11institutetext: Institute for Mechanical Systems, ETH Zürich, 8092 Zürich,
Switzerland22institutetext: Department of Theoretical Physics, TIFR,
Mumbai3,43,4institutetext: Department of Physics, IISER Mohali,
Sector 81, Knowledge City, Mohali, Punjab, 140306, India

# Revisiting the Thermal AdS partition function

Roshan Kaundinya 2 Omkar Nippanikar 3 Akash Singh 4 K. P. Yogendran
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We rewrite the worldsheet torus partition function of the Thermal AdS CFT by
isolating the boundary parameters. Using this, we show that the spectrum of
the Euclidean BTZ black hole and Lorentzian AdS3 can be extracted – the latter
as a zero temperature limit. A similar procedure recovers the Lorentzian BTZ
spectrum proposed in an earlier work. We then use our expression to construct
a boundary modular invariant expression as a Poincaré series.

###### Keywords:

BTZ, Partition function, AdS/CFT, Black hole

In this letter, we discuss the CFT worldsheet partition functions of global
$AdS_{3}$, the Lorentzian and Euclidean BTZ black holes, and Thermal
$AdS_{3}.$ These geometries are closely related to each other giving us
several reasons to compare them. While a modular invariant torus partition
function for Thermal $AdS_{3}$ (TADS) was constructed in MO2 , that of
Lorentzian $AdS_{3}$ (LADS) was proposed in Israel . Even though the relation
between the Euclidean BTZ (EBTZ) and TADS as orbifold CFTs was known from the
work of Malstrom , there remained a question of how to reinterpret the
expression in MO2  in EBTZ terms. A somewhat basic question is the relation
between the spectra of the Euclidean and Lorentzian spacetimes. The authors of
MO2  interpreted their result as a bulk free energy to obtain the spectrum of
states of the Lorentzian $AdS_{3}$ CFT proposed in MO1 . A question of
interest therefore is to ask if the spectrum of Lorentzian $AdS_{3}$ can be
obtained in a more direct manner as a zero temperature limit.

The conformal boundary of these Euclidean spacetimes is a torus with a complex
structure determined by the temperature and chemical potential. The worldsheet
partition functions, after including ghosts and integrating over the
worldsheet modular parameter will be a function of the boundary modular
parameter. In string theory, the exponentiated torus amplitude gives a one
loop contribution to the spacetime partition function which, by the logic of
the AdS/CFT correspondence, is interpreted as the partition function of the
dual CFT residing on the boundary torus. Hence, it should be invariant under
modular transformations acting on the boundary torus modular parameter.

Perhaps, the most important reason to study this system with more care is that
it is a finite temperature black hole spacetime whose various avatars can all
be exactly solved (as perturbative string theories) in principle. Since such
systems are at a premium, a careful ground work can set the stage for many
future studies.

In this work, we rewrite the partition function of MO2  in a manner that aids
the exploration of these questions. This rewriting allows us to expand the
partition function in a manner appropriate to the Euclidean BTZ black hole
thereby identifying its spectrum. This expansion also points out that while
the (Euclidean) NS-NS B-field vanishes at $r=0$ in Thermal AdS, it vanishes at
the horizon in the Euclidean BTZ black hole. We show that this translates into
the statement that the operator dual to the bulk B-field has different VEVs in
these two states.

We then show how the spectrum of Lorentzian $AdS_{3}$ may be extracted as a
zero temperature limit. In fact, the entire partition function matches the
proposal presented in Israel . We also show how to Wick rotate the Euclidean
BTZ spectrum into the Lorentzian black hole and reproduce the results in
Nippanikar . In this case though, the limit is not a zero temperature limit.
Finally, using further Poisson resummations, we present a Poincaré series for
the partition function of the boundary CFT together with a sketch of how the
Hawking-Page transition might appear.

We conclude with a summary and discuss some future questions for study.

##  1 The Euclidean Geometries

In this section, we will discuss Thermal AdS and the Euclidean BTZ black holes
from the vantage point of the AdS/CFT correspondence. The analysis in this
section is essentially a repeat of earlier work DMMV , but we will obtain a
couple of additional insights. Regarding these as solutions of a gravitational
theory will allow us to characterise the difference between these geometries
in terms applicable to a dual boundary CFT.

The bulk string theory can be viewed in three different lights: as Einstein
gravity with a cosmological constant, as a 3D string theory with a $B_{NS}$
field, or as a 10-D string theory again with $B_{NS}$ fields. In the second
case, these geometries are to be thought of as arising from a compactification
of string theory. In the last case, one views the 3D-geometry as a factor in a
product geometry defining a 10-D spacetime such as, for instance, the stringy
S-dual of the D1-D5 system.

In any case, the observation that Thermal AdS and Euclidean BTZ are both
orbifolds of the 3D upper half plane $\mathbb{H}^{3}$ affords us a solvable
worldsheet description of the 3D string theory. The orbifolding $\mathbb{Z}$
subgroup defines two real numbers which translate into the temperature and an
angular chemical potential of the dual field theory.

For setting up the AdS/CFT perspective, we will view the bulk geometries as
solutions of a (super)gravity system whose action can be taken to be

$S=\frac{1}{2\kappa_{3}^{2}}\int d^{3}x\
e^{-2\Phi}\sqrt{g}\left(R-2\Lambda+4(\nabla\Phi)^{2}-\frac{1}{2}|H|^{2}\right).$
(1)

The cosmological constant is necessary to interpret the solution as the 3D
description of the F1-NS5 system in string theory. The pure gravity viewpoint
will include only a negative cosmological constant. In either case, we are to
seek Euclidean solutions that are asymptotically AdS and whose boundary has a
torus topology with periodicities defining the temperature and chemical
potential of the dual CFT state.

A torus is defined as a quotient of the complex plane with metric
$ds^{2}=\rho_{2}dzd\bar{z}$, by a lattice:

$z\sim z+2\pi\qquad z\sim z+2\pi T.$ (2)

We can change coordinates $z=2\pi(\phi^{\prime}_{E}+Tt^{\prime}_{E})$ with
$T=T_{1}+iT_{2}$ so that the metric takes the form:

$g_{MN}=\frac{\rho_{2}}{T_{2}}\begin{pmatrix}1&T_{1}\\\
T_{1}&|T|^{2}\end{pmatrix}$ (3)

with $\rho_{2}=(2\pi)^{2}$ and the identifications become

$(\phi^{\prime}_{E},t^{\prime}_{E})\sim(\phi^{\prime}_{E}+1,t^{\prime}_{E})\,,\qquad(\phi^{\prime}_{E},t^{\prime}_{E})\sim(\phi^{\prime}_{E},t^{\prime}_{E}+1)\,.$
(4)

In the metric above, $\rho_{2}$ has dimensions $L^{2}$ while $T$ is
dimensionless and the coordinate fields
$X^{\mu}(\tau,\sigma)=(\phi^{\prime}_{E},t^{\prime}_{E})$ of the string sigma
model are dimensionless. However, two parameters $T$ and $\frac{aT+b}{cT+d}$
with $a,b,c,d$ integers satisfying $ad-bc=1$ define the same torus.

When comparing two Euclidean geometries which are being viewed as bulk saddle
points defining AdS duals corresponding to the same state of the boundary
theory, we will only need that the asymptotic torii be the same upto such
modular transformations. Thus, before comparing two such geometries as
candidate descriptions of a single state of the boundary theory, we should
first use suitable modular to transformations to bring $T$ to within the
fundamental domain of the Teichmuller space $\mathcal{H}/PSL(2,Z).$ This
fundamental domain parametrizes inequivalent torii. Further, the volume
$\rho_{2}$ of the boundary torii needs to be the same. In addition, for
Euclidean spacetimes, there is the question of choosing the time direction.

In this section, we will manage these problems by rewriting the Thermal AdS
and Euclidean BTZ geometries so that the boundary torus takes the form given
in (3). Then, we will simply require that the non-normalizable modes of all
components of the bulk metric be equal (at a particular UV cutoff).

One solution with an asymptotic boundary torus is the Thermal $AdS_{3}$ (TADS)
geometry with metric:

$ds^{2}=\frac{l^{2}}{r^{2}+l^{2}}dr^{2}+\frac{r^{2}+l^{2}}{l^{2}}dt^{2}+\frac{r^{2}}{l^{2}}dx^{2}$
(5)

where $x\sim x+2\pi R$ with R being the radius of the boundary circle. All
coordinates are dimensionful. If $l\neq R$, we have a conical singularity at
$r=0$. We will use the parameter $R$ as a book-keeping device throughout.

The thermal identification

$(t,x)\sim(t+\beta,x+\tilde{\mu}\beta)$ (6)

defines the inverse temperature $\beta$ and a dimensionless chemical potential
$\tilde{\mu}$.

We will rewrite this metric so that the identifications become as in (4).
Using $R\phi_{E}=x-\tilde{\mu}t$ and
$(\phi_{E}^{\prime},t^{\prime}_{E})\coloneqq(\frac{\phi_{E}}{2\pi},\frac{t}{\beta})$,
this leads to:

$\displaystyle ds^{2}_{E}=$
$\displaystyle\frac{l^{2}}{r^{2}+l^{2}}dr^{2}+\frac{(2\pi
R)^{2}}{l^{2}}\left[r^{2}d{\phi^{\prime}_{E}}^{2}+2r^{2}\tilde{\mu}\tilde{\beta}d\phi^{\prime}_{E}dt^{\prime}_{E}+\tilde{\beta}^{2}\left(l^{2}+(\tilde{\mu}^{2}+1)r^{2}\right)d{t^{\prime}_{E}}^{2}\right]$
(7)

with a dimensionless temperature $\tilde{\beta}=\frac{\beta}{2\pi R}$
determined in terms of the boundary radius $R$. The important observation is
the appearance of a dimensionless volume parameter $V=\frac{2\pi
R\beta}{l^{2}}$ of the boundary torus quite apart from the temperature and
chemical potential. For later use in constructing the partition function, note
that the boundary coordinates can be made dimensionful
$(l\phi^{\prime}_{E},lt^{\prime}_{E})$ by rescaling with the AdS radius.

In case we regard the above as part of a string theory compactification, we
will also have an imaginary $B_{NS}$ field with dimensions $L^{2}$ (which
arises from the WZW term in the string sigma model):

$B_{NS}=\frac{i}{l^{2}}(r^{2}-C)dt\wedge dx=i\frac{(2\pi
R)^{2}\tilde{\beta}}{l^{2}}(r^{2}-C)d\phi^{\prime}_{E}\wedge dt^{\prime}_{E}$
(8)

from which we get $H=2i\sqrt{g}$ which solves the gravity equations. Since the
$x$ circle shrinks to zero size at $r=0$ making the coordinate ill-defined,
$C=0$ for the two form to be sensible.

Another space with a torus boundary is the Euclidean BTZ black hole (EBTZ)
usually presented as the metric:

$ds^{2}_{E}=\frac{1}{l^{2}}\left[(r^{2}-r_{+}^{2}+r_{-}^{2})dt^{2}-2r_{+}r_{-}dt\
dx+r^{2}dx^{2}\right]+\frac{l^{2}r^{2}dr^{2}}{(r^{2}-r^{2}_{+})(r^{2}+r^{2}_{-})}$
(9)

where $x\sim x+2\pi R.$

At the horizon $r=r_{+}$, the $t$ coordinate shrinks to zero size defining a
plane together with the radial coordinate $r$. This forces a periodicity on
the $t$ coordinate $t\sim t_{E}+\beta_{H}$ with the temperature
$\beta_{H}=\frac{2\pi r_{+}l^{2}}{(r_{+}^{2}+r_{-}^{2})}$ from the requirement
that the geometry terminate smoothly at $r=r_{+}$. Simultaneously, the $x$
coordinate is assigned a period $x\sim x+\tilde{\mu}\beta_{H}$ with
$\tilde{\mu}=\tilde{\mu}_{H}=\frac{r_{-}}{r_{+}}$.

By defining $x=R\phi_{E}+\tilde{\mu}t$ and rescaling
$(\phi_{E}^{\prime},t^{\prime}_{E})\coloneqq(\frac{\phi_{E}}{2\pi},\frac{t}{\beta_{H}})$,
we ensure that the identifications are as in (4) while the metric becomes

$ds^{2}_{E}=\frac{(2\pi
R)^{2}}{l^{2}}\biggl{[}{\tilde{\beta}_{H}}^{2}\left(r^{2}(1+\tilde{\mu}^{2})-r_{+}^{2}+r^{2}_{-}-2\tilde{\mu}r_{+}r_{-}\right)d{t^{\prime}_{E}}^{2}+r^{2}d{\phi^{\prime}_{E}}^{2}\\\
+2\tilde{\beta}_{H}(\tilde{\mu}r^{2}-r_{+}r_{-})dt^{\prime}_{E}d\phi^{\prime}_{E}\biggl{]}+\frac{r^{2}dr^{2}}{(r^{2}-r^{2}_{+})(r^{2}+r^{2}_{-})}$
(10)

where $\tilde{\beta}_{H}=\frac{r_{+}l^{2}}{R(r_{+}^{2}+r_{-}^{2})}.$ Again, we
will have a Euclidean $B_{NS}$ field which gives $H=2i\sqrt{g}=dB$:

$B_{E}=i\frac{(2\pi
R^{2})\tilde{\beta}_{H}}{l^{2}}(r^{2}-\bar{\alpha})d\phi^{\prime}_{E}\wedge
dt^{\prime}_{E}$ (11)

where $\bar{\alpha}$ is an integration constant. Since the $t_{E}$ co-ordinate
becomes ill-defined at $r=r_{+}$, the two form $B_{E}$ must vanish at
$r=r_{+}$ which forces $\bar{\alpha}=r_{+}^{2}.$

This is in contrast with the previous geometry where it is the $x$ circle
which shrinks to zero size and that too at $r=0$. From the string theory point
of view, we can regard this difference between the two geometries as being
captured by the B-field. The B-fields in the two cases differ by a constant,
but this constant cannot be changed by a “small” gauge transformation sashok .
As we will see below, the two geometries also differ in the expectation value
of the operator dual to the B-field.

The boundary data can be obtained from the bulk fields by using the Fefferman-
Graham expansion. This is achieved by redefining the radial coordinate so that
the bulk metric takes the form of a series

$ds^{2}_{g}=\frac{l^{2}d\rho^{2}}{\rho^{2}}+\frac{l^{2}}{\rho^{2}}(G_{MN})_{g}dX^{M}dX^{N}+l^{2}(\tilde{T}_{MN})_{g}dX^{M}dX^{N}+\frac{\rho^{2}}{l^{2}}(F_{MN})_{g}dX^{M}dX^{N}+\ldots$
(12)

where $X^{1}=\phi^{\prime}_{E}$ and $X^{2}=t^{\prime}_{E}$ are dimensionless
boundary coordinates and the subscript $g$ labels the geometries. The tensor
$G_{MN}$ is the induced metric on the conformal boundary and $T_{MN}$ with
dimensions $L^{0}$ is interpreted as the (VEV of the) boundary energy-momentum
operator.

