Deriving the ${\text{AdS}}_3/{\text{CFT}}_2$ Correspondence

Abstract

It was recently argued that string theory on ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathbb{T}}^4$ with one unit ($k=1$) of NS-NS flux is exactly dual to the symmetric orbifold CFT ${\mathrm{Sym}}^N({\mathbb{T}}^4)$. 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.

arXiv comment55 pages; v2: matches JHEP version

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