Homology in Abelian Lattice Models

TL;DR

This paper explores abelian lattice gauge theories on arbitrary topologies, reformulating the models using homology classes instead of dual lattices, which simplifies the analysis of topological modes in 2D and 3D.

Contribution

It introduces a duality-free approach to abelian lattice models that highlights homology classes as topological modes, avoiding issues with irregular dual complexes.

Findings

Homology classes characterize topological modes in lattice gauge theories.

The approach simplifies the treatment of irregular dual complexes.

Detailed analysis provided for 2D and 3D cases.

Abstract

We study abelian lattice gauge theory defined on a simplicial complex with arbitrary topology. The use of dual objects allows one to reformulate the theory in terms of new dynamical variables; however, we avoid the use of the dual lattice entirely. Topological modes which are present in the transformation now appear as homology classes, in contrast to the cohomology modes found in the dual cell picture. Irregularities of dual cell complexes do not arise in this approach. We treat the two and three dimensional cases in detail.