Fermions on tori in uniform Abelian fields

We study Fermi fields defined on tori in the presence of gauge backgrounds carrying a nontrivial topology. We show that 2k-dimensional field space can alternatively be described by fields over a k-dimensional space. This dual description is particularly natural when the background is uniform and Abelian. The reduction in number of dimensions carries over to the lattice. The lattice ultraviolet regularization induces an infrared regularization of the lower dimensional representations. We focus on $k=1,2.$