Constructing compact manifolds with exceptional holonomy

TL;DR

This survey reviews methods for constructing compact 7- and 8-dimensional manifolds with exceptional holonomy groups G2 and Spin(7), highlighting techniques involving complex geometry and singularity resolution.

Contribution

It compiles and explains various construction techniques for compact G2 and Spin(7) manifolds, including those starting from orbifolds and Calabi-Yau manifolds.

Findings

Techniques using orbifold singularity resolution are effective for G2 and Spin(7) manifolds.

Starting from Calabi-Yau manifolds provides more complex construction methods.

The paper summarizes existing methods, detailed in the author's book.

Abstract

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. There are also more complicated constructions which begin with a Calabi-Yau manifold or orbifold. All the material in this paper is covered in much more detail in the author's book, "Compact manifolds with special holonomy", Oxford University Press, 2000.