Solving the Topological String on K3 Fibrations

TL;DR

This paper solves the holomorphic anomaly equations for specific Calabi-Yau manifolds, revealing how topological string amplitudes can be expanded holomorphically across moduli space, especially for K3-fibrations linked to heterotic duality.

Contribution

It provides explicit solutions to the holomorphic anomaly equations for two-parameter Calabi-Yau hypersurfaces, utilizing modular forms to encode invariants in K3-fibrations.

Findings

Holomorphic expansions of topological string amplitudes achieved across moduli space.

Explicit solutions for K3-fibrations in weighted projective hypersurfaces.

Connection between topological invariants and modular forms via heterotic duality.

Abstract

We present solutions of the holomorphic anomaly equations for compact two-parameter Calabi-Yau manifolds which are hypersurfaces in weighted projective space. In particular we focus on K3-fibrations where due to heterotic type II duality the topological invariants in the fibre direction are encoded in certain modular forms. The formalism employed provides holomorphic expansions of topological string amplitudes everywhere in moduli space.