Open Gromov-Witten invariants and SYZ under local conifold transitions

2019

Pacific J. Math.

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We prove that generalized conifolds and orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. Our work mathematically confirms the gauge-theoretic assertion of Aganagic-Karch-Lüst-Miemiec, and also provides a supportive evidence to Morrison's conjecture that geometric transitions are reversed under mirror symmetry. We now focus on two natural generalizations of the conifold: generalized conifolds and orbifolded conifolds. For integers k, l ≥ 1, a generalized conifold is given by G k,l := {(x, y, z, w) ∈ C 4 | xy − (1 + z) k (1 + w) l = 0} and an orbifolded conifold is given by O k,l := {(u 1 , v 1 , u 2 , v 2 , z) ∈ C 5 | u 1 v 1 − (1 + z) k = u 2 v 2 − (1 + z) l = 0}. (We have made a change of coordinates, namely z → 1 + z and w → 1 + w, for later convenience.) They reduce to the conifold when k = l = 1. The punctured generalized conifold is defined as G k,l := G k,l \ D G , where D G := {z = 0} ∪ {w = 0} is a normal-crossing anti-canonical divisor of G k,l , and the punctured orbifolded conifold as O k,l := O k,l \ D O , where D O := {z = 0} is a smooth anticanonical divisor of O k,l. As is the case of the conifold, their symplectic structures and complex structures are governed by the crepant resolutions and deformations respectively. The main theorem of the present paper is the following. Theorem 1.3 (Theorem 3.1). The punctured generalized conifold G k,l is mirror symmetric to the punctured orbifolded conifold O k,l in the sense that the deformed punctured generalized conifold G k,l is SYZ mirror symmetric to the resolved punctured orbifolded conifold O k,l , and the resolved punctured generalized conifold G k,l is SYZ mirror symmetric to the deformed punctured orbifolded conifold O k,l. G k,l O O

Section: Syz Mirror Constructionmentioning

confidence: 99%

Section: Conifold Pointmentioning

confidence: 99%

See 1 more Smart Citation

Geometric transitions and SYZ mirror symmetry

Kanazawa

2019

Pacific J. Math.

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“…Similar result holds for toric semi-Fano manifolds [CLLT12], in which the proof involves Seidel representation and degeneration techniques. Open mirror theorem is useful for studying global mirror symmetry, as illustrated in [CCLT14,Lau13].…”

Section: Now We Are Prepared To State the Open Mirror Theoremmentioning

confidence: 99%

Gross-Siebert’s slab functions and open GW invariants for toric Calabi-Yau manifolds

Mathematical Research Letters

2015

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This paper derives an equality between the slab functions in Gross-Siebert program and generating functions of open Gromov-Witten invariants for toric Calabi-Yau manifolds, and thereby confirms a conjecture of Gross-Siebert on symplectic enumerative meaning of slab functions. The proof is based on the open mirror theorem of Chan-Cho-Lau-Tseng [CCLT13]. It shows an instance of correspondence between tropical and symplectic geometry in the open sector.

“…When the discriminant loci of the Lagrangian fibration are relatively simple such as in the case of toric Calabi-Yau manifolds or their conifold transitions, quantum corrections by holomorphic discs have a neat expression and so the SYZ construction can be explicitly worked out [CLL12,AAK,Lau14]. In general the discriminant locus of a Lagrangian fibration is rather complicated, these holomorphic discs interact with each other and form complicated scattering diagrams studied by Kontsevich-Soibelman [KS06] and Gross-Siebert [GS11].…”

Section: Introductionmentioning

confidence: 99%

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

Cho

Hong²,

2017

J. Differential Geom.

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110

This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients. Contents 1. Introduction 1 Acknowledgement 10 2. Algebraic construction of localized mirror functor 10 3. Immersed Lagrangian Floer theory and generalized SYZ construction 17 4. Geometric construction of localized mirror functor 23 5. Generalized SYZ for a finite-group quotient 27