3d Gravity as a random ensemble

We give further evidence that the matrix-tensor model studied in \cite{belin2023} is dual to AdS$_{3}$ gravity including the sum over topologies. This provides a 3D version of the duality between JT gravity and an ensemble of random Hamiltonians, in which the matrix and tensor provide random CFT$_2$ data subject to a potential that incorporates the bootstrap constraints. We show how the Feynman rules of the ensemble produce a sum over all three-manifolds and how surgery is implemented by the matrix integral. The partition functions of the resulting 3d gravity theory agree with Virasoro TQFT (VTQFT) on a fixed, hyperbolic manifold. However, on non-hyperbolic geometries, our 3d gravity theory differs from VTQFT, leading to a difference in the eigenvalue statistics of the associated ensemble. As explained in \cite{belin2023}, the Schwinger-Dyson (SD) equations of the matrix-tensor integral play a crucial role in understanding how gravity emerges in the limit that the ensemble localizes to exact CFT's. We show how the SD equations can be translated into a combinatorial problem about three-manifolds.