Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

Abstract

We present a proof of the mirror conjecture of Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in the total space of the canonical line bundle of the projective plane (Graber-Zaslow [arXiv:hep-th/0109075]), (ii) an outer brane at arbitrary framing in the resolved conifold (Zhou [arXiv:1001.0447]), and (iii) an outer brane at zero framing in the total space of the canonical line bundle of the projective plane (Brini [arXiv:1102.0281, Section 5.3]).

39 pages, 11 figures

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• Integrality structures in topological strings I: framed unknot
• Open Gromov-Witten Theory of $K_{{\mathbb{P}}^2},K_{{\mathbb{P}}^1\times {\mathbb{P}}^1},K_{W\mathbb{P}[1,1,2]},K_{{\mathbb{F}}_1}$ and Jacobi Forms
• Open/closed Correspondence via Relative/local Correspondence
• On explicit formulae of LMOV invariants
• The SYZ mirror symmetry and the BKMP remodeling conjecture
• Open/Closed BPS Correspondence and Integrality
• $T$-equivariant disc potentials for toric Calabi-Yau manifolds
• A String Dual for Partially Topological Chern-Simons-Matter Theories
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• Type II/F-theory Superpotentials and Ooguri-Vafa Invariants of Compact Calabi-Yau Threefolds with Three Deformations
• Quantum McKay correspondence for disc invariants of orbifold vertex
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
• Enumerative geometry of surfaces and topological strings
• The log-open correspondence for two-component Looijenga pairs
• Contact Invariants, Open String Invariants and Weinstein Conjecture