Triangulating Quantum Gravity in AdS3

The path integral of pure 3D gravity is formulated on a finite region of spacetime, with boundary conditions that fix dihedral angles or geodesic lengths. This amplitude calculates statistics of black hole matrix elements 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.