  This is my Ph.D. thesis which I am going to defend in 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.
