Deriving the $\text{AdS}_{3}/\text{CFT}_{2}$ Correspondence

TL;DR

This paper establishes a direct correspondence between string theory on AdS3 with NS-NS flux and the symmetric orbifold CFT, showing their correlators are related through holomorphic covering maps, thus confirming the duality at the level of correlators.

Contribution

It demonstrates how world-sheet correlators in AdS3 string theory are localized to configurations matching symmetric orbifold CFT correlators via holomorphic covering maps, providing a concrete link between the two theories.

Findings

World-sheet correlators are delta-function localized in moduli space.

Correlators match those constructed via covering maps in symmetric orbifold CFT.

Supports the duality between AdS3 string theory and symmetric orbifold CFT at the correlator level.

Abstract

It was recently argued that string theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with one unit ($k=1$) of NS-NS flux is exactly dual to the symmetric orbifold CFT ${\rm Sym}^N(\mathbb{T}^4)$. In this paper we show how to directly relate the $n$-point correlators of the two sides to one another. In particular, we argue that the correlators of the world-sheet theory are delta-function-localised in string moduli space to those configurations that allow for a holomorphic covering map of the $\text{S}^2$-boundary of $\text{AdS}_3$ by the world-sheet. This striking feature can be seen both from a careful Ward identity analysis, as well as from semi-classically exact AdS$_3$ solutions that are pinned to the boundary. The world-sheet correlators therefore have exactly the same structure as in the Lunin-Mathur construction of symmetric orbifold CFT correlators in terms of a covering surface -- which now gets identified with the world-sheet. Together with the results of arXiv:1803.04423 and arXiv:1812.01007 this essentially demonstrates how the $k=1$ $\text{AdS}_3$ string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.