11institutetext: Indian Institute of Technology Kanpur, Kanpur 208016,
India.22institutetext: Department of Physics, Arizona State University, Tempe,
AZ 85281, USA.33institutetext: Department of Physics, University of North
Florida, Jacksonville, FL 32224, USA.44institutetext: Science Institute,
University of Iceland, Dunhaga 3, 107, Reykjavík , Iceland.

# A generalized Selberg zeta function for flat space cosmologies

Arjun Bagchi1, Cynthia Keeler2, Victoria Martin3 and Rahul Poddar4
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Flat space cosmologies (FSCs) are time dependent solutions of three-
dimensional (3D) gravity with a vanishing cosmological constant. They can be
constructed from a discrete quotient of empty 3D flat spacetime and are also
called shifted boost orbifolds. Using this quotient structure, we build a new
and generalized Selberg zeta function for FSCs, and show that it is directly
related to the scalar 1-loop partition function. We then propose an extension
of this formalism applicable to more general quotient manifolds
$\mathcal{M}/\mathbb{Z}$, based on representation theory of fields propagating
on this background. Our prescription constitutes a novel and expedient method
for calculating regularized 1-loop determinants, without resorting to the heat
kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg
zeta function, and match them to known results.

##  1 Introduction

Three dimensional (3D) spacetime has been a very useful playground for
understanding aspects of gravity and especially quantum gravity. The lack of
propagating degrees of freedom in 3D Einstein gravity was, in the early days,
thought of as an indication of the triviality of the theory. The discovery of
the Bañados-Teitelboim-Zanelli (BTZ) black holes in 3D Anti-de Sitter
spacetimes (AdS3) Banados:1992wn  made it clear that 3D gravity was rich with
structure but more analytically tractable than its higher dimensional avatars,
making it a particularly useful tool in understanding elusive quantum aspects.

Non-extremal BTZ black holes of generic mass and angular momentum make up the
most general zero mode solutions with Brown-Henneaux Brown:1986nw  boundary
conditions. BTZ black holes can be understood geometrically as orbifolds of
AdS3 Banados:1992gq . For asymptotically flat spacetimes in 3D, a very similar
story exists. The most generic zero mode solutions with boundary conditions
outlined in Barnich:2012aw  are cosmological solutions with mass and angular
momentum Cornalba:2002fi  called Flat Space Cosmologies (FSCs). FSCs can also
be understood as orbifolds of flat space. Specifically the FSCs correspond to
the shifted-boost orbifold of 3D flat space. The connection between the non-
extremal BTZs and FSCs is actually more profound. Minkowski spacetimes can of
course be reached by an infinite radius limit of AdS spacetimes. One can take
a similar infinite radius limit on BTZ black holes. This rather
straightforward exercise is made interesting by the fact that there are no
black holes in 3D flat space. What the limit does is take the outer horizon of
the non-extremal BTZ to infinity while keeping the inner horizon at a finite
value Cornalba:2002fi ; Bagchi:2012xr . We thus end up with a spacetime which
is the inside of the original black hole. The radial and temporal directions
are flipped and hence we have a cosmological spacetime instead of a black
hole.

3D quotient manifolds (such as the BTZ black hole and FSC spacetime) are ideal
for studying quantum corrections in the presence of horizons. The 3D aspect is
important, because gravity is known to be 1-loop exact in this context
Vassilevich:2003xt ; Giombi:2008vd . The quotient aspect is also important: in
recent years, a technique has been developed to expediently calculate
functional determinants of kinetic operators from the spacetime quotient
structure alone Keeler:2018lza ; Keeler:2019wsx ; Martin:2022duk . In this
technique, one utilizes the generators of the quotient group to build a
Selberg-like zeta function $Z_{\Gamma}(s)$, which is directly related to the
regularized scalar 1-loop partition function. For example, the 1-loop
partition function for a complex scalar field of mass $m$ is

$Z^{(1)}(\Delta)=\int\mathcal{D}\phi
e^{-\int\phi^{*}\left(-\nabla^{2}+m^{2}\right)\phi}\propto\text{det}\left(-\nabla^{2}+m^{2}\right)^{-1},$
(1)

where $\Delta$ is the conformal dimension of the scalar field (for AdS3, we
have $\Delta=1+\sqrt{1+L^{2}m^{2}}$). Traditional methods of computing (1)
include the heat kernel (for example, Giombi:2008vd ), which one must then
regulate. Alternatively, using the Selberg technique, we directly obtain

$Z^{(1)}_{\text{reg}}(\Delta)=\frac{1}{Z_{\Gamma}(\Delta)}.$ (2)

There is a beautiful number-theoretic aspect to the Selberg approach as well:
the zeros of Selberg zeta function correspond to the quasinormal modes (QNMs)
of the field in question. Thus, Selberg technique can be viewed as an
extension of the work of Denef, Hartnoll and Sachdev Denef:2009kn , who used
the Weierstrass factorization theorem to cast $Z^{(1)}$ as a product over its
zeros and poles. The extension is that, with the Selberg method, we seem to
get the overall factor $e^{\text{Pol}(\Delta)}$ for free, without resorting to
the heat kernel.

The Selberg technique was introduced in Keeler:2018lza  in the context the BTZ
black hole, using the results of mathematicians Perry and Williams
perry2003selberg . The approach was then extended to higher-dimensional
hyperbolic quotients in Martin:2019flv  and for higher spin fields in
Keeler:2019wsx . An important extension to this program appeared in
Martin:2022duk , where a Selberg zeta function was built and defined for a
non-hyperbolic quotient manifold (namely, the warped AdS3 black hole). In
addition to the physical results, the work Martin:2022duk  is of mathematical
interest, potentially signalling the validity of the celebrated Selberg trace
formula beyond hyperbolic quotients. Emboldened by the warped AdS3 result, in
this work we endeavor to construct a Selberg zeta function for flat space
quotients, using the FSC as a prototypical example mirroring the BTZ solution.

In both the contexts of the BTZ black hole and the FSC spacetime, we are
further able to recast the Selberg zeta function using representation theory.
We conjecture a general prescription for computing for the Selberg zeta
function on general smooth quotient manifolds $\mathcal{M}/\Gamma$, where
$\Gamma\sim\mathbb{Z}$:

$\zeta_{\mathcal{M}/\Gamma}(s)=\prod_{\text{descendants}}\expectationvalue{1-\gamma}_{\text{scalar
primary of weight }s}$ (3)

where $\gamma\in\Gamma$. We are optimistic that this expression provides a
tantalizing arena to further study the physical meaning of the so-called non-
standard representations the appear in the calculation of Wilson spools
Castro:2023dxp ; Castro:2023bvo . We will descibe this connection more fully
in the Discussion section.

The seminal analysis of Brown and Henneaux Brown:1986nw  revealed that the
asymptotic symmetries of 3D asymptotically AdS spacetimes enhanced from the
isometry algebra $so(2,2)$ to two copies of the Virasoro algebra

$\displaystyle[\mathcal{L}_{n},\mathcal{L}_{m}]=(n-m)\mathcal{L}_{n+m}+\frac{c}{12}n^{2}(n-1)\delta_{n+m,0},$
(4)
$\displaystyle[{\bar{\mathcal{L}}}_{n},{\bar{\mathcal{L}}}_{m}]=(n-m){\bar{\mathcal{L}}}_{n+m}+\frac{\bar{c}}{12}n^{2}(n-1)\delta_{n+m,0},\quad[\mathcal{L}_{n},{\bar{\mathcal{L}}}_{m}]=0.$
(5)

In the above, $c$ and $\bar{c}$ are the central charges, which for Einstein
gravity is given by

$c=\bar{c}=\frac{3L}{2G},$ (6)

where $L$ is the AdS radius and $G$ the Newton’s constant. The symmetries are
of course that of a 2D conformal field theory and the Brown-Henneaux analysis
is often look upon as a precursor to the AdS/CFT correspondence. Due to the
underlying infinite dimensional symmetries, AdS3/CFT2 has become a favourite
testing ground for the holographic principle. BTZ black holes are dual to
thermal states on the field theory side and have played (and continue to play)
a starring role in lower dimensional AdS holography.

Barnich and Compere Barnich:2006av  showed that an analysis similar to Brown
and Henneaux in 3D asymptotically flat spacetime led to what is called the 3D
Bondi-van der Burg-Metzner-Sachs (BMS3) algebra.

