Cluster deep loci and mirror symmetry

TL;DR

This paper characterizes the deep locus of affine cluster varieties as points with non-trivial stabilizers, verifies the conjecture for specific cases, and explores applications in symplectic topology and mirror symmetry.

Contribution

It proposes a conjecture describing the deep locus of cluster varieties and verifies it for certain types, linking it to symplectic and mirror symmetry results.

Findings

Deep locus characterized by non-trivial stabilizers.

Verification of the conjecture for finite cluster types and Grassmannian strata.

Applications in proving split-generation of Fukaya categories and homological mirror symmetry.

Abstract

Affine cluster varieties are covered up to codimension 2 by open algebraic tori. We put forth a general conjecture (based on earlier conversation between Vivek Shende and the last author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with non-trivial stabilizer under the action of cluster automorphisms. We use the diagrammatics of Demazure weaves to verify the conjecture for skew-symmetric cluster varieties of finite cluster type with arbitrary choice of frozens and for the top open positroid strata of Grassmannians $\mathrm{Gr}(2,n)$ and $\mathrm{Gr}(3,n)$. We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians $\mathrm{Gr}(2,2n+1)$ is split-generated by finitely many Lagrangian tori, and homological mirror symmetry holds with a Landau--Ginzburg model proposed by Rietsch. Finally, we study the geometry of the deep locus, and find that it can be singular and have several irreducible components of different dimensions, but they all are again cluster varieties in our examples in really full rank cases.