a study of non-critical strings in arbitrary dimensions

NUCLEARNuclearPhysicsB 368 (1992)98—120 P H Y S I CS BNorth-Holland _________________

A studyof non-criticalstringsin arbitrarydimensions

A.H. Chamseddine*Institutefor TheoreticalPhysics,Unitersity of ZOrich, Schönberggasse9, CH 8001 Zurich, Switzerland

Received7 June1991Acceptedfor publication30 August 1991

A new classical action for two-dimensional gravity is establishedwhere the graviton isaccompaniedby a scalar partner,a dilaton. It is shownthat the role of the dilaton field is topacify the inducedLiouville action, and to make quantumgravity easy to handle.The effectiveaction, definedas the sum of the contributionsfrom classical gravity, the ghosts,the inducedLiouville modeandthe matter,is a conformaltheory. It is shownthat this effectiveactioncan beinterpretedas a non-linear sigma model, coupled to a linear dilaton background,in a targetmanifold with two additional fields and minkowskian signature.This helpsus to identify thespectrumof non-critical strings in arbitrary dimensions.The analogousanalysisof non-criticalsuperstringsis also performed.

1. Introduction

Recently I haveshown that the problemsencounteredin the quantizationoftwo-dimensionalmatter systemsare resolved,provided that the proper classicalgravity action is added[1]. It was arguedfrom the considerationof the reducedthree-dimensionalgravity that a naturalcandidateis given by [2,3]

fd2xV~(R +A),

where4 is a scalarfield, R is the scalarcurvaturein two dimensionsand A is thecosmologicalconstant.The scalarfield 4 is a two-dimensionaldilaton, andhastheeffect of introducing,in thepathintegral,a constraintrestrictingtheèurvatureto aconstant.This fixes the Weyl scaling of the metric and reduces the inducedLiouville action to a constant.Equivalently, the threecomponentsof the metric

g~1~arematchedby threeconstraints.Two of theseconstraintsare due to gaugefixing of the diffeomorphisminvarianceand the third comesfrom the integrationof the scalar field /. In this way the Liouville problem is trivialized, with theintegrationovermetricssimply achieved,andthe difficulty in quantizingmatterin

* Supportedby theSwissNational Foundation(SNF).

0550-3213/92/$05.00© 1992 ElsevierSciencePublishersB.V. All rights reserved

All. Chamseddine/ Non-critical strings 99

arbitrary dimensionsavoided. Onecan summarizethis by saying that the consis-tency of two-dimensionalgravity requires the graviton to havethe dilaton as itspartner.This is similar to the situationencounteredin ten-dimensionalsupergrav-ity [4], the low-energy effective theory of the superstring.There, the graviton is

required to be accompaniedby two partners,the dilaton and the antisymmetrictensor. (In two dimensions the antisymmetric tensor decouplessince its fieldstrengthis zero.)

It is also possible,and useful, to understandthe resolution of the problem ofquantizing non-critical matter from the conformal field theory point of view.Defining the effective action as the sum of the actionsdue to the classicalparts,the inducedLiouville mode, and the ghostsarising from gaugefixing the diffeo-morphisminvariance,it is easily shown that this systemis conformal.

It is then an interesting problem to determine the properties and physicalspectrumof the resulting non-critical strings. A straightforwardanalysiscan beperformed in the light-cone gauge.Alternatively, it is possible to interpret theeffectiveactionof d scalarfields as a non-linearsigmamodel of d + 2 fields with aflat Minkowski backgroundmetric, and a linear dilaton coupling.This observation

will play a key role in interpretingthe physicalspectrum.The supersymmetriccasecan also be solved, and the systemhas N = 1 world-sheetsupersymmetry,butspace-timesupersymmetrycanonly occur in veryspecialcases.

The plan of this paper is as follows. In sect. 2, the essential results of thesolution to the quantizationof non-critical stringsare presented.In sect. 3, theeffective action of a d-dimensional system is rewritten in a form where it isexpressedas a (d + 2)-dimensionalsystemwith the time coordinatealwaysgener-ated. In sect. 4 the systemis quantizedin both the light-cone gaugeand in acovariantway, and the physical spectrumis determined.In sect. 5 the analysisisgeneralizedto the supersymmetriccase.Sect. 6 includessome commentsandtheconclusion.

2. Quantization of strings in arbitrary dimensions

In this section1 shall briefly presentthe importantpointsderivedin ref. [1] that

will be relevantto this paper.The first and most importantpoint that must be stressedis that the classical

gravity action mustbe given by

Ig= ~f d2x~~(R+A) +A~(M)+~fd2x~, (2.1)2~TM M

where ~4 is the scalar field, g~0is the two-dimensionalmetric tensor of the

100 A.H. Chamseddine/ Non-critical strings

manifold M, R is the scalarcurvature.The Eulercharacteristicis

x(M)=1f d2x~R=2(1-h), (2.2)2~ M

whereh is the genusof M.The action(2.1) differs from what is usuallytaken[5] by the presenceof the first

term. Although this action is simple, the presenceof the scalarfield 4 may seemto be ad-hocand not fundamental.To be convincedof the necessityof this fieldand the correctnessof the action, it is helpful to considerfirst the three-dimen-

sionalgravity action

13 = fMd3 ~/~(R +A). (2.3)

