A twisted FZZ-like dual for the 2D back hole

Gaston Giribet and Matías Leoni

Abdus Salam International Centre for Theoretical Pysics, ICTP,

Strada Costiera 11, 34014, Trieste, Italy.

Physics Department, Universidad de Buenos Aires, FCEN - UBA,

Ciudad Universitaria, pabellón 1, 1428, Buenos Aires, Argentina.

Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET,

Av. Rivadavia, 1917, 1033, Buenos Aires, Argentina.

We review and study the duality between string theory formulated on a curved exact background (the two dimensional black hole) and string theory in flat space with a tachyon-like potential. We generalize previous results in this subject by discussing a twisted version of the Fateev-Zamolodchikov-Zamolodchikov conjecture. This duality is shown to hold at the level of -point correlation functions on the sphere topology, and connects tree-level string amplitudes in the euclidean version of the 2D black hole ( time) to correlation functions in a non-linear -model in flat space but in presence of a tachyon wall potential and a linear dilaton. The dual CFT corresponds to the perturbed 2D quantum gravity coupled to matter ( time), where the operator that describes the tachyon-like potential can be seen as a momentum mode perturbation, while the usual sine-Liouville operator would correspond to the vortex sector . We show how the sine-Liouville interaction term arises through a twisting of the marginal deformation introduced here, and discuss such ’twisting’ as a non-trivial realization of the symmetries of the theory. After briefly reviewing the computation of correlation functions in sine-Liouville CFT, we give a precise prescription for computing correlation functions in the twisted model. To show the new version of the correspondence we make use of a formula recently proven by S. Ribault and J. Teschner, which connects the correlation functions in the Wess-Zumino-Witten theory to correlation functions in the Liouville theory. Conversely, the duality discussed here can be thought of as a free field realization of such remarkable formula.

This paper is an extended version of the authors’ contribution to the XVIth International Colloquium on Integrable Systems and Quantum Symmetries, held in Prague, Czech Republic, in June 2007. A brief version was published in Rep. Math. Phys. 61 2 (2008) 151-162. Part of the material presented here is based on the results that one of the authors has reported in Refs. [1, 2], and it is in some way related to the recent works [3, 4, 5, 6, 7].

## 1 Introduction

One of the most profound concepts in string theory is the suggestive idea that spacetime itself could be a mere emergent notion, a sort of effective description of a more fundamental entity [8]. This conception relies on the existence of the duality symmetries of string theory, which suggest that concepts such as the curvature and topology of the spacetime might be only auxiliary notions. This idea is particularly realized by examples that manifestly show the duality between string theory formulated on curved backgrounds (e.g. black holes) and the theory in flat space but in presence of tachyon-like potentials. This is the subject we will explore here; and we will do this by studying the worldsheet description of the 2D string theory in the black hole background (i.e. the gauged Wess-Zumino-Witten (WZW) model).

#### 1.1 The subject

The relation between string theory in the 2D black hole background and Liouville-like conformal field theories representing “tachyon wall” potentials was extensively explored in the past. One of the celebrated examples is the Mukhi-Vafa duality [9], relating a twisted version of the euclidean black hole to the matter coupled to 2D gravity. The literature on the connection between the CFT and the black hole CFT is actually quite rich; we should refer to the list of papers [10]-[32] and the references therein. Recently, a new relation between the 2D string theory in the euclidean black hole background and a deformation of the matter CFT has received remarkable attention: This is the so-called Fateev-Zamolodchikov-Zamolodchikov conjecture (FZZ), which states the equivalence between the black hole and the often called sine-Liouville field theory [33, 35]. In the last six years this FZZ duality has been applied to study the spectrum and interactions of strings in both the black hole geometry and the Anti-de Sitter space [34, 30, 29]; and the most important application of it was so far the formulation of the matrix model for the two-dimensional black hole [35]. In fact, when one talks about the “black hole matrix model” one is actually referring to the matrix model for the sine-Liouville deformation of the matter CFT, and thus the black hole description in such a framework emerges through the FZZ correspondence. This manifestly shows how useful the FZZ duality is in the context of string theory.

