Twisted Strings in Three-dimensional Black Holes

Sujay K. Ashoka and Jan Troostb

aThe Institute of Mathematical Sciences,

Homi Bhabha National Institute (HBNI),

IV Cross Road, C.I.T. Campus,

Taramani, Chennai, India 600113

b Laboratoire de Physique de l’École Normale Supérieure

CNRS, ENS, Université PSL, Sorbonne Université,

Université de Paris F-75005 Paris, France

We revisit the classical dynamics of fundamental strings in BTZ black holes
with NSNS flux. We analyze probe strings in the black hole background and
determine their classical energy using a Nambu-Goto action. Three-dimensional
gravity has a non-local effect on the metric surrounding a massive object
which can be measured by winding strings. We also study the behaviour of short
and long strings in the Wess-Zumino-Witten orbifold and show that their energy
matches the long string probe analysis. In the process, we fix subtleties in
the Lorentzian choice of NSNS flux. In Euclidean signature, we analyze the
one-loop fundamental string partition function on the BTZ black hole and
interpret the off-shell degrees of freedom as those of a coset orbifold.
Finally, we speculate on how to identify the quantum Lorentzian string
spectrum from the Euclidean orbifold partition function.

###### Contents

  1. 1 Introduction
  2. 2 String Theory on Black Hole Backgrounds
    1. 2.1 The BTZ Black Hole
    2. 2.2 The Black Hole and the Group
    3. 2.3 An Ambiguity
  3. 3 Classical Strings in the Black Hole Orbifold
    1. 3.1 The Geodesics
    2. 3.2 The Winding Strings
    3. 3.3 The Black Hole Orbifold
    4. 3.4 The String Energy and Angular Momentum
    5. 3.5 The Energy of Wound Geodesics
  4. 4 The Winding String Probing a Black Hole
    1. 4.1 The Winding String Probe
    2. 4.2 The Winding String in the Fefferman-Graham Expansion
  5. 5 The Partition Function and the Euclidean Twist
    1. 5.1 The Interpretation of the Partition Function
    2. 5.2 The Sigma Model on the Euclidean Black Hole
    3. 5.3 Expectations for Twisted Sectors
    4. 5.4 The Partition Function
  6. 6 Conclusions
  7. A The Euclidean Black Hole Background
    1. A.1 The Metric on the Euclidean Black Hole
    2. A.2 The Determination of the Partition Function
  8. B The Energy of a Twisted Short String

##  1 Introduction

Anti-de Sitter space-times play a central role in our study of holography in
quantum gravity. Moreover, string theory provides a theory of quantum gravity
in which holography is on a firm footing. Thus, understanding string theory in
asymptotically anti-de Sitter space-times even better remains a crucial task.
Analyzing the classical dynamics of fundamental strings in three-dimensional
anti-de Sitter space-time was instrumental to make headway in the
determination of the spectrum in the quantum theory [1, 2]. See e.g. [3] and
references therein for earlier work. This laid the basis for making progress
in an example of holography which is controlled even at string scale
curvature. It remains one of the best understood examples of holography in
quantum gravity. See e.g. [4, 5, 6, 8, 7] for further recent progress.

In this paper, we wish to make a contribution to our understanding of string
theory in other asymptotically $AdS_{3}$ space-times. Prominent examples of
such backgrounds are provided by black hole solutions in three-dimensional
anti-de Sitter space-time, the BTZ black holes [9, 10]. While many results on
the classical solutions of fundamental strings moving in these black holes
have been obtained, as well as some insight in the quantum spectrum [11, 12,
13, 14, 15, 16, 17, 18], we believe they are usefully extended by our results.
Indeed, any further step towards understanding a quantum theory of gravity on
a black hole background is worthwhile.

In section 2, we review the Lorentzian BTZ black hole background, its topology
as well as its relation to the $SL(2,\mathbb{R})$ group manifold. We will
recognize that string world sheet embeddings in the BTZ space-time give rise
to interesting questions as to the definition of the string world sheet two-
form potential term. In section 3 we review classical string solutions to the
extent that they were known, based on particle geodesics with added winding.
We embed the solutions in the Wess-Zumino-Witten orbifold representing the BTZ
black hole and analyze their energy in more detail. In section 4, we perform a
Nambu-Goto analysis of a long string winding the black hole. We relate the
analysis to that of a long string in a generic asymptotically $AdS_{3}$ space-
time with curved boundary. Comparing the analysis to the Wess-Zumino-Witten
results allows us to identify a preferred definition of the NSNS potential
term in the world sheet model. Section 5 is dedicated to a Euclidean
counterpart to the preceding sections. We analyze how to wind Euclidean
solutions such that one obtains classical solutions in the twisted sector of
the black hole orbifold. The path integral in the Euclidean orbifold coset
conformal field theory is computed, and we identify expected contributions to
the partition sum. Section 6 contains conclusions while Appendix A briefly
reviews relevant geometries. Appendix B ties up a loose end.

##  2 String Theory on Black Hole Backgrounds

In this section, we review the topology of the BTZ black hole and the relation
of the black hole to the $SL(2,\mathbb{R})$ group manifold [9, 10]. We stress
that the topology of the black hole leads to subtleties in the definition of
the perturbative fundamental string world sheet model.

###  2.1 The BTZ Black Hole

We study fundamental string theory on the BTZ black hole background [9, 10]
with metric

$ds^{2}=\frac{l^{2}r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}dr^{2}-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{l^{2}r^{2}}dt^{2}+r^{2}(d\phi-\frac{r_{+}r_{-}}{lr^{2}}dt)^{2}~{}\,,$
(2.1)

supplemented with an NSNS flux $H_{(3)}$ proportional to the volume of space-
time. We imagine a number of extra dimensions necessary to form a consistent
bosonic or super string background [19]. The inner horizon at radius $r_{-}$
and the outer horizon at radius $r_{+}$ are set by the black hole mass $M$ and
angular momentum $J$ through:111Our convention is that the radius $r_{-}$ is a
positive real number. The mass $M$ is dimensionless while the angular momentum
$J$ has dimension of length. The Ricci scalar equals $R=-6/l^{2}$.,222 Note
that if we make the coordinates $r,t$ dimensionless through the replacement
$(r,t)\rightarrow(lr,lt)$, then we find a factor of $l^{2}$ up front in the
metric. We will set this length scale equal to $l=\sqrt{k\alpha^{\prime}}$
later on.

$M=\frac{r_{+}^{2}+r_{-}^{2}}{l^{2}}~{},\qquad J=\pm\frac{2r_{+}r_{-}}{l}\,.$
(2.2)

We will often set the cosmological constant length scale $l$ to be equal to
one by a choice of units. The coordinate system (2.1) in which the BTZ black
hole is represented makes for an analogy to common coordinate choices for
four-dimensional black holes. However, the geometry of BTZ black holes is
quite different.

###  2.2 The Black Hole and the Group

As discussed in great detail by BHTZ [10], the BTZ black hole arises from the
$SL(2,\mathbb{R})$ group manifold as follows. The $SL(2,\mathbb{R})$ group
manifold with left-right invariant metric is the three-dimensional hyperboloid

$x_{-1}^{2}+x_{0}^{2}-x_{1}^{2}-x_{2}^{2}=1\,,$ (2.3)

with the metric induced from the flat embedding space. Topologically, there is
a circle in the $(x_{-1},x_{0})$ plane that can never shrink to zero size.
Indeed, the first homotopy group of the group manifold is non-zero:
$\Pi_{1}(SL(2,\mathbb{R})=\mathbb{Z}$. The universal covering group
$\widetilde{SL(2,\mathbb{R})}$ covers the group manifold and has a trivial
fundamental group. It has the topology of $\mathbb{R}^{3}$. The BTZ black hole
is obtained by dividing the universal covering group by a $\mathbb{Z}$
orbifold action. A generator of the group acts on a $2\times 2$ standard
matrix representation $g$ of the group elements of $SL(2,\mathbb{R})$ as:

$g\rightarrow
e^{\pi(r_{+}-r_{-})\sigma_{3}}\,g\,e^{\pi(r_{+}+r_{-})\sigma_{3}}\,,$ (2.4)

where $\sigma_{3}$ is the diagonal Pauli matrix with entries $\pm 1$.
Crucially, when both the horizon radii $r_{+}$ and $r_{-}$ are non-zero, the
orbifold action has no fixed points. This is equally true on the covering
group. Therefore, the orbifolded manifold is regular and has the topology of
$\mathbb{R}^{2}\times S^{1}$. The circle generated by the orbifold action is
the one parameterized by the coordinate $\phi$ in the patch where the BTZ
coordinates (2.1) are valid.

The BHTZ analysis carries further and involves the metric inherited from the
group manifold and the causal structure it induces [10]. They exclude from the
BTZ geometry the region of the manifold which contains circles that are null
or time-like on the physical ground that these lead to paradoxes, and that the
coupling to matter would create a singularity at the boundary surface $r=0$.
Wherever the topologically non-trivial circle becomes null, the BHTZ proposal
is to introduce a two-dimensional boundary to the BTZ space-time [10].

###  2.3 An Ambiguity

We have signaled two aspects of the Lorentzian BTZ black hole backgrounds that
make it more subtle to define a unique world sheet theory. The first is the
non-trivial topology of the orbifolded universal covering group which is
$\mathbb{R}^{2}\times S^{1}$. The second is the introduction of boundaries in
space-time. We wish to address subtleties associated to the first aspect and
will hardly remark on the second.

The world sheet theory depends on a NSNS two-form flux potential $B_{(2)}$.
One way to define the two-form flux term is as the integral of the field
strength $H_{(3)}=dB_{(2)}$ over a three-cycle $\Sigma_{3}$ with a boundary
that coincides with the fundamental string world sheet
$\Sigma=\partial\Sigma_{3}$. If the world sheet $\Sigma$ is topologically
trivial (and we have a quantized integral over three-cycles in the target
space) then that definition of the flux term in the world sheet action is
unique. However, we will study winding strings on BTZ black hole backgrounds.
These have world sheets that are rather of the form $\Sigma=\mathbb{R}\times
S^{1}$ where the $S^{1}$ winds the topologically non-trivial circle in the
background $\mathbb{R}^{2}\times S^{1}$ space-time. If the flux term were to
fall off at infinity, we could still define the world sheet flux term in terms
of the integral of the flux $H$ over a semi-infinite cylinder. However, this
fall-off condition is not satisfied in the background under consideration – it
is easy to see that the $H_{(3)}$ flux grows with radius since it is
proportional to $\sqrt{-G}$ where $G$ is the determinant of the BTZ metric
(2.1). Another attempt to define the theory could go through the
compactification of a cylindrical world sheet to a sphere, but this would lead
to a spherical world sheet in a space-time of the form $\mathbb{R}\times
S^{2}$ and would therefore give rise to the necessity to choose a gauge
invariant flux integral $\int_{S^{2}}B$ in order to define the model. Thus, we
are stuck with string world sheets that do not arise as boundaries of compact
three-cycles and thus the action is hard to define in terms of the gauge
invariant three-form field strength $H_{(3)}$ only.

