COURSES ANO PLENARY TALKS REVISTA MEXICANA DE FisICA.¡t) SUI'I.Ei\IE:'IITO 1. 1-10 JUNIO 2003

An introduction to the qllantulll slIpermembrane

Arundhati Dasgupla. Hermann Nico1ai, ami Jan PlcfkaAfax-!'Ianck-Institut jü,. GravitafiolHphysik. A lbert-Einstcin-Institut, Colm. Germany

Recibido el 12 de febrero de 200 1; aceptado el 22 tic marzo de 200 I

\Ve review aspeels of quantí7.•1tion 01'tne II-dimensional sllpermembrallc world volullle Iheory. \Ve explicitly construcl vertex operators forIhe massless slates and study interactiolls of supem1embranes. The open supermembranc and its vertex operators are discussed. \Ve showhow our results have direct applicalions to r..,,1atrixtheory by appropriate regulariz;¡tioll oflhe sllpcrmembrane.

Keyw()rds: Quantul11supenncmbrane. M theory

Revisamos algullos aspectos de la cuantiz;u.;ión de la teoria cn 11 dimensioncs dc sllp~'rn\l:ll1brallaSque determinan el volumen de mundo.Construimos explícitamente operadores de vórtices para los estados sin t1MSay cstudiamus las interacciones de supermcmbranas. Se discuteel caso dc la supcnnembrana abierta y sus operadores de vórtices. Adcmús nHlstramos cómo lluestros resultados tienen aplicaciones directasen la teoria dc t\latriccs mediante la rcgul:lrización apropiada de la supcrtnembrana.

Descriprores: Supermembranas cuúnticas, tcoría M

rAes: 11.25.Uv; 11.25.Yb

1. Introductioll

As we revicw the progrcss over Ihe last few dccadcs \Vefind thal there are unexploreJ avenues in the 1I dimensionalsupennembranc theOlY which can Iead to a better under-standing of quanlUt11 t-.1-theory and hence non-perturbativesIring/quantu111 gravity. The supermemhrane is a 2+ 1 di-mensional obJect moving in II dimcnsional space, ,•....ilh aworld volumc thcory [l] which when quantizcd will give us aglimpsc of the fundamenlal degrees of frccdom of !v1-theory.Attempts to quantize Ihe world volume theory in analogy lO10 dimensional \\'orld sheel slring theory llave revealed manyintercsting featmes which distingllish it from lhe 10 dimen-sional slring theory. anel abo make it very diflicult lo sol\'e.Thc main sourcc of dilliclllties IS the faet that, unlikc thestring. lhe 2 + I dimensional world volumc Iheory is interael-ing. Moreover il does nol have a eonformal invariance. Ho\\'-ever, certain crucial featUfes of the sllpermembrane, namelyIhe fael that il can be regularizcd lO yidd a supcrsymmelrieMatrix theory [21. and the faet that íts spcetrum is eontinu-ous [3.4], lead to a multiparticle inlerpretation of the spcc-trum [5]. and the !leed for a second quantizeJ description 01'the membrane. \Ve still do 110tha\'c a complete understand-ing ofthis, nor is the cxislcnce ofa normalizable ground stateconfirmed. 1I0,,"e\'cr, there is evidence from ~vtatri.\ theorythat such a state inoceo exists and it contains massless slalescorresponding 10 the massless multiplet of 11 dimensionalsupergravíty [G].

Givcn Ihis situalioll, we can look for 1'urther quantitics todescribe the supermembrane which will hopcfully shcd somemore Iight OIl the undcrlying qllantum thcory. The interae-tions of the masslcss sector is one intercsting avenue to ex.plore. Therc ure computalions of d = 11 seattcring ampli-ludes fOI"thc massless sector [7-0] using the quuntizcd su-pCIvarticle. llo\Vever lhe calculated amplitudes divergc, ami

the coclTIcicnts can be fixcd only by duality with 10 dimen-sional string theory. As we know, in the ease of the super-string [lO] as wdl as the superparticle [7, 9] scattering ampli-tudes are evalllalco by determining lhe vertex operators andinserting Ihcm inlo path-integral amplitudes. The vertex op-crators for lhe supcrmcmbranc derived in Rcf. 11, preeisc1yseek to achieve Ihe SJllle for a supermelllbrane scattering am.plítude [12. 13]. The operalors are determincd uniqucly andprovide the first Slcp tO\I,:ardsunderstanding rnassless interae-lions. 130th lhe superstring ano the superparticlc vertex opera-tors are cOlllainco in the sllpcrmcmbrane vertex operators. Byredllcing the two spatial dillleIlsions of the supermernbrane\Vorld volume to zcro, the sllperp.uticle is obtained. (This \Vecall lhe particle limit). On the othel" hand, wrapping one ofthedirections uf Ihe tllembranc along a cOlllpaet dircction givesus a slIperstring in d = 10. This 'dollble dimensional rcdue-tion (DDR)' 01"the d = 11 supcrmembrane vcrtex operatorsgives llS all the superstring vertex operators. Also, by virtlle01' lhe faet thal Matrix lheory is recovcred after appropriatcrcgularization, these vertex operators give linearized MatrixactioJl in \Vcak d = 11 supcrgravity background.

\Ve also discuss the vertex operators for the open mern-brane as it is closcly relaled to heterolic string thcory [14] anoin rCCl:nt times has ussumcd importance duc 10 its relation tonon-colllllllltati"c Iheories in prescIlcc of a background eon-stant three form gallge ficld slrcnglh [15]. Many olher topíesrcgarding the quantlllll sllpermembranc have been discussedin detail in prcviolls reviews [16. 17].

