A Note on Z_2 Symmetries of the KZ Equation

We continue the study of hidden Z_2 symmetries of the four-point sl(2)_k Knizhnik-Zamolodchikov equation iniciated in hep-th/0508019. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector w=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS_3 has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the non-violating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, non diagonal functional relations between different solutions of the KZ equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both WZNW and Liouville conformal theories.


Introduction
The SL(2, R) k WZNW model plays an important role within the context of string theory. This conformal model enters in the worldsheet description of the theory formulated on exact non-compact curved backgrounds, being the most celebrated examples: the string theory on AdS 3 and on the 2D black hole (the last, by means of its gauged SL(2, R) k /U(1) construction). Consequently, among the main motivations, this topic received particular attention due to its relevance for black hole physics and its relation with the AdS/CF T correspondence. Actually, this conformal field theory raises the hope to work out the details of the correspondence beyond the particle limit approximation [2,3,4]. Here, we continue the study of the observables in this theory, focussing out attention on the four-point function.

String amplitudes in AdS 3
Although the structure of the WZNW model on SL(2, R) (corresponding to strings in Lorentzian AdS 3 ) is not yet well understood, the observables of this theory are assumed to be well defined in terms of the analytic continuation of the correlation functions in the Euclidean case, i.e. in the SL(2, C) k /SU(2) WZNW gauged model. Hence, the string scattering amplitudes in Lorentzian AdS 3 are obtained by integrating over the worldsheet insertions of vertex operators in the model on SL(2, C)/SU (2) and then extending the range of the indices of continuous representations in order to include the discrete representations of SL(2, R) as well. The correlation functions defined in such a way typically develop poles in the space of representation indices and these poles are then interpreted as state conditions of bounded states (called "short strings" in [5]) and yield selection rules [6,7] for the string scattering amplitudes. Besides, the continuous representations (the "long strings" in [5]) have a clear interpretation as asymptotic states and define the S-matrix in the Lorentzian target space.
In the last decade, we gained important information on the explicit functional form of the WZNW correlation functions, and this enabled us to study the string theory on AdS 3 beyond the supergravity approximation, both at the level of the spectrum [8] and at the level of its interactions (see [9,5] and references therein). Originally, some particular cases of the two and three-point functions were explicitly computed in [10,11], and was Teschner who presented the general result in [12] and carefully described its formal aspects in [12,13,14], by employing the boostrap approach and the minisuperspace approximation. Subsequently, other approaches, like the path integral techniques [15] and the free field representation [16,17], were employed to rederive such correlation functions. The operator product expansion was studied in Ref. [6,7] in order to investigate the fusion rules, and the crossing symmetry of the correlation functions was eventually proven in Ref. [18] by making use of its analogy with the Liouville field theory [19,20]; see also [21]. The study of the correlation functions was also shown to be useful for consistency checks of the conjectured dualities between this and other conformal models [22,23,24].
After these objectives were achieved, the study of correlation functions in the SL(2, R) k WZNW model acquired a new dimension since Maldacena and Ooguri pointed out the existence of new representations of SL(2, R) k contributing to the spectrum of the string theory in AdS 3 winding three-point function and the conservative one. The way of showing such connection takes into account a recent result that presents a new map between WZNW and Liouville correlators. This map, different from that employed in [14] to prove the crossing symmetry in SL(2, C)/SU (2), was discovered by Stoyanovsky some years ago [28], and it was further developed by Teschner and Ribault in Ref. [29,30,31]. In [32,33,34,35] this map between both CFTs (henceforth denoted SRT map) was analyzed in the context of the implications it has in string theory (see also Ref. [36] for recent works on the relation between Liouville and WZNW correlators). In references [37,38], a free field realization of the SRT map was given, and it was shown to reproduce the correct three-point function for the case where one spectral flowed state (string state with winding ω = 1) is considered. Such observable had been also computed in [1] through "the other" connection to Liouville theory, discovered by Fateev and Zamolodchikov in [19] and extended in Ref. [20,14,21] to the non-compact case (henceforth denoted FZ map). We will be more precise in the following paragraph.

