nLab special holonomy

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

• simply connected

• neither locally a product nor a symmetric space

then the possible special holonomy groups are the following

classification of special holonomy manifolds by Berger's theorem:

	$\,$G-structure$\,$	$\,$special holonomy$\,$	$\,$dimension$\,$	$\,$preserved differential form$\,$
$\,\mathbb{C}\,$	$\,$Kähler manifold$\,$	$\,$U(n)$\,$	$\,2n\,$	$\,$Kähler forms $\omega_2\,$
	$\,$Calabi-Yau manifold$\,$	$\,$SU(n)$\,$	$\,2n\,$
$\,\mathbb{H}\,$	$\,$quaternionic Kähler manifold$\,$	$\,$Sp(n).Sp(1)$\,$	$\,4n\,$	$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
	$\,$hyper-Kähler manifold$\,$	$\,$Sp(n)$\,$	$\,4n\,$	$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$	$\,$Spin(7) manifold$\,$	$\,$Spin(7)$\,$	$\,$8$\,$	$\,$Cayley form$\,$
	$\,$G2 manifold$\,$	$\,$G2$\,$	$\,7\,$	$\,$associative 3-form$\,$

Relation to $G$-reductions

A manifold having special holonomy 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)

Via $\mathbb{O}$-Riemannian manifolds

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G2-manifold$\;$

(Leung 02)

Related concepts

• stable form, definite form

• connection on a bundle

  • connection on a 2-bundle, connection on an infinity-bundle

• parallel transport, higher parallel transport

• holonomy

  • holonomy group

  • special holonomy, reduction of structure groups, G-structure, exceptional geometry, Walker coordinates

References

General

The classification in Berger's theorem is due to

• Marcel 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

• Andrei Moroianu, Uwe Semmelmann, Parallel spinors and holonomy groups, Journal of Mathematical Physics 41 (2000), 2395-2402 (arXiv:math/9903062)

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

• Naichung Conan Leung, Riemannian Geometry Over Different Normed Division Algebras, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. (publisher page)

See also

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

On special holonomy orbifolds:

• Jeff Cheeger, Gang Tian, Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy, Communications in Mathematical Physics volume 255, pages 391–417 (2005) (doi:10.1007/s00220-004-1279-0)

In supergravity and string theory

Discussion of special holonomy manifolds in supergravity and superstring theory as fiber-spaces for KK-compactifications preserving some number of supersymmetries:

• Steven Gubser, Special holonomy in string theory and M-theory, In Steven Gubser, Joseph Lykken (eds.) Strings, Branes and Extra Dimensions - TASI 2001, World Scientific 2004 (arXiv:hep-th/0201114, doi:10.1142/5495)