A duality map for quantum cluster varieties from surfaces

TL;DR

This paper introduces a canonical map linking laminations on punctured surfaces to the quantized algebra of functions on cluster varieties, confirming several conjectured properties and building on the quantum trace map.

Contribution

It constructs a canonical duality map for quantum cluster varieties from surfaces, verifying key conjectures and utilizing the quantum trace framework.

Findings

The map satisfies properties conjectured by Fock and Goncharov.

The construction is based on the quantum trace map by Bonahon and Wong.

The work advances understanding of quantum cluster varieties associated with surfaces.

Abstract

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon and Wong.