Overview
Recently, a new possibility to study the four-point function involving winding states has raised because of a discovery made by Ribault [30] and Fateev [39], stating that correlators involving winding states in WZNW can be written in terms of correlators in Liouville field theory. In the case of four-point function with one state in the sector ω = −1, on which we are interested here, this turns out to correspond to the Liouville five-point function. In principle, this does not seem to imply an actual simplification since five-point function in Liouville theory cannot be simply solved either. However, we noticed that the SRT map is not the only way of mapping "the same" Liouville five-point function to a (different) four-point function of the WZNW theory. Indeed, we can also do this by employing the non-compact generalization of the FZ map [19]. If this "second map" is used, the four-point function reached in the WZNW side is one that contains four non-winding states, enabling us to relate winding violating processes in AdS 3 with their conservative analogs. However, this is not the whole story since, unlike the SRT map [28,29,30], the FZ map involves a non-diagonal correspondence between the quantum numbers of both Liouville and WZNW sides. Hence, besides the appropriate combination of the SRT and the FZ maps, a sort of a "diagonalization procedure" is also required in order to present the result in a clear form. This diagonalization is eventually achieved by using the FZ map itself and the reflection symmetry of Liouville theory. By doing something similar to that in Ref.
[1], we will first indicate how such diagonalization is realized due to identities holding between different exact solutions of the KZ equation. In fact, this paper can be regarded as a addendum to Ref. [1], being the second part of our study of hidden symmetries in the four-point sl(2) k KZ equation. In [1], a set of Z 2 symmetry transformations of the KZ equation were studied. Such involutions were realized by means of the action on the four indices of SL(2, R) representations and led to prove identities between different exact solutions of the KZ equation. The main tool for working out such identities was the FZ map, mapping four-point functions of the WZNW theory to a particular subset of five-point functions of the Liouville field theory. Following this line, here we explore the implications of new symmetry transformations on the solutions of KZ equation. The plan of the paper goes as it follows: In the next section we will review the connections between four-point functions in WZNW theory and five-point function in Liouville field theory. The fact that there is not a unique map of this kind 1 is the reasons for a non trivial relation between correlators of winding and non-winding states to exist. In section 3, as a preliminary result, we first prove a new identity between exact solutions of the KZ equation, complementing the catalog presented in Ref. [27,1]. This identity is again realized by a non diagonal Z 2 transformation of the class studied in Ref. [1], acting on the four indices of the representations of SL(2, R). Then, this leads us to show how the four-point correlation function involving one spectral flowed state in the winding sector ω = −1 can be written in terms of the correlation function of non flowed (non winding) states.

