Chiral Anomaly on Fuzzy 2-Sphere

TL;DR

This paper studies chiral anomalies for fermions on a noncommutative fuzzy 2-sphere, demonstrating that correct anomalies can be obtained without regularization, and explores different formulations of chiral currents and their associated Ward-Takahashi identities.

Contribution

It introduces gauge-invariant and gauge-covariant definitions of chiral currents on the fuzzy 2-sphere and analyzes their anomalous Ward-Takahashi identities.

Findings

Correct chiral anomaly reproduced without regularization.

Different forms of Ward-Takahashi identities for gauge-invariant and covariant currents.

Explicit nonlocality in the gauge-invariant current's Ward-Takahashi identity.

Abstract

We investigate chiral anomaly for fermions in fundamental representation on noncommutative (fuzzy) 2-sphere. In spite that this system is realized by finite dimensional matrices and no regularization is necessary for either UV or IR, we can reproduce the correct chiral anomaly which is consistent with the calculations done in flat noncommutative space. Like the flat case, there are ambiguities to define chiral currents. We define chiral currents in a gauge-invariant way and a gauge-covariant way, and show that the corresponding anomalous chiral Ward-Takahashi identities take different forms. The Ward-Takahashi identity for the gauge-invariant current contains explicit nonlocality while that for the covariant one is given by a local expression.