General Solution of the non-abelian Gauss law and non-abelian analogs of the Hodge decomposition

TL;DR

This paper provides a general solution to the non-abelian Gauss law and introduces two non-abelian Hodge decompositions, advancing understanding of gauge field dualities in three-dimensional Yang-Mills theories.

Contribution

It presents novel decompositions of non-abelian vector fields and solutions to the Gauss law, extending classical Hodge theory to non-abelian gauge contexts.

Findings

Decomposition of isotriplet vector fields into covariant curl and gradient.

A new non-abelian Hodge decomposition involving a magnetic field of a Yang-Mills potential.

Relevance of these results for duality transformations in non-abelian gauge theories.

Abstract

General solution of the non-abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-abelian analogs of the Hodge decomposition in three dimensions are addressed. i) Decomposition of an isotriplet vector field $V_{i}^{a}(x)$ as sum of covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained. ii) A decomposition of the form $V_{i}^{a}=B_{i}^{a}(C)+D_{i}(C) \phi^{a} $ which involves non-abelian magnetic field of a new Yang-Mills potential C is also presented. These results are relevant for duality transformation for non-abelian gauge fields.