CONFORMAL TECHNIQUES IN STRINGTHEORY AND STRING FIELD THEORY

StevenB. GIDDINGS

JosephHenry Laboratories, Princeton University, Princeton, NJ 08544, USAand LymanLaboratory of Physics,Harvard University, Cambridge,MA 02138, USA

ItNORTH-HOLLAND - AMSTERDAM

PHYSICSREPORTS(Review Section of PhysicsLetters)170, No. 3(1988) 167—212. North-Holland. Amsterdam

CONFORMAL TECHNIQUES IN STRING THEORY AND STRING FIELD THEORY

StevenB. GIDDINGS*JosephHenry Laboratories,Princeton University. Princeton. NJ 08544. USA

and Lyman Laboratoryof Phvszcs**.Harvard University, (‘ambridge,MA 02138. USA

ReceivedApril 1988

Contents:

I. Introduction 169 3.4. Summary 1932. Fundamentaltechniques 172 4. Equivalenceof Polyakov and light-coneformalisms I 9-1

2.1. The Polyakov path integral 172 4.1. The integrationregion 1962.2. Riemannsurfaces 176 4.2, The integrationmeasure 199

3. String field theoryand the geometryof moduli space 18)) 4.3. Wave functions,vertex operators.and physicalstates 2023.1. Basicstring field theory 181 4.4. Summary 2073.2. String field theory and the triangulation of moduli 5. Conclusion ~09

space 185 References 113.3. String field theoryand the Polyakov measure 191)

Abstract:The applicationof someconformaland Riemannsurfacetechniquesto string theoryand string field theory is described.First a brief review (it

Riemannsurfacetechniquesand of the Polyakovapproachto string theory is presented.This is followed h~’a discussionof somefeaturesof stringfield theoryandof its Feynmanrules. Specifically, it is shown that theFeynmandiagramsfor Witten’s string field theoryrespectmodularinvariance.and in particulargive a triangulationof moduli space.The Polyakovformalismis thenusedto derivetheFeynmanrules that shouldfollow from thistheoryupongauge fixing. It shouldalsobe possibleto apply this derivation to deducetheFeynmanrules for othergauge-fixedstring field theories.Following this, Riemannsurface techniquesare turnedto theproblemof proving theequivalenceof the Polyakov and light-cone formalisms. It isfirst shown that the light-conediagramstriangulatemoduli space.ThenthePolyakovmeasureis workedOut for thesediagrams,andshownto equalthat deducedfrom the light-cone gauge-fixedformalism. Also presentedis a short descriptionof the comparisonof physical statesin the twoformalisms. The equivalenceof the two formalisms in particular constitutesa proof of the unitarity of the Polyakovframework for the closedbosonicstring.

* Supportedin part by a National ScienceFoundationGraduateFellowship.and by National ScienceFoundationGrants PHYSO-t9754and

PHY82-15249.** Presentaddress.

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SB. Giddings, Conformaltechniquesin string theoryand string field theory 169

Dedicatedto myparents,and to Elizabeth

1. Introduction

String theory hasin the pastfew yearsstirred up a greatdeal of excitementin the theoreticalphysicscommunity. This excitementresults from string theory’s promiseto be a “theory of everything”, aunified theory of all matter andall forces. If this promiseholdstrue, then string theorysolvesat oneblow a largenumberof the outstandingfundamentalproblemsin field theory and particle physics.Itoffers us a meansof treatingproblemsthat many theoreticalphysicistsmight not haveexpectedto beapproachablein this century. In particular,stringtheory appearsto bring from asomewhatunexpecteddirection a solutionto a problemthat hasplaguedtwentiethcenturyphysics,namelythat of reconcilinggravity and quantum mechanics.String theory also promises to answer other deep and difficultquestions:the questionof howthe gaugegroupof natureis determined(andwhat it is); the questionofwhy the fermionsshouldfall into certainrepresentationsof this group;the questionof why we have(atleast)threegenerationsof thesefermions;andthe questionof what determinesall of the variousmassesand couplingconstantsin our low-energyworld. String theory shouldalso answerquestionsabouttheshort-distancestructureof spacetime,andaboutvery earlycosmology.It hasevenbeensuggestedthatstring theory might shedsomelight on an evenmorefundamentalquestion,by possibly providing arationalefor quantummechanics.*)

It is clear that string theory hasmuch potential. It is equally clear that thereare many currentdifficulties in stringtheory. A largenumberof thesefall underthe broadcategoryof compactificationfrom the supersymmetrictheory in ten-dimensionalflat space,with a low-energygaugegroupsuchasE8 x E8, to our non-supersymmetricworld of four dimensions,with gauge group SU(3)x SU(2)xU(1). Under this categoryare the issuesof preservationof low-energy supersymmetry[1] and ofsymmetrybreakingto the observedgaugegroup of low-energy physics[1,2]. Hereit is essentialto seehow stringtheorychoosesavacuumwith the correctpropertiesto producearealisticlow-energyworld;at the presentthe mechanismby which the theory choosesamongpossiblevacuais a mystery. Alsoassociatedwith this categoryis the difficult problemof finding a workabledescriptionof the physicsofsupersymmetrybreaking,andthe possiblyrelated(and quitevexing) questionof why the cosmologicalconstantis so miniscule.Oncethe issuesin this categoryarecompletelyunderstood,we shouldhavetheknowledgeof how, at least in principle,to calculatelow-energycouplingconstantsand the fermionicmassspectrum,and in particular we should be able to check whetherstring theory reproducesthestandardmodelat low energy.

Beyondthe problemsof relatingstringtheory to “real physics”,thereis oneBig Question:namely,whatexactly is stringtheory? Of coursethis questionis relatedto our morepracticalproblems;with adeeperknowledgeof the natureof string theory we would likely havethe tools to better addresstheissuesof compactification,etc. A full-blown non-perturbativeformulation of string theory,such asstringfield theory, is likely to shedlight on the mysteriesof the compactificationmechanism.Yet thisfundamentalquestion certainly has a great deal of validity on its own. We are supposedto bediscoveringa new fundamentaltheory of physics here.Yet sofar, all we haveis a collectionof ad hocrulesandcalculationalschemes.The “preferred” string theory,the heteroticstring,certainlyappearsto

*)Indeed in string perturbationtheoryclassical and quantum physics are united under the rubric of conformal field theory on a Riemann

surface.Thedifferencebetweendoingclassical andquantumphysicsis just thedifferencebetweenworkingon asurfaceof genuszero(the sphere),or on surfacesof highergenus.

170 SB.Giddings, Conformaltechniquesin string theory and string field theory

be constructedin a ratherartificial way. Eventhevery elegantPolyakovapproachto the calculationofstring scatteringamplitudesseemsquite ad hocand incomplete.Indeed,the Polyakovapproachtells ushow to computescatteringamplitudesin a fixed backgroundfield configuration. On the otherhand, itdoes not, except through a rather round-aboutroute (namely through the conditions of conformalinvariancefor the correspondingsigma model [3], or through the analysisof scatteringamplitudes[4]),tell us how to find the field equationsor their solutions, much less the correct non-perturbativeframeworkto describestring theory.

To be sure, despite its shortcomingson the fundamentallevel, string theory has had a numberofastoundingsuccesses,and has potential for many more. The accomplishmentthat touched offwidespread interest in string theory is of course the cancellation of anomaliesand infinities forparticular gauge groups [5]. Physicists were quite pleasedby the fact that the requirement forcancellationof anomalieshighly constrainsthe gaugegroupof the theory; in fact the gaugegroupof theten-dimensionaltheory is almostunique. Following this the study of compactificationswith low-energysupersymmetry[1, 2] hasshownthat string theoryis capableof giving a very closeto realisticpicture oflow-energyphysics, with fermions in complexrepresentationsof appropriategaugegroupsandwith anatural explanationof the occurrenceof the generations.Other successesinclude the use of presenttechnology to deduce the first terms in the string equations of motion, including corrections toEinstein’sequations[3, 4]; perhapssuchwork will shed light on aspectsof generalrelativity, suchasthesingularity theorems.Further, in addition to providing a solution to the problemof quantumgravity,string theory also tells us how to incorporateparticlesof arbitrarily high spin into consistenttheories.Finally, the greatdegreeof uniquenessand the fact that thereare no free parametersin string theoryshould hopefully allow us to go much further, perhapseven to calculation of quantitiessuch as thefine-structureconstant,the massof the electron,etc. It will certainly be interestingto see how far wecan go without a fundamentalunderstandingof string theory.

Despite the lack of a solid foundationfor string theory, thereare a numberof hints of the deeperstructurebehind string theory. Many of the currently understoodnice featuresof string theory stemfrom the fact that the stringperturbationseriesis closelyconnectedto the theory of Riemannsurfaces.For one thing, the fact that the Feynmandiagramsare two-dimensionalmanifolds is responsibleforboth the good ultraviolet behavior of string theory, and for the high degree of uniquenessof itsinteractions.Furthermore,thereare closeties betweenstring theory andthe complexgeometryof bothRiemann surfaces and of the space of Riemann surfaces (moduli space). Indeed, it seemsquiteastoundingthat physicalprinciples like gaugeinvariance,or locality, have led physiciststo deducetwoinequivalenttriangulationsof the moduli spacefor a surfaceof arbitrary genus,a by no meanstrivialtask in view of the topological complexity of moduli space. There are also close relationsbetweenquantitiesof physical interest(e.g. correlationfunctions,partition functions) andthe complexfunctiontheory on both Riemann surfaces and moduli space. The foundationsof string theory are alsoconnectedwith special propertiesof Lie groups (for example the triality of SO(8), or the specialfeaturesof E8), and also to the theory of infinite-dimensionalLie algebras.

In short, string theory has many featuresthat makeit appearthat it is likely a sensibletheory ofphysics. String theory also hasa numberof mysteriousandelegantaspectsthat makeone feel that itshould havean underlyingbasis of greatprofundity. Therearehints of deepconnectionswith variousfields of mathematics—thetheory of Riemannsurfaces,algebraicgeometry.affine Lie algebras,andnumbertheory. One has the impressionthat we are only scratchingthe surface,and that a very richmathematicalstructurelies just below.

So far Riemannsurfacetechniqueshave beenquite instrumentalin the study of string theory.Theseedsfor the developmentof Riemannsurfacetechniqueswere sown in the bronzeageof stringtheory,

S.B. Giddings, Conformaltechniquesin string theory andstring field theory 171

and havebegunto be thoroughlyexploredin the presentiron age.Whethertheywill continueto be acentralfeatureremainsto be seen.Indeed,Riemannsurfacesappearin stringperturbationtheory,butthat is certainlyno guaranteethat theyarea fundamentalelementof the deeperstructurefor whichweare hunting, any morethanthe theory of graphsis fundamentalto ordinaryfield theory.Nevertheless,it will be the goalof this report to developandutilize aspectsof the theory of Riemannsurfacesfor thestudyof someaspectsof stringtheory.Specifically, I will discusstechniquesthat havebeenusefulin thestudyof string field theory,and could conceivablybe usefulin the furtherdevelopmentof string fieldtheory.Again, string field theoriesin their presentincarnationmayonly be remotelyconnectedto theultimate formulationof string theory,but yet theymaycontaina germof truth. Furthermore,I discusswhat may be thought of as a reverseapplication, namely that of using string theory to deduceinformation aboutmoduli space:I will sketch the rediscoveryof a triangulationof moduli space,andthe discoveryof a completelydifferent triangulationof moduli space.Finally, I will outline the useofRiemannsurfacetechniquesto prove the equivalenceof two different formulationsof string perturba-tion theory,and thus to prove the unitarity of the Polyakov formalismfor string theory.

The organizationof this report is as follows.Chaptertwo beginswith someintroductory material. The first sectionconsistsof a review of the

Polyakov approachto string theory, including the gauge fixing and the expressionof the Polyakovintegrandas a measureon moduli space. The following sectioncontainsa brief review of Riemannsurfacetools that areusefulin studyingthe Polyakovpathintegralandthe geometryof theworld sheet.These include quasiconformaldeformations and Beltrami differentials, quadraticdifferentials, andAbelian differentials.

Chapterthreefocuseson string field theory,andin particularon its relationwith two aspectsof thegeometryof moduli space. The first section containsa brief review of some of the conceptsandtechniquesof string field theory. This is followed by a discussionof how Witten’s string field theorygives a single cover of moduli space. The third sectiondiscusseshow string field theory produces,through its ghostcontent,the Polyakovintegrationmeasureon moduli space;this reasoningis usedinreverseto deducefrom the Polyakovapproachthe Siegelgauge-fixedFeynmanrulesfor Witten’s stringfield theory. The fourth sectioncloseswith somecomments.

In chapter four the equivalenceof the Polyakovand light-cone formalismsis discussed.The firstsection showsthat the light-cone diagramsdo in fact provide a single cover (andin fact a previouslyunknown—atleast to mathematicians—triangulation)of moduli space.In the following sectionthePolyakov measureis worked out in the light-conecoordinatesfor moduli space,andit is shownto beequivalentto the measurededucedfrom the interactingstring picture. Sectionthreethen discussestheequivalenceof the externalstate correlationfunctions. In the processthe relation betweenvertexoperatorsandwave functionsspecifiedon a smallboundaryof a surfaceis sketched,togetherwith therelationbetweenthe statesof the Polyakovpictureand thoseof the light-conepicture.Thelast sectionendswith a discussionof prospectsfor furtherextensionsof this work.

Finally, chapterfive concludeswith someclosingremarks.In this report I havefocusedon the bosonicstring; the superstringis mentionedonly in passing.

Certainlythe bosonicstring is a good laboratoryfor the developmentof string theory techniquesandfor exploring parts of the foundation of string theory, but we would really like to attain anunderstandingof the superstringand heteroticstring. It is likely that this will involve the generalizationof Riemannsurfacesto superRiemannsurfaces.*) The theory of superRiemannsurfacesis nowherenearas highly developedasthe theoryof ordinaryRiemannsurfaces,andit remainsto be seenwhether

‘1Somepaperson superRiemannsurfacesarelisted in ref. [6].

172 S. B. Giddings, Conformaltechniquesin string theory andstring field theory

superRiemannsurfaceshavethe samepowerandutility as their ordinary brothers.It is very likely thatmanyof the Riemannsurfacetechniquesdiscussedin this reportandelsewherehavesupergeneraliza-tions, but that must be left for further work.

This report in largepart summarizeswork I havedonein collaborationwith variouspeople.Much ofchapterthreewasdone in collaborationwith E. Martinecand E. Witten, and muchof chapterfour wasdone in collaboration with S.A. Wolpert and E. D’Hoker. I would like to once againexpressmyappreciationto my collaboratorsfor fruitful collaborationsand for many valuablediscussions.

2. Fundamental techniques

This chapterwill beginwith a reviewof somebasicformalismthat will be usedin the sequel.Sectiononebriefly reviewsthe Polyakovapproachto calculationof string amplitudes,andsectiontwo containsbackgroundmaterial on Riemannsurfaces.

2.1. The Polyakovpath integral

In the perturbationtheoryapproachto string theory,an n-point scatteringamplitudeat the g-looporder for the closed,orientedbosonicstring has a Feynmandiagram representationas an orientedtwo-dimensionalsurfaceof genusg with n boundarycurves correspondingto the incoming states.Similarly for the open string, scatteringamplitudesare representedby two-dimensionalmanifoldswithboth handlesand multiple boundarycomponents.To calculate an amplitudecorrespondingto such adiagramin the Polyakovformulation[7], we performthe pathintegralover all mapsof the surfaceintothe spacetimemanifold M, as well as independentlyintegratingover all two-dimensionalmetricson thesurface.

In this integralover the mapsinto spacetime,externalstatesareprovidedby wave functionsdefinedon the boundarycomponentsof the surface.WhencalculatingS-matrixelementswe takethe incomingand outgoing statesto be on shell. This correspondsto putting the boundarycomponentsassociatedwith the statesat large spacetimedistance.From the point of view of the conformal structureof thesurface(which, as we shall see,is the structurerelevantfor string theory) this is indistinguishablefromprovidinganincoming wave function on atiny circle cut out aroundsomefinite point on a conformallyequivalentworld sheet.The two picturesarejust relatedby a conformalmappingthat locally lookslikew ez, which takes a small disk in the w plane to a cylinder approachinginfinity in the z plane.Thereforeputting an externalstateon shell correspondsto a “pinching down” of the tubesconnectingthe world sheetin questionto the restof the string world sheetrepresentingthe outsideworld. At thepinch we can summarizethe effect of the externalstate usinga vertexoperatorwhich introducestheflow of quantumnumbersonto the world sheetfrom outside. Conversely,we can view an incomingwave function correspondingto an on-shell stateas being provided by a vertex operatorat infinity.Thereforeon-shell externalstatescan alwaysbe representedby vertexoperatorsinsertedat fixed points{P~P2,. . . , P~}on a topological surface.*)In the caseof the open string, therewill still remainboundarycomponentsthat have not beenshrunkto a point; thesecorrespondto loops. Freestringboundaryconditionsare usedon such boundarycomponents.

Let .~ be a two-dimensionaltopological surfaceof genusg with n specifiedpoints(or punctures)andsboundarycomponents.In the pathintegralwe wishto considerthe space~ = {X~:.~ —~ M}

* This will be elaboratedupon in section4.3

S.B. Giddings, Conformaltechniquesin string theoryand stringfield theory 173

of all mapsof thesurfaceinto the spacetimeM, as well as the space~ of all metricsdefinedon .~ ~.

The Polyakovpath integral is thendefinedas*)

Ag(1,. . . , n) ~ ~gabJ ~Xw e~(~71 (2.1)

where ~V,are either vertex operatorsor wave functionscorrespondingto the external states. Theintegrationmeasureis inducedby the standardforms for the metricson the spaceof mapsandon thespaceof world sheetmetrics [7]:

Il~x~12=1 d2us/~G~~ ~ (2.2)

lSg~~ll d2u~g( gacgbd + Cg~gcd) ~ab ~cd (2.3)

Here u is a local coordinateon ~ G,~is the metric on the spacetimemanifold M, and C is someconstant.**)We takeas our action

= ~— d2u\,r~gabG ~aX~ôbX~

As is well known, the formal pathintegral (2.1) is infinite; this is becausethe integrandincludesafactorof thevolume of the local symmetrygroupsof the action.Theseare the groupsof Weyl rescalingsandof reparametrizations(or diffeomorphismsof the surfaceonto itself), whichacton thespace~, as

Conf(~,5)= {fl(o): ~ab ~_~eg~6}, Diff(~~)= {~a(): g0,, ‘—~ Ôa~C ôb

4~ ~

Thus we should insteaddefine the amplitudeby

Ag(1,. . . , n) J Vol(Conf)Vol(Diff) (~~ det’ ~ -13 ~ ~. , ~ (2.4)

wherewe work in 26 spacetimedimensionsand usethe notation

(Jd~~det’ . . , ~c~)=1 ~ e~*(~/t,... ~I~)

(det’ i is the determinantof the scalarLaplacianwith zeromodesexcluded).Having divided out thevolumesof Conf andDuff wearenaturally led to considerthe integralover the Teichmüllerspace~or over the moduli space.A~

5,

*) Here as always we will rotate to imaginary time.** The results in the Polyakov integral are independent [7] of C, so henceforth we drop it.

174 SB.Giddings, Conformaltechniquesin string theory and stringfield theorr

= ~.S g = S

n~ ConfDiff~‘ “~ ConfDiff

HereDiff11 representsthe diffeomorphismsconnectedto the identity; m~(= Diff/Diff() is the mappingclass group, better known to physicists (in the caseof the torus) as the modulargroup.

The Polyakovpath integralthereforeinvolvesonly metricsmoduloWeyl rescalingsanddiffeomorph-isms. Thus it is the space of conformal structures (or equivalently, in two dimensions,complexstructures)on the surface,ratherthan the spaceof metricson the surface,that is relevant.A metric ofcoursespecifiesa complexstructure;given a metric~ on the manifold ~ it is a fundamentalresultfor two dimensionsthat in aneighborhoodof any point P EI~ we can pick a parametrization~“ suchthat this metric takesconformallyflat form,

ds2 = g~

6do~do~= e’1[(d~’)2 + (d~2)2J-

Introducing complexcoordinatesz = ~‘ + i~,i= ~‘ — i~,we can rewrite this as

ds2=2g~~dzd~.

A two-manifold~ with a specifiedconformalstructureis called a Riemannsurface.To make(2.4) moreexplicit we follow the discussionof refs. [8—12].The action of the diffeomorph-

ism groupDiff0 can be describedlocally in termsof an infinitesimaldiffeomorphismgeneratedby the

globally well-defined vector field ~ This inducesthe changein metric ~ =V~aV6 + Vh ~iVa~ Thetracepart of ?Ig~6 can be eliminated by a Weyl rescaling; thus we considerthe trace-freepart

~ig~ =V~~Vh+ V~~ — ~ab V( ~ (P aV)ah.

We then havea decompositionof the cotangentspaceto ~, that is orthogonalunderthemetric (2.3),

= (~11g~~)EB(RangeP~)EB(KerPr).

The cotangentspaceto Teichmüllerspaceis the subspaceKer P~of tracelesssymmetrictensors /a6’

On such a tracelesssymmetric tensorwe have (P~c~)h= ~ or in local complex coordinates

= 0 (note that çb~= 0 by tracelessness).A tensor that transformslike aIoz0 ~9I9zunder aholomorphic change of coordinatesand that has holomorphic coefficients is called a holomorphicquadratic differential. Thereforewe see that the spaceKer P1 is precisely the spaceof holomorphicquadraticdifferentials,* which on ann-puncturedsurfaceof genusg has complexdimension3g — 3 + n.If we generalizeto the caseof surfaceswith s boundaries,n interior punctures,andm punctureson theboundaries,then the spaceof holomorphicquadraticdifferentialshasreal dimension**) 6g — 6 + 2n +

m + 3s.

* Herewe allowonly quadraticdifferentials with first-orderpolesat thepunctures.Thesearethenormalizablequadraticdifferentials[13], i.e.,they satisfy (~,~)<~in a metric that behaveslike g~,-~1I~z— at the puncture.

• * To derivethis formula for the caseof surfaceswith boundarycomponentswe take the Riemanndouble alongthesecomponentsto find a

closed surfaceof genus2g +s — 1 with 2n + m specifiedpoints.Thus we have a6g — 6 + 2n +m + 3.~complexdimensionalspaceof holomorphic

quadraticdifferentials.The numberof parametersin thegeneralvariation is thenhalved,to 6g — 6 + 2n + m + 3srealparameters,by therestrictionof symmetryof thedoubledsurfaceabouttheformerboundarycomponents.It shouldalso benotedthat if thesurface hasconformalKilling vectors(which occuronly for thesphereor diskwith fewerthan threepunctures,or thetorus or annuluswith no punctures),thenumberof realmoduli is6g — 6 +2n + m +3s +p, wherep is therealdimensionof thespaceof conformalKilling vectors. For suchsurfacesthePolyakovpathintegralmustbe slightly modified by inclusionof anextragauge fixing determinant.Thesespecialcaseswill not be explicitly treatedin this report; for furtherdetailsthe readeris encouragedto consult thereferences.

S.B. Giddings,Conformaltechniquesin string theoryand stringfield theory 175

In order to factorout the volumesof the symmetrygroupswefirst pick agaugeslice S throughthespaceof metrics~ suchthatfor everymetricg E ~ thereis auniquemetric~E Sfrom whichit canbe reachedby a combineddiffeomorphismand Weyl rescaling.This situationis picturedschematicallyin fig. 1. The slice S correspondsto a family ~a

6(T’~) of metricsparametrizedby the 6g —6 + 2n + m +

3s modularparametersre’. Supposewe pick apoint r~’; then,as we havesaid,themetric~ab(T~) definesa complexstructurez on the surface-~ ~. Let r~+ sr” be the coordinatesof aneighboringpoint on theslice S, with l~ra I ~ 1. The metric at the neighbouringpoint can be written as a small variation of theoriginal metric,

ds2(r~+ ~Ta) = 2g~Idz+ gZZ 6ra h~diI2 + 2 ~ig~dzdi,

for somecomplexh~andreal ~ The tangentvectorsto the slice S are, up to Weyl rescalings,theBeltrami differentials ~a)1 = gzzh*~*_ The parametersT on the slice S will provide the coordinatesinwhich we will computethe measurefor integrationover the moduli space.iIl.

To computethis measure,we considerthe orthogonaldecompositionof the cotangentspaceto thespaceof metrics

~~ab = (P1 ~V)ab + ~ ~Ta 4~ab+ ~ ~ab (2.5)

Here We is a continuous vector field that generatesa diffeomorphismof the surface; ~ areholomorphicquadraticdifferentialsthat serveas the basisfor thecontangentspaceto moduli space;er”parametrizethe modulardeformations;and ~u1representsthe changein conformalfactor. Using theorthogonaldecomposition(2.5) we can decomposethe measure,

~~ab = ~ ~g(d6Iff) ~g(I~od) (2.6)

This measureis invariant underreparametrizations,andwhencombinedwith the X pathintegral isalso invariant under Weyl rescalings(in D = 26) [7—9],so we can pull the volume elementsback tothoseevaluatedusingthe metrics,~E S. The integral overthe factor ~liflcancelsthe factor Vol(Conf).To cancel Vol(Diff) we change variables of integration, in the process picking up a Jacobiandeterminant,

= det’(P~P1)1’2~lJV.

,

Conf.

/~Sg

S

01ff.Fig. 1. A representativeslice S throughthespaceof metrics.This slice hasthesamedimensionasthemodulispaceandis transverseto theorbitsofthe diffeomorphismgroup andgroup of Weyl rescalings.Also shownis a tangentvectorto the slice,a Beltrami differential.

176 .S.B. Giddings. Conformaltechniquesin string theory and string field theory

Of coursewehave to be careful in defining the determinantsin this expression;for detailsseeref. [14].Notice that we have used det’, since we must omit the zero modes, which correspondto thedeformationsof moduli. The factorVol(Diff) is canceledby the integralover ~?2~V,leavinguswith theJacobiandeterminant.

To properlytreat the last factor in (2.6),noticethat the tangentvectors to the slice S in generalhave componentsboth in the direction of diffeomorphismsand in the direction of pure modulardeformations;see fig. 1. Thereforeif we chooseto integrateover the ~ to get the correctvolumeelementwe must project the tangentvectors to the slice S onto the basis covectorsfor modulardeformations,namelythe holomorphicquadraticdifferentials.The natural inner productbetweentheBeltrami differentialsis given by (seethe nextsection)

~=1 d2~~gg2~~.

Thereforefor the last factor in (2.6) we obtain

d t/ ~ \ ogo5-2n+ns-~-3*~g(mod) = e ~ ~ dr~ . (2.7)ab det(~a,~p) y1

Puttingall the piecestogether,we find for the Polyakov amplitude*)

6g 6+2n+tn--3., det1 ~ )A(1,.., n) = J fJ d~ ~ ~“ ~ ~ det’(P~P

1)’ 2y—1 [det(4,,,4~)J

x ( ~ det’ ... (2.8)Jd

whereas usual det’ excludesthe zero modes.This expressionis valid for surfaceswithout conformalKilling vectors;for surfaceswith conformalKilling vectorsanadditionalgaugefixing determinantmustbe includedfor the symmetrygeneratedby the Killing vector. Furtherdiscussionof this point maybefound in refs. [9—121.

2.2. Riemannsurfaces

We havejust seenthat Riemannsurfacesare the basic geometricalobjects in the Polyakovpath

integral, In the following chapterswe will frequentlyneedto use somebasicconceptsin the theoryof*) Note that to definethis expressionin the casewherethereare free boundarycomponentssomecare mustbe taken to enforcethe correct

boundaryconditionson thevariousfields over which thedeterminantsaretaken.The correctchoicefor Xis, of course,n,9~X= 0, wheren~is thenormal to theboundary (Neumannboundaryconditions). The boundaryconditionsfor the metric deformationsand diffeomorphismvectorfieldsare more subtle;they essentiallyarise from the condition that the adjoint of P1 is P~= —VU. with no boundary terms,and that the Neumannboundaryconditionsfor thefield X arepreserved.To actually implementtheseboundaryconditions,it is useful to thinkof just taking theRiemanndoubleof the surface,and restrictingattentionto fields with theappropriatereflectionsymmetry.The boundaryconditionsturn out to be noV” = I)for the diffeomorphismvector fields, and ntb4~,,= 0 (t is thetangentto theboundary)for the tracelessdeformationsof themetric.

S.B. Giddings,Conformaltechniquesin string theoryand stringfield theory 177

Riemannsurfaces;to start on the right foot, I begin with a short section reviewing someRiemannsurfacetechniques.Someuseful mathematicalreferencesarelisted in refs. [15,16]. Thosewhowish tosee somephysicsfirst mayskip this section,returningat anysign of impendingdifficulty with notationor concepts.

As was mentionedin the precedingsection, a metric ~ on a surface Z~ specifies a complexstructureon that surface.This is because the transition functions between the local complexvariableszinducedby the metric g~6in anytwo intersectingcoordinatepatcheswill be analytic. The surface.~ ~,

togetherwith the complex structureinduced by the metric ~ab is called a Riemannsurface;we willdenotea generalRiemannsurfaceby ~ Given the manifold .~, the set of all Riemannsurfacescorrespondingto the manifoldis the set of all metricsthat can be put on the manifold,modulogloballydefined diffeomorphisms (or reparametrizations) and local rescalings of the metric (Weyl rescalings). ‘K)

This spaceof all Riemannsurfacesbasedon themanifold ~ is knownas themodulispacefor ~,. Aclosely associatedspaceis Teichmüllerspace,which is the spaceof all metricsmoduloWeyl rescalingsanddiffeomorphismsin the connectedcomponentof the identity. Thus moduli spaceis the quotientofthe (simplyconnected)Teichmüllerspaceby the groupof all diffeomorphismsmodulothe diffeomorph-isms connectedto the identity; this latter groupis known as the mappingclass group.

Supposethat we are given anothermetric j~6that cannotbe brought to ~ab by a globally definedcoordinatetransformationandWeyl rescaling.Thenthis new metric definesanothercomplexstructurewith local complex coordinatew. Since z and w correspondto different complexstructures,thetransformationrelating the local parametersw and z is not conformal in some neighborhood.Aconformaltransformationwould havetakensmall circles in the z coordinatesto small circlesin the wcoordinates(sinceit is just a local scaling),but sincethe transformationis not conformalit takessmallcircles in z to small ellipses in w. Such a transformation(with eccentricityof the ellipses uniformlyboundedover the surface)is called a quasiconformaltransformation.*

Quasiconformaltransformationsrepresentchangesin complexstructure,i.e., theytakeus from onepoint in moduli spaceto another.Indeed,if we rewrite the metric ~ab in the local coordinatesz, we find

d~2=2j~~I8wI2~dz+ ~ di~.

The changein complexstructureis representedby

= ô~.wIe9~W.

Clearly such a ~4transformsas dz05919i under a holomorphicchangeof coordinates.Any measur-

able, essentially boundedfunction with these transformationpropertiesis known as a Beltramidifferential (see,e.g., ref. [16]).

It can beseenthat quasiconformaltransformationscanbe representedusingcoordinatedeformationsnot globally defined (i.e. not C°)over the surface2~,.To illustrate this considerthe torus. Thecomplexstructureof the torusis characterizedby two real modularparameters.Representingthetorusasa flat cylinder with oppositeendsidentifiedas in fig. 2a, it is easyto describethetwo deformationsof

*) The setof all metricson a surface,modulodiffeomorphismsandWeyl rescalings,is thesetof conformalstructuresfor thesurface;in twodimensionsspecificationof a conformalstructureis equivalentto specificationof a complexstructure.

**IA naturalquestionis whetherall diffeomorphismsarequasiconformal;theansweris no. Examplesareprovidedby mappingsfrom theupper

half plane to the full complexplane; circlesnear theboundaryof theupper half planeget infinitely stretched.

178 5. B. Giddings, Conformaltechniquesin string theory and stringfield theory

(a) (b)

Br

‘p.t.Q \\\\“~R /‘A’.Q P..N\~q

Fig. 2. (a) The torus representedas a rectanglewith oppositesidesidentified. (b) Quasiconformal“stretch” deformationrepresentedby cuttingrectangleR.stretchingit, andreapplyingit to a secondcopyof R.The resulting torus hastwo copiesof thedoubly shadedregton.andconsequentlyhas an increasedmodulus.

the complexstructurecorrespondingto the two moduli. The first, the Frenchel—Nielsentwist [17].resultsfrom holding one endof thecylinder fixed andtwisting the otherendbefore identifying the twoends.The seconddeformationresultsfrom stretchingthe cylinder,or alternatelyfrom addinga piecetothe end of the cylinder, before identifying the two ends.

