论文标题
拓扑量子场理论和同质谱系
Topological quantum field theories and homotopy cobordisms
论文作者
论文摘要
我们构建一个类别$ \ mathrm {homcob} $，其对象是{\ it均匀生成的}拓扑空间，并且其形态为{\ it同伴cospans}。 给定一个歧管submanifold Pair $（m，a）$，我们证明函数在$ \ mathrm {homcob} $中，来自映射类Groupoid $ \ mathrm {mcg} _ {m} _ {m}^{a}^{a} $的全部sub groupoid，并从全部的运动集体固定群中$ \ mathrm {mot} _ {m}^{a} $，其对象是同质性生成的。 我们还构建了一个函子系列$ \ mathsf {z} _g \ colon \ mathrm {homcob} \ to \ mathbf {vect} $，每个有限组$ g $一个。 这些概括的拓扑量子场理论先前由Yetter构建，并且是Dijkgraaf-Witten的不介于的版本。 Given a space $X$, we prove that $\mathsf{Z}_G(X)$ can be expressed as the $\mathbb{C}$-vector space with basis natural transformation classes of maps $\{π(X,X_0)\to G\} $ for some finite representative set of points $X_0\subset X$, demonstrating that $ \ mathsf {z} _g $是可以明确计算的。
We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class groupoid $\mathrm{MCG}_{M}^{A}$, and from the full subgroupoid of the motion groupoid $\mathrm{Mot}_{M}^{A}$, whose objects are homotopically 1-finitely generated. We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to \mathbf{Vect}$, one for each finite group $G$. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten. Given a space $X$, we prove that $\mathsf{Z}_G(X)$ can be expressed as the $\mathbb{C}$-vector space with basis natural transformation classes of maps $\{π(X,X_0)\to G\} $ for some finite representative set of points $X_0\subset X$, demonstrating that $\mathsf{Z}_G$ is explicitly calculable.
