×

zbMATH — the first resource for mathematics

Tropically constructed Lagrangians in mirror quintic threefolds. (English) Zbl 07289293
Summary: We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.
We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.
As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that \(>300\) mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14T20 Geometric aspects of tropical varieties
14T90 Applications of tropical geometry
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
57R17 Symplectic and contact topology in high or arbitrary dimension
53D12 Lagrangian submanifolds; Maslov index
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abreu, Miguel, ‘Kähler metrics on toric orbifolds’, J. Differ. Geom.58(1) (2001), 151-187. · Zbl 1035.53055
[2] Abreu, Miguel, ‘Kähler geometry of toric manifolds in symplectic coordinates’, in Symplectic and Contact Topology: Interactions and Perspectives, Vol. 35 of Fields Institute Communications (American Mathematical Society, Providence, RI, 2003), 1-24. · Zbl 1044.53051
[3] Paul, S. Aspinwall, Tom Bridgeland, Craw, Alastair, Douglas, Michael R., Gross, Mark, Kapustin, Anton, Moore, Gregory W., Segal, Graeme, Szendrői, Balázs and Wilson, P. M. H.. Dirichlet Branes and Mirror Symmetry, Vol. 4 of Clay Mathematics Monographs (American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009). · Zbl 1188.14026
[4] Candelas, Philip, De La Ossa, Xenia C., Green, Paul S. and Parkes, Linda, ‘A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory’, Nucl. Phys. B, 359(1) (1991), 21-74. · Zbl 1098.32506
[5] Silva, Ana Cannas Da, ‘Symplectic toric manifolds’, in Symplectic Geometry of Integrable Hamiltonian Systems(Birkhäuser,Basel, 2003), 85-173. · Zbl 1073.53102
[6] Castaño-Bernard, R. and Matessi, D., ‘Lagrangian 3-torus fibrations’, J. Differ. Geom. (81) (2009), 483-573. · Zbl 1177.14080
[7] Chantraine, Baptiste, ‘Lagrangian concordance of Legendrian knots’, Algebr. Geom. Topol.10(1) (2010), 63-85. · Zbl 1203.57010
[8] Cueto, M. A. and Deopurkar, A., ‘Anticanonical tropical cubic del pezzos contain exactly 27 lines’, (2019), arXiv:1906.08196.
[9] Delzant, T., ‘Hamiltoniens périodiques et images convexes de l’application moment’ [Periodic Hamiltonians and convex images of the moment map], Bull. Soc. Math. France116(3) (1988), 315-339. · Zbl 0676.58029
[10] Donaldson, S. K., ‘Symplectic submanifolds and almost-complex geometry’, J. Differ. Geom.44(4) (1996), 666-705. · Zbl 0883.53032
[11] Eliashberg, Y., ‘Filling by holomorphic discs and its applications’, in Geometry of Low-Dimensional Manifolds, 2, Vol. 151 of London Mathematical Society Lecture Note Series(Cambridge University Press, Cambridge, 1991), 45-67.
[12] Eliashberg, Y., ‘Contact \(3\) -manifolds twenty years since J. Martinet’s work’, Ann. Inst. Fourier (Grenoble)42(1-2) (1992), 165-192. · Zbl 0756.53017
[13] Eliashberg, Y. and Fraser, M., ‘Classification of topologically trivial Legendrian knots’, in Geometry, Topology, and Dynamics, Vol. 15 of CRM Proceedings & Lecture Notes(American Mathematical Society, Providence, RI, 1998), 17-51. · Zbl 0907.53021
[14] Etnyre, J. B., ‘Legendrian and transversal knots’, in Handbook of Knot Theory(Elsevier B. V., Amsterdam, 2005), 105-185. · Zbl 1095.57006
[15] Evans, J. D. and Mauri, M., ‘Constructing local models for Lagrangian torus fibrations’, (2019), arXiv:1905.09229.