Seelion 2 is an introduction to supermcmbrane basics.mainly to fix Ilotations. \Ve O1lsorcvicw the relation 01' super-membralle to ~1atrix lheory, ano then lhe multiparticle inter-pretation 01' Ihe sllpcrmcmbranc spcctrum. In Seco 3 we givt:thc \'cttex operators fOI"lhe m3sslcss sector ofthe theoT)'. See-tion 4 deals with scattcring amplituJc calculations with thehclp ofvcrlcx operators.

2 ARUNDHATI OASGUPTA, HERMANN NICOLAI, ANO JAN PLEFKA

FIGURE 1. Various limits ofthc supcrmembrane model.

2. Supcrmcmbrancs

gij is the world volume metric (g = Jet gij) whieh is eonsid-ered independcnt here, and Be 13.4, the three form of d = 11superspaee. T is the tension of the supermembrane, and itis the sol e length sea le in the theory. Unlike the superstringeoupling constant !/s there is no analogolls superrnembranecoupling constant. This precllldes a perturbative expansionof supcrmembranc interactions making the supermembranetheory non-pertllrbative theory from the very outset. We shallset T = 1 for convenicnce henceforth.

The action has redundant degrees of freedom due to theprcsence of thc local reparamctrizatioo freedom of the worldvolume and fcrmionic Ii symmctry of the action [1]. Thc 1\,

symmetry of the action requires one crucial fact: the back-ground field component fields of the graviton h~l'"gravitino1111 and the three form ClllIP must obey d = 11 supergravitycquations oflllotion [18]. This is similar lo the GS formula-tion 01' Ihe superstring, but quite difTerent [rom the NSR su-perstring, where one has to resort to fJ function evaluation toshow that the background fields obey the supergravity equa-tions of motion. Howcver, one aspeet of the action ( 1) is thatthough it gives the coupling ofthe sllpcrmembrane to super-space background fields, it is very difficult to extrae! the cou.pling in terms of lhe component ficlds (i.e. !l¡IV' 1J!¡¡,CllVP)'

The Illethod of gauge completion used in Ref. 19, has suc-ceedcd in dctcrmining the eouplings only to order (j3 of thetarget space ferlllions. There are other methods of cvaluatingthe aetion in cOlllponents as in Ref. 21, but, as we show inthe llcxt section to linear order in the fields, the couplingscan be detcrmined cxaetly, through lhe evaluation of vcrtexoperators.

To begin with, onc restriets oneself to quantizing the su-perlllembrane propagating in ¡lar target space. Thc coordi.nates in d = 11 fiat spacc are ZM = (XIJ. (V'), (J is a Majo-rana spinor. Thc superelfbein can be determincd [1], and wehave

<Jimen,ionalre<Juctilln

¡ul1ide límil

rc<JlIction

SupcfTTIernhr"neOpel! {"I,"cd

~-~/-r

Cm" N ""''''''''''"/

~

S=T.!d'~[ - ~h9;jn7n;¡/<.b+~h-,;Jkn~n7nfBCBA]' (1)

Superspace notation has been used wherc spacc com-prises ofboth the bosonie and fermionic coordinatcs. Explic-itly

L;upcmfing l C:;:;:--:ld~IO>Il~1T ~)'j'c 1IJ parlic\e

límíl

In analogy with the particle and the string, the action for therncmbrane is given by the 2+ 1 dimensional world volumcswcpt out in the target space. The supcnncmbranc is a ex-teosioo obtaincd by adding ferrnionic target spacc 'coordi-nates'. A supersymmctric aetion involving the bosonic andthe ferrnionic coordinates can be written down eonsistentlyonly in special dimensions, of which d = 11 is the maximalvalue.o This action in the background of el = 11 sllpergravitytakes the form

where Z M are the coordina tes of the membranc in the tar-get superspace, E~l is the super-elfbein, and ~¡, (i = 0, 1,2)denote the 2+ 1 dimensional world volume coordinates. Tan-gent space indices are denoted by A and world indices by Af.

I

In addition, we use the equation ofmotion for the world vol.utl1e mctric [lij = '/ob ni n~.Plugging this into (1) gives thesupermelllbrane in ftat space

n: = 8¡Z''H E;[,

a = O,. ,10.

ZM = (X",RO),

!' = O,. ,10.

A = (n,a),

n = 1,. ,32;

M = (¡l., ó),

0=1, ... ,32. (2)

ni = D¡(J°,

n"oQ = -i (f""R)(o ,(f"R)Q)'

BoQ, = -~ (f,,,,R)(o (f"R)Q (f"e),). (3 )

(4)

The reparametrization ofthe world volume can be used lO fix lhe bosonic dcgrces offreedom to eighl. (E.g. onc can go to slaticgaugc by identifying XO,1,2 = ~0,1,2 leaving 8 transversc degrees). The}{ sYlllmclry ofthe action is then used to reduce thcnumber of fefmionic degrces of freedom. The 1110stefficient way of dealing with the aboye action and its symmetrics howcvcris to go te lhe Iightcone gauge [1, 2]. The light cone directions X:f: = (1/ -/2)(XO ::1:::XlO) are singlcd out, and X+ is identi-

Re\'. Me.•. Fís. 49 Sl (2003) 1-10

AN INTRODUCTION TO THE QUANTUM SUI'ERMEMBRANE 3

fied with Ihe time direction of the world volume, which \Vedenote by T here (hercafter, \Ve denote the spatial directionse.2 == al,1). The!'\. symmetry also allows to set 1/2 ofthefermionic degrecs of freedom to zero. The gaugc eonditionsthus rcad:

with e = frsD.~~,sueh that are = o (~ is a scalar parame-ter), and henee arca ofthe two surface is prescrved under thistransformalíon. The arca preserving difTeomorphisms as dc-flned aboye ean be attributcd with the fol1owing Lie bracketstrueture:

1 -,H = - (P + g-) - ,"'0 \"()~"O () (7)2P+ ' .," ,~,

\\'here p+ can fixed as a constant of motion, j5 OTO':.'g = (crlDr.\ Ds.\.)2 and a, b.. . denote the transverse direc-¡iu.iS. Wc also use 50(9) I matrices henceforth. It is nole-worthy that the Hamiltonian does not depend on the centreof mass coordinates .Vo and Oo. The supennembrane massfonnula

where r~= (1/12) ([o" [10). One is left with 9 trans-verse degrces of freedom and the fennionic coordinate (J isrcduced lo having 16 degrees offreedol1l in this gauge. In ad-dilíon one has lo solve for X - which givcs an extra constraintreducing the IOtal number ofbosonic degrees of freedom to 8.The details ofthis derivation can be found ín Refs. 1 and 2.

Tho: Ilamiltonian density dcrivcd from the gaugc fixed la-grangian has the following fonn (1-'+ = 8C./DTX-; r =OC/O,",;):

( 14)

( 12)óA = {{,Al = ,"'O,{ü,A.

By dcfining a covariant derivative.

In fael, with the abovc knowledge, one can start with alagrangian invadant under APO, to yicld (13) as the Hamilto-nian. Thc APD invadant Lagrangian can be writtcn in a com.pact form by introducing an auxiliary flcld w, \Vhich trans-forms as a gauge field under APO transformalions:

H= (f.¡;_~{XO,Xb)2+0)O{x",e}). (13)

Thus {A, ll} = (r"8,.Aalill defines a Lie bracket foranytwofunctions A..f]. It shares aH the requisite propertics of Liebraeket, namely antisymmetry, assoeiativity and satisfies theJacobi identity. The Hamiltonian density can thus be rewrit-tcn as:

(5)

(6)

x+ = xt +T,

[re = O,

(8) the lagrangian becomcs [2]:

is also indepcndent of the centre of m<lss coordinates. TheHamiltonian is accompanied by a constraint (obtained by tak-ing cUTIofthe equation 8rX- = DTXl1urX a - WDrO

C = ~ (DX,,)2 _ ~ {X",X'}"" ,1

_ iODO - iO)" {X",e}. (15)

The potential in the Hamiltonian

(9)

(10)

The lagrangian is also invariant under the target space super-symmctries, 16 of which are linearly rcalizcd with para meter'7, and due 10 the gaugc fixing, anothcr 16 of which are non-lincarly realízcd with parameter €. They are ofthe fonn:

vanishes for conflgurations where olle of the directions isindependent of one of the membrane world volume coordi-nates. In othcr words there are valleys in the potential sig-nalling degenerate surfaces. The mcmbrane surfaee can bemade arbitrarily nanrow in one direetion, anJ hence a singlemembranc can be deformed ínto arbitrary numher of mem-branes without any cost of energy. This is interpreted as thenon-cxistcnee of a definite membrane numbcr, and no definitemembrane topology. A path intcgral approach 10 membranequantizatíon Iike in string theory (ti la Polyakov), where thereis a sum over different topologies for the world sheet, eannotbe achíevcd.

The Iight cone Hamiltonian is slil1 invariant under a classof diffeomorphisms, called area-preserving ditTeolllorphisms(APD), aeling as [22,2]:

Note thal the action is ¡nvariant under Ihe above trallS-formations up to lotal derivatives, and for closed supermem-brancs the total uerivatives do not makc any contribution (theficlds are assumed to vanish at T = Ioo). However, for opensupenncmbrancs, wherc there are boundaries at the end ofspatial Jircetions oflhe world volume, Dirichlet or Neumannboundary condítions havc to be imposed 10 ensure that theboundary tcrms vanish. One finds that to ensure invarianceunder supcrsymmctry, it is nceessary that the supermembraneends only on 1,5 or 9-dilllcnsional hypersurfaces. This set ofconJitions have been derived carlicr in Rcfs. 24 and 23 andby dClllanding invariance under t{ symmetry of the covariantaclion [2.'j].

ó'" = -2,e,óX" = -2,,"e,

(11 )

Re\'. Me.•. Fú. 49 SI (2003) 1-10

4 ARUNDIIATI DASGUPTA. IIERMANN NICOLAI, AND JAN PLEFKA

To check for invariallcc under supcrsymlnctry transfor-malions (16), \Ve vary lhe acrion and find lhe followingboundary tcrms:

Defining 1\ = 1/2(1 :f: ,'1+1 ... ....,10).which aet as projec-¡ion operalors for dimensions JJ = 1,2, ;l, {j, 9, we find thatthe follO\ ••..ing conditions:

(27)

(28)

(29)

(30){~2,~I}=6

OnX'" = COllstanL

(O; - O,') X " = O.

(D, - iO,W = O,

We no\\' concentrale on the rclatian of Ihe APD diffeomor-phism Iransformations to SU(}\') [01' SO(N) for Ihe openmcmbranc) gauge transformations under suitable regulariza-tions [2]. It was first discussed in Ref. 2, and led 10 the su-pcrmembranes' rclalion lO i\-fatrix theory, and M-thcory [5].In the case 01' open mcmbranes, it leads to the rclation to hel-erolle Matrix thcory for mcmbranes ending on 9-brancs [14).and to other exotic theories [15).

