}(\tau)$ is then determined by the bulk
action evaluated on the AdS soliton solution which we discuss in Section 5.2.
We end with a discussion of open questions and future directions in Section 6.

##  2 General Structure of Casimir Energy Density

###  2.1 High Temperature Expansion from Effective Field Theory

Here we derive the universal formula (1.6) for the Casimir energy density
${\mathcal{E}}(\tau)$ in the thin torus limit $\tau\to i\infty$. As already
mentioned in the introduction, this will be achieved by analyzing the
effective field theory (EFT) from KK reduction on the small circle in this
limit [20, 21].

We work with the CFT on $T^{2}\times{\mathbb{R}}^{d-2}$ and start by putting
the metric (1.1) into the KK form

$\displaystyle
ds^{2}=\beta^{2}(dt+a)^{2}+L^{2}dx^{2}+dy^{2}_{\mathbb{R}^{d-2}}\,,$ (2.1)

where $a=\mu dx$ is KK photon (graviphoton) and

$\displaystyle{\beta}=L_{1}\,,\quad\mu=\tau_{1}\,,\quad
L=L_{2}\sin\theta=\tau_{2}{\beta}\,.$ (2.2)

Treating the $t$ direction as Euclidean time, we can interpret the CFT
partition function on $T^{2}\times{\mathbb{R}}^{d-2}$ as a thermal partition
function over the Hilbert space on the base manifold
$S^{1}_{L}\times{\mathbb{R}}^{d-2}$,

$\displaystyle Z(T^{2}\times{\mathbb{R}}^{d-2})={\rm
Tr\,}_{{\mathcal{H}}_{S^{1}_{L}\times{\mathbb{R}}^{d-2}}}\left[e^{-{\beta}(H_{S^{1}_{L}\times{\mathbb{R}}^{d-2}}+i\mu
P_{S^{1}_{L}\times{\mathbb{R}}^{d-2}})}\right]\,,$ (2.3)

where $P_{S^{1}_{L}\times{\mathbb{R}}^{d-2}}$ is the generator for translation
along the base $S^{1}_{L}$ (the charges are quantized as ${n\over L}$ for
$n\in{\mathbb{Z}}$) and $\mu$ the corresponding chemical potential.

Since we can equivalently pick one of the non-compact directions in
${\mathbb{R}}^{d-2}$ to be Euclidean time, in which case the partition
function is dominated by the ground state contribution, we arrive at the
following relation

$\displaystyle e^{-V_{d-2}E_{\rm
vac}(L_{1},L_{2},\theta)}=e^{-{\beta}F_{S^{1}_{L}\times{\mathbb{R}}^{d-2}}(L,{\beta},\mu)}$
(2.4)

between the ground state energy on $T^{2}\times{\mathbb{R}}^{d-3}$ and the
free energy on $S^{1}_{L}\times{\mathbb{R}}^{d-2}$.

We now focus on the limit ${\beta}\ll L$ with $\mu$ fixed (equivalently
$\tau_{2}\to\infty$ with $\tau_{1}$ fixed). From the extensivity of the free
energy and (1.3) with $A=L{\beta}$ (the area of $T^{2}$), we obtain a relation

$\displaystyle\lim_{\tau\to i\infty}{\mathcal{E}}(\tau)=-A^{d\over
2}f({\beta},\mu)\,,$ (2.5)

between the Casimir energy density ${\mathcal{E}}(\tau)$ on
$T^{2}\times{\mathbb{R}}^{d-3}$ and the free energy density $f({\beta},\mu)$
on $S^{1}_{L}\times{\mathbb{R}}^{d-2}$. The latter is naturally captured by
the EFT from KK reduction of the CFT on $S^{1}_{\beta}$ in (2.1). Below we
will deduce universal properties of $f({\beta},\mu)$ from general EFT
considerations.

On general grounds, the $d-1$ dimensional Wilsonian EFT from the KK reduction
of a $d$ dimensional CFT on $S^{1}_{\beta}$ is obtained from integrating out
KK modes of mass $m_{\rm KK}\sim{1\over{\beta}}$. At infinite volume (on the
base manifold), it is well-known that this leads to an effective action
$S_{\rm EFT}$ that consists of local analytic functionals in the background
fields which capture the induced contact interactions. Here the relevant
background fields consist of the metric on the base manifold $g_{ij}$ and the
KK photon gauge field $a_{i}$. The possible local functionals are constrained
by invariance under background diffeomorphism and gauge transformations
(together they correspond to diffeomorphism transformations in $d$
dimensions). These functionals take the form

$\displaystyle S_{\rm EFT}\ni{\beta}^{2n+2m-d+1}\int
d^{d-1}x\sqrt{g}R[g]^{n}F[a]^{m}\,,$ (2.6)

for $n,m\in{\mathbb{Z}}_{\geq 0}$ where $R[g]$ denotes schematically the
Riemann curvature on the base manifold and $F[a]$ the field strength of the KK
photon. They constitute a derivative expansion of $S_{\rm EFT}$ where the
${\beta}$ dependence is fixed by dimensional analysis, and so equivalently
describes a high temperature (small ${\beta}$) expansion of $S_{\rm EFT}$.

Now the background we are interested in (2.1) has vanishing curvature for both
the metric $g_{ij}$ and the gauge field $a_{i}$ on the base. Consequently, the
only non-vanishing local functional among (2.6) has $n=m=0$, which corresponds
to the cosmological constant term generated by integrating out massive KK
modes. Therefore, we have

$\displaystyle S_{\rm EFT}=-{c_{1}\over{\beta}^{d-1}}\int
d^{d-1}\sqrt{g}+S_{\rm gapless}\,,$ (2.7)

where $S_{\rm gapless}$ is the action that governs the dynamics of the gapless
modes that survive the KK reduction. Note that quantum effects at finite
temperature typically generate a non-negative thermal mass squared $m^{2}_{\rm
thermal}$ for the KK zero mode. While there have been recent counter-examples
to this statement (see [22, 23]), here we assume this is true. The remaining
unlifted KK zero modes are then described by a $d-1$ dimensional CFT.

An important caveat here is that if the $d$ dimensional CFT has gravitational
anomalies, such anomalies must be matched by the EFT. In this case, local
functionals of the Chern-Simons type can appear in $S_{\rm EFT}$, which are
generally not invariant under background gauge transformations777Such Chern-
Simons type terms are either non-invariant under small gauge transformations
because of field-dependent couplings or non-invariant under large gauge
transformations because of ill-quantized constant couplings. and the non-
invariance precisely reproduces the $d$ dimensional anomalous variation upon
reduction on $S^{1}_{\beta}$. Nevertheless, the flatness of our background
ensures that they do not contribute to $S_{\rm EFT}$.888This is to be
contrasted with the cases where the base manifold of the KK reduction has
nontrivial curvature (e.g. $S^{d-1}$ or $S^{1}\times S^{d-2}$ for $d\geq 4$).

We emphasize that the simple form of the $S_{\rm EFT}$ in (2.7) completely
determines the EFT partition function, correspondingly the free energy density
$f({\beta},\mu)$, in the high temperature limit to all orders in ${\beta}$ ,

$\displaystyle\lim_{\beta\to
0}\,f(\beta,\mu)={-{c_{1}\over\beta^{d}}}+{1\over\beta}\left(-{{c_{2}\over
L^{d-1}}}\right)+f_{\rm np}(\beta,\mu)\,,$ (2.8)

up to terms that are non-perturbative in $\beta$ (i.e. exponentially small as
$\beta\to 0$) which we package together in $f_{\rm np}(\beta,\mu)$ and will
come back to shortly. The first term on the RHS of (2.8) comes obviously from
the cosmological constant term in (2.7). Comparing with (1.8) in the $\beta\ll
L$ limit, we find

$\displaystyle c_{1}=-{b_{T}\over d}$ (2.9)

is determined by one-point function of the stress tensor $T_{\mu\nu}$. The
second term on the RHS of (2.8) comes from the thermal free energy of the
$d-1$ dimensional CFT (that survives the KK reduction) on
$S^{1}_{L}\times{\mathbb{R}}^{d-2}$ (i.e. (1.8) with $\beta$ replaced by $L$
and $d$ replaced by $d-1$) and the coefficient $c_{2}$ is determined by the
one-point function of the stress tensor $\widehat{T}_{ij}$ for this $d-1$
dimensional CFT,

$\displaystyle c_{2}=-{b_{\widehat{T}}\over d-1}\,.$ (2.10)

###  2.2 Non-perturbative Corrections from Worldline Instantons

The high temperature EFT does not completely determine the free energy of the
KK-reduced theory on the base manifold away from the infinite volume limit.
The corrections are denoted by $f_{\rm np}({\beta},\mu)$ in (2.8) which are
exponentially suppressed (i.e. behave as $e^{-L/{\beta}}$). Such non-
perturbative contributions are naturally associated with worldline instantons,
namely virtual massive particles propagating around the compact circle
$S^{1}_{L}$ on the base manifold. They are most easily seen from the worldline
representation for the one-loop effective action of a free massive particle
[24, 25],999Here for simplicity we focus on scalar particles. Generalizations
to spinning particles can be found in [24] by introducing auxiliary variables
along the worldline.

$\displaystyle S_{\tilde{M},Q}=\int_{0}^{\infty}{dT\over
T}e^{-{\tilde{M}^{2}}T}\int_{x^{j}(T)=x^{j}(0)}Dx^{j}(\varphi)\exp\left[-\int_{0}^{T}d\varphi\left({1\over
4}(\dot{x}^{j})^{2}+iQa_{j}\dot{x}^{j}\right)\right]\,,$ (2.11)

where $x^{j}(\varphi)$ parametrizes the worldline of a scalar particle of mass
$\tilde{M}\sim{1\over{\beta}}$ and KK-charge $Q\in{\mathbb{Z}}$ and $T$ is the
usual Schwinger parameter. For nonzero $Q$, the particle couples to the
background KK-photon $a_{i}$. The saddle point equation for the worldline path
integral over $x^{j}(\varphi)$ is simply $\ddot{x}^{i}=0$ since the $a_{i}$ is
flat. The Euclidean solutions compatible with the periodicity condition are

$\displaystyle x^{i}(\varphi)=({n\varphi\over T},0,\dots,0)\,,$ (2.12)

where the only nonzero entry is in the $S^{1}_{L}$ direction and this solution
winds $n$ times along $S^{1}_{L}$. The one-loop effective action (2.11) is
dominated by these saddles for large mass $\tilde{M}L\sim{L/{\beta}}\gg 1$,

$\displaystyle\lim_{\tilde{M}L\gg 1}S_{M,Q}=e^{-n\tilde{M}L}e^{2\pi
inQ\mu}\,,$ (2.13)

on top of which there are perturbative contributions in $1/\tilde{M}$ coming
from the fluctuations.

The $n=0$ case of (2.12) is the trivial saddle which is insensitive to compact
directions of the spacetime manifold. The fluctuations thereof precisely
generate the perturbative terms in the effective action that are power law in
$\beta$. The case $n\neq 0$ of (2.12) corresponds to nontrivial worldline
instantons and (2.13) gives the instanton action which depends on the chemical
potential $\mu$ if the particle carries a nonzero KK charge $Q$. For
convenience we define the dimensionless mass $M$ by $\tilde{M}={2\pi
M\over{\beta}}$.

The most general contributions to $f_{\rm np}({\beta},\mu)$ come from multiple
worldline instantons, each labeled by its mass $M$, KK charge $Q$ and winding
number $n$. Moreover these worldline instantons may interact with one another,
producing corrections beyond the dilute instanton gas approximation [26].

The non-perturbative contributions to the free energy takes the following
form, as a sum over multi-instanton configurations

$\displaystyle f_{\rm
np}(\beta,\mu)=-\sum_{j=1}^{\infty}\sum_{\vec{M},\vec{Q}}e^{-{2\pi
L\over{\beta}}\sum_{i=1}^{j}M_{i}}e^{2\pi
i\mu\sum_{i=1}^{j}Q_{i}}g_{j,\vec{M},\vec{Q}}\left({{\beta}\over L}\right)\,.$
(2.14)

Here $j$ counts the number of worldline instantons each with unit winding
number. The vectors $\vec{M},\vec{Q}$ each has $j$ entries and they specify
the masses and KK charges of individual instantons in this
configuration.101010In this parametrization, a worldline instanton of charge
$Q$ and winding number $n$ is represented as $n$ identical worldline
instantons of unit winding number and the same charge $Q$. The fluctuations
around each $j$-instanton configuration and the integration over the moduli
space of such configurations are captured by the factor
$g_{j,\vec{M},\vec{Q}}\left({{\beta}\over L}\right)$ which contains power-law
terms in ${{\beta}\over L}$.

For free CFTs compactified on $S^{1}_{\beta}$, the worldline instantons coming
from massive KK modes do not interact and $g_{j,\vec{M},\vec{Q}}$ can be
computed explicitly from (2.11). See further discussions in Section 4 for the
free scalar CFT. In general CFT upon circle reduction, there are nontrivial
interactions among worldline instantons which are not taken into account by
(2.11). In Section 5.1, we will find explicit expressions of
$g_{j,\vec{M},\vec{Q}}$ for the worldline instantons in the interacting $O(N)$
CFT.

Using the relation between the free energy and the Casimir energy (2.5) (and
also (2.2)), we conclude the EFT analysis with the following universal
behavior for the Casimir energy density on $T^{2}\times{\mathbb{R}}^{d-3,1}$,

$\displaystyle\lim_{\tau\to i\infty}{\mathcal{E}}(\tau)=c_{1}\tau_{2}^{d\over
2}+c_{2}\tau_{2}^{1-{d\over
2}}+\sum_{j=1}^{\infty}\sum_{\vec{M},\vec{Q}}e^{-{2\pi\tau_{2}}\sum_{i=1}^{j}M_{i}}e^{2\pi
i\tau_{1}\sum_{i=1}^{j}Q_{i}}g_{j,\vec{M},\vec{Q}}(\tau_{2})\,,$ (2.15)

which contains two perturbative terms whose coefficients are related to finite
temperature CFT data by (2.9) and (2.10), and a tower of nonperturbative
contributions from the worldline instanton gas.

##  3 Modular Properties from $PSL(2,{\mathbb{Z}})$ Spectral Theory

As already explained in the introduction, the toroidal Casimir energy density
${\mathcal{E}}(\tau)$ is a modular (invariant) function of the complex moduli
$\tau$. Here we want to understand what types of non-holomorphic modular
functions are relevant for this CFT observable.

