# nLab special holonomy

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Differential cohomology

differential cohomology

# Contents

## Idea

For $X$ a space equipped with a $G$-connection on a bundle $\nabla$ (for some Lie group $G$) and for $x \in X$ any point, the parallel transport of $\nabla$ assigns to each curve $\Gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $hol_\nabla(\gamma) \in G$: the holonomy of $\nabla$ along that curve.

The holonomy group of $\nabla$ at $x$ is the subgroup of $G$ on these elements.

If $\nabla$ is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup $H$ of the special orthogonal group, one says that $(X,g)$ is a manifold of special holonomy .

## Properties

### Classification

Berger's theorem says that if a manifold $X$ is

then the possible special holonomy groups are the following

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
Kähler manifoldU(k)$2k$Kähler forms $\omega_2$
Calabi-Yau manifoldSU(k)$2k$
hyper-Kähler manifoldSp(k)$4k$$\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$)
quaternionic Kähler manifold$4k$$\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$
G2 manifoldG2$7$associative 3-form
Spin(7) manifoldSpin(7)8Cayley form

### Relation to $G$-reductions

A manifold having special holonmy means that there is a corresponding reduction of structure groups.

###### Theorem

Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ is conjugate to a subgroup of $G$.

Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

## References

The classification in Berger's theorem is due to

• M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955)

For more see

• Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)

• Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monographs (200o)

• Luis J. Boya, Special Holonomy Manifolds in Physics Monografías de la Real Academia de Ciencias de Zaragoza. 29: 37–47, (2006). (pdf)

Discussion of the relation to Killing spinors includes

Revised on January 22, 2015 21:14:35 by Urs Schreiber (88.100.66.95)