相对多辛几何的应用
Periods, prequantization, and rigidity in relative multisymplectic geometry
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中文总结 AI 辅助
本文研究相对多辛几何，通过构建相对积分理论，证明了关于闭相对形式、相对Weil - Kostant定理等多个结论，并展示了拟哈密顿\(G\)空间矩映射理论在融合下的典范函子性，还对相关领域进行了展望。
中文摘要 AI 辅助
相对多辛几何研究光滑映射\(F\colon M\to N\)，其在\(F\)的映射锥复形中配备闭的、非退化的相对\((n + 1)\)形式\(\varpi\)，以及作者早期工作中发展的相关相对可观测量的李\(n\)代数和相对同伦矩映射。本文旨在通过系统收集应用来展示该框架的范围，每个应用都有完整的陈述和证明。构建了相对积分理论并用于证明：闭相对形式是作用泛函的拓扑项；相对Weil - Kostant定理；相对Noether定理；相对辛情形下的刚性定理；拟哈密顿\(G\)空间矩映射理论在融合下的典范函子性。最后展望了流体动力学、狄拉克几何和约化。
英文摘要
Relative multisymplectic geometry replaces differential forms on a single manifold by cocycles in the mapping cone of a smooth map $F\colon M\to N$. Building on the relative Cartan calculus, the Lie $n$-algebras of relative observables, and the relative homotopy moment maps developed in companion work, we establish a range of applications showing that the framework is a working tool rather than a formal generalization. We first construct the integration pairing between relative differential forms and smooth relative chains, and prove two structural results that make it usable: a period criterion, reducing relative integrality to the periods of the target form together with a defect homomorphism on the classes killed by $F_*$, and a functoriality theorem for morphisms of arrows. The criterion yields a structure theorem for levels: the integers $k$ at which $k\varpi$ is relatively integral always form a cyclic group $k_0\mathbb{Z}$, with no hypothesis on $F$, and $k_0$ is computable from finitely many integrals whenever the relevant homology is finitely generated. Together these tools yield a characterization of homotopy-invariant bulk--boundary action functionals, hence a precise treatment of Wess--Zumino terms; a relative Weil--Kostant theorem, whose specialization to a Lagrangian submanifold is the Bohr--Sommerfeld condition of geometric quantization; a relative Noether identity, with a bulk--boundary splitting of the conserved charges; and a rigidity theorem making comoment maps unique, strict and equivariant, so that the Kostant--Souriau cocycle disappears. Two closing sections analyse the degenerate edges $M=\varnothing$ and $N=\mathrm{pt}$, at which the absolute theories are recovered, and separate weak from strong nondegeneracy, determining which results require which.
