# A Topological Way of Finding Solutions to Yang-Mills Equation

## TL;DR

This paper introduces a topological method based on Wightman axioms to systematically find solutions to the classical Yang-Mills equations, including new solutions, in both flat and curved spacetimes.

## Contribution

It presents a novel approach leveraging form invariance conditions to systematically derive Yang-Mills solutions with nontrivial topology.

## Key findings

- Recovered known Yang-Mills solutions as special cases
- Produced new solutions not previously reported
- Applicable to both flat and curved spacetimes

## Abstract

We propose a systematic way of finding solutions to classical Yang-Mills equation with nontrivial topology. This approach is based on one of Wightman axioms for quantum field theory, which is referred to as form invariance condition in this paper. For a given gauge group and a spacetime with certain isometries, thanks to this axiom that imposes strong constraints on the general Ansatz, a systematic way of solving Yang-Mills equation can be obtained in both flat and curved spacetimes. In order to demonstrate this method, we recover various known solutions as special cases as well as produce new solutions not previously reported in the literature.