aainstitutetext: Indian Institute of Science Education and Research Tirupati,
Karakambadi Road, Mangalam, Tirupati, Andhra Pradesh, Indiabbinstitutetext:
Indian Institute of Science Education and Research Mohali,
Sector 81, Knowledge City, Mohali, Punjab, India

# The BTZ black hole spectrum and partition function

Omkar Vinayak Nippanikar a Aditya Sharma b K. P. Yogendran
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this article, we revisit the spectrum of the Lorentzian BTZ black hole
conformal field theory. Building on a detailed analysis of geodesics, we
identify a complete set of states for the harmonic analysis. We then
demonstrate that the CFT spectrum is, plausibly, that of $AdS_{3}$ by
rewriting the partition function of the latter CFT.

###### Keywords:

BTZ black hole, geodesics, hyperbolic basis, modular invariance, partition
function, closed timelike curves, 2 WZW model, CFT

††arxiv: 2112.11253

##  1 Introduction

The BTZ black hole is a solution of 3D pure gravity with a negative
cosmological constant. It also arises in string theory as a near-horizon
region of a non-extremal system of 1- and 5- branes. This geometry has no
curvature singularity because btz  it is a discrete quotient of 2. The work of
mog ; mog2  provided a complete understanding of the spectrum of the $AdS_{3}$
(2) string sigma model. In particular, the authors argued that additional
spectrally flowed representations of the 2 Kač-Moody algebra form a necessary
part of the string spectrum. These representations were necessary to lift the
bound on the mass of string states in $AdS_{3}$, otherwise imposed by the
absence of ghosts. The ‘long strings’ were then necessary to obviate the
restriction on spacetime energy of spectrally flowed discrete series
representations of the 2 conformal field theory (CFT) (while keeping the
internal conformal weights fixed).

In kpy , the spectrum of the 2D Lorentzian black hole, which is a coset CFT of
2, was studied. In this case, even point-like states of the string theory were
obtained after spectral flow with the spectral flow parameter being identified
with the spacetime energy. The ‘spacelike’ long strings appeared when the
energy increased above the mass parameter of the state. The spectrum so
constructed was argued to be consistent with an earlier proposal Israel  for
the partition function of this Lorentzian black hole.

If we adopt the view that the black hole is a highly excited state (or density
matrix) of the global $AdS_{3}$ string theory, then all the representations of
the latter carry over as states of the black hole geometry. However, since the
time coordinates of the two spacetimes do not obviously “rotate” into each
other, this continuation is not immediate. In fact, the twisted sectors of the
BTZ orbifold arise from spectral flow along the orbifolding direction, which
is different from the direction of spectral flow in the $AdS_{3}$ theory.
Thus, it is not obvious if the twisted sector states of the BTZ black hole can
be viewed as being states of the spectrally flowed sectors of the $AdS_{3}$
theory.

We know that in the case of $AdS_{3}$, the principal continuous series arose
by quantising spacelike geodesics. In this case also, we expect to see the
same, but there is a question of representation-theoretic interest. The
principal continuous representations of 2, when written in the basis Mukunda
that diagonalizes the ‘hyperbolic’ generator $J^{(2)}$, contain two
eigenvectors
$\mathchoice{\left|\lambda,\pm\right\rangle}{|\lambda,\pm\rangle}{|\lambda,\pm\rangle}{|\lambda,\pm\rangle}$
for each real eigenvalue $\lambda$ of $J^{(2)}$. In the 2D black hole kpy ,
the need for this doubling was associated with the presence of two distinct
asymptotic regions. We can therefore try to understand the need for doubling
of the spectrum in terms intrinsic to the BTZ geometry.

Finally, the extended BTZ geometry has regions containing closed timelike
curves (CTCs). An important puzzle that is not yet resolved is how to handle
these regions in string theory. In NS , it was suggested that these regions
maybe be fully excised by using involutions in 2. However, the question of
consistency can be considered settled only if one can exhibit a modular
invariant partition function. We can expect that analysing the geodesics from
the point of view of the WZW model could shed some light on these questions.

This document is organised as follows. After a brief recap of the geometry of
the BTZ black hole as a discrete quotient of the 2 manifold, we present a
detailed study of the geodesics of the black hole. In particular, we use this
to identify a geodesically complete spacetime. We then examine the effect of
certain discrete symmetries to understand the multiplicities of
representations upon quantisation. We follow this up with a study of twisted
sectors and the Virasoro conditions, which give us information about the
possible states that make up the spectrum of this CFT. We then propose that
the $AdS_{3}$ partition function of Israel  contains all these identified
states in a manner that is consistent with the orbifold CFT. Finally, we
conclude with a summary and discussion. Special cases of the non-rotating and
extremal black holes are discussed in Appendix B. Other appendices also
contain supplementary material for a self-contained presentation.

##  2 BTZ as an orbifold

We recap results from btz ; NS ; kraus  about the BTZ geometry which is
defined as the spacetime with the metric

$ds^{2}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{\ell^{2}r^{2}}dt^{2}+\frac{\ell^{2}r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}dr^{2}+r^{2}\left(d\phi-\frac{r_{+}r_{-}}{\ell
r^{2}}dt\right)^{2}\,,$ (1)

where the coordinate $\phi$ is periodic with period $2\pi$. Henceforth, we set
the $AdS_{3}$ radius $\ell=1$. If needed, this may be restored by the
transformation $(ds,r,r_{\pm},t)\mapsto\ell(ds,r,r_{\pm},t)$. The metric above
suggests that the coordinate $r^{2}$ has the range $[0,\infty)$. However, as
we will see, most geodesics freely pass to regions beyond. Thus, we are forced
to include additional regions to define a geodesically and causally complete
manifold. The surface $r^{2}=0$ is termed the BTZ singularity even though the
entire space has constant (negative) curvature, because the periodic $\phi$
coordinate becomes timelike in the region $r^{2}<0$. Hence this region
contains CTCs.

This spacetime may also be viewed as (a part of) a discrete quotient of the 2
group manifold, which is defined as the set of real $2\times 2$ matrices with
unit determinant:

$g=\begin{pmatrix}x_{1}-x_{2}&x_{3}-x_{0}\\\
x_{3}+x_{0}&x_{1}+x_{2}\end{pmatrix}\,,\qquad\qquad\det g=1\,.$ (2)

The quotienting relation is

$g\sim P_{+}\,g\,P_{-}\qquad\forall\ g\in SL(2,\mathbb{R})\,,\qquad
P_{\pm}=e^{\pi(r_{+}\mp r_{-})\sigma^{3}}\,.$ (3)

The metric (1) is equal to the invariant metric (Killing-Cartan form) of 2
provided we choose coordinates as described below. The special case of the
extremal black hole with $r_{+}=r_{-}$ requires other forms for the charts
malstrom  that we shall discuss in Appendix B.

The 2 group manifold may be fully covered once, by matrices of the form

$g=(-\mathbb{I})^{\epsilon_{1}}\,e^{\frac{1}{2}(r_{+}-r_{-})(t+\phi)\sigma^{3}}\,(i\sigma^{2})^{\epsilon_{2}}\,{\bf
p}\,e^{-\frac{1}{2}(r_{+}+r_{-})(t-\phi)\sigma^{3}}\,,$ (4)

where $\epsilon_{1,2}\in\\{0,1\\}$ and ${\bf p}$ is one of
$e^{\rho\sigma^{1}}$, $e^{i\rho\sigma^{2}}$ or
$\left(\begin{smallmatrix}1&0\\\ \rho&1\end{smallmatrix}\right)$. In the first
and last cases $\rho\in\mathbb{R}$ while in the second case
$\rho\in[-\frac{\pi}{4},\frac{\pi}{4}]$. The third choice for ${\bf p}$ is
applicable for 2 matrices with zero entries. It parametrises the horizons
$r^{2}=r_{\pm}^{2}$. The BTZ identification (3) implies that the
$\phi$-coordinate is $2\pi$-periodic. Hence, the $(t,\phi)$ coordinates define
the “boundary” cylinder at $\rho\to\infty$.

Following NS , we shall refer to these coordinate charts using the notation
$\pm D_{i}^{\pm}$ where $i\in\\{1,2,3,4\\}$, which is to be understood as
follows. The charts $D_{1,4}$ use ${\bf p}=e^{\rho\sigma^{1}}$ while $D_{2,3}$
use ${\bf p}=e^{i\rho\sigma^{2}}$. The $D_{3,4}$ charts are obtained by
setting $\epsilon_{2}=1$ while the $D_{1,2}$ charts have $\epsilon_{2}=0$. The
sign in front of the $D_{i}$ represents $(-\mathbb{I})^{\epsilon_{1}}$ while
the sign in the superscript denotes the sign of $\rho$. When written
explicitly, the chart $+D_{1}^{+}$ of 2 has matrices of the form

$g=e^{\frac{1}{2}(r_{+}-r_{-})(t+\phi)\sigma^{3}}\begin{pmatrix}\cosh\rho&\sinh\rho\\\
\sinh\rho&\cosh\rho\end{pmatrix}e^{-\frac{1}{2}(r_{+}+r_{-})(t-\phi)\sigma^{3}}\,.$
(5)

With this parametrisation and the substitution
$\cosh^{2}\rho=\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}$, the Killing-
Cartan form $ds^{2}=-\frac{1}{2}\text{Tr}(dg^{-1}\,dg)$ becomes the BTZ metric
(1). Hence, the $+D_{1}^{+}$ chart covers the region $r^{2}>r_{+}^{2}$ outside
the outer horizon.

We note here that global $AdS_{3}$ is a cover of 2 obtained by unwrapping the
timelike elliptic (compact) direction in 2. This is, however not the time
direction of the BTZ black hole.

The other 2 charts can also be mapped to other regions of the BTZ geometry.
The region $r_{-}^{2}<r^{2}<r_{+}^{2}$ between the inner and outer horizons is
parametrised by the $D_{2,3}$ charts with
$\cos^{2}\rho=\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}$. The $D_{4}$ charts
with $\sinh^{2}\rho=-\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}$ parametrise
the region $r^{2}<r_{-}^{2}$ inside inner horizon. These charts also include
the $r^{2}<0$ regions containing CTCs. Thus, the entire BTZ geometry (i.e.,
for all $r^{2}\in\mathbb{R}$) is fully covered once, by the atlas
$\Omega_{1}=D_{1}^{+}\cup D_{2}^{-}\cup(-D_{3}^{+})\cup(-D_{4}^{-})$. This
combination of charts ensures that the vector field
$\frac{\partial}{\partial\rho}$ is continuous. The atlas
$\Omega_{2}=D_{1}^{-}\cup D_{2}^{+}\cup D_{3}^{-}\cup D_{4}^{+}$ is defined to
consist of complementary charts obtained from $\Omega_{1}$ by $\sigma^{3}$
conjugation. Together with two more atlases $-\Omega_{1,2}$, a single cover of
the 2 group manifold includes four copies of the BTZ black hole.111The way
these atlases cover 2 is described in Section 4. On the other hand, if we
consider global $AdS_{3}$ which is the universal cover of 2 and then impose
the BTZ identification, we get a geometry with infinitely many copies of the
black hole and boundary components kraus .

##  3 Geodesics of the BTZ

The motion of a string on the BTZ geometry is governed by the action

$S_{\text{WZW}}[g]=-\frac{k}{8\pi}\int
d\tau\,d\sigma\,\text{Tr}\left(\partial_{a}gg^{-1}\partial^{a}gg^{-1}\right)+k\Gamma_{\text{WZ}}[g]\,.$
(6)

The Wess-Zumino term $\Gamma_{\text{WZ}}$ represents the pullback of the NS-NS
B-field to the world volume of the string. Due to the WZ term, it can be seen
that the right and left currents, defined as

$J_{-}=kg^{-1}\partial_{-}g\,,\qquad\qquad J_{+}=-k\partial_{+}gg^{-1}$ (7)

respectively, are chirally conserved i.e., $\partial_{\pm}J_{\mp}=0$. Here,
$x^{\pm}=\tau\pm\sigma$ and
$\partial_{\pm}=\frac{1}{2}(\partial_{\tau}\pm\partial_{\sigma})$. The
worldsheet stress tensor of this theory has components

$T_{\pm\pm}=\frac{1}{2k}\text{Tr}(J_{\pm}J_{\pm})\,.$ (8)

In the parametrisations (4), the action (6) is invariant under $t$ and $\phi$
translations. Let us denote the associated conserved quantities as $E$ and $L$
respectively. These will turn out to be the energy and angular momentum from
the point of view of asymptotic observer outside the outer horizon
($r^{2}>r_{+}^{2}$). When evaluated along geodesics (i.e., ‘collapsed’
strings) these are expressible in terms of the currents as

$kE=-\Delta_{-}J^{(2)}_{+}-\Delta_{+}J^{(2)}_{-}\,,\qquad\qquad
kL=-\Delta_{-}J^{(2)}_{+}+\Delta_{+}J^{(2)}_{-}\,.$ (9)

Here, $\Delta_{\pm}=r_{+}\pm r_{-}$ and the components are computed as
$J^{(2)}_{\pm}=\frac{1}{2}\text{Tr}(\sigma^{3}J_{\pm})$. However, for stringy
configurations (extended worldsheets), an additional total derivative term is
required in the definition of the angular momentum for consistency with the
level matching condition NS ; Esko ; RR ; GPS .

The Euler Lagrange equations of motion for point-like trajectories gives us
the geodesic equations on the BTZ geometry (1). These equations Cruz ; Troost1
are

$\displaystyle\dot{y}(\tau)^{2}$
$\displaystyle=4m^{2}\left(-y^{2}+\frac{\alpha}{m^{2}}y+\frac{\beta}{m^{2}}\right)\,,$
(10a) $\displaystyle\dot{t}+\dot{\phi}$
$\displaystyle=\frac{1}{(r_{+}-r_{-})}\left(\frac{Er_{+}+Lr_{-}}{r_{+}^{2}-r^{2}}+\frac{Er_{-}+Lr_{+}}{r^{2}-r_{-}^{2}}\right)\,,$
(10b) $\displaystyle\dot{t}-\dot{\phi}$
$\displaystyle=\frac{1}{(r_{+}+r_{-})}\left(\frac{Er_{+}+Lr_{-}}{r_{+}^{2}-r^{2}}-\frac{Er_{-}+Lr_{+}}{r^{2}-r_{-}^{2}}\right)\,,$
(10c)

where $y=r^{2}$, $\alpha=E^{2}-L^{2}+Mm^{2}$ and
$\beta=L^{2}M-\frac{1}{4}m^{2}J^{2}+ELJ$. It is perhaps worth mentioning a
pedagogical point. The radial equation (10a) has solutions with constant $y$
at roots of the quadratic on the right-hand side. However, these do not solve
the second-order geodesic equations unless $\dot{t}+\dot{\phi}$ or
$\dot{t}-\dot{\phi}$ vanish.

The equations of the WZW model amount to conservation of the left and right
currents. For pointlike trajectories, these imply the geodesic equations. That
is to say, the geodesic equations can be written as

$\partial_{\tau}^{2}g-\partial_{\tau}gg^{-1}\partial_{\tau}g=0\quad\Longleftrightarrow\quad\partial_{\tau}(g^{-1}\partial_{\tau}g)=0=\partial_{\tau}(\partial_{\tau}gg^{-1})\,.$
(11)

These equations are solved by any one parameter subgroup i.e.,
$g(\tau)=A\,\exp(m\tau T)B$ is a solution for any matrices $A,B\in$ 2 and
where $T$ is an element of the Lie algebra of 2. Thus, we can rewrite the
solutions as 2 matrices using the parametrisations (4). In the process of
uplifting the geodesics as 2 matrices, several sign choices have to be made.
These choices can be interpreted as representing the same BTZ geodesics in
different 2 regions/charts. Due to the symmetries of the BTZ geometry, one can
always shift $t$ and $\phi$ in the solution, which is equivalent to
multiplying the solution matrix by constant diagonal 2 matrices on the left
and right.

