Engineering Perturbative String Duals for Symmetric Product Orbifold CFTs

Constructing a holographic string theory dual for a CFT in the perturbative, weakly coupled regime is a holy grail for gauge/string dualities that would not only open the door for proofs of the AdS/CFT correspondence but could also provide novel examples of string duals with and without supersymmetry. In this work we consider some marginal perturbation of a family of symmetric product orbifolds in two dimensions. From their correlation functions we engineer a bosonic string theory whose amplitudes are shown to reproduce the CFT correlation function order-by-order both in the coupling and in $1/N$. Our derivation does not require to compute and compare correlation functions explicitly but rather relies on a sequence of identities that can be derived using path integral methods. The bosonic string theory we engineer is based on the field content of the Kac-Wakimoto representation of strings in $AdS_3$ with $k$ units of pure NSNS flux, but the interaction terms we obtain are different. They include current algebra preserving interaction terms with one unit of spectral flow.


Introduction
About 25 years after Maldacena proposed the first concrete example of a 't Hooft-like gauge-string duality [1], the AdS/CFT correspondence has become the key source for novel insight into nonperturbative Quantum Field Theory (QFT) on the one hand and into quantum gravity on the other.
In the early years, supergravity constructions were exploited to assemble an extensive zoo of such dualities in various dimensions and with various amounts of supersymmetry.Note that the existence of some geometric supergravity regime on the string theory side was build into the search strategy from the start.This was not seen as a significant drawback but rather became the most celebrated feature of the AdS/CFT correspondence: It allowed to compute quantities in strongly coupled field theories through the geometry of the dual supergravity.On the other hand, strong/weak coupling dualities are of course notoriously difficult to verify.
In the decade after the first examples of the AdS/CFT correspondence had been proposed, testing its implications became a major focus of the field.This started with protected quantities that could be calculated exactly with the help of supersymmetry and were shown to interpolate successfully between the supergravity regime and the weakly coupled field theory.The famous circular Wilson loop [2] in N = 4 Superymmetric Yang-Mills (SYM) theory provides a prototypical example.With the rise of integrability based techniques that was initiated by Minahan and Zarembo [3], precision tests of unprotected quantities also became available.The most prominent example of this kind was uncovered in the context of the cusp anomalous dimension of N = 4 SYM theory.In a seminal paper Beisert-Eden-Staudacher [4] proposed a remarkably simple integral equation that could be used to compute the cusp anomalous dimension for any value of the coupling.In particular, the so-called BES equation gives access to both weak and strong coupling expansions and the latter were checked to very high precision against gauge theory calculations on the one hand and string calculations on the other.The form of the BES equation resembles the integral equations that are familiar from the Thermodynamic Bethe Ansatz (TBA) in 1+1-dimensional integrable quantum field theory.While the appearance of such an equation seems surprising from the gauge theory perspective, it is rather natural from the dual side.After all, the Gubser-Klebanov-Polyakov string [5] that captures the deformation of the cusp anomalous dimension away from the supergravity regime is a 1+1-dimensional quantum system.
For a gauge theory practitioner, on the other hand, the BES equation remains truly remarkable since it miraculously manages to repackage gauge theory calculations in such a way that even perturbative precision calculations can be performed on a sheet of paper or on a laptop.
Moving on from extensive tests to genuine proofs of the AdS/CFT correspondence has been a holy grail of the field ever since Maldacena proposed the first example.While this still seems out of reach for higher dimensional theories, in spite of very advanced and powerful techniques, there has been some progress in lower dimensions recently.The simplest non-trivial examples of the AdS/CFT correspondence appear for 1-dimensional systems of quantum mechanics.In this case the best studied examples involve Jackiw-Teitelboom (JT) gravity or deformations thereof on the dual side.In particular.the holographic relation between a new critical string theory, dubbed the Virasoro minimal string, and certain double scales matrix integrals was recently proven in [6].This includes the famous duality between JT gavity and matrix integrals of Saad et.al. [7] in a certain limit.Moving up in dimensions to 2-dimensional conformal field theories a few holographic dualities are on equally solid ground.In particular, it was shown in [8][9][10] that the worldsheet of superstring on AdS 3 × S 3 × T 4   with the minimal unit of NSNS-flux localizes to particular Riemann surfaces, which essentially derives the AdS 3 /CFT 2 duality between the minimal tension superstrings and the orbifold point of symmetric product orbifold T 4 /S N .
One would certainly hope that proofs of the AdS/CFT correspondence can eventually lead to derivations which engineer new examples of dual string theories directly from the perturbative formulation of the gauge theory.As we have reviewed above, the BES equation remains valid all the way to vanishing 't Hooft coupling in the gauge theory.In this sense, even the free gauge theory admits a stringy description.While the historical path has lead us to think of the gauge theory in terms of strings on AdS 5 × S 5 , it seems likely that there exists another dual description of the string theory that captures the perturbative expansion around the point where the curvature radius R of AdS 5 assumes a minimal value.If such a string dual indeed exists, it would provide a string theoretic dual of type IIB theory on AdS 5 × S 5 , somewhat akin to the string theoretic dual one can construct at the Gepner points of Calabi-Yau compactifications, see [11] and references therein.This nurtures hopes that one day we might actually be able to turn the perturbative formulation of gauge theory into the worldsheet formulation of some (non-geometric) string theory and then use the technology of 1+1-dimensional (integrable) models to perform gauge theory calculations with the astounding efficiency of BES-like equation.We refer to this kind of repackaging of gauge theory degrees of freedom as engineering of a dual string theory.Note that such an engineering process could extend the usual AdS/CFT correspondence since it does require the existence of a large volume regime in which the string theory becomes geometric.
While this may still seem like a far fetched vision for higher dimensional gauge theories, it now seems to come within reach, at least in lower dimensions.Indeed, what we will lay out below may be considered as the first concrete example of engineering a perturbative string dual in the sense we described.The context of our discussion is an interacting 2-dimensional CFT that can be obtained from special symmetric product orbifolds with parent theory M by switching on a particular marginal interaction.The duality of this theory with strings on some AdS 3 compactification with pure NSNS flux was first proposed and tested in a very remarkable paper by Eberhardt [12], see also [13][14][15] for some related work.Here we shall take this duality significantly further by rewriting the complete set of   orbifold correlators at any order in the coupling constant and in 1/N as a bosonic string theory.After some significant massaging of the orbifold correlators using techniques we had developed in [16,17], we will be able to read off the field content and the interactions of the dual string background without ever computing a single correlator explicitly.
In order to describe our main constructions and results in more detail, we will begin with a brief review of the relevant symmetric product orbifolds.Regardless of the specific parent theory M one uses, symmetric product orbifolds possess a diagrammatic representation that resembles the diagrammatic representation of gauge theories.In both cases, propagators are represented as double lines.But unlike ordinary gauge theories, the two lines representing propagators in symmetric product orbifolds need to be distinguished.Following [18] we will denote one by a solid and the other by a dashed line.The analogue of single trace operators in gauge theory are fields in single cycle twist fields of the symmetric product orbifold.The length w of the cycle controls the length of the operator, i.e. the number of Wick contractions it is involved in.We represent such operators as shown in Figure 1a.through a circle with 2w emanating propagator lines.The Feynman graphs that can be drawn by connecting the double lines at the vertices with double line propagators must satisfy two conditions that were first described in [18].The precise formulation of these constraints is not relevant for our discussion.One example of a diagram that represents one of the contributions to the 4-point function of twist fields with w ν = 2, ν = 1, . . ., 4 is depicted in Figure 1b.As one can easily see, it can be drawn on a surface of genus g = 1.More generally, the genus g of the surface is related to the length parameters w ν , ν = 1, . . ., n of n fields and the number R of closed solid loops through 2 − 2g = 2R − ν w ν + n . (1.1) Eberhardt duality proposal involves a particular marginal operator in the w = 2 twisted sector of the symmetric orbifold theory [12], see also [14].A concrete formula will be given below.The pictorial representation of this operator is given by a 4-point vertex, see Figure 2. In the perturbed symmetric orbifold theory, this operators is introduced with a parameter µ that controls the strength of the coupling.
There is a well explored way to encode various diagrams that contribute to a correlation function in the symmetric product orbifold through covering maps.These covering maps Γ : Σ g → S 2 describe an R-fold branched covering of the sphere on which we evaluate the orbifold correlators by a surface of genus g.The branching numbers of the branched cover are determined by the cycle lengths of the fields in the correlator.
We shall propose some precise (non-geometric) bosonic string theory and show that it is indeed dual to perturbed symmetric product orbifold.The string theory reproduced the CFT correlators we described in the previous two paragraphs order-by-order in the coupling constant µ.The underlying worldsheet CFT with c = 26 is a product of some unitary CFT X with some appropriate Wess-Zumino-Novikov-Witten (WZNW) model.The field content of the latter is given by a bosonic βγ system along with a linear dilaton ϕ with background charge To this free field theory we add two marginal interaction terms.The first one is given by where v (−1) denotes some (spectrally flowed) vacuum state of the βγ system, see below.The interaction term S − is a very close relative of the "screening charge of second kind" that was found by Bershadski and Ooguri in [19].Insertions of the interaction vertex (1.3) are needed in order to reproduce the free orbifold theory.In fact, the number of insertions coincides with the number of sheets in the branched covering of S 2 . 1 In order to also reproduce the interaction terms from the marginal perturbation of the orbifold theory one has to introduce a second interaction term in the worldsheet string theory.The latter is given by This interaction term seems new and has very unusual properties.In particular, it produces a trivial contribution when it is inserted in maximally winding number violating correlation functions as long as it stays away from other insertion points, It turns out, however, that non-trivial contributions of the interaction term (1.4) can arise.They stem from regions in the moduli space of field insertions in which pairs of these vertex operators collide.Once such a pair is formed it is able to reproduce the insertion of the interaction vertex in the dual symmetric product orbifold.Let us stress that the theory (1.2) along with the two interaction terms (1.3) and (1.4) differs from the sigma model on AdS 3 , even though the latter can be expressed through the same field content.The relation between our worldsheet theory and the usual worldsheet model for strings in AdS 3 is rather reminiscent of the relation between Liouville field theory and its nonperturbative dual that is obtained by sending b to 1/b.This is of course not enirely unexpected.
Our construction of this worldsheet model, including the two screening charges we have displayed above is entirely systematic.Its derivation rests on two pillars.The first pillar is a novel embedding of the symmetric product orbifolds into the a free worlsheet theory with Kac-Wakimoto field content.This embedding is realized concretely in eqs.(4.4) and (6.19) for tree level and higher genus contributions, respectively.These two formulas represent the central new results of this work. 2The relation are quite reminiscent of the relation between Liouville field theory and the H + 3 WZNW model [23].In our setup, the role of Liouville field theory is played by the perturbed symmetric product orbifold whose interaction is indeed of Liouville form, though dressed with some twist field.And indeed, the tools we use in order to proof our formulas are mostly borrowed from our path integral derivation of the Liouville-H + 3 duality, see [16], with some extensions from [17] and some new ingredients.At this point we have just rewritten correlations functions of twist fields in the symmetric product orbifolds through correlation functions of vertex operators with non-trivial spectral flow in the Kac-Wakimoto free field theory.But in comparion to the conventional rewriting in terms of higher genus correlators of the parent theory, see above, all the complicated prefactors of the latter get completely absorbed through our way of rewriting the correlations functions in terms of Kac-Wakimoto vertex operators.While conceptually, our embedding just provides an equality between sets of correlators in two different CFTs, the remarkable absorption of prefactors is decisive for our ability to uplift the the correlation functions to string theory.This uplift exploits the second important pillar of the construction, namely the observation that the string theory contains a field γ(z) which localizes to a covering map Γ : Σ g → S 2 in computations of string amplitudes.The map Γ turns out to coincide with the covering map that perturbative expansion of the symmetric product orbifold, see above.
The localization of the field γ and its relation with the branching functions Γ go back to works of Eberhardt, Gaberdiel and Gopakumar, see e.g.[8] and references therein.
Let us finish this introduction with a brief outline.In the next section we will give a rather selfcontained introduction into those symmetric product orbifolds that participate Eberhardt's duality.
In particular we shall state a precise formula that expresses correlation functions of the orbifold theory in terms of correlators of the parent theory on branched coverings of the sphere.A particular focus is on the properties of the associated covering maps Γ. Section 3 contains some background material on SL(2, R) WZNW models and their free feld realization.We will also review the importance of spectrally flowed sectors and construct the relevant vertex operators.At the end of section 3 we have thereby collected all the material that is needed to establish a new realization of orbifold correlators in terms of correlation functions of the WZNW model.Our important new formula is stated and proved in section 4, using techniques from earlier work on the H + 3 -Liouville relation [23] and the closely related Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality [24], see [16,17].This result is then applied in section 5 to argue that the string amplitudes of the bosonic string theory described through eqs.(1.2), (1.3) and (1.4) indeed reproduce the correlation functions of the orbifold theory.The arguments are based on the localization mechanism for the field γ that we briefly sketched in the previous paragraph.
In section 6 we extend all the above beyond the leading terms of the planar theory by considering coverings Γ associated with Riemann surfaces Σ g of genus g > 0, see also [9].This work concludes with an extensive list of interesting extensions as well as applications.

