# Contemts

## 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 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.

## 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.

groupsymboluniversal coversymbolhigher coversymbol
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)$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$Poincaré spin group$\widehat {ISO}(n,1)$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$
superconformal group
geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group $O(d-1,2)$$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 groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher 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