Topological string amplitudes for the local half K3 surface

TL;DR

This paper develops a method to compute higher-genus topological string amplitudes for the local half K3 surface using Seiberg-Witten curves, modular forms, and anomaly equations, covering all local del Pezzo surfaces.

Contribution

It introduces a novel approach combining Seiberg-Witten curves and modular forms to explicitly compute topological string amplitudes up to genus three.

Findings

Explicit formulas for amplitudes up to genus three

Unified framework for local del Pezzo surfaces

Solution of holomorphic anomaly and gap conditions

Abstract

We study topological string amplitudes for the local half K3 surface. We develop a method of computing higher-genus amplitudes along the lines of the direct integration formalism, making full use of the Seiberg-Witten curve expressed in terms of modular forms and E_8-invariant Jacobi forms. The Seiberg-Witten curve was constructed previously for the low-energy effective theory of the non-critical E-string theory in R^4 x T^2. We clarify how the amplitudes are written as polynomials in a finite number of generators expressed in terms of the Seiberg-Witten curve. We determine the coefficients of the polynomials by solving the holomorphic anomaly equation and the gap condition, and construct the amplitudes explicitly up to genus three. The results encompass topological string amplitudes for all local del Pezzo surfaces.