Open-Closed String Mirror Symmetry
Wolfgang Lerche
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 $\GG$. 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 $E_8\times E_8$ 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 ${\del_t}^5\GG\sim \sum g_\ell \ell^5 {q^\ell\over 1-q^\ell}$ to $K3$ surfaces. This can be seen as a novel manifestation of the mirror map, acting here between open and closed string sectors.

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