On some 3-point functions in the $W_4$ CFT and related braiding matrix

We construct a class of 3-point constants in the $sl(4)$ Toda conformal theory $W_4$, extending the examples in Fateev and Litvinov. Their knowledge allows to determine the braiding/fusing matrix transforming 4-point conformal blocks of one fundamental, labelled by the 6-dimensional $sl(4)$ representation, and three partially degenerate vertex operators. It is a $3 \times 3$ submatrix of the generic $6 \times 6$ fusing matrix consistent with the fusion rules for the particular class of representations. We check a braiding relation which has wider applications to conformal models with $sl(4)$ symmetry. The 3-point constants in dual regions of central charge are compared in preparation for a BPS like relation in the $\hat{sl}(4)$ WZW model.


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. 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. 1 In this paper we are dealing with the sl(4) Toda conformal theory W 4 . The 3-point functions known so far involve one vertex operator V β with a degenerate charge β proportional to the fundamental weight ω 1 , or ω 3 = ω * 1 , i.e., the highest weight of the 4dimensional sl(4) representation. Our focus instead is on the symmetric representations β = β * and, in particular, "scalars" with β = −kω 2 b, where ω 2 is the highest weight of the 6-dimensional fundamental sl(4) representation and k is arbitrary 2 . The real parameter b 1 Apart from these traditional 2d methods a novel approach to the computation of the 3-point constants is provided by the (5d version of the) AGT-W relation [4], [5], see [6], where the main example of [1] has been recently reproduced, as well as references therein. 2 Here "scalar" refers to the 4d context of conformal group representations, for which the components 2j i = −(β, α i )/b, i = 1, 3 label the SL(2, C) spins, while △ = (β, ω 2 )/b corresponds to the 4d conformal dimension.
parametrises Toda central charge. In section 2 we compute a 3-point OPE constant for two scalar and one symmetric representations by deriving and solving a recurrence relation for the corresponding Coulomb gas integrals along the method of [7], [1] and then analytically continue it. A slightly more general 3-point constant is given in the Appendix.
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 derived in section 3 are needed. The restriction to chiral vertex operators V β a 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 [8], [9]. 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, analogs of the two Virasoro theory ingredients of Liouville gravity, c > 25 (Liouville) and c < 1 ("matter"). We discuss a BPS-like relation for the two sets of weights and show that the product of the two 3-point constants trivialises in the semi-classical "light" charges" limit. Furthermore we discuss the possible implications for the related 3-point correlators in the corresponding WZW theories, the determination of which is still an open problem.

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) scalar product 2), as well as the two other W 4 quantum numbers, are invariant with respect to an action of the Weyl reflection group w ⋆ β = Qρ + w(β − Qρ) . (2. 3) The Coulomb gas representation of the OPE constant is defined for the screening charge conservation condition The OPE constant is given by a i s i − multiple 2d integral I s 1 ,s 2 ,s 3 (β 1 , β 2 ). We compute this integral in the particular case when β a = −k a ω 2 b , a = 1, 2 and symmetric β 3 = (−l(ω 1 + ω 3 ) − k 3 ω 2 )b = β * , so that in (2.4) s 1 = s 3 . We shall skip the detailed computation 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 [7] in order to derive recursion relations for sl(n) Toda multiple 2d integrals. 3 In our case after s steps one gets an integral of type I s 1 −s,s 2 −2s,s 3 −s (β 1 + sω 2 b, β 2 + sω 2 b) so that setting s = s 3 = s 1 the integral is reduced to a Liouville type integral which is known. In particular for β 2 = −ω 2 b the resulting formula reproduces (in agreement with the general formula (1.51) of [1]) 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 3 of the 6 points of the weight diagram Γ ω 2 , i.e., The next step is the standard analytic continuation of the constant, to be denoted , in which one first gets rid of the integers s 1 and s 2 − 2s 1 exploiting the charge conservation condition (2.4), so that the Coulomb gas OPE constant is reproduced as a double residue. 4 We shall write down the related formula for β 3 = β * 3 → 2ρQ − β 3 , equivalently obtained by multiplication with the reflection amplitude R(β 3 ) corresponding to the longest Weyl group element w 121321 [1] namely, . 3 The formula derives from the observation that for the particular value 2b 2 = −1 there are two free field representations for any N -point function in the Virasoro theory, s.t. in the second one all charges are replaced by their Weyl images, with the two numbers of screening charge operators summing up to N − 2. The proportionality constant between the two integrals is given by the product of reflection amplitudes. 4 A more general OPE constant with β 1 = β * 1 which yields a Coulomb gas correlator via three residua is given in the Appendix.
In (2.7) λ T is proportional to the Toda cosmological constant, λ T = πµ T γ(b 2 ). In the products over positive roots in (2.7) the root α 1 can be replaced with α 3 (and α 13 = α 1 +α 2 We have also used that the two weight β 1 , β 2 have zero components (β a , α i ) = 0 for i = 1, 3 in order to write (2.7) in a form which makes it explicitly symmetric when the third weight β 3 is also chosen of this type, as we shall need it below: in that particular case the ratio in the second line of (2.7) produces a finite constant for (β 3 , α 1 ) → 0, while (β, ω 2 − ω 1 ) = (β, ω 1 ) in all scalar products of this type in the denominators in the last two lines.
Using (2.3) the terms in (2.7) 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] for the FL example)

