Dynamical generalization of Yetter's model based on a crossed module of discrete groups

TL;DR

This paper introduces a new dynamical lattice model based on crossed modules of finite groups, generalizing topological theories and gauge theories, with potential insights into their phase structures and geometric properties.

Contribution

It constructs a novel dynamical lattice model using crossed modules, extending previous topological and gauge theories to include non-abelian structures and dynamics.

Findings

Model reduces to known topological theories in certain limits

Establishes a connection between crossed module geometry and gauge theory properties

Provides a framework for analyzing phase diagrams of complex gauge models

Abstract

We construct a dynamical lattice model based on a crossed module of possibly non-abelian finite groups. Its degrees of freedom are defined on links and plaquettes, while gauge transformations are based on vertices and links of the underlying lattice. We specify the Hilbert space, define basic observables (including the Hamiltonian) and initiate a~discussion on the model's phase diagram. The constructed model generalizes, and in appropriate limits reduces to, topological theories with symmetries described by groups and crossed modules, lattice Yang-Mills theory and $2$-form electrodynamics. We conclude by reviewing classifying spaces of crossed modules, with an emphasis on the direct relation between their geometry and properties of gauge theories under consideration.