Quantum Bundle Description of the Quantum Projective Spaces

TL;DR

This paper develops a quantum bundle framework for quantum projective spaces, constructing calculi, principal bundles, and strong connections, thereby advancing the geometric understanding of noncommutative quantum spaces.

Contribution

It introduces a new calculus on quantum spheres, models quantum projective spaces as base spaces of principal bundles, and constructs strong connections, enriching quantum geometry tools.

Findings

Realized Heckenberger-Kolb calculus as a quotient of standard bicovariant calculus

Presented quantum projective spaces as base spaces of two quantum principal bundles

Constructed strong connections for the quantum bundles

Abstract

We realise Heckenberger and Kolb's canonical calculus on quantum projective (n-1)-space as the restriction of a distinguished quotient of the standard bicovariant calculus for Cq[SUn]. We introduce a calculus on the quantum (2n-1)-sphere in the same way. With respect to these choices of calculi, we present quantum projective (N-1)-space as the base space of two different quantum principal bundles, one with total space Cq[SUn], and the other with total space Cq[S^(2n-1)]. We go on to give Cq[CP^n] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb's calculus as an associated vector bundle to the principal bundle with total space Cq[SUn]. Finally, we construct strong connections for both bundles.