Spectral flow and string correlators in AdS$_3\times S^3 \times T^4$

We consider three-point correlation functions for superstrings propagating in AdS$_3\times S^3 \times T^4$. In the RNS formalism, these generically involve correlators with current insertions. When vertex operators with non-trivial spectral flow charges are present, their complicated OPEs with the currents imply that standard methods can not be used to compute such correlators. Here we develop novel techniques for computing all $m$-basis correlators of the supersymmetric model. We then show how, in some cases, these results can be translated to the $x$-basis. We obtain a new family of holographic three-point functions involving spacetime chiral primaries living in spectrally flowed sectors of the worldsheet CFT. These match precisely the predictions from the holographic dual at the symmetric product orbifold point. Finally, we also consider long strings and compute the probability amplitude associated with the process describing the emission/absorption of fundamental string quanta.


Introduction
The propagation of superstrings in AdS 3 × S 3 × T 4 (or K3) backgrounds has become one of the most fruitful frameworks for exploring the AdS/CFT correspondence. Unlike for the case of AdS 5 × S 5 , perhaps the other prototypical model for studying holography, identifying the corresponding dual quantum field theory has been an elusive task 1 . Nevertheless, the AdS 3 case outperforms the higher-dimensional one in that it provides a scenario in which one can describe a wide range of exciting phenomena beyond the supergravity limit. Indeed, one has access to a worldsheet description which is, in principle, exactly solvable [3][4][5][6][7][8]. The relevance of this model has recently increased due not only to multiple advances where j a and V are the bosonic currents and spectrally flowed primaries, respectively. As already mentioned, these expressions are not only relevant from a pure AdS 3 perspective, but they can be useful in the context of other related coset models. Next, in order to deduce the corresponding results for operators in the spacetime representation, we follow a path similar to that introduced in [8] and further developed in [46]. Although we are not able to obtain the x-basis correlators in full generality, as was the case in [46], we compute this kind of three-point functions for a wide range of spectral flow assignments. More precisely, we focus on the NS-NS sector of the worldsheet theory, and compute where V ω are the supersymmetric vertex operators and h are their spacetime weights, while the superscript "(0)" stands for the ghost picture-zero version of the corresponding unflowed operator. We then show that the precise matching with the holographic CFT results holds for the associated families of chiral primary correlators as well, thus extending the analysis of [34]. Although somewhat restrictive, our results further allow us to explore, for the first time, how processes, where the background emits/absorbs a unit of fundamental string charge take place within the supersymmetric model with n 5 > 1.
The paper is organized as follows. In Section 2, after reviewing some basics about superstring theory in AdS 3 × S 3 × T 4 , we re-compute explicitly all unflowed bosonic threepoint functions involving descendant insertions, both in the m and in the x basis, obtaining the unflowed three-point functions in a way that is well suited for the generalization to the spectrally flowed cases. In Section 3 we discuss in detail the vertex algebra in spectrally flowed sectors and the so-called series identification. We define field operators for both short and long strings and their ghost picture-zero versions. We also compute the corresponding two-point functions. Then, in Section 4, we extend the computation of three-point correlators in the m basis to arbitrary spectrally flowed insertions by introducing a method for dealing with bosonic correlation functions with descendant insertions. We also compute three-point correlators in the x basis. Finally, in Section 5, we present our concluding remarks.
2 Superstring theory on AdS 3 × S 3 × T 4 We start by briefly reviewing the relevant aspects of superstring theory on AdS 3 × S 3 × T 4 , the near-horizon region of the background sourced by n 5 NS5-branes and n 1 fundamental strings. The main building block of the worldsheet theory is given by the SL(2,R)-WZW model introduced in [3,[6][7][8]. The supersymmetric extension was investigated in [4,5,32], while the corresponding extremal correlators were studied in [33,34,53].
It will also be useful to use the bosonized forms for the SL(2,R) and SU(2) fermions. We consider canonically normalized bosonic fields H I , with I = 1, . . . 5, and introducê The bosonization of ψ a and χ α then reads In the remainder of the paper we will mostly omit the hats and explicitly include the phase factors only if necessary.
The stress tensor and the supercurrent of the matter sector of the worldsheet CFT derived from the Sugawara construction are We also have the standard bc and βγ ghost systems, leading to the BRST charge (2.11) The βγ system is also bosonized as where ϕ(z)ϕ(w) − ln(z − w) has background charge 2, and ξ(z)η(w) ∼ (z − w) −1 . The spacetime supercharges can be written as where S ε are spin fields and ε I = ±1. These are constrained by ε 1 ε 2 ε 3 = ε 4 ε 5 = 1 due to BRST-invariance and mutual locality, giving the supercharges of the spacetime N = (4, 4) superconformal algebra. Moreover, the R-symmetry of the boundary theory is generated by the worldsheet SU(2) currents.

