Integrating curved Yang-Mills gauge theories

We construct a gauge theory based on principal bundles $\mathcal{P}$ equipped with a right $\mathcal{G}$-action, where $\mathcal{G}$ is a Lie group bundle instead of a Lie group. Due to the fact that a $\mathcal{G}$-action acts fibre by fibre, pushforwards of tangent vectors via a right-translation act now only on the vertical structure of $\mathcal{P}$. Thus, we generalize pushforwards using a connection on $\mathcal{G}$ which will modify the pushforward. A horizontal distribution on $\mathcal{P}$ invariant under such a modified pushforward will provide a proper notion of Ehresmann connection. For achieving gauge invariance we impose conditions on the connection 1-form $\mu$ on $\mathcal{G}$: $\mu$ has to be a multiplicative form, \textit{i.e.}\ closed w.r.t.\ a certain simplicial differential $\delta$ on $\mathcal{G}$, and the curvature $R_\mu$ of $\mu$ has to be $\delta$-exact with primitive $\zeta$; $\mu$ will be the generalization of the Maurer-Cartan form of the classical gauge theory, while the $\delta$-exactness of $R_\mu$ will generalize the role of the Maurer-Cartan equation. This introduces the notion of multiplicative Yang-Mills connections, a connection which helped classifying singular foliations and symmetry breaking. For allowing curved connections on $\mathcal{G}$ in the dynamical theory we will need to generalize the typical definition of the curvature/field strength $F$ on $\mathcal{P}$ by adding $\zeta$ to $F$. Several examples for a gauge theory with a curved $\mu$ will be provided, including the inner group bundle of the Hopf fibration $\mathbb{S}^7 \to \mathbb{S}^4$, and a classification for gauge theories with structural semisimple group bundles will be provided, including a classification for whether these theories admit a classical description.