Although at the beginning it appeared as a conjecture, a proof of the FZZ
duality was eventually given some years ago^{1}^{1}1More recently, after this paper was published, Y. Hikida and V. Schomerus
presented a proof of the FZZ conjecture [36].. This was
done in two steps: first, by proving the equivalence of the corresponding
N=2 supersymmetric extensions of both the 2D black hole -model and
the sine-Liouville theory [37]; and, secondly, by showing that the
fermionic parts of the N=2 theories eventually decouple, yielding the
bosonic duality as an hereditary property [38], see also [39, 40]. This could be done because both sine-Liouville and the black
hole theory admit a natural^{2}^{2}2The 2D black hole can be realized by means of the Kazama-Suzuki construction
[41, 42], while the sine-Liouville theory can be seen as a sector of
the N=2 Liouville theory. The bosonic version of the FZZ duality can be seen
to arise by GKO quotienting the R-symmetry of the N=2 version.
embedding in N=2 theories, where the duality can be seen as a manifestation
of the mirror symmetry. However, one could be also interested in seeing
whether a proof of such a duality exists at the level of the bosonic theory
itself. In this paper we will show how such a duality can be actually proven
(at the level of the sphere topology) without resorting to arguments based
on supersymmetry but just making use of the conformal structure of the
theory.

#### 1.2 The result

We will show that any -point correlation functions in the WZW ( ) on the sphere topology is equivalent to a -point correlation functions in a two-dimensional conformal field theory that describes a linear dilaton -model perturbed by a tachyon-like potential. This actually resembles the FZZ correspondence; however, instead of considering a vortex perturbation with winding here we will consider momentum modes of the sector . To be precise, the theory we will consider is defined by turning on the modes and in the following action

(1) |

where and . Namely, the perturbation we will consider is given by the operator

(2) |

where we denoted , which has to be distinguished from the T-dual direction . Operators )-operators with respect to the stress-tensor of the free theory are (

(3) |

so that they represent marginal deformations of the linear dilaton theory.
However, it is worth pointing out that condition (3) is
not sufficient to affirm that the theory defined by action (1) is
exactly marginal. In general, proving a theory is an exact
conformal field theory is highly non-trivial. Nevertheless, there is strong
evidence that particular perturbations belonging to those in (1) do
represent^{3}^{3}3One example of such a perturbation is sine-Liouville potential, which we
will discuss in section 3. Notice also that, at the critical value ,
the perturbations in (1) are precisely those discussed in [35]
in the context of matrix model. CFTs.

Coefficients in (1) must satisfy the condition for the Lagrangian to be real, and thus the theory results invariant under . The scaling relations between different couplings are given by standard KPZ arguments [43, 44, 45], being the scale of the theory governed by one of these constants, analogously as to how the Liouville cosmological constant introduces the scale in the matter CFT. The central charge of the theory is then obtained from the operator product expansion of the stress-tensor, yielding Eventually, we will be interested in adding a time-like free boson to the theory in order to define a Lorentzian target space of the form , so the central charge will receive an additional contribution coming from the time direction, yielding

(4) |

while the stress-tensor will result supplemented by a term . For practical purposes, this time-like
direction can be thought of as an auxiliary degree of freedom, and it does
not enter in the non-trivial part of the duality we want to discuss, being
coupled to the other directions just by the value of the central charge^{4}^{4}4In the case the theory corresponds to the product the condition demands . On the other
hand, if the space is just the coset the
corresponding condition reads . .

#### 1.3 Outline

The particular correspondence between the model (1) and the 2D black
hole we will discuss turns out to be realized at the level of -point
functions on the sphere topology, and corresponds to a twisted version of
the FZZ correspondence^{5}^{5}5In the sense that it involves a deformation of the sine-Liouville
interaction term in the action.. Consequently, we will discuss the latter
first. While being similar, the duality we will discuss herein presents two
important differences with respects to the FZZ: The first difference is that
the new duality admits to be proven^{6}^{6}6cf. Ref. [36]. in a relatively simple way without
resorting to arguments based on mirror symmetry of its supersymmetric
extension; secondly, it involves higher momentum modes () instead of
winding modes of the sector . We will make the precise statement of the
new correspondence in section 4, where we also address its proof. The paper
is organized as follows: In section 2 we review some features of the
conformal field theories that play an important role in our work. First, we
review the computation of correlation functions in Liouville field theory
with the purpose of emphasizing some features and refer to the analogy with
the Liouville case whenever an illustrative example is needed. Secondly, we
discuss some general aspects of the 2D black hole -model. Once
these two CFTs are introduced, we discuss how correlation functions in both
theories are related through a formula recently proven by S. Ribault and J.
Teschner [3, 4]. Their formula connects correlation functions in both
WZW and Liouville theory in a remarkably direct way [46], and it turns
out to be important for proving our result. In section 3 we briefly review
the FZZ dual for the 2D black hole; namely the sine-Liouville field theory.
In section 4 we introduce a “twisted” version of the
sine-Liouville theory, and we show that such “deformed”
sine-Liouville turns out to be a dual for the 2D black hole as well. A
crucial piece to show this new version of the duality is the
Ribault-Teschner formula mentioned above, for which we present a free field
realization that is eventually identified as being precisely the deformed
sine-Liouville model we want to study. Section 5 contains the conclusions.