We will now discuss the various solutions to the geodesic equations. We also
focus on conditions that isolate geodesics which do not trespass into the
regions containing CTCs.

###  3.1 Timelike geodesics

In this case, the solution to (10a) is given by

$r^{2}(\tau)=\frac{\alpha}{2m^{2}}+\frac{\gamma}{2m^{2}}\sin(2m(\tau-\tau_{0})+c)\,,$
(12)

where we have defined $\alpha=E^{2}-L^{2}+m^{2}M$,
$\beta=L^{2}M+ELJ-\frac{1}{4}m^{2}J^{2}$ and
$\gamma^{2}=\alpha^{2}+4m^{2}\beta$. Without loss of generality, we will set
the integration constant $c=2m\tau_{0}$ to simplify various expressions, in
what follows.

These trajectories, in general, oscillate from outside the outer horizon to
inside the inner horizon. However, if $\beta<0$, the trajectories do not pass
through the singularity ($r^{2}=0$) into the region containing closed timelike
curves. If $\alpha<0$, the trajectories already start behind the singularity
at $\tau=0$. Imposing $\beta<0$ then ensures that these are prevented from
entering the region $r^{2}>0$.

Using the radial solution (12), we can integrate (10c) and (10b). The
resulting solutions can lifted as 2 matrices using (5). The lifts of timelike
geodesics are

$\left(\frac{\gamma}{m^{2}(r_{+}^{2}-r_{-}^{2})}\right)^{1/2}\begin{pmatrix}\frac{\sin(m\tau+\frac{\phi_{-}}{2})}{\sqrt{\tan\frac{\phi_{-}}{2}}}&\frac{\cos(m\tau-\frac{\phi_{+}}{2})}{\sqrt{\cot\frac{\phi_{+}}{2}}}\\\
\frac{\sin(m\tau+\frac{\phi_{+}}{2})}{\sqrt{\tan\frac{\phi_{+}}{2}}}&\frac{\cos(m\tau-\frac{\phi_{-}}{2})}{\sqrt{\cot\frac{\phi_{-}}{2}}}\end{pmatrix}\,,$
(13)

where the angles $\phi_{\pm}\in[0,\pi]$ are defined by
$\cot\phi_{\pm}=\frac{2m(Er_{\pm}+Lr_{\mp})}{(E^{2}-L^{2})\mp(r_{+}^{2}-r_{-}^{2})m^{2}}$.
Note that the normalisation factor written here is proportional to
$\sqrt{\sin\phi_{-}-\sin\phi_{+}}$. In particular, it may be purely imaginary.
In these cases, the matrix is to be interpreted only after dropping the
overall $i$ and flipping the sign of exactly one row or column (to retain
$\det=1$).

The conserved quantities evaluated along the solution (13) are

$T_{\pm\pm}=-\frac{k}{4}m^{2}\,,\qquad\qquad
J^{(2)}_{\pm}=\pm\frac{k}{2}m\left[\cot\left(\frac{\phi_{+}\mp\phi_{-}}{2}\right)\right]^{\pm
1}=-\frac{k(E\pm L)}{2(r_{+}\mp r_{-})}\,.$ (14)

Since $T_{\pm\pm}<0$, upon quantisation, we can expect that these will belong
to the discrete series representation of 2.

From the definition of $\cot\phi_{\pm}$, we see that special solutions occur
when $\phi_{\pm}=0$ or $\phi_{\pm}=\frac{\pi}{2}$. The condition
$\phi_{\pm}=0$ is equivalent to $E^{2}-L^{2}=\pm m^{2}(r_{+}^{2}-r_{-}^{2})$
and can be rewritten as

$J^{(2)}_{+}J^{(2)}_{-}=\mp kT_{\pm\pm}\,.$ (15)

These solutions satisfy a relation

$\cos[2(r_{\mp}t-r_{\pm}\phi)]=\frac{1}{2}\left(\frac{r_{+}^{2}-r_{-}^{2}}{r^{2}-r_{\pm}^{2}}\right)\,.$
(16)

The second set of special solutions which satisfy $\phi_{\pm}=\frac{\pi}{2}$
obey

$J^{(2)}_{+}=\mp J^{(2)}_{-}\quad\Longleftrightarrow\quad
r_{\pm}t-r_{\mp}\phi={\text{constant}}\,.$ (17)

These geodesics touch the horizon $r^{2}=r_{\pm}^{2}$ before turning back.
Clearly, the geodesics with $\phi_{-}=\frac{\pi}{2}$ (and arbitrary
$\phi_{+}$) do not explore the region with CTCs and indeed satisfy $\beta<0$.
In the further special case $\phi_{+}=\frac{\pi}{2}=\phi_{-}$, they also
remain between the horizons. The other set $\phi_{+}=\frac{\pi}{2}$ do not
necessarily satisfy $\beta<0.$

Each time the geodesic (13) crosses a horizon, one of the matrix entries
becomes zero (see chart (4)). Thus, there are three natural time scales that
appear in the solution (13) viz., the proper time taken to traverse:  (a) from
the past outer horizon to the future outer horizon $\Delta\tau_{+}$;  (b) from
the past inner horizon to the future inner horizon $\Delta\tau_{-}$;  (c)
between the outer and inner horizons $\Delta\tau_{0}$ .

The proper time interval to traverse from past horizon to future horizon
(while outside the outer horizon) is
$|m\Delta\tau_{+}|=|\phi_{+}-\frac{\pi}{2}|$. Similarly, the proper time taken
to traverse from past horizon to future horizon (while inside the inner
horizon) is $|m\Delta\tau_{-}|=|\phi_{-}-\frac{\pi}{2}|$. Also, the proper
time taken to traverse from outer horizon to inner horizon (while between the
horizons) is the lesser of
$|m\Delta\tau_{1}|=|\frac{\pi}{2}-\frac{1}{2}(\phi_{+}+\phi_{-})|$ and
$|m\Delta\tau_{2}|=\frac{1}{2}|\phi_{+}-\phi_{-}|$. Remarkably, all these
time-scales can be expressed in terms of the currents as

$\displaystyle\tan|m\Delta\tau_{\pm}|=\left|\frac{km}{2}\left(\frac{J^{(2)}_{+}\pm
J^{(2)}_{-}}{J_{+}^{(2)}J_{-}^{(2)}\mp kT}\right)\right|\,,$ (18a)
$\displaystyle\tan|m\Delta\tau_{1}|=\left|\frac{km}{2J^{(2)}_{-}}\right|\,,\qquad\qquad\tan|m\Delta\tau_{2}|=\left|\frac{km}{2J^{(2)}_{+}}\right|\,,$
(18b)

where $T=T_{\pm\pm}=-\frac{k}{4}m^{2}$ denotes the stress tensor evaluated
along the timelike geodesic (13). The fact that all these timescales are well
defined is consistent with the fact that the timelike geodesics cross all four
horizons at finite proper times.

###  3.2 Spacelike geodesics

As in $AdS_{3}$, we see that none of the timelike geodesics reaches the
boundary observer. We may expect spacelike geodesics to include such
scattering solutions.

Spacelike geodesics of the BTZ black hole were also considered in Troost1 . We
write the solutions as

$r^{2}(\tau)=\frac{-\alpha}{2m^{2}}\pm\frac{1}{4}\left(e^{2m\tau}+\frac{\alpha^{2}-4m^{2}\beta}{m^{4}}e^{-2m\tau}\right)\,,$
(19)

where now $\alpha=E^{2}-L^{2}-m^{2}M$ and
$\beta=L^{2}M+ELJ+\frac{1}{4}m^{2}J^{2}$. If $\alpha^{2}-4m^{2}\beta\neq 0$
then, this may be rewritten upto a shift of $\tau$ as

$\displaystyle r^{2}(\tau)$
$\displaystyle=\frac{1}{2m^{2}}\left(-\alpha\pm\sqrt{\alpha^{2}-4m^{2}\beta}\cosh
2m\tau\right)$ $\displaystyle{\text{if }}\alpha^{2}-4m^{2}\beta>0\,,$ (20a)
$\displaystyle r^{2}(\tau)$
$\displaystyle=\frac{1}{2m^{2}}\left(-\alpha\pm\sqrt{|\alpha^{2}-4m^{2}\beta|}\sinh
2m\tau\right)$ $\displaystyle{\text{if }}\alpha^{2}-4m^{2}\beta<0\,.$ (20b)

First, we observe that the geodesics (20b) with $\alpha^{2}-4m^{2}\beta<0$
necessarily cross into the region containing CTCs. For (20a) with the upper
sign choice, there is a minimum distance of approach
$r_{*}^{2}=\frac{-\alpha}{2m^{2}}+\frac{1}{2m^{2}}\sqrt{\alpha^{2}-4m^{2}\beta}$.
This is positive if $\alpha<0$ or $\beta<0$, whence the geodesic stays out of
the region with CTCs. Unlike timelike geodesics, there are solutions that stay
out of this region but do not satisfy $\beta<0$. Interestingly, the constraint
$\beta<0$ does not allow for values $|E|<\sqrt{m^{2}M}$ of the energy
irrespective of the angular momentum $L$. However, these values of energy are
allowed if $\alpha<0$ instead.

With the angular equations (10b) and (10c) integrated, we can uplift the
solutions (20a) as the 2 matrices

$\begin{split}\frac{\begin{pmatrix}e^{-m\tau}A_{11}+e^{m\tau}&e^{-m\tau}A_{12}+e^{m\tau}\\\
e^{-m\tau}A_{21}+e^{m\tau}&e^{-m\tau}A_{22}+e^{m\tau}\end{pmatrix}}{\sqrt{4m^{2}(r_{+}^{2}-r_{-}^{2})}}\,,\end{split}\qquad\begin{split}\begin{aligned}
A_{11}&=(L-mr_{+})^{2}-(E+mr_{-})^{2}\,,\\\
A_{12}&=(L+mr_{-})^{2}-(E-mr_{+})^{2}\,,\\\
A_{21}&=(L-mr_{-})^{2}-(E+mr_{+})^{2}\,,\\\
A_{22}&=(L+mr_{+})^{2}-(E-mr_{-})^{2}\,.\end{aligned}\end{split}$ (21)

The lift of (20a) with lower sign choice is obtained by flipping the signs of
all $A_{ij}$ followed by sign flips of exactly one row or column. The
worldsheet stress tensor and the conserved charges evaluated along this
solution are

$T_{\pm\pm}=\frac{k}{4}m^{2}\,,\qquad\qquad J^{(2)}_{\pm}=-\frac{k(E\pm
L)}{2(r_{+}\mp r_{-})}\,.$ (22)

Since $T_{\pm\pm}>0$, we can expect these geodesics to arise from the
principal continuous series of 2 upon quantisation.

The lifted solution can also be written purely in terms of these conserved
charges using
$k^{2}A_{ij}=(r_{-}^{2}-r_{+}^{2})(2J^{(2)}_{+}-(-1)^{i}km)(2J^{(2)}_{-}+(-1)^{j}km)$.
Clearly, special solutions occur at $(J_{\pm}^{(2)})^{2}=kT_{\pm\pm}$ when
$\alpha^{2}-4m^{2}\beta=0$. This implies that $r^{2}$ asymptotes to a constant
as $\tau\to-\infty$. We can see that these solutions satisfy

$\displaystyle e^{\pm 2(r_{+}+r_{-})(t-\phi)}$
$\displaystyle=\frac{r^{2}-r_{-}^{2}}{r^{2}-r_{+}^{2}}\,,$ $\displaystyle E+L$
$\displaystyle=\pm m(r_{+}-r_{-})\,,$ $\displaystyle J^{(2)}_{+}$
$\displaystyle=\mp\frac{k}{2}m\,;$ (23a) $\displaystyle e^{\pm
2(r_{+}-r_{-})(t+\phi)}$
$\displaystyle=\frac{r^{2}-r_{-}^{2}}{r^{2}-r_{+}^{2}}\,,$ $\displaystyle E-L$
$\displaystyle=\pm m(r_{+}+r_{-})\,,$ $\displaystyle J^{(2)}_{-}$
$\displaystyle=\mp\frac{k}{2}m\,.$ (23b)

For another special case $J^{(2)}_{+}=\pm J^{(2)}_{-}$, the diagonal or off-
diagonal entries become independent of $E$ and $L$ after a shift of $\tau$.
These solutions satisfy
$\alpha^{2}-4m^{2}\beta=(\alpha+2m^{2}r_{\mp}^{2})^{2}$ and similar to the
timelike case, these spacelike geodesics touch the horizons at
$r^{2}=r_{\mp}^{2}$, as seen from (19).

As for the timelike geodesics, we can calculate the time scales associated to
a spacelike geodesic with $\alpha^{2}-4m^{2}\beta>0$ (and upper sign choice in
(20a)). As before, let us denote the proper time taken to traverse:  (a) from
the past outer horizon to the future outer horizon as $\Delta\tau_{+}$;  (b)
from the past inner horizon to the future inner horizon as $\Delta\tau_{-}$;
(c) from outer horizon to inner horizon as $\Delta\tau_{0}$ . These time-
scales can be expressed in terms of the currents (22) as

$\displaystyle\tanh|m\Delta\tau_{\pm}|$
$\displaystyle=\left|\frac{km}{2}\left(\frac{J^{(2)}_{+}\pm
J^{(2)}_{-}}{J_{+}^{(2)}J_{-}^{(2)}\pm kT}\right)\right|\,,$ (24a)
$\displaystyle\tanh|m\Delta\tau_{0}|$
$\displaystyle={\text{min}}\left\\{\left|\frac{km}{2J_{+}^{(2)}}\right|,\left|\frac{km}{2J_{-}^{(2)}}\right|\right\\}\,,$
(24b)

where $T=T_{\pm\pm}=\frac{k}{4}m^{2}$ denotes the stress tensor (22) evaluated
along the spacelike geodesic. It is to be noted that the right hand side needs
to be less than unity for the corresponding time scale to be defined. This
will be the case, whenever the geodesic is such that it crosses the
appropriate horizons. We will return to this point in Section 4 where we
discuss the geometry of the solutions.

###  3.3 Null geodesics

The most general null geodesic of the BTZ black hole is

$\displaystyle\alpha r^{2}(\tau)$ $\displaystyle=\left(\alpha\tau+\sqrt{\alpha
r_{0}^{2}+\beta}\right)^{2}-\beta$ $\displaystyle{\text{if }}\alpha\neq 0\,,$
(25a) $\displaystyle r^{2}(\tau)$ $\displaystyle=2E(r_{+}\pm
r_{-})\tau+r_{0}^{2}$ $\displaystyle{\text{if }}\alpha=0\,,$ (25b)

where $\alpha=E^{2}-L^{2}$, $\beta=L(LM+EJ)$ and $r_{0}=r(0)$. When
$\alpha>0$, these geodesics reach the boundary of $AdS_{3}$. If $\alpha<0$,
they inevitably penetrate through to the region $r^{2}<0$ which contains CTCs.
If $\alpha>0$ they always penetrate inside the inner horizon because
${\text{min}}\\{\alpha(r^{2}(\tau)-r_{-}^{2})\\}=-(Er_{-}+Lr_{+})^{2}\leq 0$.
In addition to $\alpha>0$, if $\beta<0$ these geodesics do not pass into the
region $r^{2}<0$ containing CTCs. This maybe viewed as a bound
$\frac{|E|}{M}>|\frac{L}{J}|$ on the energy $E$ which is stronger than
$E^{2}>L^{2}$ (since $|J|<M$).

