A crossed module representation of a 2-group constructed from the 3-loop group \(\Omega ^3G\)

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 \(\Omega ^3 G\) ${\mathrm{\Omega }}^{3}G$ 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 \(\widehat{\Omega ^3 G}\) $\widehat{{\mathrm{\Omega }}^{3}G}$ of this group and a group of automorphisms acting on it as explained in a recent article by Mickelsson and Niemimäki. We shall construct a representation of this crossed module in terms of a representation of \(\widehat{\Omega ^3 G}\) $\widehat{{\mathrm{\Omega }}^{3}G}$ on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of \(\widehat{\Omega ^3 G}\) $\widehat{{\mathrm{\Omega }}^{3}G}$ as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.