Double genus expansion for general $\Omega$ background

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.