# Heisenberg Modules over Quantum 2-tori are metrized quantum vector bundles

## TL;DR

This paper demonstrates that Heisenberg modules over quantum 2-tori, equipped with canonical connections, constitute a family of metrized quantum vector bundles, advancing the understanding of their geometric and metric properties.

## Contribution

It establishes that Heisenberg modules over quantum 2-tori are metrized quantum vector bundles, a key step towards showing their continuity in the modular Gromov-Hausdorff propinquity.

## Key findings

- Heisenberg modules form metrized quantum vector bundles with canonical connections.
- This work advances the understanding of quantum metric geometry of modules over quantum tori.
- It paves the way for proving the continuity of Heisenberg modules in the modular Gromov-Hausdorff propinquity.

## Abstract

The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov-Hausdorff propinquity.