Mathematics underlying the F-theory/heterotic string duality in eight dimensions

We give an analytic description of the moduli space of classical vacua for F-theory compactified on elliptic K3 surfaces, on open tubular regions near the two Type II boundary divisors. The structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors. As the two Type II divisors can be naturally identified as moduli spaces of elliptic curves and flat G-bundles with G = (E-8 x E-8) Z(2) or Spin (32)/Z(2), one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two-torus corresponding to large volumes of the torus. This provides a proof for the classical version of F-theory/Heterotic String Duality in eight dimensions. A description of the Type II boundary points in terms of elliptic stable K3 surfaces is also given.