Worldsheet four-point functions in AdS(3)/CFT(2)

We calculate some extremal and non-extremal four-point functions on the sphere of certain chiral primary operators for strings on AdS_3 x S^3 x T^4. The computation is done for small values of the spacetime cross-ratio where global SL(2) and SU(2) descendants may be neglected in the intermediate channel. Ignoring also current algebra descendants, we find that in the non-extremal case the integrated worldsheet correlators factorize into spacetime three-point functions, which is non-trivial due to the integration over the moduli space. We then restrict to the extremal case and compare our results with the four-point correlators recently computed in the dual boundary theory. We also discuss a particular non-extremal correlator involving two chiral and two anti-chiral operators.


Introduction
The AdS 3 /CFT 2 correspondence [1] is one of the most studied and tested dualities. In the last couple of years much progress has been made in identifying correlation functions. In [2,3,4] extremal and non-extremal three-point functions of chiral primary operators in the worldsheet theory for string theory on AdS 3 × S 3 × T 4 were successfully matched to the corresponding correlators in the dual boundary theory [5,6,7], see also [8,9] for correlators involving spectrally-flowed states. Later it was also shown in [10] that the cubic couplings in supergravity [11,12,13] can be brought into agreement with the symmetric orbifold correlators when mixings with multi-particle operators are taken into account. The equivalence between string theory/supergravity and field theory correlators was at first quite remarkable since the computations were preformed at different points in the moduli space. A careful analysis of the moduli dependence of the chiral ring eventually showed though that all three-point functions obey a non-renormalization theorem [14]. As a corollary, it followed that also all extremal n-point functions (n > 3) are protected along the moduli space.
In this paper we compute some extremal and non-extremal four-point functions of chiral primary operators in the worldsheet theory. The general structure of four-point functions for string theory on AdS 3 × M, where M is some compact manifold, was studied in [15], see also [16] for related work. Our goal is to apply these techniques to a more concrete case, by specializing M = S 3 × T 4 , and compare the results with expectations from the boundary conformal field theory. In this way we may test and explore the non-renormalization theorem of [14].
Apart from the question of non-renormalization, the computation of worldsheet fourpoint functions in AdS 3 is also interesting in its own right. As compared to similar computations of worldsheet two-and three-point functions [2,3,4], four-point functions are much more involved for the following reasons. First, unlike in two-and three-point functions, one cannot fix all worldsheet coordinates by modular invariance anymore. In general, four-point functions require a true integration over the worldsheet cross-ratio z, i.e. an integration over the moduli space. Second, four-point functions on AdS 3 involve also an integration over the locus of the continuous representation of the SL(2) affine algebra, i.e. along the line h = 1/2 + is (s ∈ R). Third, the integration over the SL (2) representation label h in turn requires a careful analysis of the pole structure of the fourpoint functions [15,16]. Fourth, in principle there are all sorts of states in the intermediate channel, such as primary states, descendants, single-and multi-particle states etc. To simplify the computation, one needs to find selection criteria for these states. All these questions will be addressed in some concrete examples.
We begin by computing some non-extremal worldsheet four-point functions. Here we are interested in the question of their factorization into spacetime three-point functions. Other than in the boundary conformal field theory, this question is non-trivial due to the integration over the moduli space. Next, for comparison with the corresponding boundary correlators, we then restrict the four-point functions to the extremal case and find agreement with the (single-particle contribution to the) boundary correlators, which have previously been found in [17]. We also compute a particular non-extremal worldsheet correlator and compare it with its dual boundary correlator [17], which consists of two chiral and two anti-chiral operators. We summarize our results in the conclusions.

