Vafa-Witten invariants from wall-crossing for framed sheaves
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918438647103488 |
|---|---|
| author | Arbesfeld, Noah Kool, Martijn Laarakker, Ties |
| author_facet | Arbesfeld, Noah Kool, Martijn Laarakker, Ties |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07898 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vafa-Witten invariants from wall-crossing for framed sheaves Arbesfeld, Noah Kool, Martijn Laarakker, Ties Algebraic Geometry High Energy Physics - Theory 14D21, 14C05, 16G20, 14J81 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. |
| title | Vafa-Witten invariants from wall-crossing for framed sheaves |
| topic | Algebraic Geometry High Energy Physics - Theory 14D21, 14C05, 16G20, 14J81 |
| url | https://arxiv.org/abs/2602.07898 |