$T_{MN}=\frac{2l}{16\pi G_{3}}\tilde{T}_{MN}$ (13)

For thermal AdS (5), we get relation

$\log\frac{\rho}{\rho_{A}}=\log\left(\frac{r+\sqrt{r^{2}+l^{2}}}{2l}\right)$
(14)

containing a integration constant $\rho_{A}$ with length dimensions. This
gives us the boundary metric of Thermal AdS:

$(G_{MN})_{TAdS_{3}}=\frac{\rho_{A}^{2}(2\pi
R)^{2}}{16l^{2}}\begin{pmatrix}1&\tilde{\mu}\tilde{\beta}\\\
\tilde{\mu}\tilde{\beta}&\tilde{\beta}^{2}(1+{\tilde{\mu}}^{2})\end{pmatrix}$
(15)

which is dimensionful because the coordinates are dimensionless. A similar
computation for the BTZ geometry gives:

$\log\frac{\rho}{\rho_{B}}=\log\frac{\sqrt{r^{2}+r_{-}^{2}}+\sqrt{r^{2}-r_{+}^{2}}}{2\sqrt{r_{+}^{2}+r_{-}^{2}}}$
(16)

contains the integration constant $\rho_{B}$. Using this, the boundary metric
is obtained to be:

$\displaystyle(G_{MN})_{EBTZ}=\frac{\rho_{B}^{2}(r_{+}^{2}+r_{-}^{2})(2\pi
R)^{2}}{16l^{4}}\begin{pmatrix}1&\tilde{\mu}\tilde{\beta}_{H}\\\
\tilde{\mu}\tilde{\beta}_{H}&\tilde{\beta}_{H}^{2}(1+{\tilde{\mu}}^{2})\end{pmatrix}$
(17)

We can compare the metrics above with the torus metric (3) and read off the
modular parameter. The modular parameter for thermal AdS turns out to be
$T_{A}=\tilde{\beta}(i+\tilde{\mu})$ and
$T_{B}=\tilde{\beta}_{H}(i+\tilde{\mu}_{H})$ for the black hole. Note that the
two geometries have the same chemical potential and temperature if
$\beta=\beta_{H}$ and $\tilde{\mu}=\tilde{\mu}_{H}.$ Using
$\tilde{\beta}_{H}=\frac{r_{+}l^{2}}{R(r_{+}^{2}+r_{-}^{2})}$, and
$\tilde{\mu}=\mu_{H}\equiv\frac{r_{-}}{r_{+}}$:

$T_{B}=\tilde{\beta}_{H}(i+\tilde{\mu}_{H})=-\frac{l^{2}}{R(-r_{-}+ir_{+})}.$
(18)

i.e., an S-modular transformation of a torus with modular parameter
$(-r_{-}+ir_{+}).$ Thus, one could have expected that by substituting
$\beta(i+\mu)=\frac{-l^{2}}{R(-r_{-}+ir_{+})}$ in the Thermal AdS partition
function RR , we should obtain the BTZ partition function. However, this naive
expectation needs an important modification because the boundary torii have
different volumes in the two cases. To see this, we first read off the
boundary volume parameters by comparison:

$(\rho_{2})_{TADS}=\frac{\rho_{A}^{2}R^{2}}{4l^{2}}\tilde{\beta}\pi^{2}=\frac{\pi\rho_{A}^{2}\beta
R}{8l^{2}}\qquad(\rho_{2})_{EBTZ}=\frac{\rho_{B}^{2}R^{2}\pi^{2}(r_{+}^{2}+r_{-}^{2})\tilde{\beta}_{H}}{4l^{4}}=\frac{\pi^{2}\rho_{B}^{2}Rr_{+}}{4l^{2}}$
(19)

for TAdS and EBTZ respectively. Note that the volume parameter $\rho_{2}$
depends on size of the boundary circle $R$ and has dimensions of volume
(because the coordinates are dimensionless).

The integration constants can be fixed by requiring that
$\lim_{r\to\infty}\frac{\rho}{r}=1$ for both geometries. This gives, for TADS,
$\rho_{A}=l$ and the volume of the AdS boundary becomes

$(\rho_{2})_{TADS}=\frac{\beta R\pi}{8}$ (20)

while for EBTZ, $\rho_{B}=\sqrt{r_{+}^{2}+r_{-}^{2}}$ and the boundary torus
has a different volume

$(\rho_{2})_{EBTZ}=\frac{R\pi^{2}(r_{+}^{2}+r_{-}^{2})r_{+}}{4l^{2}}=\frac{R\pi}{8|T|^{2}}\frac{\beta}{|T|^{2}}.$
(21)

rescaled by $|T|^{4}$. This observation needs to be taken into account in
attempting to extract the BTZ spectrum from the Thermal AdS partition function
of MO2 .

Alternatively, we have the option of choosing $\rho_{A,B}$ to rescale the
boundary volume to any value, for instance, $\rho_{2}.$ This will allow ensure
that the boundary torii are identical which then permits us to compare the on-
shell actions and identify a Hawking-Page phase transition if any.

The boundary EBTZ metric can also be written as

$\frac{(G_{MN})_{EBTZ}}{(2\pi
R)^{2}}=\frac{\rho_{B}^{2}}{16l^{4}}(r_{+}^{2}+r_{-}^{2})\tilde{\beta}_{H}^{2}(1+\tilde{\mu}^{2})\begin{pmatrix}\frac{1}{\beta_{H}^{2}(1+\mu^{2})}&\frac{\mu}{\beta_{H}(1+\mu^{2})}\\\
\frac{\mu}{\beta_{H}(1+\mu^{2})}&1\end{pmatrix}.$ (22)

Because 1 is in the “wrong” position in comparison to the torus metric of (3),
we can read off the modular parameter only if we rotate $\phi_{E},t_{E}$:

$(G_{MN})_{EBTZ}\propto(i\sigma_{2})\begin{pmatrix}1&\frac{2\pi\mu}{\beta_{H}(1+\mu^{2})}\\\
\frac{2\pi\mu}{\beta_{H}(1+\mu^{2})}&\frac{4\pi^{2}}{\beta_{H}^{2}(1+\mu^{2})}\end{pmatrix}(-i\sigma_{2})$
(23)

which gives $T_{B}=\frac{-1}{\beta_{H}(i+\mu)}$ (we have omitted the prefactor
in writing the above rotation). The rotation is the manifestation of the fact
that the circle that shrinks to zero size is different in the two geometries.
The form of the rotation matrix used in $G_{MN}$ implies that
$(\phi_{E},t_{E})\to(\phi_{E},t_{E}).(i\sigma_{2})=(t_{E},-\phi_{E})$ or
$z\to-iz.$

The rotation means that the identifications in the original coordinates are

$\phi_{E}+it_{E}\sim\phi_{E}+it_{E}+\frac{2\pi
i}{\beta_{H}(i+\mu_{H})}\sim\phi+it_{E}-i(r_{-}-ir_{+}).$ (24)

where the presence of $i$ signals the rotation This is the relation between
the orbifold parameters of EBTZ and TADS noted in Malstrom . In this version,
we note that the volume parameters of TADS and EBTZ are equal if
$\rho_{A}^{2}=\rho_{B}^{2}.$ It now follows that, if we replace the Thermal
AdS parameters $\beta(i+\mu)\to i(r_{-}-ir_{+})$, we will obtain the partition
function of the Euclidean BTZ black hole provided we account for the rotation
in interpreting the answer. We will use this approach in discussing the EBTZ
torus partition function in section 2.

Note that had we interchanged $x,y$ of the torus, the volume form in the bulk
geometry would have changed sign - which at least in string theory, will mean
that the B-field changes sign whereas, under rotation there is no sign change.
This is important in order to ensure that the non-normalizable modes of the
$B_{NS}$ field is the same for TADS and EBTZ geometries before we can compare
them.

The normalizable modes of the bulk metric gives us the dimensionless stress-
energy tensor of the boundary theory:

$\displaystyle(T_{MN})_{EBTZ}=\begin{pmatrix}\frac{\pi
R^{2}(r_{+}^{2}-r_{-}^{2})}{4G_{3}l^{3}}&-\frac{\pi Rr_{-}}{4G_{3}l}\\\
\frac{\pi Rr_{-}}{4G_{3}l}&-\frac{l\pi}{4G_{3}}\end{pmatrix},$
$\displaystyle(T_{MN})_{TAdS_{3}}=\begin{pmatrix}-\frac{\pi
R^{2}}{4G_{3}l}&-\frac{\tilde{\mu}\beta R}{8G_{3}l}\\\ -\frac{\tilde{\mu}\beta
R}{8G_{3}l}&\frac{\beta^{2}(1-\tilde{\mu}^{2})}{16\pi G_{3}l}\end{pmatrix}$
(25)

It should be noted that these are independent of the integration constant
$\rho_{A,B}^{2}$ reflecting scale invariance as expected of a conformal field
theory. The tracelessness of these stress-energy tensors can be verified using
the boundary metric.

###  1.1 Thermodynamics and Phase transition

If we wish to regard these two geometries as bulk saddle points corresponding
to the same boundary state, we need to ensure that the boundary metrics are
the same. Since $t_{E}$ and $\phi_{E}$ have the same periodicities, we just
need to compare the TADS metric in (15) with the first form given in (17), we
see that the two are identical but for the prefactors. To match these, we need
to set the Kahler moduli $\rho_{A}$ in terms of $\rho_{B}$ as

$\rho_{A}^{2}=\rho_{B}^{2}M=\rho_{B}^{2}(r_{+}^{2}+r_{-}^{2}).$ (26)

We also have to ensure that the non-normalizable modes of the B-field are
equal. If we regard these two geometries as solutions to pure 3D gravity, this
comparison is moot (since there is no $B_{NS}$ field in pure gravity).
Therefore, in the string theory perspective, we expect this to be
automatically satisfied provided the boundary metrics are the same. Applying
the F-G expansion to the Euclidean B-field:

$B_{EBTZ}=i(2\pi
R)^{2}\frac{\tilde{\beta}_{H}}{l^{2}}\left(\frac{(r^{2}_{+}+r^{2}_{-})\rho_{B}^{2}}{16\rho^{2}}-\frac{r^{2}_{+}+r^{2}_{-}}{2}+\ldots\right)d\phi^{\prime}_{E}\wedge
dt^{\prime}_{E}.$ (27)

and for AdS, we have the expansion

$B_{TAdS_{3}}=i(2\pi
R)^{2}\frac{\tilde{\beta}}{l^{2}}\left(\frac{l^{2}\rho_{A}^{2}}{16\rho^{2}}-\frac{l^{2}}{2}+\ldots\right)d\phi^{\prime}_{E}\wedge
dt^{\prime}_{E}.$ (28)

The non-normalizable modes (sources) $J$ and normalizable modes (VEVs)
$\langle\hat{L}\rangle$ are

$\displaystyle J_{EBTZ}$ $\displaystyle=$ $\displaystyle i(2\pi
R)^{2}\frac{r_{+}\rho_{B}^{2}}{16}=i\frac{\pi^{2}\rho_{B}^{2}}{4\beta(1+\mu^{2})}\qquad\langle\hat{L}\rangle_{EBTZ}$
$\displaystyle=$ $\displaystyle-i(2\pi)^{2}r_{+}$
$\displaystyle=-i\frac{1}{\beta_{H}(1+\mu_{H}^{2})}$ $\displaystyle
J_{TAdS_{3}}$ $\displaystyle=$ $\displaystyle i(2\pi
R)^{2}\frac{\beta\rho_{A}^{2}}{4}=i\frac{\pi\rho_{B}^{2}}{4\beta(1+\mu^{2})}\qquad\langle\hat{L}\rangle_{TAdS}$
$\displaystyle=$ $\displaystyle-i\frac{\beta(2\pi)^{2}}{4}$ (29)

where, in the second line, we have used the relation (26) and the relation
between $r_{+}$ and the BTZ temperature to show that the sources $J$ are
equal. We also note that the sources are equal to the Kahler modulus
$J=i\rho_{2}$ (19).

On the other hand, the expectation values of this operator are not equal
implying that TAdS and EBTZ are different states of the boundary theory. The
operator $\hat{L}$ in the boundary theory which couples to the bulk $B-$field
will have scaling dimension $\Delta=2.$ The relation between the non-
normalizable mode and the normalizable mode
$J=\frac{2\rho_{B}^{2}}{4\pi^{2}}\langle\hat{L}\rangle$ is an equation of
state provided the operator $\hat{L}$ defines some kind of conserved “charge”
of the dual theory.

The geometries that we are studying should be regarded as defining a grand-
canonical ensemble of a boundary theory because of the temperature periodicity
and the angular periodicity which leads to a cross term in the metric. In such
a situation, the phase of the theory is determined by the configuration
corresponding to the lowest grand potential which we will determine by using
the procedure of holographic renormalization.

We begin by regarding the two spacetimes as solutions of a gravity theory
defined by

$S=\frac{1}{16\pi G_{3}}\int d^{3}x\sqrt{g}(R-2\Lambda).$ (30)

where the cosmological constant $\Lambda=-\frac{1}{l^{2}}.$ Even if we replace
the cosmological constant with an NS-NS field $H$ or consider a 10-D bulk
gravity action as the starting point, we obtain the same results.

For the Thermal AdS geometry, the Entropy (defined as the area of a horizon)
is zero. The grand potential is evaluated by the procedure of Holographic
Renormalization Skenderis  which requires the addition of specific
counterterms to the gravitational action.

$G_{AdS}=\frac{S_{AdS}^{E}}{4\pi^{2}R\beta}=\frac{s_{ads}+s_{GH}+s_{ct}}{4\pi^{2}R\beta}$
(31)

Using these, we find that

$G_{AdS}=-\frac{1}{16\pi G_{3}l}=-p$ (32)

Note that since $G_{AdS}$ is independent of the temperature $\beta$ and
chemical potential $\mu$, the entropy $s=-\frac{\partial G}{\partial T}$ and
number density $n=-\frac{\partial G}{\partial\mu}$ evaluate to zero. The Euler
relation $\epsilon+p=\mu n+sT$ implies that $\epsilon=-p$ which agrees with
$\epsilon=\frac{\partial(\beta G)}{\partial\beta}$.

For EBTZ on the other hand, the entropy evaluated by using the area law is:

$S=\int\frac{dA}{4G_{3}}=\frac{2\pi
R}{4G_{3}l}\int_{0}^{1}d\phi_{E}r\big{|}_{r=r_{+}}=\frac{2\pi
Rr_{+}}{4G_{3}l}.$ (33)

The Grand potential evaluated in the same manner as for Thermal AdS gives

$G_{BH}=-\frac{1}{16\pi G_{3}l^{3}}(r_{+}^{2}+r_{-}^{2})$ (34)

We may determine the Entropy density from

$s=-\frac{\partial G_{BH}}{\partial
T}\Big{|}_{\tilde{\mu}}=\frac{r_{+}}{4G_{3}l}$ (35)

where $\tilde{\mu}=\frac{r_{-}}{r_{+}}.$

The number density

$n=-\frac{\partial
G_{BH}}{\partial\tilde{\mu}}\Big{|}_{T}=-\frac{r_{+}r_{-}}{8\pi G_{3}l^{3}}$
(36)

and the pressure is

$p=-G_{BH}=\frac{1}{16\pi G_{3}l^{3}}(r_{+}^{2}+r_{-}^{2})$ (37)

The Euler relation

$\epsilon+p=\tilde{\mu}n+sT$ (38)

then gives us $\epsilon=\frac{1}{16\pi G_{3}l^{3}}(r_{+}^{2}-r_{-}^{2})$ which
agrees with the thermodynamic derivative.