## 2 Conformal field theory

To begin with, let us discuss some aspects of correlation functions in Liouville field theory. The reason for doing this is that Liouville theory is the prototypical example of non-compact conformal field theory [47] and thus the techniques for computing correlation functions in this model are analogous to those we will employ in the rest of the paper. Moreover, the models we will consider here are actually deformations of the Liouville theory coupled to a matter field, so that it is clearly convenient to consider this model first.

### 2.1 Liouville theory

#### 2.1.1 Liouville field theory coupled to matter

Liouville theory naturally arises in the formulation of the two-dimensional quantum gravity and in the path integral quantization of string theory [48]. This is a non-trivial conformal field theory [49, 50] whose action reads

(5) |

where is a real positive parameter called “the Liouville cosmological constant”. The background charge parameter takes the value in order to make the Liouville barrier potential to be a marginal operator. In the conformal gauge, the linear dilaton term , which involves the two-dimensional Ricci scalar has to be understood as keeping track of the coupling with the worldsheet curvature that receives a contribution coming from the point at infinity. The theory is globally defined once one specifies the boundary conditions, and this can be done by imposing the behavior for large , that is compatible with the spherical topology. Under holomorphic transformations Liouville field transforms in a way that depends on , namely . In this paper we will be interested in the coupling of Liouville theory to a boson field represented by an additional piece in the action (5) above. Moreover, we can also include the “time” direction Then, the central charge of the whole theory is given by

where refers to the Liouville central charge. Important objects of the theory are the exponential vertex operators [51]

which turn out to be local operators of conformal dimension with respect to the stress-tensor of the free theory,

(6) |

Now, let us move on and discuss correlation functions.

#### 2.1.2 Liouville correlation functions

The non-trivial part of correlation functions in the theory (6) is given by the Liouville correlation functions [50, 53, 54, 55], and these are formally defined as follows

and, on the spherical topology, these can be written by using that

(7) |

namely,

(8) |

This permits to compute correlation functions by employing the standard Gaussian measure and free field techniques. The overall factor and the -function come from the integration over the zero-mode of the Liouville field , and it also yields the insertion of an specific amount, of screening operators in the correlator. In deriving (8), the identity and the Gauss-Bonnet theorem were used to find out the relation between , , and the momenta , which for a manifold of generic genus and punctures would yield

(9) |

So, the correlators can be computed through the Wick contraction of the operators by using the propagator which corresponds to the free theory (6) and yields the operator product expansion . In principle, this could be used to integrate the expression for explicitly. Nevertheless, it is worth noticing that the expression (8) can be considered just formally since, in general, is not an integer number. Hence, in order to compute generic correlation functions one has to deal with the problem of making sense of such integral representation. With the purpose of giving an example, let us describe below the computation of the partition function on the sphere in detail. Such case corresponds to and , and the number of screening operators to be integrated out turns out to be . That is, in order to compute the genus zero partition function we have to consider the correlation function of three local operators inserted at the points and to compensate the volume of the conformal Killing group, . This has to be distinguished from the direct computation of the three-point function [53] of three “light” states as we will discuss below.

