Prepotential, Mirror Map and F-Theory on K3

TL;DR

This paper explores one-loop corrections to F^4 couplings in heterotic string theory on T^2, their relation to F-theory via 7-brane geometry, and the mathematical connection to mirror maps and K3 surfaces, testing dualities in eight dimensions.

Contribution

It provides a detailed analysis of F^4 couplings in heterotic and F-theory frameworks, revealing new insights into mirror maps and dualities involving K3 surfaces.

Findings

Holomorphic prepotentials characterize one-loop corrections.

F-theory couplings can be derived from 7-brane geometry.

Mathematical association of couplings with K3 surfaces and mirror maps.

Abstract

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.