$\displaystyle[\mathfrak{L}_{n},\mathfrak{L}_{m}]=(n-m)\mathfrak{L}_{n+m}+\frac{c_{\mathfrak{L}}}{12}\delta_{n+m,0}n^{2}(n-1),$
(7)
$\displaystyle[\mathfrak{L}_{n},\mathfrak{M}_{m}]=(n-m)\mathfrak{M}_{n+m}+\frac{c_{\mathfrak{M}}}{12}\delta_{n+m,0}n^{2}(n-1),\quad[\mathfrak{M}_{n},\mathfrak{M}_{m}]=0.$
(8)

The central terms can again be computed and for Einstein gravity they are
given by

$c_{\mathfrak{L}}=0,\quad c_{\mathfrak{M}}=\frac{3}{G}.$ (9)

Taking a cue out of the discussion above about holography in AdS, a line of
research on the construction of holography for flat spacetimes has emerged
which currently is called Carrollian holography. The principle claim is that
the holographic dual of 3D asymptotically flat spacetimes is a 2D quantum
field theory which has the infinite dimensional BMS3 algebra as its symmetries
Bagchi:2010zz ; Bagchi:2012cy . The word Carrollian is reflective of the fact
that the 2D field theories with this symmetry can be obtained in a Carrollian
limit of 2D relativistic CFTs when one takes the speed of light to zero
Duval:2014uva ; Bagchi:2016bcd . In the above context, this is the fact that
the BMS3 algebra can be obtained from an Inönü-Wigner contraction of the two
copies of Virasoro algebra:

$\displaystyle\mathfrak{L}_{n}=\mathcal{L}_{n}-{\bar{\mathcal{L}}}_{-n},\quad\mathfrak{M}_{n}=\frac{1}{L}\left(\mathcal{L}_{n}+{\bar{\mathcal{L}}}_{-n}\right),\quad
L\to\infty.$ (10)

Here the inverse of the AdS radius plays the role of the speed of light in the
boundary theory and the $L\to\infty$ limit is analogous to the $c_{l}\to 0$
limit, where $c_{l}$ is the speed of light. More generally, the BMSd+1 algebra
which one can obtain from the asymptotic analysis of $d$-dimensional flat
spacetimes and the Conformal Carrolld algebra that arises in the $c_{l}\to 0$
limit of relativistic conformal algebra in $(d-1)$\- dimensions are isomorphic
Duval:2014uva ; Bagchi:2016bcd . The proposal that Carrollian CFTs are
putative duals of asymptotically flat spacetimes thus passes the first and
most obvious check of holography, the matching of bulk and boundary
symmetries. Carrollian holography is a co-dimension one holographic dual and
has recently also been used in the more physically interesting 4D
asymptotically flat spacetimes, starting with Donnay:2022aba ; Bagchi:2022emh
. The Carrollian approach is to be contrasted to the Celestial approach which
advocates a co-dimension two holographic dual Strominger:2017zoo ;
Pasterski:2021raf . A comparison between the two approaches has been recently
discussed in Bagchi:2023cen .

This line of enquiry has led to some interesting successes in the 3D bulk – 2D
boundary case, including the matching of the bulk entropy to a BMS-Cardy
analysis Bagchi:2012xr , matching of entanglement entropy between the bulk and
boundary theories Bagchi:2014iea ; Jiang:2017ecm , matching of stress-tensor
correlations Bagchi:2015wna . In a manner analogous to the BTZ black hole, the
FSC solutions play a central role in the construction of holography for 3D
flatspace. The entropy of the FSC is reproduced by the above mentioned BMS-
Cardy analysis, which uses a Carrollian version of modular transformation
Bagchi:2012xr . One can also look at a matching of structure constants of the
2D Carrollian CFTs with an analysis of a one-point function probe in the
background of a FSC. This was done in Bagchi:2020rwb  following methods
similar to the AdS story Kraus:2016nwo . The BMS-Cardy and the one-point
analysis has been generalised to FSCs with extra U(1) symmetries in
Bagchi:2022xug . There is a natural generalisation to Carrollian torus two
point functions and connections to bulk quasi-normal modes (QNM) in
Bagchi:2023uqm .

Our investigations in this paper would address the question of QNM of the FSC
solution from a completely different point of view, using the method of the
Selberg zeta function which we alluded to above and go on to describe in more
detail below. The most obvious way forward in computing say scalar QNM, i.e.
by solving the Klein-Gordon equation in the FSC background, runs into problems
as one now needs to put boundary conditions on the cosmological horizon as
opposed to the black hole horizon. We choose to circumvent this problem by
appealing to the quotient structure of the FSC and generalizing the
construction of the Selberg-zeta function to these orbifolds of 3D flatspace.
We will see that we will reproduce results of FSC QNM of Bagchi:2023uqm
derived earlier using Carroll modular transformation techniques.

This paper is organized as follows. In Section 2, we review the two necessary
topics for this work: the Selberg zeta function in the context of BTZ black
holes and the relevant aspects of the FSC spacetime. In Section 3 we derive
the Selberg zeta function for FSC spacetimes in two ways: (1) using
representation theory as in equation (3), and (2) from the quotient group
action, as reviewed in Section 2.1. In Section 4, we show that our FSC Selberg
zeta function does indeed reproduce the correct scalar 1-loop partition
function. In Section 5, we calculate the zeros of the FSC Selberg zeta
function, and compare them to the FSC QNMs that were calculated in
Bagchi:2023uqm . In Section 6 we review our results and discuss future
directions.

##  2 Review

We review the two main ideas that we will need to construct a Selberg-like
zeta function for FSC spacetimes. In Section 2.1, we review how a Selberg zeta
function was built for the Euclidean BTZ black hole
($\mathbb{H}^{3}/\mathbb{Z}$) perry2003selberg . In Section 2.2, we review the
geometry and quotient structure of our spacetime of interest: flat space
cosmologies ($\mathbb{R}^{3}/\mathbb{Z}$) Cornalba:2002fi ; Bagchi:2012xr ;
Bagchi:2013lma .

###  2.1 Selberg zeta function for $\mathbb{H}^{3}/\mathbb{Z}$

In this section, we review how to construct the Selberg zeta function for the
Euclidean BTZ black hole, which has quotient structure
$\mathbb{H}^{3}/\mathbb{Z}$. Many of the ideas that we will discuss in this
section were first presented in perry2003selberg .

We begin with the BTZ black hole metric in Boyer-Lindquist-like coordinates

$ds^{2}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{L^{2}r^{2}}dt^{2}+\frac{L^{2}r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}dr^{2}+r^{2}\left(d\phi-\frac{r_{+}r_{-}}{Lr^{2}}dt\right)^{2},$
(11)

where $L$ is the AdS radius, and the outer and inner horizons, $r_{+}$ and
$r_{-}$, are related to the black hole’s mass $M$ and angular momentum $J$:

$r_{\pm}=\sqrt{2GL(LM+J)}\pm\sqrt{2GL(LM-J)}.$ (12)

The Euclidean BTZ black hole is obtained from (11) via the transformations
$t\rightarrow-i\tau$, $J\rightarrow-iJ_{E}$ and $r_{-}\rightarrow-i|r_{-}|$.
The Euclidean BTZ black hole metric can be built from the Poincaré patch
metric

$ds^{2}=\frac{L^{2}}{z^{2}}(dx^{2}+dy_{E}^{2}+dz^{2})$ (13)

through a set of discontinuous coordinate transformations, valid in regions
$r>r_{+}$, $r_{+}>r>r_{-}$ and $r_{-}>r$ Banados:1992gq . For concreteness we
will focus on the coordinate transformation valid for $r>r_{+}$:

$\begin{split}&x=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}\cos\Big(\frac{r_{+}\tau}{L^{2}}+\frac{|r_{-}|\phi}{L}\Big{missing})\exp\Big(\frac{r_{+}\phi}{L}-\frac{|r_{-}|\tau}{L^{2}}\Big{missing})\\\
&y_{E}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}\sin\Big(\frac{r_{+}\tau}{L^{2}}+\frac{|r_{-}|\phi}{L}\Big{missing})\exp\Big(\frac{r_{+}\phi}{L}-\frac{|r_{-}|\tau}{L^{2}}\Big{missing})\\\
&z=\sqrt{\frac{r_{+}^{2}-r_{-}^{2}}{r^{2}-r_{-}^{2}}}\exp\Big(\frac{r_{+}\phi}{L}-\frac{|r_{-}|\tau}{L^{2}}\Big{missing}).\end{split}$
(14)

We can now view the coordinate transformation (14) through a group theoretic
lens. The identification $\phi\sim\phi+2\pi$ allows for the BTZ black hole to
be understood as a quotient of AdS3 by a discrete subgroup
$\Gamma\sim\mathbb{Z}$ of the isometry group
SL${}_{2}(\mathbb{R})\times$SL${}_{2},(\mathbb{R})$. We can study the group
action of the single generator $\gamma\in\Gamma$ by taking
$\phi\rightarrow\phi+2\pi n$ in (14). This will map a point
$(x,y_{E},z)\in\mathbb{H}^{3}$ to another point
$(x^{\prime},y_{E}^{\prime},z^{\prime})$

