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Bubble divergences from twisted cohomology

Valentin Bonzom, Matteo Smerlak

TL;DR

The paper tackles bubble divergences in flat spinfoams by introducing twisted cohomology to quantify divergence degrees, proving that the dominant, regulator-free part of the partition function is governed by the second twisted Betti number on the non-singular flat connection space. It then connects this dominant contribution to the Reidemeister torsion of the foam, extending Barrett and Naish-Guzmán’s results to more general settings and to critical (non-regular) connections under suitable assumptions. The authors develop a heat-kernel regularization to extract the divergence degree, analyze its behavior under presentations of the fundamental group (Tietze moves), and illustrate the framework explicitly in SU(2) Yang–Mills theory on closed orientable surfaces, including detailed treatment of 2D cases like the torus. They further provide explicit torsion-based expressions for the dominant amplitude, distinguishing irreducible and reducible flat connections, and discuss limitations arising from singularities of the representation variety. The work offers a principled route to defining finite topological invariants in the flat spinfoam setting and outlines future directions for connecting to 3D Ponzano–Regge and 4D spinfoam models.

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.

Bubble divergences from twisted cohomology

TL;DR

The paper tackles bubble divergences in flat spinfoams by introducing twisted cohomology to quantify divergence degrees, proving that the dominant, regulator-free part of the partition function is governed by the second twisted Betti number on the non-singular flat connection space. It then connects this dominant contribution to the Reidemeister torsion of the foam, extending Barrett and Naish-Guzmán’s results to more general settings and to critical (non-regular) connections under suitable assumptions. The authors develop a heat-kernel regularization to extract the divergence degree, analyze its behavior under presentations of the fundamental group (Tietze moves), and illustrate the framework explicitly in SU(2) Yang–Mills theory on closed orientable surfaces, including detailed treatment of 2D cases like the torus. They further provide explicit torsion-based expressions for the dominant amplitude, distinguishing irreducible and reducible flat connections, and discuss limitations arising from singularities of the representation variety. The work offers a principled route to defining finite topological invariants in the flat spinfoam setting and outlines future directions for connecting to 3D Ponzano–Regge and 4D spinfoam models.

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.

Paper Structure

This paper contains 16 sections, 3 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Discrete connections and twisted cohomology
  3. The flat spinfoam model
  4. Foams and bubbles
  5. Gauge transformations
  6. Flat connections and twisted cohomology
  7. Divergence degree away from singularities
  8. Heat kernel regularization
  9. Transformations under changes of presentation of the fundamental group
  10. Divergence degree on the space of non-singular flat connections
  11. $\mathop{\mathrm{SU}}\nolimits(2)$ Yang-Mills theory on closed orientable surfaces
  12. Relation to Reidemeister torsion
  13. Extracting the torsion
  14. Irreducible connections
  15. Reducible connections
  16. ...and 1 more sections

Key Result

Theorem 1

Whenever the ratio in (amplitude scaling) is integrable with respect to the Riemannian volume form on the set of non-singular flat connections ${\mathcal{F}}_{0}$, the divergence degree of a closed foam is given by where $b^2_{0}(\Gamma,G)$ is the value of the second twisted Betti number on ${\mathcal{F}}_{0}$.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • Theorem 2
  • Theorem 3