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 Nf flavors of massless quarks, the Euclidean Dirac equation in an instanton background has n+ positive and n−egative 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(1)0 irreps and the final one with r(2)0 irreps. We show that the index (n+−n−) in such a background is equal to a new instanton charge Tnew = ± [r(2)0−r(1)0]. Thus Tnew = (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 Nf fundamental fermions, this residual symmetry is Z2Nf, whereas for adjoint quarks it is Z4Nf.