nLab
conformal geometry
Contents
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 AdS-CFT 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 contextgauge 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
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 CR-Geometry , 2007 (pdf )

Hega Baum , Andreas Juhl , Conformal Differential Geometry: Q-curvature 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 )

Sean Curry, A. Rod Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity , 2014, (arXiv:1412.7559 )

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
Last revised on May 22, 2019 at 13:34:53.
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