nLab conformal geometry

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.

Related concepts

• conformal map, conformal transformation, conformal group

• conformally flat manifold

• causal structure

• Fefferman-Graham ambient construction

• conformal compactification

• CFT

• twistor space

• conformal cobordism category

• conformal connection

• superconformal geometry

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 $Iso(d-1,1)$	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

• David Gu, Conformal structure, slides (pdf)

• eom, Conformal structure

• Boris Apanasov, Doubles of Atoroidal Manifolds, Their Conformal Uniformization and Deformations In: J. Ławrynowicz (ed.) Deformations of Mathematical Structures Springer, Dordrecht (1989) (doi:10.1007/978-94-009-2643-1_9)

• Boris Apanasov, Conformal Geometry of Discrete Groups and Manifolds, De Gruyter Expositions in Mathematics, 32, De Gruyter (2000) (doi:10.1515/9783110808056)

Discussion of conformal structure as G-structure:

• Maks A. Akivis, Vladislav V. Goldberg, Conformal and Grassmann structures, Differential Geom. Appl. 8 (1998) no. 2 177-203 (arXiv:math/9805107)

Discussion of conformal Cartan geometry (parabolic geometry):

• 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:

• A. Rod Gover, Andreas Čap, Conformal and CR geometry: Spectral and nonlocal aspects (2007) [pdf]

Discussion with an eye towards combination with spin geometry:

• Pierre Anglès, Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics (2008)

Application to gravity (general relativity):

• Juan A. Valiente Kroon: Conformal Methods in General Relativity, Cambridge University Press (2017, 2022) [doi:10.1017/9781009291309, oapen:20.500.12657/59217, pdf]