Further advances in twistor theory / Volume I: The Penrose transform and its applications 0582004667, 9780582004665, 0470216557, 9780470216552

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L J Mason & L P Hughston mum.“ \cu ('nllcgc. (')\|nri|x‘l{1 . H- I'I'c :('.I\'I):.'IIIIH'I. i..,I'..1lIH-trn .\'

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1.3.12 The rwislor transform and t‘uufornmlly invariani uporaturs 1.3.13 On Mlclmnl Murray's Iwismr currt-spumlr-ncv

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Predictions of Pt lwistur lllUIII'I 1'1)! lI-phimh

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Foreword (H; [((Jgtl

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Preface

In 1976 a groupofus at the Mathematical Institute. Oxford. began to circulate from time to time an inl'onnal publication called Tuistor Neu slcttcr. The accumulated material gained sufficient interest to warrant its publication three years later in 1979 in a volume called showers in minor theory, edited by R.S.Ward and one of us iL.i‘.ll.). .‘s'ou. a further ten years have passed. and Twistor Newsletter has continued to appear on a regular basis. As a consequence a considerable mass of twistor lore has been stockpiled. Before the arnval ol‘tlte FIN Advances in nifsmr tlimri relatively little tvvistor material was finding1 its way into print. and T\\l:sl0l’ Newsletter ivas rather like an underground journal

Since then the picture has

changed. but Twistor Newsletter has continued to serve as a vehicle tor the informal circulation and

communication of mathematical ideas. unfettered tlor better or \vorsei h} the requirements of a fonnal presentation. the delays of fC!L‘fL‘C.\'.Jf.1lllel editors. and proper publication. But silty bring out ten years work worth of material. if much of it. on the one hand. has already now found its way into publication iii one toriii or another— and. on the other hand. quite a lot of it

hastiovv perhaps been superseded! First we touch agree \iith Goethe that. "Das bestc. was wir von dcr Gesclticte habctt. ist dcr Entliiisiasmus. den sic crrcgl" ttiic greatest [coin that we derive from

history is the eritliasi'usm that it arutiit'a' in us i. Moreover. approximately half the maten'al in these inlumes has not appeared to the literature. and often these articles are of substantial importance in understanding the subsequent development of the field; cl. lor example "l'he anti-sell dual Coulomb licld's noii- Hausdorlf tvvistor space tel (1,2).

The articles are mostlg. mitten in an informal and easy to read style. laying out the actual motivation for the ideas and addressing tvhat the author considers to be the important issues iii the area

However. many applications ol tutstor theory have non acquired the status iii a complete and

self-contained collection oi results for Much I! is often more appropriate to suppress motivation and speculation in favour of mathematical rigour and a slightly more formal 3|}lc‘ of presentation: for

csiunple. the application oftuistortheor} to representation theory l§l.3.15' 1M. and the treatment of fields with sources (chapter Lot. Nevertheless. the tivistor programme with ll\ aim of resolving problemsin basic physics remains unfultillcd. It is important that the motivation for the ideas. and the successes and difficulties encountered in reali/iiig them. are available lor those who are interested in these tundmuetttal problcttis trom physics. liven as far as the sell-contained applications of tvvistor

theory are eoneemcd. the reader may find the minimal aniclcs in these volumes an easier route into the subject than the loitgei and more rigorous accounts in the literature. .-\ lunhcr attraction oi this collection. vie tcel. l\' that the whole :s indeed greater titan its pans.

The articles iii each chapter trace the developitient ol the siibiect ol the chapter. pmvtding bath a histoij. of the piogrcss. and also a primer tor the beginner

('hapter l.l contains a discussion of the

Overall backgiound and mathematical formalism and each subsequent chapter has been supplemented

':;- an introduction uhieh should be sultieient to give the non—expert and beginning graduate student a

tcothold. Furthermore. we think that there are numerous ideas (or ideas tor itleast lurking in these rages —thougth and alter thoughts — tn;u1_\' never carried to complctzon or fruition.

The

:ei'elopment of a theory can be a gradual process. and it does make sense Irom time to time to return -~ earlier thoughts that may have been temporarily abandoned simply because some crucial idea was

:ming ornot to be tound at the time. We l’eel that these solumes will provide an excellent perspective on the current state of twislor z-sory. The substantial evolution [tom the previous \‘OlUl’llL‘. Adutrtt'cs t'n mister theory. is rminiseent of a remark once made b.) R.P.Fe)nma.n on the nature of scientific progress — "At eut'h

"z‘c'fl'llg it tthvays seems to me that very little progress is made. :\'(’l.'('ri"l('h'.\_\. iftnu look over any estimable length of time. ufew years my. you find it futttitsttt‘ priieress and u H‘ hard to understand

-.. is that am happen it! the some time that nut/tint: is happening

I think that it is something like the

mt}. t'hittib‘ change in the tit) - the) eritti’ttutit fade out here and build up there and than tool. [t'lt'ff .: it ct'hfferertt Cermm things that were brought up in the his! meeting its .sttleei‘stimts come mm focus at realities. Thei- iir'ttg atone nith them other things ithnm ii-htt'h it great iii-it! is ihti'ttssed and which «it heroine rmhttet tritium in tt'.‘¢' rtetr met-mug. " To us it seems that the tunher advtmces itt twistor meory have indeed been very much like that.

We are both pleased to ackntmledge the ltelp of Roger l’enrose. and we would also like to L‘Idllk Tsou Sheung 'l'sun t‘or help u. ith the references and Debby Morgan for her extraordinary effons :n typesetting the omate and baroque notation employed by many twistor theon’sts.

LJ. Mason &- LP. Httghstmt September 19.3“)

-\ note on global structure and cross references.

We have dixided the nevi material into two

'-olumes. Volume I' The l'enri‘ise tramfurm and its (Ippltt‘tltittnS and i'ohune II: Twistors in t'llfl't'd spitee-ti‘rm'mt‘ which this is the first. These two volumes will be referred to in the text as l and It.

Thus by §|,2.3 we mean article three of chapter 2 of volume I'. and by §|l.4.5 ‘8 we mean articles 5 to ‘t in chapter 4 of volume II. We refer to Adi-antes in minor theory as volume 0. Titus by §O.5.t we . ean article 1 in chapterS ol that book.

Chapter 1.1 contains a general discussion ot' the physical aspirations underlying twistor theory and an introduction to the basic turntalistn. The first anicles of each subsequent chapter prm tde both moral” and mathematical baekground to the material eontained to that chapter. mdes and bibliography appear at the end of volume ll.

A comprehensive

Chapter 1 An overview of twistor theory

3.11.1 General lmckground in; 1...]. Mann:

Huger i'enruae's lwislnr liieur)‘ is .3 [Mathematical framework than clarifies and. (iEFpI‘NH uur -:;-Jz:rslandmg of basic physics.

[clean and eonanueliuns from lwislor theory have a wide

.::io:-t_-.- uf itlrplit‘mious in boll! mallzrlnniics and mathematical physics.

The Ime is liml

Misti-r 3|IH(‘I' ~lmultl eventually rr-pim‘e spur-lime as lin- Menu fur physics. Much ul' lwistur liieory i:- cnnrernvd will: re‘l'orumlaliug basic physics in 10mm of

ifructures uII lwistur space. mil-idem

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physio. Ill YWir-IUI spher- thal spare—time can iIu-lf in- rightly regarded as a

:‘erivccl m wrondnry structure.

For linear fields the 'i"enruso lrausl'orm' Irmuluuw fields

0:. spare-lum- In mulpiex analylir ('ulmmuiugy ('iasaeza un :1 I'umplcx lnaniluld: Hxlstor ?;'.H.'l".

’I'ili: transform is in mine ways analogous to a llmlrm trans-form: space-Iiim- l'\'l'lli>

'17:.- represeuled by cumplex projerliw lines in lwislur space. mu] the field is (Jiliilillt'ti it} L!‘.II,‘gTiIiiUll nl' [re-e functions on Iiwsc- lines.

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The trmmlalinu of physics is sulpslunlinliy

in” there are important twismr descriprium uf mlulinus‘ to lin- l|0ll-i||l|'-’ll will

~i-;al Einstein and wlhlunl Yang-Mill.- equnlions “ilifil have Coll>idPrhbiP rumiwmniiml

fliiiry and certainly do suggest llw exisu'nrr nf « more cumpio-Ie picture. ()ilmr oxnmplu-s ul' appiil‘minns nf rwimnr theory in Irmllwnmlical physics include Ilw quasi-lurdi definitiun uf mass in general relativity (see chapter |l.l). the mnurm-Iiun of instanzmu ml 5”" and Ilmuolmles on R3 [Aliyah [979. Ililcllin i032) and lhe eonslrlu‘linu of

ronl'ormal livid tin-mics in [our cliumnsiuus lt'lmpter lull. .-\pplirnlious in pure lunllrmualin include new npprnnciles lo reprewntntiun theory lseo chupien- [.2 and i.3l. non rl‘nili'llilil' 'f'aurhy-llit'lunnn strut-lures (I’I-nrusc- 1933). and the intrmluriinu of powerful nu-tlmrle fur

:he study uf wlf-cluul 4—mruril’ulds and Ixylwr-kahler manifnlds (m- \‘niulne III.

In :nlnlnliml

rnnllwnmlih Iuinlnr thew} inn: provided a unificmimi of ”I“ lllt'lliuds of line tin-um." Hf Integrable- syslmns l\'l"ard i985. Mnmm .i.‘ Spuriing 1951)). As yel. twisl‘ur liurury does not hinge specifically an any new pllyslral proposals.

II is.

2 g 1. An m'rrvic-w of twister theory

l'i\ll|Pl'. a coherent collection of intrigumg and powerful mathematical mvtliods that haw.wplicutiun lo \x-i-lI-acrnptml physiral theories such as general rulatit'ity. vlm'trntnagnn'timi.

and Yang-Mills. Hamiltonian

Au optimistic View is that this is analoguns in rhr di-u-lnpmu-m uf

umchauirs:

originally

it

had

only

mathmuatirnl

applications.

lin'

thu-

lurninlatimi [thllllplt‘d llu- rnrrc-t-t artirulation ol' llll' new physical proposals rI-quiiml |ll quantum llii-ory.

Twister lll"t_|f_\' has bt't‘ll developed with n vivw to thi- nerds of a Ill!‘til’_'.

that combines quantum theory aml grins-ml relativity (although there is still pvrliaps mu litthr input from quantum field lhroryi. so it is liupml that it will prm‘idc- Ill" formalism in

which thr' new physical proposals of a thr-nry of quantum gravity run lu- framm‘l.

The "initiation from phustcs. iii-tail.

I shall allclltpl tn prom-m thr physical motivation in sum-3

A: with an) n'ivntiliv Pnduu'uur. twistnr Hunt} is probably lmst justiliwl in terms

of its cuncrvtr- applications: the motivating factors detailed Iii-hm .iri- instead inimi-lml h.prm'idr- an uwrt'ir-w of tlu- origins and ambitions of tho twistor programme with whirh lltllt‘ll nl' llll'Sf‘ vulumt‘s is cunu-riwd.

it should lu- empliasizi-tl that somt- ul' Ihis inmit'ntii-n

is tiriitatiVo,

Thr usr of complex slmclurr-s. \tat's in physics.

The notion of a n'uiititiuum .1!le in two lltlllt' dim-rum

Space-tinn- is modelled on a four dimr‘usimial continuum «If real nutniu‘rs.

“ha-was quantum mnrlianit‘s rt-quirvs the“ usv of lllP continuum of couipln-x numlu-t- Hdcscrilw probability amplitudes and the cliarartflrization of physical stairs as lay-s m a cumplm. llillu-rl space.

.-\n clruu'utary exaniplv of the relationship between girumrtry and thc- ifumplt-x unmln-rs of quantum mechanics is protidml by tho standard truatmr-nt of non-relativistic quantum muchunii‘al spin. 'l'lii- spin of an oln-rtmn. say. is described by the state:

It > = -\!up> +ultlown> .\.;iEC.

l‘Iarh state dcl'mI-s a utiiquv (.llff‘fllullaclhill alums, u‘liii‘h it is remain that lllt‘ c‘lr-t‘lrlm's spin is aligned. 'l'ltus thr split-ru ul' dirt‘ctions at a llilllll is idr-nlific‘d with thn spaw- ol' such shut-s.

l'his is tlu- spam- of pairs of complex numbers (Ln) under tln- I'lllllVfllF‘ElCt'

l:\.u)~[u.\.u#l. u-EC. "720. Hull} the ray doturminv-d liy IL‘) is rc-Ic-x-uut |.

Thu i-li-nu-nt.ir_\

quantum llll‘l'llillllt‘ii uf spin thrtvlinc ldt‘llllfil'\ tho? rvnl two-splmn- (if ilIrPrtious at a puiut

with HIV lliomann sphere or the i-oi‘nplcx proiu-ctit'u? line. 1wistor spacr generalities this rorrc-spotuh:-nrv.

It is a rumplPx manifold (roznplrt

51.1.1 General barkground| 3

;.-n"..=rtivp it..p:ice. CPJI and gives rise to rent space-time with remarkable economy: one rec;

needs thr- open region PT’ in twister space as a complex manifold in order to

l Err-rinstruct

Minkowskt space M Ingetl‘ter thh its eunformul structure. .

:amfold.

PT’

contain.-

.

.

.

no local information

A: a complex

.

J

theme. locally equivalent

to

C

.

in

many

l

l cite-rent

i'nyst: it

l

ts the global structure thnt

-

fill()\\'.\ one to identify those ohierts

u

-

-

I

|¢:nl-)nt(trplll(' rotnplex protective hnns) that correspond to pulltti 0t Minkowski spare. l

l

Complex lttttttlwrs also r-ntu-r into physics in classes of solutions to various. tic-ht

immtions.

For example. the varntttn equations for a plane-fronted wave in netternl

inmtivity reduce to the t'nntlitiun that one of the metric coeffirients be holntnorphir. this I ,ae be extended: the general solution of a wide Flaws of field equations can be constructed 'frorn freely specified holomorphic funrtions on twistur space.

from the point of view of

tumor theory these field equations reduce to the (”michy-ltit-nutnn equatinns.

l‘his- is not

Fcriy true for linear zero rest mass field» but also for non-linear equations .uteh as the WI!.¢:al Einstein vacuum equations. the self-dual Yang-mills equations and their background :ot tpled zero rest mass fields.

These results already have. wide application tutd hint at the

luntence of a more complete picture in which non-wif-dnal l-Il'ltlfi are described by means of

l{Emotnorphte . geomet ry. l

1'Ttuurdas u grumetn'e framework for quantum gramty.

There are" situations in whii'h it

ltnuld he tlesirnhle for the notion of an event to hernme ill defined whilst still living nhh- to 1.51»; and answer physirnl questions.

The l’enrose-ltzm'king singularity theorems imply lhnl

01.! universe contains singularittes where space-time geometry breaks down so that ennuc'mntutn theory of gravity must take over.

At length scales of the nrrlm nt’ 10—33-311. the-

Plane}: length. quantum l'lttrtualions in the metrir filiuuld be sufficiently strong to fort-e us an ahandon our conventional notion of Slttlcevlltttc geometry.

Even at length «ah-z. ot' the

writer of Ill-”PHI. attempts to localise one particle will result in the creation of many more making it effectively impossible to give an operational definition of metrical geometry at Li'h distnnres.

A: Penroae tlD-‘Stil puts it. "\\'hile it may he that only at lit-u Cltl i4 it

tic-:‘rfiut‘y to hinder a ilv-ucription of spare-time rndienlly removed from that of :1 manifold. 33‘ view was (and still is) that even at the much larger levels of elementary particles. (It

r-ethups atoms. Where quantum behaviour holds sway. the filflllllfll‘tl space-little descriptions :nve erased to be the most physically appropriate ones. and some other pirtttrr- of reality. though at

fruitful".

that

level equivalent

to the spacedinw one. should prove to he the more

I'l'lll‘tmt' suggests that one should take a positive view on the inaulequary of

-;Itt(‘f_ tum. pictures for the understanding ot quantum phenomena. and try to provide :1 gm-ture of objective reality . albeit one perhaps rndicnlly different from the Usual one.

4 . 1. An overview of twistur theory

The points of space-time. events. are realized physically in an indirect la:hion~an

('\'t'til is defined as the time and location of the interaction of two particles.

Geometry

should In- huilt from units v.hich are in some sense idealizations of particles.

.»\ twistor

provides the appropriate idealization of a particle: classically a tuistor describes the linear and angular momentum structure of a mass-less particle (with spin).

.-\ space time event

can then be defined as the two sphere of massles: particles that intersect. there. llie usual \‘lt‘\\‘ concerning the nature of quantum gravity is that one can apply some

standard quantization procedure to the. field equations of general relativity. This leads to a picture in which space-time events are well defined in the quantum theory. but the metric. and therefore the light-cone. is subject to quantum flurtuations and is therefore ill-defined.

l'his approach

leads to «_n':-t'\\'lleltriing technical and interpretational problems.

For

example. the definition of spinnrs twhich are necessary for the description of fermion-ti |~ tied to the light-cone and cannot conformal metric.

be made independently in any sensible way of a

Spinors are more fundamental than world vectors. and m the. null

directions should retain their identity and not he 'smeared'.

Instead. according to the

philosophy of the twiatur programme. the points of spaci-«tittie should be quantized—the points of twistor space retain their identity (and so in particular mill directions are well defined). but space-time puitth are derived objects so that they themselves can hecome ~nlijeet to lll'iwnlu-rg untertainty relations and quantum fluctuations.

The divergences in quantum field theory ari~e from the point-point interactions of quantum fields.

If space-time eVents can he summed. it seems likely that these divergences

might he naturally regularized.

(Indeed. there is some evidence for this from the theory of

twislur diagrams: see chapter LS). .-\t the classical level, a treatment of space-time events as secondary may facilitate the study of the structure and classification of space-time singularities.

A neighlmurlinml of a

«'ingularity would correspond to a region in twistor space in which the objects that give rise to spat‘e-llltlt' evt'ttls censu- In be defined.

It may he [tUbbll‘lll‘ to understand physical

properties of the ‘singular' regions by referring to the appropriate region in twistor space. According to this View. in lll" neighbourhood ul a singularity. space-tinn- ceases to be a

well defined notion: but the twistor space lives on. and physical properties lenergy. etc.) can he discussed in terms of it. The successes of quantum I“l'?|'ll'0ll_\'lliillllf§ indicate tha' space-time based physic: is

likely to he accurate down to at least Ill—"‘ctn. and pussrlily further towards quantum gravity length

scales.

At

these and

larger length

Sl'Hll'h. twislor theory should

mathematically consistent with conventional space-time physics.

lu-

Nevertlielesx. new insight

l.‘ perhaps l'l'qlllh'tl at length scales larger than the Planck length.

Consider length sl'alt‘\

§l.l.l General hackgrouudu 5

c—r’ the order of ltl'mern.

Since we are ten orders of magnitude larger than quantum

gravity length scales. spacrutime geometry should still provide an accurate description of ptyairs.

However. a prntuu is now a vast nebula. of magnitude l0Ill length units.

\Vith

tha picture it scents difficult to understand the prolon'a behaviour as an individual firtit‘lC—ilu quantized spin and charge. and its precise rest lllflss.

Even more difficult is

yrhaps its indistinguishability from all other protons implied by the. Pauli exclusion ;ftr.ciple.

One may hope that tnistor theory provides the tools to resolve these conceptual

chfirnltitfi.

l . . . . , J: man}; approaches to quantum gravity .1 graviton l5 represented by a linear 5P|l1-|“'t) field i . . . . . . it: Minknwskt spare. However if a graviton ts meant to represent a physical quantum of l

gravity, it

licc-c-truclion

must

carry

a

finite amount

of curvature.

l’enrnse‘s

non-linear gravitnn

offers a definition lalheit tentative) of a positive frequency graviton in a

i {bunt} cigt‘t‘tal‘itll‘ carrying a finite amount of curvature: such gravitnns are obtained |._v

lieforming a region Pl ' in twistor space.

‘l‘hese ideas have not as yet been developed into

proper theory ~hut the underlying ideas are robust and highly stlggt'ntit'e.

The positive frequency condition.

()ne of the first achievements of twistor theory was to

’rttide a geometrir picture of the positive frequency condition on lllflsaless fields.

The

-.sion of fields into their pusitiw: and negative frequency parts is an essential sic-p in the unsitinn from Classical to quantum field theory.

It was hoped that one could provide :1

:her dimensional analogue of the splitting of functions on the real line in C into a part .uniorphic on the upper half plane and one on the lower half plane. Twistor space provides the appropriate generalization.

Fields on real Mittl-towski span»

rorrespond lo cohmnulogv classes on a real coditncnsion one hypersurfaee PM in twistor

ace PT.

