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

Abstract

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.

arXiv comment25 pages, 1 figure

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• On the BPS spectrum of 5d SU(2) super-Yang-Mills
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• Attractor invariants, brane tilings and crystals