Vertex operators in the unflowed sector
Let us define the physical vertex operators. We will mostly follow the conventions of [33,34,53]. We also focus on the holomorphic part of the theory, omitting the antiholomorphic dependence in most of the expressions below. Since bosonic and fermionic currents commute, vertex operators factorize into a product of bosonic primaries and free fermions. Let V h (x, z) be an SL(2,R) k primary field of weight and spin h. It satisfies The fields V h (x, z) are defined in the so-called x-basis, the x variable being identified holographically with the complex coordinate of the boundary theory [5]. Their spacetime modes then correspond to the m-basis operators The bosonic currents OPEs with the fields V hm (z) are The relevant unflowed representations of the zero-mode algebra are -Principal series discrete representation of lowest/highest-weight: these are built by acting with j ± 0 on the state |h, ±h (created by V h,±h (0) acting on the vacuum), which is annihilated by j ∓ 0 : Note that for operators in D ± h one can invert (2.17). Indeed, the (residues of the) poles located at x = 0 give states in the highest-weight representations, namely (2.20) Conversely, operators in the lowest-weight representation give the power-series expansion around x = ∞. In particular, we have -Principal continuous series: These are given by It was shown in [6] that the spectrum of the SL(2,R) k -WZW model is built out of continuous and lowest weight representations with together with their spectrally flowed images, defined below.
The bosonic SU(2) k WZW model has primary fields W l,n with n = −l, . . . , l and weight where the spin l is bounded by [54] 0 ≤ l ≤ k 2 . (2.25) The OPEs with the currents k α are given by Similarly to the SL(2,R) case, one can make use of the isospin variables y, with W l (y, z) = l n=−l y l−n W l,n (z), (2.27) so that Eqs. (2.26) read where P − y,l = −∂ y , P 0 y,l = y∂ y − l , P + y,l = y 2 ∂ y − 2ly. (2.29) The properties of SL(2,R) and SU(2) bosonic primary operators can be written compactly in the following form: where x 12 = x 1 − x 2 , and we have introduced the currents K(y 1 , z 1 )K(y 2 , z 2 ) ∼ −n 5 y 2 12 z 2 12 + 1 z 12 y 2 12 ∂ y 2 + 2y 12 K(y 2 , z 2 ) (2.33) It will be useful to work similarly with the fermions as expected for fields with SL(2,R) spin h = −1 and SU(2) spin l = 1, respectively. In the supersymmetric theory, unflowed vertex operators that belong to the continuous series representations are projected out by GSO due to their tachyonic nature. Physical vertex operators polarized in the SL(2,R) and SU(2) directions belonging to the unflowed sector are given in [33,53,55]. We focus on operators dual chiral primaries of the holographically dual CFT 3 , namely where Φ h;l (x, y) = V h (x)W l (y) and the exponential of the free boson ϕ comes from the ghost sector of the theory. The worldsheet dependence is omitted. These operators have definite (supersymmetric) spins (H, L) = (h − 1, l) and (H, L) = (h, l + 1), and represent massless excitations polarized in the AdS 3 and S 3 directions, respectively. Recall that one must also include fermion excitations in the anti-holomorphic sector. The Virasoro conditions impose which is solved by setting h = l + 1 (the second solution falls outside of the range (2.23)). Therefore, we will omit the subscript l assuming this relation. In both cases, H gives the corresponding spacetime weight and, since H = L with L the spacetime R-charge, these are chiral primary operators of the boundary theory. From now we will focus our attention on V h . The treatment for W h is analogous. Using the relation (2.17) it is possible to express V h in the m-basis. Explicitly, we have and where we have once again ignored the anti-holomorphic sector.