By a trivial dimensionalreduction,and truncationof the two-dimensionalactionaccordingto

— g~0 0

0 ~

the reducedaction takesthe form

12 = +A). (2.4)

When this simpleprocedureis also carriedout by reducinga (D + 1)-dimensionalEinstein--Hilbertaction to D dimensions,a similar result is obtained.When D ±2it is possibleto rescalethe metric accordingto

g13 .~çf~2/(D_2)g

0,

andthe 4R term transformsto

fdDx~4R—* fd’~x~/~[R+ const.x (~In 4)2]. (2.5)

It is thenclear that when D � 2 the scalar field 4 can be truncated(i.e. set tozero)afterrescaling.This is not thecasewhen D = 2 andthe scalarfield ~ cannotbe rescaledaway. Insteadone gets,underthe rescalingg~—* ~ the trans-formation

Id x~,gdR—~ fd2x~[~R+ const.x (d4)2]

A.H. Chamseddine/ Non-criticalstrings 101

At this point thereare two choices.Either to truncatethe field ~ and the action(2.4) would vanish, or to keep the scalar field j and the correspondinggravityaction mustthen begivenby (2.4).The first possibility is the standardchoicewhich

leads to the familiar difficulties of quantizingnon-critical strings of dimensionsd> 1 [6—8].The secondpossibility correspondsto what I am now advocatingand

avoids the difficulties encounteredwith thefirst choice.Also becausethe measurein the spaceof metricsis not invariantunderrescaling,it is importantto take(2.4)as the action and not the resealedversion of it. Anotherargumentto supportthenecessityof using the scalarfield is the fact that therearecertain field theories,such as ten-dimensionalsupergravity[4] which for consistencyrequiresthe gravitonto beaccompaniedby two partners,the dilation andthe antisymmetrictensor.Thisis of coursethe samecombinationthat occursin the masslessspectraof the criticalstring, andthis will also be the casefor the non-critical stringsas formulatedhere.In otherwords, the consistencyof the two-dimensionaltheoryrequiresthe two-di-mensional dilaton to accompanythe graviton. The antisymmetric tensor is notnecessaryin two dimensionssince its field strengthvanishesidentically and thusdecouples.

Having establishedthe naturalnessof the action (2.1) it is now possible toconsiderthe partition function of a mattersystemof d-dimensionalscalarfields

= ~f d2x~g~a~XiapXi. (2.6)8~ M

Performingthe Xt integrationgives [5]

—d/2

8~.2 det’~i1~2

2x~/~

where 12 is the volume of space-timeand /~g is the laplacian.The jacobianfromthe measureof metricsto thoseof diffeomorphismandWeyl modevariablesis anexpressioninvolving determinantsof operators.Gauge fixing the diffeomorphisminvarianceis achievedwith [9,10]

~ e2°, (2.7)

where ~, is a backgroundmetric chosenso that Rg = 1 when h = 0, Rg = 0 andArea(~)= 1 when h = 1, and Rg= — 1 when h ~ 2. The curvatureRg is related tothe backgroundcurvatureby

Rg=e~2~(L1gO-+Rg). (2.8)

102 A.H. Chamseddine/ Non-criticalstrings

After transformingthe determinantsfrom functionsof the metric to functionsof ~f3’ a factor proportional to the Liouville action is generated.The c~f integra-tion is also immediate and implies the constraint ~(R + A). Thus, apartfrom afactor to be written shortly, the metric integrationsreduceto the a- integration

fDa- e _2ôL~(~go+Rg e2~’+A), (2.9)

where SL(o~)is the Liouville action [11]

SL(u) = ~f d2x~(~~3aua +Rga-+~’e2~). (2.10)

This integrationcanbe easilyperformedbecausethe delta function constraintisnothingbut the Liouville equation.The solution of a- canthenbe substitutedintothe Liouville action to give [1]

~ —h)

wherethe shift from d — 26 to d — 24 results from thejacobiandue to the changefrom the argumentof the deltafunction to that of a-. The partitionfunction for afixed genush is then

(d±2)/2f d~x~/~Zh =KA(1/6x24~x1~)f

Mh 8~r2det’zL~

det PIIFI 1/2x det(~.tJH~k>, (2.11)

detK4’1

whereMh is themoduli spacefor genush with complexdimensions3h — ~ arequadraticdifferentials,p~is dual Beltrami differentialsandP1 is the operatorthatsendsvectorsinto symmetrictracelesstensors.Notice that the dependenceon A isequivalent to an area dependence,since the delta function constraint and thedefinition of the Euler characteristic~(M) implies the consistencycondition

AA = 4~(h—1).

BecauseA is an arbitrary parameter,it is suggestivethat we integrateover A

taking the aboveconsistencycondition into account,

Z[A] = fdA ~(AA — 4ir(h — 1))Z[h].