We can take the point of view that world sheets that have no boundaries have
an action equal to the $B_{(2)}$ field integrated over the world sheet
$\Sigma$. Because the world sheet has no boundary, the action is gauge
invariant. We argue that this point of view gives rise to a related ambiguity.
To make this discussion concrete, let us focus on the BTZ black hole
background in the coordinates (2.1). The $H_{(3)}$ flux is proportional to

$H_{(3)}=rdr\wedge dt\wedge d\phi\,.$ (2.5)

We can integrate the $H_{(3)}$ flux up to a $B_{(2)}$ potential:

$B_{(2)}=\frac{1}{2}(r^{2}-c)dt\wedge d\phi\,,$ (2.6)

where we introduced a radial integration constant $c$. Suppose we have a world
sheet of topology $\mathbb{R}\times S^{1}$ stretching along the directions
$(t,\phi)$. Imagine that we wish to define the world sheet action using the
integral over a three-cycle which is a finite cylinder inside the space-time
topology $\mathbb{R}^{2}\times S^{1}$. The space-time cylinder has one
boundary equal to the world sheet that we study, and a second boundary which
we will call the cut-off surface $\Sigma_{co}$. When we integrate the
$H_{(3)}$ flux over the finite cylinder, we schematically find:

$\displaystyle\int_{\text{cylinder}}H_{(3)}$
$\displaystyle=\int_{\Sigma}B_{(2)}-\int_{\Sigma_{co}}B_{(2)}$
$\displaystyle=\frac{1}{2}\int_{\Sigma}(r^{2}-c)dtd\phi-\frac{1}{2}\int_{\Sigma_{co}}(r_{co}^{2}-c)dtd\phi$
$\displaystyle=\frac{1}{2}\int(r^{2}-r_{co}^{2})dtd\phi\,.$ (2.7)

What this calculation illustrates is that the ambiguity in the choice of cut-
off surface $r_{co}$ is directly related to the ambiguity in our choice of
integration constant $c$ in the isolated world sheet $B_{(2)}$-field term.
Therefore, either we need a cut-off surface when we work in terms of the
$H_{(3)}$-flux (and the world sheet action is then independent of the constant
term in the $B_{(2)}$-field), or we define the world sheet action in terms of
the $B_{(2)}$-field in which case it explicitly depends on the constant $c$
(namely the $B_{(2)}$-field flux along the non-trivial two-cycle). Thus, the
definition of the model requires a choice. We will encounter this subtlety in
various guises in what follows – here we have concentrated on the bare-bones
origin. For related but different discussions of the subtlety, see e.g.
section 5 of [14] and section 3 of [15].

Finally, the potential introduction of space-time boundaries (at $r=0$) [10]
is a daunting prospect from the view point of the world sheet theory [13]. We
rather imagine that a microscopic definition of fundamental string
perturbation theory in F1-NS5 backgrounds, with the addition of appropriate
string theory matter and sources will take care of the protection of
chronology near the BHTZ $r=0$ surface. This deserves to be the object of
further study.

##  3 Classical Strings in the Black Hole Orbifold

After these preliminary observations, we recall the shape of space-like and
time-like geodesics in BTZ black hole backgrounds [20, 13]. We can dress these
geodesic solutions with winding to find winding classical strings in black
hole backgrounds [11, 12, 13]. We calculate the on-shell energy generically,
and at zero mass in particular. We compare our results to the analysis of
classical strings in three-dimensional anti-de Sitter space [1]. These results
will provide useful points of reference both to analyze the Nambu-Goto action
of a long string probe in section 4 and for the analysis of the conformal
field theory partition function on the black hole background in section 5.

###  3.1 The Geodesics

To study the motion of classical fundamental strings in the black hole
background, it is useful to understand the motion of particles first [20, 13].
A relativistic particle with mass $m$ follows geodesics on the black hole. We
parameterize the geodesics using world line proper time $\tau$ such that it
satisfies the equation:

$G_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=-l^{2}m^{2}\,.$ (3.1)

The geodesics have been computed in the literature, and we briefly recall the
results [20, 13]. The energy $E$ and angular momentum $L$ of the particle are
two constants of motion associated to the Killing vectors that translate time
$t$ and the angular coordinate $\phi$ in the black hole background (2.1). The
Killing vectors give rise to the conserved quantities:

$\displaystyle E$ $\displaystyle=(−M+r^{2})\dot{t}+\frac{J}{2}\dot{\phi}~{},$
$\displaystyle L$ $\displaystyle=r^{2}\dot{\phi}−\frac{J}{2}\dot{t}\,,$ (3.2)

where we denoted by a dot the derivative with respect to world line time
$\tau$. There is one remaining differential equation in the radial variable
$y=r^{2}$. In terms of the parameters $(\alpha,\beta)$

$\displaystyle\alpha$ $\displaystyle=E^{2}-L^{2}+Mm^{2}~{},$
$\displaystyle\beta$ $\displaystyle=L^{2}M-\frac{m^{2}J^{2}}{4}-JEL\,,$ (3.3)

the differential equation reads

$\displaystyle\dot{y}^{2}$ $\displaystyle=-4m^{2}y^{2}+4\alpha y+4\beta\,.$
(3.4)

Time-like, light-like and space-like geodesics can be integrated from this
equation in terms of their energy $E$ and angular momentum $L$. We concentrate
on two classes of geodesics that are of particular interest to us. These two
classes are time-like geodesics and space-like geodesics with large enough
energy such that $\alpha^{2}+4m^{2}\beta>0$. Time-like geodesics are the
starting point for finding short winding strings [1]. Time-like geodesics have
$m^{2}>0$ and the solution to the radial equation reads:

$\displaystyle r^{2}$
$\displaystyle=\frac{\alpha}{2m^{2}}+\frac{\sqrt{\alpha^{2}+4m^{2}\beta}}{2m^{2}}\sin
2m\tau\,.$ (3.5)

The geodesic stretches over an interval in the radial direction. The minimal
and maximal radius depend on the parameters $(M,J;E,L)$. An example geodesic
in a $J=0$ zero angular momentum black hole at zero particle angular momentum
$L=0$ will have a radial evolution between $r=0$ and
$r_{\text{max}}^{2}=E^{2}/m^{2}+r_{+}^{2}$, namely from the center to outside
the outer horizon.

Space-like geodesics with large enough energy such that
$\alpha^{2}+4m^{2}\beta>0$ are useful to find long winding strings. They have
the radial trajectory:

$\displaystyle r^{2}$
$\displaystyle=\frac{\alpha}{2m^{2}}+\sqrt{\frac{\alpha^{2}+4m^{2}\beta}{4m^{4}}}\cosh\sqrt{-4m^{2}}\tau\,.$
(3.6)

At zero angular momenta $J=0=L$ and large energy, the radial geodesic for a
tachyon stretches from zero radius all the way to infinity. For a more
complete discussion of classes of geodesic solutions, we refer to [20, 13].

In string theory (or e.g. a Kaluza-Klein theory), the particle on the black
hole background has a mass squared determined by the on-shell condition which
schematically reads:

$-\frac{km^{2}}{4}+h=0\,,$ (3.7)

where $h$ is a quantity that measures a property of the internal degree of
freedom333For example, the quantity $h$ could be set by an internal conformal
dimension in a compact conformal field theory defining a compact factor of a
string background or a mass due to a Kaluza-Klein momentum. and we have
restored the cosmological length scale $l=\sqrt{k\alpha^{\prime}}$, related to
the level $k$ of the Wess-Zumino-Witten model to be discussed in subsection
3.3.

###  3.2 The Winding Strings

It is well-known how to add winding to the string consistently with the string
equations of motion in conformal gauge [1, 11, 14, 13]. The winding solutions
read:

$\displaystyle t(\tau,\sigma)$ $\displaystyle=t_{\text{geodesic}}(\tau)+w\tau$
$\displaystyle\phi(\tau,\sigma)$
$\displaystyle=\phi_{\text{geodesic}}(\tau)+w\sigma$ $\displaystyle r(\tau)$
$\displaystyle=r_{\text{geodesic}}(\tau)\,,$ (3.8)

where the lower index indicates any solution for the time evolution of a
geodesic as discussed in subsection 3.1. The winding solution is a puffed up
geodesic, which winds the angular direction $\phi$ and is appropriately
stretched in time.

We presented the classical solutions in conformal gauge. To find the physical
spectrum of winding strings, we still need to solve the on-shell conditions.
In the next subsection, we will solve these conditions, carefully embed the
solutions into an orbifold Wess-Zumino-Witten model and analyze their physical
properties in more depth.

###  3.3 The Black Hole Orbifold

In the following subsections, we discuss properties of the orbifold Wess-
Zumino-Witten model that represents the conformal field theory with the BTZ
black hole target. We refer to [11, 12, 13, 14, 15, 16] for related analyses.
Compared to these useful resources, in this section we present in more detail
properties of the classical solutions and solve more explicitly for the on-
shell energy.

In our study we concentrate on the region outside the outer horizon. That
region can be covered in terms of a group element
parameterization:444Implicitly, we always take the universal cover
$\widetilde{SL(2,\mathbb{R})}$.

$\displaystyle g$
$\displaystyle=e^{\frac{r_{+}-r_{-}}{2}(t+\phi)\sigma_{3}}\,e^{\rho\sigma_{1}}\,e^{-\frac{r_{+}+r_{-}}{2}(t-\phi)\sigma_{3}}~{},$
(3.9)

which consists of three factors equal to exponentials of hyperbolic generators
of $sl(2,\mathbb{R})$. The radial coordinate $\rho$ is related to the BTZ
radial coordinate $r$ through

$\displaystyle\cosh^{2}\rho$
$\displaystyle=\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}\,.$ (3.10)

We list the (most relevant) current components $J,\bar{J}$ in the $\sigma_{3}$
direction as well as the energy-momentum tensor in black hole coordinates –
with $x^{\pm}=\tau\pm\sigma$ –:

$\displaystyle J$
$\displaystyle=-\frac{k}{r_{+}-r_{-}}\left((r^{2}-r_{+}r_{-})\partial_{+}\phi+(-r^{2}+r_{+}^{2}+r_{-}^{2}-r_{+}r_{-})\partial_{+}t\right)$
$\displaystyle\bar{J}$
$\displaystyle=\frac{k}{r_{+}+r_{-}}\left((r^{2}+r_{+}r_{-})\partial_{-}\phi+(r^{2}-r_{+}^{2}-r_{-}^{2}-r_{+}r_{-})\partial_{-}t\right)$
(3.11) $\displaystyle T$
$\displaystyle=k\Big{(}r^{2}\,(\partial_{+}\phi)^{2}+\,(r_{+}^{2}+r_{-}^{2}-r^{2})\,(\partial_{+}t)^{2}-2\,\,r_{+}\,r_{-}\,\partial_{+}t\,\partial_{+}\phi$
(3.12) $\displaystyle\hskip
162.18062pt+\,\frac{r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}\,(\partial_{+}r)^{2}\Big{)}~{}.$
(3.13)

A similar expression holds for the left moving stress tensor component with
$\partial_{+}$ replaced by $\partial_{-}$. As discussed in section 2, the
black hole orbifold corresponds to a $\mathbb{Z}$ action on the universal
cover $\widetilde{G}$ of the group $G=SL(2,\mathbb{R})$. The generator of the
group is embedded in the left-right isometry group
$\widetilde{G}_{L}\times\widetilde{G}_{R}$ that acts on the universal cover.
The group action is generated by the zero modes of the left and right
currents. We recall that one chooses an embedding of the group $\mathbb{Z}$
such that the action of a generator of the group is [10]:

$g\longrightarrow
e^{\pi(r_{+}-r_{-})\sigma_{3}}\,g\,e^{\pi(r_{+}+r_{-})\sigma_{3}}\,,$ (3.14)

which boils down to the identification $\phi\equiv\phi+2\pi$. Thus, the action
of the group $\widetilde{G}_{L}\times\widetilde{G}_{R}$ that commutes with the
orbifold action consists of the (left and right) one parameter hyperbolic
group actions which are obtained by exponentiating the generator $\sigma_{3}$.
These unbroken symmetries give rise to the energy and angular momentum
generators.

