Prepotential, mirror map and $F$-theory on $K3$

Abstract

We compute certain one-loop corrections to JP 4 couplings of the heterotic string compactified on T 2 , and show that they can be characterized by holomorphic prepotentials G-We then discuss how some of these couplings can be obtained in F-theory, or more precisely from 7brane geometry in type IIB language. We in particular study theories with Es x E$ and 50(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 d^G ~ Y^9^ T-^? to ^3 surfaces. This can be seen as a novel manifestation of the mirror map, acting here between open and closed string sectors.