Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

Chan, Kwokwai; Lau, Siu-Cheong; Leung, Naichung Conan; Tseng, Hsian-Hua

doi:10.1215/00127094-0000003x

Cited by 35 publications

(54 citation statements)

References 24 publications

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“…While some general foundational work has been done [65,73], at this point most of the results in the theory rely on additional structure. In [21,22] Lagrangian Floer theory is employed to study the case when the boundary condition is a fiber of the moment map. In the toric context, a mathematical approach [13,33,54,68] to construct operatively a virtual counting theory of open maps is via the use of localization [3,4,8,41,59], quantum knot invariants [47,62], and ordinary Gromov-Witten and DonaldsonThomas theory via "gluing along the boundary" [2,60,63].…”

Section: 2mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-CortiIritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

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Section: 2mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Enumerative meaning of the mirror map 4 was obtained in the compact semi-Fano toric case and toric Calabi-Yau case [CLT13,CLLT11,CLLT12,CCLT]. It was shown that the mirror map equals to the so-called SYZ map, which arises from SYZ construction and is written in terms of disc invariants.…”

Section: Algorithm To Count Polygonsmentioning

confidence: 99%

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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“…In more explicit terms, coefficients of W LF are virtual counts of Maslov index 2 stable disks, or more precisely, genus 0 open Gromov-Witten invariants, and W LF is a perturbation of the Hori-Vafa potential W of the form W LF = W + correction terms because coefficients of W only encode counts of embedded disks (which is why W LF = W only when X is toric Fano). In general it is very hard to compute W LF , but explicit formulas are known in a few low-dimensional examples [2,22,6] and when X is semi-Fano [7,8,28].…”

Section: Introductionmentioning

confidence: 99%

Tropical counting from asymptotic analysis on Maurer-Cartan equations

Chan

Nikolas

2020

Trans. Amer. Math. Soc.

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Let X = XΣ be a toric surface and (X, W ) be its Landau-Ginzburg (LG) mirror where W is the Hori-Vafa potential [37]. We apply asymptotic analysis to study the extended deformation theory of the LG model (X, W ), and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W . For X = P 2 , our construction reproduces Gross' perturbed potential Wn [29] which was proven to be the universal unfolding of W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of Wn across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross [29] (in the case of X = P 2 ). 1As in [10], a solution Φ to the extended MC equation 1.3 of (G/I) * , * can be constructed using Kuranishi's method [42], namely, by summing over directed ribbon weighted d-pointed k-trees (see Definition 2.6) with input Π. The MC solution Φ can then be decomposed aswhere Ξ i,i ∈ (G/I) i,i . A major discovery of this paper is that, as ℏ → 0, the correction terms Ξ 1,1 and Ξ 0,0 give rise to tropical disks of Maslov index 0 and 2 respectively: 1 (=Theorem 4.12). Each of the terms Ξ 0,0 , Ξ 1,1 of the Maurer-Cartan solution Φ can be expressed as a sum over tropical disks Γ whose moduli space M Γ is non-empty of codimensionhere Mono(Γ) is a holomorphic function and Log(Θ Γ ) is a holomorphic vector field defined explicitly for a tropical disk Γ, and α Γ is a Dolbeault (0, 1 − M I(Γ) 2 )-form with asymptotic support along the 1 These are the only nontrivial bulk deformations.

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Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

Cited by 35 publications

References 24 publications

Crepant resolutions and open strings

Crepant resolutions and open strings

Lagrangian Floer potential of orbifold spheres

Tropical counting from asymptotic analysis on Maurer-Cartan equations

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