Monopole-branch formula for refined rank-three Vafa–Witten invariants

Let $S$ be a smooth projective surface satisfying $H_1(S,\mathbb Z)=0$ and $p_g>0$. For any polarization $H$ and first Chern class $c_1$, let $\mathsf Z_{S,H,3,c_1}^{\mathrm{mono}}(q,y)$ be the monopole-branch contribution, and let $\delta_{a,b}$ equal $1$ when $a-b\in 3H^2(S,\mathbb Z)$ and $0$ otherwise. Let $\Theta_{A_2,(0,0)}$, $\Theta_{A_2,(1,0)}$, and $W_\pm$ be the theta series and quadratic-equation solutions defined in the source. Rank-three monopole-branch conjecture. The normalized monopole contribution equals the displayed expression involving the powers of the modular factors, the sum of $\mathrm{SW}(a)\mathrm{SW}(b)\delta_{c_1+a,b}$, and the powers of $W_\pm$. This is a predicted closed formula for the monopole branch in terms of Seiberg–Witten data and modular forms; the source supplies no proof in general.