Combinatorial Invariants from Four Dimensional Lattice Models

TL;DR

This paper investigates four-dimensional lattice gauge theories with groups Z2 and Z3, demonstrating their subdivision invariance and deriving combinatorial invariants of simplicial complexes, with implications for topological invariants.

Contribution

It introduces subdivision invariance of Boltzmann weights in 4D lattice models and constructs combinatorial invariants from these gauge theories.

Findings

Boltzmann weights are invariant under subdivision moves at specific coupling values

Partition function acts as a combinatorial invariant for boundaryless complexes

An extra phase factor appears under boundary subdivision moves

Abstract

We study the subdivision properties of certain lattice gauge theories based on the groups $Z_{2}$ and $Z_{3}$, in four dimensions. The Boltzmann weights are shown to be invariant under all type $(k,l)$ subdivision moves, at certain discrete values of the coupling parameter. The partition function then provides a combinatorial invariant of the underlying simplicial complex, at least when there is no boundary. We also show how an extra phase factor arises when comparing Boltzmann weights under the Alexander moves, where the boundary undergoes subdivision.