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 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.


Of 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.

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

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

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

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

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group
geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


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

Revised on March 26, 2015 00:06:18 by David Roberts (