Note on Z 2 symmetries of the Knizhnik-Zamolodchikov equation

We continue the study of hidden Z2 symmetries of the four-point sl(2)k Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector ω=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 AdS3 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 nonviolating 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...


I. INTRODUCTION
The SL͑2,R͒ k Wess-Zumino-Novikov-Witten ͑WZNW͒ 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 noncompact curved backgrounds, being the most celebrated examples: the string theory on AdS 3 and on the two-dimensional ͑2D͒ black hole ͑the last, by means of its gauged SL͑2,R͒ k /U͑1͒ construction͒.Consequently, among the main motivations, this topic received particular attention due to its relevance for black hole physics and its relation with the anti-de Sitter space ͑AdS͒/conformal field theory ͑CFT͒ correspondence.3][4] Here, we continue the study of the observables in this theory, focusing out attention on the four-point function.

A. String amplitudes in AdS 3
Although the structure of the WZNW model on SL͑2,R͒ ͑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 SL͑2,C͒ k /SU͑2͒ 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 SL͑2,C͒ /SU͑2͒ and then extending the range of the indices of continuous representations in order to include the discrete representations of SL͑2,R͒ 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 Ref. 5͒ and yield selection rules 6,7 for the string scattering amplitudes.Besides, the continuous representations ͑the "long strings" in Ref. 5͒ 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 Refs.9 and 5 and references therein͒.Originally, some particular cases of the two-and three-point functions were explicitly computed in Refs 10 and 11, and Teschner was who presented the general result in Ref. 12 and carefully described its formal aspects in Refs.12-14, by employing the bootstrap approach and the minisuperspace approximation.Subsequently, other approaches, such as 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 Refs.6 and 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 Ref. 21͒.][24] After these objectives were achieved, the study of correlation functions in the SL͑2,R͒ k WZNW model acquired a new dimension since Maldacena and Ooguri pointed out the existence of new representations of SL͑2,R͒ k contributing to the spectrum of the string theory in AdS 3 . 5hen, the correlation functions involving these new states had to be analyzed as well.These new representations, obtained from the standard ones by acting with the spectral flow transformation, are semiclassically related to the possibility of the AdS 3 strings to have a nonzero winding number.This interpretation in terms of "winding numbers" does not regard a topological winding, but it turns out to be a consequence of the presence of a nontrivial NS-NS B background field.Then, from the beginning, this winding number, as a nontopological degree of freedom, was assumed to be possibly violated when the interactions were to take place.This violation was actually first suggested in a unpublished work by Fateev et al. for the case of the 2D black hole. 25 free field computation of three-point function of such winding states, including the violating winding case, appeared in Ref. 9, and a similar free field realization was studied in more detail in Refs.16 and 26.An impressive analysis of the correlation functions in the SL͑2,R͒ k WZNW model was then presented in the paper by Maldacena and Ooguri. 5There, the pole structure of two-, three-and four-point functions was discussed in the framework of the semiclassical interpretation and the AdS/CFT correspondence.The exact expressions of two-and three-point functions, including the violating winding three-point function, were fully analyzed.Besides, the string theory interpretation of such observables, as representing scattering amplitudes, was given with precision.The four-point function was also studied in Ref. 5, and, even though there is still no closed expression for the generic case available, our understanding of its analytic structure was substantially increased due to the results of that work.By making use of the factorization ansatz given by Teschner in Ref. 14, Maldacena and Ooguri proposed an analytic extension of the expression for the SL͑2,C͒ k /SU͑2͒ conformal blocks.Thus, they gave a precise prescription to integrate the monodromy invariant expression over the space of SL͑2,R͒ k representations and to pick up the pole contribution of the discrete states.Perhaps, the two main observations made in Ref. 5 regarding the four-point functions are the existence of additional poles in the middle of the moduli space, and the fact that the factorization of the four-point function only permits the usual interpretation in terms of physical intermediate states for particular incoming and outgoing kine-matic configurations.The analytic structure of particular four-point functions, those leading to logarithmic singularities, was studied in Ref. 27, showing that this example fits the standard structure of four-point functions in the AdS/CFT correspondence.
Despite all this information we get about four-point functions, it is worth noticing that the mentioned cases only took into account nonflowed representations ͑representing nonwinding string states͒ as those describing the incoming and outgoing states.For instance, even though the winding strings of the sector = −1 were shown to arise in the intermediate channels of four-point functions, 5 this was shown by analyzing the processes that only involve incoming and outgoing states of the sector = 0.Then, it would be interesting to extend the study to the case of four-point functions that involve external states of the sectors 0.Here is where our result enters in the game since it actually permits to get information of the four-point winding violating functions from all what is already known about the conservative case.In fact, in this note we will show how to connect the correlation functions involving one flowed state ͑winding string state͒ of the sector =−1 ͑with winding number =−1͒ to the analogous quantity that merely involves nonflowed states ͑just strings with winding number =0͒.We will also argue that this is actually analogous to the relation existing between the violating 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 Ref. 14 to prove the crossing symmetry in SL͑2,C͒ /SU͑2͒, was discovered by Stoyanovsky some years ago, 28 and it was further developed by Ribault and Teschner 29 and Ribault. 30,31In Refs.32-35 this map between both CFTs ͑henceforth denominated Stoyanovsky-Ribault-Teschner map, and denoted SRT map͒ 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 correlators͒.In Refs.37 and 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 =1͒ is considered.Such observable had been also computed in Ref. 1 through "the other" connection to Liouville theory, discovered by Fateev and Zamolodchikov in Ref. 19 and extended in Refs.14, 20, and 21 to the noncompact case ͑henceforth denominated Fateev-Zamolodchikov map, and denoted FZ map͒.We will be more precise in the following paragraph.

