Vafa-Witten invariants from wall-crossing for framed sheaves

Martijn Kool; Noah Arbesfeld; Ties Laarakker

arxiv: 2602.07898 · v2 · submitted 2026-02-08 · 🧮 math.AG · hep-th

Vafa-Witten invariants from wall-crossing for framed sheaves

Noah Arbesfeld , Martijn Kool , Ties Laarakker This is my paper

Pith reviewed 2026-05-16 06:36 UTC · model grok-4.3

classification 🧮 math.AG hep-th

keywords Vafa-Witten invariantswall-crossingframed sheavesmoduli spacesmixed Hodge moduleschi_y genusnested Hilbert schemesSU(r) partition function

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The pith

Wall-crossing identities for framed sheaves determine the vertical Vafa-Witten invariants for rank two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper rewrites the vertical contribution to the refined SU(r) Vafa-Witten partition function of a smooth projective surface with non-zero holomorphic 2-form as an expression involving the chi_y-genera of moduli spaces of framed sheaves on the projective plane. It introduces a new stable/co-stable wall-crossing formula for these framed moduli spaces, proved using the theory of mixed Hodge modules, and combines it with a known blow-up formula. These relations are applied to obtain constraints on the Vafa-Witten invariants that match predictions from existing conjectures, and they yield a proof of the vertical part of the Vafa-Witten formula when the rank r equals two.

Core claim

The vertical contribution to the refined SU(r) Vafa-Witten partition function, originally expressed via nested Hilbert schemes, equals a generating function built from chi_y-genera of framed sheaf moduli spaces on P^2; the stable/co-stable and blow-up wall-crossing identities among these spaces then constrain the invariants and determine their explicit form for r=2.

What carries the argument

The stable/co-stable wall-crossing identity for moduli spaces of framed sheaves on P^2, established using mixed Hodge modules.

If this is right

The vertical Vafa-Witten invariants for rank 2 match the explicit formula conjectured by Vafa and Witten.
The same wall-crossing relations impose constraints on the invariants for ranks r greater than 2.
The vertical contribution can be computed recursively from the chi_y-genera via the blow-up and stable/co-stable identities.
Nested Hilbert scheme expressions reduce to data from framed sheaf moduli spaces on P^2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same technique could be used to relate vertical contributions on other surfaces to framed data after blow-ups.
Analogous wall-crossing arguments might apply to Donaldson-Thomas or Pandharipande-Thomas invariants of similar moduli spaces.
The method supplies a concrete way to verify the full Vafa-Witten conjecture for small ranks on surfaces with holomorphic 2-forms.

Load-bearing premise

The theory of mixed Hodge modules applies directly to the moduli spaces of framed sheaves to establish the stable/co-stable wall-crossing identity.

What would settle it

An explicit computation of the vertical Vafa-Witten contribution for the projective plane at rank 2 that fails to match the formula obtained from the wall-crossing relations would disprove the result.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the vertical r=2 Vafa-Witten term by proving a new stable/co-stable wall-crossing formula for framed sheaves using mixed Hodge modules.

read the letter

The main takeaway is that this paper proves the vertical part of the Vafa-Witten formula for rank two by establishing a new stable/co-stable wall-crossing identity for moduli spaces of framed sheaves. They start by expressing the vertical contribution to the refined SU(r) Vafa-Witten partition function in terms of chi_y-genera of framed sheaf moduli on projective plane. Then they recall a blow-up formula from Kuhn-Leigh-Tanaka and prove a new wall-crossing formula between stable and co-stable chambers using mixed Hodge modules. Applying these gives constraints on the invariants that align with conjectures by Göttsche and others. For r=2 this yields the desired proof of the vertical term. What stands out is the clean reduction to framed sheaves and the use of mixed Hodge modules to handle the wall-crossing. This seems like a solid way to make progress on these partition functions without having to compute the moduli spaces directly. The potential soft spot is in the application of the mixed Hodge module theory. The stress test raises whether the framed sheaf moduli spaces are quasi-projective with compatible stratifications and whether the nearby and vanishing cycle functors work as expected without corrections due to the framing or behavior at infinity. If the paper verifies these conditions explicitly, then the proof goes through. Otherwise there could be a gap. Overall the argument looks internally consistent and builds on prior results without circularity. The math appears sound based on the strategy outlined. This paper is for specialists in enumerative algebraic geometry working on surface invariants and wall-crossing phenomena. A reader familiar with Vafa-Witten theory and Hodge modules will get the most out of it. I would recommend sending it to peer review. The result is significant enough to warrant referee attention even if some technical details need polishing.

Referee Report

2 major / 2 minor

Summary. The paper expresses the vertical contribution to the refined SU(r) Vafa-Witten partition function of a smooth projective surface in terms of χ_y-genera of moduli spaces of framed sheaves on P². It states a blow-up formula (Kuhn-Leigh-Tanaka) and proves a new stable/co-stable wall-crossing identity for these moduli spaces using mixed Hodge modules. These identities are applied to derive constraints on Vafa-Witten invariants and, for r=2, to prove the vertical part of the Vafa-Witten formula.