Unlike the holomorphic case where the space of modular functions is highly
constrained (to be rational functions of the modular invariant $j(\tau)$), the
space of non-holomorphic modular functions is much bigger. This is necessary
to accommodate the vast zoo of modular functions of different properties that
arise from QFT observables which are mostly non-holomorphic. Well-studied
examples of such modular functions include the torus partition functions of
$d=2$ CFTs and correlation functions in the $d=4$ ${\mathcal{N}}=4$ super-
Yang-Mills (SYM) CFT where $\tau$ is the complexified Yang-Mills coupling. An
important question there is to understand the general properties of the
relevant modular functions as dictated by the underlying CFT. Impressive
progress is made in the recent years on this question for both the $d=2$ CFTs
[27, 28] and the $d=4$ SYM [29] using the powerful tool of
$PSL(2,{\mathbb{Z}})$ spectral theory (see also previous work [30]). Here we
initiate to tackle the same question in yet another different context, that is
the toroidal Casimir energy of general $d>2$ CFTs.

###  3.1 Review of $PSL(2,{\mathbb{Z}})$ Spectral Theory

Let us start with a quick review of the necessary ingredients for the
$PSL(2,{\mathbb{Z}})$ spectral theory. See for example [31, 32] for more
details. Here we follow the conventions of mathematics literature to denote
the real and imaginary parts of the complex moduli as $\tau=x+iy$ (which we
haven written as $\tau=\tau_{1}+i\tau_{2}$ in the rest of the paper). The
standard $PSL(2,{\mathbb{Z}})$ fundamental domain is denoted by
${\mathcal{F}}=PSL(2,{\mathbb{Z}})\backslash{\mathbb{H}}$ with the hyperbolic
metric

$\displaystyle ds^{2}={dx^{2}+dy^{2}\over y^{2}}\,,$ (3.1)

and hyperbolic volume

$\displaystyle{\rm vol}({\mathcal{F}})={\pi\over 3}\,.$ (3.2)

For modular invariant square integrable functions $f(\tau),g(\tau)\in
L^{2}({\mathcal{F}})$, there is a natural inner product known as the Petersson
inner product defined by the following integral over ${\mathcal{F}}$,

$\displaystyle\langle f,g\rangle\equiv\int_{\mathcal{F}}{d^{2}\tau\over
y^{2}}f(\tau)\overline{g(\tau)}\,,$ (3.3)

Square integrability here simply means finite $\langle f,f\rangle$. The
hyperbolic Laplacian on ${\mathcal{F}}$ defined by

$\displaystyle\Delta\equiv-y^{2}(\partial_{x}^{2}+\partial_{y}^{2})\,,$ (3.4)

is self-adjoint with respect to (3.3) on $L^{2}({\mathcal{F}})$. Consequently,
any modular invariant function $f\in L^{2}({\mathcal{F}})$ admits a unique
orthogonal decomposition into eigenfunctions with respect to $\Delta$,

$\displaystyle f(\tau)=\sum_{j=0}^{\infty}\nu_{j}(\tau){\langle
f,\nu_{j}\rangle\over\langle\nu_{j},\nu_{j}\rangle}+{1\over 4\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds\langle f,E_{s}\rangle E_{s}(\tau)\,,$ (3.5)

known as the Roelcke-Selberg spectral decomposition. The eigenfunctions here
consists of a continuous family of non-holomorphic Eisenstein series
$E_{s}(\tau)$ with $s={1\over 2}+i{\mathbb{R}}$, which satisfies

$\displaystyle\Delta E_{s}=s(1-s)E_{s}\,,$ (3.6)

and a discrete family called the cusp Maass forms $\nu_{i}(\tau)$ satisfying

$\displaystyle\Delta\nu_{j}=\lambda_{j}\nu_{j}\,,\quad\lambda_{j}={1\over
4}+R_{j}^{2}\,,$ (3.7)

where $R_{j}$ are a set of sporadic positive real numbers.

Let us review some properties of these modular eigenfunctions below. The
Eisenstein series has the following explicit definition, in the form of a
Fourier decomposition,111111Note that the Eisenstein series is not in
$L^{2}({\mathcal{F}})$. Nonetheless when smeared by a function $f(t)\in
L^{2}({\mathbb{R}})$, $\int_{\mathbb{R}}dt\,f(t)E_{1/2+it}(\tau)$ defines
modular functions in $L^{2}({\mathcal{F}})$.

$\displaystyle E_{s}(\tau)=\varphi_{s}(y)+\sum_{j=1}^{\infty}4\cos(2\pi
jx){\sigma_{2s-1}(j)\over j^{s-1/2}\Lambda(s)}\sqrt{y}K_{s-{1\over 2}}(2\pi
jy)\,,\quad\varphi_{s}(y)=y^{s}+{\Lambda(1-s)\over\Lambda(s)}y^{1-s}\,,$ (3.8)

where $\sigma_{m}(n)=\sum_{d|n}d^{m}$ is the divisor function and $\Lambda(s)$
is the completed Riemann zeta function defined below,

$\displaystyle\Lambda(s)\equiv\pi^{-s}\Gamma(s)\zeta(2s)\,.$ (3.9)

It satisfies the following reflection relation,

$\displaystyle\Lambda(s)=\Lambda\left({1\over 2}-s\right)\,,$ (3.10)

and is meromorphic for $s\in{\mathbb{C}}$ with simple poles at $s=0,{1\over
2}$ and residues

$\displaystyle{\rm Res}_{s=0}\Lambda(s)=-{\rm Res}_{s={1\over
2}}\Lambda(s)=-{1\over 2}\,,$ (3.11)

$\Lambda(s)$ has zeroes at $s={\rho\over 2}$ for nontrivial zeros $\rho$ of
the Riemann zeta function (with ${\rm Re}\,\rho={1\over 2}$ according to the
Riemann hypothesis).

Correspondingly, the Eisenstein series $E_{s}(\tau)$ is meromorphic in $s$,
with a simple pole at $s=1$ and other sporadic simple poles at $s={\rho\over
2}$. In particular the residue at $s=1$ is,

$\displaystyle{\rm Res}_{s=1}E_{s}(\tau)={3\over\pi}={\rm
vol}({\mathcal{F}})^{-1}\,.$ (3.12)

For ${\rm Re}\,s>1$, the Eisenstein series is equivalently defined by the
following Poincaré series

$\displaystyle E_{s}(\tau)=\sum_{{\gamma}\in\Gamma_{\infty}\backslash
PSL(2,{\mathbb{Z}})}{\rm Im}({\gamma}\tau)^{s}={1\over
2}\sum_{m,n\in{\mathbb{Z}},(m,n)=1}{y^{s}\over|m\tau+n|^{2s}}\,,$ (3.13)

which is clearly $PSL(2,{\mathbb{Z}})$ invariant and its meromorphic
continuation to $s\in{\mathbb{C}}$ is given by (3.8). It is convenient to
define the completed (symmetric) Eisenstein series

$\displaystyle E_{s}^{*}(\tau)\equiv\Lambda(s)E_{s}(\tau)\,,$ (3.14)

which satisfies

$\displaystyle E_{s}^{*}(\tau)=E^{*}_{1-s}(\tau)\,.$ (3.15)

For ${\rm Re}\,s>1$, it can also be written as

$\displaystyle E^{*}_{s}(\tau)={1\over
2}\pi^{-s}\Gamma(s)\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}{y^{s}\over|m\tau+n|^{2s}}\,,$
(3.16)

where $\sum^{\prime}$ means the $m=n=0$ term is dropped. The overlaps in (3.5)
are explicitly given by,

$\displaystyle\langle F,E_{s}\rangle=\int_{\mathcal{F}}{d^{2}\tau\over
y^{2}}F(\tau)E_{1-s}(\tau)=\int_{0}^{\infty}dyy^{-1-s}F_{0}(y)\,,$ (3.17)

where $F_{0}\equiv\int_{-1/2}^{1/2}dxF(\tau)$ is the zero Fourier mode of
$F(\tau)$ and the last equality follows from the unfolding trick (using the
Poincaré series definition of $E_{s}$ in (3.13) and then analytic
continuation).

The Maass cusp forms $\nu_{j}(\tau)$ are eigenfunctions of $\Delta$ in
$L^{2}({\mathcal{F}})$. The special case is the constant function usually
normalized as,

$\displaystyle\nu_{0}\equiv\sqrt{3\over\pi}\,.$ (3.18)

For more general Maass cusp forms, depending on their parity under $x\to-x$,
we have

$\displaystyle\nu_{j}^{+}(\tau)=$
$\displaystyle\sum_{k=1}^{\infty}a_{k}^{(j,+)}\cos(2\pi
kx)\sqrt{y}K_{iR_{j}^{+}}(2\pi ky)\,,$ (3.19)
$\displaystyle\nu_{j}^{-}(\tau)=$
$\displaystyle\sum_{k=1}^{\infty}a_{k}^{(j,-)}\sin(2\pi
kx)\sqrt{y}K_{iR_{j}^{-}}(2\pi ky)\,,$

where the order of the Bessel functions are related to the $\Delta$ eigenvalue
$\lambda_{j}$ by

$\displaystyle\lambda_{j}^{\pm}\equiv{1\over 4}+(R_{j}^{\pm})^{2}\,.$ (3.20)

The normalization is commonly fixed by $a_{1}^{(j,\pm)}=1$. The coefficients
$a_{j}^{(j,\pm)}$ and $R_{j}^{\pm}$ are not known analytically but numerical
results are available (see for example Appendix A.2 of [27]).

###  3.2 Spectral Decomposition of Casimir Energy Density

The spectral decomposition (3.5) only applies to modular functions that are
square integrable. As we have derived from EFT arguments in Section 2.1, the
CFT toroidal Casimir energy density ${\mathcal{E}}(\tau)$ has the following
cusp behavior (see (2.15)),

$\displaystyle\lim_{\tau\to i\infty}{\mathcal{E}}(\tau)=c_{1}y^{d\over
2}+c_{2}y^{1-{d\over 2}}+{\mathcal{O}}\left(y^{\alpha}e^{-2\pi Mny}e^{2\pi
iknx}\right)\,,$ (3.21)

for $c_{1}>0$ and $c_{2}\geq 0$, thus clearly not square integrable for $d>2$.

Nevertheless there is a natural way to regularize this cusp behavior using the
hyperbolic Laplacian,

$\displaystyle{\mathcal{E}}_{\rm reg}(\tau)\equiv\left(\Delta+{d(d-2)\over
4}\right){\mathcal{E}}\,,$ (3.22)

which is modular invariant and vanishes exponentially at the cusp

$\displaystyle\lim_{\tau\to i\infty}{\mathcal{E}}_{\rm
reg}(\tau)={\mathcal{O}}\left(y^{\alpha}e^{-2\pi Mny}e^{2\pi iknx}\right)$
(3.23)

and thus now in $L^{2}({\mathcal{F}})$. Therefore we can apply the
decomposition (3.5) and write,

$\displaystyle{\mathcal{E}}_{\rm
reg}(\tau)=\sum_{j=0}^{\infty}{\langle{\mathcal{E}}_{\rm
reg},\nu_{j}\rangle\over\langle\nu_{j},\nu_{j}\rangle}\nu_{j}(\tau)+{1\over
4\pi i}\int_{{\rm Re\,}s={1\over 2}}ds{\langle{\mathcal{E}}_{\rm
reg},E_{s}\rangle}E_{s}(\tau)\,.$ (3.24)

It is straightforward to invert the differential operator in (3.22) and
obtain,

$\displaystyle\left(\Delta+{d(d-2)\over 4}\right)^{-1}{\mathcal{E}}_{\rm
reg}=\sum_{j=0}^{\infty}{\langle{\mathcal{E}}_{\rm
reg},\nu_{j}\rangle\over\lambda_{j}+{d(d-2)\over
4}}{\nu_{j}(\tau)\over\langle\nu_{j},\nu_{j}\rangle}+{1\over 4\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds{\langle{\mathcal{E}}_{\rm
reg},E_{s}\rangle\over{(d-1)^{2}\over 4}-{(2s-1)^{2}\over 4}}E_{s}(\tau)\,,$
(3.25)

which is still in $L^{2}({\mathcal{F}})$ since coefficients in the
decomposition are now divided by positive numbers that are bounded below by
${(d-1)^{2}\over 4}$.

This inversion is ambiguous up to the kernel of $\Delta+{d(d-2)\over 4}$ which
can be fixed by adding the Eisenstein series $E_{d\over 2}(\tau)$ whose
coefficient is fixed by the cusp behavior of ${\mathcal{E}}$ in (3.21). We
thus find121212In parity-invariant CFT, only the even cusp forms $\nu_{j}^{+}$
in (3.19) appears in the spectral decomposition of the toroidal Casimir
energy.

$\displaystyle{\mathcal{E}}=$ $\displaystyle\,{c_{1}}E_{d\over
2}(\tau)+\sum_{j=0}^{\infty}{\langle{\mathcal{E}}_{\rm
reg},\nu_{j}\rangle\over\lambda_{j}+{d(d-2)\over
4}}{\nu_{j}(\tau)\over\langle\nu_{j},\nu_{j}\rangle}+{1\over 4\pi i}\int_{{\rm
Re\,}s={1\over 2}}ds{\langle{\mathcal{E}}_{\rm
reg},E_{s}\rangle\over{(d-1)^{2}\over 4}-{(2s-1)^{2}\over 4}}E_{s}(\tau)\,.$
(3.26)

This decomposition is unique. Suppose there is another modular function
${\mathcal{E}}^{\prime}$ with the same cusp behavior as in (3.21) and

$\displaystyle\left(\Delta+{d(d-2)\over
4}\right){\mathcal{E}}^{\prime}={\mathcal{E}}_{\rm reg}\,.$ (3.27)

Then ${\mathcal{E}}-{\mathcal{E}}^{\prime}\in L^{2}({\mathcal{F}})$ and

$\displaystyle\left(\Delta+{d(d-2)\over
4}\right)({\mathcal{E}}-{\mathcal{E}}^{\prime})=0\,,$ (3.28)

which is not possible by the spectral decomposition (3.5), of which the
eigenspectrum for $\Delta$ is bounded below by $\frac{1}{4}$.

