Memoirs 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) MEMOIRS of the American Mathematical Society SUBMISSION. This journal is designed particylarly for long research papers (and groups of cognate papers) in pure and applied mathematics. The papers, in general, are longer than those in the TRANSACTIONS of the American Mathematical Society, with which it shares an editorial committee. Mathematical papers intended for publication in the Memoirs should be addressed to one of the editors: Ordinary differential equations, partial differential equations, and applied mathematics to JOEL A. SMOLLER. 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Box 1571. Annex Station. Providence, Rl 02901- 9930. All orders must be accompanied by payment. Other correspondence should be addressed to Box 6248. Providence. Rl 02940. MEMOIRS of the American Mathematical Society (ISSN 0065-9266) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street. Providence. Rhode Island 02904. Second Class postage paid at Provi dence, Rhode Island 02940. Postmaster: Send address changes to Memoirs of the American Mathematical Society. American Mathematical Society. Box 6248. Providence. Rl 02940. Copyright © 1987. American Mathematical Society. All rights reserved. Information on Copying and Reprinting can be found at the back of this journal. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. © 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,