A note on background independence in $\text{AdS}_3$ string theory

TL;DR

This paper discusses background independence in AdS3 string theory, showing the path integral depends only on boundary data and localizes onto holomorphic maps, implying a trivial gravitational state space.

Contribution

It provides a covariant description of the worldsheet theory at k=1 NS-NS flux and demonstrates background independence and localization properties of the path integral.

Findings

Path integral depends only on boundary conformal structure.

Localization onto holomorphic covering maps.

Gravitational path integral is trivial for closed manifolds.

Abstract

In this note, we comment on the path integral formulation of string theory on $\mathcal{M}\times\text{S}^3\times\mathbb{T}^4$ where $\mathcal{M}$ is any hyperbolic 3-manifold. In the special case of $k=1$ NS-NS flux, we provide a covariant description of the worldsheet theory and argue that the path integral depends only on the details of the conformal boundary $\partial\mathcal{M}$, making the background independence of this theory manifest. We provide a simple path integral argument that the path integral localizes onto holomorphic covering maps from the worldsheet to the boundary. For closed manifolds $\mathcal{M}$, the gravitational path integral is argued to be trivial. This implies that the bulk gravitational theory has precisely one state in its Hilbert space. Finally, we comment on the effect of continuous deformations of the worldsheet theory which introduce non-minimal string tension.