Noncommutative metric geometry of quantum circle bundles

TL;DR

This paper develops a framework for constructing quantum metrics on noncommutative circle bundles, generalizing existing results, and applies it to quantum spheres with metrics derived from quantum group deformations.

Contribution

It introduces a method to lift spectral geometric data to total algebras in quantum circle bundles, unifying and extending previous approaches in noncommutative geometry.

Findings

Spectral geometry can be lifted to total algebras under certain conditions.

The framework applies to quantum spheres, providing new quantum metrics from q-geometric data.

The approach generalizes results related to crossed products and quantum groups.

Abstract

In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital $C^*$-algebra together with a quantum metric of spectral geometric origin on the fixed point algebra. Under a few extra conditions on the spectral subspaces we show that the spectral geometric data on the base algebra can be lifted to the total algebra. Notably, the lifted spectral geometry is independent of the choice of frames and is permitted to interact with the total algebra via a twisted derivation. Under these conditions, it is explained how to assemble our data into a quantum metric on the total algebra in a way which unifies and generalizes a couple of results in the literature relating to crossed products by the integers and to quantum $SU(2)$. We apply our ideas to the higher Vaksman-Soibelman quantum spheres and endow them with quantum metrics arising from $q$-geometric data. In this context, the twist is enforced by the structure of the Drinfeld-Jimbo deformation arising from the Lie algebra of the special unitary group.