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Tropically constructed Lagrangians in mirror quintic threefolds

Part of: Surfaces and higher-dimensional varieties Differential topology Symplectic geometry, contact geometry

Published online by Cambridge University Press:  20 November 2020

Cheuk Yu Mak  and
Helge Ruddat
Cheuk Yu Mak
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD; E-mail: [email protected]
Helge Ruddat
Affiliation:
JGU Mainz, Institut für Mathematik, Staudingerweg 9, 55128 Mainz, Germany; E-mail: [email protected]

Abstract

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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 classification

Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 14J32: Calabi-Yau manifolds 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category 57R17: Symplectic and contact topology
Type
Differential Geometry and Geometric Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e58
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Abreu, Miguel, ‘Kähler metrics on toric orbifolds’, J. Differ. Geom. 58(1) (2001), 151–187.CrossRefGoogle Scholar
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.Google Scholar
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).Google Scholar
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.CrossRefGoogle Scholar
Silva, Ana Cannas da, ‘Symplectic toric manifolds’, in Symplectic Geometry of Integrable Hamiltonian Systems (Birkhäuser, Basel, 2003), 85–173.Google Scholar
Castaño-Bernard, R. and Matessi, D., ‘Lagrangian 3-torus fibrations’, J. Differ. Geom. (81) (2009), 483–573.CrossRefGoogle Scholar
Chantraine, Baptiste, ‘Lagrangian concordance of Legendrian knots’, Algebr. Geom. Topol. 10(1) (2010), 63–85.CrossRefGoogle Scholar
Cueto, M. A. and Deopurkar, A., ‘Anticanonical tropical cubic del pezzos contain exactly 27 lines’, (2019), arXiv:1906.08196.Google Scholar
Delzant, T., ‘Hamiltoniens périodiques et images convexes de l’application moment’ [Periodic Hamiltonians and convex images of the moment map], Bull. Soc. Math. France 116(3) (1988), 315–339.CrossRefGoogle Scholar
Donaldson, S. K., ‘Symplectic submanifolds and almost-complex geometry’, J. Differ. Geom. 44(4) (1996), 666–705.CrossRefGoogle Scholar
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.Google Scholar
Eliashberg, Y., ‘Contact $3$ -manifolds twenty years since J. Martinet’s work’, Ann. Inst. Fourier (Grenoble) 42(1-2) (1992), 165–192.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Etnyre, J. B., ‘Legendrian and transversal knots’, in Handbook of Knot Theory (Elsevier B. V., Amsterdam, 2005), 105–185.CrossRefGoogle Scholar
Evans, J. D. and Mauri, M., ‘Constructing local models for Lagrangian torus fibrations’, (2019), arXiv:1905.09229.Google Scholar
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).Google Scholar
Fulton, W.. Introduction to Toric Varieties. Annals Mathematics Studies, Vol. 131 (Princeton University Press, Princeton, NJ, 1993).CrossRefGoogle Scholar
Geiges, H., An Introduction to Contact Topology , Vol. 109 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2008).Google Scholar
Gromov, M., ‘Pseudo holomorphic curves in symplectic manifolds’, Invent. Math. 82(2) (1985), 307–347.CrossRefGoogle Scholar
Gross, M., ‘Topological Mirror Symmetry’, Invent. Math. 144(1) (2001), 75–137.CrossRefGoogle Scholar
Gross, M., ‘Toric degenerations and Batyrev-Borisov duality’, Math. Ann. 333 (2005), 645–688.CrossRefGoogle Scholar
Gross, M. and Siebert, B., ‘Affine manifolds, log structures, and mirror symmetry’, Turk. J. Math. 27(1) (2003), 33–60.Google Scholar
Gross, M. and Siebert, B., ‘Mirror symmetry via logarithmic degeneration data. I’, J. Differ. Geom. 72(2) (2006), 169–338.CrossRefGoogle Scholar
Guillemin, V., ‘Kaehler structures on toric varieties’, J. Differ. Geom. 40(2) (1994), 285–309.CrossRefGoogle Scholar
