发表机构
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进研究所数学与科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出高阶几何框架，通过靶空间形状与庞特里亚金对偶刻画磁性高阶形式对称性，证明纤维化与分解定理，并以Pin(2)规范理论为例展示新对称性。
AI 中文摘要
我们为预量子场论的磁性高阶形式对称性发展了一个高阶几何框架。具有层状靶空间的sigma模型的磁性（拓扑）荷由靶空间的形状编码，等价地由其场层的形状编码，该形状与控制电对称性的平坦自同构群是op-对偶的。磁性对称背景通过将这些荷与选定的拓扑场论模栈对偶化而获得，推广了庞特里亚金对偶性，并通过堆叠作用于理论。内聚模态允许具有非有限和非离散数据的靶空间。当对偶化栈是一个群状 $E_\infty$-对象时，我们证明了一个纤维化定理，将电规范理论中的磁性对称性用未规范理论和规范群的磁性对称性来表达，该定理由一个具有显式第一页的同伦不动点谱序列计算。对于带有联络的高阶 $U(1)$-gerbes 模栈，编码体拓扑局域拉格朗日量，我们证明了一个分解定理：任何连通靶空间的磁性对称背景非规范地分裂为其积分同调群的庞特里亚金对偶，并且必然是平坦的，这是将离散荷与带联络的靶空间对偶化的结果。我们用纯 $\mathrm{Pin}^\pm(2)$ 规范理论说明这两个定理：$O(2)$ 在形式度 $d-3$、$d-4$ 和 $d-5$ 上携带可逆的磁性 $\mathbb{Z}_2$ 对称性，而这些对称性对于 $\mathrm{Pin}^-(2)$ 是不存在的，前两个由谱序列的单个微分区分。规范化和物理应用将在后续工作中发展。
英文摘要
We develop a higher-geometric framework for magnetic higher-form symmetries of pre-quantum field theories. The magnetic (topological) charges of a sigma-model with stacky target are encoded by the shape of the target, equivalently of its stack of fields, which is op-dual to the flat automorphism group controlling electric symmetries. Magnetic symmetry backgrounds are obtained by dualizing these charges against a chosen moduli stack of topological field theories, generalizing Pontryagin duality, and act on the theory by stacking. Cohesive modalities allow for targets with non-finite and non-discrete data. When the dualizing stack is a grouplike $E_\infty$-object, we prove a fibration theorem expressing the magnetic symmetries of an electrically gauged theory in terms of those of the ungauged theory and of the gauge group, computed by a homotopy-fixed-point spectral sequence with an explicit first page. For moduli stacks of higher $U(1)$-gerbes with connection, encoding bulk topological local Lagrangians, we prove a factorization theorem: the magnetic symmetry backgrounds of any connected target split, non-canonically, into Pontryagin duals of its integral homology groups, and are necessarily flat, as a consequence of dualizing discrete charges against a target with connections. We illustrate both theorems with pure $\mathrm{Pin}^\pm(2)$ gauge theory: $O(2)$ carries invertible magnetic $\mathbb{Z}_2$ symmetries in form degrees $d-3$, $d-4$ and $d-5$ that are absent for $\mathrm{Pin}^-(2)$, the first two distinguished by a single differential of the spectral sequence. Gauging and physical applications are developed in upcoming work.
Comments70pp. + Appendices. Comments welcome!