Clearlythe vectorfields describingthesetwo deformationsarenot C°on thetorus.Considerthe caseof the stretchdeformation.First wecut thetorus alongtwo cycles to obtainthe rectangleR, fig. 2a. Wethen reparametrizethe surface,taking as new coordinatesthose obtainedby cutting the rectangle,stretchingone of the pieces,and reapplyingthe rectangleto a copy of the original rectangle,R’, as infig. 2b. Thus we have the samesurface,but in a new parametrizationgiven by z’ = z + ~z, where

~iz=~r— if Imz<a, ~lz0 if Imz>a . (2.9)2a

We thentakethe metric of R’ andpull it backto R by thechangeof variablez’(z, i). The effectof thisis to adda smallpiece to R sincewe havenow includedtwo copiesof the overlapregion in fig. 2b. Ifthe original metric was ds2 = 2g

2~dz d~,the new metric is

d~2= 2g~~~1+ ~zI2~dz + ~ d~.

For small ~z the Beltrami differential for the inducedchangein complexstructureis therefore

=

with ~zgiven by (2.9). A similar representationcan be takenfor the twist deformation.The pictureofquasiconformaldeformationsas arising from discontinuousreparametrizationsof the surfacewill beusefulin our laterdiscussion.

Also closely relatedto Beltrami differentials andchangesof moduli are the holomorphicquadraticdifferentials4~which transformas ~9Iôz0 dlaz under holomorphicchangeof variable. As we haveseen,thesehavea naturalcorrespondencewith small tracelessdeformationsof the metric; thus thespaceQ(R~~)of holomorphicquadraticdifferentialson a Riemannsurfaceis naturally isomorphictothe holomorphiccotangentspaceto Teichmüllerspaceat the point correspondingto the metric gab’

Oneeasilydefinesan inner productbetweenQ(R~2)andthe spaceB(R~~)of Beltrami differentials; it

is given by

SB.Giddings, Conformaltechniquesin string theory andstringfield theor,v 179

~ J d2g~~ (2.10)~

If we define Q~(R~~)C B(R~)to be the set of Beltrami differentials orthogonalto the quadraticdifferentials by this product, then the dualspaceto Q(R~~)is the spaceB(R~,)IQ~(R~

5).Thus thespaceB(R~5)!Q

1(R~)is isomorphicto the tangentspaceto Teichmüller space.There is a uniqueharmonicBeltrami differential in eachcosetof B(R~~)IQ~(R~

5),sowe can takeasour representationof the tangentspaceto Teichmüllerspacethe set of harmonicBeltrami differentials,HB(R~S).Each~4E HB(R~5)can be written in the form ~4= ~ for some4~EQ(R~~).On the tangentandcotangentspacesHB(R~S)and Q(R~2)we can also introducethe naturalinner products

(4 ~)fd2ff~~v~, (~, ~)=f~ (2.11)

Clearly theseinner productsdependnot just on the complexstructure,but also on the conformalfactor, i.e., they dependon the full metric. If we take the metric g~to be a constantcurvaturerepresentativeof its conformal class, the products(2.11) are just two different forms of the Weil—Peterssoninner product; their real parts give the Weil—Peterssonmetric andcometric.

Another extremelyuseful tool in our Riemannsurface explorationsis the Abelian differential.Abeliandifferential is just a slightly shorterway of sayingmeromorphicone-form.Besidesbeingusefulto us,Abelian differentialsarealsoquite fundamentalin the study of the theory of Riemannsurfaces.Particularlyimportantarethe holomorphicAbeliandifferentials;onecan easilyseefrom the Riemann—Roch theorem (see,e.g., ref. [18] or Farkasand Kra [15]) or by other means that the spaceofholomorphic one-forms on a genusg surface is of complex dimensiong. If we pick a canonicalhomology basis { A

1, . . . , A~,B1, . . . , Bg} for the surface, as in fig. 3, then by taking linearcombinationswe can choosea canonicalnormalization

WJ =6jJ

for our holomorphicone-forms at

1 (here I = 1,. . . , g). For the integralsaroundthe B cycleswe find

= flu

~ is called the period matrix, and is symmetric with positive definite imaginary part (for anon-degenerateRiemann surface). The elements of the period matrix can be used as complex

Fig. 3. The choice of a canonicalhomology basisfor a generalRiemannsurface.

181) SB.Giddings. Conformaltechniquesin string theory andstringfield theory

coordinatesfor moduli space[19] althoughclearly theremust be relationsamongthem,*) reducingthenumberof independentcomplexparametersto 3g — 3. We will alsomakeuse of Abeliandifferentialsthat have for singularitiesa pair of simple poles of equalbut oppositeresidues;as we shall see,theseareclosely relatedto both Green’sfunctionson Riemannsurfaces,andto theconstructionof light-conediagramsfor Riemannsurfaces.

3. String field theory and the geometry of moduli space

Much recent work on the foundations of string theory has been focused on some of the elegantfeaturesof the Polyakovapproach[7] to strings,in which the perturbationseriesappearsasa sumoverRiemannsurfaces.In this chapterI will discussa differentbut relatedapproachto stringtheory,namelystring field theory. Before beginning a discussionof string field theory, however, it is worthwhile toextendthe discussionof the introduction,and elaborateon the reasonsfor physicists’ interestin stringfield theory. In particular,we know that the string perturbationtheory treatedas a sum over Riemannsurfacesis an exceptionallybeautiful and rich structurethat can be explored by a variety of powerfulmathematicalmethods.The question is why we should be seekinga more general framework withwhich to describestring theory.

Thereare several answersto this question;here I will give one answerinvolving the issue of basicprinciples, and two answers regardingmore practical issues. On the first point physicists have anexceptionallystrongmotivation for studyingstring field theory: no one knowswhatstring theory reallyis. Of coursephysicists expect that there should be some deep principle underlying string theory,perhapsas the principle of equivalenceand Riemanniangeometryunderlie the theory of GeneralRelativity. So far string theory consistsof a numberof somewhatad hoc, yet seeminglypowerful, oreven“miraculous”, rules. It appearsthat whenwe arediscussingthesum over Riemannsurfaceswe areonly scratchingthe surface. Work on string field theory is directedtoward unearthingthis deeperstructure. There are, of course,other approachesattempting to discover the fundamentalbasis forstring theory; an example is that of Friedan andShenker[20].

On a morepractical note,there is the problemof making contactwith the physicalworld. Severalproblemsarisehere. Oneproblemis that the loop expansioncouplingconstantmay likely not be small,so that the perturbationseriesdoesnot converge. If this is the case,it is not clear that we could findsensibleanswersevenfor perturbativephysics.Secondly,evenif we couldsumthis serieswe would stillmiss non-perturbativeeffects, in which there is expectedto be important physics. Specifically, anon-perturbativeformalism appearsto be neededfor a completestudyof problemssuch ascosmologyandcompactification,supersymmetryandgaugegroupbreakingandthe determinationof the massesofthefermions,andfor a resolutionof thequestionof singularitiesin the theoryof GeneralRelativity. Inother words, it seemsthat a workable non-perturbativeformalism would be very useful, if notnecessary,to study a largesubsetof the physicalquestionsthat physicistshopestring theorywill be ableto answer.

We might illustrate one of the troubleswith perturbationtheory by way of analogy. Insteadofexploring string theory through its perturbationtheory, we could imagine exploring the theory ofgravity throughits perturbationtheory (if it hada well-definedperturbation theory). This would involvestudyingscatteringprocessesof spin-two masslessparticles. If we expectedto describethe whole theory

* I The problem of finding such relations is the Schottky problem.

SB.Giddings,Conformal techniquesin string theoryand string field theory 181

of gravity by this approachwe would in particular expectto be able to find Schwarzschild’ssolutioncorrespondingto a black hole. Studying the perturbationtheory for spin-two masslessparticles,however, is clearly not the way to find black hole solutions—theyare manifestly not perturbativephenomena.(Similarly in Yang—Mills theory, to find interestingthingslike instantonsolutionswe mustleavethe realm of perturbationtheory.) Of course,to find blackholeswe mustgo beyondperturbationtheory to invent General Relativity, the field theory for the gravitational field, in the processrecognizingthe deep relation betweengravity and Riemanniangeometry.Then we mayproceedtostudy solutionsof nontrivial geometrysuch asthe Schwarzschildsolution. Similarly, the expectationisthat to mine the manyriches of the stringwe must invent a non-perturbativeformalism suchas stringfield theory;sincestring theory is a generalizationof the theoryof gravity, we might expectstring fieldtheory to somehowinvolve somedeepergeneralizationof Riemanniangeometry.

In this chapterI will principally focuson issuesinvolving the field theory for openstrings that hasrecently beeninventedby Witten [21] in a generalizationof the BRST approachto string field theoryinitially proposedby Siegel[22].From thistheory we will seeseveralgeneralfeaturesarisingthat couldpossibly also be featuresof a closedstring field theory, if such a theory is developed.One of thesefeaturesis the cell decompositionof moduli spacein which the cells correspondto different stringFeynmandiagrams.A secondfeatureis providedby the reparametrizationghoststhat mustbe usedtodefine the string field theory; these ghosts give us the correct measure for the integration of the stringamplitude over moduli space.This chapterwill be restrictedto the caseof the bosonicstring. Wittenhas also constructeda field theory for the open superstring[23] and similar featuresshould perhapsappearfor that theory in the contextof supermodulispace[6]. I will beginby giving a brief sketchofsomebasicsof string field theory,andin particularof Witten’sstringfield theory andtheFeynmanrulesfor that theory.This will be followed by a discussionof the triangulationof moduli spaceobtainedfromtheseFeynmanrules, after which I will sketch how the reparametrizationghosts give the stringintegrationmeasure.

3.1. Basicstringfield theory

Beforelaunchinginto the constructionof stringfield theory we shouldgive a generalpicture of whatwe meanby a string field theory. It helps to first think aboutthe field theory of the masslessparticle.For the non-interacting particle we could begin with the action

S = f dr s/~gt~-~-- ~-- ~

correspondingto the lengthof the particle’strajectory. HereX~(r)gives the positionof the particle,gis the one-dimensionalmetric tensor,and is the flat backgroundmetric for aspacetimeof dimensionD. To calculateamplitudesfor this particleto propagatefrom an initial to a final statewe thencould dothe pathintegralusingthis action. This correspondsroughly to the Polyakovapproachto string theory.(Of coursefor the particlethis picturegives no naturalway to treatinteractions—theycorrespondto abreakdownof the manifold structureof the world diagram.)We can alternatelyswitch to a modifieddescriptionof the particle. This is given in termsof a quantumfield 4(X), for which we havethe action

I=fdDX(~~+ ~

182 SB. Giddings,Conformal techniquesin string theory and stringfield theors’

The variable X~labels the position of the particle, so we see that çb is a function of the possibleconfigurationsof the particle.The first termin theactionis the kineticpiece,which gives a prescriptionfor the propagationof the particle;the secondterm is the interactionpiecewhichwe could not naturallyinclude in the previous description of the particle. Herefor illustration I havetakena cubic interaction,which is shown pictorially in fig. 4a; in fig. 4b I show an exampleof a Feynmandiagramthat wouldcomefrom this action in the calculation of an amplitude for two particles to scatteroff oneanother.

Now string field theoryshouldjust be a generalizationof this. Sincein string theory we aretrying topictureelementaryparticlesas tiny strings,we know that fig. 4b shouldreally look like fig. 4c; if we hada good enoughmicroscopewe would see that the particle is in reality a string and that the particletrajectory is in reality a two-dimensionalsurface. (Of course propagationof different modes ofvibration of the string correspondsto the propagationof what we would call different “elementaryparticles”.)So as beforewe can calculatethe amplitudefor a processby doinga functionalintegraloverall different intermediateconfigurationsusingthe Polyakovaction

S= ~ ~6X”~5,. (3.1)

(Here interactionsare well definedsince they do not changethe fact that we have a manifold.) Butinsteadof taking this approachwe would like to introducea string field theory.As before,we defineaquantumfield A[X(u)]; since we are now dealing with a string we havea field functional of stringconfigurations.Once againwe want to introducean actionwith a kineticterm andaninteractionterm;because of the observation that there is really only one fundamental interaction in string theory, namelythat in which two stringsmeetto form a third string,we might expectthat we shouldbe able to find anaction with a single cubic interaction term.

In constructingthe stringfield theory it is usefulto considera string field that is not justa functionalof the physical string configurationsX~(o’),but also of the reparametrizationghosts[24] c(u), b(o);these can be expanded in normal modes with coefficients c,,. b,5 that obey canonicalanticommutationrelations{cm,b~}= ~ In the Polyakovapproachto string theory the reparametrizationghostsarejust local Grassman-valuedfields on the stringworld sheet;the pathintegralover thesefields providesthe Fadde’ev—Popov determinant from the gauge fixing of the reparametrization invariance[8].‘~)Thestring field A[X(cr), c(a), b(o)] can itself then be expandedin termsof the normal modecoefficients

(a) (b) (c)

Fig. 4. (a) The basic cubicinteraction for a particle. (b) The Feynmandiagramcorrespondingto a scatteringproces.(c) The Feynmandiagram atextrememagnification, where we find that the particle is really a string.

* I This will be further discussedin section3.

SB.Giddings, Conformaltechniquesin string theoryand string field theory 183

X~,c~,b~,which areto betreatedas operatorsactingon avacuumstatein the usualtypeof Fock spaceconstruction. Notice that the presence of the zero modes c0, b0, with { c0, b0 } = 1, indicates that thegroundstate is doubly degenerate;we call the two ground statesI~)and ~ with relations

b0I]5) =0, cCI],) = ~), c0~]’)=0, b0I~) = I]~) (PR) = 1.

Oncethe reparametrizationgaugeinvariancehasbeenfixed, the reparametrizationinvarianceis in asenserecapturedby BRST invariance[25]. The BRST invarianceof a free string is generatedby theBRST operator [24,26] Q, which takesthe form (: : denotes normal ordering)

Q = ~ (c~L~+ ~

Here L~are the generatorsof the ordinary Virasoro algebra for the D fields X~,

[L~, L~]= L~÷~+ ~Dm(m2 — 1)~m+n~,

andL~”= ~m (n — m) :bn+mcm:are the Virasorogeneratorsfor the ghostsystem.The operatorQ hasthe crucialproperty Q2 = 0; this is true only in the critical spacetime dimension [24]D = 26. Anotherimportantoperatorto keepin mindis the ghostnumberoperatorNg; this simply countsthe numberofghost excitations minus the number of antighost excitations of a state. Ng is given by

Ng = ~ (c~b~— b~c~)+ ~(c0b0— b0c0).

n>O

To write the openstring action,Witten introducedon the spaceof stringfieldsa multiplication law *,

an integrationlaw J, and a gradingof the algebraof the fields. This construction,togetherwith theBRSToperator which acts as a derivation of the *-algebra, is a concrete realization of the axioms ofConnes’non-commutativegeometry[27]. The multiplication law mapstwo string fields to a third; it isdefinedas the limit of a pathintegral on the region shownin fig. 5a. Notice that thereis a curvaturesingularity of strength R= — ii~2(o.— u’) at the interaction point. Pictorially, the expressionA = B* Ccan be thought of as an overlap integral setting the right half of string B equalto the left half of stringC; the productA thenhasleft half equalto the left half of string B andright half equalto the right halfof string C, as shown in fig. Sb. All of this can be mademuch more precise;see ref. [21] for more

(~) (b)

Fig. 5. The openstring multiplication law. (a) The world-sheetpicturein which wedo thepathintegralwith boundaryconditionsimposedby stringfields B and C to find the string field functional A(X(u)). In this path integral we take the limit as al—’ 0. (b) The interpretationof thismultiplication law in termsof strings; the right half of B is equatedto the left half of C and the leftover halvesmake up the stringA.

184 S.B. Giddings. Conformaltechniquesin string theorcand stying field theory

details. The integration J mapsstring fields into the complexnumbers.The integration j A can bethoughtof as an overlapintegralsettingthe left half of string A equalto the right half. The string fieldA is alsogiven a Z., grading (—l)~’= (_l)~/~12 dependingon ghostnumber;thephysicalstateshaveghost number —1/2 and are odd elementsof the algebra. The *-algebra is associative*)but non-commutative;however,under the integral sign we have

I- A*B=(—Y~JB*A.

Crucial in the constructionof a gauge-invariantaction is the fact that the BRST operatoracts as aderivationof the *-algebra,

Q(A*B)= QA*B+(_)AA*QB.

