The Elements of Twistor Theory

Transcription

1 The Elements of Twistor Theory Stephen Huggett 10th of January, Introduction These are notes from my lecture at the Twistor String Theory workshop held at the Mathematical Institute Oxford, 10th 14th January 005. They are almost the same as the slides I used, except that I have compressed them to save paper and corrected a couple of errors. Twistor geometry Twistor space T is C 4 with coordinates Z α = (Z 0, Z 1, Z, Z 3 ). Projective twistor space PT is CP 3. Z PT has homogeneous coordinates [Z 0, Z 1, Z, Z 3 ]. A plane in PT is a CP given by an equation of the form Z α A α = 0. In other words it is determined by a dual (projective) twistor [A 0, A 1, A, A 3 ]. A line in PT is a CP 1 given by the intersection of two planes Z α A α = Z β B β = 0. Of course there is some freedom in the choice of A α and B β. What is the space of these lines in PT? Each is determined by a skew simple (0,) twistor L αβ. The condition for simplicity can be written 3L α[β L γδ] = L αβ L γδ + L αγ L δβ + L αδ L βγ = 0, s.huggett@plymouth.ac.uk huggett/home.html 1

2 which defines a quadric Q in CP 5 (called the Klein quadric). By changing the coordinates we can see that Q is actually the space of generators of the cone in C 6. T + V W X Y Z = 0 Here is the change of coordinates: T = V = L L 01 W = L 3 1 L 01 X = i (L 0 L 13 ) Y = 1 (L 0 + L 13 ) Z = i (L 1 + L 03 ) Here is the embedding of M in the cone: i (L 03 L 1 ) x a (x 0, 1 (1 xb x b ), 1 (1 + xb x b ), x 1, x, x 3 ) Thought of in R 6 our cone is the O(, 4) null cone of ds = dt + dv dw dx dy dz. Each of its generators (except those for which W V = 0) meets the plane W V = 1 in a point, and the intersection of this plane and the cone is just Minkowski space M. So the space of generators is a compactification M c of M. It is the conformal compactification: the extra generators form a null cone at infinity. We have shown that there is a four real dimensional family of lines in PT corresponding to M, but we have not so far shown how to identify them in PT. For a twistor Z to lie on a line L it must satisfy two linear equations. Except when the line is given by Z = Z 3 = 0, these can be written ( ) Z 0 Z 1 = i ( ) ( ) x 0 + x 3 x 1 + ix Z x 1 ix x 0 x 3 Z 3

3 where x a is the space-time point corresponding to L. More concisely, if we write Z α = (ω A, π A ) we have ω A = ix AA π A. If Z also lies on the line corresponding to y AA, then x AA π A = y AA π A and so the matrix x AA y AA x a and y a are null-separated. must be singular. The condition for this is that If in ω A = ix AA π A we think of the twistor as fixed and solve for the point x AA we find that x AA = x AA 0 + µ A π A for arbitrary µ A. So Z α = (ω A, π A ) corresponds to this alpha-plane: it is a totally null two complex dimensional plane in complex Minkowski space. PT complex projective line point intersection of lines CM c point alpha-plane null-separation of points In general an alpha-plane will have no real point, but when it does it contains a whole real null ray: if x AA 0 is real then so is x AA = x AA 0 + r π A π A for any real r. If Z α is the twistor for this alpha-plane then Σ(Z) = ω A π A + ω A π A = Z 0 Z + Z 1 Z 3 + Z 0 Z + Z 1 Z 3 = 0. 3

4 This Hermitian form Σ divides PT into three regions: Σ(Z) > 0 PT + Σ(Z) = 0 Σ(Z) < 0 PN PT PN is the space of real null rays: it is a five real dimensional manifold with a C-R structure. We could imagine discovering projective twistor space this way. If x AA is real then any Z lying on the corresponding line in PT satisfies ω A = ix AA π A and hence has Σ(Z) = 0. Thus points in real (compactified) Minkowski space correspond to lines lying entirely in PN. Any two twistors on a given line in PN represent null rays through the corresponding point in M. So intrinsically the line in PN is the celestial sphere of the space-time point. Lines lying entirely in PT + correspond to points z AA = x AA iy AA with y AA timelike and future-pointing, or in other words points z AA in the future tube. This will lead later to a very elegant twistor description of positive frequency, using the fact that positive frequency fields can be characterized by having holomorphic extensions into the future tube. 3 The Penrose transform Consider the twistor function f(z α ) = 1 (A α Z α )(B β Z β ) where A α = (A A, A A ), B α = (B A, B A ) are two constant dual twistors. We aim to calculate a field at the point x AA in CM. Restrict Z α to the line L x so that A α Z α = (ia A x AA + A A )π A α A π A B α Z α = (ib A x AA + B A )π A β A π A 4

5 and consider the contour integral ϕ(x) = 1 1 πi (α A π A )(β B π B ) π C dπc. This is well defined on CP 1 since the integrand has total homogeneity zero. Further, there will exist a contour around which to do the integral provided the two poles are distinct: α A β A 0. This is satisfied unless L x intersects the line A [α B β], or in other words unless x is on the null cone of a point y in CM. The contour integral above yields ϕ(x) = A A B A (x y), which is a solution of the wave equation ϕ = 0. Solutions of the massless free field equations for spin n/ AA ϕ A...B = 0 (for positive frequency positive helicity) and AA ψ A...B = 0 (for positive frequency negative helicity) may be solved in a similar way. Let f n (Z α ) and f n (Z α ) be holomorphic and homogenous of degrees n and n. Define fields ϕ A...B (x) and ψ A...B(x) by: ϕ A...B (x) = 1 π A... π B f n (Z α )π C dπ C πi ψ A...B (x) = 1 πi ω A... ω B f n (Z α )π C dπ C where both integrands are first restricted so that Z lies on L x. The remarkable thing here is that ϕ and ψ are automatically solutions of the massless free field equations. 5

