On some 3-point functions in the W 4 CFT and related braiding matrix

,


Introduction
The 2d conformal field theories (CFT) related to the sl(2) algebra, like the Virasoro, the WZW models with the affine sl(2) KM algebra and their supersymmetric extensions, are by now well established. This includes explicit expressions for basic data as the operator product expansion (OPE) coefficients (3-point functions) and the braiding/fusing matrices transforming conformal blocks. Much less is known about these structures in the CFT with higher rank symmetries, although a considerable progress in Toda CFT [2] was made by Fateev and Litvinov (FL) [1,3]. Further advances in the field are important for the development of the higher rank 2d CFT as well as for potential applications in the string theory side of the AdS/CFT correspondence.
In the free field (Coulomb gas) approach the OPE constants are represented by complicated integrals which have to be computed explicitly before analytic continuation. The alternative derivation of functional relations arising from locality (crossing symmetry) of particular 4-point functions involving degenerate vertex operators requires the knowledge of fundamental braiding/fusing matrix elements, which in general are also part of the problem.
In [1,3] Fateev and Litvinov developed a general method of recursively computing certain class of conformal integrals and gave explicit examples of 3-point constants. 1 In the case of Toda W 3 theory they have as well computed the fundamental fusing matrix directly from the integral representations of the 4-point blocks; some partial results in the general W n case were also obtained.
In section 3 we use this data to derive the fusing matrix F transforming the corresponding 4-point conformal blocks with one fundamental vertex V −ω 2 b . Here we follow a path somewhat opposite to the standard consideration in which -given the fusing matrix, one solves for the 3-point constants the system of equations implied by locality of the 4-point function. We shall not need the explicit integral realisation of this particular Toda 4-point function with three more partially degenerate representations of the type β a = −k a ω 2 b , a = 1, 2, 3 . In the intermediate channels appear also vertex operators with symmetric weights so that in the equations the more general constants of the type (1.1) derived in section 2 are needed. The restriction to chiral vertex operators V βa of such particular highest weights effectively restricts the braiding/fusing matrix to a 3 × 3 submatrix; its matrix elements are explicitly described.
Finally in this section we check a braiding identity, which is equivalent to a standard identity for the modular group on the sphere with 4 holes. This relation imposes restrictions solely on certain products of F matrix elements and allows in principle for more general solutions for the individual F matrix elements than the ones computed in the W 4 CFT. The semi-classical "heavy charges" limit of the identity is a particular sl(4) analog of the one exploited in the strong coupling sl(2) sigma model constructions in [9,10]. This suggests that the explicit expressions for the products of the fusing matrix elements extracted from Toda CFT (or their closely related WZW model counterparts) may eventually be used as a first step in higher rank generalisations of that work.
In the last section 4 we compare the 3-point constants in two regions of the central charge, W 4 analogs of the two Virasoro theory ingredients of Liouville gravity (non-critical string theory) with c > 25 (Liouville) and c < 1 ("matter"). We discuss a BPS-like relation for the two sets of weights which is intrinsic for the vertex operators of theŝl(4) WZW models, related to Toda theory by the quantum Hamiltonian Drinfeld Sokolov (DS) reduction. Although it seems that there are no direct W 4 analogs of the physical fields of Liouville gravity, we show that the product of the two 3-point Toda constants with weights subject to the BPS constraint trivialises in the semi-classical "light" charges" limit. Furthermore we speculate on the possible implications for the related WZW 3-point 2 These highest weights correspond to scalars in the context of 4d conformal group representations, where in general the (nonnegative, integer) components 2ji = −(β, αi)/b, i = 1, 3 label the SL(2, C) spins, while = (β, ω2)/b corresponds to the 4d conformal dimension.

JHEP12(2015)079
correlators (the determination of which is still an open problem) and make a comparison with computations of related 3-point correlators in the supergravity approximations of the AdS 5 × S 5 strings [11,12]. The appendix contains some details of the computation of the 3-point functions, including one slightly more general constant not presented in section 2, as well as an alternative Coulomb like representation of the 4-point functions discussed in section 3. It reveals a connection to certain Liouville correlators.