AN. Chamseddine/ Non-critical strings 103

From this it is easyto seethat

ZEAl ~

which agreeswith the semi-classicalapproximation[12]. Of coursethe result hereis exact. The secondpoint to emphasizeis that in the metric integrationsthenumberof componentsof the metric and the numberof constraintsmatchexactly. Two constraintscomefrom gaugefixing the diffeomorphism invarianceandthe third comesfrom integratingthe scalar field. This is an indication that thepure gravity sectoris topological. Indeed,this was shown to be the case in thefirst-order formalism of such an action [3].

To seethat the familiar difficulties are avoidedin the conformal field theoryanalysis,and in analogywith the analysis of Distler and Kawai [8], considertheeffectivetwo-dimensionalaction.This is definedas the sumof the classicalaction(gravity and matter), the ghost action arising from gaugefixing, and the inducedLiouville action:

= ~fMd2x~~31~x1 + ~f d2x~~(R~+ + A e2~1)

+ ~fMd2~~ + ;fMd2x~k(~~3~ a-+a-~) +~ e~], (2.12)

where b and c are the ghostfields and a is givenby

a=~(24—d).

By varying the action (2.11) with respectto the backgroundmetric ~ gives the

energy—momentumtensor *

= 2a(~aS~a-— a~a-a0a-)+ (~j~ — 23(~çb8 /3)a-)

+ g~0[a(a~a-aya-— 2~V~a-)+ (~a~a-— ~ (2.13a)

Tatt~= — + ~ (2.13b)

T~05t= — b0)yC~’— b~(~V$)c~’— (trace). (2.13c)

wherep. and A areset to zero.Whenp. andA arenot zero thefields a- and4 canbe expressedin terms of free fields, the expressionfor a- being the familiarsolution of the Liouville equation,while that of ~ is a solution of a generalized

* In reality onehasalsoto addto theaction (2.12)a backgroundpiece(a/2 r)fMd2xy~R ~ so

that the traceof T,,,,~would be consistentwith the o- and 4 equationsof motion.


Liouville equationandis morecomplicated[3]. By settingp. and A to zeroboth a-and ~ canbe decomposedinto holomorphicandantiholomorphicpieces.The twotermsinvolving p. and A canthenbe consideredasperturbation.In this casethepropagatorscanbe read from the action

= —~‘ ln(z — w),

(a-(z)a-(w)) = 0,(2.14)

Ka-(z)4(w))= —~ln(z—w),

=a ln(z-w).

By working out the operator product expansionof the energy—momentumtensor~ where

T(z) T~=2a(ä~a-—(~~a-)2)+ (a~~4— 2a~4a~a-)

— ~X~ZX~ — 2ba~c— 0~bc, (2.15)

it canbe easilyverified that the systemis free of the conformalanomaly

2T(w) 3 T(w)T(z)T(w)= +

(z—w)2 ZW

To be able to consider the terms p. e2’ and A4 e2~as perturbations to theconformal theory without spoiling the conformal nature, the terms must beprimary fields with conformalweight (1, 1). The first term satisfiesthis property,

3 e2°©~)T(z) e2~= + W (2.16)

(z—w)2 ZW

The other termhasan ambiguitysince it correspondsto a productof two fields atthe samepoint,

fdz d2q~(z,2) e2~’©.

This can be normalordered,but is nota primary field,

e2°~~ :4(w) e2~~~:0 (:4(w) e20©~~:)T(z):4(w) e2~”~:=— 2 + 2 + —

(z—w) (z—w) Z W

(2.17)

A.H. Chamseddine/ Non-critical strings 105

However, the combination : (4 + 2cr) e2°~:behavescorrectly,

(:(~+2a-)e2a:) 0w(:(~+2a-)re2~:)T(z)(:(~+2a-)e2U:) = 2(z—w) ZW

(2.18)

but is not unique. As can be seen from eqs. (3.1) and (3.8) in sect. 3, thecombination(e2~’~— e2°)is also primary, andhas 4~’e2°asthe first term in aperturbativeexpansionin ~. This is an indication that the effective action is notrenormalizedexcept for the A term where it changesfrom Açb e2U to A(q5 +

2cr) e2°or A(e2~”~— e2~’).Only a careful analysiscan determinethe correct

renormalizationof the A-term.

3. The non-linear sigma model interpretation

In sect. 2 I haveestablishedthat the partition function for a systemof d scalarfields can always be computed,and the difficulty associatedwith the integration

over the Liouville mode easily removedby taking the classicalgravity action asgiven by eq.(2.4).The effectiveaction,after renormalizingthe A term,describesaconformalfield theory. It shouldnow be possibleto usethis conformalinvarianceto gain some insight into the structureand the physical spectrumof the theory.From this point of view it is possibleto set first p. and A to zero,andtreatthem asperturbationsto the conformaltheory.