We can relate these formulas to those obtained in subsection 3.1. For
instance, we can evaluate the currents and stress tensor on the geodesic
solutions. If we declare that all space-time coordinates are $\sigma$
independent, we can substitute the expressions for the time derivatives
$\dot{t}$ and $\dot{\phi}$ (see equation (3.2)), and $\dot{r}$ (see equation
(3.4)), which are all given in terms of the conserved particle energy $E$ and
angular momentum $L$ in the formulas (3.11) for the currents and stress
tensor. We find the relations valid for particle geodesics:

$\displaystyle{J}_{\text{particle}}$
$\displaystyle=\frac{k}{r_{+}-r_{-}}\frac{E-L}{2}~{},\quad{\bar{J}}_{\text{particle}}=\frac{k}{r_{+}+r_{-}}\frac{E+L}{2}~{},\quad
T_{\text{particle}}=\bar{T}_{\text{particle}}=-\frac{km^{2}}{4}~{}.$ (3.15)

###  3.4 The String Energy and Angular Momentum

After embedding the geodesic solutions, we can apply the twist procedure:

$\phi\rightarrow\phi+w\sigma~{},\qquad t\rightarrow t+w\tau~{}\,,$ (3.16)

to obtain the wound solutions. At the level of the group element, this amounts
to the transformation [12]:

$g\rightarrow
e^{(r_{+}-r_{-})wx^{+}\sigma^{3}/2}\,g\,e^{-(r_{+}+r_{-})wx^{-}\sigma^{3}/2}~{}.$
(3.17)

This in turn leads to the shifted currents

$\displaystyle J$
$\displaystyle={J}_{\text{particle}}-(r_{+}-r_{-})\frac{kw}{2}~{},\qquad\bar{J}={\bar{J}}_{\text{particle}}-(r_{+}+r_{-})\frac{kw}{2}\,,$
(3.18)

and the shifted stress tensor for the solution in the twisted sector:

$\displaystyle T$
$\displaystyle=T_{\text{particle}}-w(r_{+}-r_{-})J_{\text{particle}}+\frac{k}{4}(r_{+}-r_{-})^{2}w^{2}$
(3.19) $\displaystyle\bar{T}$
$\displaystyle=\bar{T}_{\text{particle}}-w(r_{+}+r_{-})\bar{J}_{\text{particle}}+\frac{k}{4}(r_{+}+r_{-})^{2}w^{2}~{}.$

These shifts implement the twisting operation in the sigma model [1, 12, 13].
Given these currents, it was observed in [11, 12] that the angular momentum
generator $L_{\text{string}}$ encoded in the level matching condition is:

$\displaystyle L_{\text{string}}$
$\displaystyle=-(r_{+}-r_{-}){J}+(r_{+}+r_{-}){\bar{J}}+\frac{k}{2}wJ_{BH}~{}.$
(3.20)

From now on, we add an explicit index to the black hole angular momentum
$J_{BH}$ to distinguish it from the current component $J$. The string energy
$E{\text{string}}$ equals:

$\displaystyle E_{\text{string}}$
$\displaystyle=(r_{+}-r_{-}){J}+(r_{+}+r_{-}){\bar{J}}~{}.$ (3.21)

In our subsequent discussions we reserve the symbol $E$ to indicate the
particle energy and $E_{\text{string}}$ to denote the energy of a winding
string solution.

###  3.5 The Energy of Wound Geodesics

Presently, we analyze the winding strings in more detail, following a similar
analysis performed in the three-dimensional anti-de Sitter space-time [1]. We
discuss two cases: a time like geodesic and a space like geodesic with large
enough energy such that $E^{2}-L^{2}+m^{2}M>0$. We parameterize the solutions
slightly differently from earlier work on the subject and we explicitly solve
the on-shell conditions. Later on, we will link up these results with an
asymptotic probe string analysis performed in section 4 and compute the Wess-
Zumino-Witten orbifold choice for the integration constant $c$ in the two-form
flux potential discussed in section 2.

####  3.5.1 Winding Time-like Geodesics

We first focus on the time-like case and assume that the initial particle
energy $E$ as well as the mass parameter $m$ are positive. The on-shell
conditions in conformal gauge on the string world sheet are

$T+h=0~{},\qquad\bar{T}+\bar{h}=0~{},$ (3.22)

where $(h,\bar{h})$ are the conformal dimensions of operators in the internal
sector of a consistent string background. The energy-momentum components $T$
and $\bar{T}$ are given in (3.19), with the particle energy-momentum
$T_{\text{particle}}$ computed in equation (3.15). Solving the on-shell
conditions for the currents and substituting them back into the expression for
the string energy we find

$E_{\text{string}}=\frac{1}{w}(h+\bar{h}-\frac{km^{2}}{2})-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})~{}\,.$
(3.23)

For time-like geodesics, since $m^{2}>0$ we find the inequality

$E_{\text{string}}<\frac{1}{w}(h+\bar{h})-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})~{}.$
(3.24)

It is useful to parameterize the particle energy $E$ as follows:

$E=m\widetilde{E}~{}.$ (3.25)

We substitute this parameterization into the on-shell conditions and solve for
the particle mass $m$ and angular momentum $L$ to find

$m=-\widetilde{E}w\pm\sqrt{\frac{2h+2\bar{h}}{k}+w^{2}(\widetilde{E}^{2}+r_{+}^{2}+r_{-}^{2})}~{},\quad
L=\frac{\bar{h}-h}{kw}+r_{+}\,r_{-}\,w~{}.$ (3.26)

We observe that if we choose the positive root for the mass $m$, then in the
limit of large $\widetilde{E}$ the mass $m$ goes to zero. We can take this
limit such that the particle energy $E$ remains finite. The string energy then
takes the form

$E_{\text{string}}=\frac{h+\bar{h}}{w}-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})+O((\frac{m}{E})^{2})~{}.$
(3.27)

At larger energy $E$, or, at fixed energy $E$ and smaller mass $m$, the time-
like particle geodesic reaches further out in the black hole space-time. The
same statement holds for the wound string. Thus, the limit we are considering
is one in which the string gets closer to the boundary of space-time. We have
computed the energy of the string in that limit.

Finally, in order to facilitate comparison with a later calculation, we
consider a short twisted string at its turning point. For simplicity we set
the angular momenta to zero: $J_{BH}=L=0$. From the geodesic solution (3.5),
we conclude that at the turning point we have a maximal radius

$r_{\text{max}}^{2}=\frac{E^{2}+Mm^{2}}{m^{2}}=\widetilde{E}^{2}+r_{+}^{2}~{}.$
(3.28)

This relates the particle energy and maximal radius of the short string. Our
final goal is rather to express the string energy $E_{\text{string}}$ in terms
of the maximum radial extent $r_{\text{max}}$. For this purpose, we
concentrate on the setting with no internal excitation, $\bar{h}=\bar{h}=0$.
We then have

$\displaystyle E_{\text{string}}$
$\displaystyle=-\frac{km^{2}}{2w}-\frac{kwr_{+}^{2}}{2}$ (3.29)

Substituting the value (3.26) for $m$, we find the energy of the string in
terms of the ratio $\widetilde{E}$. Finally, rewriting the energy
$\widetilde{E}$ in terms of $r_{\text{max}}$ using equation (3.28), we find

$\displaystyle E_{\text{string}}$
$\displaystyle=-k\,w\,r_{\text{max}}\left(r_{\text{max}}-\sqrt{r_{\text{max}}^{2}-r_{+}^{2}}\right)\,.$
(3.30)

This is the relation between the energy $E_{\text{string}}$ of the short
string and the maximal radius $r_{\text{max}}$ it attains at zero angular
momentum. The energy of a twisted short string in the background of a black
hole with angular momentum is computed in Appendix B.

####  3.5.2 Winding Space-like Geodesics

We turn to the winding space-like geodesics for which $m^{2}$ is negative. It
is convenient to define the parameter $\kappa$ through

$m=i\,\kappa~{},$ (3.31)

and choose $\kappa$ to be a positive real number. We solve the on-shell
conditions and obtain the expression for the string energy in the twisted
sector:

$E_{\text{string}}=\frac{1}{w}(h+\bar{h}+\frac{k\,\kappa^{2}}{2})-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})~{}.$
(3.32)

Since $\kappa^{2}>0$ we obtain the inequality

$E_{\text{string}}>\frac{1}{w}(h+\bar{h})-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})~{}.$
(3.33)

This inequality is complementary to the inequality (3.24) for time-like
excitations. For the space-like case we parameterize the energy as

$E=\kappa\,\widetilde{E}~{},$ (3.34)

and substitute into the on-shell conditions to solve for the imaginary mass
$\kappa$ and the angular momentum $L$:

$\kappa=\widetilde{E}w\pm\sqrt{w^{2}(\widetilde{E}^{2}-r_{+}^{2}-r_{-}^{2})-\frac{2h+2\bar{h}}{k}}~{},\quad
L=\frac{\bar{h}-h}{kw}+r_{+}\,r_{-}\,w~{}.$ (3.35)

We choose the negative root for the parameter $\kappa$ and observe as before
that one can take the limits $\widetilde{E}\rightarrow\infty$ and
$\kappa\rightarrow 0$ such that the energy $E$ remains finite. Substituting
this into the expression for the string energy we obtain

$E_{\text{string}}=\frac{h+\bar{h}}{w}-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})+O((\frac{\kappa}{E})^{2})~{}.$
(3.36)

This is identical to the result obtained for the time-like geodesic. This
suggests that there is a transition between the two solutions at
$m^{2}=0=\kappa^{2}$, while keeping the energy finite. From the point of view
of the time-like geodesic it is clear that the transition takes place at large
radius. In this respect, the analysis is similar to the one in $AdS_{3}$ [1].

There is nevertheless an important difference between the analysis of winding
strings in the BTZ black hole background as compared to the winding strings in
$AdS_{3}$. In the latter case, it was observed that as the internal conformal
dimension reached a critical value, given by $h=\bar{h}=kw^{2}/4$, the mass
$m$ (or the parameter $\kappa$) tended to zero and there was a transition
between the short string and long string solution for finite values of the
rescaled energy $\widetilde{E}$ [1]. The critical value of the internal
conformal dimension $h$ at which one obtains an exactly vanishing mass $m$
independently of the rescaled energy $\widetilde{E}$ would occur in the BTZ
background at a negative, unphysical internal conformal dimension given by
$h=\bar{h}=-kw(r_{+}^{2}+r_{-}^{2})/4$. Thus, in the BTZ background, we see
the short to long string transition only in the scaling limit.

##  4 The Winding String Probing a Black Hole

In this section, we analyze the Nambu-Goto action for a winding string near
the boundary of the three-dimensional black hole supplemented with NSNS flux.
This is a variation on the long string analysis in [21] in anti-de Sitter
space-time. It will allow us to calculate the energy of the winding string
more directly, and to understand which choice of NSNS two-form potential is
made in the Wess-Zumino-Witten model. Moreover, we will show how the choice of
definition of the model shows up in Fefferman-Graham coordinates.

###  4.1 The Winding String Probe

We study a winding string near infinity. The black hole metric asymptotes to
the $AdS_{3}$ metric to leading order. However, it is known that the leading
tension and volume contributions of a fundamental string in an asymptotically
$AdS_{3}$ background with NSNS flux cancel. The naively subdominant terms
become the leading non-zero terms and they depend on the BTZ black hole
background under consideration. This is related to the fact that a fundamental
string probe is a $1/2$ BPS object in string theory and has a charge equal to
its mass.

