Extended Holomorphic Anomaly in Gauge Theory

The partition function of an N = 2 gauge theory in the Omega-background satisfies, for generic value of the parameter beta = -epsilon(1)/epsilon(2), the, in general extended, but otherwise beta-independent, holomorphic anomaly equation of special geometry. Modularity together with the (beta-dependent) gap structure at the various singular loci in the moduli space completely fixes the holomorphic ambiguity, also when the extension is non-trivial. In some cases, the theory at the orbifold radius, corresponding to beta = 2, can be identified with an orientifold of the theory at beta = 1. The various connections give hints for embedding the structure into the topological string.