#### 2.1.3 A working example: the spherical partition function

Although it is usually said that string partition function on the spherical topology vanishes, we know that this is not necessary the case when the theory is formulated on non-trivial backgrounds. A classical example of this is the two-dimensional string theory formulated in both tachyonic and gravitational non-trivial backgrounds we will be discussing along this paper. Such models admit a description in terms of the Liouville-type sigma model actions, so that the computation of the corresponding genus zero partition functions involves the computation of spherical partition function of Liouville theory or some deformation of it. Here, we will describe a remarkably simple calculation of the Liouville partition function on the spherical topology by using the free field techniques. The free field techniques to be employed here were developed so far by Dotsenko and Fateev [56, 57], and by Goulian and Li [58] (see also [60, 61, 62]). The partition function is then given by

(10) |

with . According to the standard Wick rules, we can write

This can be explicitly solved for integer by using the Dotsenko-Fateev integral formula worked out in reference [57]. Even though we are interested in the case where is generic enough, and this can mean a negative real number, we can assume that this is an integer positive number through the integration and then try to analytically extend the final expression accordingly. In this way, we get

where, as usual, we denoted ; and we also denoted for notational convenience. Once again, this expression only makes sense for being a positive integer number, so that the non-trivial point here is that of performing analytic continuation. In order to do this, we can rewrite the expression above by taking into account that So we can expand it as

(11) |

Now, some simplifications are required. First, we can use that and to arrange the last product. Then, we can rewrite the product as

that is

and then use to write

where the identities
were also used. Again, the properties of the -function can be used
to write and . Then, once all is written in terms of the
partition function reads^{7}^{7}7Notice that we have absorbed a factor in the definition of the
measure of the path integral.

(12) |

This is the exact result for the Liouville partition function on the spherical topology, which turns out to be a non trivial function of . It oscillates with growing frequency and decreasing amplitude according approaches the values and . One of the puzzling features of the expression (12) is the fact that it does not manifest the self-duality that the Liouville theory seems to present under the transformation . In order to understand this point, it is convenient to compare the direct computation of we gave above with the analogous computation of the Liouville structure constant (three-point functions) for the particular configuration . The difference between both calculations is given by the overall factor in (10). As mentioned, this factor comes from the integration over the zero-mode of the field , but it can be also thought of as coming from the combinatorial problem of permuting all the screening operators. Actually, for integer this factor can be written as , where the divergent factor keeps track of a divergence due to the non-compactness of the Liouville direction. In fact, this yields the factorial arising in the residue corresponding to the poles of resonant correlators. On the other hand, in the case of being computing the structure constant , unlike the computation of , such overall factor should be instead of since one has to divide by the permutation of screening charges. Hence, we have . This is precisely consistent with the fact that , see Ref. [59]. Thus, this combinatorial problem appears as being the origin of the breakdown of the Liouville self-duality at the level of the partition function.

Now, let us move to study another CFT that is also a crucial piece in our discussion: the CFT that describes the 2D black hole -model.

### 2.2 String theory in the 2D black hole

#### 2.2.1 The action and the semiclassical picture

String theory in two dimensions presents very interesting properties that make of it a fruitful ground to study features of its higher dimensional analogues. One example is given by the 2D black hole solution discovered in Refs. [63, 64, 65]. This black hole solution is supported by a dilaton configuration, and it turns out to be an exact conformal background on which formulate string theory. In fact, the 2D black hole -model action corresponds to the gauged level- WZW theory [63]. An excellent comprehensive review on this model can be found in Ref. [66].

The worldsheet action for string theory in a two-dimensional metric-dilaton background, once setting , reads

(13) |

where the indices run over the two coordinates of the target space, whose metric is . This action is written in the conformal gauge, so, as we discussed before, the dilaton term has to be understood as keeping track of the coupling with the worldsheet curvature that receives a contribution coming from the point at infinity. The vanishing of the one-loop -functions demands , with being now the Ricci tensor associated to the target space metric . Since the 2D black hole string theory corresponds to the WZW model, it admits an exact algebraic description in terms of the current conformal algebra of the WZW theory; and we will comment on this in the following subsection. In the semiclassical limit, governed by the large regime, the euclidean version of the background is described by the following configurations for the metric and the dilaton ,

It is well known that the geometry of the euclidean black hole is that of a semi-infinite cigar that asymptotically looks like a cylinder. The angular coordinate of such cylinder is , while the coordinate is the one that goes along the cigar, running from (the tip of the cigar, where the string theory is strongly coupled) to (where the string coupling tends to zero). To get a semiclassical picture of this geometry, let us consider the large regime and redefine the radial coordinate as . Then, in the large approximation, and by also rescaling the angular coordinate by a factor the metric reads

