4 problems

A torsion-free ${\mathrm{Spin}}(7)$-manifold is a Riemannian manifold with torsion-free ${\mathrm{Spin}}(7)$-structure, and a smooth Cayley fibration is a smooth fibration whose fi…

Let $N_{1,0}$ be the Aloff--Wallach space occurring as the principal orbit, and let the associated $\mathrm{Spin}(7)$ cone be the conical $\mathrm{Spin}(7)$ space with link…

Let $Z_+$ be an acyl Calabi–Yau four-fold with asymptotic neck region $I_{t_+} \times {\mathbb S}^1_{\theta_+} \times X_3^+$, and let $Z_-$ be an acyl $G_2$-manifold with asymptoti…

Let $M$ be an $8$-dimensional Riemannian manifold with holonomy $Spin_7$, and let $\Omega^{ch}_M$ denote its chiral de Rham complex. Write $C^\infty(M,\Omega^{ch}_M)$ for the assoc…