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 ChernSimons theory on three-manifolds.14N35, 53D45; 57M27

Abstract. We outline a proof of a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [25] based on large N duality.

Abstract. We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms.

Motivated by the Mariño-Vafa formula of Hodge integrals and physicists’ predictions on local Gromov-Witten invariants of toric Fano surfaces in a Calabi-Yau threefold, the third author conjectured a formula of certain Hodge integrals in terms of certain Chern-Simons invariants of the Hopf link. We prove this formula by virtual localization on moduli spaces of relative stable morphisms.