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We extend the direct integration method of the holomorphic anomaly equationsto general Omega backgrounds for pure SU(2) N=2 Super-Yang-Mills theory andtopological string theory on non-compact Calabi-Yau threefolds. We find that anextension of the holomorphic anomaly equation, modularity and boundaryconditions provided by the perturbative terms as well as by the gap conditionat the conifold are sufficient to solve the generalized theory in the abovecases. In particular we use the method to solve the topological string for thegeneral Omega backgrounds on non-compact toric Calabi-Yau spaces. The conifoldboundary condition follows from that the N=2 Schwinger-Loop calculation withBPS states coupled to a self-dual and an anti-self-dual field strength. Wecalculate such BPS states also for the decompactification limit of Calabi-Yauspaces with regular K3 fibrations and half K3s embedded in Calabi-Yaubackgrounds.

Direct integration for general Omega backgrounds