TY - JOUR

T1 - Twisted sheaves and SU(r)/Z_r Vafa-Witten theory

AU - Jiang, Yunfeng

AU - Kool, Martijn

N1 - 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.

PY - 2022/2

Y1 - 2022/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85118678496&partnerID=8YFLogxK

U2 - 10.1007/s00208-021-02303-6

DO - 10.1007/s00208-021-02303-6

M3 - Article

SN - 0025-5831

VL - 382

SP - 719

EP - 743

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -