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Holomorphic anomaly equation and BPS state counting of rational elliptic surface

Holomorphic anomaly equation and BPS state counting of rational elliptic surface

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 …