Lie Algebroid Yang Mills with Matter Fields

TL;DR

This paper extends Lie algebroid Yang-Mills theories by relaxing fiber metric conditions and explores coupling to matter fields, revealing gauge invariance conditions related to covariant constancy.

Contribution

It introduces generalized conditions for gauge invariance in Lie algebroid Yang-Mills theories and discusses matter coupling with nonlinear representations.

Findings

Relaxed fiber metric conditions for gauge invariance

Coupling to scalar fields via nonlinear representations

Gauge invariance implies covariant constancy of fiber metrics

Abstract

Lie algebroid Yang-Mills theories are a generalization of Yang-Mills gauge theories, replacing the structural Lie algebra by a Lie algebroid E. In this note we relax the conditions on the fiber metric of E for gauge invariance of the action functional. Coupling to scalar fields requires possibly nonlinear representations of Lie algebroids. In all cases, gauge invariance is seen to lead to a condition of covariant constancy on the respective fiber metric in question with respect to an appropriate Lie algebroid connection. The presentation is kept in part explicit so as to be accessible also to a less mathematically oriented audience.