Vafa–Witten invariants from modular anomaly

Recently, a universal formula for non-holomorphic modular completion of the generating functions refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses through holomorphic lower ranks. Here we show that $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this can be used to extract themselves, thereby providing Betti numbers instanton moduli spaces such surfaces. As result, derive closed their completions all $N$. Besides, our construction reveals simple way instances fiber-base duality, which new non-trivial identities generalized Appell functions. also suggests existence invariants, whose meaning however remains obscure.