PN divides PT into two halves. Pf+ and PT'. PN being their common

.ndary.

Such a cohomnlogv class can then be split uniquely into a positive frequency

.-i which extends over i'f' and a negative frequency part whirh extends Itt‘t'r PT'. (PM. T' and PT‘ are I't‘pr.‘Cll\'f‘l_\

the span-.5 of spin zero. 4’: and

—h helicity Inn‘sIo-s,‘

quit-t. 1|

{fiftfl‘flluf

invariance.

[\K'ISIUI'

theory

-_:'..rniali_v invariant min-ets of physics. :Ll cour- structurr- nf space-time.

is a

natural

vehicle for

the description

nl‘

This gives precedence to null getillr'~it‘s and tlm

rI‘hese are more fundamental than the metric itrueture

., time-like and space-like geodesics: only the space of null geodesics can be given a mph-x structure (or more strictly a ('anchy-Riemann structure).

Massless fields (including Maxwell theory and linear gravity) are conformally invariant. In the standard model. the basic fields (except the. lliggs boson} are all ntassless in the litnit that interactions are switched off. and pick up their mass via their interaction with the lliggs field. Planck mass.

Indeed. the finite masses they do haw: are Very small compnred to the The mechanism by which conformal symmetry is broken is. surely. a

fundamental part of physics and an understanding of this mechanism would give insight into the nature of rest muss. its relation to gravity nnd—wc tnny hope—its spectrum under

quantization. conformal

Twister theory provides a conformally invariant formalism. so that any

symmetry

breaking is

made explicit—reduction

to

Poincare

invariance on

twistor space must be effected by introducing the ‘inftnity twistor'.

Special conmdcrahons:

Many of the triathemntical methods of twister theory can he

extended to other signatures and dimensions—indeed in many different ways. less. certain

features of twistor theory are special

Neverthe-

to Lorentzian signature in four

dimensions. and one may regard these features as sufficiently ‘essential' as to provide an explanation of the dimension

and

signature.

of space-tin‘ie.

First.

the

nonvlinenr

constructions do not appear to have nnulogties appropriate for physics in space-time dimension greater than four.

Second. the Lorentz signature of space-time arises from the

real codimension one hyper-surface PM in twister space.

llypersurfaces of this type emerge

as the natural boundaries of of the domain of definition of the cohnniology classes and nonlinear construction: used to describe physical fields on twister space. Fixing twister space as primary. rather than dunl twistor space. introduces [cf-right asymmetry.

Therefore there is the possibility of providing some explanation of parity

violating physics.

This however turns out not to he a bonus for twistor theory. hut instead

one of its most unyielding problems. it has so far heen difficult to understand satisfactorily the

wide-scale

left-right

symmetry

present

in

physics

lsuch

as

in

gravitation

and

electromagnetism) in the context of twister theory. Originally. ideas from spin networks Were inl'ltwnliad in the development of twister tlmnry.

However. their relationship to twistnr theory him since become less clear.

These

ideas are of more than mere historical interest and the reader is referred to l‘enmse tlSJTla). (lih'lh). 11972) and tlilTQ) for detailed discussions of spin networks and their

possible relationship to twister theory.

The present status of (winter theory as an approach to physics. "l‘hesr: aspirations are still it long Way from being rcnliZed. and in the most part. the twister formulation is equivalent to the standard space-time formulation.

Tutu notable exceptions are the twistor particle

§l.1.2 Mathematical background I 7

rugratntmu “here internal \yntmelrir's of elementary particles arise from ”If geometry Hf

{tr twistnr description. and twistor diagram theory in which infrared divergences .muturally regularized by inclusion nt' a deviation from the ittCidence relations [which s'rnduces uncertainty into the Incation of space-time evrntst.

The status of these ideas

remains. hnwr-vnr. far from rh-ar and then- 15 still little contact with gravitatiun. 'I‘wistur theory provides an excellent treatment of mnsslt-ss free fields (chapters l.'.2. L3

Ltd Lt).

Massive particle:- and their internal asymmetries are dir-rmsed in the twistur

puticie programme (chapter l.-l].

1hr extension of these results In non-linear firhls is still

:xuhla-nmtic: there is only a mrisfnctory troatuwnt of anti-sdfrlunl Yang-Mills fields and utti-self-dnnl gravitation (we chapters ll.'..’-II.."II.

One of the must tliffieull problems in

Tut-ssh)! tlu-ury is to generalizi- these results to general Yang-Mills and gravitational 11mm;

1:0.- first

step towards this is the so called 'googly' prolilrm [chapter ll.'.’,t.

Approaches to interactions-

There are lill'l'l"

Background coupling can be introduced in Hue non-linear

muster constructions where they exist “.0. for the case where the hackgmund field is antiMif—dunl).

Fields with snurr‘r‘s are incorporated by the introduction of ttutt-llatrstlnrt't'

tuistor spaces tclmptcr 1.0).

Scattering amplitudes as dt'st‘rlbt‘d by Feynman diagrmns :m-

calcnlaled from twistnr diagrams l'chnpter L5].

Twistur ideas can he intrm‘lnrr'd into full

u-nernl relativity by means of hypc-nurfacn uusmrs (chapter ”.31 and spare» ol' cutttplex ,—_ .ll geodesics- {chapter IL“. at twisturh nnd spinnrs in

The definition of quasi-local nuns (chapter NJ), the tlmur} higher dimensions (chapters l.'..’ and

Lil). and

the twi~tnr

approaches to representation: thenry (chapter LB). are some examples ul' the application uf "-xihtot idea: to prublem: in physics and mathematics Ism- also chapter ”.5 and chnptu-r l.-l

Fur fnrt her .1 plllit‘rli inns).

El.l.2 Mathematical background by L. I. .ttuwn

l'hu- mutation and mathematical approach used in twialnr theory.- mny hr- nnfat‘nilinr ”I part to some readers so we shall attempt to summariw thr- background material that will h4assumed.

I. Mmkouwskr space and two-component .spanors.

Mink-maid space M' is a real t'uur

dimensional affine mace (isomorphic to R‘) with the flat Lorentzinn metric:

8 | 1. An overview of twistor theory

d9: rlabdz“dr°=(d1:°)’—(dr‘)2—(dx”)'—(d1:3)3.

Here 12=(r°.rl.r2,:ra) are the standard coordinates on R4 and the Einstein summation convention is used. as it will be throughout these volumes. tangent space to M.

The vector space Tcl is the

lndices will be raised and lowered using ”ab and its inverse I)” where

nabnbc=6§ (63 is the Kronecker delta). An abstract index A,B or a on some quantity EA or vCl is regarded as a label signifying the vector space SA or T“, to which it belongs.

Only when we take a basis of the

appropriate vector space. say €Ad=(EAO,£A1) for the 2-dimensional vector space Sm can we translate the abstract indices into concrete indices which take on numerical values: £d=(£°,£l)=(£‘4€A°.EAE_41)=£A€AA. way as if the indices were concrete. abstract.

Notationally.

concrete

distinguished by underlining.

Tensor algebra is performed in precisely the same Unless otherwise stated. indices will be taken to be

indices.

which

do

take

on

numerical

values,

are

Square brackets around a clump of indices denotes skew

symmetrization and round brackets denotes symmetrization. The Lorentz group 0(1.3) is the group of group of linear transformations of the

tangent space T“ to M which preserve the inner product Val/b1)“. If we code V2 as the 2x2 matrix:

v°+v3

v1+iv2

v‘—z'v'~’

W—va

l

I

we have 2del(V5‘fl )= Val/9.

Premultiplying V44

by

unit determinant complex 2x2

matrices, and postmultiplying by the complex conjugate matrices induces all the Lorentz group. This yields the isomorphism of the complexified Lorentz group

SO(4.C)ESL(2,C)xSL(2.C)'

or. alternatively. the ‘2-1 covering of the real identity connected component of the Lorentz

group $000.3) by SL(2.C). The spin spaces. SA and 5“,, are the two dimensional complex vector spaces on which SL(2,C) and its complex conjugate act. We can write:

r ' 2del(VAA' )=€A35AIBI‘”"‘ vBB I

§I.l.2 Mathematical background I 9

shore 5AB=E[AB] and in the basis above 501:1 (similarly foreA,B,). CAB and 5.43 are :e Levi-Civita spinors. Spinor indices can be raised and lowered using 5A3 and its inverse

,43 which satisfy EBAE BC =£AC=6AC.

Since 5.43 is skew, the ordering of indices is

;;:portant and we have the following conventions:

£A=5A853a

rogether

with

their

£A=£BEBA

primed

counterparts.

If

{A

and

"A

are

proportional.

5‘4036A3=£A17A= 0; conversely EAnA=0 implies that {A and ”A are proportional. Equation (1) defines the isomorphism between Ta and the tensor product of the '2 :omplex dimensional unprimed spin space SA on which the group SL(2,C) acts and its :omplex conjugate S A], primed spin space:

T°=TAA’=SA®SA"

Complex conjugation

:efines a map from SA to SA" éAfléAl

Some useful spinor translations are:

"ab=‘AB5AIBI~

EabcdziEACEBDE,4’D’EB'C“_iEADEBCEA’C’EB'D"

Where Eabcd=6[abcd] is the Levi Civita symbol for T", for which in a right-handed time-:~riented orthonormal frame €0123=1. If Fab is antisymmetric, then

Fab=¢ABEAIBI+¢AIBIEAB

where 2¢AB=2¢MB)= AA,BB,5AIBI together with the complex conjugate relation. complex

two-form

respectively.

In

Fab

is

self-dual

spinor terms this

Fab=¢AB€A’B' for F anti-self—dual.

or

anti-self-dual

becomes

if

éeabchCd=iFab

Fab=&A’B’EAB

for

or

A

_iFab

Fab self-dual, and

The spinor translation of the source free Maxwell

equations, VIanclzozvaFab on Fab=F[ab] is:

Vfi,¢AB=0 and VfiléA,B,=0,

where VAA’ is the space-time derivative Vaza/Bz“. A vector k" is null and future-pointing if and only if it can be written as the product of I

a spinor and its complex conjugate ka=KAEA-

The future pointing null vector Ir“

determines rcA up to phase, nA—relox“. The phase of a spinor up to sign may be realized

10 | 1. An overview of twistor theory

geometrically by consideration of the two-form (flag plane):

Fab=NAKBEAIBI+RAIRBI€ABa

which satisfies b F[ach]d=0=Faib Fa '

These conditions imply that Fab is the skew product of two covectors. kazicAFcA, and ra. Fab=k[arb]’ where k2=0 and raka=0. The flag plane ‘cncodes’ the phase of re" up to sign: under NA —cl9rrA. the flag plane rotates by 20.

2. Conformal geometry.

The Poincare group is the group of diffeomorphisnis of M

preserving the Lorentz metric.

It is the semi-direct product of the Lorentz group O(1.3)

with the group of translations of R4.

It is useful to consider the Poincare group as a

subgroup of the conformal group C(1,3), the group of motions of Minkowski space preserving the metric up to scale.

The conformal group does not act on Ml as such.

One

must enlarge M' to M by inclusion of the light-cone at infinty 5. since the inversion a In

1.‘

7:51") interchanges S with the light-cone of the origin. Compactified Minkowski space M = M'US is the projective light-cone of a metric of signature (2.4) on W. In the language of projective geometry. M is a quadric hypersurface in lRlPs. Consider coordinates (u,v.1‘") on [R‘5 with the metric:

dsgzdudv+drud:r"

The quadric is the set of points in [RP5 for which uv+za1°=0.

(2).

Compactified Minkowski

space M, is thus {(u,v,.r°) such that uv+:ra:r“=0}/{(u,v,ra)~(Au,Av,AI") with AyéO, AER}. The quadric M‘ in lRlF’5 has a natural conformal structure: the points a: and y are null connected if 1: lies in the plane in RP5 that is tangent to M at y, and vice-versa.

The

identity connected component of the conformal group Co(1.3), arises from the action of SO(2.4) on the projective light-cone M, in R5. This is a 2-1 covering: SO(2,4)—rCo(l,3). We can represent MI as the subset of M coordinatised by I“ obtained by setting 11:1 and UZ—Iara.

The metric on lR‘5 reduces to the Minkowski metric on M'.

factor, is specified by the hyperplane (with null normal vector) 11:1. 11:0 with the projective light-cone is S.

The conformal

The intersection of

§I.l.2 Mathematical background | 11

The spin group of SO(2,4) is SU(2.2).

Thus SU(2,2) is a 4-] cover of Co(l.3).

The

:-asic representation space of SU(‘2,2), C4 together with a pseudo-Hermitean inner product ;:' signature (++——). is twistor space T°. the fundamental inner product being given by :l 2.2):2020.

3. Zero rest mass particles and twistors.

A twistor contains the information of the linear

W

and angular momentum of a zero rest mass particle. The linear and angular momentum of massless

particle is

represented

by a future-pointing null

vector p".

the linear

b

momentum. and a skew tensor field M”(a:)=MI‘1 1(1), the angular momentum of the system about the space-time point I“. The position dependence of Mab(1:) is:

Mab(1:)=}l-l“b(0)+2p[°1:b]. The intrinsic spin is measured by the Pauli-Lubanski spin vector:

Sa=%€adepb Mcd’

which is position independent.

For massless particles occurring in physics Sazspa. where s

:5 the helicity of the particle and |s| is the spin. Since pa is null and future-pointing, we have pa=7rA,fiA for some constant spinor n’ 4 , determined up to phase. The condition Sa=spa yields:

.

Mab=zwl

.4

B

l

’

.

A!

Tr )EAB—uil

where wA(r)=w’l(0)—ix’l“l'7rfl,.

B,

1r

)EAB

The space of such spinors (wA(0),1r-4,) are coordinates on

four dimensional complex (non-projective) vector space ll"J of twistors 2°. We write. using

abstract indices. Z°=(w'4.rr.‘|,), Z°€T°.

Note that the massless particle determines Z"

only up to a freedom of a phase. Z°—.eioZ“.

The massless particle determines also the

complex conjugate pair of spinors: _

_AI

(HA, LU

__ ):ZOGT0.

a dual or (complex conjugate) twistor. We have

I

25.,=(w3r3+7r3,03 )r'rArrA, so that the helicity is given by: A25=(U

WA+1TAID'4I)=ZOZO.

12 I 1. An overview of twistor theory

This defines the fundamental pseudo—llermitean inner product. of signature (2,2) between a twistor and its complex conjugate.

The group of conformal motions of M are realized on

T” by complex linear transformations preserving E, SU(2,2).

4. The twistor equation.

It was noted above that the spinor 9/1 has position dependence

associated to that of Mab(1:). We have:

wA(a:)=wA(0)-ia:AAI7rA,

This is the general solution of the conformally invariant twistor equation:

(A B)_ 0.

VA’“

—

This is also the (one index) ‘lx’illing spinor’ equation and the \Vess-Zumino equation—in supersymmetry this is the condition that wA defines a global supersymmetry (see chapter

1.4).

The vector field k"=wALDAI is a null conformal killing vector, and the general

conformal killing vector can be obtained as the sum of such vector fields. The relationship between solutions of the twistor equation. symmetries and conserved quantities is explored

in more detail in chapter ”.1 in connection with the quasi-local mass construction.

5. Geometrical interpretations of twistors in M, null geodesics and Robinson congruences. When

Z°Za=0,

the

massless

equivalently wA. vanishes.

particle

has a well

defined

world-line where

Mair or

This is a null geodesic aligned along (4,34 through the points

(BAA! such that wA($)=0. When the particle has non-zero spin (ZazagéO) the particle no longer has a well defined world-line.

It can nevertheless be represented by the congruence of null geodesics

determined by the vector field uAD'Alzk".

This is a Robinson congruence.

A picture of

this congruence at a given instant can be obtained by projecting the null vector k“ so that

it is tangent to a. flat space-like constant time hypersurface in M. \Nheu projected into the hypersurface, k“ is tangent to a foliation by circles which form the Ilopf fibration of 33 over 52. As time evolves, the picture moves along the axis at the speed of light.

6. Geometrical interpretation of twistors in CM and the Klein correspondence.

The

geometrical description of a. twistor simplifies in complexified Minkowski space CM.

Just

as M was defined as a quadric in lRlPs, CM] is a complex (non-degenerate) quadric in CIPF’: CM' is then C4. the complexification of Mllle4.

A twistor 2" with spinor field .41 A then

§I.1.2 Mathematical background I 13 cur-responds to the complex twodimensional plane on which w A vanishes. I

The plane

I

assists of those (complex) values of I’m for which:

I

wA(I)=w‘4(0)—iIA'4 «A,=0.

I

I

:1: these values of IAA . the twistor Z" is incident with I“.

When "A’ vanishes. the two-

;{ane lies on 3. Tangent vectors to this 2-plane are all of the form “4AA for various "A' Such planes re totally null in that all tangent vectors are null and mutually orthogonal. The two-form ::at is orthogonal to the 2-plane always“; is self-dual. Totally null self-dual 2-planes are ziied a-planes.

Dual twistors are similarly represented by totally null 2-planes that are

uti-self-dual. These are B-planes. Since Z" and AZ" (A€C*) determine the same a-plane. the space of a-planes is _;~.-ojective twistor space. PT=T°/{Z°~AZ°}. zeal twistor space. PT*.

Similarly the space of fl-planes is projective

PT is thus CPs. and PT* is its dual, also CPa. A point of non-

:rojective twistor space is represented by an o-plane together with a covariantly constant

5;.inor 7T4, aligned along it as above.

.4. point IECM is represented in projective twistor space by the Riemann sphere or CIPl of :rojective twistors that are incident with :c. This is a two plane through the origin in T". C noose an origin in CM]; then a point in CM' can be represented by its position vector. _.4.4’

a

I

I

n

n

.

I

The twistors lncxdent With this pornt are Z".=(-..2(0)'4,71'A,)=(1.1:”M 77A,, 7A,) as WA: .aries.

Non-projectively this is a complex 2-plane.

I

:aordmates on CPI.

I

Projectively. 71-4, are homogeneous I

n

I

l

I

I

Note that the space of non-pr0jective tWIstors through the pomt 1‘” l I

.s the primed spin space at 1"” .

The picture just described is the Klein correspondence in which projective lines in CP3 =PT) are represented by points on the quadric CM in CPS. :oordinatized as follows.

The correspondence can be

Lines in PT can be represented by skew two-index twistors Xafi

satisfying the simplicity condition ,\"°".\"” :0. This implies that X0“3 is the skew product

of two twistors: XOB=Al°Bm for some A".B°€T".

The line is then the set of [2°] such

'hat Z°=AA°+pBu for some A41. or alternatively such that ZlaX’hlzo. The space of skew two index twistors X05 is C5. Q(X.X)=XOBX7660576=0, where 50H76=Elafi16]'

Such X” are simple if and only if

Q defines a quadratic form on C". The

twistors X” and AX”. AEC". determine the same line in PT and conversely a line in PT

determines X” up to scale.

Thus the space of lines in PT is the quadric Q:0 in CPS,

14 | 1. An overview of twistor theory

which is CMI.

(For further discussion of the Klein representation and its applications to

twistor theory see Penrose Se Rindler 1986, pp. 307-332, and Hughston 85 Hurd 1983.)

7. Breaking conformal invariance; the infinity twister.

Conformal invariance can be

broken by introducing a skew simple two index twistor IaB-

The twistor 1m, determines

the line in IPT corresponding to the vertex of the null cone at infinity, 3. in CM].

In the coordinates introduced above on R6 we have X05105: u. and the metric can be retrieved as follows. Fix the scale on Xafiby setting Xafilap=2; then

d53=d,\"’dx”eofl,,. defines the metric on the region [Yap/03.790. which is MlI in CM. The coordinates on T°. (wA(0),7rA,). were obtained relative to a choice of origin and infinity in M.

With these coordinates, the origin is represented by the matrix 0:23 and

infinity by 105 as follows:

0

_ 5A3

“”-

0

0

O°B-

o

’

0

—

0

o e

I

_

0

’ °"_

[M— a AB

0

aA'B"

_

0

0

0

where 0032;5057607 a .IS the dual of 0°” and [05 .IS the dual of IO”. The only nontrivial components of 505.7, are 5.435 A’B’ and 5A,B,E'AB. I

I

l

I

A point in CM]I with position vector

l

I’M [5 given by the matrix )1“, l .a X’ofi_—§l

—

AB Iat-

- A II

B,

_ B

X.

9

—21 A,

_

013—

EA,B,

5.43

‘

B,

113A

‘1

—z:rB“

AIBI -

—%IaIaE

The Minkowski interval between two points. r“ and y“ is given by the expression

(Ia-y")(Ia-ya)=-4

The

point

1'

is

real

if

and

only

if.

under

the

conjugation

Z°~Zu

we

have

X°u—XGB=%SW376X15. This implies that the line {2“ such that Z°Xafi=0} lies entirely in the real hypersurface Z°Z°=0.