$\gamma^{n}\cdot(x,y_{E},z)=(x^{\prime},y_{E}^{\prime},z^{\prime})$ (15)

through

$\begin{split}x^{\prime}&=e^{2\pi nr_{+}/L}\big{(}x\cos{(2\pi
n|r_{-}|/L)}-y_{E}\sin{(2\pi n|r_{-}|/L)}\big{)}\\\ y_{E}^{\prime}&=e^{2\pi
nr_{+}/L}\big{(}y_{E}\cos{(2\pi n|r_{-}|/L)}+x\sin{(2\pi
n|r_{-}|/L)}\big{)}\\\ z^{\prime}&=e^{2\pi nr_{+}/L}z.\end{split}$ (16)

By inspecting (16), it is clear that the group action can be understood as a
dilation and a rotation:

$\gamma\begin{pmatrix}x\\\ y_{E}\\\ z\\\
\end{pmatrix}=\begin{pmatrix}e^{2a}&0&0\\\ 0&e^{2a}&0\\\ 0&0&e^{2a}\\\
\end{pmatrix}\begin{pmatrix}\cos 2b_{E}&-\sin 2b_{E}&0\\\ \sin 2b_{E}&\cos
2b_{E}&0\\\ 0&0&1\\\ \end{pmatrix}\begin{pmatrix}x\\\ y_{E}\\\ z\\\
\end{pmatrix},$ (17)

where $a=\pi r_{+}/L$ and $b_{E}=\pi|r_{-}|/L$. The matrix $\gamma$ has three
eigenvalues: $e^{2a}$ and $e^{2a\pm 2ib_{E}}$. The authors of perry2003selberg
used these eigenvalues to construct the Selberg zeta function of the BTZ black
hole:

$Z_{\Gamma}(s)=\prod_{k_{1},k_{2}=0}^{\infty}\left[1-e^{-(2a-2ib_{E})k_{1}}e^{-(2a+2ib_{E})k_{2}}e^{-2as}\right]\;.$
(18)

In Keeler:2018lza , it was shown explicitly that the function $Z_{\Gamma}(s)$
is directly related to the regularized 1-loop partition function for a complex
scalar field, obtained in a rather simple way from the spacetime quotient
structure alone. As such, the authors of Keeler:2018lza  also showed that the
zeros of $Z_{\Gamma}(s)$ are mapped to the quasinormal modes of a scalar field
propagating on a BTZ background, as expected from the work of Denef, Hartnoll
and Sachdev Denef:2009kn .

We end this section with two comments. First, another useful way to write the
zeta function (18) is

$Z_{\Gamma}(s)=\prod_{k_{1},k_{2}=0}^{\infty}\left[1-q^{k_{2}+s/2}\bar{q}^{k_{1}+s/2}\right],$
(19)

where $q=e^{2\pi i\tau}$, $\tau=\tau_{1}+i\tau_{2}$, $a=\pi\tau_{2}$ and
$b_{E}=-\pi\tau_{1}$. This has the benefit of having a symmetric contribution
of $s$. Second, it is worth mentioning that the quotient generator
$\gamma\in\Gamma$ can be written as a linear combination of embedding
generators Banados:1992gq , using the conventions in appendix A and (16)

$\partial_{\phi}=\frac{r_{+}}{L}J_{U,Y}-\frac{|r_{-}|}{L}J_{X,V_{E}}.$ (20)

Here $J_{A,B}$ are the isometries in the embedding space which preserve the
AdS3 hyperboloid, defined in (80). In Poincaré coordinates (13), it is evident
that these generate dilations
$J_{U,Y}=x\partial_{x}+y_{E}\partial_{y_{E}}+z\partial_{z}$ and rotations
$J_{X,V_{E}}=y_{E}\partial_{x}-x\partial_{y_{E}}$. Writing (20) in terms of
the SL${}_{2}(\mathbb{R})$ generators $\mathcal{L}_{n}$ and
$\bar{\mathcal{L}}_{n}$ using (88) and performing a Wick rotation, we have

$2\pi\partial_{\phi}=2\pi
i\left(-\frac{|r_{-}|}{L}(\mathcal{L}_{0}-\bar{\mathcal{L}_{0}})+i\frac{r_{+}}{L}(\mathcal{L}_{0}+\bar{\mathcal{L}}_{0})\right).$
(21)

Identifying these coefficients with $\tau_{1}=-b_{E}/\pi=-|r_{-}|/L$ and
$\tau_{2}=a/\pi=r_{+}/L$, we have

$2\pi\partial_{\phi}=2\pi
i\big{(}(\mathcal{L}_{0}-\bar{\mathcal{L}}_{0})\tau_{1}+i(\mathcal{L}_{0}+\bar{\mathcal{L}}_{0})\tau_{2}\big{)}.$
(22)

This form of writing the quotient generator is useful in generalizing the
Selberg zeta function to other quotient manifolds, as seen in section 3.1.

###  2.2 Flat Space Cosmology

As discussed in the introduction, flat space cosmologies can be obtained by
taking a particular large $L$ limit of the BTZ black hole as well as an
orbifold of 3d flat spacetimes Cornalba:2002fi ; Bagchi:2012xr . In this
section, we review the limiting construction and the quotient structure of
flat space cosmologies: $\mathbb{R}^{3}/\mathbb{Z}$ Cornalba:2002fi ;
Bagchi:2012xr ; Bagchi:2013lma . We will also discuss briefly properties of
these solutions and their importance in the context of flatspace holography.

We begin by consider again the non-extremal BTZ black hole metric (11):

$\displaystyle ds_{\textrm{\tiny BTZ}}^{2}$ $\displaystyle=$
$\displaystyle\left(8GM-\frac{r^{2}}{L^{2}}\right)dt^{2}+\frac{dr^{2}}{-8GM+\frac{r^{2}}{L^{2}}+\frac{16G^{2}J^{2}}{r^{2}}}-8GJdtd\phi+r^{2}d\phi^{2}.$
(23)

To obtain a flat space quotient geometry from (23), one rescales the horizons
and sends the AdS radius to infinity

$r_{+}\to L\sqrt{8GM}=L\hat{r}_{+},\qquad
r_{-}\to\sqrt{\frac{2G}{M}}|J|=r_{0},\qquad\frac{L}{G}\to\infty,$ (24)

where Newton’s constant in the last expression is included to make the limit
dimensionless111It is important to note here that this rescaling of the radius
with $G$ does not effect the structure of the asymptotic symmetry algebra (7)
except changing $c_{\mathfrak{M}}$ from $3/G$ to $3$.. The result is the flat
metric

$ds_{\textrm{\tiny
FSC}}^{2}=\hat{r}_{+}^{2}dt^{2}-\frac{r^{2}\,dr^{2}}{\hat{r}_{+}^{2}(r^{2}-r_{0}^{2})}+r^{2}d\phi^{2}-2\hat{r}_{+}r_{0}dtd\phi.$
(25)

This spacetime is known as a flat space cosmology (FSC), because the radial
coordinate is now timelike, and thus the horizon $r_{0}$ is a timelike (i.e.
cosmological) horizon. The limit (24) essentially pushes the outer horizon
$r_{+}$ of the BTZ black hole to infinity, so that we are now inside the black
hole, and the BTZ inner horizon $r_{-}$ becomes the cosmological horizon
$r_{0}$. The FSC metric is a solution to the vacuum Einstein equations with no
cosmological constant.

Just as the BTZ black hole is a quotient of empty AdS3, the FSC is a quotient
of empty 3D Minkowski space. That is, for the Minkowski metric

$ds^{2}=-dT^{2}+dX^{2}+dY^{2},$ (26)

there are two coordinate transformations (either valid for $r>r_{0}$ or
$r<r_{0}$) that build the FSC spacetime. The transformation valid for
$r>r_{0}$,

$T=\sqrt{\frac{r^{2}-r_{0}^{2}}{\hat{r}_{+}^{2}}}\cosh\left(\hat{r}_{+}\phi\right),\quad
X=\sqrt{\frac{r^{2}-r_{0}^{2}}{\hat{r}_{+}^{2}}}\sinh\left(\hat{r}_{+}\phi\right),\quad
Y=r_{0}\phi-\hat{r}_{+}t,$ (27)

reproduces the FSC metric (25). Under the action $\phi\sim\phi+2\pi$, the
Minkowski coordinates transform as $X^{\pm}\sim e^{\pm
2\pi\hat{r}_{+}}X^{\pm}$ and $Y\sim Y+2\pi r_{0}$, where $X^{\pm}\equiv X\pm
T$.

