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.
SO_0(1,d+1) Racah coefficients: Type I representations
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We use AdS/CFT inspired methods to study the Racah coefficients for type I representations of the Lorentz group SO_0(1,d+1) with d>1. For such representations (a multiple of) the Racah coefficient can be represented as an integral of a product of 6 bulk-to-bulk propagators over 4 copies of the hyperbolic space H_{d+1}. To compute the integrals we represent the bulk-to-bulk propagators in terms of bulk-to-boundary ones. The bulk integrals can be computed explicitly, and the boundary integrations are carried out by introducing Feynman parameters. The final result is an integral representation of the Racah coefficient given by 4 Barnes-Mellin type integrals.
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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.