It is convenient to work with the symmetric Eisenstein series
$E^{*}_{s}(\tau)$, correspondingly we define

$\displaystyle\\{f,E_{s}\\}\equiv{\langle
f,E_{s}\rangle\over\Lambda(s)}\,,\quad\\{f,E_{s}\\}=\\{f,E_{1-s}\\}\,,$ (3.29)

then we can write,

$\displaystyle{\mathcal{E}}(\tau)=$ $\displaystyle\,c_{1}E_{{d\over
2}}(\tau)+{12\over d(d-2)\pi}\int_{\mathcal{F}}{d^{2}\tau\over
y^{2}}{\mathcal{E}}_{\rm reg}(\tau)+{1\over 4\pi i}\int_{{\rm Re\,}s={1\over
2}}ds{\\{{\mathcal{E}}_{\rm reg},E_{s}\\}\over{(d-1)^{2}\over
4}-{(2s-1)^{2}\over 4}}E^{*}_{s}(\tau)$ (3.30) $\displaystyle+$
$\displaystyle\sum_{j=1}^{\infty}{\langle{\mathcal{E}}_{\rm
reg},\nu_{j}\rangle\over\lambda_{j}+{d(d-2)\over 4}}\nu_{j}(\tau)\,,$

Focusing on the zero Fourier mode of ${\mathcal{E}}$, we have

$\displaystyle{\mathcal{E}}_{0}(y)=$ $\displaystyle c_{1}\varphi_{d\over
2}(y)+{12\over d(d-2)\pi}\int_{\mathcal{F}}{d^{2}\tau\over
y^{2}}{\mathcal{E}}_{\rm reg}(\tau)+{1\over 2\pi i}\int_{{\rm Re\,}s={1\over
2}}ds{\\{{\mathcal{E}}_{\rm reg},E_{s}\\}\over{(d-1)^{2}\over
4}-{(2s-1)^{2}\over 4}}\Lambda(s)y^{s}\,,$ (3.31)

and similarly for the zero mode of ${\mathcal{E}}_{\rm reg}$

$\displaystyle({\mathcal{E}}_{\rm
reg})_{0}={3\over\pi}\int_{{\mathcal{F}}}{d^{2}\tau\over
y^{2}}{\mathcal{E}}_{\rm reg}(\tau)+{1\over 2\pi i}\int_{{\rm Re\,}s={1\over
2}}ds\\{{\mathcal{E}}_{\rm reg},E_{s}\\}\Lambda(s)y^{s}\,.$ (3.32)

We have used here that non-constant cusp forms do not have zero Fourier mode.

A major advantage of the spectral decomposition is that it isolates the
potential unknowns and packages them into the coefficients, which we define as
as

$\displaystyle h(s)\equiv\\{{\mathcal{E}}_{\rm reg},E_{s}\\}\,,\quad
p_{j}\equiv\langle{\mathcal{E}}_{\rm reg},\nu_{j}\rangle~{}{\rm for}~{}j\geq
1\,,$ (3.33)

which completely determines ${\mathcal{E}}(\tau)$. We summarize the properties
of $h(s)$ below:

  1. 1.

$h(s)=h(1-s)$,

  2. 2.

$h(s)\in{\mathbb{R}}$ for $s\in{\mathbb{R}}$,

  3. 3.

$h({1\over 2})=0$,

  4. 4.

$h(0)={6\over\pi}\int_{{\mathcal{F}}}{d^{2}\tau\over y^{2}}{\mathcal{E}}_{\rm
reg}(\tau)$,

  5. 5.

$h(s)\Lambda(s)$ is meromorphic for $s\in{\mathbb{C}}$ with a single simple
pole at $s=0$,

  6. 6.

$h({d\over 2})=(d-1)\left({c_{2}\over\Lambda({d-1\over
2})}-{c_{1}\over\Lambda({d\over 2})}\right)$.

The first three properties above follow from the definition of the overlap
$\\{{\mathcal{E}}_{\rm reg},E_{s}\\}$ and that ${\mathcal{E}}_{\rm reg}$ is
square integrable. In general the spectral overlap $\\{f,E_{s}\\}$ for modular
function $f\in L^{2}({\mathcal{F}})$ has a meromorphic analytic continuation
to $s\in{\mathbb{C}}$. Here for ${\mathcal{E}}_{\rm reg}$, we will see that
$h(s)\Lambda(s)$ cannot have any poles except $s=0$. To see this, let us study
the behavior of (3.32) in the large $y$ region. We are then instructed to
deform the contour to the left. This could potentially generate perturbative
terms in $y$ from poles in the integrand. Since ${\mathcal{E}}_{\rm
reg}(\tau)$ cannot have perturbative terms in $y$ near the cusp, the integrand
$h(s)\Lambda(s)$ must have the right pole structure to cancel the constant
term in (3.32). This comes from the pole of $h(s)\Lambda(s)$ at $s=0$. Since
$\Lambda(s)$ has a simple pole at $s=0$ (3.11), we conclude

$\displaystyle h(0)={6\over\pi}\int_{\mathcal{F}}{d^{2}\tau\over
y^{2}}{\mathcal{E}}_{\rm reg}(\tau)\,.$ (3.34)

Furthermore, there cannot be poles in $h(s)$ to the left of ${\rm Re}\,s=0$,
and then by reflection, $h(s)$ cannot have poles anywhere for
$s\in{\mathbb{C}}$ except at zeroes of $\Lambda(s)$ along ${\mathop{\rm
Re}\nolimits s}=1/4$ (and its reflection along ${\mathop{\rm Re}\nolimits
s}=3/4$). Similarly by looking at the large $y$ behavior of (3.31), and
matching with the cusp behavior in (3.21), we conclude

$\displaystyle h\left({d\over 2}\right)=h\left({2-d\over
2}\right)=(d-1)\left({c_{2}\over\Lambda({d-1\over
2})}-{c_{1}\over\Lambda({d\over 2})}\right)\,.$ (3.35)

Therefore we can equivalently write, by contour deformation,

$\displaystyle{\mathcal{E}}={1\over 2}\left(c_{1}+{\Lambda({d\over
2})\over\Lambda({d-1\over 2})}c_{2}\right)E_{d\over 2}(\tau)+{1\over 4\pi
i}\int_{{\rm Re\,}s>{d\over 2}}ds{\\{{\mathcal{E}}_{\rm
reg},E_{s}\\}\over{(d-1)^{2}\over 4}-{(2s-1)^{2}\over
4}}E^{*}_{s}(\tau)+\sum_{j=1}^{\infty}{\langle{\mathcal{E}}_{\rm
reg},\nu_{j}\rangle\over\lambda_{j}+{d(d-2)\over 4}}\nu_{j}(\tau)\,.$ (3.36)

and for the zero Fourier mode,

$\displaystyle{\mathcal{E}}_{0}=$ $\displaystyle{1\over
2}\left(c_{1}+{\Lambda({d\over 2})\over\Lambda({d-1\over
2})}c_{2}\right)\varphi_{d\over 2}(y)+{1\over 4\pi i}\int_{{\rm Re\,}s>{d\over
2}}ds{\\{{\mathcal{E}}_{\rm reg},E_{s}\\}\over{(d-1)^{2}\over
4}-{(2s-1)^{2}\over 4}}\varphi_{s}(y)\,.$ (3.37)

So far we have focused on the overlap coefficient function $h(s)$ in (3.33)
that is necessary and sufficient to reconstruct the zero Fourier mode of
${\mathcal{E}}(\tau)$. To fully determine ${\mathcal{E}}(\tau)$ we also need
the overlaps with the cusp forms (3.19) which are denoted by $p_{j}$ in
(3.33). It is widely believed (though not proven) that the cusp forms do not
have degenerate eigenvalues [33, 34, 35]. Consequently, the overlaps $p_{j}$
is determined by the $k=1$ Fourier mode of ${\mathcal{E}}(\tau)$ using the
orthogonality conditions for the Bessel functions $K_{i\nu}(2\pi y)$ [36].
Physically, the $k$-th Fourier mode ${\mathcal{E}}_{k}(y)$ of
${\mathcal{E}}(\tau)$ accounts for the contributions from worldline instantons
of total KK charge $k$ to the free energy (equivalently toroidal Casimir
energy). From the above reasoning, we have just concluded that all instanton
sectors of higher KK charges are completely determined by the $k=0$ and $k=1$
sectors! Such a general statement about CFT is of course a consequence of the
powerful modular invariance for the toroidal geometry.131313We note that this
is essentially a restatement of similar constraints discussed in [27, 29]
albeit in a different physical context. Furthermore, there are also
constraints on the $k=0$ sector (“scalar sector”) itself from modular
invariance. In [28], a closed modular bootstrap equation was derived for the
scalar sector of a class of $d=2$ CFTs. Analogously, the $k=0$ mode
${\mathcal{E}}_{0}$ (3.31) and equivalently the spectral overlap $h(s)$ in
(3.33) satisfy a similar bootstrap equation. We leave a more detailed study of
the spectral decomposition for the toroidal Casimir energy to the future.

##  4 Casimir Energy Density in Free Scalar Theories

For illustration of the general structure predicted by the EFT analysis in
Section 2 and the spectral theory analysis in Section 3, here we consider the
free CFT of a real scalar in dimension $d>2$. The computation of the toroidal
Casimir energy for the free scalar was done in [37, 38] (and more recently for
higher dimensional torus in [39]). Below we review these results and compare
with our general result (2.15) from EFT analysis.

The scalar Casimir energy density on
${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-3,1}$ is given by the following
obvious sum-integral,

$\displaystyle E_{\rm
vac}=\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}\int{d^{d-3}\vec{p}\over(2\pi)^{d-3}}{1\over
2}\omega_{m,n}(\vec{p})\,,$ (4.1)

where each individual mode contributes,

$\displaystyle\omega_{m,n}({\vec{p}})=\sqrt{\vec{p}^{2}+{(m\vec{k}_{1}+n\vec{k}_{2})^{2}}}\,,$
(4.2)

and $\vec{k}_{1}$ and $\vec{k}_{2}$ are the momentum lattice basis vectors on
the $T^{2}$,

$\displaystyle\vec{k}_{1}={2\pi\over
L_{1}}\left(1,-\frac{\tau_{1}}{\tau_{2}}\right),\quad\vec{k}_{2}={2\pi\over
L_{1}}\left(0,\frac{1}{\tau_{2}}\right)\,.$ (4.3)

Note that we have explicitly excluded the $m=n=0$ term in the sum.

The sum-integral (4.1) suffers from UV divergence. We regulate the non-compact
momentum integral by dimensional regularization,

$\displaystyle E_{\rm vac}=-{1\over(L\tau_{2})^{d-2}}{\pi^{d-2\over 2}\over
2}\Gamma\left(1-{d\over
2}\right)\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}|m\tau+n|^{d-2}\,.$
(4.4)

The residual sum over the momentum lattice is regulated by analytic
continuation using the real analytic Eisenstein series defined in (3.16).
Taking into account the rescaling (1.3), we obtain the dimensionless modular
invariant Casimir energy density for the free scalar,

$\displaystyle{\mathcal{E}}(\tau)=E^{*}_{2-d\over 2}(\tau)=E^{*}_{d\over
2}(\tau)\,,$ (4.5)

where in the last equality we have used the reflection symmetry (3.15) of the
Eisenstein series.

The free scalar Casimir energy density has a simple Fourier decomposition in
$\tau_{1}$ (see (3.8)),

$\displaystyle{\cal
E}(\tau)={\Lambda(d/2)}\tau_{2}^{d/2}+{{\Lambda(1-d/2)}}\tau_{2}^{1-d/2}+4\tau_{2}^{1/2}\sum_{k,n=1}^{\infty}\left({k\over
n}\right)^{d-1\over 2}{\cos(2\pi kn\tau_{1})K_{{d-1\over 2}}(2\pi
kn\tau_{2})}\,.$ (4.6)

This makes explicit the hierarchy between perturbative and non-perturbative
contributions in the limit $\tau\to i\infty$. The former comes from the zero
Fourier mode and comparing to (2.15), we find the corresponding coefficients

$\displaystyle c_{1}=\Lambda\left({d\over 2}\right)\,,\quad
c_{2}=\Lambda\left({d-1\over 2}\right)\,,$ (4.7)

where $\Lambda(s)$ is defined in (3.9). Indeed, the thermal one-point function
of stress tensor of a free real scalar (see for example [19]) takes the form
(1.7) with

$\displaystyle b_{T}=-{2\zeta(d)\over d{\rm vol}(S^{d})}=-{1\over
d}\,\Lambda\left({d\over 2}\right)\,,$ (4.8)

where we have used ${\rm vol}(S^{d})={2\pi^{d/2}\over\Gamma({d/2})}$. Since
the gapless sector of a free scalar upon $S^{1}$ reduction is obviously a free
scalar in one lower dimension, we see (4.8) is in agreement with (4.7) via the
general relations (2.9) and (2.10).

The non-perturbative contributions to the toroidal Casimir energy for the free
scalar come entirely from the nonzero Fourier modes in (4.6). As explained in
Section 2.2, they are associated with worldline instantons of massive
particles in the KK reduced theory. In the free scalar theory, such massive
particles are in one-to-one correspondence with the $k$-th KK modes ($k\neq
0$), whose (dimensionless) mass and KK-charge satisfy $M=Q=k$. The worldline
instanton from the $k$-th KK mode going $n$-times around the compact
$S^{1}_{L}$ is weighted by $e^{2\pi ikn\tau}$ from its worldline action (see
(2.11)). Together with fluctuations thereof, they account for the nonzero
Fourier modes in (4.6).

Finally, in terms of the spectral decomposition discussed in Section 3, the
free scalar toroidal Casimir energy is the special case where the only nonzero
term on the RHS in (3.30) is the first term. Equivalently ${\mathcal{E}}_{\rm
reg}=0$ (see (3.22)) as a consequence of the Laplace-type differential
equation (3.6) satisfied by the Eisenstein series $E^{*}_{d\over 2}$.

##  5 Casimir Energy Density in Interacting CFTs

The power of our general results from the previous sections lies in that it
applies to general CFTs. While it is very difficult to compute the toroidal
Casimir energy for interacting CFTs in general, here we discuss examples where
analytical methods are applicable. We will see how these analytical results
confirm our general formula (2.15) from EFT considerations and we will also
discuss the interesting features brought about by nontrivial interactions.

###  5.1 Critical $O(N)$ Scalar CFT

The first interacting CFT we consider is the critical $O(N)$ CFT in $d=3$
dimensions. This is perhaps the most widely studied class of CFTs in $d>2$. It
models the second order phase transitions of three-dimensional statistical
models including the uniaxial magnet ($N=1$), the XY magnet ($N=2$), the
Heisenberg magnet ($N=3$) and the spherical model ($N=\infty$), as well as the
quantum critical points of two-dimensional materials (see [40] for a recent
review). In the large $N$ limit, the singlet sector of the $O(N)$ CFT is
conjectured to have a holographic dual [15, 16, 17] described by the
Vasiliev’s higher-spin gauge theory on AdS4 [41, 42, 43] subject to suitable
boundary conditions [44, 45].141414See also [46, 47] for recent works on this
holographic duality beyond the leading large $N$ limit.

To describe the $O(N)$ CFT, we start with a scalar field theory defined by the
following $O(N)$ symmetric action on flat space ${\mathbb{R}}^{3}$,151515We
refer the readers to [48] for further details on the $O(N)$ model and its
large $N$ limit.

