Formulation of gauge theories on transitive Lie algebroids

TL;DR

This paper develops a mathematical framework for gauge theories using transitive Lie algebroids, introducing generalized connections, curvature, and action functionals, with applications to Atiyah algebroids and derivations.

Contribution

It formulates gauge theories on transitive Lie algebroids with explicit geometric structures and action functionals, extending traditional gauge theory concepts.

Findings

Established a geometric formulation of gauge theories on transitive Lie algebroids.

Derived explicit action functionals in global and local forms.

Connected the framework to non-commutative gauge theories.

Abstract

In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar non-commutative gauge theories.