B. 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 = −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 ͑different͒ four-point function of the WZNW theory.Indeed, we can also do this by employing the noncompact generalization of the FZ map. 19If this "second map" is used, the four-point function reached in the WZNW side is one that contains four nonwinding 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 nondiagonal 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 an addendum to Ref.

A. The WZNW model
We are interested in the WZNW model formulated on the SL͑2,R͒ group manifold.Its action corresponds to the non-linear model of strings in Lorentzian AdS 3 space.On the other hand, its Euclidean version is similarly given by the gauged SL͑2,C͒ /SU͑2͒ model.The states of the Euclidean model are characterized by normalizable operators on the Poincaré hyper-half-plane H 3 + =SL͑2,C͒ /SU͑2͒.These operators can be conveniently written as where the variables ␥ , ␥ ¯, and u are associated with the AdS 3 Euclidean metric in Poincaré coordinates, namely, ds 2 = k͑u −2 du 2 + u 2 d␥d␥ ¯͒, while the complex coordinates x and x ¯represent auxiliary variables that expand the SL͑2,C͒ representations as it follows for a = ͕3, + ,−͖.The dots "…" stand for "regular termsЉ in the OPE, while the differential operators D x a correspond to the realization Besides, the operators J a ͑z͒ are the local Kac-Moody currents, whose Fourier modes are defined by J a ͑z͒ = ͚ nZ J n a z −1−n and satisfy the sl͑2͒ ˆk affine Kac-Moody algebra of level k.The value of k is related to the string length and the AdS radius through k = l AdS 2 / l s 2 .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 ͑1͒ reads where the indices take the values j = 1 2 + iR Ͼ0 for SL͑2,C͒ /SU͑2͒.In order to propose a similar algebraic realization for the Lorentzian string theory, it is usually assumed that the observables of the Euclidean theory admit an analytic continuation in the variable j, now parametrizing both continuous and discrete representations of SL͑2,R͒.It is worth noticing that the transformation j → 1− j is a symmetry of formula ͑3͒; this corresponds to the so-called Weyl reflection symmetry and will be important for us in Sec.III.Operators ͑1͒ represent the vertex operators of the string theory in AdS 3 and define the correlation functions.Then, after integrating over the variables z , the functional form of the N-point correlators will still depend on the auxiliary variables x , with = ͕1,2, ... ,N͖.Within the context of the AdS 3 / CFT 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 parametrizing representations of the SL͑2,R͒ group, by using a pair of indices j , m.In this frame, the string states in AdS 3 are given by vectors ͉j , m͘ ͉j , m ¯͘ of the SL͑2,R͒ ϫ SL͑2,R͒ 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 , is associated with the winding number of strings in AdS 3 , at least in what respects to the states with a suitable asymptotic description ͑the long strings͒.From an algebraic point of view, 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 0͒ in the m basis are to be denoted ⌽ j,m,m ͑z͒ and define the correlation functions in this basis, which we will denote as ͗⌽ j 1 ,m 1 ,m ¯1͑z 1 ͒⌽ j 2 ,m 2 ,m ¯2͑z 2 ͒ , ... ,⌽ j N ,m N ,m ¯N͑z N ͒͘.The conformal dimension of the states in the m basis depends both on m and through the expression In the case = 0, operators ⌽ j,m,m =0 ͑z͒ are related to those of the x basis through the Fourier transform On the other hand, the definition of string states with 0 in the x basis was studied in Ref. 5 and 40; however, these do not have a simple expression.Now, let us discuss the correlation functions in more detail.

