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. 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_{2N_{f}}, whereas for adjoint quarks it is Z_{4N_{f}}.

Highlights

  • Published by the American Physical SocietyOf the path-integral measure, which gives the anomaly. For the fermions in the fundamental representation of SUðNÞ (i.e., quarks), there remains a residual Z2Nf symmetry, while for adjoint fermions the residual symmetry is Z4Nf

  • Introduction.—A symmetry of a classical theory that cannot be implemented in its quantum counterpart is said to be broken anomalously

  • We show that the index in such a background is equal to a new instanton charge T new 1⁄4 Æ1⁄2rð02Þ − r0ð1ފ

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Summary

Published by the American Physical Society

Of the path-integral measure, which gives the anomaly. For the fermions in the fundamental representation of SUðNÞ (i.e., quarks), there remains a residual Z2Nf symmetry, while for adjoint fermions the residual symmetry is Z4Nf. The surprise here is that the SUðNÞ matrix model is vastly different from SUðNÞ gauge field theory. The matrix model discussed here was first presented in [11,12,13], and has been shown to be an excellent candidate for an effective low-energy approximation of SUðNÞ YangMills theory on S3 × R. To serve as a correct low-energy approximation of Yang-Mills theory, this quantum-mechanical model must exhibit the axial anomaly. We need to define a gaugecovariant time derivative of Ai. For that, we introduce a set of time-dependent real functions, conveniently named as M0a, and the matrix A0 ≡ M0aTa. A0 is the parallel transporter needed to define the covariant derivative along the temporal direction and under a gauge transformation hðtÞ ∈ SUðNÞ, A0 → A00 1⁄4 hA0h−1 − h_ h−1. The Lagrangian with minimally coupled massless quarks is L 1⁄4 LYM þ LF where LF 1⁄4 R3Ψ 1⁄2iγ0Dt þ γiAi − ð3=2RÞγ5γ0ŠΨ is the gauge-covariant Dirac Lagrangian [13,19]

It is convenient to rescale to dimensionless variables
Substituting the ansatz
Nr Nrj
The solutions of
The Euclidean fermionic action can be written as
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