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Quantum geometry of elliptic Calabi-Yau manifolds. (English) Zbl 1270.81180
Summary: We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base. $$T$$-duality on the fibre implies that the topological string free energy also captures the BPS-invariants of $$D$$4-branes wrapping the elliptic fibre and a class in the base. We verify this proposal by explicit computation of the BPS invariants of 3 $$D$$4-branes on the rational elliptic surface.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T50 Anomalies in quantum field theory 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J30 $$3$$-folds 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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