In this note, we comment on the path integral formulation of string theory on \( \mathcal{M} \) × S3 × \( {\mathbbm{T}}^4 \) where \( \mathcal{M} \) is any hyperbolic 3-manifold. In the special case of k = 1 units of 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. Finally, we comment on the effect of continuous deformations of the worldsheet theory which introduce non-minimal string tension.