We consider a string that winds $w$ times at large radius $r$. We plug this
ansatz into a Nambu-Goto action supplemented with a $B_{(2)}$ flux term. We
take a $B_{(2)}$ field potential equal to – see section 2 –:

$B_{(2)}=\frac{1}{2}(r^{2}-c)\,dt\wedge d\phi\,.$ (4.1)

For starters, we concentrate on the leading and subleading terms in a
$1/r^{2}$ expansion. The BTZ metric in the $r^{-2}$ expansion reads:

$\displaystyle ds^{2}$
$\displaystyle=\frac{dr^{2}}{r^{2}}+r^{2}(-dt^{2}+d\phi^{2})$
$\displaystyle+\frac{dr^{2}}{r^{4}}(r_{+}^{2}+r_{-}^{2})+(r_{+}^{2}+r_{-}^{2})dt^{2}-2r_{+}r_{-}dtd\phi$
$\displaystyle+O(r^{-6})dr^{2}\,.$ (4.2)

All further corrections are proportional to $dr^{2}$. The action $S$ is

$S=S_{NG}+S_{B}\,,$ (4.3)

where the flux term in the action is $S_{B}$ and $S_{NG}$ is the Nambu-Goto
action. We choose the static gauge appropriate to describe a long string that
winds $w$ times around the circle:

$\tau=t~{},\qquad\phi=w\sigma~{},$ (4.4)

and consider the radius $r(\tau,\sigma)$ to be an arbitrary function of the
world sheet coordinates. We find the flux term:

$S_{B}=\frac{l^{2}}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma\int
d\tau\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{\mu\nu}=\frac{kw}{2\pi}\int
d\tau d\sigma~{}(r(\tau,\sigma)^{2}-c)\,.$ (4.5)

The Nambu-Goto action $S_{NG}$ is equal to:

$\displaystyle S_{NG}$ $\displaystyle=-\frac{l^{2}}{2\pi\alpha^{\prime}}\int
d\tau\int_{0}^{2\pi}d\sigma\sqrt{-\det\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}G_{\mu\nu}}\,.$
(4.6)

The argument of the square root is the determinant of the induced metric
$h_{\alpha\beta}$ on the worldvolume of the long string. The components of the
induced metric have the following leading terms in the large $r$ expansion:

$\displaystyle h_{\tau\tau}$
$\displaystyle=-r^{2}+r_{+}^{2}+r_{-}^{2}+\left(\frac{\dot{r}}{r}\right)^{2}~{},\quad
h_{\tau\sigma}=-r_{+}r_{-}w+\frac{\dot{r}r^{\prime}}{r^{2}}~{},\quad
h_{\sigma\sigma}=w^{2}r^{2}+\left(\frac{r^{\prime}}{r}\right)^{2}~{}.$ (4.7)

In the total action including the flux term, the leading volume and area
contributions cancel and using $l^{2}=k\,\alpha^{\prime}$, we obtain

$\displaystyle S=\frac{kw}{4\pi}\int d\tau
d\sigma\,\left(r_{+}^{2}+r_{-}^{2}-2c)+\frac{1}{r^{2}}\left(\dot{r}^{2}-\frac{1}{w^{2}}(r^{\prime})^{2}+\frac{1}{4}(r_{+}^{2}-r_{-}^{2})^{2}\right)\right)+\ldots$
(4.8)

We recognize a non-trivial kinetic term and a leading potential term that
gives the string an effective negative potential energy:

$V=-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2}-2c)\,.$ (4.9)

The final result depends on the choice of constant $c$. If we identify the
potential energy with the energy (3.27) computed in the Wess-Zumino-Witten
model for an on-shell large radius string, we find that we must set the
constant $c$ to zero. If we do set the constant $c$ to zero, then the
$AdS_{3}$ result $V=kw/2$ [1, 21] is recovered by analytic continuation in the
outer horizon radius: $r_{+}^{2}=M\longrightarrow-1$ (which indeed continues
the BTZ metric (2.1) into the global $AdS_{3}$ metric).555In the anti-de
Sitter space-time, it can be straightforwardly argued that $c=0$ is a
necessary condition for regularity in the interior. Therefore our result is
consistent with the result of [1, 21]. Our calculation extends the result to
the BTZ black hole backgrounds and provides information on the implicit choice
of flux term in the orbifold Wess-Zumino-Witten model.

####  4.1.1 The Energy of the Probe String

Finally, we perform a check of the fact that the integration constant is zero
in the Wess-Zumino-Witten model to all orders in the radius expansion, by
calculating the exact Nambu-Goto string energy and comparing with the result
obtained using the world sheet Wess-Zumino-Witten model. For this purpose let
us compute the energy near the turning point in the simple case of $r_{-}=0$.
We set the derivatives to zero, $\dot{r}=0=r^{\prime}$ and introduce
$r=r_{\text{max}}$, the maximum radius attained by the string. The string
energy is then given by the (negative) integral over the Lagrangian:

$\displaystyle E_{\text{string}}$ $\displaystyle=-\int
d\sigma~{}L=-kw\left((r_{\text{max}}^{2}-c)-r_{\text{max}}\sqrt{r_{\text{max}}^{2}-r_{+}^{2}}\right)\,.$
(4.10)

Only for zero integration constant $c$ the string energy matches the Wess-
Zumino-Witten result (3.30).The equality of the Nambu-Goto and Wess-Zumino-
Witten energies in the presence of a non-zero black hole angular momentum
$J_{BH}$ is proven in Appendix B.

####  4.1.2 The Winding String as a Probe in Three-dimensional Gravity

One may wonder how the effective tension of the string changes due to the
presence of the black hole. As a first step in an understanding, we note that
in three-dimensional gravity, a point particle creates a deficit angle. A
second static point particle far removed sees a trivial metric and does not
feel a force due to the presence of the first particle. However, a non-local
excitation like a winding string will notice that the asymptotic circle has
shrunk. We can either consider that the string winds a smaller circle, or that
its effective tension has shrunk. Our black hole set-up is different, but the
physical energy of the winding asymptotic string still depends on the mass $M$
of the black hole in the interior. That is a long range three-dimensional
gravity effect of the black hole mass on the non-local winding probe.

Finally, note that the effective tension of the winding string is zero for a
massless BTZ black hole. One way to understand this is via the interpolation
between the $AdS_{3}$ space and the massless black hole. The conical deficit
angle gradually eats up the whole space. Another way to understand the
vanishing effective tension is to recall that the massless BTZ black hole is
like three-dimensional anti-de Sitter space in the Poincaré patch, with a
cylinder boundary. The zero result we find for this case is consistent with
the result obtained in [21].666The Liouville gap in conformal dimension that
is invoked on the sphere in [21] is semi-classically absent on the cylinder.

###  4.2 The Winding String in the Fefferman-Graham Expansion

Our analysis raises the question of how the asymptotic analysis of [21] in
Fefferman-Graham coordinates accommodates these potential terms. In this
section, we make the link between the determination of the long string action
in Fefferman-Graham coordinates, and our analysis in BTZ coordinates. The link
relates to the remarks made in section 2.

The starting point for the analysis [21] is the asymptotically $AdS_{3}$
metric in Fefferman-Graham coordinates:777Note that the black hole metric
(2.1) in BTZ coordinates is of a different form.

$ds^{2}=l^{2}({d\xi^{2}}+\frac{e^{2\xi}}{4}g_{ij}dx^{i}dx^{j}-P_{ij}dx^{i}dx^{j}+\dots)\,.$
(4.11)

where the tensor $P$ has a trace determined by the curvature $R$ of the
leading boundary metric $g$

$g^{ij}P_{ij}=\frac{R}{2}\,$ (4.12)

and is otherwise undetermined by the bulk equations of motion. The tensor $P$
is closely related to the boundary energy-momentum tensor. The Fefferman-
Graham expansion is exact in two dimensions, and only needs one further
subleading correction.

When one works in the Euclidean and assumes that one can cap off the
asymptotically $AdS$ space in the interior, then the result of [21] on the
action of a long string is universal in the sense that it is independent of
the space-time boundary energy-momentum tensor. The result for the long string
action is then [21]:

$\displaystyle S$ $\displaystyle=w\,l^{2}\int
d^{2}x\sqrt{-g}(\frac{R}{4}-\xi\frac{R}{2}-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\xi\partial_{\beta}\xi+O(e^{-2\xi}))\,.$
(4.13)

Compared to [21], we have added an overall factor of $w$ for a string that
winds the angular direction $w$ times. Here, we also consider Lorentzian
space-times in which there is not necessarily a unique way to cap off the
interior (as in the Lorentzian BTZ black hole). In section 2, we have
discussed the resulting ambiguity in the long string action. It comes down to
adding a constant contribution to the flux term in the action. The resulting
final action is:

$\displaystyle S$ $\displaystyle=w\,l^{2}\int
d^{2}x\sqrt{-g}(\frac{R}{4}-\xi\frac{R}{2}-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\xi\partial_{\beta}\xi+\tilde{c}+O(e^{-2\xi}))\,.$
(4.14)

The action for the scalar $\xi$ on the long string world sheet contains a
linear dilaton deftly exploited in [21]. The linear dilaton slope is set by
the cosmological constant.

#### Discussion

To compare the approaches of subsections 4.1 and 4.2, it is useful to relate
the BTZ and Fefferman-Graham coordinate systems. The BTZ metric in Fefferman-
Graham coordinates is found through the coordinate transformation
$e^{\xi}/2=\rho(r)$ that satisfies:

$\frac{r^{2}\,dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}=\frac{d\rho^{2}}{\rho^{2}}~{}.$
(4.15)

Solving for the radial coordinate $\rho(r)$ we find the solution in the regime
$r>r_{+}$ beyond the outer horizon:

$\rho(r)=\sqrt{r^{2}-r_{+}^{2}}+\sqrt{r^{2}-r_{-}^{2}}~{}.$ (4.16)

We also define the null coordinates $w^{\pm}$:

$w^{\pm}=\frac{1}{2}(\phi\pm t)~{}.$ (4.17)

The metric in these coordinates is

$\displaystyle ds^{2}$
$\displaystyle=\frac{d\rho^{2}}{\rho^{2}}+\rho^{2}\left(dw_{+}+\frac{(r_{+}+r_{-})^{2}}{\rho^{2}}dw_{-}\right)\left(dw_{-}+\frac{(r_{+}-r_{-})^{2}}{\rho^{2}}dw_{+}\right)\,.$
(4.18)

We compare the flux terms in the respective coordinate systems. They both
start out with a quadratic divergence at large radius. At large radius the
coordinate systems are related by a shift in the radius squared. This shifts
shows up as a constant term in the flux contribution to the action. More in
detail, the transformation of the flux term is:

$\int(r^{2}-c)\,d\phi
dt=\int(\frac{\rho^{2}}{4}+\frac{r_{+}^{2}+r_{-}^{2}}{2}-c+\dots)\,d\phi
dt\,.$ (4.19)

Thus, the constant flux term $c$ in the BTZ coordinates and the constant
$\tilde{c}$ in the Fefferman-Graham coordinates are related as:

$c=\tilde{c}-\frac{r_{+}^{2}+r_{-}^{2}}{2}\,.$ (4.20)

For instance, if $c=0$, then $\tilde{c}=(r_{+}^{2}+r_{-}^{2})/2$ and the
potential term in the action (4.14) agrees with the potential energy found in
section 3. The choice of constant is reflected here in a choice of
parameterization of the radial coordinate at infinity or more precisely in a
choice of radial cut-off as discussed in section 2. This clarifies the
interplay between the asymptotic long string analysis in Fefferman-Graham
coordinates [21], in BTZ coordinates, as well as the flux term in the action.

