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The Penrose Transform
Its Interaction With Representation Theory
By Robert J. Baston, Michael G. Eastwood Dover Publications, Inc.
Copyright © 2016 Robert J. Baston and Michael G. Eastwood
All rights reserved.
ISBN: 978-0-486-81662-3
Contents
1 Introduction, 1,
2 Lie Algebras and Flag Manifolds, 11,
3 Homogeneous Vector Bundles on G/P, 23,
4 The Weyl Group, its Actions, and Hasse Diagrams, 37,
5 The Bott–Borel–Weil Theorem, 48,
6 Realizations of G/P, 58,
7 The Penrose Transform in Principle, 73,
8 The Bernstein–Gelfand–Gelfand Resolution, 81,
9 The Penrose Transform in Practice, 94,
10 Constructing Unitary Representations, 155,
11 Module Structures on Cohomology, 172,
12 Conclusions and Outlook, 206,
References, 208,
Index, 219,
CHAPTER 1
INTRODUCTION
In this first chapter we want to try to develop the Penrose transform from scratch. Penrose originally noticed that there are simple contour integral formulae for solutions of a series of interesting equations from physics, namely the zero rest mass free field equations of mathematical physics. The Penrose transform is, in one sense, a machine for showing that all solutions can be obtained in this way. It is a lot more than that, for, as we shall see, it generates the equations as geometric invariants and says much about their symmetries.
Zero rest mass fields are of fundamental importance in physics. They describe electromagnetism, massless neutrinos, and linearized gravity. They all share the property that they are conformally invariant — that is they are determined by the conformal geometry of spacetime and so depend only on knowing how to measure relative lengths and angles, not on an overall length scale. Thus the equations which specify these fields are invariant under motions which preserve this conformal geometry and so the space of fields is invariant under the group of these motions. Our ultimate aim is to develop a Penrose transform when this group is any complex semisimple Lie group and to see what the representation theory of such groups implies about the Penrose transform.
We shall begin with a short study of Maxwell's equations for electromagnetism, which, in the absence of charges, amount to a pair of zero rest mass equations on Minkowski space. The idea is first to see where they are most naturally defined, and then to understand the contour integral formulae for them geometrically.
The rest of this book does not depend on an understanding of this chapter, which may therefore be omitted at a first reading. We hope, however, that the reader will eventually take some time out to understand the origins of the transform in mathematical physics.
Minkowski space
Maxwell's equations for a free electromagnetic field (i.e., one in the absence of charges) can be succinctly written down using differential forms as follows:
dF = 0 and d*F = 0.
Here F is a two-form on R4 which represents the electric and magnetic fields, d is exterior differentiation, and * is the Hodge star operation with respect to a (flat) Lorentz metric g on R4. With indices, they may be written as
[MATHEMATICAL EXPRESSION OMITTED]
where [...] indicates that the enclosed indices are to be skewed over. Here [MATHEMATICAL EXPRESSION OMITTED] where εabcd is the volume form with respect to g and we are using the Einstein summation convention (as in [127], for example). It is easy to check from these formulae that * acting on two-forms is independent of the scale of the metric g — any metric κ2g yields the same * on two-forms. Of course, d is invariant under any diffeomorphism of R4, and it follows that Maxwell's equations are invariant under the conformal motions of R4, i.e., diffeomorphisms of R4 which preserve g up to scale. These include the Poincaré motions which are those globally defined conformal motions which preserve the scale of g. Now on two forms, *2 = -1 and so we may write
F = φ + [??]
where φ and [??] are in the ±i eigenspaces of * — in particular, they are necessarily complex two-forms (and conjugate). In terms of these, Maxwell's equations become
dφ = 0 and d[??] = 0.
There are three observations to make here. First, if the metric g is taken to be Euclidean, then *2 = 1 and there is no need to introduce complex two-forms to obtain the decomposition; on the other hand, we may simply choose to allow F to take complex values and replace R4 by C4. This turns out to be a very convenient thing to do and even the most natural thing to do when we study contour integral formulae, in a moment. It may seem rather strange physically, but even then it is a wise move, especially if we have quantum mechanics in mind — there we are actually interested in the analytic continuation of fields from R4 to tube domains in C4. There is also the bonus of not having to distinguish between different signatures for g. So, from now on, we work over C and refer to four dimensional complex Euclidean space (with its flat holomorphic metric) as affine Minkowski space.
