Holography for the Lorentz Group Racah Coefficients

TL;DR

This paper presents a novel holographic approach to representing Lorentz group Racah coefficients using boundary propagators, offering a geometric interpretation related to an extended tetrahedron.

Contribution

It introduces a new holographic construction for Racah coefficients utilizing boundary propagators, simplifying the integral representation.

Findings

Holographic realization involves only boundary objects.

The construction provides a geometric interpretation with an extended tetrahedron.

Alternative method uses boundary-to-boundary propagators instead of bulk-to-bulk.

Abstract

A known realization of the Lorentz group Racah coefficients is given by an integral of a product of 6 ``propagators'' over 4 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 propagator. 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.