3-point W 4 constants
We consider the W 4 CFT with central charge for real values of the parameter b. We shall skip the detailed presentation of the basics of Toda conformal theory and the free field (Coulomb gas) representation of the correlation functions: the reader is referred to [1], as well as to the original paper of Fateev and Lukyanov [2], formulated in the dual region of central charge with b → ib in (2.1). The OPE constant of 2d scalar vertex operators is where the conformal dimension is given by the sl(4) inner product and ρ = 3 i=1 ω i is the Weyl vector. The dimension (2.2), as well as the two other W 4 quantum numbers, are invariant with respect to an action of the Weyl reflection group so that any of the vertex operators V w β represents the same field. The Coulomb gas representation of the OPE constant is defined for the charge conservation condition The integers s i in front of the simple roots α i in (2.4) count the number of screening charge vertex operators V α i b (z,z) , i = 1, 2, 3, from the interaction term of Toda action. These operators are spinless fields of dimension (α i b) = 1. Formula (2.4) describes a generic sl(4) type fusion rule in which β 3 is obtained by a shift of, say, β 1 with the weight diagram The OPE constant is given by a i s i − multiple 2d integral I s 1 ,s 2 ,s 3 (β 1 , β 2 ) (formula (1.33) of [1] recalled in (A.1) below). We compute this integral in the particular case when the three highest weights are chosen as in (1.1). The components (β a , α 2 ) of the weights in (1.1) take arbitrary values, subject of the condition (2.4); the latter implies that s 1 = s 3 and we shall assume that the integer l := −(β 3 , α 1 )/b = s 2 − 2s 1 is nonnegative.

JHEP12(2015)079
We shall skip the detailed computation of the OPE constant since it follows straightforwardly the steps of the method explained in [1], which is based on the use of a sl(2) type duality formula [8] in order to derive recursion relations for sl(n) Toda multiple 2d integrals; see the appendix for a short summary of the procedure. In our case after s steps one gets an integral of type so that setting s = s 3 = s 1 the integral is reduced to known Liouville Coulomb integral. In particular for β 2 = −ω 2 b the resulting formula reproduces the structure constants c h := c(β 1 , −ω 2 b, 2ρQ − (β 1 − hb)) of the fusion of V β 1 with the fundamental field V −ω 2 b corresponding to three of the six points of the weight diagram Γ ω 2 , i.e., (2.5) These are the three weights of Γ ω 2 preserving the symmetric type β 3 = β * 3 with l ≥ 0. The expressions for these OPE constants reproduce special cases of the general formula (1.51) of [1] valid for arbitrary β 1 . For the partially degenerate weights of type (β 1 , α i ) = 0 , i = 1, 3 the remaining three OPE constants in [1] vanish, in agreement with the vanishing of the corresponding sl(4) tensor product decomposition multiplicities: the sl(4) (or sl(4)) Verma modules of highest weights λ = −β/b with non-negative integer components l (i) = (λ, α i ), i = 1, 3 have two singular vectors, whose factorisation imposes additional restrictions on the fundamental fusion rule. In particular in the case (λ 1 , α 1 ) = 0 = (λ 1 , α 3 ) for each of the three weights h = ±(−ω 1 + ω 3 ) , ω 2 − (ω 1 + ω 3 ) ∈ Γ ω 2 there is an odd Weyl group element w 1 , or w 3 , or both, the shifted action of which keeps λ 3 = λ 1 + h invariant, a property which does not depend on the value of (λ 1 , α 2 ), and which implies the vanishing of the corresponding fusion multiplicities. 3 Our next step is the standard analytic continuation of the OPE constant, to be denoted C(β 1 , β 2 , 2ρQ−β 3 ) for weights of the type (1.1) not restricted by (2.4), so that the Coulomb gas OPE constant is reproduced as a double residue c(β 1 , β 2 , 2ρQ − β 3 ) = res (β 3 12 ,ω 2 −2ω 1 )=−(s 2 −2s 1 )b res (β 3 12 ,ω 1 )=−s 1 b C(β 1 , β 2 , 2ρQ − β 3 ) (2.6) where s 1 and s 2 − 2s 1 are nonnegative integers. 3 The sl(4) pattern of the W4 fusion rules multiplicities is independently proved in the particular case of integer dominant weights −β/b extrapolating the rational b 2 result in [13], derived by reduction of theŝl(n) WZW Verlinde formula. The fusion rule of f = −ω2b with representations of generic highest weight can be derived algebraically accounting for the factorisation of the null states in the corresponding completely degenerate W4 Verma module. The three independent singular vectors -two at level 1 and one at level 2, inherited via the quantum DS reduction from the singular vectors ofŝl(4) module, are not sufficient. Together with the two projective Ward identities corresponding to the zero modes W (j) 0 of the spin j = 3 and j = 4 currents, they provide five relations which eliminate the 3-point matrix elements containing the negative modes W (j) −k , 1 ≤ k ≤ j − 1: the latter determine the action on the fields of all higher negative modes W (j) −n , n ≥ j. To derive the fusion rule itself one needs to explore the factorisation of three more descendent null states, presumably at levels up to 5, as suggested by the classical KZ equation for this representation; see the general discussion in [1,14,15] applied to W3 examples.
Here λ T is proportional to the Toda cosmological constant, In the products over positive roots in (2.8) the root α 1 can be replaced with α 3 (and The ratio in the second line of (2.8) produces a finite constant for (β 3 , α 1 ) → −lb , l−nonnegative integer so that (2.8) has sense for such values of β 3 whenever the components (β a , α 2 ) of the three weights are generic. We have also used that the two weights β 1 , β 2 have zero components (β a , α i ) = 0 for i = 1, 3 in order to write (2.8) in a form which makes it explicitly symmetric when the third weight β 3 is also chosen of this type, i.e., l = 0 , as we shall need it below: in that particular case (β, ω 2 − ω 1 ) = (β, ω 1 ) in all products of this type in the denominators in the last two lines.
Using (2.3) the terms in (2.8) depending on the three vertices, i.e., the eight Υ b -factors in the denominators of the last two lines, can be also written as points on an orbit of the Weyl group acting on the three weights (as discussed, e.g., in [6,7] for the FL example)

