nLab
conformal group

Intuition

A conformal transformation (conformal mapping) is a transformation of a space which preserves the angles between the curves. In other words, it preserves the angels infinitesimally. Conformal group of a space which has well defined notion of angles between the curves is the group of space automorphisms which are also conformal transformations.

Description in euclidean space

In euclidean nn-space for n>2n\gt 2 a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a n1n-1-sphere. For n=2n=2, i.e. in a complex plane, this still holds for (the group of) global conformal transformations but one also has nontrivial local automorphisms. One has in fact infinite-dimensional family of local conformal transformations, which can be described by an arbitrary holomorphic or an antiholomorphic automorphism (in fact one writes zz and z¯\bar{z} as independent coordinates in the complexification 2\mathbb{C}^2 and restricts to the real part 2\mathbb{R}^2\cong \mathbb{C}). This is important for CFT in 2d.

Revised on May 19, 2010 14:45:48 by Zoran Škoda (161.53.130.104)