Conformal field theory 2.1 The WZNW model
We are interested in the WZNW model formulated on the SL(2, R) group manifold. Its action corresponds to the non-linear σ-model of strings in Lorentzian AdS 3 space. On the other hand, its Euclidean version is similarly given by the gauged SL(2, C)/SU(2) model. The states of the Euclidean model are characterized by normalizable operators on the Poincaré hyper half-plane H + 3 = SL(2, C)/SU (2). These operators can be conveniently written as where the variables γ, γ and u are associated to the AdS 3 Euclidean metric in Poincaré coordinates; namely ds 2 = k(u −2 du 2 + u 2 dγdγ); while the complex coordinates x and x represent auxiliary variables that expand the SL(2, C) representations as it follows for a = {3, +, −}. The dots "..." stand for "regular terms" in the OPE, while the differential operators D a x correspond to the realization Besides, the operators J a (z) are the local Kac-Moody currents, whose Fourier modes are defined by J a (z) = n∈Z J a n z −1−n and satisfy the sl(2) k affine Kac-Moody algebra of level k. The value of k is related to the string length and the AdS radius through k = l 2 AdS /l 2 s . This keeps track of the conformal invariance of the WZNW theory. The Sugawara construction yields the stress-tensor that can be used to compute the central charge of this theory, being The formula for the conformal dimension of operators (1) reads where the indices take the values j = 1 2 + iR >0 for SL(2, C)/SU (2). In order to propose a similar algebraic realization for the Lorentzian string theory, it is usually assumed that the observables of the Euclidean theory admits an analytic continuation in the variable j, now parameterizing both continuous and discrete representations of SL(2, R). It is worth noticing that the transformation j → 1 − j is a symmetry of the formula (3); this corresponds to the so called Weyl reflection symmetry and will be important for us in section 3. Operators (1) represent the vertex operators of the string theory in AdS 3 and define the correlation functions. Then, after integrating over the variables z µ , the functional form of the N-point correlators will still depend on the auxiliary variables x µ , with µ = {1, 2, ...N}. Within the context of the AdS 3 /CF T 2 interpretation, those auxiliary variables are interpreted as the coordinates of the dual conformal field theory in the boundary of AdS 3 , and then acquire a geometrical meaning. This picture, employing the variable x to label the representations, is usually referred as the x-basis. On the other hand, there exists a different picture that is also convenient, and in which the quantum numbers labeling the representations permits to define string states with well defined momenta in the bulk. This is the often called m-basis, and employs the standard way of parameterizing representations of the SL(2, R) group, by using a pair of indices j, m. In this frame, the string states in AdS 3 are given by vectors |j, m > ⊗ |j, m > of the SL(2, R) × SL(2, R) representations. Furthermore, as it was mentioned in the introduction, in Ref. [5] it was shown that an additional quantum number should be included in order to fully characterize the space of states in AdS 3 . This quantum number, denoted ω, is associate to the winding number of strings in AdS 3 , at least in what respects to the states with a suitable asymptotic description (the long strings). From an algebraic point of view, ω labels the spectral flow transformation that generates new representations of the theory. This is the reason because we will refer to the "spectral flowed states" as "winding states", indistinctly. The vertex operators representing winding states (states with ω = 0) in the m-basis are to be denoted Φ ω j,m,m (z) and define the correlation functions in this basis, which we will denote as Φ j 1 ,m 1 ,m 1 (z 1 )Φ j 2 ,m 2 ,m 2 (z 2 )...Φ j N ,m N ,m N (z N ) . The conformal dimension of the states in the m-basis depends both on m and ω through the expression In the case ω = 0, operators Φ ω=0 j,m,m (z) are related to those of the x-basis through the Fourier On the other hand, the definition of string states with ω = 0 in the x-basis was studied in [5,40]; however, these have not a simple expression. Now, let us discuss the correlation functions in more detail.

The four-point KZ equation
As it was commented, the two and three-point functions in the WZNW model are known [11]; and the four-point function in the sector ω = 0 was studied in detail in Ref. [14], where a consistent ansatz was proposed based in the analogy with other CFTs; we detail such proposal below.
The four-point correlation functions on the zero-genus topology are determined by conformal invariance up to a factor f , which is a function of the cross ratio z and the variables j i , x i and x i , that label the representations; namely The function f = f j 1 ,j 2 ,j 3 ,j 4 (x, z) is then given by certain linear combination of solutions to the Knizhnik-Zamolodchikov partial differential equation (KZ); i.e. that combination which turns out to be monodromy invariant. The KZ equation in the case of the SL(2, R) k WZNW model takes the form where the differential operators are With [14], we can consider the following ansatz for the solution where the functions C(j 1 , j 2 , j 3 ) and B(j 1 ) are given by the structure constants and the reflection coefficient of the SL(2, R) k WZNW model, respectively; and where the contour of integration is defined as covering the curve C = 1 2 +iR. The integration along C turns out to be redundant for a monodromy invariant solution since such particular linear combination is invariant under Weyl reflection j → 1 −j, for which the contour transforms as the complex conjugation C → C = −C.
Here, we are interested in the relation between different solutions to (6).

