  Motivated by the topological classification of hamiltonians in condensed
matter physics (topological insulators) we study the relations between chiral
Dirac operators coupled to an abelian vector potential on a torus in 3 and 1
space dimensions. We find that a large class of these hamiltonians in three
dimensions is equivalent, in K theory, to a family of hamiltonians in just one
space dimension but with a different abelian gauge group.
  The moduli space of U(1) gauge connections over a torus with a fixed Chern
class is again a torus up to a homotopy. Gerbes over a n-torus can be realized
in terms of extensions of the lattice group acting in a real vector space. The
extension comes from the action of the lattice group (thought of as "large"
gauge transformations, homomorphisms from the torus to U(1)) in the Fock space
of chiral fermions. Interestingly, the K theoretic classication of Dirac
operators coupled to vector potentials in this setting in 3 dimensions can be
related to families of Dirac operators on a circle with gauge group the
3-torus.
