Finite Action Yang-Mills Solutions on the Group Manifold

TL;DR

This paper constructs explicit solutions to Yang-Mills equations on group manifolds using Maurer-Cartan forms, showing finiteness of the action for certain groups like SU(3).

Contribution

It introduces a method to generate Yang-Mills solutions on semi-simple Lie group manifolds using invariant Maurer-Cartan forms, including explicit examples.

Findings

Solutions exist for all semi-simple Lie groups using Maurer-Cartan forms.

The Yang-Mills action is finite for solutions on unitary unimodular groups.

Explicit SU(3) solutions demonstrate the method's applicability.

Abstract

We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable one to construct solutions of the Yang-Mills equations on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of $SU(3)$.