  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.
