Integrating curved Yang-Mills gauge theories

Abstract

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 $μ$ on $\mathcal{G}$: $μ$ has to be a multiplicative form, i.e.\ closed w.r.t.\ a certain simplicial differential $δ$ on $\mathcal{G}$, and the curvature $R_μ$ of $μ$ has to be $δ$-exact with primitive $ζ$; $μ$ will be the generalization of the Maurer-Cartan form of the classical gauge theory, while the $δ$-exactness of $R_μ$ 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 (topological) 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 $\mathcal{P}$, too, by adding $ζ$ to $F$. Concluding with a description of how redo those steps when $\mathcal G$ is a Lie groupoid, and several examples for a gauge theory with a curved $μ$ will be provided, including the inner group bundle of the Hopf fibration $\mathbb{S}^7 \to \mathbb{S}^4$, and we include a classification for whether these theories admit a classical description.