  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.
