Download - Superworldvolume dynamics of superbranes from nonlinear realizations

1 June 2000

Ž .Physics Letters B 482 2000 233–240

Superworldvolume dynamics of superbranesfrom nonlinear realizations

S. Bellucci a,1, E. Ivanov a,b,2, S. Krivonos c,3

a INFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italyb Laboratoire de Physique Theorique et des Hautes Energies, Unite associee au CNRS UMR 7589, UniÕersite Paris 7,´ ´ ´ ´

2 Place Jussieu, 75251 Paris Cedex 05, Francec BogoliuboÕ Laboratory of Theoretical Physics, JINR, 141 980 Dubna, Moscow region, Russian Federation

Received 29 March 2000; accepted 28 April 2000Editor: P.V. Landshoff

Abstract

Ž .Based on the concept of the partial breaking of global supersymmetry PBGS , we derive the worldvolume superfieldequations of motion for Ns1, Ds4 supermembrane, as well as for the space-time filling D2- and D3-branes, fromnonlinear realizations of the corresponding supersymmetries. We argue that it is of no need to take care of the relevantautomorphism groups when being interested in the dynamical equations. This essentially facilitates computations. As aby-product, we obtain a new polynomial representation for the ds3,4 Born–Infeld equations, with merely a cubicnonlinearity. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 11.17.qy; 11.30.Pb

1. Introduction

During last few years there was a considerableinterest in applying the general method of nonlinearrealizations to systems with partial breaking of global

Ž .supersymmetries PBGS , first of all to the super-Žbranes as a notable example of such systems see,

w x .e.g., 1–3 and refs. therein . On this path one meetstwo problems. The first one is purely computational.Following the general prescriptions of nonlinear real-izations, one is led to include into the coset, along-

1 E-mail: [email protected] E-mail: [email protected], [email protected] E-mail: [email protected]

side with the spontaneously broken translation andsupertranslation generators, also the appropriate partof generators of the automorphism group for the

Žgiven supersymmetry algebra including those of the.Lorentz group . This makes the computations beyond

the linearized approximation rather complicated.Moreover, sometimes these additional symmetrieswhich we should take into account at the step ofdoing the coset routine appear to be explicitly broken

Žat the level of the invariant action see, e.g., Refs.w x.4–6 , with no clear reasons for this. The second,closely related difficulty is lacking of a systematicprocedure for constructing the PBGS actions. In allthe cases elaborated so far, the PBGS Lagrangianscannot be constructed in a manifestly invariant way

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00529-3

( )S. Bellucci et al.rPhysics Letters B 482 2000 233–240234

from the relevant Cartan forms: under the brokensupersymmetry transformations they are shifted by

Žthe spinor or x-derivatives like the WZNW or.Chern–Simons Lagrangians .

In the present note we argue, on several instruc-tive examples, that the automorphism symmetriescan be ignored if we are interested only in theequations of motion for the given PBGS system.This radically simplifies the calculations, resulting inrather simple manifestly covariant equations in whichall nonlinearities are hidden inside the covariantderivatives.

2. Ns1, Ds4 supermembrane and D2-brane

To clarify the main idea of our approach, let usstart from the well known systems with partially

w xbroken global supersymmetries 7,8 . Our goal is toget the corresponding superfield equations of motionin terms of the worldvolume superfields startingfrom the nonlinear realization of the global super-symmetry group.

The supermembrane in D s 4 spontaneouslybreaks half of four Ns1, Ds4 supersymmetriesand one translation. Let us split the set of generators

Žof Ns1 Ds4 Poincare superalgebra in the ds3´. � 4notation into the unbroken Q , P and brokena ab

Ž .� 4S ,Z ones a,bs1,2 . The ds3 translation gen-a

erator P sP together with the generator Z formab b a

the Ds4 translation generator. The basic anticom-mutation relations read 4

� 4 � 4Q ,Q sP , Q ,S se Z ,a b ab a b ab

� 4S ,S sP . 1Ž .a b ab

w xIn contrast to our previous considerations 8,1,2 ,here we prefer to construct the nonlinear realization

Ž .of the superalgebra 1 itself, ignoring all generatorsŽ . Žof the automorphisms of 1 the spontaneously bro-

.ken as well as unbroken ones , including those ofŽ .Ds4 Lorentz group SO 1,3 . Thus, we put all

4 Hereafter, we consider the spontaneously broken supersym-metry algebras modulo possible extra central-charge type termswhich should be present in the full algebra of the corresponding

w xNoether currents to evade the no-go theorem of Ref. 9 along thew xlines of Ref. 10 .

generators into the coset and associate the Ns1 , d� a ab4s3 superspace coordinates u , x with Q , P .a ab