Hicks, J., Tropical Lagrangians and Homological Mirror Symmetry, PhD thesis, UC Berkeley, 2019.Google Scholar
Hicks, J., ‘Tropical Lagrangian hypersurfaces are unobstructed’, J. Topol. 13(4) (2020), 1409–1454.CrossRefGoogle Scholar
Hicks, J., ‘Tropical Lagrangians in toric del-Pezzo surfaces’, 2020, arXiv:2008.07197.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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).Google Scholar
Lerman, E. and Tolman, S., ‘Hamiltonian torus actions on symplectic orbifolds and toric varieties’, Trans. Amer. Math. Soc. 349(10) (1997), 4201–4230.CrossRefGoogle Scholar
Mak, C. Y. and Wu, W., ‘Dehn twists and Lagrangian spherical manifolds’, Selecta Math. (N.S.) 25(5) (2019), 68, 85.CrossRefGoogle Scholar
Mandel, T. and Ruddat, H., ‘Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves’, 2018, arXiv:1902.07183.Google Scholar
Mandel, T. and Ruddat, H., ‘Descendant log Gromov-Witten invariants for toric varieties and tropical curves’, Trans. Amer. Math. Soc. 373 (2020), 1109–1152.CrossRefGoogle Scholar
Matessi, D., ‘Lagrangian submanifolds from tropical hypersurfaces’, 2018, arXiv:1804.01469.Google Scholar
Matessi, D., ‘Lagrangian pairs of pants’, Int. Math. Res. Notices (2019), 50.Google Scholar
McDuff, D., ‘The structure of rational and ruled symplectic $4$ -manifolds’, J. Amer. Math. Soc. 3(3) (1990), 679–712.Google Scholar
McDuff, D. and Salamon, D., Introduction to Symplectic Topology, second edition (Clarendon Press, Oxford, 1998).Google Scholar
Mikhalkin, G., ‘Examples of tropical-to-Lagrangian correspondence’, Eur. J. Math. 5(3) (2019), 1033–1066.CrossRefGoogle Scholar
Nishinou, T. and Siebert, B., ‘Toric degenerations of toric varieties and tropical curves’, Duke Math. J. 135(1) (2006), 1–51.Google Scholar
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.Google Scholar
Panizzut, M. and Vigeland, M. D., ‘Tropical lines on cubic surfaces’, (2019), arXiv:0708.3847.Google Scholar
Ruddat, H., ‘A homology theory for tropical cycles on integral affine manifolds and a perfect pairing’, (2020), arXiv:2002.12290.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Ruddat, H. and Zharkov, I., ‘Compactifying torus fibrations over integral affine manifolds with singularities’, (2020), arXiv:2003.08521.Google Scholar
Ruddat, H. and Zharkov, I., ‘Tailoring a pair of pants,’, (2020), arXiv:2001.08267.Google Scholar
Ruddat, H. and Zharkov, I.. ‘Topological Strominger-Yau-Zaslow fibrations’, in preparation.Google Scholar
Seidel, P., ‘Graded Lagrangian submanifolds’, Bull. Soc. Math. France 128(1) (2000), 103–149.Google Scholar
Seidel, P., Fukaya Categories and Picard-Lefschetz Theory, Zurich Lectures in Advanced Mathematics (European Mathematical Society, Zürich, Switzerland, 2008), 326.CrossRefGoogle Scholar
Sheldon, K., ‘Lines on complete intersection threefolds with k=0’, Math. Z. 191(2) (1986), 293–296.Google Scholar
Sheridan, N. and Smith, I., ‘Lagrangian cobordism and tropical curves’, (2018), arXiv:1805.07924.Google Scholar
Solomon, J. P., Intersection Theory on the Moduli Space of Holomorphic Curves with Lagrangian Boundary Conditions PhD thesis, Massachusetts Institute of Technology, 2006.Google Scholar
Strominger, A., Yau, S. T. and Zaslow, E., ‘Mirror symmetry is T-duality’, Nucl. Phys. B (479) (1996), 243–259.CrossRefGoogle Scholar
Thomas, R. P., ‘Moment maps, monodromy and mirror manifolds’, in Symplectic Geometry and Mirror Symmetry (World Scientific, River Edge, NJ, 2001), 467–498.CrossRefGoogle Scholar
Thomas, R. P. and Yau, S.-T., ‘Special Lagrangians, stable bundles and mean curvature flow’, Comm. Anal. Geom. 10(5) (2002), 1075–1113.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Vigeland, M. D., ‘Smooth tropical surfaces with infinitely many tropical lines’, Ark. Mat., 48(1) (2010), 177–206.CrossRefGoogle Scholar