The Modular Gromov-Hausdorff Propinquity

TL;DR

This paper introduces a new metric for Hilbert modules with differential structures, extending Gromov-Hausdorff convergence to quantum vector bundles and providing a framework for their moduli spaces, with applications to quantum tori.

Contribution

It presents a novel metric extending Gromov-Hausdorff theory to quantum vector bundles, enabling metric analysis of their moduli spaces and continuity properties.

Findings

Defined a new metric for Hilbert modules with differential structures.

Extended Gromov-Hausdorff convergence to quantum vector bundles.

Applied to the continuity of Heisenberg modules over quantum tori.

Abstract

We introduce a metric on Hilbert modules equipped with a generalized form of a differential structure, thus extending Gromov-Hausdorff convergence theory to vector bundles and quantum vector bundles --- not convergence as total space but indeed as quantum vector bundle. Our metric is new even in the classical picture, and creates a framework for the study of the moduli spaces of modules over C*-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum $2$-tori.