JHEP12(2015)079
3 Locality, fusing matrix, braiding identity one of which is labelled by a fundamental highest weight, in our case f = −bω 2 . The most interesting for the applications are the cases in which the remaining three primary fields have highest weights β a with nonnegative integer components l 3 and generic (β a , α 2 ) (or, any of their Weyl group related values providing equivalent vertex representations). These representations arise from doubly reducible Verma modules with two singular vectors. The projective Ward identities and the factorisation of all singular vectors -as well as the descendent null states in the fundamental representation f give restrictions on the conformal blocks reducing the space of descendent states described in terms of powers of the modes W [1,16,17] for different approaches to this problem. Instead of the detailed analysis of this space one can give, as in [3], an alternative argument showing that at least a subclass of these 4-point functions admit an integral Coulomb gas like representation, see the appendix. This indicates that all descendent states of the above space are eliminated so that the fusion channels of these highly degenerated 4-point functions follow the sl(4) pattern dictated by the completely degenerate field V −ω 2 b ; in what follows we restrict to this subclass of 4-point functions. In the case when all l (i) a = 0 , i = 1, 3 , a = 1, 2, 3 -which is the main case under consideration below, the alternative Coulomb representation allows to identify the linear differential equation satisfied by the 4-point function.
The 4-point function admits different equivalent diagonal decompositions in conformal blocks. They are related by braiding transformations, i.e., matrix realisation of the braiding group with generators e i , i = 1, 2, 3 on the plane (Riemann sphere) with 4 holes; e i is exchanging the chiral vertex operators at the i-th and i+1-th points and the notation refers to the fixed ordered points, not to the labels of the concrete interchanged operators. In particular the generators e 2 (for the above order of the corresponding chiral vertex operators) is represented by non-trivial braiding matrix B proportional to the fusing matrix F while e 1 and e 3 , which exchange the operators in the first two, respectively last two, fixed points, reduce due to triviality of F , to diagonal matrices. Locality (symmetry under exchange of two 2d fields) requires that the function is invariant under such transformations relating different diagonal chiral decompositions. This results in equations involving fusing matrix elements and products of 3-point constants. In the case under consideration the equations take the form of a finite sum. E.g., for the exchange of V β 1 and V β 2 they read

