Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866917979807023104 |
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| author | Jiang, Y. Kool, M. |
| author_facet | Jiang, Y. Kool, M. |
| contents | The $\mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $μ_r$-gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong.
$S$-duality, a concept from physics, predicts that the $\mathrm{SU}(r)$ and $\mathrm{SU}(r) / \mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_10368 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory Jiang, Y. Kool, M. Algebraic Geometry High Energy Physics - Theory 14D20, 14D21, 14J60, 14J81, 14F22 The $\mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $μ_r$-gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong. $S$-duality, a concept from physics, predicts that the $\mathrm{SU}(r)$ and $\mathrm{SU}(r) / \mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$. |
| title | Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory |
| topic | Algebraic Geometry High Energy Physics - Theory 14D20, 14D21, 14J60, 14J81, 14F22 |
| url | https://arxiv.org/abs/2006.10368 |