# nLab holonomy group

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• # Contents

## Idea

For $X$ a space equipped with a $G$-connection on a bundle $\nabla$ (for some Lie group $G$) and for $x \in X$ any point, the parallel transport of $\nabla$ assigns to each curve $\Gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $hol_\nabla(\gamma) \in G$: the holonomy of $\nabla$ along that curve.

The holonomy group of $\nabla$ at $x$ is the subgroup of $G$ on these elements.

If $\nabla$ is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup $H$ of the special orthogonal group, one says that $(X,g)$ is a manifold of special holonomy .

## References

(…)

Created on August 26, 2011 at 16:36:57. See the history of this page for a list of all contributions to it.