Some four-point functions in the symmetric orbifold theory
Before turning to the worldsheet theory, we briefly review some of the results in the boundary conformal field theory. We will later compare our integrated worldsheet correlators with the four-point correlators presented in this section. The boundary theory is a symmetric product orbifold theory of the type Sym(T 4 ) N = (T 4 ) N /S N with N = 4 supersymmetry, where the coordinates of the product of N copies of T 4 are identified by the action of the permutation group S N . The operators of the theory are associated to group elements g ∈ S N , which are single cycles, (1 ... n 1 ), double cycles, (1 ... n 1 )(n 1 + 1 ... n 1 + n 2 ), etc.
The chiral primary operators are given by the single-cycle twist operators with a,ā = ±, and n = 1, ..., N denotes the length of the cycle (For a precise definition see e.g. [6,7,17]). The corresponding conformal dimensions are and similarly for the antiholomorphic sector. For the comparison with string theory computations, we will later use the label h = (n + 1)/2 instead of n such that Similarly, there are multi-cycle operators associated to multi-cycle group elements of S N . Most prominent are double-cycle operators, which appear in the intermediate channel of extremal four-point functions [17].
Correlators of single-cycle twist operators are computed on covering surfaces of different genera. Quite generally, it can be shown from the Riemann-Hurwitz formula that if the cycle lengths of a p-point correlator satisfy the sphere is the only covering surface which contributes to the correlator [17]. Pakman, Rastelli and Razamat [17] computed several correlators satisfying (2.5). Among others, they found the extremal four-point functions where the function F 4 (n i ) is given by Note that F 4 ≈ 1/N at large N and that the correlators are topological, i.e. independent of the complex coordinates x. The extremality conditions imposed on these correlators imply the condition (2.5), n 4 = n 1 + n 2 + n 3 − 2. More generally, it can be shown that all extremal correlators receive contributions only from the sphere. The converse is not true. There are correlators satisfying (2.5) which are not extremal. An example is given by the non-extremal correlator [17] O The correlator scales as 1/N at large N. The conformal dimensions are h (2.14) but nevertheless satisfies (2.5). The appearance of two anti-chiral operators in (2.12) ensures charge conservation since Extremal correlators satisfy a non-renormalization theorem [14] and are thus protected along the entire moduli space. They should therefore be reproducible by a string or supergravity computation. The non-extremal correlator (2.12) is not a priori protected by a non-renormalization theorem.

Scaling of chiral primaries in the worldsheet theory
In this section we set our notation by defining the chiral primaries of the worldsheet theory. We also review the computation of their two-point functions. The scaling of the operators will be relevant when worldsheet correlators are compared with the corresponding boundary correlators. The notation follows closely that in [3].

Chiral primary operators
In the following we summarize the chiral primaries of the worldsheet theory [18,19,3]. It is understood that all fields depend on the worldsheet coordinate z, even though this dependence will be suppressed in the notation.
The worldsheet theory is the product of an N = 1 WZW model on H + 3 , an N = 1 WZW model on S 3 ≃ SU(2) and an N = 1 U(1) 4 free superconformal field theory. This WZW model has the affine world-sheet symmetry sl(2) k × su(2) k ′ × u(1) 4 . Criticality of the fermionic string on AdS 3 × S 3 requires the identification of the levels k and k ′ [20], k = k ′ . The label k denotes the supersymmetric level of the affine Lie algebras and is identified with the bosonic levels The bosonic currents are J a for SL(2) and K a for SU (2). The free fermions of SL(2) are denoted by ψ a , those of SU(2) by χ a (a = (+, 0, −) in either case). It is convenient to split the bosonic currents as and similarly K a . Finally the u(1) 4 symmetry is described in terms of free bosons as i∂Y i , and the corresponding free fermions are λ i (i = 1, 2, 3, 4).
The chiral operators are constructed from the dimension zero operators where Φ h (x) and Φ ′ j (y) are the primaries of the bosonic SL(2) and SU(2) WZW models. The labels x and y correspond to the SL(2) and SU(2) labels m and m ′ , respectively. Our conventions for these models can be found in appendix A. Since h = j + 1, the operators O j (x, y) have vanishing conformal dimensions, ∆(h) + ∆(j) = 0.

NS sector
In the NS sector there are two families of chiral primaries. In the −1 picture they are where the fields ψ(x) and χ(y) are given by The bosonized superghost field e −φ ensures that the operators have ghost number −1. Sometimes we will also need the corresponding ghost number 0 operators, which are obtained from (3.3) by acting with the picture changing operator Γ +1 . These operators will be needed to get the correct ghost number in the correlators. The ghost number 0 operators are [3,2] where the operators D A x and P a y are Here we used again the compact notation

Full chiral operators
The full chiral primary operators are given by the product of a holomorphic with an anti-holomorphic operator, where A = 0, a, 2 andĀ =0,ā,2. When integrated over the worldsheet, these operators are dual to the chiral primary operators O (A,Ā) n in the boundary theory (n = 2j + 1).

