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Floer Homology for the Gelfand-Cetlin System

  • Yuichi Nohara
  • Kazushi Ueda
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

The Gelfand-Cetlin system is a completely integrable system on a flag manifold of type A. In contrast to the case of toric moment maps, the Gelfand-Cetlin system has non-torus Lagrangian fibers on some boundary strata of the momentum polytope. In this paper we discuss Lagrangian intersection Floer theory for torus and non-torus Lagrangian fibers of the Gelfand-Cetlin system on the three-dimensional full flag manifold and the Grassmannian of two-planes in a four-dimensional vector space.

Keywords

Maslov Index Toric Manifold Floer Homology Flag Manifold Lagrangian Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Y. N. is supported by Grant-in-Aid for Young Scientists (No. 23740055). K. U. is supported by Grant-in-Aid for Young Scientists (No. 24740043).

References

  1. 1.
    Batyrev, V., Ciocan-Fontanine, I., Kim, B., van Straten, D.: Mirror symmetry and toric degenerations of partial flag manifolds. Acta Math. 184(1), 1–39 (2000)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cho, C.-H., Oh, Y.-G.: Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math. 10, 773–814 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Eguchi, T., Hori, K., Xiong, C.-S.: Gravitational quantum cohomology. Int. J. Modern Phys. A 12(9), 1743–1782 (1997)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Evans, J. D., Lekili, Y.: Floer cohomology of the Chiang Lagrangian. arXiv:1401.4073Google Scholar
  5. 5.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory—Anomaly and Obstructions, Part I and II. IP Studies in Advanced Mathematics, vol. 46. American Mathematical Society, Providence (2009)Google Scholar
  6. 6.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds I. Duke Math. J. 151(1), 23–174 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds: survey. In: Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, pp. 229–298. International Press, Boston (2012)Google Scholar
  8. 8.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory and mirror symmetry on compact toric manifolds. arXiv:1009.1648Google Scholar
  9. 9.
    Geĺfand, I. M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. (Russian) Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950)Google Scholar
  10. 10.
    Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. In: Topics in Singularity Theory. American Mathematical Society Translations: Series 2, vol. 180, pp. 103–115, The American Mathematical Society, Providence (1997)Google Scholar
  11. 11.
    Guillemin, V., Sternberg, S.: The Gelfand-Cetlin system and quantization of the complex flag manifolds. J. Funct. Annal. 52, 106–128 (1983)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Nishinou, T., Nohara, Y., Ueda, K.: Toric degenerations of Gelfand-Cetlin systems and potential functions. Adv. Math. 224, 648–706 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Rietsch, K.: A mirror symmetric construction of \(qH_{T}^{{\ast}}(G/P)_{(q)}\). Adv. Math. 217(6), 2401–2442 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Faculty of EducationKagawa UniversityKagawaJapan
  2. 2.Department of Mathematics Graduate School of ScienceOsaka UniversityOsakaJapan

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