The Liouville field theory
Now, let us briefly review the Liouville field theory, which is the other CFT in which we are interested here. Its action reads where the background charge is given by Q = b + b −1 , and the exponential self-interaction thus corresponds to a marginal deformation. We will set the Liouville cosmological constant µ to have an appropriate value in order to make the correlation functions acquire a simple form. This is achieved by properly rescaling the zero mode of ϕ(z). The Liouville field theory is reviewed with impressive detail in Ref. [41]; see also [43,44] and the recent [45]. The central charge of the theory is given by c = 1 + 6Q 2 > 1 and the vertex operators have the exponential form [46] whose conformal dimension is given by Vertex operators (9) define the Liouville N-point correlation functions, which are to be denoted by henceforth called Liouville reflection. This symmetry induces the identification between both fields V α (z) and V Q−α (z), yielding the operator valued relation which is valid for any vertex i = {1, 2, ...N} (though we exemplified it here for i = 1), and where R b (α 1 ) represents the Liouville reflection coefficient Another important feature of the Liouville correlation functions is the fact that those involving states with momentum α = −1/2b satisfy a well known partial differential equation, called the Belavin-Polyakov-Zamolodchikov equation (BPZ). This is similar to that of the minimal models and actually holds for a wider family of correlators, namely all of those involving certain state with momentum α m,n = 1−m 2 b + 1−n 2 b −1 for any pair m, n ∈ Z >0 . In particular, here we are interested in Liouville five-point correlators of the form Now, once both WZNW and Liouville theories were introduced, we move to the following ingredient in our discussion: the close relation existing between correlation functions of each of these two conformal theories. Such relation has multiple aspects indeed; we will discuss two of them in the following two subsections.

The Fateev-Zamolodchikov identity
The often called FZ map is a dictionary that connects four-point correlation functions in WZNW models to five-point correlation in Liouville field theory. This result was developed in Ref. [18] (see also Ref. [21]) and, among other ingredients involved in its derivation, is based on the relation existing between solution of the KZ and the BPZ differential ecuations. Such relation was originally noticed by Fateev and Zamolodchikov in Ref. [19] for the case of the compact SU(2) k WZNW case and the minimal models, and it basically states that, starting from any given solution of the four-point KZ equation, a systematical way of constructing a solution of the five-point BPZ equation exists. The FZ map, or strictly speaking its adaptation to the non-compact WZNW model, was employed to investigate several properties of the SL(2, R) k conformal theory, becoming one of the most fruitful tools to this end. In particular, Teschner gave to it its closed form and used it to prove the crossing symmetry of the WZNW model by assuming that a similar relation holds for the conformal blocks. Besides, Andreev rederived the fusion rules of admissible representations of theŝl(2) k affine algebra by similar means [20], and Ponsot discussed the monodromy of the theory with such techniques [21]. In Ref. [1], the FZ map was shown to be useful to prove several identities between exact solutions of the KZ equation, and we will extend such result in the next section here. But first, let us briefly review the FZ statement: The observation made in [19] is that the KZ equation satisfied by the fourpoint functions in the WZNW model agrees with the BPZ equation satisfied by a particular set of five-point functions in Liouville field theory. More specifically, Fateev and Zamolodchikov showed that it is possible to get a solution of the KZ equation by starting with one of the BPZ system. This yields the relation where and where the function Υ b (x) is the one introduced in Ref. [42] when presenting the explicit form of the Liouville three-point function (see the Appendix for the definition and a survey of functional properties of the Υ b -function). The coefficient c b is a b-dependent factor, independent of the quantum numbers j µ and α µ , whose explicit value can be found in Ref. [18] and references therein, though its explicit value is not important for our purpose here. The relation between the indices j µ , that label the SL(2, R) k representations, and the Liouville momenta α µ involves a non-diagonal invertible transformation defined through On the other hand, the relation between the WZNW level k and the Liouville parameter b is given by Now, let us make a remark on the KPZ scaling in (13): The parameter λ is a free parameter of the WZNW theory and it can be regarded as the mass of the two-dimensional black hole when the gauged SL(2, R) k /U(1) is being considered. This is just a parameter and is free of physical significance since it can be set to any positive value by means of an appropriate rescaling of the zero-mode of the dilaton. Thus, we could absorb the factor πb 2b (14) by simply shifting λ. This freedom can be used to simplify the functional form of the correlators [23].

