Geometric S-duality for Vafa–Witten invariants

Let $\hat{\mathsf{VW}}^{\mathsf m,\mathsf a}_{\mathsf d}(B,q)_t$ denote the Vafa–Witten generating series, with indices recording degree, monopole data, and flux. Geometric S-duality. If rank ${\mathsf r}=2$, then

$$(12)\cdot[\hat{\mathsf{VW}}^{\mathsf m,\mathsf a}_{\mathsf d}(B,q)]_t =2^{-2g+1}\sum_{({\mathsf d'},{\mathsf m'},{\mathsf a'})}(-1)^{({\mathsf d},{\mathsf m},{\mathsf a})\cdot({\mathsf d'},{\mathsf m'},{\mathsf a'})}[\hat{\mathsf{VW}}^{{\mathsf a'}}_{{\mathsf d'}}(B,q)]_t.$$

The relation follows formally from the proposed transformation with insertions and the permutation of the generating series under $\tau\mapsto-1/\tau$; it is presented as a conjectural geometric form of $S$-duality.