2D discrete Yang-Mills equations on the torus

TL;DR

This paper develops a discretization scheme for 2D Yang-Mills equations on a torus using discrete exterior calculus, defining discrete operators that preserve geometric features and formulating the equations as difference systems and matrices.

Contribution

It introduces a novel discretization framework for 2D Yang-Mills equations on a torus based on discrete exterior calculus, capturing key geometric properties.

Findings

Discrete exterior covariant derivative and adjoint are defined.

Yang-Mills equations are formulated as difference equations and matrices.

Framework preserves essential geometric features of the continuous theory.

Abstract

In this paper, we introduce a discretization scheme for the Yang-Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang-Mills equations are presented in the form of both a system of difference equations and a matrix form.