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

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.

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 prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers and some other topological results are consequences of the modular invariance of elliptic operators on loop space.