论文标题
集成弯曲的杨麦尔仪表理论
Integrating curved Yang-Mills gauge theories
论文作者
论文摘要
我们构建了一个基于主体捆绑$ \ MATHCAL {p} $的量规理论，配备了正确的$ \ Mathcal {g} $ - action，其中$ \ Mathcal {g} $是一个lie group bundle，而不是lie couts。由于$ \ MATHCAL {G} $ - 动作通过光纤起作用，因此仅通过右转式ACT进行切线向量，现在仅在$ \ Mathcal {p} $的垂直结构上。因此，我们使用$ \ Mathcal {g} $上的连接概括了Pushforwards，该连接将修改PushForward。在这种修改后的PushForward下，$ \ Mathcal {P} $不变的水平分布将提供Ehresmann连接的适当概念。为了实现量规不变性，我们对连接1形$μ$施加条件，将$ \ nathcal {g} $：$μ$：$μ$必须是一种乘法形式，\ textIt { $δ$ - excant ting primitive $ζ$; $μ$将是对经典仪表理论的Maurer-Cartan形式的概括，而$r_μ$的$δ$ - 表现将推广毛rer-cartan方程的作用。这引入了乘法Yang-Mills连接的概念，该连接有助于对单数叶子进行分类和对称性破坏。为了在动态理论中允许在$ \ mathcal {g} $上进行曲面连接，我们将需要通过将$ζ$添加到$ f $上来概括曲率/场强$ f $ $ \ mathcal {p} $上的典型定义。 将提供一些具有弯曲$μ$的量规理论的示例，包括HOPF纤维$ \ MATHBB {S}^7 \ to \ Mathbb {S}^4 $的内部组捆绑，以及将提供与结构半杂音组的分类的分类，包括这些分类的分类，包括这些类别的描述。
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.
