Special geometry and automorphic forms

We consider the special geometry of the vector multiplet moduli space in compactifications of the heterotic string on K3 x T2 or the typeIIA string on K3-fibered Calabi-Yau threefolds. In particular, we constructa modified dilaton that is invariant under SO(2,n;Z) T-duality transformations at the non-perturbative level and regular everywhere on the moduli space. The invariant dilaton, together with a set of other coordinates that transform covariantly under SO(2,n;Z), parameterize the moduli space. The construction involves a meromorphic automorphic function of SO(2,n;Z), that also depends on the invariant dilaton. In the weak coupling limit, the divisor of this automorphic form is an integer linear combination of the rational quadratic divisors where the gauge symmetry is enhanced classically. We also show how the non-perturbative prepotential can be expressed in terms of meromorphic automorphic forms, by expanding a T-duality invariant quantity both in terms of the standard special coordinates and in terms of the invariant dilaton and the covariant coordinates. (orig.)