PhD Thesis

This thesis is devoted to the study of the geometry of curved Yang-Mills-Higgs gauge theory (\textbf{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 $\zeta$; there is a certain relationship between $\zeta$ and $\nabla$, for example, if $\zeta \equiv 0$, then $\nabla$ is flat. In the classical situation $E$ is an action Lie algebroid, a combination of a trivial Lie algebra bundle and a Lie algebra action, $\nabla$ is then the canonical flat connection with respect to such an $E$, and $\zeta\equiv 0$. The main results of this Ph.D.~thesis are the following:

\begin{itemize} \item Reformulating curved Yang-Mills-Higgs gauge theory, also including a thorough introduction and a coordinate-free formulation, while the original formulation was not completely coordinate-free. Especially the infinitesimal gauge transformation will be generalized to a derivation on vector bundle $V$-valued functionals. Those vector bundles $V$ will be the pullback of another bundle $W$, and the gauge transformation as derivation will be induced by a Lie algebroid connection on $W$, using a more general notion of pullbacks of connections. This also supports the usage of arbitrary types of connections on $W$ in the definition of the infinitesimal gauge transformation, not just canonical flat ones as in the classical formulation. %acting on the bundle in which a functional has values in. \item Studying functionals as parameters of the infinitesimal gauge transformation, supporting a richer set of infinitesimal gauge transformations, especially the parameter itself can have a non-trivial gauge transformation. The discussion about the infinitesimal gauge transformation is also about 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; for this flatness of the connection on $W$ is necessary and sufficient. For $W= E$ and $ W = \mathrm{T}N$ we use a Lie algebroid connection known as basic connection which is not the canonical flat connection in the classical non-abelian situation; this is not the connection normally used in the standard formulation, but it reflects the symmetries of gauge theory better than the usual connection, which is in general not even flat. For $W = \mathbb{R}$ the gauge transformation is uniquely given as the Lie derivative of a vector field on the space of fields given by the field of gauge bosons and the Higgs field, and the commutator is then just the Lie bracket of vector fields; in this case the bracket will also give again a vector field related to gauge transformations. %It will be the so-called basic connection, a generalization of Lie algebra representations. \item Defining an equivalence of CYMH GTs given by a field redefinition which is a transformation of structural data like the field of gauge bosons. 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 $\zeta$: \begin{enumerate} \item On the one hand, the equivalence class related to $E = \mathrm{T}\mathds{S}^7$, $\mathds{S}^7$ the seven-dimensional sphere, admits only representatives with non-flat $\nabla$, while locally the equivalence class of all tangent bundles admits a representative with flat $\nabla$. \item On the other hand, the equivalence class related to "$E=$ LAB" (Lie algebra bundle) has a relation with an obstruction class about extending Lie algebroids by LABs; this will imply that locally there is always a representative with flat $\nabla$ while globally this may not be the case, similar to the previous bullet point. Furthermore, a canonical construction for equivalence classes with no representative with zero $\zeta$ is given, which also works locally, and an interpretation of $\zeta$ as failure of the Bianchi identity of the field strength is provided. \end{enumerate} \end{itemize}