Convergence of Heisenberg Modules over Quantum 2-tori for the Modular Gromov-Hausdorff Propinquity

TL;DR

This paper demonstrates that Heisenberg modules over quantum 2-tori, equipped with canonical connections, form a continuous family under the modular Gromov-Hausdorff propinquity, advancing the understanding of quantum metric geometry.

Contribution

It establishes the continuity of Heisenberg modules over quantum 2-tori with respect to the modular propinquity, a key step in quantum metric geometry.

Findings

Heisenberg modules form a continuous family under the modular propinquity.

Canonical connections induce this continuity.

Advances the understanding of quantum metric space convergence.

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 continuous family for the modular propinquity.