We derive some Hodge integral identities by taking various limits of the Marino-Vafa formula using the cut-and-join equation. These identities include the formula of general λg-integrals, the formula of λg−1-integrals on -Mg,1, the formula of cubic λ integrals on -Mg, and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the g conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Marino-Vafa formula.

In this paper we first study the moduli spaces related to Calabi-Yau manifolds. We then apply the results to the following problem. Let C be a fixed Riemann surface with fixed finite number of points on it. Given a CY manifold with fixed topological type, we consider the set of all families of CY manifolds of the fixed topological type over C with degenerate fibres over the fixed points up to isomorphism. This set is called Shafarevich set. The analogue of Shafarevich conjecture for CY manifolds is for which topological types of CY the Shafarevich set is finite. It is well-known that the analogue of Shafarevich conjecture is closely related to the study of the moduli space of polarized CY manifolds and the moduli space of the maps of fixed Riemann surface to the coarse moduli space of the CY manifolds. We prove the existence of the Teichmuller space of CY manifolds together with a universal family of marked CY manifolds. From this result we derive the existence of a finite cover of the coarse moduli space which is a non-singular quasi-projective manifold. Over this cover we construct a family of polarized CY manifolds. We study the moduli space of maps of the fixed Riemann with fixed points on it to the moduli space of CY manifolds constructed in the paper such that the maps map the fixed points on the Riemann surface to the discriminant locus. If this moduli space of maps is finite then Shafarevich conjecture holds. We relate the analogue of Shafarevich problem to the non-vanishing of the Yukawa coupling. We give also a counter example of the Shafarevich problem for a class of CY manifolds.

We have developed a mathematical theory of the topological vertex-a theory that was originally proposed by M. Aganagic, A. Klemm, M. Marino, 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.

We generalize the theorems in Mirror Principle Ⅰ and Ⅱ to the case of general projective manifolds without the convexity assumption. We also apply the results to balloon manifolds, and generalize to higher genus.