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The $\mathrm{SU}(r)$ Vafa-Witten partition function, which virtually countsHiggs pairs on a projective surface $S$, was mathematically defined byTanaka-Thomas. On the Langlands dual side, the first-named author recentlyintroduced virtual counts of Higgs pairs on $\mu_r$-gerbes. In this paper, weinstead use Yoshioka's moduli spaces of twisted sheaves. Using Chern charactertwisted by rational $B$-field, we give a new mathematical definition of the$\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten partition function when $r$ isprime. 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 modulartransformation. We turn this into a mathematical conjecture, which we prove forall $K3$ surfaces and prime numbers $r$.

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