The lifts of null geodesics to 2 (upto a $\tau$ shift) are

$\displaystyle\frac{1}{\sqrt{|\alpha|(r_{+}^{2}-r_{-}^{2})}}$
$\displaystyle\begin{pmatrix}\alpha\tau-(Er_{-}+Lr_{+})&\alpha\tau+(Er_{+}+Lr_{-})\\\
\alpha\tau-(Er_{+}+Lr_{-})&\alpha\tau+(Er_{-}+Lr_{+})\end{pmatrix}$
$\displaystyle{\text{if }}\alpha>0\,,$ (26a)
$\displaystyle\frac{1}{\sqrt{|\alpha|(r_{+}^{2}-r_{-}^{2})}}$
$\displaystyle\begin{pmatrix}\alpha\tau-(Er_{-}+Lr_{+})&-\alpha\tau-(Er_{+}+Lr_{-})\\\
\alpha\tau-(Er_{+}+Lr_{-})&-\alpha\tau-(Er_{-}+Lr_{+})\end{pmatrix}$
$\displaystyle{\text{if }}\alpha<0\,.$ (26b)

The geodesics (25a) cross each of the inner and outer horizons exactly twice.
As in the case of spacelike geodesics (20a), null geodesics also reach the
boundary at a finite time ($t=0$, in particular). For this reason, we expect
that the null geodesics form a special case of spacelike geodesics.

The conserved quantities evaluated along the solution (26) are

$T_{\pm\pm}=0\,,\qquad\qquad J^{(2)}_{\pm}=-\frac{k(E\pm L)}{2(r_{+}\mp
r_{-})}\,.$ (27)

The time scales associated to these geodesics

$\displaystyle\Delta\tau_{\pm}$
$\displaystyle=\frac{k}{2}\left|\frac{J^{(2)}_{+}\pm
J^{(2)}_{-}}{J^{(2)}_{+}J^{(2)}_{-}}\right|\,,$ (28a)
$\displaystyle\Delta\tau_{0}$
$\displaystyle={\text{min}}\left\\{\left|\frac{k}{2J^{(2)}_{+}}\right|,\left|\frac{k}{2J^{(2)}_{-}}\right|\right\\}$
(28b)

are seen to be the $m\to 0$ limits of corresponding time scales (24) for
spacelike geodesics as well as (18) for timelike geodesics.

To summarise, we see that for timelike geodesics, a single condition $\beta<0$
keeps them out of the region containing CTCs. On the other hand, there are
spacelike geodesics which do not satisfy $\beta<0$ but nevertheless stay out
of this region.

##  4 Geometry of the solutions

$+D_{1}^{+}$$-D_{4}^{-}$$-D_{1}^{+}$$+D_{4}^{-}$$+D_{1}^{-}$$-D_{4}^{+}$$-D_{1}^{-}$$+D_{4}^{+}$$+D_{2}^{+}$$+D_{2}^{-}$$-D_{3}^{+}$$-D_{3}^{-}$$-D_{2}^{+}$$-D_{2}^{-}$$+D_{3}^{+}$$+D_{3}^{-}$$r^{2}=\mathrlap{r_{+}^{2}}$$r^{2}=\mathrlap{r_{-}^{2}}$$r^{2}=\mathrlap{0}$$+\Omega_{1}$$-\Omega_{2}$
Figure 1: Penrose diagram of the single cover of 2 group manifold

In this section, we shall pay close attention to the trajectories of geodesics
as they traverse various regions of the BTZ geometry. Using the matrices
representing each class of solutions, we can track their evolution through the
Penrose diagram of 2. This shows the manner in which the various regions and
copies of the BTZ geometry sit inside 2 (in this consideration, we ignore the
orientation of the affine parameter $\tau$). An issue in the physics of the
BTZ geometry is the presence of regions containing CTCs. Hence, the question
of their consistent excision is important. A second issue is that four copies
of the BTZ geometry cover 2 once. Therefore, we will attempt to gather a
minimal set of charts from $\\{+D_{i}^{\pm},-D_{i}^{\pm}\\}$ required to make
a geodesically complete spacetime by first considering the null and timelike
geodesics. Subsequently, we shall discuss spacelike geodesics since these are
important only in the string theory.

Figure 1 shows the Penrose diagram of the 2 group manifold showing the regions
covered by the various charts in the notation discussed in Section 2. Consider
the null geodesic (26a) with $J^{(2)}_{+}>J^{(2)}_{-}>0$. This geodesic for
large negative $\tau$ has all entries of the matrix negative and thus lies in
the chart $-D_{1}^{+}$. As $\tau$ increases, the number of negative entries
changes gradually from four to none and the geodesic passes through the charts

$-D_{1}^{+}\to-D_{2}^{+}\to-D_{3}^{-}\to-D_{4}^{-}\to-
D_{3}^{+}\to+D_{2}^{-}\to+D_{1}^{+}$

finally ending in the chart $D_{1}^{+}$ for large positive $\tau$. Similarly,
the order in which timelike geodesics traverse the charts can also be found.
For instance, if
$0<\phi_{+}<\pi-\phi_{-}<\frac{\pi}{2}\Leftrightarrow{\text{sgn}}(m)J^{(2)}_{\pm}<0$
then the timelike geodesic (13) traverses the charts in the order

$+D_{1}^{+}\to+D_{2}^{-}\to-D_{3}^{+}\to-D_{4}^{+}\to-D_{3}^{-}\to-
D_{2}^{+}\to-D_{1}^{+}$

as $m\tau$ varies in the range $[0,\pi]$. Evolution beyond $m\tau\in[0,\pi]$
may be found using the observation that $m\tau\mapsto m\tau+\pi$ is equivalent
to multiplication by $-\mathbb{I}$. This analysis can also be repeated for
other values of $J^{(2)}_{\pm}$. Of particular interest are the special
geodesics (17) with $\phi_{-}=\frac{\pi}{2}$ (and $\phi_{+}=\frac{\pi}{2}$).
The matrices (13) corresponding to them have equal diagonal (anti-diagonal)
entries and hence, they vanish together at some value of $\tau$. Such
geodesics pass through the intersection of the future and past horizons
$r^{2}=r_{-}^{2}$ (and $r^{2}=r_{+}^{2}$).

The spacelike geodesics however start and end in asymptotic regions. Here, we
reproduce the matrix

$\sqrt{\frac{r_{+}^{2}-r_{-}^{2}}{4m^{2}}}\left(\begin{smallmatrix}e^{m\tau}-\frac{e^{-m\tau}}{k^{2}}\left\\{(2J^{(2)}_{-}-km)(2J^{(2)}_{+}+km)\right\\}&e^{m\tau}-\frac{e^{-m\tau}}{k^{2}}\left\\{(2J^{(2)}_{-}+km)(2J^{(2)}_{+}+km)\right\\}\\\
e^{m\tau}-\frac{e^{-m\tau}}{k^{2}}\left\\{(2J^{(2)}_{-}-km)(2J^{(2)}_{+}-km)\right\\}&e^{m\tau}-\frac{e^{-m\tau}}{k^{2}}\left\\{(2J^{(2)}_{-}+km)(2J^{(2)}_{+}-km)\right\\}\end{smallmatrix}\right)\,,$
(29)

that represents the spacelike geodesic (19) with upper sign choice (upto a
$\tau$ shift). When $J^{(2)}_{+}>\frac{k|m|}{2}$ and
$J^{(2)}_{-}<-\frac{k|m|}{2}$, terms in braces in the above matrix are all
negative and none of the matrix entries change sign as $\tau$ varies. Hence,
these geodesics start and end in $+D_{1}^{+}$. Note that although the right
hand sides of (24) are less than unity, the time scales $\Delta\tau_{\pm}$ and
$\Delta\tau_{0}$ for this geodesic are not defined, because it never crosses
any horizon. On the other hand, when $|J^{(2)}_{\pm}|<\frac{k|m|}{2}$, only
the terms in braces on the diagonal are negative. When
$J^{(2)}_{-}=0=J^{(2)}_{+}$, such geodesics pass through the bifurcation point
from $+D_{1}^{-}$ to $+D_{1}^{+}$. More generally, they traverse via
$+D_{2}^{\pm}$ if $\pm m(J^{(2)}_{+}+J^{(2)}_{-})<0$. Since they cross only
the outer horizon twice, it is satisfactory that $\tanh|m\Delta\tau_{0}|>1$ in
(24), thereby rendering the time scale $\Delta\tau_{0}$ undefined. However,
although $\tanh|m\Delta\tau_{\pm}|<1$, only the time scale $\Delta\tau_{+}$ is
applicable to these geodesics. These are clearly folded strings, but rather
than ending at the horizon, these end on the boundary much as in the 2D black
hole kpy .

There are also geodesics that end in $D_{1}^{+}$ starting from one of the
$D_{4}$ charts. For instance, geodesics with $-km<2J^{(2)}_{+}<km$ and
$-2J^{(2)}_{-}<km<2J^{(2)}_{-}$ start from $-D_{4}^{+}$ and end in
$D_{1}^{+}$. Similar analyses can be repeated for spacelike geodesics (19)
with the lower sign choice.

From this analysis, it is clear that each of the
$\\{+D_{i}^{\pm},-D_{i}^{\pm}\\}$ charts can be accessed from every other
chart by traversing (piecewise) geodesic trajectories. Hence, geodesic
completeness requires that we keep all these charts. In particular, we are
forced to include the $D_{4}$ charts containing the regions with CTCs. This
will mean that the quantum wavefunctions corresponding to these geodesics will
certainly have nonzero support in these charts.

##  5 Discrete symmetries

In this section, we shall consider the action of various discrete symmetries
to get a handle on the multiplicities of solutions with the same quantum
numbers. Our approach will be in the spirit of covariant quantisation – we
will examine the effect of the symmetries on the space of solutions. Some of
these discrete symmetries, such as time-reversal, are interpretable at the
level of the BTZ geometry itself. Some arise naturally by considering the
matrix representation of the solutions.

###  5.1 BTZ symmetries

The BTZ metric exhibits several discrete symmetries. It is invariant under a
simultaneous sign flip of $t$ and $\phi$. It is also invariant under
$r_{+}\leftrightarrow r_{-}$. Since these transformations preserve the
geodesic equations (10), they act on a solution (a geodesic) and produce
another solution, in general. For instance, the transformation
$(t,\phi)\mapsto(-t,-\phi)$ when applied to a solution with energy $E$ and
angular momentum $L$ gives us a solution with energy $-E$ and angular momentum
$-L$. However, all these transformations leave the quantities $\alpha$ and
$\beta$ invariant.

The string sigma model with the BTZ geometry as target space is also invariant
under $r_{+}\leftrightarrow r_{-}$. This is because the NS B-field can be
written, up to a gauge trivial term as

$B_{NS}=(r^{2}-r_{+}^{2}-r_{-}^{2})\;d\phi\wedge dt\,.$ (30)

This transformation is akin to quotienting by the ‘vectorial’ action of
$P_{+}$ in (3) if the earlier action is termed ‘axial’.

Interchanging $E\leftrightarrow L$ together with $t\leftrightarrow\phi$ and
$r^{2}-r_{+}^{2}\leftrightarrow r_{-}^{2}-r^{2}$ also leaves the geodesic
equations invariant NS . To see this, it helps to rewrite the radial equation
(10a) as

$\frac{\dot{y}^{2}}{4m^{2}}=-\left(y-\frac{M}{2}\right)^{2}+\frac{E^{2}-L^{2}}{m^{2}}\left(y-\frac{M}{2}\right)+\frac{M(E^{2}+L^{2})+2ELJ}{2m^{2}}+\frac{M^{2}-J^{2}}{4}\,.$
(31)

The interchange $r^{2}-r_{+}^{2}\leftrightarrow r_{-}^{2}-r^{2}$ now
corresponds to $y-\frac{M}{2}\mapsto\frac{M}{2}-y$. Along with
$E\leftrightarrow L$ and $t\leftrightarrow\phi$, this leaves all the geodesic
equations (10) invariant. This transformation can be implemented on any
solution of the WZW model by right multiplying the solution matrix by
$i\sigma_{2}$. When acting on a geodesic having energy and angular momentum
$(E,L)$, it results in a new geodesic with energy and angular momentum
$(E^{\prime},L^{\prime})=(L,E)$. In fact, the matrix obtained by right
multiplying any solution of the WZW model by $i\sigma_{2}$ is also a solution.
If we replace the worldsheet $\tau$ variable by $\sigma$, this solution will
turn out to have same conserved charges as the original geodesic. This is
analogous to usual T-duality where we obtain new states of the theory by
$\tau\leftrightarrow\sigma$, thereby changing sign of the right moving current
$J_{-}$. This procedure was used to obtain new stringy states of the 2D
Lorentzian black hole kpy .

To summarise, we realised that the discrete symmetry
$(t,r_{+}^{2}-r^{2})\leftrightarrow(\phi,r^{2}-r_{-}^{2})$ of the BTZ sigma
model acts on lifts of its solutions to 2 as right multiplication by
$i\sigma^{2}$. Hence, it is part of the 2$\times$2 symmetry of the WZW model
and does not lead to multiplicities of representations upon quantisation. This
is also true about the (equivalent) discrete symmetries
$(t,\phi)\mapsto(-t,-\phi)$ and $r_{\pm}\mapsto-r_{\pm}$. Both of these act on
solution matrices as conjugation by $i\sigma^{2}$.

###  5.2 2 symmetries

When we regard the solution as an 2 matrix, several natural discrete
symmetries become visible. Given a solution, the inverse of the matrix gives a
new solution. We can also consider the transpose of the matrix or multiply by
$-\mathbb{I}$. Conjugation by Pauli matrices also produces more solutions.
These transformations are discrete isometries of 2. Therefore, they will act
as global symmetries on the space of classical geodesic solutions of the WZW
model. Of these transformations, multiplication by $-\mathbb{I}$ and
conjugation by $\sigma^{3}$ commute with the orbifolding. Hence, they will act
on the space of states of the BTZ black hole as well.

#### Conjugation by Pauli Matrices.

Conjugation by $i\sigma^{2}$ is part of the 2$\times$2 symmetry of the WZW
model and hence, does not lead to multiplicities of representations upon
quantisation. Under its action on a solution, the diagonalised components of
WZW currents evaluated along the solution transform as
$J^{(2)}_{\pm}\mapsto-J^{(2)}_{\pm}$. For a BTZ geodesic having energy $E$ and
angular momentum $L$, this implies $(E,L)\mapsto(-E,-L)$. On the other hand,
conjugation by $\sigma^{3}$ is not part of the 2$\times$2 symmetry – it is an
outer automorphism and leaves the diagonalised components $J^{(2)}_{\pm}$
invariant. Finally, conjugation by $\sigma^{1}$ is simply the conjugation by
$i\sigma^{2}\sigma^{3}$.

#### Inverse and Transpose.

The inverse $(g(\tau))^{-1}$ and transpose $(g(\tau))^{t}$ of a matrix
$g(\tau)$ obtained by lifting a BTZ geodesic also project to BTZ geodesics.
However, this is not true for generic $\sigma$-dependent solutions of the WZW
model. This is easily seen by noting that $g\mapsto g^{-1}$ and $g\mapsto
g^{t}$ do not preserve the WZW equations of motion $\partial_{\pm}J_{\mp}=0$
(see Appendix A for conventions). However, combining the two operations
preserves the equations of motion. In fact, it is equivalent to conjugation by
$i\sigma^{2}$.

