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I am looking for good and recent references to constructing twistor space for curved spacetime. This could be a general spacetime, or specific ones (say maximally symmetric spaces different from Minkowski). This could be in he context of the twistor correspondence, or the twistor transform of field equations, either subject generalized to curved spacetime.

The references I am familiar with are the standard ones from about 30-40 years ago, where most constructions involve flat spacetime. Some generalizations are mentioned, but my impression is that the community had not settled at the time on a single approach. Many things happened since, and one of the things I am hoping to get is some understanding of the landscape of current approaches to the subject.

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3 Answers 3

up vote 12 down vote accepted

I'm not sure if this is exactly what you are looking for or perhaps you already know what I am about to say.

There is a geometric notion of a twistor spinor (or conformal Killing spinor): one which is in the kernel of the Penrose operator (see below). Then one defines the twistor space as the projectivisation of the space of twistor spinors. Doing this for Minkowski spacetime recovers the usual twistor space.

Let $(M,g)$ be a riemannian spin manifold. (When I say riemannian I include also the case of a metric with indefinite signature.) Let $S$ denote the complex spinor bundle. The spin connection defines a map $$ \nabla: \Gamma(S) \to \Omega^1(S) $$ from spinor fields to one-forms with values in $S$. Now $\Omega^1(S) = \Gamma(T^*M \otimes S)$ and Clifford action of one-forms on spinors gives a map $$ \Omega^1(S) \to \Gamma(S) $$ The composition of the previous two maps is the Dirac operator. The Penrose operator is in some sense the complement of the Dirac operator $D$. The kernel of the Clifford map $T^*M \otimes S \to S$ defines a subbundle $W$, say, of $T^*M \otimes S$. Composing the covariant derivative with the projection $\Omega^1(S) \to \Gamma(W)$ defines the Penrose operator $P: \Gamma(S) \to \Gamma(W)$: explicitly, $$ P_X \psi = \nabla_X \psi + \frac1n X \cdot D\psi $$ for all vector fields $X$ and spinor fields $\psi$, and where $n = \dim M$. (My Clifford algebra conventions are $X^2 = - |X|^2$.) Notice that the "gamma trace" of the Penrose operator vanishes.

There is a sizeable literature on twistor spinors mostly in riemannian and lorentzian signatures. This is the work of Helga Baum and collaborators in Berlin. A search for "twistor spinors" in MathSciNet should give you many links.

One important property of the twistor spinor equation is that it is conformally invariant, whence the twistor spinors of conformally related riemannian spin manifolds correspond in a simple way. Since you mention maximally symmetric lorentzian manifolds, this observation might be of use because such spaces are conformally flat, hence you can write down the twistor spinors simply by rescaling the twistor spinors in Minkowski spacetime. In riemannian signature (hence for round spheres and hyperbolic spaces) this is described in the 1990 Humboldt University Seminarberichte Twistor and Killing spinors on riemannian manifolds by Baum, Friedrich, Grunewald and Kath, later published by Teubner.

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Thanks, this is useful. I am mainly looking for an entry point to the literature, or a good survey. My actual confusion is not well-formed enough to make a good question at this point. Sorry for being a bit vague. –  user566 Nov 3 '11 at 0:58
Helga Baum has written a couple of surveys about twistor spinors in lorentzian geometry, but I do not think that they address the twistor transform. The twistorial people I know work mostly in riemannian signature. –  José Figueroa-O'Farrill Nov 3 '11 at 8:48

I am not too comfortable with this subject, but anyway maybe the list may be useful

  1. R. Penrose, W. Rindler, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (1988). (certainly known, I suppose)
  2. R. S. Ward, R.O. Wells, Twistor geometry and field theory (CUP, 1990) (Chapter 9)
  3. S. A. Huggett, K. P. Tod, An introduction to twistor theory (CUP, 1994) (Chapter 13 ?)
  4. M. Dunajski, Solitons, instantons, and twistors (OUP, 2010) (Chapter 10.5 ?)
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I am not familiar with the last one, but own first three. For twistors in curved spacetime, most extensive and up to dated discussion is in Ward and Wells among them. –  Demian Cho Nov 3 '11 at 21:15

This may not be exactly what you are looking for, and I am certainly not an expert in this. But, I happened to be interested in current state of art in Penrose's non linear graviton program and did a quick (~ 30 min.) literature search last year.

My impression is that there has not been large activities nor a break through. Also, as we know, twistor community is a small group and they probably don't feel a need of review articles for wider readership. (My impression was further reinforced by chatting with Penrose few weeks later, even though he seems very excited about development in twistor string theory.)

Having said that, here are my finding:

  1. A useful overview, not so current, is Penrose's article written in 99.

The central program of twistor theory in Chaos, Solitons & Fractals Vol 10, No. 2-3. pp 581-611, 1999

or you can get it in Andrew Hodges's twistor diagram page.

Penrose devoted last few sections on basic ideas and difficulties of defining twistors in curved spacetime.

  1. A collection of article edited by Lionel Mason.

Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces probably contains much more technical developments upto 2000.

Let me know if you find any useful stuff on this.

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