JHEP12(2015)079
Here c hs (β 1 ) is a shorthand notation for the OPE constant c(−ω 2 b, β 1 , 2ρQ − (β 1 − h s b)), see the general formula (1.51) in [1]. In particular c h=ω 2 = 1. In general h s stands for the weights of the weight diagram Γ ω 2 of the 6 dimensional representation, but for our restricted set of highest weights β a = −k a ω 2 b , a = 1, 2, 3, three of the OPE coefficients c hs given in [1] vanish, as discussed above, so we are left with summation over 3 of the weights, as given in (2.5). A shorthand notation for the matrix F hs,ht = F β 1 −hsb,β 2 −htb in the last equality in (3.2) is used. As indicated in the r.h.s. of (3.2) the matrix formed by the ratio of constants times F can be identified with the inverse matrix F −1 It is furthermore required that a consequence of the pentagon relation for F (or of the normalization relation in (3.1) ). In a shorthand notation we shall denote , suppressing the dependence on the third argument β 3 . The ratios in (3.3) will be denoted and thus one needs to compute all U h,+ . We give the explicit expression of the first of these ratios, computed from (2.8) . (3.6) By analogy with the Liouville case this suggests the following ansatz for F : . (3.7) From (3.7) one computes F +,+ (β 2 , β 1 ) and confirms, using (3.3), that it indeed satisfies (3.4)
We proceed in this way to obtain F h,+ for the other two shifts of β 1 → β 1 − hb. Then with the help of simple trigonometric relations one checks and proves the first of the diagonal equations in (3.2), for h t = h u = ω 2 . Similarly one finds eight of the nine F matrix elements checking the related equations. The expression for Uh ,h , however, has a different structure, not suggesting straightforwardly an expression for Fh ,h On the other hand writing the general expression of an inverse of a 3 × 3 matrix, F −1 ij = ε ikl ε jmn 2 det F F mk F nl with i, j, k, l, m, n = + , − ,h we have, e.g., etc. From this we can determine det F : Then, e.g., from we can determine Fh ,h and check the remaining identities in (3.2). Summarising we get for the matrix elements of F starting with (3.7) The last matrix element can be written in various different ways. Let us introduce some additional notation and A, A , B, B , explicitly described above, are also written in terms of Weyl group action (3.14) -9 -

JHEP12(2015)079
Compare with the Liouville case where (β, α) = 2β L , (β, ω) = β L and (3.15) One can analogously compute the fusing matrix elements corresponding to the sl(4) fundamental weights ω 1 = ω * 3 using the 3-point constant computed by Fateev and Litvinov [1] in which two of the weights are arbitrary and the third is proportional to one of these fundamental weights; this will be a special case of the 4 × 4 F matrix. Partial data on the braiding matrices in that case is also provided (though in a different gauge) by the Boltzmann weights defining integrable A Take the trace of (3.16) The eigenvalues of the monodromy Ω are computed from the difference of Toda dimensions

(3.23)
Each of the products in (3.14) which appear in the l.h.s. of (3.22) in this case is a second order polynomial in 2 cos πb(β − 2ρQ, α 2 ), as is the expression in (3.23), so the reduced relation is checked order by order. In (3.22) F enters only through the products F hs,htFht,hs , hence it is a restriction on these products. In principle the identity (3.16) with diagonal braiding determined from (3.20) may admit more general solutions for the individual F matrix elements than the present Toda CFT solution (3.7), (3.11). Indeed the sl(2) analog of the identity (3.16) with trivial r.h.s. has been exploited in the recent papers [9,10] on AdS 3 × S 3 sigma model 3-point correlators in the semi-classical strong 't Hooft coupling λ limit with large quantum numbers. Identifying b 2 = 1/ √ λ this corresponds to the semiclassical limit b → 0 with three heavy charges β a /b = η a /b 2 , η a -finite. In the sigma model case the eigenvalues of the monodromy matrix e 2πi(η(x),h) depend on the spectral parameter x and the solution for the individual F = F (x) matrix elements depends nontrivially on the specific spectral curve. On the other hand the expression for the products F hs,htFht,hs as functions of η(x) coincides with those in the WZW model, or, up to normalization, with those of Virasoro theory (cf. (3.15) and footnote 4). One may expect that the Toda theory data and their WZW extensions for the fundamental representations f = −ω i b can similarly be used as a starting point, although in this case the equation (3.19) is less restrictive by itself, compared with the sl(2) case where it uniquely determines the fusing matrix products.

JHEP12(2015)079 4 The 3-point functions in the compact ("matter") region and BPS-like relation
By analogy with the Liouville gravity described by two dual Virasoro CFT with c > 25 (Liouville) and c < 1 ("matter") we shall extend here the results of section 2 to another region of central charge of the W 4 CFT, parametrised by the same real parameter b as (2.1), The minimal W 4 theory in the region (4.1) for rational b 2 has been discussed in [2]. Here the real parameter b is generic and we shall consider vertex operators V where λ m = πµ m γ(−b 2 ) with µ m -the analog of the cosmological constant, multiplying the interaction term in the action. The reflection amplitude corresponding to the longest

