Triangulating Quantum Gravity in AdS3

I will discuss the path integral of pure 3D gravity on a finite region of spacetime, with boundary conditions that fix dihedral angles or geodesic lengths. This amplitude calculates corrections to the Gaussian statistics of OPE coefficients in the dual CFT. The fixed-length path integral is related to Virasoro TQFT; the fixed-angle path integral is the partition function of Conformal Turaev-Viro theory, a novel topological theory based on triangulations; and the two are related by a modular S-transform. This leads to a general procedure to calculate the exact path integral on hyperbolic 3-manifolds by gluing together generalized tetrahedra. The concrete new results from this approach are in AdS3, but this point of view on statistics in the gravitational path integral can possibly be adapted to other settings like higher dimensions and de Sitter space.