Geometry of curved Yang-Mills-Higgs gauge theories

TL;DR

This thesis explores the geometric structure of curved Yang-Mills-Higgs gauge theories, generalizing classical gauge theory through Lie algebroids, and investigates their reformulation, gauge transformations, and equivalence classes.

Contribution

It introduces a coordinate-free reformulation of CYMH gauge theories, analyzes gauge transformations via Lie algebroid connections, and studies invariance under field redefinitions.

Findings

Reformulation of CYMH gauge theories with coordinate-free approach

Analysis of gauge transformation algebra via Lie algebroid connections

Identification of equivalence classes with flat or zero structures

Abstract

This is my Ph.D. thesis defended at 31 May 2021, and it is devoted to the study of the geometry of curved Yang-Mills-Higgs gauge theory (CYMH GT), a theory introduced by Alexei Kotov and Thomas Strobl. This theory reformulates classical gauge theory, in particular, the Lie algebra (and its action) is generalized to a Lie algebroid $E$, equipped with a connection $\nabla$, and the field strength has an extra term $\zeta$. In the classical situation $E$ is an action Lie algebroid, $\nabla$ is then the canonical flat connection with respect to such an $E$, and $\zeta\equiv 0$. The shortened main results of this Ph.D.thesis are the following; see the abstract in the thesis itself for more information: 1. Reformulating curved Yang-Mills-Higgs gauge theory, also including a thorough introduction and a coordinate-free formulation. Especially the infinitesimal gauge transformation will be generalized to a derivation on vector bundle $V$-valued functionals, induced by a Lie algebroid connection. 2. We will discuss what type of connection for the definition of the infinitesimal gauge transformation should be used, and this is argued by studying the commutator of two infinitesimal gauge transformations, viewed as derivations on $V$-valued functionals. We take the connection on $W$ then in such a way that the commutator is again an infinitesimal gauge transformation. 3. Defining an equivalence of CYMH GTs given by a field redefinition. In order to preserve the physics, this equivalence is constructed in such a way that the Lagrangian of the studied theory is invariant under this field redefinition. It is then natural to study whether there are equivalence classes admitting representatives with flat $\nabla$ and/or zero $\zeta$, and we will do so especially for Lie algebra bundles, tangent bundles and their direct products as Lie algebroids.