Significance. If the mixed Hodge module arguments apply without correction terms, the work supplies a rigorous algebraic proof of the vertical contribution in the r=2 case of a well-known formula, together with new wall-crossing identities that constrain the invariants for general r. The approach leverages established tools (mixed Hodge modules, prior wall-crossing results) to reduce the problem to concrete computations on framed-sheaf moduli.

major comments (2)

[proof of stable/co-stable wall-crossing formula] The section proving the stable/co-stable wall-crossing identity invokes the theory of mixed Hodge modules directly on the moduli spaces of framed sheaves on P². No explicit verification is given that these spaces are quasi-projective, admit a stability stratification compatible with the nearby-cycles and vanishing-cycles functors, or that the framing and non-properness at infinity introduce no extra correction terms; this verification is load-bearing for the identity used in the r=2 reduction.
[application to Vafa-Witten invariants (r=2 case)] The reduction to the vertical part of the Vafa-Witten formula for r=2 rests on the new wall-crossing identity together with the blow-up formula. If the hypotheses of the mixed Hodge module theorem fail for the concrete framed moduli appearing in the vertical contribution, the claimed proof is incomplete; an explicit check or reference to a theorem covering the framed case is required.

minor comments (2)

[introduction to vertical contribution] Notation for the χ_y-genera and the precise definition of the vertical contribution should be recalled or cross-referenced at the start of the wall-crossing section to improve readability.
The manuscript would benefit from a short table or diagram summarizing the two wall-crossing identities and the exact sequence of reductions used for the r=2 case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major points below and will revise the manuscript to include the requested verifications.

read point-by-point responses

Referee: The section proving the stable/co-stable wall-crossing identity invokes the theory of mixed Hodge modules directly on the moduli spaces of framed sheaves on P². No explicit verification is given that these spaces are quasi-projective, admit a stability stratification compatible with the nearby-cycles and vanishing-cycles functors, or that the framing and non-properness at infinity introduce no extra correction terms; this verification is load-bearing for the identity used in the r=2 reduction.

Authors: We appreciate the referee highlighting this point. The moduli spaces of framed sheaves on P² are quasi-projective, being realized as GIT quotients of appropriate Quot schemes incorporating the framing. The stability stratification is algebraic and compatible with the nearby-cycles and vanishing-cycles functors by the general results on mixed Hodge modules for quasi-projective varieties with algebraic stratifications. The framing condition ensures the relevant loci are smooth or have mild singularities, and non-properness is handled via compactly supported cohomology in the χ_y-genera; no additional correction terms arise in the wall-crossing between stable and co-stable loci. We will add a short verification paragraph in the revised manuscript citing the relevant general theorems. revision: yes
Referee: The reduction to the vertical part of the Vafa-Witten formula for r=2 rests on the new wall-crossing identity together with the blow-up formula. If the hypotheses of the mixed Hodge module theorem fail for the concrete framed moduli appearing in the vertical contribution, the claimed proof is incomplete; an explicit check or reference to a theorem covering the framed case is required.

Authors: We agree that an explicit check strengthens the argument. In the revision we will insert a brief check confirming that the framed moduli spaces arising in the vertical contribution satisfy the hypotheses of the mixed Hodge module theorem (quasi-projectivity, compatible stratification, and applicability to the χ_y-genera), together with a reference to the standard results on Hodge modules for moduli of sheaves on projective varieties. This completes the reduction for the r=2 case. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation uses external mixed Hodge modules and independent citations

full rationale

The paper expresses the vertical contribution to the refined SU(r) Vafa-Witten partition function in terms of χ_y-genera of moduli spaces of framed sheaves on P², invokes a blow-up formula cited from Kuhn-Leigh-Tanaka, and proves a new stable/co-stable wall-crossing identity by direct application of the general theory of mixed Hodge modules. For r=2 this yields the vertical part of the Vafa-Witten formula. No step reduces the claimed result to a parameter fitted inside the paper, a self-defined quantity, or a load-bearing self-citation chain; the mixed Hodge module argument is an external tool whose hypotheses are taken as given, and the blow-up formula is from non-overlapping authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters or invented entities; the central argument rests on the domain assumption that mixed Hodge modules apply to the framed-sheaf moduli spaces.

axioms (1)

domain assumption Theory of mixed Hodge modules applies to moduli spaces of framed sheaves to prove the stable/co-stable wall-crossing formula
Invoked to establish the new wall-crossing identity used for the Vafa-Witten constraints

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discussion (0)

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We prove the latter using the theory of mixed Hodge modules... For r=2, we obtain a proof of the vertical part of a celebrated formula by Vafa-Witten.

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