Table Of ContentMemoirs of the American Mathematical Society
Number 358
Robert Pool
Yang-Mills fields
and extension theory
Published by the
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island, USA
January 1987 • Volume 65 • Number 358 (first of 5 numbers)
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TABLE OF CONTENTS
Page
I. Introduction 1
II. Preliminaries 3
1. The Basic Fibrations 3
2. Minkowski Space 6
3. Sheaves on M 7
4. Further Notation 8
5. Local Coordinates 10
III. Basic Ambitwistor Results 12
1. Data on W 12
2. Vector bundles on J\ 14
3. The Topology of the Mapping a 14
4. The Relative deRham Sequence on E 15
5. Direct Image Sheaves 21
6. Direct Image Sheaves for Vector Bundles 23
IV. The Penrose Transform and the Ward Correspondence 24
1. The Ambitwistor Transform 24
2. The Generalized Ward Correspondence 31
V. Extensions of Bundles Over I\ 33
1. Preliminaries 33
2. Extensions of Bundles Over H 35
3. Proof of Lemma V.l 40
4. Curvature and Current of a Yang-Mills Field 54
Bibliography 62
ABSTRACT
Solutions to field equations on Minkowski space correspond to elements
of cohomology groups over twistor space - complex projective 3-space - and
to elements of cohomology groups over ambitwistor space - a compact hyper-
surface in the product of twistor space with its dual. Solutions that corres
pond to geometric objects on twistor space in general satisfy some self-
duality conditions, but these constraints vanish when the generalization to
ambitwistor space is made.
In particular, Yang-Mills fields can be shown to correspond with vector
bundles over these various twistor spaces. It is hoped that eventually all of
the theory of Yang-Mills field on Minkowski space can be transferred to
geometric statements on ambitwistor space.
Here Yang-Mills field curvature and current are described as elements of
cohomology groups on ambitwistor space, and it is shown that the Yang-Mills
current corresponds to the third-order obstruction to extending the corres
ponding vector bundle. There is also a description of the Yang-Mills field's
action density. In addition, several groups of solutions to various differ
ential equations on Miskowski space are characterized as elements of various
cohomology classes over ambitwistor space.
Subject classification: 32 L 25
Library of Congress Cataloging-in-Publication Data
Pool, Robert, 1955-
Yang-Mills fields and extension theory.
(Memoirs of the American Mathematical Society,
ISSN 0065-9266; no. 358)
"January 1987, volume 65, number 358 (first of
5 numbers)."
Bibliography: p.
1. Functions of several complex variables.
2. Twistor theory. 3. Fiber spaces (Mathematics)
4. Field extensions (Mathematics) I. Title. II. Series.
QA3.A57 no. 358 [QA331] 510s [515.9'4] 86-28809
ISBN 0-8218-2422-8
i v
I. INTRODUCTION
In [20], as part of Penrose's "twistor programme", Ward first described
how to encode self-dual gauge fields on complex Minkowski space as vector
bundles on dual projective twistor space. A generalization due independently
to Isenberg-Yasskin-Green ([12]) and Witten ([25]) removed the restriction of
self-duality by dealing with vector bundles on the space of complex null lines
in complex Minkowski space, known as ambitwistor space, rather than on projec
tive twistor space. This generalization provided the capacity for working
with gauge fields with nonzero current: self-dual gauge fields are necessarily
current!ess. Much of the twistor programme can be transferred to ambitwistor
space with a similar increased generality; recent work on ambitwistors has
dealt with, for instance, massive fields ([4]) and a generalization of the
nonlinear graviton ([11]).
It was first pointed out in [9] that extension theory for vector bundles
provides a valuable format for describing those objects on ambitwistor space
that correspond under the generalized Ward correspondence to various gauge
field quantities on Minkowski space. In particular, considering ambitwistor
space as a hypersurface in a certain compact complex manifold, the obstruction
to extending the given vector bundle to a third-order neighborhood corresponds
precisely to the axial current of the associated Yang-Mills field. This
strengthened the result of [12] and [25] that the vanishing of the obstruction
corresponded to the vanishing of the current.
In keeping with the philosophy of the twistor programme, one desires to
describe a theory on ambitwistor space which will take the place of gauge
theory on Minkowski space. First steps toward this end were taken in [14]
9
where it was indicated how to interpret the curvature of a Yang-Mills field
as an extension-theoretic object on ambitwistor space; further progress was
made with the above-mentioned identification of the current. This paper
Received by the editors January 25, 1982.
1
2 ROBERT POOL
presents a careful exposition of the framework for the Yang-Mills theory on
ambitwistor space. Included are complete descriptions of the Yang-Mills
curvature and current as elements of cohomology classes on ambitwistor space,
including a proof of the result announced but not proved in [9] concerning
the equivalence of the current with third-order obstruction to extension.
The description of the curvature is applied to obtain a description of the
action density of a Yang-Mills field. In addition to the work on Yang-Mills
fields it is shown how to extend many of the results of [5] to the ambitwistor
setting.
