conformal group

A *conformal transformation* (conformal mapping) is a transformation of a space which preserves the angles between the curves. In other words, it preserves the angles 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.

In euclidean $n$-space for $n\gt 2$ a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a $n-1$-sphere.

For $n=2$, 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 $z$ and $\bar{z}$ as independent coordinates in the complexification $\mathbb{C}^2$ and restricts to the real part $\mathbb{R}^2\cong \mathbb{C}$). This is important for CFT in 2d.

For $d,t \in \mathbb{N}$ write $\mathbb{R}^{d,t}$ for the pseudo-Riemannian manifold which is the Cartesian space $\mathbb{R}^{d+t}$ equipped with the constant metric of signature $(d,t)$. I.e. for $t = 0$ this is Euclidean space and for $t=1$ this is Minkowski spacetime.

If $d+t \gt 2$ then the conformal group of $\mathbb{R}^{d,t}$ is the orthogonal group $P(d+1, t+1)/\{\pm 1\}$. The connected component of the neutral element is the special orthogonal group $SO(d+1,t+1)$. (e.g Schottenloher 08, chapter 2, theorem 2.9).

Notice that for $t= 1$ this is also the anti de Sitter group, the isometry group of anti de Sitter spacetime of dimension $d+1+t$. This equivalence is the basis of the AdS-CFT correspondence.

group | symbol | universal cover | symbol | higher cover | symbol |
---|---|---|---|---|---|

orthogonal group | $\mathrm{O}(n)$ | Pin group | $Pin(n)$ | Tring group | $Tring(n)$ |

special orthogonal group | $SO(n)$ | Spin group | $Spin(n)$ | String group | $String(n)$ |

Lorentz group | $\mathrm{O}(n,1)$ | $\,$ | $Spin(n,1)$ | $\,$ | $\,$ |

anti de Sitter group | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |

conformal group | $\mathrm{O}(n+1,t+1)$ | $\,$ | |||

Narain group | $O(n,n)$ | ||||

Poincaré group | $ISO(n,1)$ | Poincaré spin group | $\widehat {ISO}(n,1)$ | $\,$ | $\,$ |

super Poincaré group | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |

superconformal group |

Textbook accounts include

- Martin Schottenloher,
*The conformal group*, chapter 2 of*A mathematical introduction to conformal field theory*, 2008 (pdf)

and (with an eye towards combination with spin geometry)

- Pierre Anglès,
*Conformal Groups in Geometry and Spin Structures*, Progress in Mathematical Physics 2008

See also

- Wikipedia,
*Conformal group*

Revised on June 13, 2016 13:49:00
by Eponymous Howard?
(64.134.161.106)