The second observation is that not all conformal diffeomorphisms of affine Minkowksi space are well defined everywhere. For example, if x, b [member of] C4 then the mapping
[MATHEMATICAL EXPRESSION OMITTED]
is conformal but not defined on the light cone of the point -b/
b
2. To rectify this we need to compactify affine Minkowski space. This is very similar to forming the Riemann sphere from C so that fractional linear (Möbius) transforms are globally defined. We will do this in detail in a later chapter; it turns out that the right choice is the Grassmannian Gr2( C4) of all two dimensional subspaces of C4. The embedding is given by sending the point x = (x0, x1, x2, x3) of affine Minkowski space to the subspace spanned by the vectors
(1) [MATHEMATICAL EXPRESSION OMITTED]
Denote the image of x in Gr2(C4) by x. We shall refer to the resulting conformal compactification as Minkowski space. Then the global conformal motions of Minkowski space can be realized as the group SL(4,C) (modulo its centre) with its natural action. Indeed, consider the matrix
[MATHEMATICAL EXPRESSION OMITTED]
Applying this to the two vectors representing x yields the image of y in Gr2(C4).
Homogeneous bundles on Minkowski space
This brings us to our third observation — the two-form F is a section of a homogeneous vector bundle on Minkowski space and the decomposition given above is its reduction into its irreducible components. Let us briefly recall the natural bundles on Gr2(C4). The simplest is the so–called tautological bundle, S', whose fibre S'x at x is the two dimensional subspace x [subset] C4 itself. Similarly, there is the quotient bundle S whose fibre Sx at x is C4/x. S', S are the spinor bundles on Minkowski space. In Penrose's abstract index notation [44,127] they are denoted
S' = OA' and S = OA.
Their duals are denoted
S'* = OA' and S* = OA.
(Thus the natural pairing between dual bundles is achieved by contraction between an upper and lower index.) Both S and S' are homogeneous bundles — this means that the action of SL(4,C) on Minkowski space lifts to an action on sections of these bundles. This is easy to see, since any element of SL(4,C) mapping x to y in Minkowski space is by definition a map sending S'x to S'y. Furthermore, both bundles are irreducible in the sense that the isotropy group of x acts irreducibly on Sx, S'x. Put another way, neither contains a proper homogeneous subbundle. Bundles formed from S, S' by taking tensor products, direct sums, etc., are also homogeneous, in the obvious way.
Now it is a standard fact that on any Grassmannian the tangent bundle is the tensor product of the quotient bundle and the dual of the tautological bundle; so the tangent bundle of Minkowski space is
Θ = S [cross product] S'* = OAA'
and the cotangent bundle, or bundle of one-forms, is
Ω1 = S* [cross product] S' = OAA'.
From this it is easy to compute that two-forms are sections of
[MATHEMATICAL EXPRESSION OMITTED]
where [??]kS' indicates the kth symmetric power of S'. The bundle L = [conjunction]2S is called the determinant line bundle on Minkowski space. It is convenient to fix an element of [conjunction]4C4 so that we can identify L = [conjunction]2S'*. In the notation of [44] L = O]1] and L* = [conjunction]2S* = O]-1]. Then
Ω2 = O(AB)[-1][direct sum]O(A'B')[-1]
(where (A'B') indicates that the enclosed indices are to be symmetrized). This gives the decomposition F = φ + [??]. Maxwell's equations become
[nabla]AA'φAB = 0 and [nabla]A'A[??]A'B' = 0.
To write these equations we have had to choose a metric locally on Minkowski space and form the Levi–Civita connection [nabla]AA' on spinors. We must choose a metric in the conformal class of metrics. To see what this means, notice that a metric must be a section of
(2) [MATHEMATICAL EXPRESSION OMITTED]
and is in the conformal class if its projection onto the first factor is zero. Such a metric must have the form
gab = εAB[??]A'B'
where [epsion]AB and [??]A'B' are antisymmetric and each is a square root of gab (noting [conjunction]2S* [congruent to] [conjunction]2S'). Let εAB and [??]A'B' be their inverses. [nabla]AA' is defined by the requirement that it be torsion free and preserve both ε's; define [nabla]AA' = εAB[nabla]BA' and [nabla]A'A similarly. The fact that Maxwell's equations are conformally invariant corresponds to the fact that the operators
φAB -> [nabla]AA'φAB and [??]A'B' -> [nabla]A'A[??]A'B'
do not depend on the metric g, but only on the decomposition (2).
