Bundles over Quantum Real Weighted Projective Spaces

TL;DR

This paper constructs and analyzes principal U(1)-bundles over quantum real weighted projective spaces, revealing non-trivial bundles in the negative class and trivial ones in the positive class, with implications for quantum geometry.

Contribution

It provides the first explicit construction and classification of principal bundles over quantum real weighted projective spaces, distinguishing between trivial and non-trivial cases.

Findings

Principal bundles over negative class are non-trivial.

Principal bundles over positive class are trivial.

Circle actions on quantum Seifert manifolds are almost free.

Abstract

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the {\em negative} or {\em odd} class that generalises quantum real projective planes and the {\em positive} or {\em even} class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.