Exotic Lagrangian tori in Grassmannians

TL;DR

This paper constructs new Lagrangian tori in Grassmannians using cluster algebra structures, providing examples that are non-displaceable, non-isotopic, and support diverse objects in the Fukaya category.

Contribution

It introduces an iterative method to build exotic Lagrangian tori in Grassmannians based on cluster algebra and mirror symmetry, revealing their non-displaceability and categorical distinctions.

Findings

Constructed explicit examples of non-displaceable Lagrangian tori.

Showed these tori are not Hamiltonian isotopic.

Demonstrated these tori support objects in different Fukaya category summands.

Abstract

We describe an iterative construction of Lagrangian tori in the complex Grassmannian $\operatorname{Gr}(k,n)$, based on the cluster algebra structure of the coordinate ring of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. Each torus comes with a Laurent polynomial, and local systems controlled by the $k$-variables Schur polynomials at the $n$-th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over $\mathbb{C}$.