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String Theory on AdS_3

Jan de Boer, Hirosi Ooguri, Harlan Robins, Jonathan Tannenhauser

TL;DR

The paper investigates how the Brown-Henneaux Virasoro symmetry in AdS3 gravity is realized within first-quantized string theory and how this ties to the AdS3/CFT2 duality. Through a worldsheet treatment of Euclidean AdS3, vertex operator construction, and semiclassical analysis, it derives boundary Virasoro Ward identities and identifies the GKS Virasoro generators within the worldsheet framework. It shows that the central charge arises from second-quantized (disconnected) string diagrams rather than the GKS construction, and explains the role of non-unitary SL(2,C) representations for vertex operators in non-compact backgrounds. The work clarifies aspects of non-compact sigma models and the AdS/CFT dictionary while outlining future directions for Lorentzian AdS3, spectrum, and higher-order corrections.

Abstract

It was shown by Brown and Henneaux that the classical theory of gravity on AdS_3 has an infinite-dimensional symmetry group forming a Virasoro algebra. More recently, Giveon, Kutasov and Seiberg (GKS) constructed the corresponding Virasoro generators in the first-quantized string theory on AdS_3. In this paper, we explore various aspects of string theory on AdS_3 and study the relation between these two works. We show how semi-classical properties of the string theory reproduce many features of the AdS/CFT duality. Furthermore, we examine how the Virasoro symmetry of Brown and Henneaux is realized in string theory, and show how it leads to the Virasoro Ward identities of the boundary CFT. The Virasoro generators of GKS emerge naturally in this analysis. Our work clarifies several aspects of the GKS construction: why the Brown-Henneaux Virasoro algebra can be realized on the first-quantized Hilbert space, to what extent the free-field approximation is valid, and why the Virasoro generators act on the string worldsheet localized near the boundary of AdS_3. On the other hand, we find that the way the central charge of the Virasoro algebra is generated is different from the mechanism proposed by GKS.

String Theory on AdS_3

TL;DR

The paper investigates how the Brown-Henneaux Virasoro symmetry in AdS3 gravity is realized within first-quantized string theory and how this ties to the AdS3/CFT2 duality. Through a worldsheet treatment of Euclidean AdS3, vertex operator construction, and semiclassical analysis, it derives boundary Virasoro Ward identities and identifies the GKS Virasoro generators within the worldsheet framework. It shows that the central charge arises from second-quantized (disconnected) string diagrams rather than the GKS construction, and explains the role of non-unitary SL(2,C) representations for vertex operators in non-compact backgrounds. The work clarifies aspects of non-compact sigma models and the AdS/CFT dictionary while outlining future directions for Lorentzian AdS3, spectrum, and higher-order corrections.

Abstract

It was shown by Brown and Henneaux that the classical theory of gravity on AdS_3 has an infinite-dimensional symmetry group forming a Virasoro algebra. More recently, Giveon, Kutasov and Seiberg (GKS) constructed the corresponding Virasoro generators in the first-quantized string theory on AdS_3. In this paper, we explore various aspects of string theory on AdS_3 and study the relation between these two works. We show how semi-classical properties of the string theory reproduce many features of the AdS/CFT duality. Furthermore, we examine how the Virasoro symmetry of Brown and Henneaux is realized in string theory, and show how it leads to the Virasoro Ward identities of the boundary CFT. The Virasoro generators of GKS emerge naturally in this analysis. Our work clarifies several aspects of the GKS construction: why the Brown-Henneaux Virasoro algebra can be realized on the first-quantized Hilbert space, to what extent the free-field approximation is valid, and why the Virasoro generators act on the string worldsheet localized near the boundary of AdS_3. On the other hand, we find that the way the central charge of the Virasoro algebra is generated is different from the mechanism proposed by GKS.

Paper Structure

This paper contains 10 sections, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The AdS$_3$/ CFT$_2$ Duality
  3. Worldsheet Description of Strings on AdS$_3$
  4. Action and Symmetry
  5. Vertex Operators
  6. Semi-classical Analysis
  7. The Virasoro Algebra
  8. Boundary Ward Identity
  9. $T(x)T(y)$ OPE and Central Charge
  10. Discussion

Figures (4)

  • Figure 1: Semi-classical worldsheet in the presence of vertex operators
  • Figure 2: Bulk IR cutoff versus worldsheet UV cutoff
  • Figure 3: A single string worldsheet contributing to the $\langle TTV_1 \ldots V_n\rangle$ correlator. This diagram does not contribute the central charge of the Virasoro algebra.
  • Figure 4: A multiple string worldsheet contributing to the $\langle \langle TTV_1 \ldots V_n\rangle\rangle$ correlator. The central charge $c=6Q_1Q_5$ is obtained from this diagram.