Sincc Ihis discussion appears in prcvious revicws [17).\Ve just briefly givc rhe rclation here, wilh an emphasis onthe regularization 01' the open membranc. As stated carlicr,the Af'D bracket has a Lic brackct structurc. It is also casyto check that the cOllllllutator of Iwo APDs Icads lo a thirdAPD

2.1. In the Iight of:'\httrix theor)'

(recaH that al is the tangcntial dcrivative along the spatialboundary of Ihe membrane). In Ihis way we see thal the su-pcrmcmbrane equations of motion wirh the aboye boundaryconditions induce a superstring theory on ils boundary. Therestrictions on the value of p can then be easily understood,becausc only for these valucs is it possiblc to match bosonicand rermionic dcgrees of freedom on the boundary. In partic-ular, for JJ = 9, \Ve obtain the world shcet equations for thehcterotie string, and l' = ~11l bccolllcs a chirality matrix onthe (9+ I )-dimensional branc. By (23). ooly one of the chi-ralities of (1 survivcs on the boundary. leaving an appropriatecquation ofmotion ror the chiral coordinate. \Ve had lo putallo = (J to obtain Ihe frec heterotic string cquations 01' mo-tion.

A tler we havc obtained the gauge fixed lagrangian, withthe approprlate equation of motion for the world volumefields. what rcmains is Ihe quantization of the theory. Somcattcmpts in this dircetion have bcen revie\Vcd in Rcf. 17. Here\Ve confinc oursclves to brief commenls and show ho\\! Ma-tri x n:gularization of the membranc leads to a intcrpretationorthe spectrum. \Ve describe the latter first.

Using this. \ve define r' = L:~lO"xmi"m = constan! ma-trix. The cquations obtained by puUing a = Al in (24) and(25) reduce to the following linear wave cquations on theboundary:

The last ofthese, the equation ofmorion for the auxiliarygaugc field w is same as the constraint (9). What is interest-ing is how these cquations reduce on the boundary. Takinga = 111 in (24) on the boundary, Ihe normal derivative be-comcs:( 17)

( 18)

,\/ = 2.... , JI (Ncumann) (19)

111 = fI + 1, ... , !O (Dirichlct). (20)

D/IX'\! = O for

D¡Xm = O for

wherc 11" is ¡he unir normal 00 lhe boundary and ("~'Ir Iheunir tangential vector lO lhe boundary. For the supcrmclll-branc cnding on a]J dimensional hypcrsurfacc lhe followingboundary conditions are rcquircd:

are required so that (22) is obeyed, which restrict p = 1.!j, 9.ft is interesting to see Ihat (23) results in tbe fcrmionie de-grees of freedom being reduced to 8 on lhe boundary. Forp = 9. P_ coincides with Ihe chiral operator forthe boundary!heory. This is Ihe first sign that the boundary ¡heory, whichis esscntially a string theory induccd by Ihe mcmbrane has aheterotic structure.ln fact. by looking al the equations OflllO-tion obeyed by the membranc on the boull(iary. we find thattheyare 'free' cquation ofmotion for a string. For simplicity.\Ve discuss lhe p = 9 case. and its rclalion lo heterotic Matrixthcory [26]. The p = 5 case also has many interesting appli-eations [27}. especially in the light of non~commutative openmembrane theories proposed and disCllsscd in Ref. 15. lhep = 1 case is yet to be investigated.

Once we have ensured Ihat the bOllndary lenns vanish.Ihe bulk equalion of molíon for !he fields are:

D'X"-{{X",X"},X'}-i{O,,"O} =0, (24)

DO + h. X.O} = O. (25)

{DX",X")-i{B,O}=O. (26)

To impose the bounJary conditions on thc cnds ofthc su-perrncmbranc. \Ve define Ihe normal and tangential deriva-tives on Ihe boundary:

011 imposing lhe condition (20), we finu to gel lhe tCfms lovanish alollg the NCtllllalll1 dircctions, additional conditionson () musl be imposed. These transtale as

I¡y\fe = f/,.\f,.y'\.O£f'.\f¡IIl....,xe = O. (22)

Rev. Mex. Fis. 4951 (2003) 1-10

AN lNTRODUCTION TO TIIE QUANTUM SUPERMEMARANE 5

where

H = Tr G +¡[X".Xb]2 + [X",l"J] ,,,e) (34)

{YA, Y/J} = 9A/JCYC, yA = ~A/JYB

~A/J '/nc = 68 (31)

Given a basis of orthogonal functions {Y A} on the spa-tial manifold, we can expand the coordinates as Xa(a) =X8 + LA XaAYA(a). The Lie bracketthen assumes the fol-lowing form:

2.2. Alllll"o;lching qu<tn'iz<ltion

Wherc Wl.: have distillguished XJO = AJO as it transfol"lns intlle adjoint ofthe 50(1\') group. AIso the fermions are bro-ken IIp as (-)+, (-)- whieh transrrom in the adjoint and sym-metric reprcsentaion ofthe gauge grollp. The Xi transfoTln inthe symmelric traecless representation of SO(1\r). The aboyematrix regularization is the heterotic Matrix theol)' [2Ci}. butwithollt the t\\'isted sector fields expeeted to yield the addi-tional Es ('"E8 degrees of frecuom [H}. These twisted ficldsappear at the boundaries 01' the membranes, and I¡vc only onthe 9-brancs. t\ proper melllbrane origin ofthese ficlds is yctto be uetermincd. For the case of lhe membranc ending ona fivc brane, thcre are Illany interesting possibilities [27], butmuch remains to be done.

The resultant theory is yet te be quantized fully. How-cver, from the nature uf the Hamiltonian, it can be seen thatthe supermembrane spcctrum is continuous and there is nomass gap. This points tm••..ards a multiparticle interpretationofthe spectrllm, and hcnce a second quantizcd picture ortheSllperrtlclll brane.

s =.! Tr (DX2 + DAJO + [AIO,X']' + [X',Xi]2

-i(-)+Dicl+ - iW D0- + 2ie+r'lx', W]) (35)

lÍm - (- )(l+m)}ím for 1 + 1Tl odd being the correct basis.

