# Notes on massless scalar field partition functions, modular inariance and
Eisenstein series

Francesco Alessio, Glenn Barnich, Martin Bonte

Notes on massless scalar field partition functions, modular invariance and
Eisenstein series

Francesco Alessioa,b,c, Glenn Barnichb, Martin Bonteb

a Nordita

KTH Royal Institute of Technology and Stockholm University

Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

E-mail<EMAIL_ADDRESS>
b Physique Théorique et Mathématique, Université libre de Bruxelles and

International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles,
Belgium

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
c Dipartimento di Fisica “E. Pancini” and

INFN Università degli studi di Napoli “Federico II”, I-80125 Napoli, Italy

Abstract. The partition function of a massless scalar field on a Euclidean
spacetime manifold $\mathbb{R}^{d-1}\times\mathbb{T}^{2}$ and with momentum
operator in the compact spatial dimension coupled through a purely imaginary
chemical potential is computed. It is modular covariant and admits a simple
expression in terms of a real analytic SL$(2,\mathbb{Z})$ Eisenstein series
with $s=(d+1)/2$. Different techniques for computing the partition function
illustrate complementary aspects of the Eisenstein series: the functional
approach gives its series representation, the operator approach yields its
Fourier series, while the proper time/heat kernel/world-line approach shows
that it is the Mellin transform of a Riemann theta function. High/low
temperature duality is generalized to the case of a non-vanishing chemical
potential. By clarifying the dependence of the partition function on the
geometry of the torus, we discuss how modular covariance is a consequence of
full SL$(2,\mathbb{Z})$ invariance. When the spacetime manifold is
$\mathbb{R}^{p}\times\mathbb{T}^{q+1}$, the partition function is given in
terms of a SL$(q+1,\mathbb{Z})$ Eisenstein series again with $s=(d+1)/2$. In
this case, we obtain the high/low temperature duality through a suitably
adapted dual parametrization of the lattice defining the torus. On
$\mathbb{T}^{d+1}$, the computation is more subtle. An additional divergence
leads to an harmonic anomaly.

###### Contents

  1. 1 Introduction
  2. 2 Generalities. One small spacetime dimension
    1. 2.1 Mode expansion
    2. 2.2 Canonical approach to partition function in large volume limit
    3. 2.3 Functional approach and zeta function
    4. 2.4 Functional approach to partition function in large volume limit
    5. 2.5 Schwinger proper time, heat kernel and worldline approach
    6. 2.6 Worldline approach to partition function in the large volume limit
    7. 2.7 Casimir energy on $\mathbb{R}^{d-1}\times\mathbb{S}^{1}$
  3. 3 Two small spacetime dimensions and Eisenstein series
    1. 3.1 Functional approach
    2. 3.2 Canonical approach
    3. 3.3 Worldline approach
    4. 3.4 One spatial dimension
  4. 4 Higher dimensional tori
    1. 4.1 Generalities
    2. 4.2 Functional approach
    3. 4.3 Canonical approach
    4. 4.4 Worldline approach
    5. 4.5 Partition function on $\mathbb{T}^{d+1}$ and harmonic anomaly
  5. 5 Outlook
  6. Acknowledgements
  7. A $\mathrm{GL}(n,\mathbb{R})/\mathrm{SO}(n,\mathbb{R})$ and $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n,\mathbb{R})$ Laplacians
  8. B Trigonometric expansions

##  1 Introduction

The SL$(2,\mathbb{Z})$ real analytic Eisenstein series,

$f_{s}(\tau,\bar{\tau})=\sum_{(m,n)\in\mathbb{Z}^{2}/(0,0)}\frac{\tau_{2}^{s}}{\absolutevalue{m+n\tau}^{2s}},\quad\mathfrak{Re}(s)>1,$
(1.1)

with $\tau=\tau_{1}+i\tau_{2}\in\mathbb{C}$, invariant under modular
transformations

$\tau\to\frac{a\tau+b}{c\tau+d},\quad a,b,c,d\in\mathbb{Z},\quad ad-bc=1,$
(1.2)

feature prominently in theoretical physics, for instance in the context of
string theory amplitudes, gravitational instantons and Feynman integrals, see
e.g. [1, 2, 3, 4, 5] and references therein.

The purpose of the first part of these notes is to review how such series
directly appear in the partition function of a real massless scalar field
living on a Euclidean spacetime manifold with two small dimensions with
periodic boundary conditions. One of these small dimensions is the Euclidean
time, the other being one of the spatial dimensions, while the remaining $d-1$
spatial dimensions are taken to be large. When $d=1$, one recovers the well-
known modular invariant result.

Casimir energies and partition functions for such configurations were
originally computed in [6] in terms of Epstein zeta functions (see also [7, 8]
for worldline methods and [9, 10, 11] for further developments and reviews).
This result has also been derived in the context of conformal field theories
in higher dimensions in [12]. The aim there was an attempt to generalize some
of the successes of two dimensional conformal field theories in general and of
modular invariance, and thus of high/low temperature duality, in particular
[13, 14, 15] to higher dimensions [16]. More recent investigations along these
lines include [17, 18, 19, 20].

Our motivation for re-analyzing these scalar field results comes from a recent
investigation on the implications of boundary or fall-off conditions for both
the degrees of freedom and the partition functions of gauge and gravitational
theories [21, 22]. More concretely, we have studied how a modular covariant
partition function can be constructed for electromagnetism in the context of
the Casimir set-up, that is to say for a slab geometry with two perfectly
conducting parallel and infinitely extended metal plates separated by a
distance $a$. We have shown how the well-known $E$ and $H$ modes (see e.g.
[23] for an early review) may be combined into a single massless scalar field
with periodic boundary conditions on an interval of length $2a$. Modular
covariance for electromagnetism then follows from the scalar field result when
understanding what electromagnetic operator corresponds to scalar field
momentum in the compact dimension [24]. Moreover, this analysis may be
generalized directly to the case of gravitons [25].

In the second part of these notes, we compute the partition function on the
flat Euclidean spacetime manifold $\mathbb{R}^{d-q}\times\mathbb{T}^{q+1}$ in
terms of a SL$(q+1,\mathbb{Z})$ Eisenstein series with $s=\frac{d+1}{2}$. On a
skewed torus, the periodicities of the scalar field are described by general
lattice vectors. A key point is to introduce adapted coordinates in which
these periodicities become equal to unity along each direction. In these
coordinates, the information on the torus is encoded in the coupling of the
scalar field to a flat background metric that contains the information on the
geometry of the torus. The more skewed the torus, the more observables are
turned on in the partition function. High/low temperature duality follows by
two complementary ADM parametrizations of this metric. For the partition
function on $\mathbb{T}^{d+1}$, the result is divergent. As understood in [1],
using dimensional regularization, the subtraction of the pole leads to an
harmonic anomaly.

The literature on partition functions for massless scalar fields and its
dependence on boundary conditions on the one hand, and on physical
applications of Eisenstein series on the other, is substantial. Hence, the
question on the added value of the current discussion, which necessarily has
considerable overlaps with existing literature, is legitimate.

(i) From the viewpoint of the finite temperature Casimir effect, our
discussion and techniques follow closely the approach developed in [26, 27,
28, 29] in order to derive and understand temperature inversion symmetry,
originally discovered in [30] through the method of images, in terms of
functional methods and Epstein zeta functions. Our addition here consists in
including a chemical potential for linear momentum in the compact dimension
which brings one from Epstein zeta functions with temperature inversion
symmetry to real analytic SL$(2,\mathbb{Z})$ Eisenstein series with modular
covariance.

(ii) From the viewpoint of modular transformations in higher dimensions, as
compared to the analysis in [12], our derivation provides the full analytic
expression for the partition function in the case of the simplest model of a
free massless scalar field on the flat Euclidean manifold
$\mathbb{R}^{p}\times\mathbb{T}^{q+1}$ for $p+q=d$, in terms of
SL$(q+1,\mathbb{Z})$ Eisenstein series, with an explicit proof of
SL$(q+1,\mathbb{Z})$ invariance built in. As compared to [18], for the model
under consideration, there is full control on finite-size corrections, and
also a new formula, valid now at low rather than at high temperature, that
relates the leading contribution to the entropy to the Casimir energy density
of a massless scalar field in one dimension lower. With respect to [1], the
high temperature/ large volume discussion is extended from a factorized torus
$\mathbb{T}^{q}\times\mathbb{S}^{1}$ to a general torus $\mathbb{T}^{q+1}$
through a dual ADM parametrization.

(iii) From the viewpoint of the relevance of Eisenstein series in physical
applications, we show that, in the context of quantum statistical physics,
partitions functions of massless scalar fields on Euclidean spacetime
manifolds of the form $\mathbb{R}^{p}\times\mathbb{T}^{q+1}$ are among the
simplest physical observables that are directly expressed through such series.
Various approaches to computing these partition functions, such as functional
integrals, canonical quantization or proper-time/heat kernel/world-line
methods illustrate complementary aspects of Eisenstein series.

The plan of the paper is the following. In the next section, we briefly review
how to derive the scalar black body result, i.e., the partition function for a
massless scalar field in $d$ large spatial dimensions by canonical, path
integral and proper time methods. At the end of the section, we recall how
this black body partition function can be used to derive the Casimir energy at
zero temperature in the case of one small spatial dimension.

We then turn to the case of finite temperature with one small spatial
dimension where the partition function can be simply expressed in terms of the
Eisenstein series $f_{(d+1)/2}$ when turning on a purely imaginary chemical
potential for the momentum operator in the small dimension. Our discussion
covers the Casimir effect at finite and zero temperature.

In the last section, we generalize the discussion to $p$ large and $q=d-p$
small spatial dimensions. We clarify how the partition function depends on the
full geometry of the $q+1$ dimensions torus, including the case $q=1$, and
express the result in terms of an SL$(q+1,\mathbb{Z})$ invariant partition
function. All results of the previous sections may be obtained as particular
cases, so that one may start directly with this section if one is interested
in the final results and the general structure, but not necessarily the way
our understanding of these matters has evolved from studying the simpler
particular cases first.

##  2 Generalities. One small spacetime dimension

###  2.1 Mode expansion

Consider a massless scalar field $\phi$ in $d$ spatial dimensions, whose
Lagrangian action reads

$S[\phi]=-\frac{1}{2}\int\text{d}x^{0}\int_{V_{d}}\text{d}^{d}x\
\partial_{\mu}\phi\partial^{\mu}\phi,$ (2.1)

while the first order Hamiltonian action is

$S_{H}[\phi,\pi]=\int\text{d}x^{0}[\int_{V_{d}}\text{d}^{d}x\
\dot{\phi}\pi-H],\quad
H=\frac{1}{2}\int_{V_{d}}\text{d}^{d}x\big{(}\pi^{2}+\partial_{i}\phi\partial^{i}\phi\big{)},$
(2.2)

with associated canonical Poisson brackets

$\\{\phi(x^{0},x),\pi(x^{0},x^{\prime})\\}=\delta^{d}(x-x^{\prime}).$ (2.3)

Our conventions for indices and their ranges are as follows: the spacetime and
spatial indices are $\mu=0,...,d$, and $i=1,...,d$, respectively. Besides the
Hamiltonian, the other observables that are relevant for us here are linear
momenta in the $x^{i}$ direction,

$P_{i}=-\int_{V_{d}}\text{d}^{d}x\ \pi\partial_{i}\phi.$ (2.4)

Our aim is to discuss analytic expressions in terms of Eisenstein series and
associated modular properties of partition functions of the type

$Z_{d}(\beta,\mu^{j})={\rm Tr}\ e^{-\beta(\hat{H}-i\mu^{j}\hat{P}_{j})},$
(2.5)

and generalizations thereof. In order to take advantage of techniques in
complex analysis, we use purely imaginary chemical potentials
$i\mu^{j},\mu^{j}\in\mathbb{R}$, as in 2-dimensional conformal field theory
and also for instance in [31, 32, 33].

In order to evaluate such partition functions, a standard procedure is to put
the system in a hyperrectangular box of volume $V_{d}=\prod^{d}_{i=1}L_{i}$
and to choose periodic boundary conditions in all spatial dimensions,

$\phi(x^{0},x^{1},...,x^{i},...,x^{d})=\phi(x^{0},x^{1},...,x^{i}+L_{i},...,x^{d}).$
(2.6)

The appropriate orthonormal basis vectors $\\{e_{k_{i}}\\}$ are plane waves,

$e_{k_{i}}(x)=\frac{1}{\sqrt{V_{d}}}e^{ik_{i}x^{i}},\quad k_{i}=\frac{2\pi
n_{i}}{L_{i}},\quad\left(e_{k_{i}},e_{k^{\prime}_{i}}\right)=\int_{V_{d}}\text{d}^{d}x\
e^{*}_{k_{i}}(x)e_{k^{\prime}_{i}}(x)=\prod_{i}\delta_{n_{i},n^{\prime}_{i}}.$
(2.7)

The fields $\phi,\pi$ admit the mode decomposition

$\phi(x)=\sum_{n_{i}\in\mathbb{Z}^{d}}\phi_{k_{i}}e_{k_{i}}(x),\quad\pi(x)=\sum_{n_{i}\in\mathbb{Z}^{d}}\pi_{k_{i}}(x)e_{k_{i}},$
(2.8)

with reality conditions
$\phi_{k_{i}}=\phi^{*}_{-k_{i}},\pi_{k_{i}}=\pi^{*}_{-k_{i}}$. In terms of the
Fourier components, the observables are

$H=\frac{1}{2}\sum_{n_{i}\in\mathbb{Z}^{d}}(\pi_{k_{i}}\pi^{*}_{k_{i}}+\omega^{2}_{k_{i}}\phi_{k_{i}}\phi^{*}_{k_{i}}),\quad
P_{i}=-i\sum_{n_{i}\in\mathbb{Z}^{d}}k_{i}\phi_{k_{i}}\pi^{*}_{k_{i}},$ (2.9)

where $\omega_{k_{i}}=\sqrt{k_{j}k^{j}}$, while the non-vanishing canonical
Poisson brackets become

$\\{\phi_{k_{i}},\pi^{*}_{k^{\prime}_{i}}\\}=\prod_{i}\delta_{n_{i},n^{\prime}_{i}}.$
(2.10)

The zero mode is denoted by $\phi_{0}=q,\pi_{0}=p$ and, for the Fourier
components with $n_{i}\neq(0,\dots,0)$, one defines standard oscillator
variables,

$a_{k_{i}}=\sqrt{\frac{\omega_{k_{i}}}{2}}\big{[}\phi_{k_{i}}+\frac{i}{\omega_{k_{i}}}\pi_{k_{i}}\big{]}.$
(2.11)

In these terms,

$\begin{split}\phi&={V_{d}}^{-\frac{d-1}{2d}}q+\frac{1}{\sqrt{V_{d}}}{\sum_{n_{i}\in\mathbb{Z}^{d}}}^{\prime}\frac{1}{\sqrt{2\omega_{k_{i}}}}(a_{k_{i}}e^{ik_{j}x^{j}}+{\rm
c.c.}),\\\
\pi&={V_{d}}^{-\frac{d+1}{2d}}p-\frac{i}{\sqrt{V_{d}}}{\sum_{n_{i}\in\mathbb{Z}^{d}}}^{\prime}\sqrt{\frac{\omega_{k_{i}}}{2}}(a_{k_{i}}e^{ik_{j}x^{j}}-{\rm
c.c.}),\end{split}$ (2.12)

where the prime indicates that the term with $n_{i}=(0,\dots,0)$ is excluded
from the sum and $q,p$ are dimensionless. The observables are given by

$H={V_{d}}^{-\frac{1}{d}}\frac{p^{2}}{2}+\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\omega_{k_{i}}(a^{*}_{k_{i}}a_{k_{i}}+a_{k_{i}}a^{*}_{k_{i}}),\quad
P_{i}=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}k_{i}(a^{*}_{k_{i}}a_{k_{i}}+a_{k_{i}}a^{*}_{k_{i}}),$
(2.13)

while the non-vanishing Poisson brackets become

$\\{q,p\\}=1,\quad\\{a_{k_{i}},a^{*}_{k_{i}}\\}=-i\prod_{i}\delta_{n_{i},n^{\prime}_{i}}.$
(2.14)

###  2.2 Canonical approach to partition function in large volume limit

Throughout this section, where we will consider all spatial dimensions to be
large, we will also take the chemical potentials $\mu^{j}$ to vanish. How the
final result changes when they are turned on will be discussed from a more
general perspective later.

####  2.2.1 Zero mode contribution

It follows from the expression of the Hamiltonian (2.13) that the zero mode
sector of the theory corresponds to a free particle with Hamiltonian
$H_{0}={V_{d}}^{-\frac{1}{d}}\frac{1}{2}p^{2}$. Therefore one may quantize the
full theory in a Fock space for which the vacua $|p\rangle$ are labeled by the
continuous eigenvalues $p$ of $\hat{p}$,

$\hat{p}|p\rangle=p|p\rangle,\quad\hat{a}_{k_{i}}|p\rangle=0.$ (2.15)

It follows that

$Z_{0}(\beta)={\rm
Tr}\,e^{-\beta\hat{H}_{0}}=\int^{+\infty}_{-\infty}\text{d}p\,e^{-\frac{b}{2}p^{2}}=\sqrt{\frac{2\pi}{b}},\quad
b={V_{d}}^{-\frac{1}{d}}\beta.$ (2.16)

Note that if one would choose as a basis position eigenstates $|q\rangle$
instead of momentum eigenstates $|p\rangle$, the result of the above trace
would be divergent. However the finite part will still be given by the result
in (2.16).

####  2.2.2 Oscillator contribution

For the oscillator modes quantized in Fock space, one may directly evaluate
the trace using the normal ordered Hamiltonian $:\hat{H}^{\prime}:$ so that no
divergences occur. If $N_{k_{i}}$ denotes the occupation number of the
oscillator associated to $k_{i}$, we have,

$Z^{\prime}_{d}(\beta)={\rm Tr}\
e^{-\beta:\hat{H}^{\prime}:}=\sideset{}{{}^{\prime}}{\prod}_{n_{i}\in\mathbb{Z}^{d}}\sum_{N_{k_{i}}\in\mathbb{N}}e^{-\beta\omega_{k_{i}}N_{k_{i}}}=\sideset{}{{}^{\prime}}{\prod}_{n_{i}\in\mathbb{Z}^{d}}\frac{1}{1-e^{-\beta\omega_{k_{i}}}},$
(2.17)

and thus

$\ln
Z^{\prime}_{d}(\beta)=-\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\ln(1-e^{-\beta\omega_{k_{i}}}).$
(2.18)

For large $L_{i}$, the sums can be approximated by integrals using
$\text{d}k_{i}=\frac{2\pi}{L_{i}}\text{d}n_{i}$. After performing the
integrals over the angles in hyperspherical coordinates, one finds

$\ln
Z^{\prime}_{d}(\beta)=-\frac{V_{d}}{(2\pi)^{d}}\int\text{d}^{d}k\ln(1-e^{-\beta\omega_{k_{i}}})\\\
=-\frac{V_{d}}{(2\pi)^{d}}{\rm
Vol}(\mathbb{S}^{d-1})\int^{\infty}_{0}\text{d}kk^{d-1}\ln(1-e^{-\beta k}).$
(2.19)

Since the contribution of the zero mode $\ln Z_{0}(\beta)\sim\ln V_{d}$ is
subdominant in the large volume limit, it may be neglected in most of the
paper, unless otherwise specified.

On the one hand, ${\rm
Vol}(\mathbb{S}^{d-1})={2\pi^{\frac{d}{2}}}/{\Gamma(\frac{d}{2})}$ while, on
the other, the integral becomes after the change of variables $x=\beta k$ and
a standard integrations by parts,

$\int^{\infty}_{0}\text{d}kk^{d-1}\ln(1-e^{-\beta
k})=-\frac{1}{d\beta^{d}}\int^{\infty}_{0}\text{d}x\frac{x^{d}}{e^{x}-1}=-\frac{1}{d\beta^{d}}\Gamma(d+1)\zeta(d+1).$
(2.20)

One thus ends up with the “scalar black body” partition function

$\ln
Z_{d}(\beta)=\frac{\Gamma(d)\zeta(d+1)}{2^{d-1}\pi^{\frac{d}{2}}\Gamma(\frac{d}{2})}\frac{V_{d}}{\beta^{d}}.$
(2.21)

Furthermore, using the reduplication formula

$\Gamma(z)\sqrt{\pi}=2^{z-1}\Gamma(\frac{z}{2})\Gamma(\frac{z+1}{2}),$ (2.22)

at $z=d$, and the definition of the completion of the zeta function,

$\xi(z)=\frac{\Gamma(\frac{z}{2})\zeta(z)}{\pi^{\frac{z}{2}}},$ (2.23)

at $z=d+1$, the result can be written as

$\boxed{\ln
Z_{d}(\beta)=\xi(d+1)\frac{V_{d}}{\beta^{d}}}=\left\\{\begin{array}[]{ll}\frac{\pi}{6}\frac{L_{1}}{\beta}&{\rm
for}\ d=1\\\ \frac{\zeta(3)}{2\pi}\frac{L_{1}L_{2}}{\beta^{2}}&{\rm for}\
d=2\\\ \frac{\pi^{2}}{90}\frac{L_{1}L_{2}L_{3}}{\beta^{3}}&{\rm for}\ d=3\\\
&\vdots\end{array}\right..$ (2.24)

Remarks:

(i) This result for the partition function is independent of the choice of
boundary conditions in the large volume limit. For instance, for Dirichlet
conditions
$\phi(t,x^{1},...,0,...,x^{d})=0=\phi(t,x^{1},...,L_{i},...,x^{d})$, the
appropriate basis is $e_{k_{i}}=\sqrt{\frac{2^{d}}{V_{d}}}\prod_{i}\sin
k_{i}x=e^{*}_{k_{i}}$, $k_{i}=\frac{\pi n_{i}}{L_{i}}$. In particular, there
is no mode at $n_{i}=(0,\dots,0)$ that has to be dealt with. All mode
expansions are the same except that sums and products are restricted to
$n_{i}>0$. In the evaluation of the partition function after (2.18), this is
compensated by the fact that $\text{d}k_{i}=\frac{\pi}{L_{i}}\text{d}n_{i}$.
Equivalently, the sums can be extended to $\mathbb{Z}^{d}/(0,\dots,0)$ while
dividing by $2^{d}$, because the summands are even functions of the $n_{i}$,
and the computation proceeds as before.

(ii) For $d=3$, the standard argument to get the partition functions for the
electromagnetic field is to multiply the scalar field result by $2$ in order
to take into account the $2$ independent polarizations of the photon. On
account of the equivalence established in [24] of the electromagnetic field
with Casimir boundary conditions and a scalar field on the double interval
with periodic boundary conditions in the direction $x^{3}$, together with the
independence of the results on boundary conditions in the large volume limit,
this can also be obtained by the replacement replacement $L_{3}=2a$, where $a$
is the distance between the two parallel plates. In the case where the $x^{3}$
dimension is “small”, the former argument no longer applies but the latter
still does.