Penrose's contour integrals
Consider the second of these equations. Penrose has given a contour integral formula for its solutions [119,128]:
[MATHEMATICAL EXPRESSION OMITTED]
To interpret this formula, let x be fixed and let πA' [member of] S'x determine a vector z = η(π, x) [member of] C4. f is a holomorphic function on an appropriate region of C4 which is homogeneous of degree -4 so that f(λz) = λ-4f(z) for λ [member of] C. Let πD' = εD'E' πE'. It is easy to see that the integrand is independent of the scale of π (because of the homogeneity of f.) It is therefore well defined on a domain contained in the projective line PS'x of one dimensional subspaces of x. By requiring f to have appropriately situated singularities, we may suppose that this domain is not simply connected and choose to evaluate the integral over a non-trivial contour. We may also suppose that this prescription may be carried out smoothly as we vary x.
To check that [??] is a solution we confine ourselves to affine Minkowksi space X. Then [nabla]AA' = [partial derivative]/[partial derivative]xAA', where xAA' is the matrix
[MATHEMATICAL EXPRESSION OMITTED]
and the index A labels rows whilst A' labels columns. Following (1), x is the column span of
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
It follows that we may write
[partial derivative]/[partial derivative]xCC'f(z) = πC'fC(z),
and so
[MATHEMATICAL EXPRESSION OMITTED]
(by the antisymmetry of εA'C').
We required that f be homogeneous of degree -4; it is clear from this calculation that if f is homogeneous of degree -n - 2 and if
[MATHEMATICAL EXPRESSION OMITTED]
then
[MATHEMATICAL EXPRESSION OMITTED]
where ψ [member of] [??]nS' [cross product] L* = O(A'B' ...C')[-1]. Solutions of these equations are called free zero rest mass fields of helicity n/2. The operator
[MATHEMATICAL EXPRESSION OMITTED]
is conformally invariant and is nothing more than the Dirac–Weyl operator of helicity n/2.
Let us try to give a geometrical interpretation of these integrals. Notice that f should be thought of as a section of a homogeneous line bundle over CP3. The natural homogeneous line bundle on CP3 is again a tautological bundle, H, whose fibre at a point is the one dimensional subspace of C4 specified by that point. We claim that if f is homogeneous of degree -1 on C4 then f determines a section of H. This is easy to see: for Z [member of] C4 consider f(Z)Z [member of] C4. This is constant along any one dimensional subspace of C4 and defines an element of that subspace. Moreover, if f is homogeneous of degree -k< 0 then f defines a section of H[cross product]k. (This bundle is usually denoted by O(-k).)
Notice also that f may not be defined over all of CP3, for otherwise, by Cauchy's theorem, ψ would be zero. To avoid this, f should have singularities arranged in such a way that f(η(π, x)) is non-singular over, say, an annulus in PS'x for each x [member of] X. There is also a natural freedom to change f without affecting [psu]; by Cauchy's theorem, again, if we add to f a function [??] with [??](η(π, x)) non-singular over a contractible region of PS'x for each x [member of] X, ψ is unchanged.
Penrose recognized that this freedom is exactly the freedom of a Cech representative of a first cohomology class with values in O(-n-2) defined over the region Z in CP3 swept out by all the lines PS'x. So contour integration gives a map
P : H1 (Z, O(-n - 2) -> zero rest mass fields ψA'B' ...C' on X.
(It has to be checked that the domain of P is all of the cohomology group.)
We shall see that P is an isomorphism. The general machine which has been developed to prove this is the Penrose transform. We can construct this machine from the contour integral formula by trying to give it a geometrical interpretation.
(Continues...)
Excerpted from The Penrose Transform by Robert J. Baston, Michael G. Eastwood. Copyright © 2016 Robert J. Baston and Michael G. Eastwood. Excerpted by permission of Dover Publications, Inc..
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