8. The future tube and positive frequency massless fields. The open region in €err in CM whose points have position vectors with past-pointing time-like imaginary parts is called

§I.1.2 Mathematical background I 15

‘:-= forward or future tube. Similarly CM- is the region in CM whose points have position -e-:-tors with future-pointing time-like imaginary parts. The Fourier transform of a massless field is supported on the light-cone p220 in T ‘vtnentnm space.

A massless field is said to be of positive frequency when its Fourier

transform is supported on the future light-cone in momentum space, and negative "nguency when its Fourier transform is supported on the past light-cone.

It is clear from

';-:- Fourier representation that a (normalizable) massless field has a unique splitting into a

:~;-sitive and a negative frequency part. A field defined on Ml is of positive frequency if and only if it has an analytic extension .:—:r Civil“.

This can be seen from the fact that the e-ip'I kernel in the Fourier transform

9 exponentially small for p future-pointing null. and the imaginary part of .1: past-pointing. ». for IECM+.

Points of CM+ can be characterized by the property that the corresponding line in ;:-Jjective twistor space lies entirely in PT+. This follows from the fact that 2-? restricted

a line L;- is given by Z-Z=i(IAAI—EA’lI)1r .4 ,TrA. This is positive for all 71", if and only .:' the imaginary part of :r is past~pointing time-like. We will see that positive frequency massless fields are in 1-1 correspondence with :ohomology classes on P'll'+.

'3. Twistor quantization and the zero rest mass field formulae. Twistor space restricted to 32:5 modulo the phase Z—eloZ. T°l2.2_s/{Z~eloZ}, is the phase space of zero rest :zass particles with helicity s.

The standard quantum mechanical commutation relations

5;».- (PmMub), i.e. those of fix the Poincaré Lie algebra. is implied by the following:

[2“, z"]=o=[2a, 25], and [2°, 251:7:63. This suggests that one should treat the coordinates 2" as position coordinates, and the 2° as momentum coordinates, and consider wave functions depending holomorphically on Z“. On such wave functions the operator Z“ is represented by multiplication and 20 is :epresented by -h%.

s:

AI—

We represent the helicity operator symmetrically so that:

(z-Z+Z-z)=_gh(z°6%a+2)

Thus a. state function flZ) is in an eigenstate of helicity iffis an eigenfunction of the Euler 'homogeneity’

operator

T=Z°8/BZ°.

Eigenfunctions

of T

with

eigenvalue

:1 are

16 | 1. An overview of twistor theory homogeneous functions of homogeneity degree n.

For integer values of 11 they can be

thought of as sections of the line bundle C(71) on projective twistor space PT. bundle

0(n)

is

the

11“1

power

of the

hyperplane

section

bundle

on

The line

CP3=PT,

or

alternatively the —n”‘ power of the line bundle T—~|PT and has first Chern class 11. From here on we shall put h=1. For helicity s>0, wave functions should be defined as

solutions of (T+2+2s)f=0 on

limb-2:200 /{Z~eloZ}

O(—2—‘25) on lPT+={[Z]€lPll', Z-Z>0}.

and therefore as sections o:-'

However, there are no global holomorphic

sections of O(—‘2—25) on PT+. Instead the wave functions are cohomology classes on |P'll’+: they are elements of the sheaf cohomology group H1(IPT+.O(—2—2.s). Elements of H‘( U,O(n)), with U some open set in PT, can be represented in a variety of ways.

The most popular methods are (Tech. where the cohomology classes are

represented by equivalence classes of collections of holomorphic functions on Stein subsets in U, and Dolbeault where the classes are represented by (TD-closed (0.1) forms on U modulo 5-exact forms. We shall use Cech representatives here (see Woodhouse 1985 for a discussion of twistor methods using Dolbeault cohomology).

Consider the open neighbourhood U in [PT that is

swept out by the lines corresponding to points of an open ball U, in CM'. be covered by two Stein open sets, U0 and U1.

The set U can

Typically U0 and U1 can be chosen to be

the complements of rro,=0. and 71'1,=0 in U respectively for some choice of primed spin

frame.

A Cech cohomology class [flEH1(U,O(n)) is an equivalence class of holomorphic

sections f0, of C(11) defined on Uon Ul modulo the addition of holomorphic sections go and 571 defined on U(J and UI respectively.

For the purposes of the integral formulae below we

shall take fto be some representative of a fixed class. It is possible to make contact with the standard quantization of massless particles by means of an integral formula.

Recall first that on space-time, wave functions for massless

particles are represented by means of solutions of the conformally invariant zero rest mass

A'B’ ' ‘ ‘ LI

where the spinor field ‘pA’B’---L’=9°(A'B’---L’) has ‘23 indices and conformal weight —1. Helicity zero wave functions are represented by solutions of 059:0 (again «,9 has conformal weight —1), and negative helicity wave functions are solutions of the complex conjugate wave equation with —23 unprimed indices. For [flEH1(U,O(—2—23)) with 520, we restrict fto the line L.— in L' corresponding to the point $6 U, by writing flwA.1rA,)ILx=j(z':rAAI7rA,,7rA,).

The integral formula is as

§I.1.2 Mathematical background I 17

'r-Iiows (Penrose 1968, 1969: Penrose and MacCallum 1972):

_

l

’ AA,

‘P(I)AIBI___LI — fifwA'wB'lqu'flu

Tue integrand has homogeneity degree zero.

M, WAI‘WA’)WIVI’dTr

.

That 99A,“. solves the massless field

—.._uations can be demonstrated by differentation under the integral sign.

The contour of

:‘egration links the intersection of the line L: with U00 U] which is an annulus.

If one

mooses a different Cech representative, f+go say, then the difference is the integral of go.

Tnis difference is zero since the contour of integration can be contracted to a point within "IL; (although not within U00 U1). For s=(A-.\')—

with A non-

simple, i.e. A-AgéO. A 15-parameter class of solutions is generated by putting

, a flax".6 ) J2, ¢(X‘"’)=(.xafio

where Don” is any 0(6,C) pseudo-orthogonal matrix, satisfying 00376076,, 2 6E: ‘23].

References Dirac, P.A.M. (1936) Ann. of Maths. 37, No. 2. 429-442

Eastwood, M.G. & Hughston, L.P. (1979) §O.‘2.15. Hurd. T.R. (1981) Conformal geometry and massive fields. unpublished essay. Mathematical Institute, Oxford.

§I.2.6 Conformal weight and spin bundles I 31

51.2.6 Conformal weight and spin bundles by L.P.Hughston 8.: T. R. Ilurd (TN 12, July

2931) Conformal weights.

An important simplification that arises from the CIP5 approach to

space-time is the natural description of the various conformally invariant space-time

:undles.

Recall that Xafl = —Xp° (01,13 =0,1.2,3) are homogenous coordinates on Cl“5 and

:izat Minkowski space-time may be regarded as the 4-quadric defined by the relation .‘i'z'fll’wl = 0. The conformal group is the subgroup of the projective group that preserves this quadric.

We shall call the Cl”5 sheaf C(11): = 001,0) the sheaf of germs of holomorphic sections with primed conformal weight n.

Consider now the trivial twistor bundle 0°. all its

powers, and all tensor products with 0(n) (e.g.

O[°5](l): =O°A05®O(l)).

Because of the

.. . . 6 . . mmensron of twrstor space (4). the bundles O and Ohfli] are lSOl‘nOl‘plllC: but not. 1

naturally so; a choice of a. global section of Olan] must be made. Let Ohm-'61:: O(—1’.1) be called the bundle with ‘primed weight” —1 and ‘unprimed weight’ ].

Spinor sheaves on CP5.

One can proceed to define what we shall call spinor sheaves.

Consider the following infinite Cfli’5 sheaf sequence (which is exact only when restricted to .Q :

)

x” A” X‘" ---——.0w,,,(—2)—o0,,(—1)—o0°(0)—o 09:57]“

(1)

4Y6” Ola576]a(2) x97 o[0516][opr](3)

and in particular examine the images of these maps

---:fg(—1). T"(0). flaawlfl).

which

are defined by exact sequences:

x"

'

,afi

06(4) A—+:r°(0)——.o

(2)

61

mm L 3[°“”’(1)—.0

These image sheaves (not locally free), tensored with powers of O(1'.0) and O(—l',l). will be called spinor sheaves. There is some redundancy in their definition because of the many canonical isomorphisms, such as

32 I 2. Concrete approaches to the Penrose transform

if“ .3 ”(i)®0(—1’.—1):rfl(—1). Thus, a primed spinor field. for example, is represented on CP5 as a (wislm‘ valued I

function

d3°(X) that is of the form .flmétfiX) for some function £5.

But does this really

make sense? To see that it does. restrict attention to the quadric Q.

Primed and unprimcd conformal weight. The space-time conformally weighted bundles are

given by On[n]'[m]: = pn0(n’,m). where

O(n’,m): = [0(1’.0)®"+’"]®[0(—1'.1)®’"]. When

restricted

to .0. the sequence (1) becomes exact

(since. on {2. a)“ =Afiaffi

¢¢I°XM = 0. etc.), allowing images to be identified with kernels.

In other words.

sequence (1) splits into a series of short exact sequences on .0:

: '05 O—opnh°B(—l)—opnOB(—1)—0pn:l’°(0)—v0

(3a)

457

0—‘Pnya(ol——’Pnoa(0)l—"Pn

3‘°"“(1)—oo

(3b)

Note too that the sequence (33.) tensored with [190(1) is the natural dual of (3b).

Infinity.

The usual picture of spin bundles requires a. breaking of conformal invariance by

the choice of infinity twistor

I

at?

—

0

0

0

II 5,43

and the restriction of attention to affine Minkowski space.

(We do not fix [as since this

would give us a section of olefifl” A section of pnif°(0) is a twist 0 twistor valued field ¢"(X) such that oInXM = 0. Let

¢"(X)—o¢"’(X) = ¢°(X)Ia"' 131]

Then, since qtlaX

= 0, the map is invertible on 0—3:

§I.2.6 Conformal weight and spin bundles | 33

-2,\'°A,¢A' ¢A’(X)

:

—2x‘”1.,‘,¢" _

B

-

X" 1M,

_

13

0

x“ 1mg

We can therefore make the identification pnh’°(0) = SA'[0]'[0]. The dual of pn3°(0) is [193.3(1), a section of which may now be thought of as an

equivalence

class

of

twist

0

dual

twistor

valued

fields

[En(.\’)] = {0(X)

mod

._x;~“n[a‘,,](X)}. On 0-: let: -2,\"°A,5a(x) a—+ EA() II": [E]

A

IaB

'0

‘

aixich is independent of the choice for [fin]. Moreover, we. can recover [Ea] from £4, since: .

4‘7

—2J

own—dfl'étl =

lpwéa

\robl

’

=£0(1\’)

"10‘! 4‘nwnla‘37].

a0

The inner product between d5°and [E0] is: r ' 1 $1 _¢ a A,-I3'7 15,50 _ —_. :05 1,3745%0 _ -2.\°A,£OI,A a _ E ¢A,

a-

X511

'— 51

Yaw] ’

—

A’

'

01

The relation between the primed bundle and its dual is as follows:

I

I

sA[01’[01:= pgrm) 2 5A,[—1]’[01 and 5A,[01’[01; = 129340): 5*“ [WM

A similar (but slightly trickier) exercise leads to the identifications:

SA[0]’[0]= = pnmomsnorl—u. and

mom]: = p{2:rc.(1)=s.‘,‘[o]’[1].

The canonical isomorphism SAI[0]'[0];SA,[—l]'[0] may alternatively be regarded as a consequence of the existence of a canonical isomorphism

between

S” B][0]'[0] and

On[—l]'[0]. This isomorphism is executed by multiplication by X03 (the canonical section I

I

of S” B ][1]'[0]) and is a direct reinterpretation of index raising using 5N3].

The choice of ‘cpsilon’.

Care has been exercised throughout this note to avoid the

introduction of can”, this being a. choice of global section of Ohm-761'

The lack of an 6

requires the distinction between primed and unprimed conformal weights: a choice of 5

34 | 2. Concrete approaches to the Penrose transform

affords their identification. The distinction between ‘primed’ and ‘unprimed' conformal weights is a subtle one. am: may form suggestively the basis—We are tempted to speculate—for an identification of some of the quantum numbers of elementary particle physics.

§I.2.7 Mini—twistors by P.E.Joncs (IN 14. July 1982)

Mini-twistor space MT is a two dimensional complex manifold that arose in connection with solving the Bogomoln’yi equations on R3. Hitchin (1982).

This note will explain its

relationship to the standard twistor picture and also how stationary zero rest mass fields can be described in mini-twister terms.

Consider a unit time-like vector field % on CMI'.

Its integral curves correspond to

quadrics in IP'll'l each having an car-generator (the line ‘I’) removed.

Q induces a vector

a: field 7 on each such quadric.

The integral curves of 7 are the ,G-generators of the quadric.

In homogeneous coordinates ‘I 7211.4" 7',

IL A awA

where til/l, = diag (1,1).

MT is obtained from lli'll'I on factoring out by the [B-generators.

resulting in a rank 1 holomorphic fibration pzlPTl—hMl'll'. Explicitly .f'

0 1 w 1 ._ ("—w’flov’w—I)—'°-(C~Z°) l w0 w 1

p77,“) A

7Ir!

71' ;

o

0

w

1

7r 1

w

o

0

l u) 7I' I

w

0

-

._

('fi‘W—I'w—I)—'(w—.w—,— l 1 (#1,) 2 )

1

-(C~Z)

Thus 6: % zl : (42°. so MT is the 0(2) line bundle on [P]. As far as stationary systems on CMI are concerned. one can consider them as existing on C3. which is equipped with coordinates

XAIBI_2

Hl(U”,O(—m,—n)):kerEl:I‘(U,O(—n)[—1])~I‘(U.O(—n—2)[—'2][—l]'i

m+n=2

11‘( U~,0(—m,—n))=kerVAA':r( U,OA,(—l—n)[—1])-—.I‘( U.O"‘( —-2— m[—2][—1]’) m+n=1

H‘(U"0( ,

m,n

)) Marv/“2mU.offi”'L“’(n—1)[—1])_.r(v.0“"Lm'wn_2)[—2][1]’) 2 Im{ v” A ,:I" ( U.0"‘"'L’ (an —.r ( up A ,"4"'“"'«n —1)l—ll ._ .>} M7120

51.2.13 Cohomology of the quadric and homogeneous massless fields I 51

figures in round brackets are homogeneities in 1:). For the case m: n = 0. one can then. as in the time-translation case. take the potential

;-3du|o gauge and construct. from this a. solution of the C* Bogomolny equations

*dw = (145

: L'. where w and d5 are respectively a one-form and a scalar on U. l

m is represented by a I

;:ction wA4,(:L') which is homogeneous of degree —1 and satisfies 1AA UAA,: . :z'en by a. function 45(1) homogeneous of degree zero.

I

45 IS

The Bogomolny equations can be

titten as

9. B c' _ -1133, VM “’B)C"‘VAB’¢'

\ solution is defined up to gauge transformations of the form wA 4,D—wAA,+VA4,f where f s homogeneous of degree zero.

The solution can be constructed from a potential-modulo-

'auge UJAA,(1:) (homogeneous of degree —1) according to

as: —i(:r-w)

w AA’:

(W) AA’ "(1.1-)3

The Bogomolny equations then follow from the anti self duality equations on “£44"

Than ks to M.G.Eastwood.

ieference

1 Jones, P.E. ‘Mini-twistors’ §I.2.7. n-

Eastwood, M.G., Penrose. R. 8.: Wells, [1.0. (1981) Cohomology and massless fields, Commun. Math. Phys. 78 305-51.

3, Jones, PE. (1984) ‘Mini-twistors’, D.Phi| thesis, Oxford.

52 l 2. Concrete approaches to the Penrose transform

§I.2.14 A note on background-coupled massless fields: formal neighbourhoods, a contour integral, and the Fierz—Pauli and Buchdahl conditions by A.Helfer (TN ‘20. September 1985)

Recall that if an anti-self—dual electromagnetic field is described by a Ward line bundle I. over a region U in PT. the space of massless fields for charge q and helicity s over the corresponding region in CM is

zq,,=H1( U.L"®0(—2—2s)).

It is known that. if 520 and fis a. representative twistor function in the group on the right. the massless field is given by an integral of the form

1

mfflAI'HFBIG

g(Is7r'4l)

.

.

f(zr 1r,1r)A1r.

(*)

Two problems arise: (1) Give a contour integral for s: —%, and (2) How does the twistor description relate to the Fierz-Pauli conditions?

In this note, we give an answer to the

first, and show that it has something to do with the second, the link being the theory of formal neighbourhoods in twistor theory.

Similar analyses will hold for Yang-Mills field

and the non-linear graviton. The idea is this: To evaluate the field due to an element fof Hl(---) at 1:" for 520. we do the following: (1) Restrict fto the line L; in PT. (2) Trivialize I. over L;. (This amounts to choosing a gauge at 1°.) (3) Use this trivialization to regard fle as an element of 111(LI,O(—2—?.s)). (4) Do the usual contour integral (Serre duality) on L: = I?! to get the field. As we shall see, this agrees with (*). For 5: —%, replace the above with: (1') Restrict fto the first formal neighbourhood of L, (call this F3). (2') Trivialize I. over the first formal neighbourhood (F1). (3’) Use this trivialization to regard leI as an element of ‘H'(F,,O(—1))‘ (the quote marks are because this is unorthodox notation) (4’) Do the usual integral with one B/c‘hu’1 derivative.

We will be informal with formal neighbourhoods. treatment).

(See [1] & [‘2] for a rigorous

For the present purposes. it suffices to know that a function defined on the

first formal neighbourhood F: of L; is a Taylor series in a neighbourhood of L, in which one sets to zero all terms of higher than first order off L:- That is, if

$512.14 A note on background-coupled massless fieldsl 53 I

DA(.1:°) = M4 —iIAA WA,

:;en the functions are of the form

f(1rA,)+E)AfA(7rA,)+nothing else.

Geometry of the Bundles.

We call the line bundle on CM on which the electromagnetic

;~;»tentia.l is a connection the em bundle; its fibre at I“ the em space at I“.

The Ward

i'jndle L is the em bundle translated into twistor terms. and elements of the em space at :‘ are sections of the Ward bundle over L,. To derive some formulae from this. let U: UluUz. and the fibre coordinate of I. over

7: be C3" The patching is

C1=CchP{-icF(Z°)} .vhere e is the unit of electric charge and I" is the twistor function representing the electromagnetic field.

we know there are splitting functions gj(:r“.7r4,) holomorphic over

J With F(i1:7r,7r) = gl(r,7r) - 92(I,7l').

The splitting functions are unique up to

gj(z.«)—.gj(x.w)+a(x)

(..)

and satisfy I

I

.

«‘4 VAA'gj : 77A (16¢AA,).

.\"ote that the effect of (u) on (*Ihk) is a gauge transformation.

(tint!)

The choice of splitting is

the choice of gauge at 1:“. Let a section Cj(1rA,) of L over LI be given. Then

(107,1!) = (2(7r’1,)erp{—ie(g1(1:,1r)—gg(1‘.7r))} whence

(161:1){ieg1(17,7r)} : Czexp{iegg(z,7r)}.

The left-hand side is holomorphic over U1, the right-hand side over U2, and both are homogeneous of degree zero. therefore both must be equal to some constant. say (0- Then

g+5f where

f=f(wATrA,7rA,,7‘rA) where f has homogeneity degree (—n—2,0), i.e. free functions of three variables.

However, given such a form. 9. we can read off the derivative of the

cnaracteristic initial data, this being just the coefficient of Zolapdza.

There can be no

gauge freedom in this quantity. The resolution lies in the fact that the Kirchoff-d’Adhemar integral formula is also well defined for evaluating the field at points of the initial data hypersurface.

So the form 9

must be regular over all lines in twistor space corresponding to points of 3. This eliminates the gauge freedom for representatives of negative homogeneity. and reduces the gauge freedom for representatives of zero and positive homogeneity to 5f where f=j(7r,7‘r).

The

gauge freedom in y then corresponds precisely to that in the chosen forms of the rharacteristic data. For fields which do not extend to 3. the above representatives provide a Kirchoff-

d‘Adhémar type integral formula using ‘virtual‘ characteristic initial data which does have the above mentioned gauge freedom. 4) Conformal invariance and an inversion in the origin will provide similar results for initial data prescribed on a light-cone in conformally flat space-times.