By examiningthe effective action it is evident that the kinetic terms for the a-and ~ fields are mixed, and that it is possible to diagonalizethe corresponding2 x 2 kinetic matrix. The diagonalizationdependson the threepossiblerangesof

the parametera: positive, negativeor zero, correspondingto whether d is lessthan, larger than or equal to 24. The diagonalizedfields, in the threecasesarerespectively

X1=\/~(2a-+~), X0=-ç~=~ (d<24), (3.la)

Xd+1= ~, X0=V~~(2a-+~) (d>24), (3.lb)

Xd+1=cr_~, X°=cr+4 (d=24). (3.lc)

Whenexpressedin termsof the newfields, the effective actionwith p. and A set


to zero, takesthe familiar form

‘~ = ~f~dx g00~~0

4X~ + ~ (3.2)

where X~Lis a (d + 2)-dimensionalvector,

X~=(X°,x1, xd~), i=1,2,...,d, (3.3)

and is flat backgroundspace-timemetricin d + 2 dimensionswith Minkowskisignature

~=(—1, 1,...,1). (3.4)

The vector is a constantvectorsatisfying

= a = ~(24 —d), (3.5)

andwhosespecific form dependson the sign of a,

~ a>0,

~ a<0, (3.6)

nM2(1,0,...,1), a=0.

TheX~propagatorstake now the simplecannonicalform

KX~(z)X~(w))= —sr’ ln(z — w), (3.7)

correspondingto a stringfield in a D = d + 2 targetspace-timemanifold,with flatbackgroundmetric, anda linear dilaton backgroundcT = ~

This phenomenonresemblesthe situation encounteredin the quantizationof

the inducedgravity action where the Liouville mode was effectively describedbyan additional dimension[13,14].The differencehere is that eachof the Liouvillemode a- andthe field ~ describesan additional dimension,andthat the signatureof the targetmanifold is always minkowskian for all valuesof the dimension d.This is so providedthat the d-dimensionalfields X’ haveeuclideansignature.Thefields a- and4, alwaysconspireto generateonespace-likeandonetime-like targetmanifold coordinate.Accepting the naturalassumptionthat the signatureof X’ iseuclidean, the conclusion is then that the space-timegeneratedwill have aMinkowski signature.

All. Chamseddine/ Non-critical strings 107

In termsof the fields X~,the energy—momentumtensortakesthe simpleform

T(Z)matt~~ltY= —13X~3X”~ + ~ (3.8)

In this form, andoncethe ghostpartis also included,it is verysimpleto verify thatthe theory is free of the conformal anomaly.

We havethusshownthat the effectiveaction takesa form of a non-linearsigmamodel with a targetmanifold whose coordinatesare X’~ in a backgroundflatmetric G~jXt~]= ij.~, a linear dilaton coupling cP[X’~] = ~ and a zeroantisymmetricfield backgroundB,~~[Xt2]= 0. From thework of Callan et al. [15] itis easy to check that the /3-function equationsare satisfied for this background.The equationsare

I3G,,. = R~— + 2S7~V,4)+ O( a’),

/3B~—V5H~—2V5tJtH

51~,+O(a’), 39

1 D—262 ~ +O(a’),16~r 3a

where a’ is the string tension takento be equal to 2 in our normalization, andH,~5is the B~field strength.Onecaneasily verify that all the above/3-functionsvanish for the backgroundin the action (3.2) by noting that D = d + 2 and n1~n,2= ~(24—d).

The /3-functionsin eq.(3.7) canbe derivedfrom the effectivefield theoryaction

in D-dimensionalspace-time[15]

1D = fdDX~e2~[3, + (R + 4(V~)2— + O(a’)]. (3.10)

The class of non-critical strings obtained as solutions of the non-linear sigmamodel in a linear dilaton backgroundhas beenknown for some time [13]. Thedifferencehereis that we startedwith matterin a flat euclideanbackground,andonly the effective theory correspondsto flat minkowskian space-timewith acompletelyspecifiedlinear dilatonbackground.The cosmologicalconsiderationsofthis classof theorieshasbeenextensivelystudied[13,14].

Rememberingthat the space-timemetricof the D-dimensionalmanifold is notG,~but ratherthe resealedtensorG,~= G~e4’~2~~”,then the stringmodelunder considerationcorrespondsto a target manifold with an actual metricG~= i~, ~ This metric dependson eitherthe time or the extraspace-like coordinateor both. With this interpretation,the spectrumof the theorycanbeseento be a graviton, an antisymmetrictensorand a dilaton in a D-dimensional

108 A.H. Chamseddine/ Non-criticalstrings

space-time.Theseare masslessbut in a linear dilaton background.By this it ismeantthat the fields G~,By,, and c1 haveto be expandedas

~ = +

B~=B~, (3.11)

+cP.

To extract a sensibledefinition of massis such a background,it is helpful tostudy the momentum of the string fields excitations. To do this expand theequationof motion of the field B~keepingonly linear terms to get

(0~— 2n5)H”~ = 0. (3.12)

If the gaugefreedomassociatedwith the invarianceof the effective action underB~—* B~,+ 0[~A~]is fixed by the condition

(0~— 2n5)B~= 0,

eq. (3.12) simplifies to

(o~o5— 2n505)B~= 0. (3.13)

As shall be seenlater, theseare the sameconditions as the ones obtainedbyrequiring that the vertexoperatorswhich producethesestatesareprimary fields.Also by examining the equationsof motion of a massivescalar field S in the

presenceof a linear dilaton coupling of the form

f~ + m252),

we obtain

(o~o’~— 2n~0’1— m2)S= 0. (3.14)

All theseconsiderationssuggestthat the correctdefinition of the massoperatorofthe string statesfor the systemunderconsiderationis given by

m2 = —p~p~— 2in,~p~. (3.15)

To concludethis sectionwe also considerthecasewhenp. and A arenot set tozero. From the expressionof the partition function it is clearthat the dependenceof Z on p. is of the form e~ where A is the area,andthe dependenceon A has


beenalreadydetermined.Since thesearenot dependenton the modularparame-ter, one may concludethat the natureof the physical spectrumis independentofthese two parameters.To simplify the analysis in the next sectionswe shallcontinueto set both p. and A to zero.