Finally, we can discuss the Hawking-Page transition for this system. This
involves comparing the pressures

$p_{A}=\frac{1}{16\pi G_{3}l^{3}}\qquad p_{B}=\frac{1}{16\pi
G_{3}l^{3}}(r_{+}^{2}+r_{-}^{2})$ (39)

which tells us that when
$r_{+}^{2}+r_{-}^{2}=\frac{l^{2}}{\beta^{2}(1+\mu^{2})}>1$, i.e., at high
temperatures, the black hole, having the higher pressure, is the preferred
description of the phase. This condition agrees with the condition obtained in
Malstrom , of course.

##  2 Partition functions

In this section, we will manipulate the CFT torus partition functions for
these geometries. We will first rewrite the torus partition function of
Thermal AdS by isolating a ‘torus’ factor. This factor in the partition
function will be modular invariant by itself and will contain all the
dependence on the spacetime parameters $T$ and $\rho$. We will then expand
these partition functions in q-series which will allow us to read off the
spectrum. In particular, we read off the spectrum of the Euclidean BTZ black
hole. We also show that the spectra agrees with semi-classical analysis and
the orbifold interpretations. Finally, we will compare the partition functions
themselves.

The partition function of thermal $AdS_{3}$ was computed in MO2 , by
orbifolding a WZW model with target space $\mathbb{H}^{3}/SU(2).$ In the
previous section, we introduced the TADS geometry which had a boundary circle
of radius $R$ which defined the Kahler parameter $\rho_{2}.$

This torus amplitude computed in MO2  is:

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\frac{\beta\sqrt{k-2}}{2\pi\sqrt{\tau_{2}}}\sum_{n,m\in\mathbb{Z}}\frac{e^{-\frac{k\beta^{2}}{4\pi\tau_{2}}|m-n\tau|^{2}+\frac{2\pi(\mathfrak{I}(U_{n,m}))^{2}}{\tau_{2}}}}{|\vartheta_{1}(U_{n,m}|\tau)|^{2}},$
(40)

where111Our $U_{n,m}$ is the complex conjugate of that used in MO2  because we
work in conventions where the left factor of the solution of the WZW model is
holomorphic (left-moving on Lorentzian worldsheet). See Appendix A for
details. $U_{n,m}\coloneqq\frac{\tilde{\beta}}{2\pi}(\tilde{\mu}-i)(n\tau-m)$
and $k$ is the level of the WZW model. We will henceforth omit the tilde in
the temperature and chemical potential. Since the Kahler parameter $\rho_{2}$
contains the boundary radius $R$, and in MO2 , the level $k$ sets the size of
the boundary radius $l^{2}=k\alpha^{\prime}$, the parameter $k$ in the above
partition function is to be identified with $\rho_{2}$. This fact will be
important when we try to compare the partition functions of EBTZ and TADS.

###  2.1 q-Expansion

This partition function has only two integers, although we might expect to see
four integers related to winding and momenta of the torus submanifold fibred
over the radial direction. We will therefore apply some preliminary massaging.
First rewrite the partition function using delta functions as

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\frac{\beta\sqrt{k-2}}{2\pi\sqrt{\tau_{2}}}\sum_{n,m\in\mathbb{Z}}e^{-\frac{k\beta^{2}}{4\pi\tau_{2}}|m-n\tau|^{2}}\\\
\times\int_{-\infty}^{\infty}\mkern-18.0mudz_{1}\int_{-\infty}^{\infty}\mkern-18.0mudz_{2}\int_{-\infty}^{\infty}\mkern-18.0mudk_{1}\int_{-\infty}^{\infty}\mkern-18.0mudk_{2}\;\frac{e^{\frac{2\pi
z^{2}_{2}}{\tau_{2}}}e^{2\pi
i\bigl{[}k_{1}(z_{1}-\mathfrak{R}(U_{n,m}))+k_{2}(z_{2}-\mathfrak{I}(U_{n,m}))\bigr{]}}}{|\vartheta_{1}(z_{1}+iz_{2}|\tau)|^{2}}\,.$
(41)

Then, perform a Poisson Resummation in $m$ to obtain

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\sqrt{\frac{k-2}{k}}\sum_{m^{\prime},n\in\mathbb{Z}}\int
d^{2}z\;d^{2}k\;\frac{e^{\frac{2\pi}{\tau_{2}}z_{2}^{2}}\;e^{2\pi
i\left[k_{1}\left(z_{1}-\frac{\beta}{2\pi}n\tau_{2}\right)+k_{2}\left(z_{2}-\frac{\beta}{2\pi}\mu
n\tau_{2}\right)\right]}}{|\vartheta_{1}(z_{1}+iz_{2}|\tau)|^{2}}\\\
\times\exp\left\\{2\pi
i\tau_{1}nm^{\prime}-\pi\tau_{2}\left[\frac{1}{k}\left(\frac{2\pi}{\beta}m^{\prime}-k_{2}+k_{1}\mu\right)^{2}+k\left(\frac{\beta
n}{2\pi}\right)^{2}\right]\right\\}.$ (42)

We can now integrate over $k_{2}$ and perform a change of variables
$z_{2}=v_{1}\tau_{2}$ and $z_{1}=v_{1}\tau_{1}-v_{2}$ which results in

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\sqrt{\frac{k-2}{\tau_{2}}}\sum_{m^{\prime},n\in\mathbb{Z}}\tau_{2}\int_{-\infty}^{\infty}\mkern-18.0mudv_{1}\int_{-\infty}^{\infty}\mkern-18.0mudv_{2}\int_{-\infty}^{\infty}\mkern-18.0mudk_{1}\\\
\times\frac{e^{(2-k)\pi v_{1}^{2}\tau_{2}}\;e^{2\pi
i\left[k_{1}(v_{1}\tau_{1}-v_{2})+v_{1}\tau_{2}\left(k_{1}\mu+\frac{2\pi}{\beta}m^{\prime}-\frac{i\mu
k\beta n}{2\pi}\right)\right]}}{|\vartheta_{1}(v_{1}\tau-v_{2}|\tau)|^{2}}\\\
\times\exp\left\\{2\pi
i\tau_{1}nm^{\prime}-\pi\tau_{2}\left[k\left(\frac{\beta
n}{2\pi}\right)^{2}(1+\mu^{2})+2i\mu m^{\prime}n+\frac{i\beta
nk_{1}}{\pi}(1+\mu^{2})\right]\right\\}\,.$ (43)

We split $v_{1}=s_{1}+w_{1}$ and $v_{2}=s_{2}+w_{2}$, where
$w_{1},w_{2}\in\mathbb{Z}$ and $s_{1},s_{2}\in[0,1)$. This results in

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\sqrt{\frac{k-2}{\tau_{2}}}\sum_{\begin{subarray}{c}m^{\prime},n,\\\
w_{1},w_{2}\in\mathbb{Z}\end{subarray}}\tau_{2}\int_{0}^{1}\mkern-9.0muds_{1}\int_{0}^{1}\mkern-9.0muds_{2}\int_{-\infty}^{\infty}\mkern-18.0mudk_{1}\;\frac{e^{2\pi
i\left[\tau_{1}(nm^{\prime}+k_{1}(s_{1}+w_{1}))-k_{1}(w_{2}+s_{2})\right]}}{|\vartheta_{1}(s_{1}\tau-
s_{2}|\tau)|^{2}}\\\ \times
e^{-\pi\tau_{2}\left[-2i(s_{1}+w_{1})\left(k_{1}\mu+\frac{2\pi}{\beta}m^{\prime}-\frac{i\mu
k\beta
n}{2\pi}\right)+(k-2)s_{1}^{2}+k(2s_{1}w_{1}+w_{1}^{2})+k\left(\frac{\beta
n}{2\pi}\right)^{2}(1+\mu^{2})+2i\mu m^{\prime}n+\frac{i\beta
nk_{1}}{\pi}(1+\mu^{2})\right]}\,,$ (44)

upon using the following identity for the $\vartheta_{1}$ function

${\frac{e^{(2-k)\pi(s_{1}+w_{1})^{2}\tau_{2}}}{|\vartheta_{1}((s_{1}+w_{1})\tau-
v_{2}|\tau)|^{2}}=\frac{e^{(2-k)\pi
s_{1}^{2}\tau_{2}}}{|\vartheta_{1}(s_{1}\tau-
v_{2}|\tau)|^{2}}\;e^{-k\pi\tau_{2}(2s_{1}w_{1}+w_{1}^{2})}}\,.$ (45)

We can get rid of the integral in $k_{1}$ by replacing the sum
$\sum_{w_{2}}e^{-2\pi iw_{2}k_{1}}$ with $\sum_{a}\delta(k_{1}-a)$, resulting
in

$Z_{TAdS_{3}}(\beta,\mu;\tau)=\sqrt{\tau_{2}(k-2)}\sum_{\begin{subarray}{c}m^{\prime},n,\\\
w_{1},a\in\mathbb{Z}\end{subarray}}\int_{0}^{1}\mkern-9.0muds_{1}\int_{0}^{1}\mkern-9.0muds_{2}\;\frac{e^{-2\pi
ias_{2}}e^{2\pi\tau_{2}s_{1}^{2}}}{|\vartheta_{1}(s_{1}\tau-
s_{2}|\tau)|^{2}}\;e^{2\pi
i\tau_{1}\left[nm^{\prime}+a(s_{1}+w_{1})\right]}\\\ \times
e^{-\pi\tau_{2}\left[-2i(s_{1}+w_{1})\left(a\mu+\frac{2\pi}{\beta}m^{\prime}-\frac{i\mu
k\beta n}{2\pi}\right)+k(s_{1}+w_{1})^{2}+k\left(\frac{\beta
n}{2\pi}\right)^{2}(1+\mu^{2})+2i\mu m^{\prime}n+\frac{i\beta
na}{\pi}(1+\mu^{2})\right]}\,.$ (46)

At this stage, it is worth pointing out that in the above partition function,
we have isolated all dependence on the external parameters $\beta,\mu$ in the
second line.

We will now expand the above expression following the ideas of MO2 ; Israel
so that we can compare the conformal weights with expectations based on
classical analysis of strings moving on the Thermal AdS geometry. The details
of this expansion are presented in Appendix B and the result takes the form:

$\displaystyle Z(\beta,\mu;\tau,\bar{\tau})$
$\displaystyle=\sum_{m^{\prime},n,w_{1}\in\mathbb{Z}}\sum_{q^{\prime},\bar{q}^{\prime},N,\bar{N}}\Biggl{\\{}\int_{-\infty}^{\infty}\frac{ds}{\pi}\left[\sum_{P_{\pm,0}=-\infty}^{0}\tfrac{1}{2is-
kw_{1}+\left(i(q-\bar{q})\mu+\frac{2\pi i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+1+q+\bar{q}}\right.$ (47)
$\displaystyle\left.-\sum_{P_{\pm,0}=0}^{\infty}\tfrac{1}{2is-
kw_{1}+\left(i(q-\bar{q})\mu+\frac{2\pi i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)-1+q+\bar{q}}\right]+\sum_{\begin{subarray}{c}P_{\pm,0}=-\infty\\\
\frac{1}{2}+is=j_{q,\bar{q},w_{1}}^{n,m^{\prime}}\\\
-\frac{(k-2)}{2}<\mathfrak{I}(s)<0\end{subarray}}^{0}\Biggr{\\}}$
$\displaystyle\times e^{2\pi
i\tau_{1}\left[w_{1}(q-\bar{q})+nm^{\prime}+N-\bar{N}\right]}\times
e^{-2\pi\tau_{2}\left[\frac{2s^{2}}{k-2}-\frac{1}{4}+N+\bar{N}\right]}$
$\displaystyle\times
e^{-2\pi\tau_{2}\left[-w_{1}\left(i(q-\bar{q})\mu+\frac{2\pi
i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+\frac{k}{2}w_{1}^{2}+\frac{k}{2}\left(\frac{\beta
n}{2\pi}\right)^{2}(\mu^{2}+1)+i\mu m^{\prime}n+\frac{i\beta
n(q-\bar{q})}{2\pi}(1+\mu^{2})\right]}\,,$

where the discrete series states have quadratic Casimir $j(1-j)$ with $j$
taking the values

$j_{q,\bar{q},w_{1}}^{n,m^{\prime}}\coloneqq\frac{k}{2}w_{1}-\frac{1}{2}\left(i(q-\bar{q})\mu+\frac{2\pi
i}{\beta}m^{\prime}+\frac{\mu k\beta n}{2\pi}\right)-\frac{1}{2}(q+\bar{q})$
(48)

and $\sum_{q^{\prime},\bar{q}^{\prime},N,\bar{N}}$ is shorthand for a sum over
infinitely many variables
$P^{+}_{\pm,p},P^{-}_{\pm,p},P_{\pm,p}\in\\{0,1,2,\ldots\\}$ (for each
$p\in\\{1,2,3,\ldots\\}$) with $(q,\bar{q})$ and $(N,\bar{N})$ defined as

$\displaystyle q$ $\displaystyle\coloneqq P_{+,0}+q^{\prime}\coloneqq
P_{+,0}+\sum_{p=1}^{\infty}(P^{+}_{+,p}-P^{-}_{+,p})\,,$ (49a)
$\displaystyle\bar{q}$ $\displaystyle\coloneqq
P_{-,0}+\bar{q}^{\prime}\coloneqq
P_{-,0}+\sum_{p=1}^{\infty}(P^{+}_{-,p}-P^{-}_{-,p})\,,$ (49b) $\displaystyle
N$
$\displaystyle\coloneqq\sum_{p=1}^{\infty}p(P_{+,p}+P^{+}_{+,p}+P^{-}_{+,p})\,,$
(49c) $\displaystyle\bar{N}$
$\displaystyle\coloneqq\sum_{p=1}^{\infty}p(P_{-,p}+P^{+}_{-,p}+P^{-}_{-,p})\,.$
(49d)

It is important to point out that the quantities in braces in (47) are to be
understood as a degeneracy of states with the specified values of
$L_{0},\bar{L}_{0}$. For the continuous series, the first two terms are
combined into a density of states MO2 . The summations in the last term will
give us the degeneracy of the states of the Discrete series which is clearly a
positive integer.

