Anomaly freedom in Seiberg-Witten noncommutative gauge theories

TL;DR

This paper demonstrates that Seiberg-Witten noncommutative gauge theories share the same one-loop anomalies as their commutative versions, with the standard anomaly canceled by usual conditions, ensuring anomaly freedom.

Contribution

It explicitly calculates the one-loop anomaly in noncommutative gauge theories and shows it matches the commutative case, confirming anomaly cancellation via standard conditions.

Findings

One-loop anomalies in noncommutative theories are identical to commutative ones.

The only true anomaly is the standard Bardeen anomaly, which cancels under usual conditions.

Noncommutativity does not introduce new anomalies at one loop.

Abstract

We show that noncommutative gauge theories with arbitrary compact gauge group defined by means of the Seiberg-Witten map have the same one-loop anomalies as their commutative counterparts. This is done in two steps. By explicitly calculating the $\epsilon^{\m_1\m_2\m_3\m_4}$ part of the renormalized effective action, we first find the would-be one-loop anomaly of the theory to all orders in the noncommutativity parameter $\theta^{\m\n}$. And secondly we isolate in the would-be anomaly radiative corrections which are not BRS trivial. This gives as the only true anomaly occurring in the theory the standard Bardeen anomaly of commutative spacetime, which is set to zero by the usual anomaly cancellation condition.