More effort would

be

null

required

to

generalise

these

considerations

to

more

general

initial

hypersurfaces in conformally flat space-times or H-spaces. This article is based on earlier work with Kozameh and Newman, §l.2.15.

Reference

[1] Woodhouse. N.M.J. (1985) Real methods in twistor theory, Classical and Quantum gravity, ‘2, 257-291.

[‘2] Penrose. R. 8.: Rindler W. (1984, 6) Spinors and space-time Vols 1 & ll, C.U.P.

data

62 | 2. Concrete approaches to the Penrose transform

§I.2.17 Local twistors and the Penrose transform for homogeneous bundles by L.J.Mason (TN 23, February 1987)

This note explains how the Penrose transform for the cohomology of homogeneous bundles on twistor space can be realized explicitly using the minimum of technology. The principal innovation is that the space-time fields are local twistor valued solutions of the zero rest mass field equations possibly satisfying further algebraic and differential constraints.

The

Penrose transform for homogeneous bundles has already been treated in much greater

generality by Eastwood [1]. However, it will emerge that the treatment provided here has various advantages, especially for explicit computations. curved

twistor

spaces

and

conformally

anti-self-dua]

provides the Ricci curvature correction terms.

This approach also works for space-times and

automatically

It is likely that the ideas generalize to

twistor spaces of different dimension. I will first give a brief account of the Penrose transform for tensors 9:? of homogeneity degree n, and then give various methods for deducing the cohomology of the irreducible bundles on twistor space from this.

Reducible homogeneous bundles.

I shall build the homogeneous bundles from the trivial

rank-four vector bundle, T°leT—PT.

Local sections of this bundle can be defined

geometrically as homogeneity degree —-1 vector fields. A°a/6Z°

[zr’a/az", Aha/{92"} = _A°a/aZ° on the appropriate region in T over PT. This rank four bundle is trivial; choose a basis in T,

69°,g=0,...,3.

then

age/62°

is

a global

holomorphic frame.

However,

this

geometrical definition determines the (non-trivial) action of the conformal group SL(4,C) on local sections. This is also a good definition when twistor space is curved although T° is then no longer trivial.

For both flat and curved twistor spaces T° is the Ward transform of the

local twistor bundle on space-time.

(see pp. 275 of [5]. or ['2] which uses the alternate

definition, (JlO(—1))*, of T"; see [4] for basic facts about local twistors). in the flat case we can evaluate a tensor 93:22 with homogeneity degree (71—12) as a field on Ml by first expressing it in terms of components with respect to the global frame.

"' instead of ‘converging to’ in (13).

This statement summarises

the sense in which cohomology classes in H*(Y,g'10) can be represented in terms of elements of the groups Hq(Y..Qg).

In general, as the reader will have gathered, this

representation is rather complicated but in practice. if one is concerned with Hp( Yuri—[0) for small p, or if one has vanishing results for some of the groups Hq( KHZ) then drastic simplifications can result.

In any case. it is in terms of the groups Hq( KHZ) that we shall

formulate our ‘integration over the fibres of f'. For this. one requires, in general, another spectral sequence: the Leray spectral sequence.

Let us begin with a locally finite cover ‘l’=(V,.) of X by Stein open subsets:

then f‘1°l‘=(flV,,) is a locally finite cover of Y by open subsets which are in general not Stein. Consider now the following éech-Dolbeault augmented double complex

§I.3.2 What is the Penrose transform?| 83

l

T

Y.s°"(n)

c°(r‘v.s°"m)

_6. C‘s-Waffle)

rA§,+} (where all the maps, including Pn —'F'n, are induced from those in (0.1)). With these definitions. the exactness of (0.1) automatically implies that of

o—T, —.A% —'Pn—Fn—v0

(1.1),

which simply states that fields are locally isomorphic to potentials/gauge. and that solutions of the dual twistor equation symmetric dual twistors.

(for arbitrarily many indices) are essentially

(Note incidentally that (1.1) is the local version of sequence (5)

in Ward 50.2.3.)

Next, observe that the surjectivity of El:0—~O easily implies that the sequence

At—AEH—o

(1.2)

is exact. The exactness of (0.1), together with this fact. then implies that of

o—Ln—A%_—..4?.-.A;‘.—o

(1.3),

which is just the definition of Ln extended to a resolution.

Conversely, it is elementary to check that the exactness of (1.1), (1.2) and (1.3) together imply that of (0.1) - the exactness of the above three sequences can all be proved separately, the only difficult part being that of A9,—.P,,—+F,,—.0.

However, it would be

nice to have a simple proof of the exactness of (0.1) since (1.1) and (1.3) follow so easily.

2.

Because all the maps in (0.1) are defined globally, one can take sections of it over

§I.3.4 A generalized de Rham sequence | 87

arbitrary open subsets UCMl; thus, with just a. formal change of notation. one can emulate Eastwood’s analysis in §O.2.10 to obtain an exact sequence

r( U,A3.+) F(U.P,,) ‘ — 0—171‘(U,T,,)—» —_——._5 Fl U,F,,) —-§ H'2 (U111) (2.1) U.A%)—'F(U.Pn) imr( imI‘( U,A;.)—.r(U,A?,+ where

1‘1”(U.T ):= n

kerF( U.Afi)—r( mt”) lmF( U,A£_1)—4F( Uu‘lrpi)

The reason for this notation is that when U is Stein, (0.1) is an acyclic resolution of 3"_2T* = T", so that f!P( U,Tn) = Hp( U, T") for all p by the abstract de Rham theorem. From (2.1) one sees that a potential/gauge description of helicity lzln fields on U is equivalent to the classical description if fII( U, Tn) = 1-f2( U, Tn) = 0. Alternatively, one can break (1.1) into pair of short exact sequences and consider the resulting long exact cohomology sequences.

This has the effect of splitting the conditions Hp( U.Tn) :0 into

HP(U.T,1)=0 plus a holomorphic condition which is trivially satisfied when U is Stein and which is vacuous for 11:1 . ln [2] there is a proof of the existence of contractiblc domains Uct4 which do not satisfy this holomorphic condition for any n>1. thereby demonstrating its necessity in general.

3.

An interesting result is obtained if one applies the EP-transform of Eastwood, Penrose

8.: Wells’ article on cohomology and massless fields: the open set UCM is said to be 9N (N: 0.1....) if the intersection of U with every a-plane is connected and has vanishing j—th Betti numbers for j=1,'2....,N. plane’.

Similarly, ‘EPTV is defined with ‘a-plane‘ replaced by ‘fl-

Denoting by U", U“, the subsets of . PT. PT" respectively associated to U by the

twistor correspondence, it can be shown that

(a) U is 9,»,

rw’P")

~ 1

H

_

:mr(U.A?,)—r(U,P,)‘H(U .O(n 2)) (b) U is 91=>r( U.F,,)=H‘( u*”.0(—n—2))

(c) Uis 6.15:»

I‘(U,A;.+)

2H9 u”,o

zmr(U,A:.)—.r(U,A?,,)

(

—2

(n

l)

(The proof of (a) and (b) is in the paper mentioned above, and a more complicated proof

of all three is in [2].) The last group on the right of (c) above can be shown to vanish always because of the

form of U"; thus if, for example, U is an open Stein subset of Ml which is both ‘EP’: and 92,

88 | 3. Abstract approaches to the Penrose transform then inserting the above into (2.1) gives the exact sequence

0—H‘(U,Tn)—-H‘(U".O(n—2))—~H"’(U*".O(—11—2))—H2(U,T,,)~0

(3.1

a generalization of the twistor transform isomorphism

”Renew—2)): H1(Prt.O(—n—2)).

In particular every point in MI has arbitrarily small neighbourhoods U for which

H‘( U“.O(n—‘2))— H‘( (1*".0(—n—2))

is an isomorphism for all 11.

Reference

[1] Buchdahl, N.P. (1980) D.Phil. Qualifying dissertation. Oxford.

§I.3.5 The inverse twistor function revisited by N.P.Buchdahl (WIN 11. February 198'!)

In [2] Penrose gave an explicit integral formula for generating homogeneous functions on parts of projective twistor space from

massless fields defined on certain regions of

compactified, complexilied Minkowski space, such that. the procedure provided an inverse to the usual €P-transf0rm taking the homogeneous functions to fields.

The purpose of this

article is to give a cohomological interpretation of this inverse transform (the inverse twister function). The notation used here will follow that in [1].

In particular. le=projective twistor

space, |F:=projective primed spin bundle. M:=compactified, complexified Minkowski space‘ and yle—olP. u:F—.Ml are the standard ‘projections'. resolution

0—+S—’R°—.R'—rR2—»--

For a sheaf S on a space X. a

51.3.5 The inverse twistor function revisited | 89

is denoted by X:0—~S—~ R. and the groups

(kerl’(X,RP)—-r(XsRPH)) (imF(/\',RP_1)-‘F(X«Rp))

by HP(F(X._R°)).

Before proceeding, some general definitions and results from sheaf theory will first be collected together in the next section.

1.

If X, Y are topological spaces. f;,\’_. Y is a mapping and S is a. sheaf on Y. the

topological inverse image of S, f—‘S. is the sheaf on X characterized by (f'15);::5f(r) for all IEX. (i.e.

If X and Y are smooth manifolds and f is a smooth mapping of maximal rank

rL~(clf)= dim

stim X at every

point of X). then

there exists a resolution

X: O—rIEY—»E; (where {Y is the sheaf of germs of smooth functions on Y).

I'Iere E} is

the sheaf of germs of relative q—forms on X: locally a relative q-form ‘looks like’ a q—form on a fibre of f parameterised by the variables transverse to the fibre.

The differentials

df:£}’—~£}'+l are just differentiation along the fibres. If B is a smooth bundle on Y. then because the transition functions of f*B are constant along the fibres of I. one can tensor through the above resolution by £X(f* B) to obtain a resolution X20—vf_l£Y(B)—E}(B), where £}(B):=£}®£X£X(f*8).

Also. in complete

analogy to the smooth case, if X and Y are complex manifolds and fis holomorphic (and of maximal rank), then there is a. resolution X:0-—'f-IOY( V)—-.Q;(V) for any holomorphic vector bundle Von Y. Finally, the following theorem from [3] will play a central role in subsequent sections of this article:

Theorem: Let X. Y be paracompact complex manifolds and f:X—. Y be a surjective holomorphic mapping of maximal rank.

for each

Let V be a holomorphic vector bundle on Y. If.

ye Y, f'l(y) is connected and

Hp(f'l(y),C)=0 for p:1....,N, then the

canonical mapping

HP( Y,O( V))—. H”(XJ-10( m)

is an isomorphism for p=0,1....,N. and a monomorphism for p: N+1.

2.

Let B::PxP\diagonal, and let rrzfl—‘lP be the projection onto the first factor.

An

element of B is a pair of linear 1-subspaces of if which. being non-coincidental. uniquely determine a. linear 2-subspace of if, i.e.

an element of M; let 0' denote this map B—Ml.

Similarly, there is a mapping r:B—~|F which is defined so that the following diagram commutes:

90 l 3. Abstract approaches to the Penrose transform

B a]

1,.

M] 7

\1r

IF 7

(2.1

P

If UCM is open. let U02=a-'(U), (where. as usual Ul=u"l( U) and U"=p( U’)).

If.

moreover. UCM', then U0: leD. where D: =CP‘xCP'\diagonal. In this case one can use either the homogeneous coordinates (U°.V‘G) on U0 (with IGDU“ V3960), or the space-timelike coordinates (:"B',7rA,,LA,) where I°B'=(i:rABI.6§:) are ‘homogeneous‘ coordinates on MII and NAHLA, are homogeneous coordinates on their respective ClPI's.

In terms of these

coordinates, the various mappings of (2.1) are

I

Al

(xAA ,WAI,1A,)°—( U",V°)=(I°

‘7 Al

1:“

l'

LA,)

U0

”\

A!

ou—(ra

Al

7A,,r“

5’ 1’ Y

0A,

Jud—Tor

__

a

I H UoV—UTOU

7rA,—U

It will be shown subsequently that the inverse twistor function naturally gives rise to elements of the cohomology group H‘(Uo,rr'10P(—n—2)), which, under the conditions for which the 9-transform is an isomorphism explicitly generate representative cocycles for elements of the group H‘( U",OP(—n—2)).

3.

The space lD=CP1xClP1\diagonal introduced above is of considerable importance

and deserves Special attention. D is a fibration over CP, with fibre CP,\{point} =C. and is therefore of the same homotopy type as CPl.

Moreover, D is Stein (since it can be

embedded as a closed submanifold of C“). so that the line bundles on it are completely determined by H2(D,l) :2.

It is not too difficult to deduce that the restriction mapping

[12(ClPGClPhZ)—oH2(D,Z):Z$l~2 (7(m,n)=0(m+1,n+l) on D.

is

given

by

(m,n)Hm—n.

so

it

follows

that

(The same isomorphism is thus also true on U0 for any

topologically trivial Stein UCMll.) By using the fact that every section fEF(D,O(m.n)) can be expressed as a holomorphic function on C‘xC“. a direct argument with power series shows that f may be expressed (uniquely) in the form

I...A'

p-JCA I

f(7'|'A,,l.A,) =

sgma:{—m.-n}

B""BI

"n+5

n+5"

51.3.5 The inverse twistor function revisited | 91 I...3I e pzzwAnA' and C"1 1

,4! ...3I "+3: C(

‘

a").

It follows that every section feI'( U0,OB(m.n)) can be expressed in the form (3.1) but ‘

a

' h C4 l

I

A’ ---A'

B "Hnow belonging to I"( U,0( 1 A

III!

A

Suppose now that C 1 _

,

I

A’ ---A'

" €F( U,O( l ---A' A'

f,;=p ‘CAI

m+n+2')[-s]’).

"

1

")[—s)'); then

"'A'n

""1

1r

4A,”

r=0,...,n

A'1"'"A’r‘A'r+1” is a well-defined element of I‘( UO.O(r-—s,n-r—s)), and a direct but tedious calculation gives

5 8

. _ ’yfiVAA’wl ’ ---.4’ 7| _ 1 A’ "A'1.,.TAIP‘AI

%+%=lp

8

~-‘ '

A I n—I

r+1

(4.1) I

—s +p

where y5:=(6g,—i1:3‘4') and

...A'

1

A!

11-1

(7"l'i'].'—'5')EGA'CJ

"A"

503,:=(0,6‘3:).

(Since

'WAprlAIHI...LAln_I

U and

V are homogeneous

coordinates. equation (4.1) does not really make sense, and moreover, the equation tacitly assumes that UCM'; however, there is an obvious meaning to the equation and the conclusions to be drawn from it will be manifestly invariant, so these ambiguities will be ignored for the sake of transparency and simplicity). Taking a: n+1 in (4.1), one sees that each of the equations

afr 0fr+1_ FwdWU _0

— r—O,

,n _. 1

(4-2)

is equivalent to CAT-IA,“ satisfying the zero-rest-mass equations.

Furthermore, (4.2)

implies 62f?

=0

r=0,---.n

(4.3)

0 (71°19 v“ I ”,AI

(In fact. each of the n+1 equations (4.3) is also equivalent to Cf1 1

" satisfying the zero

rest mass equations, although to see this directly requires a. very lengthy calculation.)

The section foel‘( UD.O(—n—l,—1)) is particularly important—it is the null datum for the field Cull-‘14,".

Setting f:=§+2p

l

B: "'B' s-n-2

CA

l

'

"I": _

“A',_n_2‘3'1

.

‘33

B' ”.5!

and using the operator R defined above. it 18 easy to check that C

l

'

n must satisfy the

zero-rest-mass equations, and that g.=8h/0V° ——_1*_—IaC°/0U° , where Co is the null 3mg!" n

datum for the field CB 5.

.

The isomorphism (4.4) is the first in a chain of such, relating cohomolog'y classes on

U0, U’ and U. in the appropriate circumstances. on U".

The next step relates classes on

U0 and U'. The mapping 1': Uo-o U’ defines a. fibration over (1' with fibre CIP1\{point} =C.

Thus

by the theorem of Section 0. one has Hp( Uo,r‘lflz(m)):Hp( U’,.QZ(m)) for all m and all

p.920.

Since

the

resolution

lF:0—~p'lOP(m)—o9fi(m)

generates

a

resolution

B:0—or'lp"OP(m)—or"lf2:(m), one may apply these isomorphisms to the (two) pairs of long exact cohomology sequences which these resolutions induce.

Then a multiple

application of the Five Lemma shows that

Hp( Uovr‘lfl-lop(m)): Hp( y’all-10'“ 171))

for all p20.

But por=1r, so r'lp'10(m)=x'1OP(m). and one therefore concludes that

H’( Uo,1r"OP(m)):Hp( u',p-‘oP(m)) for all p20, 77162. It is a relatively simple exercise to check that the diagram

H‘(r(Uo.n: / \. 0_’C(.q

r_.) 0(p q r)_. —90(

q+r+1__p+q+1

P+q+1 _q-q+:+1)

9

\

O( -p— q-r—4

/ p+q+r+a q C( 0—} -°) -q-r—3

) o-n-v-rJ—lo'

§I.3.15 The Penrose transform for complex homogeneous spaces l 135

Again, for p: q: r=0 this is the holomorphic deRham complex with two-forms split into (anti-) self-dual parts:

.93.

For p=q=0 this is Buchdahl's generalized deRham sequence [3].

It relates the field

versus potential/modulo gauge description of massless field on M and is therefore of fundamental importance in twistor theory.

The general case is used extensively in [6] for

the same reason. Notice that the pattern of the BGG resolution 0—K:>H is independent

of original choice of A. In fact, this pattern may be recognized as describing the standard cell structure of M (dots representing cells of even dimensions, lines recording attaching maps).

This is not a coincidence! This always happens and is known as the Bruhat

decomposition.

When A :0 (i.e. the resolution starts with C) the BGG resolution may be

realized by those holomorphic forms orthogonal to such typical cells. The BGG resolution is often far more efficient than the deRham resolution.

Consider, for example the case of

F1,,,3(C4) = x—x—x: without going into any detail. the general pattern is:

This is the classical BGG case when P is minimal (i.e. a Borel): the general (Lepowsky) case may be deduced from this and the BBW theorem for G/B—» G/P.

In this case all

irreducible bundles are line bundles so 1 3 5 6 5 3 1 give the dimension pattern for the

BGG resolution (#W=24(=4!)) so already much more efficient than the deRham resolution of C where the dimensions are 1 6 15 20 15 6 1.

When the tangent bundle on

G/P is irreducible. the BGG resolution of C coincides with the deRham resolution, so for

136 l 3. Abstract approaches to the Penrose transform

general A on such a P one may reasonably maintain the terminology of ‘generalized deRham sequence'.

.

As a final example one can write down the deRham sequence on M5 = x—cfi as follows:

2 o

-

o

x—.( o

o

o

0—.C—.§—.°< o

1

f

—r-x—l(—o_x—2{ oo 01

\ e

1 -x—2(

0/.

—o

O!

\ 'x‘_2( 2

° 2 _.-,HI{ _.-x_2< _.o .0

0

The Penrose Transfonn: We assume familarity with the general approach to constructing the Penrose transform for a given correspondence say

Y

"J

\#

X

Z

as described, for example, in [4].

Thus, one supposes U to be a suitably shaped open

subset of X and the Penrose transform interprets the cohomology Hk(p(u_1( U)),O( V)) in terms of differential equations on U determined by choice of the vector bundle V on Z. To simplify notation we shall remove U from explicit mention throughout.

There are four

steps in constructing the transform (1) Hk(Z,O( V)) = H"(Y,p‘10(V)) (technical conditions here);

(2)

Use

the

relative

deRham

complex

0—‘p’lO(V)—+Q:(V)

to

compute

Hk( Y,[.l-IO(V)) in terms of analytic cohomology Hq( Y,.Q,’:( V)); (3) Use the BBW theorem (and, usually degenerate, Leray spectral sequence) to intepret Hq(Y,.QB( V)) down on X; often this is the end result but sometimes: (4) reinterpret the equations on X (e.g. potential/gauge=field).

For complex homogeneous spaces as discussed earlier one may

replace (2) by (2)’ and use the ECG resolution along the fibres of p.

Step (3) is then

easier in general. Step (4) now uses the ECG sequences on X. This approach is also useful

in the ‘curved’ case (cf. this machine in action:

[1] and M.G.Eastwood in chapter 11.5).

It is about time to see

51.3.15 The Penrose transform for complex homogeneous spaces | 137

Examples:

1) HRH—3?) (This is H‘(O(4|—1,0,0)) discussed in [5]): The BGG resolution along the fibres of o—x—x—oo—o—x is WisaLfi—fi'éL-og—fi.

