Determinants of the Calabi-Yau Metrics on K3 Surfaces, Discriminants, Theta Lifts and Counting Problems in the A and B Models

Abstract

The Dedekind eta functions plays important role in different branches of Mathematics and Theoretical Physics. One way to construct Dedekind Eta function to use the explicit formula (Kroncker limit formula) for the regularized determinants of the Laplacian of the flat metric acting of (0,1) forms on elliptic curves. The holomorphic part of the regularized determinant is the Dedekind eta functions. In this paper we generalized the above approach to the case of K3 surfaces. We give an explicit formula of the regularized determinants of the Laplacians of Calabi Yau metrics on K3 Surfaces, following suggestions by R. Borcherds. The holomorphic part of the regularized determinants will be the higher dimensional analogue of Dedekind Eta function. We give explicit formulas for the number of non singular rational curves with a fixed volume with respect to a Hodge metric in the case of K3 surfaces with Picard group unimodular even lattice by using the holomorphic part exp_{3,19} of the regularized determinants det_{(0,1)}. We gave the combinatorial interpretation of the restriction of the automorphic form exp_{3,19} on the moduli of K3 surfaces with unimodular Picard lattices in the A and B models. The results obtained in this paper are related to some results of Bershadsky, Cecotti, Ouguri and Vafa. See BCOV.