#### Multiplication by $-\mathbb{I}$.

This transformation acts differently on lifts of different kinds of BTZ
solutions. Consider a solution $g(\tau)$ of the WZW model obtained by lifting
a timelike BTZ geodesic. Multiplying this by $-\mathbb{I}$ leaves it unchanged
because such a solution is periodic, and its sign can be flipped by a
translation $m\tau\mapsto m\tau+\pi$ of the affine parameter. However, for the
spacelike geodesics, $g(\tau)$ and $-g(\tau)$ represent two distinct solutions
of the 2 WZW model (which project to the same BTZ geodesic, albeit in
different charts). We propose that this is the classical version of the
statement that for each value $s^{2}+\frac{1}{4}$ of the quadratic Casimir,
the corresponding principal continuous series representation (in the
hyperbolic basis) has two states with identical energy and angular momentum
quantum numbers Mukunda . Such a doubling is of course, absent for the
discrete series.

#### Summary.

We summarise the discussion by asking how many distinct geodesics exist for
given values $(E,L,m^{2})$ of energy, angular momentum and quadratic Casimir.
For the sake of concreteness, we focus on the spacelike geodesic (29) with
$J^{(2)}_{+}>\frac{k|m|}{2}$ and $J^{(2)}_{-}<-\frac{k|m|}{2}$, which remains
entirely in the $+D_{1}^{+}$ chart. Conjugation by $\sigma^{3}$ transforms
this to a geodesic lying entirely in $+D_{1}^{-}$. Multiplying these two
solutions by $-\mathbb{I}$ results in geodesics lying entirely in the charts
$-D_{1}^{+}$ and $-D_{1}^{-}$ respectively. Hence, upon quantisation, we
expect 4 states corresponding to spacelike geodesics for given quantum numbers
$(E,L,s^{2}+\frac{1}{4})$. On the other hand, because multiplication by
$-\mathbb{I}$ leaves timelike geodesics invariant, we expect only 2 states
corresponding to them for a given $(E,L,-j(j+1))$. The states of the string
theory will be obtained by quantizing these geodesics and winding strings
obtained by twisting them.

##  6 Twisted sectors

In this section, we shall study the twisted sector solutions following NS ;
kraus ; Esko . The twisted sector solutions are defined by the equation

$g_{\Omega}(\tau,\sigma+2\pi)=e^{\Omega\pi(r_{+}-r_{-})\sigma^{3}}g_{\Omega}(\tau,\sigma)e^{\Omega\pi(r_{+}+r_{-})\sigma^{3}}\,,$
(32)

with $\Omega$ defining the twisting parameter. It maybe observed – in the
parametrisation (5), for example – that the right hand side translates into
the shift $\phi\mapsto\phi+2\pi\Omega$. Since $\phi\sim\phi+2\pi$ (due to the
BTZ identification), the twisted solutions will be compatible with the
orbifolding if $\Omega$ is an integer. More explicitly, given a solution
$\tilde{g}$ of the WZW model such that
$\tilde{g}(\tau,\sigma+2\pi)=\tilde{g}(\tau,\sigma)$, the following is a
twisted sector solution:

$g_{\Omega}(\tau,\sigma)=e^{\frac{\Omega}{2}(r_{+}-r_{-})\sigma^{3}(\tau+\sigma)}\tilde{g}(\tau,\sigma)e^{-\frac{\Omega}{2}(r_{+}+r_{-})\sigma^{3}(\tau-\sigma)}$
(33)

i.e., $g_{\Omega}$ satisfies the orbifolding condition (32).

The components of currents evaluated along the twisted solution (33) are

$\displaystyle J^{(2)}_{\pm}$
$\displaystyle=\tilde{J}^{(2)}_{\pm}-\frac{k}{2}\Omega(r_{+}\mp r_{-})\,,$
(34a) $\displaystyle J^{(+)}_{\pm}$ $\displaystyle=e^{-\Omega(r_{+}\mp
r_{-})(\tau\pm\sigma)}\tilde{J}^{(+)}_{\pm}\,,$ (34b) $\displaystyle
J^{(-)}_{\pm}$ $\displaystyle=e^{+\Omega(r_{+}\mp
r_{-})(\tau\pm\sigma)}\tilde{J}^{(-)}_{\pm}\,.$ (34c) Here, $\tilde{J}_{\pm}$
denote currents evaluated on the solution $\tilde{g}$ before twisting. These
relations imply that the Poisson brackets (81) are invariant under twisting.
The stress tensor of the twisted solution evaluates to: $\displaystyle
T_{\pm\pm}$
$\displaystyle=\tilde{T}_{\pm\pm}+\frac{k}{2}\Omega(\tilde{E}\pm\tilde{L})+\frac{k}{4}\Omega^{2}(r_{+}\mp
r_{-})^{2}$ (34d) $\displaystyle=\tilde{T}_{\pm\pm}-\Omega(r_{+}\mp
r_{-})\tilde{J}^{(2)}_{\pm}+\frac{k}{4}\Omega^{2}(r_{+}\mp r_{-})^{2}$

In the quantum theory, this implies transformations of the corresponding zero-
mode eigenvalues

$L_{0}=\tilde{L}_{0}-\Omega(r_{+}\mp
r_{-})\tilde{J}^{(2)}_{\pm,0}+\frac{k}{4}\Omega^{2}(r_{+}\mp r_{-})^{2}+N\,,$
(35)

where the level $N$ is a non-negative integer. This differs from the $AdS_{3}$
case in an important way for the discrete series representations. Consider the
spectrally flowed lowest weight discrete series $\hat{\mathcal{D}}_{j}^{+,w}$
in elliptic basis, for which

$L_{0}=\frac{-(j+\frac{1}{2})^{2}}{k-2}+\frac{1}{4(k-2)}+w\tilde{J}^{(0)}_{\pm,0}-\frac{k}{4}w^{2}+N\,,\qquad\tilde{J}^{(0)}_{\pm,0}\in\\{j,j+1,j+2,\ldots\\}\,.$
(36)

We have used $w$ to denote the $AdS_{3}$ spectral flow parameter. As is well
known, this representation is identified with a highest weight discrete series
having one less unit of spectral flow i.e.,

$\hat{\mathcal{D}}_{j}^{+,w}=\hat{\mathcal{D}}_{-\frac{k}{2}-j}^{-,w-1}\,.$
(37)

This may be verified by shifting $w$ in (36) and completing a square. However,
this is not possible in (35). Hence, there is no such identification in the
BTZ case.

The spacetime energy was defined in equation (9); using the above, we see that
the winding number $\Omega$ adds to the spacetime energy as

$E=\tilde{E}+\Omega M\,.$ (38)

On the other hand, for the spacetime angular momentum, it was pointed out by
Hemming and Keski-Vakkuri Esko  that for consistency with level matching on
the worldsheet, we should use

$Q_{\phi}=\int_{0}^{2\pi}\frac{d\sigma}{2\pi}\left(-\Delta_{-}J_{+}^{(2)}+\Delta_{+}J_{-}^{(2)}+\frac{k}{2}J\partial_{\sigma}\phi\right)$
(39)

to define the $\phi$ translation operator. For the twisted solution, the
angular momentum $kL=Q_{\phi}$ evaluates to

$L=\tilde{L}-\frac{\Omega}{2}J\,.$ (40)

Note that only $t$ and $\phi$ coordinates of the worldsheet change under
twisting but the radial extent (in the BTZ geometry) of the worldsheet is not
affected:

$t(\tau,\sigma)\mapsto
t(\tau,\sigma)+\Omega\tau\,,\qquad\phi(\tau,\sigma)\mapsto\phi(\tau,\sigma)+\Omega\sigma\,,\qquad
r^{2}(\tau)\mapsto r^{2}(\tau)\,.$ (41)

This implies that if a geodesic $\tilde{g}(\tau)$ lies entirely in the region
$r^{2}>0$ without CTCs, then the twisted (winding) string $g_{\Omega}$ also
remains in the region without CTCs.

The definition of $\beta$ appearing in the geodesic equations (10) may be
extended to the twisted sector by replacing the $-\frac{1}{4}m^{2}$ appearing
in it by $\frac{1}{2k}(T_{++}+T_{--})$. Rather remarkably, the condition
$\beta<0$ is now seen to be invariant under twisting:

$\beta=ML^{2}+ELJ+(T_{++}+T_{--})\frac{J^{2}}{2k}=M\tilde{L}^{2}+\tilde{E}\tilde{L}J+(\tilde{T}_{++}+\tilde{T}_{--})\frac{J^{2}}{2k}=\tilde{\beta}\,.$
(42)

At this stage, we are left with a somewhat mixed situation. As far as the
timelike geodesics are concerned, there is a single condition $\beta<0$, which
eliminates the need to consider the regions with CTCs. It is also satisfactory
that this condition is invariant under the spectral flow which produces the
twisted sectors. However, there is an entire family of spacelike geodesics
with $\beta>0$ which never penetrate the region with CTCs (those with
$\alpha<0$ and upper sign choice in (20a)). Thus, it may be incorrect to
impose $\beta<0$ as a condition on all geodesics to keep them out of the
region with CTCs. Hence, a natural question is whether the Virasoro
constraints remove geodesics that enter the region with CTCs.

##  7 Virasoro conditions

In this section, we will explore if a consistent excision of the states
penetrating the unphysical region can occur by using the Virasoro conditions.
Physical states in string theory, at the classical level, require the
vanishing of the worldsheet stress tensor

$T_{++}+h_{+}=0=T_{--}+h_{-}\,,$ (43)

where $h_{\pm}$ is the contribution from the ‘internal CFT’. We assume that
the internal CFT is unitary and hence, $h_{\pm}\geq 0$. Clearly, only timelike
geodesics (and possibly null) with $T_{++}=-\frac{k}{4}m^{2}$ can be physical
in the untwisted sector. However, the Virasoro constraints
$-\frac{k}{4}m^{2}+h_{\pm}=0$ on timelike geodesics constraint only the
quadratic Casimir $m^{2}$ but not their energy and angular momentum. Hence,
they do not already prevent them from exploring the region with CTCs.

In the twisted sectors with twist $\Omega$, the Virasoro conditions become

$\tilde{T}_{\pm\pm}-\Omega(r_{+}\mp
r_{-})\tilde{J}^{(2)}_{\pm}+\frac{k}{4}\Omega^{2}(r_{+}\mp
r_{-})^{2}+h_{\pm}=0\,.$ (44)

By adding the Virasoro conditions, we get a relation for the energy

$k\Omega\tilde{E}=-2\tilde{T}_{\pm\pm}-\frac{kM\Omega^{2}}{2}-(h_{+}+h_{-})=k\Omega(E-\Omega
M)\,.$ (45)

We see that for fixed $h_{\pm}$ and $\Omega>0$, the energies of spectral flows
of timelike geodesics ($\tilde{T}_{\pm\pm}<0$) are bounded below

$E>\frac{M\Omega}{2}-\frac{h_{+}+h_{-}}{k\Omega}$ (46)

as in the case of $AdS_{3}$ studied in mog . Clearly, states of the continuous
series ($\tilde{T}_{\pm\pm}>0$) will step in to obviate this restriction on
energy. The difference with the $AdS_{3}$ case is the presence of the black
hole mass factor. As in $AdS_{3}$, the states of the continuous series
(spacelike geodesics) scatter out to the boundary. Thus, these can play the
role of operators of the dual CFT Ooguri .

The solution obtained by $\Omega$ units of spectral flow on a timelike
geodesic does not enter the region with CTCs, if it satisfies $\beta<0$.
Rewriting this condition using the Virasoro constraints (44), we find

$\left(h_{+}-h_{-}-\frac{k}{2}J\Omega^{2}\right)\left[(h_{+}-h_{-})M+J(h_{+}+h_{-})\right]<\frac{k}{2}m^{2}J(h_{+}-h_{-})\,.$
(47)

Clearly, this condition is not automatic for a generic internal CFT. In fact,
we find that physical states of the string theory which satisfy222This was
derived assuming $(h_{+}-h_{-})\Omega>0$.

$E<\frac{M}{k\Omega}(h_{+}-h_{-})+\frac{J}{k\Omega}(h_{+}+h_{-})-\frac{J\Omega
M}{2}-\frac{J^{2}\Omega}{2}\left(\frac{h_{+}+h_{-}}{h_{+}-h_{-}}\right)-\frac{h_{+}+h_{-}}{k\Omega}+\frac{M\Omega}{2}\,,$
(48)

do pass into the region with CTCs. However, it may be possible to choose an
internal CFT so that $\beta<0$ is automatic. For example, $h_{+}=h_{-}$
automates $\beta<0$ but this is perhaps, too strong a requirement. In any
case, (47) does not prevent physical timelike geodesics from entering the
region with CTCs. It only ensures that physical winding strings do not explore
this region. Since $\beta$ is invariant under twisting, it also prevents
geodesics that lead to physical winding strings from exploring these regions.

All these considerations do not apply to spacelike geodesics since there are
spacelike geodesics that exist entirely in the region with CTCs. Thus, in
conclusion, the Virasoro conditions do not help truncate the space of states
to those that remain in a causally sensible region of the BTZ geometry.

##  8 Torus partition function and modular invariance

In this section, we show that the modular invariant partition function for
global $AdS_{3}$ constructed in Israel  contains all the states of the BTZ
black hole that were discussed previously.333A similar remark has already been
made by Eberhardt Eberhardt  in the tensionless limit. The fact that these are
two different spacetimes arises when we try to identify the spacetime quantum
numbers from the partition function.

We start with the partition function for $AdS_{3}$ as written in Israel  (see
Appendix C for notation):

$\displaystyle Z$
$\displaystyle=4\sqrt{\tau_{2}}(k-2)^{\frac{3}{2}}\int_{0}^{1}d^{2}s\;\int_{0}^{1}d^{2}t\;\frac{e^{\frac{2\pi}{\tau_{2}}({\text{Im}}(s_{1}\tau-
s_{2}))^{2}}}{|\vartheta_{1}(s_{1}\tau-s_{2}|\tau)|^{2}}$ (49)
$\displaystyle\quad\times\sum_{m,w,m^{\prime},w^{\prime}\in\mathbb{Z}}\zeta{\biggl{[}\begin{array}[]{c}w+s_{1}-t_{1}\\\
m+s_{2}-t_{2}\end{array}\biggr{]}}(k)\;\zeta{\biggl{[}\begin{array}[]{c}w^{\prime}+t_{1}\\\
m^{\prime}+t_{2}\end{array}\biggr{]}}(-k)$

In what follows, we first expand this partition function as a power series in
$q$ to identify the states. This allows us to identify the spacetime isometry
currents for these states and make contact with the analysis of NS . Finally,
we compare this identification of the currents with the corresponding relation
for $AdS_{3}.$

###  8.1 Series expansion of the partition function

We follow the procedure detailed in Israel  to expand the partition function
(49). First, we expand the factor $|\vartheta_{1}|^{-2}$ as explained in
Appendix C and trade off $(m,m^{\prime})$ for $(n,n^{\prime})$ by Poisson
resummation. The exponent in $s_{1}$ can be linearised by introducing a
Gaussian integral over $s$ allowing us to integrate $s_{1}$ out and obtain:

$\displaystyle Z$
$\displaystyle=\frac{4(k-2)}{\pi}\sum_{\begin{subarray}{c}w,w^{\prime},n,n^{\prime}\in\mathbb{Z}\\\
q,\bar{q},N,\bar{N}\end{subarray}}\int_{0}^{1}dt_{2}\int_{0}^{1}ds_{2}\;\exp\left[2\pi
it_{2}(n-n^{\prime})+2\pi is_{2}(\bar{q}-q-n)\right]$ (50)
$\displaystyle\quad\times\int_{0}^{1}dt_{1}\int_{-\infty}^{\infty}ds\left[\frac{1}{2is+q+\bar{q}+1+k(w-t_{1})}-\frac{e^{-2\pi\tau_{2}(2is+q+\bar{q}+1+k(w-t_{1}))}}{2is+q+\bar{q}+1+k(w-t_{1})}\right]$
$\displaystyle\quad\times\exp\left[2\pi
i\tau_{1}(n(w-t_{1})+n^{\prime}(w^{\prime}+t_{1})+N-\bar{N})\right]$
$\displaystyle\quad\times\exp\left[-2\pi\tau_{2}\left(\frac{2s^{2}}{k-2}-\frac{1}{4}+\frac{n^{2}}{2k}-\frac{n^{\prime
2}}{2k}+\frac{k(w-t_{1})^{2}}{2}-\frac{k(w^{\prime}+t_{1})^{2}}{2}+N+\bar{N}\right)\right]$