Symmetric Product Orbifolds and Their Correlations
The starting point of this work is provided by a certain class of symmetric product orbifolds.Here we shall provide a short self-contained introduction on their field content and correlation functions.With a focus on the particular parent theory that appears in the context of holography we will construct and discuss the fields that are dual to one-particle states on the string theory side of the correspondence.These include one particular marginal operator that we will use to deform the symmetric product orbifolds.Correlation functions in symmetric product orbifolds can be related to correlators of the parent theory through so-called covering maps.This will be reviewed in the second subsection.

Symmetric product orbifolds and their perturbation
Symmetric product orbifolds of the form M N /S N can be associated with any 2-dimensional conformal field theory M. We refer to the latter as the parent theory an denote its central charge by c M .The central charge of the associated symmetric product orbifold is given by c = N c M .
Even though much of what we shall review below is largely independent of the particular choice of the parent CFT M we shall introduce the relevant one right away.Following [25] we consider products M = X ⊗ R φ of some unitary conformal field theory X and a (free) linear dilaton φ.This means that the state space of the parent theory splits as Here, the first factor H X is spanned by the states ψ of the CFT X .We shall denote the left-and right-moving conformal weights of the states ψ by h ψ and hψ , respectively.The central charge c X of the CFT X is supposed to be of the form with some parameter k.In order to motivate this parametrization we anticipate that the worldsheet theory of the dual string background will turn out to be a product of the CFT X and an SL(2, R) WZNW model at level k.The latter has central charge c k = 3k/(k−2).Hence, eq.(2.2) will eventually ensure that the worldsheet CFT gives rise to a consistent critical bosonic string theory.
The second factor in the product on the right hand side of equation (2.1) refers to the state space of a linear dilaton CFT, i.e. the free conformal field theory for a single bosonic field φ with background charge Note that the background charge of the linear dilaton is tied to the parameter k that we introduced in eq.(2.2).In our conventions, the choice of background charge implies that the associated central charge of the linear dilaton theory is given by Following standard conventions, the vertex operators of the linear dilaton CFT will be denoted by V α and their left-and right-moving weights by h α , hα , with To be quite precise, the usual normal ordering prescriptions must be applied in order for these vertex operators to be well defined in the quantum theory.Such a prescription is simply assumed throughout the entire text.
Putting the two factors together, we obtain the parent CFT M with combined central charge given by and states/fields of the form and a similar expression for hα (ψ) with h ψ replaced by hψ .This concludes our brief description of the relevant parent CFT M and we can now turn to the associated symmetric product orbifold.
In order to construct the symmetric product orbifold Sym N (M) = M N /S N we prepare N of identical copies of the parent CFT M and then perform an orbifold construction with respect to the action of the symmetric group S N that permuted the N identical copies.According to the standard rules, the resulting orbifold conformal field theory has twisted sectors that are labeled by the conjugacy classes of the symmetric group.Let ω ∈ S N be any permutation of N objects.We shall denote by [ω] the associated conjugacy class, i.e. the set of elements that are related to ω by conjugation with any other element, by [ω].As is well known, any permutation can be decomposed into a product of single cycle permutations.In the dual string theory, twisted sector states in a single cycle permutation are represented by single particle states.Therefore, our discussion will focus on the twisted sectors that are associated with conjugacy classes of single cycle permutations.There is one such conjugacy class [w] for each cycle length w ≤ N .We shall denote the ground states of the associated twisted sectors by O w = O [w] (p).According to standard results form the theory of orbifolds, these ground states have conformal weight By acting with the fields (2.5) from the parent theory M we can obtain the following set of twisted sector states/fields for all w ≤ N, α ∈ Q φ /2 + iR and ψ ∈ H X .Here we assumed the twist fields O w to be canonically normalized and included a w-dependent prefactor by hand that will turn out to be convenient later on.The conformal weights of these fields are given by and a similar expression for the weight h(w) α (ψ).Let us stress that the first term is obtained from the conformal weights in the parent theory, see eq. (2.5), by division with the cycle length w.This reflects the fact that in the twisted sector the boundary conditions are changed so that the field only comes back to itself after circling w times around the twist field O w .
We are now finally prepared to identify the marginal operator we want to deform the symmetric orbifold theory by, see [12], and to define the set of correlation functions we will analyse throughout the remainder of this work.Given the formulas we displayed in the previous paragraph it is easy to verify that the following operator is indeed marginal.Let us stress that we have chosen the state ψ of the CFT X to be given by the vacuum state ψ = |0⟩.With the parameter α = 1/2b and the cycle length w = 2, our formula (2.8) indeed evaluates to h 1/2b = 1.In the process of this calculation one needs to insert the expression (2.3) for the screening charge of the linear dilaton theory.The operator (2.9) resembles the interaction term of Liouville field theory.Indeed, it involves a single exponential of the linear dilaton field φ which is now dressed with the twist field O 2 .As in Liouville field theory, one can expect the resulting model to be conformal.
Up to now we have avoided to specify the insertion point of our fields.Throughout this work we will deal with correlation functions in which the fields of the symmetric product orbifold are inserted on a sphere.Insertion points in the sphere will be denoted by the letters x, y, to distinguish them for insertion points on higher genus surfaces that will enter our discussion soon.The subject of our interest are n-point functions of fields in the perturbed symmetric product orbifold on the sphere,3 where ψ ν ∈ H X and the subscript 0 indicates that the correlation function in the integrand on the right hand side is evaluated by the undeformed symmetric product orbifold.We note that for correlation functions in the linear dilaton theory to be non-zero, the fields must satisfy the charge condition This is entirely analogous to Liouville field theory, where correlation functions depend meromorphically on the parameters α ν and develop poles whenever these parameters satisfy charge conditions.It is the residues of these poles that can be computed in perturbation theory.The parameter λ we have written into the expansion is more of a bookkeeping parameter than a true coupling since, as in Liouville field theory, it can be absorbed in a redefinition of the zero mode of φ.

Correlations functions and covering maps
It has been appreciated for a long time that correlation functions of twist fields in orbifold theories on a sphere can be evaluated in terms of correlation functions of the parent theory on some appropriate branches cover.We will first review the general construction in the case of symmetric product orbifolds (see, e.g.[18,26,27]) before discussing some specific aspects that concern the specific correlations functions (2.10) that arise in the context of our perturbative deformation.
So, let us consider some set of fields V for the moment that is irrelevant.As stated in the previous section, all these fields are inserted on a Riemann sphere, i.e. a surface of genus g = 0.It turns out that the correlation function of these fields can be evaluated by considering correlation functions of the associated fields V α A (ψ A ; z A ) in the parent theory.The latter, however, must now be inserted at the branching points of an R-sheeted branched covering over the sphere.The order of the m branch points is given by the integers w A that specify the relevant twist fields in the orbifold theory.According to the Riemann-Hurwitz formula, the number R of sheets, the ramification numbers w A and the genus g of the covering surface Σ g are constrained by the equation4 For the time being we want to focus on the contributions from covering surfaces of genus g = 0, leaving the extension to coverings of higher genus to section 6.