Two-point functions
The string theory two-point function factorizes into bosonic, fermionic and ghost contributions. The latter is given by e −ϕ(z 1 ) e −ϕ(z 2 ) = z −1 12 . The SL(2,R) two-point function is given by [56] There are additional families of physical operators in the NSNS sector which are not dual to chiral primaries. We will not discuss them in this paper due to the fact that their three-point functions are not protected by supersymmetry. In any case, it would be interesting to use the techniques developed here to compute the corresponding three-point functions in order to compare them with the correlators in the recent proposal of [1] for the holographic dual. Indeed, and as opposed to the usual symmetric orbifold theory considered for instance in [33,53], this is conjectured to be the dual CFT at the same point in the moduli space where the worldsheet theory is defined, so that one should also be able to match non-protected quantities.
where [3,8] and ν is an arbitrary constant. As for the fermion propagator, we recall that the x-and zdependence of the propagator is fixed by worldsheet and spacetime Ward identities. ψ(x, z) is a primary field of spin h = −1 and worldsheet weight ∆ ψ = 1/2 with respect to the SL(2, R) −2 algebra generated by the currentsĵ a . Making use of Eq. (2.21), we have as expected. On the other hand, the SU(2) contributions read and The full two-point functions then take the form

Picture changing and three-point functions
In order to compute string three-point functions in the NS sector of the theory it is necessary to obtain the ghost picture-zero version of the above vertex operators. As usual, the picturechanging operator is defined in terms of the total (matter) supercurrent G, so that where O stands for a generic vertex operator. For unflowed states, this was derived in [33,53], giving Due to the second term appearing on the RHS of Eqs. (2.49), three-point correlators generically involve bosonic correlators with descendant insertions. More precisely, and focusing on the SL(2,R) case, besides the usual primary correlators one needs (2.50) These correlators were computed in [33,53] directly in the x-basis from the OPEs between currents and primaries by means of the usual contour integration techniques. However, and as will be discussed below, this method is unavailable for vertex operators with arbitrary spectral flow charges since the corresponding OPEs contain many unknown terms [20] (except for the singly-flowed case [8,57]). We now derive (2.50) in an alternative way, best suited for the generalization to the spectrally flowed cases. Similar to what was done in [46] for primary correlators, the idea is to relate it with some m-basis correlator, and read out the corresponding structure constant. We first determine the z-and x-dependence the correlator. This is fixed by the action of the global currents. We know that V h is a primary field of spin h and conformal weight ∆ h = −h(h − 1)/n 5 . On the other hand, (jV h ) is a descendant that has well-defined spin h − 1 and weight ∆ h + 1. Indeed, using (2.32) and (2.30a) we have Hence, .
(2.52) The goal is to derive the relation between the structure constants C(h i ) and those of the primary three-point functions , (2.53) derived in [3,8] in terms of Barnes double Gamma functions. For this, we note that the identities in Eqs. (2.17) and (2.21) imply Using (2.52) on the first line of (2.54) it turns out that, up to the z-dependence, We are then interested in m-basis correlators of the form and reversing the contour, the OPEs in (2.30a) imply Consequently, given that (jV where we have used the relation Applying the same strategy we can re-derive the relevant m-basis three-point function [58] as The x-basis three-point function is given by where, again, we have ignored the z-dependence. Therefore Note that the delta function coincides with that of (2.56). Hence, we find In other words, which agrees with the results of [33]. This result can also be used to compute the fermionic correlator with an extraĵ(x) insertion. Setting h 1 = 0 (for the identity operator) and Analogous expressions hold for the SU(2) correlators.
We have now obtained all the necessary ingredients of the (unflowed) supersymmetric three-point function, which gives As it was discussed in [33,53], the shifts in the bosonic levels of SL(2,R) n 5 +2 and SU(2) n 5 −2 conspire precisely so that, for operators in the discrete representations with h i = l i + 1, the product of the corresponding three-point functions simplify considerably. Explicitly, and omitting the x and y dependence, Consequently, one gets where, as before, One can compute all other NSNS correlators involving V and W insertions analogously, see [33,53]. In order to identify the precise boundary duals one has to integrate over the worldsheet insertion points. It was shown in [8,33] that this procedure removes the divergence appearing in the two-point functions (2.46) and (2.47) coming from the δ(h 1 − h 2 ) factors, but it also generates a finite multiplicative factor 1 − 2h. This, together with the correct normalization for the spacetime operators, can be derived by making use of the spacetime Ward identities. We review this computation in Sec. 4.2.2 below, where we further provide the extension to the spectrally flowed sectors.
Finally, in order to compare the three-point worldsheet three-point functions with those of the holographic CFT we also need to include the usual factors associated with the string coupling g s and the volume of the T 4 . As discussed in [33,53], this leads to a precise matching with the correlators of the symmetric product orbifold CFT. At this point of the moduli space, the operators defined above correspond to families of chiral primaries of twist n = 2h − 1 and weights H = (n ± 1)/2, respectively.
However, due to Eq. (2.23), operators in the unflowed sector of the worldsheet theory only account for chiral primaries with twists n < n 5 . Except for the rather special cases where n ∈ n 5 N [59] (see also [60]), all primaries with higher twists live in the spectrally flowed sectors of the worldsheet theory [34], which now turn to.