The Stoyanovsky-Ribault-Teschner identity
Analogously as to the case of the FZ map, the SRT map, discovered by Stoyanovsky in [28], translates solutions of the KZ equation into solutions of the BPZ equation. But this is, in some sense, more general. Unlike the FZ map, the SRT map presents two advantages on its partner: the first is that it involves a diagonal transformation between the quantum numbers j µ and α µ ; the second advantage, and more important indeed, is that it holds for the case of the N- where the normalization factor is given by while the z-dependent function is Here, the Liouville cosmological constant µ has been set to the value µ = b 2 π −2 for convenience; this is to make the KPZ scaling of both sides of (16) match. It is also important to keep in mind the presence of the free parameter λ of the WZNW theory, and the fact that this can still be set to an appropriate value in order to absorb the powers of Γ(b 2 )/Γ(1 + b 2 ) in the KPZ overall factor of WZNW correlators [23]. For our purpose, we do not need to focus the attention on the specific value of the j-independent normalization factor, which we wrote above just for completeness, being equal to 2π 3−2N b/M! c M +2 k and where c k represents a k-dependent factor whose exact value is discussed in [30] but, again, is not relevant for us.
As in the case of the FZ map, the relation between the level k of the WZNW theory and the parameter b of the Liouville theory is given by while the quantum numbers labeling the states of both theories are related through We also have and, then, the total winding number is given by This manifestly shows that scattering processes leading to the violation of the total winding number conservation can occur in principle. In particular, Ribault formula states that the four-point function involving one flowed state of the sector ω = −1 obeys where c b is, again, a J-independent factor whose specific value, c b = 2π −5 bc N −2−M k , is known in terms of c k , though not important for us. Notice that we have changed the notation here, where we replaced j µ by J µ ; this will be convenient later. Regarding the relation between J µ and α µ , let us make some remarks here: Notice that, according to (18), the Liouville reflection α µ → Q−α µ translates in terms of the WZNW correlators in doing the Weyl reflection j → 1−j. This is a simple comment, but is also important. In particular, because of (11), this implies that the correlation function involving the field Φ j (x|z) and the one involving the field Φ 1−j (x|z) are connected one to each other by the overall factor R b (b(k/2 − j)). It is not difficult to see that this simple affirmation leads directly to the exact expression for the WZNW two-point function, for instance. This comment will become important below.

The four-point function of winding strings 3.1 The idea
The idea is to use Eq. (13) and Eq. (20) in order to relate the four-point function Φ . The nexus will be the Liouville five-point function 4 µ=1 V αµ (z µ )V −1/2b (x) , appearing in the right hand side of both expressions: this is connected to the first through the SRT map while is connected to the second through the FZ map. After doing this, the second step would be to "diagonalize" the relation between indices j µ and indices J µ . We will do this by using the preliminary result of section 3.2. Notice that, according to Eq. (13) and Eq. (20), the relation between momenta J µ and j µ is given by (29), below. The third step would be to perform a Fourier transform in order to translate the operators Φ jµ (x µ |z µ ) of the x-basis into operators Φ jµ,mµ,mµ (z µ ) of the m-basis.

Non diagonal symmetry of the KZ equation
As mentioned, the purpose here is to prove a functional relation obeyed by two solutions of the KZ equation that will be useful further. Specifically, we will see that four-point correlation functions with momenta j 1 , ...j 4 have a simple expression in terms of the analogous quantity with momenta j 1 , ... j 4 , being related through To see this, let us begin by observing that, according to (15), the fact of performing the change j µ → j µ for all the indices µ = {1, 2, 3, 4} translates in terms of the Liouville momenta α µ in doing the change α i → Q − α i for i = {2, 3, 4}, while keeping α 1 unchanged. Then, by taking into account the Liouville reflection symmetry (11) and the dictionary (15), we get 2 The factor F k (j 1 , j 2 , j 3 , j 4 )/F k ( j 1 , j 2 , j 3 , j 4 ) involves a quotient of products of Υ b -functions that, by making use of the formulae in the Appendix (see (35) below), can be substantially simplified once one takes into account the presence of the three factors R b (α i ) (i = 2, 3, 4). Then, the results reads is a polynomial in the indices j µ , namely P(j) = −3j 1 + j 2 + j 3 + j 4 , though is not actually relevant here. Since j 1 + j 2 = j 3 + j 4 and j 1 + j 3 = j 2 + j 4 , one immediately notices that the Z 2 -invariant form under the involution j µ → j µ is given by Formula (22) will be useful in the next section. Similar functional relations were studied in Ref. [1,20,47]. Now, let us move to the case of winding strings.

