标题： 一个从$3$-循环群$Ω^3 G$构造的$2$-群的交叉模表示 显示英文标题

标题： A crossed module representation of a $2$-group constructed from the $3$-loop group $Ω^3 G$

摘要： 在外部规范势中对3流形上的手征费米子进行量化已知会导致规范群的阿贝尔扩张。 在本文中，我们关注的是基于光滑映射在3球面上取值于紧致李群$G.$的情况$\Omega^3 G$。从该群的一个阿贝尔扩张$\widehat{\Omega^3 G}$构造了一个交叉模，并且该群的一个自同构群作用如Mickelson和Niemimäki最近文章中所述。 我们将以$\widehat{\Omega^3 G}$在具有值于费米子福克空间的规范势函数空间上的表示，以及$\widehat{\Omega^3 G}$自同构群作为福克空间中规范反对易关系代数的外自同构的表示形式来构造这个交叉模的表示。 显示英文摘要

摘要： The quantization of chiral fermions on a 3-manifold in an external gauge potential is known to lead to an abelian extension of the gauge group. In this article we concentrate on the case of $\Omega^3 G$ of based smooth maps on a 3-sphere taking values in a compact Lie group $G.$ There is a crossed module constructed from an abelian extension $\widehat{\Omega^3 G}$ of this group and a group of automorphims acting on it as explained in a recent article by Mickelson and Niemim\"aki. We shall construct a representation of this crossed module in terms of a repesentation of $\widehat{\Omega^3 G}$ on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of $\widehat{\Omega^3 G}$ as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.