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# 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

#### 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.