Diffeomorphisms and spin foam models

TL;DR

This paper identifies a residual diffeomorphism-related (translational) gauge symmetry that survives discretization in 3D spin foam models, showing that the commonly encountered divergences originate from the infinite volume of this gauge group. By performing a precise Fadeev–Popov gauge fixing, the authors demonstrate that dividing by the gauge volume yields the Ponzano–Regge amplitudes and explains the renormalization historically used in these models. They connect the discrete translational symmetry to a discretized Bianchi identity and show how a maximal-tree gauge fixing collapses the triangulation while preserving the overall partition function, suggesting that the continuum limit respects diffeomorphism symmetry once gauge fixed. The discussion extends to higher dimensions, highlighting how residual diffeomorphisms might act on spin foam labels and how a positive cosmological constant (via Turaev–Viro) yields finite models, whereas some finite models may effectively break diffeomorphism invariance. These insights clarify the role of gauge symmetry in spin foams and have implications for constructing finite, physically meaningful models in 4D.

Abstract

We study the action of diffeomorphisms on spin foam models. We prove that in 3 dimensions, there is a residual action of the diffeomorphisms that explains the naive divergences of state sum models. We present the gauge fixing of this symmetry and show that it explains the original renormalization of Ponzano-Regge model. We discuss the implication this action of diffeomorphisms has on higher dimensional spin foam models and especially the finite ones.