(14) |

that asymptotically looks like the cylinder of radius . The parameter is related to the mass of the black hole, and it can be fixed to any positive value by shifting . Considering finite- corrections leads to a shifting in and then the metric and the dilaton result corrected. In such case, the dilaton reads

Thus, the 2D string theory in the euclidean black hole background can be semiclassically described by a deformation of the linear dilaton theory

(15) |

and, according to (14) and taking into account the finite- corrections, such “deformation” corresponds to perturbing the action (15) with the graviton-like operator [19]

(16) |

this is true up to a BRST-trivial^{8}^{8}8That means that it is pure gauge in the BRST cohomology. operator of the
form In these terms, the theory can be in
principle solved (e.g. its correlation functions can be computed) by using
the free field approach and the Coulomb-like correlators . Operator (16) is usually called the “black hole mass
operator”. The inclusion of this operator in the action has to be thought of
as being valid in a semiclassical picture and can be shown to be equivalent
to the free field representation of the WZW model.

In the large region of the space (where the theory turns out to
be weakly coupled) we have that the non-linear -model of strings in
the black hole seems to coincide with the action . Furthermore, there is a way of seeing that operator (16) actually describes the dilatonic black hole -model
beyond the semiclassical picture. To do so, it is necessary to argue that
such an action unambiguously describes the full theory beyond the weak limit
region [24, 5] and, for instance, reproduces the exact correlation
functions. This seems to be hard to be proven in general; nevertheless,
there is a nice way of showing that the perturbation (16)
corresponds to the theory on the black hole background. This relies on the
algebraic description of the
WZW theory and is quite direct: The point is that the action , once supplemented with the BRST-trivial
operator and a free time-like boson , can be shown to be related to
the well known free field realization of the WZW
action through a -boost given by^{9}^{9}9Please, do not mistake the time-like coordinate for the notation used
for the stress-tensor. Excuse us for this overlap in the notation.

and the standard bosonization^{10}^{10}10It is usually convenient to use a different bosonization, expressing the
field as an exponential function. This would lead to a
Liouville-like interaction in the action. , , with and with [67]. In
fact, this leads to the Wakimoto free field description of the current algebra in terms of the linear dilaton field and
the ghost system [68]. In Wakimoto variables one
identifies the theory as being the WZW model formulated on
with the elements of the group written in the Gauss parameterization. Then,
the coset theory is obtained by simply taking
out the time-like direction which realizes the current^{11}^{11}11Alternatively, an additional free boson, analogous to can be added in
order to relize the gauging, see [21, 22] and referenctes therein.

recall that this is a time-like direction so that the corresponding correlator flips its sign and thus turns out to be .

On the other hand, let us mention that the dual theory (i.e. the sine-Liouville theory) is also defined as a perturbation of (15); see (34) below. According to this picture, it is possible to consider FZZ duality as a relation between different marginal deformations of the same free linear dilaton background. This was the philosophy in Ref. [5], where the FZZ correspondence was seen from a generalized perspective, considering it as an example of a set of connections existing between different marginal deformations of (15). Here, we will be discussing a similar correspondence; we will consider perturbations carrying momentum modes of the tachyon potential and discuss how it describes WZW correlation functions. We will dedicate some effort to understand the relation between such perturbation and the standard FZZ duality (that involves modes). But, first, let us continue our description of the theory in the black hole background with appropriate detail.

#### 2.2.2 String spectrum in the 2D black hole and its relation to strings

The spectrum of the 2D sting theory in the black hole background corresponds
to certain sector of the Hilbert space of the gauged WZW model, and is thus given in terms of certain representations
of . The string
states are thus described by vectors which are associated to vertex operators where , and are indices that label the
states of the representations of the group. In order to define the string
theory, it is necessary to identify which is the subset of representations
that have to be taken into account. Such a subset has to satisfy several
requirements^{12}^{12}12For an interesting discussion on non-compact conformal field theories see
[47].. In the case of the free theory these requirements are
associated to the normalizability and unitarity of the string states. At the
level of the interacting theory, additional properties are requested, like
the closure of the fusion rules, the factorization properties of -point
functions, etc.