Further confirming the analogywith the d and f operatorsacting on ordinary differential forms on amanifold is the identity (“integration by parts”)

f QA =0,

which is true for any field A. The action itself is written

I=J(A*QA+ ~A*A*A): (3.2)

it looks like a Chern—Simonsform. Using the aboveidentitiesoneeasilychecksthe invarianceof thisaction underthe gaugetransformation

~A=Qe+A*e—e*A. (3.3)

To calculate scatteringamplitudeswith the action (3.2) one must first fix a gauge;otherwisethekinetic operatorcannotbe invertedto define a propagatoranda set of Feynmanrules.The gaugefixingfor thefull-nonlinear invariance(3.3) appearsto be non-trivial [29]. However,thereis an ansatzfor thenon-lineargaugefixing motivatedby the gaugefixing in the non-interactingcase:this ansatzshouldgivethe correctanswerfor the tree diagrams**) and moregeneralarguments[32] that will be sketchedinthe sequelindicatethat this ansatzshould also be correct for loop diagrams.We begin with the gaugefixing for the linear invarianceof the non-interactingtheory,

= QE.

A suitablegaugechoiceprovesto be the Siegelgaugechoice [22]b~A= 0, which implies A =

where A is a stateexcluding the ghost zero modes. On such a state the kinetic part of the actionbecomes

~‘ Associativity doesnot necessarilyhold for fields outside the open string Fock space;sec ref. (28].** I This hasbeenexplicitly checkedfor the four-point tachyonscatteringamplitude in ref. [30]. It has alsobeen shown [31)that thevertex in

this formulation of field theory reproducestheusual three-point couplings betweenstates.

S.B.Giddings, Conformaltechniquesin string theory andstring field theory 185

JA*QA= (AI(~IbOQbOI~)IA).

We thenusethe relationb0Qb0= b04,where4 = L~+ LO~’~is the ordinaryworld sheetHamiltonian,toreduce the kinetic term to t~A141A). Now the kinetic operatordoeshavean inverse;it is given by1/4 = J~°dr e~Since4 is the world sheetHamiltonian,the operatore~just tells usto propagateastring freely for a time r; this builds up a flat rectangularworld sheetof width ir and length r. Theintegralover T tells us to sumoverall suchworld sheets.Of course,insteadof completelydroppingthezero-mode part of the string field we could have left it in; then the b0 insertion would be represented asa line integral $ do b(cr) across the rectangle.

Whenwe introduceinteractions,they just tell us to tie togethersuch flat rectanglesthreeat a timeusingverticeslike thatshownin fig. 5a. Let me onceagainemphasizethat for sucha picturethe entiresurface is flat except at the interaction point where there is a curvaturedelta function, and theboundariesare all straight lines in this flat metric. Thus in the Feynmanrulesof Witten’s string fieldtheory we encounter all Riemann surfaces obtained by gluing together such flat strips of any givenlength in the mannerprescribedby the vertex.

3.2. Stringfield theory and the triangulation of modulispace

With the geometricalFeynmanrulesfor the string field theory in handI will first tackle the issue ofmodular invariance. If the string field theory actually is a sensibleextrapolationof the Venezianomodel, it should somehowincorporatethis fundamentalfeatureof the Venezianomodel. Indeed,thestring field theory shouldthrough its propagatorvariablesgive us a simple parametrizationof modulispace,a by no meanstrivial taskin view of the topologicalcomplexityof moduli space.’K~In thissectionI will discuss this parametrizationof moduli spacefor the general orientedopen string Feynmandiagram.

Supposewe considera diagramlike that shownin fig. 6a. It is not hard to convinceoneselfthat thisdiagram is topologically a torus with a disk removed. (To see this note that the surface hasoneboundarycomponentand is orientable;one can then look at the homology basis,or betteryet, countthe curvaturesingularitiesand use Gauss_Bonnet,**)to concludethat the surfacehasgenusone.)Furthermore,we see that we have a three-parameterfamily of such surfaces,correspondingto the

(a) (b)

Fig. 6. (a) An openstring vacuumdiagramwith threetwisted propagators;topologically this is equivalentto a torus with one hole. (b) A vacuumdiagram with 2g + 1 twisted propagators;topologicallythis surfaceis a surfaceof genusg with a disk removed.

*) Indeed,theEuler characterfor themodulispacefor aoncepuncturedsurfaceof genusg is givenby x(-4I~)= ~(1— 2g),where~is Riemann’s

zetafunction. (x is non-integralbecausemoduli spaceis an orbifold.)**) Noticethat my units aresuchthatR= 1 for theunit sphere;this differsby a factorof two fromotherconventions.Thereaderis warnedto

keeptheextra factor of two in mind, to get the correct Eulercharacter!

186 S.B. Giddings. Conformaltechniquesin string theory and stringfield theory

different values of the three propagatorvariables r1, i-2, i-s. But notice that three is precisely thedimensionof the moduli spacefor the toruswith a disk removed.Therefore,asthe threeparameters~vary over the interval (0, ~c) we sweepout someopen set in the moduli space.If we are interestedinobtaininga modular invariant descriptionof string theory, then the question is whetherthis open setdelineatedby the field theory Feynmandiagramsis in fact a one-to-onecoverof moduli spacefor thetorus with one hole.

More generally,one can consider diagramssuch as the one shown in fig. 6h. Such a diagram istopologically a surfaceof genusg with one boundarycomponent.Of coursetherewill be many otherFeynmandiagramsfrom our Feynmanrules that will give surfacesof genusg with one boundary:distinct diagramsjust correspondto differentcombinatorialgluing patternsfor the propagators.We canin asimilar manneralsoobtainsurfacesof arbitrary genuswith morethan oneboundarycomponent,orwith statespropagatingoff to infinity. Thereforewe would like to checkwhetherfrom such diagramsweget a one-to-onecover for the moduli spaceof a surface of genusg with one or more boundarycomponents.

To explorethe question of whetheror not we have a cover of moduli spacewe will first put theFeynmandiagramsin a simpler form. This is doneby cutting eachpropagatorin half alongits midline;sincethe cutsnevertouchthe boundarythe Feynmandiagramthenfalls apartinto acylinder (or, in thecase where there are several boundary components,into several cylinders*)), Such a cylinder ispictured in fig. 7, with the gluing pattern necessaryto return to the original surface indicated. Apresentationof a Riemann surface assuch a collection of cylinders (or possiblybi-infinite strips),all ofequalheight, and with the boundaryidentificationsmarkedalong one boundaryof eachcylinder (orstrip), will be called a canonicalpresentationof the Riemannsurface.To each Feynmandiagram wehavea uniquecanonicalpresentation,andconversely,to eachcanonicalpresentationwehavea uniqueFeynmandiagram. (Noticethat we have to be carefulto forbid gluing patternssuch as thoseshowninfig. 8a, b; the former give curvaturesingularitiesthat cannotcomefrom our Feynmanrules, and thelatter are redundantspecifications.)The fact that we have an injective cover of moduli spacethencomesfrom the theoremthat given a Riemannsurfacewith one or moreboundarycomponents,thereexists a unique(up to combinatorial factors) canonical presentationof that surface.**) ThereforeforeachRiemannsurfacethereis a uniquestring field theory Feynmandiagram[34].This maystrike one

ac b

b a C a a a bI I

a b(a) (b)

Fig. 7. The cylinder obtainedby cutting a Feynmandiagram apart Fig. 8. Gluing patternsthat are forbidden in consideringcanonicalalong the midlines of the propagators;the lower boundaryof the presentations.(a) A pattern that doesnot contributeto thetype ofcylinder correspondsto the original boundaryof the surface, and Feynmandiagram we are considering;(h) a redundantpattern.alongthe upper boundaryI have indicated the gluing pattern neces-sary to restorethe original surface.(Shown is the caseof the torus,fig. 6a.)

In the casewhere we have externalstates.therewill also be bi-infinite strips in the decomposition.** I This theoremis basedon an idea dueto W. Thurston(unpublished).with detailsprovided by Bowditch and Epstein;thefirst completeproof

was by D. Mumford (unpublished)and J. Harer. See ref. ]33).

SB.Giddings, Conformaltechniquesin string theory andstringfield theory 187

as surprising—modularinvariancewas not an ingredientof Witten’s string field theory. Rather,gaugeinvarianceservedas an input and the single cover of moduli spacelater appearedin a somewhatmysteriousfashion.

Of course in assertingthat we have a single cover of moduli spacewe must be mindful of theFeynmansymmetryfactorsfor the graphs;for example,in the caseof the torus with aholewe really geta six-fold coverof the moduli spacewhenwe allow Tt, r2, r~to eachrangeoverthe interval (0,cx). Thisis compensated by the Feynman symmetry factor 3! for the string field theory graph. A relation betweenthe problemof computing the symmetry factorsfor the cell decompositionof moduli spaceand thecombinatoricsof field theory was in fact obtainedby Pennerbefore the correspondencebetweenthestring field theory Feynman rules and the cell decomposition of moduli space was known [35]; thesymmetryfactorswerefound by consideringan interactingfield theoryof Hermitianmatricesthat canbethoughtof as roughly correspondingto theChan—Patonfactorsusedto introducenon-Abeliangaugesymmetriesinto the theoryof openstrings.

Outlineproof—triangulationtheoremFor completenessI will give a brief sketchof the proofof the abovetheoremthat to eachRiemann

surfacewith oneor moreboundariestherecorrespondsa uniqueFeynmandiagramfrom Witten’sstringfield theory.For more detailsand rigor, the readeris referredto refs. [33,36, 37]. First considerthecasewherewe havea Riemannsurfacewith a single boundarycomponent.We will embedin thisRiemannsurfacea cylindersuchthat oneboundaryof the cylinder is identified with the boundaryof thesurface.This cylinderwill be takento cover the Riemannsurface,soalongthe otherboundaryof thecylinder identifications of boundary arcs are made. Other than these identificationswe take theembeddingto be one-to-one.This situationis presentedfor the genusonecasein fig. 9. Theembeddedcylinder is of courseconformallyequivalentto a “standard” right circular flat cylinder of height b,circumferencea, and modulusm= b/a. However, in general the conformal rescalingsnecessarytomakethe metric of the embeddedcylinder flat can be differentas we approachthe identified boundaryarcs from differentsides.Thereforein generalwe cannotglue togetherthe boundarysegmentson theconformally equivalentflat cylinder; generically the boundarysegmentsto be identified will havedifferent lengthsin the new Euclideanmetric.

To avoid this problem we use our freedom to adjust the embedding of the cylinder. It is not hard toconvince oneself on grounds of compactness that there ought to be at leastone embeddedcylinder forwhich the modulusm attainsalocal maximum.Considerthis cylinder andits conformallyequivalentflatcylinder. Supposethat on the flat cylinder the lengthsdo not matchup. Thenthe conformalrescalingfactorsmust be different as we approachthe boundaryof the cylinder from oppositesides;call thesefactorsp, p’. Onecan imaginevarying the identified boundarysegmentsof theembeddedcylinder suchthat an amountof area~IAis movedfrom oneside of theboundaryto the other;seefig. 10. Then in the

Fig. 9. The embeddingof acylinder in thetoruswith onehole; oneboundaryof thecylinder is identified with the boundaryof thetorus, andtheotherboundary justtouchesitself so that we get aone-to-onecover of thetorus exceptalongthe identified segments.The identified segmentsaremarkeda, b, c; thesecorrespondto thesegmentsa, b and c of fig. 7.

188 S.B. Giddings. Conformaltechniquesin string theory and string field theory

‘pM’

~p~I

Fig. 10. A small variationof theboundaryof the embeddedcylinder suchthat anarea IA is movedfrom onesideof theboundary to the other;onthe conformally equivalentcylinder p IA is addedto one part and p IA is subtractedfrom anotherpart.

conformallyequivalentflat cylinder we haveaddedan amountof areap ~iA to onepart of the cylinderandsubtractedan amountof areap’ ~iA from anotherpart,for a total changeLIA = (p — p’) ~iA in theareaof the cylinder. Therefore if we use a conformalmappingto return us onceagainto a standardright circular flat cylinder, we will makea changein the modulusthat is first order in ~A. This is notpossible if we are locatedat a local maximum in modulus, so p must equalp’ and consequentlyidentified boundarysegmentsmusthaveequallengths.This showsthat if we takean embeddedcylinderof maximalmoduluswe obtain a canonicalpresentationof the surface.

For the secondpart of the proofwe mustshow that sucha canonicalpresentationis unique.Supposewe had two such canonicalpresentations,which we will representas rectangleswith oppositesidesidentified as in fig. 11. Of courseif we havedifferentpresentationsof the Riemannsurfacethenthe topboundariesof the rectanglesdo not correspond.We can by an overall scalingset the heightsof both RandR’ equalto the commonvalueb; the circumferenceswill thenbe a anda’. Considera verticalcurveC on R andits imageC’ on R’ (seefig. 11). Thesecurvesbegin andendon the lower boundariesof R,R’; correspondingpointson the upperboundariesare connectedby the boundaryidentifications.Letp(x, y) bethe conformalscalingnecessaryto passfrom the metric of R to that of R’; here(x, y) arethecoordinateson R. Clearly_curveC’ must have lengthgreaterthan 2b; in the coordinateson R this iswritten b ~ ~ j~.dy ~p(x, y). Applying the Cauchy—Schwarz inequality, we see that b <

~ ~ dy p(x, y). We then integratethis inequality over x, finding (since we cover the rectangletwice)ab ~ $R dx dy p(x, y); however, this latter expressionis just the area in the metric of R’. Thereforeab ~ a’b or a � a’. But we could have just as well reversedthe roles of R, R’ to concludea’ < a, so amust equala’. This meanswe must haveobtainedequality in the above inequalities,and this in turnimplies that the two canonicalpresentationsareone and the same.

R p(x,y) R’I {~c~cJ ~C~C’~

Fig. 11. Two canonicalpresentationsof a given Riemannsurface,representedas two rectanglesR,R’ with verticalsides identified. The lowerboundariescorrespondto theboundaryof the Riemannsurface,and the identificationsof arcsaremadeon theupperboundaries.Also shownis acurve C and its image C’.

S.B. Giddings, Conformaltechniquesin string theory andstring field theor 189

The prooffor the casewith multiple boundarycomponents,or with boundarycomponentsandstatesasymptoticallyapproachinginfinity is justa slight variation of this. First considerthecaseof sboundarycomponentsandno externalstates,i.e. openstring vacuumdiagrams.We nowembedin the surfacescylinderssuchthat oneboundaryof eachcylinder is a boundaryof the Riemannsurface,andsothat thecylinderscover thesurfacewithout overlapping.Onceagain,eachof thesecylindersR,, i = 1,. . . , s, isconformally equivalentto a standardcylinder R, with modulus m,. Clearly we can maximize themodulusof a given cylinder by taking the moduli of all othercylindersto zero; it is equallyclearthatthis is not what we want. Instead,fix the ratiosof the moduli of the cylinders; saym1 = c1m for somefixed constantsc. Now we vary the embeddingto find a maximum in the variable m. This will give apresentationof the surfacein terms of flat cylindersglued togetheron their boundaries,by a slightlyfancierversionof the argumentusedabove.The generalidea is that if the conformalrescalingfactorsdo not matchacrossboundaries,then,if wemakea small variation of the boundariesof the embeddedcylindersthat preservesthe ratio of the moduli, this will lead to a first-ordervariation in the parameterm. This in turn contradictsthe assumptionthat the variable m is at a local extremum.~

The resulting presentationof the surfacein termsof flat cylinders is not, however, a canonicalpresentation;in a canonicalpresentationthe heightsof all of the cylinderswouldbe equal.Now we useour freedomto vary the constantsc5, in the processvarying the presentationof the surface.Clearly, aswe vary an individual c, from zero to infinity, while holding the othersfixed, the height b~of thecorrespondingstandardcylinderwill alsovary from zeroto infinity. Thus the mapfrom { c1 } to { b1 } is acontinuousboundarypreservingmap from (0,os)’ to (0, cs~)~,sowe areguaranteedto be able to find aset of ce’s such that all the bk’s take on the samevalue, say ir/2. This gives us our canonicalpresentation.To show it is unique,we justaddsomebells andwhistles to our aboveuniquenessproof.More explicitly, supposewe havetwo canonicalpresentationswith cylindersR,, R . ConsideraverticalcurveC~that startson alower boundaryof anR., continuesthrough_boundaryidentificationonto anR.,andendson the boundaryof R.. Thenas above,b ~ ~ dy ~p(x, y) (here,as before,the b,, b areallfixed to a commonvalue b). Now just apply the Cauchy—Schwarzinequality,and thenintegrateoverxto find a,b ~ ~ f,~dx dy p(x, y). Here S1 is the region on the surface swept out by all verticaltrajectoriesoriginatingon the boundaryof R.. We thensumthis inequality over all of the cylindersR1theright-handside givesthe areain the new metric (sincewe coverthe surfacetwice), and asbeforeweconcludethat a1 + + a~� a + + a. Reversereasoningreversesthe inequality, so the inequalityis in fact an equality.This can only be true if all of the contributinginequalitieswere in fact equalities,andthis in turn implies that the presentationsareone andthe same,establishinguniqueness.

