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# Advances in Theoretical and Mathematical Physics

## Volume 16 (2012)

### Number 3

### Direct integration for general Ω backgrounds

Pages: 805 – 849

DOI: http://dx.doi.org/10.4310/ATMP.2012.v16.n3.a2

#### Authors

#### Abstract

We extend the direct integration method of the holomorphic anomalyequations to general $\Omega$ backgrounds $\epsilon_1\neq -\epsilon_2$for pure SU(2) $N=2$ Super-Yang–Mills theory and topological stringtheory on non-compact Calabi–Yau threefolds. We find that anextension of the holomorphic anomaly equation, modularity andboundary conditions provided by the perturbative terms as well as bythe gap condition at the conifold are sufficient to solve the generalizedtheory in the above cases. In particular, we use the method to solvethe topological string for the general $\Omega$ backgrounds onnon-compact toric Calabi–Yau spaces. The conifold boundarycondition follows from that the $N=2$ Schwinger-loop calculationwith Bogomol'nyi-Prasad-Sommerfield (BPS) states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the large basedecompactification limit of Calabi–Yau spaces withregular K3 fibrations and half $K$3s embedded in Calabi–Yau backgrounds.