S-duality conjecture for SU(r) Vafa–Witten partition functions

Let $S$ be a smooth projective surface and let $r\in{\mathbb Z}_{>0}$. For $L\in\operatorname{Pic}(S)$, define the $\operatorname{SU}(r)$ Vafa–Witten partition function by

$$Z_{r,L}(S,\operatorname{SU}(r);q)=\sum_{c_2}\operatorname{VW}_{(r,L,c_2)}(S)q^{c_2}.$$

Define $Z_{r,L}(S,\operatorname{SU}(r)/{\mathbb Z}_r;q)$ as the partition function obtained by summing the twisted invariants over $\mu_r$-gerbes with the prescribed character weights. S-duality conjecture. The partition functions $Z_{r,L}(S,\operatorname{SU}(r);q)$ and $Z_{r,L}(S,\operatorname{SU}(r)/{\mathbb Z}_r;q)$ satisfy the S-duality formula stated above. The conjecture is the paper’s central comparison between ordinary and gerbe-twisted Vafa–Witten invariants; it is proved only in specified cases later in the paper, not for arbitrary smooth projective surfaces and ranks.