$\displaystyle S[\phi]=\int
d^{3}x\left[\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}+\frac{1}{2}r\phi_{i}\phi^{i}+\frac{1}{4!}\frac{u}{N}(\phi_{i}\phi^{i})^{2}\right]\,,$
(5.1)

where $\phi_{i}$ with $i=1,2,\dots N$ denotes a scalar field in the vector
representation of $O(N)$ and $u,r$ are the coupling constants. The quartic
interaction is relevant in $d=3$ and triggers an renormalization group (RG)
flow from the free $O(N)$ symmetric theory to a nontrivial fixed point in the
infra-red (IR) described by the critical $O(N)$ CFT. Near the critical point,
the mass term is relevant and thus needs to be tuned $r\to r_{c}$ to ensure a
vanishing mass gap and furthermore we take $u\to\infty$ to reach the CFT.

Despite the explicit UV Lagrangian, there is no small parameter in (5.1) and
thus it is not useful in practice to extract CFT data in general. Nonetheless,
in the large $N$ limit, as is suggested by the form of (5.1) with $u$ fixed,
we can make use of the $1/N$ expansion.

Here the standard procedure starts with the Hubbard-Stratonovich
transformation to replace the quartic interaction term by introducing an
auxiliary scalar field $\lambda$,

$\displaystyle S[\phi,\lambda]=\int
d^{3}x\left[\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}+\frac{1}{2}r\phi_{i}\phi^{i}+\frac{6N\lambda^{2}}{u}+i\lambda\phi_{i}\phi^{i}\right]\,,$
(5.2)

and path integrate over real $\lambda$. Since $\phi$ appears quadratically in
$S[\phi,\lambda]$, we can integrate it out and obtain an action for $\lambda$,

$\displaystyle S[\lambda]=\frac{N}{2}{\rm
Tr\,}\ln(-\partial^{2}+r+2i\lambda)+\frac{6N\lambda^{2}}{u}\,.$ (5.3)

In the large $N$ limit, from the saddle point approximation, we obtain the
following equation for $\lambda$,

$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{k^{2}+r+2i\lambda}=\frac{12i\lambda}{u}\,.$
(5.4)

This is known as the gap equation because it determines the mass gap
$m_{{\mathbb{R}}^{3}}^{2}=r+2i\lambda$.161616As we will see, for the critical
$O(N)$ model on ${\mathbb{R}}\times T^{2}$, the saddle point of $\lambda$ is
imaginary and the mass gap $m_{{\mathbb{R}}^{3}}^{2}>0$. The critical coupling
is determined by

$\displaystyle r_{c}=-{u\over 6}\int{d^{3}k\over(2\pi)^{3}}{1\over k^{2}}\,.$
(5.5)

It is convenient to work with the dimensional regularization so that
$r_{c}=0$. Therefore to study the large $N$ $O(N)$ CFT, it suffices to focus
on the following action

$\displaystyle S[\phi,\lambda]=\int
d^{3}x\left[\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}+i\lambda\phi_{i}\phi^{i}\right]\,,$
(5.6)

where we have sent $u\to\infty$ to decouple the irrelevant deformation
$\lambda^{2}$. Similarly the action for $\lambda$ simplifies to,

$\displaystyle S[\lambda]=\frac{N}{2}{\rm
Tr\,}\ln(-\partial^{2}+2i\lambda)\,.$ (5.7)

On ${\mathbb{R}}^{3}$, the CFT is clearly described by the $\lambda=0$ saddle
point, similarly for other conformally flat spacetime manifolds such as
$S^{3}$.

####  5.1.1 Casimir Energy from Gap Equation

We now study the $O(N)$ CFT on ${\mathbb{R}}\times T^{2}$ to determine the
toroidal Casimir energy (see also previous work [49]). This boils down to
solving the path integral for the action (5.6) (equivalently (5.7)) on
${\mathbb{R}}\times T^{2}$. The saddle point (gap) equation for $\lambda$
becomes,

$\displaystyle
0=\frac{1}{A}\sum_{\vec{k}}\int\frac{d\omega}{2\pi}{1\over\omega^{2}+\vec{k}^{2}+2i\lambda}\,,$
(5.8)

where $A$ is the area of $T^{2}$ and $\vec{k}$ is a lattice vector with the
basis vectors in (4.3). We define

$\displaystyle 2i\lambda\equiv\left(2\pi\over L_{1}\right)^{2}\Delta^{2}$
(5.9)

and integrate over $\omega$ to obtain,

$\displaystyle
0=\sum_{m,n\in{\mathbb{Z}}}\frac{1}{\sqrt{|m\tau+n|^{2}+\tau_{2}^{2}\Delta^{2}}}$
(5.10)

The gap equation (after regularization) will determine $\Delta$ as a function
of the complex moduli $\tau$. Physically, $2\pi\Delta$ is the induced
dimensionless mass of the $\phi^{i}$ fields due to the nontrivial geometry,
generalizing the thermal mass in the case of
$S^{1}_{\beta}\times{\mathbb{R}}^{2}$. Equivalently, it determines the one-
point function of the $O(N)$ invariant operator $\phi_{i}\phi^{i}$ on
${\mathbb{R}}\times T^{2}$. In the limit the $T^{2}$ area $A\to\infty$, the
induced mass $m^{2}=2i\lambda$ vanishes which is consistent with vanishing gap
in the CFT on ${\mathbb{R}}^{3}$.

The large $N$ Casimir energy is then given by the following mode sum as in the
theory of $N$ free scalars of mass $\Delta$ (see around (4.1)),

$\displaystyle E_{\rm vac}={N\over 2}{2\pi\over
L_{1}\tau_{2}}\sum_{m,n\in{\mathbb{Z}}}\sqrt{|m\tau+n|^{2}+\tau_{2}^{2}\Delta^{2}}\,.$
(5.11)

Equivalently, this follows from evaluating the CFT partition function on
$S^{1}_{\beta}\times T^{2}$

$\displaystyle E_{\rm vac}=-\lim_{{\beta}\to\infty}{1\over{\beta}}\log
Z={N\over 2}\sum_{\vec{k}}\int{d\omega\over
2\pi}\log(\omega^{2}+\vec{k}^{2}+2i\lambda)\,,$ (5.12)

after integrating over $\omega$ and dropping extensive terms on $T^{2}$ (which
can be absorbed by the $d=3$ cosmological constant counterterm).

After a rescaling (see (1.3)), we obtain the dimensionless Casimir energy
density,

$\displaystyle{\mathcal{E}}(\tau)=-{\pi N}\tau_{2}^{-{1\over
2}}\sum_{m,n\in{\mathbb{Z}}}\sqrt{|m\tau+n|^{2}+\tau_{2}^{2}\Delta(\tau)^{2}}\,.$
(5.13)

To regularize the momentum lattice sum that appears in the gap equation (5.10)
and the Casimir energy (5.13), we define the following generalized Eisenstein
series,

$\displaystyle G_{s}(\tau,\Delta)={1\over
2}\pi^{-s}\Gamma(s)\sum_{m,n\in{\mathbb{Z}}}{\tau_{2}^{s}\over(|m\tau+n|^{2}+\tau_{2}^{2}\Delta(\tau)^{2})^{s}}.$
(5.14)

The real analytic Eisenstein series defined in (3.16) is recovered as a
special case at $\Delta=0$,

$\displaystyle E_{s}^{*}(\tau)=\lim_{\Delta\to
0}\left(G_{s}(\tau,\Delta)-{1\over
2}{\Gamma(s)\over(\pi\tau_{2}\Delta^{2})^{s}}\right)\,.$ (5.15)

More generally, $G_{s}(\tau,\Delta)$ is obviously modular invariant if
$\tau_{2}\Delta(\tau)^{2}$ is modular invariant (e.g. a constant), since the
above can be written as a sum of Poincaré series,

$\displaystyle
G_{s}(\tau,\Delta)=\pi^{-s}\Gamma(s)\sum_{k=1}^{\infty}\sum_{{\gamma}\in\Gamma_{\infty}\backslash
PSL(2,{\mathbb{Z}})}{{\rm Im}({\gamma}\tau)^{s}\over\left(k^{2}+{\rm
Im}({\gamma}\tau)\tau_{2}\Delta(\tau)^{2}\right)^{s}}+{1\over
2}{\Gamma(s)\over(\pi\tau_{2}\Delta^{2})^{s}}.$ (5.16)

Here for the application to the CFT on ${\mathbb{R}}\times T^{2}$, as a
consequence of large diffeomorphisms on $T^{2}$, we see indeed that the
induced mass normalized by the $T^{2}$ area
${2i\lambda}A=(2\pi)^{2}\tau_{2}\Delta^{2}$ is modular invariant and therefore
so is $G_{s}(\tau,\Delta)$.

Similar to the Eisenstein series $E_{s}(\tau)$, the function
$G_{s}(\tau,\Delta)$ admits an analytic continuation in $s$ that is finite at
$s=-1/2$ and $s=1/2$ which will be relevant for the gap equation and the
Casimir energy respectively. We note that the function $G_{s}(\tau,\Delta)$ is
a slight modification of the function $g_{s}(\Delta,\tau)$ introduced in [49]
which they used to analyze the spectrum on $T^{2}$ numerically. There the
analytic continuation in $s$ is provided by an integral formula. Here we will
provide an alternative but equivalent formula in the form of Poincaré series
(more precisely sum of them) which makes the modular invariance
manifest,171717The special case of the modular invariant function
$G_{s}(\tau,\Delta)$ when $\tau_{2}\Delta^{2}$ is a constant has also recently
shown up in a different context [50].

$\displaystyle G_{s}(\tau,\Delta)=$
$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}\left(\frac{\tau_{2}\Delta}{|n+m\tau|}\right)^{1-s}K_{1-s}(2\pi\Delta|n+m\tau|)+{\Gamma(s-1)\over
2}\left(\pi\tau_{2}\Delta^{2}\right)^{1-s}\,,$ (5.17)

and is well defined for $\Delta$ away from the branch cut $\Delta<0$. The
details of the derivation and the comparison to the formulae in [49] are given
in Appendix A.

In terms of the regulated generalized Eisenstein series $G_{s}(\tau,\Delta)$
in (5.17), the gap equation becomes181818In other words, $\Delta$ is the fixed
point of the $\tau$-dependent lattice sum.

$\displaystyle 0=G_{1\over
2}(\tau,\Delta)\quad\Leftrightarrow\quad\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}\frac{1}{|n+m\tau|}e^{-2\pi\Delta|n+m\tau|}=2\pi\Delta\,,$
(5.18)

and the Casimir energy is

$\displaystyle{\mathcal{E}}(\tau,\Delta)={N}G_{-{1\over
2}}(\tau,\Delta)=\frac{N\tau_{2}^{\frac{3}{2}}}{4\pi}\sideset{}{{}^{\prime}}{\sum}_{m,n\in{\mathbb{Z}}}\frac{1+2\pi\Delta|n+m\tau|}{|n+m\tau|^{3}}e^{-2\pi\Delta|n+m\tau|}+\frac{2N\pi^{2}\Delta^{3}\tau_{2}^{\frac{3}{2}}}{3}\,.$
(5.19)

Note the close resemblance to the corresponding expression for the free scalar
(4.4). The toroidal Casimir energy for the $O(N)$ CFT is obtained from solving
the gap equation (5.18) for the induced mass $\Delta(\tau)$ and then plugging
it in (5.19). The gap equation says that the CFT Casimir energy extremizes
${\mathcal{E}}(\tau,\Delta)$ with respect to $\Delta^{2}$. The induced mass
squared $\Delta^{2}$ must be non-negative to avoid an instability on $T^{2}$
(and thus divergent partition function on $T^{3}$). For our choice of branch
cut in (5.17), this means the saddle point has $\Delta(\tau)\geq 0$, which is
clearly consistent with (5.18). In fact, the saddle point is a local minimum
of ${\mathcal{E}}(\tau,\Delta)$, since

$\displaystyle{\partial^{2}{\mathcal{E}}(\tau,\Delta)\over\partial(\Delta^{2})^{2}}={N}\pi^{2}\tau_{2}^{2}G_{{3\over
2}}(\tau,\Delta)=\frac{N\tau_{2}^{\frac{3}{2}}}{32\pi^{2}\Delta}\sum_{m,n\in{\mathbb{Z}}}e^{-2\pi\Delta|n+m\tau|}\,,$
(5.20)

which is positive for $\Delta\geq 0$.

We solve the gap equation (5.18) numerically and then evaluate the toroidal
Casimir energy (5.19).191919For the numerical evaluation, we find another form
(A.6) for $G_{s}(\tau,\Delta)$ to be more useful, as the sum of Bessel
functions converges very fast on the standard fundamental domain. In Figure 1,
we present a plot of ${\mathcal{E}}(\tau)$ for the $O(N)$ CFT in the standard
$PSL(2,{\mathbb{Z}})$ fundamental domain. We see the growth behavior at large
$\tau_{2}$ as $\tau_{2}^{\frac{3}{2}}$, and the $\tau_{1}$ dependence is
almost negligible in this region. In Figure 2, we present a plot of
$\Delta(\tau)$ in the standard $PSL(2,{\mathbb{Z}})$ fundamental domain. The
induced mass tends toward a constant at large $\tau_{2}$. The dependence on
$\tau_{1}$ is also very weak. In the next section, we discuss analytic results
for ${\mathcal{E}}(\tau)$ in the thin torus (high temperature) limit, which
will explain these observations.

$\frac{{\mathcal{E}}(\tau)}{N}$$\tau_{2}$$\tau_{1}$

$\frac{{\mathcal{E}}(\tau)}{N}$$\tau_{2}$

Figure 1: Toroidal Casimir energy of the $O(N)$ scalar CFT as a function of
$\tau$, computed in the standard $PSL(2,\mathbb{Z})$ fundamental domain. In
the left figure, we zoom into the region around $\tau_{2}=1$ to show the
$\tau_{1}$ dependence of the Casimir energy, which is more prominent at
smaller $\tau_{2}$. In the right figure, we project the 3D plot along the
$-\tau_{1}$ direction to highlight the suppression of $\tau_{1}$ independence
in this fundamental domain. ${\mathcal{E}}(\tau)$ grows at large $\tau_{2}$ as
$\tau_{2}^{3/2}$, which becomes dominant as early as $\tau_{2}\sim 1$ in the
$O(N)$ scalar CFT. The $\tau_{1}$ dependence is negligible in this domain,
except for minor dependence near the boundary at $|\tau|=1$, which is mostly
contributed by the lowest nonzero KK mode.

$\Delta(\tau)$$\tau_{2}$$\tau_{1}$

Figure 2: Rescaled induced mass $\Delta$ of the $O(N)$ scalar CFT as a
function of $\tau$, computed in the standard $PSL(2,\mathbb{Z})$ fundamental
domain. The induced mass hardly depends on $\tau_{1}$ in this
$PSL(2,{\mathbb{Z}})$ frame, and approaches a constant at large $\tau_{2}$.