There are two noteworthy features: the location of the poles involve
$q+\bar{q},$ but $q+\bar{q}$ does not appear in the continuous series
$L_{0}+\bar{L}_{0}$. And, the discrete series states have both real and
imaginary parts in the radial momentum $s.$ The imaginary parts will localize
the wavefunctions in the interior of the geometry as expected since massive
particles cannot reach the boundary of AdS.

###  2.2 Interpretation – thermal $AdS_{3}$ viewpoint

The action for the SL(2,$\mathbb{R}$) WZW model at level $k$ written in the
global coordinates
$g=e^{\frac{i}{2}(t_{\text{global}}+\phi)\sigma_{2}}e^{\rho\sigma_{3}}e^{\frac{i}{2}(t_{\text{global}}-\phi)\sigma_{2}}$
is

$S_{\text{WZW}}[g]=\frac{-k}{2\pi}\int
dx^{+}dx^{-}\;\Big{[}\partial_{+}\phi\partial_{-}\phi\sinh^{2}\rho+\partial_{+}\rho\partial_{-}\rho-\partial_{+}t_{\text{global}}\partial_{-}t_{\text{global}}\cosh^{2}\rho\\\
{+(\sinh^{2}\rho+C)}\left(\partial_{-}t_{\text{global}}\partial_{+}\phi-\partial_{-}\phi\partial_{+}t_{\text{global}}\right)\Big{]}\,,$
(50)

where $C$ is a gauge “ambiguity” in the B-field. Note that this action is
manifestly Lorentzian. Thermal AdS is obtained from this by a Wick rotation
and orbifolding. In thermal $AdS_{3}$, $\rho=0$ is the origin of a polar
coordinate system. The 1-form $d\phi$ and hence, the 2-form $dt\wedge d\phi$
are not defined at $\rho=0$. So, it is expected that $C=0$ to ensure that the
B-field vanishes at $\rho=0$ where the geometry “terminates”. The relation
between these coordinates and those of the previous section are $r=\sinh\rho$
and the AdS radius is related to $l^{2}=k\alpha^{\prime}$

The Virasoro generators $L_{0},\bar{L}_{0}$ for $AdS_{3}$ are constructed out
of the currents of SL(2,$\mathbb{R}$) WZW model. The energy and angular
momentum of TAdS are both obtained from the elliptic generator of
SL(2,$\mathbb{R}$). Thus, we can express MO1

$\displaystyle L_{0}+\bar{L}_{0}$
$\displaystyle=\frac{1}{k}\biggl{[}(J^{(0)}_{+,0})^{2}-(J^{\prime(0)}_{+,0})^{2}+(J^{(0)}_{-,0})^{2}-(J^{\prime(0)}_{-,0})^{2}\biggl{]}+\frac{2s^{2}+\frac{1}{2}}{k-2}\,,$
(51a) $\displaystyle L_{0}-\bar{L}_{0}$
$\displaystyle=\frac{1}{k}\biggl{[}(J^{(0)}_{+,0})^{2}-(J^{\prime(0)}_{+,0})^{2}-(J^{(0)}_{-,0})^{2}+(J^{\prime(0)}_{-,0})^{2}\biggl{]}\,,$
(51b)

where $J^{\prime(0)}_{\pm,0}$ denote eigenvalues of currents in the sector
twisted by the orbifolding. The signs appearing in the above expression depend
on the elliptic nature of the diagonalised generator. Wick rotation from
Lorentzian AdS to Thermal AdS does not affect the elliptic nature of the
generator, but it makes their eigenvalues complex.

We can verify that the choice

$\displaystyle J^{\prime(0)}_{\pm,0}$
$\displaystyle=\pm\frac{1}{2}\left[-\frac{2\pi
im^{\prime}}{\beta}+(q-\bar{q})(-i\mu\pm 1)\right]-\frac{k}{2}\frac{i\beta
n}{2\pi}\,,$ (52a) $\displaystyle J^{(0)}_{\pm,0}$
$\displaystyle=J^{\prime(0)}_{\pm,0}\mp\frac{k}{2}\left(-w_{1}-\frac{\beta
n}{2\pi}(-\mu\pm i)\right)\,,$ (52b)

ensures that $L_{0}\pm\bar{L}_{0}$ read off from (47) satisfy (51). We can now
compare the currents with what is expected for thermal $AdS_{3}$ with inverse
temperature $\beta$ but chemical potential $-\mu$. The Noether charge
corresponding to $\phi$ and $t_{\text{global}}$ translations calculated from
the action $S_{\text{WZW}}=\int d\tau d\sigma\mathcal{L}$ are

$\displaystyle\Pi_{\phi}$
$\displaystyle\coloneqq-\int_{0}^{2\pi}\mkern-18.0mud\sigma\;\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\phi)}=J^{\prime(0)}_{+,0}+J^{\prime(0)}_{-,0}-\frac{ik\beta
n}{2\pi}(C-1)=q-\bar{q}-kC\frac{i\beta n}{2\pi}\,,$ (53a)
$\displaystyle\Pi_{t_{\text{global}}}$
$\displaystyle\coloneqq-\int_{0}^{2\pi}\mkern-18.0mud\sigma\;\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}t_{\text{global}})}=J^{\prime(0)}_{+,0}-J^{\prime(0)}_{-,0}+kw_{1}C-kC\frac{\mu\beta
n}{2\pi}$ (53b) $\displaystyle=-\frac{2\pi
im^{\prime}}{\beta}-i\mu(q-\bar{q})+kw_{1}C-kC\frac{\mu\beta n}{2\pi}\,.$
(53c)

The target space momenta are correctly quantised for $C=0$ i.e.,

$\Pi_{\phi}=q-\bar{q}\in\mathbb{Z}\,,\qquad\frac{i\beta}{2\pi}\Pi_{t_{\text{global}}}-\frac{\mu\beta}{2\pi}\Pi_{\phi}=m^{\prime}\in\mathbb{Z}\,.$
(54)

The relation between $J^{\prime(0)}_{\pm,0}$ and $J^{(0)}_{\pm,0}$ means that
(52) are the eigenvalues evaluated in the twisted sector with
$(t_{\text{global}},\phi)\xmapsto{\sigma\mapsto\sigma+2\pi}(t_{\text{global}},\phi)-n(i\beta,-\mu\beta)-w_{1}(0,2\pi)$.
In particular, the integers $q-\bar{q}$ and $m^{\prime}$ are (resp.) momenta
along the $\phi$ direction and the orbifold direction, while $w_{1}$ and $n$
are (resp.) the winding numbers along these directions. Note the presence of
Matsubara terms in the energy

$\Pi_{t_{\text{global}}}=\frac{2\pi}{i\beta}m^{\prime}-i\mu(q-\bar{q})$ (55)

as appropriate to a thermal system.

Before moving on, a representation theoretic comment is in order. In the case
of Lorentzian $AdS_{3}$, it is well known that highest and lowest weight
discrete series with amounts of spectral flow differing by 1 and the radial
momenta related by reflection about $\frac{k}{2}$ are identified. This also
holds in thermal $AdS_{3}$ and is clear from the interpretation of the
partition function discussed so far. To see this, note that

$J^{(0)}_{\pm,0}=\pm(j+q_{\pm})\,,\qquad
J^{\prime(0)}_{\pm,0}=\pm(j+q_{\pm})\pm\frac{k}{2}\Omega_{\pm}\,,$ (56)

where we have defined $(q_{+},q_{-})\coloneqq(q,\bar{q})$,
$\Omega_{\pm}\coloneqq-w_{1}+\frac{\beta n}{2\pi}(\mu\mp i)$ and $j\coloneqq
j^{n,m^{\prime}}_{q,\bar{q},w_{1}}$ from (48), for convenience of notation. In
this notation, (51) implies

$\displaystyle L_{0}$
$\displaystyle=\frac{-(j-\frac{1}{2})^{2}+\frac{1}{4}}{k-2}-\Omega_{+}(j+q)-\frac{k}{4}\Omega_{+}^{2}+N$
(57a)
$\displaystyle=\frac{-(j^{\prime}-\frac{1}{2})^{2}+\frac{1}{4}}{k-2}+\Omega^{\prime}_{+}(j^{\prime}-q)-\frac{k}{4}{\Omega^{\prime}_{+}}^{2}+N+q$
(57b)

(and similarly for $\bar{L}_{0}$), where the radial momenta $j$ and
$j^{\prime}$ are related by a reflection about $\frac{k}{2}$ i.e.,
$j^{\prime}=\frac{k}{2}-j$ and the spectral flow parameters are related by
$\Omega^{\prime}_{\pm}=\Omega_{\pm}+1$ i.e, by a shift of the $\phi$ winding
$w_{1}$ by 1. This shows that the lowest weight discrete series is identified
with the highest weight discrete series with reflected radial momentum and
shifted spectral flow, exactly as in Lorentzian $AdS_{3}$. This is a
consequence of the interpretation (52) that states are labelled by eigenvalues
of the elliptic generators $J^{\prime(0)}_{\pm,0}$. As we shall see, this
identification will fail in the EBTZ intepretation of the same partition
function where the states are labelled by eigenvalues of the hyperbolic
generators $J^{\prime(2)}_{\pm,0}$ instead.

###  2.3 Interpretation – Euclidean BTZ viewpoint

The Euclidean BTZ black hole is also an orbifold of the $\mathbb{H}^{3}$ in
the same manner as Thermal AdS Malstrom . Through (24), we observed that the
EBTZ can be viewed as an orbifold of $\mathbb{H}^{3}$ with a modular parameter
$\tilde{T}_{B}=\pm(ir_{-}+r_{+})$ provided we also rotate $t_{E}$ and
$\phi_{E}$ by $\frac{\pi}{2}.$

In addition, we noted that using this second form of the boundary metric, the
boundary volumes will be the same if $\rho_{A}=\rho_{B}.$ This means that by
setting $(\beta,\mu\beta)=(2\pi r_{+},2\pi r_{-})\in\mathbb{R}^{2}$, the same
partition function (46) may be interpreted as the Euclidean BTZ partition
function.

Thus, we begin with (47) where we replace $(\beta,\mu\beta)=(2\pi r_{+},2\pi
r_{-}):$

$\displaystyle Z(\beta,\mu;\tau,\bar{\tau})$
$\displaystyle=\sum_{m^{\prime},n,w_{1}\in\mathbb{Z}}\sum_{q^{\prime},\bar{q}^{\prime},N,\bar{N}}\Biggl{\\{}\Biggr{\\}}\times
e^{2\pi i\tau_{1}\left[w_{1}(q-\bar{q})+nm^{\prime}+N-\bar{N}\right]}\times
e^{-2\pi\tau_{2}\left[\frac{2s^{2}}{k-2}+N+\bar{N}\right]}$ (58)
$\displaystyle\times
e^{-2\pi\tau_{2}\left[-\frac{1}{4}-w_{1}\left(i(q-\bar{q})\frac{r_{-}}{r_{+}}+\frac{im^{\prime}}{r_{+}}+knr_{-}\right)+\frac{k}{2}w_{1}^{2}+\frac{k}{2}n^{2}(r_{-}^{2}+r_{+}^{2})+i\frac{r_{-}}{r_{+}}m^{\prime}n+\frac{in(q-\bar{q})}{r_{+}}(r_{-}^{2}+r_{+}^{2})\right]}\,,$

where the factors with braces account for the degeneracy as in that formula.
We will focus on the second line in the following since that piece involves
the currents of the underlying WZW model.

In this section, we will study the conformal weights obtained from the above
q-series and show that it agrees with the expected structure for the BTZ black
hole NS  once we take into account Wick rotation.

The spectrum of the Lorentzian BTZ black hole was first explored in NS , where
the conformal weights $L_{0},\bar{L}_{0}$ where shown to be expressed in terms
of the hyperbolic generators of SL(2,$\mathbb{R}$). Thus, for the Euclidean
BTZ as well, we expect

$\displaystyle L_{0}+\bar{L}_{0}$
$\displaystyle=\frac{1}{k}\biggl{[}-(J^{(2)}_{+,0})^{2}+(J^{\prime(2)}_{+,0})^{2}-(J^{(2)}_{-,0})^{2}+(J^{\prime(2)}_{-,0})^{2}\biggl{]}+\frac{2s^{2}+\frac{1}{2}}{k-2}+N+\bar{N}\,,$
(59a) $\displaystyle L_{0}-\bar{L}_{0}$
$\displaystyle=\frac{1}{k}\biggl{[}-(J^{(2)}_{+,0})^{2}+(J^{\prime(2)}_{+,0})^{2}+(J^{(2)}_{-,0})^{2}-(J^{\prime(2)}_{-,0})^{2}\biggl{]}+N-\bar{N}\,.$
(59b)

with different sign choices compared to Thermal AdS (51). The Euclidean nature
will again mean that the eigenvalues $J^{(2)}_{\pm,0}$ and
$J^{\prime(2)}_{\pm,0}$ will not be real.

A choice of current eigenvalues satisfying these relations with
$L_{0}\pm\bar{L}_{0}$ read off from (47) is

$\displaystyle\begin{aligned} J^{(2)}_{\pm,0}&=\left[\frac{\pm
m^{\prime}+i(q-\bar{q})\Delta_{\pm}}{2r_{+}}\right]\pm\frac{k}{2}(-ir_{-}n+iw_{1})\\\
&=\frac{-1}{2\Delta_{\mp}}\left[\frac{\mp
m^{\prime}\Delta_{\mp}-i(q-\bar{q})(r_{+}^{2}+r_{-}^{2})}{r_{+}}\right]\pm\frac{k}{2}(-ir_{-}n+iw_{1})\,,\\\
\end{aligned}$ (60a) $\displaystyle\begin{aligned}
J^{\prime(2)}_{\pm,0}&=J^{(2)}_{\pm,0}-\frac{k}{2}\left(\pm
iw_{1}-n\Delta_{\mp}\right)\\\ &=\frac{-1}{2\Delta_{\mp}}\left[\frac{\mp
m^{\prime}\Delta_{\mp}-i(q-\bar{q})(r_{+}^{2}+r_{-}^{2})}{r_{+}}\right]+\frac{k}{2}r_{+}n\,,\end{aligned}$
(60b)

where $\Delta_{\pm}\coloneqq r_{+}\mp ir_{-}$. At this stage, there are four
integers: $m^{\prime}$, $(q-\bar{q})$, $n$ and $w_{1}$. We expect these to be
related to the winding and momentum quantum numbers along the $t_{\text{BTZ}}$
and $\phi_{\text{BTZ}}$ directions of the Black hole geometry.