B. 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 = 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, and where The function f = f j 1 ,j 2 ,j 3 ,j 4 ͑x , z͒ is then given by certain linear combination of solutions to the Knizhnik-Zamolodchikov ͑KZ͒ partial differential equation; i.e., that combination which turns out to be monodromy invariant.The KZ equation in the case of the SL͑2,R͒ k WZNW model takes the form where the differential operators are With Ref. 14, we can consider the following ansatz for the solution: where the functions C͑j 1 , j 2 , j 3 ͒ and B͑j 1 ͒ are given by the structure constants and the reflection coefficient of the SL͑2,R͒ 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 Eq. ͑6͒.

C. 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 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 ͑z͒.
whose conformal dimension is given by Vertex operators ͓Eq.͑9͔͒ define the Liouville N-point correlation functions, which are to be denoted by ͗V a 1 ͑z 1 ͒V a 2 ͑z 2 ͒ , ... ,V a N ͑z N ͒͘.Notice that formula ͑10͒ remains invariant under ␣ → Q − ␣, henceforth called Liouville reflection.This symmetry induces the identification between both fields V ␣ ͑z͒ and V Q−␣ ͑z͒, yielding the operator valued relation which is valid for any vertex i = ͕1,2, ... ,N͖ ͑though we exemplified it here for i =1͒, and where R b ͑␣ 1 ͒ represents the Liouville reflection coefficient, Another important feature of the Liouville correlation functions is the fact that those involving states with momentum ␣ =−1/2b satisfy a well known partial differential equation, called the Belavin-Polyakov-Zamolodchikov ͑BPZ͒ equation.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 ␣ m,n = ͓͑1−m͒ /2͔b + ͓͑1−n͒ /2͔b −1 for any pair m , n Z Ͼ0 .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 Secs.II D and II E.

D. 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. 21͒ and, among other ingredients involved in its derivation, is based on the relation existing between solution of the KZ and the BPZ differential equations.Such relation was originally noticed by Fateev and Zamolodchikov in Ref. 19 for the case of the compact SU͑2͒ 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 noncompact WZNW model, was employed to investigate several properties of the SL͑2,R͒ 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 the sl͑2͒ ˆk affine algebra by similar means, 20 and Ponsot discussed the monodromy of the theory with such techniques. 21In 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 Ref. 19 is that the KZ equation satisfied by the four-point 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 ⌼ b ͑x͒ 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 ⌼ b function͒.The coefficient c b is a b-dependent factor, independent of the quantum numbers j and ␣ , 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 , that label the SL͑2,R͒ k representations, and the Liouville momenta ␣ involves a nondiagonal invertible transformation defined through On the other hand, the relation between the WZNW level k and the Liouville parameter b is given by b Now, let us make a remark on the KPZ scaling in Eq. ͑13͒: The parameter 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 SL͑2,R͒ k /U͑1͒ 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 b 2b 2 ⌫͑1−b 2 ͒ / ⌫͑1+b 2 ͒ in Eq. ͑14͒ by simply shifting .This freedom can be used to simplify the functional form of the correlators. 23

E. The Stoyanovsky-Ribault-Teschner identity
Analogously as to the case of the FZ map, the SRT map, discovered by Stoyanovsky in Ref. 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 and ␣ ; the second advantage, and more important indeed, is that it holds for the case of the N-point KZ equation for an arbitrary N. One of the original versions of the SRT map states the correspondence between N-point functions in the WZNW theory and the 2N − 2-point function of the Liouville theory.Furthermore, this was generalized by Ribault where the normalization factor is given by while the z-dependent function is

