Yang-Mills Fields and Riemannian Geometry

TL;DR

This paper introduces a geometric, gauge-invariant variable framework for Yang-Mills theories that explicitly enforces Gauss' law and reveals additional constraints, with a focus on the $SU(2)$ gauge group and extensions to $SU(N>2)$.

Contribution

It develops a Riemannian geometric formalism for Yang-Mills fields using gauge-invariant variables, extending to static sources and larger gauge groups.

Findings

Gauge-invariant variables satisfy Gauss' law explicitly

The underlying geometry is Riemannian based on $GL(3)$ for $SU(2)$

Formalism extends to include static color sources and larger gauge groups

Abstract

It is possible to define new, gauge invariant variables in the Hilbert space of Yang-Mills theories which manifestly implement Gauss' law on physical states. These variables have furthermore a geometrical meaning, and allow one to uncover further constraints physical states must satisfy. For gauge group $SU(2)$, the underlying geometry is Riemannian and based on the group $GL(3)$. The formalism allows also for the inclusion of static color sources and the extension to gauge groups $SU(N>2)$, both of which are discussed here.