Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lorenz Eberhardt, Matthias R. Gaberdiel, Rajesh Gopakumar

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.

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

Abstract

It was recently argued that string theory on with one unit () of NS-NS flux is exactly dual to the symmetric orbifold CFT . In this paper we show how to directly relate the -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 -boundary of 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 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 string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.

Paper Structure

This paper contains 37 sections, 176 equations, 3 figures.

Table of Contents

  1. Introduction
  2. From Worldsheet CFT to Spacetime CFT
  3. Goal and Roadmap
  4. Review of the Lunin-Mathur Construction
  5. The basic worldsheet CFT setup
  6. The Ward identities
  7. The constraint equations
  8. The fusion rules
  9. The recursion relations
  10. Comparison to Maldacena-Ooguri
  11. A simple solution
  12. The covering map
  13. The 3-point case
  14. The proof of the 3-point solution
  15. The solution in the general case
  16. ...and 22 more sections

Figures (3)

  • Figure 1: A cartoon of the semiclassical string solution in the example of a 4-point function. The worldsheet is pinned to the boundary sphere via the covering map and has ramification indices $w_i$ at the insertion points.
  • Figure 2: A covering surface with two branch points of order $2$. We plot the real part of the inverse of the covering map, which has two sheets. We show the two sheets from a side view (left) and a top view (right). There are two branch points in the picture and monodromy around them interchanges the two sheets.
  • Figure 3: The spectrally flowed representation of $\mathfrak{sl}(2,\mathds{R})_{k+2}$ in the example $w=2$. The 'edge states' (inside the red strip) are the spectrally flowed affine primaries and they form an $\mathfrak{sl}(2,\mathds{R})$ representation under $J^\pm_{\pm w}$ and $J^3_0-\frac{(k+2)w}{2}$, see eq. \ref{['zeromode']}. The states away from the edge have higher multiplicity and can be reached with the affine oscillators. The global $\mathfrak{sl}(2,\mathds{R})$ zero-mode algebra acts horizontally and is resummed in the $x$-basis.