Lattice perturbation theory in noncommutative geometry and parity anomaly in 3d noncommutative QED

We formulate lattice perturbation theory for gauge theories in noncommutativegeometry. We apply it to three-dimensional noncommutative QED and calculate theeffective action induced by Dirac fermions. In particular "parity invariance"of a massless theory receives an anomaly expressed by the noncommutativeChern-Simons action. The coefficient of the anomaly is labelled by an integerdepending on the lattice action, which is a noncommutative counterpart of thephenomenon known in the commutative theory. The parity anomaly can also beobtained using Ginsparg-Wilson fermions, where the masslessness is guaranteedat finite lattice spacing. This suggests a natural definition of thelattice-regularized Chern-Simons theory on a noncommutative torus, which couldenable nonperturbative studies of quantum Hall systems.Comment: 31 pages. LaTeX, feynmf. Minor changes, references added and typos corrected. Final version published in JHE