This givcs ¿¡"-JI = N(N - 1)/2 as lhe numbcrofgcner-ators 01' the finite group. This as we know is the numbcr ofgencrators of SO(N). For the Neumann directions (18), weget these directions to transform :-,)'mmetríc tensor reprcsen-tations ofSO(N) [23, :2.1].In fact, the regularized action hasthe following fOTln(in 1I dimensions):

(33 )lim fAlle = {,All(.'\ .\ 9•.•...•00

with f;....wC the structurc constanl ofsomc finite dimensionalgroup labelcd by ;\. For c10sed mcmbranes of arbitrary topol-ogy [2,28] lhis finile dimensional group is SU(N), and lheHamiltonian of the regulated membrane turns out to pre-cisely coincide with rhe Hamiltonian of dimcnsionally re-dueed SU(¡\') super-Yang Mill's lheory. This can easily beseen by substituting the regularized coordinates in (13), andreplacing the APOs by appropriate commurators. The non-zero X A'S Iransform in the adjoint representation of thcSU(N) group.

YAJJC =!(e. atrSYADrY13Ds}/c,

11AH = J ti"}, a VII Vn. (32)

Given the aboye, the basis which is illfinite dimensionalfor a continuum manifold, can be reslricted to so me finitcA = 1, ... ,A sueh lhal

In the aboye (X, (-)) denote matrices. The same Hamiltonianwas used in Ref. 5 lo describe l',,' DO-brancs in the infinitemomentum framc, with thcir Illomcnta along the II-th diree-tion of d = 11 space. Hencc supermcmbranes through Matrixtheory are intimately related to D-branes and M-theory.

For open supermcmbranes, the matrix regularizationyields difTerent finile dimensional groups, dependent on thetopology of the continuum membrane: for the disc D:2, thecylindcr, and the Mobius strip, we get the groups SO(N),whereas for lhe projeelive plane we gel U Sp(2N) [29].Moreover, depending whether the X a are Neumann orOirichlet, they either transform in the symmetric or adjointrepresentation of the gauge group. \Ve shall illustrate herewith the cxample of the disc, the regularizarion of the openmembrane. Consider YA = YIIll{1~,tP), or spherical harmon-ics, \vilh 111 :S lil. The restriction that / :S N - 1 for the spher-ical membrane leads to a basis with 'Li' -1 (2/ + 1) = .\,2 - 1independcnt componclIls, and the X transform in the ad-joint of tile SU(N) gauge group. However, when the lllelll.brane is stllck on a hypcrsurface, it essentially corrcspondsto a disc topology. with boundal)' conditions imposcd on theX1Jl (1/1 = IJ + 1, ... 10), lhis would translate as J\"I~II =

The matrix rcgularization ofrhe supermembrane proves veryuseflll, as one can use matrix quantum meehanics and inter-pre! the N -t 00 as the quanrurn supermembrane. However,the N -t 00 limit is very subtle [rnernbranes with difTerenltopology and hence difTerent APO are approximated by thesame SU(N) regularization]. Out the multiparticle interpre.tation uf the supermembrane spectrum comes entircly fromits rclation to Matrix theory. It was shown in Rcf. 3 that thesupcrmembrane spectnllTI is continuous and thcre is no.massgap. The continuous speetrum ofthc supermembrane was in.tcrpretcd initially as 3 signature of instability. However, no\\'due to its relation to matrix theory, this is attributed to thepresence ofmulti.particle states. In the original conjecture ofRef. 5, lhe diagonal c1ements of SU(N) matrices are posi.lions of DO branes, and in lhe case of block diagonal ma.trices, e3ch block corresponds to Ni coincident DO.branes,wherc ,Vi is the dill1cnsion ofthe i.th block. Each ofthesc canbe thought as separate cntitics, and in rhe infinite N limit, asseparare Il1cll1branes linked by thin tubes. This multiparticleinterpretation of rhe supcrmcmbranc spcctrum implies, that\Ve should esscntially treat the supermembrane \••..orld volullleas a sceond quantized theory.

Rev. Me .•. Fí.\'. 49 SI (2003) 1- 10

6 ARUNDHATI DASGUPTA, IIERMANN NICOLA1, AND JAN PLEFKA

(41 )

(38 )

1Jho+ = - 101/;0']'v2.

l/lila/¡ = haa = O = kacahe,Ir¡ ;¡'a = "/'kb~a = ka ifia = O = kfll/Ja,

""'-t 'l/'a = ;jJ+,kblb1/;a=k-1j~a' (40)

does 110t have conformal invariance we have to look elsc-where lO determine O. However, the requirernent that thevertex operators transform into one another under supersym-metry transformation eompetely determines their structure.In addition the technique of double dimensional reductionto give the supcrstring in lower dimensions gives us an ad-ditional check for Ihe supermembrane vertex operators. Thesuperparticlc vertex operators for d = 11 were determined inRef. 7, and also serve as a useful guide in our calculations.