For the casewherethere are externalstates, similar argumentshold. Heuristically, we can justobservethat by a Weyl rescalinglocalizedaroundthe externalstatethe stripgoing off to infinity can beprojecteddown to a point on the boundaryof a diagrampreciselyequivalentto one of the vacuumdiagramsthat we havealready treated.Alternately we can run this in reverseto go from a vacuumdiagramwith a point specified on its boundaryto a diagramwith an externalstrip shooting off toinfinity. Since all of this is localized, we would not expectit to changethe fact that we havea singlecoverof moduli space.For thosewho areskeptical,an existenceanduniquenessproof can be carriedout by the samekind of extremalargumentswe usedabove. A slight variation occursbecausesomeofthe cylinders arenow replacedby bi-infinite strips, but the reasoningis in the samespirit.* ‘5)

* I A careful proof of the fact that the above construction givesa presentation of the surface in terms of Euclidean cylinders is apparently a bitmore complicated than this; for details of the proof, see, for example,Strebel’s book Quadratic Differentials [37], Ch. VI.

**) A third alternative is to begin with a diagram with no external states, and let some of the propagator variables go to infinity, creating

infinite-length external states. This corresponds to passing to a boundary of moduli space; the moduli space for the boundary configuration iscoveredbecausethe original moduli spacewas,and the limiting configurations are well defined.

191) SB. Giddings,Conformaltechniquesin string theory andstringfield theory

This argumentcompletesthe demonstrationthat conformal structureson a given topologicalsurfaceare in one-to-onecorrespondencewith string field theory Feynmandiagrams.We see that we haveaconstructionthat is manifestlyinvariant underthe action of the mappingclassgroup; at no point in ourdiscussionhavewe madereferenceto the choice of marking on the surface.Thereforethe Feynmanrules give a cell decompositionof the moduli spaceof a given Riemannsurfacewith at least oneboundarycomponent;the cells correspondto the different combinatorialgluing patterns,i.e. to thedifferentFeynmandiagramsthat onecan constructto representthe given topological surface.Whenweaddup the differentFeynmandiagramswith their appropriateFeynmansymmetryfactorswe thereforeobtain a one-to-onecover of moduli space.Furtherdiscussioncan be found in ref. [34].

3.3. Stringfield theory and the Polyakovmeasure

Another featureof Witten’s string field theory for the open bosonicstring is that it should providethe Polyakovmeasurefor integrationovermoduli space.’51This hasbeencheckedexplicitly in the caseof the tree amplitudewith four externaltachyons[30], and is expectedto extendwithout difficulty toarbitrarytreeamplitudes.For the loop amplitudesthe problemis moredifficult. To seethis notethat infixing the lineargaugeinvarianceof the free theory we encounterghosts-for-ghosts.* *1 ~ The reasonforthis is that in the linear casewe haveinvarianceunderchanges~A = Qe,but once this is usedto fix aparticular gaugefor A we still have an invarianceunder the transformation~e = QA, necessitatinganothergauge-fixingdeterminant.This processcontinuesindefinitely, leading to an infinite numberofghosts-for-ghosts.In the interacting theory the samething also happenson shell, i.e., when we areworking with a field configurationthat satisfiesthe equationsof motion QA + A * A = 0, but off-shellthings are moresubtle. It is of coursenecessaryto havea gauge-fixingprocedurethat works evenoffshell since in doing the path integral for string fields we include off-shell string fields; therewill beghostspropagatingin loops. There certainly appearto be redundantgaugeinvariances,but they arecomplicated.In tree diagramsthe ghosts decouple,so this is not much of a problem,but for loops asatisfactoryresolutionof this question is necessaryto get the proper integrationmeasureon modulispace.

It turns out that the problem can be attackedfrom a different direction: we can start from thePolyakov measureand try to work back to somethingthat looks like it should comefrom the fullgauge-fixedstring field theory. Remarkably,by this route it looks like one can comeratherclose towhat shouldbe the stringfield theory result [32]. Onecould alsohopethat this proceduremayprovidecluesto formulation of otherstring field theories.

We begin with the familiar Polyakovpath integral for an n-point correlationfunction (n could ofcoursebe zero, in which casewe would be computingthe partition function), on a surfaceof genusg

with sboundarycomponents.In section2.1 wefoundan expressionfor the amplitude,eq. (2.8). In thatexpressionit is usefulto representthe determinantdet~(P~Pi)t/2in termsof pathintegrals[8] overthelocal anticommutingghostand antighostfields c~z,~Z b,, b~

2,

* I The Polyakov measure can of course be thought of as arising from the Weil—Peterssonmeasureon moduli space,but since the

Weil—Peterssonmeasureis not invariant underWeyl rescalings.it is more natural to treat thestring measureas a whole without referenceto theWeil—Peterssonmeasure.The full Polyakov measureis, of course, invariantunderweyl rescalings.

*~> This fact has been noticedby severalphysicists.~The ghostswe considerhere are distinct from thereparametrizationghostsc(cr), b(cr); they areperhapsmore aptly referred to as ‘ghost

strings”since they areentire string fields thatplay therole of Fadde’ev—Popovghostsin fixing thegauge invarianceof theclassicalactionfor thestring field.

S. B. Giddings, Conformal techniques in string theory and string field theory 191

detl(P~Pt)v2 = J~c’ ~‘ ~b’ ~b’ e~°, (3.4)

where

~= ~—Jd2oVgb~V1c?2g~+c.c.~~~— (b~,V1c~z)+c.c.

In this integral we of course omit the zero modes of ~ b~,which are the holomorphicquadraticdifferentials and their complexconjugates.This is indicatedby the prime; since eachsuch non-zeromode of ~ is in one-to-onecorrespondencewith a continuousvectorfield on the surface,we alsoput aprime on the ghost field c to indicate the correspondence.

As it turnsout,we can alsorepresentthe finite-dimensionaldeterminantin (2.7) as a pathintegralby a slight extensionof the ghostandantighostfields. Of courseit is easyto extendthe antighosts;wejust include Grassmanfields b~correspondingto the holomorphicquadraticdifferentials.To extendthe ghost fields, recall that we can representa tangentvector to S by V1V for somediscontinuousvector field V. (If V werecontinuouson the Riemannsurfaceit would generateapurediffeomorph-ism ratherthana deformationof complexstructure;as discussedin section2.2, we obtaina changeinthe modulus of the surface if we take V to be a discontinuousvector field on the surface.)Correspondingly,we introducethe extraghostfields c,, which arediscontinuouson the surface.Thenthe finite-dimensionaldeterminantin (2.7) can be written

deta,~p~=JJJ~ca~~bbaexp(_~((b;z,vsc+(b~i,vz~:))). (3.5)

The two pathintegrals(3.4), (3.5) can be combinedinto a single pathintegralwith action

SG= 2 (b:z,v5c’~)+ ~— ~ (b~,V~c)+c.c.,

wherec.c. indicatesthat we addalsothe actionfor the complexconjugatefields. We can lump thezeromodesand the non-zeromodestogetherinto a single field by defining c = c’ + ~a ca, b = b’ + Ea b~.Thenthe action can be rewritten

S0 = (b5~,V5c5) — ~ (bk,V

5c~)+ c.c.

wherewe usethe fact that (b5,VsC~z)= 0, as can easily be seenby integrationby parts. To get anon-zeroanswerfor the integral over the b and c fields we have to saturateall modesin the Berezinintegral.The only way to saturatethe integralis to pair termsfrom b’Vca with termsfrom b’~Vc’,or topair termsfrom b’Vc’ with termsfrom b’~Vca,as the readercan easilyverify. The crossterm involvingb’ andca will thereforenot contributeto the pathintegral becauseof thevanishingof the baVcP term;thus it can be droppedfrom the aboveexpression,leavingthe action

SG = ~ Vscz)+ c.c.

192 S.B. Giddings, Conformal techniquesin string theory andstringfield theor~

After assemblingthe abovepieces,we find for the Polyakov path integral

r (~g—6+2n+*n+3.s I

A(1 n) = j II dr,, ,j ~2~ci~b ~T~b

ti

~ ... ~ (3.6)

Here the coordinatesT~provide the coordinatesfor the moduli space.ii1~,.We should also include in our discussionof the ghostcontributionsto the Polyakovpathintegral

somecommentson the ghostinsertionsdueto the curvatureof the Riemannsurfaceandat the externalstates.Theseghost insertionsmust be includedwheneverwe work in the bosonicformulationfor theghosts,so that we get the correctcorrelationfunctions. In fact, when we actually perform the ghostpath integral in (3.6) we will encounterthe anomalyin the conservationof ghostnumber[381.This ismost transparentwhenwe work with bosonizedghosts (cf. refs. [22]; see also refs. [39]), cZ = e”~’~,~ = ~ In suchexpressionsthe coefficient of i~in the exponentcorrespondsto the ghostnumberN0 of the operator.In the action for the bosonizedghostthereareexplicit termscouplingçb(z) to theEulercharacterdensityon the Riemannsurface[40,32],

S~=~J d2ff~gga~6~+~(f d2~R~+ J ~ (3.7)

(Here k is the extrinsic curvatureof ~ and are the interior angles of the cornerson c3.~,.)

Thereforewe see that Eulerdensity servesas a sourcefor ghostnumber.~ In particular,this actionthen gives a term e3’~~2(correspondingto a ghostnumber~) at eachof the curvaturesingularitiesatthe interaction points in the Feynmandiagramsfor Witten’s string field theory. Furthermore,in thecomputationof theghostcorrelationfunction the externalstatesmustalsoserveas sourcesfor the ghostfields. For the externalstatesconsideredas strips coming in from largetime, the ghostnumberis justthe ghostnumberfor physicalstates,N

0 = — ~. In actuallycomputingthe ghostcorrelationfunction itprovesusefulto conformally mapthe semi-infinite stripsto finite points; for openstring diagramstheconformal mapping is usually takenso that thesefinite points lie on a straight boundaryof the newworld sheet.Fromthe action(3.7) it is clear that this conformalmappingchangesthe ghostnumberby

+ ~sincewe aremappingthe endof along rectangleinto a straightboundary,in the processflatteningout the two right angles.Therefore the vertex operatorthat shouldbe used to representthe externalstateon a straightboundaryhas ghostnumber +1. The natural (BRST invariant) vertexoperatorstouse are thus operatorsof the form czV~.,where V~.is one of the usual vertex operatorsin theKoba—Nielsenformulation [42]. This argument,whenmadeprecise,gives a relation betweenon-shellstatesin string field theoryand vertex operators(including ghostfactors) in the Polyakovapproach.

We are now quite closeto having the Feynmanrules that would comefrom gaugefixing the stringfield theory. To see this, notice that the moduli for the field theory Feynmandiagramsare theSchwinger parametersTa~ A deformation of a particular Ta just correspondsto a deformation

* The curvature terms in (3.7) haveessentiallytwo roles: they lead to cancelationof the conformalanomaly in thePolyakov integrand,and

they ensurethat thefields e’~1.e*o) transform with the correctconformalweight. Thesestatementsareelaboratedupon in ref. [41].

S. B. Giddings, Conformal techniques in string theory and string field theory 193

lengtheningthe rectangularstrip that representsthe propagatorby an amount ~‘r. The discontinuousvectorfield that accomplishesthis can easilybe written in termsof the naturalcomplexparameterz onthe strip. The parameterz = x+ iy is chosensuchthat x rangesfrom 0 to ir andy rangesfrom 0 to T onthe propagator.Then the desiredvectorfield is

~Z~TVZ~T2 ify<a,

SZ=~TVZ=0 ify>a.

This vectorfield hasits discontinuity alongthe line y = a. Notice that the vectorfield is takento havesupportonly on the propagatorwhosemoduluswe arechanging;it is zeroelsewhereon the Feynmandiagram.Furthermore,V is designedto only havea discontinuityalongy = a; it goes continuouslytozero at eitherendof the propagator.The ghost field correspondingto this deformationvectorfield issimply c = OnV~,whereOn is someconstantbasiselementof the Grassmanalgebra.In the pathintegralwe can do the Grassrnanintegralover the ca, which pulls down into the correlationfunction aproductof terms5 d~crs/ggZZb5,V5V.We can nowintegrateby parts;althoughb~2is not holomorphic,the fullcorrelationfunction with b5~insertedis, as a function of z, holomorphicaway from externalsourcesofghostcharge(of which thereare nonewithin the supportof Va), so upon integrationby partswe getzerofor the term with V5 actingon b55. However,this is not the end of the story; sincethe field V isdiscontinuouswe get from the integration by parts a “surface term” that is proportional to thediscontinuity in V~.This term takesthe form Sc dzb55, whereCa is the contouralongwhich Va isdiscontinuous.Thereforewe find preciselythe ghostline integral insertionsthat appearin the gaugefixing ansatzmotivatedby the non-interactingtheory,namely b0 = Sc dz b~5.

For the Feynmanrules of the gauge-fixednonlinearaction we expect to find all valuesof ghostnumbercirculating in loops(we havein the Polyakovapproachnothingrestrictingthe ghostnumberinloopsto that of physical states,Ng = — ~). We havefor the propagatorthe ordinarykinetic operator4correspondingto free propagationof the string,togetherwith a b0 insertion.The gauge-fixedactionforthe string field theorycan thenbe written schematicallyas

~ (3.8)

where .~ is a string field with no restrictionon its ghostnumber.Recentwork indicates that such anactionhas a secondquantizedBRST invarianceand furtherappearsto reproducethe Feynmanrulesdeducedhere from the Polyakovapproach[43]. Furthermore,a generalgaugefixing procedurehasbeen applied to the action (3.2) and from this procedureit appearspossibleto derive a gauge-fixedaction’5~of the form (3.8) by passingto the Siegel gauge[29].

3.4. Summary

I havedescribedhow Witten’s open string field theory capturestwo aspectsof the geometryof themoduli spaceof Riemannsurfaces.Thefirst is the topology of moduli spacethat is describedby the cell

* This derivation apparentlyrelies on the fields being associative;however, when considering loop diagrams closedstrings are also involved,

and theproduct * is non-associativefor closedstring fields [28].Therefore thingsmay be evenmore subtle than they first appear in the derivation ofthe gauge-fixedFeynman rules for witten’s string field theory. I thank A. Strominger for pointing these facts out.

194 S.B. Giddings.Conformaltechniquesin string theory and string field theory

decompositionof moduli spacein which eachdistinct string field theorydiagramcorrespondsto a cell.The secondfeature is the expressionof an integrationmeasureover moduli spaceprovidedby thereparametrizationghosts. This measurecan optionally be thought of as having its origin in theWeil—Peterssonmeasureon moduli space.Using the rules I havesketchedhere,it shouldbe possibletodeducethe ghost insertions(up to termsthatdecouplein the path integral) that would comefrom anarbitrarygauge-fixedstring field theory constructedusing the ghost fields.

A big open problemis that of inventing a field theory structurethat describesclosed strings. Thelocal measurefor integration in such a theory would also be provided by the ghostsin the mannerdiscussedabove,but it is not known how to invent a gauge-invariantstructurefor the closedstring thatgives the expectedghost insertionsand geometricalFeynmanrules. It might be hopedthat a closedstring field theory gives a natural cell decompositionof moduli space.One problemwith this is thenaturalappearanceof closedRiemannsurfacesin the closedstring Feynmandiagrams.*)It is presentlynot knownhow to triangulatethe moduli spaceof a Riemannsurfacewithout boundariesor punctures.Furthermore,closed string structuresappearto be closely tied to the complexstructureof modulispace,so we would hopeto havea cell decompositionof moduli spacein which the complexstructureappearsin a more transparentguise.

One possible approachto the construction of a closed string field theory is the light-cone-likeapproach[451.Furtherencouragementthat the light-cone-likeapproachmaybe relevantis had onceone realizesthat the light-cone Feynmandiagramsdo in fact provide a cell decompositionof modulispace(with a single top dimensionalcell!), andthat this cell decompositionis extremelynatural fromthe Riemannsurfacepoint of view, and also is closely relatedto the complexcoordinateson modulispace[461.This construction,which is also instrumentalin proving the equivalenceof light-conegaugestring theory and Polyakov string theory (and hence unitarity of Polyakov string theory) will bediscussedin the next chapter.

4. Equivalence of Polyakov and light-cone formalisms

Historically there have been two functional integral approachesdeveloped to treat the loopexpansionin string theory.The interactingstring picture[47], whichwas developedfirst, is closelytiedto the physical picture of strings propagatingin spacetimeand undergoingoccasionalinteractions.Incontrast,the Polyakovapproach[71involves a sum over geometricalsurfacesand hencethe physicalpicture of string propagationis more obscured.A long-standingquestionhasbeenwhetherthesetwoformalismsare in fact equivalent;in this chapterI will summarizesomerecentwork establishingtheequality of amplitudescalculatedby either prescription[46,48]. For simplicity I will only discussthecaseof closed strings, exceptin section4.4, wheresomecommentsabout the moduli spacefor openstrings aremade.