###  2.3 Functional approach and zeta function

The functional evaluation of partition functions [34] (see also e.g. [35, 36]
for earlier connected work, [37] for the inclusion of chemical potentials and
[10] for a review) starts from the Hamiltonian path integral representation

$Z_{d}(\beta,\mu^{j})=\int\prod_{x^{a}}\text{d}\phi(x^{a})\prod_{x^{a}}\frac{\text{d}\pi(x^{a})}{2\pi}\
e^{-S^{E}_{H}},$ (2.25)

where $x^{a}=(x^{1},\dots,x^{d+1})$ and the first order Euclidean action is

$S^{E}_{H}[\phi,\pi]=\int^{\beta}_{0}\text{d}x^{d+1}\big{[}\int_{V_{d}}\text{d}^{d}x(-i\pi\partial_{d+1}\phi)+(H-i\mu^{j}P_{j})\big{]}.$
(2.26)

The sum is over periodic phase space paths of period $\beta=L_{d+1}$ in
Euclidean time $x^{d+1}$,

$\phi(x^{i},x^{d+1}+\beta)=\phi(x^{i},x^{d+1}),\quad\pi(x^{i},x^{d+1}+\beta)=\pi(x^{i},x^{d+1}).$
(2.27)

After integration over the momenta, this leads to

$Z_{d}(\beta,\mu^{j})=\big{(}{\rm
det}[2\pi\delta^{d+1}(x^{\prime},x)]\big{)}^{-\frac{1}{2}}\int\prod_{x^{a}}d\phi(x^{a})\
e^{-S^{E}_{L}}$ (2.28)

with

$S^{E}_{L}[\phi]=\frac{1}{2}\int_{V_{d+1}}\text{d}^{d+1}x[(\partial_{d+1}\phi-\mu^{j}\partial_{j}\phi)^{2}+\partial_{j}\phi\partial^{j}\phi],$
(2.29)

where the integration is now over a $(d+1)$-dimensional hyperrectangle in
Euclidean space of volume $V_{d+1}=V_{d}\beta=\prod^{d+1}_{a=1}L_{a}$. Except
for the replacement $\partial_{d+1}\to\partial_{d+1}-\mu^{j}\partial_{j}$, the
operator in the action is minus the Laplacian in $d+1$ dimensions with
periodic boundary conditions in all dimensions. Since the eigenfunctions are
$\frac{1}{\sqrt{V_{d+1}}}e^{ik_{a}x^{a}}$ with $k_{d+1}=\frac{2\pi
n_{d+1}}{\beta}$, the eigenvalues are

$\lambda_{n_{a}}=\Big{[}\Big{(}\frac{2\pi
n_{d+1}}{\beta}-\sum_{j}\mu^{j}\frac{2\pi
n_{j}}{L_{j}}\Big{)}^{2}+\sum_{i}\Big{(}\frac{2\pi
n_{i}}{L_{i}}\Big{)}^{2}\Big{]}.$ (2.30)

In order to define the partition function, formally given by

$Z_{d}(\beta,\mu^{j})=\sideset{}{{}^{\prime}}{\prod}_{n_{a}\in\mathbb{Z}^{d+1}}\sqrt{\frac{2\pi}{\lambda_{n_{a}}}}=\Big{[}{\rm
det}(\frac{-\Delta_{\mu}}{2\pi})\Big{]}^{-\frac{1}{2}},$ (2.31)

one introduces a parameter $\nu$ of dimension inverse length and considers the
dimensionless operators $A_{\mu}=-\Delta_{\mu}/2\pi\nu^{2}$. The associated
zeta function is defined by

$\zeta_{A_{\mu}}(s)=\sideset{}{{}^{\prime}}{\sum}_{n_{a}\in\mathbb{Z}^{d+1}}\lambda_{n_{a}}^{-s}(2\pi\nu^{2})^{s}.$
(2.32)

In these terms, the partition function is given by

$\ln
Z_{d}(\beta,\mu^{j})=\frac{1}{2}\zeta^{\prime}_{A_{\mu}}(0)=\frac{1}{2}\zeta^{\prime}_{-\Delta_{\mu}}(0)+\frac{1}{2}\ln(2\pi\nu^{2})\zeta_{-\Delta_{\mu}}(0).$
(2.33)

###  2.4 Functional approach to partition function in large volume limit

Let us recover the canonical result of section 2.2 through the functional
approach. In this case, in the absence of chemical potentials, $\mu^{j}=0$,
$A_{0}=-\frac{\Delta_{0}}{2\pi\nu^{2}}$ we again turn the sums over the
$n_{i}$ into integrals,

$\zeta_{A_{0}}(s)=(2\pi\nu^{2})^{s}\frac{V_{d}}{(2\pi)^{d}}\int\text{d}^{d}k\
\sum_{n_{d+1}\in\mathbb{Z}}[(\frac{2\pi
n_{d+1}}{\beta})^{2}+k_{i}k^{i}]^{-s},$ (2.34)

From the Euclidean spacetime point of view, there is thus only one small
dimension $x^{d+1}$ of length $\beta$. After performing the integrals over the
angles in hyperspherical coordinates, we have

$\zeta_{A_{0}}(s)=(2\pi\nu^{2})^{s}\frac{V_{d}2\pi^{\frac{d}{2}}}{(2\pi)^{d}\Gamma(\frac{d}{2})}\int^{\infty}_{0}\text{d}k\
k^{d-1}\sum_{n_{d+1}\in\mathbb{Z}}[(\frac{2\pi
n_{d+1}}{\beta})^{2}+k^{2}]^{-s}$ (2.35)

The divergent integral contained in the term at $n_{d+1}=0$ is regulated
through an infrared cut-off $\epsilon$,

$\int^{\infty}_{\epsilon}\text{d}k\
k^{d-1-2s}=-\frac{\epsilon^{d-2s}}{d-2s},\quad\mathfrak{Re}(s)>\frac{d}{2}.$
(2.36)

This expression and its derivative with respect to $s$ both vanish at $s=0$ in
the limit when $\epsilon\to 0$ and can thus be discarded. In the remaining
terms, the integrals converge for $\mathfrak{Re}(s)>\frac{d}{2}$. After the
change of variables, $k=\frac{2\pi}{\beta}y$, we are left with

$\zeta_{A_{0}}(s)=(2\pi\nu^{2})^{s}\frac{V_{d}2\pi^{\frac{d}{2}}}{(2\pi)^{d}\Gamma(\frac{d}{2})}(\frac{2\pi}{\beta})^{d-2s}\int^{\infty}_{0}\text{d}y\
y^{d-1}\sideset{}{{}^{\prime}}{\sum}_{n_{d+1}\in\mathbb{Z}}[(n_{d+1})^{2}+y^{2}]^{-s}.$
(2.37)

The additional change of variables $y=n_{d+1}\sinh x$ together with

$\int^{\infty}_{0}\text{d}x\frac{\sinh^{d-1}(x)}{\cosh^{2s-1}(x)}=\frac{1}{2}\frac{\Gamma(\frac{d}{2})\Gamma(s-\frac{d}{2})}{\Gamma(s)},$
(2.38)

then yields

$\zeta_{A_{0}}(s)=(2\pi\nu^{2})^{s}\frac{V_{d}2\pi^{\frac{d}{2}}}{(2\pi)^{d}}(\frac{2\pi}{\beta})^{d-2s}\frac{\Gamma(s-\frac{d}{2})\zeta(2s-d)}{\Gamma(s)},$
(2.39)

which can be written more compactly as

$\zeta_{A_{0}}(s)=\frac{\nu^{2s}V_{d}}{\beta^{d-2s}}2^{1-s}\frac{\xi(2s-d)}{\Gamma(s)}.$
(2.40)

Since $\frac{1}{\Gamma(s)}=s+{O}(s^{2})$, it follows that $\zeta_{A_{0}}(0)=0$
and

$\ln Z_{d}(\beta)=\xi(-d)\frac{V_{d}}{\beta^{d}}.$ (2.41)

When using the reflection formula

$\pi^{-\frac{z}{2}}\Gamma(\frac{z}{2})\zeta(z)=\pi^{\frac{z-1}{2}}\Gamma(\frac{1-z}{2})\zeta(1-z)\iff\xi(z)=\xi(1-z).$
(2.42)

at $z=-d$, this agrees with (2.24).

###  2.5 Schwinger proper time, heat kernel and worldline approach

The quadratic Lagrangian path integral (2.28) is formally solved by the one-
loop contribution

$\ln Z_{d}(\beta,\mu^{j})=-\frac{1}{2}{\rm Tr}\
\ln\frac{\delta^{2}S^{E}_{L}}{\delta\phi\delta\phi},$ (2.43)

which can be written as an integral over Schwinger proper time $t$ as

$\ln
Z_{d}(\beta,\mu^{j})=\frac{1}{2}\int^{\infty}_{0}\frac{\text{d}t}{t}\,{\rm
Tr}\ e^{-t\hat{\rm H}_{\mu}},$ (2.44)

after having removed the zero modes. Here

$\hat{\rm
H}_{\mu}=(\hat{p}_{d+1}-\mu^{j}\hat{p}_{j})^{2}+\hat{p}_{j}\hat{p}^{j},$
(2.45)

is not the field theory Hamiltonian but the one of a free particle of mass
$\frac{1}{2}$ in $d+1$ (Euclidean) dimensions described by the canonical
variables $(x^{a},p_{a})$. In terms of the operator

$\hat{T}_{\mu}(t)=e^{-t\hat{\rm
H}_{\mu}},\quad-\frac{d}{dt}\hat{T}_{\mu}(t)=\hat{\rm
H}_{\mu}\hat{T}_{\mu}(t),$ (2.46)

the relation to the zeta function is

$\zeta_{\hat{\rm H}_{\mu}}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\text{d}t\
t^{s-1}{\rm Tr}\ (\hat{T}_{\mu}(t)-\hat{\Pi}),$ (2.47)

where $\hat{\Pi}$ denotes the orthogonal projector onto ${\rm Ker}\ \hat{\rm
H}_{\mu}$, ${\rm Tr}\,\hat{\Pi}={\rm dim}\,{\rm Ker}\,\hat{\rm H}_{\mu}$.

Remarks:

(i) If one takes vanishing chemical potentials $\mu^{j}=0$ and considers
infinite Euclidean spacetime, which means in particular zero temperature,
$\beta=L_{d+1}\to\infty$, the matrix element

$K_{-\Delta_{0}}(x^{\prime},x;t)=\langle x^{\prime}|\hat{T}(t)|x\rangle$
(2.48)

is called the heat kernel since it satisfies the heat equation

$\frac{d}{dt}K_{-\Delta_{0}}(x^{\prime},x;t)=\Delta_{0}K_{-\Delta_{0}}(x^{\prime},x;t),\quad
K_{-\Delta_{0}}(x^{\prime},x;0)=\delta^{d+1}(x^{\prime}-x).$ (2.49)

The solution is well-known and can be obtained for instance by working out the
(Euclidean) path integral for the transition amplitude of a particle of mass
$m=\frac{1}{2}$ in $d+1$ dimensions:

$K_{-\Delta_{0}}(x^{\prime},x;t)=\frac{1}{(4\pi
t)^{\frac{d+1}{2}}}e^{-\frac{1}{4t}(x^{\prime}_{a}-x_{a})(x^{\prime
a}-x^{a})}.$ (2.50)

(ii) If one takes vanishing chemical potentials $\mu^{j}$ and only one
spacetime dimension with finite $\beta$ instead, the periodic boundary
condition, $\phi(0)=\phi(\beta)$, requires one to solve the heat kernel for a
particle on a circle. The first form of the solution may be obtained by
inserting a resolution of the identity in terms of eigenstates of the
Laplacian.

$\varphi_{n}(x)=\langle
x|n\rangle=\frac{1}{\sqrt{\beta}}e^{i\omega_{n}x},\quad\omega_{n}=\frac{2\pi}{\beta}n,$
(2.51)

so that

$K_{-\Delta_{P}}(x^{\prime},x;t)=\sum_{n\in\mathbb{Z}}e^{-t\omega^{2}_{n}}\varphi_{n}(x^{\prime})\varphi_{n}^{*}(x)$
(2.52)

and may be expressed in terms of the Jacobi theta function defined by

$\vartheta_{3}(z|\tau)=\sum_{n\in\mathbb{Z}}e^{\pi in^{2}\tau+2\pi inz},$
(2.53)

as follows

$K_{-\Delta_{P}}(x^{\prime},x;t)=\frac{1}{\beta}\sum_{n\in\mathbb{Z}}e^{-t(\frac{2\pi
n}{\beta})^{2}+2\pi
in\frac{x^{\prime}-x}{\beta}}=\frac{1}{\beta}\vartheta_{3}(\frac{x^{\prime}-x}{\beta}|i\frac{4\pi}{\beta^{2}}t).$
(2.54)

The second form of the solution follows by considering a path integral
including paths that wind $m$ times around the circle of length $\beta$,

$K_{-\Delta_{P}}(x^{\prime},x;t)=\frac{1}{\sqrt{4\pi
t}}\sum_{m\in\mathbb{Z}}e^{-\frac{1}{4t}(x^{\prime}-x+\beta m)^{2}}.$ (2.55)

Both expressions are related through the Poisson summation formula,

$\sum_{m^{\alpha}\in\mathbb{Z}^{n}}e^{-\pi(m^{\alpha}+y^{\alpha})A_{\alpha\beta}(m^{\beta}+y^{\beta})+2\pi
im^{\alpha}z_{\alpha}}\\\
=\frac{1}{\sqrt{\mathrm{det}\,A}}\sum_{n_{\alpha}\in\mathbb{Z}^{n}}e^{-\pi(n_{\alpha}+z_{\alpha})(A^{-1})^{\alpha\beta}(n_{\beta}+z_{\beta})-2\pi
i(n_{\alpha}+z_{\alpha})y^{\alpha}},$ (2.56)

in the simple case where $n=1$. For later use, one also defines

$\theta(t)=\vartheta_{3}(0|it)=\sum_{n\in\mathbb{Z}}e^{-t\pi n^{2}},$ (2.57)

with modular transformation

$\vartheta_{3}(0|-\frac{1}{\tau})=\sqrt{-i\tau}\vartheta_{3}(0,\tau)\Longrightarrow\theta(\frac{1}{t})=\sqrt{t}\theta(t),$
(2.58)

in terms of which both expressions for the trace of the heat kernel,

$K_{-\Delta_{P}}(t)=\int^{\beta}_{0}\text{d}x\,K_{-\Delta_{P}}(x,x;t)$ (2.59)

of a particle on the circle may be written compactly as

$K_{-\Delta_{P}}(t)=\theta(\frac{4\pi}{\beta^{2}}t)=\frac{\beta}{\sqrt{4\pi
t}}\theta(\frac{\beta^{2}}{4\pi t}).$ (2.60)

###  2.6 Worldline approach to partition function in the large volume limit

The full heat kernel in the large volume limit is obtained by combining the
heat kernel in (2.50) with $d\to d-1$ for the large spatial dimensions with
that in (2.54),

$\langle x^{\prime}|\hat{T}_{0}(t)|x\rangle=\frac{1}{\beta(4\pi
t)^{\frac{d}{2}}}e^{-\frac{1}{4t}(x^{\prime}_{i}-x_{i})(x^{\prime
i}-x^{i})}\vartheta_{3}(\frac{x^{\prime}_{d+1}-x_{d+1}}{\beta};i\frac{4\pi}{\beta^{2}}t).$
(2.61)

According to (2.44), the partition function in the large volume limit is

$\ln
Z_{d}(\beta)=\frac{V_{d}}{2^{d+1}\pi^{\frac{d}{2}}}\sideset{}{{}^{\prime}}{\sum}_{n_{d+1}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}t\
t^{-\frac{d}{2}-1}e^{-t(\frac{2\pi n_{d+1}}{\beta})^{2}}.$ (2.62)

After the change of variables $t\to(\frac{2\pi}{\beta})^{2}t$, this becomes

$\ln
Z_{d}(\beta)=\frac{V_{d}}{\beta^{d}}\frac{\pi^{\frac{d}{2}}}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{d+1}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}{t}\
{t}^{-\frac{d}{2}-1}e^{-tn_{d+1}^{2}},$ (2.63)

which may be written in terms of the theta function as

$\ln Z_{d}(\beta)=\frac{V_{d}}{\beta^{d}}\int^{\infty}_{0}\text{d}{t}\
{t}^{-\frac{d}{2}-1}\frac{1}{2}(\theta(t)-1).$ (2.64)

Going back to (2.63) and assuming that $d<-2$, one may use

$\frac{1}{\Gamma(s)}\int^{\infty}_{0}\text{d}{t}\
{t}^{s-1}e^{-{t}\lambda}=\frac{1}{\lambda^{s}},\quad\mathfrak{Re}(s)>0,$
(2.65)

with $\lambda>0$, so that

$\frac{\pi^{\frac{d}{2}}}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{d+1}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}{t}\
{t}^{-\frac{d}{2}-1}e^{-{t}n_{d+1}^{2}}=\pi^{\frac{d}{2}}\Gamma(-\frac{d}{2})\zeta(-d)=\xi(-d).$
(2.66)

It follows that (2.63) agrees with (2.41). Again, the result for positive
integer $d$ is obtained after using the reflection formula.

When using (2.55) instead of (2.54), the full heat kernel in the large volume
limit is instead

$\langle x^{\prime}|\hat{T}_{0}(t)|x\rangle=\frac{1}{(4\pi
t)^{\frac{d+1}{2}}}e^{-\frac{1}{4t}(x^{\prime}_{i}-x_{i})(x^{\prime
i}-x^{i})}\sum_{m^{d+1}\in\mathbb{Z}}e^{-\frac{1}{4t}(x^{\prime}-x+\beta
m^{d+1})^{2}}.$ (2.67)

For the partition function, after dropping the $m^{d+1}=0$ term, this gives

$\ln
Z_{d}(\beta)=\frac{V_{d}\beta}{2^{d+2}\pi^{\frac{d+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{m^{d+1}\in\mathbb{Z}}\int^{\infty}_{0}dt\
t^{-\frac{d+3}{2}}e^{-\frac{1}{4t}(m^{d+1}\beta)^{2}}.$ (2.68)

After the change of variables $t\rightarrow\frac{\beta^{2}}{4t}$, one now gets
directly

$\ln
Z_{d}(\beta)=\frac{V_{d}}{\beta^{d}}\frac{1}{2\pi^{\frac{d+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{m^{d+1}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}{t}\
{t}^{\frac{d+1}{2}-1}e^{-t(m^{d+1})^{2}}=\frac{V_{d}}{\beta^{d}}\xi(d+1),$
(2.69)

in agreement with the result obtained from canonical quantization in (2.24).

In terms of the theta function, the first expression for the partition
function in (2.69) is

$\boxed{\ln
Z_{d}(\beta)=\frac{V_{d}}{\beta^{d}}\int^{\infty}_{0}\text{d}t\,t^{\frac{d+1}{2}-1}\frac{1}{2}\big{(}\theta(t)-1\big{)}}.$
(2.70)

In other words, equivalence of the canonical and worldline approach for the
partition function shows for $z=d+1$ that the completion of the zeta function
is the Mellin transform of $\frac{1}{2}(\theta(t)-1)$,

$\xi(z)=\frac{1}{2}\int^{\infty}_{0}\text{d}t\,t^{\frac{z}{2}-1}\big{(}\theta(t)-1\big{)}.$
(2.71)

###  2.7 Casimir energy on $\mathbb{R}^{d-1}\times\mathbb{S}^{1}$

Suppose now that the spatial dimension $x^{d}$ is small, while all other
spatial dimensions $x^{I}$, for $I=1,\dots,d-1$ are large, $L_{I}\gg L_{d}$.
In other words, one considers a slab geometry with two infinite parallel
hyperplanes separated by a distance $L_{d}$, or because of the periodic
boundary conditions, a scalar field on the spatial manifold
$\mathbb{R}^{d-1}\times\mathbb{S}^{1}_{L_{d}}$. Suppose also that one uses
symmetric instead of normal ordering, so that

$\hat{H}=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\omega_{k_{i}}(\hat{a}^{\dagger}_{k_{i}}\hat{a}_{k_{i}}+\hat{a}_{k_{i}}\hat{a}^{\dagger}_{k_{i}})=\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\omega_{k_{i}}(\hat{a}^{\dagger}_{k_{i}}\hat{a}_{k_{i}}+\frac{1}{2}).$
(2.72)

The divergent zero-point energy is

$E^{d}_{0}=\langle
0|\hat{H}|0\rangle=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\omega_{k_{i}}.$
(2.73)

Transforming $d-1$ sums into integrals, and using $\zeta$ function
regularization to give a meaning to this divergent quantity, we need to
evaluate

$E^{d}_{0}(s)=\frac{1}{2}\nu^{2s}\frac{V_{d-1}}{(2\pi)^{d-1}}\sum_{n_{d}\in\mathbb{Z}}\int{\text{d}^{d-1}k}\
[(\frac{2\pi n_{d}}{L_{d}})^{2}+k_{I}k^{I}]^{-s+\frac{1}{2}},$ (2.74)

where $V_{d-1}=\prod_{I=1}^{d-1}L_{I}$. This is the same computation as in
(2.34), up an overall factor of $(2\pi\nu^{2})^{s}$ versus
$\frac{1}{2}\nu^{2s}$ and the replacements $d\to d-1$, $s\to s-\frac{1}{2}$,
$\beta=L_{d+1}\to L_{d}$. The result can thus be read off (2.39),

$E^{d}_{0}(s)=\frac{1}{2}\nu^{2s}\frac{V_{d-1}}{L_{d}^{d-2s}}2^{2-2s}\pi^{\frac{d+1}{2}-2s}\frac{\Gamma(s-\frac{d}{2})\zeta(2s-d)}{\Gamma(s-\frac{1}{2})},$
(2.75)

which yields at $s=0$,

$E_{0}^{d}(0)=-\xi(-d)\frac{V_{d-1}}{L_{d}^{d}},$ (2.76)

and thus, after using the reflection formula,

$\boxed{E^{d}_{0}(0)=-\xi(d+1)\frac{V_{d-1}}{L_{d}^{d}}}=\left\\{\begin{array}[]{ll}-\frac{\pi}{6}\frac{1}{L_{1}}&{\rm
for}\ d=1\\\ -\frac{\zeta(3)}{2\pi}\frac{L_{1}}{L_{2}^{2}}&{\rm for}\ d=2\\\
-\frac{\pi^{2}}{90}\frac{L_{1}L_{2}}{L_{3}^{3}}&{\rm for}\ d=3\\\
&\vdots\end{array}\right..$ (2.77)

Remarks:

(i) For later purpose and to make contact with the literature, one may write

$E_{0}^{d}=-\varepsilon^{d}_{\rm vac}\frac{V_{d-1}}{L_{d}^{d}},$ (2.78)

where $\varepsilon^{d}_{\rm vac}$ is a dimensionless number characterizing the
Casimir energy whose value is

$\varepsilon^{d}_{\rm vac}=\xi(d+1)$ (2.79)

in the current model.

The relation between the partition function (2.24) in the large volume limit
and the Casimir energy density (see e.g. [40, 6, 26, 41, 27, 28]) may then be
written as

$\ln Z_{d}(\beta)=\varepsilon^{d}_{\rm vac}\frac{V_{d}}{\beta^{d}}.$ (2.80)

(ii) On account of the previously discussed equivalence, the correct
electromagnetic results with perfectly conducting plates at $x^{3}=0$ and
$x^{3}=a$ are again obtained through the replacement $L_{3}=2a$.

