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Toric degenerations of integrable systems on Grassmannians and polygon spaces

Published online by Cambridge University Press:  11 January 2016

Yuichi Nohara  and
Kazushi Ueda
Yuichi Nohara
Affiliation:
Yuichi Nohara Faculty of Education Kagawa University TakamatsuKagawa 760-8522Japannohara@ed.kagawa-u.ac.jp
Kazushi Ueda
Affiliation:
Department of Mathematics Graduate School of Science Osaka UniversityToyonaka Osaka, 560-0043Japankazushi@math.sci.osaka-u.ac.jp
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Abstract

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We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and we compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

Keywords

37J3553D37
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 214 , June 2014 , pp. 125 - 168
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[A1] Auroux, D., Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 2007, 51–91. MR 2386535.Google Scholar
[A2] Auroux, D., “Special Lagrangian fibrations, wall-crossing, and mirror symmetry” in Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry, Surv. Differ. Geom. 13, International Press, Somerville, Mass., 2009, 1–47. MR 2537081.Google Scholar
[B] Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 1994, 493–535. MR 1269718.Google Scholar
[BZ] Berenstein, A. and Zelevinsky, A., Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 2001, 77–128. MR 1802793. DOI 10.1007/s002220000102.Google Scholar
[CO] Cho, C.-H. and Oh, Y.-G., Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 2006, 773–814. MR 2282365. DOI 10.4310/AJM.2006.v10.n4.a10.Google Scholar
[EHX] Eguchi, T., Hori, K., and Xiong, C.-S., Gravitational quantum cohomology, Inter-nat. J. Modern Phys. A 12 1997, 1743–1782. MR 1439892. DOI 10.1142/ S0217751X97001146.CrossRefGoogle Scholar
[Fo] Foth, P., Moduli spaces of polygons and punctured Riemann spheres, Canad. Math. Bull. 43 2000, 162–173. MR 1754021. DOI 10.4153/CMB-2000–024-1.Google Scholar
[FO+1] Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Lagrangian Intersection Floer Theory: Anomaly and Obstruction, AMS/IP Stud. Adv. Math. 46, Amer. Math. Soc., Providence, 2009. MR 2553465.Google Scholar
[FO+2] Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151 2010, 23–174. MR 2573826. DOI 10.1215/00127094–2009-062.Google Scholar
[FO+3] Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Lagrangian Floer theory and mirror symmetry on compact toric manifolds, preprint, arXiv:1009.1648v2 [math.SG].CrossRefGoogle Scholar
[FO+4] Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Toric degeneration and non-displaceable Lagrangian tori in S2 × S2 , preprint, arXiv:1002.1660v1 [math.SG].Google Scholar
[GM] Gel’fand, I. M. and MacPherson, R. D., Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math. 44 1982, 279–312. MR 0658730. DOI 10.1016/0001-870882)90040–8.Google Scholar
[GS] Guillemin, V. and Sternberg, S., The Gel’fand–Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal. 52 1983, 106–128. MR 0705993. DOI 10.1016/0022-1236(83)90092–7.Google Scholar
[HK] Hausmann, J.-C. and Knutson, A., Polygon spaces and Grassmannians, Enseign. Math. (2) 43 1997, 173–198. MR 1460127.Google Scholar
[HMM] Howard, B., Manon, C., and Millson, J. J., The toric geometry of triangulated polygons in Euclidean space, Canad. J. Math. 63 2011, 878–937. MR 2849001. DOI 10.4153/CJM-2011–021-0.Google Scholar
[J] Jeffrey, L. C., Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann. 298 1994, 667–692. MR 1268599. DOI 10.1007/BF01459756.Google Scholar
[KY] Kamiyama, Y. and Yoshida, T., Symplectic toric space associated to triangle inequalities, Geom. Dedicata 93 2002, 25–36. MR 1934683. DOI 10.1023/A: 1020393910472.Google Scholar
[KM] Kapovich, M. and Millson, J. J., The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 1996, 479–513. MR 1431002.Google Scholar
[Kl] Klyachko, A. A., “Spatial polygons and stable configurations of points in the projective line” in Algebraic Geometry and Its Applications (Yaroslavl’, 1992), Aspects Math. E25, Vieweg, Braunschweig, 1994, 67–84. MR 1282021.CrossRefGoogle Scholar
[Ko] Kostant, B., “Orbits, symplectic structures and representation theory” in Proceedings of the U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), Nippon Hyoronsha, Tokyo, 1966, 71. MR 0213476.Google Scholar
[MP] Millson, J. J. and Poritz, J. A., Around polygons in R3 and S3 , Comm. Math. Phys. 218 2001, 315–331. MR 1828984. DOI 10.1007/PL00005557.Google Scholar
[NNU1] Nishinou, T., Nohara, Y., and Ueda, K., Toric degenerations of Gelfand– Cetlin systems and potential functions, Adv. Math. 224 2010, 648–706. MR 2609019. DOI 10.1016/j.aim.2009.12.012.Google Scholar
[NNU2] Nishinou, T., Nohara, Y., and Ueda, K., Potential functions via toric degenerations, Proc. Japan Acad. Ser. A Math. Sci. 88 2012, 31–33. MR 2879356. DOI 10.3792/pjaa.88.31.Google Scholar
[R1] Ruan, W.-D., “Lagrangian torus fibration of quintic hypersurfaces, I: Fermat quintic case” in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, Mass., 1999), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, 2001, 297–332. MR 1876075.Google Scholar
[R2] Ruan, W.-D., Lagrangian torus fibration of quintic Calabi–Yau hypersurfaces, II: Technical results on gradient flow construction, J. Symplectic Geom. 1 2002, 435–521. MR 1959057.Google Scholar
[Ru] Rusinko, J., Equivalence of mirror families constructed from toric degenerations of flag varieties, Transform. Groups 13 2008, 173–194. MR 2421321. DOI 10.1007/s00031–008-9008-y.Google Scholar
[SS] Speyer, D. and Sturmfels, B., The tropical Grassmannian, Adv. Geom. 4 2004, 389–411. MR 2071813. DOI 10.1515/advg.2004.023.Google Scholar
[T] Treloar, T., The symplectic geometry of polygons in the 3-sphere, Canad. J.Math. 54 2002, 30–54. MR 1880958. DOI 10.4153/CJM-2002–002-1.CrossRefGoogle Scholar