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

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 ${\mathbb{Z}}_{4N_f}$.