4. Light-cone and covariant quantization

When examiningthe genus-onepartitionfunction of d scalarfields (in this caseA is naturallyzero, and p. can be tuned to zero) it is immediate to see that thecontributionsof the ghostsarisingfrom gaugefixing the diffeomorphisminvarianceexactlymatchthe contributionsof the fields a- and 4,. This is so because[9,10]

~ (4.1)

where T is the modular parameterT = + ir2, and 77(T) is the Dedekind etafunction. The genus-onepartition function of d scalarfields X’ is then

r U T ~ —d/2Z1 = K) 2 2 (4~-

2T2I ~(r) ) - (4.2)

8~-T2

This can be thoughtof as the light-coneexpressionof the d + 2 scalarfields X~.Notice that when d = 24 the aboveexpressioncoincideswith that of the criticalstring. However, becauseof the linear dilaton backgroundthe interpretationwillbe different, althoughthe physical spectrumis identical. Denotingby ~ and 0 asthe coordinateson the cylinder (the use of theoverlinein this notationis to avoidconfusion with the modular parameterT and the scalar field a-), the light-conegauge is defined by [16]

X~=2P~+X~°, (4.3)

where X~=(1/V~)(X0±X’~1).The X~(x°,x’) splits into left-moving and

right-moving partsX~(x~)and X~(x), where x ~= ~±0. The mode expansionof the left-moversis given by

X~(x~) = ~ +P~x~+i~ ~ e’~. (4.4)n~O

From this it is easyto find that 0÷X~’= ~ e°’~, where a~’ P~.By applying the transformationfrom the coordinateson the planeto thoseon

the cylinder, z = ex°~’,anddroppingT~0~tsincewe areworking in the light-conegauge,the energy—momentumtensorbecomes

T~~=— ~0~X~0÷X~ + n~(0~X~+ i0+X~)+ ~ (4.5)

110 A.H. Chamseddine/ Non-critical atrings

This can be seenfrom the transformationproporties

T(z) = e2zT(ez), 0X~(ez)= e~(0X~L(z)—np), (4.6)

and the i factor results from the Wick rotation ~ —s ir. Substituting the modeexpansionof X~into ~ we obtain

T~~=—~ ~ a~a~em~1*+ ~ (P~—imn)a~e_1m~

m,p

+ in÷P~+in La~ e~m~+1 — ~d. (4.7)

The zero componentof the constraintT~~=0 is then

— ~~ +P~P+in~P~-l-inP+ 1— = 0. (4.8)m

After quantizingthe X’ the term Lma-~ma~mustbe normal orderedto removethe ambiguity,

~~ = fr’P’ + ~. : a~~a~:+ ~(— ~yd), (4.9)m n>O

usingthe zetafunction regularization.Substitutingeq.(4.9) in (4.8) gives

— ~ + in~P~= : a~ma~:—1. (4.10)m>O

Adopting the definition of massarrivedat in sect. 3 implies

~ a’,~a~:—1 (4.11)m>O

andsimilarly for right-moversin termsof c~.After usingthe relation

m~= m~=

we obtain anexpressionfor the massspectrum,

m2= ~ (:a~ma~:+:ã’mã~:)—2=Lo+Lo—2. (4.12)m>O

This formula is identical to the familiar formula encounteredin non-criticalstrings,andwith identicalspectrum[16]. Only the definition of massin relationtomomentais changedbecauseof the presenceof the linear dilaton coupling. The


groundstateis the tachyon,and the masslessstatesare the graviton, the antisym-metric tensor and the dilaton. We conclude that the main differencebetweencritical strings andnon-critical ones is that the latter correspondto an expandedtargetmanifold with the vacuum correspondingto a linear dilaton backgroundn,,X~’.

It is also possibleto perform the quantizationin a covariantway keepingtheghosts.If L~are the Fourier componentsof T(z),

dwL~=~—w”~’T(w), (4.13)

2~i

the BRST-invariant states must satisfy [17]

L~Ic~)=L~I~i)=0, n~1,(4.14)

(Lv— 1)I~>=(Lv— 1)I~)=0,

X” - matter+gravity ik X’~where L0 is the L0 componentof T . The operatore ‘~ 10) gener-atesa statewith momentumk. This operatoris a primaryfield provided

~ +in~k’1= 1, (4.15)

which can be derivedfrom the operator product expansionof T(z) with ~With our definition of the massrelationfor a statewith momentum k, this statecorrespondsto the tachyon

m2= — (k,,k~+ 2in,~k’~)= —2. (4.16)

The next state is � ~ e~kXandhas the operatorproductexpansionwith

T(z)