Taking direct images under o—x—x—oo—x—o

using BBW gives a. spectral sequence

converging to Hp+q(g—3—'i£") whose El-level is

0

-3

3

L—S

0

1

l

0

-5

D

0—

Le.

I VAA' VB3'

V BC’

0(A'B'c')—° 0(A3)c'(_2)—*°A(—4) | o

o

o ——

Thus HRH—3(5) = {Solutions of Vfi'V§¢A,B,C, = 0, ¢A'B’C’ = ¢(A'B'C’) of weight zero} as in [5].

2) ”*(3—2—9‘): BGG resolution:

3—2—9: —0 Li; —. Lg—i‘t

(I) El-level:

so BBW gives

0

0

l. O 0 2 -2 l 0 -4 3 o+o—o—*—o — o—-)(~—o—o

However, the ECG resolution of 3-3—3 on o—u—o is:

21.2 /

\

loo_.3_;3_g

1-52—00-5E

\

l 4—.

138 | 3. Abstract approaches to the Penrose transform

1

So

l

D

2 -2 1 o 4 -3 _lcer: o—*—o—oo—x—o

0

H (°-°-X)——.—1—W zmm—x—o—n—x—o

=ke1(o—-x-3 -3—2—vo—«x-l "5—3)

A B _ kedvA,vB,.-om30,—.00(A,B,)(—2)). _

In other words. this is a potential/gauge description of the ‘dual’ field of 1916—5—35). Thus we have the twister transform:

H‘(§—2—3):H‘(2—$—3 :0. For example, the action on a oneparticle state is obtained as follows:

Q,R°|¢>=0,3"-g/daaM’ds-(I)VM,¢(2)|0>

(definition of |¢>)

198 I4. Twistor theory and elementary particle physics

= g/daaAA’[Q.R°,¢-]VM,¢|0> = g/daaM’-(—qB’(z)sr3.(x))'6M,¢|0> = —i/daa'AAloVAA,(qB’(z)W3,)¢|0>

(integrate by parts)

= -i/d3aAAl[i6A,B'qAW3,+ qBI(1:)VAB,WA,']¢|O> = —fdsaAA'WZ,[qA¢—iqB'(:r)VAB,¢]|0>

(vmyyl = 0)

(integrate by parts)

= I—iq5’(z)v33,¢+qa¢> The natural action of R“ on states containing helicity —§ particles is determined if we impose the anticommutation relation:

{R°,WZ,} = 0.

The anticommutation relations

{12".}2”} = o are now derived from (2) and the assumed statistics of the field operators. The adjoint operator Tia , defined such that =(E) for all [45> and lip), maps states as follows:

PE=I¢A(=)>~I—p‘(=)¢A(=)> PE;|¢(;)>_.0

for fixed P°HpA(1:). Summarizing, we have In!

[R°,d5‘] = -X°

B’

E,

where X‘lY

irA B,

E

3' CA;

{nan} =0

§1.4.9 Axisymmetric stationary fields | 199

together with the adjoint equations [R—pflfl = 73AWA+ etc. The conformal group SU(2,2) is generated by the tracefree part of the anticommutator: “AB+%6BA(s+d)

KABI

PA’B

I fiB’A,+%6AIB(s_d)

12{ R0,]?fl} =E°= [3 One can check that

[193, R7] = 65:12“ and [150, R7] = 4a,)!1] and from these we derive the commutator for the generators of U(2,2):

I53 .531 = ~6zzr+'6:5'zThe trace .9: %E‘; is the total hellcity operator, and is changed by :l:% by 12° and 725. Finally we note that the algebra of Poincaré supersymmetry is obtained from the above by forgetting the wAE'JA' parts of R" and Ti, and the k“ and d parts of E3.

51.4.9 Axisymmetric stationary fields by R.S. Ward (TN 11. February 1981)

Twistor theory can be applied to several stationary axisymmetric field theories.

I should

like to mention three of these: the 3-dim Laplace equation, the Yang-Mills-Higgs equations and Einstein’s equations.

One can generate axisymmetric solutions of the 3-dim Laplace

equation by specializing the usual contour integral formula and arriving at the following.

Define 7 and C by 7=iw°/1ro,—iul/1r1, and C: 1l'o,/1I",. Then

Mas/.2) = fi{1(7) % f holomorphic, is a solution of the Laplace equation, axisymmetric about the z-axis.

(A) (I

think it is the general such solution). The inverse twistor function appears to be extremely simple here, because of the fact that 7: —2z on the z-axis* so that K7) =¢(0,0.—7/2).

For example, the ‘Schwarzschild—Weyl' [1] solution

200 I4. Twistor theory and elementary particle physics

is easily seen to arise from f: %log(%).

Someone might like to investigate all this in

more detail.

Notes:

(i) oi is real if fsatisfies the reality condition f(—7) =fl7). (ii) The formula (A) is

essentially due to Whittaker [2]. Now the problem of finding stationary ‘multi-monopole‘ solutions of the Yang-MillsHiggs equations (see [3] for details). Twistor methods give (at least implicitly) the general such solution, as follows. Take a. 2x2 unimodular matrix F'(7.C) of holomorphic functions, satisfying the reality condition

[Ki—Z") = conjugate transpose of F(7,C).

This gives rise to a Yang-Mills-Higgs field in space time which is stationary, real, and a solution of the field equations. The problem (not yet solved) is to ensure that this field has the correct global properties.

The only explicit cases so far understood are axisymmetric

ones, where the {-dependence of F is rather special. The two simplest ones are [4]

F:

7'l(e"-e"') -Ce"’ _ _ (-18 1

‘78 1

H'1(e7+e_7)

(26-7

C-ze-v

Hos—7

where H: 72+1rz/4. Finally, stationary axisymmetric vacuum space-times. As pointed out by L.Witten [5], this problem can also in principle be solved by twistor methods: the matrices F above ‘encompass' all such space-times.

The only case as yet understood (by me) is that of the

Weyl solutions, which correspond to F: diag (”fix-fl”).

References

[1] Kramer, D. et. a]. (1980) Exact solutions, p. 201, C.U.P.

[2] Whittaker, E.T. 8.: Watson, G.N. (1984) A course of modern analysis, 518.3, C.U.P. [3] Jaffe, A. 8; Taubes, C. (1980) Vortices and monopoles. Progress in physics 2, Birkhauser.

[4] Ward, RS. (1981) Yang-Mills-Higgs monopole of charge 2, Comm. Math. Phys. 79 317-325.

[5] Witten, L., Phys. Rev. D19. 718. (thanks to K.P.Tod for pointing this paper out.)

51.4.10 Extended Regge trajectories updated | 201

.My conventions are too = i(t+z)1ro,+i(:r—iy)1r1,, u1 = i(:r+iy)ro,+i(t— z)1r1,. Hystery:

How are the twistor methods related to ‘Backlund‘ methods; cf. Forgacs et. al..

Phys. Lett. 998 (1981), 232-236; Cosgrove, JMP 21 2417.

‘Note added, 14 April 1989:

Many of the problems raised in this article have been solved.

.For stationary axisymmetric vacuum space-times, see: “'ard, R.S. (1983) Gen. Rel. Grav. 15, 105-109.

“'oodhouse, N.M.J. (1987) Class. Quant. Grav. 4 799-814. “'oodhouse, N.M.J. & Mason, L.J. (1988) Nonlinearity, 1, 73-114.

For monopoles. there are too many papers to list; but see:

Hitchin, N.J. (1983) Comm.

Math. Phys. 83, 579-602; 89 (1983), 145-190; 114 (1988), 463-474.

51.4.10 Extended Regge trajectories updated by Tsou Shcung Tsun (TN 12. July 1981)

A novel way of looking at Regge trajectories was proposed which was consistent with the twistor outlook.” Data from the 1976 Rosenfeld Tables” were used. Since then two new issues of these tables have appeared. We think it appropriate to update the trajectories of

[1] using the most recent data” , including the K‘ resonance marked D found in 4). The resultant trajectories follow. obtained. dots.

‘Remarkably straight’n extended Regge trajectories are still

Notations are the same as in [1] except that open circles are replaced by small

Any change in mass value, spin-parity assignment and/or ‘star-status' is indicated

by underlining the resonance concerned. Notable changes ‘in our favour' are i) upgrading and discovery of several resonances in the ‘extended' part of the nucleon trajectory, ii) upgrading of two of the four resonances in the [:0 unnatural parity meason trajectory, and iii) most remarkably a still doubtful 1:; unnatural parity resonance I. which when assigned the value 1': —5, lies almost exactly on the much extended part of the trajectory. The only ‘unfavourable' change we observe is the disappearance of some resonances previously tentatively assigned to the extended part of the [:1 unnatural parity meson trajectory. Thanks are due to L.P.Hughston, T.R..Hurd and R.Penrose for discussions.

202 I4. Twistor theory and elementary particle physics

I 3 . ' ..

‘I

Nolan! Pull! Unllll'll Plrhy

3*

21

‘I

-5

.3

-I

2

I

5

5

[-11

-24

g

J

I 3 I

JJ Ulllllfll Pauly 2" A,

‘1

5

0'

-b

4

¥

5

3

-14 o.

-3.

'3-

l = In Nllnnl Full,

-E .‘l "

I g I” -2. 4

Unllnnl .3‘

3

Puily

L

24

-

-‘4 0'

‘1

02

(J-

I!

M-

-9

51.4.10 Extended Regge trajectories updated | 203

8.

\ I:IIZ Unnatural

5:0

A 1%

Parity

Nalunl

I=JI2 5:0 Unauunl

Pull;

8‘

Nnnml Fully

IEO '

Fully

10‘

4.4

8‘

‘A

~12 -10 1t:

l: Unnlunl

|=l

h=PI

9

41

3‘

Fully

5' Nallnl Pnrlly 1/»

W

U

Unnnlnrll Plrily 2.

”

-2mo

OH‘M

‘ “gum 3 6

-12

3 I=II2 Unnalunl Parl l1

s=-2

B

204 I4. Twistor theory and elementary particle physics

In the diagrams the s—axis is em2 in G'eV2 and the y—axis is j, c and 1' being defined in Ref.

1. Notice that the baryons are drawn in one scale, and the mesons in another (different) scale.

References

[1] Penrose R., Sparling G.A.J. and Tsou Sheung Tsun. J. Phys. A. 11 (1978) . L231-L235 [2] ‘Reviews of Particle Properties'. 1976, Rev. Mod. Phys. 48 No. 2 Part 11 SI [3] ‘Reviews of Particle Properties‘. 1980, Rev. Mod. Phys. 52 No. ‘2 Part I] [4] Aston D. et al, Phys. lett. 99B (1981) 502-506

Most recent update:

There are ten resonances (5 mesons, 5baryons) whose JP were

predicted in 19761) and confirmed (with one wrong and one unknown parity) in 1986 5). Of these five (four with certianty ) were predicted to have negative j.

Especially good are

the f1(1285) and f1(1420) with negative j. and N(2700). (M2200) and .1(2350) with positive 3'. Further details will appear elsewhere.

[5] ‘Reviews of particle properties‘ (1986) Phys. Lett. 170B.

§1.4.11 How are Bose and Fermi statistics expressed in twistor theory? by L.P.Hughston & T.R.Hurd (TN 13, December 1981)

Too much attention is devoted to the antique spin and statistics ‘theorem’ of quantum field theory (Pauli 1940, Schwinger 1981. Streater & Wightman 1964). which takes the form

that

‘when such-and-such

conditions are imposed on

the quantum

fields an

inconsistency emerges if the “incorrect” statistics for the fields are assumed'. It would seem much more natural to look for a geometrical construction that produces the correct spin and statistics relations directly in a neat and tidy way. We present such a construction here for the special case of non-interacting zero-rest-mass fields. Although the case is very special, nevertheless the result is sufficiently'pretty that we

would expect some analog of the idea to go through even when interactions are incorporated.

Rather than wagering on the spin and statistics results by means of a

reductio ad absurdum, we simply build them in from the outset.

51.4.11 How are Bose and Fermi statistics expressed in twistor theory? I 205

. Wave Functions For non-interacting fields it pays to forget the quantum field operators and look directly at the n—particle wave functions.

For two scalar particles, for example,

one wants a two point function ll'l(:c,y) which (for zero rest mass in each variable) satisfies a:

U

0!?(z,y) = 0,

DW(:,y) = 0.

The condition of Bose statistics is that the wave function by symmetric, i.e.:

Way) = Wm)In the case of two zero rest mass particles each with helicity s (3 negative, say) we have the field equations:

‘ A’A WA...3,p...Q(z.y)=0.V 9 PP wA---B.P---Q('t‘y)=0'

V

The condition of Bose (or Fermi) statistics is then:

WA...B'p...Q(-Tay)=imp...Q'A...3(!/»3),

(*)

the plus sign being taken when sis integral (Bose statistics), and the minus sign when sis é, odd-integral (Fermi statistics). The relation (as) above is equivalent to the statement one frequently sees in quantum theory books requiring the wave function for two-particle states

to be symmetric (resp. antisymmetric) under ‘simultaneous interchange of the space and spin coordinates’ for Bose (resp. Fermi) statistics.

2. Main Result for Base Statistics.

We assume the reader is familiar with the standard

twistor isomorphism (Penrose 1979, §0.2.2 84 §0.2.6):

Theorem. H1(P§,,O(—2s—2)):{future analytic z.r.m. fields of helicity s},

where ‘future-analytic’ means analytic on CM+ (open future tube), the term ‘positive frequency’ being reserved for fields analytic on the closure C—W’ (terminology suggested by Toby Bailey). Consider the space of symmetrized pairs of twistors X“ Y3).

This space is a 6-

dimensional algebraic variety Va sitting in the 9-dimensional projective space of symmetric twistors A”. The equation for V6 is AMAMAVEOMJ =0. By requiring both X“ and Y“ to lie in P: (top half of twistor space) we get a region Vi+C V6.

Let OV(m) denote that

standard sheaf of twisted holomorphic functions on V6. obtained from 0(m) on P9 by

206 |4. Twistor theory and elementary particle physics

restriction. Then we have the following result:

Theorem: H’( VihoV(—2n—2))z{two-particle states of future analytic z.r.m. fields of helicity n (n integral) satisfying Bose statistics}.

The corresponding result for fermions is more complicated (it is given below), but what is remarkable is that the same underlying ‘symmetrized twistor space‘ V5,... can be used for both fermions and bosons, the only difference being a different choice of sheaf. This is nice.

because it would be very awkward if systems of fermions and systems of bosons had to live on different ‘kinds’ of twistor spaces.

3. Symmetric Functions. To get a picture of why the isomorphism described above works one must consider properties of symmetric functions of two twistors. Suppose fl.\'°,Y°) is homogeneous of the same degree (m, say) in each variable and is symmetric, i.e.

flX",Ya) =flY°,X°). Then there exists a function F'(A°fl) homogeneous of degree m such that flX°,Y°fl) = F(X(° Y”). This is perhaps not entirely obvious, since one might imagine other symmetric combinations of X" and Y“ could be manufactured that were in some sense ‘independent’ of X“: Ya). A plausible candidate might be XI°Y51XhY“, for example, which

is

certainly

symmetric,

but

not

manifestly

related

to

XmYm.

However

XlaYGI/‘flfil’n can in fact be constructed by applying a Riemann tensor type Young tableau symmetry operator to XWYNXU Ya); following this line of reasoning one concludes that any symmetric homogeneous function of two twistors can be expressed as a function F(X(aY‘a)), Le.

a. function on a region of V6, and it is not difficult to check (by use of

twistor contour integral formulae) that symmetric two-twistor functions do indeed give rise to two-particle states satisfying Bose statistics. Geometrically the situation is summarized as follows. twistors T” is P”.

The space of projective rank-2

This space contains three important subspaces: P9, the space of

projective symmetric twistors A”; P5, the space of projective skew-symmetric twistors; and Q6, a 6-dimensional variety that is the Segre embedding of Pale3 in P15 given by

T05=X°Y6.

The equation for Q6 is INST“: T°6Tw.

earlier, a subvariety of P9.

Note that Pgan=

twistor space, appearing with ‘multiplicity 2‘ Le.

The variety V‘5 is. as noted

V6005=(P3)2, where P3 is projective Top = 2°28 = A”.

For any point in 93:: Ve—(Pa)2 there exists a unique line in F’115 through it that intersects Q6 precisely twice.

If A°B=X(°Y5) is the point in \75, then X°Y0 and Y‘Xfi

are the two points in Q‘3 that the special line intersects. The line then goes on to intersect

51.4.11 How are Bose and Fermi statistics expressed in twistor theory? I 207

P5 at the point X“. Ya], which lies on

the Klein quadric 0‘

(space-time) in P5.

Compactified complex space-time Q4 is the quadric cap." Tinfl =0. with T” = 71”]. It is interesting to note in our picture that both twistor space and space-time appear.

This

leads to a very direct geometrical ‘construction’ of the Klein representation for lines in P3. Now consider a set '7' in V6.

For each point in V we construct the corresponding

‘special line’, and by intersecting this set of lines. with 0° we obtain a pair of regions 0, and Q, in 0°. If F(X(° Y”) is a function defined on V, then by mapping along the special

lines a symmetric function is defined on Q,uQ,cP3xP3.

4. Main Result for Fermi Statistics.

Here we require antisymmetric functions.

Such a

function necessarily satisfies flX,X) =0, and so is of the form Xia Y‘BIGOAX“ Y“) for some

COAX“ Y“) on V6. For a fermion state we want flX,Y) to have odd homogeneity in each variable, therefore G much have even homogeneity.

Since we are interested in Gm, only

modulo terms annihilated by X“ Ya] the relevant sheaf 3'(m) for fermions is given by the following exact sequence:

(v6)

o§(m—1)£olam(m)—.sv(m)—.o

where p is O‘A'v—oOi'X" Ynepnp; the cohomology result is:

Theorem: H2( V3,“?! V(—2n—2))={two-particle states of future-analytic z.r.m. fields of helicity n—g (n integral) satisfying Fermi statistics}.

For further details see Hughston and Hurd 1983.

References Hughston, L.P. 8: Hurd, T.R.. (1983) A geometrical approach to Base and Fermi statistics, Phys. Lett. 1278, no. 3,4, 201-203.

Penrose, R. (1979) §O.2.2 and 50.2.6. Pauli, W. 1940 Phys. Rev. 58. 716-722.

Schwinger, J. 1951 Phys. Rev. 82, 914-927. Streater, R..F. & Wightman, AS. 1964 PCT, Spin and statistics, and all that, Reading Massachusetts: W.A. Benjamin, Inc.

208 I4. Twistor theory and elementary particle physics

§1.4.12 Higher dimensions by 12.5. Ward (TN 17, January 1984)

In §I.2.10, I raised the question of whether the twistor treatment of self-dual Yang-Mills generalizes to dimensions greater than four. solvable

non-linear

hyperbolic/elliptic

I conjectured that there are no twistor-

equations

in

dim

greater

than

four.

(By

‘hyperbolic/elliptic’ I mean that there are as many equations as unknowns, and that the characteristic surfaces are quadratic cones.

The self-dual Yang-Mills (SDYM) equations

are hyperbolic-elliptic in this sense. if one takes gauge freedom into account.

Also, they

imply the Yang-Mills equations.) But there is a lot one can do in higher dimensions, and I

want to report on some work in this direction. More details may be found in [1]. The SDYM equations are integrability conditions for a linear system

«A'DAA,,¢=0,

(t)

where Da = 8a+Aa is the gauge-covariant derivative. Thinking of them in this way leads directly to the twistor construction. dimension.

So one should try to generalize (t) to higher

This can be done in many different ways;

I shall now give two examples in

dimension eight.

Example A. Let unprimed capital indices be 4-dimensional, while primed indices remain 2dimension. So 1:“ =IAAI are now coords on C3.

Let us restrict to the Euclidean slice R8.

The metric is gab: EABEA’B' as usual, and the symmetry group of the setup is

[Sp(2)xSp(1)]/Zz. a subgroup of 80(8).

(The Sp’s occur because they are what preserve

the 5's.) The integrability of (at) then gives us a system of equations on A0.

These (it

turns out) imply the YM equations, but are not hyperbolic/elliptic, being overdetermined (18 equations in 7 unknowns). The picture here is that twistor space is CPS, the lines in which correspond to points in

compactified C8. The basic equation is M4 =izAA'1rA as usual. H1(O(—2)) corresponds to solutions of 0503M,¢=0 which implies, but is stronger than, 041:0.

Vector bundles

over the twistor space correspond to solutions of the system of 18 equations mentioned above.

Example B. symmetric

Both sorts of indices remain 2-dimensional, but now :r:"=:r'4“wg'cl (totally in

A’B’C‘)

are

coordinates

on

R3.

