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conformal connection in nLab

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Idea

A conformal connection is a (H↪G)(H \hookrightarrow G)-Cartan connection for G=O +(n,1)G= O^+(n,1) an orthochronous Lorentz group and HH the stabilizer subgroup of a lightlike line in Minkowski space ℝ n,1\mathbb{R}^{n,1}. Here G/HG/H is the celestial sphere of dimension n−1n-1.

An example of parabolic Cartan 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 GG	subgroup (monomorphism) H↪GH \hookrightarrow G	quotient (“coset space”) G/HG/H	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group Iso(d)Iso(d)	rotation group O(d)O(d)	Cartesian space ℝ d\mathbb{R}^d	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group Iso(d−1,1)Iso(d-1,1)	Lorentz group O(d−1,1)O(d-1,1)	Minkowski spacetime ℝ d−1,1\mathbb{R}^{d-1,1}	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group O(d−1,2)O(d-1,2)	O(d−1,1)O(d-1,1)	anti de Sitter spacetime AdS dAdS^d				AdS gravity
	de Sitter group O(d,1)O(d,1)	O(d−1,1)O(d-1,1)	de Sitter spacetime dS ddS^d				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group O(d,t+1)O(d,t+1)	conformal parabolic subgroup	Möbius space S d,tS^{d,t}		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group GG	subgroup (monomorphism) H↪GH \hookrightarrow G	quotient (“coset space”) G/HG/H	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime ℝ d−1,1\|N\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 GG	2-monomorphism H→GH \to G	homotopy quotient G//HG//H	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) H→GH \to G	homotopy quotient G//HG//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

Wikipedia, Conformal connection
Felipe Leitner, Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry, 2007 (pdf)
Andreas Cap, Jan Slovák, sections 1.1.5, 1.6 of Parabolic Geometries I – Background and General Theory, AMS 2009

Last revised on March 17, 2015 at 22:20:39. See the history of this page for a list of all contributions to it.