Bubble divergences from twisted cohomology

TL;DR

This paper introduces twisted cohomology to analyze bubble divergences in lattice topological field theories, extending previous cohomological methods and relating divergences to Reidemeister torsion, with applications to 2D and 3D models.

Contribution

It develops a new framework using twisted cohomology to compute divergence degrees in models where cellular cohomology is insufficient, generalizing prior results.

Findings

Twisted cohomology effectively captures bubble divergences.

Divergence degree relates to Reidemeister torsion.

Limitations arise from singularities in the representation variety.

Abstract

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as `bubble divergences'. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant {\it twisted} cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of Barrett and Naish-Guzman. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.