In Chapter II the basic background and notation will be introduced with
references to the relevant literature. Several double fibrations of compact
complex manifolds will be introduced with the most important being
IG
/ \
where M is compactified complex Minkowski space, J\ denotes ambitwistor space
and E is a flag manifold serving as the correspondence space. In Chapter
III we examine how geometric objects on J\ can give rise to solutions of
differential equations on W: Briefly, exterior differentiation along the
fibres of a gives rise to various differential operators on W, and sections
of sheaves which are pulled back from 1A to IG and then pushed down to 1M
automatically are constant with respect to these operators. The necessary
tools include an isomorphism theorem concerning pull backs along a of cohomo
logy groups, a relative deRham sequence on IG, direct images of sheaves along
p, and a spectral sequence relating cohomology on £ to cohomology on 1M.
Generalizations of the Penrose transform ([16]) and the Ward correspondence
([20]) are set forth in Chapter IV. Chapter V is devoted to the description
of the curvature and current of a Yang-Mills field in terms of extension
theory on ambitwistor space. The extension theory is developed carefully
and a vital lemma is proved in detail, and then constructive proofs of the
identification of the curvature and current with certain cohomology classes on
W are given.
II. PRELIMINARIES
The purpose of this chapter is to introduce the background material nec
essary to the development of the ideas in this paper and to fix the notation
which shall be used throughout. For the most part we shall follow the notation
of [5], which is the basic reference necessary for an understanding of the con
cepts presented here. There will also be included a summary of the local
coordinates which shall be used on the various complex manifolds which appear.
1. The Basic Fibrations. The fundamental geometric data which shall concern
us here are contained in several double fibrations of (open subsets of) cer
tain complex manifolds. Let TT denote twistor space, a four-dimensional com
plex vector space equipped with a nondegenerate Hermitian bilinear form $
of signature +,+,-,- (cf. [22] [23]). For the purposes of this paper, we
9
will choose coordinates Z = (Z ,Z ,Z ,Z ) on T such that o has the form
$(Za) = Z°Z2 + z ¥ + z ¥ + Z3!1.
We denote by TT* the dual to TT and choose dual coordinates W = (W ,W,,
Q
Wp,W ) so that the induced form $ on TT* has the form
3
$(w ) = w w + w w + w W + w ^ .
Q 2 1 3 2 0
The alternate notation
/A x , 0 1 x . ,70 71 72 73N
(u> ,7^,) = (O) ,U> ,TTg,,TT^,) 1= (Z ,Z ,Z ,Z )
(n^') = (n ,n £°V) : = (W,WW,W)
0 r 0 r 2 3
will often be more convenient, and will be used interchangeably with the
first.
A flag manifold on a complex vector space V is a compact complex mani
fold which consists of nested sets of linear subspaces of V. That is, if
1 1 h K i? < ••• < ^n < ^m ^ are inte9ers> ^en a ^a9 manifold of V is
defined by
3
4 ROBERT POOL
F. . . (V) := {L. c L c , c L : L. is a (complex) linear
subspace of V of dimension i.}.
j
For the case n = 1, F..(V) is the Grassmannian of i-planes in V, and if
i = 1 as well, then F-j(V) = P(V), the projective space of complex lines in
V. See [24] for details and a proof that F. . . (V) is actually a com-
pact complex manifold.
The twistor correspondence is the following double fibration of flag
manifolds on TT:
F := F (TT)
12
/ \
T
P r-F^TT) ^ M %F (¥).
2
The fibrations y and v are defined by
y(L L ) = L
r 2 1
v(L-| ,L) = L ,
2 2
and the correspondence T is defined by
x(Z) = voy-^Z) =P (C),
2
T'^Z) = yov-](z) ^P^C).
The manifold P, projective twistor space, is isomorphic to PJc), by the
comments on flag manifolds above. For details on the twistor correspondence,
see [22].
The dual twistor correspondence is
F* :- F Cm
23
P* :=F C1T) -*" W =F (IT)
3 2
where \i*, v*, and T* are defined analagously to the twistor case. Note
that there is a natural isomorphism P* =P(JT ).
This work will be concerned mainly with the ambitwistor transform:
B := F,„ar)
/ . " \
;
n :-F (IT) -- w = F (IT).
13 2
YANG-MILLS FIELDS AND EXTENSION THEORY 5
Here T(X) =P,(C) and T" (X) = P, xp (Normally there will be no chance
of confusion arising from the same notation T being used for the correspon
dence mapping in the twistor case as well as in the ambitwistor case because
the context will distinguish them. When the need arises to distinguish them
explicitly, the ambitwistor correspondence will be written with a subscript J\,
T^.) It is important to note that ft is naturally embedded in P xp*; that
is, we have
P\ = {(L L ) € P xp* L c L L
19 3 : 1 3
In homogeneous twistor coordinates on P xp*,
» = {([Za]> [W]) : ZaW = 0},
a a
as can be easily checked; thus P\ is a quadratic hypersurface in P xP*.
We will refer to F\ as ambitwistor space. (The reason for this terminology
lies in the fact that twistor constructions seem to have an inherent handed-
ness--in the sense of physics—either right-handed or left-handed, while the
more general constructions are ambidextrous, so to speak.) We note too that
G =^i23 can &e considered to be a codimension-four submanifold of F x F* =
F-J2 x F 3 in a natural way.
2
We can also consider J\ as belonging to another double fibration in the
following way. If p, p are the projections of P x P* onto P and P*,
respectively, we define
IT := p| R : R +P
* := p| ^ : A -*P*.
That is, if L c L defines a point in A, then
TT(L1 ,L3) = L-j,
ft(LL ) = L .
r 3 3
The double fibration appears then as
P P*
where the diagram is commutative. A similar relationship holds for the various
correspondence spaces,
