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Disk potential functions for quadrics

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Abstract

We compute the disk potential of Gelfand–Zeitlin monotone torus fiber in a quadric hypersurface by exploiting toric degenerations, Lie theoretical mirror symmetry, and the structural result of the monotone Fukaya category.

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References

  1. Auroux, D.: Mirror symmetry and \(T\)-duality in the complement of an anticanonical divisor. J. Gökova Geom. Topol. GGT 1, 51–91 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Auroux, D.: Special Lagrangian fibrations, wall-crossing, and mirror symmetry, Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, Surv. Differ. Geom., Int. Press, Somerville. 13: 1–47 (2009)

  3. Biran, P., Cornea, O.: Lagrangian topology and enumerative geometry. Geom. Topol. 16(2), 963–1052 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Batyrev, V.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)

  5. Berenstein, A., Zelevinsky, A.: Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143(1), 77–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cho, C.-H.: Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus. Int. Math. Res. Not. 35, 1803–1843 (2004)

  8. Cho, C.-H.: Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle. J. Geom. Phys. 58(11), 1465–1476 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cho, Y., Kim, Y.: Monotone Lagrangians in flag varieties. Int. Math. Res. Not. IMRN. 18, 13892–13945 (2021)

  10. Cho, Y., Kim, Y.: Lagrangian fibers of Gelfand-Cetlin systems of \({SO}(n)\)-type. Transform. Groups 25(4), 1063–1102 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cho, Y., Kim, Y., Lee, E., Park, K.-D.: Small toric resolutions of toric varieties of string polytopes with small indices. Commun. Contemp. Math. 25(1), 2150112 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cho, Y., Kim, Y., Oh, Y.-G.: Lagrangian fibers of Gelfand-Cetlin systems. Adv. Math. 372, 107304 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chan, K., Lau, S.-C., Leung, N. C.: SYZ mirror symmetry for toric Calabi-Yau manifolds. J. Differential Geom. 90(2), 177–250 (2012)

  14. Chan, K., Lau, S.-C., Leung, N. C., Tseng, H.-H.: Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds. Duke Math. J. 166(8), 1405–1462 (2017)

  15. Cho, C.-H., Oh, Y.-G.: Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math. 10(4), 773–814 (2006)

  16. Cho, C.-H., Poddar, M.: Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds. J. Differential Geom. 98(1), 21–116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Charest, F., Woodward, C.T.: Floer cohomology and flips. Mem. Amer. Math. Soc. 279(1372), 166 (2022)

    MathSciNet  MATH  Google Scholar 

  18. Eliashberg, Y., Polterovich, L.: The problem of Lagrangian knots in four-manifolds. Geometric Topol. Athens, GA 1993 2, 313–327 (1993)

    MathSciNet  MATH  Google Scholar 

  19. Fukaya, K., Oh, Y-G., Ohta, H., Ono, K.: Canonical models of filtered \(A_\infty \)-algebras and Morse complexes, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI. 49: 201–227 (2009)

  20. Fukaya, K., Oh, Y-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol 46. American Mathematical Society, Providence, RI; International Press, Somerville, MA, (2009)

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds II: bulk deformations. Selecta Math. (N.S.) 17(3), 609–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Toric degeneration and nondisplaceable Lagrangian tori in \(S^2\times S^2\). Int. Math. Res. Not. IMRN 13, 2942–2993 (2012)

    Article  MATH  Google Scholar 

  24. Fukaya, K.: Cyclic symmetry and adic convergence in Lagrangian Floer theory. Kyoto J. Math. 50(3), 521–590 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gonciulea, N., Lakshmibai, V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Guillemin, V., Sternberg, S.: The Gel’fand-Cetlin system and quantization of the complex flag manifolds. J. Funct. Anal. 52(1), 106–128 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Guillemin, V., Sternberg, S.: On collective complete integrability according to the method of Thimm. Ergodic Theory Dynam. Systems 3(2), 219–230 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hausmann, J.-C., Knutson, A.: Polygon spaces and Grassmannians. Enseign. Math. 43(1–2), 173–198 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Harada, M., Kaveh, K.: Integrable systems, toric degenerations and Okounkov bodies. Invent. Math. 202(3), 927–985 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hong, H., Kim, Y., Lau, S-C.: Immersed two-spheres and SYZ with application to grassmannians, preprint, arXiv:1805.11738 (2019)

