Extended Holomorphic Anomaly in Gauge Theory

TL;DR

This paper explores the extended holomorphic anomaly equations in N=2 gauge theories within the Omega-background, revealing how modularity and gap structures determine ambiguities and connecting to topological string theories.

Contribution

It introduces an extended holomorphic anomaly framework for N=2 gauge theories in the Omega-background, including beta-dependent structures and their implications for topological string embedding.

Findings

Holomorphic anomaly equations are extended and beta-independent.

Modularity and gap structures fix holomorphic ambiguities.

Connections to topological string theory are suggested.

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.