Schwinger-Dyson equation in three-dimensional simplicial quantum gravity

TL;DR

This paper explores the Schwinger-Dyson equation within three-dimensional simplicial quantum gravity, introducing boundary operators linked to surfaces and proposing conditions that select specific solutions related to triangulations of the 3-sphere.

Contribution

It introduces boundary operators in 3D simplicial quantum gravity and demonstrates their amplitudes satisfy the Schwinger-Dyson equation, proposing conditions to isolate solutions for the 3-sphere topology.

Findings

Amplitudes of boundary operators satisfy Schwinger-Dyson equations.

Factorization conditions select solutions related to $S^3$ triangulations.

The model extends Boulatov's framework with boundary considerations.

Abstract

We study the simplicial quantum gravity in three dimensions. Motivated by the Boulatov's model which generates a sum over simplicial complexes weighted with the Turaev-Viro invariant, we introduce boundary operators in the simplicial gravity associated to compact orientable surfaces. An amplitude of the boundary operator is given by a sum over triangulations in the interior of the boundary surface. It turns out that the amplitude solves the Schwinger-Dyson equation even if we restrict the topology in the interior of the surface, as far as the surface is non-degenerate. We propose a set of factorization conditions on the amplitudes which singles out a solution associated to triangulations of $S^3$.