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

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.