Non Abelian Dual Maps in Path Space

TL;DR

This paper extends duality transformations to non-abelian gauge theories using a path space approach, defining a non-local map that generalizes Hodge duality and reproduces Yang-Mills equations.

Contribution

It introduces a novel path space formulation for non-abelian dual maps, generalizing abelian duality and providing a new framework for non-abelian gauge theories.

Findings

Defines a pre-dual functional in path space.

Introduces a non-local map reducing to Hodge-* duality.

Derives equations representing Yang-Mills and Bianchi identities.

Abstract

We study an extension of the procedure to construct duality transformations among abelian gauge theories to the non abelian case using a path space formulation. We define a pre-dual functional in path space and introduce a particular non local map among Lie algebra valued 1-form functionals that reduces to the ordinary Hodge-* duality map of the abelian theories. Further, we establish a full set of equations on path space representing the ordinary Yang Mills equations and Bianchi identities of non abelian gauge theories of 4-dimensional euclidean space.