The relevant R-sheeted branched coverings of the Riemann sphere by a surface Σ 0 of genus g = 0 can be nicely encoded in a meromorphic covering map Γ : Σ 0 → S 2 with the following behavior near the branch points at z = z A of the surface Σ 0 .The constant term x A in the expansion is determined by the insertion point of the twisted sector field in the orbifold correlator.
The form (2.13) of the local expansions near the branch points imposes very strong constraints on Γ which eliminate any continuous parameters.This is best analysed by looking at the derivative ∂Γ.The expansions (2.13) obviously imply that ∂Γ has a zero of order w A − 1 at z = z A .Hence, the total order of zeroes is given by 2R (0) − 2. Since ∂Γ is meromorphic of weight one, we conclude that it must have R = R (0) second order poles, i.e. it must be of the form This expression contains a total number of m + R + 1 parameters, namely the positions z A of zeroes, the positions v j of poles and the constant prefactor q.Upon passing from ∂Γ to Γ we gain one more integration constant.Now let us look at the constraints.The total residue of ∂Γ vanishes by construction, but since ∂Γ is a total derivative, all the local residues at v j must vanish.This imposes R − 1 extra conditions.After integration, we also need to impose the relations Γ(z A ) = x A which are clearly not built into eq.(2.14).This gives m conditions and leaves us with just 3 parameters.The latter are of course associated with the global conformal symmetry of the orbifold correlation function we consider.These may be fixed by requiring that z 1 = 0, z 2 = 1 and z 3 = ∞.From now on we shall assume this choice.We have now shown what we anticipated, namely that the covering map Γ is fully determined by the insertion points x A , up to some remaining discrete choices that are associated with different Feynman graphs.This also means that the positions z A of the branch points, the positions v j of the poles, the prefactor q as well as the coefficients a Γ A in eq.(2.13) should be considered as functions of the insertion points x B , i.e.
In some simple cases the dependence can be calculated explicitly, but in general this poses a very challenging problem one cannot hope to solve, especially for larger number of insertion points in the orbifold theory.
Given the notion of the covering map it is now possible to give a precise formula for the relation between the correlation function of the symmetric product orbifold on the sphere and the correlation function of the parent theory on the covering surface Σ 0 .With a proper regularization of divergences, see e.g.[22,26], the final result is (2.16) Here the sum runs over the discrete set of covering maps.For the leading planar contributions, these covering maps satisfy eq.(2.13).The exponents in the summands contain the weights that were defined previously in eq.(2.8) and we have introduced functions . (2.17) Note that the positions of zeroes z A and poles v j of ∂Γ are functions of the insertion points x B on the left hand side, as we explained above.We also note that the left hand side is the leading contribution to the correlator on the right hand side in the 1/N expansion.Correction arise from branched coverings of higher genus.
We conclude this subsection with a few specific comments on the integrated correlation functions (2.10) we are going to analyse below.To apply the formulas (2.16) and (2.17), we set m = n + s and and set w A = 2, α A = 1/2b, ψ A = |0⟩ and x A = y A−n for all A > n.By plugging eq. ( 2.16) into eq.
(2.10) we obtain where we introduces bΓ a = ãΓ n+a for a = 1, . . ., s.Let us note that the number R of poles of the relevant covering map Γ does depend on the number s of insertion of the marginal operator.Indeed, according to formula (2.12) we find Of course, the number of poles must be an integer.As was discussed in some detail also in [12], this condition is only satisfied for s even in case the quantity R 0 is integer. 5he expression we displayed still contains an integration over the insertion points y a of the perturbing operator of the orbifold model.The integrand, on the other hand, does no longer depend on the y a explicitly, but only implicitly through the locations z ν and u a of the branch points on Σ 0 as well as through the functions ãΓ ν , bΓ a and ξ j .It is certainly natural to change the integration variables from the parameters y a on the original sphere S 2 to the parameters u a on the covering surface Σ 0 , i.e. to the insertion points of the fields V 1/2b on the right hand side.After this change of variables we have Here we used that the Jacobian for the change of variables is such that We stress that this formula looks simpler than it is.In particular we stress that all the branch points z ν were functions of the y a and hence are now functions of u a , i.e. while we integrate over the insertion points of V 1/2b , the other vertex operators are also moved around.In addition, the various prefactors contain a complicated dependence on the data of the orbifold correlator.For later use, let us spell these factors out explicitly, see eq. (2.17), (2.20)While we have now indeed rewritten original correlators of the perturbed orbifold theory in terms of integrated correlation functions of the parent theory, the relation is very hard to use for explicit calculations, especially for higher number of field insertions or higher orders of perturbation theory, simply because explicit formulas for the covering maps are very hard to come by.In this sense, the calculation of orbifold correlations remains a rather difficult problem.Fortunately, our rewriting in terms of a dual string theory will not require such an explicit solution.
3 Some Background on SL(2, R) WZNW Models Studies of the WZNW model with non-compact target group SL(2, R) have a long history, see in particular [28][29][30] and references to earlier papers therein.In the so-called Kac-Wakimoto representation their field content is given by a linear dilaton field ϕ and a bosonic βγ system.Here we shall briefly review how these give rise to representations of affine Kac-Moody algebras, including those that are obtained by action of the spectral flow automorphism.Such representations are known from [28][29][30] to play a crucial role in holography.We will also construct vertex operators, including those carrying non-vanishing spectral flow index, in terms of the Kac-Wakimoto fields.The second subsection contains a technical result that collects all spectral flow into a single insertion point by exploiting a relationship with non-compact parafermions.This will play an important role in our subsequent analysis of orbifold correlators, see section 4.
3.1 Free field realization of the SL(2, R) WZNW model SL(2, R) WZNW models are known to possess a very useful first order formulation in terms of a scalar field ϕ with background charge Q ϕ and a bosonic βγ system of central charge c βγ = 2.The action of this system is given by where the first non-interacting terms reads (see eq. (1.2)) Here the curvature R is computed from the worldsheet metric ds 2 = |ρ(z)| 2 dzdz as We choose the background charge Q ϕ of the linear dilaton conformal field theory to be given by where b is the same parameter we used to parametrize the background charge Q φ of the linear dilaton φ of the parent theory M in the previous section.Note, however, that the two fields have different background charge.The value we choose for Q ϕ here ensures that the central charge c k of our model (3.2) is given by Hence, once we add the conformal field theory X , the total central charge assumes the critical value that is necessary for the associated bosonic string theory to be consistent.
For the time being we are entirely agnostic about the form of the interaction terms except that we require it to be marginal and to commute with the current algebra symmetry of the free theory.
The latter is famously given by (