####  5.1.2 High Temperature Expansion and Worldline Instantons

We now study the toroidal Casimir energy ${\mathcal{E}}(\tau)$ for the $O(N)$
CFT in the thin torus $\tau\to i\infty$ limit, to compare with the universal
formula (2.15) derived from high temperature EFT and the numerical results
obtained above. For this purpose it is useful to use another form of the
regulated generalized Eisenstein series (5.17). Specifically, we have

$\displaystyle{\mathcal{E}}(\tau,\Delta)=N{\tau_{2}^{3\over
2}f(\Delta)}+2N\tau_{2}^{1\over
2}\sum_{n\in{\mathbb{Z}}}\sum_{m=1}^{\infty}{1\over m}e^{2\pi
inm\tau_{1}}\sqrt{n^{2}+\Delta^{2}}K_{1}(2\pi\sqrt{n^{2}+\Delta^{2}}m\tau_{2})\,,$
(5.21)

where

$\displaystyle
f(\Delta)\equiv\frac{1}{6\pi}\left(4\pi^{3}\Delta^{3}+6\pi\Delta\text{Li}_{2}\left(e^{-2\pi\Delta}\right)+3\text{Li}_{3}\left(e^{-2\pi\Delta}\right)\right)\,.$
(5.22)

The derivation is given in Appendix A. The gap equation, which locates the
local minimum of ${\mathcal{E}}(\tau,\Delta)$ in $\Delta$, becomes

$\displaystyle\pi\Delta+\log(1-e^{-2\pi\Delta})=2\sum_{n\in{\mathbb{Z}}}\sum_{m=1}^{\infty}e^{2\pi
inm\tau_{1}}K_{0}(2\pi\sqrt{n^{2}+\Delta^{2}}m\tau_{2})\,.$ (5.23)

Let us first analyze the gap equation (5.23) and Casimir energy (5.21) at
infinite $\tau_{2}$. From positivity and asymptotic behavior the Bessel
function $K_{0}(x)$, it is clear that the saddle point $\Delta$ that solves
(5.23) approaches a constant $\Delta_{0}\equiv\Delta(\tau=i\infty)$ which
satisfies

$\displaystyle\pi\Delta_{0}+\log(1-e^{-2\pi\Delta_{0}})=0\quad\Rightarrow\quad\Delta_{0}=\frac{1}{\pi}\log(\frac{1+\sqrt{5}}{2})\,.$
(5.24)

Consequently, the Casimir energy evaluates to

$\displaystyle{\mathcal{E}}(\tau=i\infty)=\frac{N\tau_{2}^{3\over
2}}{6\pi}\left(4\pi^{3}\Delta_{0}^{3}+6\pi\Delta_{0}\text{Li}_{2}\left(e^{-2\pi\Delta_{0}}\right)+3\text{Li}_{3}\left(e^{-2\pi\Delta_{0}}\right)\right)={2N\zeta(3)\over
5\pi}\tau_{2}^{3\over 2}\,.$ (5.25)

This determines the coefficient $c_{1}$ in (2.15) for the $O(N)$ CFT in the
large $N$ limit to be

$\displaystyle c_{1}^{O(N)}={2N\zeta(3)\over 5\pi}\,.$ (5.26)

Here $2\pi\Delta_{0}$ coincides with the dimensionless thermal mass [13] on
$\mathbb{R}^{2}\times S^{1}_{\beta}$ and $c_{1}$ agrees with the thermal free
energy [12] via the relation (2.9).

A main advantage of the equations (5.23) and (5.21) is that it can be solved
recursively in a large $\tau_{2}$ expansion. First we note that by dropping
all exponentially suppressed terms in (5.23) and (5.21), there is no
perturbative (i.e. power law in $\tau_{2}$) correction to (5.24) and (5.25).
Consequently we conclude that $c_{2}$ in (2.15) vanishes for the $O(N)$ CFT to
the leading order in the large $N$ limit,

$\displaystyle c_{2}^{O(N)}=0\,.$ (5.27)

This is consistent with the fact that the circle reduced theory (keeping only
zero KK modes), namely the $O(N)$ vector model in $d=2$ (equivalent to the
$O(N)$ $\sigma$-model in the IR) famously has a mass gap for $N\geq 3$ [51,
52, 53, 54, 55, 56, 57, 58].

Given the general discussion in see Section 2.2, we are then led to the
following expansion of the solution $\Delta(\tau)$ for the induced mass,

$\displaystyle\Delta(\tau)=\Delta_{0}+\sum_{j=1}^{\infty}\sum_{\begin{subarray}{c}\vec{m}\in\mathbb{Z}^{j},\\\
m_{1}\leq m_{2}\leq\cdots\leq
m_{j}\end{subarray}}e^{-2\pi\tau_{2}\sum_{i=1}^{j}\sqrt{m_{i}^{2}+\Delta_{0}^{2}}}e^{2\pi
i\sum_{i=1}^{j}m_{i}\tau_{1}}\Delta_{j,\vec{m}}(\tau_{2})\,.$ (5.28)

The second term on the RHS above keeps track of the contributions from
multiple worldline instantons in the EFT from KK reduction. Here $j$ counts
the number of worldline instantons and each individual instanton labeled by
$i=1,\dots,j$ has KK charge $Q=m_{i}$ and mass
$M=\sqrt{m_{i}^{2}+\Delta_{0}^{2}}$.

Similarly for the Casimir energy, we have the following expansion,

$\displaystyle{\mathcal{E}}(\tau)=c_{1}\tau_{2}^{\frac{3}{2}}+\sum_{j=1}^{\infty}\sum_{\begin{subarray}{c}\vec{m}\in\mathbb{Z}^{j},\\\
m_{1}\leq m_{2}\leq\cdots\leq
m_{j}\end{subarray}}e^{-2\pi\tau_{2}\sum_{i=1}^{j}\sqrt{m_{i}^{2}+\Delta_{0}^{2}}}e^{2\pi
i\sum_{i=1}^{j}m_{i}\tau_{1}}{\mathcal{E}}_{j,\vec{m}}(\tau_{2})\,.$ (5.29)

Here the coefficient functions $\Delta_{j,\vec{m}}(\tau_{2})$ and
${\mathcal{E}}_{j,\vec{m}}(\tau_{2})$ are perturbative in $\tau_{2}$ and
account for fluctuations on the multi-instanton background and integrations
over moduli of such configurations. Furthermore, parity invariance of the
$O(N)$ CFT implies that

$\displaystyle\Delta_{j,\vec{m}}(\tau_{2})=\Delta_{j,-\vec{m}}(\tau_{2})\,,\quad{\mathcal{E}}_{j,\vec{m}}(\tau_{2})={\mathcal{E}}_{j,-\vec{m}}(\tau_{2})\,.$
(5.30)

One can solve for $\Delta_{j,\vec{m}}$ from (5.23) recursively starting from
$j=1$, and then evaluate (5.21) on the solution. Below we denote
$\mathcal{K}_{\nu}(z)\equiv K_{\nu}(z)e^{z}$ and
$M_{i}\equiv\sqrt{m_{i}^{2}+\Delta_{0}^{2}}$ for convenience, and list the
solutions that account for the single-instanton effects,

$\displaystyle\Delta_{1,m_{1}}=$
$\displaystyle\,\frac{2}{\sqrt{5}\pi}\mathcal{K}_{0}\left(2\pi
M_{1}\tau_{2}\right)=\frac{1}{\pi}\sqrt{\frac{1}{5M_{1}\tau_{2}}}\left(1-\frac{1}{16\pi
M_{1}\tau_{2}}+{\mathcal{O}}(\tau_{2}^{2})\right)\,,$ (5.31)

the two-instanton contributions,

$\displaystyle\Delta_{2,m_{1},m_{2}}=$
$\displaystyle\,\frac{2\pi}{\sqrt{5}}\Delta_{1,m_{1}}\Delta_{1,m_{2}}+\frac{2}{\sqrt{5}\pi}\mathcal{K}_{0}\left(4\pi
M_{1}\tau_{2}\right)\delta_{m_{1},m_{2}}$ (5.32) $\displaystyle-$
$\displaystyle\,\frac{8\Delta_{0}\tau_{2}}{5\pi
M_{1}}\mathcal{K}_{1}\left(2\pi M_{1}\tau_{2}\right)\mathcal{K}_{0}\left(2\pi
M_{2}\tau_{2}\right)+\text{distinct perm. of }\\{m_{i}\\}$ $\displaystyle=$
$\displaystyle-\frac{2\Delta_{0}}{5\pi}\frac{1}{M_{1}^{\frac{3}{2}}M_{2}^{\frac{1}{2}}}+\frac{\delta_{m_{1},m_{2}}}{\pi\sqrt{10M_{1}}}\tau_{2}^{-\frac{1}{2}}+{\mathcal{O}}(\tau_{2}^{-1})+\text{distinct
perm. of }\\{m_{i}\\}\,,$

and three-instanton contributions,

$\displaystyle\Delta_{3,m_{1},m_{2},m_{3}}=\,\frac{4\pi}{\sqrt{5}}\Delta_{1,m_{1}}\Delta_{2,m_{2},m_{3}}-\frac{4\pi^{2}}{3}\Delta_{1,m_{1}}\Delta_{1,m_{2}}\Delta_{1,m_{2}}+\frac{2}{\sqrt{5}\pi}\mathcal{K}_{0}\left(6\pi
M_{1}\tau_{2}\right)\delta_{m_{1},m_{2}}\delta_{m_{1},m_{3}}$ (5.33)
$\displaystyle+$
$\displaystyle\,\frac{2\tau_{2}}{\sqrt{5}M_{1}^{3}}\left[2\pi\tau_{2}\Delta_{0}^{2}M_{1}\,\mathcal{K}_{0}\left(2\pi
M_{1}\tau_{2}\right)+(-m_{1}^{2}+\Delta_{0}^{2})\mathcal{K}_{1}\left(2\pi
M_{1}\tau_{2}\right)\right]\Delta_{1,m_{2}}\Delta_{1,m_{3}}$ $\displaystyle-$
$\displaystyle\,\frac{8\Delta_{0}\tau_{2}}{\sqrt{5}}\frac{\mathcal{K}_{1}\left(4\pi
M_{1}\tau_{2}\right)}{M_{1}}\delta_{m_{1},m_{2}}\Delta_{1,m_{3}}-\frac{4\Delta_{0}\tau_{2}}{\sqrt{5}}\frac{\mathcal{K}_{1}\left(2\pi
M_{1}\tau_{2}\right)}{M_{1}}\Delta_{2,m_{2},m_{3}}+\text{distinct perm. of
}\\{m_{i}\\}$ $\displaystyle=$
$\displaystyle\,\frac{2\Delta_{0}^{2}\tau_{2}^{\frac{1}{2}}}{5\sqrt{5}\pi}\left(\frac{1}{M_{1}^{\frac{5}{2}}M_{2}^{\frac{1}{2}}M_{3}^{\frac{1}{2}}}+\frac{2}{M_{1}^{\frac{3}{2}}M_{2}^{\frac{3}{2}}M_{3}^{\frac{1}{2}}}\right)-\frac{\sqrt{2}\Delta_{0}}{5\pi}\left(\frac{2}{M_{1}^{\frac{3}{2}}M_{3}^{\frac{1}{2}}}+\frac{1}{M_{1}^{\frac{1}{2}}M_{3}^{\frac{3}{2}}}\right)\delta_{m_{1},m_{2}}$
$\displaystyle+$
$\displaystyle\,{\mathcal{O}}(\tau_{2}^{-\frac{1}{2}})+\text{distinct perm. of
}\\{m_{i}\\}\,.$

Similarly for the toroidal Casimir energy of the $O(N)$ CFT, we find the
following one-instanton contributions,

$\displaystyle{\mathcal{E}}_{1,m_{1}}=$
$\displaystyle\,2\tau_{2}^{\frac{1}{2}}M_{1}\,\mathcal{K}_{1}\left(2\pi
M_{1}\tau_{2}\right)=\sqrt{M_{1}}\left(1+\frac{3}{16\pi
M_{1}\tau_{2}}+{\mathcal{O}}(\tau_{2}^{-2})\right)\,,$ (5.34)

two-instanton contributions,

$\displaystyle{\mathcal{E}}_{2,m_{1},m_{2}}=$
$\displaystyle-\sqrt{5}\pi^{2}\Delta_{0}\tau_{2}^{\frac{3}{2}}\Delta_{1,m_{1}}\Delta_{1,m_{2}}+\tau_{2}^{\frac{1}{2}}M_{1}\,\mathcal{K}_{1}\left(4\pi
M_{1}\tau_{2}\right)\delta_{m_{1},m_{2}}+\text{distinct perm. of }\\{m_{i}\\}$
(5.35) $\displaystyle=$
$\displaystyle-\frac{\tau_{2}^{\frac{1}{2}}\Delta_{0}}{\sqrt{5M_{1}M_{2}}}+\frac{\sqrt{2M_{1}}}{4}\delta_{m_{1},m_{2}}+{\mathcal{O}}(\tau_{2}^{-\frac{1}{2}})+\text{distinct
perm. of }\\{m_{i}\\}\,,$

and three-instanton contributions,

$\displaystyle{\mathcal{E}}_{3,m_{1},m_{2},m_{3}}=$
$\displaystyle\frac{2\pi^{2}(\sqrt{5}-2\pi\Delta_{0})}{3}\tau_{2}^{3/2}\Delta_{1,m_{1}}\Delta_{1,m_{2}}\Delta_{1,m_{3}}+\frac{2\tau_{2}^{\frac{1}{2}}}{3}M_{1}\,\mathcal{K}_{1}\left(6\pi
M_{1}\tau_{2}\right)\delta_{m_{1},m_{2}}\delta_{m_{1},m_{3}}$ (5.36)
$\displaystyle+$
$\displaystyle\,2\tau_{2}^{\frac{1}{2}}\left[\frac{1+2\pi^{2}\tau_{2}^{2}\Delta_{0}^{2}}{M_{1}}\,\mathcal{K}_{0}\left(2\pi
M_{1}\tau_{2}\right)-\pi\tau_{2}\mathcal{K}_{2}\left(2\pi
M_{1}\tau_{2}\right)\right]\Delta_{1,m_{2}}\Delta_{1,m_{3}}$ $\displaystyle-$
$\displaystyle\,4\pi\Delta_{0}\tau_{2}^{\frac{3}{2}}\mathcal{K}_{0}\left(4\pi
M_{1}\tau_{2}\right)\delta_{m_{1},m_{2}}\Delta_{1,m_{3}}+\text{distinct perm.
of }\\{m_{i}\\}$ $\displaystyle=$
$\displaystyle\,\frac{2\Delta_{0}^{2}\tau_{2}}{5}\frac{1}{M_{1}^{\frac{3}{2}}M_{2}^{\frac{1}{2}}M_{3}^{\frac{1}{2}}}-\sqrt{\frac{2}{5}}\frac{\Delta_{0}}{\sqrt{M_{1}M_{3}}}\delta_{m_{1},m_{2}}\tau_{2}^{\frac{1}{2}}+{\mathcal{O}}(1)+\text{distinct
perm. of }\\{m_{i}\\}\,.$

$\frac{{\mathcal{E}}(\tau)}{N}$$\tau_{1}$

Figure 3: Casimir energy of $O(N)$ scalar CFT as a function of
$\tau_{1}\in[-1/2,1/2]$ at fixed $\tau_{2}=1$.

