Homotopy lattice gauge fields 1: The fields and their properties

TL;DR

This paper introduces homotopy lattice gauge fields (HLGFs), enriching traditional lattice gauge theory with higher-dimensional parallel transport to capture homotopies, and explores their topological and quantum field theory implications.

Contribution

It develops a new framework for gauge fields incorporating higher homotopy data, providing a refined approach to lattice gauge theory without requiring higher category theory background.

Findings

Defines HLGFs with higher parallel transport

Establishes formulas for topological charge in 2D

Refines standard lattice gauge theory with higher homotopy data

Abstract

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.