Cluster deep loci and 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 fourth author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with nontrivial 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 G r ( 2 , n ) Gr(2,n) and G r ( 3 , n ) Gr(3,n) . We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians G r ( 2 , 2 n + 1 ) 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.
Related
BibTeX
@article{castronovo2026cluster,
title = {{Cluster deep loci and mirror symmetry}},
author = {Marco Castronovo and Mikhail Gorsky and José Simental and David E Speyer},
journal = {Communications of the American Mathematical Society},
year = {2026},
month = aug,
doi = {10.1090/cams/76},
url = {https://doi.org/10.1090/cams/76},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - Cluster deep loci and mirror symmetry
AU - Marco Castronovo
AU - Mikhail Gorsky
AU - José Simental
AU - David E Speyer
PY - 2026
DA - 2026/08/05
JO - Communications of the American Mathematical Society
DO - 10.1090/cams/76
UR - https://doi.org/10.1090/cams/76
AB - 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 fourth author) characterizing their deep locus, i.e. the complement of all cluster charts, as the locus of points with nontrivial 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 G r ( 2 , n ) Gr(2,n) and G r ( 3 , n ) Gr(3,n) . We illustrate how this already has applications in symplectic topology and mirror symmetry, by proving that the Fukaya category of Grassmannians G r ( 2 , 2 n + 1 ) 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.
SN - 2692-3688
ER -