Dirac Operator on Fuzzy AdS_2

TL;DR

This paper constructs the Dirac and chirality operators on noncommutative AdS_2, analyzes their spectra, and discusses implications for spectral triples and projective modules in noncommutative geometry.

Contribution

It introduces a method to define Dirac and chirality operators on noncommutative AdS_2 and explores their spectral properties, advancing noncommutative geometric analysis.

Findings

Spectrum degeneracy is lifted in the noncommutative case.

Discrete spectrum of the Dirac operator is derived.

Construction of chirality operator suggests a way to define projective modules.

Abstract

In this article we construct the chirality and Dirac operators on noncommutative AdS_2. We also derive the discrete spectrum of the Dirac operator which is important in the study of the spectral triple associated with AdS_2. It is shown that the degeneracy of the spectrum present in the commutative AdS_2 is lifted in the noncommutative case. The way we construct the chirality operator is suggestive of how to introduce the projector operators of the corresponding projective modules on this space.