The

Euclidean

drAAIB’ddrAA,B,C, and the symmetry group is SO(4)CSO(8). system

line-element

is

This time take as linear

§I.4.13 Super-twistors | 209 I

I

«A 1r

«CIDAAIBICHIJ = 0.

The integrability conditions now form a well-determined system (7 equations in 7 unknowns), but the characteristic cone ist of degree >2 (i.e.

not quadratic).

The YM

equations are not automatically implied. The twistor space is 0’", with basic equation wAA'B'=i1:AA'B'c’1rC,.

H’(O(—2))

corresponds to solutions of 845:0. where B isl a scalar differential operator of order greater than 2.

(The two statements marked with 1' are tentative, as I have not worked

through the algebra.)

. References [1] Ward. RS. (1984) Completely-solvable gauge-field equations in dimension greater than four, Nucl. Phys. B236, 381-396.

Note added. 14 April 1989: see also the papers: Corrigan. E., Goddard, P. & Kent, A. (1985) Some comments on the ADHM construction in 4k dimensions, Comm. Math. Phys. 100. 1-14.

Marnone Capria, M. & Salamon, S.M. (1988) Yang-Mills fields on quaternionic spaces, Nonlinearity 1, 517-530.

§I.4.13 Super—twistors by A.M.Pilato (1I’N 20, September 1985)

Introduction. Kostant’s supermanifolds [6] consist of an underlying manifold together with an enlarged sheaf of functions. Locally the latter is required to have the form 0(A'RN) so

that one has the following terminating series for a super function

1w")=x=>+f.-(z)1 it is still possible to obtain, without much effort. classical field equations

implied by the super Yang-Mills (super) field equations.

To obtain an equivalence the

algebra gets out of hand and some clever inductive method is required.

Using recursive

relations generated by a kind of Euler homogeneity operator, Harnad et a], [2], showed that integrability on super null lines for N =3 is equivalent to a system of equations which they claim are the super Yang-Mills equations.

Adopting our definition of super Yang-

Mills the claim is indeed true in the abelian case.

But in the general non-abelian case

[v.r‘"} :0 implies the vanishing of certain (anti-)commutators of the equations in [2], Le. it implies stronger equations.

References [1] Eastwood, M.G. (1984) Supersymmetry, twistors and the Yang-Mills equations, preprint. [2] Harnad, J., Hurtubise, J., Legare, M. and Schnider. S. (1985) Constraint equations and

{51.4.15 Extended Regge trajectories and baryoniums | 217

field equations in supersymmetric N: 3 Yang-Mills theory, Nucl. Phys. B256. :3: Kostant, B. (1977) Graded manifolds, graded Lie theory and prequantization. in Differential geometric methods in mathematical physics, LNM 570, Springer. J-_ Leites, D.A. (1980) Introduction to the theory of supermanifolds, Russian Math. surveys, 35 1-64.

:3] Pilato. A.M. (1986) Elementary states, supergeometry and twistor theory. D.Phi|. thesis, Oxford. 6] Rocek, M. (1981) in ‘Superspace 8.: supergravity’, 71-131, C.U.P. :7] Sohnius, M. (1978) Bianchi identities for supersymmetric gauge theories. Nucl. Phys.

B133, 275. )3] Witten, E. (1978) An interpretation of classical Yang-Mills theory, Phys. Lett. 7713,

394. 9] Manin, Yu. I. (1988) Gauge field theory and complex geometry, Springer GMW 289.

51.4.15 Extended Regge trajectories and baryoniums by Tsou Sheung Tsun (TN 17. January 1984)

Extended Regge trajectories (short for: theory with infinitely straight trajectories of

Regge) were introduced in [1] to make use of the extra freedom arising from the fact that the spin J of a particle is not a good quantum number. Only 12 can be measured. Thus, with some inspiration from twistor theory, one can define a new quantum number j by

j+§=e(1+;),e=¢1. Remarkably straight Regge trajectories were obtained by plotting j versus cmz, instead of the usual J versus m2.

For baryons. a definite assignment of 6 seems to be indicated,

whereas for ordinary mesons no definite assignment can be made to cover all cases. In [2] it was suggested that we bring in the exotic mesons. Whereas the ordinary mesons belong to either the octet or the singlet representation of SU3 (in quark language, qfi states), the exotic mesons belong to the 27-dimensional representation (which can be interpreted as qqirq states).

These latter occur mostly as baryon-antibaryon bound states, and are

sometimes called baryoniums. In [2] it was shown that with the assignment

218 I4. Twistor theory and elementary particle physics

+1 for 9

—

—1 for 27

we obtained straight extended Regge trajectories for isospins I: 0,§,1. Recently, several new resonances were observed in the Afi system [3] in the reaction

K‘p-rAip. at energies of 8.25 GeV/c, 18.5 GeV/c and 50 GeV/c. As can be seen in the figure. they sit comfortably on our extended I =% trajectory.

A conventional trajectory for these

resonances would give a. slope of 1.7 Gel/"2, very different from the classical Regge trajectorim all of slope ~lGeV_2.

We also note that for the non-strange baryoniums

trajectories slope values of 0.63 Ge V”: and 0.85 Ge V”2 have been suggested.

Thus our

scheme has the advantage that there is just one universal slope for all the trajectories. This will certainly merit further study once the data on baryoniums become more conclusive, for example, from high intensity machines like the LEAR.

1 5 (-2“;th

-2.

Minn.

(A?) 14"

§1.4.16 Twistors and minimal surfaces | 219 References

[1] Penrose, R., Sparling, G.A.J. and Tsou Sheung Tsun (1978) J. Phys. A11 L231. [2] Tsou Sheung Tsun in TN 8; Hughston. L.P. & Tsou Sheung Tsun in 50.4.4.

[3] Cleland, W.E. et. al. (1981) Nucl. Phys. B184 1; Baubillier. M. et al. (1981) Nucl. Phys. B183 1; Armstrong, T. et. al. (1983) Nucl. Phys. B227 p.365.

Additional note: The U particles exhibited in [2] which form the same family as the ones mentioned here have been further confirmed by recent experiments at CERN. Details will be reported elsewhere.

51.4.16 Twistors and minimal surfaces by W. T.Shaw (TN 20, September 1985)

It has been known since the work of Weierstrass (1866) that the equations of a. minimal surface in R3 can be solved in terms of a single holomorphic function of a single variable. This can be cast into a result from twistor theory in three dimensions (Hitchin, 1982),

since a. minimal surface in R3 corresponds to the real part of a null holomorphic curve in

C3, which corresponds to a holomorphic curve in II", the twistor space associated with R3 or C3.

One can re-derive these results in a. simple way by considering the problem of

finding minimal 2-surfaces in four dimensions, using ‘normal’ twistor theory.

One can

solve the minimal 2-surface equations in four dimensions in terms of two holomorphic functions of one complex variable.

Looking for solutions confined to a ‘t=0’ hyperplane

leads immediately to the Weierstrass construction.

As usual, let projective twistor space

be given homogeneous coordinates

Z°=[u4,1rA,]

(1)

and consider the curve in P3 defined by

w" =r‘lm.

(2)

where f is homogeneous of degree +1 in 1rA" Now consider the space-time points satisfying

220 l4. Twistor theory and elementary particle physics

21"":A, = f4[7rN]; 3"“ = —i0f4/01A,. Note that this only makes sense if f is homogeneous of degree +1.

(3a,b) Also, by taking the

exterior derivative of

1r#81“ /81rA, = f"

(4)

3f4

we see that

3

_ A’ A

awAlafl'BrdTB’ _ 7r

W (5)

for some differential form WA.

On taking the derivative of (3b) and using (5) it follows

that z“(1r) is a null curve. If we introduce a coordinate

C= wrap/«1,

(6)

then we see that 3““) is also holomorphic in C.

Thus the curve :“(()=X“+iY“ is a

complex null holomorphic curve in C‘. Its real part X“ is therefore a minimal 2-surface S in Minkowski space-time (see e.g. Lawson, 1980: the real part of a null holomorphic curve is a minimal 2-surface in all dimensions).

The coordinate C is related to isothermal

coordinates (u,v) for S by C = u+ iv, and the metric of S is in the form

«Is2 =f(u,v)[du2+ dv’].

(7)

Now consider explicit coordinates. Let

my] = «,.F"[c1

(8)

where F"[c]=f“((,1)= moan]. The conditions (3a,b) become

is“ = 5“,;

is“ = IPA-(F‘s

(9)

(10)

In terms of the usual vector-spinor correspondence, this becomes (with ’ denoting v(=a/a()

~15 t= —if—ig+i1). Our ‘quantization’ treated all fields of any .9 as charged Fermions. usual spin-statistics relations.

In particular no attention has been paid to the

There exist variants of the above construction which would

treat the fields as charged Bosons, or in the case 3 = 0, as uncharged Bosons. However it is not yet known whether the additional technical conditions that enter in these variants are satisfied in the present setting.

(iii) Note that although our classical fields were all right-handed automatically

brings in

the

Fock space built on

(3)1),

the formalism

positive-frequency fields of both

handednesses. (iv) It is obviously important to determine what relation the amplitudes determined by a twistor conformal field theory have to twistor diagram theory.

Any possible link with twistor diagram theory is obscure to the author (but see Hodges in TN 26).

Notable by its absence from the above is dual twistor space. It may be, however.

that the proposed extensions to four dimensions should be modified to allow glueing of twistor space to dual twistor space, even though the two-dimensional theory is formulated solely in terms of one twistor space.

One reason for believing such a thing is the

differences in the actions of the reality sturctures alluded to under (3) above.

At that

level, the two 2-dimensional twistor spaces are, after all, completely unrelated to each other, whereas the twistor spaces are linked by the conjugation in four dimensions. I acknowledge many useful talks with R.J.Baston, E.Dunne, A.P.Hodges, R.Penrose,

246 I4. Twistor theory and elementary particle physics

G.B.Segal, Tsou Sheung Tsun.

References [1] Penrose, R. and Rindler, W. (1986) Spinors and space-time, vol2, Appendix, O.U.P. [2] Eastwood, M.G. and Pilato, A.M., 0n the density of elementary states, §I.3.10. [3] Segal, GB. (1989) The definition of conformal field theory, to appear. Witten, E. (1988) Quantum field theory, grassmanians and algebraic currves, Comm.

Math. Phys., 113, 529-600. [4] Pressley, A. & Segal, GB. (1986) Loops groups, O.U.P.

[5] Woodhouse (1981) Geometric quantization and the Bogoliubov transformation, Proc. Roy. Soc. London, A378, 119-139.

§I.4.22 Pretzel twistor spaces by R.Penrose (TN 26, March 1988)

In the previous article §I.4.21, Singer has provided a very appealing suggestion for combining some of the attractive features of string theory/conformal field theory with

twistor theory.

As an

analogue of the (bounded)

Riemann surfaces of string

theory/conformal field theory (pretzels), one imagines a complex manifold—say of 3 dimensions or 4 dimensions, according to whether we are considering projective or nonprojective twistors—with a boundary consisting of a number of copies of PN or N, as the case may be, see figure 1.

(See also Tsou S.T.'s following article 51.4.23 for a suggestion

{vhereby these PN’s shrink down to lines.)

The most appropriate formulation of this idea

remains somewhat unclear as of now, but we might think in terms of some sort of analogue of the Fock spaces that in string theory/conformal field theory are to be specified at each connected portion of the boundary—say, at each ‘end‘.

Then amplitudes could be

obtained when elements of the appropriate Fock spaces are specified at each end. From the point of view of twistor theory, a Fock space (in its normal physical interpretation) might seem inappropriate.

For reasons of wishing to tie up this idea with

twistor diagram theory—ideas due to Hodges (see TN 26 and next chapter)—and of exploiting the relevant crossing symmetry, spin-statistics, duality, etc., we might prefer to think of the ‘ends’ as corresponding to individual particles, rather than states involving an

§I.4.22 Pretzel twistor spaces | 247

indefinite number of particles as described by Fock space elements.

Figure 1

The twistorial description of Fock space elements would be by certain elements of

c ea H‘(T+,O) e H2(T+xT+,O) e H3(T+xT+xT+.0) e

i.e. by collections of ‘twistor functions’

f0: fi(Z°)s f2(Z°,Y°), fs(zarya»xa)r“

which are, respectively ‘O-functions’, 1-functions, 2-functions, 3-functions, etc., where the nfunction f" is the twistor wave function for a state consisting of n rnassless particles. (The ‘O-function’ f0 would simply be an element of C, defining the amplitude for zero particles.) Instead of this, we might well prefer to think of f0. f1, f3, f3,

as referring to the

functions of several twistors which come up in the twister particle programme. Thus. to a Ifirst approximation‘, f1 might describe the amplitude for the state of a. massless particle, f2

the state of a lepton ('2’), f3 the state of an ‘ordinary’ hadron (??), f4, etc.. for more ‘exotic' hadrons (7??) (and f0 for a ‘nothing’), but an actual massive particle might be imagined to have small contributions all indefinitely far along the sequence.

We expect f1 to be a. 1-

function1 and suspect that I, might perhaps be a relative 1-function (or 2-function?) relative to some sort of diagonal locus in TGT, conformal invariance being broken at this stage (so mass and [cm come in).

Perhaps all the L are relative 1-functions (or 2-

functions) or something (whence foEO).

Perhaps, as with strings, the higher fs describe

higher ‘modes’, where oscillation of strings are replaced by deformations of PN.

Perhaps,

in accordance with an idea floated by G.B.Sega.l, they refer to different jet bundles over

(P)N. 1'I use the term 'n-function' to denote an element of n"I (holomorphic) sheaf cohomology.

248 I4. Twistor theory and elementary particle physics

All this is extremely vague and speculative, as yet. Let us backtrack a little and try to see whether there are, indeed, any interesting ‘pretzel’

twistor spaces.

As with

string/conformal field theories, closed pretzel spaces are of particular interest.

The

simplest way to construct such a twistor space (considering the case of pretzelized ETs only) would seem to be the following. Consider first the two limiting projective motions of IPT:

Ea

’ 6‘]

E >0 small

Figure 2

These may be achieved by

«do

1

0

t

0

o

0

1

0

t

w]

"0'

t

0

1

0

"'0;

7r

0

t

0

1

7r .

w‘

= z?

—=i

no

«N

ll

H

Z? =

with t€(—1,1), the two limiting cases being t—rl, t—o—l, respectively.

The 5-surface into which PN is carried (i.e. given by Zia?” =0) is another copy of PN (identical with it as a CR-manifold)—call it PM.

line L"'Cl"T+ given by

As t-1, this 5-surface closes in on the

A Z9 =

f: p

, A

and as t-—*—l it closes in on L—CIPT’, given by 29 = f" .

—p

§I.4.22 Pretzel twistor spaces | 249

Figure 3

Note that if we define [212:|w°|2+|w1|2+l7ro,|2+|1rl,|2, then |§|2=(1+t2)|Z|2+2tZ°Z. and 24“?“ = (1+t2)Z°Za+2t|Z|2. Hence PN, is defined by

2"z¢.:|z12 = 2t:1+t2.

PNPE (resp. PNE_1) is the boundary of a tubular neighbourhood of L+ (resp. L‘). The simplest pretzel space 9, is given by identifying 2" with g" for some fixed T in (0,1) (and hence, up to proportionality, 2" is also identified with g“ where t=tanh nr, for

all n62).

This can also be achieved by drilling out the regions (inside) above lPNt, and

below (inside) PN_t, (where t’ =tanhér) and identifying the boundaries. There is actually some freedom in how such an identification is made, and this allows different pretzel twistor spaces to be built in this way.

identify boundaries:

gives one ‘handle’ PN_¢’

Figure 4

The spaces EPI are the analogues of toruses, i.e. of Riemann surfaces of genus 1.

For

higher genus (genus y), we can carry out several such identifications simultaneously to give a ‘CIP3 with g handles’. Perhaps easier to visualize, but equivalent, is to take two copies of CP3 and to identify across from one to the other.

250 |4. Twistor theory and elementary particle physics

CP3 '3

We could also do

Figure 6

etc., but this gives us no more generality.

(To see this, clip all the tubes necessary to

make a tree; then fit all the CPas inside one of them and we are back with the case of

‘handles’ as before, when we re-glue where we had clipped.) The analogies between these spaces and Riemann surfaces are quite striking.

In

particular, we can ask for the dimension m of moduli space for pretzel twistor spaces of genus p (i.e., how many complex parameters are needed to characterize a ‘EPg, where a 99 is a CP3 with g ‘handles’ of this type).

(I do not know whether this ‘genus' corresponds to

something standard in algebraic geometry.)

Recall that for Riemann surfaces the answer

(due to Riemann) is 39—3, except when 9:0 or 1—where for g: 0 the answer is 0 and for g: 1 it is 1. For ‘5", the answer turns out to be precisely 5 times as large—with the single exception that for 91, m=3 (instead of 5). surfaces.

Think first of a labelled CPS.

The proof is similar to that for Riemann

The labelling can be achieved by specifying 5

points in general position on CPa.

sanded/H . . paint:

0 .

Figure 7

51.4.22 Pretzel twistor spaces | 251

ere are 15 complex degrees of freedom in the specification of each pipe.

(Or do it with

Lth 2 copies of CP3 as in Figure 5 which may be a. little easier to visualize.) comes from the size of the projective group on CP3 (15:42—1).

member of parameters needed to define a labelled 99.

The 15

We have 159 for the

We have to factor out by the

,Eecdom in doing the labelling—which is 15 parameter’s worth unless, in the generic case,

there are continuous (holomorphic) symmetries of 9,. notions of the labelling corresponding to a symmetry.

We do not factor out by the Suppose that there are d

dimensions of symmetries. Then we get

m = 159—(15—d).

“'hen g=0 then clearly d: 15, so m=0. When g: 1 then a direct argument shows that tn :3 (whence d=3 in the generic case). (This argument is: 91 is defined by Z°'=‘T°BZ‘5, up to proportionality. for some fixed T“,.

The ratios of the eigenvalues of T“, give the

moduli of “.Pl.) It is not hard to see that d=0 whenever 921 (as with Riemann surfaces), so 171: 159—15 in these cases. I should remark that there is a subtlety involved in the glueing of the pipes together

l‘handles’). When we think of PN1_€ surrounding L+, we think of it as an S’xS3 for which

the 33’s surround the various points of U) and where the different points of L+ give the 52:

Figure 8

If we glue this to PN¢_1 we must do so in such a way that the small 335 of PN1_¢ are stretched the length of PN1_¢.

Figure 9

252 |4. Twistor theory and elementary particle physics

and vice versa. This is possible because of a topological relation which I write symbolically as szxsazsaxs’, each side being a circle bundle over 32x52, but where the twist can be transferred from being over the second S2 factor to over the first. Hopf bundle of SI over S2 to give 33.)

(This is the Clifford-

If we take planes through L’, they sweep out one

family of 338 on each IPNt (‘small 338. when 1: 1—6, and running the ‘length’ of PN, when l=c—1); planes through L+ sweep out the other family of 538 (‘long’ when 1: 1—6 and

‘small’ when t: (—1). The relation ‘S’xSazsssz' can also be seen by examining pairs of orthogonal ‘unit vectors at the origin of R4.

Fix attention on one vector: it sweeps out an

53 while the other gives a. (trivial) bundle over it (trivial since 33) is parallelizable). Then think of the vectors in the other order.

(Ti UP“

can”

ClPa, so long as tube is not pinched Figure 10

If we consider a thin tube, where the tube narrows down to nothing, we get in the limit the space (considered by twistorians in connection with null lines on C3, and by Donaldson in a context similar to the present one) which is two CPSs joined along a quadric, which is a blown—up line in each, and identified with generator systems reversed:

blown— u P

blown- up lino,

line

Figure 11

The topology of ‘59, is an S'2 bundle over .S'4 with g handles (SaxR handles).

51.4.22 Pretzel twistor spaces | 253

C|P3

CP3

General

plane in

one

CP3 emerges as plane through the line in the other—and vice- versa.

Figure 12

51 «fibres

and

5" with handles

..

-.---

-r.g‘-i

CP3 with tubes Figure 13

These spaces EP, are ones where there are CPls with neighbourhoods which are identical with portions of CPa—and are twistor spaces of conformally flat 4-spaces.

Much more can be said.

Of course. in line with non-linear graviton constructions we

shall want to deform these spaces; also to deform the CRvstructure of IPN, etc. progress.

Work in

Thanks to M.A.Singer. A.P.Hodges, G.B.Segal, T.S.Tsou, E.Dunne, FLBaston,

and S.K.Donaldson.

254 I4. Twistor theory and elementary particle physics

51.4.23 A possible role of vertex operators in Singer’s picture of four-dimensional conformal field theory by Tsou Sheung Tsun (TN 26, March 1988)

Singer proposes (see §I.4.21) that one can View the Riemann surfaces of two-dimensional conformal field theory as a two-dimensional analogue of twistor space.