The four-point function of winding strings
In this subsection, as an application of our formula (22), we will employ it to show that, by making use of both FZ and SRT maps, it is feasible to write down a formula that expresses the winding violating four-point functions in terms of the zero-winding four-point function. To do this, let us begin by considering the FZ identity where the quantum numbers α µ and j µ are then related as it follows Now, let us also define quantum numbers J µ as for µ = {1, 2, 3, 4}. Then, by taking into account (11), we have Consequently, we can write the following SRT identity Hence, plugging (25) into (27), we find where we simply defined and where, according to the definitions above, the indices J µ and j µ turn out to be related by for i = {2, 3, 4}. Then, it is feasible to show that (30) where the symbol ∼ stands just because we are not writing the k-dependent overall factor and the precise KPZ scaling here (this can be directly read from Eq. (26) if necessary). For short, in the last equation we have employed the notation and one could also find convenient to define Identity (28) does already express the winding violating correlators in terms of the conservative ones. However, such expression is still not fully satisfactory since the right hand side of (28) contains a four-point correlator that, instead of involving states with momenta J µ , involves states with momenta j µ , µ = {1, 2, 3, 4}. Here is where our result of subsection 3.2. becomes useful, since the following step would be to relate the correlator in the j µ -basis with the one that is diagonal in the J µ -basis. This can be simply achieved by making use of the equation (22) and by taking into account (29), yielding with the normalization factor being here, P(j) is again a polynomial in the indices j µ that is related to the P(j) in (22) through the replacement j i → J i , i = {2, 3, 4}, j 1 → k/2 − J 1 . In writing down (34) we used the fact that, as in the previous subsection, the combination of the Υ b -functions and the Γ-functions in R b (α i ) leads to a very simple expression. In this case, it comes from Hence, up to the Weyl reflection j 1 → 1 − j 1 that we discuss below, equations (33) and (28) represent the result we wanted to prove: the four-point function involving one winding state of the sector ω = −1 admits to be expressed in terms of the four-point function of non-winding states. Let us discuss the final form of this result in the following paragraphs (see (37) below), where we address the comparison with the case of the three-point function. We will also comment on possible applications and conclude with some remarks in the following section.

Analogy with the case of the three-point function
The three-point function that includes one spectral flowed state of the sector ω = −1 can be written in terms of the structure constant of three non-flowed states [25,5]. In the subsection above we proved a similar relation at the level of the four-point functions. Then, the natural question we want to address now is whether both relations are connected in some way. As we of the KZ equation, was studied in the first part of our work, [1]. As it was detailed in Ref. [5], when presenting the original derivation of [25], the proportionality factor connecting both violating winding and non-violating winding three-point correlators is basically given by the reflection coefficient B(J 1 ) ∼ B −1 (k/2 − J 1 ) of the SL(2, R) k WZNW model (see the formula above). This quantity has the form and therefor This permits to show that the picture for the three-point functions turns out to be similar to the one we are obtaining here for the four-point functions. Actually, this is not only manifested in the shifting J 1 → k/2 − J 1 , but also in the fact that, when combining both (28) and (33), a factor Υ b (2b(k/2−J 1 )) Υ b (2b(k/2−J 1 )−b) also arises in the product between A k (J 1 , J 2 , J 3 , J 4 |x, z) and B k (J 1 , J 2 , J 3 , J 4 |x, z). This factor is precisely proportional to (2J − 1)B(J); thus, this is actually in equal footing as to how the WZNW reflection factor stands in the relation between violating and non-violating three-point correlators.
On the other hand, some nice cancellations occur when combining expressions (28) and (33). First, we observed that, instead of the operator Φ j 1 (x|z), the correlator in the right hand side of (28) involves its Weyl reflected operator Φ 1−j 1 (x|z); and, as we commented at the end of section 2, this implies that a factor R −1 b (b(k/2 − j 1 )) has to be included as well in order to plug (28) into (33). Then, such reflection coefficient is eventually simplified due to another contribution standing when multiplying A k (J 1 , J 2 , J 3 , J 4 |x, z) and B k (J 1 , J 2 , J 3 , J 4 |x, z); namely, Moreover, these are not all the cancellations that take place. Let us also observe that three factors of the form where B(J 1 ) is the WZNW reflection coefficient (36), while the symbol ∼ stands for some kdependent overall factor. For short, in this expression we denoted x 2 = z 2 = 0 and x 3 = z 3 = 1, and exponents ∆ 1,2,3 and ∆ 2,3 refer to J-dependent (m-independent) linear combinations that are directly given by the exponents arising in (28) and (33). Expression (37) represents the main result here, and it turns out to be actually analogous to the formula connecting winding violating three-point functions to those involving merely three non-winding states. Namely, we showed that the SL(2, R) k WZNW four-point function while involving one state of the spectral flowed sector ω 1 = −1, can be expressed in terms of the integral of the four-point function , defined in terms of merely non-spectral flowed states ω µ = 0, µ = {1, 2, 3, 4}. The integration over the complex variable x clearly stands for the requirement of the Fourier transform when changing to the m-basis. Furthermore, it is worth pointing out that the factor is also present here, completing the analogy with the three-point function case. The fact that a similar pattern is found at the level of the four-point correlators opens a window that would permit to gain information about the violating winding four-point functions by making use of all what is known about the WZNW conservative amplitudes.