[16] Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part II, Vol. 46 of AMS/IP Studies in Advanced Mathematics(American Mathematical Society, Providence, RI, 2009). · Zbl 1181.53003
[17] Fulton, W.. Introduction to Toric Varieties. Annals Mathematics Studies, Vol. 131 (Princeton University Press, Princeton, NJ, 1993). · Zbl 0813.14039
[18] Geiges, H., An Introduction to Contact Topology, Vol. 109 of Cambridge Studies in Advanced Mathematics(Cambridge University Press, Cambridge, 2008).
[19] Gromov, M., ‘Pseudo holomorphic curves in symplectic manifolds’, Invent. Math.82(2) (1985), 307-347. · Zbl 0592.53025
[20] Gross, M., ‘Topological Mirror Symmetry’, Invent. Math.144(1) (2001), 75-137. · Zbl 1077.14052
[21] Gross, M., ‘Toric degenerations and Batyrev-Borisov duality’, Math. Ann.333 (2005), 645-688. · Zbl 1086.14035
[22] Gross, M. and Siebert, B., ‘Affine manifolds, log structures, and mirror symmetry’, Turk. J. Math.27(1) (2003), 33-60. · Zbl 1063.14048
[23] Gross, M. and Siebert, B., ‘Mirror symmetry via logarithmic degeneration data. I’, J. Differ. Geom.72(2) (2006), 169-338. · Zbl 1107.14029
[24] Guillemin, V., ‘Kaehler structures on toric varieties’, J. Differ. Geom.40(2) (1994), 285-309. · Zbl 0813.53042
[25] Hicks, J., Tropical Lagrangians and Homological Mirror Symmetry, PhD thesis, UC Berkeley, 2019.
[26] Hicks, J., ‘Tropical Lagrangian hypersurfaces are unobstructed’, J. Topol.13(4) (2020), 1409-1454. · Zbl 07262224
[27] Hicks, J., ‘Tropical Lagrangians in toric del-Pezzo surfaces’, 2020, arXiv:2008.07197.
[28] Hitchin, N., ‘Lectures on special Lagrangian submanifolds’, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds Vol. 23 of AMS/IP Studies in Advanced Mathematics (American Mathematical Society, Providence, RI, 2001), 151-182. · Zbl 1079.14522
[29] Joyce, D., ‘On counting special Lagrangian homology 3-spheres’, in Topology and Geometry: Commemorating SISTAG, Vol. 314 of Contemporary Mathematics(American Mathematical Society, Providence, RI, 2002), 125-151. · Zbl 1060.53059
[30] Joyce, D., ‘Lectures on special Lagrangian geometry’, in Global Theory of Minimal Surfaces, Vol. 2 of Clay Mathematics Proceedings(American Mathematical Society, Providence, RI, 2005), 667-695. · Zbl 1102.53037
[31] Joyce, D., ‘Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow’, EMS Surv. Math. Sci.2(1) (2015), 1-62. · Zbl 1347.53052
[32] Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B.. Toroidal Embeddings. I, Vol. 339 of Lecture Notes in Mathematics(Springer-Verlag, Berlin, 1973).
[33] Lerman, E. and Tolman, S., ‘Hamiltonian torus actions on symplectic orbifolds and toric varieties’, Trans. Amer. Math. Soc.349(10) (1997), 4201-4230. · Zbl 0897.58016
[34] Mak, C. Y. and Wu, W., ‘Dehn twists and Lagrangian spherical manifolds’, Selecta Math. (N.S.)25(5) (2019), 68, 85. · Zbl 1436.53063
[35] Mandel, T. and Ruddat, H., ‘Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves’, 2018, arXiv:1902.07183.
[36] Mandel, T. and Ruddat, H., ‘Descendant log Gromov-Witten invariants for toric varieties and tropical curves’, Trans. Amer. Math. Soc.373 (2020), 1109-1152. · Zbl 1442.14166
[37] Matessi, D., ‘Lagrangian submanifolds from tropical hypersurfaces’, 2018, arXiv:1804.01469.
[38] Matessi, D., ‘Lagrangian pairs of pants’, Int. Math. Res. Notices (2019), 50.
