Recent Advances in the Theory of Holonomy

TL;DR

This paper reviews recent progress in classifying holonomy groups of torsion-free affine connections, highlighting completed classifications, existence results, and special holonomy cases like G_2 and Spin(7).

Contribution

It summarizes the classification of irreducibly acting subgroups of GL(n,R) as holonomy groups and discusses their realization and special cases.

Findings

Complete classification of irreducible holonomy groups by Chi, Merkulov, and Schwachhofer.

Exterior differential systems confirm all classified groups occur as holonomy.

Discussion of Joyce's results on compact manifolds with special holonomy G_2 and Spin(7).

Abstract

This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account of the completion of the classification of these groups by Chi, Merkulov, and Schwachhofer as well as of the exterior differential systems analysis that shows that all of these groups do, in fact, occur. Some discussion of the results of Joyce on the existence of compact examples with holonomy G_2 or Spin(7) is also included.