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

Abstract

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- 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 r0(1) irreps and the final one with r0(2) irreps. We show that the index (n+-n-) in such a background is equal to a new instanton charge Tnew=�[r0(2)-r0(1)]. 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. � 2021 authors. Published by the American Physical Society. Funded by SCOAP3.