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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          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) 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

          • Roger Penrose, Relativistic Symmetry Groups, in A.O.Barut (ed.), Group Theory in Non-Linear Problems: Lectures Presented at the NATO Advanced Study Institute on Mathematical Physics (1974)

          • Valentina, Conformal compactifications (pdf)

          • Jörg Frauendiener, Conformal Infinity (living reviews)

          • Nikolay M. Nikolov, Ivan T. Todorov, Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory, arXiv:math-ph/0412039 (inspirehep pdf).

          • R. Fioresi, M. A. Lledo, Veeravalli Varadarajan, The Minkowski and conformal superspaces, J.Math.Phys.48:113505,2007 (arXiv:math/0609813)

          • Willie Wong, Conformal compactification of space-time (web) (2009).

          Last revised on August 2, 2016 at 02:24:08. See the history of this page for a list of all contributions to it.