Non-associative magnetic translations from parallel transport in projective Hilbert bundles

The non-associativity of translations in a quantum system with magnetic field background has received renewed interest association topologically trivial gerbes over $\mathbb{R}^n.$ is described by 3-cocycle the group $\mathbb{R}^n$ values unit circle $S^1.$ space $M$ are classified Dixmier-Douady class which an element $\mathrm{H}^3(M,\mathbb{Z}).$ However, there finer description terms local differential forms degrees $d=0,1,2,3$ and case for $n=3$ 2-form part $B$ non zero divergence. In this paper we study theoretic construction $n$-component fermions on circle. associativity arises when trying to lift translation action 1-particle second quantized system.