(iii) Note that, even with symmetric ordering,

$P^{0}_{i}=\langle 0|\hat{P}_{i}|0\rangle=0,$ (2.81)

irrespective of the existence or not of a small spatial dimension, because
either $\sum_{n_{i}\in\mathbb{Z}}k_{i}=0$ or $\int\text{d}k_{i}k_{i}=0$.

(iv) The computation of the Casimir energy can also be done using a suitable
integral representation. Starting from the representation (2.65) at
$s=-\frac{1}{2}$ (to be understood in the sense of analytic continuation), the
zero point energy in (2.73) is written as

$E^{d}_{0}=\frac{1}{2\Gamma(-\frac{1}{2})}\sideset{}{{}^{\prime}}{\sum}_{n_{i}\in\mathbb{Z}^{d}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{3}{2}}e^{-t\omega^{2}_{k_{i}}}$
(2.82)

After changing sums to integrals for the $d-1$ large dimensions and
integrating over the angles, one gets

$E^{d}_{0}=-\frac{1}{4\sqrt{\pi}}\frac{V_{d-1}}{(2\pi)^{d-1}}\frac{2\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})}\sum_{n_{d}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{3}{2}}\int^{\infty}_{0}dkk^{d-2}e^{-t\big{[}(\frac{2\pi}{L_{d}}n_{d})^{2}+k^{2}\big{]}}.$
(2.83)

The change of variables $k=\frac{2\pi}{L_{d}}\sqrt{y}$, $t\to
t=(\frac{L_{d}}{2\pi})^{2}t$ then leads to

$E^{d}_{0}=-\frac{V_{d-1}}{L_{d}^{d}}\frac{\pi^{\frac{d}{2}}}{2\Gamma(\frac{d-1}{2})}\sum_{n_{d}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{3}{2}}e^{-tn_{d}^{2}}\int^{\infty}_{0}\text{d}yy^{\frac{d-1}{2}-1}e^{-ty}.$
(2.84)

Performing the integral over $y$ using (2.65) again gives

$E^{d}_{0}=-\frac{V_{d-1}}{L_{d}^{d}}\frac{\pi^{\frac{d}{2}}}{2}\sum_{n_{d}\in\mathbb{Z}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{d}{2}-1}e^{-tn_{d}^{2}}$
(2.85)

The integral term with $n_{d}=0$ is divergent,
$\int^{\infty}_{\epsilon}\text{d}t\,t^{-\frac{d}{2}-1}=\frac{2\epsilon^{-\frac{d}{2}}}{d}$
and neglected, while the terms with $n_{d}\neq 0$ are evaluated using (2.65)
to yield

$E^{d}_{0}=-\frac{V_{d-1}}{L_{d}^{d}}{\pi^{\frac{d}{2}}}\Gamma(-\frac{d}{2})\zeta(-d),$
(2.86)

in agreement with (2.76).

##  3 Two small spacetime dimensions and Eisenstein series

###  3.1 Functional approach

We now consider the case where only the first $d-1$ spatial dimensions are
large, while $L_{d}$ is small, and where a chemical potential for linear
momentum in the small dimension $x^{d}$ is turned on. We start here with the
functional approach because it yields directly the result in terms of the
Eisenstein series in (1.1). In this case $I=1,\dots,d-1$,
$V_{d-1}=\prod_{I=1}^{d-1}L_{I}$ and, when taking into account (2.30), we get
instead of (2.34),

$\zeta_{-\Delta_{\mu}}(s)=\frac{V_{d-1}}{(2\pi)^{d-1}}\int\text{d}^{d-1}k\
\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}[(\frac{2\pi
n_{d+1}}{\beta}-\mu\frac{2\pi n_{d}}{L_{d}})^{2}+(\frac{2\pi
n_{d}}{L_{d}})^{2}+k_{I}k^{I}]^{-s}.$ (3.1)

The computation then proceeds as in section 2.4, up to the replacements $d\to
d-1$ and of the zeta function by a suitable Eisenstein series. More precisely,
instead of (2.37), we now get

$\zeta_{-\Delta_{\mu}}(s)=\frac{V_{d-1}}{(2\pi)^{d-1}}\frac{2\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})}(\frac{2\pi}{\beta})^{d-1-2s}\\\
\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\int_{0}^{\infty}\text{d}y\
y^{d-2}[(n_{d+1}-n_{d}\frac{\beta\mu}{L_{d}})^{2}+n_{d}^{2}(\frac{\beta}{L_{d}})^{2}+y^{2}]^{-s},$
(3.2)

For later convenience, we also change parametrization and use
$\alpha=\beta\mu$ so that we are now computing

${\rm Z}_{d,1}(\beta,\alpha)={\rm Tr}\ e^{-\beta\hat{H}+i\alpha\hat{P}_{d}}.$
(3.3)

Introducing the modular parameter

$\tau=\frac{\alpha+i\beta}{L_{d}},$ (3.4)

the sum simplifies to

$\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\int_{0}^{\infty}\text{d}y\
y^{d-2}[\absolutevalue{n_{d+1}+n_{d}\tau}^{2}+y^{2}]^{-s}.$ (3.5)

The integral is convergent for $\mathfrak{Re}(s)>\frac{d-1}{2}$ and can be
performed through the change of variables
$y=\absolutevalue{n_{d+1}+n_{d}\tau}^{2}\sinh x$ which gives

$\zeta_{-\Delta_{\alpha}}(s)=\frac{V_{d-1}}{(2\pi)^{d-1}}\frac{\pi^{\frac{d-1}{2}}\Gamma(s-\frac{d-1}{2})}{\Gamma(s)}(\frac{2\pi}{\beta})^{d-1-2s}\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\absolutevalue{n_{d+1}+n_{d}\tau}^{d-1-2s}.$
(3.6)

In terms of the real analytic Eisenstein series (1.1), this can be written as

$\zeta_{-\Delta_{\alpha}}(s)=\frac{V_{d-1}}{(2\pi)^{d-1}}\frac{\pi^{\frac{d-1}{2}}\Gamma(s-\frac{d-1}{2})}{\Gamma(s)\tau_{2}^{s-\frac{d-1}{2}}}(\frac{2\pi}{\beta})^{d-1-2s}f_{s-\frac{d-1}{2}}(\tau,\bar{\tau}),$
(3.7)

When using the functional relation for the analytically continued Eisenstein
series,

$\pi^{-z}\Gamma(z)f_{z}(\tau,\bar{\tau})=\pi^{z-1}\Gamma(1-z)f_{1-z}(\tau,\bar{\tau}).$
(3.8)

for $z=s-\frac{d-1}{2}$, and also $\tau_{2}=\frac{\beta}{L_{d}}$, we end up
with

$\zeta_{-\Delta_{\alpha}}(s)=\frac{\Gamma(\frac{d+1}{2}-s)}{2^{2s}\pi^{\frac{d+1}{2}}\Gamma(s)}\frac{V_{d-1}}{L_{d}^{d-1-2s}}\frac{f_{\frac{d+1}{2}-s}(\tau,\bar{\tau})}{\tau_{2}^{\frac{d-1}{2}-s}}.$
(3.9)

Again, since $\Gamma(s)\approx\frac{1}{s}+{O}(s^{0})$,
$\zeta_{-\Delta_{\alpha}}(0)$, if $\mathcal{Z}_{d,1}(\tau,\bar{\tau})={\rm
Z}_{d,1}(\beta,{\alpha})$, the partition function is

$\ln{\mathcal{Z}}_{d,1}(\tau,\bar{\tau})=\frac{\Gamma(\frac{d+1}{2})V_{d-1}}{2\pi^{\frac{d+1}{2}}L_{d}^{d-1}\tau_{2}^{\frac{d-1}{2}}}f_{\frac{d+1}{2}}(\tau,\bar{\tau}).$
(3.10)

In terms of the completion of the real analytic Eisenstein series,

$\xi(z;\tau,\bar{\tau})=\frac{\Gamma(\frac{z}{2})f_{\frac{z}{2}}(\tau,\bar{\tau})}{2\pi^{\frac{z}{2}}},$
(3.11)

which satisfies the reflection formula

$\xi(z;\tau,\bar{\tau})=\xi(2-z;\tau,\bar{\tau}),$ (3.12)

this gives

$\boxed{\ln{\mathcal{Z}}_{d,1}(\tau,\bar{\tau})=\frac{V_{d-1}}{L_{d}^{d-1}\tau_{2}^{\frac{d-1}{2}}}\xi(d+1;\tau,\bar{\tau})}.$
(3.13)

Since $\xi(z;\tau,\bar{\tau})$ is invariant under modular transformations
(1.2) while $\tau_{2}$ transforms as

$\tau_{2}\rightarrow\frac{\tau_{2}}{\absolutevalue{c\tau+d}^{2}},$ (3.14)

the partition function (3.11) transforms as

$\boxed{\ln{\mathcal{Z}}_{d,1}(\tau^{\prime},\bar{\tau}^{\prime})=\absolutevalue{c\tau+d}^{d-1}\ln{\mathcal{Z}}_{d,1}(\tau,\bar{\tau})}.$
(3.15)

Remarks:

(i) The above result holds for $d>1$ because if $d=1$, i.e., for the free
boson on $\mathbb{T}^{2}$,
$\ln{\mathcal{Z}}_{1,1}(\tau,\bar{\tau})=\frac{1}{2\pi}f_{1}(\tau,\bar{\tau})$,
is not convergent and more care is needed to derive the correct result. This
is reviewed in section 3.4.

(ii) As discussed in the introduction, for $d=3$, the result is directly
relevant for the electromagnetic field with Casimir boundary conditions after
the replacement $L_{3}=2a$.

(iii) If the chemical potential vanishes, ${\alpha}=0$, $\tau=i\tau_{2}$, with
$\tau_{2}=\frac{\beta}{L_{d}}$ and
$\mathcal{Z}_{d,1}(i\tau_{2},-i\tau_{2})=\mathcal{Z}_{d,1}(\tau_{2})=Z_{d,1}(\beta)$,
the result can be written in terms of an Epstein zeta function,

$\zeta(s;1,\tau_{2}^{2})=\sideset{}{{}^{\prime}}{\sum}_{(m^{d+1},m^{d})\in\mathbb{Z}^{2}}\frac{1}{[(m^{d})^{2}+(m^{d+1})^{2}\tau_{2}^{2}]^{s}},$
(3.16)

as

$\ln{\mathcal{Z}}_{d,1}(\tau_{2})=\frac{\Gamma(\frac{d+1}{2})}{2\pi^{\frac{d+1}{2}}}\frac{V_{d-1}\tau_{2}}{L_{d}^{d-1}}\zeta(\frac{d+1}{2};1,\tau_{2}^{2}).$
(3.17)

If $a=0=d$, $b=1=-c$, the modular transformation reduces to temperature
inversion $\tau_{2}\to\frac{1}{\tau_{2}}$ with

$\ln{\mathcal{Z}}_{d,1}(\frac{1}{\tau_{2}})=\tau_{2}^{d-1}\ln{\mathcal{Z}}_{d,1}(\tau_{2}).$
(3.18)

Some properties and a trigonometric expansion for the Epstein zeta function
are reviewed in Appendix B.

(iv) Setting $m^{d}=0$ in (3.17) gives

$\ln{\mathcal{Z}}^{\rm
high}_{d,1}(\tau_{2})=\xi(d+1)\frac{V_{d-1}}{L_{d}^{d-1}\tau_{2}^{d}},$ (3.19)

which is the black body result (2.24). From the analysis of section 2, we know
that this is the dominant contribution in the high temperature/large $L_{d}$
limit $\tau_{2}\ll 1$. This justifies a posteriori the use of a normal ordered
Hamiltonian in section 2.2.

(v) From equation (3.18), it follows that in the low temperature/small $L_{d}$
limit $\tau_{2}\gg 1$, the dominant contribution is given by

$\ln{\mathcal{Z}}^{\rm
low}_{d,1}(\tau_{2})=\xi(d+1)\frac{V_{d-1}\tau_{2}}{L_{d}^{d-1}}.$ (3.20)

The result for the Casimir energy with one small spatial dimension in (2.77),
then follows by taking $-\frac{\partial}{\partial\beta}$ of this dominant
contribution.

(vi) Without necessarily implying the existence of an underlying infinite-
dimensional Virasoro algebra, one may write the partition function in a form
suggested by conformal field theories in $2$ spacetime dimensions as

${\mathcal{Z}}_{d,1}(\tau,\bar{\tau})={\rm Tr}\ e^{2\pi
i\tau(\hat{L}_{0}-\frac{c_{d}}{24})-2\pi
i\bar{\tau}(\hat{\bar{L}}_{0}-\frac{c_{d}}{24})}.$ (3.21)

When taking the original expression of the partition function (3.3) and the
expression of the modular parameter (3.4) into account, it follows that

$\hat{L}_{0}=\frac{L_{d}}{4\pi}(\hat{H}+\hat{P}_{d})+\frac{c_{d}}{24},\quad\hat{\bar{L}}_{0}=\frac{L_{d}}{4\pi}(\hat{H}-\hat{P}_{d})+\frac{c_{d}}{24},$
(3.22)

or, equivalently,

$\hat{H}=\frac{2\pi}{L_{d}}(\hat{L}_{0}+\hat{\bar{L}}_{0}-\frac{c_{d}}{12}),\quad\hat{P}_{d}=\frac{2\pi}{L_{d}}(\hat{L}_{0}-\hat{\bar{L}}_{0}).$
(3.23)

Note that in terms of the energy-momentum tensor,

$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\eta_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi,$
(3.24)

these observables are given by

$L_{0}=\frac{L_{d}}{2\pi}\int_{V_{d}}\text{d}^{d}x(T_{--}+T_{+-}),\quad\bar{L}_{0}=\frac{L_{d}}{2\pi}\int_{V_{d}}\text{d}^{d}x(T_{++}+T_{+-}),$
(3.25)

where $x^{\pm}=x^{0}\pm x^{d}$,
$T_{\pm\pm}=\partial_{\pm}\phi\partial_{\pm}\phi$ and
$T_{+-}=\frac{1}{4}\partial_{I}\phi\partial^{I}\phi$ no longer vanishes for
$d>1$. As follows from Remark (v) and will be worked out in detail in section
3.2, in a low temperature expansion, the leading contribution to the partition
function is

$\ln{\rm Z}^{\rm low}_{d,1}(\beta,0)=-\beta E^{d}_{0},$ (3.26)

with $E_{0}^{d}$ given in (2.77). Identifying this contribution with that
involving the central charge $c_{d}$ in (3.21), it follows that

$E_{0}^{d}=-\frac{\pi c_{d}}{6L_{d}},$ (3.27)

and then, when taking (2.77) into account, that

$\boxed{c_{d}=\frac{6\xi(d+1)}{\pi}\frac{V_{d-1}}{L_{d}^{d-1}}}.$ (3.28)

In terms of the central charge, the partition function becomes

$\ln{\mathcal{Z}}_{d,1}(\tau,\bar{\tau})=\frac{\pi
c_{d}}{12\zeta(d+1)}\tau_{2}^{-\frac{d-1}{2}}f_{\frac{d+1}{2}}(\tau,\bar{\tau}),$
(3.29)

while

$\varepsilon^{d}_{\rm vac}=\frac{\pi c_{d}}{6}\frac{L_{d}^{d}}{V_{d}}.$ (3.30)

It also follows from Remark (iv) and will be worked out in detail in section
3.2, that the leading contribution at high temperature/ large $L_{d}$ is

$\ln{\rm Z}^{\rm high}_{d,1}(\beta,0)=\frac{\pi c_{d}L^{d}_{d}}{6\beta^{d}}.$
(3.31)

Following [18], the associated asymptotic micro-canonical density of states is
then

$\rho_{d}(E)=\int^{\infty}_{0}\text{d}\beta\,Z^{\rm
high}_{d,1}(\beta,0)e^{\beta E}=\int_{0}^{\beta}\text{d}\beta\,e^{\frac{\pi
c_{d}L^{d}_{d}}{6\beta^{d}}+\beta E},$ (3.32)

which yields in the saddle point approximation,

$\ln\rho_{d}(E)\approx(d+1)(\frac{EL_{d}}{d})^{\frac{d}{d+1}}(\frac{\pi
c_{d}}{6})^{\frac{1}{d+1}}.$ (3.33)

(vii) Real analytic Eisenstein series are eigenfunctions of the Laplace-
Beltrami operator on the upper half-plane $\mathbb{H}$ (see e.g. [4]),

$\Delta_{\mathbb{H}}=\tau_{2}^{2}(\partial_{\tau_{1}}^{2}+\partial^{2}_{\tau_{2}}),\quad\Delta_{\mathbb{H}}f_{s}(\tau,\bar{\tau})=s(s-1)f_{s}(\tau,\bar{\tau}).$
(3.34)

In terms of $\ln{\mathcal{Z}}_{d,1}(\tau,\bar{\tau})$, one finds

$\Delta_{\mathbb{H}}\ln{\mathcal{Z}}_{d,1}=(d-1)\Big{[}\ln{\mathcal{Z}}_{d,1}-\tau_{2}\partial_{\tau_{2}}\ln{\mathcal{Z}}_{d,1}\Big{]}.$
(3.35)

When taking into account that

$\langle\hat{H}\rangle=-\partial_{\beta}\ln{\rm
Z}_{d,1}(\beta,\alpha),\quad\langle\hat{P}_{d}\rangle=-i\partial_{\alpha}\ln{\rm
Z}_{d,1}(\beta,\alpha),$ (3.36)

as well as

$\Delta\hat{O}=\hat{O}-\langle\hat{O}\rangle,\quad\langle(\Delta\hat{O})^{2}\rangle=\langle\hat{O}^{2}\rangle-(\langle\hat{O}\rangle)^{2},$
(3.37)

the interpretation of equation (3.35) in quantum statistical terms is

$\boxed{\beta^{2}\Big{[}\langle(\Delta\hat{H})^{2}-(\Delta\hat{P}_{d})^{2}\rangle\Big{]}=(d-1)\Big{[}\ln{\rm
Z}_{d,1}+\beta\langle\hat{H}\rangle\Big{]}}.$ (3.38)

Alternatively, using
$\Delta_{\mathbb{H}}=-(\tau-\bar{\tau})^{2}\partial_{\tau}\partial_{\bar{\tau}}$,
this may be written as

$\boxed{-(2\pi(\tau-\bar{\tau}))^{2}\Big{[}\langle\hat{L}_{0}\hat{\bar{L}}_{0}\rangle-\langle\hat{L}_{0}\rangle\langle\hat{\bar{L}}_{0}\rangle\Big{]}}\\\
=\boxed{(d-1)\Big{[}\ln{\mathcal{Z}}_{d,1}-i\pi(\tau-\bar{\tau})\langle\hat{L}_{0}+\hat{\bar{L}}_{0}-\frac{c_{d}}{12}\rangle\Big{]}}.$
(3.39)

(viii) As in section 2.7, the regularized Casimir energy on the spatial
$\mathbb{R}^{d-2}\times\mathbb{T}^{2}$, where $\mathbb{T}^{2}$ is rectangular,
can be obtained from the zeta function in (3.9), by multiplying the result by
$\frac{1}{2}\nu^{2s}$ and the replacements $\tau\to i\frac{L_{d}}{L_{d-1}}$,
$d\to d-1$, $s\to s-\frac{1}{2}$. If $x^{I}$, for $I=1\dots d-2$, denote the
large dimensions and $V_{d-2}=\prod_{I=1}^{d-2}L_{I}$,

$E_{0}(s)=\frac{1}{2}\nu^{2s}\frac{\Gamma(\frac{d+1}{2}-s)}{2^{2s-1}\pi^{\frac{d}{2}}\Gamma(s-\frac{1}{2})}\frac{V_{d-2}L_{d}}{L_{d-1}^{d-2s}}\zeta(\frac{d+1}{2}-s;(\frac{L_{d}}{L_{d-1}})^{2},1),$
(3.40)

so that

$E_{0}(0)=-\frac{1}{2}\frac{\Gamma(\frac{d+1}{2})}{\pi^{\frac{d+1}{2}}}\frac{V_{d-2}L_{d}}{L_{d-1}^{d}}\zeta(\frac{d+1}{2};(\frac{L_{d}}{L_{d-1}})^{2},1).$
(3.41)

###  3.2 Canonical approach

In this section, we evaluate the partition function

${\rm
Z}_{d,1}(\beta,\alpha)=\mathrm{Tr}\,e^{-\beta\hat{H}+i\alpha\hat{P}_{d}},$
(3.42)

directly in the operator formalism where $\hat{H}$ and $\hat{P}_{d}$ are given
by

$\hat{H}=\sum_{n_{i}\in\mathbb{Z}^{d}}\omega_{k}\hat{a}^{\dagger}_{k_{i}}\hat{a}_{k_{i}}+E_{0}^{d},\qquad\hat{P}_{d}=\sum_{n_{i}\in\mathbb{Z}^{d}}k_{d}\hat{a}^{\dagger}_{k_{i}}\hat{a}_{k_{i}},$
(3.43)

and $E_{0}^{d}$ is given in (2.77). It follows that

${\rm Z}_{d,1}(\beta,\alpha)=e^{-\beta
E_{0}^{d}}\sideset{}{{}^{\prime}}{\prod}_{n_{i}\in\mathbb{Z}^{d}}\sum_{N_{k_{i}}\in\mathbb{N}}e^{(-\beta\omega_{k}+i\alpha
k_{d})N_{k_{i}}}=e^{-\beta
E_{0}^{d}}\sideset{}{{}^{\prime}}{\prod}_{n_{i}\in\mathbb{Z}^{d}}\frac{1}{1-e^{-\beta\omega_{k}+i\alpha
k_{d}}}.$ (3.44)

Turning again the sums into integrals in the large dimensions gives

$\begin{split}\ln{\rm Z}_{d,1}(\beta,\alpha)&=-\beta
E_{0}^{d}-\frac{V_{d-1}}{(2\pi)^{d-1}}\int_{-\infty}^{\infty}\text{d}^{d-1}k\sum_{n_{d}\in\mathbb{Z}}\ln\left[1-e^{-\beta\omega_{k}+i\alpha
k_{d}}\right]\\\ &=-\beta
E_{0}^{d}-\frac{V_{d-1}}{(2\pi)^{d-1}}\frac{2\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})}\sum_{n_{d}\in\mathbb{Z}}\int_{0}^{\infty}\text{d}k\,k^{d-2}\ln[1-e^{-\beta\sqrt{k^{2}+k^{2}_{d}}+i\alpha
k_{d}}],\end{split}$ (3.45)

where $k=k_{I}k^{I}$, the range of $I$ is from $1$ to $d-1$ and
$V_{d-1}=\prod_{I=1}^{d-1}L_{I}$. For the sum over $n_{d}$, it is convenient
to split $n_{d}=0$ from the other terms. When using (2.20) for $d\to d-1$ for
the former term, we get

