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Let (X, ω) be a symplectic manifold, J be an ω-tame almost complex structure, and L be a Lagrangian submanifold. The stable compactifi-cation of the moduli space of parametrized J-holomorphic curves in X with boundary in L (with prescribed topological data) is compact and Hausdorff in Gromov's C ∞-topology. We construct a Kuranishi structure with corners in… (More)

In " The Yang-Mills equations over Riemann surfaces " , Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. I will review the work of Atiyah and Bott and describe generalization to closed nonorientable surfaces based on joint work with Nan-Kuo Ho and Daniel Ramras. I will also survey some results on… (More)

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… (More)

In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to… (More)

We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of Kac-Moody algebras. Such generating series appear in calculations of Gromov-Witten in-variants by localization techniques. It generalizes a formula conjectured by Mariño and Vafa, recently proved in joint work with Chiu-Chu Melissa… (More)

We prove the following stronger version of the positivity of quasi-local mass stated in [22]: the quasi-local energy (mass) of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface and the boundary of the… (More)

We outline a proof of a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality.

We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [24] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.

We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons.… (More)