Communications in Number Theory and Physics

Volume 6 (2012)

Number 4

Quantum geometry of elliptic Calabi–Yau manifolds

Pages: 849 – 917

DOI: http://dx.doi.org/10.4310/CNTP.2012.v6.n4.a5

Authors

Albrecht Klemm (Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Germany)

Jan Manschot (Max Planck Institute for Mathematics, Bonn, Germany)

Thomas Wotschke (Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Germany)

Abstract

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.

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