Thus for the superrnembrane the vertex operator for thegravilon should be ofthe form (36)

V" = hab /dT({'aOr,b[XO(T,<Y;),B(T,<Y;)]e".x, (37)

Since we work in light cone gauge lhe polarisations arescparaled as Iransvcrse (with indices a, b) and longitudinal(\Vith ¡ndices +, -). Furthcr the gravilino is split up into two50(9) spinors 4)0., '~~'b.To solvc for the aboye vertex opera-tors, we have to resort lo rl = 11 onshell conditions. Notethar these conditions are cnsured by the f',: invariance of thesupermcmbranc action [1]. These conditions are given by

Bccause the polarisations are given in light cone gaugc, Ihesupcrsymmetry transformations (38) and (39) are separatedinlo 161inearly realized (1]) and 16 non-linearly realized (1').Howcver for the sakc ofsimplicity, \Ve shall here record onlyon the linear supcrsymmetry lransformations listed bclow

\••..here Ó!/Jh is the transformed gravitino polarisalion vector,we can solve for Ví,. And "'¡p, Ve which have the followingtransformalions under supersymmetry:

hal, denotes the polarisation tensor ofthe graviton, and oab isa local operalor ofthe supermembranc light cone coordinates.¡;denotes the lllomcntum oflhe graviton, and this operator inthe superslring case creates a graviton state when acting onthe string ground state. Using Ihe faet that under supersym-mctry transformalions

oooo

C:LS, __,

'..'L,

N,- CXJ

N,

u

oooooo

~, N,

FIGURE 2. Matrix-membranc corrcspondencc.

Though it still remains lO be proven that the supermembranehas massless states, there is suffieient evidcncc that they exist,and as discussed in previous sections, should belong to theel = 11 supergravity multiplet, obeying appropriate onshcllconditions. This then leads to the question whether therc ex-ists vertex operators for the supermembrane in analogy withthe superstring vertex operators. The supcrstring vertex oper-ators prove to be extremelyuseful in the calculation ofstringinteractions due to a number ofreasons. The supcrstring is afirst quantized thcory, and hence vertcx operalors which arelocal operators act as 'creation' operators. Moreover, the coo-fonnal symme!ry ofthe string allows the asymptotic statcs ofthe spectrum to be mapped to local operators at finite pointson the world sheet. The vertex operators can thcn be insertedin the path integral, and of scattering amplitudes evaluatedusing them. Further, conformal invariance al so restricts thenature of the vertex operators which can are uniquely deter~mined.

Vertex operalOrs by definition are ofthe fol1owing form:

VI. = J d' < h ' O" [X, BJc".' (36)

where h denotes the polarisation ofthe state, and O the localoperator corresponding lO that state whieh has a momentumF Given the faet that supermembrane world volurne theory

3. Vertex operators

Howcvcr, Dne crucial qucstion still rcmains: can one finda nonnalizable ground state for the theory? And if so, docsthe spcctrum havc masslcss states? By just looking al thezcro mode sector of the theory, one can build statcs whichtransform as 44 tB 88 of SO(9) in the basaníc sector and128 ofSO(9) in the fermionic sector, Since the Hamiltonianand hencc mas s (7) does no! dcpcnd on the zera modes, thisshould givc the massless sector of the thcory, provided thegroundstatc corresponding lo the non-zero modes transformsas a SO(9) singlct. Attempts to prove this have no! seeo muchsuccess up lo 1100\!, However, in Matrix theory, considerableprogre ss has be en made in etTorts to prove the existen ce ofnormalizable ground states [6] in the case ofSU(2) or SU(3).Further infonnation can be found in the review ofRef. 17 andwe refrain from giving the details here. In the nex! section,we discuss vertex operators for the supermembrane, which isone of the directions which can shed more light on this sideofM-lhcory,

Rev. Mex. Fís. 49 SI (20(H) 1-10

AN INTRODUCTION TO nlE QUANTUM SUPERMEMBRANE 7

We now state the vertex operators, and shall explicitly check for thc linear supersymmtery ofthe graviton vertex operator[Rabe = (1/12)8,a',0, Ra' = (1/4)0,a'0].

V" = ha' [DXa DX' - {Xa,X'} {X',X'} _;O,a {X',O} - 2DXa R" k,

" _ Oh (D ,'a Rab>.) -ik.XVh+ - - •• a+ .•'1. - ""b e ,

Vh++ = h++ e-ik.X ,

Ve = -Cabc DXa {Xb, XC} e-ik.X

(44)

(45)

(46)

+ Fa'," [(DXa - ~R"' k, )Rb'd - ~{xa, X'}R'" - ~6{X', X !}o,a',",! O]e-ik.X (47)

Ve+ = Cab+({Xa,X'} + 3R"" k,)e-iL< (48)

V~ = ,p" [(DXa - 2R"' k, +" {X', Xa}) O] e-ikX

+ 0, [1. DX (DXa - 2W' k, + ,,{X', Xa}) O+ ~" {X', X'} (DXa - {Xa, X"h")O

5+ 8"B {X', X'} ne"" k" + 31"B {X', X'}R"" kd + t ",O ({ Xa, X') R'" + {X', X"}Ra') k"

+} (rbB{Xa,O)¡'B-e{X",O}B) + ~'eR"' R'''k, k,,] e-ik.x,

V~+ = - [1/J+0+ 0+ (,a DXa + ~"'{xa, X'}) e] e-ik.X

(49)

(50)

To see that the linear supcfsymmctry is realized, wc implemcnt lhe following transformation in the graviton vertex (44):óXa = Ów = 0, óe = 1}. In other words only the tenns proportional to e shall contribute to the variation, which is written thus:

The tcnns in the first ¡¡ne can be grouped together to yieId the gravitino polarisation as per (43). The tcnns in the sccond ¡inehave to vanish c1early, and by a partial integration cancel each other. Note that, in the case ofthe open membrane we have to becareful in orderto ensure the vanishingofthe aboye. In fact, we find that the additional condition of OtwlvG = O, hmMloG = Ohas to be imposcd to cnsure the vanishing of the boundary terms. The condition can be understood easily as thc residualsymmctry ofthc string worldshcet is just a constant shift in the coordinates and thcrc is no analogous APD gaugc transformationThe second one implies that there are no R 0 R one forms on the boundary string thcory. In case ofthe membrane ending on9-branes this is easy to undcrstand as the boundary string theory is heterotic string theory.

