Rank $N$ Vafa–Witten invariants, modularity and blow-up

We derive explicit expressions for the generating functions of refined Vafa-Witten invariants $\Omega(\gamma,y)$ of $\mathbb{P}^2$ of arbitrary rank $N$ and for their non-holomorphic modular completions. In the course of derivation we also provide: i) a generalization of the recently found generating functions of $\Omega(\gamma,y)$ and their completions for Hirzebruch and del Pezzo surfaces in the canonical chamber of the moduli space to a generic chamber; ii) a version of the blow-up formula expressed directly in terms of these generating functions and its reformulation in a manifestly modular form.