Holonomy Spin Foam Models: Definition and Coarse Graining

TL;DR

This work presents a holonomy-based unification of spin foam models, casting current theories on arbitrary 2-complexes and, importantly, enabling straightforward coarse graining via lattice-gauge methods. By introducing edge functions $E$ and face weights $oldsymbol{\omega}$, the authors define a state sum that encompasses BF, Barrett–Crane, EPRL, and FK-type models, with parameterizations through irreps and subgroup invariances. The paper clarifies the relations to Operator Spin Foam Models and Group Field Theory, and shows how boundaries induce a universal boundary Hilbert space, facilitating boundary spin-network interpretations. In 2D and 3D, the coarse graining analysis—via exact recursions, Migdal–Kadanoff schemes, and Pachner moves—reveals fixed points akin to BF and high-temperature regimes, with finite-group examples (e.g., $S_3$) illustrating how the E-function space organizes the RG flow and how truncations can capture essential dynamics. Overall, the framework provides a practical and flexible platform for exploring continuum limits, renormalization, and the role of diffeomorphism-like symmetries in spin foam quantum gravity.

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.

Figures (12)

• Figure 1: Holonomy variables associated to half edges ($g_{ve}$) and wedges ($g_{ef}$) of a face.
• Figure 2: Holonomy variables in the dual description of operator spin foam models.
• Figure 3: Graphical notation of the relation between the OSFM version of the edge propagator $P_e$ and the one used in (\ref{['Eq:HolonomyStateSum']}), which is also used in the BC and EPRL models.
• Figure 4: Variables in the group field theory formulation.
• Figure 5: Holonomy variables in the vertex amplitude formulation.