- (ik’2 — 2n~)e(aX~e~’~’)T(z)E~0~X~0,,,X~etkX= — ________________________

(z — w)

~k k~+in kM+1 -

+ (ox’~ox~�et~~X)2 P~’ w(z—w)

o(OX’-’aX”E eu/~X)+ ‘i” w. (4.17)

From thiswe deducethat this operatoris a primary field providedthat

(ik’~— 2n~)c~= 0, m2= + in,~k~= 0. (4.18)

112 AK. Chamseddine/ Non-criticalstrings

Comparing eq. (4.18) with eqs. (3.12) and (3.13) derived from the betaequations,we see that the first equationcorrespondsto fixing the gaugeinvarianceofwhile the secondstatesthat the operatorcorrespondsto a masslessstate.Thisstatethencorrespondsto the graviton,the dilatonandthe antisymmetrictensor.Itis easyto checkthat e ~ correspondsto a massivestate,m2= 2,and satisfiestransversalityandtracelessnessconditions.

5. Superstrings

It hasnow beenshownthat the bosonicstringsin an arbitrarydimensiond has

the same familiar spectrum as the critical case, although with a different interpre-tation. It thus shares the problem of the tachyon. This one hopes to avoid by

considering strings with world-sheetsupersymmetry,i.e. superstrings[18].All the stepstaken in the case of the bosonic strings can be generalizedto

superstrings.First, the classicalsupergravityactionmustbe takento be

‘sugra = ~_fd2zE~(R±+K), (5.1)

where d2zE is the supervolumeelementd2x dO do sdetE~, 1 is the dilatonsuperfield, and R~_is the supercurvature.(The componentform of this actioncan be found in ref. [19] andthe notationis the sameas ref. [9].) The advantageofintroducing the scalar superfield 1~is that when integratedin the path integralgives the constraint

(5.2)

restricting the supercurvatureto a constantvalue. The spaceof supergeometries

canbe parametrizedby

V ~ L~AM—e e e

where V, ~ and L are the fields of superdiffeomorphism,superWeyl andsuperLorentz transformations.E~is backgroundsuperzweibein in a slice S of dimen-sion 6h — 6, whereh is the genusof of the surface.

If the matteraction is takento be

1jmatter = ____fd2zED_xzD+xt, (5.3)

4ir

then the X’ integrationcanbe performedimmediately.Also thejacobianresultingfrom the changeof variablesof E~to E~is known. The resulting expression is


independentof V and L but dependson ~. (The independenceon L reflectstheabsenceof Lorentz anomalies.)

The .~ dependenceis determinedby heat kernel methods,and the genus-hpartition function is given by

—d/2

sdet<p.JIcI3~) 1/2 8ir2 sdet’i0

Zh=17f fldmj ,. (sdetP~Pi) F(K),sM5 i sdet(~JHIK) ifd2zE

(5.4)

wherep.j are the super Beltrami differentials, “~K their dualquadraticdifferentialsand F(K) is the integral

F(K) =fD~(R+_+K) exp[~(d—10)SSL(~)I. (5.5)

The expression ~~L(~) appearing in eq. (5.5) is the induced super Liouville action,

= ~fd2zE(ññ+~+iR±~). (5.6)

The presenceof the deltafunction constraintin eq. (5.5) makes the ~ integrationtrivial. The reason is that the constraint restricts ~ to obey the differentialequation

Y= —2iD~D_~+R~+K e~=0, (5.7)

and this is the familiar super Liouville equation. The solution of this equation canbe substituted into the super Liouville action, and this reducesto a constantdependingon K [19],

8~2sdet’S0F(K) = (exp[~(d—8)SSL(~)})YO. (5.8)

fi d2zE

The shift of d — 10 to d — 8 in the coefficient of the superLiouville action is dueto the changeof variablesfrom .~ to Y in eq. (5.6). With this, the expressionforF(K) reduces to

8~2sdet’S0

F(K) cx K(8’©X/4. (5.9)

fi d2zE

114 AK. Chamseddine/ Non-critical strings

The K dependencecan be changedto a length dependence,through the consis-tency condition

KL+4~(1—h)=0, (5.10)

where L is defined to be

L = if d2zE.

As in the bosonic caseit is difficult to determinethe spectrumby examiningonly the expressionof the partition function. To do this it is helpful to constructthe effective conformal field theory. The effective action is definedas the sumofthe classical gravity action, the matter action, the super ghosts action due to gaugefixing, andthe inducedsuperLiouville action.This is written as

,eff = jmatter + jsugra + jsgh (5.11)

wherethe matteraction is givenby eq. (5.2), but usinga backgroundE~insteadofE~.The supergravity and super ghosts parts are given by

,Sugra _fd2zE[a( ~ñ+~+~~+_~)+p. e~

+ (i~++2~~D_~+ K e’)], (5.12)

jsgh = fd2zEBDC+ c.c.,

where

a = 1 — ~d.

When setting p. and K to zero, all fields becomefree fields which split intoholomorphicandantiholomorphicpartswith the propagators

(X~(z1,O1)X~(z2,02)) —~° In z12,

(~(z1,01)Z(z2, 02)) = 0,

K~(z1,O1)~(z2,02)) = —~ln(z12), (5.13)

<1(z1, O1)cI’(z2, 02)) =a In z12,

(B(z1, 01)C(z2, 02)) = 012/z12,

where z12= — z2 — 0102, and 012 = UI — 02.


