nLab conformal compactification

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A conformal compactification is an embedding of a non-compact Lorentzian manifold into a compact Lorentzian manifold as a dense open subspace, such that the embedding is a conformal map .

Example

The (or rather, any) conformal compactification M¯\overline{M} of (flat) Minkowski space M= n,1M = \mathbb{R}^{n,1} is important in various treatments of behaviour at infinity of physics thereon (see e.g. Penrose-Hawking theorem).

Section 4.2 of Nikolov & Todorov 2004 gives a thorough geometric discussion of the construction and gives explicit expressions in coordinates, and shows that (the underlying manifold of) this M¯\overline{M} is in fact (S n×S 1)/{±1}(S^n \times S^1)/\{\pm 1\}, for a specific embedding of S n×S 1S^n\times S^1 in n+1,2\mathbb{R}^{n+1,2}, and the action by scalar multiplication.

Let q(x,t)q(x,t) be the standard indefinite quadratic form of signature (n,1)(n,1) on n,1\mathbb{R}^{n,1} and define the following map n,1 n+1,2 n,1× 1,1\mathbb{R}^{n,1} \to \mathbb{R}^{n+1,2} \simeq \mathbb{R}^{n,1}\times \mathbb{R}^{1,1}:

C˜:(x,t)(x,t,1q(x,t)2,1q(x,t)2) \tilde{C}\colon (x,t) \mapsto (x,t,\frac{1-q(x,t)}{2},\frac{-1-q(x,t)}{2})

This is a diffeomorphism on its image, which can be described as the intersection of the hyperplane vw=1v-w=1 and the quadric q(x,t)+q(v,w)=0q(x,t) + q'(v,w)=0, where (v,w)(v,w) are coordinates on 1,1\mathbb{R}^{1,1} with the quadratic form q(v,w)=v 2w 2q'(v,w) = v^2-w^2. In particular, the image of this map avoids the origin in n+1,2\mathbb{R}^{n+1,2}, and rearranging the defining equation for the quadric we get |x| 2+v 2=t 2+w 2|x|^2 + v^2 = t^2 + w^2 =K=K, say, where K0K\neq 0. We can then scale C˜(x,t)\tilde{C}(x,t) to C(x,t)C(x,t) so that K=1K=1, which means C(x,t)S n×S 1C(x,t) \in S^n\times S^1. [TODO: calculate this normalisation] Then the final quotient by {±1}\{\pm 1\} gives the desired dense embedding n,1(S n×S 1)/{±1}\mathbb{R}^{n,1} \to (S^n\times S^1)/\{\pm 1\}.

One could also skip the normalisation step if desired, and pass directly to the quotient by × >0 *×{±1}\mathbb{R}^\times \simeq \mathbb{R}_{\gt0}^* \times \{\pm 1\}, treating C˜(x,t)\tilde{C}(x,t) as homogeneous coordinates.

The blog post (Wong 2009)) gives a discussion of comformal compactification in general, with Minkowski space as an example. It describes the underlying manifold of (the ‘usual’ construction of) M¯\overline{M} is S n×S 1S^n\times S^1, but then clarifies that this is a ‘double cover’.

A conformal compactification, of complexified Minkowski spacetime 3,1\mathbb{R}^{3,1}, is given by the Klein quadric. (eg, Fioresi-Lledo-Varadarajan 07, section 2). This plays a key role in the twistor correspondence.

References

On conformal compactification via Cartan geometry:

  • Daniel Platt: Conformal Geodesics in Cartan Calculus, MSc thesis (2017) [pdf, pdf]

On conformal boundaries, mostly in the context of the AdS/CFT correspondence:

Last revised on July 2, 2024 at 09:41:10. See the history of this page for a list of all contributions to it.