The Euclidean FSC solution is constructed by performing the Wick rotation
Bagchi:2013lma :

$\hat{r}_{+}=-i\tilde{r}_{+},\qquad t=i\tau,\qquad T=-iT_{E}.$ (28)

The quotient $\phi\to\phi+2\pi$ is generated by the following Killing vector
Bagchi:2012xr :

$2\pi\partial_{\phi}=2\pi\big{(}r_{0}\partial_{Y}+\hat{r}_{+}(X\partial_{T}+T\partial_{X})\big{)}=2\pi\big{(}r_{0}\partial_{Y}+\tilde{r}_{+}(X\partial_{T_{E}}-T_{E}\partial_{X})\big{)},$
(29)

written in terms of the Lorentzian and Euclidean coordinates. In terms of the
BMS generators ((91), after performing a Wick rotation), the quotient
generator reads

$2\pi\partial_{\phi}=2\pi
i\big{(}\mathfrak{L}_{0}\eta+i\mathfrak{M}_{0}\rho\big{)},$ (30)

where $\eta=\tilde{r}_{+},~{}\rho=-\frac{r_{0}}{G}$ are the modular parameters
of the flat space quotient Bagchi:2012xr . One can also start with the
quotient structure of the non-extremal BTZ and take the limit directly on the
generator of the quotient to obtain the above.

FSCs have many interesting properties some of which we briefly mention below.

  * $\star$

The thermodynamics of FSC solutions have been studied in the literature. One
can derive a first law of thermodynamics for FSC and this has peculiar
negative signs Bagchi:2012xr . The reason behind this is the cosmological
horizon can be thought of as descending from the inner horizon of the non-
extremal BTZ, which itself comes with a peculiar “wrong-sign” first law
Castro:2012av ; Detournay:2012ug .

  * $\star$

As mentioned in the introduction, one can compute the entropy of the FSC as
the area of the cosmological event horizon. This can then be matched with a
Cardy-like state counting computation in the boundary 2D Carrollian CFT
Bagchi:2012xr  222The initial works use Galilean CFTs instead of Carrollian
CFTs. In $D=2$, the Galilean and Carrollian (conformal) algebras are
isomorphic and hence computationally there is no distinction between the two.
It was later appreciated that calling the algebra Carrollian is more
appropriate since the isomorphism, in its most primitive version, does not
hold beyond $D=2$ and the connection between Carrollian CFTs and asymptotic
symmetries of flat spacetimes extend to all dimensions.. (See also
Barnich:2012xq .) This could be thought of as a first validation of the
Carrollian holography programme. Logarithmic corrections of the FSC entropy
were addressed in Bagchi:2013qva . The BMS-Cardy formula was re-derived from
the limit in Riegler:2014bia ; Fareghbal:2014qga .

  * $\star$

There are phase transitions between hot flat space and the FSC solutions that
can be thought of as analogue of the Hawking-Page phase transitions in AdS3
Bagchi:2013lma . These cosmological phase transitions takes a time independent
solution of 3d Einstein gravity to a time dependent one.

  * $\star$

There are generalisations of FSCs to include additional $U(1)$ charges and
higher spins Gary:2014ppa ; Ammon:2017vwt ; Basu:2017aqn .

Apart from the above, FSCs have been used in other holographic contexts e.g.
in studies of entanglement Bagchi:2014iea ; Basu:2015evh ; Jiang:2017ecm ;
Apolo:2020qjm  and chaos Bagchi:2021qfe  in flat spacetimes, for verifying the
construction of an asymptotic formula for BMS structure constants derived from
modular properties of the BMS torus one-point function Bagchi:2020rwb , and
similar studies for torus two-point functions Bagchi:2023uqm . In general,
FSCs can be thought of as the bulk duals of thermal states in the 2d
Carrollian CFT. So, in conclusion, FSCs play a vital role in understanding
various aspects of gravity in 3d asymptotically flat spacetimes and especially
are important in understanding holography in this set-up.

##  3 Deriving the generalized Selberg zeta function for
$\mathbb{R}^{3}/\mathbb{Z}$

In this section, we derive a Selberg-like zeta function for the FSC spacetime
in two different ways:

  1. 1.

a novel method using representation theory

  2. 2.

direct examination of the quotient group action, as done for the BTZ black
hole in Section 2.1.

In both cases, the Selberg-like zeta function we build is precisely related to
the regularized 1-loop complex scalar partition function Barnich:2015mui  via:

$Z_{\Gamma}(\Delta)=\frac{1}{Z^{(1)}_{\text{reg}}(\Delta)},$ (31)

where $\Delta$ labels the representation of the field. Thus our formalism
constitutes a new and easy way to derive 1-loop partition functions of scalar
fields (as well as higher spin fields Keeler:2018lza ) on quotient manifolds.

###  3.1 Selberg-like zeta function from representation theory

We propose the following construction for a Selberg-like zeta function
generalized to non-hyperbolic quotient manifolds $\mathcal{M}/\Gamma$, where
$\Gamma\sim\mathbb{Z}$, motivated by representation theory333Since we consider
only $\Gamma\sim\mathbb{Z}$, $\Gamma$ has only one generator. For more general
subgroups of SL${}_{2}(\mathbb{C}$), equation (32) will also contain a product
over generators $\gamma$.:

$\zeta_{\mathcal{M}/\Gamma}(s)=\prod_{\text{descendants}}\expectationvalue{1-\gamma}_{\text{scalar
primary of weight }s}$ (32)

where $\gamma\in\Gamma$. In the examples we consider, the quotient $\Gamma$ is
generated by the identification of the angular coordinate $\phi\sim\phi+2\pi$,
so the discrete subgroup

$\Gamma=\\{\gamma^{n}|n\in\mathbb{Z}\\}\subset\text{Isom}(\mathcal{M})$ (33)

is generated by $\gamma=e^{2\pi\partial_{\phi}}$. Therefore, the zeta function
will be expressed as

$\zeta_{\mathcal{M}/\Gamma}(s)=\prod_{\text{descendants}}\expectationvalue{1-e^{2\pi\partial_{\phi}}}_{\text{scalar
primary of weight }s}$ (34)

in the cases we consider. We will now demonstrate that this construction
reproduces the Selberg zeta function for the BTZ black hole, as well as the
scalar 1-loop partition function for FSCs Barnich:2015mui .

####  3.1.1 BTZ black hole

Fields in AdS3 form representations of the asymptotic symmetries of AdS3, i.e.
two copies of the Virasoro algebra. We will restrict our attention on the
global subgroup of the algebra, which forms the AdS isometry group
SL${}_{2}(\mathbb{R})\times$SL${}_{2}(\mathbb{R})$. The generators of the
isometry group are $\mathcal{L}_{n},\bar{\mathcal{L}}_{n}$, where $n=\pm 1,0$
(for explicit definitions of $(\mathcal{L}_{n},\bar{\mathcal{L}}_{n})$, please
see Appendix A, and in particular (88)).

All states $\ket{h,\bar{h}}$ of the above theory are labelled by their
conformal dimensions $(h,\bar{h})$:

$\mathcal{L}_{0}\ket{h,\bar{h}}=h\ket{h,\bar{h}},\quad\bar{\mathcal{L}}_{0}\ket{h,\bar{h}}={\bar{h}}\ket{h,\bar{h}}.$
(35)

A primary state $\ket{h,\bar{h}}_{p}$ with respect to
SL${}_{2}(\mathbb{R})\times$SL${}_{2}(\mathbb{R})$ (and a “quasi”-primary with
respect to the two copies of the Virasoro algebra) is further annihilated by
both $\mathcal{L}_{1}$ and $\bar{\mathcal{L}}_{1}$:

$\mathcal{L}_{1}\ket{h,\bar{h}}_{p}=0,\quad\bar{\mathcal{L}}_{1}\ket{h,\bar{h}}_{p}=0.$
(36)

We will drop the subscript $p$ on the primary state from now on. The
descendants of this primary state can be written as

$\begin{split}\left(\mathcal{L}_{-1}\right)^{k_{2}}\ket{h,\bar{h}}=\ket{h,\bar{h},k_{2}},\quad\left(\bar{\mathcal{L}}_{-1}\right)^{k_{1}}\ket{h,\bar{h}}=\ket{h,\bar{h},k_{1}},\end{split}$
(37)

with $(k_{1},k_{2})\in\mathbb{Z}_{\geq 0}$.