The energy—momentumtensorT is definedby

1~= __fd2zE(E~!6E~)T. (5.14)

2ir

By varying the effective action (5.13), the energy—momentum tensor is then

calculated to be

T = ~ + Tmatter+ ~sgh

wherethe different partsare

= 2a(_~±~~~+ i5~) + (n~-ñ~~i5~- I5~I5+~

= — ~ñ÷xiñ~+x1, (5.15)

T~= -~B~D~C+~D~B- C-D~B~C.

With the propagatorsin (5.13), it is possibleafter some algebra,to show that thetheory is superconformalwith the operatorproductexpansion

~D2T(z2, 02) 012T(z1, 01)T(z202) = —~---T(z2,02) + + —02T(z2,02), (5.16)

z12 z12

where D2 = 0/002 + 020/0z2.The term p.fd2zE e~is an interactionterm in the

super-conformaltheorysinceit is a primary field of weight ~

-~D e~2,°2) 0T( z

1, 0~)e~2°2) = ~ ~ + 2 2 + ~ e~2S2) (5.17)z12 zi2 zi2

However, andas in the bosoniccase,the term Kfd2zE~e~is not conformal and

mustbe renormalizedto Kfd2z E(~P+ 2) e~or to Kfd2z~ — e~)whichcan be seento be primary operatorswith weight (-i-, ~). The correct choice canonly be madeafterfurther study.

To understandthe meaningof this conformal theory, and to get an insight intothe resulting spectrum, it is useful to realize this superconformaltheory as anon-linear sigmamodel in two dimensionshigher. First diagonalizethe fields .~

and D sincetherekinetic termsare mixed. The diagonalizedsuperfieldsX° andX’~1 are defined according to the three possibilities, a > 0, a <0, and a = 0,respectively,by

X0=~(2~+~), X~~1=—)=-P (d<8), (5.18a)

116 A.H. Chamseddine/ Non-criticalStrings

x°= ~ Xt=V~~(2~+~) (d>8), (5.18b)

X°=.~’+cP, X”~1=Z—1 (d=8). (5.18c)

In all threecasesthe effective action of matter, and gravity can be rewritten intermsof the X~as

5sugra+matter = ~fd2zE(n x~ñ+x~77~~+ 2i(n~X~)~+), (5.19)

where n~is the vector given in eq. (3.6) and 77~,,is the Minkowski flat metricgivenin eq.(3.4). The energy—momentumtensoralso simplifies to

T= ~ ~ (5.20)

In this form it is straightforwardto verify that the effective theory is superconfor-mal after addingthe contributionsof the superghosts.

Unlike the bosonic case,it is more difficult here to extract the spectrum.Thereasonis that thebackgroundcouplinginvolvesbosons,andworld-sheetfermions,while the spectrumis expectedto havebosonsandspace-timefermions. The betaequationsdo indeedvanishfor the backgroundappearingin the action(5.19). Thebosonic part would naturally include in the masslesssector, the graviton, theantisymmetrictensorandthe dilaton, providedonemodifies the definition of massto accommodatethe presenceof the linear dilatonbackgroundasin eq.(3.15). Forthe restof the spectrumtherearepossibleGSO projectionsoperatorsthat mustbetaken into account.In this respectthe best strategyis to adoptthe result that thematterinteractionwe startedwith correspondto an effective interactionwith two

extra fields, modify the definition of mass as in eq. (3.15) and then go to alight-cone gaugewhere the X~ and X are eliminated. This will give all thespectra,but to this the GSO conditions obtained from the modular invariantpartition function mustbe applied.

The light-conegaugein this caseis given by (in componentform)

X~=2P~, ~I’i~i=0. (5.21)

Decomposingthe energy—momentumtensorT(x, 0) into componentswe cansolvefor 0~X and ~-I’i~iin termsof X’ and ~

(n_0~+ 2P~)0~X~=~(0+Xb0+Xl—(5.22)

(n_0±+P~)Wi4i=~,~0+X1.

AN. Chamseddine/ Non-critical strings 117

Using the oscillator expansion for X’ and ~ give the mass formulas in theNeveu—Schwarz and the Ramond sectors, respectively,

m2= ~ :a~a~,,:+ ~ :rbL~b~:—~ (NS), (5.23a)m>t) r>O

m2= ~ :ci~a~:+ ~ :rb~b,’: (R), (5.23b)m>O r>O

where m a ~ and r a Z + ~. Theseresults,just like thoseof the bosoniccase,arein completecorrespondencewith the critical string theory. At this point it isappropriate to discuss the different possible boundary conditions. The bosons X’must have periodic boundary conditions if one wishes to preserve the space-timestructure of the (d + 2)-dimensional space. The fermions can be periodic orantiperiodic.If the left andright moversarenot independent,then it canbe easilyseen that the resulting partition function is modular invariant, as there is acomplete balance between left and right movers. Indeed the partition function

takes the form

Zl=Kf~r(T2)2I77(T)I3d(I02I2d+ 0312d~ 0412d~ IO1I2d), (5.24)

where the 0~,i = 1,.. . , 4, are the Jacobitheta functions. In this casethereis noGSO condition and no states are projected out. This can change if one allows theleft and right moversto haveindependentboundaryconditions.However,this canonly be arrangedin specialdimensions,where dL and dR must be multiples of 8.This can be modified if shifted or twisted boundaryconditionsfor the X’ areallowed. A complete analysis is possible using the techniques of the unifiedconstructionof modular invariantpartition function [20].