$\ln{\rm Z}_{d,1}(\beta,\alpha)=-\beta
E_{0}^{d}+\frac{\Gamma(d-1)\zeta(d)}{2^{d-2}\pi^{\frac{d-1}{2}}\Gamma(\frac{d-1}{2})}\frac{V_{d-1}}{\beta^{d-1}}-\frac{V_{d-1}}{2^{d-2}\pi^{\frac{d-1}{2}}\Gamma(\frac{d-1}{2})}\sideset{}{{}^{\prime}}{\sum}_{n_{d}\in\mathbb{Z}}I_{d}(\beta,\alpha;n_{d}),$
(3.46)

where

$I_{d}(\beta,\alpha;n_{d})=\int_{0}^{\infty}\text{d}k\,k^{d-2}\ln[1-e^{-\beta\sqrt{k^{2}+k^{2}_{d}}+i\alpha
k_{d}}].$ (3.47)

Introducing the variable $z$ as

$\displaystyle z=\beta\sqrt{k^{2}+k^{2}_{d}},\quad
k^{d-2}\text{d}k=\frac{1}{\beta^{2}\beta^{d-3}}(z^{2}-\beta^{2}k^{2}_{d})^{\frac{d-3}{2}}z\text{d}z,$
(3.48)

we get

$I_{d}(\beta,\alpha;n_{d})=-\frac{1}{\beta^{d-1}}\int_{\beta\absolutevalue{k_{d}}}^{\infty}\text{d}z\,z(z^{2}-\beta^{2}k^{2}_{d})^{\frac{d-3}{2}}\ln\left[1-e^{-z+i\alpha
k_{d}}\right].$ (3.49)

Expanding the logarithm as

$\ln\left[1-e^{-z+i\alpha
k_{d}}\right]=-\sum_{l\in\mathbb{N}^{*}}\frac{e^{-lz+il\alpha k_{d}}}{l},$
(3.50)

we get, after the change of variables $z^{\prime}=lz$,

$I_{d}(\beta,\alpha;n_{d})=-\frac{1}{\beta^{d-1}}\sum_{l\in\mathbb{N}^{*}}\frac{e^{il\alpha
k_{d}}}{l^{d}}\int_{l\beta\absolutevalue{k_{d}}}^{\infty}\text{d}zz(z^{2}-l^{2}\beta^{2}k^{2}_{d})^{\frac{d-3}{2}}e^{-z},$
(3.51)

The integrals are given in terms of a modified Bessel function of the second
kind as

$\int_{l\beta\absolutevalue{k_{d}}}^{\infty}\text{d}zz(z^{2}-l^{2}\beta^{2}k^{2}_{d})^{\frac{d-3}{2}}e^{-z}=2^{\frac{d-2}{2}}\frac{\Gamma(\frac{d-1}{2})}{\sqrt{\pi}}(l\beta\absolutevalue{k_{d}})^{\frac{d}{2}}K_{\frac{d}{2}}(l\beta\absolutevalue{k_{d}}),$
(3.52)

so that

$\ln{\rm Z}_{d,1}(\beta,\alpha)=-\beta
E_{0}^{d}+\frac{\Gamma(d-1)\zeta(d)}{2^{d-2}\pi^{\frac{d-1}{2}}\Gamma(\frac{d-1}{2})}\frac{V_{d-1}}{\beta^{d-1}}\\\
+2\frac{V_{d-1}}{L_{d}^{\frac{d}{2}}\beta^{\frac{d-2}{2}}}\sideset{}{{}^{\prime}}{\sum}_{n_{d}\in\mathbb{Z}}\sum_{l\in\mathbb{N}^{*}}(\frac{\absolutevalue{n_{d}}}{l})^{\frac{d}{2}}K_{\frac{d}{2}}(2\pi
l\absolutevalue{n_{d}}\frac{\beta}{L_{d}})e^{2\pi
iln_{d}\frac{\alpha}{L_{d}}}.$ (3.53)

Remarks:

(i) At low temperature/small distance $\frac{\beta}{L_{d}}\gg 1$, the leading
term in the expansion of the partition function is directly related to the
Casimir energy. The leading correction is the contribution of the modes with
spatial frequencies $n_{d}=0$. It coincides with the black body result (2.24)
of a massless scalar field in $d-1$ spatial dimensions. On account of the
equivalence of this expression with (2.82), it can also be written more
compactly as $\xi(d)\frac{V_{d-1}}{\beta^{d-1}}$. It is independent of $L_{d}$
and thus does not contribute to the Casimir pressure,

${\rm p}_{d,1}(\beta,\alpha)=\frac{1}{V_{d-1}}\frac{\partial(\beta^{-1}\ln{\rm
Z}_{d,1}(\beta,\alpha))}{\partial L_{d}}.$ (3.54)

The asymptotic expansion

$K_{\nu}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}(1+O(x^{-1})),$ (3.55)

for large $x$, implies that all other terms are exponentially suppressed. It
follows that low-temperature/small distance expansion of the Casimir pressure
is

$\boxed{{\rm p}^{\rm
low}_{d,1}(\beta,\alpha)=-d\xi(d+1)\frac{1}{L^{d+1}_{d}}+\dots},$ (3.56)

where the dots denote exponentially suppressed terms. In the low-
temperature/small distance expansion of the entropy,

${\rm S}_{d,1}(\beta,\alpha)=(1-\beta\partial_{\beta})\ln{\rm
Z}_{d,1}(\beta,\alpha),$ (3.57)

on the other hand, the first term in (3.53) proportional to the Casimir energy
in $d$ spatial dimensions drops out since it is linear in $\beta$ and the
leading term now comes from the lower dimensional scalar field, i.e., the
modes with $n_{d}=0$,

$\boxed{{\rm S}^{\rm
low}_{d,1}(\beta,\alpha)=d\xi(d)\frac{V_{d-1}}{\beta^{d-1}}+\dots}.$ (3.58)

It may be written in terms of the dimensionless number characterizing the
Casimir energy density defined in (2.78) in $d-1$ spatial dimensions as

${\rm S}^{\rm low}_{d,1}(\beta,\alpha)=d\varepsilon_{\rm
vac}^{d-1}\frac{V_{d-1}}{\beta^{d-1}}+\dots.$ (3.59)

This is a novel low-temperature formula for the model under consideration,
that relates the leading contribution to the entropy to the Casimir energy
density of a massless scalar field in one dimension lower. Even though it
looks similar, it is not the higher dimensional generalization of the Cardy
formula [14, 16] discussed in [18], which is valid at high temperature (see
below). Since the two leading terms in the low temperature/small intervall
expansion of (3.53) do not depend on $\alpha$ they are the same when starting
from $\ln Z_{d,1}(\beta,\mu)$. It follows that the leading terms in the
expansion of

$p_{d,1}(\beta,\mu)=\frac{1}{V_{d-1}}\frac{\partial(\beta^{-1}\ln{Z}_{d,1}(\beta,\mu))}{\partial
L_{d}},\quad
S_{d,1}(\beta,\mu)=(1-\beta\partial_{\beta})\ln{Z}_{d,1}(\beta,\mu),$ (3.60)

are still given by the right hand sides of (3.56) and of (3.58).

(ii) When changing the double sum in the last term to a sum over
$m=ln_{d}\in\mathbb{Z}^{*}$ and introducing the divisor sum

$\sigma_{s}(m)=\sum_{n|m}n^{s},$ (3.61)

the exponentially suppressed terms, i.e., the last line of (3.53), may be
written as

$2\frac{V_{d-1}}{L_{d}^{\frac{d}{2}}\beta^{\frac{d-2}{2}}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\sigma_{-d}(m)\absolutevalue{m}^{\frac{d}{2}}K_{\frac{d}{2}}(2\pi
m\frac{\beta}{L_{d}})e^{2\pi im\frac{\alpha}{L_{d}}}.$ (3.62)

When taking into account the explicit expression for the vacuum energy
$E_{0}^{d}$ in (2.77), the partition function may be re-written as

$\boxed{\ln{\rm
Z}_{d,1}(\beta,\alpha)=\frac{V_{d-1}}{L_{d}^{d-1}}\Big{[}\xi(d+1)\frac{\beta}{L_{d}}+\xi(d)(\frac{L_{d}}{\beta})^{d-1}}\\\
\boxed{+2(\frac{L_{d}}{\beta})^{\frac{d-2}{2}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\sigma_{-d}(m)\absolutevalue{m}^{\frac{d}{2}}K_{\frac{d}{2}}(2\pi
m\frac{\beta}{L_{d}})e^{2\pi im\frac{\alpha}{L_{d}}}\Big{]}}.$ (3.63)

The equivalence of the functional Lagrangian and the canonical approaches to
computing the partition function then implies that this last result is the
same than (3.10). For $s=\frac{d+1}{2}$, this shows

$f_{s}(\tau,\bar{\tau})=2\zeta(2s)\Big{[}\tau_{2}^{s}+\frac{\xi(2s-1)}{\xi(2s)}\tau_{2}^{1-s}\\\
+\frac{2}{\xi(2s)}\tau_{2}^{\frac{1}{2}}\sideset{}{{}^{\prime}}{\sum}_{m}\sigma_{1-2s}(m)\absolutevalue{m}^{\frac{2s-1}{2}}K_{\frac{2s-1}{2}}(2\pi
m\tau_{2})e^{2\pi im\tau_{1}}\Big{]},$ (3.64)

which is the Fourier expansion of $f_{s}(\tau,\bar{\tau})$, traditionally
derived using Poisson resummation (see e.g. [4], Appendix A).

(iii) When expressed in terms of inverse temperature and chemical potential,
the generating set of transformations of the modular group become

$\tau^{\prime}=\tau+1\iff\left\\{\begin{array}[]{l}(\frac{\beta}{L_{d}})^{\prime}=\frac{\beta}{L_{d}}\\\
(\frac{\alpha}{L_{d}})^{\prime}=\frac{\alpha}{L_{d}}+1\end{array}\right.\quad\tau^{\prime}=-\frac{1}{\tau}\iff\left\\{\begin{array}[]{l}(\frac{\beta}{L_{d}})^{\prime}=\frac{L_{d}\beta}{\alpha^{2}+\beta^{2}}\\\
(\frac{\alpha}{L_{d}})^{\prime}=-\frac{L_{d}\alpha}{\alpha^{2}+\beta^{2}}\end{array}\right..$
(3.65)

Under the first of these transformations, $\ln{\rm Z}_{d,1}(\beta,\alpha)$ in
(3.63) is manifestly invariant, as required by (3.15) for $c=0,a=b=d=1$. For
the second of these transformations, we get

$\ln{\rm
Z}_{d,1}(\beta^{\prime},\alpha^{\prime})=\frac{(\alpha^{2}+\beta^{2})^{\frac{d-1}{2}}}{L_{d}^{d-1}}\ln{\rm
Z}_{d,1}(\beta,\alpha).$ (3.66)

When transposing and using the explicit expression in (3.63) for the LHS, this
gives

$\ln{\rm
Z}_{d,1}(\beta,\alpha)=\frac{V_{d-1}}{(\alpha^{2}+\beta^{2})^{\frac{d-1}{2}}}\Big{[}\xi(d+1)\frac{L_{d}\beta}{\alpha^{2}+\beta^{2}}+\xi(d)(\frac{\alpha^{2}+\beta^{2}}{L_{d}\beta})^{d-1}\\\
+2(\frac{\alpha^{2}+\beta^{2}}{L_{d}\beta})^{\frac{d-2}{2}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\sigma_{-d}(m)\absolutevalue{m}^{\frac{d}{2}}K_{\frac{d}{2}}(2\pi
m\frac{L_{d}\beta}{\alpha^{2}+\beta^{2}})e^{-2\pi
im\frac{L_{d}\alpha}{\alpha^{2}+\beta^{2}}}\Big{]}.$ (3.67)

For a high temperature/large interval expansion, it is more convenient to use
$\mu$ rather than $\alpha$, in terms of which the previous expression becomes

$\ln{Z}_{d,1}(\beta,\mu)=\frac{V_{d-1}}{L_{d}^{d-1}(1+\mu^{2})^{\frac{d-1}{2}}}\Big{[}\frac{\xi(d+1)}{1+\mu^{2}}(\frac{\beta}{L_{d}})^{-d}+\xi(d)(1+\mu^{2})^{d-1}+\\\
+2(\frac{\beta}{L_{d}})^{-\frac{d}{2}}(1+\mu^{2})^{\frac{d-2}{2}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\sigma_{-d}(m)\absolutevalue{m}^{\frac{d}{2}}K_{\frac{d}{2}}(2\pi
m\frac{L_{d}}{\beta}\frac{1}{1+\mu^{2}})e^{-2\pi
im\frac{L_{d}}{\beta}\frac{1}{1+\mu^{2}}}\Big{]}.$ (3.68)

The high temperature/large interval expansion is then defined by
$\frac{\beta}{L_{d}}\ll 1$ at fixed $\mu$. The leading contributions are given
by the first two terms, while the others are exponentially suppressed,

$\ln Z^{\rm
high}_{d,1}(\beta,\mu)=\frac{V_{d-1}L_{d}}{\beta^{d}}\xi(d+1)(1+\mu^{2})^{-\frac{d+1}{2}}+\frac{V_{d-1}}{L_{d}^{d-1}}\xi(d)(1+\mu^{2})^{\frac{d-1}{2}}+\dots.$
(3.69)

The associated expansion of the Casimir pressure is

$p_{d,1}^{\rm
high}(\beta,\mu)=\frac{\xi(d+1)}{\beta^{d+1}}(1+\mu^{2})^{-\frac{d+1}{2}}-\frac{(d-1)\xi(d)}{\beta
L_{d}^{d}}(1+\mu^{2})^{\frac{d-1}{2}}+\dots,$ (3.70)

while for the entropy, one finds

$S^{\rm
high}_{d,1}(\beta,\mu)=\frac{V_{d-1}L_{d}}{\beta^{d}}(d+1)\xi(d+1)(1+\mu^{2})^{-\frac{d+1}{2}}+\frac{V_{d-1}}{L_{d}^{d-1}}\xi(d)(1+\mu^{2})^{\frac{d-1}{2}}+\dots.$
(3.71)

This can be written as

$S^{\rm high}_{d,1}(\beta,\mu)={(d+1)\varepsilon^{d}_{\rm
vac}}\frac{V_{d}}{\beta^{d}}(1+\mu^{2})^{-\frac{d+1}{2}}+\varepsilon^{d-1}_{\rm
vac}\frac{V_{d-1}}{L_{d}^{d-1}}(1+\mu^{2})^{\frac{d-1}{2}}+\dots.$ (3.72)

The first term, which can also be written as,

$S^{\rm high}_{d,1}(\beta,\mu)\approx{(d+1)}\frac{\pi
c_{d}L^{d}_{d}}{6\beta^{d}}(1+\mu^{2})^{-\frac{d+1}{2}},$ (3.73)

generalizes formula (8) of [18] to the case of a non-vanishing chemical
potential. The associated asymptotic density of states becomes

$\ln\rho_{d}(E,\mu)\approx(d+1)(\frac{EL_{d}}{d})^{\frac{d}{d+1}}(\frac{\pi
c_{d}}{6})^{\frac{1}{d+1}}\frac{1}{\sqrt{1+\mu^{2}}}.$ (3.74)

###  3.3 Worldline approach

Consider a particle on a torus with

$\hat{\rm
H}_{\beta,\alpha}=(\hat{p}_{d+1}-\frac{\alpha}{\beta}\hat{p}_{d})^{2}+\hat{p}_{d}^{2}.$
(3.75)

The associated heat kernel is

$\langle x^{\prime}|e^{-t\hat{\rm H}_{\beta,\alpha}}|x\rangle=\frac{1}{\beta
L_{d}}\sum_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}e^{-tf},$ (3.76)

where

$f=|\frac{2\pi n_{d+1}}{\beta}+\frac{2\pi(\alpha+i\beta)n_{d}}{\beta
L_{d}}|^{2}+i2\pi[n_{d+1}\frac{x^{\prime}_{d+1}-x_{d+1}}{\beta}+{n_{d}}\frac{x^{\prime}_{d}-x_{d}}{L_{d}}],$
(3.77)

which can again be shown by inserting a resolution of identity. When combining
this heat kernel with the heat kernel (2.50) for $d-1$ large dimensions, one
gets, after dropping the $n_{d+1}=0=n_{d}$ term,

$\ln{\rm
Z}_{d,1}(\beta,\alpha)=\frac{V_{d-1}}{2^{d}\pi^{\frac{d-1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{d+1}{2}}e^{-t(\frac{2\pi}{\beta})^{2}|n_{d+1}+\tau
n_{d}|^{2}}.$ (3.78)

Using again (2.65),

$\frac{1}{2^{d}\pi^{\frac{d-1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{d+1}{2}}e^{-t(\frac{2\pi}{\beta})^{2}|n_{d+1}+\tau
n_{d}|^{2}}\\\
=\frac{\Gamma(\frac{1-d}{2})}{2^{d}\pi^{\frac{d-1}{2}}}(\frac{2\pi}{\beta})^{d-1}\sideset{}{{}^{\prime}}{\sum}_{(n_{d+1},n_{d})\in\mathbb{Z}^{2}}\frac{1}{|n_{d+1}+\tau
n_{d}|^{1-d}},$ (3.79)

which can be expressed in terms of the Eisenstein series (1.1) to yield

$\ln\mathcal{Z}_{d,1}(\tau,\bar{\tau})=\frac{\pi^{\frac{d-1}{2}}\Gamma(\frac{1-d}{2})}{2}\frac{V_{d-1}}{L_{d}^{d-1}}\frac{f_{\frac{1-d}{2}}(\tau,\bar{\tau})}{\tau_{2}^{\frac{d-1}{2}}}.$
(3.80)

This agrees with (3.10) after using the reflection formula for Eisenstein
series (3.8) at $z=\frac{d+1}{2}$.

Another way to obtain the heat kernel for a particle on a torus is by a
standard path integral computation, in which case one finds instead

$\langle x^{\prime}|e^{-t\hat{\rm H}_{\beta,\alpha}}|x\rangle=\frac{1}{4\pi
t}\sum_{(m^{d},m^{d+1})\in\mathbb{Z}^{2}}e^{-\frac{1}{4t}g},$ (3.81)

where

$g={(\alpha^{2}+\beta^{2})}(\frac{x^{\prime}_{d+1}-x_{d+1}}{\beta}+m^{d+1})^{2}+2\alpha
L_{d}(\frac{x^{\prime}_{d+1}-x_{d+1}}{\beta}+m^{d+1})(\frac{x^{\prime}_{d}-x_{d}}{L_{d}}+m^{d})\\\
+L_{d}^{2}(\frac{x^{\prime}_{d}-x_{d}}{L_{d}}+m^{d})^{2}.$ (3.82)

These two expressions for the heat kernel of a particle on a torus, (3.76) and
(3.81) are related by the Poisson summation formula. When combining now the
expression in (3.81) with the heat kernel in the $d-1$ large dimensions, one
gets, after dropping the $m^{d}=0=m^{d+1}$ term,

$\ln{\rm
Z}_{d,1}(\beta,\alpha)=\frac{V_{d-1}L_{d}\beta}{2^{d+2}\pi^{\frac{d+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{(m^{d},m^{d+1})\in\mathbb{Z}^{2}}\int^{\infty}_{0}\text{d}t\,t^{-\frac{d+3}{2}}e^{-\frac{1}{4t}L_{d}^{2}|m^{d+1}\tau+m^{d}|^{2}}.$
(3.83)

In this case, the change of integration variable $t\to\frac{1}{t}$ gives

$\ln{\rm
Z}_{d,1}(\beta,\alpha)=\frac{V_{d-1}L_{d}\beta}{2^{d+2}\pi^{\frac{d+1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{(m^{d},m^{d+1})\in\mathbb{Z}^{2}}\int^{\infty}_{0}\text{d}t\,t^{\frac{d+1}{2}-1}e^{-\frac{t}{4}L_{d}^{2}|m^{d+1}\tau+m^{d}|^{2}}.$
(3.84)

Using now (2.65), the result can be shown directly to agree with (3.10)
without using a reflection formula. Alternatively, the equivalence of both
expressions (3.78) and (3.83) can be used to prove the reflection formula
(3.8).

Remarks:

(i) For a particle on a torus as in (3.75), one notes that under the canonical
transformation

$\left\\{\begin{array}[]{l}p^{\prime}_{d+1}=p_{d+1}-\frac{\alpha}{\beta}p_{d},\\\
p^{\prime}_{d}=p_{d}\end{array}\right.\quad\left\\{\begin{array}[]{l}x^{\prime}_{d+1}=x_{d+1},\\\
x^{\prime}_{d}=x_{d}+\frac{\alpha}{\beta}x_{d+1},\end{array}\right.$ (3.85)

the Hamiltonian is diagonalized, $\hat{\rm
H}_{\beta,\alpha}=(\hat{p}^{\prime}_{d+1})^{2}+(\hat{p}^{\prime}_{d})^{2}$,
while the periodicities

$x_{d+1}\sim x_{d+1}+m^{d+1}\beta,\quad x_{d}\sim x_{d}+m^{d}L_{d},$ (3.86)

become

$x^{\prime}_{d+1}\sim x^{\prime}_{d+1}+m^{d+1}\beta,\quad x^{\prime}_{d}\sim
x^{\prime}_{d}+m^{d}L_{d}+m^{d+1}\alpha.$ (3.87)

When $\alpha=0$, the role played by the canonical pairs
$(x^{\prime}_{d+1},p^{\prime}_{d+1})$ and $(x^{\prime}_{d},p^{\prime}_{d})$ in
the computation of ${\rm Tr}\ e^{-t\hat{\rm H}_{\beta,0}}$ is symmetric if one
exchanges at the same time the periodicities.

(ii) The Riemann theta function is defined by

$\vartheta_{n}(z|g)=\sum_{m^{\alpha}\in\mathbb{Z}^{n}}e^{\pi
im^{\alpha}g_{\alpha\beta}m^{\beta}+2\pi
im^{\alpha}z_{\alpha}},\qquad\alpha,\beta=1,...,n,$ (3.88)

for $g=g_{1}+ig_{2}\in\mathbb{H}_{n}=\\{g\in
M_{n}(\mathbb{C})|g=g^{T},g_{2}>0\\}$ and $z\in\mathbb{C}^{n}$. Since
$\vartheta_{n=1}(z|g)|_{g=\tau}=\vartheta_{3}(z|\tau)$, the Riemann theta
function is the higher dimensional generalization of the Jacobi theta
function. One can also define

$\theta_{n}(g)\equiv\vartheta_{n}(0|ig)=\sum_{m^{\alpha}\in\mathbb{Z}^{n}}e^{-\pi
m^{\alpha}g_{\alpha\beta}m^{\beta}}.$ (3.89)

Consider now the matrix

$\hat{g}_{\alpha\beta}=\frac{1}{\tau_{2}}\left(\begin{matrix}1&\tau_{1}\\\
\tau_{1}&\absolutevalue{\tau}^{2}\end{matrix}\right),$ (3.90)

with unit determinant. After the change of variables
$\frac{\tau_{2}L^{2}_{d}}{4\pi t}=t^{\prime}$, the partition function in
(3.83) can be written as

$\boxed{\ln\mathcal{Z}_{d,1}(\tau,\bar{\tau})=\frac{V_{d-1}}{2L^{d-1}_{d}\tau_{2}^{\frac{d-1}{2}}}\int_{0}^{\infty}\text{d}t\,t^{\frac{d+1}{2}-1}\big{(}\theta_{2}(t\hat{g})-1\big{)}}.$
(3.91)

Note that this suitably generalizes (2.70) to the case where the Euclidean
spacetime manifold is $\mathbb{R}^{d-1}\times\mathbb{T}^{2}$. This integral
expression makes modular covariance of the partition function transparent in
the sense that the full behavior is governed by the prefactor alone since
$\theta_{2}(\hat{g})$ is modular invariant. Indeed, since
$m^{\alpha}\hat{g}_{\alpha\beta}m^{\beta}=\frac{1}{\tau_{2}}|m^{d+1}\tau+m^{d}|^{2}$,
the transformation $\tau^{\prime}=\tau+1$, can be absorbed by shifting the
sum, $m^{d+1}\to m^{d+1}-1$, the transformation
$\tau^{\prime}=-\frac{1}{\tau}$ results in the replacement of $\hat{g}$ by
$\hat{g}^{-1}=\epsilon^{T}\hat{g}^{T}\epsilon$, with $\epsilon$ completely
antisymmetric, and $\epsilon^{12}=1$, which can again be absorbed by re-
arranging the sums.