This leads to the

further proposal that four-dimensional conformal field theory can be obtained through

replacing the Riemann surface (with boundary a collection of 81's) by a complex 3-dim manifold (with boundary a collection of lPN‘s). Now one way to make contact with the physical world of interactions in the 2-dim theory is to introduce vertex operators. bounding circles.

For simplicity, consider a cylinder with two

If we think of these two circles as representing an incoming and an

outgoing state (just as in string theory). then by conformal invariance the bounding circles can be shrunk to points (with their coordinate neighbourhoods). and the Riemann surface becomes a sphere with two punctures (see figure 1). The ‘physics' is then represented by local operators called vertex operators inserted at these points.

These vertex operators keep track of the momenta, positions and quantum

numbers of the particle states. For spinless particles one can take e.g. V°(k,z) = e ik-X'

and

for spin 2, V2(Ic,z) = aaxbawceik'x. where k: momentum and X: X(z) = coordinate. To get the N-point amplitude one then takes the vaccuum expectation value of the product of N such vertex operators: N A = .

i=1

Depending on the particular problem, these functions of k and 2 can be integrated with respect to either variable, and the resultant functions are also called vertex operators.

Figure 1

The twistor picture is tantalizingly similar.

If we take a line in PT we can choose a

51.4.23 A possible role of vertex operators in Singer’s picture of 4-dim CFT | 255

tubular neighbourhood of it to make it look like a standard PM. (See R. Penrose’s previous article). Since these PN are to play the role of the S1 in the 2-dimensional theory, the lines those neighbourhoods they are then correspond to the points (or vertices) at which one can insert vertex operators.

But a line in PT corresponds to a point in 4-dimensional

”space-time’, and since such a line is in a. PM the point is ‘real’. object to which one can attach vertex operators. dementary states.

80 this looks the right

The obvious suggestion is to consider

Unfortunately I do not know how to do this concretely at present.

Perhaps one can think of a different kind of ‘Penrose transform'. A different way to look at vertex operators is in the representation theory of Diff SI. They correlate different Verma modules rather like the way Clebsch-Gordan coefficients connect different spins.

In fact, I think that it is ‘pictorially’ correct to say that vertex

operators are souped-up continuum versions of Clebsch-Gordan coefficients which are just numbers.

Roughly, let I be a non-negative integer and j a half-integer such that 052351.

Consider the affine Lie algebra of SI(2,C), and denote by VJ- the subspace of the integrable highest weight module corresponding to j, determined by a certain vaccuurn condition. These Vj are irreducible Sl(2,C)-modules of dimension 2j+1.

Then the Virasoro alegra

(centrally extended Lie algebra of Diff .5") acts on each of these Vj. Given a vertex

.71

in

satisfying Ijl—j,|gj_0.

When r>0 the integration is performed over a closed contour surrounding Z- W:0 usually with contour .S'lxsgk—l where k is the multiplicity of the line.

When r50 the contour is

§I.5.l Perturbation theory | 261

taken to have boundary in Z- W: 0.

Wavy lines indicate that the boundary of the contour

must lie in the subspace on which the inner product between the twistor and the dual twistor connected by the wavy line vanishes (the form defined by the diagram should vanish on this subspace).

The boundary contours can be replaced by closed contours by

use of logarithms (see Hodges this volume and 1985b).

Thus. the scalar product diagram. (1), leads to the following integral:

1

DZADW

(2in (A-ZXB-ZXC- WxD- W> as well. It follows from this that

o!””limo (y)

¢B "n =ZW9 ¢BB’--.DDI(I,!J) 9qu

pqrs

and if qur,(Z°) are the twistor functions corresponding to 1.63: __D,(:L‘),

¢n(Z°: We) = Zqurs(W°)qu1-J(Z°)P‘Ir‘

§I.5.2 The twistor transform and propagators | 269

Sparling has evaluated the expression on the right, using as a complete set of orthonormal states

i

r

1

Z-

pZ-D"

f”°"(za) : (pi?!>i(z(.,4)€lliz.sg‘“

_

s l a.

’D-

'

qura( We) = (;i;1)2(2$ MVQIEP' man

where p+q+n= r+s,

A-Z=B-§=—C-c_:=—D-E=1 and A-E=A-E=A-D=B.5‘= 3-25: 05:0.

In his thesis Sparling shows that this leads to

II _

a

¢n0 to derive a sequence (non-exact) of maps

(f)

H”(M.n") = H”(M,d "'1)—’Hp+1(M,d "-2)...“ ---—.HP+""(M,dr2°)_.HP+"(M,c)‘_/.c the last map being ordinary (Cech) cohomology evaluation on the cycle V.

This

incorporates two special cases: (i) good old-fashioned contour integration (given when p=0 and V: the contour); (ii) the original evaluation map (a) (given when p=n and V: M

(compact)). One powerful advantage of this is that it enables us to treat cases when M is noncompact.

(Previously we had had to resort to using compact cohomology for (a) when M

is non-compact, but this is not so transparent.) The disadvantage, of course, is that we need to specify V.

Two examples of the use of (e) spring to mind: (I) non-projective

twistor integrals, e.g. the

1

2

2

Wf...fl%).ud fA'HAd 1’:

that fit in with the twistor particle programme.

(I have checked this explicitly for the

normal l-twistor function evaluation at a field point; here V533).

(II) Evaluation of a

twistor function (H1) at a googly field point—which amounts effectively to M, where =PQE] is the cohomology class (H1) which arises from a. googly map 1::an (ca-spin space),

=§@ being

We have £512] = E; in ordinary twistor diagrams, the line a: in PT being the intersection

of planes A and g.

Here ‘5' is asymptotic twistor space, which need not be compact.

Deforming slightly away from the flat case, we find that a suitable V exists1 with

V=53x33. Finally, it is not hard to see that any evaJuation using a multiply map can also be achieved using a cup product, the specific V for ‘u’ arising as the intersection of the V for

‘o’ with 6‘11, or, homologously, with 67'; ‘u' is more flexible, though ‘0’ is more unique.

1Note added, July 1989: The situation turns out to be considerably more complicated than is suggested here. The Saints3 contour does not actually survive when the googly maps are deformed from those of flat space to those arising in Specific self-dual curved models. It appears that contours of a Pochhammer type are likely to be needed.

51.5.5 Three channels for the box diagram | 275 References

Ginsberg, M.L. 86 Huggett, S.A., §O.5.12 in Advances in twistor theory Penrose. R. (1975) in Quantum gravity; an Oxford symposium. eds. C.J.Isham, R.Penrose and D.W.Sciama. O.U.P. Penrose. R. §O.3.9.

Postcn’pt: We have the following obvious (l) generalization of (e):

H’(M,dn"‘)Kc,

(s)

where M is complex n—dimensional and V is real (p+q)-dimensional.

(dflq-l=sheaf of

holomorphic closed q—forms.) Case q: n gives (e). Proof, the same. ‘Application’: ordinary

twistor integral §%§---flZ)1rA,d1rA' viewed as $1( = V) in C2, p=0, q: 1, n=2.

51.5.5 Three Channels for the box diagram by S.A.Huggett 85 R.Penrose (TN 10I July

1980) Introduction.

In this article we show that all three channels for 8. ¢‘ integral do in fact

survive translation into twistors.

(We do not use any cohomological techniques, and nor

do we blow the box diagram up.) This result refutes the ancient belief that the box

diagram only has one of these channels and vindicates the hope that twistor diagrams incorporate crossing symmetry [1].

Translation Procedure. The massless scalar 45" integral is /¢1¢2¢3¢4d‘z

(1)

S where S is a 4 real dimensional contour in CM.

are elementary states.

Suppose the zero rest mass fields ¢1,---,¢4

The first step in the translation is to choose the following twistor

it

all: ll

H¢2v

V'lTH‘Wp

ll EF

N—Q N-D:

WLWH¢P

IH

functions to generate these elementary states

H454

276 I5. Twistor diagrams and scattering

Given A,B,~--,H in a particular position move them until one of the following conditions holds:

channel : ¢1,¢2 negative frequency and ¢3,¢4 positive channel : ¢1,¢3 negative frequency and ¢2,¢4 positive

channel : ¢1a¢4 negative frequency and 452,453 positive

Now (i) draw in the contour S=M and (ii) continuously move A,B,---,H and the contour until A,B,---,H are in their original positions. We now have three space-time contours, one for each channel. We use the notation We. =

WA ,WA’

so that

1

/WA deAxA' dXA,AYB dYBAZB' dZB,AcI“J:

becomes

(2)

WW 0 D Y Y GH

P:( l I I l

I >

A B XX

where p, restricts all the twistors to go through 1:“. This integral is over a. contour in the 8 complex dimensional space B. B is a bundle over CM with fibre

{( WA,XA,,YA,ZA,)6(CP1)‘}. We define two subbundles of B.

||= :{( WA,XA,,YA,ZA,,1:“)EB:WA0< YA and XA,o0=0_ 15.

(5)

Using (5) again, we

conclude that

d(vi) 2 14.

(6)

The next steps are indicated in Figure 3.

We note that the restriction of (4) exhibits

each of (viii) and (x) as fibrations over 1(C) with contractible fibres. Thus d(viii) = d(x) = d(C) = 7. From these it follows that d(vii) Z 10 and so (6) would imply d(ix)

2 11

and so

d(xi) 2 3.

(7)

A moment's reflection will reveal the result expressed in Figure 4. The map

(Z, W, X, Y) _.. (Z, PV)

expresses (xi) as a fibre bundle over

S2

with contractable fibre (cf. the calculation

involved in (viii) and (x)). Thus d(xi) = d(S’) = 2. This contradiction with (7) completes the non-existence proof.

51.5.15 The topology of the box diagram | 337

338 I5. Twistor diagrams and scattering

K

K

L

=

K

L

Figure 4 3.

The generic box. The same dissection of the box diagram can be used to calculate the

cohomology in the case where K, L. M. N are in general position.

The result is that the

dimension of H16(D) is two, confirming that the contours found by Sparling [4] are the only ‘philosophy two’ contours for the box diagram.

Let us outline the new complications

that arise in this calculation. (a) The space (v) is no longer fibred over (C).

One can proceed either by using

Eastwood’s proposed method or by using the following variant.

One wants to use the

same map (4) to break up the cohomology but the fibres change homotopy type over the parts of the base where Y is incident with K, where X is incident with L, or where both these conditions happen together. We therefore use the LES indicated in Figure 5 to break

(v) up into pieces (v)°, (v)l and (v)2 on which (4) is a. fibration.

51.5.15 The topology of the box diagram | 339

It is relatively easy to compute d of each of these spaces; when the answers are combined

we find d(v) = 14.

The generator is mapped to the generator of H’°(iii) so one concludes that

H160) = H“(vi).

(b) This strategy of ‘filtering’ the space by pieces on which (4) induces a fibration is also used in the analysis of (viii) and (x). The conclusion is that

H”(vi) = Ha(xi).

(c) The result expressed by Figure 4 is clearly false for generic lines K, L, M, N so a further collection of LES is required to make the remaining pair of external lines springy. It follows from the analysis of these that

”160) = 32(P’ X P’- Q1- Q2. C) where Q’- is the zero-set of

0,-(1. a) = QjM’wAaA. for two non-degenerate matrices QjAA'.

This is precisely the space discussed in [4]

We end by noting that the conclusions arrived at here are confirmed (with considerably

less effort) by the general theory of projective twistor diagrams that is developed in [3]. The status of ‘philosophy zero’ hard contours of A.P.Hodges remains obscure.

References. Eastwood. M.G, ‘Some comments on the topology of twistor diagrams’ §I.5.9 Huggett. S.A. and Singer, M.A. ‘Two philosophies for twistor diagrams’ §I.5.12 Huggett, S.A. and Singer, M.A. (1989) ‘Relative cohomology and projective twistor diagrams’ Trans. A.M.S. Sparling, G.A.J. (1975) in Quantum gravity. an Oxford symposium, eds. C.J.lsham. R..Penrose and D.W.Sciama, O.U.P.

340

Chapter 6 Sources and currents; relative cohomology and non-Hausdorff twistor spaces

§I.6.1 Twistor blisters by L.J.Mason

This chapter arose from the study of fields with sources. one must introduce non-Hausdorff twistor spaces.

To understand sources correctly

Relative cohomology is then required in

order to represent cohomology groups of non-Hausdorff spaces in terms of familiar objects on Hausdorff spaces. The relative cohomology classes describe the source structure. Non-Hausdorff twistor spaces and relative cohomology have application in other branches of twistor theory: see, for example, the treatment of stationary axisymmetric solutions of certain field equations in the appendix to this introduction, Woodhouse 31. Mason (1987) and Baston 51.3.17. On a more speculative note, in 50.3.9 (cf. the comments at the end of 51.6.7). Penrose suggests that non-Hausdorff twistor spaces may be required for the ‘googly graviton'.

In chapter 11.2 Penrose proposes a deformation for googly

gravitons whose linearized limit requires the use of relative cohomology classes.

It is also

likely that relative cohomology will be needed for a cohomological interpretation of twistor diagrams; in particular the Feynman propagator is a sourced field and should therefore require some of the following ideas.

In §I.6.12 Penrose explains how non-Hausdorff

manifolds can encode the information of chaotic structures such as the Mandelbrot set giving hope to the possibility of a holomorphic description of chaotic systems such as the

full Einstein or Yang-Mills equations.

Sources and currents. they give rise.

Sources and currents are analysed via the external fields to which

We take the sources to lie on a time-like world-line in Minkowski space.

The description presumably generalizes to sources contained within a ‘worId-tube’ of finite radius. For the purposes of this introduction we shall consider only the straight world-line.

”={(t:3ilhz)€Mlz-=p=z=o}The simplest example is the static solution p1 of the wave equation.

An important

§I.6.1 Twistor blisters | 341

feature is that it is double valued in CM, returning to —% as one analytically continues

around a loop surrounding )3. This can be seen from the ambiguity in the sign of the square root used to define r=‘l1'2+y2+zz. The twistor description involves a quadric Q in PT. This arises as follows.

The Coulomb field and the quadric. Each point of 0' determines the line in twistor space of twistors incident with that point. The union of these lines is a quadric Q in PT given by the equation: Q=wA fiIWAI=Os

where 'I" is the time-like tangent vector to a.

If Z°=(w‘4,7rA,) is incident with rr,

wA=itTAA'1rA, for some t. and the identity TAAITIB:=EAIBI gives Q=0 as required. (We

have normalized T‘ so that T“T¢.=2.) We can set TAAI=0A0AI+LALAI so that Q becomes ulvro,—u°7rl,=0.

(Note that

wl=quA etc.) A nondegenerate quadric. Q=0. in PT is biholomorphic to ClP'xCPl, and is thus ruled by two families of lines.

Two members of the same family are always disjoint. and each

member of one family meets each member of the other in one point.

The points of Q, as

given above, can be parametrized by ([t,z\],1rA,)—o(ilTAA'wAHAvr-AJ where [LA] and :rA, are both homogeneous coordinates for CPGCP'.

The lines t=constant, Azl. correspond to

points in CM] on the (complexified) world-line a. The lines [7A,]=constant correspond to points in complex null infinity. The self-dual Coulomb field has twistor function 1/622, and the ‘unit charge’ solution of the wave equation has twistor function 1/62. The interpretation of these twistor functions and those for more general sourced fields is the subject of the following material.

Double valued fields and a non-Hausdorff twistor space. Fields with sources on a' are fields defined on the complement of or in M.

Let us consider analytic fields. so that we can

complexify to obtain fields holomorphic on a region U containing a in CM; such fields will

only be defined on the complement of the hypersurface 2={r2=1:2+y2+22=0} in U. If a field is single valued in U, then it extends over r2=0, and is therefore source free. In general, fields with a source on 0' are double valued—as one analytically continues the field around a closed path surrounding 13:0 in CM]. it comes back to a. different value, returning to its original value only after continuing it around the closed path a second time.

The analytic continuation of the advanced field from a source is minus the retarded

field (see 511.6.5). For the solution l/r of the wave equation, this double valued behaviour

342 I6. Sources and currents can be seen from the ambiguity in the sign of the square root used to define r=.|:2+y2+32. Let V be the double cover of U—E. It would seem that all fields on U with sauce on a are given by regular fields on V (cf. 51.6.11). Such fields can all be obtained from a twistor description using a non-Hausdorff twistor space. In §II.6.2 (see also 51.6.11 & Bailey 1985) the twistor space V0 for V is the nonl-Iausdorff manifold obtained as follows.

Let Q be the quadric in PT associated to a and

let U be the subset of PT consisting of twistors incident with the region U in CM.

Then

the twistor space, 170, for V is the non-Hausdorff space obtained by glueing together two copies of U everywhere except on Q. The space Va is non-Hausdorff since two points in Va that lie over the same point of Q in U cannot be separated by disjoint open sets each containing one point and not the other (see figure 1). The region V can be recovered as the space of lines in VQ which intersect each copy of

the quadric, Q. in V0 once at different points of Q. For each line in U which intersects Q in two distinct points we obtain two lines in

V0 which intersect both copies of the

quadric. These two lines can be continuously deformed into each other whilst remaining in V0. The space of such lines in VQ is therefore a double cover of the space of those lines in U which intersect Q in two distinct points. to points of 2.

However lines in U touching Q correspond

Thus we see that the space of such lines in VQ is a double cover of U-E

as required and is therefore V. See figure 1.

\

PVQ

Two lieu Co rrupondms *\ a fa ‘HM sang point a} U.

Figure 1

In §I.6.2 Penrose and Sparling show how the Ward twisted photon associated to the antiself-dual Coulomb field is a line bundle on PTQ‘ the space formed by taking two copies of PT glued together on the complement of the quadric. Q. The

Penrose

transform

can

be

generalized

to

give

a

correspondence

between

H1(VQ,0(—n)) and helicity n—2 fields on Vand thus fields with sources on a (§l.6.ll). It turns out to be relatively straightforward to understand the cohomology of non-

§I.6.1 Twistor blisters | 343

Hausdorff spaces obtained in this way. In 51.6.8 T.N.Bailey proves that:

H1(VQ)=H‘(t'/)eH‘Q(f/)

where [110(0) is a relative cohomology group (see also Singer 51.6.10). (The formula is true for cohomology defined by flabby resolutions, care is required when using Cech and, in particular, Dolbeault cohomology. [n §I.6.4 relative cohomology is briefly described.) This decomposition can be interpreted as follows (see §l.6.7).

The group 111(0)

provides the possible source free fields that one can add on to a field with source on 0' and the group HIQU'J) encodes the possible source structures on the world-line a (e.g. all the multipole moments, see 51.6.6).

Hb(l"ll’,O(—4)).

From the long exact relative cohomology sequence we find that

Hb(l"ll',0(—4))EH°(P‘I1—Q,O(—4)). which

is

easily

The Coulomb solution is given by an element of

checked

The simplest element of this second group is 1/03

to correspond

to the self-dual

Coulomb

field.

Similarly

l/QE HIQ(PT,O(—2)) corresponds to the static solution of the wave equation l/r. In §I.6.4 T.N.Bailey shows how to obtain a relative cohomology element for the basic solution of the Maxwell equations with unit charge on a general world-line.

ln §1.6.5

R.Penrose shows how to obtain the advanced and retarded fields associated to this source,

and shows that the advanced field is the analytic continuations of minus the retarded field. In §I.6.6 T.N.Bailey produces formulae for the relative cohomology element corresponding to sources (in a general world-line with arbitrary multipole structure, in particular he shows how to obtain a twistor integral for the right handed monopole (charge) of a source.

In

51.6.7 R.Penrose shows how to interpret relative cohomology in terms of line bundles on the non-Hausdorff twistor space.

Appendix to 51.6.1: Non-Hausdorff reduced twistor space for stationary axisymrnetric fields

The material in this appendix arose from constructions in Woodhouse and Mason (1988). That paper was concerned with stationary axisymmetric self-dual Yang-Mills and its application to stationary axisymmetric solutions of the Einstein vacuum equations (see Ward 1983 and 51.4.9). Here I will only discuss massless scalar fields. A typical situation where a non-Hausdorff space arises is when one is considering the quotient space of a foliation in which there exists families of leaves with one connected

344 |6. Sources and currents

component whose limit is a leaf with two connected components. (See Woodhouse 51.6.3.) Fields invariant

under symmetries on

a region

U in space-time correspond

to

cohomology classes which are invariant under the action of the symmetries on the appropriate region (7 in twistor space.

Invariant cohomology classes can be obtained by

pulling back cohomology classes from the reduced twistor space R( U), the space of orbits

of the (complex) symmetries in (7.

