Integrating curved Yang-Mills gauge theories
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912356449124352 |
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| author | Fischer, Simon-Raphael |
| author_facet | Fischer, Simon-Raphael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02924 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Integrating curved Yang-Mills gauge theories Fischer, Simon-Raphael Mathematical Physics High Energy Physics - Theory Primary 53D17, Secondary 81T13, 17B99 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. |
| title | Integrating curved Yang-Mills gauge theories |
| topic | Mathematical Physics High Energy Physics - Theory Primary 53D17, Secondary 81T13, 17B99 |
| url | https://arxiv.org/abs/2210.02924 |