Extended Holomorphic Anomaly in Gauge Theory

TL;DR

The paper demonstrates that the partition function of ${\cal N}=2$ gauge theories in the $\Omega$-background obeys an extended holomorphic anomaly equation for general $\beta=-\epsilon_1/\epsilon_2$, with boundary data governed by Schwinger-type expansions. By combining modularity with a $\beta$-dependent gap structure, the holomorphic ambiguity is fully fixed, and the authors identify an orientifold-like relation at $\beta=2$ connecting to the theory at $\beta=1$, while also linking these results to the topological string via geometric engineering. The analysis across SU(2) theories with $N_f=0,1,2,3$ reveals how the extension manifests in different matter content, including open-period interpretations of ${\cal T}$ and a rich pattern of even/odd amplitudes and gap structures. These results provide evidence that mirror symmetry and holomorphic anomaly techniques extend beyond the self-dual case and hint at a broader embedding into refined topological string theory. The work lays a foundation for incorporating orientifolds and real topological string data into the gauge theory/topological string correspondence.

Abstract

The partition function of an N=2 gauge theory in the Omega-background satisfies, for generic value of the parameter beta=-eps_1/eps_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.