Topological string amplitudes for the local half K3 surface

TL;DR

This work develops a universal, SW-curve–based framework to compute topological string amplitudes on the local $\tfrac{1}{2}$K3 surface, expressing higher-genus amplitudes as polynomials in a finite set of generators tied to the Seiberg–Witten curve and $E_8$-invariant Jacobi forms. By combining holomorphic (modular) anomaly equations with a gap condition, the authors obtain closed-form expressions for $F_g$ up to genus three and show that these results extend to all local del Pezzo surfaces via a simple curve replacement. The approach yields explicit genus-zero through genus-three data, including detailed genus-two and genus-three formulas and GV invariants, and it reproduces known results for massless ${\cal B}_8$, ${\mathbb P}^2$, and ${\mathbb P}^1\times {\mathbb P}^1$. The paper also presents multiple equivalent representations of the amplitudes (in $\phi$, $u$, and direct-integration forms) and discusses potential extensions, including connections to topological recursion and further explorations of higher-genus structure. Overall, the results provide a unified, symmetry-driven method to access all-genus topological string data for a broad class of non-compact Calabi–Yau threefolds.

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.