Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface

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).

27 pages, uses harvmac.tex; published form with note added in proof

References in corpus (2)

• Counting higher genus curves in a Calabi-Yau manifold
• Hodge integrals and degenerate contributions

Cited by in corpus (1)

• Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions