Homotopy classes of gauge fields and the lattice

TL;DR

This paper introduces extended lattice gauge (ELG) fields, a topological framework that generalizes lattice gauge theory by capturing principal bundle data through homotopy classes of gauge fields on manifolds.

Contribution

It defines ELG fields using homotopical equivalence of parallel transport maps, extending lattice gauge theory to include topological information about principal bundles.

Findings

ELG fields form a covering space over lattice gauge fields.

Connected components of ELG fields classify principal G-bundles.

A criterion is provided for equivalence of bundles over different cotriangulations.

Abstract

For a smooth manifold $M$, possibly with boundary and corners, and a Lie group $G$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of finite-dimensional families of paths relative to $\mathscr{C}$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG fields of a given pair $(M,\mathscr{C})$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $G$-bundles on $M$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.