Integrating curved Yang-Mills gauge theories
Simon-Raphael Fischer
TL;DR
The paper extends traditional Yang–Mills gauge theory by promoting the structure group to a Lie group bundle and developing a gauge-theoretic framework with a modified pushforward, multiplicative total Maurer–Cartan forms, and a generalized field strength. Central constructions include the Darboux derivative $ riangle$, the total Maurer–Cartan form $oldsymbol{ u_G}^{ ext{tot}}$, and the generalized curvature $F = d^{ abla^{ ext{YM}}}A(oldsymbol{H},oldsymbol{H}) + oldsymbol{ ho}$, together with a multiplicative Yang–Mills connection on the LGB that satisfies a generalized Maurer–Cartan equation. The work provides a concrete CYM theory, with explicit examples such as the inner group bundle of the Hopf fibration, and a classification for gauge theories based on semisimple LABs, illustrating non-classical descriptions that cannot be flattened by field redefinitions. It also connects these geometric structures to singular foliations and symmetry breaking, suggesting rich mathematical and physical applications beyond classical YM theory. Overall, the framework systematizes curved Yang–Mills gauge theory on LGBs and lays groundwork for further generalizations and applications in gauge theory and differential geometry.
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, \textit{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 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 $ζ$ to $F$. 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 a classification for gauge theories with structural semisimple group bundles will be provided, including a classification for whether these theories admit a classical description.