A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

TL;DR

This paper develops a framework for analyzing spectral triples in noncommutative geometry, applies it to quantum projective space, and constructs a q-deformed Dirac operator with a non-trivial K-homology class.

Contribution

It introduces a framework for the spectral analysis of covariant K"ahler structures and constructs a new spectral triple for quantum projective space with non-trivial K-homology.

Findings

Constructed an even spectral triple for quantum projective space.

Demonstrated the spectral triple satisfies Connes' axioms up to compact resolvent.

Extended the approach to quantum Hermitian symmetric spaces.

Abstract

The notion of a K\"ahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite K\"ahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant K\"ahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.