In terms of the SL${}_{2}(\mathbb{R})$ generators and the modular parameter
for the boundary torus $\tau=\tau_{1}+i\tau_{2}$ we have the group element
$\gamma$ in terms of the generator of the quotient (22):

$\gamma=e^{2\pi\partial_{\phi}}=e^{2\pi
i((\mathcal{L}_{0}-\bar{\mathcal{L}}_{0})\tau_{1}+(\mathcal{L}_{0}+\bar{\mathcal{L}}_{0})i\tau_{2})}=q^{\mathcal{L}_{0}}\bar{q}^{\bar{\mathcal{L}}_{0}},$
(38)

where $q=e^{2\pi i\tau}$. If we consider a primary scalar field of conformal
dimension $\Delta=h+\bar{h}=2h=2\bar{h}$ and its descendents, the eigenvalues
of the SL${}_{2}(\mathbb{R})$ generators will be

$\begin{split}\mathcal{L}_{0}\ket{h,\bar{h},k_{1},k_{2}}&=\left(h+k_{2}\right)\ket{h,\bar{h},k_{1},k_{2}}=\left(\frac{\Delta}{2}+k_{2}\right)\ket{h,\bar{h},k_{1},k_{2}}\\\
\bar{\mathcal{L}}_{0}\ket{h,\bar{h},k_{1},k_{2}}&=\left(\bar{h}+k_{1}\right)\ket{h,\bar{h},k_{1},k_{2}}=\left(\frac{\Delta}{2}+k_{1}\right)\ket{h,\bar{h},k_{1},k_{2}}.\end{split}$
(39)

Plugging (39) into the zeta function prescription (32), we recover the Selberg
zeta function for the BTZ black hole reported in perry2003selberg , after
identifying $\Delta$ with the parameter $s$:

$\begin{split}\zeta_{\mathbb{H}^{3}/\mathbb{Z}}(s)&=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2\pi
i((k_{1}-k_{2})\tau_{1}+(k_{1}+k_{2}+s)i\tau_{2})}\right)\\\
&=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2ib_{E}(k_{1}-k_{2})-2a(k_{1}+k_{2}+s)}\right).\end{split}$
(40)

Recall that, for the BTZ black hole, the parameters $a,b$ take the values
$a=\pi r_{+}/L,\,b_{E}=\pi|r_{-}|/L$ as discussed in Section 2.1.

####  3.1.2 Flat Space Cosmology

Now we apply the prescription (32):

$\zeta_{\mathbb{R}^{3}/\Gamma}(s)=\prod_{\text{descendants}}\expectationvalue{1-\gamma}_{\text{scalar
primary of weight }s}$ (41)

to the FSC spacetime, where $\Gamma\sim\mathbb{Z}$. The group element $\gamma$
in terms of the BMS3 generators is

$\gamma=e^{2\pi\partial_{\phi}}=e^{2\pi
i(\mathfrak{L}_{0}\eta+i\mathfrak{M}_{0}\rho)},$ (42)

where $\eta$ and $\rho$ are the modular parameters of the flat space quotient
defined in the Section 2.2.

Since fields in flat space must be representations of the global symmetry
group, we can label a field with their mass $m$ and spin $k_{1}-k_{2}$:

$\mathfrak{M}_{0}\ket{m,k_{1}-k_{2}}=m\ket{m,k_{1}-k_{2}},\quad\mathfrak{L}_{0}\ket{m,k_{1}-k_{2}}=(k_{1}-k_{2})\ket{m,k_{1}-k_{2}}.$
(43)

A primary scalar field will therefore have spin $k_{1}-k_{2}=0$. We have
labelled descendents with $k_{1}-k_{2}$ because of $\mathfrak{L}$’s
relationship with the SL${}_{2}(\mathbb{R})$ $\mathcal{L}$’s (90) and the
alternate construction of the zeta function discussed in the following section
3.2. Now we can plug this into (41) and obtain the zeta function for FSC from
just considering the quotient, once again identifying the mass $m$ with the
parameter $s$:

$\zeta_{\mathbb{R}^{3}/\mathbb{Z}}(s)=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2\pi
i(\eta(k_{1}-k_{2})+i\rho s)}\right),$ (44)

where $\eta=\tilde{r}_{+}$ and $\rho=-r_{0}/G$ for FSC.

As a check, we will see that this is precisely related to the 1-loop scalar
partition function on FSC Barnich:2015mui , and will obtain it as a careful
limit from the BTZ zeta function below.

###  3.2 FSC zeta function from quotient group action

Another useful way to construct the zeta function is by looking at the
eigenvalues of the quotient group element as an action on the coordinates (as
in (17)). We begin by revisiting this technique for the BTZ black hole,
including some further details regarding how this construction works in terms
of prime geodesics, in Section 3.2.1. We then use this approach to build the
Selberg zeta function for FSC in Section 3.2.2.

####  3.2.1 BTZ black hole

Let us consider the group action of $\phi\sim\phi+2\pi$ on $\mathbb{H}^{3}$,
as constructed in (17):

$\gamma\begin{pmatrix}x\\\ y_{E}\\\ z\\\
\end{pmatrix}=\begin{pmatrix}e^{2a}&0&0\\\ 0&e^{2a}&0\\\ 0&0&e^{2a}\\\
\end{pmatrix}\begin{pmatrix}\cos 2b_{E}&-\sin 2b_{E}&0\\\ \sin 2b_{E}&\cos
2b_{E}&0\\\ 0&0&1\\\ \end{pmatrix}\begin{pmatrix}x\\\ y_{E}\\\ z\\\
\end{pmatrix}.$ (45)

The eigenvalues of $\gamma$ are $e^{2a},e^{2(a\pm ib_{E})}$.

It will be important to define the prime geodesic for the following
discussion. The quotient manifold $\mathcal{M}/\Gamma$ will have closed
geodesics, which are geodesics that that are periodic due to the action of the
quotient group. A prime geodesic is a geodesic which only requires a singular
action of the group to trace out its path. In this example, the geodesic going
radially outward on the $z$ axis is the prime geodesic, since it is invariant
under the rotation in the $x,y$ plane. The length of the prime geodesic is
therefore given by the eigenvalue $e^{2a}$.

The zeta function can be constructed by an Euler product over the eigenvalues
of $\gamma$. The weights of the eigenvalues are non-negative integers, except
for the eigenvalue corresponding to the length of the prime geodesic, whose
weight is the argument of the zeta function, and is not restricted to
integers. The resulting zeta function is therefore

$\zeta_{\mathbb{H}^{3}/\mathbb{Z}}(s)=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2(a+ib)k_{1}}e^{2(a-ib)k_{2}}e^{2as}\right),$
(46)

which reproduces the zeta function for $\mathbb{H}^{3}/\mathbb{Z}$. This
method has the advantage of not constructing representations of fields on the
manifold, but the previous approach gives a direct relationship to the 1-loop
partition function of the scalar field on the corresponding background.

####  3.2.2 Flat Space Cosmology

Similarly, let us construct the zeta function for $\mathbb{R}^{3}/\mathbb{Z}$.
The group action on the coordinates of flat space can be expressed as

$\gamma\begin{pmatrix}T_{E}\\\ X\\\
Y\end{pmatrix}=\begin{pmatrix}\cos(2\pi\eta)&\sin(2\pi\eta)&0\\\
-\sin(2\pi\eta)&\cos(2\pi\eta)&0\\\ 0&0&e^{-2\pi\rho G\partial_{Y}}\\\
\end{pmatrix}\begin{pmatrix}T_{E}\\\ X\\\ Y\end{pmatrix}.$ (47)

To calculate the eigenvalue of the transformation corresponding to the
translation along $Y$, it will be convenient to perform the coordinate
transformation $Y=\log Z$. The flat space metric is

$ds^{2}=dT_{E}^{2}+dX^{2}+\frac{dZ^{2}}{Z^{2}},$ (48)

and hence translations along $Y$ are now scale transformations along $Z$:

$Y\to Y+a\implies Z\to e^{a}Z.$ (49)

The group action can be expressed as a matrix acting on the coordinates

$\gamma\begin{pmatrix}T_{E}\\\ X\\\
Z\end{pmatrix}=\begin{pmatrix}\cos(2\pi\eta)&\sin(2\pi\eta)&0\\\
-\sin(2\pi\eta)&\cos(2\pi\eta)&0\\\ 0&0&e^{-2\pi G\rho}\\\
\end{pmatrix}\begin{pmatrix}T_{E}\\\ X\\\ Z\end{pmatrix},$ (50)

which allows us to calculate the eigenvalues of $\gamma$ as
$e^{-2\pi\rho},e^{\pm 2\pi i\eta}$. The prime geodesic is along the $Y$ axis,
which means the length of the prime geodesic is the eigenvalue $e^{-2\pi
G\rho}$, and will be weighted by the argument of the zeta function. The zeta
function, following the prescription described before for the
$\mathbb{H}^{3}/\mathbb{Z}$ case can now be constructed:

$\begin{split}\zeta_{\mathbb{R}^{3}/\mathbb{Z}}=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{-2\pi\rho
s}e^{2\pi i\eta k_{1}}e^{-2\pi i\eta
k_{2}}\right)=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2\pi
i(\eta(k_{1}-k_{2})+i\rho s)}\right),\end{split}$ (51)

which reproduces the zeta function constructed via our representation theory
method (44).

