Spectral flow in AdS 3 /CFT 2

Journal of High Energy Physics

We study AdS3× S1× Y supersymmetric string theory backgrounds with Neveu-Schwarz-Neveu-Schwarz flux that are dual to $$ \mathcal{N} $$ N = 2 superconformal theories on the boundary. We classify all worldsheet vertex operators that correspond to space-time chiral primaries. We compute space-time chiral ring structure constants for operators in the zero spectral flow sector using the operator product expansion in the worldsheet theory. We find that the structure constants take a universal form that depends only on the topological data of the $$ \mathcal{N} $$ N = 2 superconformal theory on Y.

Journal of High Energy Physics, 2021

Recently an exact worldsheet description of strings propagating in certain black hole microstate geometries was constructed in terms of null-gauged WZW models. In this paper we consider a family of such coset models, in which the currents being gauged are specified by a set of parameters that a priori take arbitrary values. We show that consistency of the spectrum of the worldsheet CFT implies a set of quantisation conditions and parity restrictions on the gauging parameters. We also derive these constraints from an independent geometrical analysis of smoothness, absence of horizons and absence of closed timelike curves. This allows us to prove that the complete set of consistent backgrounds in this class of models is precisely the general family of (NS5-decoupled) non-BPS solutions known as the JMaRT solutions, together with their various (BPS and non-BPS) limits. We clarify several aspects of these backgrounds by expressing their six-dimensional solutions explicitly in terms of fi...

Physical Review D, 2016

We discuss a Coulomb gas realization of n-point correlation functions in the SL(2, R) Wess-Zumino-Witten (WZW) model that is suitable to compute scattering amplitudes of winding strings in 3-dimensional Anti-de Sitter space at tree-level and one-loop. This is a refined version of previously proposed free-field realizations that, among other features, accomplishes to make the H + 3 WZW-Liouville correspondence manifest. 1 Understanding the physics of winding string states in three-dimensional Anti-de Sitter (AdS 3) Neveu-Schwarz-Neveu-Schwarz (NS-NS) backgrounds was crucial to establish the consistency of the theory at quantum level. The unitarity of the spectrum [1], the modular invariance of the one-loop partition function [2], and the consistency of scattering amplitudes [3] were not well understood until the spectrally flowed sectors of the Hilbert space, associated to winding degrees of freedom, were discovered by Maldacena and Ooguri in Ref. [1]. In the context of AdS/CFT correspondence, winding sectors were also very important to complete the dictionary between bulk and boundary three-point correlation functions [4, 5]. The string scattering amplitudes involving winding string states in AdS 3 were first computed in Ref. [6] by means of the Coulomb gas representation proposed in Ref. [7]. The result was later reproduced by other methods [3, 8, 9]. One of the salient features of these scattering processes has to be with the conservation of the total winding number: Being not of topological origin, but due to the presence of the B-field in the background, the string winding number in AdS 3 space is not necessarily conserved by interactions. Nevertheless, this non-conservation exhibits a peculiar feature: As originally observed by Fateev, Zamolodchikov, and Zamolodchikov, the violation of the winding number in a tree-level amplitude is bounded from above. More precisely, for an nstring scattering amplitude to be non-vanishing, the total winding number ∆ω = ω 1 + ω 2 + ...ω n has to obey |∆ω| ≤ n − 2. The purpose of this paper is to present a formalism to compute treelevel and one-loop n-string winding violating scattering amplitudes in AdS 3 NS-NS background. This formalism is a refined version of the one proposed in [7, 6], extending the latter in two directions: On the one hand, the prescription we propose here is suitable for describing oneloop amplitudes; namely, to compute Wess-Zumino-Witten (WZW) correlation functions on the torus. On the other hand, this version of the Coulomb gas realization accomplishes to make the correspondence between WZW and Liouville correlation functions manifest, and it represents an advantage. The paper is organized as follows: In Section 2, after a brief introduction to string theory in AdS 3 NS-NS backgrounds, we will study the worldsheet conformal field theory, which corresponds to the H + 3 = SL(2, C)/SU (2) WZW theory. We will describe in detail the free field approach of the theory as originally proposed in [7, 6]. In Section 3, we will review the so-called H + 3 WZW-Liouville correspondence. This is a remarkable tool that permits to write the string amplitudes in AdS 3 as a convolution of correlation functions of Liouville field theory. This will serve us as guideline in the construction. In Section 4, we will discuss the tree-level string scattering amplitudes. This will allow us to present our Coulomb gas formalism in a constructive way, first considering the maximally winding violating amplitudes, then analyzing the next-to-maximally winding violating process, and so on. In Section 5, we will discuss the one-loop scattering amplitudes. We will adapt our Coulomb gas method to the case of genus-1 Riemann surfaces, discussing the characteristic features of it, such as the existence of twisted sectors.