Alternatively, the equivalence of the functional and world-line approaches to
computing the partition function, i.e., the equality of the right hand sides
of (3.13) and (3.91), shows that, at $z=d+1$, the completed real analytic
Eisenstein series is directly related to the Mellin transform of the Riemann
theta function minus one,

$\xi(z;\tau,\bar{\tau})=\frac{1}{2}\int_{0}^{\infty}\text{d}t\,t^{\frac{z}{2}-1}\big{(}\theta_{2}(t\hat{g})-1\big{)},\quad\mathfrak{Re}(z)>1.$
(3.92)

###  3.4 One spatial dimension

In this section, we review for completeness the well-known result [42, 43]
(see e.g. [44, 45, 46] for reviews) for a torus in Euclidean spacetime, that
is to say we derive the partition function of a massless scalar field in a one
spatial dimension, $d=1$, $L_{1}=L$, with periodic boundary conditions. As we
have seen in section 3.1, a naive application of the functional approach leads
one to the divergent expression

$\ln{\mathcal{Z}}_{1}(\tau,\bar{\tau})=\frac{1}{2\pi}f_{1}(\tau,\bar{\tau}).$
(3.93)

####  3.4.1 Canonical approach

No such divergences occur in the canonical approach. In the analysis of
section 2.1, the full Hamiltonian, including the $n=0$ mode, which is a free
particle, is now given by

$H=\frac{1}{L}\frac{p^{2}}{2}+\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n\in\mathbb{Z}}\omega_{n}(a^{*}_{n}a_{n}+a_{n}a^{*}_{n}),\qquad\omega_{n}=\frac{2\pi\absolutevalue{n}}{L}.$
(3.94)

After the conventional redefinition $p=\sqrt{{4\pi}}a_{0}$, the quantum
Hamiltonian including the zero point energy $E^{1}_{0}$ in (2.77) can be
written as

$\hat{H}=-\frac{\pi}{6L}+\frac{2\pi}{L}\big{(}\hat{a}_{0}^{2}+\sideset{}{{}^{\prime}}{\sum}_{n\in\mathbb{Z}}\absolutevalue{n}\hat{a}^{\dagger}_{n}\hat{a}_{n}\big{)}.$
(3.95)

Using (2.16), we get

${\rm
Z}_{1}(\beta,\alpha)=\sqrt{\frac{L}{2\pi\beta}}e^{\frac{\pi\beta}{6L}}\sideset{}{{}^{\prime}}{\prod}_{n\in\mathbb{Z}}\sum_{N_{n}\in\mathbb{N}}e^{\frac{2\pi}{L}(-\beta\absolutevalue{n}+i\alpha
n)N_{n}}\\\
=\sqrt{\frac{L}{2\pi\beta}}e^{\frac{\pi\beta}{6L}}\sideset{}{{}^{\prime}}{\prod}_{n\in\mathbb{Z}}\frac{1}{1-e^{\frac{2\pi}{L}(-\beta\absolutevalue{n}+i\alpha
n)}}.$ (3.96)

Defining

$q=e^{2\pi i\tau},\quad\bar{q}=e^{-2\pi i\bar{\tau}},$ (3.97)

where $\tau$ and $\bar{\tau}$ are given in (3.4), we have

$\ln\mathcal{Z}_{1}(\tau,\bar{\tau})=-\frac{1}{2}\ln(2\pi)+\frac{\pi\tau_{2}}{6}-\frac{1}{2}\ln{(\tau_{2})}-\sum_{n\in\mathbb{N}^{*}}\ln{[(1-q^{n})(1-\bar{q}^{n})]}.$
(3.98)

In terms of Dedekind’s eta function,

$\eta(q)=q^{\frac{1}{24}}\prod_{n\in\mathbb{N}^{*}}(1-q^{n}),$ (3.99)

and up to irrelevant numerical factors, the modular invariant partition
function can then be written compactly as

$\boxed{\mathcal{Z}_{1}(\tau,\bar{\tau})=\frac{1}{\sqrt{\tau_{2}}\absolutevalue{\eta(q(\tau))}^{2}}}.$
(3.100)

Note that in this case, since $\ln\mathcal{Z}_{1}(\tau,\bar{\tau})$ is the sum
of a holomorphic and anti-holomorphic part, up to the contribution from the
free particle,

$\Delta_{\mathbb{H}}\ln\mathcal{Z}_{1}(\tau,\bar{\tau})=\frac{1}{2}.$ (3.101)

This is not the formal continuation of (3.35) to $d=1$ because of the
contribution of the free particle, which gives rise to the factor $1/2$ on the
right hand side. Hence, in this case,

$\langle(\Delta\hat{H})^{2}-(\Delta\hat{P})^{2}\rangle=\frac{1}{2\beta^{2}},\quad\langle\hat{L}_{0}\hat{\bar{L}}_{0}\rangle-\langle\hat{L}_{0}\rangle\langle\hat{\bar{L}}_{0}\rangle=-\frac{1}{8\pi^{2}(\tau-\bar{\tau})^{2}}.$
(3.102)

####  3.4.2 Dimensional continuation

A different way to give a meaning the non-convergent expression of the
partition function (3.93) obtained in the functional or heat kernel approach
is to study the limit $s\to 1$ of the real analytic Eisenstein series
$f_{s}(\tau,\bar{\tau})$ by starting from its Fourier expansion [4] given in
equation (3.64),

$f_{s}(\tau,\bar{\tau})=2\zeta(2s)\tau_{2}^{s}+2\sqrt{\pi}\frac{\Gamma(s-\frac{1}{2})\zeta(2s-1)}{\Gamma(s)}\tau_{2}^{1-s}\\\
+2\frac{\pi^{s}}{\Gamma(s)}\tau_{2}^{\frac{1}{2}}\sideset{}{{}^{\prime}}{\sum}_{n\in\mathbb{Z}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\absolutevalue{\frac{n}{m}}^{s-\frac{1}{2}}K_{s-\frac{1}{2}}(2\pi\absolutevalue{nm}\tau_{2})e^{2\pi
in\absolutevalue{m}\tau_{1}}.$ (3.103)

Only the second term diverges in the limit $s\rightarrow 1$, while the first
term on the right hand side becomes

$\displaystyle\frac{\pi^{2}}{3}\tau_{2}=-\frac{\pi}{12}\ln\absolutevalue{q}^{2}.$
(3.104)

Using $K_{\frac{1}{2}}(z)=e^{-z}\sqrt{\frac{\pi}{2z}}$, the term on the last
line becomes

$\pi\sideset{}{{}^{\prime}}{\sum}_{n\in\mathbb{Z}}\sideset{}{{}^{\prime}}{\sum}_{m\in\mathbb{Z}}\frac{1}{\absolutevalue{m}}e^{-2\pi\absolutevalue{nm}\tau_{2}+2\pi
in\absolutevalue{m}\tau_{1}}=2\pi\sum_{m\in\mathbb{N^{*}}}\sum_{n\in\mathbb{N^{*}}}\frac{1}{m}(e^{2\pi
inm\tau}+e^{-2\pi inm\bar{\tau}})\\\
=-2\pi\sum_{n\in\mathbb{N}^{*}}\big{[}\ln(1-e^{2\pi in\tau})+\ln(1-e^{-2\pi
in\bar{\tau}})\big{]}\\\
=-2\pi\ln\prod_{n\in\mathbb{N}^{*}}(1-q^{n})\prod_{n\in\mathbb{N}^{*}}(1-\bar{q}^{n}).$
(3.105)

These terms combine into

$-2\pi\ln\prod_{n\in\mathbb{N}^{*}}q^{\frac{1}{24}}(1-q^{n})\prod_{n\in\mathbb{N}^{*}}\bar{q}^{\frac{1}{24}}(1-\bar{q}^{n})=-2\pi\ln\absolutevalue{\eta(q)}^{2}.$
(3.106)

For the divergent term, we consider the expansion around $s=1$ using

$\zeta(s)=\frac{1}{s-1}+\gamma+{O}(s-1),$ (3.107)

so that

$2\sqrt{\pi}\frac{\Gamma(s-\frac{1}{2})\zeta(2s-1)}{\Gamma(s)}\tau_{2}^{1-s}=\frac{\pi}{s-1}+\pi\bigg{(}3\gamma+\frac{\Gamma^{\prime}(\frac{1}{2})}{\Gamma(\frac{1}{2})}-\ln\tau_{2}\bigg{)}+{O}(s-1)\\\
=\frac{\pi}{s-1}+\pi\bigg{(}2\gamma-2\ln 2-\ln\tau_{2}\bigg{)}+{O}(s-1).$
(3.108)

We thus get

$\lim_{s\to
1}\bigg{(}f_{s}(\tau,\bar{\tau})-\frac{\pi}{s-1}\bigg{)}=2\pi\Big{(}\gamma-\ln
2-\ln\sqrt{\tau_{2}}\absolutevalue{\eta(q)}^{2}\Big{)}.$ (3.109)

Therefore, once we remove the divergence, the real analytic Eisenstein series
for $s=1$ contains the result for the partition function of the $d=1$ case.
Note that the divergence in the second term of the expansion in (3.103), comes
from the $n_{1}=0$ term in the mode expansion of the field (at fixed time), as
shown in the canonical approach in the previous section. In $d>1$ this mode is
a scalar field in $d-1$ spatial dimensions. For $d=1$, it is a free particle,
which gives the factor $\sqrt{\tau_{2}}$ needed for modular invariance.
Associated with the divergence, the finite, $\tau_{2}$ independent term, is
undetermined and needs to be fixed by a normalization condition.

How this computation appears from the viewpoint of the integral representation
connected to the world-line approach is discussed in [43].

##  4 Higher dimensional tori

###  4.1 Generalities

We now consider the more general case of the Euclidean spacetime manifold
$\mathbb{R}^{p}\times\mathbb{T}^{q+1}$ [6, 12], with $p+q=d$. Coordinates on
$\mathbb{R}^{p}$ are denoted by $x^{I}$, $I=1,...,p$. The skewed torus
$\mathbb{T}^{q+1}$ is defined to be the quotient of $\mathbb{R}^{q+1}$ with
coordinates $x^{a}=(x^{i},x^{d+1})$, $i=p+1,\dots d$, by the lattice
$\Lambda^{d+1}$ generated by a set of $q+1$ constant frame vectors
${e}\indices{{}^{a}_{\alpha}}\in\mathbb{R}^{d+1}$,

$\mathbb{T}^{q+1}=\mathbb{R}^{q+1}/\Lambda^{q+1},\quad\Lambda^{q+1}=\big{\\{}m^{\alpha}{e}\indices{{}^{a}_{\alpha}}|\,m^{\alpha}\in\mathbb{Z}^{q+1}\big{\\}},$
(4.1)

with $\alpha=1,\dots,q+1$.

Functions on this manifold satisfy the periodicity conditions

$\phi(x^{I},x^{a})=\phi(x^{I},x^{a}+{e}\indices{{}^{a}_{\alpha}}),\quad\forall\alpha$
(4.2)

In $x^{a}$ coordinates, the information on the geometry of the torus is thus
encoded in the skewed periodicities determined by the frame vectors
${e}\indices{{}^{a}_{\alpha}}$ whereas the Lagrangian Euclidean action is

$\displaystyle
S^{E}_{L}[\phi]=\frac{1}{2}\int_{V_{d+1}}\text{d}^{d+1}x\,[\partial_{I}\phi\partial^{I}\phi+\delta^{ab}\partial_{a}\phi\partial_{b}\phi].$
(4.3)

Associated to the $q+1$ linearly independent frame vectors, there is a co-
frame ${e}\indices{{}_{a}^{\alpha}}$ such that

${e}\indices{{}_{\alpha}^{a}}{e}\indices{{}_{b}^{\alpha}}=\delta^{a}_{b},\quad{e}\indices{{}_{\alpha}^{a}}{e}\indices{{}_{a}^{\beta}}=\delta^{\beta}_{\alpha}.$
(4.4)

This allows one to define coordinates
$y^{\alpha}={e}\indices{{}_{a}^{\alpha}}x^{a}$ in which periodicities of the
field become simply unity in all these directions separately,

$\phi(x^{I},y^{\alpha})=\phi(x^{I},y^{\alpha}+\delta_{\beta}^{\alpha}),\quad\forall\beta.$
(4.5)

In turn, the Cartesian metric $\delta_{ab}$ becomes

$g_{\alpha\beta}={e}\indices{{}^{a}_{\alpha}}\delta_{ab}{e}\indices{{}^{b}_{\beta}},$
(4.6)

When working in $y^{\alpha}$ coordinates, the information on the non-trivial
geometry of the torus gets encoded in the Lagrangian action through the
coupling of the scalar field to the metric $g_{\alpha\beta}$,

$S^{E}_{L}[\phi;g]=\frac{1}{2}\int_{V_{d+1}}\text{d}^{p}x\text{d}^{q+1}y\,\sqrt{g}[g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+\partial_{I}\phi\partial^{I}\phi].$
(4.7)

Note that since the metric is constant and flat, there is no need here to
consider the conformally coupled scalar field with its improved energy-
momentum tensor, as usually done in the case of a general curved background
(see e.g. [47, 35, 12]). The volume of the torus is
$\text{V}_{q+1}=\mathrm{det}\,{e}\indices{{}^{a}_{\alpha}}=\sqrt{{g}}$. For
later use, we define the re-scaled frame, co-frame, metric and inverse metric,

$\begin{split}\hat{e}\indices{{}^{a}_{\alpha}}=(\text{V}_{q+1})^{-\frac{1}{q+1}}{e}\indices{{}^{a}_{\alpha}},\quad\hat{e}\indices{{}_{a}^{\alpha}}=(\text{V}_{q+1})^{\frac{1}{q+1}}{e}\indices{{}_{a}^{\alpha}},\\\
\hat{g}_{\alpha\beta}=(\text{V}_{q+1})^{-\frac{2}{q+1}}g_{\alpha\beta},\quad\hat{g}^{\alpha\beta}=(\text{V}_{q+1})^{\frac{2}{q+1}}g^{\alpha\beta},\end{split}$
(4.8)

all of which have unit determinant.

Even though we will not need this construction here, we note that for the dual
lattice, the periodicities in $x_{a}$ coordinates are defined instead by the
vectors $e\indices{{}_{a}^{\alpha}}$. The coordinates in which these
periodicities all become unity separately in all directions are
$y_{\alpha}=x_{a}e\indices{{}^{a}_{\alpha}}$. The metric associated with the
dual torus is $g^{\alpha\beta}$, the inverse of the metric associated with the
original torus.

###  4.2 Functional approach

The aim is now to compute the partition function of the massless scalar field
in this flat background,

$Z_{d,q}(g)=\int\prod_{x^{I},y^{\alpha}}d\phi(x^{I},y^{\alpha})\,e^{-S^{E}_{L}[\phi;g]}.$
(4.9)

As before, we treat $\mathbb{R}^{p}$ by first considering
$\mathbb{S}^{1}_{L_{1}}\times\dots\times\mathbb{S}^{1}_{L_{p}}$ with
coordinates $x^{I}$, $I=1,\dots p$, so that the fields satisfy the periodicity
conditions $\phi(x^{I}+L_{I},y^{\alpha})=\phi(x^{I},y^{\alpha})$. The passage
to $\mathbb{R}^{p}$ is done by replacing sums by integrals at the expense of a
factor $(2\pi)^{-p}V_{p}$, $V_{p}=\prod^{p}_{I=1}L_{I}$. On
$\mathbb{S}^{1}_{L_{1}}\times\dots\times\mathbb{S}^{1}_{L_{p}}\times\mathbb{T}^{q+1}$,
an orthonormal basis of functions is

$e_{n_{I},n_{\alpha}}(x,y)=\frac{1}{\sqrt{V_{d+1}}}e^{k_{I}x^{I}+2\pi
in_{\alpha}y^{\alpha}},$ (4.10)

with $k_{I}=\frac{2\pi n_{I}}{L_{I}}$, $n_{I}\in\mathbb{Z}^{p}$,
$n_{\alpha}\in\mathbb{Z}^{q+1}$, and $V_{d+1}=V_{d-q}\text{V}_{q+1}$. The
orthonormality condition is

$(e_{n_{I},n_{\alpha}},e_{n^{\prime}_{I},n^{\prime}_{\alpha}})=\int_{V_{d+1}}\text{d}^{p}x\text{d}^{q+1}y\,\sqrt{g}e^{*}_{n_{I},n_{\alpha}}(x,y)e_{n^{\prime}_{I},n^{\prime}_{\alpha}}(x,y)\\\
=\frac{1}{V_{d+1}}\int_{V_{d+1}}\text{d}^{p}x\text{d}^{q+1}y\,\sqrt{g}e^{i(k^{\prime}_{I}-k_{I})x^{I}}e^{2\pi
i(n^{\prime}_{\alpha}-n_{\alpha})y^{\alpha}}=\prod_{I,\alpha}\delta_{n_{I},n^{\prime}_{I}}\delta_{n_{\alpha},n^{\prime}_{\alpha}}.$
(4.11)

These basis functions are eigenfunctions of the Laplacian,

$\Delta=\partial_{I}\partial^{I}+g^{\alpha\beta}\partial_{\alpha}\partial_{\beta},$
(4.12) $\Delta\
e_{n_{I},n_{\alpha}}=-\lambda^{2}_{n_{I},n_{\alpha}}e_{n_{I},n_{\alpha}},\quad\lambda^{2}_{n_{I},n_{\alpha}}=k_{I}k^{I}+(2\pi)^{2}n_{\alpha}g^{\alpha\beta}n_{\beta}.$
(4.13)

When taking into account that the $p$ first dimensions become large, the
associated zeta function is

$\zeta_{-\Delta}(s)=\frac{V_{p}}{(2\pi)^{p}}\int\text{d}^{p}k\
\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{q+1}}\big{[}(2\pi)^{2}n_{\alpha}g^{\alpha\beta}n_{\beta}+k_{I}k^{I}\big{]}^{-s}=\frac{V_{p}}{(2\pi)^{p}}\frac{2\pi^{\frac{p}{2}}}{\Gamma(\frac{p}{2})}\\\
\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{q+1}}\int_{0}^{\infty}\text{d}k\,k^{p-1}\big{[}(2\pi)^{2}n_{\alpha}g^{\alpha\beta}n_{\beta}+k^{2}\big{]}^{-s}=\frac{V_{p}\pi^{\frac{p}{2}-2s}\Gamma(s-\frac{p}{2})}{2^{2s}\Gamma(s)}\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{q+1}}\big{[}n_{\alpha}g^{\alpha\beta}n_{\beta}\big{]}^{\frac{p}{2}-s}\\\
=\frac{V_{p}\pi^{\frac{p}{2}-2s}\Gamma(s-\frac{p}{2})}{2^{2s}\Gamma(s)(\text{V}_{q+1})^{\frac{p-2s}{q+1}}}\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{q+1}}\big{[}n_{\alpha}\,\hat{g}^{\alpha\beta}n_{\beta}\big{]}^{\frac{p}{2}-s}=\frac{V_{p}\pi^{\frac{p}{2}-2s}\Gamma(s-\frac{p}{2})}{2^{2s}\Gamma(s)(\text{V}_{q+1})^{\frac{p-2s}{q+1}}}f_{s-\frac{p}{2}}(q+1;\hat{g}^{-1}).$
(4.14)

In the last line, we have identified the sum in terms of the
SL$(n,\mathbb{Z})$ Eisenstein series in equation (2.6) of [1] and denoted for
notational coherence here by $f_{s}(n;\hat{g}^{-1})$,

$f_{s}(n;\hat{g}^{-1})=\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{n}}[n_{\alpha}\,\hat{g}^{\alpha\beta}n_{\beta}]^{-s},\quad\mathfrak{Re}(s)>\frac{n}{2}.$
(4.15)

As shown in [1], this series satisfies the reflection formula

$\pi^{-s}\Gamma(s)f_{s}(n;\hat{g}^{-1})=\pi^{s-\frac{n}{2}}\Gamma\bigg{(}\frac{n}{2}-s\bigg{)}f_{\frac{n}{2}-s}(n;\hat{g}),$
(4.16)

where

$f_{s}(n;\hat{g})=\sideset{}{{}^{\prime}}{\sum}_{m^{\alpha}\in\mathbb{Z}^{n}}[m^{\alpha}\,\hat{g}_{\alpha\beta}m^{\beta}]^{-s},\quad\mathfrak{Re}(s)>\frac{n}{2}.$
(4.17)

When applied to (4.14), we get

$\zeta_{-\Delta}(s)=\frac{\Gamma(\frac{p+q+1}{2}-s)V_{p}}{2^{2s}\Gamma(s)\pi^{\frac{p+q+1}{2}}(\text{V}_{q+1})^{\frac{p-2s}{q+1}}}f_{\frac{p+q+1}{2}-s}(q+1;\hat{g}),$
(4.18)

and then, using (2.33) together with $p+q=d$,

$\ln
Z_{d,q}(g)=\frac{\Gamma(\frac{d+1}{2})V_{d-q}}{2\pi^{\frac{d+1}{2}}(\text{V}_{q+1})^{\frac{d-q}{q+1}}}f_{\frac{d+1}{2}}(q+1;\hat{g}),$
(4.19)

In terms of the completed SL($n,\mathbb{Z}$) Eisenstein series

$\xi(z,n;\hat{g})=\frac{\Gamma(\frac{z}{2})f_{\frac{z}{2}}(n;\hat{g})}{2\pi^{\frac{z}{2}}},$
(4.20)

which satisfies the reflection formula

$\xi(z,n;\hat{g}^{-1})=\xi(n-z,n;\hat{g}),$ (4.21)

the partition function becomes

$\boxed{\ln{Z}_{d,q}(g)=\frac{V_{d-q}}{(\text{V}_{q+1})^{\frac{d-q}{q+1}}}\xi(d+1,q+1;\hat{g})}.$
(4.22)

Remarks:

(i) The above derivation is only valid for $p>1,q<d$. Indeed, in the case
where there are no large spatial dimensions, $p=0,q=d$, the expression

$\ln{Z}_{d,d}(g)=\xi(d+1,d+1;\hat{g}),$ (4.23)

diverges. As emphasized in section 2.2.1, this is due to the presence of the
zero mode, whose contribution cannot be neglected when there are no large
dimensions. A renormalized expression, relying on the canonical approach of
section 4.3, is provided in section 4.5.

