Prepared for submission to JHEP Higher genus correlators for tensionless AdS3 strings Bob Knighton Institut f¨urTheoretische Physik, ETH Z¨urich Wolfgang-Pauli-Straße 27, 8093 Z¨urich,Switzerland E-mail: [email protected] Abstract: It was recently shown in [1] that tree-level correlation functions in tensionless 3 4 string theory on AdS3 × S × T match the expected form of correlation functions in the 4 symmetric orbifold CFT on T in the large N limit. This analysis utilized the free-field realization of the psu(1; 1j2)1 Wess-Zumino-Witten model, along with a surprising identity directly relating these correlation functions to a branched covering of the boundary of AdS3. In particular, this identity implied the unusual feature that the string theory correlators localize to points in the moduli space for which the worldsheet covers the boundary of AdS3 with specified branching near the insertion points. In this work we generalize this analysis past the tree-level approximation, demonstrating its validity to higher genus worldsheets, and in turn providing strong evidence for this incarnation of the AdS=CFT correspondence at all orders in perturbation theory. arXiv:2012.01445v3 [hep-th] 6 Jul 2021 Contents 1 Introduction2 2 The worldsheet theory3 2.1 The free field construction of sl(2; R)1 ⊕ ud(1)4 2.2 Highest-weight representations5 2.3 Spectral flow6 2.4 Vertex operators8 2.5 Picture changing and correlation functions9 3 The worldsheet as a covering space 11 3.1 Covering maps 11 3.2 The incidence relation 12 3.3 Localization 12 3.4 Non-renormalization 13 3.5 Proof of the incidence relation 14 3.6 Constraining the correlators 15 4 The covering map from local Ward identities 17 4.1 Constraints on the covering map 17 4.2 The local Ward identities 19 4.3 Delta function localization 21 5 Discussion 22 A Some Riemann surface theory 24 A.1 Meromorphic forms 25 A.2 Divisors 25 A.3 Existence of forms and Abel's theorem 27 A.4 Spinors and spin structures 29 A.5 Constructing functions and the prime form 29 B Localization of the W fields 31 C The g=2-form 32 { 1 { 1 Introduction Recently, progress has been made in providing a derivation of a special case of the AdS=CFT 3 4 correspondence [2] relating type IIB string theory on AdS3 × S × T with k = 1 units of N 4 pure NS-NS flux to the symmetric orbifold Sym (T ) in the large N limit. It has been shown that the full tree-level spectra of both theories match [3,4]. It was also known that the correlation functions of the symmetric orbifold theory obey nontrivial Ward identities for strings on AdS3 both at tree level [5] and at higher genus [6] in the RNS formalism. However, it was not shown that the symmetric orbifold correlators were the unique solutions to the Ward identities. Other recent approaches toward the realization of this duality have been explored in [7{9]. The central property that relates the correlation functions of tensionless string theory to those of the symmetric orbifold is that both are related to a certain meromorphic function Γ:Σ ! S2 which covers S2 by the worldsheet. From the perspective of the symmetric orbifold, Γ plays the role of a conformal transformation under which the pullback of the twisted fields inserted into correlators become single-valued [10{12], as shown in Figure 1. On the string theory side, this holomorphic function is interpreted as a map from the string worldsheet to the boundary of AdS3, for which the incoming and outgoing string states asymptotically wind wi times at the points xi on the boundary. The interpretation of this incarnation of the AdS3=CFT2 duality is, then, that the string worldsheet is exactly the covering space used in the calculation of symmetric orbifold correlation functions, as was originally proposed in [13]. As such, one would expect the worldsheet correlators to localize to those points in the moduli space for which the worldsheet is such a covering space. That such a solution was permitted by the worldsheet Ward identities in the RNS formalism was the primary result of [5,6]. In [1], another approach for constraining correlation functions for the tensionless (k = 1) string was employed, similar in spirit to that of [5,6]. This approach was based on the fact that, at k = 1, the hybrid formalism of Berkovits, Vafa and Witten [14] simplifies into (a quotient of) a free field theory on the worldsheet, consisting of four spin-1=2 Bosons and four spin-1=2 Fermions. The Ward identities, when expressed in these variables, turn out to be enough to essentially fix the form of the correlation functions of highest-weight states in the tensionless string at tree level. The crux of this analysis was a striking