Self-dual and anti-self-dual solutions of discrete Yang-Mills equations on a double complex

TL;DR

This paper constructs discrete self-dual and anti-self-dual solutions for SU(2) Yang-Mills equations on a combinatorial model of four-dimensional space, using double complex and quaternionic methods, highlighting analogies with continuous solutions.

Contribution

It introduces a novel discrete model for Yang-Mills equations and constructs explicit self-dual and anti-self-dual solutions using innovative mathematical techniques.

Findings

Discrete solutions analogous to continuous instantons and anti-instantons.

Use of double complex and quaternionic approaches to solve discrete Yang-Mills equations.

Discussion of similarities between discrete and continuous self-dual solutions.

Abstract

We study a discrete model of the SU(2) Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$. Self-dual and anti-self-dual solutions of discrete Yang-Mills equations are constructed. To obtain these solutions we use both techniques of a double complex and the quaternionic approach. Interesting analogies between instanton, anti-instanton solutions of discrete and continual self-dual, anti-self-dual equations are also discussed.