(ii) The result (4.22) can be expressed more simply in terms of the completed
GL$(n,\mathbb{Z})$ Eisenstein series $\xi(z,n;g)$, which has the same
definition as the SL$(n,\mathbb{Z})$ one in equation (4.20) but with
$\hat{g}_{\alpha\beta}$ replaced by $g_{\alpha\beta}$, and satisfies the
inversion formula

$\xi(z,n;g^{-1})=\text{V}_{n}\xi(n-z,n;g),$ (4.24)
$\ln{Z}_{d,q}(g)=V_{d-q}\text{V}_{q+1}\xi(d+1,q+1;g).$ (4.25)

(iii) The SL$(n,\mathbb{Z})$ Eisenstein series is invariant under
SL$(n,\mathbb{Z})$ transformations of the lattice vectors defining
$\mathbb{T}^{n}$: if $S\indices{{}_{\alpha}^{\beta}}\in{\rm SL}(n,\mathbb{Z})$
and

${e^{\prime}}\indices{{}^{a}_{\alpha}}=S\indices{{}_{\alpha}^{\beta}}{e}\indices{{}^{a}_{\beta}},\qquad
g^{\prime}_{\alpha\beta}=S\indices{{}_{\alpha}^{\gamma}}g_{\gamma\delta}S\indices{{}_{\beta}^{\delta}},$
(4.26)

then

$f_{s}(n;\hat{g}^{\prime})=\sideset{}{{}^{\prime}}{\sum}_{m^{\alpha}\in\mathbb{Z}^{n}}[m^{\alpha}\,\hat{g}^{\prime}_{\alpha\beta}m^{\beta}]^{-s}=\sideset{}{{}^{\prime}}{\sum}_{m^{\alpha}\in\mathbb{Z}^{n}}[m^{\alpha}S\indices{{}_{\alpha}^{\gamma}}\hat{g}_{\gamma\delta}S\indices{{}_{\beta}^{\delta}}m^{\beta}]^{-s}\\\
=\sideset{}{{}^{\prime}}{\sum}_{m^{\prime\alpha}\in\mathbb{Z}^{n}}[m^{\prime\gamma}\hat{g}_{\gamma\delta}\,m^{\prime\delta}]^{-s}=f_{s}(n;\hat{g}),$
(4.27)

where $m^{\prime\gamma}=m^{\alpha}S_{\alpha}{}^{\gamma}$. A similar proof
holds for $f_{s}(n;\hat{g}^{\prime-1})=f_{s}(n;\hat{g}^{-1})$. Since both the
volume of $\mathbb{T}^{q+1}$ and $\xi(d+1,q+1;\hat{g})$, are separately
SL$(q+1,\mathbb{Z})$ invariant, so is the full partition function in (4.22).

(iv) In order to make the connection with the discussion in section 2.3, one
should introduce, instead of $y^{\alpha}$, coordinates
$y^{\prime\alpha}=L_{(\alpha)}y^{\alpha}$ with $L_{d+1}=\beta$, so that the
periodicities in (4.5) get replaced by

$\phi(y^{\prime\alpha})=\phi(y^{\prime\alpha}+L_{\alpha}),\quad\forall\alpha.$
(4.28)

In $y^{\prime\alpha}$ coordinates, the Euclidean action in (2.29) corresponds
to the particular choice of metric given by ${g}^{\prime ij}=\delta^{ij}$ and
${g}^{\prime d+1,i}=-\mu^{i}$, where $i=1,\dots,d$.

(v) For $p=d$ and $q=0$,

$f_{s}(1;\hat{g})=2\zeta(2s),\quad\xi(z,1;\hat{g})=\xi(z),\quad\hat{g}=1,$
(4.29)

and (4.22) reduces to the scalar black body result (2.24).

For $p=d-1$ and $q=1$, (4.22) reduces to the partition function on
$\mathbb{R}^{d-1}\times\mathbb{T}^{2}$ in (3.10) covariant under the
SL$(2,\mathbb{Z})/\mathbb{Z}^{2}$ transformations of the modular parameter in
(1.2). Indeed, when using the parametrization

$\begin{split}\hat{e}\indices{{}^{a}_{\alpha}}&=\frac{1}{\sqrt{L_{d}\beta}}\begin{pmatrix}L_{d}&\alpha\\\
0&\beta\end{pmatrix}=\frac{1}{\sqrt{\tau_{2}}}\begin{pmatrix}1&\tau_{1}\\\
0&\tau_{2}\end{pmatrix},\\\
\hat{e}\indices{{}_{a}^{\alpha}}&=\frac{1}{\sqrt{L_{d}\beta}}\begin{pmatrix}\beta&0\\\
-\alpha&L_{d}\end{pmatrix}=\frac{1}{\sqrt{\tau_{2}}}\begin{pmatrix}\tau_{2}&0\\\
-\tau_{1}&1\end{pmatrix},\\\
\hat{g}_{\alpha\beta}&=\frac{1}{L_{d}\beta}\begin{pmatrix}L_{d}^{2}&L_{d}\alpha\\\
L_{d}\alpha&\alpha^{2}+\beta^{2}\end{pmatrix}=\frac{1}{\tau_{2}}\begin{pmatrix}1&\tau_{1}\\\
\tau_{1}&\absolutevalue{\tau}^{2}\end{pmatrix},\\\
\hat{g}^{\alpha\beta}&=\frac{1}{L_{d}\beta}\begin{pmatrix}\alpha^{2}+\beta^{2}&-L_{d}\alpha\\\
-L_{d}\alpha&L_{d}^{2}\end{pmatrix}=\frac{1}{\tau_{2}}\begin{pmatrix}\absolutevalue{\tau}^{2}&-\tau_{1}\\\
-\tau_{1}&1\end{pmatrix},\end{split}$ (4.30)

so that

$\displaystyle
f_{s}(2;\hat{g})=\sideset{}{{}^{\prime}}{\sum}_{(m^{1},m^{2})\in\mathbb{Z}^{2}}\frac{\tau_{2}^{s}}{\absolutevalue{m^{2}\tau+m^{1}}^{2s}}=f_{s}(\tau,\bar{\tau}),$
(4.31)

is the real analytic Eisenstein series of (1.1). In addition, since

$\tau=\frac{e\indices{{}^{d}_{d+1}}+ie\indices{{}^{d+1}_{d+1}}}{e\indices{{}^{d}_{d}}+ie\indices{{}^{d+1}_{d}}},$
(4.32)

the modular transformation (1.2) originates from an SL$(2,\mathbb{Z})$
transformation of the lattice vectors generating $\mathbb{T}^{2}$ of the form

$S\indices{{}_{\alpha}^{\beta}}=\begin{pmatrix}d&c\\\ b&a\end{pmatrix}.$
(4.33)

(vi) The reflection formula (4.16) reduces to the one satisfied by the real
analytic Eisenstein series $f_{s}(\tau,\bar{\tau})$ in (3.8) for $n=2$.
Indeed, in this case, since $\hat{g}^{-1}=\epsilon^{T}\hat{g}^{T}\epsilon$,
with $\epsilon\in$ SL$(2,\mathbb{Z})$ completely antisymmetric, and
$\epsilon^{12}=1$, it follows in particular that
$f_{s}(2;\hat{g})=f_{s}(2;\hat{g}^{-1})$. For $n=1$, formula (4.16) reduces to
the reflection formula (2.42) for the Riemann zeta function.

(vii) As shown in Appendix A, the SL$(n,\mathbb{Z})$ and GL$(n,\mathbb{Z})$
Eisenstein series are eigenfunctions of suitable Laplacians. In order to
understand the interpretation of these relations in terms of the partition
function, we introduce suitable observables, an integrated energy momentum
tensor,

${\mathcal{T}}_{\alpha\beta}=\frac{2}{\sqrt{g}}\frac{\partial
S^{E}_{L}[\phi;g]}{\partial
g^{\alpha\beta}}=\int\text{d}^{p}x\text{d}^{q+1}y\Big{[}\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{1}{2}g_{\alpha\beta}(\partial_{\gamma}\phi\partial^{\gamma}\phi+\partial_{I}\phi\partial^{I}\phi)\Big{]},$
(4.34)

as well as integrated bilinears in first order derivatives of the field in the
small direction,

${\mathcal{D}}_{\alpha\beta}=\int\text{d}^{p}x\text{d}^{q+1}y\,\partial_{\alpha}\phi\partial_{\beta}\phi.$
(4.35)

When using $\sqrt{g}=t^{\frac{q+1}{2}}=\text{V}_{q+1}$, these definitions
imply that

$\begin{split}\frac{\partial
S^{E}_{L}[\phi;g]}{\partial\hat{g}_{\alpha\beta}}&=-\frac{\sqrt{g}}{2}tJ\indices{{}_{\gamma\delta}^{\alpha\beta}}\mathcal{T}^{\gamma\delta}=-t^{\frac{q-1}{2}}\hat{J}\indices{{}^{\alpha\beta\gamma\delta}}{\mathcal{D}}_{\gamma\delta},\\\
t\frac{\partial S^{E}_{L}[\phi;g]}{\partial
t}&=-\frac{1}{2}t^{\frac{q+1}{2}}g^{\alpha\beta}\mathcal{T}_{\alpha\beta},\end{split}$
(4.36)

where the tensor $\hat{J}\indices{{}^{\alpha\beta\gamma\delta}}$ is defined in
Appendix A, with $n=q+1$ here. It then follows from (A.18) and (4.22) that

$(q+1)t^{q-1}\hat{J}\indices{{}^{\alpha\beta\gamma\delta}}\langle\Delta\hat{\mathcal{D}}_{\alpha\beta}\Delta\hat{\mathcal{D}}_{\gamma\delta}\rangle-q(q+3)t^{\frac{q-1}{2}}\hat{g}^{\alpha\beta}\langle\hat{\mathcal{D}}_{\alpha\beta}\rangle\\\
=q(d-q)(d+1)\ln Z_{d,q}(g),$ (4.37)

while

$g^{\alpha\beta}\langle\hat{\mathcal{T}}_{\alpha\beta}\rangle=(q-d)\frac{1}{\text{V}_{q+1}}\ln
Z_{d,q}(g).$ (4.38)

(viii) It is of interest to see whether $\mathrm{SL}(q+1,\mathbb{Z})$
invariance of the partition function in (4.22) could have been anticipated
from the path integral definition (4.9) before doing the actual computation.
From the viewpoint of Ward identities, when concentrating on transformations
of the coordinates $y^{\alpha}$ associated with the torus alone, a possible
argument goes as follows. The action $S^{E}_{L}[\phi;g]$ is invariant under
diffeomorphisms, $y^{\prime\alpha}=y^{\prime\alpha}(y^{\beta})$,
$S^{E}_{L}[\phi^{\prime},g^{\prime}]=S^{E}_{L}[\phi,g]$, when $\phi$
transforms as a scalar and $g_{\alpha\beta}$ as a tensor field. A standard
change of integration variables in the path integral then would imply that
$Z[g^{\prime}]=Z[g]$ provided that the path integral measure is invariant.
This formal argument does however not take the boundary conditions into
account. Besides the action $S^{E}_{L}[\phi;g]$, the theory is also defined by
the boundary conditions (4.5). When neglecting the zero mode, the path
integral measure is defined in terms of the Fourier coefficients for the
periodic paths. It thus stays invariant if the paths stay periodic with the
same periodicities,
$\phi^{\prime}(y^{\prime\alpha}+\delta^{\alpha}_{\beta})=\phi^{\prime}(y^{\prime\alpha})$,
or equivalently,
$\phi(y^{\alpha}(y^{\prime\beta}+\delta^{\beta}_{\gamma}))=\phi(y^{\alpha}(y^{\prime\beta}))$.
In turn, this requires
$y^{\alpha}(y^{\prime\beta}+\delta^{\beta}_{\gamma})=y^{\alpha}(y^{\prime\beta})+m^{\alpha}$,
with $m^{\alpha}\in\mathbb{Z}$, and then that
$y^{\alpha}=\sum_{n^{\prime}_{\beta}\in\mathbb{Z}^{q+1}}Y^{\alpha}_{n^{\prime}_{\beta}}e^{2\pi
in^{\prime}_{\beta}y^{\prime\beta}}+S\indices{{}^{\alpha}_{\gamma}}y^{\prime\gamma}$
with $S\indices{{}^{\alpha}_{\gamma}}\in\mathrm{SL}(q+1,\mathbb{Z})$. When
restricting to
$y^{\alpha}=Y^{\alpha}_{0}+S\indices{{}^{\alpha}_{\gamma}}y^{\prime\gamma}$
guarantees that the associated transformation of the metric remains constant.

###  4.3 Canonical approach

In order to set up the Hamiltonian analysis, we choose the ADM parametrization
(see [48] for a review) of the metric adapted to Euclidean signature,

$g^{\rm ADM}_{\alpha\beta}=\begin{pmatrix}h_{\iota\kappa}&N_{\kappa}\\\
N_{\iota}&N^{2}+N_{\lambda}N^{\lambda}\end{pmatrix},\quad g^{\alpha\beta}_{\rm
ADM}=\begin{pmatrix}h^{\iota\kappa}+\frac{N^{\iota}N^{\kappa}}{N^{2}}&-\frac{N^{\kappa}}{N^{2}}\\\
-\frac{N^{\iota}}{N^{2}}&\frac{1}{N^{2}}\end{pmatrix},$ (4.39)

where $\iota=d-q+1,\dots,d$ and $N_{\iota}=h_{\iota\kappa}N^{\kappa}$, so that

$S^{E}_{L}[\phi;g^{\rm
ADM}]=\frac{1}{2}\int_{V_{d+1}}\text{d}^{p}x\text{d}^{q+1}y\,\Big{[}\frac{\sqrt{h}}{N}(\partial_{d+1}\phi-N^{\iota}\partial_{\iota}\phi)^{2}\\\
+N\sqrt{h}(h^{\iota\kappa}\partial_{\iota}\phi\partial_{\kappa}\phi+\partial_{I}\phi\partial^{I}\phi)\Big{]},$
(4.40)

with associated first order action

$S^{E}_{H}[\phi,\pi,g^{\rm
ADM}]=\int_{V_{d+1}}\text{d}^{p}x\text{d}^{q+1}y\,\Big{[}-i\pi\partial_{d+1}\phi+N\mathcal{H}(h)+iN^{\iota}\mathcal{H}_{\iota}\Big{]},$
(4.41)

where

$\mathcal{H}(h)=\frac{1}{2\sqrt{h}}\pi^{2}+\frac{1}{2}\sqrt{h}(h^{\iota\kappa}\partial_{\iota}\phi\partial_{\kappa}\phi+\partial_{I}\phi\partial^{I}\phi),\quad\mathcal{H}_{\iota}=\pi\partial_{\iota}\phi.$
(4.42)

A moving frame associated to the ADM form of the metric is

${e}\indices{{}_{\rm
ADM}^{a}{}_{\alpha}}=\begin{pmatrix}\theta\indices{{}^{i}_{\iota}}&\theta\indices{{}^{i}_{\kappa}}N^{\kappa}\\\
0\ldots 0&N\end{pmatrix},\qquad{e}\indices{{}^{\rm
ADM}_{a}{}^{\alpha}}=\begin{pmatrix}\theta\indices{{}_{i}^{\iota}}&\begin{matrix}0\\\
\vdots\\\ 0\end{matrix}\\\ -\frac{N^{\iota}}{N}&\frac{1}{N}\end{pmatrix},$
(4.43)

where $\theta\indices{{}^{i}_{\iota}}$ is a moving frame for the spatial
metric,

$h_{\iota\kappa}=\theta\indices{{}^{i}_{\iota}}\delta_{ij}\theta\indices{{}^{j}_{\kappa}}.$
(4.44)

In this parametrization, if we define
$\mathrm{det}\,\theta\indices{{}^{i}_{\iota}}=\sqrt{h}=\text{V}_{q}$, we have
${\rm det}\,{e}\indices{{}_{\rm ADM}^{a}{}_{\alpha}}=N\text{V}_{q}$ and ${\rm
det}\,g^{\mathrm{ADM}}_{\alpha\beta}=N^{2}\text{V}^{2}_{q}$.

If an explicit parametrization of the spatial metric is needed as well, one
may continue to parametrize both the spatial metric and the associated moving
frame in ADM form with lapse and shift $N_{(q)},N^{\iota_{d-1}}_{(q)}$, with
$\iota_{d-1}=d-q+1,\dots,d-1$ until one reaches $(q)=d-q+1$, in which case
only the lapse $N_{(d-q+1)}$ remains. Note that this yields a parametrization
of $e\indices{{}_{\rm ADM}^{a}{}_{\alpha}}$ of Iwasawa type,

$e\indices{{}_{\rm
ADM}^{a}{}_{\alpha}}=(\mathcal{K}\mathcal{A}\mathcal{N})\indices{{}^{a}_{\alpha}},$
(4.45)

where the SO$(q+1,\mathbb{R})$ element $\mathcal{K}$ is the unit matrix,
$\mathcal{K}=\mathbf{1}$, $\mathcal{A}={\rm diag}(N_{(d-q+1)},\dots
N_{(d)},N=N_{(d+1)})$, while $\mathcal{N}$ is upper diagonal with 1’s on the
diagonal. In this parametrization, (and also with a more general Iwasawa
parametrization including a non-trivial SO$(q+1,\mathbb{R})$ element that does
not contribute to the ADM metric by construction), the volume of the torus
$\mathbb{T}^{q+1}$ is simply the product of the lapses,
$\text{V}_{q+1}=\prod_{\alpha=d-q+1}^{d+1}N_{(\alpha)}$.

Let us define

$H(h)=\int_{V_{d}}\text{d}^{p}x\text{d}^{d}y\,\mathcal{H}(h).$ (4.46)

When using that $\pi$ is a tensor density under spatial coordinate
transformations,

$\pi(x^{I},y^{\iota},y^{d+1})=\sqrt{h}\pi(x^{I},x^{i},y^{d+1}),$ (4.47)

it follows that $H(h)$ is the standard Hamiltonian when expressed in $x$
coordinates,

$H(h)=\frac{1}{2}\int_{V_{d}}\text{d}^{d}x\big{(}\pi^{2}+\partial_{I}\phi\partial^{I}\phi+\partial_{i}\phi\partial^{i}\phi\big{)}.$
(4.48)

Furthermore, if we define

$\displaystyle
H_{\iota}(h)=\int_{V_{d}}\text{d}^{p}x\text{d}^{d}y\,\mathcal{H}_{\iota}(h),$
(4.49)

using again (4.47), we have in $x$ coordinates

$H_{\iota}(h)=-\theta^{i}{}_{\iota}P_{i},\quad
P_{i}=-\int_{V_{d}}\text{d}^{d}x\,\pi\partial_{i}\phi.$ (4.50)

It now follows from the usual phase space path integral arguments that, in
operator formalism, the path integral weighted by the first order Euclidean
action (4.41), corresponds to

${\rm Z}_{d,q}(\beta,\alpha^{j},h)={\rm
Tr}\,e^{-\beta\hat{H}(h)+i\alpha^{j}\hat{P}_{j}}={\rm
Tr}\,e^{-\beta(\hat{H}(h)+i\mu^{j}\hat{P}_{j})},$ (4.51)

provided that

$\beta=N,\quad\alpha^{j}=\theta\indices{{}^{j}_{\iota}}N^{\iota},\quad\mu^{j}=\theta\indices{{}^{j}_{\iota}}\frac{N^{\iota}}{N}.$
(4.52)

Note that this result holds for general lattice vectors
$e\indices{{}^{a}_{\alpha}}$, so that the Iwasawa decomposition (4.45), may
also contain a non-trivial rotation matrix.

We now proceed to directly evaluate the trace (4.51) in the operator approach
on the spatial manifold $\mathbb{R}^{p}\times\mathbb{T}^{q}$, with $d=p+q$. As
before, in section 4.2, we start by replacing $\mathbb{R}^{p}$ by
$\mathbb{S}^{1}_{L_{1}}\times\dots\times\mathbb{S}^{1}_{L_{p}}$. An
orthonormal basis of functions is

$e_{n_{I},n_{\iota}}(x)=\frac{1}{\sqrt{V_{d}}}e^{ik_{I}x^{I}}e^{2\pi
in_{\iota}\theta\indices{{}_{i}^{\iota}}x^{i}},\qquad k_{I}=\frac{2\pi
n_{I}}{L_{I}},\qquad n_{I}\in\mathbb{Z}^{p},\qquad
n_{\alpha}\in\mathbb{Z}^{q+1},$ (4.53)

where $V_{d}=V_{d-q}\text{V}_{q}$. The orthonormality condition is

$(e_{n_{I},n_{\iota}},e_{n^{\prime}_{I},n^{\prime}_{\iota}})=\int_{V_{d}}\text{d}^{d}x\,e^{*}_{n_{I},n_{\iota}}(x)e_{n^{\prime}_{I},n^{\prime}_{\iota}}(x)\\\
=\frac{1}{V_{d}}\int_{V_{d-q}}\text{d}^{p}x\text{d}^{q}y\sqrt{h}\,e^{i(k^{\prime}_{I}-k_{I})x^{I}}e^{2\pi
i(n^{\prime}_{\iota}-n_{\iota})y^{\iota}}=\prod_{I,\iota}\delta_{n_{I},n^{\prime}_{I}}\delta_{n_{\iota},n^{\prime}_{\iota}}.$
(4.54)

The field $\phi$ and its momentum $\pi$ satisfy the following periodicity
conditions along the $x^{i}$ coordinates

$\phi(x^{I},x^{i},x^{d+1})=\phi(x^{I},x^{i}+\theta\indices{{}^{i}_{\iota}},x^{d+1}),\quad\pi(x^{I},x^{i},x^{d+1})=\pi(x^{I},x^{i}+\theta\indices{{}^{i}_{\iota}},x^{d+1}),$
(4.55)

$\forall\iota=d-q+1,...,d$, and therefore they admit the mode expansions

$\phi(x^{I},x^{i})=\sum_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}\phi_{n_{I},n_{\iota}}e_{n_{I},n_{\iota}}(x),\quad\pi(x^{I},x^{i})=\sum_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}\pi_{n_{I},n_{\iota}}e_{n_{I},n_{\iota}}(x),$
(4.56)

where the Fourier components satisfy the reality conditions
$\phi_{n_{I},n_{\iota}}=\phi^{*}_{-n_{I},-n_{\iota}}$ and
$\pi_{n_{I},n_{\iota}}=\pi^{*}_{-n_{I},-n_{\iota}}$. Note that, according to
the transformation law (4.47), the Fourier components of
$\pi(x^{I},y^{\iota})$ are $\sqrt{h}\pi_{n_{I},n_{\iota}}$. Inserting the mode
expansions of $\phi(x^{I},y^{\iota})$ and $\pi(x^{I},y^{\iota})$ and the
ansatz for the metric into the Hamiltonian (4.46), we get

$H(h)=\frac{1}{2}\sum_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}(\pi_{n_{I},n_{\iota}}\pi^{*}_{n_{I},n_{\iota}}+\omega^{2}_{k_{i}}\phi_{n_{I},n_{\iota}}\phi^{*}_{n_{I},n_{\iota}}),$
(4.57)

with

$\omega_{k_{i}}=\sqrt{k_{I}k^{I}+(2\pi)^{2}n_{\iota}h^{\iota\kappa}n_{\kappa}}.$
(4.58)

Similarly, using (4.50), the mode expansion of the momenta is

$P_{i}=-i\theta_{i}{}^{\iota}\sum_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}(2\pi
n_{\iota})\pi^{*}_{n_{I},n_{\iota}}\phi_{n_{I},n_{\iota}}.$ (4.59)

Introducing oscillator variables

$a_{n_{I},n_{\iota}}=\sqrt{\frac{\omega_{k_{i}}}{2}}\big{[}\phi_{n_{I},n_{\iota}}+\frac{i}{\omega_{k_{i}}}\pi_{n_{I},n_{\iota}}\big{]},$
(4.60)

the quantized Hamiltonian without the $(n_{I},n_{\iota})=(0,\dots,0)$ mode and
the linear momenta are given by

$\begin{split}\hat{H^{\prime}}(h)&=\sideset{}{{}^{\prime}}{\sum}_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}\omega_{k_{i}}\hat{a}^{\dagger}_{n_{I},n_{\iota}}\hat{a}_{n_{I},n_{\iota}}+E^{d,q}_{0}(h),\\\
\hat{P}_{i}&=\theta_{i}{}^{\iota}\sideset{}{{}^{\prime}}{\sum}_{(n_{i},n_{\iota})\in\mathbb{Z}^{d}}(2\pi
in_{\iota})\hat{a}^{\dagger}_{n_{I},n_{\iota}}\hat{a}_{n_{I},n_{\iota}},\end{split}$
(4.61)

where $E^{d,q}_{0}(h)$ is the Casimir energy on
$\mathbb{R}^{p}\times\mathbb{T}^{q}$. Introducing
$\hat{h}_{\iota\kappa}=(\text{V}_{q})^{-\frac{2}{q}}h_{\iota\kappa}$ and
$\hat{\theta}^{i}{}_{\iota}=(\text{V}_{q})^{-\frac{1}{q}}\theta^{i}{}_{\iota}$
with unit determinants, it is explicitly given by

$E^{d,q}_{0}(h)=-\frac{1}{2}\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q+1}{q}}}\frac{\Gamma(\frac{d+1}{2})}{\pi^{\frac{d+1}{2}}}f_{\frac{d+1}{2}}(q;\hat{h})=-\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q+1}{q}}}\xi(d+1,q;\hat{h}),$
(4.62)

and reduces to (3.41) in the case where $q=2$ and where $h$ is diagonal.

