Spin Foam Diagrammatics and Topological Invariance

TL;DR

This paper introduces a diagrammatic approach to prove the topological invariance of the Turaev-Viro model, extending the proof to arbitrary cellular decompositions and broad classes of quantum groups, with implications for quantum gravity models.

Contribution

It provides a novel diagrammatic proof of topological invariance for the Turaev-Viro model and generalizes it to arbitrary cellular decompositions and quantum groups.

Findings

Proved topological invariance using diagrammatic identities.

Extended invariance to arbitrary cellular decompositions.

Applicable to a broad class of quantum groups.

Abstract

We provide a simple proof of the topological invariance of the Turaev-Viro model (corresponding to simplicial 3d pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF-theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, being proposed as models of quantum gravity in 4d, under refinement of the cellular decomposition.