  31. Karshon, Y.: Periodic Hamiltonian flows on four-dimensional manifolds. Mem. Amer. Math. Soc. 141(672), viii+71 (1999)

    MathSciNet  MATH  Google Scholar 

  32. Kapovich, M., Millson, J.J.: The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44(3), 479–513 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lane, J.: The geometric structure of symplectic contraction. Int. Math. Res. Not. IMRN. 12, 3521–3539 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145–179 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nishinou, T., Nohara, Y., Ueda, K.: Potential functions via toric degenerations, preprint , arXiv:0812.0066 (2010)

  36. Nishinou, T., Nohara, Y., Ueda, K.: Toric degenerations of Gelfand-Cetlin systems and potential functions. Adv. Math. 224(2), 648–706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nohara, Y., Ueda, K.: Toric degenerations of integrable systems on Grassmannians and polygon spaces. Nagoya Math. J. 214, 125–168 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Oh, Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Comm. Pure Appl. Math. 46(7), 949–993 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  39. Oh, Y-G .: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks II. \(({\bf C}{\rm P}^n,{\bf R}{\rm P}^n)\), Comm. Pure Appl. Math. 46(7): 995–1012 (1993)

  40. Oh, Y.-G.: Relative Floer and quantum cohomology and the symplectic topology of Lagrangian submanifolds, Contact and symplectic geometry, vol. 8, pp. 201–267. Cambridg: Publications of the Newton Institute, Cambridge University Press, Cambridge (1994)

  41. Oakley, J., Usher, M.: On certain Lagrangian submanifolds of \(S^2\times S^2\) and \({C}{{\rm P}}^n\). Algebr. Geom. Topol. 16(1), 149–209 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pabiniak, M.: Hamiltonian torus actions in equivariant cohomology and symplectic topology, Thesis (Ph.D.)–Cornell University, 125 pp (2012)

  43. Pech, C., Rietsch, K., Williams, L.: On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections. Adv. Math. 300, 275–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Przhiyalkovskiĭ, V.V.: Weak Landau-Ginzburg models of smooth Fano threefolds. Izv. Ross. Akad. Nauk Ser. Mat. 77(4), 135–160 (2013)

    MathSciNet  Google Scholar 

  45. Rietsch, K.: A mirror symmetric construction of \(qH^\ast _T(G/P)_{(q)}\). Adv. Math. 217(6), 2401–2442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ruan, W.-D.: Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces. II. Technical results on gradient flow construction. J. Symplectic Geom. 1(3), 435–521 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sheridan, N.: On the Fukaya category of a Fano hypersurface in projective space. Publ. Math. Inst. Hautes Études Sci. 124, 165–317 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Thimm, A.: Integrable geodesic flows on homogeneous spaces. Ergodic Theory Dynam. Systems. 1(4), 495–517 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  49. Woodward, C.T.: Gauged Floer theory of toric moment fibers. Geom. Funct. Anal. 21(3), 680–749 (2011)

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Acknowledgements

The author express his deep gratitude to Yunhyung Cho, Hansol Hong and Siu-Cheong Lau. The paper grows out from the collaborations and discussions with them. The author would like to thank the anonymous referees for the helpful comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government NRF-2021R1F1A1057739 and NRF-2020R1A5A1016126, and Pusan National University Research Grant, 2021.

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Kim, Y. Disk potential functions for quadrics. J. Fixed Point Theory Appl. 25, 46 (2023). https://doi.org/10.1007/s11784-023-01049-9

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