Normalized Vafa–Witten partition functions satisfy S-duality

Let $S$ be a smooth projective surface, let $q=e^{2\pi i\tau}$, and let $\eta(q)=q^{1/24}\prod_{k\geq 1}(1-q^k)$ be the Dedekind eta function. Define the normalized partition function by

$$\widehat{Z}(\operatorname{SU}(r);q)=\eta(q)^{-w}Z(\operatorname{SU}(r);q).$$

Define the analogous normalized partition function for $\operatorname{SU}(r)/{\mathbb Z}_r$. Normalized S-duality conjecture. The normalized partition functions satisfy

$$\widehat{Z}\left(\operatorname{SU}(r);-\frac{1}{\tau}\right)=\pm r^{-\frac{\chi}{2}}\widehat{Z}(\operatorname{SU}(r)/{\mathbb Z}_r;\tau).$$

This is the eta-normalized form of the Vafa–Witten S-duality prediction, in which the modular-weight factor is removed. The paper relates these functions to invariants of moduli spaces of Gieseker-stable Higgs sheaves but leaves the conjecture open in general.