A crossed module representation of a $2$-group constructed from the $3$-loop group $Ω^3 G$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910499203973120 |
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| author | Mickelsson, Jouko |
| author_facet | Mickelsson, Jouko |
| contents | 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 $Ω^3 G$ 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 $\widehat{Ω^3 G}$ of this group and a group of automorphims acting on it as explained in a recent article by Mickelson and Niemimäki. We shall construct a representation of this crossed module in terms of a repesentation of $\widehat{Ω^3 G}$ on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of $\widehat{Ω^3 G}$ as outer automorphisms of the canonical anticommutation relations algebra in the Fock space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04995 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A crossed module representation of a $2$-group constructed from the $3$-loop group $Ω^3 G$ Mickelsson, Jouko Mathematical Physics Representation Theory 22E67, 81T50 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 $Ω^3 G$ 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 $\widehat{Ω^3 G}$ of this group and a group of automorphims acting on it as explained in a recent article by Mickelson and Niemimäki. We shall construct a representation of this crossed module in terms of a repesentation of $\widehat{Ω^3 G}$ on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of $\widehat{Ω^3 G}$ as outer automorphisms of the canonical anticommutation relations algebra in the Fock space. |
| title | A crossed module representation of a $2$-group constructed from the $3$-loop group $Ω^3 G$ |
| topic | Mathematical Physics Representation Theory 22E67, 81T50 |
| url | https://arxiv.org/abs/2311.04995 |