JHEP12(2015)079
Weyl group element w 121321 is the analytic continuation of (2.7) (written first as a finite ratio of γ-functions and then rewritten in terms of Υ b -functions) . (4.5) Analogously to (2.10) the eight three charge factors in (4.4) can be written as points on an orbit with respect to the shifted Weyl action (4.3). The F -matrix elements are obtained by the same analytic continuation of the Toda ones in (3.7), e.g., etc.. The W 4 CFT is described alternatively as the (principal) quantum DS reduction of â sl(4) WZW model (or its dual). With the parametrisation in (2.1) and (4.1) in the noncompact and compact WZW analogs the corresponding Sugawara dimensions are given by  For any pair of weights β and e related by an element w of the Weyl group one has a BPS-like relation In particular there is only one nontrivial element of Weyl group, w 2132 , s.t. its shifted action preserves the sl(4) representations of type (λ, α 1 ) = 0 = (λ, α 3 ), namely, w 2132 · λ = −λ − 4ω 2 , so that the first line of (4.9) reads (4.10) 5 On the level of 2-and 3-point functions the reduction amounts (up to a constant) to a "x → z" limit of the sl(4) isospin variables, see [20] and references therein. In particular, for the vertex highest weights of type (λ, α1) = 0 = (λ, α3) , to which we shall restrict in what follows, they are described by a 4d vector xµ and the "x → z" limit reads x 2 ij → |zij| 4 . E.g., applied to the WZW 2-point functions G
Recall that in the sl(2) case the relation in the first line of (4.9) and its dual yield for the Virasoro dimension ( e + αb )+ m (e) = 1 , = ±1. Accordingly the products cc V β V , describe BRST invariant operators -the tachyons of the Liouville gravity. They have trivial, up to leg factors, 3-point function [21,22]. Apparently unlike the Virasoro case one cannot realise W 4 analogs of such operators through products of vertex operators from the two regions of the theory.
Nevertheless in view of the relation between theŝl(4) WZW and the W 4 conformal theories we may expect that the 3-point constants in the two W 4 regions are closely related. Indeed, take all e a = (0, r a , 0)b and impose (4.10), i.e., C(β 1 , β 2 , β 3 ) = C(e 1 + 4ω 2 b, e 2 + 4ω 2 b, e 3 + 4ω 2 b). One then has for the product of the two related constants This is achieved by the use of one of the functional relations (2.9) and produces finite products of γ-functions for each of the two constants, that are furthermore compensated in the product CC m up to the factor A(β 1 , β 2 , β 3 ) in (4.11) and a φ((β a , α 2 )), the explicit expression of which we skip. As clear from (4.11) the productCC m itself is trivial up to field renormalisation: the modified denominator from the third (fourth) line in (2.8) cancels the modified numerator from the fourth (third) line in (4.4) respectively. One may expect that the two constantsC m (e 1 , e 2 , e 3 ) andC(β 1 , β 2 , β 3 ) will describe the corresponding 3-point constants of the compact and noncompact WZW model. This conjecture remains to be checked. In any case the triviality of the productCC m whenever the relation (4.10) is imposed is a property expected for the correlators of BRST invariant states in the non-critical string model described by a G/G topological CFT, see, e.g., [23].
In the semi-classical limit b → 0 with "light" charges, i.e., (β a , α 2 )/b = σ a are assumed finite, the factor in (4.11) which depends nontrivially on the three charges goes to a numerical constant, A(σ 1 b, σ 2 b, σ 3 b) → 1/9. In other words in this limit the cancellation expected for the WZW counterparts of the W 4 constants holds true for the Toda constants themselves.

JHEP12(2015)079
We conclude with a remark about the "light-charge" limit of each of the constants C(β 1 , β 2 , β 3 ) andC m (e 1 , e 2 , e 3 ) computed using the asymptotics of Υ b (x) As explained above in these constants compared to the initial Toda ones one replaces Q → b and e 0 → −b. All weights are taken to be proportional to the second fundamental weight ω 2 , β a = σ a ω 2 b , e a = r a ω 2 b. We have in the limit b → 0 with finite σ a , r ā . (4.13) One recognizes in the Γ-function ratios of the first lines in (4.12) and (4.13) precisely the expressions of the 3-points constants of scalar 4d fields computed by integrating the bulkbounday kernels (classical vertex operators) over the cosets AdS 5 and S 5 , respectively [11,12]. In this comparison we identify the charges (β a , α 2 )/b -with the 4d scalar field conformal dimensions a and the weights e a /b (taking nonzero integer values) with the 4d isospins given by the SU(4) representation (0, J a , 0). 6 The condition (4.10) for which the product of (4.12) and (4.13) trivialises implies with such identification J a = a − 4.
On the other hand we can identify (β a , α 2 )/b = (β a , ω 2 )/b with a + 4 instead. Then neither of the two factors inC (4.12) reproduces the AdS 5 result, but the trivialisation of the fullCC m (and, in this limit, of the Toda constants product CC m itself) due to (4.10) holds true for J a = a , which is the actual 4d supersymmetric BPS condition for the given class of representations; the second line in (4.9) is equivalent to the vanishing of the second Casimir of the superconformal algebra sl(2, 2|4). 7 Note that Toda light charge classical correlators can be computed alternatively by integrals of the exponential fields over the "bulk" SL(4) group, as shown on examples in [1], generalising the computation [25] in the Liouville case.

