We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles -including any direct sum of line bundles -on P n . This includes proving the formula of Candelas-de la Ossa-Green-Parkes for the instanton prepotential function for quintic in P 4 . We derive, among many other examples, the so-called multiple cover formula for GW invariants of P 1 . We also prove a formula for enumerating Euler classes which arise in the so-called local mirror symmetry for some noncompact Calabi-Yau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma model.

No abstract

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

We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle.

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

We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles -including any direct sum of line bundles -on P n . This includes proving the formula of Candelas-de la Ossa-Green-Parkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential function for the quintic in P 4 . We derive, among many other examples, the multiple cover formula for Gromov-Witten invariants of P 1 , computed earlier by Morrison-Aspinwall and by Manin in different approaches. We also prove a formula for enumerating Euler classes which arise in the so-called local mirror symmetry for some noncompact Calabi-Yau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma model. 11/97

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.

No abstract