##  4 The 1-loop partition function and the Selberg zeta function

It was shown in Keeler:2018lza 444For example, see equation (47) of
Keeler:2018lza . that the regularized 1-loop partition function of a real,
massive scalar field propagating on a BTZ black hole background is directly
related to the BTZ Selberg zeta function of perry2003selberg :

$Z^{\text{1-loop}}_{\text{regularized}}(\Delta)=\frac{1}{Z_{\Gamma}(\Delta)}.$
(52)

In the above expression, $\Delta=1+\sqrt{1+m^{2}L^{2}}$, the function
$Z_{\Gamma}(s)$ was presented in (18), and the subscript “regularized” is to
differentiate it from the full, divergent 1-loop partition function arising
from the heat kernel calculation Giombi:2008vd :

$Z^{\text{1-loop}}(\Delta)=\text{Vol}\left(\mathbb{H}^{3}/\mathbb{Z}\right)+Z^{\text{1-loop}}_{\text{regularized}}(\Delta).$
(53)

Equation (52) makes it clear that the poles of
$Z^{\text{1-loop}}_{\text{regularized}}$ correspond to the zeros of
$Z_{\Gamma}$.

The 1-loop partition function of a real, massive scalar field propagating on
an FSC background was calculated by Barnich:2015mui . See also Merbis:2019wgk
. In that case, equation (53) is replaced by

$Z^{\text{1-loop}}(m)=\text{Vol}\left(\mathbb{R}^{3}/\mathbb{Z}\right)+Z^{\text{1-loop}}_{\text{regularized}}(m),$
(54)

where $m$ is the scalar mass. In this section, we employ our generalized
Selberg zeta function (32) to obtain the interesting result

$Z^{\text{1-loop}}_{\text{regularized}}(m)=\frac{1}{\zeta(m)},$ (55)

in direct analogy to the BTZ black hole. Thus, equation (53) provides a
remarkably easy way to compute the regularized 1-loop partition function,
eschewing heat kernel techniques. From equation (99), the 1-loop scalar
partition function is

$Z^{\text{1-loop}}_{\text{flat, scalar}}(m)=(\det\nabla^{2}_{\text{flat,
scalar}})^{-\frac{1}{2}}=\exp(\sum_{n=1}^{\infty}\frac{e^{-2\pi
mn\rho}}{n|(1-e^{2\pi in\eta})|^{2}}).$ (56)

This can be rewritten to fit the form of the zeta function,

$\begin{split}Z^{\text{1-loop}}_{\text{flat,
scalar}}(m)&=\exp(\sum_{n=1}^{\infty}\frac{e^{-2\pi\rho mn}}{n|(1-e^{2\pi
i\eta n})|^{2}})\\\
&=\exp(\sum_{n=1}^{\infty}\sum_{k_{1},k_{2}=0}^{\infty}\frac{1}{n}(e^{-2\pi\rho
m}e^{2\pi i\eta(k_{1}-k_{2})})^{n})\\\
&=\prod_{k_{1},k_{2}=0}^{\infty}\frac{1}{1-e^{2\pi
i(\eta(k_{1}-k_{2})+im\rho)}}.\end{split}$ (57)

We see that the generalized Selberg zeta function constructed for
$\mathbb{R}^{3}/\mathbb{Z}$ reproduces the regularized scalar 1-loop partition
function:

$\zeta_{\mathbb{R}^{3}/\mathbb{Z}}(s)=(\det\nabla^{2}_{\text{flat,
scalar}})^{\frac{1}{2}}=\prod_{k_{1},k_{2}=0}^{\infty}\left(1-e^{2\pi
i(is\rho+(k_{1}-k_{2})\eta)}\right)=\left(Z^{\text{1-loop}}_{\text{flat,
scalar}}(m)\right)^{-1}\bigg{|}_{m=s}.$ (58)

##  5 The Selberg zeros as quasinormal modes

It was shown in Keeler:2018lza  that the zeros of the BTZ Selberg zeta
function give the BTZ quasinormal modes. That is, given the zeros $s_{\star}$
(defined by $Z_{\Gamma}(s_{\star})=0$), the following statements are
equivalent:

$s_{\star}=\Delta\qquad\leftrightarrow\qquad\omega_{QN}=\omega_{n},$ (59)

where $\Delta$ is the conformal dimension of the field, $\omega_{n}$ are the
thermal Matsubara frequencies defined by regularity at the horizon, and
$\omega_{QN}$ are the QNMs. As discussed in Keeler:2018lza , equation (59)
constitutes an alternative method of finding quasinormal modes: given
$(s_{\star},\omega_{n},\omega_{QN})$, we can use (59) to determine
$\omega_{QN}$ (up to an overall function). In this section, we first calculate
the thermal frequencies in FSC spacetimes, and then move on to study FSC QNMs
via (59).

###  5.1 Thermal Frequencies in FSC

Thermal frequencies are calculated by demanding regularity of the Euclidean
metric at the horizon, and then imposing the conditions on to the scalar field
solution Castro:2017mfj . The FSC metric is

$ds^{2}=-\frac{1}{\hat{r}_{+}}\frac{r^{2}}{r^{2}-r_{0}^{2}}dr^{2}+\frac{\hat{r}_{+}^{2}(r^{2}-r_{0}^{2})}{r^{2}}dt^{2}+r^{2}\left(d\phi-\frac{\hat{r}_{+}r_{0}}{r^{2}}dt\right)^{2}.$
(60)

Performing the Wick rotation:

$t=i\tau,\quad\hat{r}_{+}=-i\tilde{r}_{+},$ (61)

we get the Euclidean FSC metric

$ds^{2}=\frac{1}{\tilde{r}_{+}}\frac{r^{2}}{r^{2}-r_{0}^{2}}dr^{2}+\frac{\tilde{r}_{+}^{2}(r^{2}-r_{0}^{2})}{r^{2}}d\tau^{2}+r^{2}\left(d\phi-\frac{\tilde{r}_{+}r_{0}}{r^{2}}d\tau\right)^{2}.$
(62)

We now require the FSC metric to not have any conical singularities near the
horizon. The near horizon metric can be explored using the coordinate
$r^{2}=r_{0}^{2}+\epsilon\rho^{2}$ for small $\epsilon$, which yeilds

$ds^{2}=r_{0}^{2}\left(d\phi-\frac{\tilde{r}_{+}}{r_{0}}d\tau\right)^{2}+\frac{\epsilon}{\tilde{r}_{+}^{2}}\left(d\rho^{2}+\tilde{r}_{+}^{2}\rho^{2}d\phi^{2}\right)+\mathcal{O}(\epsilon^{2}).$
(63)

For there to be no conical singularities in the subleading term, we have to go
around the subleading term in $\phi$, while keeping the transverse direction
$\phi-\frac{\tilde{r}_{+}}{r_{0}}\tau$ fixed. Therefore,

$\phi\sim\phi+\frac{2\pi}{\tilde{r}_{+}},\quad\tau\sim\tau+\frac{2\pi
r_{0}}{\tilde{r}_{+}^{2}}.$ (64)

The scalar field ansatz can be written as

$\Phi(t,r,\phi)=e^{i(\omega t-k\phi)}f(r).$ (65)

Performing the Wick rotation (61) and applying the regularity conditions at
the horizon (64) to the scalar field, we have the condition

$e^{\frac{2\pi}{\tilde{r}_{+}^{2}}(i\tilde{r}_{+}k+r_{0}\omega)}=1$ (66)

so that the scalar field is also regular at the horizon. Solving the above
condition for $\omega$, we obtain the thermal frequencies in the spacetime:

$\omega_{n}=i\frac{\tilde{r}_{+}}{r_{0}}(n\tilde{r}_{+}-k),\quad
n\in\mathbb{Z}.$ (67)

###  5.2 Quasinormal modes in FSC

We would now like to implement the condition555Note that in flat space
$\Delta=m$.:

$s_{\star}=m\implies\omega_{n}=\omega_{QN}.$ (68)

That is, tuning the Selberg zeros $s_{\star}$ to the mass $m$ is equivalent to
equating the thermal frequencies to the quasinormal modes. We can write this
condition as

$s_{\star}-m+Q\left(\omega_{n}-\omega_{QN}\right)=0,$ (69)

where $Q$ is an undetermined function. For the BTZ case, we know how to
determine $Q$: we know that the BTZ QNMs should not contain the thermal
integer $n$, so we can choose $Q$ to eliminate $n$. In the FSC case it is not
so simple, because (in analogy to de Sitter QNMs) we expect $\omega_{QN}$ to
contain thermal and angular quantum numbers (but not radial ones).