The integrals over $t_{2}$ and $s_{2}$ lead to the constraints
$n=n^{\prime}=\bar{q}-q$. Of the two integrals in the second line, we change
variables in the second as

$\tilde{s}=s+\frac{i}{2}(k-2)\,,\qquad\quad\tilde{w}=w+1\,,\qquad\quad\tilde{N}=N+q\,,\qquad\quad\tilde{\bar{N}}=\bar{N}+\bar{q}\,.$
(51)

This change of variables corresponds to a spectral flow if (49) is interpreted
as the global $AdS_{3}$ partition function. Hence, the sum over
$(q,\bar{q},N,\bar{N})$ differs from that over
$(q,\bar{q},\tilde{N},\tilde{\bar{N}})$ only in the signs of zero mode
contributions mog2  (denoted $P_{\pm,0}$ in (94a)) to $(q,\bar{q})$. The
factor $e^{-2\pi\tau_{2}(2is+q+\bar{q}+1+k(w-t_{1}))}$ is now completely
absorbed into the new variables. However, the $\tilde{s}$ integral now runs
along $\mathbb{R}+\frac{i}{2}(k-2)$. This differs from the integral in the
first term by residues of poles in the strip enclosed by
$\mathbb{R}+\frac{i}{2}(k-2)$ and $\mathbb{R}$. Collecting these residues, we
obtain the net result:

$\displaystyle Z$
$\displaystyle=8i(k-2)\sum_{\begin{subarray}{c}w,w^{\prime},n,n^{\prime}\in\mathbb{Z}\\\
N,\bar{N}\end{subarray}}\delta_{n,n^{\prime}}\int_{0}^{1}dt_{1}\Biggl{\\{}\int_{0}^{\infty}ds\;\rho(s)\exp\left[-2\pi\tau_{2}\left(\frac{2s^{2}+1/2}{k-2}\right)\right]$
(52)
$\displaystyle\quad+\sum_{q,\bar{q}}\delta_{n,\bar{q}-q}\exp\left[-2\pi\tau_{2}\left(\frac{-2j(j+1)}{k-2}\right)\right]_{\frac{1}{2}<-j=\frac{q+\bar{q}}{2}+\frac{k}{2}(w-t_{1})<\frac{k-1}{2}}\Biggr{\\}}$
$\displaystyle\quad\times\exp\left[2\pi
i\tau_{1}(n(w-t_{1})+n^{\prime}(w^{\prime}+t_{1})+N-\bar{N})\right]$
$\displaystyle\quad\times\exp\left[-2\pi\tau_{2}\left(\frac{n^{2}}{2k}-\frac{n^{\prime
2}}{2k}+\frac{k(w-t_{1})^{2}}{2}-\frac{k(w^{\prime}+t_{1})^{2}}{2}-\frac{3k}{12(k-2)}+N+\bar{N}\right)\right]$

Here, the second line corresponds to states of the discrete series. The first
line is obtained by combining the $s$ and $\tilde{s}$ integrals over
$\mathbb{R}$ and further combining the contributions from positive and
negative values of the integration variable. This denotes contribution from
the continuous series representations, whose density of states for a given
quadratic Casimir $s^{2}+\frac{1}{4}$ is

$\rho(s)=\frac{1}{\pi
i}{\text{Re}}\left[\sideset{}{{}^{+}}{\sum}_{q,\bar{q}}\frac{\delta_{n,\bar{q}-q}}{2is+q+\bar{q}+1+k(w-t_{1})}-\sideset{}{{}^{-}}{\sum}_{q,\bar{q}}\frac{\delta_{n,\bar{q}-q}}{2is+q+\bar{q}-1+k(w-t_{1})}\right]\,,$
(53)

where the superscripts on sums indicate signs of zero mode contributions to
$(q,\bar{q})$. They lead to divergences that need to be regularised mog2  to
obtain

$\sum_{N,\bar{N}}\rho(s)=\sum_{\begin{subarray}{c}P_{\pm,p},P^{+}_{\pm,p},P^{-}_{\pm,p}=0\\\
p\in\\{0,1,2,3,\ldots\\}\end{subarray}}^{\infty}\Biggl{\\{}\frac{\ln\epsilon}{\pi
i}+\frac{1}{4\pi
i}\frac{d}{ids}\ln\Biggl{[}\frac{\Gamma(\frac{1}{2}-is+\frac{|n+q^{\prime}-\bar{q}^{\prime}|+q^{\prime}+\bar{q}^{\prime}}{2}+\frac{k}{2}(w-t_{1}))}{\Gamma(\frac{1}{2}+is+\frac{|n+q^{\prime}-\bar{q}^{\prime}|+q^{\prime}+\bar{q}^{\prime}}{2}+\frac{k}{2}(w-t_{1}))}\\\
\times\frac{\Gamma(\frac{1}{2}-is+\frac{|n+q^{\prime}-\bar{q}^{\prime}|-q^{\prime}-\bar{q}^{\prime}}{2}-\frac{k}{2}(w-t_{1}))}{\Gamma(\frac{1}{2}+is+\frac{|n+q^{\prime}-\bar{q}^{\prime}|-q^{\prime}-\bar{q}^{\prime}}{2}-\frac{k}{2}(w-t_{1}))}\Biggr{]}+\mathcal{O}(\epsilon)\Biggr{\\}}\,,$
(54)

where $(q^{\prime},\bar{q}^{\prime})$ denote the expressions (94a) for
$(q,\bar{q})$ void of the zero mode contributions (i.e., $P_{+,0}=0=P_{-,0}$).

We can now identify the $J^{(2)}_{\pm,0}$ eigenvalues $\tilde{J}_{\pm}$ (and
$J_{\pm}$) before (and after) spectral flow respectively as

$\begin{pmatrix}\tilde{J}_{+}\\\ J_{+}\\\ \tilde{J}_{-}\\\
J_{-}\end{pmatrix}=\frac{k}{2}\begin{pmatrix}-\cosh\theta_{+}&\sinh\theta_{+}&0&0\\\
-\sinh\theta_{+}&\cosh\theta_{+}&0&0\\\
0&0&-\cosh\theta_{-}&\sinh\theta_{-}\\\
0&0&-\sinh\theta_{-}&\cosh\theta_{-}\end{pmatrix}\begin{pmatrix}w^{\prime}+t_{1}-\frac{n^{\prime}}{k}\\\
w-t_{1}+\frac{n}{k}\\\ w^{\prime}+t_{1}+\frac{n^{\prime}}{k}\\\
w-t_{1}-\frac{n}{k}\end{pmatrix},$ (55)

where $e^{-\theta_{\pm}}=\Delta_{\mp}=r_{+}\mp r_{-}$. This identification
ensures that $L_{0}\pm\bar{L}_{0}$ for primaries (i.e., $N=0=\bar{N}$) read
off from (52) have exactly the forms

$\begin{split}\begin{aligned}
L_{0}-\bar{L}_{0}&=\frac{1}{k}(-\tilde{J}_{+}^{2}+J_{+}^{2})-\frac{1}{k}(-\tilde{J}_{-}^{2}+J_{-}^{2})\,,\\\
L_{0}+\bar{L}_{0}&=\frac{1}{k}(-\tilde{J}_{+}^{2}+J_{+}^{2})+\frac{1}{k}(-\tilde{J}_{-}^{2}+J_{-}^{2})+\frac{2(s^{2}+\frac{1}{4})}{k-2}\,,\end{aligned}\end{split}\qquad\quad\begin{split}J_{\pm}&=\tilde{J}_{\pm}-\frac{k}{2}\Omega\Delta_{\mp}\,,\end{split}$
(56)

which are as expected from the classical result (34d) for a spectrally flowed
primary. The same expressions for $L_{0}\pm\bar{L}_{0}$ were also obtained in
the quantised CFT by Natsuume and Satoh NS . Along with the expected form of
$L_{0}\pm\bar{L}_{0}$, this identification also ensures the following:

  * •

The spectral flow parameter turns out to be $\Omega=-(w+w^{\prime})$ as is
clear from

$J_{+}-\tilde{J}_{+}=\frac{k}{2}\left(w+w^{\prime}+\frac{n-n^{\prime}}{k}\right)\Delta_{-}\,,\quad
J_{-}-\tilde{J}_{-}=\frac{k}{2}\left(w+w^{\prime}-\frac{n-n^{\prime}}{k}\right)\Delta_{+}\,,$
(57)

using the constraint $n=n^{\prime}$ appearing in (52). This constraint ensures
that the winding number along the non-compact $t$ direction of the BTZ
geometry is zero.

  * •

While analysing geodesics on the BTZ geometry, we expressed the energy and
angular momentum of a geodesic as linear combinations (9) of WZW currents.
Evaluating these using the identification (55), we find that

$\displaystyle k\tilde{L}$
$\displaystyle=-\Delta_{-}\tilde{J}_{+}+\Delta_{+}\tilde{J}_{-}=-n-\frac{k}{2}(w+w^{\prime})J\,,$
(58a) $\displaystyle k\tilde{E}$
$\displaystyle=-\Delta_{-}\tilde{J}_{+}-\Delta_{+}\tilde{J}_{-}=-\frac{k}{2}(w-w^{\prime}-2t_{1})+\frac{k}{2}(w+w^{\prime})M\,,$
(58b)

where $J=2r_{+}r_{-}$ and $M=r_{+}^{2}+r_{-}^{2}$ are the black hole angular
momentum and mass, respectively. Setting $\Omega=-(w+w^{\prime})=0$ we observe
that angular momentum is quantised, while the energy is an arbitrary real
number. This is exactly as expected for geodesics on the BTZ geometry because
the $\phi$ coordinate is $2\pi$-periodic while the $t$ coordinate is non-
compact. So, we may identify the states appearing in (52) with
$w+w^{\prime}=0$ as those obtained by quantizing the geodesics of the BTZ
black hole.

  * •

As discussed previously, the energy and angular momentum of a solution twisted
by $\Omega$ units are related to those before twisting as in (38) and (40).
The identification (55) also reproduces these results

$\displaystyle kL$
$\displaystyle=-\Delta_{-}J_{+}+\Delta_{+}J_{-}+\frac{k}{2}\Omega
J=-n=k\left(\tilde{L}-\frac{\Omega}{2}J\right)\,,$ (59a) $\displaystyle kE$
$\displaystyle=-\Delta_{-}J_{+}-\Delta_{+}J_{-}=-\frac{k}{2}(w-w^{\prime}-2t_{1})-\frac{k}{2}(w+w^{\prime})M=k(\tilde{E}+\Omega
M)\,.$ (59b)

Notice that it is the generator $Q_{\phi}$ defined in (39) that is quantised
rather than the combination $-\Delta_{-}J_{+}+\Delta_{+}J_{-}$. Hence, our
identification (55) is consistent with the Noether ambiguity pointed out by
Hemming and Keski-Vakkuri Esko .

  * •

From our analysis of discrete symmetries in Section 5, we concluded a doubling
of continuous series representations relative to the discrete series. However,
considering winding strings and descendants as in Section 6, an identification
similar to (37) was found to be missing in the BTZ case. This revealed a
doubling of discrete series representations in BTZ relative to $AdS_{3}$.
Hence, it is satisfactory that our BTZ interpretation of the $AdS_{3}$
partition function follows without changing the relative degeneracy between
discrete and continuous series contributions in (52).

Having identified the physical quantum numbers $(E,L)$, we can rewrite the
density of states (54) in terms of them as

$\sum_{N,\bar{N}}\rho(s)=\sum_{\begin{subarray}{c}P_{\pm,p},P^{+}_{\pm,p},P^{-}_{\pm,p}=0\\\
p\in\\{0,1,2,3,\ldots\\}\end{subarray}}^{\infty}\frac{1}{4\pi
i}\frac{d}{ids}\ln\Biggl{[}\frac{\Gamma(\frac{1}{2}-is+\frac{|kL-q^{\prime}+\bar{q}^{\prime}|+q^{\prime}+\bar{q}^{\prime}}{2}-\frac{k}{2}(E-\frac{\Omega}{2}(M-1)))}{\Gamma(\frac{1}{2}+is+\frac{|kL-q^{\prime}+\bar{q}^{\prime}|+q^{\prime}+\bar{q}^{\prime}}{2}-\frac{k}{2}(E-\frac{\Omega}{2}(M-1)))}\\\
\times\frac{\Gamma(\frac{1}{2}-is+\frac{|kL-q^{\prime}+\bar{q}^{\prime}|-q^{\prime}-\bar{q}^{\prime}}{2}+\frac{k}{2}(E-\frac{\Omega}{2}(M-1)))}{\Gamma(\frac{1}{2}+is+\frac{|kL-q^{\prime}+\bar{q}^{\prime}|-q^{\prime}-\bar{q}^{\prime}}{2}+\frac{k}{2}(E-\frac{\Omega}{2}(M-1)))}\Biggr{]}\,,$
(60)

where we have set $\epsilon=0$ after omitting the $\ln\epsilon$ divergence for
the sake of brevity. It is interesting that this expression does not involve
the black hole angular momentum $J$.

To summarise, we have shown that the primaries (states with $N=0=\bar{N}$)
read off from the partition function (49) are in one-to-one correspondence
with quantised geodesics on the BTZ black hole and string worldsheets winding
around its $\phi$ direction, obtained by twisting the geodesics.

Now, let us consider the states with non-zero $(N,\bar{N})$. As described in
Appendix C, the quantum numbers $(N,\bar{N})$ and $(q,\bar{q})$ arise from
expansion of the factor $|\vartheta_{1}|^{-2}$ present in (49). This factor
includes six terms – three holomorphic and three anti-holomorphic – along with
two factors that contribute only to $(q,\bar{q})$ but not $(N,\bar{N})$. In
what follows, we attempt to understand these quantum numbers using the vertex
operator constructions proposed in Satoh ; Hemming .

###  8.2 Vertex operators

We begin this section with a brief review of the free field realisation of
$\hat{\mathfrak{sl}}(2)_{k}$ introduced by Satoh Satoh . Then, we describe the
vertex operators written by Satoh Satoh  and Hemming Hemming , corresponding
to the primary states read off from the partition function. We shall use these
vertex operators to propose an interpretation for the quantum numbers
$(q,\bar{q},N,\bar{N})$ and therefore the factor $|\vartheta_{1}|^{-2}$
appearing in (49).

