Exotic Lagrangian tori in Grassmannians

Abstract

We describe an iterative construction of Lagrangian tori in the complex Grassmannian $\mathrm{Gr}(k,n)$, 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 $\mathbb{C}$.

arXiv comment26 pages, 5 figures. Simplified exposition; expanded discussion of generation of the Fukaya category

References in corpus (12)

• Total positivity, Grassmannians, and networks
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Weak Separation and Plabic Graphs
• Minkowski Polynomials and Mutations
• Cluster varieties from Legendrian knots
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• On Exotic Lagrangian Tori in CP^2
• Infinitely many exotic monotone Lagrangian tori in CP^2
• Invariants for Lagrangian tori
• Infinitely many monotone Lagrangian tori in ${\mathbb{R}}^6$
• Low-area Floer theory and non-displaceability
• Fukaya category of Grassmannians: rectangles

Cited by in corpus (2)

• Fukaya category of Grassmannians: rectangles
• Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs