A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

Abstract

The notion of a Kähler 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 Kähler 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 Kähler 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 Dolbeault-Dirac operator of complex projective space. Finally, the extension of this approach to a certain canonical class of irreducible quantum flag manifolds is discussed in detail.