Special Geometry and Automorphic Forms

TL;DR

This paper explores the special geometry of heterotic and type IIA string compactifications, constructing a T-duality invariant dilaton and expressing the non-perturbative prepotential using automorphic forms.

Contribution

It introduces a T-duality invariant dilaton and formulates the non-perturbative prepotential with automorphic forms in string compactifications.

Findings

Invariant dilaton is regular and invariant under T-duality.

Automorphic forms encode gauge symmetry enhancements.

Prepotential expressed via automorphic functions.

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.