Gromov-witten Invariants of Toric Calabi-yau Threefolds

A Mathematical Theory of the Topological Vertex

We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.

Integrality of Gopakumar–vafa Invariants of Toric Calabi–yau Threefolds

The Gopakumar–Vafa invariant is a number defined as certain linear combination of the Gromov–Witten invariants. We prove that the GV invariants of the toric Calabi–Yau threefold are integers and that the invariants at high genera vanish. The proof of the integrality is the elementary number theory and that of the vanishing uses the operator formalism and the exponential formula.

Flops, Type Iii Contractions and Gromov–witten Invariants on Calabi–yau Threefolds

Zero dimensional Donaldson–Thomas invariants of threefolds

Ever since the pioneer work of Donaldson and Thomas on Yang–Mills theory over Calabi–Yau threefolds [5, 13], people have been searching for their roles in the study of Calabi–Yau geometry and their relations with other branches of mathematics. The recent results and conjectures of Maulik, Nekrasov, Okounkov and Pandharipande [10, 11] that relate the invariants of the moduli of ideal sheaves of ...

Open Gromov-Witten invariants and SYZ under local conifold transitions

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-YauZaslow (SYZ) mirror. Moreover, it proves that the SYZ mirrors and disk potentials vary smoothly under conifold transitions...