Axial Anomaly in the $SU(N)$ Gauge Matrix Model

TL;DR

This paper investigates the axial anomaly in the $SU(N)$ gauge matrix model, demonstrating how instantons affect zero modes, measure invariance, and residual symmetries, revealing the anomaly's role in symmetry breaking.

Contribution

It establishes the relation between the Dirac operator index and instanton charge, and characterizes the residual chiral symmetry after anomaly-induced breaking.

Findings

Index equals instanton charge.

Path integral measure is not invariant under chiral rotation.

Residual symmetry is a finite cyclic group, $bZ_{2N_f}$ or $bZ_{4N_f}$.

Abstract

The $SU(N)$ Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to $N_f$ flavors of massless quarks, the Euclidean Dirac equation in an instanton background has $n_+$ positive and $n_-$ negative chirality zero modes. We show that the index $(n_+ - n_-)$ is equal to a suitably defined instanton charge. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the non-invariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For $N_f$ fundamental fermions, this residual symmetry is $\mathbb{Z}_{2N_f}$, whereas for adjoint quarks it is $\mathbb{Z}_{4N_f}$.