The free field realisation of the $\hat{\mathfrak{sl}}(2)_{k}$ algebra
provided by Satoh Satoh  consists of three chiral bosonic fields $X_{a}$ (with
$a\in\\{0,1,2\\}$) satisfying OPEs

$X_{a}(z)X_{b}(w)\sim-\eta_{ab}\ln(z-w)\,,\qquad\qquad\eta_{ab}={\text{diag}}(-1,1,1)\,.$
(61)

The holomorphic $\hat{\mathfrak{sl}}(2)_{k}$ algebra in hyperbolic basis (see
Appendix A) and its Sugawara stress tensor is realised in terms of these
fields as

$\displaystyle iJ_{+}^{(\pm)}$
$\displaystyle=e^{\mp\sqrt{\frac{2}{k}}(X_{0}-X_{1})}\left(\sqrt{\frac{k}{2}}\partial
X_{0}\mp\sqrt{\frac{k-2}{2}}\partial X_{2}\right)\,,$ (62a) $\displaystyle
iJ_{+}^{(2)}$ $\displaystyle=\sqrt{\frac{k}{2}}\partial X_{1}\,,$ (62b)
$\displaystyle T_{++}$ $\displaystyle=-\frac{1}{2}\eta^{ab}\partial
X_{a}\partial X_{b}-\frac{1}{\sqrt{2(k-2)}}\partial^{2}X_{2}\,.$ (62c)

Using this free field realisation, we may verify that the holomorphic vertex
operator

$V^{C}_{j,\tilde{J}_{+},J_{+}}=\exp\left[\left(i\tilde{J}_{+}X_{0}-iJ_{+}X_{1}\right)\sqrt{\frac{2}{k}}+jX_{2}\sqrt{\frac{2}{k-2}}\right]$
(63)

has $L_{0}$ eigenvalue
$\frac{-j(j+1)}{k-2}+\frac{1}{k}(-\tilde{J}_{+}^{2}+J_{+}^{2})$ and
$J^{(2)}_{+,0}$ eigenvalue $J_{+}$. Thus, the vertex operator
$V^{C}_{-\frac{1}{2}+is,\tilde{J}_{+},\tilde{J}_{+}-\frac{k}{2}\Omega\Delta_{-}}$
corresponds to the holomorphic part of a state obtained by a twist of $\Omega$
units from a continuous series state with $J^{(2)}_{\pm,0}$ eigenvalues
$\tilde{J}_{\pm}$.

On the other hand, the interchange $X_{0}\leftrightarrow iX_{2}$ yields
another free field realisation of the same $\hat{\mathfrak{sl}}(2)_{k}$
algebra. It was pointed out by Hemming Hemming  that this realisation can be
used to write the vertex operators $V^{D}_{j,\tilde{J}_{+},J_{+}}$
corresponding to holomorphic parts of discrete series states. These are
obtained from (63) by the same interchange $X_{0}\leftrightarrow iX_{2}$.

Similarly, we may use anti-holomorphic parts $\tilde{X}_{a}$ (with
$a\in\\{0,1,2\\}$) of the bosonic fields to obtain a free field realisation of
the anti-holomorphic current algebra and corresponding vertex operators. The
vertex operators corresponding to states read off from the partition function
(49) are then $V_{j,\tilde{J}_{+},J_{+}}\tilde{V}_{j,\tilde{J}_{-},J_{-}}$
with

$\tilde{J}_{\pm}=\frac{\frac{k}{2}(w-w^{\prime}-2t_{1})\pm
n}{2\Delta_{\mp}}-\frac{k}{4}(w+w^{\prime})\Delta_{\mp}\,,\qquad\quad
J_{\pm}=\tilde{J}_{\pm}+\frac{k}{2}(w+w^{\prime})\Delta_{\mp}\,.$ (64)

We shall use this to propose an interpretation of the denominator
$|\vartheta_{1}|^{2}$ appearing in (49).

To begin with, the factors in $|\vartheta_{1}|^{-2}$ do not represent the
modes $(J^{(2)}_{\pm,n\leq 0},J^{(+)}_{\pm,n<0},J^{(-)}_{\pm,n<0})$. If this
were the case, then the hyperbolic basis commutation relations (81b) would
imply that they contribute $(iq,i\bar{q})$ additively to
$(J^{(2)}_{+,0},J^{(2)}_{-,0})$ eigenvalues, without any
$(\Delta_{-},\Delta_{+})$ factors. Instead they contribute
$\pm\frac{\bar{q}-q}{2\Delta_{\mp}}$ to $J^{(2)}_{\pm,0}$ eigenvalues as is
clear from (64) and the constraint $n=\bar{q}-q$ from (52). Furthermore, the
second line of (52) makes it clear that the factor $|\vartheta_{1}|^{-2}$ also
contributes $-\frac{q+\bar{q}}{2}$ to $j$-values of the discrete series.
Hence, the denominator $|\vartheta_{1}|^{2}$ has to be interpreted differently
for states of the discrete and continuous series.

An interpretation of the factors in $|\vartheta_{1}|^{-2}$ that is consistent
with these observations is as follows. In terms corresponding to the
continuous series in (52), the six factors in the infinite product from
$|\vartheta_{1}|^{-2}$ can be attributed to the operators

$\displaystyle J^{(2)}_{+,-p}\,,$ $\displaystyle A^{\pm}_{+,-p}$
$\displaystyle=J^{(2)}_{+,-p}V^{C}_{0,\frac{\mp 1}{2\Delta_{-}},\frac{\mp
1}{2\Delta_{-}}}\tilde{V}^{C}_{0,\frac{\pm 1}{2\Delta_{+}},\frac{\pm
1}{2\Delta_{+}}}\,,$ (65a) $\displaystyle J^{(2)}_{-,-p}\,,$ $\displaystyle
A^{\pm}_{-,-p}$ $\displaystyle=J^{(2)}_{-,-p}V^{C}_{0,\frac{\pm
1}{2\Delta_{-}},\frac{\pm 1}{2\Delta_{-}}}\tilde{V}^{C}_{0,\frac{\mp
1}{2\Delta_{+}},\frac{\mp 1}{2\Delta_{+}}}\,,$ (65b)

for each $p\in\\{1,2,3,...\\}$. These operators contribute to both
$(q,\bar{q})$ and $(N,\bar{N})$. On the other hand, the two factors in
$|\vartheta_{1}|^{-2}$ that diverge as $s_{1},s_{2}\to 0$ are due to

$A^{+}_{+,0}=V^{C}_{0,\frac{-1}{2\Delta_{-}},\frac{-1}{2\Delta_{-}}}\tilde{V}^{C}_{0,\frac{1}{2\Delta_{+}},\frac{1}{2\Delta_{+}}}\,,\qquad\qquad\qquad
A^{+}_{-,0}=V^{C}_{0,\frac{1}{2\Delta_{-}},\frac{1}{2\Delta_{-}}}\tilde{V}^{C}_{0,\frac{-1}{2\Delta_{+}},\frac{-1}{2\Delta_{+}}}\,.$
(66)

These operators contribute only to $J^{(2)}_{\pm,0}$ (via $q$ and $\bar{q}$)
and not to $L_{0}\pm\bar{L}_{0}$, just like the corresponding factors in
$|\vartheta_{1}|^{-2}$. This interpretation reproduces the exact degeneracies
of $(q,\bar{q})$ and $(N,\bar{N})$ as they appear in (52). To see this, note
that the contribution of these operators to $\tilde{J}_{\pm}$ and $J_{\pm}$ is
precisely $\mp\frac{q-\bar{q}}{2\Delta_{\mp}}$ where

$\displaystyle q$
$\displaystyle=\\#A^{+}_{+,0}+\sum_{p=1}^{\infty}(\\#A^{+}_{+,-p}-\\#A^{-}_{+,-p})\,,$
(67a) $\displaystyle\bar{q}$
$\displaystyle=\\#A^{+}_{-,0}+\sum_{p=1}^{\infty}(\\#A^{+}_{-,-p}-\\#A^{-}_{-,-p})\,,$
(67b)

and their contribution to $(N,\bar{N})$ is also as expected from the expansion
of $|\vartheta_{1}|^{-2}$ (see Appendix C) viz.,

$\displaystyle N$
$\displaystyle=\sum_{p=1}^{\infty}p(\\#J^{(2)}_{+,-p}+\\#A^{+}_{+,-p}+\\#A^{-}_{+,-p})\,,$
(68a) $\displaystyle\bar{N}$
$\displaystyle=\sum_{p=1}^{\infty}p(\\#J^{(2)}_{-,-p}+\\#A^{+}_{-,-p}+\\#A^{-}_{-,-p})\,.$
(68b)

Comparing expressions (67) and (68) with (94) confirms that our interpretation
reproduces degeneracies appearing in the partition function.

As already explained, the same interpretation does not apply to
$|\vartheta_{1}|^{-2}$ multiplying terms corresponding to the discrete series
in (52). The various factors in $|\vartheta_{1}|^{-2}$ are now to be
interpreted as arising due to the following operators

$\displaystyle J^{(2)}_{+,-p}\,,$ $\displaystyle A^{\pm}_{+,-p}$
$\displaystyle=J^{(2)}_{+,-p}V^{D}_{\frac{\mp 1}{2},\frac{\mp
1}{2\Delta_{-}},\frac{\mp 1}{2\Delta_{-}}}\tilde{V}^{D}_{\frac{\mp
1}{2},\frac{\pm 1}{2\Delta_{+}},\frac{\pm 1}{2\Delta_{+}}}\,,$ (69a)
$\displaystyle J^{(2)}_{-,-p}\,,$ $\displaystyle A^{\pm}_{-,-p}$
$\displaystyle=J^{(2)}_{-,-p}V^{D}_{\frac{\mp 1}{2},\frac{\pm
1}{2\Delta_{-}},\frac{\pm 1}{2\Delta_{-}}}\tilde{V}^{D}_{\frac{\mp
1}{2},\frac{\mp 1}{2\Delta_{+}},\frac{\mp 1}{2\Delta_{+}}}\,,$ (69b)
$\displaystyle A^{+}_{\pm,0}$ $\displaystyle=V^{D}_{\frac{-1}{2},\frac{\mp
1}{2\Delta_{-}},\frac{\mp
1}{2\Delta_{-}}}\tilde{V}^{D}_{\frac{-1}{2},\frac{\pm
1}{2\Delta_{+}},\frac{\pm 1}{2\Delta_{+}}}\,,$ (69c)

for each $p\in\\{1,2,3,...\\}$. This interpretation correctly ensures the
appearance of $\frac{q+\bar{q}}{2}$ in the discrete series $j$-values.

In summary, we have shown that the $AdS_{3}$ partition function (49) contains
quantised geodesics of the BTZ black hole and states obtained by twisting
them. The same partition function also captures the doubling of the continuous
series and the non-degeneracy of the discrete series. In addition, we have
also shown that the spectrum of primaries agree with vertex operator
constructions. All in all, this makes a strong case that the $AdS_{3}$ states
carry over into the BTZ spectrum.

###  8.3 Comparison with $AdS_{3}$

The partition function (49) is exactly the one written by Israel et al. Israel
for $AdS_{3}$. So, let us examine the difference in interpretation that also
revels the BTZ spectrum it encodes.

The conformal weights for global $AdS_{3}$, where the elliptic timelike
component $J^{(0)}_{\pm,0}$ (see Appendix A) is diagonalised, can be written
as

$\begin{split}\begin{aligned}
L_{0}-\bar{L}_{0}&=\frac{1}{k}(\tilde{J}_{+}^{2}-J_{+}^{2})-\frac{1}{k}(\tilde{J}_{-}^{2}-J_{-}^{2})\,,\\\
L_{0}+\bar{L}_{0}&=\frac{1}{k}(\tilde{J}_{+}^{2}-J_{+}^{2})+\frac{1}{k}(\tilde{J}_{-}^{2}-J_{-}^{2})+\frac{2(s^{2}+\frac{1}{4})}{k-2}\,,\end{aligned}\end{split}\qquad\quad\begin{split}J_{\pm}&=\tilde{J}_{\pm}\pm\frac{k}{2}\Omega\,,\end{split}$
(70)

where $\tilde{J}_{\pm}$ (and $J_{\pm}$) now denote $J^{(0)}_{\pm,0}$
eigenvalues before (and after) spectral flow by $\Omega$ units. Comparing this
with the coefficients of $2\pi i\tau_{1}$ and $-2\pi\tau_{2}$ in the exponent
of (52) we find:

$\displaystyle\tilde{J}_{+}$
$\displaystyle=\frac{k}{2}\left(w-t_{1}+\frac{n}{k}\right)\,,$ (71a)
$\displaystyle J_{+}$
$\displaystyle=-\frac{k}{2}\left(w^{\prime}+t_{1}-\frac{n^{\prime}}{k}\right)\,,$
(71b) $\displaystyle\tilde{J}_{-}$
$\displaystyle=-\frac{k}{2}\left(w-t_{1}-\frac{n}{k}\right)\,,$ (71c)
$\displaystyle J_{-}$
$\displaystyle=\frac{k}{2}\left(w^{\prime}+t_{1}+\frac{n^{\prime}}{k}\right)\,.$
(71d)

This is the correct identification because it reproduces the correct
quantisation of spacetime currents. The spectral flow parameter as calculated
from both $J_{+}-\tilde{J}_{+}$ and $J_{-}-\tilde{J}_{-}$ match and turn out
to be $\Omega=-(w+w^{\prime}).$ This follows after using the constraint
$n=n^{\prime}$, which implies that there is no winding along the time
direction of $AdS_{3}$ in global coordinates. Now, we may verify that for
states with $w+w^{\prime}=0$, the $AdS_{3}$ energy
$\tilde{J}_{+}-\tilde{J}_{-}$ is an arbitrary real number while the angular
momentum $\tilde{J}_{+}+\tilde{J}_{-}$ is quantised.

It is important that the direction conjugate to the current
$(J^{(0)}_{+},J^{(0)}_{-})$ in the $AdS_{3}$ geometry is not the direction
conjugate to $(J^{(2)}_{+},J^{(2)}_{-})$, along which the BTZ orbifolding has
been performed. We claim that this is reflected in the fact that there is no
choice of $\theta_{+},\theta_{-}\in\mathbb{C}$ that can be used in (55), to
get the $AdS_{3}$ currents (71). Instead, the matrix in (55) needs to be
replaced by a rotation matrix as in (71), to obtain the $AdS_{3}$ currents.

##  9 Summary and Discussion

In this work, we revisited the spectrum of the BTZ black hole CFT first
discussed in NS , starting with a careful study of the geodesics. The space of
classical solutions, appropriately quantised, will provide a basis of states
to construct the current algebra representations.

We first identified conditions on solutions that restrict them to regions of
the BTZ geometry without CTCs. Such a condition should be necessary because in
regions with CTCs, the time coordinate ceases to be single-valued on the
worldsheet. This will prevent a lightcone gauge choice and hence lead to
difficulties in constructing a well-defined spacetime spectrum.

The continuous series states include worldsheets that reach the boundary of
$AdS_{3}$ and thus represent operators of the dual CFT. It will be interesting
to explore these operators in a putative dual such as the symmetric product
CFT. The classical analysis revealed natural time scales associated with these
geodesics, which could be expressed in terms of the currents. Therefore, we
can expect that these time scales appear in the correlators of the
corresponding operators in the dual CFT. This may lead to insights about the
black hole interior vijay ; HongLiu .

Lifting the geodesics to classical solutions of the 2 WZW model revealed more
discrete symmetries of the space of geodesics than the ones obvious from the
BTZ point of view. The study of discrete symmetries was used to predict
multiplicities of quantum numbers in the quantised sigma model on the BTZ
background. Among the discrete symmetries was multiplication by $-\mathbb{I}$.
It revealed a classical version of the doubling Mukunda  of continuous series
representations of 2 in the hyperbolic basis. A similar doubling was also
found in the $2$-dimensional Lorentzian black hole kpy . In that context,
however, it was time-reversal symmetry that motivated this doubling. Another
important discrete symmetry was
$(r^{2}-r_{+}^{2},t)\leftrightarrow(r_{-}^{2}-r^{2},\phi)$. As proposed by
Natsuume and Satoh NS , orbifolding with respect to this T-duality could be
interpreted as truncating the BTZ geometry to exclude the region with closed
timelike curves.

We also studied solutions of the sigma model obtained by twisting the
geodesics. We noted that solutions obtained by twisting geodesics that do not
explore the region with CTCs remain in the causally sound region. Also, the
$\beta<0$ condition was invariant under twisting once the Noether ambiguity
was taken into account. However, $\beta<0$ is too strong a constraint to
impose on all solutions because many winding strings obtained by twisting
spacelike geodesics do not satisfy $\beta<0$; yet, they stay out of the region
with CTCs and may also satisfy Virasoro conditions.

