Geometric transitions with Spin(7) holonomy via a dynamical system

TL;DR

This paper explores the global geometry of specific Spin(7) holonomy metrics, revealing transitions analogous to conifold transitions in Calabi-Yau spaces, using a dynamical system approach.

Contribution

It provides a detailed analysis of two families of Spin(7) metrics with known singular and regular orbits, and introduces new transition phenomena between these families.

Findings

Identification of two families of Spin(7) metrics with specific orbit structures.

Discovery of a new transition between Spin(7) families with similar asymptotics.

Application of a dynamical system approach to analyze Spin(7) equations.

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 $N(1,-1) \cong \mathrm{SU}(3)/\mathrm{U}(1)$ 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.