Thereareboth physical andmathematicalmotivationsfor wantingto know that theseformalismsareequivalent.On the physicalside, thisresult is oneway to establishtheunitarity of the PolyakovAnsatz.Since the interactingstring picture originatesfrom a treatmentof strings in a physical gauge,it gives

* I This is potentially not a problemif thePolyakov pathintegralsoverthe closedsurfacesall vanish; this has beenshownto happenin certain

cases(seeref. [44]). Nevertheless,we shouldprobablyhave in string field theorya meansto computethese,to showthat they are zero.

SB.Giddings,Conformal techniquesin string theoryand stringfield theory 195

manifestlyunitary amplitudes;hencethe equivalencewith the Polyakovamplitudesestablishesformalunitarity of thoseamplitudes.In fact, sincethe amplitudesof the interactingstringpicture areunitaryby construction,we shouldbe suspiciousof the Polyakov approachif it doesnot reproducethe sameamplitudes.Furthermorethis constructioninvolvesa treatmentof the light-coneworld sheetsfrom thePolyakovpoint of view; this can in turnbe usedto gain insight [32] into possibleformulationsof stringfield theory involving the light-cone world sheets[45].

On the mathematicalside, equivalenceof thesetwo formalisms could hardly be expectedif thelight-coneworld sheetsdid not provide a single coverof moduli space.However, this is a non-trivialstatement.Moduli spaceis topologically quite complicated(its Euler characteris x = ~(1— 2g) forgenusg, with onepuncture[33,36]), andso in generalit is hardto find simple parametrizationsof it.Until it was shownthat physicistshadheld in their possessionfor severalyearsthe explicit light-coneparametrizationof moduli space,mathematiciansonly knew of essentiallyone explicit parametrization[33] (and still none for the moduli space of a surface without marked points). The light-coneparametrizationin fact providesa cell decompositionof moduli space, and thus may be useful forextractingtopologicalinformationaboutmoduli space.Finally, the Polyakovintegrationmeasureovermoduli spacehasbeenshownto be the squareof a uniqueholomorphicform on moduli space,anditsequivalencewith the light-conemeasuregives asimpleexpressionfor this form which involvesonly thedeterminantof the scalarLaplacian.

There are two main steps necessaryin establishingthe equivalenceof the interactingstring andPolyakovamplitudes.Both approachesinvolve the integral of some integrationmeasureover somefinite-dimensionalspace. Hence equality is establishedif we (a) demonstratethe equivalenceof theintegrationregionsand (b) demonstratethe equivalenceof the integrationmeasures.

As we saw in section2.2, a Polyakov amplitudefor a processwith n externalstatesandg loopsiswritten abstractlyas a functional integralover the spaceof all metrics; this is gaugefixed by picking aslice to which any metric can be brought by a combinedWeyl rescalingand diffeomorphism.Thefunctional integral is then (in D = 26)

det( 4, ~ -t3A~(1,. ,n)J[dm] det(4,,~~)~2(det~p~pt)t12(JMd2~~det14)~ ira). (4.1)

Recall that At~denotesthe moduli space, which is representedby a gauge fixing slice and hascoordinatesm. The Beltrami differentials /~aarethe tangentvectorsto the slice correspondingto unitdeformationsin the coordinatesm, andthe quadraticdifferentials4,~arethe basiscovectorsfor modulispace.Thus the expressioninvolving ~ and 4, representsthe “volume element”on moduli spacein thecoordinatesm. The term det’(P~P~)is the gaugefixing determinant.The determinantof the Laplaciancomes from the functional integral of the maps of the world sheet into spacetime,as does thecorrelation function involving the external states (here representedby wave functionals ‘V). Thisfunctional integral over X’~is computedusing the Polyakov action5~= ~ SM d

2~s/ggm’ ef1,~X~’~In the interactingstring picturethe analogousamplitude is

A1(1,.. . , n) = f [ds] (SM ~ det’ 4) -12 ~y ... ~ ~. (4.2)

196 S.B. Giddings. Conformal techniquesin string theory and stringfield theory

Here the integration is over the parameterss parametrizingthe light-cone diagrams(to be describedshortly); from the functional integral over the 24 physical degreesof freedom we find the factorinvolving the determinantof the Laplacian and also the correlationfunction from the externalstatewave functionals ~W.This functional integral is computed using the light-cone action 5~=~ 5 dT du [(~~X) + (t9~X)]. The externalstatecorrelationfunctionsareequal in the two approaches(this can be checked explicitly for tachyonic external states [47], and follows in general byfactorization).Thereforeto showequivalenceI will first, in section1, arguethatthe light-conediagramscover moduli spaceexactly once [46]; then computationof the Polyakovmeasurein the light-conecoordinatesfor moduli space,demonstratedin section2, will show the equalityof the two integrationmeasures[48]. Section 3 will sketchthe relationshipbetweenthe externalstate correlationfunctions.

4.1. Theintegration region

Light-cone diagrams are constructedby connecting together free string propagatorsusing thefundamentallocal closedstring interactionshownin fig. 12. A typical light-conediagramis shownin fig.13. In a light-cone diagramthe radiusa of any string is proportionalto its light-cone momentumPwhich is conserved;thuswhentwo strings join to form a third the radiusof the third string is equaltothe sum of the radii of the first two. The parameterslabeling such a diagram (with g loops and nexternalstates)are the radii of the externalstrings a•, i = 1 n, a set of g independentradii ofinternal strings, a1, I = 1, . . . , g, the interaction times (one sets the first interaction time to zero) r,,,a = 1,.. . , 2g + n —3, and the twist angles of the internal strings, Q, a = 1,. . . , 3g + n —3. For agiven scatteringprocessthe a,, which aretakenproportionalto the P’ momentaof theexternalstrings,are held fixed. The otherparametersareintegratedover; sincethereare6g — 6 + 2n of them,which ispreciselythe real dimensionof moduli space,it is at least conceivablethat as theseparametersrangefreely we could cover moduli space.The basic idea behindthe proofthat we do in fact cover modulispaceonceis that for a given Riemannsurfacewith somespecifiedpoints, i.e. for a given point inmoduli space,and for a given choice of a, (i.e. external P~)there is a unique canonical Abeliandifferential w. Furthermore,such an Abelian differential is in one-to-onecorrespondencewith alight-cone diagram; as we will see,w inducesthe metric on the diagram.This will showthat light-conediagramsarein one-to-onecorrespondencewith pointsin moduli space,and sowe havea single cover.

To see how this works, supposewe aregiven any Abelian differential cv = w dz on someRiemannsurface;herez is a local complexparameter.Define anew complexcoordinatew = f~w,so dw = cv, andtakethe flat metric in the coordinatew,

ds2 = dw d~= kv~Vdzdi.

This metric is conformalto the original metricdz d~,with conformalfactor wj2, Furthermore,whereasthe original coordinatez was in generalonly definedin somesmallneighborhood,the coordinatewandthe metric dw di~areessentiallyglobally definedbecausecv was, althoughw is multivaluedby integrals

-00Fig. 12. The fundamentallocal closed string interaction.

SB. Giddings, Conformal techniques in string theory and string field theory 197

(“~ (I 4 (

U ~ 1 (84 a~ ~i(V I 182 ~‘‘~ ~85 \ a1v / I I s.~ I I8~

~ ‘~ iI~6

I \~ ~‘‘ v ~r-O ‘r1 ‘i~2 1~4 T5 T3

Fig. 13. A light-cone diagram with threeincoming and one outgoingstrings, and two loops.

of w aroundcycles on the surface.The only placeswherethis all breaksdown are at the zeroesandpolesof w. Supposecv hasasimplezeroat a pointz0, so that dw -~ (z — z0)dz,or w — w0 -~ (z — z0)

2. Ifwe circle the point z

0 (in the locally flat metric dz di), the argumentof z — z0 increasesby 2~r,so theargumentof w — w0 increasesby 4ir. But thisis preciselythechangein anglefound in the flat metric ona light-cone diagram if we circle one of the interaction points. So simple zeroes of an Abeliandifferential correspondto light-cone interaction points. Further, if w has a simple pole, so thatdw — dzl(z — z0), then w — w0 ‘-~ ln(z — z0). The logarithm induces a conformal transformation takingus from apuncturedneighborhoodof the point z0 to a cylinder shootingoff to infinity, with z0 beingmappedto infinity. So we find that simplepolesin anAbeliandifferentialcorrespondto externalstateson a string diagram.

In generaltherearemanyAbeliandifferentialson a Riemannsurface;clearlywe must look for morerestrictions. On a light-cone diagram let w = T + io be the natural complex parameter;T is thelight-conetime, ando labelspositionalong thestring at fixed time. Now considera loop a1 circling oneof the internalstrings on the light-cone diagram;the integral faj dw clearlygives i timesthe lengthofthe internalstring (seefig. 14). Similarly for the integralaroundloopsC,, Cf about externalstrings we

~ Re(w) 01

Im (w) i~)~ I

(a)

(b)Fig. 14. (a) Various generatorsof the first homology of the light conediagram; (b) the corresponding cycleson a general conformally equivalentsurface.

198 S.B. Giddings. Conformaltechniquesin string theory and stringfield theory

just get i times the lengthof the externalstring. Now considerthe integralaroundaloop b1 circling oneof the internalloopsof the stringdiagram,as shownin fig. 14. Sincethe net changein light-conetime iszerofor a closedloop, Sb dw againgives a purely imaginarynumberrelatedto the twistsof the internalstrings. Sinedw = cv, and sincethe setof all loops a, b andC form a homology basisfor the Riemannsurface,we see that we require an Abelian differential with simple poles of fixed real residues(i.e.external P~) andpurely imaginary periods aroundany homology cycle.

It is easy to show that such a differential always exists and is unique. In fact, the space ofmeromorphic Abelian differentials with simple poles at two specified points P and Q is (g +1)-dimensionalby the Riemann—Rochtheorem.By addingsuch differentials we can always makeadifferential ,u with simplepolesat any specifiedpointsand with anygiven residues(as long as theysumto zero, in correspondencewith P~conservation). The most general such differential is thencv = j.~+ ~ E1w1,whereE1 areg arbitrary complexconstants,and cv,, arethe g linearly independentholomorphicdifferentials on the surface. The condition of purely imaginary periods gives

2g realconditions(g for the a curvesandg for the b curves)that uniquely fix the constantsE,. All we havetodo to solve for theE,, is invert thismatrix, which turns out to be the imaginarypartof the period matrix.This is positivedefinite andthereforeinvertible for a non-singularRiemannsurface,andthus we obtainthe uniquecanonicaldifferential cv.

We should also verify thatour uniquecanonicalAbeliandifferential cv doesin fact give a light-conediagram.To see that it does, define the complex coordinatew by dw = cv, and define a metric byds2 = dw~As mentionedabove,this metric is flat where it is well defined;the only placesit is not flatis at zeroesandpoles of cv. Furthermore,in the neighborhoodof a polethe metric looks like that of atubepropagatingoff to plus or minus infinity (dependingon the sign of the residueat the pole). Linesof constantT = Re w are the crosssectionsof the cylinder(s),and correspondto the string(s) at fixedlight-cone time T. In the neighborhoodof a zero,the metric providesa geometrythat lookslike that inthe vicinity of an interactionpoint in a light-conediagram. So locally everythinglooks fine. To see whatcould go wrongglobally, considertheAbeliandifferential r’ = e’4 dz on the squaretorus. If the angle4,correspondsto a line of irrational slope,then the lines of constantT = Im i.’ will display ergodic-typebehavior;theyneverclose.We shouldbe carefulto rule out suchbehaviorin the trajectorystructureofour canonicaldifferential; in the light-cone picture the trajectoriesof constant ‘r correspondto acollection of closed strings at fixed time T. If we did encountersituations where the constant T

trajectoriesexhibitedergodic-typebehavior, then we would have points in moduli spacethat do notcorrespondto light-cone diagrams.

Certainlythe constant-Ttrajectoriesarewell behavedin the vicinity of the poles,but it is not obviousthat they staywell behavedafter the first interaction.The key to ruling out ergodic-typebehaviorandinsuring closedtrajectoriesat fixed time is the fact that cv hasimaginary periods.To see this, supposethat cv did havea trajectorythat doesnot close (or endon azero). Thenby compactnessof the surface,theremust be somepoint of the trajectorysuch that the trajectory returnsto a small neighborhoodofthat point. (In fact, the trajectorymust havea limit cycle.) If thishappens,wecan makea closedcurveby addinga small pieceof an orthogonaltrajectory,i.e. one alongwhich Re cv is constant;seefig. 15.But thenthe integralof cv aroundthe resulting closedcurvewould havea real part,which comesfromthe small added piece! This contradictsthe fact that cv haspurely imaginary periods,so ergodic-typebehavioris ruled out. Lines of constantr arein fact closedstrings, andthe differential cv does in factinduce a light-cone diagramstructureon a generalRiemannsurface.

~ Suchdifferentialswere in fact known to theancients;for a moremoderndiscussionsee e.g. Farkasand Kra [t5[.

S. B. Giddings, Conformal techniques in string theory and string field theory 199

C ~

I ~ J’Re(w)~CONSTANT

/ ~r~r

/

Fig. 15. A hypotheticalnon-closedtrajectory is closed by addinga small segmentof an orthogonaltrajectory.

From this argumentwe see that the light-cone diagramscertainly cover moduli space.That theycovermoduli spaceonly once follows from the fact that the specificationof a light-cone diagram(withalmost everywhere flat metric) clearly determines a Riemann surface and a canonical Abeliandifferential cv = dw, andthis differentialmust be unique.So we find that the light-cone diagramsdo infact give usa single coverof the moduli spacefor a Riemannsurfacewith two or moremarkedpoints.Further, sincethe coordinates(T, a, 0) aregiven by the periodsof the canonicalAbeliandifferentialorby its integral between its zeroes,these coordinatesare nicely related to the natural complexcoordinateson moduli space.For more details the readershould see ref. [46].

4.2. The integration measure

Havingdispensedwith the issueof the integrationregion,we nowturn to the issueof the measure.Iwill discussthe computationof the Polyakovmeasurein the light-cone coordinatesfor moduli space,and show that it is equivalentto the light-cone measurein (4.2). The basic idea here is that thePolyakovmeasureinvolves the determinantof the vector Laplacian,P~P1 however,on a flat surfacethereis little to distinguishvectorsfrom scalars.The light-conesurfaceis flat exceptfor isolatedpoints,so onemight expectthe determinantof the vectorLaplacian(to the onehalf power, sincea vectorhastwo components)to be equalto the determinantof the scalarLaplacian,up to a correctionfactorduetothe curvaturesingularities.The latter factor will cancel the finite-dimensionaldeterminantin (4.1)involving ~ and 4,, and the determinant of the scalar Laplacian will cancel one power of thedeterminantin (4.1), leavingus with the determinantto the minus twelfth power as in (4.2).

To actually show that this works we must computethe variousfactorsin (4.1). I will sketch thiscalculation; further details can be found in ref. [48]. In calculatingthe finite-dimensionaldeterminant,we will needa basisfor the quadraticdifferentials (i.e. the cotangentvectorsto moduli space).Theseareconstructedusingthe Abeliandifferentialswe alreadyhavein hand.A set of g linearly independentquadraticdifferentialsis { ~ } = { w~w,,~}. The other 2g + n — 3 differentials arefound by noting thatthe canonical differential has 2g + n — 2 simple zeroes(light-cone interactionpoints) P0, Pr,. . . , P,~.Thereexist

2g + n — 3 independentAbeliandifferentials that havesimple polesat pointsP0 and

~a then the set { ~ } = { w~w,,~} consistsof quadraticdifferentials holomorphic on the Riemannsurface(minusthe externalstatepoints)andlinearly independentof eachotherandof the 4,,,.With this

2(X) S.B. Giddings. Conformal technique.sin string theorc andstring field theory

form for the quadraticdifferentialsthe computationof the overlapmatrix (4,,,, 4,~)becomesparticularlysimple; we have

(4,,,, 4, )f d2~~(g~)24, 2iJw,, A Wd.

wherewe usethe fact that the metric on the light-conediagram is ds = cv~.Notice that the canonicalAbelian differential hascompletelydisappearedfrom this formula. From this simple expressionthesematrix elementscan be computedusingRiemann’sbilinear relations(see,e.g.. Farkasand Kra [15]),andwe find after a bit of matrix algebra

[det(cb,,, 4,~ t ‘~ = 2c~(4~~)2~f det(Im .12) det[ GP(P~,P,,)] (4.3)

whereQ,,~is the periodmatrix for the Riemannsurface,and G,,(P,,, P,,) is a Green’sfunction for thescalar Laplacian, zi~G~(P,Q) = —2i~[6(P,Q) — 6(P, P

11)]. This Green’s function is constructedbytaking the real part of the integralof the Abelian differential with simplepoles at Q and P15 andwithpurely imaginary periods, G,,(P, Q) = Re J’~WQp.