Physical Review D, 2019

The insertion of the spectral flow operator in a string scattering amplitude on AdS 3 × N produces a change in the winding number of one of the incoming (or outgoing) states, making it possible to compute amplitudes of processes in which winding number in AdS 3 is not conserved. The insertion of such operator, however, might seem artificial from the worldsheet theory perspective, as it appears as an unintegrated vertex operator of conformal dimension zero that does not represent any normalizable state. Here, we show that the spectral flow operator naturally emerges in the Liouville field theory description of the WZW correlation functions once it is combined with a series of duality relations among conformal integrals. By considering multiple insertions of spectral flow operators, we study the dependence on the moduli for arbitrary number of them, and we show explicitly that the amplitude does not depend on the specific locations of the accessory insertions in the worldsheet, as required by consistency. This generalizes previous computations in which particular cases were considered. This can also be thought of as an alternative proof of the WZW-Liouville correspondence in the case of maximally winding violating correlators.

Journal of High Energy Physics, 2018

We study superstring theory in three dimensional Anti-de Sitter spacetime with NS-NS flux, focusing on the case where the radius of curvature is equal to the string length. This corresponds to the critical level k = 1 in the formulation as a Wess-Zumino-Witten model. Previously, it was argued that a transition takes place at this special radius, from a phase dominated by black holes at larger radius to one dominated by long strings at smaller radius. We argue that the infinite tower of modes that become massless at k = 1 is a signal of this transition. We propose a simple two-dimensional conformal field theory as the holographic dual to superstring theory at k = 1. As evidence for our conjecture, we demonstrate that our putative dual exactly reproduces the full spectrum of the long strings of the weakly coupled string theory, including states unprotected by supersymmetry.

International Journal of Modern Physics D, 2020

We review old and recent results on a special limit of string theory on [Formula: see text] with pure NS–NS fluxes: the limit in which the string length [Formula: see text] equals the [Formula: see text] radius [Formula: see text]. At this point of the moduli space, the theory exhibits special properties, which we discuss. Special attention is focused on features of correlation functions that are related to the noncompactness of the boundary CFT target space, and on how these features change when the point [Formula: see text] is approached. Also, we briefly review the recent proposals for exact realizations of AdS/CFT correspondence at this special point.

Journal of High Energy Physics, 2017

Lorentzian continuation of the Sine-Liouville model describes nonhomogeneous rolling closed string tachyon. Via T-duality, this relates to the gauged H 3 + Wess-Zumino-Witten model at subcritical level. This model is exactly solvable. We give a closed formula for the 3-point correlation functions for the model at level k within the range 0 < k < 2, which relates to the analogous quantity for k > 2 in a similar way as how the Harlow-Maltz-Witten 3-point function of timelike Liouville field theory relates to the analytic continuation of the Dorn-Otto-Zamolodchikov-Zamolodchikov structure constants: we find that the ratio between both 3-point functions can be written in terms of quotients of Jacobi's θ-functions, while their product exhibits remarkable cancellations and eventually factorizes. Our formula is consistent with previous proposals made in the literature.

Journal of High Energy Physics, 2018

A solvable irrelevant deformation of AdS 3 /CFT 2 correspondence leading to a theory with Hagedorn spectrum at high energy has been recently proposed. It consists of a single trace deformation of the boundary theory, which is inspired by the recent work on solvable TT deformations of two-dimensional CFTs. Thought of as a worldsheet σ-model, the interpretation of the deformed theory from the bulk viewpoint is that of string theory on a background that interpolates between AdS 3 in the IR and a linear dilaton vacuum of little string theory in the UV. The insertion of the operator that realizes the deformation in the correlation functions produces a logarithmic divergence, leading to the renormalization of the primary operators, which thus acquire an anomalous dimension. We compute this anomalous dimension explicitly, and this provides us with a direct way of determining the spectrum of the theory. We discuss this and other features of the correlation functions in presence of the deformation.