Vafa-Witten invariants from wall-crossing for framed sheaves

TL;DR

This paper derives new wall-crossing formulas for moduli spaces of framed sheaves and applies them to compute refined Vafa-Witten invariants on surfaces, confirming conjectured formulas in special cases.

Contribution

It introduces a new stable/co-stable wall-crossing formula for framed sheaves and applies it to verify predictions of Vafa-Witten invariants.

Findings

Derived explicit expressions for the vertical contribution in terms of nested Hilbert schemes.

Proved a new stable/co-stable wall-crossing formula using mixed Hodge modules.

Confirmed the vertical part of Vafa-Witten's formula for rank 2 cases.

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.