Spectrally flowed states
We now discuss states belonging to the spectrally flowed representations in the NSNS sector of the worldsheet theory. These include both short strings, which encompass most of the spacetime chiral primaries, and also long strings which remain at finite energy when reaching the asymptotic boundary of AdS 3 , and for which the spectral flow charge is understood as a winding number.

Vertex operators for short strings
When considering the discrete representations it is useful to perform spectral flow not only in the SL(2,R) sector but also in SU(2) [34]. For the latter, the spectral flow maps different standard affine representations into each other, while for SL(2,R) it generates new inequivalent representations, for which the conformal weight is not bounded from below [6].
The relevant spectral flow isomorphisms act on the current modes as while the fermions transform as with ω ∈ Z. The currents j a , k a ,ĵ a andk a transform similarly with n 5 replaced by the corresponding levels. The Virasoro modes shift according to while for the (total) matter supercurrent we have Let us focus on the SL(2,R) sector. Flowed vertex operators in the m-basis are constructed upon the spectrally flowed primaries, whose bosonic part we denote by V ω hm . These are primary fields with respect to the flowed currents j a,ω . Hence, for ω > 0 they satisfy the following OPEs: and it is the lowest-weight state in a representation of the zero-mode algebra with spin Similarly, V −ω h,m corresponds to a highest-weight state with spin h ω = −m + kω/2. Note also that V ω h,m and V −ω h,−m (with ω > 0) have the same spin; they will contribute to the same x-basis operator.
Flowed fermionic fields conveniently expressed in terms of their bosonized form as These correspond to lowest-weight states in spin a − ω representations, respectively, with a = 0, +, −. Note that, for any ω > 0, the flowed fermions have either negative or zero spin, and belong to a finite representation of the global part of the symmetry algebra. Also, −,ω . Moreover, the fermionic identity is mapped to a field of spin −ω and conformal weight ω 2 /2 given by Combining the above expressions, we see that flowed primary states of the complete supersymmetric SL(2,R) n 5 algebra are of the form By including the analogous SU(2) factors, this leads to vertex operators given by where we have included the analogous construction based upon the unflowed states polarized in the SU(2) directions.
The corresponding x-basis operators are constructed analogously to those of the unflowed sector, that is, by constructing the appropriate linear combination of all fields in the associated global multiplet. These are obtained by acting freely with J ± 0 on the highest/lowest-weight states defined above. In the bosonic SL(2,R) sector, we have Note, however, that for ω = 0, the modes U ω h,m withm = ±(h + kω/2) have no simple m-basis expressions. As anticipated above, the x-basis field contains both the lowest-and highest-weights representations, i.e. both those with positive and negative spectral flow charges. It follows from Eqs. (3.5) that the zero-mode currents still act on the flowed xbasis operators through the differential operators D a x,h+kω/2 defined in (2.16). An analogous construction can be done for the fermions and the fermionic currents: The supersymmetric x-basis vertex operators come in four families, depending on the sign choice in Eq. (3.12) and also on the polarization. On the one hand, we have where we defined W l (y) and χ ω (y) analogously to their SL(2,R) counterparts, and used ψ +,ω = ψ −,ω−2 . Here V ω ±h (x) are the x-basis bosonic primary fields of spin h ω = ±h+kω/2, defined in such a way that the corresponding lowest-and highest-weight modes can be extracted of the vertex as follows: Similarly, we also have However, and as we discuss next, only two of these four families of vertex operators are actually independent.

