Tropically constructed Lagrangians in mirror quintic threefolds
Tropically constructed Lagrangians in mirror quintic threefolds
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.
Tropical Lagrangian, Calabi-Yau 3 fold, Constructing Lagrangian submanifolds
Mak, Cheuk Yu
49c234b8-842f-4cda-b082-d36505c24626
Ruddat, Helge
c750ab41-018d-4053-b0e7-0593fb4d5296
20 November 2020
Mak, Cheuk Yu
49c234b8-842f-4cda-b082-d36505c24626
Ruddat, Helge
c750ab41-018d-4053-b0e7-0593fb4d5296
Mak, Cheuk Yu and Ruddat, Helge
(2020)
Tropically constructed Lagrangians in mirror quintic threefolds.
Forum of Mathematics, Sigma, 8, [e58].
(doi:10.1017/fms.2020.54).
Abstract
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.
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tropically-constructed-lagrangians
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Accepted/In Press date: 9 October 2020
Published date: 20 November 2020
Keywords:
Tropical Lagrangian, Calabi-Yau 3 fold, Constructing Lagrangian submanifolds
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Local EPrints ID: 478671
URI: http://eprints.soton.ac.uk/id/eprint/478671
ISSN: 2050-5094
PURE UUID: b148c3a8-bd6a-4f3e-b000-1e91bd3c8e17
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Date deposited: 07 Jul 2023 16:30
Last modified: 17 Mar 2024 04:17
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Author:
Cheuk Yu Mak
Author:
Helge Ruddat
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