While studying twisted sectors, we also noted that the highest weight discrete
series does not map to another lowest weight discrete series under spectral
flow in the hyperbolic basis. This is unlike the case in $AdS_{3}$ and implies
a doubling of the discrete series, just like the continuous series. This fits
nicely with the fact that BTZ states may be read off from the $AdS_{3}$
partition function without changing the relative contributions to the
partition function from continuous and discrete series representations.

The quantum spectrum of the BTZ geometry is expected to contain states
obtained by quantising the geodesics. These were already identified through a
coset construction by NS . We then showed that the partition function of the
$AdS_{3}$ sigma model as constructed in Israel  contains all the states
identified classically. Expanding this partition function in a $q$-series, we
showed that it is possible to reinterpret the quantum numbers of primaries
read off from this expansion to match those calculated by Natsuume and Satoh
NS . This interpretation reproduced correct geometric features of the
spectrum, such as angular momentum quantisation and the absence of winding
around the non-compact direction. We also verified its consistency with the
Noether ambiguity Esko  encountered in the study of twisted sectors. The
descendants of these primaries and their degeneracies could also be understood
using free field realisations of the $\hat{\mathfrak{sl}}(2)_{k}$ algebra
introduced by Satoh and Hemming Satoh ; Hemming .

The fact that the same partition function could describe two distinct
spacetimes deserves some comment. The BTZ interpretation required the
introduction of spacetime moduli $r_{\pm}$ defining the black hole. This
differed from its usual $AdS_{3}$ interpretation in an essential way – the
target space isometry currents were related to quantum numbers appearing in
the partition function by a boost in the former case and by a rotation in the
latter.

However, the partition function includes discrete series primaries (even in
the untwisted sector) corresponding to geodesics that explore the region with
CTCs. This unsavoury situation could mean that the actual set of states is a
subset, perhaps restricted by the $\beta<0$ condition. If so, the internal CFT
will have to be chosen so that primaries with $\beta>0$ are not physical. In
this context, it is instructive to examine the wavefunctions for scalar fields
near the BTZ singularity $r^{2}=0$. It is easy to show that the relevant
eigenvalue problem reduces to
$(\frac{d^{2}}{dy^{2}}-\frac{4M}{J^{2}}\frac{d}{dy}+\frac{4}{J^{4}}\beta)\Phi=0$
(where $y=r^{2}$) near $r^{2}=0$. If $\beta<0$, this only has exponential
solutions $\Phi_{\pm}(y)\propto\exp[\frac{2y}{J^{2}}(M\pm\sqrt{M^{2}-\beta})]$
which can be interpreted to mean that there is no probability current (at the
level of spacetime QFTs) into or from the region $r^{2}<0$. One can expect
that the tensor product of such exponential wavefunctions will give only
exponentials. Thus, the OPE of such representations in the CFT will also close
among themselves. This is perhaps satisfactory. Of course, we have to check
that the level matching condition suffices to ensure that OPEs are single-
valued on the worldsheet.

Alternatively, we may orbifold the spectrum read off here by the T-duality
symmetry as proposed in NS . It will be interesting to study the target space
interpretation of primaries twisted under T-duality. We also wonder if this
orbifold involves a projection onto discrete energy (and angular momentum)
eigenvalues, similar to those appearing in Satoh’s proposal Satoh . This
proposal was constructed by starting with states having $J_{+}=\pm J_{-}$. As
we already observed, one set of such timelike geodesics (17) does not
necessarily satisfy $\beta<0$ and hence, may pass into the region with CTCs.
Finally, a more radical conclusion could be that the regions containing CTCs
are tolerated in string theory.

A more technical question is whether it is possible to rewrite the proposed
partition function (49) (or propose another modular invariant spectrum) in a
manner that replaces $|\vartheta_{1}|^{-2}$ with its analytic continuation
natural to $\hat{\mathfrak{sl}}(2)_{k}$ in the hyperbolic basis.
Interestingly, one possible answer to this question arises from a double Wick
rotation from $AdS_{3}$ to BTZ.

Consider the global parametrisation
$\exp[\frac{i\sigma_{2}}{2}(t_{\text{g}}+\phi_{\text{g}})]\exp(\rho\sigma_{3})\exp[\frac{i\sigma_{2}}{2}(t_{\text{g}}-\phi_{\text{g}})]$
of $AdS_{3}$. Here, $\phi_{\text{g}}$ denotes a $2\pi$-periodic angular
coordinate and $t_{\text{g}}$ denotes a non-compact timelike coordinate. Now,
decompactify the $\phi_{\text{g}}$ direction444The $AdS_{3}$ metric
$ds^{2}=-dt_{\text{g}}^{2}+d\rho^{2}+\rho^{2}d\phi_{\text{g}}^{2}$ for small
$\rho$ is that of plane parametrised by polar coordinates
$(\rho,\phi_{\text{g}})$. Hence, decompactifying $\phi_{\text{g}}$ maps the
single point $\rho=0$ to an entire line parametrised by $\phi_{\text{g}}$. and
then perform the double Wick rotation:

$\begin{split}(\phi_{\text{g}},t_{\text{g}})=(i\hat{t},i\hat{\phi})\,,\end{split}\qquad\qquad\qquad\begin{split}\begin{aligned}
\hat{t}&=r_{+}t-r_{-}\phi\,,\\\ \hat{\phi}&=r_{+}\phi-
r_{-}t\,.\end{aligned}\end{split}$ (72)

Under this double Wick rotation, the action (hence, metric and $B$-field) of
the 2 WZW model transforms into that of the BTZ sigma model (75). Also, the
$2\pi$-periodicity of the BTZ angular coordinate $\phi$ implies the
compactification (equivalently, a $\mathbb{Z}$ orbifold)

$(\hat{\phi},\hat{t})\sim(\hat{\phi},\hat{t})+(2\pi r_{+},-2\pi r_{-})\,.$
(73)

As a result, one may expect that the BTZ partition function is obtained by
setting the temperature and chemical potential in the thermal $AdS_{3}$
partition function obtained by Maldacena et al. mog2  to be
$(\beta,\mu\beta)=(2\pi r_{+},-2\pi ir_{-})$. This results in the following
modular invariant

$Z_{\text{WickRotated}}=r_{+}\sqrt{\frac{k-2}{\tau_{2}}}\sum_{m,n\in\mathbb{Z}}\frac{e^{-\frac{\pi
k}{\tau_{2}}r_{+}^{2}|n\tau-m|^{2}+\frac{2\pi}{\tau_{2}}\left(r_{+}(n\tau_{1}-m)-ir_{-}n\tau_{2}\right)^{2}}}{\vartheta_{1}(-i\Delta_{-}(n\tau-m)|\tau)\bar{\vartheta}_{1}(i\Delta_{+}(n\bar{\tau}-m)|\bar{\tau})}\,.$
(74)

In contrast to the partition function (49), notice that the denominator
$\vartheta_{1}$ and $\bar{\vartheta}_{1}$ have purely imaginary arguments for
$\tau\in\mathbb{R}$. This is natural to the hyperbolic basis for the following
reason. Just like the global coordinates
$(t_{\text{g}}+\phi_{\text{g}},t_{\text{g}}-\phi_{\text{g}})$, the arguments
of $\vartheta_{1}$ and $\bar{\vartheta}_{1}$ are chemical potentials conjugate
to zero modes $(J_{+,0}^{(0)},J_{-,0}^{(0)})$ of the elliptic generator. So,
the double Wick rotation (72) maps them to chemical potentials conjugate to
the hyperbolic generator. This allows us to interpret the sum in (74) as a sum
over twists and shifts, that would naturally arise from the orbifolding
condition (3) that defines the BTZ black hole.

Exactly as in the case of thermal $AdS_{3}$ mog2 , the contribution
$-\frac{\pi k}{\tau_{2}}r_{+}^{2}|n\tau-m|^{2}$ to the exponential is the zero
mode WZW action for the sector with
$\phi(z+2\pi,\bar{z}+2\pi)=\phi(z,\bar{z})+2\pi n$ and
$\phi(z+2\pi\tau,\bar{z}+2\pi\bar{\tau})=\phi(z,\bar{z})+2\pi m$. We may
attempt to expand this modular invariant as a $q$-series to read off the
spectrum it encodes. If $r_{-}$ were purely imaginary, this simply reveals the
spectrum of a CFT with thermal $AdS_{3}$ target space having
$(\beta,\mu\beta)=(2\pi r_{+},-2\pi ir_{-})$. However, for purely real
$r_{-}$, it is not clear to us how an expansion that reveals the Lorentzian
BTZ spectrum can be performed.

Finally, it will of be some interest to study the spacetime modular
transformation malstrom  between Euclidean BTZ and thermal $AdS_{3}$ in terms
of the partition functions (49) and (74).

##  Appendix A The 2 WZW model

In this appendix, we present the details of the 2 WZW model to clarify our
conventions. The action for this model is given by

$\displaystyle S_{\text{WZW}}[g]$ $\displaystyle=\frac{k}{8\pi}\int
d\tau\,d\sigma\,\sqrt{-h}\,\text{Tr}\left(\partial_{a}\,g^{-1}\partial^{a}\,g\right)+k\,\Gamma_{\text{WZ}}[g]\,,$
(75) $\displaystyle\Gamma_{\text{WZ}}[g]$
$\displaystyle=-\frac{1}{12\pi}\int\epsilon^{abc}\,\text{Tr}\left(\partial_{a}g\,g^{-1}\partial_{b}g\,g^{-1}\partial_{c}g\,g^{-1}\right)\,.$
(76)

Here, the trace is calculated in the 2-dimensional representation of 2.

We chose the normalisation of 2 generators such that
$\text{Tr}(\tau^{a}\tau^{b})=\frac{1}{2}\eta^{ab}$ and where
$\eta_{ab}={\text{diag}}(-1,1,1)$. More specifically,
$\tau^{0}=\frac{i\sigma^{2}}{2}$, $\tau^{1}=\frac{\sigma^{1}}{2}$ and
$\tau^{2}=\frac{\sigma^{3}}{2}$. The WZW currents are defined as
$J_{+}=-k\partial_{+}g\,g^{-1}$ and $J_{-}=kg^{-1}\,\partial_{-}g$. Here,
$\partial_{\pm}$ denote derivatives with respect to worldsheet lightcone
coordinates $x^{\pm}=\tau\pm\sigma$. The equations of motion that follow from
(75) are $\partial_{\pm}J_{\mp}=0$. The components $J^{(a)}_{\pm}$ of currents
are defined by $J^{(a)}_{\pm}=\text{Tr}(\tau^{a}J_{\pm})$ and
$J^{(\pm)}_{+}=J^{(0)}_{+}\pm J^{(1)}_{+}$. The stress tensor (defined as
$T_{ab}=\frac{-4\pi}{\sqrt{-h}}\frac{\delta S_{\text{WZW}}}{\delta h^{ab}}$)
turns out to be $T_{\pm\pm}=\frac{1}{k}\eta_{ab}J^{(a)}_{\pm}J^{(b)}_{\pm}$ in
terms of the currents. In the $+D_{1}^{+}$ charts (5), the WZW action
evaluates to

$S_{\text{WZW}}[g]=\frac{-k}{2\pi}\int
dx^{+}\,dx^{-}\,\Bigl{[}(r_{-}^{2}+r_{+}^{2}-r^{2})\partial_{+}t\partial_{-}t+\frac{r^{2}\partial_{+}r\partial_{-}r}{\left(r^{2}-r_{-}^{2}\right)\left(r^{2}-r_{+}^{2}\right)}+r^{2}\partial_{-}\phi\partial_{+}\phi\\\
-r_{-}r_{+}(\partial_{-}t\partial_{+}\phi+\partial_{-}\phi\partial_{+}t)+\left(r^{2}-r_{+}^{2}\right)(\partial_{+}\phi\partial_{-}t-\partial_{+}t\partial_{-}\phi)\Bigr{]}\,,$
(77)

and the components of currents to be diagonalised upon quantisation are
expressed as

$\displaystyle J_{+}^{(2)}$
$\displaystyle=\frac{-k}{r_{+}-r_{-}}\biggl{[}\partial_{+}\phi\left(r^{2}-r_{+}r_{-}\right)+\partial_{+}t\left(r_{+}^{2}+r_{-}^{2}-r_{+}r_{-}-r^{2}\right)\biggr{]}\,,$
(78a) $\displaystyle J_{-}^{(2)}$
$\displaystyle=\frac{k}{r_{+}+r_{-}}\biggl{[}\partial_{-}\phi\left(r^{2}+r_{+}r_{-}\right)+\partial_{-}t\left(r^{2}-r_{+}^{2}-r_{-}^{2}-r_{+}r_{-}\right)\biggr{]}\,.$
(78b)

From the equations of motion, it follows that the currents $J_{+}$ and $J_{-}$
evaluated along a classical solution are purely left- and right- moving,
respectively. Hence, they admit the following mode expansions in terms of
worldsheet lightcone coordinates:

$J_{+}^{(a)}=\sum_{n\in\mathbb{Z}}J^{(a)}_{+,n}\,e^{-inx^{+}}\qquad\qquad
J_{-}^{(a)}=\sum_{n\in\mathbb{Z}}J^{(a)}_{-,n}\,e^{-inx^{-}}$ (79)

Expressing the components in terms of the coordinates $(r,t,\phi)$ and their
corresponding momenta (each, indexed by the worldsheet coordinate $\sigma$)
allows us to compute the following equal-$\tau$ Poisson brackets

$\displaystyle\left\\{J_{+}^{(2)}(\cdot,\sigma),J_{+}^{(2)}(\cdot,\sigma^{\prime})\right\\}$
$\displaystyle=\pi k\partial_{\sigma}\delta(\sigma-\sigma^{\prime})\,,$ (80a)
$\displaystyle\left\\{J_{+}^{(2)}(\cdot,\sigma),J_{+}^{(\pm)}(\cdot,\sigma^{\prime})\right\\}$
$\displaystyle=\mp 2\pi
J_{+}^{(\pm)}(\cdot,\sigma^{\prime})\delta(\sigma-\sigma^{\prime})\,,$ (80b)
$\displaystyle\left\\{J_{+}^{(+)}(\cdot,\sigma),J_{+}^{(-)}(\cdot,\sigma^{\prime})\right\\}$
$\displaystyle=4\pi
J_{+}^{(2)}(\cdot,\sigma^{\prime})\delta(\sigma-\sigma^{\prime})-2\pi
k\partial_{\sigma}\delta(\sigma-\sigma^{\prime})\,.$ (80c)

We may Fourier transform these Poisson brackets evaluated along a classical
solution and then promote them to commutators via
$[\cdot,\cdot]=i\\{\cdot,\cdot\\}$. This canonical quantisation results in the
$\hat{\mathfrak{sl}}(2)_{k}$ algebra, expressed in the hyperbolic basis as

$\displaystyle\left[J^{(2)}_{+,n},J^{(2)}_{+,m}\right]$
$\displaystyle=\frac{k}{2}n\delta_{n+m,0}\,,$ (81a)
$\displaystyle\left[J^{(2)}_{+,n},J^{(\pm)}_{+,m}\right]$ $\displaystyle=\mp
iJ^{(\pm)}_{+,n+m}\,,$ (81b)
$\displaystyle\left[J^{(+)}_{+,n},J^{(-)}_{+,m}\right]$
$\displaystyle=2iJ^{(2)}_{+,n+m}-kn\delta_{n+m,0}\,.$ (81c)

The right-moving current modes $J^{(a)}_{-,n}$ also satisfy the same algebra.

