Lattice Perturbation Theory in Noncommutative Geometry and Parity Anomaly in 3D Noncommutative QED

TL;DR

This paper develops lattice perturbation theory for noncommutative gauge theories, applies it to 3D noncommutative QED, and explores the parity anomaly and its relation to noncommutative Chern-Simons action, enabling nonperturbative studies.

Contribution

It introduces a lattice perturbation framework for noncommutative gauge theories and analyzes the parity anomaly in 3D noncommutative QED, connecting it to noncommutative Chern-Simons theory.

Findings

Parity anomaly is expressed by noncommutative Chern-Simons action.

The anomaly coefficient depends on the lattice action.

Ginsparg-Wilson fermions preserve masslessness at finite lattice spacing.

Abstract

We formulate lattice perturbation theory for gauge theories in noncommutative geometry. We apply it to three-dimensional noncommutative QED and calculate the effective action induced by Dirac fermions. In particular "parity invariance" of a massless theory receives an anomaly expressed by the noncommutative Chern-Simons action. The coefficient of the anomaly is labelled by an integer depending on the lattice action, which is a noncommutative counterpart of the phenomenon known in the commutative theory. The parity anomaly can also be obtained using Ginsparg-Wilson fermions, where the masslessness is guaranteed at finite lattice spacing. This suggests a natural definition of the lattice-regularized Chern-Simons theory on a noncommutative torus, which could enable nonperturbative studies of quantum Hall systems.