Concluding remarks
We have constructed 3-point functions in the W 4 Toda theory and have used them to derive novel data on a fundamental braiding/fusing matrix extending the rank 1 results. The solution described by a 3 × 3 matrix applies to a particular class of representations JHEP12(2015)079 arising from partially degenerate Verma modules with highest weights proportional to the sl(4) fundamental weight ω 2 . The examples of OPE structure constants computed here and in [1] are still quite simple and need to be extended to positive integer "4d spin" components l (i) a = −(β a , α i )/b , i = 1, 3. For that purpose the AGT-W approach [6,7] might be more constructive. On the other hand one can try to exploit the pentagon equation for the 6 × 6 F matrix as a recursive relation given the initial data computed here and in [1].
We have analysed a higher rank analog of the braiding relation which played a basic role in the construction of the semi-classical limit of a class of 3-point functions on AdS 3 × S 3 [9,10] and have identified it with a standard identity in the modular group on the plane with four holes. The explicit data for the solutions of the braiding identity provided by Toda CFT, in particular their "heavy charge" limit, may thus find application to the quasiclassics of conformal sigma models described by compact and noncompact forms of SL(4, C), generalising the SL(2, C) results. Here again for a realistic application one needs first to extend the result beyond the particular class of representations.
More precisely, for this application one needs the extension of the Toda modular data to that of its WZW model counterpart; we hope to return to this problem. The computation of the correspondingŝl(4) WZW 3-point functions is important also in view of the possible application to the G/G models. As we have observed, the affine sl(4) WZW theories can alternatively describe the simplest BPS states in the "light charge" classical limit by a different mechanism than the one provided by the supergravity approximation. The 2d CFT expected to describe the worldsheet realisation of the N = 4 YM theory lacks the affine symmetry of the (super)conformal WZW models. Nevertheless further development of the latter may provide some inside on the structure of the former.

Acknowledgments
We thank Ivan Todorov for a useful discussion concerning (3.18). VBP acknowledges the hospitality of the Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Italy. This work is partially supported by the Bulgarian NSF Grant DFNI T02/6 and by the COST action MP-1405 QSPACE.

A Details on the calculation of the Coulomb integrals
We start with briefly recalling the technique [1] of computation of some multiple integrals generalising Selberg integrals. The Toda 3-point Coulomb integral (with one type of screening charges) is given by I s 1 ,s 2 ,s 3 (β 1 , β 2 ) = 3 k=1 dµ s k (t (k) ) D −2b 2 s k (t (k) )× (A.1)
The integral can be computed recursively for particular sets of weights β 1 , β 2 , exploiting a duality formula [8] originating from the Virasoro theory of central charge c = −2 dµ n (y)D n (y) This formula results from two alternative Coulomb gas representations of the n + m + 2point function, obtained by replacing each vertex with its dual of the same conformal dimension; the compatibilty of the two charge conservation conditions, involving different numbers of screening charges, fixes the parameter b parametrizing the central charge. The two integral representations coincide up to a constant C n ({p j }) = C −1 m ({−1−p j }), indicated in the r.h.s. of (A.2), which is given by a product of reflection amplitudes.

JHEP12(2015)079
In writing (A.5) we have used the functional relation (2.9) to replace products of γfunctions with ratio of Υ b − functions. The ratio of (regularized) Υ b -functions in the r.h.s. of the first line is a finite product of γ's, written in a compact form. This factor can be rewritten getting rid of the nonnegative integers s 1 , s 2 using (2.4) and then can be continued for arbitrary β a of the type in (1.1) without the restrictions implied by (2.4), thus giving

JHEP12(2015)079
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.