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Toric degenerations of Gelfand-Cetlin systems and potential functions. (English) Zbl 1221.53122
A flag manifold of type \(A\) admits a special completely integrable system, the Gelfand-Cetlin system [V. Guillemin and S. Sternberg, J. Funct. Anal. 52, 106–128 (1983; Zbl 0522.58021)]. The paper studies these systems via toric geometry. The central result of the paper is the existence of a toric degeneration of a Gelfand-Cetlin system to an integrable system given by a toric variety and its moment map.
The authors apply their result to compute the potential function for Lagrangian torus fibers of the Gelfand-Cetlin system, following closely K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono [Duke Math. J. 151, No. 1, 23–175 (2010; Zbl 1190.53078)]. The results are used to prove the existence of a non-displacable Lagrangian torus in the flag manifold, an analogue of Theorem 1.5 in [loc. cit.]. The last section relates the potential function for Lagrangian torus fibers of the classical Gelfand-Cetlin system on the full flag manifold to the phase function for an integral representation of the solution to the quantum Toda lattice.

MSC:
53D40 Symplectic aspects of Floer homology and cohomology
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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