Fermions on tori in uniform Abelian fields

Abstract

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 kdimensional 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.

Chiral anomalies in the reduced model

“…[31] in the context of the overlap. 15 Under the infinitesimal gauge transformation, δ η U µ = [ω, U µ ], δ η ψ = ωψ and δ η ψ = −ψω, one can show that…”

“…Namely, the assumed form (2.11) is preserved under the gauge transformation in the reduced model 14) provided that Ω ∈ U(N) or Ω ∈ SU(N) is a diagonal matrix. This transformation induces a transformation on u µ 15) that is nothing but the conventional U(1) gauge transformation due to eq. (2.9).…”

“…15 The above is for the reduced model. The correspondence to the U(1) theory in section 2 is applied also to this system of Weyl fermion, because couplings to 14 The above U(1) bundle, the connection and the curvature were first addressed in ref.…”

“…The field u 14) and it has vanishing field strength f µν (x) = 0 and carries the Wilson (or Polyakov) line, 15) and carries a constant field strength 16) where the "magnetic flux" m µν is an integer bounded by…”

“…Using the abelian background of ref. [15], they explicitly demonstrated that certain configurations in the reduced model lead to Q = 0 for d = 2 and d = 4. They also argued that there may exist some remnant of the gauge anomaly in reduced chiral gauge theories.…”