##  Appendix B Special cases

In this appendix, we shall consider some special cases of previous
considerations, namely the non-rotating $(r_{-}=0)$ and the extremal
$(r_{+}=r_{-})$ black holes.

In the case of the non-rotating black hole, the BTZ identification becomes the
axial action by $e^{\pi r_{+}\sigma^{3}}$. At the level of geodesics, we see
that $\beta=L^{2}r_{+}^{2}>0$. Therefore, all the timelike geodesics enter the
region with closed timelike curves. We also observe that
$\Delta_{+}=\Delta_{-}$ which implies $\theta_{+}=\theta_{-}$ in the
identification (55) of currents. With this substitution, the interpretation of
quantum numbers appearing in the partition function follows without any
change.

However, this is not the case with the extremal limit. Naive substitution of
$r_{-}=r_{+}$ in the parametrisation (5) gives singular matrices. Further,
substituting $r_{+}=r_{-}$ in the currents evaluated along geodesics (say
(14)) makes them singular. So, we need to proceed with a little care. To
realise the left and right 2 symmetries of the extremal BTZ black hole, we
need to parametrise 2 by charts such that the BTZ identification
$\phi\sim\phi+2\pi$ takes the form discussed in the work of Maldacena and
Strominger malstrom . Such a chart is

$e^{\frac{1}{2}(t+\phi)\sigma^{+}}\begin{pmatrix}-\frac{r_{+}}{\rho}&\frac{1}{2\rho}\\\
-\rho&-\frac{\rho}{2r_{+}}\end{pmatrix}e^{-r_{+}(t-\phi)\sigma^{3}}\,,$ (82)

where $\sigma^{\pm}=\sigma^{1}\pm i\sigma^{2}$ and $\rho^{2}=r^{2}-r_{+}^{2}$.
With this parametrisation, the energy $E$ and angular momentum $L$ of a
geodesic may be expressed in terms of the left parabolic generator
$J_{+}^{(+)}=\frac{1}{2}\text{Tr}(\sigma^{+}J_{+})$ and the right hyperbolic
generator $J_{-}^{(2)}$ evaluated along the geodesic as

$J_{+}^{(+)}=-\frac{k}{2}(E+L)\,,\qquad\qquad
J_{-}^{(2)}=-\frac{k}{2}\frac{E-L}{2r_{+}}\,.$ (83)

Twisted sectors of the extremal case are also different from those of generic
BTZ black holes. The analogue of twisting (33) in the extremal case is

$g_{\Omega}(\tau,\sigma)=e^{\frac{\Omega}{2}x^{+}\sigma^{+}}\tilde{g}(\tau,\sigma)e^{-\Omega
r_{+}x^{-}\sigma^{3}}\,,\qquad\qquad x^{\pm}=\tau\pm\sigma\,,$ (84)

and components of the left current evaluated along the twisted solution are
related to those before twisting in a manner different from the $r_{+}\neq
r_{-}$ case. In particular, the component of the left current to be
diagonalised upon quantisation retains its value after twisting

$\displaystyle J_{+}^{(+)}$ $\displaystyle=\tilde{J}_{+}^{(+)}\,,$
$\displaystyle J_{-}^{(+)}$
$\displaystyle=e^{-2r_{+}x^{-}\Omega}\tilde{J}_{-}^{(+)}\,,$ (85a)
$\displaystyle J_{+}^{(-)}$ $\displaystyle=\tilde{J}_{+}^{(-)}+k\Omega+2\Omega
x^{+}\tilde{J}_{+}^{(2)}+(\Omega x^{+})^{2}\tilde{J}_{+}^{(+)}\,,$
$\displaystyle J_{-}^{(-)}$
$\displaystyle=e^{+2r_{+}x^{-}\Omega}\tilde{J}_{-}^{(-)}\,,$ (85b)
$\displaystyle J_{+}^{(2)}$ $\displaystyle=\tilde{J}_{+}^{(2)}+\Omega
x^{+}\tilde{J}_{+}^{(+)}\,,$ $\displaystyle J_{-}^{(2)}$
$\displaystyle=\tilde{J}_{-}^{(2)}-kr_{+}\Omega\,.$ (85c)

We may verify that the currents evaluated along the twisted solution also
satisfy the same Poisson brackets (81) as those evaluated along the solution
before twisting.

The relation between values of the stress tensor evaluated along twisted and
untwisted solutions follows from (85); it is

$T_{++}=\tilde{T}_{++}-\Omega\tilde{J}_{+}^{(+)}\,,\qquad\qquad
T_{--}=\tilde{T}_{--}-2r_{+}\Omega\tilde{J}_{-}^{(2)}+kr_{+}^{2}\Omega^{2}\,.$
(86)

This agrees with $L_{0}\pm\bar{L}_{0}$ read off from the expansion (52), if we
identify eigenvalues $(J_{+},J_{-})$ and $(\tilde{J}_{+},\tilde{J}_{-})$ of
$(J_{+}^{(+)},J_{-}^{(2)})$ after and before spectral flow respectively as

$\begin{pmatrix}\tilde{J}_{+}\\\ J_{+}\\\ \tilde{J}_{-}\\\
J_{-}\end{pmatrix}=\frac{k}{2}\begin{pmatrix}\frac{1}{2}&-\frac{1}{2}&0&0\\\
\frac{1}{2}&-\frac{1}{2}&0&0\\\ 0&0&-\cosh\theta_{-}&\sinh\theta_{-}\\\
0&0&-\sinh\theta_{-}&\cosh\theta_{-}\end{pmatrix}\begin{pmatrix}w^{\prime}+t_{1}-\frac{n^{\prime}}{k}\\\
w-t_{1}+\frac{n}{k}\\\ w^{\prime}+t_{1}+\frac{n^{\prime}}{k}\\\
w-t_{1}-\frac{n}{k}\end{pmatrix}\,,$ (87)

where $e^{-\theta_{-}}=\Delta_{+}=2r_{+}$. Notice that the matrix used has
zero determinant. In this sense, it is different from the generic BTZ case
(55) as well as the $AdS_{3}$ case (71). Exactly as in those cases, we can
verify that with this identification, the spectral flow parameter is
$\Omega=-(w+w^{\prime})$. The energy and angular momentum before and after
spectral flow have the same expressions (58) and (59) as the generic case, but
with $M=2r_{+}^{2}=J$. Hence, this identification is consistent with angular
momentum quantisation and allows the energy to have an arbitrary real value.
It is also quantitatively consistent with the Noether ambiguity (39) discussed
earlier.

##  Appendix C Free boson conformal blocks and the $\vartheta_{1}$-function

The free boson conformal blocks appearing in the partition function (49) are
defined as

$\zeta{\biggl{[}\begin{array}[]{c}w\\\
m\end{array}\biggr{]}}(k)=\sqrt{\frac{k}{\tau_{2}}}\exp\left(-\frac{\pi
k}{\tau_{2}}|w\tau-m|^{2}\right)\,.$ (88)

The $\vartheta_{1}$-function is defined as

$\vartheta_{1}(v|\tau)=\sum_{p\in\mathbb{Z}}e^{\pi
i\tau(p+\frac{1}{2})^{2}+2\pi i(v+\frac{1}{2})(p+\frac{1}{2})}\,.$ (89)

It can also be expressed (letting, $q=e^{2\pi i\tau}$) as the infinite product

$\vartheta_{1}(v|\tau)=-2q^{1/8}\sin\pi v\prod_{p=1}^{\infty}(1-e^{2i\pi
v}q^{p})(1-q^{p})(1-e^{-2i\pi v}q^{p})\,.$ (90)

To expand $|\vartheta_{1}|^{-2}$ as a power series to obtain (50), we start
with the product representation

$\frac{1}{|\vartheta(v|\tau)|^{2}}=\frac{(q\bar{q})^{-1/8}}{4\sin(\pi
v)\sin(\pi\bar{v})}\prod_{p=1}^{\infty}\frac{1}{(1-q^{p})(1-\bar{q}^{p})}\\\
\times\frac{1}{(1-e^{2iv\pi}q^{p})(1-e^{-2iv\pi}q^{p})(1-e^{-2i\bar{v}\pi}\bar{q}^{p})(1-e^{2i\bar{v}\pi}\bar{q}^{p})}\,.$
(91)

Each factor in the product may be expanded into a geometric series. However,
for the series to converge, a different expansion needs to performed depending
on ${\text{Im}}(v)$. In the partition function (49), we have $v=s_{1}\tau-
s_{2}$ with $s_{1},s_{2}\in(0,1)$ and hence $0<{\text{Im}}(v)<\tau_{2}$. A
convergent series expansion of the prefactor from (91) in this domain is

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

Its convergence follows from $|e^{\pm 2i\pi(s_{1}\tau_{1}-s_{2})}e^{-2\pi
s_{1}\tau_{2}}|<1$ (for $\tau_{2}>0$). A similar expansion can be used for
each factor in the infinite product part of (91). These expansions will
converge because $s_{1}<p\in\\{1,2,3,\ldots\\}$. The resulting expansion is

$\frac{1}{|\vartheta(s_{1}\tau-
s_{2}|\tau)|^{2}}=\sum_{q,\bar{q},N,\bar{N}}e^{2\pi is_{2}(\bar{q}-q)+2\pi
i\tau_{1}[s_{1}(q-\bar{q})+N-\bar{N}]-2\pi\tau_{2}[s_{1}(1+q+\bar{q})-\frac{1}{4}+N+\bar{N}]}\,,$
(93)

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

$\displaystyle q$
$\displaystyle=P_{+,0}+\sum_{p=1}^{\infty}(P^{+}_{+,p}-P^{-}_{+,p})\,,$
$\displaystyle\bar{q}$
$\displaystyle=P_{-,0}+\sum_{p=1}^{\infty}(P^{+}_{-,p}-P^{-}_{-,p})\,,$ (94a)
$\displaystyle N$
$\displaystyle=\sum_{p=1}^{\infty}p(P_{+,p}+P^{+}_{+,p}+P^{-}_{+,p})\,,$
$\displaystyle\bar{N}$
$\displaystyle=\sum_{p=1}^{\infty}p(P_{-,p}+P^{+}_{-,p}+P^{-}_{-,p})\,.$ (94b)

###### Acknowledgements.

O.N. and A.S. thank IISER Mohali for providing hostel accommodation for part
of their MS thesis work. They were supported, in part, through the INSPIRE
SHE. All of us thank Jitsi and Overleaf for providing valuable and free
services which enabled us to engage in a fruitful collaboration.

#### Note added.

As this manuscript was being finalised, the paper ST  appeared on the arXiv
where the modular invariant (74) was presented as the partition function of
the Euclidean BTZ black hole.

## References

  * (1) M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, _Geometry of the (2+1) black hole_ , _Phys. Rev. D_ 48 (1993) 1506. Erratum: [Phys. Rev. D 88, 069902 (2013)] doi:10.1103/PhysRevD.48.1506, 10.1103/PhysRevD.88.069902 [gr-qc/9302012].
  * (2) J. M. Maldacena and H. Ooguri, _Strings in AdS(3) and SL(2,R) WZW model 1.: The Spectrum_ , _J. Math. Phys._ 42, (2001) 2929. doi:10.1063/1.1377273 [hep-th/0001053].
  * (3) J. M. Maldacena, H. Ooguri and J. Son, _Strings in AdS(3) and the SL(2,R) WZW model. Part 2. Euclidean black hole_ , _J. Math. Phys._ 42 (2001) 2961. doi:10.1063/1.1377039 [hep-th/0005183].
  * (4) K. P. Yogendran, _Closed Strings in the 2D Lorentzian Black Hole_ , arXiv:1808.10109 [hep-th].
  * (5) D. Israel, C. Kounnas and M. P. Petropoulos, _Superstrings on NS5 backgrounds, deformed AdS(3) and holography_ , _JHEP_ 10 (2003) 028. doi:10.1088/1126-6708/2003/10/028 [arXiv:hep-th/0306053 [hep-th]].
  * (6) J. G. Kuriyan, N. Mukunda, E. C. G. Sudarshan, _Master Analytic Representation: Reduction of $O(2,1)$ in an $O(1,1)$ Basis_, _J. Math. Phys._ 9, (1968) 2100. doi: 10.1063/1.1664551
  * (7) M. Natsuume and Y. Satoh, _String theory on three-dimensional black holes_ , _Int. J. Mod. Phys. A_ 13 (1998) 1229. doi:10.1142/S0217751X98000585 [hep-th/9611041].
  * (8) S. Hemming, E. Keski-Vakkuri and P. Kraus, _Strings in the extended BTZ space-time_ , _JHEP_ 0210 (2002) 006. doi:10.1088/1126-6708/2002/10/006 [hep-th/0208003].
  * (9) J. M. Maldacena and A. Strominger, _AdS(3) black holes and a stringy exclusion principle_ , _JHEP_ 9812, (1998) 005. doi:10.1088/1126-6708/1998/12/005 [hep-th/9804085].
  * (10) S. Hemming and E. Keski-Vakkuri, _The Spectrum of strings on BTZ black holes and spectral flow in the SL(2,R) WZW model_ , _Nucl. Phys. B_ 626, (2002) 363. doi:10.1016/S0550-3213(02)00021-4 [hep-th/0110252].
  * (11) M. Rangamani and S. F. Ross, _Winding tachyons in BTZ_ , _Phys. Rev. D_ 77 (2008) 026010. doi:10.1103/PhysRevD.77.026010 [arXiv:0706.0663 [hep-th]].
  * (12) S. B. Giddings, J. Polchinski and A. Strominger, _Four-dimensional black holes in string theory_ , _Phys. Rev. D_ 48 (1993) 5784-5797. doi:10.1103/PhysRevD.48.5784 [arXiv:hep-th/9305083 [hep-th]].
  * (13) N. Cruz, C. Martinez and L. Pena, _Geodesic structure of the (2+1) black hole_ , _Class. Quant. Grav._ 11 (1994) 2731-2740. doi:10.1088/0264-9381/11/11/014 [arXiv:gr-qc/9401025 [gr-qc]].
  * (14) J. Troost, _Winding strings and AdS(3) black holes_ , _JHEP_ 0209 (2002) 041. doi:10.1088/1126-6708/2002/09/041 [hep-th/0206118].
  * (15) J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser, _String theory on AdS(3)_ , _JHEP_ 12 (1998) 026. doi:10.1088/1126-6708/1998/12/026 [arXiv:hep-th/9812046 [hep-th]].
  * (16) L. Eberhardt, _Partition functions of the tensionless string_ , _JHEP_ 03 (2021), 176. doi:10.1007/JHEP03(2021)176 [arXiv:2008.07533 [hep-th]].
  * (17) Y. Satoh, _Ghost - free and modular invariant spectra of a string in SL(2,R) and three-dimensional black hole geometry_ , _Nucl. Phys. B_ 513 (1998), 213-228. doi:10.1016/S0550-3213(97)00701-3 [arXiv:hep-th/9705208 [hep-th]].
  * (18) S. Hemming, _On free field realizations of strings in BTZ_ , _Int. J. Mod. Phys. A_ 19 (2004), 1579-1588. doi:10.1142/S0217751X04017859 [arXiv:hep-th/0304009 [hep-th]].
  * (19) V. Balasubramanian, A. Kar and G. Sárosi, _Holographic Probes of Inner Horizons_ , _JHEP_ 06 (2020), 054. doi:10.1007/JHEP06(2020)054 [arXiv:1911.12413 [hep-th]].
  * (20) G. Festuccia and H. Liu, _Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I._ , _JHEP_ 04 (2006), 044. doi:10.1088/1126-6708/2006/04/044 [arXiv:hep-th/0506202 [hep-th]].
  * (21) S. K. Ashok and J. Troost, _Twisted Strings in Three-dimensional Black Holes_ , arXiv:2112.08784.