3.3)
There are two well known examples of such interaction terms, but as we shall see later, the duality with the orbifold model makes a very particular and rather unconventional choice.
Spectral flow and winding number.According to [28] the so-called spectral flow automorphism of the sl(2) current algebra plays a key role in holography.The spectral flow automorphism is easy to spell out.On the modes of the three currents it acts according to The automorphism can be used to construct new spectrally flowed representations of the current algebra from highest weight representations by concatenation.These contain spectrally flowed vacuum states |w⟩ satisfying for n ≥ 0 and a = 3, ±.The spectral flow automprphim actually descends from an automorphism of the βγ system and the free scalar field ϕ.By definition, it acts as where a ϕ n denote the Fourier modes of the field b −1 ∂ϕ = a ϕ n z −n−1 .It is easy to verify that this action induces the action (3.4) on the currents (3.3).In terms of the free fields, we can now characterize the spectrally flowed vacua |w⟩ as where the ghost contribution satisfies for n ≥ 0 and the second factor |w⟩ ϕ is given by In the following we shall frequently pass between states and fields of our WZNW model.In order to do so, we shall often denote the field ψ(z, z) := [ψ](z, z) by placing the associated state ψ in rectangular brackets [•].For the field that is associated to the spectrally flowed vacuum |w⟩ we use the following shorthand, The conformal weight of these spectrally flowed vacua is given by This concludes our brief review of spectral flow and winding number in the Kac-Wakimoto free field representation.
Free field realization of vertex operators.The standard vertex operators of the SL(2, R) WZNW model in the so-called m-basis are given by Here j is a complex parameter and m, m are real.In the definition of Φ we have included a nontrivial normalization This will be convenient later on.As was first explained in [28] in the context of the AdS/CFT correspondence it is not sufficient to consider these standard operators.Instead, Maldacena and Ooguri argued that the spectrum of string theory on AdS 3 with pure NSNS flux includes additional states that are obtained by applying the action of the spectral flow automorphisms.In other words, they suggested to study the following larger class of vertex operators that is labeled by an additional spectral flow index w ∈ Z, It is indeed obtained by acting with the spectral flow automorphism ρ w on the standard vertex operators.The latter maps the usual vacuum |0⟩ of the theory to the spectrally flowed vacua |w⟩ and replaces the zero modes γ 0 , γ0 by their spectral flow image γ −w , γ−w .In addition, it generates the additional term −w/2b 2 in the exponent.
For our analysis below it will be crucial to switch to the µ-basis.In the free field representations, the latter are given by Φ j (µ|z) = |µ| 2j e µγ(z)−μγ(z) e 2bjϕ(z,z) . (3.12) From these we can recover the vertex operators (3.10) in m-basis through the following simple integral transform We can extend this relation to operators with non-trivial spectral flow index w as The operator on the left hand side of this equality was defined independently through equation (3.11).

Correlators and parafermionic representation
Our discussion below will involve various correlation functions of both integrated and unintegrated vertex operators in the free Kac-Wakimoto field theory.The standard correlation function of our vertex operators (3.11) possesses a path integral representation of the form Here we have placed a superscript Σ on the expectation value to refer to the surface on which the local operators are inserted.Alternatively, we can write these correlation functions in terms of vertex operators in the µ-basis as For the time we shall restrict to surfaces of genus g = 0, leaving the discussion of g > 0 to section 6.
We want to introduce a second set of correlation functions that are obtained from the standard ones through the insertion of a δ-function constraint in the µ-integral on the right hand side of the previous equation where, for a surface Σ = Σ 0 of genus g = 0, the variable û is given by û = J µ J z J .Here Ω is some complex parameter that we chose when we wrote the argument of the δ-function.Since the restricted correlation functions (3.17) depend on Ω, we have displayed this parameter as a subscript on the right hand side.But the dependence is rather mild.In fact, as one can easily see, upon rescaling of Ω we have Manipulating (restricted) correlation functions of vertex operators with non-vanishing winding numbers w is assisted by the use of the parafermionic representation.In particular it allows to derive some very useful identities that we shall review before we conclude this section.
Parafermionic representation of vertex operators.Following e.g.[13,22]  The free boson that comes with the factor U( 1) is denoted by χ(z, z) = χ(z) + χ(z).This field is normalized such that and it gives rise to the u(1) current With the parafermionic vertex operators and the free boson, the vertex operators (3.11) of the spectrally flowed states can be written as What makes this parafermionic representation of the SL(2, R) WZNW model so useful is the fact that the parafermionic primary fields Ψ j m, m do not depend on the spectral flow number w.Instead, all the dependence on w is in the U(1) vertex operator, i.e. in a free field theory where it is rather easy to deal with.
Concentrating spectral flow.The parafermionic representation of the WZNW model may be employed to show that within correlation functions all spectral flow can be concentrated in a single insertion point ξ on the worldsheet, where the winding number w of the vertex operator (3.8) we place at the new insertion point ξ is given by The relation we displayed here is certainly also valid for the standard correlation function (3.15).The version we stated here, however, used the restricted expectation value we defined in eq.(3.17).The prefactor, finally, is given by Strictly speaking, the expectation values should be computed without integration over the zero mode χ 0 to avoid U(1) charge conservation.But since the total charge within the correlators in the numerator and the denominator is the same, it is sufficient to take the ratios of the coefficients in front of the δ functions.These factors drop out form the quotient anyway.As a consequence, the functions Θ is the same for standard and restricted correlators, and it does not depend on the parameter Ω.

