Metric Tannakian Duality

TL;DR

This paper extends Tannaka-Krein duality to metric groups, establishing a dual metric on representations and exploring conditions for metric recovery and applications to T-duality and quantum Gromov-Hausdorff distance.

Contribution

It introduces a framework for incorporating metrics into Tannaka-Krein duality, defining dual metrics on representation categories and analyzing metric recovery conditions.

Findings

Defined dual metrics on representation categories for metric groups.

Characterized conditions for recovering the original metric from the dual.

Explored applications to T-duality and quantum Gromov-Hausdorff distance.

Abstract

We incorporate metric data into the framework of Tannaka-Krein duality. Thus, for any group with left invariant metric, we produce a dual metric on its category of unitary representations. We characterize the conditions under which a "double-dual" metric on the group may be recovered from the metric on representations, and provide conditions under which a metric agrees with its double-dual. We also consider some applications to T-duality and quantum Gromov-Hausdorff distance.