͑17͒
Here, the Liouville cosmological constant has been set to the value = b 2 −2 for convenience; this is to make the KPZ scaling of both sides of Eq. ͑16͒ match.It is also important to keep in mind the presence of the free parameter of the WZNW theory, and the fact that this can still be set to an appropriate value in order to absorb the powers of ⌫͑b 2 ͒ / ⌫͑1+b 2 ͒ in the KPZ overall factor of WZNW correlators. 23For 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 2 3−2N b / M!c k M+2 where c k represents a k-dependent factor whose exact value is discussed in Ref. 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 b 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 = −1 obeys where c ˆb is, again, a J-independent factor whose specific value, c ˆb =2 −5 bc k N−2−M , is known in terms of c k , though not important for us.Notice that we have changed the notation here, where we replaced j by J ; this will be convenient later.Regarding the relation between J and ␣ , let us make some remarks here: Notice that, according to Eq. ͑18͒, the Liouville reflection ␣ → Q − ␣ 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 Eq. ͑11͒, this implies that the correlation function involving the field ⌽ j ͑x ͉ z͒ and the one involving the field ⌽ 1−j ͑x ͉ z͒ are connected one to each other by the overall factor R b ͑b͑k /2− j͒͒.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.

A. The idea
The idea is to use Eqs.͑13͒ and ͑20͒ in order to relate the four-point function The nexus will be the Liouville five-point function ͗͟ =1 4 V ␣ ͑z ͒V −1/2b ͑x͒͘, appearing in the right hand side of both expressions: this is connected to the first through the SRT map while it is connected to the second through the FZ map.After doing this, the second step would be to "diagonalize" the relation between indices j and indices J .We will do this by using the preliminary result of Sec.III B. Notice that, according to Eqs. ͑13͒ and ͑20͒, the relation between momenta J and j is given by Eq. ͑29͒ below.The third step would be to perform a Fourier transform in order to translate the operators ⌽ j ͑x ͉ z ͒ of the x basis into operators ⌽ j ,m ,m ¯͑z ͒ of the m basis.

B. Nondiagonal 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 Eq. ͑15͒, the fact of performing the change j ˜ → j for all the indices = ͕1,2,3,4͖ translates in terms of the Liouville momenta ␣ in doing the change ␣ i → Q − ␣ i for i = ͕2,3,4͖, while keeping ␣ 1 unchanged.Then, by taking into account the Liouville reflection symmetry ͓Eq.͑11͔͒ and the dictionary ͓Eq.͑15͔͒, we get ͑I thank Garraffo for discussions on this formula.͒ The factor F k ͑j 1 , j 2 , j 3 , j 4 ͒ / F k ͑j ˜1 , j ˜2 , j ˜3 , j ˜4͒ involves a quotient of products of ⌼ b functions that, by making use of the formulas in the Appendix ͓see Eq. ͑35͒ below͔, can be substantially simplified once one takes into account the presence of the three factors R b ͑␣ i ͒ ͑i =2,3,4͒.Then, the results reads where P͑j͒ is a polynomial in the indices j , namely, P͑j͒ =−3j 1 + j 2 + j 3 + j 4 , though it 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 → j ˜ is given by where Formula ͑22͒ will be useful in the next section.Similar functional relations were studied in Refs. 1, 20, and 47.Now, let us move to the case of winding strings.

C. The four-point function of winding strings
In this section, as an application of our formula ͓Eq.͑22͔͒, 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 ␣ ˆ and j are then related as it follows Now, let us also define quantum numbers J as b͑k/2 − J ͒ = ␣ ˆ, ͑24͒ for = ͕1,2,3,4͖.Then, by taking into account Eq. ͑11͒, we have Consequently, we can write the following SRT identity: Hence, plugging Eq. ͑25͒ into Eq.͑27͒, we find where we simply defined and where, according to the definitions above, the indices J and j turn out to be related by for i = ͕2,3,4͖.Then, it is feasible to show that where the symbol "ϳ" 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. ͑26͒ if necessary͔.In short, in the last equation we have employed the notation and one could also find it convenient to define Identity ͑28͒ 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 Eq. ͑28͒ contains a four-point correlator that, instead of involving states with momenta J , involves states with momenta j , = ͕1,2,3,4͖.Here is where our result of Sec.III B becomes useful, since the following step would be to relate the correlator in the j basis with the one that is diagonal in the J basis.This can be simply achieved by making use of Equation ͑22͒ and by taking into account Eq. ͑29͒, yielding with the normalization factor being Here, P ˆ͑j͒ is again a polynomial in the indices j that is related to the P͑j͒ in Eq. ͑22͒ through the replacement j i → J i , i = ͕2,3,4͖, j 1 → k /2−J 1 .In writing down Eq. ͑34͒ we used the fact that, as in Sec.III B, the combination of the ⌼ b functions and the ⌫ functions in R b ͑␣ i ͒ leads to a very simple expression.In this case, it comes from .

