Bubble divergences: sorting out topology from cell structure

TL;DR

This work derives an exact powercounting formula for bubble divergences in the flat spinfoam model by employing twisted cohomology, showing the divergence degree ${\Omega(\Gamma,G)}$ equals the second twisted Betti number ${b^2_\phi(\Gamma,G)}$ and, for 2-skeletons of cell decompositions ${\Delta_M}$ of a pseudomanifold ${M}$, decomposes as ${\Omega(\Gamma,G)=I(M,G)+\omega(\Delta_M,G)}$ with a topology term ${I(M,G)}$ and a cellular term ${\omega(\Delta_M,G)}$. The paper develops discrete connections, twisted cohomology, and Laplace-approximation techniques to compute divergences, and demonstrates how the approach reproduces known results from Pachner moves, bubble counting, and jacket-based analyses while clarifying when ${\Omega}$ need not be an integer multiple of ${\dim G}$. Through three-dimensional examples, it illustrates how the topology of ${M}$ (via ${\pi_1(M)}$ and ${\chi(M)}$) governs the divergent behavior, and it connects to Gurau’s colored tensor models and the associated ${1/N}$ expansion. Overall, the framework provides a unified, topologically aware powercounting method that encompasses prior results and offers algebraic-topological insight into divergences in group field theories.

Abstract

We conclude our analysis of bubble divergences in the flat spinfoam model. In [arXiv:1008.1476] we showed that the divergence degree of an arbitrary two-complex Gamma can be evaluated exactly by means of twisted cohomology. Here, we specialize this result to the case where Gamma is the two-skeleton of the cell decomposition of a pseudomanifold, and sharpen it with a careful analysis of the cellular and topological structures involved. Moreover, we explain in detail how this approach reproduces all the previous powercounting results for the Boulatov-Ooguri (colored) tensor models, and sheds light on algebraic-topological aspects of Gurau's 1/N expansion.