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Vafa-Witten invariants from wall-crossing for framed sheaves

Noah Arbesfeld, Martijn Kool, Ties Laarakker

TL;DR

This work develops a refined understanding of SU$(r)$ Vafa-Witten invariants on surfaces with $p_g>0$ by isolating the vertical contribution and expressing it through nested Hilbert schemes and moduli of framed sheaves on $\mathbb{P}^2$. It introduces two wall-crossing identities for framed sheaves—the Kuhn-Leigh-Tanaka blow-up formula and a new stable/co-stable wall-crossing—proving the latter via equivariant mixed Hodge modules. These tools yield universal formulae for the vertical part, connect them to framed-sheaf generating series, and constrain VW invariants in line with Göttsche's conjectures; in rank $r=2$ they establish the vertical component of the classic VW formula. The approach combines toric reductions, localization on Hilbert schemes, and Nakajima quiver-variety techniques to produce symmetry and blow-up relations that underpin S-duality predictions. Overall, the paper provides a rigorous framework for expressing VW invariants in terms of framed moduli and nested Hilbert schemes, with broad implications for wall-crossing phenomena in gauge-theoretic invariants.

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 $χ_y$-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 Göttsche 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.

Vafa-Witten invariants from wall-crossing for framed sheaves

TL;DR

This work develops a refined understanding of SU Vafa-Witten invariants on surfaces with by isolating the vertical contribution and expressing it through nested Hilbert schemes and moduli of framed sheaves on . It introduces two wall-crossing identities for framed sheaves—the Kuhn-Leigh-Tanaka blow-up formula and a new stable/co-stable wall-crossing—proving the latter via equivariant mixed Hodge modules. These tools yield universal formulae for the vertical part, connect them to framed-sheaf generating series, and constrain VW invariants in line with Göttsche's conjectures; in rank they establish the vertical component of the classic VW formula. The approach combines toric reductions, localization on Hilbert schemes, and Nakajima quiver-variety techniques to produce symmetry and blow-up relations that underpin S-duality predictions. Overall, the paper provides a rigorous framework for expressing VW invariants in terms of framed moduli and nested Hilbert schemes, with broad implications for wall-crossing phenomena in gauge-theoretic invariants.

Abstract

We consider the refined 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 -genera of moduli spaces of framed sheaves on . 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 Göttsche and the second- and third-named authors. For , we obtain a proof of the vertical part of a celebrated formula by Vafa-Witten.
Paper Structure (32 sections, 11 theorems, 177 equations, 1 figure)

This paper contains 32 sections, 11 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vertical contribution
  3. Framed sheaves
  4. Wall-crossing for framed sheaves
  5. A blow-up formula
  6. Stable/co-stable wall-crossing
  7. Relations among the universal series
  8. Nested Hilbert schemes
  9. K-theory preliminaries
  10. Reduction to $\operatorname{Hilb}^{{\boldsymbol{n}}}(S)$
  11. A class on the Hilbert scheme
  12. Universality
  13. Reduction to $\operatorname{Hilb}^{{\boldsymbol{n}}}(S)$ for toric $S$
  14. Reduction to $\operatorname{Hilb}^{{\boldsymbol{n}}}(\mathbb C^2)$
  15. Moduli of framed sheaves
  16. ...and 17 more sections

Key Result

Theorem 1.1

For any $r>1$, there exist universal generating series $A$, $B$, $\{C_{ij}\}_{1 \leq i \leq j \leq r-1}$ with the following property.Up to an explicit normalization term, given in Section sec:GT, the universal series $A,B,C_{ij}$ lie in $1+ q \, \mathbb Q(y^{\frac{1}{2}})[[q]]$. They only depend on

Figures (1)

  • Figure 1: The two toric charts of $\widehat{\mathbb C}^2$.

Theorems & Definitions (22)

  • Theorem 1.1: Laarakker
  • Remark 1.2
  • Proposition 1.3
  • Theorem 1.4: Kuhn-Leigh-Tanaka
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 2.1: Gholampour-Thomas
  • Remark 2.2
  • Lemma 2.3
  • ...and 12 more