Special Geometry and Automorphic Forms

Abstract

We consider special geometry of the vector multiplet moduli space in compactifications of the heterotic string on $K3\times T^2$ or the type IIA string on $K3$-fibered Calabi-Yau threefolds. In particular, we construct a 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.

21 pages, plain LaTeX. Minor changes, references added

References in corpus (1)

• Perturbative gravitational couplings and Siegel modular forms in $D=4,N=2$ string models

Cited by in corpus (4)

• Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections
• (0,2) Heterotic Gauge Couplings and their M-Theory Origin
• The low-energy effective action for perturbative heterotic strings on $K_3\times T^2$ and the d=4 N=2 vector-tensor multiplet
• Special Geometry and Compactification on a Circle