To interpret these currents, we use coordinates $(\hat{t},\hat{\phi})$ which
are rotated with respect to the TAdS coordinates
$({\color[rgb]{1,0,0}-i}t_{\text{global}},\phi)=(-\hat{\phi},\hat{t})$.
Euclidean BTZ coordinates are defined by the requirement that the
identifications inherited from TAdS map to the EBTZ identifications:

$\displaystyle\hat{t}$ $\displaystyle\sim\hat{t}+2\pi$
$\displaystyle\Longleftrightarrow$
$\displaystyle(t_{\text{EBTZ}},\phi_{\text{BTZ}})$
$\displaystyle\sim(t_{\text{EBTZ}},\phi_{\text{BTZ}})+\frac{2\pi(r_{+},r_{-})}{r_{+}^{2}+r_{-}^{2}}\,,$
(61a) $\displaystyle(\hat{t},\hat{\phi})$
$\displaystyle\sim(\hat{t},\hat{\phi})-2\pi(r_{-},r_{+})$
$\displaystyle\Longleftrightarrow$
$\displaystyle(t_{\text{EBTZ}},\phi_{\text{BTZ}})$
$\displaystyle\sim(t_{\text{EBTZ}},\phi_{\text{BTZ}})-(0,2\pi)\,.$ (61b)

The EBTZ coordinates $(t_{\text{EBTZ}},\phi_{\text{BTZ}})$ are then obtained
as

$\hat{t}\coloneqq
r_{+}t_{\text{EBTZ}}+r_{-}\phi_{\text{BTZ}}\,,\qquad\hat{\phi}\coloneqq
r_{+}\phi_{\text{BTZ}}-r_{-}t_{\text{EBTZ}}\,.$ (62)

The eigenvalues (60) are evaluated in the spectrally flowed sector with
$(\hat{t},\hat{\phi})\xmapsto{\sigma\mapsto\sigma+2\pi}(\hat{t},\hat{\phi})+w_{1}(2\pi,0)-n(2\pi
r_{-},2\pi r_{+})$. This follows from the relation between
$J^{\prime(2)}_{\pm,0}$ and $J^{(2)}_{\pm,0}$. In this sector, we may evaluate
the target space momenta using the action (50) with
$({\color[rgb]{1,0,0}-i}t_{\text{global}},\phi)=(-\hat{\phi},\hat{t})$ (which
equals the EBTZ sigma model action) as before:

$\displaystyle-\Pi_{\hat{\phi}}$
$\displaystyle\coloneqq\int_{0}^{2\pi}\mkern-18.0mud\sigma\;\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\hat{\phi})}=-J^{\prime(2)}_{+,0}+J^{\prime(2)}_{-,0}-kC(iw_{1}{\color[rgb]{1,0,0}-i}r_{-}n)$
(63a)
$\displaystyle=-\frac{m^{\prime}}{r_{+}}-\frac{{\color[rgb]{1,0,0}i}r_{-}}{r_{+}}i(q-\bar{q})-kCiw_{1}{\color[rgb]{1,0,0}+i}kCr_{-}n\,,$
(63b) $\displaystyle{\color[rgb]{1,0,0}-i}\Pi_{\hat{t}}$
$\displaystyle\coloneqq
i\int_{0}^{2\pi}\mkern-18.0mud\sigma\;\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\hat{t})}=-J^{\prime(2)}_{+,0}-J^{\prime(2)}_{-,0}-k(C-1)r_{+}n=-i(q-\bar{q})-kCr_{+}n\,.$
(63c)

We may now verify that target space momenta are correctly quantised for $C=0$
i.e.,

$\Pi_{\hat{t}}=q-\bar{q}\in\mathbb{Z}\,,\qquad\Pi_{\phi_{\text{BTZ}}}=-r_{-}\Pi_{\hat{t}}+r_{+}\Pi_{\hat{\phi}}=m^{\prime}\in\mathbb{Z}\,.$
(64)

with $m^{\prime}$ being the angular momentum as appropriate for a point-
particle.

The choice $C=0$ for the EBTZ action matches what would be obtained from the
Thermal $AdS_{3}$ action by
$({\color[rgb]{1,0,0}-i}t_{\text{global}},\phi)=(-\hat{\phi},\hat{t})$. It
implies that $B_{\hat{t}\hat{\phi}}\propto r^{2}-r_{+}^{2}$ and hence vanishes
at $r^{2}=r_{+}^{2}$ as expected.

The winding number along the angular direction $\phi_{\text{BTZ}}$ is $n$ and
that along the Euclidean time direction $\hat{t}$ is $w_{1}$. As in
${\text{TAdS}}_{3}$, this is clear from the $\sigma\mapsto\sigma+2\pi$
transformation mentioned below (62). This means that $n$ labels the twisted
sectors of the orbifolding that produces the Euclidean BTZ from
$\mathbb{H}^{3}$ CFT since the orbifolding identification leads to the
periodicity of the $\phi_{\text{BTZ}}$-coordinate.

One key difference between the EBTZ and thermal $AdS_{3}$ interpretations of
the same partition function (47) is that the EBTZ interpretation (60) involves
the hyperbolic generators $J^{(2)}_{\pm,0}$ rather than the elliptic ones.
Because of this, (59) implies

$L_{0}=\frac{-(j-\frac{1}{2})^{2}+\frac{1}{4}}{k-2}-i\Omega_{+}(j+q)+\frac{k}{4}\Omega_{+}^{2}+N$
(65)

(and similarly $\bar{L}_{0}$) using the facts that

$J^{(2)}_{\pm,0}=\pm i(j+q_{\pm})\,,\qquad J^{\prime(2)}_{\pm,0}=\pm
i(j+q_{\pm})-\frac{k}{2}\Omega_{\pm}\,,$ (66)

where $(q_{+},q_{-})\coloneqq(q,\bar{q})$, $\Omega_{\pm}\coloneqq\pm
iw_{1}-n\Delta_{\mp}$ and $j\coloneqq j^{n,m^{\prime}}_{q,\bar{q},w_{1}}$ with
$(\beta,\mu\beta)=(2\pi r_{+},2\pi r_{-})$ i.e,

$j\coloneqq
j^{n,m^{\prime}}_{q,\bar{q},w_{1}}=\frac{k}{2}w_{1}-\frac{1}{2}\left(i(q-\bar{q})\frac{r_{-}}{r_{+}}+\frac{im^{\prime}}{r_{+}}-knr_{-}\right)-\frac{1}{2}(q+\bar{q})\,.$
(67)

Here, (65) is different from (57) in the sign of the last term
$+\frac{k}{4}\Omega_{\pm}^{2}$ in $L_{0}$ and $\bar{L}_{0}$, which is a
consequence of the hyperbolic generators being diagonalised. This prevents us
form reinterpreting a shift of the spatial winding number $n$ as a change of
$j$ and $N$. Hence, differently spectral flowed discrete series
representations are not to be identified in EBTZ. This agrees with our
previous analysis Nippanikar  for Lorentzian BTZ, where this representation
theoretic fact could also be seen at the classical level through discrete
symmetries of BTZ geodesics.

###  2.4 Comparison of Partition functions

In this section, we will compare the torus partition functions of Thermal AdS
and that of Euclidean BTZ at the same temperature, chemical potential and
boundary volume. We will do this by first writing the ‘torus factor’ in the
partition function appearing in the second line of (46) in terms of the the
boundary modular parameter $T$ and the Kahler parameter $\rho_{2}.$ For the
comparison to be meaningful, we will ensure that the boundary metrics are
equal. To this end, we must use the first form for the boundary metric of the
EBTZ. As we have shown, this means that the boundary torus has a rescaled
volume eqn 1.20. We will see below that this rescaling can be obtained from
the partition function as well. We begin by reminding ourselves that the
partition function of MO2  has been computed assuming $\rho_{A}=l.$

We will rewrite the partition function eqn.(46) of TADS by expressing the
chemical potential and temperature in terms of the Modular parameter $T_{A}$
and a complex Kahler parameter defined by $\rho=\rho_{1}+i\rho_{2}$. Here,
$\rho_{1}=J_{A}$ is determined by the non-normalizable mode of the B-field and
$\rho_{2}$ is the volume parameter of the boundary torus as determined from
the non-normalizable modes of the metric. From the earlier results (29), we
see that $\frac{\rho_{1}}{\rho_{2}}=i$ for both TADS and EBTZ.

Thus, substituting $\frac{\beta\mu}{2\pi}=\tilde{\mu}\tilde{\beta}=T_{1}$ and
$\frac{\beta}{2\pi}=\tilde{\beta}=T_{2}$ in eqn.(46), we obtain:

$\Sigma_{TADS}(T)=\sum_{m^{\prime},n,w_{1},a}\tau_{2}\exp\biggl{\\{}-\pi\tau_{2}\bigg{[}-\frac{2\rho_{1}}{\rho_{2}}\left(\frac{T_{1}}{T_{2}}\left((s_{1}+w_{1})a-m^{\prime}n\right)+\frac{m^{\prime}(s_{1}+w_{1})}{T_{2}}-\frac{|T|^{2}}{T_{2}}na\right)\\\
+\frac{\rho_{2}}{T_{2}}\left|(s_{1}+w_{1})-nT\right|^{2}\bigg{]}+2\pi
i\left[\tau_{1}(nm^{\prime}+a(s_{1}+w_{1}))-as_{2})\right]\biggl{\\}}$ (68)

Similarly, we can obtain the EBTZ partition function in terms of the modular
parameter. To this end, we first replace the orbifold parameters
$(\beta,\mu\beta)\to(r_{+},r_{-})$ in the TADS partition function (46) which
gives us the EBTZ partition function. We then re-express $r_{\pm}$ using the
modular parameter ${T}_{B}=-1/(-r_{-}+ir_{+})$ and $\rho$ as determined
earlier (18) to get:

$\Sigma_{EBTZ}(T)=\exp\biggl{\\{}-\pi\tau_{2}\bigg{[}\frac{2\rho_{1}}{\rho_{2}}\left(\frac{\tilde{T}_{1}}{\tilde{T}_{2}}\left({\color[rgb]{0,0,1}-}(s_{1}+w_{1})a+m^{\prime}n\right){\color[rgb]{0,0,1}-}\frac{m^{\prime}(s_{1}+w_{1})|\tilde{T}|^{2}}{\tilde{T}_{2}}+\frac{na}{\tilde{T}_{2}}\right)\\\
+\frac{\rho_{2}}{\tilde{T}_{2}}|(s_{1}+w_{1}){\color[rgb]{0,0,1}-}\frac{n}{\tilde{T}}|^{2}\bigg{]}\bigg{]}+2\pi
i\left[\tau_{1}(nm^{\prime}+a(s_{1}+w_{1}))-as_{2})\right]\biggl{\\}}.$ (69)

We can compare the TAdS and EBTZ partition functions. As shown in Malstrom ,
EBTZ partition function $\Sigma_{EBTZ}(T)$ with modular parameter $T$ must be
compared with $\Sigma_{TADS}(-\frac{1}{T})$. However, if we make the
replacement, $T\to-\frac{1}{T}$ in the TADS partition function 68, the
expression we get is not the same as the one above 69:

$\Sigma_{TADS}(-\frac{1}{T})=\exp\biggl{\\{}-\pi\tau_{2}\bigg{[}{\color[rgb]{1,0,0}-}\frac{2\rho_{1}}{\rho_{2}}\left(\frac{T_{1}}{T_{2}}\left(-(s_{1}+w_{1})a+m^{\prime}n\right)+\frac{m(s_{1}+w_{1})|T|^{2}}{T_{2}}-\frac{na}{T_{2}}\right)\\\
+\frac{\rho_{2}{\color[rgb]{1,0,0}|T|^{2}}}{T_{2}}\left|(s_{1}+w_{1}){\color[rgb]{1,0,0}+}\frac{n}{T}\right|^{2}\bigg{]}+2\pi
i\left[\tau_{1}(nm^{\prime}+a(s_{1}+w_{1}))-as_{2})\right]\biggl{\\}}$ (70)

where the differences are highlighted in red. On the other hand, as shown
previously, the volume parameters of TADS and EBTZ are related as
$\rho\to\frac{\rho}{|T|^{2}}$. In addition, recall that the TADS partition
function of (46) was for a chemical potential $-\mu$ (section 2.3). This
requires us to flip the sign of $T_{1}$ in the BTZ expression before
comparing. Making this change, and flipping the signs of $n,m^{\prime}$, we
get the EBTZ partition function.

This agrees with the observations made earlier that the EBTZ geometry requires
a $\pi/2$ rotation, since that amounts to a sign flip of the $J_{-}$ current.
This does not affect $L_{0},\bar{L}_{0}.$ Thus, the BTZ black hole is thus not
merely the S-dual of the AdS but it also requires a rotation and an additional
rescaling of the volume $\rho\to\frac{\rho}{|T|^{2}}$. This agrees with our
analysis in the first section.

##  3 The Lorentzian spacetimes

The spectrum of states for strings on the Lorentzian spacetimes of $AdS_{3}$
and BTZ black holes have been presented in MO1  and Nippanikar . In this
section, we will show that the CFT spectra can be obtained from the Euclidean
partition functions of the previous section by Wick rotating and
decompactifying the time circle. It is not surprising that we are able to
decompactify – after all $AdS_{3}$ itself requires us to decompactify the
SL(2,$\mathbb{R}$) WZW model. Further motivation to expect these Wick
rotations arises from our observations in Sections 2.2 and 2.3. In particular,
the differently spectral flowed discrete series representations are identified
in thermal $AdS_{3}$ and not identified in EBTZ, in agreement with their
Lorentzian versions. The surprise is that the Wick rotation proceeds so
smoothly and gives us the Lorentzian spectrum quite painlessly.

