A localising AdS$_3$ sigma model

TL;DR

The paper constructs a locally defined AdS$_3$ sigma-model through a CFT with $rak{sl}(2,R)_k$ symmetry at the tensionless point $k=3$, distinct from the standard SL$(2,R)_{k=3}$ WZW model. It shows that three-point functions become momentum-conserving deltas and higher-point correlators localise to covering-map configurations, with the four-point function obtained by summing conformal blocks from the three-point data, establishing crossing symmetry. The approach is motivated by holography and yields simple dualities for AdS$_3$ backgrounds, including bosonic AdS$_3$ theories dual to undeformed symmetric orbifolds, and it extends to various superstring and heterotic setups with NS-R formulations. The work clarifies obstructions in literature by demonstrating the existence of a second, delta-localised SL$(2,R)$ theory at $k=3$ and provides a framework to understand tensionless AdS$_3$/CFT$_2$ dualities via covering-space methods, with potential tachyon-free bosonic string realizations and higher-genus generalisations.

Abstract

We construct a CFT with $\mathfrak{sl}(2,\mathbb{R})_k$ symmetry at the `tensionless' point $k=3$, which is distinct from the usual $\mathrm{SL}(2,\mathbb{R})_{k=3}$ WZW model. This new CFT is much simpler than the generic WZW model: in particular its three-point functions feature momentum-conserving delta functions, and its higher-point functions localise to covering map configurations in moduli space. We establish the consistency of the theory by explicitly deriving the four-point function from the three-point data via a sum over conformal blocks. The main motivation for our construction comes from holography, and we show that various simple supersymmetric holographic dualities for $k_{\rm s}=1$ ($k=3$) can be constructed by replacing the $\mathrm{AdS}_3$ factor on the worldsheet with this alternative theory. This includes in particular the prototypical case of $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$, as well as the recently discussed example of $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{S}^3 \times \mathrm{S}^1$. However, our analysis does not require supersymmetry and also applies to bosonic ${\rm AdS}_3$ backgrounds (at $k=3$).