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

We derive explicit expressions for the generating functions of refined Vafa-Witten invariants f/(7, y) of P2 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 f/(7, 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.