Refined modularity conjecture for Vafa–Witten generating functions

Let $S$ be a smooth projective surface satisfying $H_1(S,\mathbb Z)=0$ and $p_g>0$. Let $H$ be a polarization, let $r=1$ or a prime, and let $c_1\in H^2(S,\mathbb Z)$. Let $\mathsf Z_{S,H,r,c_1}(q,y)$ be the refined Vafa–Witten generating function, with $q=e^{2\pi i\tau}$ and $y=e^{2\pi iz}$. Refined modularity conjecture. The function depends only on $[c_1]\in H^2(S,\mathbb Z)/rH^2(S,\mathbb Z)$, is the Fourier expansion of a meromorphic function on $\mathfrak H\times\mathbb C$, and satisfies the stated $T$- and $S$-transformation laws. The conjecture is motivated by $S$-duality; the paper gives evidence for ranks $1$, $2$, and $3$, but does not establish it in general.