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Holomorphic anomaly equation and BPS state counting of rational elliptic surface. (English) Zbl 1062.14504
Summary: 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 R. Gopakumar and C. Vafa [http://xxx.lanl.gov/abs/hep-th/9812127].

MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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