Dirac Operators on Quantum Projective Spaces

TL;DR

This paper constructs a family of self-adjoint operators on quantum projective spaces that form equivariant spectral triples, advancing the understanding of noncommutative geometric structures in quantum spaces.

Contribution

It introduces a new family of operators D_N with specific spectral properties, providing explicit examples of spectral triples on quantum projective spaces.

Findings

Operators D_N have compact resolvent and bounded commutators

They form 0^+ dimensional equivariant even spectral triples

Spectral triples are real with KO-dimension 2l mod 8 for specific N and l

Abstract

We construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral triple is real with KO-dimension 2l mod 8.