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Mirror Symmetry, Borcherd-Harvey-Moore Products and Determinants of the Calabi-Yau Metrics on K3 Surfaces

Based on the work of Borcherds we construct on the moduli space of K3 surfaces with B-field an automorphic form exp_{4,20} which vanishes on the totally geodesic subspaces orthogonal to -2 vectors of the even, unimodular lattice of signature (4,20). 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. The counting of rational curves on special K3 surfaces using the regularized determinants of the Laplacian of CY metrics is related to some results of Bershadsky, Cecotti, Ouguri and Vafa.