We compute certain one-loop corrections to F^4 couplings of the heterotic
string compactified on T^2, and show that they can be characterized by
holomorphic prepotentials. We then discuss how some of these couplings can be
obtained in F-theory, or more precisely from 7-brane geometry in type IIB
language. We in particular study theories with E_8 x E_8 and SO(8)^4 gauge
symmetry, on certain one-dimensional sub-spaces of the moduli space that
correspond to constant IIB coupling. For these theories, the relevant geometry
can be mapped to Riemann surfaces. Physically, the computations amount to
non-trivial tests of the basic F-theory -- heterotic duality in eight
dimensions. Mathematically, they mean to associate holomorphic 5-point
couplings of the form (del_t)^5 G = sum[ g_l l^5 q^l/(1-q^l) ] to K3 surfaces.
This can be seen as a novel manifestation of the mirror map, acting here
between open and closed string sectors.Comment: 36 pages, 2 figures (published version