We can now evaluate the partition function (4.51),

${\rm
Z}_{d,q}(\beta,\alpha^{j},h)=\mathrm{Tr}\,e^{-\beta\hat{H^{\prime}}(h)+i\alpha^{j}\hat{P}_{j}}\\\
=e^{-\beta
E^{d,q}_{0}(h)}\sideset{}{{}^{\prime}}{\prod}_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}\sum_{N_{k_{i}}}e^{(-\beta\omega_{k_{i}}+2\pi
i\alpha^{j}\theta_{j}{}^{\iota}n_{\iota})N_{k_{i}}}\\\
=\sideset{}{{}^{\prime}}{\prod}_{(n_{I},n_{\iota})\in\mathbb{Z}^{d}}\frac{e^{-\beta
E^{d,q}_{0}(h)}}{1-e^{-\beta\omega_{k_{i}}+2\pi
i\alpha^{j}\theta_{j}{}^{\iota}n_{\iota}}}.$ (4.63)

Taking the logarithm and turning the sums over $n_{I}$ into integrals gives

$\ln{\rm Z}_{d,q}(\beta,\alpha^{j},h)=-\beta E^{d,q}_{0}(h)\\\
-\frac{V_{d-q}}{(2\pi)^{p}}\frac{2\pi^{\frac{p}{2}}}{\Gamma(\frac{p}{2})}\sum_{n_{\iota}\in\mathbb{Z}^{q}}\int_{0}^{\infty}\text{d}k\,k^{p-1}\ln[1-e^{-\beta\sqrt{k^{2}+(2\pi)^{2}n_{\iota}h^{\iota\kappa}n_{\kappa}}+2\pi
i\alpha^{j}\theta_{j}{}^{\iota}n_{\iota}}].$ (4.64)

The rest of the computation proceeds as in section 3.2 with the substitutions
$d\rightarrow p+1$,
$k^{2}_{d}\rightarrow(2\pi)^{2}n_{\iota}h^{\iota\kappa}n_{\kappa}$ and $\alpha
k_{d}\rightarrow 2\pi\alpha^{j}\theta_{j}{}^{\iota}n_{\iota}$. After using the
reduplication formula (2.22), the result is

$\ln{\rm Z}_{d,q}(\beta,\alpha^{j},h)=-\beta
E^{d,q}_{0}(h)+\xi(d-q+1)\frac{V_{d-q}}{\beta^{d-q}}\\\
+2\frac{V_{d-q}}{\beta^{\frac{d-q-1}{2}}}\sideset{}{{}^{\prime}}{\sum}_{n_{\iota}\in\mathbb{Z}^{q}}\sum_{l\in\mathbb{N^{*}}}\bigg{(}\frac{\sqrt{n_{\iota}h^{\iota\kappa}n_{\kappa}}}{l}\bigg{)}^{\frac{d-q+1}{2}}K_{\frac{d+1-q}{2}}(2\pi
l\beta\sqrt{n_{\iota}h^{\iota\kappa}n_{\kappa}})e^{2\pi
iln_{\iota}\alpha^{j}\theta_{j}{}^{\iota}}.$ (4.65)

When taking into account (4.62), equation (4.65) becomes, in terms of the
dimensionless parameter $b=(\text{V}_{q})^{-\frac{1}{q}}\beta$,

$\boxed{\ln{\rm
Z}_{d,q}(\beta,\alpha^{j},h)=\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q}{q}}}\bigg{[}b\xi(d+1,q;\hat{h})+\xi(d+1-q)b^{-(d-q)}}\\\
\boxed{+2b^{-\frac{d-q-1}{2}}\sideset{}{{}^{\prime}}{\sum}_{n_{\iota}\in\mathbb{Z}^{q}}\sum_{l\in\mathbb{N}^{*}}\bigg{(}\frac{\sqrt{n_{\iota}\hat{h}^{\iota\kappa}n_{\kappa}}}{l}\bigg{)}^{\frac{d+1-q}{2}}K_{\frac{d+1-q}{2}}(2\pi
lb\sqrt{n_{\iota}\hat{h}^{\iota\kappa}n_{\kappa}})e^{2\pi
iln_{\iota}N^{\iota}}\bigg{]}}.$ (4.66)

Remarks:

(i) For $q=1$, this expression correctly reduces to (3.63) with the
identifications $V_{1}=L_{d}$, $b=\tau_{2}$ and $N^{1}=\tau_{1}$.

(ii) Using the ADM parametrization, $\text{V}_{q+1}=\text{V}_{q}\beta$ and
hence the result for the partition function in (4.22) reads

$\ln{\rm
Z_{d,q}}(\beta,\alpha^{j},h)=\frac{V_{d-q}}{(\text{V}_{q}\beta)^{\frac{d-q}{q+1}}}\xi(d+1,q+1;\hat{g}^{\mathrm{ADM}}),$
(4.67)

where

$\hat{g}^{\mathrm{ADM}}_{\alpha\beta}=b^{-\frac{2}{q+1}}\left(\begin{matrix}\hat{h}_{\iota\kappa}&\hat{h}_{\kappa\lambda}N^{\lambda}\\\
\hat{h}_{\iota\lambda}N^{\lambda}&b^{2}+\hat{h}_{\lambda\xi}N^{\lambda}N^{\xi}\end{matrix}\right),$
(4.68)
$\hat{g}^{\alpha\beta}_{\mathrm{ADM}}=b^{-\frac{2q}{q+1}}\left(\begin{matrix}b^{2}\hat{h}^{\iota\kappa}+N^{\iota}N^{\kappa}&-N^{\kappa}\\\
-N^{\iota}&1\end{matrix}\right).$ (4.69)

Comparing (4.67) and (4.66), suggests that the Fourier transform of the
completed SL($n,\mathbb{Z}$) Eisenstein series for an $n$-dimensional metric
$\hat{g}^{\rm ADM}_{\alpha\beta}$ with unit determinant parametrized in ADM
form is

$\xi(z,n;\hat{g}^{\rm
ADM})=b^{\frac{z}{n}}\xi(z,n-1;\hat{h})+b^{\frac{(n-z)(n-1)}{n}}\xi(z-n+1)\\\
+2b^{\frac{2(z-n)-(z-n-1)n}{2n}}\sideset{}{{}^{\prime}}{\sum}_{n_{\iota}\in\mathbb{Z}^{n-1}}\sum_{l\in\mathbb{N}^{*}}\bigg{(}\frac{\sqrt{n_{\iota}\hat{h}^{\iota\kappa}n_{\kappa}}}{l}\bigg{)}^{\frac{z-n+1}{2}}\\\
K_{\frac{z-n+1}{2}}(2\pi
lb\sqrt{n_{\iota}\hat{h}^{\iota\kappa}n_{\kappa}})e^{2\pi
iln_{\iota}N^{\iota}}.$ (4.70)

in the sense that the equality of (4.67) and (4.66) corresponds to the case
$z=d+1$ and $n=q+1$.

(iii) At low temperature/small box $b\gg 1$, the leading term in the expansion
to the partition function is directly related to the Casimir energy. The
leading correction is the contribution of the modes with spatial frequencies
$n_{\iota}=0$. It coincides with the black body result (2.24) of a massless
scalar field in $d-q$ spatial dimensions. It is independent of $\text{V}_{q}$
and does not contribute to the Casimir pressure,

${\rm
p}_{d,q}(\beta,\alpha^{j},h)=\frac{1}{V_{d-q}}\frac{\partial(\beta^{-1}\ln{\rm
Z}_{d,q}(\beta,\alpha^{j},h))}{\partial\text{V}_{q}}.$ (4.71)

The asymptotic behavior of the modified Bessel functions in (3.55) implies
that all other terms are exponentially suppressed. The low-temperature/small
box expansion of the Casimir pressure is thus,

${\rm p}^{\rm
low}_{d,q}(\beta,\alpha^{j},h)=-\frac{d-q+1}{q}\frac{1}{\text{V}_{q}^{\frac{d+1}{q}}}\xi(d+1,q;\hat{h}+\dots,$
(4.72)

where the dots denote exponentially suppressed terms. For the low-
temperature/small box expansion of the entropy

${\rm S}_{d,q}(\beta,\alpha^{j},h)=(1-\beta\partial_{\beta})\ln{\rm
Z}_{d,q}(\beta,\alpha^{j},h),$ (4.73)

on the other hand, the first term in (4.66) proportional to the Casimir energy
drops out since it is linear in $\beta$, while the leading term now comes from
the lower dimensional scalar field, i.e., the modes with $n_{\iota}=0$,

${\rm S}^{\rm
low}_{d,q}(\beta,\alpha^{j},h)=(d+1-q)\xi(d+1-q)\frac{V_{d-q}}{\beta^{d-q}}+\dots.$
(4.74)

Since the leading terms in the low temperature expansions (4.72), (4.74) do
not depend on $\alpha^{j}$, they are the same when starting from $\ln
Z_{d,q}(\beta,\mu^{j},h)$ with $\alpha^{j}=\beta\mu^{j}$.

(iv) In order to discuss the high temperature/large box limit, we start by
using the reflection formula (4.21) on the partition function in (4.22) with a
metric in ADM parametrization gives

$\displaystyle\ln
Z_{d,q}({g}^{\mathrm{ADM}})=\frac{V_{d-q}}{(\beta\text{V}_{q})^{\frac{d-q}{q+1}}}\xi(q-d,q+1;\hat{g}^{-1}_{\mathrm{ADM}}).$
(4.75)

For the Fourier transform in (4.70), we need to parametrize
$\hat{g}^{-1}_{\mathrm{ADM}}$ in ADM form of a metric with unit determinant
rather than the ADM form of an inverse metric with unit determinant. This
implies that one now parametrizes the frame and co-frame as

${e}\indices{{}^{\rm
ADM}_{a}{}^{\alpha}}=\begin{pmatrix}\theta\indices{{}^{D}_{i}{}^{\iota}}&\theta\indices{{}^{D}_{i}{}^{\kappa}}N^{D}_{\kappa}\\\
0\ldots 0&N_{D}\end{pmatrix},\qquad{e}\indices{{}_{\rm
ADM}^{a}{}_{\alpha}}=\begin{pmatrix}\theta\indices{{}_{D}^{i}{}_{\iota}}&\begin{matrix}0\\\
\vdots\\\ 0\end{matrix}\\\
-\frac{N^{D}_{\iota}}{N_{D}}&\frac{1}{N_{D}}\end{pmatrix},$ (4.76)

and one parametrizes the associated metric and its inverse as

$g_{\rm
ADM}^{\alpha\beta}=\begin{pmatrix}h\indices{{}_{D}^{\iota\kappa}}&N^{\kappa}_{D}\\\
N^{\iota}_{D}&N^{2}_{D}+N^{D}_{\lambda}N^{\lambda}_{D}\end{pmatrix},\quad
g^{\rm
ADM}_{\alpha\beta}=\begin{pmatrix}h\indices{{}^{D}_{\iota\kappa}}+\frac{N_{\iota}^{D}N^{D}_{\kappa}}{N^{2}_{D}}&-\frac{N^{D}_{\kappa}}{N^{2}_{D}}\\\
-\frac{N^{D}_{\iota}}{N^{2}_{D}}&\frac{1}{N^{2}_{D}}\end{pmatrix},$ (4.77)

with $N^{\iota}_{D}=h\indices{{}_{D}^{\iota\kappa}}N_{\kappa}^{D}$ and
$h\indices{{}_{D}^{\iota\kappa}}=\theta\indices{{}^{D}_{i}{}^{\iota}}\delta^{ij}\theta\indices{{}^{D}_{j}{}^{\kappa}}$.
Comparing to the usual ADM parametrization in (4.39), it follows that the dual
and standard ADM parameters are related as

$\begin{split}h\indices{{}_{D}^{\iota\kappa}}&=h\indices{{}^{\iota\kappa}}+\frac{N^{\iota}N^{\kappa}}{N^{2}},\quad\frac{1}{N_{D}^{2}}=N^{2}+N_{\lambda}N^{\lambda},\quad
N^{\iota}_{D}=-\frac{N^{\iota}}{N^{2}},\\\
h\indices{{}^{D}_{\iota\kappa}}&=h\indices{{}_{\iota\kappa}}-\frac{N_{\iota}N_{\kappa}}{N^{2}+N_{\lambda}N^{\lambda}},\end{split}$
(4.78)

and then also as,

$\frac{1}{N^{2}}=N^{2}_{D}+N^{D}_{\lambda}N^{\lambda}_{D},\quad
N_{\iota}=-\frac{N_{\iota}^{D}}{N_{D}^{2}},\quad
h\indices{{}_{\iota\kappa}}=h\indices{{}^{D}_{\iota\kappa}}+\frac{N_{\iota}^{D}N^{D}_{\kappa}}{N^{2}_{D}}.$
(4.79)

From (4.77), it follows that the determinant of ${g}^{\rm ADM}_{\alpha\beta}$
which is $N^{2}\text{V}_{q}^{2}$ in the standard parametrization, is given by
$\mathrm{det}\,{g}^{\rm ADM}=N^{-2}_{D}\mathrm{det}\,h^{D}\equiv
N^{-2}_{D}(\text{V}_{q}^{D})^{2}$ in the dual parametrization. As a
consequence, $\text{V}_{q}^{D}=\text{V}_{q}NN_{D}$, and using (4.78),

$\displaystyle\text{V}_{q}^{D}=\text{V}_{q}\frac{N}{\sqrt{N^{2}+N_{\lambda}N^{\lambda}}}.$
(4.80)

Introducing the adimensional parameter
$b_{D}=N_{D}(\text{V}^{D}_{q})^{\frac{1}{q}}$, we have for the normalized
metric,

$\displaystyle\hat{g}^{\mathrm{ADM}}_{\alpha\beta}=(b_{D})^{\frac{-2q}{q+1}}\left(\begin{matrix}b_{D}^{2}\hat{h}^{D}_{\iota\kappa}+N^{D}_{\iota}N^{D}_{\kappa}&-N_{\kappa}^{D}\\\
-N^{D}_{\iota}&1\end{matrix}\right),$ (4.81)
$\displaystyle\hat{g}^{\alpha\beta}_{\mathrm{ADM}}=(b_{D})^{-\frac{2}{q+1}}\left(\begin{matrix}\hat{h}^{\iota\kappa}_{D}&N^{\kappa}_{D}\\\
N^{\iota}_{D}&b_{D}^{2}+\hat{h}^{D}_{\iota\kappa}N^{\iota}_{D}N^{\kappa}_{D}\end{matrix}\right).$
(4.82)

In terms of the dual chemical potentials
$\mu^{j}_{D}=\frac{\theta\indices{{}_{D}^{j}{}_{\iota}}N^{\iota}_{D}}{N_{D}}=\frac{\alpha^{j}_{D}}{N_{D}}$,
it follows that equations (4.78) are equivalent to

$\begin{split}b_{D}&=\frac{1}{b}\frac{1}{(1+\mu_{i}\mu^{i})^{\frac{q+1}{2q}}},\quad\hat{\theta}^{j}_{D}{}_{\iota}b_{D}\mu^{D}_{j}=-\frac{1}{b}\frac{1}{(1+\mu_{i}\mu^{i})}\hat{\theta}^{j}{}_{\iota}\mu_{j},\\\
\hat{h}_{\iota\kappa}^{D}&=(1+\mu_{i}\mu^{i})^{\frac{1}{q}}(\hat{h}_{\iota\kappa}-\hat{\theta}^{j}{}_{\iota}\hat{\theta}^{k}{}_{\kappa}\frac{\mu_{j}\mu_{k}}{1+\mu_{i}\mu^{i}}),\end{split}$
(4.83)

and that $\text{V}_{q}^{D}=\text{V}_{q}(1+\mu_{i}\mu^{i})^{-\frac{1}{2}}$.
Note in particular that in the case $q=1$, the first two equations reduce to
the $S$ transformation of (3.65) while the last equation reduces to a trivial
identity.

We are now in a position to apply the Fourier transform (4.70) at $z=q-d$,
$n=q+1$, to (4.75),

$\ln
Z_{d,q}({g^{\mathrm{ADM}}})=\frac{V_{d-q}}{(\beta\text{V}_{q})^{\frac{d-q}{q+1}}}\bigg{[}b_{D}^{\frac{q-d}{q+1}}\xi(q-d,q;\hat{h}^{-1}_{D})+b_{D}^{\frac{q(d+1)}{q+1}}\xi(-d)\\\
+2b_{D}^{\frac{dq+2q-d}{2(q+1)}}\sideset{}{{}^{\prime}}{\sum}_{m^{\iota}\in\mathbb{Z}^{q}}\sum_{l\in\mathbb{N}^{*}}\bigg{(}\frac{\sqrt{m^{\iota}\hat{h}^{D}_{\iota\kappa}m^{\kappa}}}{l}\bigg{)}^{-\frac{d}{2}}K_{-\frac{d}{2}}(2\pi
lb_{D}\sqrt{m^{\iota}\hat{h}_{\iota\kappa}^{D}m^{\kappa}})e^{2\pi
ilm^{\iota}N_{\iota}^{D}}\bigg{]},$ (4.84)

or, after using the reflection formulas on the first two terms,
$K_{\nu}(z)=K_{-\nu}(z)$, and simplifying,

$\boxed{\ln
Z_{d,q}({g^{\mathrm{ADM}}})=\frac{V_{d-q}}{(\text{V}^{D}_{q})^{\frac{d-q}{q}}}\bigg{[}b_{D}^{d}\xi(d+1)+\xi(d,q;\hat{h}_{D})}\\\
\boxed{+2b_{D}^{\frac{d}{2}}\sideset{}{{}^{\prime}}{\sum}_{m^{\iota}\in\mathbb{Z}^{q}}\sum_{l\in\mathbb{N}^{*}}\bigg{(}\frac{\sqrt{m^{\iota}\hat{h}^{D}_{\iota\kappa}m^{\kappa}}}{l}\bigg{)}^{-\frac{d}{2}}K_{\frac{d}{2}}(2\pi
lb_{D}\sqrt{m^{\iota}\hat{h}_{\iota\kappa}^{D}m^{\kappa}})e^{2\pi
ilm^{\iota}N_{\iota}^{D}}\bigg{]}},$ (4.85)

where $N^{D}_{\iota}=\hat{\theta}^{i}_{D}{}_{\iota}b_{D}\mu_{i}^{D}$. The
high-temperature/large volume limit is now defined by $b\ll 1$ in agreement
with the discussion after (3.67) in the case $q=1$. It follows that $b_{D}\sim
b^{-1}\gg 1$. The asymptotic behaviour of the Bessel function in (3.55)
implies that all the terms appearing in the sums in (4.85) are exponentially
suppressed. We thus get in the high-temperature/large volume limit,

$\ln
Z_{d,q}^{\mathrm{high}}(\beta,\mu^{i},h)=\frac{V_{d-q}\text{V}_{q}}{\beta^{d}}\xi(d+1)(1+\mu_{i}\mu^{i})^{-\frac{d+1}{2}}\\\
+\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q}{q}}}(1+\mu_{i}\mu^{i})^{\frac{d-q}{2q}}\xi(d,q;\hat{h}_{D})+\dots,$
(4.86)

The first term of this expression is the black body term, suitably corrected
by the coupling of the linear momenta, while the second reduces, in the $q=1$
case, to the second of (3.69). Note that, while in the $q=1$ case, the high-
temperature limit of the partition function could be simply obtained by using
modular covariance (3.15) of $\ln\mathcal{Z}_{d}(\tau,\bar{\tau})$ under the
S-transformation $\tau^{\prime}=-\frac{1}{\tau}$ as shown in (3.67) and
(3.69), in the case $q>1$, we had to use a construction involving the dual
parametrization in this derivation here. The associated expansion of the
Casimir pressure and the entropy are

$p^{\rm
high}_{d,q}(\beta,\mu^{j},h)=\frac{\xi(d+1)}{\beta^{d+1}}(1+\mu_{i}\mu^{i})^{-\frac{d+1}{2}}\\\
-\frac{(d-q)\xi(d,q;\hat{h}_{D})}{q\beta(\text{V}_{q})^{\frac{d}{q}}}(1+\mu_{i}\mu^{i})^{\frac{d-q}{2q}}+\dots,$
(4.87)

and

$S^{\rm
high}_{d,q}(\beta,\mu^{j},h)=\frac{V_{d-q}\text{V}_{q}}{\beta^{d}}(d+1)\xi(d+1)(1+\mu_{i}\mu^{i})^{-\frac{d+1}{2}}\\\
+\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q}{q}}}(1+\mu_{i}\mu^{i})^{\frac{d-q}{2q}}\xi(d,q;\hat{h}_{D})+\dots,$
(4.88)

respectively. The associated asymptotic micro-canonical density of states is
now

$\ln\rho_{d}(E,\mu^{j},h)\approx(d+1)(\frac{E}{d})^{\frac{d}{d+1}}[\xi(d+1)V_{d-q}\text{V}_{q}]^{\frac{1}{d+1}}\frac{1}{\sqrt{1+\mu_{i}\mu^{i}}},$
(4.89)

that can be also written as

$\ln\rho_{d}(E,\mu^{j},h)\approx(d+1)[\frac{E(\text{V}_{q})^{\frac{1}{q}}}{d}]^{\frac{d}{d+1}}(\frac{\pi
c_{d,q}}{6})^{\frac{1}{d+1}}\frac{1}{\sqrt{1+\mu_{i}\mu^{i}}},$ (4.90)

where

$\displaystyle
c_{d,q}=\frac{6\xi(d+1)}{\pi}\frac{V_{d-q}}{(\text{V}_{q})^{\frac{d-q}{q}}},$
(4.91)

reduces to (3.28) for $q=1$. Note however that, unlike what happens in the
$q=1$ case, here $c_{d,q}$ cannot be related to the Casimir energy
$E_{0}^{d,q}(h)$.

