Bubble divergences from cellular cohomology

TL;DR

The paper investigates bubble divergences in spinfoam-type path integrals for lattice topological gauge theories with flat $G$-connections. It introduces a homological notion of bubbles via the second Betti number and analyzes the divergence degree $\Omega(\Gamma,G)$ using heat-kernel regularization and Laplace methods. In the special cases where the foam is simply connected or the gauge group is Abelian, it proves $\Omega(\Gamma,G)=(\dim G)\,b_2(\Gamma)$; for a single degenerate flat connection it provides bounds involving $b_2(\Gamma)$ and the Euler characteristic $\chi(\Gamma)$. The paper argues that, in general, the divergence degree depends on twisted cohomology around flat connections and not merely on the topology, motivating a refined local analysis for non-Abelian cases.

Abstract

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, 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 in general, because of a phenomenon called `bubble divergences'. A common expectation is that the degree of these divergences is given by the number of `bubbles' of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian -- in both cases, the divergence degree is given by the second Betti number of the 2-complex.