nLab
conformal geometry
Context
Riemannian geometry
Contents
Idea
A conformal structure on a manifold is the structure of a Riemannian metric modulo rescalings of the metric tensor by some real valued function on the manifold. The homomorphisms of conformal structures are called conformal transformations.
In terms of Cartan geometry conformal structure is expressed by conformal connections, conformal geometry is a special case of parabolic geometry and hence of Cartan geometry.
In the context of quantum field theory conformal structure underlies the formulation of conformal field theory. Due to the coincidence of the conformal group of Minkowski spacetime of dimension $d$ with the anti de Sitter group of anti de Sitter spacetime in dimension $d+1$ there is a close relation between certain conformal field theories and certain theories of gravity. This is the content of the AdSCFT correspondence. This works most accurately in the context of supergeometry, hence for superconformal groups acting on the asymptotic boundary of super anti de Sitter spacetimes.
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 $Iso(d1,1)$  Lorentz group $O(d1,1)$  Minkowski spacetime $\mathbb{R}^{d1,1}$  Lorentzian geometry  pseudoRiemannian geometry  spin connection  Einstein gravity 
 anti de Sitter group $O(d1,2)$  $O(d1,1)$  anti de Sitter spacetime $AdS^d$     AdS gravity 
 de Sitter group $O(d,1)$  $O(d1,1)$  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 $\mathbb{R}^{d1,1\vert N}$  Lorentzian supergeometry  supergeometry  superconnection  supergravity 
 super anti de Sitter group   super anti de Sitter spacetime     
higher differential geometry  smooth 2group $G$  2monomorphism $H \to G$  homotopy quotient $G//H$  Klein 2geometry  Cartan 2geometry   
 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
Discussion of conformal Cartan geometry (parabolic geometry) includes

Andreas Čap, Jan Slovák, sections 1.1.5, 1.6 of Parabolic Geometries I – Background and General Theory, AMS 2009

Felipe Leitner, part 1, section 6 of Applications of Cartan and Tractor Calculus to Conformal and CRGeometry, 2007 (pdf)

Hega Baum, Andreas Juhl, Conformal Differential Geometry: Qcurvature and Conformal Holonomy, Oberwolfach Seminars, vol. 40, Birkhäuser, 2010, 165pp.

Andree Lischewski, section 2 of Conformal superalgebras via tractor calculus, Class.Quant.Grav. 32 (2015) 015020 (spire, arXiv:1408.2238)
A survey of the field as of 2007 is in
 A. Rod Gover, Andreas Čap, Conformal and CR geometry: Spectral and nonlocal aspects pdf
Discussion with an eye towards combination with spin geometry is in
 Pierre Anglès, Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics 2008
Revised on March 26, 2015 10:03:18
by
Urs Schreiber
(195.113.30.252)