Further applications
So far, we presented a concise application of our results of [1], by showing that the symmetries of the KZ equation, that were inferred by means of its relation to the BPZ equation in Liouville theory, lead to a relation between violating and conserving winding amplitudes in AdS 3 string theory. Besides, there exist some further application of the formula (28) we obtained here: One of these feasible applications concerns the integral representation of the four-point conformal blocks and the factorization ansatz. Actually, let us return to (7) and expand the chiral conformal blocks as it follows where, now, j acts as an internal index that labels different solutions to the differential equation.
In the stringy interpretation it is feasible to assign physical meaning to such index: this is the one parameterizing the intermediate states interchanged in a given four-point scattering process. Then, the integration over the internal index j stands in order to include all the contributions of the conformal blocks (7) to the four-point amplitude (5). By replacing (38) into the KZ equation one finds that the leading term contribution in the z-power expansion obeys the hypergeometric differential equation. For instance, this permits to analyze the monodromy properties at x = z for leading orders in the large 1/x limit. Studying this regime turns out to be important to fully understand the analytic structure of the four-point function [5]. For such purpose, it is useful to study the solution in the vicinity of the point x = z. By assuming the extension of the expressions above to the SL(2, R) case, and by deforming the contour of integration as C → k 2 − C = k−1 2 − iR, one can see that, once the integral over j is performed, the solution is actually monodromy invariant at z = x. Then, the leading contribution of the solution can be written as it follows [5] where F (a, b, c; d) is the hypergeometric function, and where {x i } refers to the set of poles located in the region 1 < Re(2j) < k − 1. These poles take the form j − j 1 − j 2 ∈ N if the constraint 4 i=1 j i < k is assumed (see [5] for the details of the construction and the issue of the integration over the complex variables z and x). Taking all this into account, one could try to plug the expression for the conformal blocks (7) in the equation we wrote in (28). This would lead to an integral realization of the winding violating four-point function. Moreover, such a realization could be then evaluated in a particular case J 4 = k/2 and, by means of the prescription of [25], be used to compute the three-point function Φ ω 1 =−1 ,m 3 ,m 3 , with two winding states of the sectors ω = +1 and ω = −1. This idea does deserve to be explored in future work 3 .
Other possible analysis that can be done regards the Coulomb gas-like integral representation recently presented in [37,38]. There, it was proven that the SRT map can be thought of as a free field representation of the SL(2, R) k model in terms of the action where the specific model representing the "matter sector" S M corresponds to a c < 1 conformal field theory defined by the action This free field representation leads to an integral expression for the four-point function when the winding number conservation is being maximally violated. On the other hand, for the cases where the winding is conserved, the action S M has to be perturbed by introducing a new term of the form This is a perturbation, represented by a primary operator of the matter sector and properly dressed with the coupling to the Liouville field in order to turn it into a marginal deformation. It could be interesting to check the formulas (28)- (33) in terms of this representation.