6 It is easy to write down examples of such functions by choosing rational functions, and we can easily check when their singularities are such that the contour integral is well defined. In general this can be fiddly, though. Also, it is clear that many twistor functions will give the same space-time field. Define Then F 1 = {L 1 : L 1 is a one-dimensional subspace of T} F = {L : L is a two-dimensional subspace of T} F = F 1, = {(L 1, L ) : L 1 and L as above with L 1 a subspace of L }. F 1 = PT, F = CM c and F can be identified with the primed spin bundle, with (local) coordinates (x a, π A ). We have the double fibration F µ ν PT CM where ν : (x a, π A ) x a µ : (x a, π A ) (ix AA π A, π A ) The inverse image of a point in PT under µ is the whole alpha-plane in F. So a function f(x a, π A ) on F pushes down to a function on PT if it is constant on alpha-planes: π A AA f = 0. We may restrict the double fibration to the future tube CM + and corresponding regions F + and PT +. Let U be a cover of F + by open sets U i. An element of C 0 (U, S) is a function f i (of type S, defined on U i ) for each i. An element of C 1 (U, S) is a function f ij (of type S, defined on U i U j ) for each i, j. 6

7 We have maps δ 0 : C 0 C 1 {f i } {ρ i f j ρ j f i } and δ 1 : C 1 C {f ij } {ρ i f jk + ρ j f ki + ρ k f ij } In our example we had just two sets: U 0 = {Z α : Z α A α 0}, U 1 = {Z α : Z α B α 0} and our function was actually f 01 defined on U 0 U 1. Let Z n(m) be the sheaf of germs of symmetric n-index primed spinor fields (x, π) which are ϕ A...B holomorphic on F +, homogeneous of degree m in π A, solutions of A A ϕ A...B = 0. Consider the onto map We have the short exact sequence π C : Z n+1(m 1) Z n(m) ϕ A...B C ϕ A...B C πc. 0 T i Z n+1( 1) πc Z n(0) 0 where T is just the kernel of the map π C. and hence ψ A...B C T ψ A...B C πc = 0, ψ A...B C = π A... π B π C f(x, π). Further, the zero rest mass equation on ψ A...B C satisfy π A AA f = 0. It is therefore a twistor function homogeneous of degree n. implies that f(x, π) must 7

8 We may now write our short exact sequence as 0 T ( n ) π A...π B π C Z n+1( 1) πc Z n(0) 0. In the following array the squares commute and the rows are exact, but in the columns we only have Imδ 0 Kerδ 1, in general. However, in the central column Imδ 0 = Kerδ 1. 0 C 0 (U, T ) C 0 (U, Z n+1( 1)) C 0 (U, Z n(0)) 0 h ia...b C h ia...b C πa δ 0 δ 0 δ 0 0 C 1 (U, T ) C 1 (U, Z n+1( 1)) C 1 (U, Z n(0)) 0 f ij π A... π B π C f ij δ 1 δ 1 δ 1 h ia...b C = 1 λ A... λ B λ C f ij λ D dλ πi γi D π E λ E h ia...b C πa = ρ i ϕ A...B We started with f ij in the kernel of δ 1, but for any f ij in the image of δ 0 we would have obtained zero, so we actually want the class of f ij in Kerδ 1 /Imδ 0 = H 1 (U, T ( n )). To see that this procedure leads to an isomorphism we proceed as follows. From our short exact sequence there is a corresponding long exact sequence of cohomology groups, which includes this part: Ȟ0 (F + ; Z n+1( 1)) Ȟ0 (F + ; Z n(0)) Ȟ1 (F + ; T ( n )) Ȟ 1 (F + ; Z n+1( 1)) The first term is global sections of Z n+1( 1) over F +. For fixed x a, such a section would give a global section homogeneous of degree 1 on CP 1, and there are no such sections. Thus Ȟ0 (F + ; Z n+1( 1)) = 0. A similar argument, slightly more technical, shows that Ȟ 1 (F + ; Z n+1( 1)) = 0 8

9 so that we are left with an isomorphism Ȟ 0 (F + ; Z n(0)) δ = Ȟ 1 (F + ; T (n )) The term on the left represents solutions of the zero rest mass equations on F +, homogeneous of degree zero in π A. They must therefore be independent of π A, and defined on CM +. The term on the right is a cohomology group on F + but the coefficients are twistor functions, so Ȟ 1 (F + ; T ( n )) = Ȟ1 (PT + ; O( n )). We have shown that the space of zero rest mass fields ϕ A...B (x) of helicity n, holomorphic on CM +, is isomorphic to Ȟ1 (PT + ; O( n )). In fact this works for other open subsets and the corresponding spaces and X CM c Y = ν 1 (X) F Z = µ(y ) PT, as long as the fibres of µ Y are connected and simply connected. We have glossed over the details of how to relate the various sheaves on X, Y, and Z. 4 References M G Eastwood, R Penrose, and R O Wells 1981 Cohomology and Massless Fields Commun. Math. Phys S A Huggett and K P Tod 1994 An introduction to twistor theory, Cambridge University Press. R Penrose and W Rindler 1984 & 1986 Spinors and space-time, volumes 1 &, Cambridge University Press. R Ward and R O Wells 1990 Twistor Geometry and Field Theory, Cambridge University Press. 9