The fixed-angle gravitational path integral in AdS3 equals a Conformal Turaev-Viro partition function, the fixed-length path integral equals a Virasoro TQFT amplitude squared, and the semiclassical geometries are built from generalized hyperbolic tetrahedra.
A volume formula for generalised hyperbolic tetrahedra,
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Triangulating quantum gravity in AdS$_3$
The fixed-angle gravitational path integral in AdS3 equals a Conformal Turaev-Viro partition function, the fixed-length path integral equals a Virasoro TQFT amplitude squared, and the semiclassical geometries are built from generalized hyperbolic tetrahedra.