nLab holonomy

Idea

Given connection on a bundle $\nabla$ over a space $X$, its parallel transport around some loop $\gamma : [0,1] \to X$, $\gamma(0) = \gamma(1) = x_0$ yields an element

$$hol_\nabla(\gamma) \in G$$

in the automorphism group of the fiber $P_{x_0}$ of the bundle. This is the holonomy of $\nabla$ around $\gamma$.

Fixing a connection $\nabla$ and a point $x \in X$ the collection of all elements $hol_\nabla(\gamma)$ for all loops $\gamma$ based at $x$ forms a subgroup of $G$, called the holonomy group.

If the Levi-Civita connection on a Riemannian manifold $(X,g)$ has a holonomy group that is a proper subgroup of the special orthogonal group one says that $(X,g)$ is a manifold with special holonomy. (More precise would be: “with special holonomy group for the Levi-Civita connection”.)

Properties

Proposition. If on a smooth principal bundle $P\to X$ there is a connection $\nabla$ whose holonomy group is $G$ then the structure group can be reduced to $G$.

(…)

Theorem

(Ambrose-Singer) Ambrose-Singer theorem: the Lie algebra of the holonomy group of a connection on a bundle $\nabla$ on $X$ at a point $x \in X$ is spanned by the parallel transport $Ad_{tra_\nabla(\gamma)}(F_A(v \vee w))$ of the curvature $F_A$ evaluated on any $v \vee w \in \wedge^2 T_y X$ at $y \in X$ along any path $\gamma$ from $x \to y$.

We may think of $Id + \Ad_{tra_\nabla(\gamma)}(F_A(\phi))$ as being the holonomy around the loop obtained by

1. going along $\gamma$ from $x$ to $y$

2. going around the infinitesimal parallelogram spanned by $v$ and $w$;

3. coming back to $x$ along the reverse path $\gamma$.

Applications

(…)

Higher holonomy

The higher holonomy (see there) of circle n-bundles with connection is given by fiber integration in ordinary differential cohomology.

Related concepts

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

• parallel transport, higher parallel transport

• holonomy

  • holonomy group

  • special holonomy

  • Wilson loop

  • nonabelian Stokes theorem

• fiber integration in differential cohomology

  • fiber integration in ordinary differential cohomology

References

With an eye towards application in mathematical physics:

• Mikio Nakahara, Section 10.2 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)