special holonomy (changes) in nLab

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Contents

Idea

For XX a space equipped with a GG-connection on a bundle ∇\nabla (for some Lie group GG) and for x∈Xx \in X any point, the parallel transport of ∇\nabla assigns to each curve γ:S 1→X\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

• 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\,
ℂ\,\mathbb{C}\,	\,Kähler manifold\,	\,U(n)\,	2n\,2n\,	\,Kähler forms ω 2\omega_2\,
	\,Calabi-Yau manifold\,	\,SU(n)\,	2n\,2n\,
ℍ\,\mathbb{H}\,	\,quaternionic Kähler manifold\,	\,Sp(n).Sp(1)\,	4n\,4n\,	ω 4=ω 1∧ω 1+ω 2∧ω 2+ω 3∧ω 3\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
	\,hyper-Kähler manifold\,	\,Sp(n)\,	4n\,4n\,	ω=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) manifold\,	\,Spin(7)\,	\,8\,	\,Cayley form\,
	\,G₂ manifold\,	\,G₂\,	7\,7\,	\,associative 3-form\,

Relation to GG-reductions

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

This appears as ( Joyce 2000, prop. 3.1.8 ) .

Via 𝕆\mathbb{O}-Riemannian manifolds

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

(Leung 02)

• 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 expressed byBerger's theorem is due to 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 see:

• Simon Salamon, Riemannian Geometry and Holonomy Groups, Research Notes in Mathematics 201, Longman (1989)

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

• Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monographs (2000) [[ISBN:9780198506010](https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=us&lang=en&)]

• 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. Diff. Geom. 61:2 (2002) 289–333(euclid)

See also

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

On special holonomy orbifolds:

• Jeff Cheeger, Gang Tian, Anti-self-duality of curvature and degeneration of metrics with special holonomy, Commun. Math. Phys. 255, 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)