(v) The partition function contains additional integrated observables through
the dependence of the Hamiltonian on the parameters encoded in
$h^{\iota\kappa}$. Defining,

${}^{(q)}{\mathcal{T}}_{\iota\kappa}=\frac{2}{\sqrt{h}}\frac{\partial
S^{E}_{H}[\phi,\pi;g^{\mathrm{ADM}}]}{\partial h^{\iota\kappa}}\\\
=\int_{V_{d-q}}\text{d}^{p}x\text{d}^{q+1}yN\Big{[}\partial_{\iota}\phi\partial_{\kappa}\phi-\frac{1}{2}h_{\iota\kappa}\big{(}-\frac{1}{h}\pi^{2}+h^{\lambda\xi}\partial_{\lambda}\phi\partial_{\xi}\phi+\partial_{I}\phi\partial^{I}\phi\big{)}\Big{]},$
(4.92)

which correspond to

${}^{(q)}{\mathcal{T}}_{ik}=\sqrt{h}\theta\indices{{}_{i}^{\iota}}\theta\indices{{}_{k}^{\kappa}}{}^{(q)}{\mathcal{T}}_{\iota\kappa}\\\
=\int_{V_{d+1}}\text{d}^{d+1}x\Big{[}\partial_{i}\phi\partial_{k}\phi-\frac{1}{2}\delta_{ik}\big{(}-\pi^{2}+\partial_{i}\phi\partial^{i}\phi+\partial_{I}\phi\partial^{I}\phi\big{)}\Big{]},$
(4.93)

in $x$ coordinates, it follows that

$\langle{}^{(q)}\widehat{{\mathcal{T}}}_{\iota\kappa}\rangle=-\frac{2}{\sqrt{h}}\frac{\partial}{\partial
h^{\iota\kappa}}\ln Z_{d,q}(g).$ (4.94)

As in the covariant case, one may consider instead the variable $v$ defined by
$h_{\iota\kappa}=v\hat{h}_{\iota\kappa}$ with
$\text{V}_{q}=\sqrt{h}=(v)^{\frac{q}{2}}$,
$N=b(\text{V}_{q})^{\frac{1}{q}}=b(v)^{\frac{1}{2}}$, and the integrated
observables

${}^{(q)}{\mathcal{D}}_{\iota\kappa}=\int\text{d}^{p}x\text{d}^{q+1}y\,\partial_{\iota}\phi\partial_{\kappa}\phi,$
(4.95)

such that

$\begin{split}\frac{\partial
S^{E}_{H}[\phi,\pi;g^{\mathrm{ADM}}]}{\partial\hat{h}_{\iota\kappa}}&=-\frac{\sqrt{h}}{2}v{}^{(q)}J\indices{{}_{\xi\lambda}^{\iota\kappa}}\mathcal{}^{(q)}\mathcal{T}^{\xi\lambda}=-\frac{1}{2}(v)^{\frac{q-1}{2}}b{}^{(q)}\hat{J}\indices{{}^{\iota\kappa\xi\lambda}}{}^{(q)}{\mathcal{D}}_{\xi\lambda},\\\
v\frac{\partial S^{E}_{H}[\phi,\pi;g^{\mathrm{ADM}}]}{\partial
v}&=-\frac{1}{2}(v)^{\frac{q}{2}}h^{\iota\kappa}{}^{(q)}\mathcal{T}_{\iota\kappa}.\end{split}$
(4.96)

Notice that the last observable is directly related to the Casimir pressure.
It then follows from (A.18) that

$h^{\iota\kappa}\langle{}^{(q)}\hat{\mathcal{T}}_{\iota\kappa}\rangle=(q-d)\frac{1}{\text{V}_{q}}\ln
Z_{d,q}(g^{\mathrm{ADM}}).$ (4.97)

More generally, the variables of the Hamiltonian formalism may be chosen as
$v$, $\hat{h}_{\iota\kappa}$,$N^{\iota}$, $b$, while those of the Lagrangian
formalism as $t$, $\hat{g}_{\alpha\beta}$. When using the ADM parametrization
(4.68) together with $t=v(b)^{\frac{2}{q+1}}$, the equivalence of the
Lagrangian and Hamiltonian path integrals implies not only

$g^{\alpha\beta}\langle\hat{\mathcal{T}}_{\alpha\beta}\rangle=\frac{1}{(v)^{\frac{1}{2}}b}h^{\iota\kappa}\langle{}^{(q)}\hat{\mathcal{T}}_{\iota\kappa}\rangle,$
(4.98)

in agreement with (4.97) and (4.38), but yields more generally the relations
between Lagrangian and Hamiltonian observables inside correlation functions.
In particular, when expressed in terms of Hamiltonian observables, the
relation (4.37) will be equivalent to applying the ADM form of the SL$(q+1)$
Laplacian in (A.28) to (4.67). Indeed, to use this Laplacian, it is the
spacetime volume $\text{V}_{q+1}$ that is kept fixed, so that this Laplacian
acts on (4.67) with fixed pre-factor
${V_{d-q}}(\text{V}_{q+1})^{-\frac{d-q}{q+1}}\xi(d+1,q+1;g^{\mathrm{ADM}})$
and the powers of $b$ are those in (4.70) with $z=d+1$, $n=q+1$ rather than
those in (4.66).

(vi) In the case $q=1$, the issue of modular covariance of
$\ln\mathcal{Z}_{d}(\tau,\bar{\tau})$ in (3.13) versus SL$(2,\mathbb{Z})$
invariance of $\ln Z_{d}(g)$ in (4.22) can be clarified when working with
homogeneous coordinates. A general vielbein characterizing $\mathbb{T}^{2}$
and its Iwasawa decomposition may be written in terms of two complex variables
$z_{1}=x_{1}+iy_{1}$, $z_{2}=x_{2}+iy_{2}$ as

${e}\indices{{}^{a}_{\alpha}}=\begin{pmatrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{pmatrix}\\\
=\begin{pmatrix}\frac{x_{1}}{|z_{1}|}&-\frac{y_{1}}{|z_{1}|}\\\
\frac{y_{1}}{|z_{1}|}&\frac{x_{1}}{|z_{1}|}\end{pmatrix}\sqrt{{\mathfrak{Im}}(\bar{z}_{1}z_{2})}\begin{pmatrix}\frac{|z_{1}|}{\sqrt{\mathfrak{Im}(\bar{z}_{1}z_{2})}}&0\\\
0&\frac{\sqrt{\mathfrak{Im}(\bar{z}_{1}z_{2})}}{|z_{1}|}\end{pmatrix}\begin{pmatrix}1&\frac{{\mathfrak{Re}}(\bar{z}_{1}z_{2})}{|z_{1}|^{2}}\\\
0&1\end{pmatrix},$ (4.99)

where we assume ${\mathfrak{Im}}(\bar{z}_{1}z_{2})>0$. This gives

$g_{\alpha\beta}=\begin{pmatrix}\absolutevalue{z_{1}}^{2}&\mathfrak{Re}(\bar{z}_{1}z_{2})\\\
\mathfrak{Re}(\bar{z}_{1}z_{2})&\absolutevalue{z_{2}}^{2}\end{pmatrix},\quad
g^{\alpha\beta}=\frac{1}{(\mathfrak{Im}(\bar{z}_{1}z_{2}))^{2}}\begin{pmatrix}\absolutevalue{z_{2}}^{2}&-\mathfrak{Re}(\bar{z}_{1}z_{2})\\\
-\mathfrak{Re}(\bar{z}_{1}z_{2})&\absolutevalue{z_{1}}^{2}\end{pmatrix},$
(4.100)

where the relation

$\absolutevalue{z_{2}}^{2}=\frac{(\mathfrak{Re}(\bar{z}_{1}z_{2}))^{2}+(\mathfrak{Im}(\bar{z}_{1}z_{2}))^{2}}{\absolutevalue{z_{1}}^{2}},$
(4.101)

holds. Note that we have added the matrix element
$y_{1}=e\indices{{}^{d+1}_{d}}$, which was absent in (4.30), so that the
earlier discussion is recovered when $x_{1}=L_{d}$, $y_{1}=0$, $x_{2}=\alpha$,
$y_{2}=\beta$.

Under an SL$(2,\mathbb{Z})$ transformation as in (4.33),
$(e^{\prime})\indices{{}^{a}_{\alpha}}=e\indices{{}^{a}_{\beta}}S\indices{{}_{\alpha}^{\beta}}$,
we have

$\begin{pmatrix}x^{\prime}_{1}&x^{\prime}_{2}\\\
y^{\prime}_{1}&y^{\prime}_{2}\end{pmatrix}=\begin{pmatrix}dx_{1}+cx_{2}&bx_{1}+ax_{2}\\\
dy_{1}+cy_{2}&by_{1}+ay_{2}\end{pmatrix},$ (4.102)

while the modular parameter, which according to (4.32) is now given by

$\tau=\frac{z_{2}}{z_{1}},$ (4.103)

continues to transform as in (1.2). The ADM form of the metric (4.100) is
obtained from the change of variables

$N=\frac{\mathfrak{Im}(\bar{z}_{1}z_{2})}{\absolutevalue{z_{1}}},\quad
N^{1}=\frac{\mathfrak{Re}(\bar{z}_{1}z_{2})}{\absolutevalue{z_{1}}^{2}},\quad
h_{11}=N^{2}_{(1)}=\absolutevalue{z_{1}}^{2}.$ (4.104)

When using (4.52), it follows that temperature and the chemical potential are
given in terms of the parameters $z_{1},z_{2}$ characterizing the torus by the
non-linear relations

$\beta=N=\frac{\mathfrak{Im}(\bar{z}_{1}z_{2})}{\absolutevalue{z_{1}}},\quad\alpha=\sqrt{h_{11}}N^{1}=\frac{\mathfrak{Re}(\bar{z}_{1}z_{2})}{\absolutevalue{z_{1}}}.$
(4.105)

For notational convenience, we also introduce

$\gamma=\sqrt{h_{11}}=\absolutevalue{z_{1}}.$ (4.106)

In these terms,

$H(\gamma)=\frac{1}{2}\int_{V_{d-1}}\text{d}^{p}x\text{d}^{2}y\Big{[}\frac{1}{\gamma}(\pi^{2}+\partial_{1}\phi\partial_{1}\phi)+\gamma\partial_{I}\phi\partial^{I}\phi\Big{]},$
(4.107)

and

${}^{(1)}{\mathcal{T}}_{11}=-\frac{1}{2}\int_{V_{d-1}}\text{d}^{p}x\text{d}^{2}y\,\beta\Big{[}\pi^{2}+\partial_{1}\phi\partial_{1}\phi-\gamma^{2}\partial_{I}\phi\partial^{I}\phi\Big{]}.$
(4.108)

Writing

${\rm Z_{d}}(\beta,\alpha,\gamma)={\rm Tr}\,e^{-\beta\hat{H}(\gamma)+i\alpha
P_{d}},$ (4.109)

it follows from (4.67) that

$\boxed{\ln{\rm
Z_{d,1}}(\beta,\alpha,\gamma)=\frac{V_{d-1}}{(\gamma\beta)^{\frac{d-1}{2}}}\xi(d+1,2;\hat{g}^{\mathrm{ADM}})},$
(4.110)

where the ADM metric with unit determinant is expressed in terms of the
chemical potentials $\beta,\alpha,\gamma$ as

$\hat{g}^{\mathrm{ADM}}_{\alpha\beta}=\begin{pmatrix}\displaystyle{\frac{\gamma}{\beta}}&\displaystyle{\frac{\alpha}{\beta}}\\\
\displaystyle{\frac{\alpha}{\beta}}&\displaystyle{\frac{\beta}{\gamma}+\frac{\alpha^{2}}{\beta\gamma}}\end{pmatrix}.$
(4.111)

The transformation laws of the chemical potentials under SL$(2,\mathbb{Z})$
transformation can be obtained from those of $z_{1},z_{2}$ in (4.102) and are
explicitly given by

$\begin{split}\gamma^{\prime}&=\sqrt{d^{2}\gamma^{2}+2cd\gamma\alpha+c^{2}(\alpha^{2}+\beta^{2})},\\\
\beta^{\prime}&=\frac{\beta\gamma}{\sqrt{d^{2}\gamma^{2}+2cd\gamma\alpha+c^{2}(\alpha^{2}+\beta^{2})}},\\\
\alpha^{\prime}&=\frac{db\gamma^{2}+ac(\alpha^{2}+\beta^{2})+(ad+bc)\alpha\gamma}{\sqrt{d^{2}\gamma^{2}+2cd\gamma\alpha+c^{2}(\alpha^{2}+\beta^{2})}}.\end{split}$
(4.112)

It follows that (4.110) is fully SL$(2,\mathbb{Z})$ invariant under these
transformations, whereas it is covariant under modular transformations of the
modular parameter $\tau$ alone.

Finally, we note that in terms of the modular parameter (4.103), the ADM
metric with unit determinant keeps the same form as in (4.30),

$\hat{g}^{\mathrm{ADM}}_{\alpha\beta}=\frac{1}{\tau_{2}}\begin{pmatrix}1&\tau_{1}\\\
\tau_{1}&\absolutevalue{\tau}^{2}\end{pmatrix}.$ (4.113)

In the particular case where $x_{1}=L_{d}=\gamma$, $y_{1}=0$, $x_{2}=\alpha$,
$y_{2}=\beta$, the partition function in (4.110), when expressed in terms of
$\tau$ and $\bar{\tau}$, reduces to (3.13) as it should.

###  4.4 Worldline approach

In the world-line approach, one considers a particle on the skewed Euclidean
torus ${\mathbb{T}}^{q+1}$. The Hamiltonian is

$\hat{\mathrm{H}}_{\mathbb{T}^{q+1}}=\delta^{ab}\hat{p}_{a}\hat{p}_{b}={g}^{\alpha\beta}\hat{p}_{\alpha}\hat{p}_{\beta},$
(4.114)

$\hat{p}_{\alpha}=-i\partial_{\alpha}$. The basis of eigenfunctions of the
Laplacian on $\mathbb{T}^{q+1}$ are the $e_{n_{\alpha}}(y)=\langle y|p\rangle$
given in (4.10), where $|y\rangle\equiv|y^{\alpha}\rangle$ and
$|p\rangle\equiv|p_{\alpha}\rangle$ with $p_{\alpha}=2\pi n_{\alpha}$. The
heat kernel is

$K_{\mathbb{T}^{q+1}}(y^{\prime\alpha},y^{\alpha};t)=\langle
y^{\prime}|e^{-t\hat{\mathrm{H}}_{\mathbb{T}^{q+1}}}|y\rangle=\sum_{n_{\alpha}\in\mathbb{Z}^{d+1}}\sum_{n^{\prime}_{\alpha}\in\mathbb{Z}^{d+1}}\langle
y^{\prime}|p^{\prime}\rangle\langle
p^{\prime}|e^{-t\hat{\mathrm{H}}_{\mathbb{T}^{q+1}}}|p\rangle\langle
p|y\rangle\\\
=\frac{1}{\text{V}_{q+1}}\sum_{n_{\alpha}\in\mathbb{Z}^{d+1}}e^{2\pi
in_{\alpha}(y^{\prime\alpha}-y^{\alpha})}e^{-t(2\pi)^{2}{g}^{\alpha\beta}n_{\alpha}n_{\beta}}=\frac{1}{\text{V}_{q+1}}\vartheta_{d+1}(y^{\prime\alpha}-y^{\alpha}|it4\pi{g}^{-1}),$
(4.115)

where a resolution of the identity in terms of momentum eigenstates has been
used and the result has been expressed in terms of the Riemann theta function
in (3.88). Including also the contribution of the large $d-q$ dimensions, the
heat kernel on $\mathbb{R}^{p}\times\mathbb{T}^{q+1}$ is

$\boxed{K_{\mathbb{R}^{p}\times\mathbb{T}^{q+1}}(x^{\prime
I},y^{\prime\alpha},x^{I},y^{\alpha};t)}\\\ \boxed{=\frac{1}{(4\pi
t)^{\frac{p}{2}}\text{V}_{q+1}}e^{-\frac{1}{4t}(x^{\prime}_{I}-x_{I})(x^{\prime
I}-x^{I})}\vartheta_{q+1}(y^{\prime\alpha}-y^{\alpha}|it4\pi{g}^{-1})}.$
(4.116)

Its trace reduces to

$\mathrm{Tr}\,e^{-t\hat{\mathrm{H}}_{\mathbb{R}^{p}\times\mathbb{T}^{q+1}}}=\frac{V_{p}}{(4\pi
t)^{\frac{d-q}{2}}}\sum_{n_{\alpha}\in\mathbb{Z}^{q+1}}e^{-t(2\pi)^{2}{g}^{\alpha\beta}n_{\alpha}n_{\beta}}=\frac{V_{p}}{(4\pi
t)^{\frac{d-q}{2}}}\theta_{q+1}(t4\pi{g}^{-1}),$ (4.117)

while the partition function becomes

$\boxed{\ln
Z_{d,q}({g})}=\frac{V_{d-q}}{2}\sideset{}{{}^{\prime}}{\sum}_{n_{\alpha}\in\mathbb{Z}^{q+1}}\int_{0}^{\infty}\frac{\text{d}t}{t}\frac{1}{(4\pi
t)^{\frac{d-q}{2}}}e^{-t(2\pi)^{2}{g}^{\alpha\beta}n_{\alpha}n_{\beta}}\\\
\boxed{=\frac{1}{2}\frac{V_{d-q}}{(\text{V}_{q+1})^{\frac{d-q}{q+1}}}\int_{0}^{\infty}\text{d}t\,t^{\frac{q-d}{2}-1}(\theta(t\hat{g}^{-1})-1)}.$
(4.118)

Alternatively, a path integral computation for the heat kernel on
$\mathbb{T}^{q+1}$ that includes paths winding around the cycles of the torus
yields

$K_{\mathbb{T}^{q+1}}(y^{\prime\alpha},y^{\alpha};t)=\frac{1}{(4\pi
t)^{\frac{q+1}{2}}}\sum_{m^{\alpha}\in\mathbb{Z}^{q+1}}e^{-\frac{1}{4t}{g}_{\alpha\beta}(y^{\prime\alpha}-y^{\alpha}+m^{\alpha})(y^{\prime\beta}-y^{\beta}+m^{\beta})},$
(4.119)

which can be shown to be equivalent to (4.115) by using the Poisson summation
formula (2.56). For $\mathbb{R}^{p}\times\mathbb{T}^{q+1}$, one finds now

$\boxed{K_{\mathbb{R}^{p}\times\mathbb{T}^{q+1}}(x^{\prime
I},y^{\prime\alpha},x^{I},y^{\alpha};t)=}\\\
\boxed{\sum_{m^{\alpha}\in\mathbb{Z}^{q+1}}\frac{1}{(4\pi
t)^{\frac{d+1}{2}}}e^{-\frac{1}{4t}[(x^{\prime}_{I}-x_{I})(x^{\prime
I}-x^{I})+(y^{\prime\alpha}-y^{\alpha}+m^{\alpha}){g}_{\alpha\beta}(y^{\prime\beta}-y^{\beta}+m^{\beta})]},}$
(4.120)

with trace

$\displaystyle\mathrm{Tr}\,e^{-t\hat{{\rm
H}}_{\mathbb{R}^{p}\times\mathbb{T}^{q+1}}}=\frac{V_{d-q}\text{V}_{q+1}}{(4\pi
t)^{\frac{d+1}{2}}}\sum_{m^{\alpha}\in\mathbb{Z}^{q+1}}e^{-\frac{1}{4t}m^{\alpha}{g}_{\alpha\beta}m^{\beta}},$
(4.121)

while the partition function, after changing the integration variable
$t^{\prime}=\frac{1}{4\pi t}(\text{V}_{q+1})^{\frac{2}{q+1}}$, is given by

$\boxed{\ln{Z}_{d,q}(g)=\frac{V_{d-q}}{2(\text{V}_{q+1})^{\frac{d-q}{q+1}}}\int_{0}^{\infty}\text{d}t\,t^{\frac{d+1}{2}-1}\big{(}\theta_{q+1}(t\hat{g})-1\big{)}}.$
(4.122)

When comparing this expression with equation (4.22), it is consistent with the
suggestion that the completed SL$(n,\mathbb{Z})$ Eisenstein series is related
to the Mellin transform of the Riemann theta function $\theta_{n}(t\hat{g})$
minus 1,

$\xi(z,n;\hat{g})=\frac{1}{2}\int_{0}^{\infty}\text{d}t\,t^{\frac{z}{2}-1}\big{(}\theta_{n}(t\hat{g})-1\big{)},\quad\mathfrak{Re}(z)>1.$
(4.123)

in the sense that the equality of (4.22) and (4.122) corresponds to the case
$z=d+1$, $n=q+1$.

From the world-line approach, the Fourier transform and the high
temperature/large box expansion may also be obtained directly without
necessarily passing through canonical quantization by the following
generalization of the approach of [1], section 2.4. The argument of the
exponential, $-\pi tn_{\alpha}\hat{g}^{\alpha\beta}n_{\beta}$ in (4.118), when
using the dual ADM parametrization in (4.82) together with
$N^{\iota}_{D}=\hat{\theta}^{D}_{i}{}^{\iota}b_{D}\mu_{D}^{i}$, becomes

$-\pi
tb_{D}^{-\frac{2}{q+1}}\big{[}n_{\iota}\hat{h}^{\iota\kappa}_{D}n_{\kappa}+2n_{\iota}N^{\iota}_{D}n_{d+1}+(b^{2}_{D}+N^{\iota}_{D}N_{\iota}^{D})n_{d+1}^{2}\big{]}\\\
=-\pi
tb_{D}^{-\frac{2}{q+1}}\big{[}(n_{\iota}+N_{\iota}^{D}n_{d+1})\hat{h}^{\iota\kappa}_{D}(n_{\kappa}+N^{D}_{\kappa}n_{d+1})+b^{2}_{D}n_{d+1}^{2}\big{]}.$
(4.124)

One then splits the sum over $n_{\alpha}\in\mathbb{Z}^{q+1}/(0,\dots,0)$. The
first part is given by $n_{d+1}=0$. After the change of variables
$t^{\prime}=(b_{D})^{-\frac{2}{q+1}}t$ and using