Twisted sheaves and $$\mathrm {SU}(r) / {\mathbb {Z}}_{r}$$ Vafa–Witten theory

Abstract

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 \(\mu _r\)-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 \(\mathrm {SU}(r) / {\mathbb {Z}}_r\) 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.