Twisted sheaves and SU(r)/Z_r Vafa-Witten theory

Yunfeng Jiang; Martijn Kool

doi:10.1007/s00208-021-02303-6

Twisted sheaves and SU(r)/Z_r Vafa-Witten theory

Yunfeng Jiang, Martijn Kool

University of Kansas

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

The 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 SU (r) / 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 SU (r) and SU (r) / Z _r partition 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.

Original language	English
Pages (from-to)	719-743
Number of pages	25
Journal	Mathematische Annalen
Volume	382
Issue number	1-2
Early online date	9 Nov 2021
DOIs	https://doi.org/10.1007/s00208-021-02303-6
Publication status	Published - Feb 2022

Bibliographical note

Funding Information:
We thank Amin Gholampour, Lothar Göttsche, and Ties Laarakker for helpful discussions related to this paper. Special thanks go to Richard Thomas, who suggested looking at Brauer classes during discussions on Remark 1.3. The authors would like to thank the Institute of Mathematical Sciences at ShanghaiTech, where most of this work was carried out. Y.J. is partially supported by NSF DMS-1600997. M.K. is supported by NWO grant VI.Vidi.192.012.

Funding Information:
We thank Amin Gholampour, Lothar Göttsche, and Ties Laarakker for helpful discussions related to this paper. Special thanks go to Richard Thomas, who suggested looking at Brauer classes during discussions on Remark . The authors would like to thank the Institute of Mathematical Sciences at ShanghaiTech, where most of this work was carried out. Y.J. is partially supported by NSF DMS-1600997. M.K. is supported by NWO grant VI.Vidi.192.012.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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Jiang-Kool2022_Article_TwistedSheavesAndMathrmSURMathFinal published version, 488 KBLicence: Taverne

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abstract = "The 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{\textquoteright}s moduli spaces of twisted sheaves. Using Chern character twisted by rational B-field, we give a new mathematical definition of the SU (r) / 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 SU (r) and SU (r) / Z r partition 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.",

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Twisted sheaves and SU(r)/Z_r Vafa-Witten theory

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