##  5 The Partition Function and the Euclidean Twist

In this section, we analyze properties of string theory in Euclidean BTZ black
holes in three-dimensional anti-de Sitter space-time. We highlight the
differences between the one loop vacuum amplitude in the black hole background
and the thermal $AdS_{3}$ vacuum free energy. We compute the world sheet
conformal field theory partition function on the BTZ background. It can be
interpreted as the partition function corresponding to the $\mathbb{Z}$
orbifold that gives rise to the black hole. We describe the twisted sectors of
the conformal field theory and how they fit into the classical theory. We
comment on how to fix the flux term in the Euclidean theory. We refer to [17,
18] for interesting and closely related results. We believe our analysis
further clarifies the relation between the spectrum and the partition
function. We stress the important differences that appear in the analytic
continuation of the Euclidean results to their Lorentzian signature
counterparts between the case of the anti-de Sitter and the black hole space-
times.

###  5.1 The Interpretation of the Partition Function

To interpret the black hole partition function, it is useful to contrast it
with the one loop, finite temperature free energy of string theory in three-
dimensional anti-de Sitter space. The latter is obtained by compactifying the
$AdS_{3}$ space-time on a thermal circle [22, 2]. The resulting partition
function depends on two parameters (apart from the cosmological constant),
which are the inverse temperature $\beta$, and the chemical potential for
angular momentum $\mu\beta$. We thus have a multi-string free energy, related
to the one-loop vacuum energy $Z_{EAdS_{3}}(\beta,\mu\beta)$ [22, 2]. It is
obtained by thermal compactification, which is implemented through the trace:

$Z_{EAdS_{3}}(\beta,\mu\beta)=Tr_{{\cal H}}e^{-\beta H+i\mu\beta L}\,.$ (5.1)

The Hamiltonian $H$ implements time translation, the operator $L$ corresponds
to the rotation generator and we trace over the Hilbert space ${\cal H}$ to
make this a one-loop vacuum amplitude. To lay bare the relation between the
thermal amplitude and the Lorentzian spectral interpretation requires further
extensive analysis [22, 2, 23].

The nature of the BTZ one-loop vacuum amplitude is different. It is a function
of two parameters, both specified by the Euclidean black hole background
itself. They are the mass and angular momentum of the black hole or the inner
and outer horizon radii. The partition function takes the form of an orbifold
partition function with twisted sectors labelled by the winding number $w$ and
a projection operator that guarantees invariance under the orbifold group:

$Z_{BTZ(M,J)}=\sum_{w}Tr_{{\cal H}_{w}}e^{2\pi iwL}\,.$ (5.2)

The Hilbert space splits up into twisted sectors labelled by a winding number
$w$. The orbifold projection operator that identifies the geometry is related
to the angular translation generator $L$ in the Euclidean coset conformal
field theory on $H_{3}^{+}=SL(2,\mathbb{C})/SU(2)$.

We conclude that we are performing a trace in which we glue by a complexified
angular momentum (instead of gluing by a complexified energy). While for anti-
de Sitter space we have a partition function with two parameters, for each BTZ
background we have a single partition function. In the first case, we can
couple two parameters to two charges. In the second case, we can analyze the
trace as a function of the (two parameter) background. The BTZ Hilbert space
depends on the radii $r_{\pm}$ and it is this dependence that is measured by
the trace.

We refer to Appendix A for a more elaborate discussion of the geometry, the
angular Killing vector and the identification in the Euclidean model.

###  5.2 The Sigma Model on the Euclidean Black Hole

It is useful to specify a few expectations we have for the orbifold spectrum,
directly in the Euclidean context. The Euclidean coset model is often
presented in Poincaré coordinates $(\xi,\gamma,\bar{\gamma})$:

$\displaystyle ds^{2}$ $\displaystyle=d\xi^{2}+e^{2\xi}d\gamma d\bar{\gamma}$
(5.3)

with the two-form potential $B_{(2)}$

$B_{(2)}=\frac{1}{2}e^{2\xi}\,d\gamma\wedge d\bar{\gamma}~{}.$ (5.4)

The $B$-field is imaginary and the worldsheet theory is not reflection
positive. We will be concerned with Euclidean worldsheets for which the
$B$-field contribution to the worldsheet action is real [24]:

$\displaystyle S=\frac{k_{b}}{\pi}\int
d^{2}z~{}(\partial\xi\bar{\partial}\xi+e^{2\xi}\bar{\partial}\gamma\partial\bar{\gamma})~{}.$
(5.5)

We have defined the complex world sheet coordinates $z=i\tau_{E}+\sigma$, and
$\bar{z}=-i\tau_{E}+\sigma$. In the Appendix we show how to relate the
$(\xi,\gamma,\bar{\gamma})$ coordinates to the Wick rotated BTZ coordinates
$(r,t_{E},\phi)$. Using the explicit change of variables in (A.3), one can
check that the $B_{(2)}$ -field is strictly gauge equivalent to:

$B_{(2)}=\frac{i}{2}\,(r^{2}-r_{+}^{2})dt_{E}\wedge d\phi~{}.$ (5.6)

The integration constant $c$, which was discussed in detail in our discussion
of the Lorentzian theory, now takes the non-zero value $c=r_{+}^{2}$. In the
Euclidean, the time circle pinches off at the outer horizon and the demand to
avoid a source term at the tip fixes the value of the integration
constant.888The demand that the three-form flux be regular at the tip is not
necessarily compatible with other reasonable demands. See e.g. [25] for a
discussion in the context of matching probe and gravity free energies.

###  5.3 Expectations for Twisted Sectors

The classical coset sigma-model has solutions to the equations of motion that
follow from the factorized solutions for the model on $SL(2,\mathbb{C})$ (see
e.g. [26]):

$\displaystyle\xi(z,\bar{z})$
$\displaystyle=\log(1+b(z)\bar{b}(\bar{z}))+\rho(z)+\bar{\rho}(\bar{z})$
$\displaystyle\gamma(z,\bar{z})$
$\displaystyle=a(z)+\frac{e^{-2\rho(z)}\bar{b}(\bar{z})}{1+b(z)\bar{b}(\bar{z})}$
$\displaystyle\bar{\gamma}(z,\bar{z})$
$\displaystyle=\bar{a}(\bar{z})+\frac{e^{-2\bar{\rho}(\bar{z})}{b}(z)}{1+b(z)\bar{b}(\bar{z})}\,,$
(5.7)

where the functions on the right hand side are holomorphic or anti-
holomorphic.

####  5.3.1 The Twisted Euclidean Classical Solutions

As we saw in Section 3.2, given a Lorentzian classical string solution to the
equations of motion, it is possible to find a winding string solution by
performing a twist. In the Euclidean, there is an analogous operation that
maps one solution into another:

$t_{E}\rightarrow t_{E}+w\tau_{E}~{},\qquad\phi\rightarrow\phi+w\sigma~{}.$
(5.8)

In the Appendix A.1, we relate the BTZ coordinates to coordinates
$(\varphi,v,\bar{v})$ suitable for world sheet path integration [27]. See
equation (A.2). Those coordinates we can in turn relate to the Euclidean
Poincaré coordinates:

$\gamma=e^{\phi}\,v~{},\quad\bar{\gamma}=e^{\phi}\,\bar{v}~{},\quad\xi=-\varphi~{}.$
(5.9)

The Euclidean twist action (5.8) then can be translated to the
$(\varphi,v,\bar{v})$ variables, and subsequently to the
$(\xi,\gamma,\bar{\gamma})$ coordinates:

$\displaystyle\gamma$ $\displaystyle\longrightarrow
e^{w(r_{+}+ir_{-})z}\,\gamma~{},\quad\bar{\gamma}\longrightarrow
e^{w(r_{+}-ir_{-})\bar{z}}\,\bar{\gamma}~{},$ $\displaystyle\xi$
$\displaystyle\longrightarrow\xi-\frac{w}{2}\left((r_{+}+ir_{-})z+(r_{+}-ir_{-})\bar{z}\right)~{}.$
(5.10)

Comparing with the general classical solutions (5.7), we find that the twisted
solutions are obtained from the untwisted ones by the action:

$\displaystyle a(z)$ $\displaystyle\longrightarrow
e^{iw(r_{-}-ir_{+})z}~{}a(z)~{},\quad b(z)\longrightarrow
b(z)~{},\quad\rho(z)\longrightarrow\rho(z)-i\frac{w}{2}(r_{-}-ir_{-})z~{},$
(5.11)

and analogously for the complex conjugate functions $\bar{a}$, $\bar{b}$ and
$\bar{\rho}$.

####  5.3.2 The Euclidean Coset Currents and Their Twists

As in the Lorentzian model, we wish to understand the action of twisting on
the classical currents and energy momentum tensors. The $sl(2,\mathbb{C)}$
current generators in the $(\xi,\gamma,\bar{\gamma})$ coordinates are (see
e.g. [28]):

$\displaystyle J^{-}$ $\displaystyle=i\,ke^{2\xi}\partial\bar{\gamma}$
$\displaystyle J^{3}$ $\displaystyle=-i\,k(\partial\xi-\gamma
e^{2\xi}\partial\bar{\gamma})$ $\displaystyle J^{+}$
$\displaystyle=-i\,k(\partial\gamma+2\gamma\partial\xi-\gamma^{2}e^{2\xi}\partial\bar{\gamma})\,.$
(5.12)

Thus, the action of twisting on the Euclidean current generators is:

$\displaystyle J^{-}$ $\displaystyle\rightarrow e^{-w(r_{+}+ir_{-}){z}}J^{-}$
$\displaystyle J^{3}$ $\displaystyle\rightarrow
J^{3}+i\frac{kw}{2}(r_{+}+ir_{-})$ $\displaystyle J^{+}$
$\displaystyle\rightarrow e^{w(r_{+}-ir_{-}){z}}J^{+}\,.$ (5.13)

For the barred $sl(2,\mathbb{C})$ currents, similar twisting rules hold. The
energy-momentum tensor component in terms of these currents is

$\displaystyle T$
$\displaystyle=-\frac{1}{2k}(J^{-}J^{+}+J^{+}J^{-})+\frac{1}{k}(J^{3})^{2}$
$\displaystyle=-k((\partial\xi)^{2}+e^{2\xi}\partial\gamma\partial\bar{\gamma})~{}.$
(5.14)

The anti-holomorphic stress tensor can be similarly written with the
substitution $\partial\rightarrow\bar{\partial}$. The twisting action on the
Euclidean energy momentum tensor is:

$\displaystyle T$ $\displaystyle\longrightarrow
T+w(r_{+}+ir_{-})(iJ^{3})+\frac{kw^{2}}{4}(r_{+}+ir_{-})^{2}\,,$ (5.15)
$\displaystyle\bar{T}$
$\displaystyle\longrightarrow\bar{T}+w(r_{+}-ir_{-})(-i\bar{J}^{3})+\frac{kw^{2}}{4}(r_{+}-ir_{-})^{2}\,.$
(5.16)

We observe that the Lorentzian twists (3.19) are obtained by analytic
continuation of the inner horizon radius $r_{-}$. By the same analytic
continuation procedure, or by analysing level matching directly in the
Euclidean, one can write down the Euclidean angular momentum generator:

$\displaystyle L_{\text{string}}^{E}$
$\displaystyle=-(iJ^{3})(r_{+}+ir_{-})+(-i\bar{J}^{3})(r_{+}-ir_{-})+kw(-ir_{-}r_{+})\,.$
(5.17)

We have analyzed classical properties of the Euclidean twisted sectors. These
classical properties will be reflected in the world sheet conformal field
theory partition function.

