Conformal group

The conformal group of a semi- Riemannian manifold is the (component of one of the ) Lie group of conformal mappings of the manifold in itself It is thus a subset of the Diffeomorphismengruppe and contains the isometry group of the manifold.

For the physics are particularly the conformal groups of manifolds with flat metric of importance. D For the Euclidean space of dimension is the conformal group is isomorphic to the group SO ( d 1,1 ). In solid-state physics and string theory occur to systems that are scale invariant at least a good approximation. These systems are described quantum physics with conformal quantum field theories which are invariant under the conformal group.

Particularly the two-dimensional case is interesting for string theory, the space then represents the world sheet of a string. In the two-dimensional plane case with the Minkowski metric, the Lie algebra for the conformal group contains the infinite-dimensional Witt algebra of polynomial vector fields on the unit circle (see Conformal Mapping ).