Holonomy

The mathematical notion of holonomy of a connection of a vector or principal bundle over a differentiable manifold (abbreviated simply holonomy ) referred to in the differential geometry, the group of linear transformations which is induced by the parallel transport of vectors along curves closed. Carries a manifold M a Riemannian metric, then its holonomy group is given by that of the Levi -Civita connection on the tangent bundle of M.

Importance in physics

Holonomiegruppen play an important role in theoretical physics, both in quantum field theory (see Wilson loop), and in particular in string theory. Here is the holonomy group of a compact six - and seven-dimensional manifolds of interest, since in a compactification of the theory on these spaces, the number of preserved supersymmetry of the covariant constant spinors depends on the maximum number, which is in turn determined by the holonomy. Manifolds of special interest are six-dimensional Calabi -Yau manifolds with SU (3 ) holonomy, and siebendimensionale manifolds with G2 holonomy.