Geometric Transitions with Spin(7) Holonomy via a Dynamical System

Abstract

We clarify the global geometry of two 1-parameter families of cohomogeneity one Spin(7) holonomy metrics with generic orbit the Aloff–Wallach space \(N(1,-1) \cong \text {SU}(3)/\text {U}(1)\) and singular orbits \(S^5\) and \({\mathbb {C}}P^{2}\), which at short distance were shown to exist by Reidegeld. The two families fit into the geography of previously known families of cohomogeneity one metrics with exceptional holonomy and provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi–Yau 3-folds. Furthermore, we discover that there is another transition to families of Spin(7) holonomy metrics which have a similar asymptotic behaviour on one end, but are singular on the other end. We obtain our results by relating the Spin(7)-equations to a simple dynamical system on a 3-dimensional cube.