###  5.4 The Partition Function

In this section we calculate the one loop string amplitude in the Euclidean
black hole background and in particular the BTZ world sheet partition function
factor. Furthermore we speculate on its Lorentzian interpretation.

The partition function is a weighted sum over the quantum spectrum. It can be
computed by adapting the calculation of the thermal $AdS_{3}$ partition
function [2]. For the purpose of computing the partition function, it is most
convenient to write the worldsheet action for strings propagating in the coset
orbifold model in the $(\varphi,v,\bar{v})$ coordinates – we refer to Appendix
A and [2, 23] for the details –:

$\displaystyle S=\frac{k_{b}}{\pi}\int
d^{2}z~{}(\partial\varphi\bar{\partial}\varphi+(\partial\bar{v}+\bar{v}\partial\varphi)(\bar{\partial}v+v\bar{\partial}\varphi))~{},$
(5.18)

where we have denoted the quantum level of the bosonic $sl(2,\mathbb{R})$
current algebra by $k_{b}$. As shown in the Appendix, the coordinates undergo
a non-trivial identification in the black hole orbifold – see equation (A.9):

$(\varphi,v,\bar{v})\sim(\varphi+2\pi r_{+},e^{2\pi ir_{-}}v,e^{-2\pi
ir_{-}}\bar{v})~{},$ (5.19)

where $r_{\pm}$ are the inner and outer horizon radii. The one loop string
amplitude on the background is

$Z=\int_{{\cal
F}_{0}}\frac{d^{2}\tau}{4\tau_{2}^{2}}Z_{BTZ}(r_{\pm})~{}Z_{gh}~{}Z_{int}~{},$
(5.20)

where the ghost and internal sector partition functions are given by

$\displaystyle Z_{\text{gh}}$ $\displaystyle=\tau_{2}~{}|\eta(\tau)|^{4}~{},$
(5.21) $\displaystyle Z_{\text{int}}$
$\displaystyle=(q\bar{q})^{-\frac{c_{\text{int}}}{24}}\sum_{h}d(h,\bar{h})~{}q^{h}{\bar{q}}^{\bar{h}}~{},$
(5.22)

$q=e^{2\pi i\tau}$ is the elliptic nome and $d(h,\bar{h})$ denotes a
degeneracy of the states of the internal conformal field theory. Given the
coordinate identifications (5.19) and comparing with the thermal $AdS_{3}$
calculation, one can write down the Euclidean conformal field theory partition
function by a suitable replacement of the fugacities $(\beta,\mu\beta)$ with
the background horizon radii $(r_{+},r_{-})$ – see Appendix A.2 –:

$Z_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\sum_{w,m}\frac{e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}+\frac{2\pi}{\tau_{2}}\text{Im}(\bar{U}_{m,w})^{2}}}{|\theta_{1}(\bar{U}_{m,w},\tau)|^{2}}\,.$
(5.23)

We have defined the holonomy:

$\bar{U}_{m,w}=(r_{-}-ir_{+})(m-w\tau)\,.$ (5.24)

Note that the holonomy $\bar{U}$ transforms as
$\bar{U}\rightarrow\bar{U}/(c\tau+d)$ under world sheet modular
transformations $\tau\rightarrow(a\tau+d)/(c\tau+d)$ which renders the
partition function modular invariant.999 Either from this transformation
property or from the definition of Ray-Singer torsion [29] – see also [27] –
it is clear that we must have the $\tau$ dependent holonomy $\bar{U}=\nu$ for
the partition function factor $\theta_{1}(\nu,\tau)$. This corrects an
inaccuracy in [2, 23] which is of no further consequence there.

####  5.4.1 An Analysis of the Spectrum

In the remainder of this section we will develop arguments on how to interpret
the Euclidean and Lorentzian spectral content of the partition function. Our
arguments and analysis are speculative – we hope they will be good guides for
a more rigorous analysis. See also [17, 18] for a related but different
discussion.

We expand the $\theta_{1}$-function in the denominator of the partition
function (5.23) in a series expansion using $z=e^{2\pi i\nu}$:

$\frac{1}{\theta_{1}(\nu,\tau)}=i\frac{1}{\eta^{3}}\sum_{r\in\mathbb{Z}}z^{r+\frac{1}{2}}S_{r}(q)~{},\quad\text{with}\quad
S_{r}(q)=\sum_{n\geq 0}(-1)^{n}q^{\frac{n}{2}(n+2r+1)}~{}.$ (5.25)

This expansion is valid only when $|q|<z<1$. We ignore this substantial
difficulty here and develop a further formal argument (similar in spirit to
Appendix B of [1]). The $\nu$ variable is given by the holonomy
$\bar{U}_{m,w}$. As a result of our formal expansion, we miss out on certain
contributions to the partition function. We will suggest an interpretation of
some of these contributions in terms of continuous representations later on.
At the moment we concentrate on contributions of the twisted discrete states
to the partition function. Keeping this caveat in mind, we continue with the
series expansion. We obtain

$\displaystyle Z_{BTZ}$
$\displaystyle=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\frac{1}{|\eta|^{6}}\sum_{w,m,r,\bar{r}}S_{r}S_{\bar{r}}~{}e^{2\pi
imr_{-}(r-\bar{r})}~{}e^{2\pi mr_{+}(1+r+\bar{r})}$ (5.26) $\displaystyle
q^{-(r+\frac{1}{2})w(r_{-}-ir_{+})}~{}\bar{q}^{-(\bar{r}+\frac{1}{2})w(r_{-}+ir_{+})}~{}e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}+\frac{2\pi}{\tau_{2}}\text{Im}(\bar{U}_{m,w})^{2}}\,.$
(5.27)

We use an integral representation that introduces a formal radial momentum
$s$:

$\int_{-\infty}^{+\infty}ds~{}e^{-4\pi\tau_{2}\frac{s^{2}}{k_{b}-2}}e^{-4\pi
is\text{Im}(\bar{U}_{m,w})}=\frac{1}{2}\frac{\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}e^{-\frac{(k_{b}-2)\pi\text{Im}(\bar{U}_{m,w})^{2}}{\tau_{2}}}\,.$
(5.28)

We then obtain

$\displaystyle Z_{BTZ}$
$\displaystyle=\frac{2r_{+}}{|\eta|^{6}}\sum_{w,m,r,\bar{r}}S_{r}S_{\bar{r}}~{}e^{2\pi
imr_{-}(r-\bar{r})}~{}e^{2\pi
mr_{+}(1+r+\bar{r})}q^{-(r+\frac{1}{2})w(r_{-}-ir_{+})}~{}\bar{q}^{-(\bar{r}+\frac{1}{2})w(r_{-}+ir_{+})}$
(5.29)
$\displaystyle~{}e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}}~{}e^{\frac{k_{b}\pi\text{Im}(\bar{U}_{m,w})^{2}}{\tau_{2}}}\int_{-\infty}^{+\infty}ds~{}e^{-4\pi\tau_{2}\frac{s^{2}}{k_{b}-2}}e^{-4\pi
is\text{Im}(\bar{U}_{m,w})}$

After simplification, we find

$\displaystyle Z_{BTZ}$
$\displaystyle=\frac{2r_{+}}{|\eta|^{6}}\sum_{w,m,r,\bar{r}}S_{r}S_{\bar{r}}~{}e^{2\pi
imr_{-}(r-\bar{r})}~{}e^{2\pi
mr_{+}(1+r+\bar{r})}q^{-(r+\frac{1}{2})w(r_{-}-ir_{+})}~{}\bar{q}^{-(\bar{r}+\frac{1}{2})w(r_{-}+ir_{+})}$
(5.30) $\displaystyle~{}e^{2\pi k_{b}mwr_{+}r_{-}}~{}e^{-\pi
k_{b}w^{2}((r_{+}^{2}-r_{-}^{2})\tau_{2}+2r_{+}r_{-}\tau_{1})}~{}\int_{-\infty}^{+\infty}ds~{}e^{-4\pi\tau_{2}\frac{s^{2}}{k_{b}-2}}e^{-4\pi
is(r_{+}(w\tau_{1}-m)-r_{-}w\tau_{2})}$ (5.31)

We observe that in the exponent, only the linear term in the wrapping number
$m$ remains. Formally, the coefficient of the linear term in $m$ in the
exponent needs to be an integer times $2\pi i$ in order to pass Dirac’s comb
and we find the constraint:

$r(r_{-}-ir_{+})-\bar{r}(r_{-}+ir_{+})-ik_{b}wr_{-}r_{+}-2ir_{+}(\frac{1}{2}+is)\in\mathbb{Z}\,.$
(5.32)

Defining a Casimir variable $j=\frac{1}{2}+is$, we can rewrite the constraint
as

$(j+r)(r_{-}-ir_{+})-(j+\bar{r})(r_{-}+ir_{+})-ik_{b}wr_{-}r_{+}\in\mathbb{Z}\,.$
(5.33)

We think of this quantization equation as indicating poles in the complex $j$
plane where discrete contributions to orbifold invariant correlation functions
reside. We observe that this is precisely the condition obtained by analytic
continuation of the Lorentzian quantization condition (3.20), provided we
identify

$J^{3}=j+r~{},\quad\bar{J}^{3}=j+\bar{r}~{}.$ (5.34)

The partition function becomes

$\displaystyle
Z_{BTZ}=-\frac{2ir_{+}}{|\eta|^{6}}\sum_{w,r,\bar{r}}S_{r}S_{\bar{r}}~{}\int_{-i\infty}^{+i\infty}~{}dj~{}$
$\displaystyle
q^{-\frac{(j-\frac{1}{2})^{2}}{k_{b}-2}-(j+r)(r_{-}-ir_{+})w-\frac{k_{b}w^{2}}{4}(r_{-}-ir_{+})^{2}}$
(5.35)
$\displaystyle\bar{q}^{-\frac{(j-\frac{1}{2})^{2}}{k_{b}-2}-(j+\bar{r})(r_{-}+ir_{+})w-\frac{k_{b}w^{2}}{4}(r_{-}+ir_{+})^{2}}\,.$
(5.36)

Finally, we combine this result with the internal and ghost contributions to
write down the one loop string amplitude. Firstly we rewrite the contribution
of the internal conformal field theory and the descendants:

$\displaystyle\frac{Z_{\text{int}}}{|\eta(\tau)|^{2}}~{}S_{r}~{}S_{\bar{r}}$
$\displaystyle=(e^{4\pi\tau_{2}})^{(1-\frac{1}{4(k_{b}-2)})}\sum_{N,h,\bar{N},\bar{h}}d_{r,h,N}~{}q^{h+N}~{}q^{\bar{h}+\bar{N}}~{}.$
(5.37)

Lastly, we rotate the $j$ integration contour to coincide with the real line
(up to an implicit Feynman type regularization). This allows us to write down
the one loop string amplitude in the form

$\displaystyle Z_{BTZ}$ $\displaystyle=2\pi r_{+}\int_{{\cal
F}_{0}}\frac{d^{2}\tau}{\tau_{2}}\sum_{w,r,h,N}d_{r,h,N}\int_{-\infty}^{+\infty}\frac{dj}{\pi}~{}q^{L_{0}-1}\bar{q}^{{\bar{L}}_{0}-1}~{},$
(5.38)

where the worldsheet Virasoro generators $L_{0}$ and $\bar{L}_{0}$ are given
by

$\displaystyle L_{0}$
$\displaystyle=-\frac{j(j-1)}{k_{b}-2}+i(j+r)(r_{+}+ir_{-})w+\frac{k_{b}w^{2}}{4}(r_{+}+ir_{-})^{2}+h+N~{},$
(5.39) $\displaystyle\bar{L}_{0}$
$\displaystyle=-\frac{j(j-1)}{k_{b}-2}-i(j+\bar{r})(r_{+}-ir_{-})w+\frac{k_{b}w^{2}}{4}(r_{+}-ir_{-})^{2}+\bar{h}+\bar{N}~{}.$
(5.40)

We note that this precisely matches with the Virasoro generators (5.15) for
twisted short strings, provided we make the identification of charges (5.34).
After continuation to the Lorentzian, the speculation is that these discrete
representation excitations are the quantum counterpart to the classical short
winding strings based on time-like geodesics.