Two-point functions and normalized operators
The two-point functions of the above chiral primary operators are worked out in [2,3,8].
In order to set the notation, we briefly review the computation here.
In the NS sector the two-point function is (h = j + 1) where we defined φ i = φ(z i ) and used The two-point function scales as |x 12 | −4h (0) with h (0) as in (2.3), which agrees with the scaling of the dual boundary operator.
In the Ramond sector we get the two-point function (h = j + 1) where we used Note that one primary is in the −1/2 picture while the other one is in the −3/2 picture such that the total ghost number is −2, as required on the sphere. The two-point function scales as |x 12 | −4h (a) with h (a) as in (2.3).
In order to obtain the corresponding boundary correlators we need to integrate the above two-point functions over the worldsheet coordinates z 1 and z 2 . Equivalently, we may fix z 1 = 1 and z 2 = 0 and divide the correlator by the volume of the conformal group V conf which keeps the two points fixed. As shown in appendix A in [15], this removes the divergence coming from δ(0) and introduces the factor . (3.20) We observe that other than the operators in the boundary conformal field theory, the chiral primaries are not normalized to unity. We therefore rescale the operators as

Four-point function in the NS sector
In this section we compute a four-point correlator which involves only chiral primary operators of the NS sector. In particular we are interested in computing the correlator 2 where we choose the m-labels as (d ≥ 0) The worldsheet coordinates are fixed as z 1,2,3,4 = 0, z, 1, ∞, where z is the cross-ratio z = z 12 z 34 /(z 13 z 24 ) on the worldsheet. Similarly, the continuous SL(2) representation labels are chosen as x 1,2,3,4 = 0, x, 1, ∞. Later, these labels will be identified with the complex coordinates in the boundary conformal field theory [21] and x becomes the spacetime cross-ratio. The correlator G N S 4 (x,x) involves two ghost number zero and two ghost number −1 operators,Õ (0,0) j and O (0,0) j , respectively. Note that the total ghost number of a correlator on a genus-g surface must be −χ = −(2 − 2g), which is −2 on the sphere.
is a free parameter in the H + 3 model. As in [3], we leave c ν (and thus ν) undetermined for the moment. c ν will later be fixed, when we compare the bulk and boundary correlators. The correlator (4.1) is called extremal, if the spacetime scalings of the operators in 3 (These are the scalings in x, defined as the power of the term |x 12 | −4h (0) in (3.16)). Using (2.3) and h i = j i + 1 (i = 1, ..., 4), this translates into the condition or d = 0. We will first consider the non-extremal case d > 0 and come back to the extremal case d = 0 in section 4.4. Substituting the explicit expressions for these operators, as given by (3.3) and (3.6), we get 3 The actual computation of G N S 4 (x,x) will be done along the lines of [15].

