Instantons and Chiral Anomaly in Fuzzy Physics

TL;DR

This paper develops a noncommutative geometric approach to discretize quantum field theories on fuzzy manifolds, effectively capturing topological features like instantons and axial anomalies that are challenging for traditional lattice methods.

Contribution

It introduces discrete representations of topological terms and derives the axial anomaly on fuzzy spheres, advancing the treatment of topological aspects in fuzzy physics.

Findings

Gauge field action bounded by instanton number magnitude

Successful representation of theta-terms and topological susceptibility

Derivation of axial anomaly on fuzzy sphere

Abstract

In continuum physics, there are important topological aspects like instantons, theta-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we present discrete representations of theta-terms and topological susceptibility for gauge theories and derive axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions is bounded by the modulus of the instanton number as in the continuum.