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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: cheukyu.mak@ed.ac.uk
Helge Ruddat
Affiliation:
JGU Mainz, Institut für Mathematik, Staudingerweg 9, 55128 Mainz, Germany; E-mail: ruddat@uni-mainz.de

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
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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

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