Holography for the Lorentz group Racah coefficients

A known realization of the Lorentz group Racah coefficients is given by an integral of a product of six 'propagators' over four copies of the hyperbolic space. These are 'bulk-to-bulk' propagators in that they are functions of two points in the hyperbolic space. It is known that the bulk-to-bulk propagator can be constructed out of two bulk-to-boundary ones. We point out that there is another way to obtain the same object. Namely, one can use two bulk-to-boundary and one boundary-to-boundary propagators. Starting from this construction and carrying out the bulk integrals we obtain a realization of the Racah coefficients that is 'holographic' in the sense that it only involves boundary objects. This holographic realization admits a geometric interpretation in terms of an 'extended' tetrahedron.