Series Identifications
As is well known, for short strings the independence of the spectrally flowed representations holds only up to the identifications [8] V where the proportionality factor for the SL(2,R) case takes the form as can be derived from the two-point function, using that In other words, In the supersymmetric theory, this translates into [18] V (3.21) As it was shown recently in [11] for the bosonic SL(2,R) case, these identifications also hold in for x-basis operators involving the global descendants. This can be understood intuitively as follows. Vertex operators in the x-basis can be thought of as a translated version of the m-basis flowed primaries associated to their values at the origin x = 0, see Eqs. (3.16) [20]. Indeed, translations in the complex x-plane are generated by the action of A similar discussion holds for the SU(2) primaries and the respective fermions. As a consequence, we find that the supersymmetric vertex operators satisfy the following identities:

Two-point functions
With the definitions given in the previous section, it is possible to compute the two-point functions straightforwardly as in the unflowed sector. The bosonic two-point functions can be computed by taking the limits where h ω i = h i + kω i /2. The m-basis two-point function must conserve the total amount of spectral flow, ω 1 = ω 2 [8]. Additionally, and as can be shown by using the parafermionic decomposition, the m-basis correlators only depend on the total spectral flow charge, but not on the specific assignment of spectral flow charges. This is up to the shifts in the usual powers of z ij . Then with ∆ ω h defined as in Eq. (3.6). Similarly, this can be done for the other sectors and we have where we have used that W ω l (y 1 )W ω l (y 2 ) = y 2l+k ω 12 z 2l(l+1)/n 5 +2lω+k ω 2 /2 12 , (3.27) and (3.28) The correlators involving the vertex operators V ω −h and W ω −h can be derived by the series identification given in Eq. (3.22).

Picture Changing
As in the unflowed case, in order to compute string three-point we will need the picture-zero version of the spectrally flowed vertex operators [34]. We first discuss the flowed primaries and then consider the x-basis operators.
The SL(2,R) relevant contributions to the supercurrent G can be expressed in terms of the H I bosons as where H ω = h − 1 + n 5 ω/2 total spacetime weight. In the x-basis, this reads Here W ω h−1,−(h−2) (y) are the sum over the multiplet constructed from the flowed primary W ω h−1,−(h−2) . Note that, both in the ψ and χ sectors, the fermion number on each factor of A 1 differs in one unit with respect to those in A 2 . Hence, whenever A ω 1 leads to a nonzero contribution in a given correlator, that of A ω 2 will vanish, and vice-versa. Finally, an analogous discussion gives It follows from the above expressions that generic NS sector three-point functions will involve bosonic correlator of the form To the best of our knowledge, these have not been computed in the literature. One of the main results of this paper is to obtain closed-form expressions for all such m-basis correlators. We will also compute several families of their x-basis counterparts. These computations are performed in section 4.

