2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/51371We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).AMS-Latex, 30 pages; Table 3,4 and typos corrected and references addedHigh Energy Physics - TheoryAlgebraic Geometrytext