Complete non-compact $\mathrm{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds

We develop a new construction of complete non-compact 8-manifolds with Riemannian holonomy equal to $\mathrm{Spin}(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds, and the result is generalized to orbifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($A{T}^{2}C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry, analogously to how ALC metrics generalize ALF metrics. As an application of this construction, we produce infinitely many diffeomorphism types of $A{T}^{2}C$ $\mathrm{Spin}(7)$-manifolds and the first known examples of complete toric $\mathrm{Spin}(7)$-manifold.