The remaining coset parameters are Goldstone super-a aŽ . Ž .fields, c 'c x,u , q'q x,u . A coset element

g is defined by

gse x ab Pabeu aQ aeqZec aSa . 2Ž .As the next step of the coset formalism, one con-structs the Cartan 1-forms

gy1dgsv a Q qv abP qv Zqv aS , 3Ž .Q a P ab Z S a

v sdqqc du a ,Z a

1 1ab ab Ža b. Ža b.v sdx q u du q c dc ,P 4 4

v a sdu a , v a sdc a , 4Ž .Q S

and define the covariant derivatives

cdy1DD s E E ,Ž . abab cd

1 1b c b cDD sD q c D c DD sD q c DD c E ,a a a bc a a bc2 2

5Ž .

where

E1 b � 4D s q u E , D , D sE , 6Ž .a ab a b ab2aEu

1 1cd c d d c c d d cE s d d qd d q c E c qc E c .Ž . Ž .ab a b a b ab ab2 4

7Ž .

They obey the following algebra:

w x f gDD , DD syDD c DD c DD ,ab cd ab cd f g

w x f gDD , DD sDD c DD c DD ,ab c ab c f g

� 4 f gDD , DD sDD qDD c DD c DD . 8Ž .a b ab a b f g

Not all of the above Goldstone superfieldsq x ,u ,c a x ,u must be treated as independent.� 4Ž . Ž .

Indeed, c appears inside the form v linearly anda Z

so can be covariantly eliminated by the manifestlyŽ w x.covariant constraint inverse Higgs effect 11

v s0´c sDD q , 9Ž .Z a adu

<where means the ordinary du-projection of thedu

Ž .form. Thus the superfield q x,u is the only essen-tial Goldstone superfield needed to present the par-tial spontaneous breaking Ns1 , Ds4´ Ns1 , ds3 within the coset scheme.

( )S. Bellucci et al.rPhysics Letters B 482 2000 233–240 235

Now we are ready to put additional, manifestlyŽ .covariant constraints on the superfield q x,u , in

order to get dynamical equations. The main idea is tocovariantize the ‘‘flat’’ equations of motion. Namely,we simply replace the flat covariant derivatives inthe standard equation of motion for the bosonicscalar superfield in ds3

DaD qs0 10Ž .a

Ž .by the covariant ones 5

DD a DD qs0 . 11Ž .a

Ž .The equation 11 coincides with the equation ofmotion of the supermembrane in Ds4 as it was

w xpresented in 8 . Thus, we conclude that, at least inthis specific case, additional superfields-parametersof the extended coset with all the automorphismsymmetry generators included are auxiliary and canbe dropped out if we are interested in the equationsof motion only.

w x Ž .Actually, in 8 Eq. 11 was deduced, proceedingfrom the Ds4 Lorentz covariant coset formalismwith preserving all initial symmetries. This means

Ž .that 11 , having been now reproduced from thecoset involving only the translations and supertrans-lations generators, possesses the hidden covarianceunder the full Ds4 Lorentz group. On the otherhand, one more automorphism symmetry of the Ns1, Ds4 supersymmetry algebra, ‘‘g ’’ symmetry, is5

Ž .explicitly broken in Eq. 11 , and there is no way tokeep it. In the ds3 notation this symmetry is

Ž .realized as an extra SO 2 with respect to which thegenerators Q and S and, respectively, the coseta b

parameters u a,c a form a 2-vector. This symmetry isspontaneously broken at the level of the transforma-

Ž .tion laws, with the auxiliary field of q x,u beingŽ .the relevant Goldstone field. From Eq. 11 we con-

clude that it cannot be preserved even in this sponta-neously broken form when q is subjected to thedynamical equation: one can preserve the sponta-neously broken Ds4 Lorentz symmetry at most.

Ž .This U 1 is explicitly broken in the off-shell PBGSw xaction of Ref. 8 , as well as in the corresponding

w xGreen–Schwarz action 7 . A similar phenomenonw x Ž .was observed in Refs. 4,5 for the Ns 1,0 , Ds6

3-brane. There, the auxiliary fields of the basicworldvolume Ns1, ds4 Goldstone chiral super-multiplet are the Goldstone fields parameterizing the

Ž . Ž . Ž .coset SU 2 rU 1 of the automorphism SU 2A A AŽ .group of Ns 1,0 , Ds6 Poincare superalgebra,´

Ž .and the coset part of SU 2 is realized as nonlinearA

shifts of these fields. In the superfield equations ofmotion of the 3-brane and the corresponding off-shell

Ž .action this SU 2 is explicitly broken down toAŽ .U 1 , though the spontaneously broken Ds6A

Lorentz symmetry is still preserved.As a straightforward application of the idea that

the automorphism symmetries are irrelevant whendeducing the equations of motion, let us consider the

Žcase of the ‘‘space-time filling’’ D2-brane i.e. hav-ing no scalar fields in its worldvolume multiplet thefield content of which is that of Ns1, ds3 vector

.multiplet . The main problem with the description ofD-branes within the standard nonlinear realizationapproach is the lack of the coset generators to whichone could relate the gauge fields as the coset parame-ters 5. So we do not know how interpret the gaugefields as coset parameters in this case 6. Let us showhow these difficulties can be circumvented in thepresent approach.