͑35͒
Hence, up to the Weyl reflection j 1 → 1− j 1 that we discuss below, Eqs.͑33͒ and ͑28͒ represent the result we wanted to prove: the four-point function involving one winding state of the sector = −1 admits to be expressed in terms of the four-point function of nonwinding states.Let us discuss the final form of this result in the following paragraphs ͓see Eq. ͑37͒ below͔, 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.

D. Analogy with the case of the three-point function
The three-point function that includes one spectral flowed state of the sector = −1 can be written in terms of the structure constant of three nonflowed states. 5,25In Sec.III C 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 will see below, these are indeed analogous.Actually, expressions ͑28͒ and ͑33͒, once considered together, correspond to the generalization of the formula that holds for the three-point structure constants.First, one of the aspects one notices in the formulas above is that the first operator in the right hand side of Eq. ͑33͒ represents the state of momentum ͑k /2͒ − J 1 , instead that of momentum J 1 .Actually, this should not be a surprise since it precisely resembles what happens at the level of the threepoint function, where the violating winding correlator ͑ϱ͒͘ turns out to be proportional to the integral of the conservative structure constant that corresponds to The same association between momenta J 1 and k /2−J 1 occurs for the four-point functions here.Actually, such relation between states with quantum numbers =0, J and those with = ±1, k /2−J is understood from the algebraic point of view: This concerns the identification between the discrete series of the SL ͑2,R͒ k representations by means of the spectral flow automorphism.This is related to the fact that both states with = 0 and ͉͉ = 1 satisfy similar KZ equations.Besides, this is related to the fact that the OPE between the vertex operator ⌽ J ͑z͒ and the spectral flow operator ⌽ k/2 ͑w͒, which is necessary to provide the winding 1 = 1 to the first, yields the single string contribution ⌽ k/2−J ͑z͒.The involution J → k /2−J, as a symmetry of the KZ equation, was studied in the first part of our work. 1As it was detailed in Ref. 5, when presenting the original derivation of Ref. 25, the proportionality factor connecting both violating winding and nonviolating winding three-point correlators is basically given by the reflection coefficient B͑J 1 ͒ ϳ B −1 ͑k /2−J 1 ͒ of the SL ͑2,R͒ k WZNW model ͑see the formula above͒.This quantity has the form and therefore 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 Eqs.͑28͒ and ͑33͒, a factor ⌼ b ͑2b͑k /2−J 1 ͒͒ / ⌼ b ͑2b͑k /2−J 1 ͒ − b͒ also arises in the product between A k ͑J 1 , J 2 , J 3 , J 4 ͉ x , z͒ and B k ͑J 1 , J 2 , J 3 , J 4 ͉ x , z͒.This factor is precisely proportional to ͑2J −1͒B͑J͒; thus, this is actually in equal footing as to how the WZNW reflection factor stands in the relation between violating and nonviolating three-point correlators.
On the other hand, some nice cancellations occur when combining expressions ͑28͒ and ͑33͒.First, we observed that, instead of the operator ⌽ j 1 ͑x ͉ z͒, the correlator on the right hand side of Eq. ͑28͒ involves its Weyl reflected operator ⌽ 1−j 1 ͑x ͉ z͒, and, as we commented at the end of Sec.II, this implies that a factor R b −1 ͑b͑k /2− j 1 ͒͒ has to be included as well in order to plug Eq. ͑28͒ into Eq.͑33͒.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 , z͒ and B k ͑J 1 , J 2 , J 3 , J 4 ͉ x , z͒, namely, Moreover, these are not all the cancellations that take place.Let us also observe that three factors of the form and their respective inverses are mutually canceled as well.These come from the second line of Eq. ͑30͒ and the second line of Eq. ͑34͒, respectively.Finally, Eqs.͑28͒ and ͑33͒, considered together, lead to the following expression: where B͑J 1 ͒ is the WZNW reflection coefficient ͓Eq.͑36͔͒, while the symbol ϳ stands for some k-dependent 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-independent͒ linear combinations that are directly given by the exponents arising in Eqs.͑28͒ and ͑33͒.Expression ͑37͒ 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 nonwinding states.Namely, we showed that the SL ͑2,R͒ k WZNW four-point function, ͑ϱ͒͘, while involving one state of the spectral flowed sector 1 = −1, can be expressed in terms of the integral of the four-point function defined in terms of merely nonspectral flowed states =0, = ͕1,2,3,4͖.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 that is known about the WZNW conservative amplitudes.

