Extended holomorphic anomaly in gauge theory - Krefl, Daniel; Walcher, Johannes (Letters in mathematical physics, 2010)
The partition function of an N=2N=2{\mathcal {N}=2} gauge theory in the Ω-background satisfies, for generic value of the parameter β=−ϵ1/ϵ2β=−ϵ1/ϵ2{β=-{ε_1}/{ε_2}} , the, in general extended, but otherwise β-independent, holomorphic anomaly equation of special geometry. Modularity together with the (β-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 β = 2, can be identified with an “orientifold” of the theory at β = 1. The various connections give hints for embedding the structure into the topological string.