Some correlators inside
Following [15], we write the SL(2) four-point function in terms of the factorization ansatz [22] where the normalization C(h) is given by . The functions B(h) and C(h 1 , h 2 , h 3 ) are the scaling of the SL(2) two-point function and the SL(2) structure constants, respectively. They are given by (A.3) and (A.5) in appendix A. As in [15], we change variables from z to u by defining u = z/x and consider the case |x| < 1. We may then perform an expansion of F h (x; u) in powers of x as Substituting this expansion into the KZ equation for SL(2) [22], one finds that the first term obeys the hypergeometric equation in u, i.e.
We will sometimes use the shorthand In what follows we will focus on the leading term in the x expansion, where the ellipsis represents higher order terms in x. Such terms correspond to descendants under the global SL(2) algebra [15], which do not play a role in the small x region. It is convenient to write F h (u) as a power series in u, .
A similar factorization ansatz can be found for the SU(2) four-point function. As shown in appendix C, at small z the SU(2) four-point function with m-values as in (4.2) can be expanded as 4 are the inverse of the binomial coefficients, (4.14) The δ-function reflects the charge conservation m 1 +m 2 +m 3 +m 4 = 0. The normalization C ′ (j) is given by C ′ (j) = C ′ j,j 1 ,j 2 C ′ j,j 3 ,j 4 (no summation over j). The SU(2) structure constants C ′ j 1 ,j 2 ,j 3 and the functions D(j 1 , j 2 , J) are given by (A.20) and (C.8) in the appendix, respectively.
We will also need some other four-point correlators for G N S 4 (x,x). For the following, it is useful to define the n-point correlators with k, m = 1, ..., n, in which one or two bosonic currents j(x) act on the product of n SL(2) functions Φ h (x). As shown in appendix B, such correlators can entirely be expressed in terms of derivatives of the SL(2) n-point function. In particular, the functions d 4 and d (4) 24 appearing in (4.4) can be computed by means of (B.6) and (B.7). Using only the first term in the small x expansion (4.9) of the SL(2) four-point function (4.5) (and and H(a, b, c, n) as in (4.11). For k = 4, 2, 1, the coefficients are given by 5 2,n = z 34 z 24 z 23 1,n = z 34 z 14 z 13 (4.20) 5 Here we also list the coefficientd 1,n for later use.
Finally, the correlator d (4) 24 is given by which, for brevity, is expanded around z = 0 (the ellipses denote further terms subleading in z). Also the x-and z-dependence is already fixed as above. Note that the above expressions for the d (4) correlators are only valid for small x.
We will also need the fermionic correlators which have been computed using (B.8) in appendix B.

Moduli integration and integral over h
We now perform the integrals over the worldsheet cross-ratio u and the SL(2) representation label h. We wish to do the u-integral before the integral over h but need to be careful about the occurrence of divergences. Following [15,16], we therefore regularize the u-integral by introducing a cut-off parameter ε and divide the range of u into two regions: region I: |u| < ε region II: |u| > ε .
In region I there are only operators in the intermediate channel whose SL(2) part is associated with short strings with winding number w = 0 [15]. In region II there can be long strings with w = 1 and two-particle states [15]. The representation theory of SL (2) does not allow any other spectrally-flowed states in the intermediate channel.
An important observation is that "single-cycle" operators in the spacetime CFT arise locally on the worldsheet, i.e. in the small u region, while "multi-cycle" operators correspond to non-local contributions coming from the large u region [15,16]. 6 Since at large N multi-particle contributions are suppressed in non-extremal correlators [17], we may restrict to the one-particle contributions to the four-point correlator. We therefore consider only region I and ignore possible two-particle contributions coming from region II.
Formally, the one-particle contributions are taken into account by first integrating over the small u region, |u| < ε, and then taking the limit ε → 0. This is the limit where the operators approach each other in their worldsheet coordinates. For |u| < ε, we may then expand G N S 4 (x,x) in powers of u as where we display only the most singular term in the square brackets. Subleading terms are summarized in O(u 0 ).
We now turn to the integration over h. The h-integral is defined along the line h = k−1 2 + is (s ∈ R), away from the locus of the continuous representation of SL(2), h = 1 2 + is. The reason for the deformation is that only there the integrand is equivalent to a monodromy invariant solution, cf. (4.34) in [15]. It is possible to shift the integration contour back to h = 1 2 + is. However, in general, the integral picks up pole residues when the poles cross the integration contour. At small u there are altogether four types of poles of the h-integral which may contribute to the integral. These are [15]: The poles of type II-IV are poles in the structure constants C(h, h 1 , h 2 ). As discussed extensively in [16], none of these poles contributes to the integral. Even though naively one might interpret the contributions from the poles of type II as "double-cycle" operators in the spacetime CFT, such contributions go to zero in the ε → 0 limit [16]. Type III poles do not appear if h 1 + h 2 < k+1 2 [15]. The contribution coming from poles of type IV was found to be canceled by the same contribution from crossing the integration contour [16].
We are left with poles of type I. These poles correspond to short string representations (with zero winding number) in the SL(2) WZW model [15]. The condition A particular solution is n + n ′ = 0 and h = j + 1. Since n and n ′ are both positive, n = n ′ = 0 and we recover the on-shell condition for chiral primaries in the intermediate channel. As such they map to single-cycle chiral primary operators in the spacetime CFT.
For n+n ′ = 0, we generically do not get a rational conformal weight h. Substituting the condition (4.27) into (4.25), we find that the correlator depends on x as x h−n−h 1 −h 2 . This should be compared with the x dependence of the corresponding boundary four-point function, which is x H−H 1 −H 2 (see e.g. (4.2) in [15]), where H denotes the corresponding spacetime conformal weights. Since H = h − n with h as in (4.28), one therefore identifies this contribution as coming from SL(2) short string descendants (of the type (J − −1 ) n (J − −1 )n|h, m =m = h ) in the intermediate channel [15]. These states have a continuous spectrum for h > 0, if one chooses the universal cover of SL(2) as the target space. Since H = h − n is generically irrational, it is not clear to us which boundary states can be identified with the current algebra descendants. In the following we therefore restrict to the case n = n ′ = 0 (h = j + 1), for which there are only chiral primary operators in the intermediate channel, and ignore possible contributions from current algebra descendants.
This leads to some simplification of the product C(h)C ′ (j). Recall the following relation between the structure constants of SL(2) and SU(2) found in [2,3], which holds for h i = j i + 1 (i = 1, 2, 3) and k b = k ′ b − 4. From this we find the identity since h = j + 1. In other words, the poles of the SL(2) structure constants cancel against the zeros of the SU(2) structure constants.
With these identities, we may now return to G N S 4 (x,x). Applying the residue theorem 7 and taking the limit ε → 0, we get The factor ∂ h (∆(h))| h=j+1 /(2π 2 ) = (2j + 1)/(2π 2 k) in the denominator is precisely the factor (3.20). It is related to the fact that we need to integrate over the conformal group on the worldsheet when comparing two-point functions on the worldsheet to two-point functions in spacetime. Recall that spacetime four-point functions can be considered as a sum over the product of two three-point functions divided by the two-point function.
We must still normalize the four-point function with respect to the scaling of the twopoint functions. For the four-point function of the corresponding normalized operators (3.21), we then find where we introduced the factor

Factorization into three-point functions
It is possible to rewrite G N S 4 (x,x) as the product of two three-point functions. For that, we label the state in the intermediate channel by j and set its m quantum number as m = j. 8 Then, the charge conservation m = m 1 + m 2 selects the term with Let us denote the r.h.s. of (4.26) by f (h) such that for with h 0 = j + 1.
8 More generally, one could have set m = j −d withd ≥ 0. Each term in G N S 4 (x,x) would then scale as |x| 2(j−j1−j2) = |x| 2(−d+d) . Since at small x the leading term in the sum over j is that ford = 0, we may neglect global SU (2) descendants. Note that we have already ignored global SL(2) descendants in (4.9).
in the sum over j. For this particular value of j, or d = j 1 + j 2 − j, G j,0 reduces to and G N S 4 (x,x) becomes However, this is nothing but the expected factorization in terms of three-point functions, The ellipsis indicates terms subleading in x. The x-dependence |x| −2d is now contained in the left three-point function.

The extremal case and comparison with the boundary theory
So far, general non-extremal four-point functions have not been considered in the dual symmetric orbifold theory. For comparison with the results in the boundary conformal field theory, we therefore specialize now to the extremal case j 4 = j 1 + j 2 + j 3 , for which the dual boundary correlator is known [17]. As we can see from (4.35) for d = 0 (i.e. j = j 1 + j 2 ), G j,0 = δ 2 j 1 +j 2 +j 3 ,j 4 , and hence The result is independent of the cross-ratio x, as expected for extremal correlators. Changing variables from j to n by setting n i = 2j i + 1 (i = 1, 2, 3, 4), we get G N S 4 (x,x) = 1 N n 5/2 4 (n 1 n 2 n 3 ) 1/2ñ n 4 (4.40) withñ = n 1 + n 2 − 1. In the large N limit, this is in agreement with the single-cycle contribution to the boundary correlator (2.6), which is given by (2.6) times the factor n/n 4 [17]. This is the contribution coming from single-cycle operators in the intermediate channel.
As argued in [17], in the extremal case contributions coming from double-cycle operators in the intermediate channel are not suppressed at large N. It was found that the combined effect of single-and double-cycle operators is given by the single-cycle contribution times the factor n 4 /ñ, symbolically: full extremal correlator = single-+ double-cycle contribution = n 4 n · (single-cycle contribution) .
Clearly, it would be desirable to reproduce this factor in the worldsheet theory. Doublecycle terms in the spacetime OPE arise nonlocally on the worldsheet and are presently not very-well understood.

