Exact N=4 correlators of AdS(3)/CFT(2)

We extend to chiral N=4 operators the holographic agreement recently found between correlators of the symmetric orbifold of M^4 at large N and type IIB strings propagating in AdS(3) x S^3 x M^4, where M^4=T^4 or K3. We also present expressions for some bulk correlators not yet computed in the boundary.


I. INTRODUCTION
One of the simplest realizations of the AdS n+1 /CF T n duality [1,2,3,4] is the duality between type IIB string theory in AdS 3 × S 3 × M 4 , where M 4 is either a torus T 4 or a K3 surface, and a two-dimensional N = 4 superconformal field theory in the moduli space of the symmetric product of M 4 .This duality can be derived by considering the near horizon of a system of Q 1 D1 branes and Q 5 D5 branes wrapping M 4 .
The bulk and the boundary theories have equivalent moduli spaces [5,6], and on both sides of the duality there are special points where the theory has a solvable description.In the bulk, the special point corresponds to a supergravity frame without RR flux [7], where the string worldsheet is described, for Euclidean AdS 3 , by H + 3 and SU (2) WZW models at level k = Q 5 .The second special point corresponds in the boundary to the symmetric orbifold of Recently, progress was made in checking the duality of the two theories at the dynamical level [8,9] by comparing correlators at these solvable points.It was shown there that the large N limit of certain three-point functions of chiral fields computed earlier in the symmetric product CFT agree precisely with string theory threepoint functions computed in the sphere.This verification of the AdS 3 /CF T 2 duality is surprising because the computations are carried out at different points in the moduli space, thus suggesting a non-renormalization theorem.
In [8] it was shown that computations in the bulk reproduce one of the correlators of chiral SU (2) multiplets computed in the boundary in [10,11].In [9] it was shown that, for M 4 = T 4 , the fusion rules and the structure constants of the complete N = 2 chiral ring in the bulk are in precise agreement with the boundary results of [12].
In this note we show that a simple computation allows to extend the comparison to those cases not considered in [8,9].We will show that the agreement of correlators for chiral N = 4 multiplets holds for all the boundary correlators computed in [11].In addition, we will give expressions for three-point functions in the bulk for M 4 = T 4 which were not yet computed in the boundary.

II. BULK-BOUNDARY AGREEMENT
Chiral SU (2) multiplets in AdS 3 × S 3 × T 4 are operators satisfying H = J, where J is the SU (2) spin and H is the SL(2, R) spin, which is interpreted as the conformal dimension in the dual theory.Physical string operators of this kind are given, in the holomorphic sector, by three families [13], as shown in the table below.
Here a = 1, 2 correspond to the two holomorphic oneforms in T 4 .The numbers h, j are the spins of the operators under the bosonic SL(2, R) k+2 and SU (2) k−2 which appear in the decomposition of the supersymmetric WZW models into bosonic WZW models and free fermions (see [9] for details).The number h takes the k−1 values h = 1, 3 /2 . . .k /2.For a given h, each operator has also an anti-holomorphic label, so the full operators are, e.g., O (0,2) h , etc.The same families of operators appear in the boundary theory [12], but the range of H there is larger.It is expected that additional operators in the bulk come from including spectrally flowed sectors of SL(2, R) [14,15,16], which we will not consider here.
The operators depend on the variables x, x, which are interpreted as the local variables of the boundary theory, and on y, ȳ, which are isospin SU (2) variables [17].They are normalized as and can be expanded into modes with definite J 3 0 , J3 0 eigenvalues, where The modes V h,M, M are normalized as where we have taken The string theory three-point functions for chiral operators were shown in [9] to be
In [9] only N = 2 chiral states were considered, so the relation J 3 = J 2 + J 1 was imposed and only the M 1,2 = J 1,2 , M 3 = −J 3 members of the SU (2) multiplet were retained by taking the limits y 1,2 → 0, y 3 → ∞.Here we will keep both J i 's and y i 's arbitrary, the only restriction coming from the SU (2) fusion rules applied to the j i 's and U(1) R-charge conservation M 1 +M 2 +M 3 = 0 (and similarly for the Mi 's).This case was considered in [8] for correlators with α i = ᾱi = 0, and M = M .In this note, we consider arbitrary α i , ᾱi = 0, 2 and M, M .Our results will thus be valid for both M 4 = T 4 and M 4 = K3, since only for operators with α, ᾱ = a these two cases differ.Correlators involving α, ᾱ = a N = 4 chiral primaries with J 3 < J 1 + J 2 were not computed yet in the boundary conformal field theory, so for these cases we will present the predictions from the bulk for M 4 = T 4 .
Let us express the operators in terms of where, in the symmetric orbifold, n is the length of the permutation cycle in the corresponding operator.Let us label also the two types of operators by ǫ = −1 for α = 0 and ǫ = +1 for α = 2.The spins are given now by Remarkably, all the correlators with α = 0, 2, which were computed in [9] separately for each case, can be expressed in terms of n i , ǫ i in a symmetric form as To compare with the results of [11] we should recast this expression in the M, M basis.Expanding ( 9) using ( 2), it is easy to read out the term , where we have used (−1) 2(J3+ J3) = 1, as follows from (8).The general correlator in the M, M basis follows from the Wigner-Eckart theorem and is given by are the SU (2) 3j symbols.Using now where Eq.( 14), which is the main result of this note, coincides precisely with eq.( 6.47) of [11], with the identifications Correlators involving operators with α, ᾱ = a are expressed similarly in the M, M basis using (6).There are essentially three classes of such correlators, given by It would be interesting to extend the computations of [11] in order to verify the holographic agreement for these correlators.

III. DISCUSSION
The bulk-boundary agreement found is surprising because the computations are done at two largely sep-arated points in the moduli space, suggesting a nonrenormalization theorem which should be investigated.Since the agreement found here is valid at large N , the question arises whether such non-renormalization theorem would hold also at finite N , and if so, how the finite N corrections should be obtained in the bulk [18].Among other interesting open questions, in [9] it was pointed out that for chiral operators in the boundary there are several ways of combining the fermions which multiply the twist fields.It would be interesting to understand what these options correspond to in the bulk.