The Noncommutative Geometry of the Quantum Projective Plane

TL;DR

This paper explores the spectral geometry of the quantum projective plane CP^2_q, constructing a Dirac operator that reveals its noncommutative geometric structure and spectrum.

Contribution

It introduces a Dirac operator on CP^2_q that forms a 0^+ summable spectral triple, advancing the understanding of noncommutative spin^c manifolds.

Findings

Constructed a Dirac operator D for CP^2_q

D is a 0^+ summable spectral triple

Explicit spectrum computation of D

Abstract

We study the spectral geometry of the quantum projective plane CP^2_q, a deformation of the complex projective plane CP^2, the simplest example of a spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0^+ summable spectral triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.