Similarly Iincar supersymmetry transformations can be performed on the rest ofthc vertex operators to see that they respectthe algebra. Details can be found in Ref. 11. The non-linear supersymmetry transfonnations are more complicated and difficultto check, and an attempt in that direction is given in Ref. 11. However, OUT vertex operators are further confinned by a series ofother consistency checks, namely their invariance under space-time gauge transfonnations and double dimensional rcductionto yield superstring vertex operators in Ref. 11. To check for the invariancc undcr óhah = k{fl~b) ofthe graviton vertex, we find

<IV, = [D(k X) (D(~ X) - Rab~ak,,) - {k X, X'}{~ .X, X'}

+ 3{ k X, X'} R""~akb - ~0~a1"{ k . X, &} - ~O{~ . X, e}] e-;kX (52)

By partiaIly integrating the first two terms, and using tile equations of motion for Xa and o (24,25), the terms completcly canceleach other. Note however for the open super membrane, at the boundaries we have to ¡mpose km~M = O, which is perfectlyconsistent with the earlier condition hmAf = O. For details of the other vertex operators, we rcfer to Ref. 11. We review thereduction to superstring vertex operators for the graviton brícfly here (we denote the string coordiantes as Xi). The doubledimensional reduction consists ofthe identification OfOllC ofthe supennembranc dircctions say X9 = rr'J. as compact, to get a

Rev. Mex. Fh. 49 SI (2003) \-10

8 ARUNDHATI DASGUPTA, IIER~lANN NICOLAI. AND JAN PI.EFKA

string in 10 dimcnsions. In the first approximatioll l!lis mcans tha! X ¡are indcpcndcnt of (72. and henec {X<i. X'I} f- O onlywhcn /l or b = 9. Thus:

{XU,X/;} = D¡X[IIJ:;J

lhe SO(9) spinors also dccomposc into two SO(8) spinors (Su, Sil). Thus [i. j = 1, .... S and rl are SO(8) matrices]:

11'" = Lsris2 '

(53)

(54)

(55)

Using Ihe aboye in lhe gravitan vcrtex, one finds tha! lhe !tjj should givc Ihe SS ~ ,\.S vertex opcrator of typc IIAsupcrstring, whilc lhe hjr¡ should givc lhe n 0 R one fonn vertex in 10 dimcnions. DesrilC lhe apparent diffcrcnccs in lhestmcturc 01' lhe vertex opcrators, \Ve rccavcr thClll pcrfcctly starting from (44) and using (53), (54), and 55). The vertexopcrator in 10 dimcnsions is:

Far the ",,), \ve gel:

¡ [ "11 j'l fI + ')é) ,.¡ Ola')j' D YJ ¡l'jlll/, ,) oim o!)llj. /. ] -,-lk ..'('i!) -tu, (JtU _ 0.\ 11 '111+ 1.\ . '11I + _11 11 '/11 "11 ( . (57)

Again the quartic tcrms are casily seen to agrce. To get rid orthe dcriv<ltivcs on H, whieh are aosent in the sllperstring ven ices,\Ve make use of the sllperstring cqualions of motion DIS ;; DoS and DIS = -DoS, anó integrate the resulting express ion bypan s to find

(58)

The same can be repeated for the three form vertcx operalor and the gravitino vcrtcx operalors. Thus, \Ve have the completesupenncmorane \'ertex operalors bOlh for closcó as \Vell as open sllpel"lnembranes. Jt should be menlioned that in Ihe caseof open Illembranes cnding on 9-branes, heterolic string vcrlcx opcralors can be rccovcred on the boundary using Ihe sametechniques 01"double dimensional reduction. EXCCpl,the additional massless states of the EH x E8 gaugc ficlds are not there. Itshall be an intercsting exercise lo look for them in the open supenncmbrane speetrum, Abo, Ihe dimensional redllction for theopen supcnncmbranc lO Typc l' theory [2G]. This is a manifcstation ol' the \Veb of dualilies rclating lhe variolls string thcories.

3.1. l\1alrix tltCOf)' veftCX Opl'r:ltofs

Since Ihe !\1alrix-regularizalion of the supenncmbrane is a straightforward procedure describcd in section 2.2. thc venexoperalOrs can be easily applied to Matrix theol)'. The coordinales X, O transform in the aojoint of SU(N) ano hence are(.\,2 _ 1) X (X2 - 1) matrices (for Ihe dosed Illelllbranc). Thc continulllll integral is replaced by a Trace operation, Bence thegravitan vel1cx (writtcn in configllration space) shall be ofthe 1'orl11:

V" = TI [{ XX + [X", Xl']' + O," [X",fl] - 2X'nn.'" D~<

-G[X" X<ln.""'~ +2(8~"Cf»)((-h""8)~~}J¡ ,(X)]. (59). DXd I 'DxcDXtl (ju

This ngrccs wilh previous calculalions of linearized Matrix currenl in arhitrary backgrounds dctcrmincd up lOO(02) [30]. Notetha! our vcrtcx opel"ators are known to al! orders in () amlunJikc as expectcd (i.e. terms up lo ,(12), they eonlain lenns only up to0(85). Note fm the opcn-mcmbrane also \Ve can suilably rcgularize rClllcmoering to rcplace the Dirichlct dircetion by SO(N)adjoint matrices and the resl by symmctric tracclcss ones. It remains no\V to implelllent the aboye in a scaltcring amplitudecaIculation.

