Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface

TL;DR

This paper derives and applies the holomorphic anomaly equation to compute Gromov-Witten invariants and BPS states of a rational elliptic surface, confirming recent theoretical proposals with explicit calculations.

Contribution

It introduces a holomorphic anomaly equation for all genera and uses it to determine higher genus Gromov-Witten invariants and BPS states, extending previous results.

Findings

Derived recursion relation for genus 0 and 1 invariants

Proposed and applied holomorphic anomaly equation for all genera

Confirmed agreement with Gopakumar-Vafa conjecture

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 G\"ottsche'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).