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

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, first-named author recently introduced virtual of $$\mu _r$$ -gerbes. In this paper, we instead use Yoshioka’s moduli spaces twisted sheaves. Using Chern character rational B-field, give new mathematical definition {SU}(r) / {\mathbb {Z}}_r$$ Vafa-Witten function when r is prime. Our uses period-index theorem de Jong. S-duality, concept from physics, predicts that and functions are related modular transformation. We turn into conjecture, prove for all K3 surfaces prime numbers r.