Twistor and Killing spinors on Lorentzian manifolds and their relations to CR and Kaehler geometry
by
Helga Baum
Humboldt-University Berlin, Germany

Oral Communication

Twistor spinors were introduced by R. Penrose in General Relativity as solutions j of the conformally covariant twistor equation

ÑX j = - \frac1n X ·Dj ,

where Ñ is the spinor derivative and D the Dirac operator. Parallel and Killing spinors are twistor spinors which satisfy the Dirac equation. We are interested in the following geometric problems:

1. Which semi-Riemannian geometries admit solutions of the twistor equation?
2. How the properties of twistor spinors are related to the geometric structures where they can occur.

1. First we describe a relation between a special class of Lorentzian twistor spinors and CR geometry. Let (N^2m+1, H, J) be a CR manifold with pseudohermitian form \theta and positive definite Levi form L_\theta. Let Ñ^W denote the Webster-Tanaka connection of the pseudohermitian manifold (N, H, J, \theta). For a fixed spin structur of the Riemannian manifold (N, g_\theta=L_\theta+\theta o \theta) we consider the S¹-principal bundle (M, \pi, N), which is associated to the square root of the canonical bundle of the CR manifold defined by the spin structure. In TM we fix a horizontal space using the connection form

A\theta = AW - \fraci4(m+1) RW ·\theta  ,

where A^W is the connection form on M defined by Ñ^W and R^W denotes the Webster-Tanaka scalar curvature. Then

h\theta : = \pi*L\theta + i \frac8m+1 \pi*\theta o A\theta

is a Lorentzian metric on M, whose conformal class is an invariant of the CR structur. (M^2m+2, h_\theta) is called Fefferman space of the strictly pseudoconvex spin manifold (N, g_\theta). Then the following result holds :
Let (N, J, \theta) be a strictly pseudoconvex spin manifold with the Fefferman space (M, h_\theta). Then there exist at least 2 (explicitly constructed) twistor spinors j on (M, h_\theta), such that
1.         V_j is a lightlike regular Killing field.
2.         V_j ·j = 0.
3.         Ñ_Vjj = i c j  ,       c in R\{0}.
On the other hand, if (B^2m+2, h) is a Lorentzian spin mannifold with twistor spinors satisfying 1.-3., then (B, h) is a S¹-bundle over a strictly pseudoconvex spin manifold N and (B, h) is locally isometric to its Fefferman space.
2. For real Killing spinors j on a Lorentzian manifold the function   Q_j = <j, j>² +g(V_j, V_j)   is an 1. integral. The local structure of Lorentzian manifolds M with real Killing spinors can be completly described. Such manifolds are folliated by hypersurfaces and consist out of warped products over Riemannian manifolds with real or imaginary Killing spinors or with parallel spinors (depending on the sign of Q_j). In particular, each Lorentzian manifold with real Killing spinors is an Einstein space of positive scalar curvature, and in case of dimension 4 of constant sectional curvature.

3. Contrary to that, there are non-Einstein Lorentzian manifolds with imaginary Killing spinors. For example, let (B, h) be a Riemannian 1-connected Ricci-flat Kaehler manifold (or any other Riemannian manifold with parallel spinors) and let   f in C^\infty(R ×B)   be a function. Then (M, g), where

M=R3 ×B     ,       g(u, s, t, x)=e2u (-2dtds + f(s, x)ds2 +hx) + du2

is a Lorentzian manifold admitting imaginary Killing spinors, which is Einstein iff f(s, ·) is harmonic on (F, h).
A Lorentzian manifold (M, g) has imaginary Killing spinors iff the cone C_-(M)=(M ×R⁺, g_C = t² g - dt²)   has parallel spinors. There is no irreducible holonomy representation for 1-connected manifolds of index 1 admitting parallel spinors and exactly one for index 2 (namly SU(1, m)). The holonomy SU(1, m) on the cone corresponds to Lorentzian Einstein-Sasaki structures on M. Regular Lorentzian Einstein-Sasaki structures can be defined on circle bundles over Riemannian Kaehler-Einstein manifolds of negative scalar curvature. As the Fefferman geometry, the Lorentzian Einstein-Sasaki geometry can be characterized by special solutions of the twistor equation. Further classes of Lorentzian manifolds with imaginary Killing spinors can be obtained by the study of indecomposable but non-irreducible holonomy representations for metrics of index 1 or 2.

4. Contrary to the Riemannnian case, where each homogeneous space with parallel spinors is flat, there are nonflat Lorentzian homogeneous spaces with parallel spinors. We describe some examples.

http://www-irm.mathematik.hu-berlin.de/~baum

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-76.