nLab conformal group

Idea

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. The 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.

Examples

Of euclidean space

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.

Of $\mathbb{R}^{d,t}$

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.

Cosets

(…) Möbius space (…)

(Arutyunov-Sokatchev 02)

(…)

Related concepts

• conformal invariance and scale invariance

• Möbius group

• conformal geometry

• conformal connection

• conformal field theory

• conformal symplectic group

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

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

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

Details on the conformal Lie algebra of conformal Killing vectors? acting on 3+1 dimensional Minkowski spacetime are spelled out for instance in

• Matthias Blau, section 9.3 of Lecture notes on general relativity (web)

Discussion in conformal field theory

• G. Arutyunov, E. Sokatchev, Conformal fields in the pp-wave limit, JHEP 0208 (2002) 014 (arXiv:hep-th/0205270)

See also

• Wikipedia, Conformal group