Embedding Orbifold Correlators into the WZNW Model
In this section we derive a remarkable formula that expresses the orbifold correlation functions we introduced in section 2 in terms of correlators of the WZNW theory we reviewed in the previous section.We will write down and comment on the relevant embedding formula in the first subsection before providing a proof in the second.

The embedding formula
Before we can state the result, we need to combine the Kac-Wakimoto free field theory with the CFT X .The product CFT will be denotes by W. Its state space is a product of the state space for the theory X with the state spaces of the βγ ghost system and linear dilaton ϕ, By construction, this state space carries an action of the Virasoro algebra with central charge c W = 26.
Let us introduce the following vertex operators for all primary fields ψ of the CFT X .These states satisfy the physical state condition of bosonic string theory provided that and similarly for m, but with h ψ replaced by hψ .Since the physical state condition determines m and m in terms of the other parameters j, w and h ψ we shall often omit the subscripts m, m when dealing with fields that satisfy the physical state condition.
Our main goal in this section is to merely rewrite the orbifold correlation functions we studied in section 2 in terms of correlation functions of vertex operators in the WZNW model.More precisely we shall prove that for any of the discrete choices of the covering map Γ the following remarkable formula holds q with the following relation between the parameters α ν and j ν , Note that we have omitted the labels m, m on the vertex operators that are inserted at z = z ν since the physical state condition fixes theses to coincide with the conformal weights of the fields in the symmetric product orbifold, i.e.
as one can check by combining eqs.(4.3), (4.5) with (2.8).In comparison to the earlier formula (2.18) that expressed the orbifold correlators in terms of correlations function of the parent theory M, the correlators on the right hand side of formula we stated here are evaluated in the free Kac-Wakimoto field theory we reviewed in the previous section.The subscript q instructs us to use the restricted correlation function we defined in eq.(3.17) with a parameter Ω = q = q Γ that is given by the constant factor q that multiplies the leading term of the numerator potential in the derivative ∂Γ of the covering map, see eq. (2.14).
We note that charge conservation for the linear dilaton ϕ with background charge Q ϕ requires that the insertions in the correlator on the right hand side of eq.(4.4) must obey Using eq.(4.5) one can easily verify that this constraint on the parameters j ν coincides with the constraint (2.11) on the parameters α ν .
There are two aspects of formula (4.4) that deserve to be stressed.On the one hand, this formula now contains vertex operators insertion at the R = R 0 + s/2 poles v j of the covering map, in contrast to eq. (2.18).On the other hand, the complicated prefactors that multiplied the correlation functions of the parent theory M in eq.(2.18) do not show up any longer in eq.(4.4).We will understand both aspects through a detailed calculation below.Later they will then turn out to be decisive for our ability to rewrite the orbifold correlators in terms of a dual string theory.

Proof of the embedding formula
In order to prove this results we will start with the correlator in the integrand on the right hand side of eq.(4.4).Let us define Our goal now is to show that this correlator agrees with the integrand on the right hand side of eq.
(2.19).The correlator C is computed in the free field theory (1.2) that contains a βγ system in addition to the linear dilaton ϕ.The correlator in the integrand of eq.(2.19), on the other hand, is evaluated in the linear dilaton theory of the field φ.To relate the two expressions, including all the prefactors in (2.19), we need to integrate out the βγ system and pass from the linear dilaton ϕ of the WZNW model to the linear dilaton φ of the orbifold.In doing so, we shall eventually follow the strategy that we used in [17] to prove the FZZ duality conjecture.But before we can do so, we need to massage the starting point of the calculation.
Step 1: Collecting the spectral flow.In a first step we apply the parafermionic representation, see subsection 3.2, in order to collect all the spectral flow in a single insertion point which we denote by ξ.The combined spectral flow from all the individual vertex operators adds up to Hence, by inserting our general formula (3.23), we can write our correlator in the from where C denotes the correlation function and the factor Θ is given by the following ratio of correlation functions of the free field χ

.13)
Here we have combined the labels µ = 1, . . ., n, a = 1, . . ., s and i = 1, . . ., R into a single label J that runs through J = 1, . . ., n + R + s and we set z n+a = u a and z n+s+j = v j .At the same time we also set and similarly for mJ .Note that the denominator in the second line contains trivial insertions at the points z = z J , J > n, on the worldsheet since m J + w J k/2 = 0 for all J > n.For later use we also introduce After moving all the spectral flow into one single insertion points at z = ξ through the parafermionic representation our task now is to compute the correlation function (4.11).
Step 2: Inverting the spectral flow.The correlator (4.11) is still not quite in the form of the correlation function that we dealt with in [17].In the case at hand, the insertion of the operators v (w) (ξ) forces β(z) to possess a pole of order w at the point z = ξ on the worldsheet.The analysis in [16,17], on the other hand, deals with the case where β(z) has a zero at z = ξ.We can relate the two different signs of the spectral flow through the following non-linear transformation of the βγ system, One can verify directly that the new fields γ and β satisfy the same operator products as the original fields γ and β.Alternatively, we can also arrive at the new fields γ and β through the bosonization of the βγ system.Let us recall that the original βγ system can be written as in terms of two decoupled linear dilaton fields χ 0 , χ 1 with operator products and background charges Q 0 χ = − 1 2 and Q 1 χ = 1 2 , respectively.In terms of the bosonic fields x, y the spectrally flowed vacua of the βγ system can be expressed as We can now pass to the new fields γ and β through a simple reflection of the fields χ 0 , χ 1 , i.e. in the bosonized variables the fields β and γ are given by β(z) ≃ −e χ 0 (z)−χ 1 (z) ∂χ 1 (z) , γ(z) ≃ e −χ 0 (z)+χ 1 (z) .(4.20) If we apply the same reflection to the bosonization formula (4.19) for the spectrally flowed vacua we find that In conclusion we have shown that the correlator (4.11) can be rewritten as where the vertex operators Φ have the same form as in eq.(3.10) but with γ replaced by γ−1 .The spectral flow carrying operator v is Step 3: Integrating out the βγ system.After completing the previous two steps, we are precisely in the setup we addressed in [17].The analysis of [16,17] is based on passing from the m-basis to the so-called µ-basis, see our discussion in section 3.1 and in particular eq.(3.Putting all this together we arrive at the following expression for our correlation function with the parameters j I as introduced in eq.(4.15).The δ-function in the integrand implements the construction of the restricted correlation function, see eq. (3.17).The variable û is given by û = I z I /µ I .Following [16,17] we perform the path integral over γ to obtain the a constraint on ∂ β.After integration with respect to the worldsheet coordinate this constraint reads Note that the residues are given by 1/µ I instead of µ I because of the way we have introduced our µ-basis.The second equality arises from the insertion of the operator v(w) (ξ) which forces β to have a zero of order w at z = ξ.But this equality can only hold provided that the parameters µ I satisfy the following n + R + s − 1 constraint equations n+R+s I=1 1/µ I (ξ − z I ) p = 0 , p = 0, . . ., w . (4.26) Since we have integrated β(z), γ(z) the correlation function is now obtained as a path-integral only over the field ϕ(z, z).As in [16], we shift the field ϕ(z, z) using the prescription The correlation function (4.24) is now given by, see [16] for details, where V α denotes the vertex operators in the linear dilaton theory that were defined in eq. ( 2.4) and the correlation function on the right hand side is to be evaluated in the linear dilaton theory of the field φ that we introduced in our discussion of the parent CFT, see section 2. In particular, the shift that we defined in eq.(4.27) implies that the new field φ has background charge Q = Q φ , just as in eq.(2.3).
In order to explicitly pass back form the µ-to the m-basis, we want to perform the integrals over all the parameters µ I .This is helped by the fact that the n + R + s integrations are constrained by n + R + s conditions in the arguments of the δ-functions.Using the following result for the Jacobian, see [17,23,31]) Thus we find with a factor θ that is given by Here we have converted the product over vertex operators in the linear dilaton theory into three separate products over ν = 1, . . ., n, a = 1, . . ., s and j = 1, . . ., R. Since the parameters α J = 0 for n + s < J, see eq. (4.15), the vertex operator insertions at the poles z = z j of the covering map Γ are trivial.We note that the resulting correlator coincides with the correlator on the right hand side of eq.(2.19).
In order to compare the coefficients, we exploit the parameters µ I can be read off from eq. ( 4.25) to take the form where we applied the regularization scheme that sets lim z→w ln |z − w| 2 = − ln |ρ(w)| 2 = 0.After multiplication with the factor Θ that was defined in eq. ( 4.12) we finally arrive at with a prefactor ϑ that is given by This is a fairly remarkable result that finally establishes the formula (4.4) for the correlators of the perturbed symmetric product orbifold in terms of a correlator in the WZNW model.Let us stress however, that our formula is purely within CFT.In fact, on both sides of the equation the fields are inserted at specific points on the worldsheet.This is to be contrasted with expressions for string amplitudes in which insertion points of worldsheet fields are integrated over.It is the task of the next section to uncover a relation of the CFT correlators with string scattering amplitudes.Our formula (4.4) holds the key for such a string theoretic formulation.

