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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.

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
Full Text: DOI arXiv
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