Mixed NS and R four-point function
The computation of the previous section can easily be adapted to other four-point functions. As a further example, we next compute a four-point function which involves two chiral primaries in the NS sector and two in the R sector. Such a four-point function is given by with m-values as in (4.2). The first two operators are Ramond chiral primaries with ghost number −1/2. The third and fourth operators are NS chiral primaries with ghost number −1 and 0. The total ghost number is therefore again −2, as required on the sphere. For the computation, we will need the fermionic correlators Proceeding as before, we use again the factorization ansatz (4.6) and get where the first term in the four-point function d with z = ux, as before. The structure of G R 4 (x,x) is similar to that of G N S 4 (x,x) as given, for instance, by (4.24). The only change is the terms in the second line.
We now perform the u-and h-integrals. In the region |u| < ε we expand (5.4) as and do the u-integral as in (4.26). Performing also the h-integral and taking the ε → 0 limit we get (5.7) As argued above, there are only chiral primary states in the intermediate channel (with h = j + 1), which allows us to use (4.30).
With the above value for c ν , c ν = 1/(2π 4 k 3 ), the corresponding rescaled correlator is Note here the difference in the scaling of R and NS operators. As argued in the previous section, at small x the leading term in the sum over j is that for j = j 1 + j 2 − d. Recalling now (4.35), G R 4 (x,x) can be rewritten as with j = j 1 + j 2 − d = j 4 − j 3 . Ellipses represent again subleading terms in x. After comparing with the three-point functions, we get the factorization with the left three-point function as in (4.38) and the right one given by [3] O For comparison with the corresponding boundary correlator, we restrict again to the extremal case, d = 0 or j 4 = j 1 + j 2 + j 3 . Then, the only non-vanishing term in the sum over j is that for j = j 1 + j 2 (with G j,0 = δ 2 j 1 +j 2 +j 3 ,j 4 ) and G R 4 (x,x) as given by (5.8) becomes independent of x, The result precisely coincides with the one-particle contribution to (2.8) upon identifying n i = 2j i + 1. At large N it is given by 9 G R 4 (x,x) = δ ab δāb 1 N (n 4 n 3 ) 1/2 (n 1 n 2 ) 1/2ñ (5.13) withñ = n 1 +n 2 −1. The result does not include possible contributions from the exchange of two-particle states.
We expect that the remaining extremal spacetime four-point correlators (2.7) and (2.9) can be reproduced by a similar worldsheet computation.

A particular non-extremal four-point function
In this section we consider the non-extremal four-point function for, at first, arbitrary j-values. Later we will fix the j-labels in order to compare the correlator with the corresponding boundary correlator (2.12). We begin by substituting the explicit expressions for the chiral primary operators, Keeping only the nonvanishing terms, we get This can be simplified by means of the identity 2χ a P a y = χ(y)∂ y − j∂ y χ(y) , (6.4) which is obtained from the expansion of χ in the y-basis, Eq. (3.5), and χ ± = χ 1 ± iχ 2 . We will also need the correlators given by (4.16) with (4.19) and (4.18), and the relations Substituting everything back into (6.3), we get where the ellipsis indicates terms subleading in x (In particular, at small x we may neglect the third and fourth term in (6.3)). As before, we use the factorization ansatz (4.6) and change variables, z = ux. At small u and small x, we obtain (6.11) At this point we need to specify the chirality of the operators in the dual boundary correlator. For this, we assign labels a 1,2,3,4 ∈ {0, 1} to the boundary operators. The label a i is zero (one), if the dual operator is chiral (antichiral). Then, U(1) charge conservation, yields the following relation among the j-values, In view of the boundary correlator (2.12) let us consider the case a 1 = a 3 = 0 (chirals) and a 2 = a 4 = 1 (antichirals) and fix the j-labels as j 1 = n−1 2 , j 2 = j 3 = 1 2 and j 4 = as required in (2.12). Using the relations (2.3) and h i = j i + 1, we find that the nonextremality condition (2.14) translates into j 4 = j 1 + j 2 + j 3 . Since j 2 = 1/2, this relation is equivalent to the U(1) charge conservation relation j 4 = j 1 − j 2 + j 3 + 1.
For the above values of j i and a i (i = 1, 2, 3, 4), it was found in [17] that in the boundary theory O with j = j 1 + 1 − j 2 = j 1 + 1/2. If we assume that the one-to-one correspondence between worldsheet and boundary operators also holds in the intermediate channel, then the sum over j reduces to a single term for which j = j 1 + 1/2.
Proceeding as before, we get The corresponding rescaled correlator is 10 At large N this agrees with the non-extremal correlator (2.12). 10 The operator O . As compared to the corresponding chiral operator, it is rescaled by an additional factor |y| −4j2 [3], which cancels |y| 2 in the numerator.