The superalgebra we start with is the same alge-Ž .bra 1 , but now without the central charge

Zs0 .

The coset element g contains only one Goldstonesuperfield c a which now must be treated as theessential one, and the covariant derivatives coincide

Ž .with 5 . Bearing in mind to end up with the irre-ducible field content of Ns1, ds3 vector multi-plet, we are led to treat c a as the correspondingsuperfield strength and to find the appropriate co-variantization of the flat irreducibility constraint andthe equation of motion. In the flat case the ds3vector multiplet is represented by a Ns1 spinor

5 For the covariant field strengths as Goldstone fields suchgenerators can still be found in the automorphism symmetry

w xalgebras 3,12 .6 It seems that the existing interpretation of gauge fields as the

w xcoset fields 13 can be generalized to the PBGS case only on theway of non-trivial unification of the gauge group algebra with thatof supersymmetry, so that the gauge group transformations ap-peared in the closure of supersymmetries before any gauge-fixingas a sort of tensorial central charges.


superfield strength m subjected to the Bianchi iden-aw xtity 14 :

D2m syE mba abaD m s0´ . 12Ž .a a b½ 5E D m s0ab

Ž .This leaves in m the first fermionic Goldstonea

component, together with the divergenceless vector< Ž .F 'D m i.e., just the gauge field strength .us0ab a b

The equation of motion reads

D2m s0 . 13Ž .a

In accordance with our approach, we propose thefollowing equations which should describe the D2-brane:

a DDac s0 , b DD2c s0 . 14Ž . Ž . Ž .a a

Ž .The equation a is a covariantization of the irre-Ž . Ž .ducibility constraint 12 while b is the covariant

equation of motion.In order to see which kind of dynamics is encodedŽ .in 14 , we considered it in the bosonic limit. We

found that it amounts to the following equations for<the vector V 'DD c :us0ab a b

E qV mV n E V c s0 . 15Ž .Ž .ac a c m n b

One can wonder how these nonlinear but polynomialequations can be related to the nonpolynomialBorn–Infeld theory which is just the bosonic core ofthe superfield D2-brane theory as was explicitly

w xdemonstrated in 8 . The trick is to rewrite the partsŽ .of Eq. 15 , respectively antisymmetric and symmet-

� 4ric in the indices a,b , as follows:

V ab

E s0 , 16Ž .ab 2ž /2yV

V c V cb a

E qE s0 , 17Ž .ac bc2 2ž / ž /2qV 2qV

where V 2 'V m nV . After passing to the ‘‘genuine’’m n

field strength

2V abab abF s ´E F s0 , 18Ž .ab22yV

Ž .the equation of motion 17 takes the familiarBorn–Infeld form

F c F cb a

E qE s0 . 19Ž .ac bc2 2ž / ž /' '1q2 F 1q2 F

Thus we have proved that the bosonic part of ourŽ .system 14 indeed coincides with the Born–Infeld

equations. One may explicitly show that the fullŽ .equations 14 are equivalent to the worldvolume

superfield equation following from the off-shell D2-w x Žbrane action given in 8 augmented with the Bianchi

Ž ..identity 12 . An indirect proof is based on the factŽ .that 14 is an Ns1 extension of the bosonic ds3

Born–Infeld equations, such that it possesses onemore nonlinearly realized supersymmetry completing

Ž .the explicit one to Ns2, ds3 superalgebra 1with Zs0. On the other hand, the Ns1, ds3

w xsuperfield action of 8 is uniquely specified byrequiring it to possess this second supersymmetry.Hence both types of equations should be equivalent.

In closing this Section, it is worth mentioning thatŽ .Eqs. 15 which equivalently describe the bosonic

Born–Infeld dynamics in ds3, look much simplerŽ . Ž .than the standard ones 18 , 19 .

3. D3-brane

As another interesting application of the proposedapproach, we shall consider the space-time fillingD3-brane in ds4. This system amounts to thePBGS pattern Ns2, ds4™ Ns1, ds4, with anonlinear generalization of Ns1, ds4 vector mul-

w xtiplet as the Goldstone multiplet 15,6 . The off-shellsuperfield action for this system and the related

w xequations of motion are known 15 , but the latterhave never been derived directly from the cosetapproach.

