  We study Fermi fields defined on tori in the presence of gauge backgrounds
carrying non-trivial 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.