###  3.1 Wick Rotation of the TADS partition function

In this section, we will see that we can directly Wick rotate the Thermal AdS
partition function to the Lorentzian AdS partition function. The partition
function for the universal cover of SL(2,$\mathbb{R}$) as written by Israel
et.al. Israel  is:

$\displaystyle Z_{SL(2,\mathbb{R})}\left(\tau,\bar{\tau}\right)$
$\displaystyle=$ $\displaystyle
4\sqrt{\tau_{2}}(k-2)^{3/2}\int_{0}^{1}d^{2}s\,\int_{0}^{1}d^{2}t\frac{\mathrm{e}^{\frac{2\pi}{\tau_{2}}(\mathrm{Im}(s_{1}\tau-
s_{2}))^{2}}}{\left|\vartheta_{1}(s_{1}\tau-s_{2}|\tau)\right|^{2}}\times$
(71)
$\displaystyle\times\sum_{m,w,m^{\prime},w^{\prime}\in\mathbb{Z}}\zeta{w+s_{1}-t_{1}\atopwithdelims[
]m+s_{2}-t_{2}}\left(k\right)\zeta{w^{\prime}+t_{1}\atopwithdelims[
]m^{\prime}+t_{2}}\left(-k\right).$

where

$\zeta{\omega\atopwithdelims[
]\mu}\left(k\right)=\sqrt{\frac{k}{\tau_{2}}}\exp\left[-\frac{\pi
k}{\tau_{2}}\left|\omega\tau-\mu\right|^{2}\right]$ (72)

The above partition can be expanded (following Israel ) into the same form as
that in Eqn(47) so that we can compare the spectrum. This is

$\displaystyle Z$
$\displaystyle=\frac{4}{\pi}\left(k-2\right)\int_{-\infty}^{\infty}ds\int_{0}^{1}d^{2}t\
\sum_{w,w^{\prime}}\sum_{n,n^{\prime}}\sum_{\begin{subarray}{c}q,\bar{q}\\\
N,\bar{N}\end{subarray}}\delta_{q-\bar{q},n}\mathrm{exp}\left[2\pi
i\tau_{1}\left(n\left(w-t_{1}\right)+n^{\prime}\left(w^{\prime}+t_{1}\right)+N-\bar{N}\right)\right]$
(73) $\displaystyle\mathrm{exp}\left[2\pi
t_{2}(n-n^{\prime})\right]\left[\frac{1}{2is+q+\bar{q}+1+k\left(w-t_{1}\right)}-\frac{1}{2is+q+\bar{q}-1+k\left(w-t_{1}\right)}\right]$
$\displaystyle\mathrm{exp}\left[-2\pi\tau_{2}\left(\frac{2s^{2}+1/2}{k-2}+\frac{n^{2}}{2k}-\frac{n^{\prime
2}}{2k}-\frac{k}{2}\left(w^{\prime}+t_{1}\right)^{2}+\frac{k}{2}\left(w-t_{1}\right)^{2}+N+\bar{N}-\frac{3k}{12\left(k-2\right)}\right)\right]$

We will compare this Lorentzian spectrum with the Euclidean Thermal AdS
spectrum (47). To begin, we list the coefficients multiplying $\tau_{1}$ in
the exponent

$\displaystyle TADS$ $\displaystyle:$
$\displaystyle\left[w_{1}(q-\bar{q})+nm^{\prime}+N-\bar{N}\right]$ (74)
$\displaystyle LADS$ $\displaystyle:$
$\displaystyle\left([n\left(w-t_{1}\right)+n^{\prime}\left(w^{\prime}+t_{1}\right)+N-\bar{N}\right]$
(75)

and $\tau_{2}$

$\displaystyle TADS$ $\displaystyle:$
$\displaystyle\left[\frac{2s^{2}}{k-2}-\frac{1}{4}-w_{1}\left(i(q-\bar{q})\mu+\frac{2\pi
i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+\frac{k}{2}w_{1}^{2}+\frac{k}{2}\left(\frac{\beta
n}{2\pi}\right)^{2}(\mu^{2}+1)\right]$ (76) $\displaystyle LADS$
$\displaystyle:$
$\displaystyle\left[\frac{2s^{2}}{k-2}-\frac{1}{4}+\frac{n^{2}}{2k}-\frac{n^{\prime
2}}{2k}-\frac{k}{2}\left(w^{\prime}+t_{1}\right)^{2}+\frac{k}{2}\left(w-t_{1}\right)^{2}\right]$
(77)

In the Lorentzian ADS, we need to set $n=n^{\prime}$ because of the constraint
arising from $t_{2}$ integration.

Starting with the TADS expression, we will take the zero temperature
$\beta\to\infty.$ From the last term in TADS we see that $n$ must be set to
zero. Also we replace $\frac{2\pi i}{\beta}m^{\prime}=E$ a real number which
involves a Wick rotation. Then, omitting obviously common terms in the
$\tau_{2}$ expression, we compare

$\displaystyle TADS$ $\displaystyle:$
$\displaystyle\left[-w_{1}\left(E+\frac{k}{2}w_{1}^{2}\right)\right]$ (78)
$\displaystyle LADS$ $\displaystyle:$
$\displaystyle\left[-\frac{k}{2}\left(w^{\prime}+t_{1}\right)^{2}+\frac{k}{2}\left(w-t_{1}\right)^{2}\right]$
(79)

Let us now consider the $\tau_{1}$ factors which requires us to compare

$w_{1}(q-\bar{q})\ {\rm and}\ n(w+w^{\prime})$ (80)

Thus, we can set $w_{1}=-(w+w^{\prime})$ since we know that $(\bar{q}-q)$ is
the angular momentum $n$. Finally, if $E=-k(w^{\prime}+t_{1})$, the exponents
in the partition function of Thermal AdS and Lorentzian AdS will be identical.
Thus, starting with TADS, we determine $w^{\prime}$ from $E$ as the integral
part. Then we write $-w_{1}=w^{\prime}+w$ \- and we get LADS.

Using these identifications, we can compare the density of states of the two
systems. The expression for the factor that gives rise to the density of
states in TADS rewritten using the above identifications is

$\Biggl{[}\int_{-\infty}^{\infty}\biggl{(}\tfrac{ds/\pi}{2is-
kw_{1}+E+1+q+\bar{q}}-\tfrac{ds/\pi}{2is-
kw_{1}+E-1+q+\bar{q}}\biggr{)}+\sum_{s\in
D_{q,\bar{q},w_{1}}^{n,m^{\prime}}}\Biggr{]}$ (81)

which we compare with

$\Biggl{[}\frac{1}{2is+q+\bar{q}+1+k\left(w-t_{1}\right)}-\frac{1}{2is+q+\bar{q}-1+k\left(w-t_{1}\right)}\Biggr{]}$
(82)

which are equal once we use the identfications. Because the denominators
become the same, the discrete series poles are at the correct location. Note
that because of the real number $t_{1}$ in the denominator, the j-values of
the discrete series are not integral (as is correct for the universal cover).

Thus, in conclusion we have shown that the spectrum of Lorentzian $AdS_{3}$
could have been extracted from the partition function of Thermal $AdS_{3}$ as
a zero temperature limit and Wick rotating.

###  3.2 The other Wick rotation

In this section, we will try to show that the spectrum of the Lorentzian BTZ
black hole as worked out in Nippanikar  can be obtained from the Euclidean BTZ
black hole by Wick rotation. In this section, we will explicitly use suffixes
$L$ for Lorentzian BTZ quantities and $E$ for those of the Euclidean BTZ.

Starting with the EBTZ currents (60), we first Wick rotate $r_{-}\to ir_{-}$
and set $w_{1}=0$, which is the winding number along the Euclidean time
direction $\hat{t}$, to get

$\displaystyle J^{(2)}_{\pm,0}$
$\displaystyle=\frac{-1}{2\Delta_{\mp}}\left[\frac{m^{\prime}r_{-}+kr_{+}r_{-}^{2}n_{E}-i(q-\bar{q})_{E}(r_{+}^{2}-r_{-}^{2})}{r_{+}}\pm\left(-m^{\prime}-kr_{+}r_{-}n_{E}\right)\right]\,,$
(83a)
$\displaystyle=\frac{-1}{2r_{+}\Delta_{\mp}}\left[(m^{\prime}+kr_{+}r_{-}n_{E})(r_{-}\mp
r_{+})-i(q-\bar{q})_{E}(r_{+}^{2}-r_{-}^{2})\right]$ (83b) $\displaystyle
J^{\prime(2)}_{\pm,0}$
$\displaystyle=J^{(2)}_{\pm,0}+\frac{k}{2}n_{E}\Delta_{\mp}\,,$ (83c)

where $\Delta_{\pm}\coloneqq r_{+}\pm r_{-}$ and the subscript $E$ is placed
to help differentiate the EBTZ quantum numbers from those of Lorentzian BTZ in
what follows.

These may be compared with the form of current eigenvalues for Lorentzian BTZ
presented in Nippanikar , which is

$\displaystyle\tilde{J}^{(2)}_{\pm,0}$
$\displaystyle=\frac{-k}{2\Delta_{\mp}}\left(\tilde{E}\pm\tilde{L}\right)\,,$
(84a) $\displaystyle J^{\prime(2)}_{\pm,0}$
$\displaystyle=\tilde{J}^{(2)}_{\pm,0}-\frac{k}{2}\Omega\Delta_{\mp}$ (84b)

where $k\tilde{E}$ and $k\tilde{L}$ are related to the spacetime energy
$kE\in\mathbb{R}$ and angular momentum $kL\in\mathbb{Z}$ of the string via

$\displaystyle E$ $\displaystyle=\tilde{E}+(r_{+}^{2}+r_{-}^{2})\Omega\,,$
$\displaystyle L$ $\displaystyle=\tilde{L}-\Omega r_{+}r_{-}\,.$ (85)

The comparison makes it clear that the Wick rotation maps the Euclidean
quantum numbers to the Lorentzian ones via

$\displaystyle m^{\prime}$ $\displaystyle=-kL\,,$ $\displaystyle n_{E}$
$\displaystyle=-\Omega\,,$ $\displaystyle-i(q-\bar{q})_{E}$
$\displaystyle=\frac{k(\tilde{E}r_{+}+r_{-}\tilde{L})}{r_{+}^{2}-r_{-}^{2}}=-\tilde{J}^{(2)}_{+}-\tilde{J}^{(2)}_{-}$
(86)

It satisfactory that the Euclidean momentum $\Pi_{\hat{t}}=(q-\bar{q})_{E}$
Wick rotates to a (purely imaginary) continuous variable, while the angular
momentum $\Pi_{\phi_{\text{BTZ}}}=m^{\prime}$ and the spectral flow $n_{E}$
remain discrete after Wick rotation.

The Wick rotation of the density of states on the other hand does not proceed
in an as intuitively reasonable manner:

$\displaystyle(q+\bar{q})_{E}+ik[\frac{(\tilde{E}r_{-}+\tilde{L}r_{+})}{r_{+}^{2}-r_{-}^{2}}]$
$\displaystyle=(q+\bar{q})_{L}-k(\tilde{E}+\frac{1}{2}(M+1)\Omega)$ (87)
$\displaystyle(q+\bar{q})_{E}+i(\tilde{J}^{(2)}_{+}-\tilde{J}^{(2)}_{-})$
$\displaystyle=(q+\bar{q})_{L}-\frac{k}{2}(M+1)\Omega+\tilde{J}^{(2)}_{+}\Delta_{-}+\tilde{J}^{(2)}_{-}\Delta_{+}$
(88)

##  4 The B-field

If we have a gauge field $A_{a}$ in the bulk, its boundary value acts as a
source to a boundary operator $J^{\mu}$ in the gauge $A_{r}=0.$ The non-
normalizable modes of $A_{\mu}$, thus appear in the form $\int
d^{d}xA_{\mu}J^{\mu}$ in the boundary theory Hamiltonian. Under the residual
gauge symmetry, the boundary term changes as $\int\partial_{\mu}\Lambda
J^{\mu}=-\int\Lambda\partial_{\mu}J^{\mu}.$ Since gauge invariance ought to be
a symmetry, requiring the above variation to vanish obtains local current
conservation. Note that this involves a gauge transformation parameter
$\Lambda(x)$ which does not vanish at infinity
$\Lambda(x)=\Lambda(r=\infty,x).$

If we have a B-field in the bulk, then it can couple to two kinds of boundary
operators - to vector operators in the manner of $B_{\rho\mu}J^{\mu}$ and to a
rank two tensor operator in the CFT as $B_{\mu\nu}J^{\mu\nu}.$

The first kind of terms gauge transform as
$(\partial_{\rho}\Lambda_{\mu}-\partial_{\mu}\Lambda_{r})J^{\mu}$. If
$\partial_{\rho}\Lambda_{\mu}$ survives at the boundary, this means that
$\Lambda_{\mu}(\rho)$ does not vanish at the boundary, i.e., it is not a small
gauge transformation, and hence is not an allowed gauge transformation in the
bulk. Thus, the residual gauge symmetry of the B-field is $\Lambda_{r}(x)$
which gives rise to current conservation as in the case of the bulk vector
field.

Under a gauge transformation of $B$ the total gauge variation of the tensor
coupling evaluates to

$\int d^{d}x\Lambda^{\nu}\partial^{\mu}(J_{\mu\nu})=0$ (89)

where $\Lambda$ is the gauge parameter. Requiring that the above variation
vanishes gives an identity satisfied by the dual operator

$\partial_{\mu}J^{\mu\nu}=0$ (90)

Since $J^{\mu\nu}$ is antisymmetric, this equation cannot be interpreted as a
continuity equation and therefore it does not give a regular conservation law.
However, this is a generalized symmetry as presented in say, McGreevy .

We write out this integral for the boundary of Lorentzian $AdS_{3}$ to get

$\int d\phi(\Lambda_{\phi}J^{t\phi})|_{t=a}^{t=b}+\int dt(\Lambda_{t}J^{\phi
t})|_{\phi=0}^{\phi=2\pi}$ (91)

This quantity must vanish for all choices of $\Lambda_{t,\phi}.$ Requiring
this quantity to vanish in Lorentzian BTZ geometry forces $J_{t\phi}=0$ \-
simply take $\Lambda_{\phi}=\delta(\phi-\phi_{0}).$

In the Euclidean BTZ case, the above integral is over a torus on which the
vector field $\Lambda_{\nu}$ is not allowed to be topologically non-trivial
since it is a gauge tranformation parameter. This certainly seems to allow
$J_{\mu\nu}=J_{BH}\neq 0$ thereby explaining the non-normalizable mode
computed from the bulk.

We also note that in the Lorentzian case, the B-field vanishes at $r^{2}=M$
Esko . From the Lorentzian BTZ point of view, the finite gauge transformation
that is required to take us from one gauge $C=0$ to another $C\neq 0$ will
translate into either a time dependent source $A_{\phi}(t)$ or a non-trivial
(i.e, non-periodic) boundary Wilson line $A_{t}(\phi).$ Thus, we seem to be
driven to the conclusion that Euclidean and Lorentzian BTZ are unrelated from
the Boundary theory viewpoint.

##  5 Boundary Modular Invariance

In this section, we will discuss the boundary modular invariance of BTZ and
Thermal AdS partition functions; that is, the modular invariance of (47) with
respect to the parameter $T$. Since the dual theory is a CFT which lives on a
torus defined by $T$ and $\rho$, we can expect, based on the general
properties of 2D CFTs, that physical quantities of the boundary theory are
invariant under modular transformations of $T.$ While the worldsheet torus
partition function of the bulk is not a physical object in string theory, it
is nevertheless interesting to ask if we have modular invariance with respect
to the spacetime parameters $T.$

We recall that a spacetime torus compactification of string theory is
characterized by two modular parameters representing coordinates on the
$O(d,d,R)$ moduli space Polchinski . The first parameter is the complex
structure of the torus $T=T_{1}+iT_{2}.$ The second parameter is a Kahler
modulus which involves the volume of the torus and a NS-NS B-field
$\rho=B+i\rho_{2}.$ The T-duality group $O(2,2,Z)$ then appears as separate
modular invariances for $\rho$ and $T.$ However, the TADS and EBTZ geometries
are warped compactifications of the torus in which the torus volume varies
along the AdS radial coordinate. Thus, we expect only a $SL(2,Z)$ Buscher
Buscher  duality which changes the complex structure of the torus. This
T-duality of the bulk theory is valid order by order in the string coupling
$g_{s}$ and hence, we expect that the one loop amplitude which is the
integrated torus partition function (after including ghosts and internal CFTs)
to be invariant under full modular group action on $T$. On the other hand, the
bulk is a warped compactification on a torus and hence we do not expect
modular invariance under $\rho.$

We start with our observation that the TAdS and the EBTZ partition function
(46) can be expressed in the form

$Z_{g}(\rho,T;\tau)=\tau_{2}\sqrt{(k-2)}\int_{0}^{1}\mkern-18.0muds_{1}\int_{0}^{1}\mkern-18.0muds_{2}\frac{e^{\frac{2\pi}{\tau_{2}}Im[s_{1}\tau-
s_{2}]^{2}}}{\sqrt{\tau_{2}}|\vartheta_{1}(s_{1}\tau-s_{2}|\tau)|^{2}}\
\Sigma_{g}[s_{2},s_{1},T,\rho],$ (92)

where $\Sigma_{g}$ contains all the parameters referring to the boundary torus
and the subscript $g$ refers to the geometries - $TAdS_{3}$ or $EBTZ$.