####  5.4.2 Continuous Contributions

The $\theta_{1}$ function in the denominator in the partition function has a
pole at holonomy $U_{m,w}=0$ among others. Expanding the $\theta_{1}$ function
in disregard of the range of validity of the expansion is formally similar to
neglecting such pole contributions [1]. In this subsection, we wish to show
that the pole at $U_{m,w}=0$ allows for an interpretation in terms of
characters of continuous representations. Inspiration for this identification
is drawn from [1].

We start once again from the initial partition function and make a sketch of
the pole contribution:

$Z^{div}_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\sum_{w,m}\frac{1}{|\eta(\tau)|^{6}}\delta(\bar{U}_{m,w})\delta(U_{m,w})$
(5.41)

What we shall do now is to rewrite this modular invariant, divergent
contribution in terms of the continuous characters. For this purpose we insert
particular prefactors that vanish at the locations of the poles:

$Z^{div}_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\sum_{w,m}e^{-\frac{\pi
k_{b}r_{+}^{2}}{\tau_{2}}|m-w\tau|^{2}}e^{\frac{\pi
k_{b}}{\tau_{2}}(\text{Im}U_{m,w})^{2}}\frac{1}{|\eta(\tau)|^{6}}\delta(\bar{U}_{m,w})\delta(U_{m,w})\,.$
(5.42)

We wish to write this delta-function contribution as an integral over twisted
continuous characters. To that end we insert the integral representation:

$\displaystyle Z^{div}_{BTZ}$
$\displaystyle=2r_{+}\int_{-\infty}^{+\infty}ds~{}e^{-4\pi\tau_{2}\frac{s^{2}}{k_{b}-2}}$
$\displaystyle\sum_{w,m}e^{-\frac{\pi
k_{b}r_{+}^{2}}{\tau_{2}}|m-w\tau|^{2}}e^{\frac{\pi
k_{b}}{\tau_{2}}(\text{Im}U_{m,w})^{2}}\frac{1}{|\eta(\tau)|^{6}}\delta(\bar{U}_{m,w})\delta(U_{m,w})\,.$
(5.43)

The discrete summation variable $m$ is Poisson dual to the quantized angular
momentum. To obtain a quantized angular momentum in the continuous
representations, the corresponding left and right generator eigenvalues need
to be locked yet allow for an intercept that is tuned between the left and the
right sector. To introduce the intercept and decouple the left and right
quantum numbers, we introduce the integral:

$\int_{0}^{1}~{}d\alpha e^{2\pi i\alpha(m-\bar{m})}=\delta_{m,\bar{m}}~{}.$
(5.44)

We then use the variable $\bar{m}$ for the right-movers and rewrite the
divergent part, by expanding the exponents and collecting terms in the powers
of the nome $q$:

$Z^{div}_{BTZ}(r_{\pm})=\frac{2r_{+}}{|\eta(\tau)|^{6}}\int_{-\infty}^{+\infty}ds~{}~{}q^{\frac{s^{2}}{k_{b}-2}-\frac{k_{b}w^{2}}{4}(r_{-}-ir_{+})^{2}}~{}\bar{q}^{\frac{s^{2}}{k_{b}-2}-\frac{k_{b}w^{2}}{4}(r_{-}+ir_{+})^{2}}\cr\sum_{w,m,\bar{m}}\int_{0}^{1}~{}d\alpha~{}e^{2\pi
im(\alpha-\frac{ik_{b}w}{2}r_{+}r_{-})}e^{-2\pi
i\bar{m}(\alpha+\frac{ik_{b}w}{2}r_{+}r_{-})}~{}\delta(\bar{U}_{m,w})\delta(U_{m,w})$
(5.45)

We dualize the $(m,\bar{m})$ variables by using the formula:

$\frac{2\pi}{r_{-}-ir_{+}}\sum_{m}\delta(m-w\tau)\,e^{2\pi
i\tilde{\alpha}m}=\frac{1}{r_{-}-ir_{+}}\sum_{n}e^{-2\pi
iw\tau(n-\tilde{\alpha})}~{}.$

For the case at hand we have
$\tilde{\alpha}=\alpha-\frac{ik_{b}w}{2}r_{+}r_{-}$. We have a similar
calculation for the right-movers giving rise to an independent dual variable
$\bar{n}$. The variable $\alpha$ remains shared between left- and right-
movers. We find:

$Z^{div}_{BTZ}=\frac{r_{+}}{2\pi^{2}(r_{+}^{2}+r_{-}^{2})}\int_{-\infty}^{+\infty}ds~{}\sum_{w,n,\bar{n}}q^{\frac{s^{2}}{k_{b}-2}-\frac{k_{b}w^{2}}{4}(r_{-}-ir_{+})^{2}}\bar{q}^{\frac{s^{2}}{k_{b}-2}-\frac{k_{b}w^{2}}{4}(r_{-}+ir_{+})^{2}}\\\
\frac{1}{|\eta(\tau)|^{6}}\int_{0}^{1}d\alpha e^{-2\pi
iw\tau({n-\alpha+i\frac{kw}{2}r_{+}r_{-}})}e^{2\pi
iw\bar{\tau}({\bar{n}-\alpha-i\frac{kw}{2}r_{+}r_{-}})}\,.$ (5.46)

Once more collecting the powers of the nome $q$, we obtain

$Z^{div}_{BTZ}=\frac{r_{+}}{2\pi^{2}(r_{+}^{2}+r_{-}^{2})}\frac{1}{|\eta(\tau)|^{6}}\int_{-\infty}^{+\infty}ds~{}\int_{0}^{1}d\alpha\\\
\sum_{w,n,\bar{n}}q^{\frac{s^{2}}{k_{b}-2}-w(n-\alpha+\frac{ik_{b}w}{2}r_{+}r_{-})-\frac{k_{b}w^{2}}{4}(r_{-}-ir_{+})^{2}}\bar{q}^{\frac{s^{2}}{k_{b}-2}-w(\bar{n}-\alpha-\frac{ik_{b}w}{2}r_{+}r_{-})-\frac{k_{b}w^{2}}{4}(r_{-}+ir_{+})^{2}}\,.$
(5.47)

We recognize the expected shifts in the generalized $J^{3}$ eigenvalues of the
continuum sector characters, respecting the quantization condition of the
angular momentum (5.17). In summary, we have succeeded in identifying a
modular invariant pole contribution with an integral over twisted sector
continuous characters. In the process, we have glued the continuous characters
consistently with angular momentum quantization. It provides an indication of
how one can consistently identify Lorentzian long winding strings that respect
angular momentum quantization in the Euclidean black hole background. These
would provide a quantum counterpart to the long winding strings solutions
based on space-like geodesics, discussed in section 3.

#### Discussion

Our analysis of the Euclidean partition function and its Lorentzian
representation theory content is illuminating but it remains partial. A more
careful treatment of the expansion is necessary as well as a more rigorous
treatment of all poles. The techniques that were used for thermal $AdS_{3}$
were based on the elliptic spectral flow property of affine $sl(2)$
representations. The hyperbolic nature of the energy and angular momentum
generators in the Lorentzian BTZ black hole render a direct extension of these
techniques challenging.

We found the angular momentum quantization equation (5.33). This quantization
condition is more readily interpreted in the Lorentzian where it corresponds
to the quantization of a combination of hyperbolic eigenvalues. In the
Euclidean, we can propose that if one defines Lorentzian BTZ amplitudes by
analytic continuation from the Euclidean $H_{3}^{+}$ model, that the poles
that capture the physics of the discrete modes reside at the complex value of
the parameter $j$ given by the Euclidean quantization rule (5.33).

In our interpretation of the partition function, we tend towards an
interpretation that matches the Lorentzian. In anti-de Sitter space, as in
ordinary quantum field theory, this entails insisting on a spectrum of
continuous energy eigenvalues, despite the fact that they are quantized in
Euclidean thermal $AdS_{3}$. In the case of the black hole space-time, we
similarly allowed for continuous energy eigenvalues in the Lorentzian black
hole interpretation of the Euclidean partition function.

We hope our analysis will be a useful stepping stone to reach a more rigorous
treatment of the partition function and its Lorentzian reading.

##  6 Conclusions

Building on earlier work [11, 12, 13, 14, 15, 16, 17, 18], we have further
analyzed the dynamics of fundamental strings in three-dimensional anti-de
Sitter black holes with Neveu-Schwarz-Neveu-Schwarz flux. We have stressed a
physical ambiguity in the definition of the model and have shown how it is
resolved in the Lorentzian Wess-Zumino-Witten model. We have computed explicit
expressions for the classical energy of winding strings based on time-like or
space-like geodesics. We supplemented the analysis with a direct analysis in
terms of the Nambu-Goto plus flux action of a long string probe. Our results
for the classical dynamics of twisted strings are more explicit then those
hitherto obtained.

We have also analyzed the Euclidean coset partition function and have shown
that it should be interpreted directly as an orbifold conformal field theory
partition function. While the Euclidean anti-de Sitter partition function
contains two fugacities corresponding to the temperature and the chemical
potential for angular momentum, the Euclidean black hole partition function
has two parameters that are fixed by the mass and angular momentum of the
black hole. Moreover, we partially identified the quantum spectral content of
the partition function in this spirit. While our analysis is preliminary, we
could identify the analytically continued discrete part of the Lorentzian
spectrum as well as a modular invariant contribution from the continuous
representations.

It is clear that much further work is needed before declaring that fundamental
string perturbation theory is well-defined and understood on the three-
dimensional black hole backgrounds under consideration. Concrete, important
and intriguing questions remain largely open. For example, the hyperbolic
orbifold may induce divergences in perturbation theory, or may be well-defined
through a specific analytic continuation procedure starting from the Euclidean
coset model. A more careful decomposition of the partition function may teach
us valuable lessons about the perturbative spectrum. The definition of the
background in terms of microscopic objects in string theory may inform us how
to treat the proposed cut-off surface at $r=0$. A long list of open and
solvable questions invites further work.

## Acknowledgements

We would like to thank Nemani V. Suryanarayana for helpful discussions and our
colleagues in general for creating a stimulating research environment.

##  Appendix A The Euclidean Black Hole Background

In this section, we briefly review the Euclidean BTZ black hole geometries in
various coordinate systems. The review serves as a reference for statements
made in the bulk of the paper. First of all, we recall that the Euclidean
black hole, like the thermal AdS geometry, has the topology of a solid torus
[30, 31]. These geometries have a constant negative curvature metric and are
locally Euclidean AdS. There is a choice of which cycle of the boundary torus
is filled in and there are a number of inequivalent choices. The Euclidean AdS
and the black hole choices are related by an S-duality transformation on the
boundary torus [30, 31].

