Prepotential, Mirror Map and F-Theory on K3

TL;DR

This work tests the eight-dimensional heterotic–F-theory duality by computing one-loop corrections to F^4 couplings in heterotic string on $T^2$ and showing they organize into a holomorphic prepotential ${ m math{G}}$, whose fifth derivative yields quintic open/closed string couplings via $(\partial_T)^5 {\cal G} = \sum_{(k,l)>0} k^5 g(kl)\frac{q_T^k q_U^l}{1-q_T^k q_U^l}$. It then matches these results to F-theory on K3 by exploiting constant IIB coupling subspaces and Seiberg–Witten–type curves, where the holomorphic data are encoded in ${\cal G}$ and cusp forms through modular functions like $J(T)$ and $\,\eta$. The two explicit realizations, $E_8\times E_8$ and $SO(8)^4$, show exact agreement between heterotic one-loop results and K3 geometry, illustrating a mirror-map between open (brane) and closed (bulk) sectors. Overall, the paper provides a quantitative bridge between heterotic calculations and F-theory geometry, revealing a novel open–closed mirror phenomenon with potential implications for curve counting and higher-dimensional holomorphic couplings.

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.