Axial Anomaly in SU(N) Yang-Mills Matrix Models

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. The vacua of the gauge theory are N-dimensional representations of SU(2), and the (anti-) self-dual instantons tunnel between two commuting representations, the initial one composed of r(0)((1)) irreps and the final one with r(0)((2)) irreps. We show that the index (n(+) - n(-)) in such a background is equal to a new instanton charge T-new = +/-[r(0)((2)) - r(0)((1))]. Thus T-new = (n(+) - n(-)) is the matrix model version of the Atiyah-Singer index theorem. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the noninvariance 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 Z(2Nf), whereas for adjoint quarks it is Z(4Nf).