Seiberg Witten Map and the Axial Anomaly in Noncommutative Field Theory

TL;DR

This paper calculates the axial anomaly in noncommutative field theories using point-splitting regularisation, demonstrating gauge invariance and relating it to the standard anomaly via the Seiberg Witten map, including all orders of the noncommutative parameter.

Contribution

It provides a gauge-invariant expression for the axial anomaly in noncommutative space-time and connects it to the classical anomaly through the Seiberg Witten map, valid to all orders of the noncommutative parameter.

Findings

Derived gauge-invariant axial anomaly in NC space-time.

Connected NC anomaly to the standard Adler-Bell-Jackiw anomaly.

Obtained all-order expression for constant fields in NC space.

Abstract

Using the point-splitting regularisation, we calculate the axial anomaly in an arbitrary even dimensional Non-Commutative (NC) field theory. Our result is (star) gauge invariant in its {\it unintegrated} form, to the leading order in the NC parameter. Exploiting the Seiberg Witten map, this result gets transformed to the familiar Adler-Bell-Jackiw anomaly in ordinary space-time. Furthermore, using this map, we derive an expression for the unintegrated axial anomaly for constant fields in NC space-time, that is valid to all finite orders of the NC parameter.