Locality, fusing matrix, braiding identity
Consider the local 4-point function V f V β 1 V β 2 V β 3 of primary spinless operators V β (z,z). The 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 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 which results in equations involving fusing matrix elements and products of 3-point constants. In the case under consideration f = −ω 2 b is a degenerate field, so the equations take the form of a finite sum, e.g., for the exchange of V β 1 and V β 2 they read Here c h s (β 1 ) is a shorthand notation for the OPE constant given by a residue of , 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, the three of the OPE coefficients c h s vanish so we are left with summation over 3 of the weights, as given in (2.5). A shorthand notation for the matrix in the last equality in (3.2) is used. As indicated in the r.h.s. the matrix formed by the ratio of constants times F can be identified with the inverse matrix It is furthermore required that a consequence of the pentagon relation for F (or of the normalization relation in (3.1)). In , 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 (2.7 ) . (3.6) By analogy with the Liouville case this suggests the following ansatz for F : .
We proceed in this way to obtain F h,+ for the other two shifts of β 1 → β 1 − hb.
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).
• Summarizing we get for the matrix elements of F starting with (3.7) The last matrix element can be written in various different ways.

(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. 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 where in general h ∈ Γ ω 2 is a weight in the 6-dim weight diagram Γ ω 2 of the fundamental is proportional to the character χ ω 2 (µ) of the fundamental representation ω 2 = (0, 1, 0) evaluated at the "angle" µ = 2πb(β − ρQ). Denote by q(β) the normalized diagonal matrix In terms of F and its inverseF the relation (3.19) reads (collecting the three overall terms e 4iπb 2 in the r.h.s.) and footnote 5), taken in the semi-classical limit b 2 = 1/ √ λ → 0 with three heavy charges In this limit the phase in the r.h.s. of (3.16) reduces to 1. 6 One may expect that the Toda theory data (3.14) 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 products FF .

The 3-point functions in the compact ("matter") region. Comparisons.
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 for rational b 2 has been discussed in [2]. Here the parameter b is arbitrary and we shall consider vertex operators V e with symmetric charges e = (rω 2 + s(ω 1 + ω 3 ))b = e * . Such 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 Weyl group element w 121321 is the analytic continuation of (2.6) (rewritten first as a finite ratio of γ-functions and then rewritten in terms of Υ b -functions) .  For any element w of the Weyl group one has a BPS-like relation where (4.10) The γ-factors in the second line of (4. The productCC m itself is trivial up to field renormalization: roughly the factor in the third (fourth) line in (2.7) cancels the one in the fourth (third) line in (4.4) respectively. In view of the relation of the WZW and Toda theory one may expect that the two modified constantsC m (e 1 , e 2 , e 3 ) andC(β 1 , β 2 , β 3 ) will describe the main structure (up to some field normalization) of the corresponding 3-point constants of the compact and noncompact WZW model. 8 This conjecture remains to be checked. In any case the triviality (up to a an 7 The same constant appears for the dual transformation △(− 1 b w 2132 · (−be)) + △ m (e) = 8. Recall that in the Virasoro case the two relations yield △(ǫe + αb ǫ ) + △ m (e) = 1 , ǫ = ±1 and describe the tachyons of the Liouville gravity [12], [13]. 8 A heuristic argument in support: the integrands of the Coulomb gas representations of the WZW 3-point correlators (accounting for their isospin SL(4) invariance) differ from the integrands of Toda ones by rational function of differences of coordinate and integration variables; this effectively modifies the weights β in the powers by factors k/b. overall field renormalisation) of the productCC m whenever the relation (4.9) 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., [14].
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 (4.10) 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 these constants themselves.
We conclude with a remark about this "light-charge" limit of each of the constants C m (e 1 , e 2 , e 3 ) andC(β 1 , β 2 , β 3 ) computed using the asymptotics of Barnes function Take the limit of the subfactors inC andC m corresponding to those given by the third line of (2.7) and (4.4) respectively: recall that Q → b and e 0 → −b and all weights are taken to be proportional to the second fundamental weight ω 2 . One recognizes in the resulting Γ-function ratios precisely the expressions of the 3-points constants of scalar 4d fields computed by integrating the boundary-bulk kernels over AdS 5 and S 5 , respectively [15], [16]. 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). 9 The cancellation due to (4.9) implies J a = △ a −4. On the other hand we can identify (β a , α 2 )/b = (β a , ω 2 )/b with △ a +4. Then neither of the two factors inC 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 (4.9) holds true for J a = △ a , which is the actual 4d supersymmetric BPS condition for the given class of representations. Different identifications for the three weights are also possible (reminiscent of the mixed correlators discussed in [17]). The compensation mechanism is due to the integration over the full "bulk" group variables rather than the coset ones, cf.
[18] for a direct classical computation of the Liouville correlator with light charges and [1] for generalisations to Toda theory. 9 These are the AdS 5 × S 5 free field ingredients of the 3-point function of "chiral primary operators" with △ a = J a studied in [16]: the compensation trivializing the full constant is due to the additional factor in the coupling constant of the supergravity cubic interaction term.

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 partially degenerate representations with highest weights proportional to the sl(4) fundamental weight ω 2 . To the best of our knowledge these higher rank quantum "6j-symbols" are new. The examples of OPE structure constants computed here are still quite simple and need to be extended which would allow to derive by the same method the full 6 × 6 fusing matrix. For that purpose the AGT-W approach [6] to the computation of Toda 3-point functions might be more constructive.
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 the worldsheet AdS 3 × S 3 3-point functions [8], [9] and have identified it with a standard identity in the modular group on the plane with four holes. The explicit data (3.14) for the solutions of the braiding identity found in 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 pointed out, there are indications that the affine sl(4) WZW theories may 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.