Because there is only one perturbative term in ${\mathcal{E}}(\tau)$ for the
$O(N)$ CFT, it is the dominant contribution on the standard fundamental domain
even for not very large $\tau_{2}$. This explains why there is little
$\tau_{1}$ dependence for ${\mathcal{E}}(\tau)$ in this domain (see Figure 1).
Furthermore, among the instanton contributions, the leading $\tau_{1}$
dependence is exponentially suppressed with a larger exponent than the leading
$\tau_{1}$ independent exponential term, due to the higher mass of instanton
with KK charge. We have checked these leading instanton contributions
presented above numerically at large $\tau_{2}$ and they agree with the
numerical solution from directly solving (5.18) and then evaluating (5.19) on
the saddle. Specifically, we plot the Casimir energy density
${\mathcal{E}}(\tau)$ for the $O(N)$ CFT as a function of $\tau_{1}$ at
$\tau_{2}=1$ in Figure 3. One can compute the leading contribution to the
amplitude of the first Fourier term in (5.29) using (5.34) which gives
$2{\mathcal{E}}_{1,1}(\tau_{2}=1)e^{-2\pi\sqrt{1+\Delta_{0}^{2}}}\approx
0.0035$, while the amplitude shown in Figure 3 is about 0.0033. We see that
even for $\tau_{2}=1$ (and it gets better for large $\tau_{2}$), the
exponential suppression is prominent and the one-instanton coefficient already
works well (about 5% error) for estimating $\tau_{1}$ dependence.

###  5.2 Holographic CFTs with Einstein Gravity Duals

Let us now consider the case of general $d\geq 3$-dimensional large $N$ CFTs
that are dual to Einstein gravity on AdSd+1 at strong coupling [15, 16, 17].
While the ground state of the CFT on $S^{d-1}$ is dual to the empty global
AdSd+1, the CFT ground state on ${\mathbb{R}}^{d-3}\times T^{2}$ is expected
to be dual to certain AdS soliton solution [59, 60, 11] . For rectangular
$T^{2}=S^{1}_{L_{1}}\times S^{1}_{L_{2}}$, the AdS soliton solution is given
in [59],

$\displaystyle ds^{2}=-{r^{2}\over\ell^{2}}dt^{2}+{\ell^{2}dr^{2}\over
r^{2}(1-(r_{0}/r)^{d})}+{r^{2}\over\ell^{2}}(1-(r_{0}/r)^{d})L_{1}^{2}dx_{1}^{2}+{r^{2}\over\ell^{2}}L_{2}^{2}dx_{2}^{2}+{r^{2}\over\ell^{2}}dy_{i}^{2}$
(5.37)

which is related by a double Wick rotation to the black $d$-brane solution.
Here $\ell$ denotes the AdS radius. The AdS radial coordinate is $r$ which
ends at $r=r_{0}$ in the interior and approaches $r=\infty$ at the asymptotic
boundary. The boundary $T^{2}$ coordinates are $x_{1},x_{2}$ which obey the
identification $x_{1}\sim x_{1}+1\,,~{}x_{2}\sim x_{2}+1$. The remaining
boundary noncompact spatial directions are $y_{i}$ with $i=1,\dots,d-3$ and
the time direction is $t$.

The circle in the $x_{1}$ direction shrinks in the bulk as $r$ decreases. To
avoid a singularity at $r=r_{0}$, this requires

$\displaystyle r_{0}={4\pi\ell^{2}\over dL_{1}}\,,$ (5.38)

so that the $x_{1}$ circle caps off smoothly. Meanwhile the circle in the
$x_{2}$ direction remains non-contractible. The energy density (in the non-
compact ${\mathbb{R}}^{d-3}$ directions) for the AdS soliton is [59]

$\displaystyle E_{\rm soliton}=-{r_{0}^{d}L_{1}L_{2}\over 16\pi
G_{N}\ell^{d+1}}=-{(4\pi)^{d-1}\ell^{d-1}\over 4d^{d}G_{N}}{L_{2}\over
L_{1}^{d-1}}\,,$ (5.39)

where $G_{N}$ denotes the Newton’s constant.

The solution of [59] for rectangular boundary $T^{2}$ has a straightforward
generalization for general $T^{2}$ with complex moduli
$\tau=\tau_{1}+i\tau_{2}$,

$\displaystyle ds^{2}=-{r^{2}\over\ell^{2}}dt^{2}+{\ell^{2}dr^{2}\over
r^{2}(1-(r_{0}/r)^{d})}+{r^{2}L_{1}^{2}\over\ell^{2}}\left((1-(r_{0}/r)^{d})(dx_{1}+\tau_{1}dx_{2})^{2}+\tau_{2}^{2}dx_{2}^{2}\right)+{r^{2}\over\ell^{2}}dy_{i}^{2}\,,$
(5.40)

where $r_{0}$ is determined by the same relation (5.38) so that the circle in
the $x_{1}$ direction caps off smoothly in the bulk.202020Topologically, the
AdS soliton is a fibration of $S^{1}\times{\mathbb{R}}^{d-2}$ over a two-disk
(parametrized by $r$ and $x_{1}$).

The energy density for this general AdS soliton solution follows from a
similar calculation,

$\displaystyle E_{\rm soliton}=-{r_{0}^{d}A\over 16\pi
G_{N}\ell^{d+1}}=-{(4\pi)^{d-1}\ell^{d-1}\over 4d^{d}G_{N}}{\tau_{2}^{d\over
2}\over A^{d-2\over 2}}\,,$ (5.41)

where we have used $A=L_{1}^{2}\tau_{2}$ as the area of the $T^{2}$.

For a fixed boundary geometry $T^{2}\times{\mathbb{R}}^{d-3}$ (i.e. fixed
complex moduli $\tau_{1}$), clearly there are multiple AdS soliton solutions
related by $PSL(2,{\mathbb{Z}})$ transformations, corresponding to filling in
different one cycles on the $T^{2}$ in the bulk. The CFT ground state then
naturally corresponds to the AdS soliton that has the minimal energy density.
From (5.41), this comes from minimizing the length $L_{1}$ of a non-
contractible cycle on the boundary $T^{2}$ that becomes contractible in the
bulk. The minimal length of a non-contractible cycle on a compact manifold
$\Sigma$ is known as the systole and denoted by ${\rm sys}(\Sigma)$. Therefore
the solution (5.40) is the bulk dual for the CFT ground state when the $x_{1}$
direction is along the shortest geodesic on the $T^{2}$. The rescaled
dimensionless Casimir energy density (see (1.3)) is determined by the systole
on $T^{2}$,

$\displaystyle{\cal E}(\tau)={(4\pi)^{d-1}\ell^{d-1}\over
4d^{d}G_{N}}{A^{d\over 2}\over{\rm sys}(T^{2})^{d}}\,.$ (5.42)

which is manifestly modular invariant since ${\rm sys}(T^{2})$ is invariant.

In the limit of large $\tau_{2}$, we see (5.42) matches onto the universal
formula (2.15) from EFT analysis with

$\displaystyle c_{1}={(4\pi)^{d-1}\ell^{d-1}\over 4d^{d}G_{N}}\,,\quad
c_{2}=0\,,$ (5.43)

where $c_{1}$ is proportional to the thermal free energy (1.8) (see (2.9)),
which is natural from the bulk due to the relation between the black brane
(which dominates the canonical ensemble) and the AdS soliton by double Wick
rotations [59]. Furthermore, the vanishing $c_{2}$ implies that upon circle
reduction, the boundary CFT is completely gapped in the large $N$ limit.
Finally, there are no non-perturbative contributions in the standard
fundamental domain, which is related to the non-smooth feature of
(5.42).212121See [61] for another perspective on the restricted dependence
(5.42) of ${\mathcal{E}}(\tau)$ on the boundary geometry.

The AdS/CFT dictionary determines the bulk parameters $\ell,G_{N}$ in terms of
the boundary CFT data. While the detailed relations depend on the specific
AdS/CFT dual pairs, the general property is that in the large $N$ limit, the
$1/N$ corrections in the CFT correspond to higher derivative interactions in
the bulk quantum gravity. When there is another tunable parameter in the large
$N$ CFT, such as the marginal coupling $g_{\rm YM}$ in the ${\mathcal{N}}=4$
super-Yang-Mills theory (SYM), one may further take the ’t Hooft limit where
the ’t Hooft coupling $\lambda$ (e.g. $\lambda=Ng_{\rm YM}^{2}$ in the SYM) is
fixed as $N\to\infty$. This is possible for CFTs with string theory duals,
where non-planar (higher-genus) and nonperturbative (in string coupling)
effects are suppressed in the ’t Hooft limit. In Table 1, we gather the
relevant AdS/CFT dictionary for the following well-known AdS/CFT dual pairs,
in $d=3$ between type IIA string theory on $AdS_{4}\times\mathbb{CP}^{3}$ and
the $U(N)_{k}\times U(N)_{-k}$ ABJM theory [62], in $d=4$ between type IIB
string theory on $AdS_{5}\times S^{5}$ and the ${\mathcal{N}}=4$ $SU(N)$
super-Yang-Mills theory [15], in $d=5$ between type I’ string theory (type IIA
with O8 orientifold) on the warped background $AdS_{6}\times_{\rm w}{\rm
HS}^{4}$ and the 5d ${\mathcal{N}}=1$ rank $N$ Seiberg theories with $N_{f}<8$
fundamental flavors [63, 64], and in $d=6$ between M-theory on $AdS_{7}\times
S^{4}$ and the 6d ${\mathcal{N}}=(2,0)$ theories [15]. Also included in Table
1 is the coefficient $c_{1}$ that determines the toroidal Casimir energy for
these large $N$ CFTs to the leading order in the large $N$ limit via (5.42)
and (5.43) which is valid at strong coupling in the CFT.222222These AdS/CFT
dual pairs as defined are fermionic and thus require a spin structure on the
boundary manifold that extends into the bulk. Here we have implicitly summed
over the spin structures on ${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-2}$
to produce a bosonic CFT on the boundary so that the general modular-invariant
large $N$ formula (5.42) still applies. If we choose to work directly in the
fermionic CFT with a bounding spin structure $\rho$, then ${\rm sys}(T^{2})$
in (5.42) needs to be replaced by the shortest geodesic length on $T^{2}$
along an anti-periodic cycle for the fermion (see general comments in Section
6 on toroidal Casimir energy in fermionic CFT).

At subleading orders in $1\over N$, there are corrections to (5.42) and (5.43)
coming from higher derivative interactions in the bulk. In type IIA/IIB string
theory and M-theory, the leading higher derivative term takes the schematic
form ${\mathcal{R}}^{4}$ which is a quartic term in the Riemann curvature
tensor with a unique supersymmetric completion [65]. The ${\mathcal{R}}^{4}$
term modifies the AdS soliton solution and also the energy density. Using the
AdS/CFT dictionary (see Table 1), this leads to a correction to $c_{1}$ in
(5.43) at ${\mathcal{O}}(N^{1\over 2})$ for the 3d ABJM and the 4d
${\mathcal{N}}=4$ SYM, and at ${\mathcal{O}}(N)$ for the 5d ${\mathcal{N}}=1$
Seiberg theories232323The calculation in the AdS6 dual of the 5d Seiberg
theories will be more subtle than the other cases because of divergences in
the 10d background at the locus of the flavor D8 branes and the O8 orientifold
plane (see for example [66] where these divergences cancel for certain
physical observables in the 5d CFT). and the 6d ${\mathcal{N}}=(2,0)$
theories.

The precise coefficient for this correction has been computed explicitly for
the ${\mathcal{N}}=4$ SYM in [67] in the ’t Hooft limit with large ’t Hooft
coupling $\lambda=Ng^{2}_{\rm YM}$,242424More precisely, [67] analyzed the
corrections to the black brane solution in AdS5 and changes to the thermal
free energy due to the ${\mathcal{R}}^{4}$ interaction. The corrections for
the AdS soliton is then obtained from a double Wick rotation. It would be
interesting to generalize this analysis to other holographic duals in string
or M-theory.

$\displaystyle c_{1}^{\rm{SYM}}={\pi^{2}\over 8}N^{2}\left(1+{15\zeta(3)\over
8}\lambda^{-{3\over 2}}+{\mathcal{O}}(\lambda^{-{5\over
2}})\right)+{\mathcal{O}}(N^{0})\,,$ (5.44)

where the first term on the RHS captures all the planar contributions and the
further subleading terms in $1\over\lambda$ come from bulk interactions at
even higher derivative orders such as $D^{4}{\mathcal{R}}^{4}$ [68]. The non-
planar contributions are contained in the second term on the RHS. Using the
fact that the ${\mathcal{R}}^{4}$ interaction depends on the type IIB axion-
dilaton, equivalently the complexified Yang-Mills coupling $\tau_{\rm
YM}\equiv{4\pi i\over g_{\rm YM}^{2}}+{\theta\over 2\pi}$, through the real
analytic Eisenstein series $E_{3\over 2}(\tau_{\rm YM})$ [65], the large $N$
expansion of $c_{1}^{\rm SYM}$ at fixed $\tau_{\rm YM}$ is given by,

$\displaystyle c_{1}^{\rm{SYM}}={\pi^{2}\over
8}N^{2}+{15\zeta(3)\sqrt{\pi}\over 512}\sqrt{N}E_{3\over 2}(\tau_{\rm
YM})+{\mathcal{O}}(N^{0})\,,$ (5.45)

which also makes manifest the $SL(2,{\mathbb{Z}})$ duality invariance in
$\tau_{\rm YM}$ of the type IIB string theory and the ${\mathcal{N}}=4$ SYM.

