Holonomy

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 just holonomy group of the Levi-Civita connection on the tangent bundle of M.

Contents

1 On vector bundles 1.1 Riemannian holonomy groups 1.2 Special holonomy manifolds in string theory 2 On principal bundles 3 References and external links

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 <math>P_\gamma\colon E_x \to E_x<math>. This map is both linear and invertible and so defines an element of GL(E_x). The holonomy group of ∇ based at x is defined as

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

The local holonomy group based at x is the subgroup <math>\mbox{Hol}^0_x(\nabla)<math> 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

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

Choosing different identifications of E_x with R^k 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:

• Hol⁰(∇) is a connected, Lie subgroup of GL(k, R).
• Hol⁰(∇) is the identity component of Hol(∇).
• There is a natural, surjective group homomorphism π₁(M) → Hol(∇)/Hol⁰(∇) where π₁(M) is the fundamental group of M which sends the homotopy class [γ] to the coset P_γ·Hol⁰(∇).
• If M is simply connected then Hol(∇) = Hol⁰(∇).
• ∇ is flat (i.e. has vanishing curvature) iff Hol⁰(∇) is trivial.

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	generic
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.

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

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.

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 <math>\tilde\gamma\colon [0,1] \to P<math> such that <math>\tilde\gamma(0) = p<math>. The end point of the horizontal lift, <math>\tilde\gamma(1)<math>, 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

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

The local holonomy group based at p is the subgroup <math>\mbox{Hol}^0_p(\omega)<math> 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,

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

Moreover if p~q the Hol_p(ω) = Hol_q(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:

• Hol⁰(ω) is a connected, Lie subgroup of G.
• Hol⁰(ω) is the identity component of Hol(ω).
• There is a natural, surjective group homomorphism π₁(M) → Hol(ω)/Hol⁰(ω).
• If M is simply connected then Hol(ω) = Hol⁰(ω).
• ω is flat (i.e. has vanishing curvature) iff Hol⁰(ω) is trivial.

References and external links

• Chi, Merkulov, and Schwachhöfer, On the incompleteness of Berger's list, arXiv:dg-ga/9508014 (http://arxiv.org/abs/dg-ga/9508014).
• Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. ISBN 0-19-850601-5.