Gauge theory on graphs

TL;DR

This paper develops a discrete gauge theory framework on graphs by defining connection and curvature forms, establishing a Weitzenb"ock formula, and analyzing a Yang-Mills functional and its equations.

Contribution

It introduces novel notions of connection and curvature on graphs, along with a Weitzenb"ock formula and a discrete Yang-Mills functional, advancing discrete gauge theory.

Findings

Established a Weitzenb"ock formula for connection Laplacians on graphs

Defined a discrete Yang-Mills functional and derived its Euler-Lagrange equations

Provided a mathematical foundation for gauge theories in discrete graph settings

Abstract

In this paper, we provide the notions of connection $1$-forms and curvature $2$-forms on graphs. We prove a Weitzenb\"ock formula for connection Laplacians in this setting. We also define a discrete Yang-Mills functional and study its Euler-Lagrange equations.