Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

Chan, Kwokwai; Lau, Siu-Cheong; Tseng, Hsian-Hua

doi:10.1016/j.aim.2013.05.018

Cited by 26 publications

(39 citation statements)

References 28 publications

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“…This can be rephrased as the conjecture that the SYZ map, defined in terms of orbi-disk invariants, is inverse to the toric mirror map of X (cf. [55, Conjecture 0.2], [16,Conjecture 1.1] and [18,Conjecture 2]). To prove this, knowledge about the orbi-disk invariants is absolutely crucial.…”

Section: 2mentioning

confidence: 99%

“…On the other hand, their conjecture is much more general and expected to hold even when X is a compact CY manifold. See [16,Conjecture 1.1] (also [18,Conjecture 2]) for a more precise formulation of the Gross-Siebert conjecture in the case of toric CY manifolds. Corollary 1.7 proves a weaker form of [16, Conjecture 1.1], which concerns periods over integral cycles inX (while here the cycles Γ 1 , .…”

Section: 2mentioning

confidence: 99%

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Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Chan¹,

Cho²,

Lau³

et al. 2016

J. Differential Geom.

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For a toric Calabi-Yau (CY) orbifold X whose underlying toric variety is semiprojective, we construct and study a non-toric Lagrangian torus fibration on X , which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of X to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in X bounded by fibers of the Gross fibration.We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (partial) compactifications of X . Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of X -this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions -an open analogue of Ruan's crepant resolution conjecture. arXiv:1306.0437v4 [math.SG] 1 Apr 20151 As explained in [18, Section 5.2], to prove the original stronger form of the conjecture, we need integral cycles whose periods have specific logarithmic terms. Such cycles have been constructed by Doran and Kerr in [33, Section 5.3 and Theorem 5.1] when X is the total space of the canonical line bundle K Y over a toric del Pezzo surface Y . Doran suggested to us that it should not be difficult to extend their construction to general toric CY varieties. Hence the stronger form of the conjecture should follow from Corollary 1.7 and their construction; cf. the discussion in [34, Section 4]. In the recent paper [74], Ruddat and Siebert gave yet another construction of such integral cycles by tropical methods. Though they worked only in the compact CY case, Ruddat pointed out that the method can be generalized to handle the toric CY case as well.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Chan¹,

Cho²,

Lau³

et al. 2016

J. Differential Geom.

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“…If this is the case, then an analogous construction in the case of toric Calabi-Yau manifolds would give an answer to the above question. [6,4], it was proved that the so-called SYZ map, which is defined in terms of generating functions of genus 0 open Gromov-Witten invariants, coincides with the inverse of the toric mirror map for any semi-projective toric Calabi-Yau manifold. The analogue of the SYZ map in the quasimap setting would just be the identity map t(q) = q.…”

Section: 1mentioning

confidence: 99%

SYZ mirror symmetry for toric Calabi-Yau manifolds

Chan¹,

Lau²,

Leung³

2012

J. Differential Geom.

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138

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In this note, we study the SYZ mirror construction for a toric Calabi-Yau manifold using instanton corrections coming from Woodward's quasimap Floer theory [40] instead of Fukaya-Oh-Ohta-Ono's Lagrangian Floer theory [18, 19, 20, 21]. We show that the resulting SYZ mirror coincides with the one written down via physical means [33,30,29] (as expected). Date: February 12, 2018. 2010 Mathematics Subject Classification. 53D37, 14J33 (primary) and 53D40, 53D45, 14N35 (secondary). Key words and phrases. Mirror symmetry, SYZ conjecture, quasimap, Lagrangian Floer theory, Calabi-Yau manifold, toric variety. 45. F. Ziltener, A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane, Mem.

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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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Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

Cited by 26 publications

References 28 publications

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds

SYZ mirror symmetry for toric Calabi-Yau manifolds

Lagrangian Floer potential of orbifold spheres

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