论文标题
通过降低TQFTS的链接运动组表示
Representations of Motion Groups of Links via Dimension Reduction of TQFTs
论文作者
论文摘要
三个球体中链接的运动组$ \ mathbb {s}^3 $是编织组的概括，它们是磁盘$ \ mathbb {d}^2 $中的点运动组。运动组的表示形式可用于建模扩展对象的统计信息，例如物理中的封闭字符串。每个$ 1 $ - 扩展的$（3+1）$ - 拓扑量子场理论（TQFT）将提供运动组的表示，但是通常很难明确计算此类表示。在本文中，我们计算$ \ mathbb {s}^3 $中链接的运动组的表示组，并从Dijkgraaf-Witten（DW）TQFTS中具有广义轴，灵感来自尺寸。一种简洁的方法来陈述我们的结果是迈向DW降低DW理论的扭曲概括（Cundure \ Ref {MainConjecture}）的一步\ textrm {dw}^{2+1} _ {c（g）} $，其中总和在所有共轭类$ [g] $ of $ g $ of $ g $ of $ g $ of $ g $和$ c（g）$ in [g] $ in [g] $中的中心化。我们证明了构图\ ref {mainconjecture}的版本，用于映射封闭歧管的类组和由纯通量标记的圆环链接的情况。
Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes.
