nLab special holonomy

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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-structure\,\,special holonomy\,\,dimension\,\,preserved differential form\,
\,\mathbb{C}\,\,Kähler manifold\,\,U(n)\,2n\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 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 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.

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 algebra\;𝔸\;\mathbb{A}\;\;Riemannian 𝔸\mathbb{A}-manifolds\;\;special Riemannian 𝔸\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\;𝕆\;\mathbb{O}\;\;Spin(7)-manifold\;\;G2-manifold\;

(Leung 02)

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

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, 1 (1965) 145–190 (dspace)

On special holonomy orbifolds:

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:

Last revised on June 15, 2023 at 13:43:54. See the history of this page for a list of all contributions to it.