E. Further applications
So far, we presented a concise application of our results of Ref. 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 formula ͑28͒ 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 Eq. ͑7͒ 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 parametrizing 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 ͓Eq.͑7͔͒ to the four-point amplitude ͓Eq.͑5͔͒.By replacing ͓Eq.͑38͔͒ 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. 5For 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,R͒ case and by deforming the contour of integration as C → ͑k /2͒ − C = ͑k −1͒ /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 ͉f j 1 ,j 2 ,j 3 ,j 4 ͑x,z͉͒ 2 = 1 where F͑a , b , c ; d͒ is the hypergeometric function, and where ͕x i ͖ refers to the set of poles located in the region 1 Ͻ Re͑2j͒ Ͻ k − 1.These poles take the form j − j 1 − j 2 N if the constraint ͚ i=1 4 j i Ͻ k is assumed ͑see Ref. 5 for the details of the construction and the issue of the integration over the complex variables z and x͒.Taking all this into account, one could try to plug the expression for the conformal blocks ͓Eq.͑7͔͒ in the equation we wrote in Eq. ͑28͒.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 Ref. 25, be used to compute the three-point function ͗⌽ J 1 ,m 1 ,m ⌽ J 3 ,m 3 ,m ¯3 3 =+1 ͘, with two winding states of the sectors = + 1 and = −1.This idea does deserve to be explored in future work ͑I thank Minces for pointing out this possible application.͒.
Other possible analysis that can be done regard the Coulomb gaslike integral representation recently presented in Refs.37 and 38.There, it was proven that the SRT map can be thought of as a free field representation of the SL ͑2,R͒ k model in terms of the action S = 1 4 ͵ d 2 z͑− ‫ץץ‬ ¯ + QR + e ͱ 2b ͒ + S M ,

͑39͒
where the specific model representing the "matter sector" S M corresponds to a c Ͻ 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

͑40͒
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 formulas ͑28͒-͑33͒ in terms of this representation.

IV. CONCLUDING REMARKS
As an application of our Ref. 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,R͒ k WZNW model͔ and the four-point function of nonwinding strings ͓nonflowed SL ͑2,R͒ k states͔.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 nondiagonal Z 2 symmetry of the KZ equation realized by Eq. ͑22͒; second, 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 = −1 spectral flowed state to the four-point function of nonspectral flowed states.This consequently relates the scattering amplitude involving one winding string state in AdS 3 with the analogous observable for nonwinding strings.Such relation is realized by Eq. ͑37͒, and is reminiscent of the relation obeyed by the SL ͑2,R͒ k structure constants.On the other hand, it is likely that the four-point function involving one state in the sector = −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 Eqs.͑28͒ and ͑33͒ could then be obtained in an alternative way; for instance, by directly studying the operator product expansion of operators ⌽ J 1 ͑x 1 ͉ z 1 ͒⌽ 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 Eqs.͑28͒ and ͑33͒ 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 operator͒ 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 Ref. 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 was already encoded in the Ribault formula ͓Eq.͑20͔͒.Our hope is that the result of this brief paper might be useful in working out the details of the SL ͑2,R͒ k WZNW four-point functions, which is our main challenge within this line of research.
in order to connect any N-point functions in WZNW to M-point functions in Liouville CFT with N ഛ M ഛ 2N − 2. Let us describe such generalized form below.