VafaWitten S-duality conjecture for SU(r) and SU(r)/Z_r

Let $(S,H)$ be a smooth polarized surface with $H_1(S,\mathbb Z)=0$ and $p_g(S)>0$, and let $r$ be prime. Let $c_1\in H^2(S,\mathbb Z)$ be algebraic. Write $\mathsf{Z}^{\mathrm{SU}(r)}_{c_1}(q)$ and $\mathsf{Z}^{\mathrm{SU}(r)/\mathbb Z_r}_{c_1}(q)$ for the corresponding VafaWitten partition functions, and set $q=\exp(2\pi i\tau)$ for $\tau\in\mathfrak H$. The associated meromorphic functions on $\mathfrak H$ are denoted by $\mathsf{Z}^{\mathrm{SU}(r)}_{c_1}(\tau)$ and $\mathsf{Z}^{\mathrm{SU}(r)/\mathbb Z_r}_{c_1}(\tau)$.

VafaWitten S-duality conjecture. The two partition functions are Fourier expansions of these meromorphic functions, which satisfy

$$\mathsf{Z}^{\mathrm{SU}(r)}_{c_1}(-1/\tau)=(-1)^{(r-1)\chi(\mathcal O_S)}\left(\frac{r\tau}{i}\right)^{-\frac{e(S)}{2}}\mathsf{Z}^{\mathrm{SU}(r)/\mathbb Z_r}_{c_1}(\tau).$$

This is the mathematical formulation of the VafaWitten S-duality prediction for these partition functions; the conjecture concerns the modular transformation under $\tau\mapsto-1/\tau$.