The Uplift to String Theory
After rewriting the correlators of the symmetric product orbifold in terms of correlation functions of operators in some c W = 26 CFT we are now well prepared uplift the latter to scattering amplitudes in a bosonic string theory.In the first subsection we take a closer look at certain vertex operators of W.
Most importantly, we shall discuss two integrated vertex operators that commute with the currents and the stress tensor of the Kac-Wakimoto free field theory.These can be added as interaction terms to the free field theory while preserving both current algebra and Virasoro symmetry.The action of this dual string theory will be spelled out in the second subsection.For this theory we shall then analyse tree level scattering amplitudes and show that they agree with the correlation functions of the symmetric product orbifold in the planar limit.

Screening charges and vertex operators
Screening charge in the w = −1 sector.A screening charge is a (scale invariant) integrated vertex operator that commutes with all three currents.The conventional screening charges of the SL(2, R) WZNW model are constructed from fields in the vacuum sector with winding number w = 0 of the current algebra.But these won't play any role below.Instead we shall now discuss screening charges that carry non-vanishing winding number, starting with w = −1.It is not difficult to see that the following object preserves the current algebra symmetry On the right hand side we have written out the construction of the relevant field Φ, see eq. (3.11).
Recall that γ 1 is a Fourier mode of the field γ(z).In order to verify that the integrated operator is The interaction term we displayed in eq. ( 5.1) is a close relative of a more conventional screening charge that was first found by Bershadsky and Ooguri [19], Indeed, the dependence on the linear dilaton ϕ is the same in both screening charges.In the ghost sector, the states γ Before we conclude this short discussion of the screening charge (5.1) we want to briefly display the operator product of the integrand with the field γ which reads i.e. the field γ has a first order pole at the insertion point (v, v).We have already pointed at some similarities between γ and the covering map at the end of the previous paragraph.Since the covering map does indeed possess poles, we expect the screening charge S − to be relevant for our theory.And indeed one can see that the corresponding local field does appear inside the correlator on the right hand side of our equation (4.4).We will get back to this point in the next subsection.
Screening charge in the w = 1 sector.Having discussed a screening charge with winding number w = −1 it seems natural to turn to the opposite direction and look for a screening charge with w = 1.
Such a screening charge does indeed exist and is given by Note that this operator does not depend on the bosonic field ϕ.Since the spectrally flowed vacuum |w = 1⟩ has vanishing conformal weight, the integral is obviously invariant under scale transformations.
In addition, it also commutes with the currents.Since, to the best of our knowledge, the screening charge S + has not been thoroughly discussed in the literature, we want to prove this statement here.
The most important observation concerns the operator product expansion of the field J + (w) = −β(w) with the operator in the integrand of the screening charge, or rather its chiral half, In the first term we have used that β n |w = 1⟩ = 0 for n > 0 along with the usual commutation relations in the βγ system.To show that the second term involves a derivative we used along with γ n |w = 1⟩ = 0 for n ≥ 0. The final result of this short calculation can be expressed more elegantly as In addition, the exponent of |a Γ 0 | is trivial.Let us stress that in the expressions for a Γ ν and ξ Γ j all products run over the indices ν and µ start at ν, µ = 1, i.e. they do not include ν, µ = 0.This establishes the triviality property (5.8) of our second screening charge S − .Before we close this subsection let is spell out the following variant of formula (5.8) This equation will play a very crucial role in the next subsection where we finally engineer the perturbative string dual.
Deforming the vertex operators.In order to pass from unintegrated vertex operators of the CFT to the integrated vertex operators of the string theory we will need to deform the vertex operators we introduced in eq.(3.11).The deformed operators depend on two 'deformation parameters' x, x as follows The operator products in the βγ system state that the field β is conjugate to the field γ.In particular, the zero mode β 0 is conjugate to the constant mode γ 0 of γ.Hence, the new x, x-dependent factors in the definition of the deformed vertex operators modify the operator product with the fields γ, see [8], In order to evaluate this further we use Consequently, the operator product between the field γ and the deformed vertex operators reads In order to appreciate this formula, we stress the striking similarity with the behavior (2.13) of the covering map Γ.The comparison shows in particular that the deformation parameter x plays the role of the insertion point in the dual CFT.This interpretation of the deformation parameter x is also natural since the zero modes of the sl(2) currents generate global conformal transformations of the CFT on the boundary, with the two modes J + 0 = −β 0 and J+ 0 = − β0 corresponding to 2-dimensional infinitesimal translations.