Next we computethe determinantK /~a’ 4,~).To do this we mustconstructthe Beltrami differentialsPa correspondingto deformationsof the light-cone diagraminducedby unit changesof the coordinates‘r, a and aO. As discussedin section 2.2, the Beltrami differentials correspondingto deformationsofcomplexstructureareinducedby discontinuousdeformationvectorfields. The generaldeformationof alight-cone diagram is generatedby deformationsof the Euclideancylinders comprisingthe diagram.Therearethreefundamentaldeformationsto consider:the twistdeformations(twisting a cylinder); thestretchdeformations(lengtheningthe cylinder); and the shift deformations(changingthe radiusof thecylinder). These changethe coordinates0, T and a respectively.These deformationvectorfields areeasilywritten down, andthe Beltrami differentialcorrespondingto the (discontinuous)deformationV~

is given by ~4 = ~ The overlap of such a differential with a holomorphicquadraticdifferential is

4,) = 2i fdz A d~(V;V2)4,, = —2i ~dz (disc Vz)4,,~:

hereI haveintegratedby partsand usedthe fact that ~ is holomorphic.We areleft with a boundaryterm, given by the integral over the contourC alongwhich Vz is discontinuous.For unit deformationsof r, a and aO the discontinuity in V’ will be unity in the w coordinates;using V = V”/co.. we find

(~‘4,~) 2i~cv,,.

Thus we reduce the evaluationof the (~,4,) overlap to the problem of integratingthe differentialscv,,, ~ over various homology cycles on the surface.The requiredintegrals are simple to perform;theyjust give elementsof the period matrix or residuesat the poles of cv,,,,. The determinantis theneasilycalculated,and the answeris I

detK p.o, q5~)= (8 )g(4 )

4g+2n—6 det(Im (2)

S. B. Giddings, Conformal techniques in string theory and string field theory 201

Combining this with (4.3) then gives the completefinite-dimensionaldeterminant,

et p~,4,p 3g+n—3 —1

A ~ \112 = (4ir) det {Gp0(P~,Pb)) . (4.4)uet~.Pa,9’~~)

Next we mustcomputedet(P~P1)1’2wenow know that weshould find the determinantof the scalar

Laplaciantimes a factor canceling(4.4). The vector and scalar Laplaciansclearly act on differentspaces,so to comparethe two operatorswe mustsupply a prescriptionto convertvectorsinto scalars.Given a vector Vz, the obviousway to find a pair of scalarsis simply to take the componentsof thevector in the light-cone coordinates;thus for eachreal vector we haveone complexscalargiven by

~Z_çj,/~~, V5=çfr/w5. (4.5)

The spaceof vectorson which thedeterminantof P~P1 is takenis the spaceof regular diffeomorph-isms of the surface.Sincecv vanishesat the points P0, ~a’ we musttake i/i to vanishtherealsoso that Vzis regular.It is not difficult to showthat on a vector field suchas (4.5), P~P1just reducesto the scalarLaplacianactingon i/i, plus correctiontermsdueto the curvaturesingularitiesatthe interactionpoints.The correctionterms involve delta functions at the points P0, ~a’ however,so they drop out whenmultiplying a function that vanishesat thesepoints. The result is that det(P~P1)~

2= det*4, wheredet* is the determinanttakenover the spaceof scalarfunctionsvanishingat the points P

0,~ Thisdeterminantin turn can be expressedas a path integral,

(det* ~~112 =J~e’° ô[~(P0)]~[~(P1)]..~~[~(P2g+~3)],

with action

1~=~

The deltafunctionscan be representedby their Fourier transforms,

3[cli(Pa)] = ~

With theseinsertedin the path integral, we can complete the square[using the Green’s functionG,,(P, Q)]; we then integrateover the variablesw to find

(detP~P1)t12 det* 4= (sM~~ det’ 4) det{Gpo(Pa,Pb)}. (4.6)

Finally we combine (4.1), (4.4), and (4.6) to find the Polyakov amplitude in the light-conecoordinateson moduli space,

A~(1,...,n)=J[dr][da][adO}(5~ det’4)(~1.~~ç).

202 S.B. Giddings. Conformaltechniquesin string theory andstring field theory

This showsus that the Polyakovmeasurefor integrationover moduli spacein fact equalsthe measure

deducedfrom the light-cone approach.

4.3. Wavefunctions.vertex operators, and physicalstates

I havenot quite shownthat the amplitudesareequivalent;we muststill comparethe momentumandpolarization dependentX correlation functions in the two approaches.It is fairly easy to see thisequivalence.In fact, as mentionedabove,thesecorrelationfunctionsagreewhenall externalstatesaretachyons[47]. The equivalenceof amplitudeswith generalphysical statesthen follows by factorizationof amplitudeswith purely tachyonicexternalstates.To makethis equivalenceslightly more concretethereare several things that should be done. First of all, on the light-cone world sheets,the externalstatesare representedby wave functionson the boundariesapproachinginfinity. However, in the usualPolyakov approachexternal states come from vertex operators.Furthermore,we know that theamplitudes of the two approachesare the same for external tachyons, and that therefore byfactorizationtheyare the samefor arbitraryamplitudes,but it would be nice to seea bit moreexplicitlythe equivalencebetweenlight-cone and Polyakovstates.In this section 1 will briefly discussboth oftheseissues.For more detail the readeris againreferredto ref. [48].

4.3.1. Wavefunctionsand vertex operatorsFirst I discussthe relationof wave functionsandvertexoperators.Oncewe havedonethis, we can

use the languageof vertex operatorsto comparephysical statesin the two approaches.Thereare twostepsin relatingthe typeof prescriptionwe havein the light-conecaseto the usualPolyakovformalismwith vertex operators.First one showshow to interchangewave functionswith local operatorson theworld sheet.Following this, it must also be shown how to split the integration over the 3g — 3 + nmoduli of the non-compact,puncturedRiemannsurfaceinto the integrationoverthe 3g — 3 moduli ofthe unpuncturedsurface,togetherwith the usualintegrationover the positionsof the vertexoperators.

In converting the wave functionsinto vertex operators,the basic observationis that a Riemannsurface with a tube traveling off to infinity is conformally equivalent to a finite surfacewith anextremelysmallholecut out of it. In a conformally invariant theory we can thereforeinterchangestatesthat propagateto or from infinity with local vertex operators.Conversely,starting with a vertexoperatoron a surface,we can cut out a small disc on the surfacearoundthe vertex operator;the pathintegralon the disc, as a functional of the valueof the field on the boundaryof the disc, gives a wavefunction. This wave function thenprovidesthe boundaryconditions for the surfaceminus the smalldisc. This is oneway of viewing the equivalencebetweenwave functionsand vertex operators,andisillustrated in fig. 16. A vertexoperatortakesthe generalform (for simplicity we choosethe origin ofour coordinatesas the location of the vertex operator)

V(r, k, X) = P[E, ~X](0) e’~°~. (4.7)

whereP[r, aX] is a polynomial in derivativesof X. Correspondingto this is the wave function

X(~r)

W[X(~)]= f ~ V(E, k, X) e’~. (4.8)

S. B. Giddings, Conformaltechniquesin string theoryand string field theory 203

= X (cr)

Fig. 16. The path integral over a surface with a vertexoperatoris split into thepath integralover thesurface minus a small disccontainingthevertex operator,convolutedwith the pathintegral over thesmall disc. The path integralover the small disc thereforeplaysthe role of a wavefunction.

To evaluatethiswave function, wewrite X~’(z,i) = X~(z, i) + Y~’(z,~), whereXeL is a solutionof theclassicalequationof motion, 4X” = 0, andhasboundaryvaluefixed by X~(u),andY~is a fluctuationfield obeying the Dirichlet boundaryconditionY~,SD = 0. SinceX’~satisfiestheequationsof motion,itcan be written

Xt’(z, i) = X~+ ~ (X~~?+ X~~f’).n>0

Here z = ~ and X~,,,X~nare the (normalized) coefficients in the Fourier expansionof theboundaryvalueX~(o)(3 is the coordinateradiusof the disc),

XlL(o) = X~’+ ~ (X~~3”e”~”+ X~~,ô”e”°).n>0

We find for the wave function

W[X(u)] = J ~ P[r, ôX~+ aY~]e1~Ik~

xexp(_~fd2~ômY~amY~)exp(_~~dnmX~ am~ç). (4.9)

Now since vertex operatorsare constructedso that they are normal ordered [49], we should notcontracttwo legs on the samevertex.*) This implies that

W[X(~)] = P[r, a~](O)e’~°~exp(_~ dnmX~amx~). (4.10)

In the limit wherethe size of the disc tendsto zero, the Gaussianfactor in (4.10) tendsto unity, andupon normalorderingwe find the expressionfor thewavefunction at thepuncturecorrespondingto theoriginal vertex operator.For example,for the graviton we have

WG[X(0)] = e~X~1X~1eu1~~’0A.

~ Notice, however, that the Dirichiet Green’s function for the disc differs from the Green’s function for the full Riemann surface, so we havedifferent normalordering prescriptions. However, in the limit 5—sO the difference can be neglected.

204 S.B. Giddings.Conformaltechniquesin string theory and stringfield theory

Notice that in comparingthe wave function and vertex operatorexpressionswe haveabsorbedsomefactors; thesefactorscontributeto fixing the relative normalizationsof the wave functionsandvertexoperators.

Having sketchedthe relationbetweenon-shellwave functionalsprovidedon boundariesapproachinginfinity andlocal operatorson the world sheet,we next turn to theissue of splitting the measureinto apiececorrespondingto integrationover the unpuncturedRiemannsurfaceand a piece giving the localintegrationmeasurefor integrationover the positionsof the punctures.In a generalset of coordinatesm for the 3g —3 + n dimensionalmoduli spacethe measureis [seeeq. (4.1)]

det(u 4,[dm1det(~,4,~)t2(det* P~P

5)’2. (4.11)

(Here the * on det remindsus that det is takenoverthe spaceof diffeomorphismsthat do not movethepunctures.)We would like to evaluatethis measurein a set of coordinatesm = (M, i,), whereM aresomecoordinatesfor the moduli space of the unpuncturedsurface, and ~, are coordinatesfor thepunctureson the surface.Correspondingto the split betweencoordinatesfor the unpuncturedsurfaceandfor the punctures,we havea splitting of the quadraticdifferentialsandof the Beltrami differentials.The set of 3g — 3 + n quadraticdifferentials { 4,~} naturally splits into the 3g — 3 differentials 4,~thatare holomorphic on the full (unpunctured)surface, and a set {4,,} of n meromorphic quadraticdifferentialswith simplepoles at the punctures.Quadraticdifferentials of the latterset aredeterminedonly up to addition of an elementof the former set; this allows us to chooseour meromorphicdifferentials such that (4,A’ 4,.) = 0. Similarly we take for our Beltrami differentials a set {~A}

correspondingto the coordinatesM, anda set { ~r~}correspondingto thecoordinates~. The formerareconstructedfrom discontinuousvectorfields that do not movethe punctures;the latter areconstructedfrom continuousvector fields i’ that do move the punctures.The overlap of one of the ~, with anarbitrary quadraticdifferential is then given by

= 2iJ dz A di (V;V~)4,~~,= _2if dz A di ~ (4.12)

Thereforesuch Beltrami differentials are orthogonalto the 4,A but not to the 4,.With the aforementionedsplitting of the differentials,we can rewrite the measure(4.11) as

[dM][d2~1] ~ ~t~)

t~2 (det* F~P1)”

2. (4.13)

Now, the determinantof the operatorP~P1can be written

[detKPV 4, )j2det ~ = det(4,,, det(Va,Vh) (4.14)

where 1’~,,4,,, are arbitrary basesfor the vectorfields andquadraticdifferentials,respectively.We canalsosplit thisexpression.We takean orthogonaldecompositionof the set {4,,,} into differentials4,, thatare meromorphic, with only simple poles at the puncturesallowed, together with the orthogonalsubspace.We alsosplit the vectorsVa into thosethat vanishat the puncturesandthosethat do not; the

S.B.Giddings, Conformaltechniquesin string theoryand stringfield theory 205

latter groupis takento be a setof 2n vectorfields Vi., eachwith supportonly in a neighborhoodofradius C(6) of the punctureQ1, andwith unit value(in the ~, coordinates)at the puncture.The innerproduct betweenone of the V, and one of the vectors that vanish at Q, is then C(S’), whereas(V~,1’) -~ 3

26~~,so this also accomplishesan orthogonaldecompositionfor small 6. Therefore(4.14)becomes

[det(P1V.,4,

det ~ = det(4,5,4,) det(V~.,~ det*P~P1. (4.15)

Of coursethe factor (~~V, 4,~)is identical to (/2,, 4,,,). Furthermore,we have

det(V,, V)L~2= 32n det[\/g(Q

1)g5,,(Q1)6~1]+ O(62~1~)

32nfl g(Q

1)+O(32~). (4.16)

Puttingall of this into (4.13), we find that the measureis

[dM][d2~~](fl\r~-~))det(~ 4,)~2 (det p±p)t12(fl62\1Th) (4.17)

The last factor is a coordinateinvariant cutoff factor, and is absorbedinto therelativenormalizationofthe vertexoperatorsandwave functions(which mustbenormalordered).Oncethis is donewe seethateq. (4.17) is precisely the desired measurefor independentintegration over the moduli of theunpuncturedsurfaceand over the positionsof the vertexoperators.This completesthe sketchof therelationbetweenthe vertexoperatorapproachandthe wave function approach.

4.3.2. Dictionary betweenphysical statesIn the previoussections I have argued the equivalence of the scatteringamplitudes in the

Mandelstamand Polyakovapproachesfor processesinvolving only tachyonsas externalstates.Theequivalenceof the amplitudeswith arbitraryexternalstatesfollows by factorization:We know that theinteracting string formalism gives factorizable amplitudes, which also happento be equal to thePolyakovamplitudes.Comparisonof the two approachesthengives afactorizationprescriptionfor thePolyakovamplitudes.Sincearbitrary statescan be built up by factorizingtachyonamplitudes,we havethus shown the equivalenceof Polyakov and interacting string picture amplitudes for arbitraryprocesses.Of course we would like to provide a “dictionary” between the states in the light-conepictureandthe physicalstatesin the Polyakovpicture.This is providedby theDDF construction[50];Isketchsomerelevantfeaturesof this for completeness.

As statedabove,to calculatethe scatteringamplitudefor an arbitraryprocesswe simply factorize aprocesswith externaltachyons.This procedureis most easily describedusing conformalfield theoryoperatorproductexpansions.For example, if we havea two-tachyonprocess,we use the operatorproductexpansion(herewe decomposeX(z,~) = X(z)+ X(i) inside the X correlation function; forsimplicity we only write the analytichalf in the following expressions)

e”~’t~~ = (z — w)k.0141T[l + i(z — w)k.o~x— ~(z — w)2k a~x+ . . .] e’~~”~

206 S.B. Giddings.Conformaltechniquesin string theory and string field theory

To pick out a state,we put k+ p on the state’s mass shell; the pole originates from the resulting(z— w)’ term in the aboveexpansion.For example,at mass zero we find the (z — w)’ coefficient

ik~a~,X(w)et~~X(w) = ~i(k — p)~a~,xe’~~ + total derivative

(the total derivativeterm will vanish underthe integral over w). Combiningthis with the anti-analyticpiece,we find the vertex operatorfor a transversegraviton.

