# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Batyrev, Victor V., Quantum cohomology rings of toric manifolds, Journées de Géométrie algébrique d’orsay, Orsay 1992, Astérisque, 218, 9-34, (1993) · Zbl 0806.14041 [2] Batyrev, Victor V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. algebraic geom., 3, 3, 493-535, (1994) · Zbl 0829.14023 [3] Batyrev, Victor V.; Ciocan-Fontanine, Ionuţ; Kim, Bumsig; van Straten, Duco, Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nuclear phys. B, 514, 3, 640-666, (1998) · Zbl 0896.14025 [4] Batyrev, Victor V.; Ciocan-Fontanine, Ionuţ; Kim, Bumsig; van Straten, Duco, Mirror symmetry and toric degenerations of partial flag manifolds, Acta math., 184, 1, 1-39, (2000) · Zbl 1022.14014 [5] Cho, Cheol-Hyun, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, J. geom. phys., 58, 11, 1465-1476, (2008) · Zbl 1161.53076 [6] Cho, Cheol-Hyun; Oh, Yong-Geun, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. math., 10, 4, 773-814, (2006) · Zbl 1130.53055 [7] Fukaya, Kenji, Morse homotopy, $$A^\infty$$-category, and Floer homologies, (), 1-102 · Zbl 0853.57030 [8] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono, Lagrangian intersection Floer theory, preprint, 2006 [9] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru, Lagrangian Floer theory on compact toric manifolds I, (2008) · Zbl 1190.53078 [10] Gel’fand, I.M.; Cetlin, M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. akad. nauk SSSR (N.S.), 71, 825-828, (1950) · Zbl 0037.15301 [11] Givental, Alexander, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, (), 103-115 · Zbl 0895.32006 [12] Givental, Alexander; Kim, Bumsig, Quantum cohomology of flag manifolds and Toda lattices, Comm. math. phys., 168, 3, 609-641, (1995) · Zbl 0828.55004 [13] Gonciulea, N.; Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties, Transform. groups, 1, 3, 215-248, (1996) · Zbl 0909.14028 [14] Guillemin, Victor; Sternberg, Shlomo, The moment map and collective motion, Ann. physics, 127, 1, 220-253, (1980) · Zbl 0453.58015 [15] Guillemin, V.; Sternberg, S., The Gel’fand-cetlin system and quantization of the complex flag manifolds, J. funct. anal., 52, 1, 106-128, (1983) · Zbl 0522.58021 [16] Hori, Kentaro; Vafa, Cumrun, Mirror symmetry, (2000) · Zbl 1044.14018 [17] Joe, Dosang; Kim, Bumsig, Equivariant mirrors and the Virasoro conjecture for flag manifolds, Int. math. res. not. IMRN, 15, 859-882, (2003) · Zbl 1146.14302 [18] Kim, Bumsig, Quantum cohomology of flag manifolds $$G / B$$ and quantum Toda lattices, Ann. of math. (2), 149, 1, 129-148, (1999) · Zbl 1054.14533 [19] Kogan, Mikhail; Miller, Ezra, Toric degeneration of Schubert varieties and Gelfand-tsetlin polytopes, Adv. math., 193, 1, 1-17, (2005) · Zbl 1084.14049 [20] Nishinou, Takeo; Siebert, Bernd, Toric degenerations of toric varieties and tropical curves, Duke math. J., 135, 1, 1-51, (2006) · Zbl 1105.14073 [21] Ruan, Wei-Dong, Lagrangian torus fibration of quintic hypersurfaces. I. Fermat quintic case, (), 297-332 · Zbl 1079.14526 [22] Ruan, Wei-Dong, Lagrangian torus fibrations and mirror symmetry of Calabi-Yau manifolds, (), 385-427 · Zbl 1079.14527 [23] Ye, Rugang, Gromov’s compactness theorem for pseudo holomorphic curves, Trans. amer. math. soc., 342, 2, 671-694, (1994) · Zbl 0810.53024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.