Curved Yang-Mills-Higgs gauge theories in the case of massless gauge bosons

TL;DR

This paper explores a covariantized formulation of Yang-Mills-Higgs theories using Lie algebroids, classifies connections, and investigates obstructions related to flatness and field redefinitions in the context of massless gauge bosons.

Contribution

It introduces a classification of connections in covariantized Yang-Mills-Higgs theories and links obstructions to Mackenzie's extension theory, providing insights into the structure of massless gauge theories.

Findings

Obstruction class relates to Mackenzie's extension theory.

Field redefinitions can alter the curvature and $ abla$ without changing physics.

The tensor $ ext{d}^ abla ext{zeta}$ measures Bianchi identity failure.

Abstract

Alexei Kotov and Thomas Strobl have introduced a covariantized formulation of Yang-Mills-Higgs gauge theories whose main motivation was to replace the Lie algebra with Lie algebroids. This allows the introduction of a possibly non-flat connection $\nabla$ on this bundle, after also introducing an additional 2-form $\zeta$ in the field strength. We will study this theory in the simplified situation of Lie algebra bundles, hence, only massless gauge bosons, and we will provide a physical motivation of $\zeta$. Moreover, we classify $\nabla$ using the gauge invariance, resulting into that $\nabla$ needs to be a Lie derivation law covering a pairing $\Xi$, as introduced by Mackenzie. There is also a field redefinition, keeping the physics invariant, but possibly changing $\zeta$ and the curvature of $\nabla$. We are going to study whether this can lead to a classical theory, and we will realize that this has a strong correspondence to Mackenzie's study about extending Lie algebroids with Lie algebra bundles. We show that Mackenzie's obstruction class is also an obstruction for having non-flat connections which are not related to a flat connection using the field redefinitions. This class is related to $\mathrm{d}^\nabla \zeta$, a tensor which also measures the failure of the Bianchi identity of the field strength and which is invariant under the field redefinition. This tensor will also provide hints about whether $\zeta$ can vanish after a field redefinition.