Vertex operators for long strings
The long string sector of the model is built upon vertex operators belonging to the continuous representation of the SL(2,R) k algebra. The simplest vertex operator that can be constructed by this is the non-excited state given by This field does not satisfy the GSO projection. However, we will consider the flowed versions For odd ω this is a physical field allowed by the GSO projection. (A similar construction can be carried out for even spectral flow charges.) The Virasoro condition reads For states in the continuous representation, the SL(2,R) k projection m can take any real value. Therefore, the Virasoro condition can be satisfied for all values of h, l and ω. Note that, unlike the short-string states of the previous section, long strings do not require a relation between the SL(2,R) k and SU(2) k spin. The vertex operator given by Eq. (3.40) has spacetime weight H ω = m + n 5 ω/2 (ω > 0). The x-basis field is built as Note that we must keep track of the value of m even in the x-basis since the spacetime weight depends directly on it. As before, the x-basis field contains both positive and negative flow representations and satisfies that The picture zero version of the long strings operator is given by where we assume an implicit sum over a, b and α, β, and with g β ln = (l + 1 − n, n, l + 1 + n) , (3.47) where b = −, 0, +. Note that both terms in (3.45) have the same total fermion number ω + 1, but they differ in one unit with respect to fermion numbers in the SL(2,R) and SU (2) sectors. Consequently, similarly to the short strings picture zero operators of the previous section, both terms are mutually exclusive inside a correlator.

Correlators, currents and spectral flow
As discussed above, the computation of the three-point functions in the NSNS sector of the worldsheet theory involves bosonic correlation functions with descendant insertions. These are non-trivial in the presence of spectrally flowed vertex operators. Due to the complicated OPEs between the latter and the currents, these correlators cannot be obtained by the standard contour integration methods. In this section, which contains our main results, we present a strategy to compute some families of such correlators.

Bosonic correlators with current insertions and general m-basis results
Let us focus on the SL(2,R) case. Our goal is to compute In order to do so, we will follow a similar strategy to that used in the unflowed case, see Sec. 2.4. The x-and z-dependence of (4.1) is fixed by the global Ward identities. Indeed, the zero-mode currents act on the operator (j ω V ω h ) in terms of its worldsheet conformal dimension and spin, namelỹ where ∆ ω h and h ω were defined in Eqs. (3.6) and (3.7) with m = h, respectively, and we have used (3.34). Hence, we can write such that we only need to determine the structure constants C ω i (h i ).
As in the unflowed case, the goal is to compute these constants by relating the correlator in Eq. (4.3) with some m-basis three-point functions. Generically, we are interested in correlators of the form where, from now on, ω i is allowed to be positive or negative. These have not been computed in the literature. When the current involved is j 0 , one can proceed analogously to the unflowed case. By writing the normal-ordered operator j a,ω 1 V ω 1 h 1 ,m 1 as a Cauchy integral and inverting the contour, we obtain 5) showcasing the expected eigenvalues as shifted by spectral flow. On the other hand, for the currents j ± case one needs to take into account the pole structure appearing in the OPEs of Eqs. (3.5). V ω i h i ,m i and the currents j a,ω 1 , that are non-trivial whenever a = ± and ω i = ω 1 . In these cases, the normal-ordered products appearing in (4.5) are given by In order to compute (4.5) for a = ±, and for a given correlator, we define The powers of (z − z i ) ±ω i ensure that the modified currents J ±,ω i (z) behave near each flowed primary insertion analogously to unflowed currents near unflowed vertex operators. In particular, using the OPEs (3.5b) and (3.5c), we find that when the contour C encircles all three insertion points. To be precise, this holds only when there is no pole at infinity, but this happens for any configuration which satisfies the so-called m-basis spectral flow violation rules, i.e. whenever On the other hand, we have Note that we have picked up a contribution from the first two non-trivial terms in the corresponding OPEs (3.5) due to the extra (z − z 1 ) −1 factor in the integrand. Proceeding similarly with the other insertions, we find that (4.8) implies the following identity: leads to consider correlators of the form Here we have chosen to write the unflowed operator in its ghost picture-zero version, given in (2.49). Then we have to compute Given that both vertex V ω h 2 and V ω h 3 have the same SL(2,R) and SU(2) fermion number only the terms with even fermion number of V 0 h 1 will be non-zero. Hence, the computation of (4.17) involves that of the bosonic correlator Given that in the m-basis the primary structure constants only depend on the total amount of spectral flow, we get (4.25) The m-basis three-point function is derived from the x-basis ones by taking the limits (4.26) Comparing this with Eq. (4.22) we get where C H (h i ) is the unflowed three-point function. In other words, We also need the fermionic correlator ĵ (x 1 )ψ ω (x 2 )ψ ω (x 3 ) . Fortunately, this can be obtained using the same methods as in the bosonic case. Indeed, we have mentioned that ψ ω are flowed primary fields of spinĥ ω = −(1 + ω) in the SL(2, R) −2 WZW model. Hence, (4.12) implies where we have set h 1 = ∆ 1 = 0.
We have now obtained all necessary ingredients for computing the supersymmetric correlator (4.17). All the correlators that involve current insertions are expressed in terms of primary ones and therefore Using the worldsheet and conformal symmetry to trivialize the z-and x-dependence by fixing z 1 = x 1 = 0, z 2 = x 2 = 1 and z 3 = x 3 = ∞ as usual, we finally obtain where H ω i = h i − 1 + n 5 ω i /2 are the spacetime weights, D(h i ) is the SL(2,R) and SU(2) three-point function product given by the Eq. (2.68), namely This generalizes the results of [34] and [57]. Non-trivial current insertions also appear in similar NSNS supersymmetric three-point functions involving short strings. We compute them analogously, our results can be summarised as follows: Moreover, with the series identifications discussed in Sec. 3.2 all correlators of the form can be obtained from those of Eq. (4.33) by using the relations given in Eq. (3.21).

