Holonomy

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In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a connection is intimately associated with the curvature of that connection.

The holonomy group of a Riemannian manifold M is the holonomy group of the Levi-Civita connection on the tangent bundle of M.

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[edit] On vector bundles

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ : [0,1] → M based at x in M, the connection defines a parallel transport map P_\gamma \colon E_x \to E_x. This map is both linear and invertible and so defines an element of GL(Ex). The holonomy group of ∇ based at x is defined as

\mbox{Hol}_x(\nabla) = \{P_\gamma \in \mbox{GL}(E_x) \mid \gamma \mbox{ is a loop based at } x\}.

The local holonomy group based at x is the subgroup \mbox{Hol}^0_x(\nabla) coming from contractible loops γ.

If M is connected then the holonomy group depends on the basepoint x only up to conjugation in GL(k, R). Explicitly, if γ is a path from x to y in M then

\mbox{Hol}_y(\nabla) = P_\gamma \mbox{Hol}_x(\nabla) P_\gamma^{-1}.

Choosing different identifications of Ex with Rk also gives conjugate subgroups. It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation.

Some important properties of holonomy group include:

See also Wilson loop.

[edit] Riemannian holonomy groups

The holonomy of a Riemannian manifold (M, g) is the just holonomy group of the Levi-Civita connection on the tangent bundle to M. A 'generic' n-dimensional Riemannian manifold has an O(n) holonomy, or SO(n) if it is orientable. Manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) have special properties.

In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a Riemannian symmetric space). Berger's list is as follows:

Hol(g) dim(M) Type of manifold Comments
SO(n) n orientable
U(n) 2n Kähler manifold Kähler
SU(n) 2n Calabi-Yau manifold Ricci-flat, Kähler
Sp(n)·Sp(1)  4n  quaternionic Kähler manifold Einstein
Sp(n) 4n hyperkähler manifold Ricci-flat, Kähler
G2 7 G2 manifold Ricci-flat
Spin(7) 8 Spin(7) manifold Ricci-flat

It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find.

Note that Sp(n) \subset SU(2n) \subset U(2n) \subset SO(4n), so every hyperkähler manifold is a Calabi-Yau manifold, every Calabi-Yau manifold is a Kähler manifold, and every Kähler manifold is orientable.

The strange list above was explained by Simons's proof of Berger's theorem. One first shows that if a Riemannian manifold is not a locally symmetric space and the reduced holonomy acts irreducibly on the tangent space, then it acts transitively on the unit sphere. The Lie groups acting transitively on spheres are known: they consist of the list above, together with 2 extra cases: the group Spin(9) acting on R16, and the group T.Sp(m) acting on R4m. Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the Cayley projective plane), and the second does not occur at all as a holonomy group.

Riemannian symmetric spaces, which are locally isometric to homogeneous spaces G / H have local holonomy isomorphic to H. These too have been completely classified.

[edit] Special holonomy manifolds in string theory

Riemannian manifolds with special holonomy play an important role in string theory compactifications. This is because special holonomy manifolds admit covariantly constant (parallel) spinors and thus preserve some fraction of the original supersymmetry. Most important are compactifications on Calabi-Yau manifolds with SU(2) or SU(3) holonomy. Also important are compactifications on G2 manifolds. See G(2) holonomy on arxiv.org

[edit] On principal bundles

The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let P be a principal G-bundle over a smooth manifold M for some Lie group G and let ω be a connection on P. Given a piecewise smooth loop γ : [0,1] → M based at x in M and a point p in the fiber over x the connection defines a unique horizontal lift \tilde\gamma\colon [0,1] \to P such that \tilde\gamma(0) = p. The end point of the horizontal lift, \tilde\gamma(1), will not generally be p but rather some other point p·g in the fiber over x. Define an equivalence relation ~ on P by saying that p~q if they can be joined by a piecewise smooth horizontal path in P.

The holonomy group of ω based at p is then defined as

\mbox{Hol}_p(\omega) = \{g \in G \mid p \sim p\cdot g\}.

The local holonomy group based at p is the subgroup \mbox{Hol}^0_p(\omega) coming from horizontal lifts of contractible loops γ.

If M and P are connected then the holonomy group depends on the basepoint p only up to conjugation in G. Explicitly,

\mbox{Hol}_{p\cdot g}(\omega) = g^{-1} \mbox{Hol}_p(\omega) g.

Moreover if p~q the Holp(ω) = Holq(ω). It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation.

Some important properties of holonomy group include:

[edit] Reference

  • Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. ISBN 0-19-850601-5.

[edit] External link

  • Chi, Merkulov, and Schwachhöfer, On the incompleteness of Berger's list, arXiv:dg-ga/9508014.
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