Correlators for tensionless strings on ${\rm AdS}_3$ orbifolds

TL;DR

The paper addresses the problem of matching correlators between tensionless string theory on $({\rm AdS}_3\times{\rm S}^3)/\mathbb{Z}_k\times {\rm T}^4$ and a subsector of the symmetric orbifold of ${\rm T}^4$ built on a twisted $k$-cycle reference state. The authors compute tree-level worldsheet correlators for untwisted-sector operators, show that the relevant coverings satisfy $\Gamma(z)=(\Gamma_0(z))^k$, and use the Lunin-Mathur covering-map framework to express the worldsheet result as a sum over image insertions that matches the symmetric orbifold correlators on the covering surface. On the CFT side, they show that, in the large-$N$ limit, spherical covering maps with $\Gamma(z)=\Gamma_0(z)^k$ dominate and derive a conformal-factor relation $C_{\Gamma_0^k}=\left|\tfrac{\partial x_{i,R}}{\partial x_i}\right|^{-h_i-\bar h_i} C_{\Gamma_0}$, establishing exact agreement with the worldsheet calculation. They also discuss marginal deformations away from the orbifold point, arguing that the leading correlator structure persists and can be extended to deformed backgrounds, while noting the expected subtleties at subleading $1/N$ and higher genus.

Abstract

The CFT dual of string theory on $({\rm AdS}_3 \times {\rm S}^3)/\mathbb{Z}_k\times \mathbb{T}^4$ is believed to be described by the subspace of the symmetric orbifold of $\mathbb{T}^4$ that comprises the low-lying excitations on top of a certain reference state. (This `non-perturbative' reference state lies in the twisted sector associated to the conjugacy class consisting of only $k$-cycles.) In a recent paper we confirmed this picture by analysing the worldsheet theory of the orbifold at the tensionless NS-NS point, and by showing that the perturbative worldsheet spectrum reproduces precisely the single particle excitations on top of this reference state. In this paper we explain that this identification also holds on the level of the correlators, at least to leading order in $1/N$.

Figures (2)

• Figure 1: The diagram representing $(234)_\sigma[(123)(45)] = (13)(245)$.
• Figure 2: A tree-like diagram for $\sigma_2\sigma_3\sigma_2$ with three initial and final cycles of length $k$.