Double genus expansion for general $\Omega$ background

TL;DR

This paper explores a two-parameter genus expansion of the Nekrasov partition function in the refined holomorphic anomaly equation, providing physical interpretation and Gopakumar-Vafa formulation insights.

Contribution

It introduces a novel two-parameter genus expansion compatible with the refined holomorphic anomaly equation and analyzes the underlying worldsheet theory with physical and quantitative evidence.

Findings

Compatibility of the two-parameter expansion with the anomaly equation

Physical interpretation of the worldsheet theory

Quantitative description via Gopakumar-Vafa formulation

Abstract

We will show how the refined holomorphic anomaly equation obeyed by the Nekrasov partition function at generic $\epsilon_1,\epsilon_2$ values becomes compatible, in a certain two parameters expansion, with the assumption that both parameters are associated to genus counting. The underlying worldsheet theory will be analysed and constrained in various ways, and we will provide both physical interpretation and some alternative evidence for this model. Finally we will use the Gopakumar - Vafa formulation for the refined topological string in order to give a more quantitative description.