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arXiv · 2602.07898

Vafa-Witten invariants from wall-crossing for framed sheaves

Noah Arbesfeld·Martijn Kool·Ties Laarakker
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Abstract

We consider the refined $\mathrm{SU}(r)$ Vafa-Witten partition function of a smooth projective surface with non-zero holomorphic 2-form. This partition function has a vertical contribution, expressible in terms of nested Hilbert schemes. First, we write the vertical contribution in terms of $\chi_y$-genera of moduli spaces of framed sheaves on ${\mathbb P}^2$. Then, we state two wall-crossing identities for moduli spaces of framed sheaves: a blow-up formula due to Kuhn-Leigh-Tanaka and a new stable/co-stable wall-crossing formula. We prove the latter using the theory of mixed Hodge modules. We apply these identities to obtain constraints on Vafa-Witten invariants predicted by conjectures of G\"ottsche and the second- and third-named authors. For $r=2$, we obtain a proof of the vertical part of a celebrated formula by Vafa-Witten.

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Noah Arbesfeld, Martijn Kool, Ties Laarakker. 2026-02-08. Vafa-Witten invariants from wall-crossing for framed sheaves. https://arxiv.org/abs/2602.07898

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