Note that equation (44) is the Euclidean Selberg zeta for FSC. The zeros are
given by the condition

$\begin{split}is_{\star}\rho+(k_{1}-k_{2})\eta=\ell\end{split}$ (70)

with $l\in\mathbb{Z}$, yielding

$s_{\star}=\frac{\ell-\eta(k_{1}-k_{2})}{i\rho}.$ (71)

Since in the exponent of the zeta function we have the vector field generating
the quotient $\partial_{\phi}$, we should identify $\ell$ with the angular
quantum number $k$, so that $\ell=\pm k$. The zeros become

$s_{\star}=\frac{\pm k-\eta(k_{1}-k_{2})}{i\rho}.$ (72)

The values of $\rho$ and $\eta$ for FSC are

$\rho=-\frac{r_{0}}{G},\quad\eta=\tilde{r}_{+},$ (73)

as calculated in Section 2.2. The thermal frequencies calculated by imposing
regularity at the horizon are (67)

$\begin{split}\omega_{n}=i\frac{\tilde{r}_{+}}{r_{0}}(n\tilde{r}_{+}-k).\end{split}$
(74)

Substituting equations (72), (73) and (74) into (69), we get:

$\begin{split}\omega&=\omega_{n}-\frac{1}{Q}\left(s_{\star}-m\right)\\\
&=\frac{i\tilde{r}_{+}}{r_{0}}(n\tilde{r}_{+}-k)-\frac{1}{Q}\left(-G\frac{\pm
k-\tilde{r}_{+}(k_{1}-k_{2})}{ir_{0}}-m\right)\\\ &=\frac{m}{Q}\mp
i\frac{k}{Qr_{0}}(G\pm
Q\tilde{r}_{+})+i\frac{\tilde{r}_{+}}{Qr_{0}}\big{(}G(k_{1}-k_{2})+Qn\tilde{r}_{+}\big{)}\end{split}$
(75)

Since we expect the FSC quasinormal modes to make sense as a limit from the
BTZ quasinormal modes, we choose that the coefficient of $k$ to be zero, since
the angular quantum number does not appear in the BTZ quasinormal modes. This
implies $Q=\mp G/\tilde{r}_{+}$. Thus, the quasinormal mode reads

$\omega=\mp\tilde{r}_{+}\left(\frac{m}{G}+i\big{(}(k_{1}-k_{2})\mp
n\big{)}\frac{\tilde{r}_{+}}{r_{0}}\right)$ (76)

Taking inspiration from the BTZ case again, we can make the identification:
$k_{1}-k_{2}=\pm n$, which recovers the leading quasinormal mode in the FSC
spacetime as computed in Bagchi:2023uqm .

$\omega=\mp\frac{m}{G}\tilde{r}_{+}=\mp i\frac{m}{G}\hat{r}_{+}.$ (77)

These correspond to the poles of a 2-point function of temporaly separated
probes in the FSC boundary theory Bagchi:2023uqm .

##  6 Discussion

In this work, we have built a Selberg-like zeta function for flat space
quotients $\mathbb{R}^{3}/\mathbb{Z}$, with the FSC as an interesting and
concrete example. We showed that this zeta function correctly gives the FSC
scalar 1-loop partition function and the dominant QNM. These results extend
the previously established Selberg method of calculating 1-loop determinants
beyond negatively-curved spacetimes. In addition, we have reinterpreted how to
construct the Selberg zeta function based on representation theory of fields
propagating on a given background, giving us more insight into building the
Selberg zeta function in more general quotient scenarios $\mathcal{M}/\Gamma$.

There are many future directions for this work. Most straightforwardly, it
would be interesting to apply the Selberg formalism to study quantum effects
in other quotient spacetimes such as k-boundary wormholes Chandra:2022fwi ;
Balasubramanian:2014hda ; Skenderis:2009ju ; Aminneborg:1997pz , and those
appearing in the context of holographic entanlgement entropy Barrella:2013wja
; Faulkner:2013yia . It would be interesting to see the implications to
corrections to black hole entropy using this formalism Banerjee:2010qc .

Perhaps the most intriguing quotients of all are the Lens spaces
$S^{3}/\mathbb{Z}_{p}$ (see for example the de Sitter Farey Tail Castro:2011xb
). This is because de Sitter admits a so-called non-standard represntation of
matter, which turns out to be more closely related to quasinormal modes
Castro:2020smu . Further, these non-standard representations are instrumental
in the recently constructed Wilson spools Castro:2023dxp ; Castro:2023bvo .
Although the non-standard representations appear to be physically relevant,
their precise physical description remains unknown. It would be interesting to
use (31) to gain more insight into these non-standard representations. This
work is currently under way.

Getting back to the specific case of 3d flat spacetimes and the FSC solutions,
while we have matched up the QNM answer (77) that was obtained in
Bagchi:2023uqm  for temporally separated probes, Bagchi:2023uqm  also obtained
an answer for spatially separated modes which was more reminiscent of answers
from a 2d CFT calculation. This was then curiously obtained from BTZ QNM as
the sub-leading term in a $L\to\infty$ expansion. It would be of interest to
see if there is a way to also obtain this curious result from our Selberg zeta
function.

## Acknowledgements

We would like to thank the Niels Bohr Institute for hospitality during the
beginning of this collaboration. We thank Daniel Grumiller for comments on the
manuscript.

AB is partially supported by a Swarnajayanti Fellowship of the Science and
Engineering Research Board (SERB) and also by the following SERB grants
SB/SJF/2019-20/08, CRG/2020/002035. CK is supported by the U.S. Department of
Energy under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.
VM and RP are supported by the Icelandic Research Fund under grants 195970-053
and 228952-051 and by the University of Iceland Research Fund. VM and RP would
like to thank the Isaac Newton Institute for Mathematical Sciences for support
and hospitality during the programme “Black holes: bridges between number
theory and holographic quantum information” when work on this paper was
undertaken. This work was supported by: EPSRC Grant number EP/R014604/1.

## APPENDICES

##  Appendix A Generators and Embedding

In this appendix, we review some of the details of Cornalba:2002fi . We
describe how 3D flat space can be embedded in $\mathbb{R}^{2,2}$ as a limit of
AdS3. We then work out the six isometry generators $J_{A,B}$ in a series of
useful coordinate systems, ultimately defining the Virasoro and BMS3
generators.

We embed the hyperboloid

$-U^{2}+X^{2}+Y^{2}-V^{2}=-L^{2}$ (78)

in $\mathbb{R}^{2,2}$ with the metric

$ds^{2}=-dU^{2}+dX^{2}+dY^{2}-dV^{2}.$ (79)

The isometries of (79) that preserve the hyperboloid are

$J_{A,B}=X_{A}\partial_{B}-X_{B}\partial_{A},$ (80)

where $(A,B)$ take values $(U,V,X,Y)$.

To embed flat space as a limit of AdS3 in $\mathbb{R}^{2,2}$, we parametrize
$U$ and $V$ with the compact coordinate $T/L$ by employing the following
embedding:

$U=\sqrt{L^{2}+X^{2}+Y^{2}}\cos\left({\frac{T}{L}}\right),\qquad
V=\sqrt{L^{2}+X^{2}+Y^{2}}\sin\left(\frac{T}{L}\right).$ (81)

To obtain AdS3, we should consider the universal covering space
$T/L\in\mathbb{R}$, where the metric is

$ds^{2}=-\frac{(L^{2}+X^{2}+Y^{2})}{L^{2}}dT^{2}+\frac{L^{2}(dX^{2}+dY^{2})+(XdY-
YdX)^{2}}{L^{2}+X^{2}+Y^{2}}.$ (82)

It is clear to see that in the limit $L\to\infty$ we recover three dimensional
flat space:

$ds^{2}=-dT^{2}+dX^{2}+dY^{2}.$ (83)

The rotation generators (80) in these “flat” coordinates are

$\begin{split}J_{U,X}&=\frac{XL}{\sqrt{L^{2}+X^{2}+Y^{2}}}\sin(\frac{T}{L})\partial_{T}-\sqrt{L^{2}+X^{2}+Y^{2}}\cos(\frac{T}{L})\partial_{X},\\\
J_{U,Y}&=\frac{YL}{\sqrt{L^{2}+X^{2}+Y^{2}}}\sin(\frac{T}{L})\partial_{T}-\sqrt{L^{2}+X^{2}+Y^{2}}\cos(\frac{T}{L})\partial_{Y},\\\
J_{X,Y}&=X\partial_{Y}-Y\partial_{X},\\\ J_{U,V}&=-L\,\partial_{T},\\\
J_{X,V}&=\sqrt{L^{2}+X^{2}+Y^{2}}\sin(\frac{T}{L})\partial_{X}+\frac{XL}{\sqrt{L^{2}+X^{2}+Y^{2}}}\cos(\frac{T}{L})\partial_{T},\\\
J_{Y,V}&=\sqrt{L^{2}+X^{2}+Y^{2}}\sin(\frac{T}{L})\partial_{Y}+\frac{YL}{\sqrt{L^{2}+X^{2}+Y^{2}}}\cos(\frac{T}{L})\partial_{T}.\end{split}$
(84)

We now present the generators $J_{A,B}$ in Poincaré patch coordinates

$ds^{2}=\frac{L^{2}(dx^{2}-dy^{2}+dz^{2})}{z^{2}}.$ (85)

The Poincaré coordinates in terms of the embedding coordinates are:

$x=\frac{X}{U+Y},\quad y=\frac{-V}{U+Y},\quad z=\frac{L}{U+Y},\quad
u=y+x,\quad v=y-x.$ (86)

The generators (80) in Poincaré coordinates are

$\begin{split}J_{U,X}&=\frac{1}{2}\left((-1+u^{2}-z^{2})\partial_{u}+(1-v^{2}+z^{2})\partial_{v}+(u-v)z\partial_{z}\right)\\\
J_{U,Y}&=u\partial_{u}+v\partial_{v}+z\partial_{z}\\\
J_{X,Y}&=\frac{1}{2}\left((-1-u^{2}+z^{2})\partial_{u}+(1+v^{2}-z^{2})\partial_{v}-(u-v)z\partial_{z}\right)\\\
J_{U,V}&=\frac{1}{2}\left((1+u^{2}+z^{2})\partial_{u}+(1+v^{2}+z^{2})\partial_{v}+(u+v)z\partial_{z}\right)\\\
J_{X,V}&=-u\partial_{u}+v\partial_{v}\\\
J_{Y,V}&=\frac{1}{2}\left((-1+u^{2}+z^{2})\partial_{u}+(-1+v^{2}+z^{2})\partial_{v}+(u+v)z\partial_{z}\right).\end{split}$
(87)

If we take a particular set of linear combinations of the generators (87), we
obtain 6 generators that form two commuting copies of the
SL${}_{2}(\mathbb{R})$ algebra. Explicitly, one can construct

$\begin{split}\mathcal{L}_{-1}&=-\partial_{u}=\frac{1}{2}(J_{U,X}-J_{U,V}+J_{X,Y}+J_{Y,V})\\\
\mathcal{L}_{0}&=-\left(u\partial_{u}+\frac{1}{2}z\partial_{z}\right)=\frac{1}{2}(J_{X,V}-J_{U,Y})\\\
\mathcal{L}_{1}&=-\left(u^{2}\partial_{u}+z^{2}\partial_{v}+uz\partial_{z}\right)=\frac{1}{2}(-J_{U,X}-J_{U,V}+J_{X,Y}-J_{Y,V})\\\
\bar{\mathcal{L}}_{-1}&=-\partial_{v}=\frac{1}{2}(-J_{U,X}-J_{U,V}-J_{X,Y}+J_{Y,V})\\\
\bar{\mathcal{L}}_{0}&=-\left(v\partial_{v}+\frac{1}{2}z\partial_{z}\right)=\frac{1}{2}(-J_{U,Y}-J_{X,V})\\\
\bar{\mathcal{L}}_{1}&=-\left(z^{2}\partial_{u}+v^{2}\partial_{v}+vz\partial_{z}\right)=\frac{1}{2}(J_{U,X}-J_{U,V}-J_{X,Y}-J_{Y,V})\end{split}$
(88)

which obey the commutation relations

$[\mathcal{L}_{m},\mathcal{L}_{n}]=(m-n)\mathcal{L}_{m+n},\quad[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}]=(m-n)\bar{\mathcal{L}}_{m+n},\quad[\mathcal{L}_{n},\bar{\mathcal{L}}_{m}]=0,$
(89)

where $(m,n)$ take values $(0,\pm 1)$. For the Euclidean generators, one has
to Wick rotate $V\to V_{E}=iV$,
$\partial_{V}\to\partial_{V_{E}}=-i\partial_{V}$ and $y\to y_{E}=iy$. In the
linear combinations of the embedding space generators above, this implies one
must replace $J_{A,V}\to-iJ_{A,V_{E}}$.

The BMS3 algebra (the algebra satisfied by the asymptotic symmetry group of 3D
flat space) can be recovered as a limit of (89) by making the following
definitions

$\mathfrak{L}_{n}=\lim_{\epsilon\to
0}\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\quad\mathfrak{M}_{n}=\lim_{\epsilon\to
0}\epsilon(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}),\quad\epsilon=\frac{G}{L}.$
(90)

The BMS3 generators in $T,X,Y$ coordinates read

$\begin{gathered}\mathfrak{L}_{\pm 1}=Y(-\partial_{X}\mp\partial_{T})+(X\mp
T)\partial_{Y},\quad\mathfrak{L}_{0}=(X\partial_{T}+T\partial_{X}),\\\
\mathfrak{M}_{\pm
1}=G(\partial_{T}\pm\partial_{X}),\quad\mathfrak{M}_{0}=G\partial_{Y}.\end{gathered}$
(91)

These obey the BMS3 algebra without central extension

$[\mathfrak{L}_{m},\mathfrak{L}_{n}]=(m-n)\mathfrak{L}_{m+n},\quad[\mathfrak{L}_{m},\mathfrak{M}_{n}]=(m-n)\mathfrak{M}_{m+n},\quad[\mathfrak{M}_{m},\mathfrak{M}_{n}]=0.$
(92)

##  Appendix B FSC partition function as a limit of the BTZ partition function

We will see that when we carefully take this limit, the partition function
poles do not accumulate into a branch cut. This allows us to construct a
meromorphic Selberg-like zeta function for FSC (for scalars, it is essentially
the FSC partition function itself). This will in turn allow us to predict FSC
QNMs which are essentially guaranteed to be correct via an appropriate
application of the Denef-Hartnoll-Sachdev method of computing QNMs (which is
now applicable since we’ve established meromorphicity) Denef:2009kn .

First, let us label the representations of the global BMS algebra as a limit
of the SL${}_{2}(\mathbb{R})\times$SL${}_{2}(\mathbb{R})$ algebra, as
discussed in the appendix (90),

$\mathfrak{L}_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\quad\mathfrak{M}_{n}=\lim_{\frac{G}{L}\to
0}\frac{G}{L}(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}).$ (93)

where $\mathfrak{L}$ are the diffeomorphisms of the spatial circle at null
infinity, and $\mathfrak{M}$ are supertranslations, and we take the
dimensionless limit $\frac{G}{L}\to 0$, and $\mathcal{L},\bar{\mathcal{L}}$
are Virasoro generators. Fields are representations of the global symmetry
algebra are labelled by the zero modes of these two sets of generators.

$\mathfrak{L}_{0}\ket{j,m}=j\ket{j,m},\quad\mathfrak{M}_{0}\ket{j,m}=m\ket{j,m}.$
(94)

We can write these eigenvalues in terms of the conformal dimensions
$h,\bar{h}$ of the bulk field with the appropriate limit:

$j=h-\bar{h},\quad m=\lim_{\frac{G}{L}\to 0}\frac{G}{L}(h+\bar{h}).$ (95)

Now let us consider the functional determinant of the scalar laplacian on
AdS3:

$-\log\det\nabla^{2}_{\text{AdS,
scalar}}=2\sum_{n=1}^{\infty}\frac{q^{nh}\bar{q}^{\,n\bar{h}}}{n|1-q^{n}|^{2}},$
(96)

where $h=\bar{h}$ since this is a scalar field and therefore has spin $j=0$,
$2h=\Delta$, and $q=e^{2\pi i\tau}$, $\tau=\tau_{1}+i\tau_{2}$. To take the
flat space limit, we have to take a limit in the modular parameter:

$\eta=\frac{\tau+\bar{\tau}}{2}=\tau_{1},\quad\frac{G}{L}\rho=\left(\frac{\tau-\bar{\tau}}{2i}\right)=\tau_{2}.$
(97)

Now, rewriting the 1-loop scalar partition function in terms of the new
modular parameter and the eigenvalues of the BMS algebra, we have

$\begin{split}-\log\det\nabla^{2}_{\text{AdS,
scalar}}&=2\sum_{n=1}^{\infty}\frac{e^{\pi
i(\eta+i\epsilon\rho)n\frac{m}{\epsilon}}e^{-\pi
i(\eta-i\epsilon\rho)n\frac{m}{\epsilon}}}{n(1-e^{2\pi
in(\eta+\epsilon\rho)})(1-e^{-2\pi in(\eta-\epsilon\rho)})},\\\
&=2\sum_{n=1}^{\infty}\frac{e^{-2\pi mn\rho}}{n(1-e^{2\pi
in(\eta+\epsilon\rho)})(1-e^{-2\pi in(\eta-\epsilon\rho)})}.\end{split}$ (98)

where $\epsilon=\frac{G}{L}$. We can easily take the limit $\epsilon\to 0$
now, to obtain

$-\log\det\nabla^{2}_{\text{flat, scalar}}=2\sum_{n=1}^{\infty}\frac{e^{-2\pi
mn\rho}}{n|(1-e^{2\pi in\eta})|^{2}}.$ (99)

which is the functional determinant of the scalar Laplacian derived in
Barnich:2015mui  using the heat kernel method, in which they use the notation
$(\eta,\rho)\rightarrow\left(\frac{\theta}{2\pi},\frac{\beta}{2\pi}\right)$.

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