Classification of irreducible holonomies of torsion-free affine connections

TL;DR

This paper classifies all possible irreducible holonomy groups of torsion-free affine connections on manifolds, completing Berger's classification from the 1950s using advanced mathematical theories.

Contribution

It provides a complete classification of irreducible holonomies for torsion-free affine connections, including exotic cases not covered by Berger's original work.

Findings

Complete list of irreducible holonomy groups

Inclusion of exotic holonomy cases

Connections to supersymmetric constructions

Abstract

The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original lists. Some remarks involving an interpretation of some of the examples in terms of supersymmetric constructions are also included.