Chiral Anomaly on Fuzzy 2-Sphere

Abstract

We investigate chiral anomaly for fermions in fundamental representation on noncommutative (fuzzy) 2-sphere. In spite that this system is realized by finite dimensional matrices and no regularization is necessary for either UV or IR, we can reproduce the correct chiral anomaly which is consistent with the calculations done in flat noncommutative space. Like the flat case, there are ambiguities to define chiral currents. We define chiral currents in a gauge-invariant way and a gauge-covariant way, and show that the corresponding anomalous chiral Ward-Takahashi identities take different forms. The Ward-Takahashi identity for the gauge-invariant current contains explicit nonlocality while that for the covariant one is given by a local expression.

latex 42 pages, references added, typo corrected, discussion added, the final version to appear in Phys.Rev.D

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• Ginsparg-Wilson Relation, Topological Invariants and Finite Noncommutative Geometry
• Monopole Bundles over Fuzzy Complex Projective Spaces
• Ginsparg-Wilson Relation and 't Hooft-Polyakov Monopole on Fuzzy 2-Sphere
• Dynamical aspects of the fuzzy CP${}^2$ in the large $N$ reduced model with a cubic term
• Monte Carlo Simulation of a NC Gauge Theory on The Fuzzy Sphere
• Dynamical generation of a nontrivial index on the fuzzy 2-sphere
• Fuzzy Complex Quadrics and Spheres
• Perturbative versus nonperturbative dynamics of the fuzzy S^2*S^2
• Perturbative dynamics of fuzzy spheres at large N
• Ginsparg-Wilson Dirac operator in the monopole backgrounds on the fuzzy 2-sphere
• Impact of supersymmetry on the nonperturbative dynamics of fuzzy spheres
• A New Noncommutative Product on the Fuzzy Two-Sphere Corresponding to the Unitary Representation of SU(2) and the Seiberg-Witten Map
• Absence of a fuzzy $S^4$ phase in the dimensionally reduced 5d Yang-Mills-Chern-Simons model
• Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator
• Lattice Perturbation Theory in Noncommutative Geometry and Parity Anomaly in 3D Noncommutative QED
• The One-Plaquette Model Limit of NC Gauge Theory in 2D
• UV/IR Mixing and Anomalies in Noncommutative Gauge Theories
• Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere
• Towards Noncommutative Fuzzy QED
• On the Anomalies and Schwinger Terms in Noncommutative Gauge Theories
• Matrix Model and Ginsparg-Wilson Relation