The Tulczyjew triple for gauge field theories

TL;DR

This paper develops a new geometric framework based on the Tulczyjew triple for gauge field theories, simplifying the description by focusing on connection and curvature, and applies it to Yang-Mills theory.

Contribution

It introduces a novel Tulczyjew triple formalism tailored for gauge theories, including new geometric structures and a reduction method for systems depending on connection and curvature.

Findings

Constructed a Tulczyjew triple for gauge field theories.

Reduced the formalism to systems depending on connection and curvature.

Applied the framework to Yang-Mills theory.

Abstract

In this article we construct and discuss a new rigorous geometric formalism for gauge field theories. The basis of our work is the notion of the Tulczyjew triple, a geometric structure which successfully solved numerous problems in mathematical description of mechanics and classical field theory. In particular, we construct a Tulczyjew triple for gauge field theories and reduce it for systems that depend only on the value of a connection and curvature instead of the entire first jet of the gauge field. We also introduce new geometric structures such as the vector-affine product of bundles and analyse the connection bundle from a new perspective. Finally, we apply the derived formalism to Yang-Mills theory.