If the orbits are contractiblc, we obtain all the

invariant cohomology classes. In the case of a time translation and a rotation about an axis. the space of orbits R( U) is non-Hausdorff for all reasonable choices of U. This non-Hausdorff feature is essential—if we identify the disconnected orbits together. we obtain CPI, a Hausdorff reduced twistor space. The cohomology on CP1 is finite dimensional or zero, depending on the sheaf and so cannot describe general stationary axisymmetric solutions of field equations—these depend

on free functions of one variable.

As will be seen. the cohomology of the non-Hausdorff

reduced twistor space is infinite dimensional and provides the general solution of the stationary axisymmetric wave equation.

The action of (complex) time translation on twistor space is generated by the holomorphic vector field:

T=«A,7"""a/aw and the action of rotation (again complexified) about an axis is given by the vector field: I

where: I

I

7'°=0A6A'+tATA'. ¢AB=oMtBl and ¢AlBl=5M EB l.

I

There are two invariant homogeneous functions. somll=¢A

I

I

"NIB" and p=uA7fi "A"

These can be used to parametrize the orbits of the symmetry group. The two dimensional orbits can be parametrized by 7: p=7rro,1r',.

For 76C these are nondegenerate quadrics.

However for 7:00, the quadric degenerates

into a pair of planes. 1r ol— —0="1" and 1r‘,=0=1r 0" each of which is a distinct orbit. So the space of leaves. 12, is non-Hausdorff, being a Riemann sphere with two points at 00. [There are three one dimensional orbits and a couple of zero dimensional orbits which we can ignore for present purposes.] Cohomology classes on R can be pulled back to PT-I to provide invariant cohomology

§I.6.1 Twistor blisters | 345

classes.

Cohomology classes on R can be described very simply.

it is first, perhaps worth

noting that the pullback of 0(—1) from R to PT—I is 0(—2). This can be seen from the fact that [p.7r0nr1,] are homogeneous coordinates on R.

A function of these coordinates of

homogeneity —1. has homogeneity -2 with respect to homogeneous twistor coordinates. The non-Hausdorff reduced twistor space is of the type covered by Bailey‘s theorem 51.5.8. This implies:

H‘(R,0(—1))=H},.(CP‘,0(—1))eH1(CP'.0(—1)).

Note that H1(C|P1,O(-1))=0.

From the long exact relative cohomology sequence we find

that:

H°(CP‘.0(—1)) —. H°(CIP1—oo,O(-1)) — H;(CP1,0(—1)) — o

and since H°(C|F",O(-1))=0. we have:

H'(R,O(—1)) = H°(C|P'—oo,O(—l)) .

Thus. elements of H1(R,O(—l)) can be represented by entire holomorphic functions. K7). To evaluate this cohomology class we can pull it back to twistor space. and insert it into the usual twistor integral formula. Since f(7) should have homogeneity —l on [2—00. on twistor space it will be represented by the twistor function 1(7)/(7r0,1r1,). If we choose affine coordinates on twistor space such that 1': l ,=1. 1ro,=C. then the twistor integral formula becomes:

Am 9‘ A—=¢ —1r'4,,d1r ( )= 1(7) (Tm— f(7) dCg

which is the integral formula in R.S.Ward’s article §I.4.9. This integral formula can be found in Whittaker 8.: Watson §18.3. W'hen j(7)=cu7. the

formula yields eazJo(ap) using a classical integral formula for Bessel functions of the first kind, and when fl7)=7" it produces the axisymmetric solution of Laplace‘s equation

r"P,.(cos 0) where P,I is the n lh Legendre polynomial.

(Here 13=1A2+y2+22. p2=$2+y2,

and p=r cos 0, where (1:,y,z) are rectangular coordinates on 3-space.)

The local reduced twistor correspondence.

The reduced twistor space R, used above. is

only appropriate for stationary axisymmetric fields which are regular over the whole of affine Minkowski space. This excludes fields which tend to zero as one goes to 00 as these in general have singularities on the interior of affine Minkowski space.

If we wish to

346 |6. Sources and currents

consider smaller domains we must examine the correspondence between space-time and the reduced twistor space more carefully. The reduced correspondence can be obtained as follows.

Minkowski space.

Consider a region U. in affine

A point IEU corresponds a map from a fixed CP1=PS’ (projective

primed spin space) into a line L, in PT-I given by WA,—o(wA,wA,)=(i:AAI7rA,,1rA,). CPl can then be projected down onto R, the reduced twistor space.

This

In order to express

this in terms of coordinates set 1rl,=1. r0,=( and

A,

B,

’

A

274A 1: ) =(—2zo('4 t

B,

I

'

.

I

I

,

)—re"LA (.8 +re"'o" 03 ),

where (r,0.z) are cylindrical coordinates on the 3-spaces orthogonal to T“, and T“ is normalized so that T"T4=—l, so that T“ is imaginary.

With these coordinates the map from P5, - R is:

'OC) ,1l’l,=—z _5(_L+e—I (z’r’C) —' 7=Il/7l'o Bloc

This map is 2-1 except where the discriminant:

A(7,r,z)=.l 7+z+ir)(7+z — ir .

vanishes. where it

is

1.1.

The map is 2-1

where the line

L:

intersects the orbit

corresponding to 7 in two points, and 1-1 where it touches the orbit at a double point.

K‘— l’omb

3': 3‘?”

for (1")611

V RM

“1501“,;

y; gait“

for

(I,Y)£U

Figure2 We can now identify the reduced twister space, R( U), corresponding to an arbitrary region. U. in M. If U' is the region in PT consisting of twistors incident with U. then R( U)

51.6.1 Twistor blisters | 347

is the non-Hausdorff space of (2-dimensional) orbits of the symmetry group in U'. orbits which are disjoint in U’ being counted separately. It can be seen that an orbit ”=71ronrl, intersects U’ in just one piece if the discriminant A(7,r,z) vanishes for some 1', z in the region U.

If U is simply connected, all other orbits will intersect U' in two pieces. Thus.

for each '7, there are two points in R( U) unless 7=—z+ ir for (r,z) contained in U.

This is sufficient to characterize R( U) if U is topologically trivial and contains a piece of the axis r=0; R( U) is obtained by taking two Riemann spheres with 7 an affine coordinate on each sphere and glueing them together over the regions where 7=z+ir for

(z,r)€ U. The arguments used previously can be applied again to represent the cohomology on R(U) in terms of holomorphic functions, 1(7) defined for 7=z+ir with (z.r)EU; thus. for example. 7'" give the axisymmetric multipole solutions of the Laplacian, r'""Pn(cosfl).

If U excludes the whole axis, the region on which R is double covered by R( U) is not simply connected and it turns out that the double covering is twisted.

This can be seen

from the fact that if (r.0,0)€ U. the associated map PS’ -. R( U) is onto. so that

7=(-r/2)(C - 1/0

7:7. (=3

defines a 1-1 map from the real axis. (:2 in PS, which is an S). to the region in R( U) over the circle 7:7 in R.

Such regions.

U. which exclude the axis, are required

for the description of

axisymmetric solutions of the Laplace equation such as Log r, and those based on Bessel functions of the second kind, which have logarithmic singularities on the axis.

Bailey’s

theorem no longer applies. and one must compute the cohomology by hand using an explicit cover for R( U). In particular the function log 7 can be interpreted, with respect to

a suitable cover, as an element of H1(R( U).0(—1)). This gives rise to the solution log r.

Further applications. The motivation in Woodhouse 8.: Mason (1988) for this construction was the description of stationary axisymmetric anti-self-dual Yang-Mills fields and. by imposition of further realtity and symmetry conditions. stationary axisymmetric solutions of the full Einstein equations. We obtain the result that stationary axisymmetric anti-selfdual Yang-Mills fields on U are in 1-1 correspondence with rank-2 holomorphic vector bundles on R( U) that are trivial when pulled back to the spin bundle.

These bundles can

be described explicitly using patching functions with respect to a Stein cover of R( U). may prove interesting to study deformations of this correspondence.

It

348 I6. Sources and currents

References Bailey, T.N. (1985) Twistors and fields with sources on world-lines, Proc. Roy. Soc. A397 143—55. Bailey, T.N. 8.5 Singer, M.A. (1989) On the twistor description of sourced fields, Proc. Roy. Soc. Lond. A422 pp. 367-385. Ward, RS. (1983) Stationary axisymmetric space-times: a new approach. Gen. Rel. Grav. 15 no. 2, 105-9.

Woodhouse, N.M.J & Mason, 1...]. (1988) The Geroch group and non-Hausdorff twistor spaces, Nonlinearity 1, 73.

§I.6.2 The anti-self-dual Coulomb field's non-Hausdorff twistor space by R.Penrose $5 G.A.J.Sparling (TN 9. November 1979)

The global twistor description of the Coulomb field (or, rather, of its anti-self—dual part) in CM (complex compactified Minkowski space) encounters certain difficulties owing to the fact that it is a 2-valued field: the field changes sign as 3 is crossed.

Furthermore. deleting

3 or C3 does not help since, within CM, C3 can be ‘dodged around‘. A further difficulty is that the charge world-line q. which is the only invariant structure available, defines merely a quadric Q in PT. and the difference, PT—Q, being Stein, has no holomorphic (coherent) cohomology and no useful holomorphic line bundles over it.

Moreover the line bundles

over PT itself are simply powers of the Hopf bundle H.

Yet, over any reasonable

neighboughood of a general line in PT there will be a ‘twisted photon’ line bundle representing the field. It would seem that one ought to be able to piece these line bundles together to obtain a line bundle representing the field globally.

What can this global line

bundle be? The answer is that it is a bundle over the non-Hausdorff complex manifold PTQ which is obtained by taking two copies of PT and identifying them along corresponding points of PT—Q.

Thus, PTQ is like PT except that each point of Q in PT becomes a pair of

Hausdorffly non-separated points in PTQ. To construct the Coulomb bundle C over PTq, we patch together the trivial 0-bundle Ho, over one copy of PT with the (—‘2)-bundle H2 over the other copy of PT as follows.

§I.6.2 The anti-self-dual Coulomb field's non-Hausdorff twistor space I 349

Let [Z°] denote the projective twistor

corresponding to Z“.

Then the points of the

product bundle H0 are simply pairs (A,[Z°]) with AEC. [Z°]EIPT, whereas the points of H2 can be represented as (z°z”.[z*]). Let the equation of Q be QaBZ°Zp=O. Then the fibres

over points of IP'IT-Q (i.e. where QafiZ°Zagé0) are obtained by simply identifying (A,[Z°])

with (z°z",[z*]) where A=Q052°z". (This is clearly linear along the fibre as is required). Over Q itself we retain two separate fibres over each point, so the result is a line bundle over PlI’Q. To work out the field at a point p of CM]. take the corresponding line .2. in lPT. Provided that p is not zero perpendicular distance from q, .2. will meet Q in two distinct points. Consider one of these points to be taken from one copy of Q in lPlI'Q and the other to be taken from the other copy. The part of the bundle C which lies above I. described in this way is a 1-bundle (H1), and the same holds if we vary 1. a little bit. This non-trivial

structure over a neighbourhood of L gives the (anti-self-dual) Coulomb field (for unit charge) in the corresponding

neighbourhood of p.

intersections of L with Q coincide. intersections are interchanged.

The field blows up when the two

The field changes sign when the roles of the two

This may be achieved continuously, varying L with the

points never coinciding.

To construct the charge n (anti-self-dual) Coulomb field. identify lPT—Q, where

(A.IZ°1)E(Z°‘Z°2-i-Z°’".[Z“1) with A=(QapZ°Zp)" (Tensor through by H"" at the end, if desired).

H0 with H?" over

350 l6. Sources and currents

§I.6.3 Non-Hausdorff manifolds by N.M.J.Woodhouse (TN 9. November 1979)

Recent developments (cf. §l.6.2) suggest that we may soon have to learn about analysis on non-Hausdorff manifolds.

Fortunately. this may not be as difficult as it seems at first

sight.

One way non-Hausdorff manifolds can arise is as the space of maximal foliation in some higher dimensional Hausdorff manifold M'.

leaves of a

Consider, for example, the

manifold illustrated as follows:

r%+-l Figure 1 Here M, is R2 less the set {(1.0);1:20} and the foliation is tangent to 0/61}. The space of leaves is a one dimensional manifold in which the two points corresponding to the two halves of the y—axis cannot be separated. Another way they can arise is by piecing together coordinate patches.

Suppose that

{Vi} is a collection of open subsets of R" and that {(vijifijn is a collection of transition functions.

That is. each

V“- is a (possibly empty) open subset of VI- and fij is '1

diffeomorphism from VI.)- onto an open subset fill V”) of Vi, with

(l) Vii=Vh f“ is the identity on Vi

(2) (vjiin)=(jij(Vij)Ji—jl) (3) fij( Vijn Va): Vjin ij and fjkofij=flk on Vijn Vii--

Let K be the disjoint union of the Vi‘s and let ~ be the equivalence relation:

”16V,- and ye V-. then r~y whenever 1:6 V”- and y=f.-j(x).

Then M=K/~ is an n-dimensional manifold (with the obvious definitions).

§I.6.3 Non-Hausdorfl manifolds | 351

However. M need not be Hausdorff.

For example. if V, and V, are copies of R, with

Vu=(0,00)C Vx, V21=(0.oo)C V2. and f” equal to the identity. then M looks like this:

Figure 2 where we have two copies of the origin which cannot be separated. Clearly there is a sense in which the first type is easier to deal with, since we can use ‘Hausdorff’ theorems in M'. However. if only a finite number (N say) of Vi‘s are involved. then the second type can also be recovered from a foliation of a Hausdorff manifold M’ with n+2 dimensions. The construction of M' is very simple.

We simply take a. polygon P with N sides

(labelled by i,j,k...: we assume that N23) and we form the disjoint union of the sets Vix P (including the sides of P, but omitting the vertices). Then for each a: in each V”, we glue

together the two copies {1:}xP and {fiJ-(rnxP of P by identifying the l-Slde of {Lj(r)}xP with the j-side of {1:}xP, again leaving out the vertices.

identify

1

1

k

k

{13XP

{rump P Figure 3

The result may still be non-Hausdorff; but if we delete the edge points of the polygons which have not been subject to identifications, then the result is a Hausdorff manifold M'. The copies of P in M, are tangent to a foliation and the space of maximal leaves is simply M. it is easy to extend this to complex manifolds (taking P to be a. subset of the Argand plane). One can also deal with the case in which N is countably infinite. Although it is rather trivial, this may give a way of extending Hausdorff theorems to non-Hausdorff manifolds.

352 [6. Sources and currents 51.6.4 Relative cohomology and sources on a line by T.N.Bailey (TN 14. July 1982)

Introduction.

If we have a piece of analytic world-line in

real

Minkowski space

parametrised by its proper time, we can thicken it a little into the complex to obtain a map y“(s) from a complex neighbourhood of a real line segment into complexified Minkowski space. In this article we give a construction for a twistor function which encodes the field of a scalar charge moving on the world-line and derive from this a twistor function for the right-handed part of the electromagnetic field of a unit charge on the world-line.

Finally,

we interpret these twistor functions as elements of relative cohomology groups. although a full understanding of what these groups represent has not been achieved (however see the subsequent articles etc.).

The Scalar field. Given an analytic world-line y“(s) and Z=(w‘4.7rA,), we define:

EAzwA—iyAAl(s)7rA,

Leaving questions about contours aside for the moment. we define a twistor function flZ" ) by:

where 0A and LA are fixed spinors and 0'L=OAI.A etc. The space-time field corresponding to flZ") is given in

the usual way (up to

multiplicative constants) by:

50(3) = f A "foAL'B—EASHAA I "A, OILLB(1:—y(s))B ' B1r BI

and doing the 1r integral first we have

”. but not over ‘ER. itself.

For

this to be possible. of course. the parts of 9b and '53, which lie over N—%=N'—€R= must be

isomorphic as bundles. Now instead of partially identifying the line bundles “.B and ‘3', let us consider them as distinct bundles over .N' and divide one by the other. This operation is to provide the map HER, -o H‘(.N').

Indeed. it evidently yields the kernel of the following

map H1(N) —r H1(.N'—€Ra) since these are the line bundles which trivialize over N—‘Jt. which is precisely the condition that ‘38 and 53’ should be isomorphic over .N'—‘.R:, as required

above. Note that Hick does not actually classify line bundles over if. but measures only the resultant ‘twist' between the two branches of the bundle as we pass across ‘32:. This is as it should be because Him, only measures the source structure and does not tell us how much radiation is present in the field.

The latter information resides in H‘(.N'). and we would

expect to pick that out from the remaining

information which resides in the line bundle

51.6.8 Why relative cohomology describes sources | 363

over .N'. Indeed. this seems to work. It is also quite easy to see that H°(.N'—€Ra)/H°(.N') provides ‘twists’ between the branches. which are those yielding zero in ”'15P. -0 H1(.N'). consistent with (A). Now this idea seems to have wide application in other contexts and particularly to the suggestion (cf. 51.3.9) that googlics might be described by non-Hausdorff structures.

In

this case it would be the line II (or some blown up version of l) which might be doubled up in some suitable kind of neighbourhood.

For the googly graviton it would not be line

bundles but deformations that would concern us.

Generally, we would consider replacing

0* by some sheaf of deformations and it appears analogously to the above discussion that the relative cohomology class would measure the ‘degree of blistering’ (say infinitesimally) over the doubled-up subspace.

A;

v k.—

blister

This is just what is needed for the (weak) googly graviton.

Thanks to T.N.Bailey and M.G.Eastwood.

51.6.8 Why relative cohomology describes sources by T.N.Bailey (TN 15, January 1983)

The results and notation of §l.6.3-§I.6.7 are assumed thoughout. copies of a neighbourhood of a piece of ruled identifying U and U' except on L.

surface 1..

Let U and U’ be two

Let U be the space obtained by

Suppressing the sheaf, the Mayer-Victoris sequence for

U, U, looks like

-- -. [1"(0) -. H”(U)eH”(U’) -+ H'w—L) —. H"+‘(U) -. which we can write as the direct sum of the relative cohomology seqence

364 I6. Sources and currents

.. —. Hy U) -. H”( U) -. 11”( (1—1.) -. Hi“(U) —. with the nearly trivial sequence

-. H'w') -. H'w’) -. 0-. H’“(U’) -. Thus we see (e.g. by the Five lemma) that

H'( U)=H1(U)eHP( u').

The relevance of all this is that fields with sources on y‘(s) have branch singularities at places where the advanced and retarded points on the world-line coincide.

They are. thus

fields on a double covering of a neighbourhood of the world-line with these points deleted.

being (say). the advanced field on one sheet and minus the retarded on the other. The space (7, as defined above. is precisely the ‘twistor space‘ corresponding to this double covering (provided one considers only lines which intersect once in each copy) and

so one should expect that fields on the double cover correspond to elements of {11(17). But as we saw above

H‘(U)=H‘(U)$H'(U') which gives a decomposition into sources on the world-line represented by H}! U) and free fields (which don’t ‘need‘ the double cover) represented by H’( U'). I think this viewpoint clarifies the role of H.“ U) in describing sources.

§I.6.8 Why relative cohomology describes sources | 365

51.6.9 Cohomological interpretation of f(Z)=QlogQ by T.N.Bailey (TN18. July 1984)

In §I.6.6 it was shown how f(Z)=log{Q/(0A'1A,)2}, Q=Qapz°ZB, which is known to be a twistor function for the Coulomb field of a charge moving on the straight world-line corresponding to the quadric Q: QOBZ°Z‘B, could be described cohomologically.

Briefly.

one exponentiatcs and uses Q/(OA'WA,)2 etc. as a representative cocycle for an element of HIQ(lPl.O).

The analogous spin-2 field is the linearized Schwarzschild solution,

the left

handed part of which is generated by the twistor function Qlog{Q/(0A'1I'A,)2}. where 0", is any constant spinor. We now give a cohomological interpretation of this. Define the quotient sheaf C(2) by the exactness of:

A05

—. Aaazozfi

o —. Tm” —. 0(2) —. (3(2) —. 0 We note first that first cohomology with coefficients in 6(2) generates massless spin~2 fields just as 0(2) does, because the 0 A . s in the contour integral are (more than) sufficient to kill off functions of the form AO:Z°Z‘,.

If we are dealing with ordinary cohomology on a

region for which the Penrose transform is an isomorphism. then we don’t describe any new

fields by using 6(2) in place of 0(2). In the case of H'Q(Pl.C(2)) (to which Qlog{Q/(o"’7rA,)2} will

shortly be shown to

belong) the long exact sequence corresponding to the defining sequence of 6(2) contains the segment:

0 _. H'q(P'.0(2)) —' ”b(P'-C