Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere

TL;DR

This paper constructs a Dirac operator on the standard Podles quantum sphere, linking spectral triples, differential calculus, and cyclic cocycles, advancing noncommutative geometric analysis of quantum spaces.

Contribution

It introduces a Dirac operator that defines a spectral triple on the Podles sphere and connects it with the differential calculus and cyclic cocycles, providing new tools for quantum geometry analysis.

Findings

|D|^{-z} is trace class for Re z>0

Commutators with D produce the covariant differential calculus

Twisted cyclic cocycle is expressed via the Dirac operator

Abstract

A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.