The string theory dual
We are finally in a position to construct the string theory dual to the perturbed symmetric product orbifold.In order to do so, we go now back to equation (4.4).As we pointed out before, the correlator on the right hand side is a CFT correlator and not yet a string amplitude.It involves two types of vertex operators that are associated with physical states, namely the operators Φ jν ,wν and Φ 1/2b 2 ,−1 , but these are not integrated over as it is done in string amplitudes.On the other hand, the operators −k,−k are integrated over but they do not correspond to physical states.Nevertheless we claim that the right hand side of eq.(4.4), and hence the correlators of the deformed symmetric product orbifold, are just a few steps away from an interpretation as a string amplitude in a bosonic string theory that we are about to construct.
The relevant worldsheet theory is generated by the fields β, γ and ϕ with two interaction terms that preserve the current algebra symmetries.The action of this theory is given by (5.17) Here, S 0 is the action of the free theory that we spelled out eq. ( 1.2) already.The two interaction terms involve the two screening charges we introduced in the previous subsection.Note that we have absorbed a possible coupling constant that one could have expected in front of the first interaction term by shifting the zero model of the linear dilaton ϕ.In the second interaction term, on the other hand, we kept the coupling constant µ.It follows form the discussion in the previous subsection that the action S defines an interacting worldsheet CFT with Virasoro central charge c = 26.The interaction terms do not only preserve the Virasoro symmetry but even the full current algebra we constructed through eqs.(3.3).By the usual constructions, such a worldsheet CFT gives rise to consistent (but non-geometric) bosonic string theory.
Within this bosonic string background we can now construct string amplitudes.To this end, we pick a set of n physical states of the worldsheet theory.We will take these to be given by where m ν and mν are determined by the weights of the state ψ ν in the CFT X through eq.(4.6) and the x-dependence of Φ was introduced in eq.(5.13).Our claim now is that the associated string amplitude which is obtained by integrating out the worksheet positions of these vertex operators, reproduces the correlators of the deformed orbifold theory, i.e.
The correlator of integrated vertex operators on the right hand side is computed in the theory (5.17) with coupling parameter µ related to the deformation parameter λ of the dual CFT by λ = µ 2 .As usual, making sense of the right hand side requires to fix a frame in which three insertion points are fixed to (0, 1, ∞) so that the integration only extends over the remaining n−3 insertion points.Finally, we put a ≃ instead of = because we will prove this equality only up to a numerical factor of the form for some complex parameter η.On the right hand side we have expressed the exponents in terms of the comformal weights (4.6) of the field in the symmetric product orbifold.This shows that any factor for the form Λ(η) can be absorbed into a rescaling of the insertion points x ν of the symmetric product orbifold.As such, it does now contain any dynamical information about the CFT beyond the scaling weight.Consequently, by working only modulo factors of the form (5.19) we do not loose any non-trivial content of the relation.
The proof of formula (5.18) relies heavily on the localization properties of the path integral over γ in the worldsheet correlation functions.These assert that in correlators with maximal violation of winding number, the worldsheet field γ localizes to the branching function Γ that we introduced in eq.(2.13) while discussing the pertubative expansion of the symmetric product orbifold, provided that R and the winding numbers are related by (w A − 1) + 1 and w A > 1 . (5.21) Note that the first three fields are inserted at (z ν , ν = 1, 2, 3) = (0, 1, ∞).As long as the condition (5.21) is satisfied, we sum over covering maps Γ which are discrete and the positions of all fields are localized at specific points on the worldsheet.Note that insertions of (integrated) vertex operators with w A = 1 do not contribute to the left hand side of the balancing condition eq.(5.21).On the other hand they force Γ to assume the value x A at the insertion point of the field.So, if we allow for insertions with w A = 1 the integration over their insertion points is replaced by a discrete sum over the positions u at which Γ(u) = x A .
We will not discuss the evidence or derivation of the localization formula (5.20) any further but simply refer to the original literature on the subject, see in particular [8]. 6Once we accept the validity of eq. ( 5.20) we can now explain our central formula (5.18).We will do so in two steps, starting with the case in which only the interaction term S − contributes.
Analysis of R 0 insertions of S − .The string amplitude on the right hand side of equation (5.18) may be computed by expanding the exponentials of the two interaction terms that appear in the theory (5.17).If we have less than R 0 insertions of S − the amplitude vanishes.Indeed, we note that the behavior of γ near our n vertex operators is given by eq. ( 5.16) while the integrands of S − create first order poles in γ.If the number of such poles is too small, there simply is no covering map Γ satisfying eq.(2.13).So, the first terms that can actually contribute arise when we have R 0 insertions of the first interaction term.In this case, there exists a discrete set of covering maps Γ with R 0 poles or, put differently, a discrete set of branched coverings of genus g = 0 with R 0 sheets.Hence the amplitude receives a discrete set of contributions that are in one-to-one correspondence with the genus g = 0 contributions to the free symmetric product orbifolds, Note that the localization property of the path integral over γ forces all the worldsheet fields to be inserted at the zeroes and poles of ∂Γ.Comparison with formula (4.4) shows that we have now almost reached the desired result, except that the vertex operators Φ ν still depend on the deformation parameter x ν while those on the right hand side of eq.(4.4) do not.But the difference is easy to remove using the Ward identities of the zero models J 3 0 and J3 0 .These generate rescalings of the parameters x ν and the associated Ward identities read One might naively expect that we can use this simple behaviour under rescaling of the variables x ν to send all of them to zero.But this turns out to be a bit too quick.As long as at least one of the variables x ν is nonzero, it determines the scale for the overall prefactor q that multiplies the derivative of the covering map Γ, see eq. (2.14).This ceases to be the case when all x ν vanish.In this sense, the limit u → 0 is not a smooth limit.
Since we cannot recover the prefactor q of the covering map after we have sent u to zero, we should make sure that we fix it before taking the limit.This is what our definition (3.17) of restricted correlations functions was designed for.A short analysis along these lines shows that there indeed exists some parameter q ′ = ηq such that q . (5.22) To go to the second line we have used the scaling behavior (3.18) of the reduced correlator and we dropped the factor Λ(η) in front of the correlator which is why we wrote ≃ rather than an equal sign.
Note that the explicit x ν dependence has now disappeared from the field insertions on the right hand side.Nevertheless, the correlator certainly continues to depend on x ν through the dependence of the insertion points z ν and v j on x ν .We recall that the latter are branch points and poles of the covering map Γ whose defining property (2.13) does depend on x ν .Thereby we have now expressed the first non-vanishing term of the original string amplitude (5.18) through the same restricted correlator that appears on the right hand side of our formula (4.4) for the correlation function in the symmetric product orbifold.
So far we have not considered the effect of inserting the second interaction term S + .So, let us now imagine that we bring down some S + insertions from the exponential while keeping the number of S − insertions fixed at the minimal value R 0 .As we have saw in eq.(5.8), insertions of S + into such a correlator are actually trivial.Hence the terms with R 0 insertions of S − reproduce the correlators of the free symmetric product orbifold.This is quite nice already, but things become truly remarkable once we increase the number of S − insertions beyond the smallest value R 0 .
More than R 0 insertions of S − .To see this, let us first consider the case in which we insert just one addition screening charge S − , i.e. we have R = R 0 + 1. 7 The additional insertion of S − creates one more pole in γ or, equivalently, an additional second order pole with vanishing residue in ∂γ.
Consequently, ∂γ is now allowed to have two additional zeroes beyond those that are associated with the n vertex operators Φ ν .Since our second interaction term has winding number w = 1, it creates a single zero in γ, see eq. (5.7), which is of course no longer visible in ∂γ.So, at first sight it might seem that we cannot get the desired zeroes from insertions of S + .But this is not true.Let us assume that we brought down four interaction terms of the second kind.Since we integrate over the insertion points of these interaction terms, our integration includes regions in which two of the vertex operators with w = 1 come close to each other.If that is the case, they generate an operator with winding The vertex operator with winding number w = 2 that appears in the leading term of the operator product expansion creates a zero of order two in γ and hence a single zero in ∂γ.Hence, if the four screening charges we have brought down from the second interaction term form two such pairs, ∂γ will have two additional zeroes, just as desired, and the correlation function is no longer zero.In other words, with R = R 0 + 1 insertions of the screening charge S − , our amplitude receives nontrivial contributions from the boundary of the integration region of four S + insertions where these collide pairwise.Since the leading term in the operator product (5.23)contains the very same operator that was required in our formula (4.4) in order to reproduce insertions of the perturbing operator of the symmetric product orbifold, we conclude that these boundary contributions are indeed given by the right hand side of our formula (4.4).The only difference is that the vertex operators at z ν are deformed by x ν .
The arguments outlined in the previous paragraph obviously rely on the localization (5.20).The latter involves operators with positive winding that are deformed.For the vertex operator insertions with winding number w = 1 that arise from the second interaction term S + , the deformation can be introduced through the following identity If we insert the our interaction term at some point u on the worldsheet, there exists some value of the deformation parameter ξ for which this insertion is non-zero, namely the value ξ = Γ(u) the branching function Γ asumes at u. Since the right hand side of the identity (5.24) involves an integration over all ξ, we can pick up some non-zero contribution for any value of u.Consequently, the integration over the insertion point of the interaction vertex remains intact while the integration over the associated deformation parameters localizes.By this mechanism, all operators with positive winding number are indeed deformed.
symmetric product orbifold that arise from branched coverings of genus g.The main goal here is to extend formula (2.10) to g > 0. Then we turn our attention to the WZNW model and establish the higher genus analogue of our formula (4.4) in the second subsection.The uplift to pure bosonic string theory will then work in precisely the same as as in the previous section.

Symmetric orbifolds beyond the planar limit
As argued in section 2, the correlation functions of twisted sector fields in the symmetric product orbifold M N /S N on a sphere can be computed as correlation functions of single CFT M on a Riemann surface defined by the covering map x = Γ(z).Here Γ(z) is required to satisfy the condition (2.13).In section 2, we assumed that the covering map defines a Riemann surface Σ = Σ 0 of genus zero.In this subsection, we generalize the analysis to the case where the covering map defines a Riemann surface Σ = Σ g of generic genus g ≥ 1. Recall that the genus g is related to the number R of closed solid loops of the diagram through eq.(1.1).
We start by reviewing some useful facts on higher genus Riemann surface.Our exposition follows [16,17,32], see also [33][34][35].On the Riemann surface Σ g , we first introduce a complex structure so that we have a notion of holomorphicity.It is well known that a complex surface of genus g admits g independent holomorphic one-forms ω a .We choose a basis in the space of holomorphic one-forms as Here α a , β a , a = 1, . . ., g, denote a basis of holomorphic cycles on the surface and τ ab is the period matrix of Σ g .In order to construct some coordinates on Σ, we utilize the Abel map where z 0 is an arbitrary reference point in Σ g .We shall need a number of basic function on our surface.
All of these are constructed in one way or another form the theta functions where δ a = (δ 1a , δ 2a ) with δ 1a , δ 2a = 0, 1/2 defines the spin structure along the homology cycles α a , β a .
The theta function is actually not periodic along the holomorphic cycles.Instead, under shifts of the form m + τ n, it behaves as According to the Riemann vanishing theorem, the theta function vanishes at a point z if and only if one can find g − 1 points p i (i = 1, . . ., g − 1) such that z can be written as Here, ∆ denotes the Riemann class.This concludes our description of background material on higher genus Riemann surfaces and it suffices to construct the relevant covering maps.
The covering map Γ : Σ g → S 2 is defined such as to satisfy the conditions (2.13) near the branch points at z = z A of the surface Σ g .As in the case of genus zero, it is convenient to consider the derivative ∂Γ.The meromorphic one-form ∂Γ has a zero of order w A − 1 at z = z A , thus the total order of zeros is 2R (g) − 2 + 2g, where R (g) is defined by (2.12).The covering map Γ has R (g) poles of the first order, whose locations are denoted by v j , as before, and hence the derivative ∂Γ should have R (g) poles of the second order at z = v j .From this information of zeros and poles, the derivative ∂Γ can therefore be written as The function E(z, w) that appears in the numerator and denominator of the formula for ∂Γ is a prime form which is defined as , where h δ (z) 2 = g a=1 ∂ a θ δ (0|τ )ω a (z) .(6.7) In this expression the spin structure δ is assumed to be odd.The prime form has weight (−1/2, 0) both for z and w and has a zero of the first order at z = w, i.e. it behaves as E(z, w) ∼ z − w.In addition, the prime form E is periodic along α a cycles, but it is not so along β a cycles.More precisely, upon shifts of z by τ a the prime form behaves as In order to turn ∂Γ into a proper (double periodic) one-form on the surface, we put the factor σ(z) 2 in the numerator on the right hand side of eq.(6.6).The object σ(z) is a g/2-form that is defined by Upon shifts of z by τ a , the g/2-form σ behaves as .10)and show that it reproduces the integrand in the right hand side of eq.(6.17) for the correlation function in the symmetric product orbifold.Here, the formula (2.12) leads to The charge conservation of the linear dilaton ϕ requires that which coincides with the momentum conservation condition (6.15) if we apply the substitution (4.5).
We evaluate the quantity (6.20) by following the same strategy as in subsection 4.2.Namely, in a first step we utilize the parafermion representation to collect all the spectral flow in a single insertion point.For surfaces of higher genus, the parafermionic representation requires a little more thought since there is a subtlety associated with phases (twists) which arise when going along non-trivial cycles of Riemann surfaces.As reviewed in appendix A of [17] (see also [36][37][38][39]), the correlation functions of the parafermionic primary fields are given by where τ ab is the period matrix of the surface Σ that we introduced previously in eq.(6.1).
From the definition of correlation functions in the paraferemionic theory (6.22), we can obtain the relation, by utilizing the fact that the parafermionic fields do not depend on the winding number w as in eq.
(3.22).Furthermore, | det ′ ∂| 2 arises from the integration over Fadeev-Popov ghosts, with the prime indicating that the zero mode contribution is removed.The correlators A Λ M and C Λ M that appear in the integrands on the right hand side of eq.(6.24) are defined by Φ jν ,0 m I , mI (z I ) ⟩ Λ,q .(6.25) They are evaluated with the SL(2, R) WZNW model in the presence of twists Λ that were introduced in eqs.(6.23).The dependence on the twist parameters is indicated by the subscript Λ on the correlators.This is not to be confused with the index q we introduced before to denote the restricted correlation functions of the Kac-Wakimoto free field theory.We also introduces the integer w = I w I by summing over all the individual winding numbers, as in eq.(3.24).The other correlators that appear in the integrands of eq. ( 6.24) are computed within the free bosonic theory as In the case of genus zero, there is no non-trivial cycle and twist, thus the correlators of parafermionic primary fields can be written by the products of those of the WZNW model and the free boson theory.
However, in the case of higher genus, the relation between the two different correlation functions of the WZNW model is not so simple and twists along non-trivial β cycles need to be introduced.We are interested in the correlation functions of the WZNW model of the form of A Λ M , without any twists.In our special type of correlation functions, the derivative ∂γ(z) of the ghost field γ can be identified woth the derivative ∂Γ(z) of the covering map.This implies that the integration over the twists Λ a is localized at Λ a = 0.For this reason, we can ignore the additional complications that arise from non-trivial twists.
Given eq. ( 6.24), we can compute the desired quantity A Λ M from the correlations function C Λ M in the integrand on the right hand side.As explained in subsection 4.2, it is convenient to convert βγ system to βγ system via the transformation (4.16)In order for eq.( 6.30) to be satisfied, the following w constraint equations need to be satisfied ) and (6.17).
There are a number of very interesting extensions of our work that would be interesting to work out.The first one is to include boundaries and interfaces in the symmetric product orbifold.In this string theory this amounts to studying branes and open strings.Building on previous work, see in particular [40,41] (and also [42]) it should be possible to derive a AdS/BCFT correspondence [43].For previous works on brans and open strings on AdS 3 , see, e.g.[44][45][46].Another interesting direction would be to study the duality relation in the presence of N = 2 supersymmetry using the NSR-formalism for superstrings.In this case one should be able to make contact with a proposal by [14,15].Still within the context of superstrings it would also seem worthwhile to consider AdS 3 × S 3 and to compare with a symmetric orbifold M N /S N , where M includes N = 4 super Liouville theory as in [25].Last but not least, one can also study holography for symmetric product orbifolds on surfaces of higher genus, and in particular on the torus with genus g = 1 see e.g.[47] and further references therein.
Given that we have identified a worksheet model that is dual to the perturbative expansion of the symmetric product orbifold, it would be very interesting to explore integrability of this theory.
Following standard lore, the screening charges S ± we introduces in writing down (5.17) are expected to generate an interesting quantum algebra that one expects to control the integrable structure of the worldsheet model.It would be very interesting to exploit such structures to write down e.g.TBAlike equations that would allow to compute the anomalous dimensions of operators in the symmetric product orbifold along the line of CFTs that is generated by the marginal operator (2.9).Given the somewhat unusual properties of our screening charge S + , this direction seems particularly interesting.
If successful, it could also make contact with a regime in which our string theory admits a sigma model description as a string theory on AdS 3 .
Finally, it would certainly be extremely interesting to extend this approach to the engineering of perturbative string theory duals to higher dimensional CFTs, and in particular to N = 4 supersymmetric Yang-Mills theory for which some preliminary explorations can be found in [48,49]. 8

Figure 1a .
Figure 1a.Graphical representation of a twist field with cycle length w = 3.

Figure 1b .
Figure 1b.Feynman diagram for a 4-point function of fields with w = 2 drawn on a surface of g = 1 with 2 solid loops (and same number of dashed ones).

Figure 2 .
Figure 2. Graphical representation of marginal operator in the twisted sector with w = 2.
we shall represent the SL(2, R) WZNW model in terms of non-compact parafermions and a free boson χ.One may think of the parafermionic representation as arising from the decomposition SL(2) ∼ SL(2)/U(1) × U(1) of the target space.The first factor SL(2)/U(1) is a geometric realization of parafermions.We denote the associated parafermionic primary fields by Ψ j m, m(z, z) .(3.19)

1 −k 1 |w
= −1⟩ that describe the chiral half of the integrand in eq.(5.1) are close relatives of the states β k−2 −1 |w = 0⟩ that appear as chiral half of the ghost sector in the integrand of screening charge (5.2).