formula, reminiscent of a twistorial identity, relating the worldsheet correlation functions to the covering map Γ : Σ ! S2. This identity directly realizes the localization of the worldsheet correlation functions, and allows the interpretation of the string worldsheet as the covering space in Figure1, at the level of string perturbation theory. The main advantage of the analysis of [1] was that this localizing solution was found to be the only one consistent with the Ward identities in the free field description. The goal of this paper is to extend the results of [1] to worldsheets of higher genus. On the string theory side, this corresponds to considering correlation functions at higher- order in perturbation theory. On the side of the dual CFT, this corresponds to considering correlation functions at higher order in the 1=N expansion. We find that an identity analogous to the main result of [1] is satisfied on higher genus worldsheets, and thus that { 2 { x1 z1 z2 z4 Γ x3 z3 x2 x4 Figure 1. A cartoon representation of a holomorphic covering map Γ : Σ ! S2 mapping the string theory worldsheet to the dual CFT sphere. Correlation functions in both tensionless string theory and the symmetric product CFT are determined by the data encoded in Γ. the conclusions of [1] holds at every order in perturbation theory. This represents highly 3 4 nontrivial evidence that tensionless string theory on AdS3 × S × T is exactly dual to the 4 symmetric product CFT on T . This paper is organized as follows. Section2 introduces the relevant details of the 3 4 free field construction of tensionless string theory on AdS3 × S × T . In particular, we formulate string theory on this spacetime in terms of free Bosons generating the algebra sl(2; R)1 ⊕ ud(1). We introduce the vertex operators corresponding to spectrally-flowed highest-weight states of this algebra, and define their correlation functions after taking into account picture-changing. In Section3, we prove an identity relating worldsheet correlation functions to the covering map used to construct correlation functions in the symmetric product CFT, which generalizes the identity found in [1] to higher genus worldsheets. This identity is essentially what allows us to show that the worldsheet correlation functions localize on the moduli space, and thus interpret the string worldsheet as the covering surface appearing in the construction of the symmetric orbifold correlation functions [10, 11]. In Section4, we show that the constraints on correlation functions arising from taking various OPE limits (reminiscent of Ward identities) are algebraically identical to the constraints used to construct covering a covering map Γ : Σ ! S2, thus providing a second method of directly relating the string correlation functions to the covering space. We then conclude the main text with a discussion in Section5. Additionally, we include a brief introduction to the relevant results from the theory of Riemann surfaces in AppendixA which are used heavily throughout the main text. 2 The worldsheet theory 3 4 In the RNS formulation of strings on AdS3 × S × T with k unites of pure NS-NS flux, one identifies the AdS3 factor with the N = 1 supersymmetric Wess-Zumino-Witten model on 3 the Lie group SL(2; R) at level k [15{17], while identifying the S factor as a similar WZW model on SU(2). Symbolically, the tensionless string can be formulated in terms of the worldsheet WZW model (1) (1) 4 sl(2; R)k ⊕ su(2)k ⊕ T : (2.1) { 3 { The so-called \tensionless string" is identified with the limit in which the level k is taken to be minimal, i.e. k = 1. However, the description of the worldsheet theory in the RNS formalism breaks down for the tensionless limit, due to the decomposition [18] (1) ∼ sl(2; R) = sl(2; R)k+2 ⊕ (3 free Fermions) k (2.2) (1) ∼ su(2)k = su(2)k−2 ⊕ (3 free Fermions) ; Thus, at k = 1, the su(2)k−2 factor is non-unitary. Thus, if one wants to study the worldsheet theory in the tensionless limit, one needs to look for an alternative formalism in which the worldsheet theory is well-defined. This is accomplished by the hybrid formalism 3 of Berkovits, Vafa and Witten [14], in which one replaces the factor AdS3 × S with its supergroup analogue psu(1; 1j2)k, which remains well-defined at level k = 1. As was noted in [4] and further explored in [1], the hybrid formalism in the tensionless limit has a particularly simple construction in terms of free fields living on the world- sheet. In particular, one constructs the u(1; 1j2)1 WZW model from four two pairs of free \symplectic" Bosons [19] and two pairs of free Fermions, and then quotients out two ud(1) subalgebras to recover the model on psu(1; 1j2)1.