The WZW model is behind the description of string theory in both the 2D black hole background (through the coset construction) and in space. These two models are closely related indeed, but still different. In the case of the black hole, the states of the spectrum are labeled by the index of the representations with the indices and falling in the lattice

(17) |

with and being integer numbers, and the conformal dimension of the vertex operators is given by

(18) |

On the other hand, string theory can be described in terms of the
WZW model on the product between the coset and a
time-like free boson [69], so that the worldsheet theory turns out to
be the product between the time and the euclidean black hole. This can be
realized by adding the contribution^{13}^{13}13Besides, one can represent string theory in space in terms of the
Wakimoto free field realization mentioned above. In terms of these fields
the metric reads . to the action (15) and by supplementing the
vertex operators with a factor . Thus, the vertex operators
on have conformal dimension given by that carries the charge under the field

(19) |

which corresponds to adding the conformal dimension of the time-like part to the coset contribution (18). In some sense, the string theory in the 2D black hole can be thought of as having constrained the states of the theory in to have vanishing bulk energy, . In this way, one has the theory on the background as an appropriate realization of sting theory in space [71, 74, 73, 75]. However, before going deeper into the string interpretation of the WZW model, some obstacles have to be overcame. In fact, even in the case of the free string theory, the fact of considering non-compact Lorentzian curved backgrounds is not trivial at all. The main obstacle in constructing the space of states is the fact that, unlike what happens in flat space, in curved space the Virasoro constraints are not enough to decouple the negative-norm string states. In the early attempts for constructing a consistent string theory in , additional ad hoc constraints were imposed on the vectors of the representations in order to decouple the ghosts. The vectors of representations are labeled by a pair of indices and , and thus such additional constraints (demanded as sufficient conditions for unitarity) imply an upper bound for the index of certain representations, and consequently an unnatural upper bound for the mass spectrum. The modern approaches to the “negative norm states problem” also include such a kind of constraint on , although this fact does not imply a bound on the mass spectrum as in the old versions it did [71]. The upper bound for the index of discrete representations, often called “unitarity bound”, reads In the case of Euclidean , the spectrum of string theory is just given by the continuous series of , parameterized by the values with and by real . On its turn, the case of string theory in Lorentzian is richer and its spectrum is composed by states belonging to both continuous and discrete series. The continuous series have states with with and , with (as in , obviously). On the other hand, the states of discrete representations satisfy with . Other important ingredient for constructing the Hilbert space is the index labeling the operators . In the black hole background, turns out to be given by (17). In , the quantum number is independent of the bulk kinetic energy and the bulk angular momentum contributing to the total energy as . Then, the question arises as to how the index appears in the Hilbert space of the WZW theory. The answer is that in order to fully parameterize the spectrum in we have to introduce the “flowed” operators (with ) which are defined through the spectral flow automorphism [71]

(20) |

acting of the original generators , which satisfy the Lie product that define the affine algebra

(21) |

Then, states belonging
to the flowed discrete representations are those obeying^{14}^{14}14or analogous relations for the Weyl reflected representations, namely .

(22) |

and being annihilated by the positive modes, namely

(23) |

States with represent highest (resp. lowest) weight states, while primary states of the continuous representations are annihilated by all the positive modes. On the other hand, the excited states in the spectrum are defined by acting with the negative modes () on the Kac-Moody primaries ; these negative modes play the role of creation operators (i.e. creating the string excitation). The “flowed states” (namely those being primary vectors with respect to the defined with ) are not primary with respect to the algebra generated by , and this is clear from (20). However, highest weight states in the series are identified with lowest weight states of , which means that spectral flow with is closed among certain subset of Kac-Moody primaries.

The states belonging to discrete representations have a discrete energy spectrum and represent the quantum version of those string states that are confined in the centre of space; these are called “short strings” and are the counterpart of those states that are confined close to the tip of the cigar geometry. On the other hand, the states of the continuous representations describe massive “long strings” that can escape to the infinity, where the theory is weakly coupled. In the case of the 2D black hole, the index of these long strings has a clear interpretation as an “asymptotically topological” degree of freedom (is not a topological one though). Because of the euclidean black hole has the geometry of a semi-infinite cigar and thus looks like a cylinder very far from the tip, the states in the asymptotic region have a winding number around such cylinder. However, this is not strictly a cylinder but has topology instead of , so that, as it happens in , the winding number conservation can be in principle violated. Of course, this feasibility of violating is not evident from the background (15)-(16), which is reliable only far from the tip of the cigar, but the phenomenon can occur when string interactions take place. Instead, in the sine-Liouville theory, the violation of the winding number is understood in a clear way, as due to the explicit dependence on the T-dual direction . We will return to this point later. Now, let us discuss the string interactions in the black hole geometry.

