Lagrangian fibers of Gelfand-Cetlin systems

Abstract

Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of Gelfand-Cetlin systems over complex partial flag manifolds, we provide a complete description of the topology of Gelfand-Cetlin fibers. We prove that all fibers are \emph{smooth} isotropic submanifolds and give a complete description of the fiber to be Lagrangian in terms of combinatorics of Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian fibers. After a few combinatorial and numercal tests for the displaceability, using the bulk-deformation of Floer cohomology by Schubert cycles, we prove that every full flag manifold $\mathcal{F}(n)$ ($n\ge 3$) with a monotone Kirillov-Kostant-Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathcal{F}(3)$ is non-displaceable the question of which was raised by Nohara-Ueda who computed its Floer cohomology to be vanishing.

84pages, lots of figures

References in corpus (3)

• Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle
• On the f-vectors of Gelfand-Cetlin polytopes
• Generating the Fukaya categories of Hamiltonian G-manifolds

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• Potential functions on Grassmannians of planes and cluster transformations
• A Leray-Serre spectral sequence for Lagrangian Floer theory
• Lagrangian fibers of Gelfand--Cetlin systems of $\mathrm{SO}(n)$-type
• Rigid fibers of spinning tops