conformal geometry



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 dd with the anti de Sitter group of anti de Sitter spacetime in dimension d+1d+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 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


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

Revised on December 21, 2016 03:51:02 by Urs Schreiber (