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Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface

S. Hosono·M. -H. Saito·A. Takahashi
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Abstract

We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certain recursion relation, holomorphic anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly equation for all genera and apply it to determine higher genus Gromov-Witten invariants and also the BPS states on the surface. Generalizing Göttsche's formula for the Hilbert scheme of $g$ points on a surface, we find precise agreement of our results with the proposal recently made by Gopakumar and Vafa(hep-th/9812127).

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S. Hosono, M. -H. Saito, A. Takahashi. 1999-12-29. Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface. https://arxiv.org/abs/hep-th/9901151

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