Formulation of gauge theories on transitive Lie algebroids
Cédric Fournel, Serge Lazzarini, Thierry Masson
TL;DR
The paper develops a gauge-theoretic framework on transitive Lie algebroids by introducing generalized connections and curvatures equipped with metrics, an inner integration along the algebraic kernel, and a Hodge star to define gauge-invariant actions. It shows how to decompose connections into ordinary and Higgs-like sectors via a reduced kernel endomorphism $\tau$, yielding a Yang–Mills–Higgs type action that reduces to ordinary Yang–Mills when $\tau=0$ and generates mass terms for gauge and matter fields when $\tau$ is nontrivial. The formalism is explicitly connected to Atiyah algebroids and to derivation algebroids of vector bundles, establishing isomorphisms between fibre-based and base-based calculi and clarifying relations with noncommutative gauge theory and Kaluza–Klein constructions. These results provide a geometrically natural, Lie-algebroid-based generalization of gauge theories that encompasses ordinary gauge theories as a special case and suggests a structured path to physical applications in forthcoming work.
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.