nLab
special holonomy

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Differential cohomology

Contents

Idea

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

The holonomy group of \nabla at xx is the subgroup of GG on these elements.

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

Properties

Classification

Berger's theorem says that if a manifold XX is

then the possible special holonomy groups are the following

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
\mathbb{C}Kähler manifoldU(k)2k2kKähler forms ω 2\omega_2
Calabi-Yau manifoldSU(k)2k2k
\mathbb{H}quaternionic Kähler manifoldSp(k)Sp(1)4k4kω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3
hyper-Kähler manifoldSp(k)4k4kω=aω 2 (1)+bω 2 (2)+cω 2 (3)\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2 (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\mathbb{O}Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG277associative 3-form

Relation to GG-reductions

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

Theorem

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

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

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

This appears as (Joyce prop. 3.1.8)

Via 𝕆\mathbb{O}-Riemannian manifolds

normed division algebra𝔸\mathbb{A}Riemannian 𝔸\mathbb{A}-manifoldsSpecial Riemannian 𝔸\mathbb{A}-manifolds
real numbers\mathbb{R}Riemannian manifoldoriented Riemannian manifold
complex numbers\mathbb{C}Kähler manifoldCalabi-Yau manifold
quaternions\mathbb{H}quaternion-Kähler manifoldhyperkähler manifold
octonions𝕆\mathbb{O}Spin(7)-manifoldG2-manifold

(Leung 02)

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 (2000)

  • 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

Discussion in terms of Riemannian geometry modeled on normed division algebras is in

See also

  • Hans Freudenthal, Lie groups in the foundations of geometry, Advances in Mathematics, volume 1, (1965) pp. 145 - 190 (dspace)

Revised on January 25, 2016 11:27:55 by Urs Schreiber (193.136.135.42)