Normalization and matching with the symmetric orbifold CFT
Let us go back to the correlators we have obtained in (4.33). In this section, we describe the matching with the corresponding chiral primary three-point functions in the symmetric orbifold CFT. As in the unflowed case, for this, we first need to find the correct normalization for relating worldsheet operators with their holographic counterparts. More precisely, local operators of the CFT living on the AdS 3 boundary are given by vertex operators such as V(x, z) integrated over their worldsheet insertion point z. Indeed, note that the worldsheet two-point functions contain a divergent factor δ(h 1 − h 2 ). As discussed in [8,34], this divergence is cancelled by the z-integrations, i.e. by fixing the insertion points at 0 and ∞ and dividing by the remaining conformal volume. However, this cancellation produces an additional finite but non-trivial multiplicative factor, which depends on the h and ω.
This constant factor can be obtained by using the spacetime Ward identities associated with the R-symmetry currents. It was shown in [5] that the operators provide the worldsheet representation for the corresponding currents. Then a generic vertex operator of the form V ω h (x)Φ int , where Φ int stands for the internal, fermionic and ghost contributions must satisfy where q 3 = −q 2 denote the corresponding R-charges. Using the methods derived in this paper, we can evaluate both sides of Eq. (4.38) independently. Using (4.28) the LHS becomes −q 2 n 5 c νx 12x13 x 23 with h ω = h + kω/2. Moreover, using Eq. (4.32), we have Hence, we find that the string two-point function differs from the spacetime one by a factor This shows that the canonically normalized spacetime operators are given by . (4.43) The factor 2h ω − 1 − 2ω is consistent with the unflowed result obtained in [8,33,53].
Although the generalization to the flowed case was anticipated in [8,34], to the best of our knowledge, we have presented the first formal proof available in the literature.
Having fixed the normalization in Eq. (4.43), we obtain the following spacetime threepoint functions: where we call H ω i to the spacetime weight of each field, being H ω = h − 1 + n 5 ω/2 for V ω h and H = h + n 5 ω/2 for W ω h , and we identify where we have inserted the factor of g s appearing in the definition of the three-point function, and the volume of the T 4 (which should also be included in the above normalizations). Since g s = v 4 n 5 /n 1 and N = n 1 n 5 , this fixes the value of ν as in the unflowed sector [33] 4 . Our results then match exactly with the corresponding predictions from the symmetric orbifold CFT, see for instance [33,34].