For $TAdS$, we have:

$\Sigma_{TADS}(T)=\sum_{m^{\prime},n,w_{1},a}\tau_{2}\exp\biggl{\\{}-\pi\tau_{2}\bigg{[}-\frac{2\rho_{1}}{\rho_{2}}\left(\frac{T_{1}}{T_{2}}\left((s_{1}+w_{1})a-m^{\prime}n\right)+\frac{m^{\prime}(s_{1}+w_{1})}{T_{2}}-\frac{|T|^{2}}{T_{2}}na\right)\\\
+\frac{\rho_{2}}{T_{2}}\left|(s_{1}+w_{1})-nT\right|^{2}\bigg{]}+2\pi
i\left[\tau_{1}(nm^{\prime}+a(s_{1}+w_{1}))-as_{2})\right]\biggl{\\}}$ (93)

Before we study modular properties with respect to $T$, it helps to perform
Poisson resummation on $w_{1},n$ in that order, with $y,u$ being the
respective dual variables. This gives us the following expression

$\displaystyle\Sigma_{TADS}$ $\displaystyle=$
$\displaystyle\frac{1}{T_{2}k}e^{[2\pi
i(as_{2}+s_{1}y)-\frac{\pi}{k\tau_{2}T_{2}^{2}}(\bar{Y}T+\bar{U})(Y\bar{T}+U)]}$
(94) $\displaystyle=$ $\displaystyle\frac{\rho^{2}_{A}}{4\rho_{2}}e^{2\pi
i(as_{2}+s_{1}y)}\,\,e^{-\frac{\rho^{2}_{A}\pi}{4\rho_{2}\tau_{2}T_{2}}((\bar{Y}T+\bar{U})(Y\bar{T}+U))}$
(95)

where $Y=y+a\tau$ and $U=u+m^{\prime}\tau$ and $T=T_{1}+iT_{2}$ and
$\tau=\tau_{1}+\frac{\rho_{1}}{\rho_{2}}\tau_{2}$ and
$\bar{\tau}=\tau_{1}-\frac{\rho_{1}}{\rho_{2}}\tau_{2}.$

In this manner of writing

$Z_{TADS}(T)=\left\\{\frac{\sqrt{k-2}\ e^{\frac{2\pi}{\tau_{2}}Im[s_{1}\tau-
s_{2}]^{2}}}{\sqrt{\tau_{2}}\ |\vartheta_{1}(s_{1}\tau-
s_{2}|\tau)|^{2}}\right\\}\,e^{2\pi
i(as_{2}+s_{1}y)}\,\left\\{\frac{(2\pi)^{2}}{\rho_{2}}e^{-\frac{\pi(2\pi)^{2}}{\rho_{2}T_{2}\tau_{2}}(YT+U)(\bar{Y}\bar{T}+\bar{U})}\right\\}$
(96)

where each factor in braces is modular invariant on the worldsheet by itself.
The second factor is the partition function of a spacetime torus with modular
parameter $T$ and Kahler parameter $\rho.$ It is coupled to the “radial
coordinate ” which is represented by the first factor via the Lagrange
multipliers $s_{1,2}.$ In particular, the modes $a,y$ that couple to
$s_{1},s_{2}$ in the Thermal AdS case 2.2 are related to the momentum and
winding along the $\phi$ circle which contracts to zero size at $r=0.$

###  5.1 Poincaré series

Under $T\to T+1$, the above expression is easily seen to be invariant if we
also replace $w_{1}\to w_{1}+n,m^{\prime}\to m^{\prime}-a.$ Under a S-modular
transformation $T\to-\frac{1}{T}$, we get

$Z_{TADS}(T)=\sqrt{k-2}\frac{e^{\frac{2\pi}{\tau_{2}}Im[s_{1}\tau-
s_{2}]^{2}}}{\sqrt{\tau_{2}}|\vartheta_{1}(s_{1}\tau-
s_{2}|\tau)|^{2}}\,e^{2\pi
i(ms_{2}+s_{1}u)}\,\frac{(2\pi)^{2}}{\rho_{2}}e^{-\frac{\pi(2\pi)^{2}}{\rho_{2}T_{2}\tau_{2}}(Y\bar{T}+U)(\bar{Y}T+\bar{U})}.$
(97)

As discussed in 2.4, under this transformation - the partition function we
obtain is interpreted as that of a Euclidean BTZ black hole. In particular,
the time circle contracts to zero size. From this partition function, we can
deduce, by expanding in a q-series, that $m^{\prime},u$ are related to the
energy and momenta along the time circle. This motivates us to interpret the
modes which couple to $s_{1,2}$ as defining a contractible cycle in the bulk.

Under a general modular transformation $T\to\frac{AT+B}{CT+D}$, the torus
factor in the partition function becomes

$\displaystyle=\frac{1}{\rho_{2}}e^{2\pi
i(as_{2}+s_{1}y)}\,\,e^{-\frac{\pi}{\rho_{2}\tau_{2}T_{2}}((Y(A\bar{T}+B)+U(C\bar{T}+D))(\bar{Y}(AT+B)+\bar{U}(CT+D)))}$
(98) $\displaystyle=\frac{1}{\rho_{2}}e^{2\pi
i(as_{2}+s_{1}y)}\,\,e^{-\frac{\pi}{\rho_{2}\tau_{2}T_{2}}(((AY+CU)\bar{T}+BY+UD)((A\bar{Y}+C\bar{U})T+(B\bar{Y}+\bar{U}D))}$
(99)

We can define a new U and a new Y variable $Y^{\prime}=AY+CU\qquad
U^{\prime}=BY+UD$ which written in matrix form

$\begin{pmatrix}a^{\prime}\\\ y^{\prime}\\\ m^{\prime}\\\
u^{\prime}\end{pmatrix}=\begin{pmatrix}A&0&C&0\\\ 0&A&0&C\\\ B&0&D&0\\\
0&B&0&D\end{pmatrix}\begin{pmatrix}a\\\ y\\\ m\\\ u\end{pmatrix}$ (100)

allows us to see that the partition function changes to

$e^{2\pi
i\left[(a^{\prime}D-Cm^{\prime})s_{2}+s_{1}(-Cu^{\prime}+Dy^{\prime})\right]}e^{-\frac{\pi}{\rho_{2}\tau_{2}T_{2}}((Y^{\prime}\bar{T}+U^{\prime})(\bar{Y^{\prime}}T+\bar{U^{\prime}}))}$
(101)

Note that note that only $C,D$ elements of the modular matrix appears in the
partition function. To obtain a boundary modular invariant partition function,
we can sum over the integers $A,B,C,D$ \- but this naive sum will be infinite.

We therefore proceed as follows. It is manifest that we have modular
invariance with respect to $T.$ Performing $T\to T+1$ can be cancelled by

$U\to U-Y,\ \ \ \begin{pmatrix}A&B\\\
C&D\end{pmatrix}\to\begin{pmatrix}A&B+A\\\ C&D+C\end{pmatrix}$ (102)

while $T\to-\frac{1}{T}$ requires

$U\to Y\ \ Y\to-U,\ \ \ \begin{pmatrix}A&B\\\
C&D\end{pmatrix}\to\begin{pmatrix}-B&A\\\ -D&C\end{pmatrix}$ (103)

Suppose $C,D$ are mutually prime. Then, Bezout’s identity

$1=C(x-kD)+D(y-kC)$ (104)

states that given one solution $x,y$ of $1=Cx+Dy$ (which certainly exists) all
solutions are of the above form. We can interpret $A=x-kD$ and $B=y-kC$ giving
us an $SL(2,\mathbb{Z})$ matrix. Since $A,B$ do not appear in the partition
function, summing over $k$ will give us an infinity. Thus, we can simply set
$k=0$ and sum over $C,D$ mutually prime to obtain an invariant expression.
From eqns (102) and (103), we see that this set is closed under $T\to T+1$ and
$T\to-\frac{1}{T}.$

The string theory torus amplitude is obtained by including an internal CFT and
the worldsheets ghosts and finally integrating over the worldsheet torus
moduli.

At this stage, we should review our expectations about boundary modular
invariance. We expect the following two properties to hold for the boundary
partition function

  * •

It should be invariant under $T\to\frac{AT+B}{CT+D}$

  * •

It should exhibit a factorization property

$Z(T)=\sum_{n}F_{N}(T)\bar{F}_{N}(\bar{T}).$

These expectations originate from the conformal invariance of the dual.

The string theory torus amplitude can be regarded as the one-loop contribution
to the free energy of the boundary theory. This is because, by analogy with
field theory, it is the sum over one loop, connected, vacuum diagrams.

Thus, we may write a boundary modular invariant expression for the partition
function by first exponentiating the integrated amplitude and then summing
over $SL(2,\mathbb{Z})$ images:

$\displaystyle Z_{CFT}(T,\rho)=e^{-\beta
F_{CFT}(T,\rho)}=\sum^{\prime}_{C,D}\exp\biggl{[}\int\frac{d^{2}\tau}{\tau_{2}}\int_{0}^{1}d^{2}s\frac{(2\pi)^{2}\sqrt{k-2}}{\rho_{2}\sqrt{\tau_{2}}}\frac{e^{\frac{2\pi}{\tau_{2}}Im[s_{1}\tau-
s_{2}]^{2}}}{|\vartheta_{1}(s_{1}\tau-s_{2}|\tau)|^{2}}\times$ (105)
$\displaystyle
Z_{int}(\tau)\,Z_{gh}(\tau)\sum_{a^{\prime}m^{\prime},y^{\prime},u^{\prime}}e^{2\pi
i\left[(a^{\prime}D-Cm^{\prime})s_{2}+s_{1}(-Cu^{\prime}+Dy^{\prime})\right]}e^{-\frac{\pi}{\rho_{2}\tau_{2}T_{2}}((Y^{\prime}\bar{T}+U^{\prime})(\bar{Y^{\prime}}T+\bar{U^{\prime}}))}\biggl{]}$

where the prime on the summation restricts $C,D$ to mutually prime values.
Each term in this summation is an AdS background or BTZ type background which
depends on the torus mode that couples to the radial mode $s$. This series is
akin to the Farey-tail expansion of the elliptic genus discussed in DMMV  (see
also MM ).

While this expression has the desired invariance, it does not manifestly
exhibit the factorization property. Indeed, it is not clear that we even have
a factorization property - especially since the summation and integration
involves delicate issues of regularization and order of limits.

The quantity obtained is the stringy one loop contribution to the CFT
effective action (at finite temperature). We can divide the log of the
partition function by the temperature $T_{2}$ to obtain the Free energy. We
emphasize that this answer is obtained by keeping the prefactor $\rho_{2}$
untouched. If we unpack $\rho_{2}=kT_{2}$ and keep $k$ fixed, then it is not
clear that we can achieve modular invariance.

###  5.2 Phase transition

Since the $Z_{CFT}$ is boundary modular invariant, we may interpret the
partition function along the lines of DMMV . Increasing the temperature, while
keeping the chemical potential fixed, amounts to rescaling $T\to\lambda T.$
Beyond a certain value of $\lambda$, we will exit a fundamental domain for
$T.$ We then have to perform a modular transformation to return to the
fundamental domain. In general, this changes the contractible cycle. If the
starting point was understood as a tower of excitations over a TADS geometry
(say, defined by the A-cycle of the boundary torus being contractible), we
will transit to a black hole interpretation since the contractible cycle will
be a linear combination of the A- and B-cycles. Thus, phase transitions, if
any, can be expected to appear at the boundary of the fundamental domain of
the modular parameter $T$.

This condition agrees with that we obtained from regular thermodynamics 1.1.
The Hawking-Page transition occurred past
$\frac{\beta^{2}(1+\mu^{2})}{4\pi^{2}}=1$ which expressed in terms of the
modular parameter is precisely $|T|^{2}=1.$ This is one of the boundaries of
the usual ‘keyhole’ fundamental domain for the modular parameter $T$ of a
torus in the UHP. On further increasing the temperature, the further modular
transformations will almost never bring us back to the situation where the
spatial circle contracts (except maybe some number theoretic cases). So, there
are no further phase transitions.

This view of the above partition function is complementary to the usual
approach, wherein for each value of $T$, presumably a particular term in the
summation over $C,D$ contributes dominantly. As we vary $T$, the same dominant
term continues until we reach the edge of the fundamental domain. Then the
dominant term changes and we get a phase transition. In this view, we regard
$T$ as a complex parameter (unrestricted by identifications).

In either view, the black hole appears because we modified our interpretation.
In the former approach the contractible changed its meaning while in the
latter view the dominant term changed. This does not give us a physical
(dynamical) picture of what produces the phase transition similar to the
manner in which Landau-Ginzburg free energies carry no information about the
physical interaction mechanisms driving a phase transition. Of course, since
we are describing equilibria, a dynamical reason is not to be expected.

##  6 Summary

In this article, we began by reworking the analysis in Malstrom  along the
conventional lines of AdS/CFT correspondence. This approach highlights the
role played by the volume parameter $\rho_{2}$ of the boundary torus. We
showed that the AdS orbifolding with parameter $\frac{-1}{T}$ produces a BTZ
black hole with a rescaled boundary volume. This also allowed us to observe
that the EBTZ and Thermal AdS differ in the sense that the VEV of the operator
dual to the $B_{NS}$ field are not equal Berkooz:2007fe . Then, we revisited
the CFT torus partition functions of Thermal AdS. We first showed how to
manipulate the partition function so that all dependence on the temperature
and chemical potential can be isolated. This form of the partition function
then allowed us to show how the same partition function with reinterpreted
parameters encodes the states of the Euclidean BTZ CFT. We also showed how
Wick rotations can be performed on the Thermal AdS partition function which
directly gave us the partition function of the Lorentzian AdS worked out
earlier in Israel . Somewhat surprisingly, the analogous Wick rotation of the
BTZ works out painlessly and gives the spectrum argued for in Nippanikar . We
were then able to write down a formula for the bulk partition function that is
actually invariant under boundary modular transformations.

The latter allowed us to conjecture that the Hawking-Page transition arises at
the boundary of a fundamental domain for the boundary modular parameter. This
transition seems to be quite insensitive to the internal CFT required in a
critical string theory as well. It must be noted that the backgrounds
appearing in the Poincaré series cannot be interpreted as microstates since
most terms are Euclidean black holes with finite entropy.

