Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

Cho, Cheol-Hyun; Poddar, Mainak

doi:10.48550/arxiv.1206.3994

Cited by 13 publications

(26 citation statements)

References 15 publications

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“…In [CHL], a geometric construction of mirror geometry based on (immersed) Lagrangian Floer theory was proposed. It automatically 1 P 1 a,b,c is toric when c = 1, and the Lagrangian Floer potential in this case was known by [CP12]. 2 P 1 a,b,c and its covering Riemann surface Σ have the same expression of mirror superpotential (defined over the domains C 3 and C 3 /G for a finite group G respectively).…”

Section: Introductionmentioning

confidence: 98%

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Lagrangian Floer potential of orbifold spheres

Cho

Hong

Kim

et al. 2017

Advances in Mathematics

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Abstract. For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov-Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.

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Section: Introductionmentioning

confidence: 98%

“…The A n case is c = 1. The orbifold sphere P 1 a,b,1 is toric, and the open Gromov-Witten potential (taking L to be the Lagrangian torus fibers of the moment map) is known by [CP12]. Namely,…”

Section: Introductionmentioning

confidence: 99%

Lagrangian Floer potential of orbifold spheres

Cho

Hong

Kim

et al. 2017

Advances in Mathematics

Self Cite

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“…However, it applies also in the orbifold and noncompact cases and in those cases can give open sets of fibers that are nondisplaceable because they have nonvanishing quasi-map invariants (called qW invariants, for short). See also the orbifold version of the standard approach by Cho and Poddar [CP12]. Figure 4.6.1 illustrates the current knowledge about the displaceability of points in P(1, 3, 5).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Displacing Lagrangian toric fibers by extended probes

Abreu

Borman

McDuff

2014

Algebr. Geom. Topol.

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Abstract. In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1, 3, 5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

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“…This gives the conifold limit point which is also an orbifold limit in this case. We can study SYZ via orbidisc invariants (defined in [15]) of the orbifold X instead of its smoothing. This was done in [9] and one obtains a different flat coordinates around the conifold limit.…”

Section: Open Gromov-witten Invariants and Syz Under Local Conifold T...mentioning

confidence: 99%

Open Gromov-Witten invariants and SYZ under local conifold transitions

Lau

2014

Journal of the London Mathematical Society

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For a local non‐toric Calabi–Yau manifold which arises as a smoothing of a toric Gorenstein singularity, this paper derives the open Gromov–Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its Strominger‐Yau‐Zaslow (SYZ) mirror. Moreover, it proves that the SYZ mirrors and disk potentials vary smoothly under conifold transitions, giving a global picture of SYZ mirror symmetry.

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Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

Cited by 13 publications

References 15 publications

Lagrangian Floer potential of orbifold spheres

Lagrangian Floer potential of orbifold spheres

Displacing Lagrangian toric fibers by extended probes

Open Gromov-Witten invariants and SYZ under local conifold transitions

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