Holonomy Spin Foam Models: Definition and Coarse Graining

TL;DR

This paper introduces a new holonomy formulation for spin foam models that extends their applicability to arbitrary complexes and finite groups, enabling numerical coarse graining and connecting with lattice gauge theories.

Contribution

It defines a holonomy-based spin foam framework that generalizes existing models and facilitates coarse graining, especially for finite groups, bridging spin foams with lattice gauge theories.

Findings

New holonomy formulation for spin foams

Extension to arbitrary two-complexes and finite groups

Enables numerical coarse graining methods

Abstract

We propose a new holonomy formulation for spin foams, which naturally extends the theory space of lattice gauge theories. This allows current spin foam models to be defined on arbitrary two-complexes as well as to generalize current spin foam models to arbitrary, in particular finite groups. The similarity with standard lattice gauge theories allows to apply standard coarse graining methods, which for finite groups can now be easily considered numerically. We will summarize other holonomy and spin network formulations of spin foams and group field theories and explain how the different representations arise through variable transformations in the partition function. A companion paper will provide a description of boundary Hilbert spaces as well as a canonical dynamic encoded in transfer operators.