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

TL;DR

This paper analyzes transition amplitudes in a 3D quantum gravity model using coloured tensor graphs, revealing topological and holographic properties, especially for simple boundary topologies like the 2-sphere.

Contribution

It introduces a systematic topological expansion of transition amplitudes in the coloured Boulatov model and explores their factorization and leading order behavior.

Findings

Transition amplitudes factorize for the 2-sphere boundary.

Leading order graphs correspond to the 3-ball topology.

First step towards holographic understanding of coloured GFT models.

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.