New construction of vector bundles on Riemann surfaces and Verlinde's formula

点击翻译按钮获取中文摘要

1. We showed that any stable vector bundle of degree 0 on a Riemann surface close to a maximally degenerate curve is obtained from a linear representation of the Schottky group uniformizing the Riemann surface. Further, we described the relationship between the representation space of the Schottky group and the moduli space of the vector bundles by using Abel-Jacobi's theorem and Verlinde's formula.2. We described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6 extending Igusa's result. Further, we extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms to the case of Siegel modular forms.3. We gave a mathematical rigorous model of the one loop approximation of the perturbative Chern-Simons integral in an abstract Wiener space setting, and by appealing to the Malliavin-Taniguchi formula of changing variables on the abstract Wiener space, we derived the asymptotic expansion for the Chern-Simons integral with respect to the charge parameter.