带角点的哈密顿规范理论：约束约化与通量超选择
Hamiltonian gauge theory with corners: constraint reduction and flux superselection
AI总结：
本文提出分阶段 Fréchet 约化方法，将带角点规范理论的动量映射分解为约束映射和通量映射，分别实现约束约化与通量超选择，并揭示约化相空间的部分泊松结构及其辛叶与量子超选择的经典对应。
摘要：
我们研究时空流形上具有余维 $1$ 带边子流形的规范理论。当理论由规范群 $\mathcal{G}$ 作用的局部动量映射描述时，我们通过分阶段 Fréchet 约化来刻画其约化相空间。动量映射分解为一个称为约束映射的体项，它定义了一个余迷向约束集，以及一个称为通量映射的边界项。在第一阶段，即约束约化中，约束集是正规子群 $\mathcal{G}_\circ\subset\mathcal{G}$（称为约束规范群）的动量映射的零点。在第二阶段，即通量超选择中，通量映射是剩余通量规范群 $\underline{\mathcal{G}}\doteq\mathcal{G}/\mathcal{G}_\circ$ 作用的动量映射，该群也控制等变性。理论的约化相空间在光滑时只是一个部分泊松流形 $\underline{\underline{\mathcal{C}}}\simeq \underline{\mathcal{C}}/\underline{\mathcal{G}}$。其辛叶被称为“通量超选择扇区”，因为它们为量子超选择现象提供了经典类比和路线图。对于角点，我们进一步赋予一个泊松流形上的辛李代数胚 $\mathsf{A}_{\partial} \to \mathcal{P}_{\partial}$，并展示在壳构型 $\mathcal{C}_{\partial}\subset\mathcal{P}_{\partial}$ 也是泊松的。$\mathcal{C}_{\partial}$ 和 $\underline{\underline{\mathcal{C}}}$ 都纤维化于一个共同的超选择空间上，标记着两个泊松结构的 Casimir 函数。我们通过明确计算一大类 Yang--Mills 理论的第一和第二阶段约化来展示该形式化框架，其中发现 $\underline{\underline{\mathcal{C}}}$ 是一个 Weinstein 空间，并讨论了对拓扑理论的进一步应用。
英文摘要：
We study gauge theories on spacetime manifolds with a codimension-$1$ submanifold with boundary. We characterise the reduced phase space of the theory whenever it is described by a local momentum map for the action of the gauge group $\mathcal{G}$, by means of Fréchet reduction by stages. The momentum map decomposes into a bulk term called constraint map, defining a coisotropic constraint set, and a boundary term called flux map. In the first stage, constraint reduction, the constraint set is the zero of a momentum map for a normal subgroup $\mathcal{G}_\circ\subset\mathcal{G}$, called constraint gauge group. In the second stage, flux superselection, the flux map is the momentum map for the residual action of the flux gauge group $\underline{\mathcal{G}}\doteq\mathcal{G}/\mathcal{G}_\circ$, which also controls equivariance. The reduced phase space of the theory, when smooth, is then only a partial Poisson manifold $\underline{\underline{\mathcal{C}}}\simeq \underline{\mathcal{C}}/\underline{\mathcal{G}}$. Its symplectic leaves are called \emph{flux superselection sectors}, for they provide a classical analogue of, and a road map to, the phenomenon of quantum superselection. To corners, we further assign a symplectic Lie algebroid over a Poisson manifold, $\mathsf{A}_{\partial} \to \mathcal{P}_{\partial}$, and show how on-shell configurations $\mathcal{C}_{\partial}\subset\mathcal{P}_{\partial}$ are also Poisson. Both $\mathcal{C}_{\partial}$ and $\underline{\underline{\mathcal{C}}}$ fibrate over a common space of superselections, labeling the Casimirs of both Poisson structures. We showcase the formalism by explicitly working out the first and second stage reductions for a broad class of Yang--Mills theories, where $\underline{\underline{\mathcal{C}}}$ is found to be a Weinstein space, and discuss further applications to topological theories.
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