[39] Mcduff, D., ‘The structure of rational and ruled symplectic \(4\) -manifolds’, J. Amer. Math. Soc.3(3) (1990), 679-712. · Zbl 0723.53019
[40] Mcduff, D. and Salamon, D., Introduction to Symplectic Topology, second edition (Clarendon Press, Oxford, 1998). · Zbl 1066.53137
[41] Mikhalkin, G., ‘Examples of tropical-to-Lagrangian correspondence’, Eur. J. Math.5(3) (2019), 1033-1066. · Zbl 1425.53102
[42] Nishinou, T. and Siebert, B., ‘Toric degenerations of toric varieties and tropical curves’, Duke Math. J.135(1) (2006), 1-51. · Zbl 1105.14073
[43] Oda, T., ‘Convex bodies and algebraic geometry—toric varieties and applications. I’, in Algebraic Geometry Seminar(Singapore, 1987)(World Scientific Publishing, Singapore, 1988), 89-94.
[44] Panizzut, M. and Vigeland, M. D., ‘Tropical lines on cubic surfaces’, (2019), arXiv:0708.3847.
[45] Ruddat, H., ‘A homology theory for tropical cycles on integral affine manifolds and a perfect pairing’, (2020), arXiv:2002.12290.
[46] Ruddat, H., ‘Mirror duality of Landau-Ginzburg models via discrete Legendre transforms’, in Homological Mirror Symmetry and Tropical Geometry, Vol. 15 of Lect. Notes Unione Mat. Ital.(Springer, Cham, Switzerland, 2014), 377-406. · Zbl 1317.53117
[47] Ruddat, H. and Siebert, B., ‘Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations’, Pub. math. IHES (2020), 82.
[48] Ruddat, H. and Zharkov, I., ‘Compactifying torus fibrations over integral affine manifolds with singularities’, (2020), arXiv:2003.08521.
[49] Ruddat, H. and Zharkov, I., ‘Tailoring a pair of pants,’, (2020), arXiv:2001.08267. · Zbl 1444.51001
[50] Ruddat, H. and Zharkov, I.. ‘Topological Strominger-Yau-Zaslow fibrations’, in preparation.
[51] Seidel, P., ‘Graded Lagrangian submanifolds’, Bull. Soc. Math. France128(1) (2000), 103-149. · Zbl 0992.53059
[52] Seidel, P., Fukaya Categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics(European Mathematical Society, Zürich, Switzerland, 2008), 326. · Zbl 1159.53001
[53] Sheldon, K., ‘Lines on complete intersection threefolds with k=0’, Math. Z.191(2) (1986), 293-296. · Zbl 0563.14020
[54] Sheridan, N. and Smith, I., ‘Lagrangian cobordism and tropical curves’, (2018), arXiv:1805.07924.
[55] Solomon, J. P., Intersection Theory on the Moduli Space of Holomorphic Curves with Lagrangian Boundary Conditions PhD thesis, Massachusetts Institute of Technology, 2006.
[56] Strominger, A., Yau, S. T. and Zaslow, E., ‘Mirror symmetry is T-duality’, Nucl. Phys. B (479) (1996), 243-259. · Zbl 0896.14024
[57] Thomas, R. P., ‘Moment maps, monodromy and mirror manifolds’, in Symplectic Geometry and Mirror Symmetry(World Scientific, River Edge, NJ, 2001), 467-498. · Zbl 1076.14525
[58] Thomas, R. P. and Yau, S.-T., ‘Special Lagrangians, stable bundles and mean curvature flow’, Comm. Anal. Geom.10(5) (2002), 1075-1113. · Zbl 1115.53054
[59] Van Garrel, M., Peter Overholser, D. and Ruddat, H., ‘Enumerative aspects of the Gross-Siebert program’, in Calabi-Yau Varieties: Arithmetic, Geometry and Physics, Vol. 34 of Fields Institute Monographs(Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2015), 337-420. · Zbl 1329.14104
[60] Vigeland, M. D., ‘Smooth tropical surfaces with infinitely many tropical lines’, Ark. Mat., 48(1) (2010), 177-206. · Zbl 1198.14061
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.