We note that this Boundary modular invariance can be regarded in two distinct
ways. From the perspective of pure 3D gravity, which could have a boundary
dual, the NS-NS B-field does not exist as a degree of freedom. Thus, the
modular invariance proof should not depend on the parameter $\rho_{1,2}.$ In
fact, there will be no complex boundary degree of freedom dual to $\rho.$ On
the other hand, if we regard the geometries as part of a string theory, then
the normalizable mode of the B-field becomes a boundary datum and the boundary
modular invariance will involve the Kahler modulus $\rho=B_{NS}+i\rho_{2}$ as
well. These two points are nicely compatible with the Poincaré series form for
the bulk partition function.

From the vantage of AdS/CFT it has been argued that we could/should treat the
cosmological constant as a boundary datum as well - dual to pressure in the
boundary theory Karch . We note that the the volume parameter $\rho_{2}$ is
the natural thermodynamic dual of pressure and is also proportional to the
level $k$. We have seen that to ensure modular invariance, boundary
S-transformations on the modular parameter $T$ must be supplemented by a
rescaling of the cosmological constant as well. This is consistent with the
(19) where we see that the volume parameters of EBTZ and TADS are rescaled by
$|T|^{2}.$

As we have noted, the Lorentzian and the Euclidean BTZ differ not only by the
Wick rotation but also by a large gauge transformation of the B-field. The
bulk B-field gives rise to a generalized symmetry of the boundary system. In
particular, the D1/D5 system which is a gauge theory living in 1+1 dimensions
with fundamental matter must possess such symmetries and therefore interesting
phases. The 1-form gauge symmetry of the B-field gives rise to a conservation
law of a stringy operator in the boundary theory McGreevy . It will be
interesting to understand the difference between the states of the boundary
theory corresponding the Lorentzian and Euclidean BTZ black holes from this
viewpoint.

Another interesting calculation is to perform the integration over $\tau$ in
the boundary partition function (105) and thereby determine the dominant
saddle, if any. This will however not answer why we need to change the
interpretation of the geometry (in terms of the contractible cycle) beyond the
Hawking-Page transition. An idea in this context will be to consider the
nature of the statistical fluctuations using a statistical analogy. In the
Ising model, at high temperatures the low lying excitations are described by
spin waves whereas the low temperature excitations are individual spins. In
terms of these effective degrees of freedom, we have statistically small
fluctuations around the dominant saddle (‘ground state’) - which is a
“condensate” of spins at low temperature and a condensate of “spin-waves” at
high temperature. Perhaps a similar expectation is reasonable - the effective
excitations change across the Hawking-Page transition. At low temperatures,
the momentum modes around the thermal circle are the important fluctuations
while at high temperatures, the effective degrees of freedom involve some
winding along a spatial direction as well. This picture will clearly not be
relevant for the Lorentzian spacetimes.

Finally, yet another way of writing the torus factor (93) is of some interest.
String theory suggests that the last factor can be rewritten

$e^{-\frac{\pi}{\rho_{2}\tau_{2}T_{2}}((Y^{\prime}\bar{T}+U^{\prime})(\bar{Y^{\prime}}T+\bar{U^{\prime}}))}=e^{-V^{T}MV}$
(106)

where the metric $M$ (on the “ charge lattice”) transforms covariantly under
the boundary modular transformations. $M$ in our case, will be a $4\times 4$
dimensional matrix. We can imagine Wick rotating this matrix corresponding to
Wick rotating the boundary time variable. It will be of interest to ask
whether the Poincaré series survives the Wick rotation and explore its
spacetime meaning.

##  Appendix A Convention used in the thermal $AdS_{3}$ partition function

The aim of this appendix is to justify the choice
$U_{n,m}=\frac{\beta}{2\pi}(\mu-i)(n\tau-m)$ in (40) which differs from that
in MO2  by complex conjugation. We begin with the action (50) for the
$SL(2,\mathbb{R})$ WZW model and perform the Wick rotation
$-it_{\text{global}}=t\in\mathbb{R}$ to obtain the Euclidean $AdS_{3}$ action.
Next, we switch from Lorentzian to Euclidean worldsheet using
$(z,\bar{z})=(e^{ix^{+}},e^{ix^{-}})$. These changes result in
$iS_{\text{WZW}}[g]=-S_{E}[g]$ where

$S_{E}[g]=\frac{k}{\pi}\int
d^{2}z\;\bigl{[}\partial\rho\bar{\partial}\rho+\partial
t\bar{\partial}t\cosh^{2}\rho+\partial\phi\bar{\partial}\phi\sinh^{2}\rho+i\sinh^{2}\rho\left(\bar{\partial}t\partial\phi-\bar{\partial}\phi\partial
t\right)\bigr{]}$ (107)

is the Euclideanised worldsheet action. We have set $C=0$ because of arguments
presented in Sections 2.2 and 2.3. Also,
$d^{2}z\coloneqq\frac{dzd\bar{z}}{(-2i)}$ so that $\int
d^{2}z=4\pi^{2}\tau_{2}$ is the area of the worldsheet torus defined by
$(z,\bar{z})\sim(z+2\pi,\bar{z}+2\pi)$ and
$(z,\bar{z})\sim(z+2\pi\tau,\bar{z}+2\pi\bar{\tau})$. The integrand in (107)
may be rewritten in terms of

$v=e^{i\phi}\sinh\rho\,,\qquad\bar{v}=e^{-i\phi}\sinh\rho\,,\qquad\xi=t-\ln\cosh\rho\,,$
(108)

upto a divergence of the vector field
$(X^{z},X^{\bar{z}})=(-i\bar{\partial}\phi\ln\cosh\rho,i\partial\phi\ln\cosh\rho)$
as

$\partial\xi\bar{\partial}\xi+\left(\bar{\partial}\bar{v}+\bar{\partial}\xi\bar{v}\right)\left(\partial
v+\partial\xi v\right)\,.$ (109)

Note that this expression for the Euclideanised worldsheet action differs from
that of MO2  by $\partial\leftrightarrow\bar{\partial}$. This is equivalent to
a sign flip of the B-field, thereby leading to opposite chiralities of the WZW
currents relative to that in MO2 . With this difference, the same derivation
as in MO2  leads to the partition function (40) with the aforementioned choice
of $U_{n,m}$.

##  Appendix B Expansion of the partition function

To rewrite the form (46) of the thermal ${\text{AdS}}_{3}$ partition function
in a manner that displays the conformal dimensions, we begin by expanding the
$|\vartheta_{1}|^{-2}$ factor. The infinite product representation

$\vartheta_{1}(v|\tau)=-2q^{1/8}\sin\pi v\prod_{p=1}^{\infty}(1-e^{2i\pi
v}q^{p})(1-q^{p})(1-e^{-2i\pi v}q^{p})\,,$ (110)

where $q=e^{2\pi i\tau}$, is useful to perform this expansion. Consider

$\frac{1}{|\vartheta(v|\tau)|^{2}}=\frac{(q\bar{q})^{-1/8}}{4\sin(\pi
v)\sin(\pi\bar{v})}\prod_{p=1}^{\infty}\frac{1}{(1-q^{p})(1-\bar{q}^{p})}\\\
\times\frac{1}{(1-e^{2iv\pi}q^{p})(1-e^{-2iv\pi}q^{p})(1-e^{-2i\bar{v}\pi}\bar{q}^{p})(1-e^{2i\bar{v}\pi}\bar{q}^{p})}\,,$
(111)

where $v=s_{1}\tau-s_{2}$ with $s_{1},s_{2}\in(0,1)$. Noting that
$0<{\text{Im}}(v)<\tau_{2}$, each factor in this product can be expanded into
a converging geometric series. For example, the prefactor expands to

$\displaystyle\frac{1}{4\sin(\pi v)\sin(\pi\bar{v})}$
$\displaystyle=\frac{e^{-2\pi
s_{1}\tau_{2}}}{(1-e^{2i\pi(s_{1}\tau_{1}-s_{2})}e^{-2\pi
s_{1}\tau_{2}})(1-e^{-2i\pi(s_{1}\tau_{1}-s_{2})}e^{-2\pi s_{1}\tau_{2}})}$
(112) $\displaystyle=e^{-2\pi
s_{1}\tau_{2}}\sum_{P_{+,0},P_{-,0}=0}^{\infty}e^{2i\pi(s_{1}\tau_{1}-s_{2})(P_{+,0}-P_{-,0})}e^{-2\pi
s_{1}\tau_{2}(P_{+,0}+P_{-,0})}\,.$

Similarly, the subsequent factors may be expanded to obtain

$\frac{1}{|\vartheta(s_{1}\tau-
s_{2}|\tau)|^{2}}=\sum_{q,\bar{q},N,\bar{N}}e^{2\pi is_{2}(q-\bar{q})+2\pi
i\tau_{1}[-s_{1}(q-\bar{q})+N-\bar{N}]-2\pi\tau_{2}[s_{1}(1-q-\bar{q})-\frac{1}{4}+N+\bar{N}]}\,,$
(113)

by introducing six summation variables
$P^{+}_{\pm,p},P^{-}_{\pm,p},P_{\pm,p}\in\\{0,1,2,\ldots\\}$ for each term in
the product labelled by $p\in\\{1,2,3,\ldots\\}$. For compactness of notation,
these infinitely many sums (together with those over $P_{\pm,0}$) are denoted
by $\sum_{q,\bar{q},N,\bar{N}}$ where $(q,\bar{q})$ and $(N,\bar{N})$ are
defined as

$\displaystyle q$
$\displaystyle=P_{+,0}+\sum_{p=1}^{\infty}(P^{+}_{+,p}-P^{-}_{+,p})\,,$
$\displaystyle\bar{q}$
$\displaystyle=P_{-,0}+\sum_{p=1}^{\infty}(P^{+}_{-,p}-P^{-}_{-,p})\,,$ (114a)
$\displaystyle N$
$\displaystyle=\sum_{p=1}^{\infty}p(P_{+,p}+P^{+}_{+,p}+P^{-}_{+,p})\,,$
$\displaystyle\bar{N}$
$\displaystyle=\sum_{p=1}^{\infty}p(P_{-,p}+P^{+}_{-,p}+P^{-}_{-,p})\,.$
(114b)

Substituting this expansion (113) into (46) and integrating over $s_{2}$
simply introduces a Kronecker delta $\delta_{a,q-\bar{q}}$ that collapses the
sum over $a\in\mathbb{Z}$. At this stage, the identity

$e^{B^{2}/4A}=\sqrt{\frac{A}{\pi}}\int_{-\infty}^{\infty}\mkern-18.0muds\;e^{-As^{2}+Bs}$
(115)

with $A=\frac{4\pi\tau_{2}}{k-2}$ and $B=4\pi is_{1}\tau_{2}$ may be used to
introduce an integral over the variable $s$. (This variable shall label the
quadratic Casimir of continuous series representations.) This linearises the
exponent in $s_{1}$ facilitating its integration:

$2\pi\tau_{2}\int_{0}^{1}ds_{1}\;e^{-2\pi\tau_{2}s_{1}\left[-2is+kw_{1}-i(q-\bar{q})\mu-\frac{2\pi
im^{\prime}}{\beta}-\frac{\mu k\beta n}{2\pi}+1-q-\bar{q}\right]}\\\
=\tfrac{e^{-2\pi\tau_{2}\left[-2is+kw_{1}-i(q-\bar{q})\mu-\frac{2\pi
im^{\prime}}{\beta}-\frac{\mu k\beta n}{2\pi}+1-q-\bar{q}\right]}}{2is-
kw_{1}+i(q-\bar{q})\mu+\frac{2\pi im^{\prime}}{\beta}+\frac{\mu k\beta
n}{2\pi}-1+q+\bar{q}}-\tfrac{1}{2is-kw_{1}+i(q-\bar{q})\mu+\frac{2\pi
im^{\prime}}{\beta}+\frac{\mu k\beta n}{2\pi}-1+q+\bar{q}}\,.$ (116)

Using this result in the expansion of the partition function and then
performing the change of variables

$\tilde{w}_{1}=w_{1}+1\,,\qquad\tilde{N}=N-q\,,\qquad\tilde{\bar{N}}=\bar{N}-\bar{q}\,,\qquad\tilde{s}=s-\frac{i(k-2)}{2}$
(117)

only in the first term arising from (116), results in the expression

$\displaystyle Z(\beta,\mu;\tau,\bar{\tau})$
$\displaystyle=\sum_{m^{\prime},n,w_{1}\in\mathbb{Z}}\sum_{q^{\prime},\bar{q}^{\prime},N,\bar{N}}\Biggl{\\{}\int_{-\infty-\frac{i(k-2)}{2}}^{\infty-\frac{i(k-2)}{2}}\frac{ds}{\pi}\sum_{P_{\pm,0}=-\infty}^{0}\tfrac{1}{2is-
kw_{1}+\left(i(q-\bar{q})\mu+\frac{2\pi i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+1+q+\bar{q}}$ (118)
$\displaystyle-\int_{-\infty}^{\infty}\frac{ds}{\pi}\sum_{P_{\pm,0}=0}^{\infty}\tfrac{1}{2is-
kw_{1}+\left(i(q-\bar{q})\mu+\frac{2\pi i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)-1+q+\bar{q}}\Biggr{\\}}$ $\displaystyle\times e^{2\pi
i\tau_{1}\left[w_{1}(q-\bar{q})+nm^{\prime}+N-\bar{N}\right]}$
$\displaystyle\times
e^{-2\pi\tau_{2}\left[\frac{2s^{2}}{k-2}-\frac{1}{4}-w_{1}\left(i(q-\bar{q})\mu+\frac{2\pi
i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+\frac{k}{2}w_{1}^{2}+\frac{k}{2}\left(\frac{\beta
n}{2\pi}\right)^{2}(\mu^{2}+1)+i\mu m^{\prime}n+\frac{i\beta
n(q-\bar{q})}{2\pi}(1+\mu^{2})+N+\bar{N}\right]}\,,$

for the partition function. The fact that the sum over $P_{\pm,0}$ in the
first line runs over non-positive integers is a consequence of the
aforementioned change of variables. Finally, adding and subtracting

$\int_{-\infty}^{\infty}\frac{ds}{\pi}\sum_{P_{\pm,0}=-\infty}^{0}\frac{1}{2is-
kw_{1}+\left(i(q-\bar{q})\mu+\frac{2\pi i}{\beta}m^{\prime}+\frac{\mu k\beta
n}{2\pi}\right)+1+q+\bar{q}}$ (119)

in the braces results in an integral over a closed contour that collects
residues of the poles located at
$\frac{1}{2}+is=\frac{k}{2}w_{1}-\frac{1}{2}\left(i(q-\bar{q})\mu+\frac{2\pi
i}{\beta}m^{\prime}+\frac{\mu k\beta n}{2\pi}\right)-\frac{1}{2}(q+\bar{q})$
provided that they lie inside to contour of $s$-integration i.e.,
$-\frac{(k-2)}{2}<\mathfrak{I}(s)<0$. The sum over these residues shall count
the degeneracies of discrete series representations and the rest of the terms
in braces will become the density of states for continuous series
representations. This is precisely the claimed expansion (47) that allows us
to read off the conformal dimensions.

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