Dim | ${\ell^{d-1}/G_{N}}$ | $\ell$ | $\lambda$ | $c_{1}$
---|---|---|---|---
$d=3$ | ${2\sqrt{2}\over 3}k^{1/2}N^{3/2}$ | $(\pi^{2}/2)^{1/6}N^{1/6}k^{1/6}\ell_{s}$ | ${N\over k}$ | ${8\sqrt{2}\pi^{2}\over 81}k^{1/2}N^{3/2}$
$d=4$ | ${2\over\pi}N^{2}$ | ${g_{\rm YM}^{1/2}N^{1/4}}\ell_{s}$ | ${Ng_{\rm YM}^{2}}$ | ${\pi^{2}\over 8}N^{2}$
$d=5$ | ${8\sqrt{2}\over 15\pi}\sqrt{8-N_{f}}N^{5/2}$ | $(18\pi^{2})^{1/4}(8-N_{f})^{-1/4}N^{1/4}\ell_{s}$ |  | ${512\sqrt{2}\pi^{3}\over 46875}{\sqrt{8-N_{f}}N^{5/2}}$
$d=6$ | ${16\over 3\pi^{2}}N^{3}$ | $(8\pi)^{1/3}N^{1/3}\ell_{11}$ |  | ${64\pi^{3}\over 2187}N^{3}$
Table 1: The AdS/CFT dictionary involving the Newton’s constant $G_{N}$ and
the AdS scale $\ell$ on AdSd+1 for the $d=3$ ABJM CFT, the $d=4$
${\mathcal{N}}=4$ $SU(N)$ SYM, the $d=5$ rank $N$ Seiberg theories and the
$d=6$ $SU(N)$ ${\mathcal{N}}=(2,0)$ theories. Here $\ell_{s}$ is the string
length and $\ell_{11}$ is the Planck length in M-theory. The ’t Hooft
couplings $\lambda$ are listed for the $d=3,4$ examples where an ’t Hooft
limit exists.

Contrary to $c_{1}^{\rm SYM}$ which receives a ${\mathcal{O}}(N^{1/2})$
correction, the coefficient $c_{2}^{\rm SYM}$ in (5.43) stays zero at
${\mathcal{O}}(N^{1/2})$.252525See Section 3 of [67] for the modified black
brane solution after taking into account the ${\mathcal{R}}^{4}$ interaction.
The modified AdS soliton (with rectangular $T^{2}$ on the boundary) is
obtained from double Wick rotation and compactifying one of the remaining
noncompact boundary directions. Using the results there, it is easy to see
that the energy density for the AdS soliton still depend on $L_{2},L_{1}$ via
$L_{2}/L_{1}^{d-1}$ as in (5.39) (i.e. extensive in $L_{2}$). Consequently
$c_{2}^{\rm SYM}=0$ to this order in the large $N$ expansion (otherwise it
would lead to non-extensive $L_{2}$ dependence in $E_{\rm soliton}$). More
generally, we expect $c_{2}^{\rm SYM}=0$ to all orders in $1\over N$ (at
nonzero $g_{\rm YM}$). This is because the SYM reduced on a circle with
thermal boundary condition262626As explained in footnote 22, we are summing
over spin structures $\rho$ on $T^{2}$. Out of the four spin structures, three
of them are bounding (even) and the remaining one is nonbounding (odd). Only
the odd spin structure (periodic boundary condition for the fermions along all
cycles) is compatible with supersymmetry (SUSY) and the SUSY algebra ensures
the ground state energy in this sector vanishes exactly. Therefore the actual
ground state of the full theory is defined in an even spin structure (which
has negative energy). For a similar reason (SUSY is preserved by periodic
boundary conditions for fermions), in the thin torus limit, out of the three
even spin structures, the one with anti-periodic (thermal) boundary condition
along the small cycle dominates.  for the fermions is described at low energy
by the pure $d=3$ Yang-Mills theory, which confines and has a mass gap [69,
70, 18]. Nonetheless, we expect the Casimir energy ${\mathcal{E}}^{\rm SYM}$
to behave drastically different at ${\mathcal{O}}(N^{0})$ where one-loop
effects in gravity enter. In particular, the one-loop determinants for the
bulk fields will depend nontrivially on the complex moduli $\tau$ of the
torus, producing non-perturbative terms in $\tau_{2}$ that modifies (5.42) at
${\mathcal{O}}(N^{0})$.

If we work in the ’t Hooft limit and focus on the planar contribution to
${\mathcal{E}}^{\rm SYM}$, it suffices to consider tree-level string theory in
the bulk. In this case, we expect the $\tau$ dependence in (5.42) for the
large $N$ Casimir energy to persist to all orders in $1\over\lambda$,272727We
have already discussed why $c_{2}^{\rm SYM}=0$ from the field theory side (at
nonzero coupling). This can also be seen concretely from the bulk string
theory in the ’t Hooft limit. At tree level, the translation symmetry along
$T^{2}\times{\mathbb{R}}^{d-2}$ is preserved in the (deformed) AdS soliton and
the metric only depends on the size of the contractible direction (e.g.
$x_{1}$ direction in (5.40)) through the regularity condition. Therefore
$E_{\rm soliton}^{\rm SYM}$ is proportional to $A$ and then by dilatation
symmetry its $T^{2}$ dependence (after rescaled by (1.3)) takes the same form
as in (5.42).

$\displaystyle\lim_{\stackrel{{\scriptstyle N\to\infty}}{{\lambda\,{\rm
fixed}}}}{1\over N^{2}}{\mathcal{E}}^{\rm SYM}={\pi^{2}\over 8}f^{\rm
SYM}(\lambda){A^{2}\over{\rm sys}(T^{2})^{4}}\,,\quad f^{\rm
SYM}(\lambda)=1+{15\zeta(3)\over 8}\lambda^{-{3\over
2}}+{\mathcal{O}}(\lambda^{-{5\over 2}})\,,$ (5.46)

where $f^{\rm SYM}(\lambda)$ receives contributions from tree-level higher
derivative interactions in type IIB string theory.282828A priori one may
wonder if there are worldsheet instanton corrections that depend on the
complex moduli $\tau$ of the $T^{2}$. However there are no nontrivial second
homology class in the AdS soliton geometry that can support worldsheet
instantons of finite actions. This does not completely rule out worldsheet-
instanton-like contributions that are non-perturbative in $\ell_{s}^{2}$,
equivalently $1\over\sqrt{\lambda}$ (see for example [71]). Here we have
suppressed such potential contributions in (5.46). As the consequence of the
relation (2.9), the same function $f(\lambda)$ determines the thermal free
energy of the SYM in the ’t Hooft limit which has been studied extensively in
the literature since [67].

We end by noting the sharp distinction between the $\tau$ dependence of the
toroidal Casimir energy ${\mathcal{E}}(\tau)$ in CFTs with matrix large $N$
limits discussed here which have Einstein gravity duals, and those with vector
large $N$ limits which have higher-spin gravity duals. In particular, at
$d=3$, for the ABJM theory and the $O(N)$ CFT in the leading large $N$ limit,
we have

$\displaystyle{\cal E}_{{\rm ABJM}_{N,k}}(\tau)={8\sqrt{2}\pi^{2}\over
81}k^{1/2}N^{3/2}{A^{3/2}\over{\rm sys}(T^{2})^{3}}\,,$ (5.47)
$\displaystyle{\cal E}_{O(N)}(\tau)=N\left({2\zeta(3)\over
5\pi}\tau_{2}^{3/2}+e^{-2\pi\Delta_{0}\tau_{2}}\sqrt{2\pi\Delta_{0}}\left(1+{3\over
16\pi\tau_{2}\Delta_{0}}+{\mathcal{O}}(\tau_{2}^{-2})\right)+{\mathcal{O}}(e^{-4\pi\Delta_{0}\tau_{2}})\right)\,.$

The expression for the $O(N)$ CFT is smooth in $\tau\in{\mathbb{H}}$ and has
an infinite tower of instanton contributions, both of which are features
absent in the ABJM case. This signals the different natures of their
holographic duals and it would be interesting to investigate this further in
the context of ABJ triality [66].

##  6 Discussions

In this paper we have studied general properties of $d>2$ CFT on
${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-3,1}$ (or
${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-2}$ in the Euclidean signature)
which is one of the simplest spacetime manifold of nontrivial topology and a
natural generalization of the thermal background
$S^{1}\times{\mathbb{R}}^{d-1}$. We focused on a basic observable on this
geometry, namely the Casimir energy (ground state energy) on the spatial
manifold ${\mathcal{M}}_{d-1}=T^{2}\times{\mathbb{R}}^{d-3}$. The
dimensionless Casimir energy density ${\mathcal{E}}(\tau)$ is a nontrivial
modular invariant function of the complex moduli $\tau$ of the torus $T^{2}$
and appears to be independent from conventional CFT data. In the thin torus
limit $\tau_{2}\to\infty$, we derived a simple universal formula (2.15) for
${\mathcal{E}}(\tau)$ which shows that the toroidal Casimir energy is
controlled by two perturbative terms in $\tau_{2}$ up to non-perturbative
corrections. This was accomplished via an effective field theory (EFT)
argument by compactifying the $d$ dimensional CFT on the small cycle of the
thin torus. The coefficients $c_{1},c_{2}$ of the perturbative terms are
proportional to familiar finite temperature observables in the CFT and its
circle reduction. The remaining non-perturbative terms in (2.15) are accounted
for by worldline instantons associated with massive particles in the $d-1$
dimensional EFT going around the remaining cycle in the base manifold.
Combining with $PSL(2,{\mathbb{Z}})$ spectral theory which is a powerful
framework to study modular invariant functions, we translated the EFT
constraints into a set of stringent conditions on the spectral overlap in the
decomposition of ${\mathcal{E}}(\tau)$ with respect to eigenfunctions of the
$PSL(2,{\mathbb{Z}})$ invariant Laplacian. This spectral decomposition makes
more explicit the class of modular invariant functions that are physically
relevant for describing the toroidal Casimir energy ${\mathcal{E}}(\tau)$ in
CFT. It also explains intricate relations between EFT data that enters into
the universal formula for ${\mathcal{E}}(\tau)$ (2.15). For example, although
${\mathcal{E}}(\tau)$ in general receives worldline instanton contributions in
the EFT from all KK charges $Q$, effects at higher total KK charges are
completely determined by the $Q=0$ and $Q=\pm 1$ sector as a consequence of
the modular invariance. We illustrate these universal properties of
${\mathcal{E}}(\tau)$ in concrete CFT examples including the free scalar CFT,
the critical $O(N)$ model and holographic CFTs with Einstein gravity duals.
Below we discuss a number of open questions and future directions.

#### Sign of Casimir Energy and Universal Bounds

One intriguing question is the sign of the Casimir energy density and possible
universal bounds on its magnitude. From the universal behavior of the toroidal
Casimir energy (2.15), it is clear that ${\mathcal{E}}(\tau)$ is positive for
large $\tau_{2}$ and unbounded from above. This translates to a negative
vacuum energy density $E_{\rm vac}$ that can be arbitrarily negative by tuning
$\tau$.292929Recall the negative sign in the relation (1.3) between $E_{\rm
vac}$ and ${\mathcal{E}}(\tau)$. Furthermore, we observe in all the examples
studied here ${\mathcal{E}}(\tau)$ is positive everywhere on the upper half-
plane. We are thus led to the following conjecture,

###### Conjecture 1

For any $d\geq 3$ unitary bosonic CFT (that is not a TQFT), the dimensionless
modular-invariant toroidal Casimir energy density ${\mathcal{E}}(\tau)$ as
defined in (1.3) is strictly positive.

For $\tau_{1}=0$, this positivity was proven in [11]. In fact in that case a
stronger result holds [11, 72], stating that along $\tau_{1}=0$,
${\mathcal{E}}(\tau)$ is convex and monotonically increasing in
$\tau_{2}\in(0,\infty)$. For holographic $d=3$ CFTs, a gravity argument was
given in [73] for the non-positivity of vacuum energy density $E_{\rm vac}\leq
0$ on ${\mathcal{M}}_{3}={\mathbb{R}}\times\Sigma$ with a general closed
spatial two-dimensional manifold $\Sigma$ (see also [74]). It would be
interesting to find a field theoretic proof that applies at general $\tau$
(and potentially for more general closed spatial manifolds303030Let us collect
some field theory evidence for this statement in $d=3$ when the CFT is defined
on ${\mathbb{R}}\times\Sigma$. When $\Sigma$ is a sphere, we have $E_{\rm
vac}(S^{2})=0$ which follows from the operator-state correspondence and the
absence of Weyl anomaly in $d=3$ (for the same reason the spherical Casimir
energy vanishes for unitary CFT in all odd spacetime dimensions). For free
scalar CFT when $\Sigma$ is a compact hyperbolic surface (see [75] for more
general discussions for free massless scalars), the regulated and rescaled
vacuum energy is determined by the Selberg zeta function for the hyperbolic
Laplacian $\Delta_{\Sigma}$ as $E_{\rm vac}(\Sigma)={1\over
2}\left.\zeta_{\Delta_{\Sigma}}(s)\right|_{s=-{1\over 2}}$. This zeta function
is negative at $s=-{1\over 2}$ as a consequence of the Selberg trace formula
(see Corollary 3.4 in [76]). Moreover, the genus two Bolza surface is an
extrema (a consequence of its large discrete isometry group [77]) for $E_{\rm
vac}(\Sigma)$ over the Teichmüller space (moduli space of hyperbolic
structures) with $E_{\rm vac}\approx-0.325003$ [78]. It would be interesting
to show if this is the global maximum (e.g. by generalizing the bootstrap
analysis in [79, 80] which found that the Bolza surface maximizes the spectral
gap).).

As a consequence of the modular invariance, the special points $\tau=i$ and
$\tau=e^{\pi i/3}$ on the standard fundamental domain which are preserved
respectively by ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ subgroups of
$PSL(2,{\mathbb{Z}})$ are extrema of ${\mathcal{E}}(\tau)$. From the CFT
examples we have studied, we notice that the ${\mathbb{Z}}_{3}$-symmetric
point $\tau=e^{2\pi i\over 3}$ is always the global minimum of
${\mathcal{E}}(\tau)$ while the ${\mathbb{Z}}_{2}$-symmetric point $\tau=i$ is
a saddle point. For free scalar theories, this is a property of the real
analytic Eisenstein series $E_{d\over 2}(\tau)$.313131To see this, we note
that the Eisenstein series $E_{s}(\tau)$ is related to the Epstein series
$E(\Lambda,s)$ on the two dimensional lattice $\Lambda$ that defines the flat
torus via $T^{2}={\mathbb{C}}/\Lambda$ as below,  $\displaystyle
E(\Lambda,s)\equiv\sideset{}{{}^{\prime}}{\sum}_{v\in\Lambda}\langle
v,v\rangle^{-s}\,,\quad E_{s}(\tau)={1\over 2\zeta(2s)}E(\Lambda,s)\,,$ (6.1)
where $v=m\tau+n$ and the quadratic form is defined by $\langle
v,v\rangle\equiv{|m\tau+n|^{2}\over\tau_{2}}$ such that the corresponding Gram
matrix has determinant one. It is well-known that the minimization of the
Epstein series on a rank $d$ lattice is related to finding the densest sphere
packing in ${\mathbb{R}}^{d}$ (more precisely lattice type packing). For
$d=2$, the minimum of $E(\Lambda,s)$ (and correspondingly the densest sphere
packing in ${\mathbb{R}}^{2}$) is achieved by the triangular lattice which
corresponds to $\tau=e^{2\pi i/3}$ [81, 82, 83, 84, 85].  For the critical
$O(N)$ model, these structures are clear from Figure 1. For holographic CFTs,
it follows from Loewner’s inequality for the systole on $T^{2}$ [86],

$\displaystyle{\rm sys}(T^{2})\leq{2\over\sqrt{3}}{\rm vol}(T^{2})\,,$ (6.2)

which is saturated for flat torus with precisely $\tau=e^{2\pi i\over 3}$. We
thus propose the following conjecture,

###### Conjecture 2

For any $d\geq 3$ unitary bosonic CFT (that is not a TQFT), the dimensionless
modular-invariant toroidal Casimir energy density ${\mathcal{E}}(\tau)$ has a
global minimum at $\tau=e^{2\pi i\over 3}$ on the standard
$PSL(2,{\mathbb{Z}})$ fundamental domain.