Themost realisticstring modelswill resultfrom heteroticconstructions.In thiscase,although thereare no restrictionscoming from conformal invariance,therewill be onescoming from requiringthe absenceof Lorentz anomalies.Thiscanbeinsured by requiring the modular invariance of the partition function. From theanalysis of ref. [20], one finds an overall phase resulting from the modular

transformations T —~ 1~+ 1, which is of the form exp[21Ti~y(~d~— dL)], implying thecondition

-j~(~dR—dL)=0(modi). (5.25)

This conditioncan be derivedimmediatelyby noting that out of the dL bosons thephaseof dR space-timeleft-moving bosonsbalancethe phaseof dR space-timeright-moving bosons,leaving dL — dR left-moving bosonswith shifted boundaryconditionsto balancethe phasecoming from dR fermions. Usingthe equivalence

118 AK. Chamseddine/ Non-critical strings

of every two fermionsto oneshifted boson,andthe oppositenessof the phasesofleft-movers to right-movers,we obtain (dL — dR) — = 0 (mod 12), where themod (12) results from the transformationof the theta function under modulartransformations.From the level matchingconditionsonefinds the condition

dR—dL)=0 (modi) (5.26)

(In the notation of ref. [20], this results from the condition V0~l/~= 0 (mod 1)).

These conditions are satisfied provided that

dL = 12m dR = 8n, (5.27)

where m and n are integers. The effective dimensions of space-time is then

DL= 12m+2, DR=8n +2. (5.28)

The simplestexamplecorrespondsto the critical case,althoughthe interpretation

here is different. Many interestingmodelscould be built for different choicesofthe values of m and n and by allowing for non-trivial boundaryconditionsfor

some of the coordinates.Becauseof the large class of possibilities this will betreatedelsewhere.

6. Conclusion

In this paperI haveshown that one mustconsiderseriouslythe possibility thatthe two-dimensionalgravitondoesnot exist independentlyof the two-dimensionalscalar field. Such a possibility doesappearin the caseof critical strings, whereinthe masslessspectrumthe graviton is accompaniedby two partners,the antisym-metric tensorandthe dilaton. (In two dimensionsthe action for the antisymmetrictensor vanishes).Once this possibility is accepted,a natural action for classicalgravity interactingwith this scalarcanbe written. This in turn makesevaluatingthepath integral, and integration over the spaceof metrics, a simple process.Theproblem of criticality of matter coupling disappears.To understandthe spectrumone looks at the effective action, where it is found that the scalar field and theLiouville mode conspireto add two dimensionsto the targetmanifold in such away that the signatureis alwaysminkowskian.Thus if onestarts from a euclideand-dimensionaltargetmanifold, the effective targetmanifold will be minkowskianof dimension d + 2 and thus predicting the signatureof space-time.From thispointof view it becomespossibleto understandthe spectrumof the theory, whichin the bosonic caseis identical to that of the critical string. The supersymmetriccaseis richer as onemustalso take the differentpossibleboundaryconditionsintoaccount.The most interestingpossibility is in the constructionof heteroticmodels


of effective dimensions8m + 2 for right moversand 12n + 2 for left movers.A lotremains to be done. Computationsof correlationfunctions and scatteringampli-tudesmustbe performed.For exampleif one considersthe N-tachyonamplitudefor the d-dimensional bosonic matter, one writes the vertex operators as

exp(ik~X~(z’)), it is possibleto recombinethe 4, terms which also occurlinearly. Integrating the field 4, gives a constraint which is of the form of aLiouville equation in the presenceof delta function sources.By substitutingthesolution of this equation into the a- termsof the action, the problemsimplifies toevaluatingthe momentumintegrals.This computationis rather involved andwillbe treated somewhereelse. The main point to emphasizeis that there is no

fundamental difficulty in such a program. Finding the analogue of this constructionin the matrix model approach[21,22], is a very interestingproblem. The analysis

performedby Ambjorn et al. [23] is useful in this caseas the R2 term they addtothe gravity action as a cut off, is equivalent to the delta function constraintresulting from the 4, integration.Another importantproblem is to investigatethepropertiesof the effectiveconformal theorywhenp. andA arenot zero.Finally, it

is an interestingpossibility to searchfor realisticmodels basedon this approach,and especiallythe oneswith N = 1 space-timesupersymmetry.

I would like to thank J. Fröhlich for very helpful discussions,and G. Felderforexplanationson conformalfield theory.

Noteadded to proof

The discussionof the renormalizationof the A-term has been recently ad-dressedby Förste [24].

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a study of non-critical strings in arbitrary dimensions

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