S-duality conjecture for SU(r) Vafa–Witten invariants on surfaces

Let $S$ be a smooth projective surface, let $r$ be a positive integer, and let $L\in\operatorname{Pic}(S)$. Define

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

and let $Z_{r,L}(S,\operatorname{SU}(r)/{\mathbb Z}_r;q)$ be the partition function of the corresponding $\operatorname{SU}(r)/{\mathbb Z}_r$-Vafa–Witten invariants.

S-duality conjecture for Vafa–Witten invariants. The two partition functions satisfy the S-duality transformation formula

$$Z_{r,L}\left(S,\operatorname{SU}(r);-\frac{1}{\tau}\right)=\pm r^{-\frac{\chi(S)}{2}}\left(\frac{\tau}{i}\right)^{\frac{\omega}{2}}Z_{r,L}(S,\operatorname{SU}(r)/{\mathbb Z}_r;\tau).$$

Here the formula is the specialization of the preceding conjectural instanton partition-function identity to Vafa–Witten invariants. The paper proves this statement for K3 surfaces in all ranks, while the formulation is presented for a smooth projective surface.