Cluster varieties from Legendrian knots

TL;DR

This paper demonstrates a systematic method to establish cluster structures on moduli spaces related to Legendrian knots by using sheaf quantizations of exact Lagrangian fillings, linking cluster algebra results to Legendrian knot theory.

Contribution

It introduces a uniform approach to derive cluster structures on moduli spaces associated with Legendrian links via sheaf quantizations of Lagrangian fillings.

Findings

Cluster structures can be systematically constructed for moduli spaces of sheaves.

Cluster algebra results can distinguish Legendrian link fillings.

The method applies to spaces like positroid strata and wild character varieties.

Abstract

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.