Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

Abstract

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.

arXiv comment42+15 pages, 28+14 figures; revised version matching article published in Annales Henri Poincaré

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• Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model