Twistor Theory

Stephen Huggett

• Editore: Routledge
• Anno: 2017
• ISBN:
  9781351406543
• Edizione: 1
• Argomento: Mathematics
• Categoria: Mathematics General

Presents the proceedings of the recently held conference at the University of Plymouth. Papers describe recent work by leading researchers in twistor theory and cover a wide range of subjects, including conformal invariants, integral transforms, Einstein equations, anti-self-dual Riemannian 4-manifolds, deformation theory, 4-dimensional conformal structures, and more.;The book is intended for complex geometers and analysts, theoretical physicists, and graduate students in complex analysis, complex differential geometry, and mathematical physics.

1
Thomas’s D-Calculus, Parabolic Invariant Theory, and Conformal Invariants

T. N. Bailey Department of Mathematics, University of Edinburgh, Edinburgh, Scotland

1 Introduction

In this talk, I want to draw attention to the connections between, and consequences of, several pieces of recent work concerning conformally invariant differential operators.

A conformal manifold is a smooth n-dimensional manifold M equipped with an equivalence class of metrics [g], the elements of which are related by conformal rescalings g ↦ Ω²g, where Ω is a nowhere vanishing smooth function. For definiteness, we shall assume that our metrics are Riemannian, although everything we say is true also in the pseudo-Riemannian case. On a conformal manifold, one has line bundles E[w] where w ∈ ℝ, sections of which are conformally weighted functions—i.e., functions f which transform according to f↦f^:=Ωwf under the conformal rescaling g↦g^:=Ω2g. I will use the abstract index notation [PRJ for tensor fields, so that for example, Ea denotes the tangent bundle of M, and U^a a section thereof. I will write Ea[w] for the tensor product of Ea with E[w], etc.

The problem which I wish to address is that of finding conformally invariant differential operators between the line bundles of conformally weighted functions—i.e. operators L:E[w]→E[q] satisfying LF^=Lf^. I will sometimes refer to such operators as invariants. One can restrict one’s attention to linear differential operators, but even then there is no complete theory. A simpler problem is to consider the special case of the “flat model” for conformal geometry, and there the case of linear operators is well understood, and there has been much progress recently on the general problem.

2 The flat model

Consider ℝⁿ⁺², where n > 1, together with a signature (n + 1, 1) bilinear form g˜. Write X^I for coordinates on ℝⁿ⁺². Let e₀ be a non-zero element of ℝⁿ⁺², null with respect to g˜. Let G denote the identity-connected component of the pseudo-orthogonal group which preserves g. Let Q denote the connected component of the set of non-zero null vectors which contains e₀.

The group G acts linearly on ℝⁿ⁺² preserving Q, and hence it acts on the space of generators of Q. This space of generators is an n-dimensional sphere, which comes equipped with a conformal structure, with G acting by conformal automorphisms (“Möbius transformations”).

Sections of the line bundles E[w] over S associated with the conformal structure can be identified with functions on Q homogeneous of degree w, in the sense that

f(λX)=λwf(X), for X∈Q, 0<λ∈ℝ.

The condition that L:E[w]→E[q] be conformally invariant is equivalent to its being equivariant with respect to the G-action. The problem of finding all conformally invariant linear operators is completely solved: it is a fairly standard exercise in representation theory (see [BE] for a review).

Let us now turn to the general problem of invariant differential operators. One way of constructing these is to use the manifestly G invariant operations of differentiating with respect to coordinates on ℝⁿ⁺², and forming contractions of the resulting tensors with g˜, the volume form ε˜, and the coordinate vector X^I. To do this however one needs a way of taking the homogeneous functi...