2 problems

Let $(\widehat{X},\widehat{\Phi})$ be an SYZ-fibred $\operatorname{Spin}(7)$-manifold, let $\tau$ be fibrewise multiplication by $-1$, and let $S$ be the compact four-manifold base…

Let $Q \simeq \mathbb{R}^4$, let $\hat{\Gamma}$ act on $Q$, and let $\Gamma \hookrightarrow Sp(1)$ be nontrivial. Write $T^{\vee}_{\Lambda}$ for the relevant dual space, $F$ for th…