Absence of spectral flow violation
According to the spectral flow selection rules for the three-point function, the most general non-conserving spectrally flowed correlators with only two winding states are together with similar ones with W (0) . Note that these correlators can not be identified, via the Eq. (3.21), with those derived in the previous section. In this sense, they allow us to test for genuine spectral flow violations in the supersymmetric model.
As an example, we will compute the three-point function while the rest can be treated similarly. The only possibly non-vanishing contribution is  The relevant fermionic three-point functions are written as x 2 13 , (4.50) χ(y 1 )χ ω−1 (y 2 )χ ω (y 3 ) = C χ y 2ω 23 y 2 13 . (4.51) By taking the appropriate limits on the x i and y i variables it is possible to identify these x and y basis correlators with the m basis ones to determine the structure constants C ψ and C χ , giving Then, inserting the operator (4.49) in a fermionic three-point function gives andĵ(x 1 ) in V The primary non-conserving spectral flow three-point function of (4.68) was derived in [8].
For the x-basis we have , (4.71) and we can obtain the m-basis function from Finally, combining all the above results we get so that the supersymmetric correlator reads Where H 1 = h 1 − 1, H 2 = h 2 − 1 and H 3 = m 3 + n 5 /2 are the spacestime weights of each field. Recall that for long strings the SU(2) and SL(2,R) spins l and h are not related and therefore there is no cancellation scheme for the product of the bosonic three-point functions involved in this result. This is in stark contrast with the short-string correlators considered above. Spectral flow, as a generator of new string states, is a distinctive aspect of the bosonic theory in AdS 3 , as is the potential non-conservation of the total spectral flow charge in a correlation function. Strangely, this non-conservation of spectral flow seems somewhat elusive in the context of the supersymmetric model, so much so that it was assumed that spectral flow non-conserving processes could even not take place, at least when discussing short strings. The results reported in this section show that the violation of the conservation of the total flow charge is possible in the supersymmetric context, although restricted to the dynamics of long strings.

Discussion
This paper focuses on three-point functions as computed from NSNS sector of the worldsheet description of type IIB superstrings propagating on AdS 3 ×S 3 ×T 4 . Although some of them were studied in [33,34,53], a large class of such correlators were not computed before. This is because the RNS formalism used to define the supersymmetric model conspires with the usual complications coming from the presence of spectrally flowed vertex operators, and introduces several technical complications. In particular, the picture-changing procedure needed for correlators with three or more insertions generates the appearance of extra current insertions in the bosonic SL(2,R) sector. In this context, the usual strategies for obtaining descendant correlators fail due to the highly non-trivial nature of OPEs between currents and spectrally flowed operators. These OPEs contain a set of higher-order poles whose precise form is mostly unknown [20].
We have presented an explicit method for overcoming these obstacles, based on the use of the generalized currents defined in Eq. (4.6). We first focused on the m-basis, obtaining closed-form results for all three point functions involving SL(2,R) spectrally flowed fields with arbitrary spectral flow charges, together with the relevant current insertions. The resulting expression, given in Eq. (4.11), takes the form of a linear combination of primary correlators. This allows one to compute all m-basis three-point functions of the supersymmetric model.
We then turned to x-basis three-point functions, best suited for the comparison with the dual holographic CFT. Following the techniques of [8] and [46], we have been able to use our m-basis results to obtain all supersymmetric three-point functions where at least one of the vertex operators is unflowed. For short strings, we considered spacetime chiral primary operators. Using the language of [8,46], we showed that all the so-called spectral flow violating correlators vanish in the supersymmetric theory, while all the spectral flow conserving ones match exactly with the predictions of the holographic CFT at the orbifold point. In doing so, we derived the normalization appropriate for short string operators in the flowed sectors of the theory, which, to the best of our knowledge had only been discussed heuristically so far. This significantly extends the results of [33,34].
Furthermore, we have also considered interactions involving long strings, which do allow for spectral flow violation. More precisely, we have obtained the first exact non-vanishing supersymmetric amplitude for a process that can be interpreted as the absorption/emission of a long string by the background geometry, which then increases/decreases its fundamental string charge by one unit [61].
We leave for the near future the computation of the remaining supersymmetric threepoint functions in the x-basis with arbitrary spectral flow charges, and the corresponding matching with the orbifold CFT predictions. For this, it will be necessary to extend the methods recently developed in [2,9] (see also [11]) to the relevant descendant correlators. Moreover, it would also be interesting to study non-protected three-point functions [62] and to compare them with those of the proposed holographically dual theory of [1]. Finally, it would also be important to consider four-point functions in the supersymmetric context, where complications similar to those we have tackled in this paper appear as well [10,63].