In practice,it is easiestto constructthe physical statesby combiningstatesat masslevel zero with asingle tachyon—thisis the DDF construction.We pick an arbitrary light-like vector k. and then let

I ~ —, — — , nkX(,~)A,,(z)= - dw d,.~Xe , A,,(z)= dw ~ e

where the integrals are along small contourssurroundingthe point z, and where i, j, / are indicestransverseto k. These operatorsact on a ground state (tachyon) of momentump, and we takek~p = 4ir. Using theseoperatorswe can converta light-cone state

a,~ ~ exp{i[p + (~(—n1) + ~ (_~))k]. x}I0)

into a covariantstate(or vertex operator)

7i..’/~’~ni~” A”,,A’~~. . . A ~ et~)~~(2). (4.18)

Such a statecan be constructedexplicitly by factorizing amplitudesinvolving eitheronly tachyons,orinvolving gravitonsand a single tachyon but it is easierto describethesestatesin termsof the DDFoperators.It is easilyshown that the A~,,A~commutewith the Virasoro generatorsL~,L,, for n > 0,and that the states (4.18) satisfy the mass shell and twist conditions L55 + L55 — 2 = 0, L55 — L0 = 0(assumingwe havetakenthe momentumon massshell andhaveenforced~, n, = ~ n-). To seethis onecomputesthe operatorproduct of the energy—momentumtensorT,,~,,T,~with the operator(4.18);recall that the L,,, L~arecoefficients in the Laurentexpansionof T,,~,,T,,,,c, aboutthe point z. Further,the A~,,A~satisfy the commutationrelations

[A~, A~,,}= [A~, A~,,J= ~

and for n >0 theygive zeroin the presenceof a momentumsourcee’~.V(z) introducedat the pointz, ascan be easilycheckedby examiningthe relevantoperatorproductexpansion.Thesefactsarein precisecorrespondencewith the analogousrelations for a~,c~,.Thus they producethe samecorrelationfunctions. Furthermore,each state (4.18) clearly correspondsto a Polyakov vertex operator, andconversely,any Weyl and reparametrizationinvariant vertex operatorin the Polyakov formalism isgaugeequivalentto one of the form (4.18) in the conformalgaugeon the world sheet.

Finally, for an example exhibiting evenmore concretelyhow the DDF operatorsmap light-conestatesinto the covariantstates,we return to the caseof the graviton,

E~1a1

1~ el k)X10)

S.B.Giddings, Conformaltechniquesin string theoryand string field theory 207

Using the operatorproductexpansion,we find

A11 e1’~’~°1= dz ~x1e”~ etPXtO)

= dz~ + aX’(O) +...) ~ ~ izk~aX(0) + . . . )e’~°~

— ~ pik~)aX~(0)e1X(0),

so the resultinggraviton vertexoperatoris (up to normalization)

~ a~x~o5xv eP~X,

where

LV = E.(6’ — ~ p’k)(61 — ~— p’k~)

is the manifestly transversecovariantpolarization tensor. At higher mass levels we usethe sameprescriptionto mapthe light-conepolarizationtensorsinto covariantpolarizationtensors:we usetheoperatorproductexpansionto express(4.18) in termsof thefield X at the point in question.Sinceinthis mannerwe find vertex operatorssatisfying the Virasoro conditions,the polarization tensorsaregaugefixed and hencesatisfy the stringy higher-spin generalizationof the Lorentz gauge conditionk~es” = 0.

4.4. Summary

In this chapterI havesketchedhowthe Polyakovamplitudesreduceto the interactingstringpictureamplitudesif we adoptthe light-cone coordinatesfor moduli space.As afinal stepit shouldbe arguedthat unitarity is compatiblewith the integrationoveronecopyof moduli space.Therearetwo pointstocheck. The first is that I haveshown equivalenceof the two expressions(4.1), (4.2) only up to aconstantfactorb

2’~2c~.However,the factorb, sinceits power is the Eulercharacterof the surface,canbe absorbedby a redefinition of the string loop coupling constant.The factor c is thenabsorbedindefining the relativenormalizationof statesin the light-conepictureand in the Polyakovpicture. Sincec is a constantuniversalto all amplitudesit can be fixed onceandfor all at tree level, andthusdoesnotspoil unitarity. The secondpoint is that if we integrateT, a, and 0 over their completeranges[i.e.,T E (—°~, ~), aE (0, atotat), 0 E(0, 2i~)],we cover the moduli space for the correspondingsurfaceseveraltimes. An exampleof this is illustratedin fig. 17; herewe seetwo different parametrizationscorrespondingto the samegeometricalsurface.The keyto avoidingthis problemis to divide out by adiscretesymmetry factor to ensurethat we counteachgeometricallight-conediagramonly once. ThesameFeynmansymmetryfactor shouldalsoappearin (4.2),sounitary amplitudesarein fact obtainedfrom the integrationover a single copy of moduli space.

I havearguedthatthe Polyakovandlight-coneformalismsareequivalent,andin particularthat thePolyakovamplitudesare unitary. There are several possibleextensionsof this work. It would seempossible to show the equivalenceof the Polyakov and light-cone amplitudes for the open and

208 S.B. Giddings. Conformaltechniquesin string theory andstring field theory

O~r~O O-~HOT

1 ~2 T3T4 t3 t4 ~I ~2

Fig. 17. An exampleof theovercountingproblem. The diagram shown would he countedmore than oncebecauseof the inequivalentways oIlabeling thesame geometricalconfiguration. This type of overcountingis canceledby Feynmansymmetry factors.

unorientedbosonic strings. Such a proof could be made in the fashion describedabove, takingadvantageof the close relationshipbetweenthe world sheetsfor the closedand open or unorientedstrings. In fact, it is a corollary of the theoremof ref. [46]that the light-conediagramsfor the orientedopenstring (including alsothe closedstring andclosedstring interactions,aswell as the four-pointopenstring interaction) provide a single cover of moduli spacefor a Riemannsurfacewith the appropriatenumberof boundarycomponents,and with at least two punctures.These puncturescorrespondtoeither open or closed externalstates.The general ideafor the proof of this corollary is that given aRiemannsurfacewith somenumberof boundarycomponentsand externalstates(i.e. a point in theappropriatemoduli space),thereis a uniqueRiemanndoubleof the surface.*) The Riemanndoubleis a

closedsurface,and will, by the theoremof ref. [46], correspondto a uniquelight-cone diagram. Thislight-conediagramhasthe samesymmetriesas the original doubledsurface,so it can be slicedin half tofind the uniquelight-conediagram for the original surfacewith boundary.It is clear that the mostgeneral diagram of this type consists of both open and closed string intermediatestates. Theinteractions are the usual three- and four-point interactions of the open string theory, the usualthree-stringinteraction of the closed string theory, the interactionwheretwo endsof an open stringtouch to form a closedstring (or vice versa),and the interactionwherean interior point of an openstringtouchesaclosedstringandabigger openstring is formed(or the otherway around).To finish theproofof the equivalenceof the Polyakovand light-coneopenstring amplitudesit remainsto check theequivalenceof the determinantsin the two measures;it seemsquite plausiblethat they are equal.

An evenmore importantproblemis to prove the unitarity of the superstringandof the heteroticstring. This could perhapsalso be done along the lines discussedin this chapter,providedone had asufficient understandingof the superstringmeasure(in both the light-cone andcovariantformalisms).Now one might use a theorem regardinga single cover of supermodulispace using a superspaceextensionof the light-cone diagrams.Becausethe topologyof asuperRiemannsurfacecomesfrom thebody directions,it seemsthat it shouldnot be difficult to extendthe coverof moduli spaceto a coverofsupermodulispace.Lastof all, it wouldbe desirable(andlikely plausible)to find a manifestlycovariantproofof the unitarity of the Polyakovformalism.

Applications of this work could possibly be madealong several lines. Since we have an explicitparametrizationof moduli space,togetherwith a fairly simple expressionfor the Polyakov integrationmeasure,it mayprove possibleto do explicit calculationsfor Polyakovscatteringamplitudesusingthelight-conecoordinatesfor moduli space.Eventhoughthereis no knownexplicit parametrizationfor themoduli space of an unpuncturedsurface, it is conceivablethat even vacuum amplitudescould becomputedusingthe light-coneparametrization.To do this onewould put wavefunctionscorresponding

*) The Riemanndouble of a surface is theclosed surfacegotten by taking two identicalcopies of the Riemannsurfaceand connectingthem

togetheralong correspondingboundaries.

S. B. Giddings, Conformal techniques in string theory and string field theory 209

to the SL2 invariantvacuumon the punctures.*) Such statesdo not care about their position on theworld sheet,so the integralover the light-coneparameterswould reduceto an integraloverthe modulifor the unpuncturedRiemannsurface,timesa simple volume factor for eachexternalstate.Finally, thiswork can be appliedalongthe linesdiscussedin chapter3 to infer the ghostinsertionsthat shouldcomefrom a gauge-fixedstring field theory basedon the light-cone world sheets.These ghost insertionsshoulddiffer from thosein ref. [45] by piecesthat decouplein the pathintegral,andcould perhapslendfurther insight into string field theory constructions.

5. Conclusion

In this report I havedescribedthe applicationof conformaland Riemannsurfacetechniquesto afewcurrentproblemsin stringtheory. Specifically, I havediscussedthe role of modularinvariancein twodifferent formulations of string theory, namely the open string field theory and the light-conegauge-fixedstring theory. It hasbeenshown that in both casesphysicists’ physicalintuition hasbeenextremelysuccessfulat perceivingfundamentalmathematicalconstructions.This, of course,is part ofthe magicof string theory.Concretely,physicists’ studyof constructionsnaturalto stringtheory hasledto the discoveryof one triangulationof moduli space.A slight perturbation(of only a few years)ofmathematicalhistorywould havemeantthat physicistshad discovereda secondtriangulationof modulispace;insteadit was only rediscovered.Generalschemesfor triangulatingmoduli spaceareof coursequite non-trivial, both becauseof the high degreeof topological complexity of moduli space,andbecausethe triangulation must work for a collection of an infinite number of spaces,one for eachtopology of the surface.

I have furthermoredescribedtechniquesfor the calculation of the Polyakov measureon modulispace.These havebeenused in two contexts.They were first of all usedto deducethe gauge-fixedFeynmanrules for Witten’s string field theory,andmoreoverin a frameworkwith generalapplicabilityto other string field theories.This is also non-trivial; indeed,the derivationof theseFeynmanrulesdirectly from the gauge-invariantactionis, as yet, not completelyunderstood.Thesetechniqueshavealsobeenusedto derive the Polyakovmeasurein thelight-cone coordinatesfor moduli space.In thesecoordinatesthe Polyakovmeasureindeedequalsthe measurededucedfrom the light-conegauge-fixedtheory; onceonealsoexaminesthe externalstatecorrelationfunctionsin the two approaches,oneis ledto the conclusionthat the Polyakovandlight-cone formalismsareindeedequivalent.This hasbeenalong-standingproblem in string theory, whose resolution establishesthe formal unitarity of thePolyakov approachto string theory.The techniquesI havedescribedshouldhavefurtherapplicabilityin finding explicit representationsfor the Polyakovmeasurein other coordinatesystemson modulispace.

It shouldbe quite clearby now that afew verybasicRiemannsurfacetechniqueshavea greatdealofpower and utility in string perturbationtheory. However,whetherRiemannsurfacescontinueto becentralto stringtheory in its ultimate formulationremainsto be seen.Indeed,it would seemthat stringfield theory shouldmeanmorethanjust finding away to slice up Riemannsurfaces.Whenwe do find anon-perturbativeformalismfor string theory,Riemannsurfaceswill certainlyappearin the studyof the

*5 Actually all we would need is a vertexoperator whose integral over the surfacegives a constant (independentof moduli). This shouldwork

for the dilaton one-point function, soit might be possibleto makeexplicit computationsusing the diagrams from theopen string field theory. On theother hand, if we look at a dilaton two-point function, which is all we can get from the light-cone diagrams, then it is not obvious that this wouldwork becauseof the short-distancesingularity asthe two vertex operators approach eachother. (I thank G. Moore for a discussionon this.)

210 .S.B. Giddings, Conformaltechniquesin string theory and stringfield theory

perturbationtheory for that formalism. However, we only arrive at perturbationtheory after gaugefixing the full gauge-invarianttheory. This is an inherent limitation to using the techniquesof chapterthreeto try to deducea gauge-invariantstring field theory: thesetechniquesgive only the gauge-fixedFeynmanrules. Thesecan of coursegive hints aboutthe structureof possiblestring field theories,butthese are only hints. This becomesparticularly clear when studying the light-cone-like string fieldtheoriesof ref. [451; thereis a greatdealof confusionaboutthe problemof the string lengthparameterand its possible connectionswith additional gauge invariance.It appearsthat very little about thisproblem can be deducedfrom the study of the light-cone world sheets.

One can only speculatethat there is some more general formalism, with more general gaugeinvariances,andin which the string lengthpossiblyentersas a gaugeparameter.It is alsoplausiblethatin this theory therearemanygaugefixing schemeswhich leadto actions that haveinteractiontermsofarbitrarily high order in the string fields.*) Indeed, if we arbitrarily pick somecubic action, oddsarethat it will not provide a coverof moduli space. In this case,when we study higher-pointscatteringamplitudes,we will be forced to introducenew interaction terms to compensatefor our errors.Thiscould in principle continueto arbitrarily high-orderinteractions;certainly it would seemdifficult, butperhaps possible, to get a consistent scheme. So with poor choice of gauge we might have anon-polynomialaction. It is quite remarkable,then, that thereexist schemes,so far only one for theopenstringandone for theclosedstring, in which thecompleteaction is no higher thandubic.**) Theseschemesare of coursethe onesI havediscussedin this report. It would bequite interestingto know ifthereareotherschemesthathaveno higher thancubicactions.One could speculatethat therearenot,at least in a simple propagator—vertex-typeformalism. In that casethe light-cone formalism for theclosedstring and Witten’s formalism for the open string are quite specialamonggaugefixings of thegreatertheory.This certainlywould explainwhy theyhavebeendiscoveredand foundto beworkable.But perhapswe should be looking beyondRiemannsurfacesfor somedeeperstructure.

Despitethe possibility that the ultimate theoryof string mayhavelittle to do with Riemannsurfaces,Riemannsurfacescan perhapsprovide cluesto the deeperstructure,and will no doubt remainusefulastools for the study of string perturbationtheory. Physicistsare of courseprincipally interestedin thesuperstringand heterotic string. The theory of ordinaryRiemannsurfacesis certainly useful for thestudyof the bosonicstring,andalsoof the superstringandheteroticstring, but it might be guessedthata more powerful framework for studyingthe latter two string theoriesis the theory of superRiemannsurfaces[6]. Unfortunatelythe theory of superRiemannsurfacesis still in its infancy; it remainsto beseenif the theory of superRiemannsurfaceshas anywherenearthe powerandutility that the theory ofordinary Riemannsurfaceshas. It would be quite interestingto see a further developmentof superRiemannsurfacetechniques;with a few of the basictools in hand it would perhapsbe possibleto gothrough thisentire reportandaddthe prefix “super” to eachword. Suchanendeavourwould of coursebe relevantto superstringfield theory andto the unitarity of the superstringand heteroticstring. Butfirst the basic technology must be developed.This could also provide clues to non-perturbativeformulations.

Beyondthis it is anyone’sguessas to what string theoryreally is. It will probably involve a rich andvery profound mathematicalstructure.All that we know for certainis that whateveris behindstringtheory is boundto be fascinating.

~‘ An idea alongtheselines hasbeen enunciatedby M. Kaku. CCNY preprint (1986).~ Therealsoexistsa schemefor theopenstring, namelythe light-conescheme,in whichtheactionhasno higherthanquartic terms.However.

it appearsnecessaryto introducealso a field for the closedstring in this formalism.

SB. Giddings,Conformal techniquesin string theoryand stringfield theory 211

Acknowledgements

This report is a slightly modified version of my PrincetonUniversity Ph.D. thesis.First andforemostI would like to thankmy advisor,EdwardWitten, for his suggestions,comments,

patientexplanations,encouragement,and support.I would also like to thankmy othercollaborators,E. D’Hoker, E. Martinec, and S.A. Wolpert. I have had many exciting andilluminating discussionswith them, as well as productivecollaborations.

Therearealsomanyothersfrom whomI havelearnedagreatdeal in conversations;apartial list is J.Bagger,C. Callan, L. Dixon, D. Gross,J. Harvey,D. Kazhdan,J. Lykken, S. Mandelstam,G. Moore,M. Muller, P. Nelson,C. Poor, S. Raby,A. Strominger,C. Thorn, W. Thurston, C. Vafa, andotherswho I haveno doubt forgotten to mention. I would also like to thank my fellow graduatestudents—including but not limited to K. Aoki, C. Crnkovic, Z. Gan, M. Goodman,I. Klebanov,D. Kusnezov,U. Lindqwister, P. Mende,J. Minahan,R. Myers, S. Naculich,V. Periwal, J. Segert,M. Selen,J.Sloan, and X.G. Wen—for many interestingdiscussionson string theory and other topics. Specialthanksto Bill Somskyfor computeradvice.Thanksare alsodueto the N.S.F.for threeyearsof supportthrough a graduatefellowship, and to Liz and my family for encouragementandsupport.

Finally, I would like to thank J. Shapiro for his many useful commentsand suggestionson thismanuscript.

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