#### 2.2.3 String amplitudes and correlation functions in the WZW theory

The string scattering amplitudes in the 2D black hole background are given by (the integration over the inserting points of) correlation functions in the WZW theory. The first exact computation of such WZW three and two-point functions was performed by K. Becker and M. Becker in Refs. [20, 22], and it was subsequently extended and studied in detail in Refs. [78]-[80] by J. Teschner. The interaction processes of winding string states were studied later in [72, 73], after J. Maldacena and H. Ooguri proposed the inclusion of spectral flowed states in the spectrum of the theory [71]. Moreover, several formalisms were employed to study the correlators in this non-compact CFT [84]-[92]. One of the most fruitful tools to work out the functional form of these WZW correlators was the analogy between these and Liouville correlators [83, 50, 82, 84]. Another useful approach to compute the exact correlation functions is the free field representation [20, 22, 89, 90, 73, 74, 75], which for the WZW model turns out to be similar to what we discussed for the Liouville theory. Let us briefly describe how this “free field computation” works for the case of the two-point function: Consider the correlation functions of exponential operators

(24) |

Then, written in terms of the Wakimoto free fields^{15}^{15}15Please, do not mistake the Wakimoto field (which is a local
function on the variable ) for the Euler -function introduced in
Eq. (11) (which is defined by ). That is, the fields in (25) have to be
distinguished from the function in (26). We preferred
to employ the standard notation here. , , and ,
such correlators read^{16}^{16}16In order to compare with the original computation in Ref. [20] it is
necessary to consider the Weyl reflection , which is a
symmetry of the formula for the conformal dimension, actually.

(25) |

where the screening inserted at is then taken to be fixed at
infinity , while and as usual
(this is analogous to what we did when discussed the case of Liouville
partition function). It is easy to see that this can be solved by using the
(analytic extension of) Dotsenko-Fateev integrals, and one eventually finds^{17}^{17}17For instance, compare with formula (49) in Ref. [73], after the Weyl
reflection.

(26) |

where the -dependent -functions stand from the combinatorial problem of counting the different ways of (Wick) contracting the -functions with the -functions in (25). Expression (26) is the so called WZW reflection coefficient and corresponds to the exact results for the two-point function. Notice that, in particular, (26) contains the factor that keeps track of finite- effects. Analogously, the expression of the three-point functions can be found by these means [22].

Also, some features of the four-point function are known, as the physical interpretation of its divergences [72], and the crossing symmetry [50]. In fact, our understanding of correlation functions in both the 2D black hole and backgrounds has substantially increased recently, and we have a relatively satisfactory understanding of these observables. Nevertheless, some features remain still open questions: One puzzle is the factorization properties of the generic four-point function and the closure of the operator product expansion of unitary states. Addressing these questions would require a deeper understanding of the analytic structure of the four-point function. The general expression for the -point functions for is not known; however, a new insight about its functional form appeared recently due to the discovery of a new relation between these and analogous correlators in Liouville field theory [46, 3, 4]. This relation between WZW and Liouville correlators is one of the key points for what we are going to study in this paper. Let us give some details about it.

### 2.3 A connection between Liouville and WZW correlation functions

Let us comment on the particular connection that exists between the correlation functions of the two conformal theories we discussed above; namely, between Liouville and WZW correlation functions. This relation is a result recently obtained by S. Ribault and J. Teschner, who have found a direct way of connecting correlators in both WZW and Liouville conformal theories [3, 4]. The formula they proved is an improved version of a previous result obtained by A. Stoyanovsky some years ago [46]. The Ribault-Teschner formula (whose more general form was presented by Ribault in Ref. [4]) connects the -point tree-level scattering amplitudes in Euclidean string theory to certain subset of -point functions in Liouville field theory, where the relation between and is determined by the winding number of the interacting strings. Even though this formula was proven for the case of the Euclidean target space, it is likely that an analytic continuation of it also holds for the Lorentzian model. The Ribault-Teschner formula reads as follows: If represent the vertex operators in the WZW model, and represent the vertex operators of Liouville theory, then it turns out that

(27) | |||||

with the normalization factor given by