Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Bardwell-Evans, Sam; Cheung, Man-Wai; Hong, Hyunkee; Lin, Yu-Shen

doi:10.48550/arxiv.2110.15234

Cited by 2 publications

(3 citation statements)

References 45 publications

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“…Their superpotential is equal to the disk potential function of the caterpillar bending system in (1.2). Recently, the scattering diagram and the disk potential of a special Lagrangian fibration in a non-monotone del Pezzo surface were computed by Bardwell-Evans-Cheung-Hong-Lin in [BECHL21], which agrees with the scattering diagram of Gross-Pandharipande-Siebert and Gross-Hacking-Keel [GPS10,GHK15]. Their chamber structure agrees with the chambers structure given by various bending systems.…”

Section: Introductionsupporting

confidence: 64%

Disk potential functions for polygon spaces

Kim¹,

Lau²,

Zheng³

2022

Preprint

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We derive a Floer theoretical SYZ mirror for an equilateral and generic polygon space. The disk potential function of the monotone torus fiber of the caterpillar bending system is calculated by computing non-trivial open Gromov-Witten invariants from the structural result of the monotone Fukaya category, the topology of fibers of completely integrable systems, and toric degenerations. Then, combining the result with the work of Nohara-Ueda [NU20] and Marsh-Rietsch [MR20], we obtain the disk potential functions of bending systems and produce a mirror cluster variety of type A without frozen variables via Lagrangian Floer theory.

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Section: Introductionsupporting

confidence: 64%

Disk potential functions for polygon spaces

Kim¹,

Lau²,

Zheng³

2022

Preprint

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“…3.26]. This was later generalized to the case when (B, P) is only simple (instead of strongly simple) 3 and further to toroidal crossing spaces in Felten-Filip-Ruddat [14] using different methods.…”

Section: 4mentioning

confidence: 99%

“…At the same time, Fukaya's Correspondence I suggests that these gradient Morse flow trees arise as adiabatic limits of loci of those Lagrangian torus fibers which bound nontrivial (Maslov index 0) holomorphic disks. This can be reformulated as a holomorphic/tropical correspondence, and much evidence has been found [15,17,33,34,10,9,32,8,3].…”

Section: Introductionmentioning

confidence: 98%

Smoothing, scattering, and a conjecture of Fukaya

Chan¹,

Leung²,

Ma³

2022

Preprint

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In 2002, Fukaya [16] proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture [41] by introducing two correspondences: one between the theory of pseudoholomorphic curves on a Calabi-Yau manifold X and the multi-valued Morse theory on the base B of an SYZ fibration p : X → B, and the other between deformation theory of the mirror X and the same multi-valued Morse theory on B. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base B is replaced by tropical geometry on the Legendre dual B. In the proof, we apply techniques of asymptotic analysis developed in [6,7] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety 0 X † introduced in [5]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part X sf ⊆ X allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

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Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Cited by 2 publications

References 45 publications

Disk potential functions for polygon spaces

Disk potential functions for polygon spaces

Smoothing, scattering, and a conjecture of Fukaya

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