Bubble divergences from cellular cohomology

TL;DR

This paper investigates bubble divergences in lattice topological field theories, showing that divergence degrees relate to the second Betti number in specific cases like simply connected complexes or Abelian groups.

Contribution

It clarifies the conditions under which bubble divergence degrees correspond to the second Betti number in certain lattice topological field theories.

Findings

Divergence degree equals the second Betti number for simply connected 2-complexes.

Divergence degree equals the second Betti number for models with Abelian structure groups.

The general expectation of divergence degree being the number of bubbles is not always valid.

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.