Concluding remarks
As an application of our [1], in this brief note we proved a new relation existing between the four-point function of winding strings in AdS 3 (spectral flowed states of the SL(2, R) k WZNW model) and the four-point function of non-winding strings (non-flowed SL(2, R) k states). We showed that the former admits a simple expression in terms of the last, which was studied in more detail in the literature. The fact that such a simple expression exits is mainly due to two facts: first, the non-diagonal Z 2 symmetry of the KZ equation realized by (22); secondly, to the fact that correlators of the WZNW model are connected to those of the Liouville theory in more than one way.
To be more precise, here we have shown the relation that connects the four-point WZNW correlation function involving one ω = −1 spectral flowed state to the four-point function of non-spectral flowed states. This consequently relates the scattering amplitude involving one winding string state in AdS 3 with the analogous observable for non-winding strings. Such relation is realized by Eq. (37), and is reminiscent of the relation obeyed by the SL(2, R) k structure constants. On the other hand, it is likely that the four-point function involving one state in the sector ω = −1 corresponds to a limiting procedure of a WZNW five-point function involving one spectral flow operator of momentum J 5 = k/2. Consequently, it is plausible that the connection manifested by equations (28) and (33) could then be obtained in an alternative way; for instance, by directly studying the operator product expansion of operators Φ J 1 (x 1 |z 1 )Φ k/2 (x 2 |z 2 ) in the coincidence limit in the five-point conformal blocks. In such case, our result would be seen from a different perspective since, as far as our derivation of (28) and (33) is invertible, this would lead to the possibility of deriving the FZ map from a particular case of the SRT map. This was, indeed, one of the main motivations we had for studying this, though the operator product expansion involving an additional spectral flow operator (a fifth operator) turns out to be complicated enough and thus we leave it for the future. Conversely, as far as the relation between both maps is not trivial, as it is emphasized in [29], our way of proving the connection between violating winding correlators and conserving winding correlators turns out to be an ingenious trick. The conciseness of such a deduction becomes particularly evident once one gets familiarized with the complicated definition of the action of spectral flow operator in the x-basis. Moreover, even in the m-basis the computation turns out to be simplified by our method because no explicit reference to the decoupling equation of the degenerate state J = k/2 was needed at all, since it is was already encoded in the Ribault formula (20). Our hope is that the result of this brief paper might be useful in working out the details of the SL(2, R) k WZNW four-point functions, which is our main challenge within this line of research.
Acknowledgement: First, I would like to thank Cecilia Garraffo for collaboration in the computations of the section 3. I am also grateful to Université Libre de Bruxelles, and specially thank Glenn Barnich, Frank Ferrari, Marc Henneaux and Mauricio Leston for their hospitality during my stay, where the first part of this work was done. I also thank Pablo Minces for his interest on this work and for very interesting suggestions about possible applications. Last, I thank the Physics Department of New York University for the hospitality during my stay, where the revised version of the papers was finished. This work was partially supported by Universidad de Buenos Aires and CONICET, Argentina.

Appendix: The function Υ b (x)
The function Υ b (x) was introduced by Zamolodchikov and Zamoldchikov in Ref. [42], and is defined as it follows where Q = b + b −1 , being b ∈ R >0 . It also admits a definition in terms of a limiting procedure involving the double Barnes Γ 2 -function, though we did not find such a relation necessary here. This function has its zeros at the points x = mb + nb −1 x = −(m + 1)b − (n + 1)b −1 for any pair of positive integers m, n ∈ Z >0 . From (41), it turns out to be evident that this function is symmetric under the inversion of the parameter b → 1/b, namely This is the first of a list of nice functional properties of this function. The second identity we find useful is the reflection property which is keeps track of the reflection symmetry of Liouville structure constants when these are written in terms of Υ b (x). Besides, (41) also presents the following properties under fixed translations The above identities are then gathered in the equation which, in particular, includes (43). The relations above also permits to prove that the following relation holds All these relations were employed through our computation.