Re\'. /l.lex. Fis. 49 SI (20(3) 1-10

AN INTRODUCTlON TO TIIE QU¡\NTUM SUPERMEMBRt\NE 9

4. Seattering amplitudes

\Ve nO\\' turn 10 Ihe discussion of thn:c point Iree level scat-tering amplitudes. For this it is advantagcous lO work in lhefrarncwork orlhe flnílc N matrix thcory. In order lo define aIrce Icvcl amplitudc wc tirsl split off Ihe center of mass dc-grecs of frccdom of lhe matrices by writing

11_ r' +X" ll- p2 + {S,S}'FIGUKE 3. Thc Vcnczi;¡no amplitudc for strings and mcmbrancs.

\Vith tracclcss matrices X(I ami é. An asymptotic l-gravitan5tatc in Matrix lhcory is Ihen givcn by

(60)ha,,(GSISTr[ (1;" J;" + IX", X'IIX', X'])eikX] IGS)

x '"'"'(k<l klJ + C6(lh) h(l1J = O.

(61 )

wher:: Ik¡ , ¡,1) is Ihe gravitan slalC of Ihe supcrparticlc [8, 7]and IGS) denotes Ihe exucl SU(N) nonnalized zero cncrgygroundstatc, \I/hose cxplicit form is 1Il1known bUl is bc1icvcdlo exisl [6J. Thc In:c level Il1rec point amplitudc is then de-fincd by

Bul as Ihe firsl corrdalor in (64) is nothing but the bosonicconlribulion to lhe 3-point d = 11 superpartide ampli-ludc [71 and as lhe fel'lllionic tel"lllS \vork out in a similarfashiol1 we se(; thal OUl" 3-poinl lree leve! amplitude

One may wonder ha\\' onc eould ever evaluate (62) upon in-scrting (63) wilholll the knowledge DI' IGS). The firsl conlrl-bulion to (62) takes the fonll

(kl, hdp" p" ril'" Ik", h,) h;;/ (GS IS Tr eik.X IGS ) (64)

Now by SO(9) eovariance (GSls Tr (-,ik-X IGS) = S, as Iheon\y SO(9) scalar it could dcpcnd on would be k'2 which van-ishcs on shell. 11Illusl Ihen be a constanl which is fixed tobe N by considcring Ihe J/' -jo O limit. Remarkably Ihe re-maining Iwo terms in (62) upon inserting (63) vanish by acombination 01'SO(9) covarianee and on-shell argulllcnts:

A'e"'" = ((J 11', 13)).\vhcre one inserls the graviton vertcx operator

V2 = h~~),~rS Tr [(1/ plJ + :2pU plJ + f){l plJ

IX'" x. '1 [X' /, x. '1) ".x] ikI r .+ , " e e + lem1l0ns.

(62)

(63)

agrees wilh the 3-point amplilude of el = 11 supergravity!Ckarly the nexl step wOllld bc to sllldy n-point tree levcl

ampliludes which should be given by

where ~ dcnoles Ihe propagator 1/( ~ Po'2 + Ú) built fromIhe ¡"tl'l"uctillg Illclllbrane Hallliltonia~ Ú. Ho\\'c\"Cr, no\\' \Veexpect Ihe details 01' the groundslate IGS) to enter the com-pUlalion. Developing some perturbalive scheme for calculat-ing (67) would be highly uesirable, but is conceivably verycOlllplicalcd as it lllusl involve an expansion in both lhe prop-agalor ~ ami lhe groundstate IGS). In other words, eventhough \Ve now have the vertex operators for the masslcssstalcs, \Ve are 110t able \vilh present teehniques lo calculalethe (sllpcr)lllclllbranc analogue of the Veneziano <llllpliluue.Fig. 3. The complicalions are easy to understand frolll theI1gure: \••..hcreas rOl'Ihe slring, Ihe intermediate slales are jusIIhe massivc string states, anu thcrefore completcly under con-trol, \Ve do not kno\V ho\V lo interprct (and to manipulate) lhesta tes thal propaga te bctween IwO venices in !he case 01' themembrane. On Ihe olhcr hand, ir\Ve could master Ihis calcu-lation \ve might be able to obtain Ihe full result lo all orders"in olle go": there \vould be no need for unitarity correeliansand lhe like in thls membranc amplitude!

Instcad we shall brictly eommenl on attcmpts to compute loop nmplitudes wilhin Ihis scennrio. Here, Icd by Ihe fonnalism inIight con e superslring and sllpcrparticlc theory, \Veproposc 10 define a membranc "-poinl one-Ioop amplitude by the cxpression

A,.''''".'e ••'= / dll¡>o '1'1'("'1'1 "'\, "'\, "'\:1 ... "'1';,), (68)

where the trace is over Ihe Hilbert space of Ú. Again lhis appcars as a dallnting lask, ho\Vever Ihe zcro mode sector of (68)already yie!ds saine amollnt of informalion. In particular lhe trace over lhe fcrmionic zcro mode () 01'(60) tclls us that all 2 and3 particle amplitudcs vanish at one loop, as at least four verlex operalors (:s: 16 O's) are needed 10 saluratc Ihe fcrmion zeromode trace

(69)

Re\'. Mex. riso 49 SI (2003) 1-10

10 ARU~DHATI DASGUPTA, HERMASN SICOLAl, ANDJAI" PLEFKA

In me rUTe graviton sector lhe first non-vanishing amplitude is then lhe 4-gravilon amplitudc whosc leading momentum dcpcn-dcncc is givcn by

(70)

\Ve lhus seo !he emergenee of !he expccled 1<.' lerm [9,31] in lhe kinemalieal seclOr. For lhe remaining traee il is uscful loilllroduee a Sehwinger time parametrizalion of lhe propagator ~ = Ja'" dt exp[ -t(P5/2 + iI)J. Then !he remaining traee fae-torizes ioto the product of the classical bosonic partition function and the intcracting contribution of lhe quantum nuctuations.For an evalualion of lhe elassieal eontrihulionlo lhis traee see ReL 13.

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