Noncommutative SU(N) theories, the axial anomaly, Fujikawa's method and the Atiyah-Singer index

TL;DR

This paper uses Fujikawa's method to compute the axial anomaly in noncommutative SU(N) gauge theories at first order in the noncommutative parameter, connecting it with the Atiyah-Singer index theorem.

Contribution

It provides the first-order computation of the $U(1)_A$ anomaly in noncommutative SU(N) theories using a general Seiberg-Witten map and relates the results to the Atiyah-Singer index theorem.

Findings

Computed the $U(1)_A$ anomaly at first order in noncommutative parameter.

Established connection between noncommutative anomaly and Atiyah-Singer index theorem.

Generalized results to nonsemisimple gauge groups with symmetric Seiberg-Witten map.

Abstract

Fujikawa's method is employed to compute at first order in the noncommutative parameter the $U(1)_A$ anomaly for noncommutative SU(N). We consider the most general Seiberg-Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, $\theta^{\mu\nu}$, which is of ``magnetic'' type. Our results for SU(N) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg-Witten map is used. Connection with the Atiyah-Singer index theorem is also made.