#### Generalizations to Fermionic CFTs

Thus far we have mostly focused on bosonic CFTs (except for comments in the
holographic examples). In the presence of fermionic matter, to define the CFT
on a general spacetime manifold ${\mathcal{M}}_{d}$ requires a spin structure.
For ${\mathcal{M}}_{d}=T^{2}\times{\mathbb{R}}^{d-2}$, there are four spin
structures that correspond to either periodic or anti-periodic boundary
conditions for the fermions along each of the two independent cycles on
$T^{2}$. The choice of the spin structure $\rho$ is a part of the data that
specifies the fermionic CFT on this geometry (in addition to the metric). The
ground state and the corresponding Casimir energy ${\mathcal{E}}_{\rho}(\tau)$
depend on $\rho$ in addition to the complex moduli $\tau$. Relatedly, for a
fixed spin structure $\rho$, the full $PSL(2,{\mathbb{Z}})$ invariance of the
bosonic Casimir energy ${\mathcal{E}}(\tau)$ is broken to congruence subgroups
preserving $\rho$ for the fermionic counterpart ${\mathcal{E}}_{\rho}(\tau)$.
Consequently, ${\mathcal{E}}_{\rho}(\tau)$ will have different modular
properties. There is also a simple generalization of the EFT analysis in
Section 2.1 for fermionic CFT with an analogous universal formula as (2.15).
In subsequent work [9], we study the toroidal Casimir energy for fermionic
CFTs systematically with concrete examples including the Gross-Neveu model and
Chern-Simons-Matter CFTs.

#### Excited States and Refinement by Symmetries

It is also interesting to study excited states in the Hilbert space on
${\mathcal{M}}_{d-1}=T^{2}\times{\mathbb{R}}^{d-3}$ and the energy spectrum.
In particular, in the similar way that the asymptotic density of high energy
states in $d=2$ CFT on $S^{1}$ is determined by the CFT central charge via the
Cardy formula [87], the asymptotic density of states on $T^{d-1}$ is
determined by the Casimir energy using modular invariance on $T^{d}$
[11].323232The case ${\mathcal{M}}_{d-1}=T^{2}\times{\mathbb{R}}^{d-3}$ can be
thought of a limit of ${\mathcal{M}}_{d-1}=T^{d-1}$. It would be interesting
to compute this asymptotic density explicitly in $d\geq 3$ CFTs, such as the
3d $O(N)$ critical model on ${\mathbb{R}}\times T^{2}$ (see [49]).

In CFTs with global symmetries, it is natural to refine the observables by
including symmetry twists (insertions of topological defects representing the
symmetry), which organize the states into representations of the symmetry and
also give rise to twisted sectors in the Hilbert space. In particular, the
Cardy formula in $d=2$ has a symmetry-refined version [88, 89] that
characterizes the asymptotic growth of operator degeneracies in a fixed charge
or twist sector. The generalization to higher $d$ CFTs with zero-form
symmetries was recently explored in [90]. Symmetries are particularly
interesting when they carry ’t Hooft anomalies, which are often reflected by
degeneracies in the twisted Hilbert space [91]. For $d=3$ CFT on
${\mathcal{M}}_{3}={\mathbb{R}}\times T^{2}$, the spatial manifold supports
both nontrivial zero-form and one-form symmetry defects and thus provides an
ideal playground to investigate consequences of anomalies for generalized
symmetries (and gravity) in CFT.

#### State-operator Correspondence on $T^{2}$ and Line Defects

In contrast to the familiar correspondence between states on $S^{2}$ and local
operators in a 3d CFT, the state-operator correspondence on a spatial $T^{2}$
is much more mysterious. In particular, as explained in [8], a basic
difficulty comes from the obstruction in realizing the ground state on $T^{2}$
via a Euclidean path integral over a compact three-manifold that fills in the
$T^{2}$. It is natural to expect line defects to play an important role in the
construction of states on $T^{2}$. After all, this is how states on $T^{2}$ in
a 3d TQFT are constructed. They come from anyons threading the non-
contractible cycle of the solid torus that fills in the $T^{2}$ [92]. For
general 3d CFTs, while it is not clear what states on $T^{2}$ can be
constructed by threading a non-topological line defect in the solid torus, one
can turn the question around and deduce constraints on the line defects in the
CFT. This is currently under investigation.

## Acknowledgments

We thank Nathan Benjamin, Scott Collier, Himanshu Khanchandani, Zohar
Komargodski, Juan Maldacena, and Edgar Shaghoulian for interesting
discussions. We are also grateful to Nathan Benjamin and Edgar Shaghoulian for
helpful comments on a draft. The work of YW was supported in part by NSF grant
PHY-2210420.

##  Appendix A Regularization of Lattice Sum on Torus

In the main text, we have encountered the following sum over lattice momentum
vectors on a torus of complex moduli $\tau$,

$\displaystyle\sum_{m,n\in{\mathbb{Z}}}{1\over(|m\tau+n|^{2}+\tau_{2}^{2}\Delta(\tau)^{2})^{s}}\,.$
(A.1)

The sum with $s=-\frac{1}{2}$ appears in the expressions for the Casimir
energy on $T^{2}$, and the sum with $s=\frac{1}{2}$ appears in the gap
equation (e.g. for the $O(N)$ model at large $N$). This sum is convergent for
$s>1$, and will require regularization otherwise. In this appendix we discuss
regularization of such sums using a generalization of the real analytic
Eisenstein series.

In [49], a regularization of (A.1) is provided by analytic continuation in
both $s$ and $d$ (the rank of the lattice) via the integral representation in
the last equality below,

$\displaystyle
g_{s}^{(d)}(\Delta,\tau,a_{1},a_{2})=\sum_{n,m\in\mathbb{Z}^{d/2}}\frac{1}{\left(\left|m+a_{2}+\left(n+a_{1}\right)\tau\right|^{2}+\gamma^{2}\right)^{s}}$
(A.2)
$\displaystyle=\frac{\pi^{s}}{\Gamma(s)}\int_{0}^{\infty}d\lambda\lambda^{s-1}\exp\left[-\pi\lambda\gamma^{2}-\frac{d\pi\lambda}{2}\left(\left(a_{1}\tau_{2}\right)^{2}+\left(a_{2}+a_{1}\tau_{1}\right)^{2}\right)\right]\Theta\left(\lambda,\bm{\Omega}(\tau),\mathbf{v}_{1}\right)^{d/2}$
$\displaystyle=\frac{\pi^{s}}{\Gamma(s)}\left\\{\int_{1}^{\infty}d\lambda\lambda^{s-1}\exp\left[-\frac{\lambda\tau_{2}^{2}L^{2}\Delta^{2}}{4\pi}-\frac{d\pi\lambda}{2}\left(\left(a_{1}\tau_{2}\right)^{2}+\left(a_{2}+a_{1}\tau_{1}\right)^{2}\right)\right]\Theta\left(\lambda,\bm{\Omega}(\tau),\mathbf{v}_{1}\right)^{d/2}\right.$
$\displaystyle+\tau_{2}^{-d/2}\int_{1}^{\infty}d\lambda\lambda^{d/2-s-1}\left[\exp\left(-\frac{\tau_{2}^{2}L^{2}\Delta^{2}}{4\pi\lambda}\right)\Theta\left(\lambda,\bm{\Omega}(\tau)^{-1},\mathbf{v}_{2}\right)^{d/2}-1+\frac{\tau_{2}^{2}L^{2}\Delta^{2}}{4\pi\lambda}\right]$
$\displaystyle\left.+\frac{\tau_{2}^{-d/2}}{s-d/2}-\frac{L^{2}\Delta^{2}}{4\pi}\frac{\tau_{2}^{2-d/2}}{1+s-d/2}\right\\}\,,$

where

$\displaystyle\gamma=\frac{\tau_{2}L\Delta}{2\pi},\quad\bm{\Omega}(\tau)=\left(\begin{array}[]{cc}|\tau|^{2}&-\tau_{1}\\\
-\tau_{1}&1\end{array}\right),\quad\bm{\Omega}(\tau)^{-1}=\frac{1}{\tau_{2}^{2}}\left(\begin{array}[]{cc}1&\tau_{1}\\\
\tau_{1}&|\tau|^{2}\end{array}\right),$ (A.3)
$\displaystyle\mathbf{v}_{1}=t\left(\begin{array}[]{c}\tau_{1}\left(-a_{2}+a_{1}\tau_{1}\right)+a_{1}\tau_{2}^{2}\\\
a_{2}-a_{1}\tau_{1}\end{array}\right),\quad\mathbf{v}_{2}=-i\left(\begin{array}[]{l}a_{1}\\\
a_{2}\end{array}\right),$
$\displaystyle\Theta(t,\bm{\Omega},\mathbf{u})\equiv\sum_{\mathbf{n}\in\mathbb{Z}^{2}}\exp\left(-\pi
t\mathbf{n}^{\top}\cdot\bm{\Omega}\cdot\mathbf{n}-2\pi\mathbf{n}^{T}\cdot\mathbf{u}\right).$

Here $a_{1},a_{2}\in[0,1)$ keep track of possible twisted boundary conditions
for a complex scalar along the two independent cycles of $T^{2}$.

The regulated expression above from [49] is finite at $d=2$ and $s=\pm 1/2$
which are the cases relevant for the Casimir energy and gap equation in the
$O(N)$ scalar CFT. However we find the integral formula cumbersome to deal
with both analytically and numerically.

Here we derive alternative but equivalent representations of the regulated
lattice sum which will be more efficient for our analysis and will also make
more explicit the modular properties. We start by defining the following
generalized Eisenstein series,

$\displaystyle
G_{s}(\tau,\Delta,a_{1},a_{2})=\frac{1}{2}\pi^{-s}\Gamma(s)\sum_{n,m\in\mathbb{Z}}\left(\frac{\tau_{2}}{|(m+a_{2})-(n+a_{1})\tau|^{2}+\Delta^{2}\tau_{2}^{2}}\right)^{s}\,,$
(A.4)

where the extra $\tau_{2}^{s}$ factor compared to (A.2) is introduced so that
$G_{s}$ transforms nicely under $PSL(2,{\mathbb{Z}})$.333333Note that in going
between (A.2) from [49] and our (A.4), there is a notation change $L^{\rm
there}=L_{1}^{\rm here}$. In this paper, we only make use of
$G_{s}(\tau,\Delta,a_{1},a_{2})$ with $a_{1}=a_{2}=0$ (5.14) which is a
modular invariant function as long as $\tau_{2}\Delta^{2}(\tau)$ is modular
invariant. For general $a_{i}$, $G_{s}(\tau,\Delta,a_{1},a_{2})$ transforms as
a non-holomorphic Jacobi form. In the following we keep $a_{i}$ general since
the manipulations we present work uniformly regardless of the values of
$a_{i}$ and the general case will be useful when incorporating symmetry twists
in subsequent work.

We will offer two different regulated expressions for the generalized
Eisenstein series $G_{s}$ defined in (A.4), which are complementary and will
both be useful for different situations. One is given by a sum of Poincaré
series,

$\displaystyle G_{s}(\Delta,\tau,a_{1},a_{2})=$
$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{m,n\in\mathbb{Z}}(\frac{\tau_{2}\Delta}{|n+m\tau|})^{1-s}K_{1-s}(2\pi\Delta|n+m\tau|)e^{2\pi
i(a_{1}n+a_{2}m)}$ (A.5)
$\displaystyle+\frac{\Gamma(s-1)}{2}(\pi\tau_{2}\Delta^{2})^{1-s}\,,$

and the other in terms of a Fourier series in $\tau_{1}$,343434Note that
$\Delta$ still has dependence on $\tau_{1}$ in general.

$\displaystyle G_{s}(\Delta,\tau,a_{1},a_{2})=$
$\displaystyle\sum_{n\in\mathbb{Z}+a_{1}}\sum_{m\neq 0}e^{2\pi
im(n\tau_{1}-a_{2})}m^{s-{1\over 2}}\tau_{2}^{1\over
2}\left(\Delta^{2}+n^{2}\right)^{\frac{1-2s}{4}}K_{\frac{1}{2}-s}(2\pi\sqrt{\Delta^{2}+n^{2}}m\tau_{2})$
(A.6) $\displaystyle+\tau_{2}^{1-s}\sum_{n\neq
0}\left(\frac{\Delta}{n}\right)^{1-s}K_{1-s}(2\pi\Delta n)e^{-2\pi
ina_{1}}+\frac{\Gamma(s-1)}{2}(\pi\tau_{2}\Delta^{2})^{1-s}\,.$

We first derive (A.5) from the expression in the second line of (A.2) (see
footnote 33 for the notation change) by performing Poisson resummations in
both $n$ and $m$ for the modes of the Riemann theta function $\Theta$ labeled
by ${\bf n}=(n,m)$, and then further splitting the new sum into the nonzero
modes (${\bf n}\neq(0,0)$) and the zero mode (${\bf n}=(0,0)$),

$\displaystyle\,G_{s}(\Delta,\tau,a_{1},a_{2})=\,\frac{\tau_{2}^{s-1}}{2}\int_{0}^{\infty}dtt^{-s}\exp\left(-\frac{\pi\tau_{2}^{2}\Delta^{2}}{t}\right)\left(\Theta\left(t,\bm{\Omega}(\tau)^{-1},\mathbf{v}_{2}\right)-1\right)$
(A.7) $\displaystyle+$
$\displaystyle\frac{\tau_{2}^{s-1}}{2}\int_{0}^{\infty}dtt^{s-2}e^{-\pi
t\tau_{2}^{2}\Delta^{2}}\,.$

The integral of the nonzero modes above produce the sum of Bessel functions in
(A.5). The integral of the zero mode is regulated by analytic continuation in
$s$, yielding the remaining term in (A.5).

The other regulated formula (A.6) for the generalized Eisenstein series is
derived by Poisson resummation in a different way. We split the Riemann theta
function $\Theta$ into modes ${\bf n}=(n,m)$ with $m=0$ or $m\neq 0$, and
perform Poisson resummation with respect to $n$. The zero mode is regulated by
analytic continuation in $s$ as above. The resulting expression in (A.6) then
follows.

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