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

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.