Conclusions
We discussed extremal and non-extremal four-point correlators in the worldsheet theory for AdS 3 ×S 3 ×T 4 . The computations were done at small cross-ratios x where we were allowed to ignore subleading contributions from global SL(2) and SU(2) descendants in the intermediate channel (In the boundary theory this corresponds to neglecting spacetime descendants.) For simplicity, we also ignored possible contributions from current algebra descendants. This is certainly allowed for extremal correlators, for which the N = 2 chiral ring structure ensures that there are only chiral primary operators in the intermediate channel. For non-extremal correlators, however, there are in principle further contributions coming from current algebra descendants, which we have not computed, but should be studied in more detail in the future.
We obtain the following results: i) We found that the integrated non-extremal correlators G N S 4 (x,x) and G R 4 (x,x), as defined in (4.1) and (5.1), factorize into the product of two spacetime three-point functions composed out of chiral primaries, see (4.37) and (5.10). Other than in the spacetime CFT, the factorization is non-trivial in the worldsheet theory because of the integration over the moduli space. If there were only chiral primary operators running in the intermediate channel, the factorization property would imply the non-renormalization of the correlator, at least at small x. However, as just stated, there can be additional terms coming from current algebra descendants, which would renormalize the four-point function. ii) We then evaluated G N S 4 (x,x) and G R 4 (x,x) for the extremal case and find agreement with the single-particle contribution to the corresponding extremal boundary correlators computed in [17]. This has been expected from the non-renormalization theorem of [14]. Note that in contrast to their non-extremal cousins, extremal four-point correlators also have two-particle states in the intermediate channel, whose contribution to the correlator is not suppressed at large N. In the boundary theory, the inclusion of the two-particle contribution amounts to multiplying the single-particle contribution by a simple factor, n 4 /ñ [17]. Clearly, it would be desirable to also derive this universal factor in the worldsheet theory by taking into account nonlocal contributions on the worldsheet. Such contributions are presently not very well understood. iii) We also computed a particular non-extremal four-point correlator, defined in (6.1), whose dual correlator in the boundary theory contains two chiral and two anti-chiral operators. This correlator is not covered by the non-renormalization theorem of [14] and therefore need not necessarily agree with its boundary counterpart. Nevertheless, we find exact agreement, cf. our result (6.16) or (6.17) with (2.12), again under the premise that we may ignore possible contributions from current algebra descendants in the intermediate channel.

Appendix
A Correlators in SL(2) k and SU (2) k ′ WZW models A.1 Two-and three-point functions in the SL(2) k WZW model The chiral primaries of the SL(2) WZW model are denoted by 11 where k is the level of the affine Lie algebra. In the current context only half-integer h will be relevant.

A.2 Four-point function in the SL(2) k WZW model
The four-point function of the SL(2) chiral primary Φ h i (z,z; x,x) is given by The function F SL(2) (z,z; x,x) is given by where and The normalization is 15) A.3 Two-and three-point functions in the SU (2) k ′ WZW model The chiral primaries of the SU(2) k ′ WZW model are denoted by 16) and have conformal dimension where j is the SU(2) representation label and k ′ the level of the affine Lie algebra.

B Some correlators
In this appendix we give some more details on the computation of some correlators used in the main text. For the computation of these correlators we will need the following OPEs (the depen-dence of the fields on z is suppressed):

C Comments on SU (2) four-point function
In this appendix we derive the factorization (4.12) of the SU(2) four-point function.