###  A.1 The Metric on the Euclidean Black Hole

More concretely, starting from the Lorentzian black hole metric (2.1), we
analytically continue the inner horizon $r_{-}$ and time coordinate $t$ as
$r_{-}\rightarrow ir_{-}$ and $t=-it_{E}$ to obtain the Euclidean black hole
metric:

$ds^{2}=\frac{r^{2}}{(r^{2}-r_{+}^{2})(r^{2}+r_{-}^{2})}dr^{2}+\frac{(r^{2}-r_{+}^{2})(r^{2}+r_{-}^{2})}{r^{2}}dt_{E}^{2}+r^{2}(d\phi-\frac{r_{+}r_{-}}{r^{2}}dt_{E})^{2}~{}.$
(A.1)

We demand regularity of the Euclidean black hole metric at the outer horizon
where the Euclidean time circle pinches off. Along with the usual periodicity
condition on the $\phi$ coordinate we therefore have the identifications:

$\displaystyle(t_{E},\phi)$ $\displaystyle\sim(t_{E}+\frac{2\pi
r_{+}}{r_{+}^{2}+r_{-}^{2}}~{},\phi+\frac{2\pi
r_{-}}{r_{+}^{2}+r_{-}^{2}})\,,\qquad\text{and}\qquad(t_{E},\phi)\sim(t_{E},\phi+2\pi)~{}.$
(A.2)

In the main text we used Poincaré coordinates to describe the Euclidean black
hole, in terms of which the solutions to the classical equations of motion
were easily written down. These are related to the $(r,t_{E},\phi)$
coordinates by the change of variables:

$\displaystyle\gamma$
$\displaystyle=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}+r_{-}^{2}}}e^{(r_{-}-ir_{+})(i\phi-
t_{E})}~{},\quad\bar{\gamma}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}+r_{-}^{2}}}e^{(r_{-}+ir_{+})(-i\phi-
t_{E})}$ (A.3) $\displaystyle\xi$
$\displaystyle=-\frac{1}{2}\log\left(\frac{r_{+}^{2}+r_{-}^{2}}{r^{2}+r_{-}^{2}}\right)-r_{+}\phi+r_{-}t_{E}\,.$
(A.4)

The metric in Poincaré coordinates takes the form:

$ds^{2}=d\xi^{2}+e^{2\xi}\,d\gamma\,d\bar{\gamma}~{}.$ (A.5)

The coordinate identifications (A.2) become:

$\displaystyle(\xi,\gamma,\bar{\gamma})\sim(\xi,e^{2\pi i}\gamma,e^{-2\pi
i}\bar{\gamma})~{},\quad\text{and}\quad(\xi,\gamma,\bar{\gamma})\sim(\xi-2\pi
r_{+},e^{2\pi i(r_{-}-ir_{+})}\gamma,e^{-2\pi
i(r_{-}+ir_{+})}\bar{\gamma})~{}.$

One of the identifications is trivialized in these coordinates.

###  A.2 The Determination of the Partition Function

In the evaluation of the one loop amplitude for string theory on the black
hole background, it turns out to be useful to work in a different set of
coordinates that allows one to render the path integration free [27]. These
new coordinates are obtained from the $(\xi,\gamma,\bar{\gamma})$ Poincaré
coordinates by a rescaling:

$v=e^{\xi}\,\gamma~{},\quad\bar{v}=e^{\xi}\,\bar{\gamma}~{},\quad\varphi=-\xi~{}.$
(A.6)

By composing the maps one finds a direct relation between these and the
$(r,t_{E},\phi)$ coordinates:

$\displaystyle v$
$\displaystyle=\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}+r_{-}^{2}}}~{}e^{ir_{+}t_{E}+ir_{-}\phi}~{},\qquad\bar{v}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}+r_{-}^{2}}}~{}e^{-ir_{+}t_{E}-ir_{-}\phi}~{}$
$\displaystyle\varphi$
$\displaystyle=\frac{1}{2}\log\left(\frac{r_{+}^{2}+r_{-}^{2}}{r^{2}+r_{-}^{2}}\right)+r_{+}\phi-
r_{-}t_{E}\,.$ (A.7)

In terms of these coordinates the metric takes the form

$\displaystyle
ds^{2}=d\varphi^{2}+(dv+vd\varphi)(d\bar{v}+\bar{v}d\varphi)~{},$ (A.8)

and the identifications (A.2) become:

$\displaystyle(\varphi,v,\bar{v})$ $\displaystyle\sim(\varphi,e^{2\pi
i}v,e^{-2\pi
i}\bar{v})~{}\,\qquad\text{and}\qquad(\varphi,v,\bar{v})\sim(\varphi+2\pi
r_{+},e^{2\pi ir_{-}}v,e^{-2\pi ir_{-}}\bar{v})~{},$ (A.9)

The identification due to regularity at the outer horizon becomes trivial in
these coordinates. At the same time, the periodicity of the original angular
coordinate in the black hole turns into a non-trivial identification. The
vector field that generates the translation in the identified direction can be
expressed as:

$\frac{\partial}{\partial\phi}=r_{+}\frac{\partial}{\partial\varphi}+i\,r_{-}\left(v\frac{\partial}{\partial
v}-\bar{v}\frac{\partial}{\partial\bar{v}}\right)~{}.$ (A.10)

Rewriting the BTZ black hole metric in terms of the $(\varphi,v,\bar{v})$
coordinates makes it possible to fluently write down the Euclidean partition
function. The metric has the same form as in thermal $AdS_{3}$. The
identifications of the coordinates are also as in the Euclidean thermal
$AdS_{3}$ space, provided one makes an analogy between the parameters of the
thermal $AdS_{3}$ space and the background mass and angular momentum of the
black hole:

$\displaystyle\beta\longleftrightarrow 2\pi
r_{+}~{},\qquad\beta\mu\longleftrightarrow 2\pi r_{-}\,.$ (A.11)

Given the calculation of [27, 2] for the partition function of thermal
$AdS_{3}$, the partition function of the conformal field theory on the
Euclidean black hole background is then obtained from that of thermal
$AdS_{3}$ by using the substitutions (A.11). One finds the result

$Z_{BTZ}=\frac{r_{+}\sqrt{k_{b}-2}}{\sqrt{\tau_{2}}}\sum_{w,m}\frac{e^{-\pi\frac{k_{b}}{\tau_{2}}r_{+}^{2}|m-w\tau|^{2}+\frac{2\pi}{\tau_{2}}Im(\bar{U}_{m,w})^{2}}}{|\theta_{1}(\bar{U}_{m,w},\tau)|^{2}}\,,$
(A.12)

where the holonomy $\bar{U}$ is given by

$\bar{U}_{m,w}=(r_{-}-ir_{+})(m-w\tau)\,.$ (A.13)

The spectral content of the partition function in the Euclidean and its
analytic continuation to the Lorentzian is analyzed in the bulk of the paper.

##  Appendix B The Energy of a Twisted Short String

In this section we show that the energy of a twisted short string in the
background of a rotating black hole computed in conformal gauge matches the
energy of a string calculated using the static gauge in a Nambu-Goto action.
We first of all write down the Nambu-Goto plus flux action in static gauge:

$t=\tau~{},\quad\phi=w\sigma~{}.$ (B.1)

The action for the twisted or winding string then takes the form

$S=k\int
dtd\phi\left(wr^{2}-\sqrt{w^{2}(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})+r^{\prime
2}-\frac{(wr^{2}\dot{r}^{2}+r_{+}r_{-}r^{\prime})^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}\,\right)~{}.$
(B.2)

To compute the energy of the string, we work near the turning point, at which
$\dot{r}=0$, we have $r^{\prime}=0$ and $r=r_{\text{max}}$, the maximum radial
extent of the string. Then, the energy of the string is entirely given by the
potential energy:

$\displaystyle E_{\text{string}}$
$\displaystyle=kw\left(\sqrt{(r_{\text{max}}^{2}-r_{+}^{2})(r_{\text{max}}^{2}-r_{-}^{2})}-r_{\text{max}}^{2}\right)~{}.$
(B.3)

We now compare this with the energy calculated from the orbifold Wess-Zumino-
Witten model (3.23),

$E_{\text{string}}=\frac{h+\bar{h}}{w}-\frac{k}{2w}m^{2}-\frac{kw}{2}(r_{+}^{2}+r_{-}^{2})~{},$
(B.4)

where the on-shell conditions fix $m$ and the particle angular momentum $L$ to
be (see equation (3.26))

$m=-\widetilde{E}w+\sqrt{\frac{2h+2\bar{h}}{k}+w^{2}(\widetilde{E}^{2}+r_{+}^{2}+r_{-}^{2})}~{},\quad
L=\frac{\bar{h}-h}{kw}+r_{+}r_{-}w~{}.$ (B.5)

Recall here that $\widetilde{E}=\frac{E}{m}$, the rescaled energy of the
particle. To compare with the Nambu-Goto calculation which did not assume any
internal directions in the target space, we set $h=\bar{h}=0$, which leads to

$m=w\left(-\widetilde{E}+\sqrt{\widetilde{E}^{2}+r_{+}^{2}+r_{-}^{2}}\right)~{},\quad
L=r_{+}r_{-}w~{}.$ (B.6)

Substituting these equalities into the expression for the energy we obtain

$E_{\text{string}}=-kw(r_{+}^{2}+r_{-}^{2}+\widetilde{E}^{2})+kw\widetilde{E}\sqrt{(r_{+}^{2}+r_{-}^{2}+\widetilde{E}^{2})}~{}.$
(B.7)

To compare the two expressions for the energy of the string, one needs to
relate the dimensionless energy $\widetilde{E}$ with the maximal radial extent
$r_{\text{max}}$ of the geodesic. The relation can be read off from the
geodesic to be

$r_{\text{max}}^{2}=\frac{\alpha+\sqrt{\alpha^{2}+4m^{2}\beta}}{2m^{2}}~{},$
(B.8)

with $\alpha$ and $\beta$ given in equation (3.3). In those expressions we set
$E=m\widetilde{E}$ and $L=r_{+}r_{-}w$, substitute for the particle mass $m$
using equation (B.6) and find that $\beta=0$. Therefore, we have

$\displaystyle r_{\text{max}}^{2}$
$\displaystyle=\frac{\alpha}{m^{2}}=\widetilde{E}^{2}+r_{+}^{2}+r_{-}^{2}-\frac{r_{+}^{2}r_{-}^{2}}{(r_{+}^{2}+r_{-}^{2})^{2}}\left(\widetilde{E}+\sqrt{\widetilde{E}^{2}+r_{+}^{2}+r_{-}^{2}}\right)^{2}.$
(B.9)

Inverting the expression, we obtain

$\widetilde{E}=\frac{1}{r_{+}^{2}-r_{-}^{2}}\left(r_{+}^{2}\sqrt{r_{\text{max}}^{2}-r_{+}^{2}}+r_{-}^{2}\sqrt{r_{\text{max}}^{2}-r_{-}^{2}}\right).$
(B.10)

Using the relations (B.9) and (B.10) between the maximal radius
$r_{\text{max}}$ and the rescaled particle energy $\widetilde{E}$, one can
prove that the energies (B.3) and (B.7) obtained from the Nambu-Goto and the
conformal gauge approach agree.

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