Our starting point is the Ns2, ds4 Poincaresuperalgebra without central charges:

Q ,Q s2 P , S ,S s2 P . 20Ž .� 4 � 4a a a a a a a a˙ ˙ ˙ ˙

Assuming the S ,S supersymmetries to be sponta-a a

neously broken, we introduce the Goldstone super-


a aŽ . Ž .fields c x,u ,u , c x,u ,u as the correspondingŽparameters in the following coset we use the same

w x.notation as in 15 :

aa a a a a˙ ˙ ˙i x P iu Q qiu Q ic S qic Saa a a a a˙ ˙ ˙gse e e . 21Ž .With the help of the Cartan forms

y1 a a a a a˙ ˙g dgs iv P q iv Q q iv Q q iv Saa Q a Q a S a˙ ˙

aq iv S ,S a

aa aav sdx

a a a a a a a a˙ ˙ ˙ ˙y i u du qu du qc dc qc dc ,ž /a a a a a a a a˙ ˙ ˙ ˙v sdu , v sdu , v sdc , v sdc ,Q Q S S

22Ž .

one can define the covariant derivatives˙bby1DD s E E ,Ž . ˙aaa a bb˙˙

˙ ˙b b b bDD sD y i c D c qc D c DD ,˙ž /a a a a bb

˙ ˙b b b bDD sD y i c D c qc D c DD , 23Ž .˙a a a a bbž /˙ ˙ ˙ ˙

where

˙ ˙ ˙ ˙bb b b b b b bE sd d y ic E c y ic E c , 24Ž .aa a a a a a a˙ ˙ ˙ ˙

and the flat covariant derivatives are defined asfollows:

E Ea a˙D s y iu E , D sy q iu E .a a a a a a˙ ˙ ˙a aEu Eu

25Ž .

Now we are ready to write the covariant version ofa athe constraints on c , c which define the super-

brane generalization of Ns1, ds4 vector multi-plet, together with the covariant equations of motionfor this system.

w xAs is well-known 16 , the Ns1, ds4 vectormultiplet is described by a chiral Ns1 field strengthW ,a

D W s0 , D W s0 , 26Ž .a a a a˙ ˙

Žwhich satisfies the irreducibility constraint Bianchi.identity

a aD W qD W s0 . 27Ž .a a

The free equations of motion for the vector multipletread

a aD W yD W s0 . 28Ž .a a

w xIt was shown in 15 that the chirality constraintsŽ .26 can be directly covariantized

DD c s0 , DD c s0 . 29Ž .a a a a˙ ˙

These conditions are compatible with the algebra ofŽ .the covariant derivatives 23 . This algebra, with the

Ž . w xconstraints 29 taken into account, reads 15

DD , DD s DD , DD s0 ,� 4 ˙½ 5a b a b˙

g gDD , DD s2 i DD y2 i DD c DD c DD ,˙ ˙ ˙½ 5 ž /a b a b a b gg

˙b bDD , DD sy2 i DD c DD c DD . 30Ž .� 4 ˙ž /a gg a gg bb˙ ˙

Ž .The first two relations in 30 guarantee the consis-tency of the above nonlinear version of Ns1, ds4chirality. They also imply, like in the flat case,

33DD s DD s0 . 31Ž . Ž . Ž .

Ž .The second flat irreducibility constraint, Eq. 27 ,is not so simple to covariantize. The straightforward

Ž .generalization of 27 ,

a aDD c qDD c s0 , 32Ž .a a

Ž .2is contradictory. Let us apply the square DD to theŽ .left-hand side of 32 . When hitting the first term in

Ž .the sum, it yields zero in virtue of the property 31 .However, it is not zero on the second term. Tocompensate for the resulting non-vanishing terms,and thus to achieve compatibility with the algebraŽ . Ž . Ž .30 and its corollaries 31 , one should modify 32

w xby some higher-order corrections 15 .Ž .Let us argue that the constraints 27 together

Ž .with the equations of motion 28 can be straightfor-wardly covariantized as

a aDD c s0 , DD c s0 . 33Ž .a a

Firstly, we note that no difficulties of the abovekind related to the compatibility with the algebraŽ . Ž .30 arise on the shell of Eq. 33 . As a consequence

Ž . Ž .of 33 and the first two relations in 30 we get

2 2DD c s0, DD c s0 . 34Ž .a a


This set is a nonlinear version of the well-knownreality condition and the equation of motion for theauxiliary field of vector multiplet. Then, applying,

Ž .e.g., DD to the second equation in 33 and makinga

Ž .use of the chirality condition 29 , we obtain thenonlinear version of the equation of motion forphotino

a g g a˙ ˙ ˙DD c y DD c DD c DD c s0 . 35Ž .ž /aa a a gg˙ ˙ ˙

Acting on this equation by one more DD and takinga

Ž .advantage of Eqs. 34 , we obtain:

a a g a g a˙ ˙ ˙DD , DD c yDD c DD , DD c DD c� 4aa a a gg˙ ˙ ˙

g g a a˙ ˙yDD c DD c DD , DD c s0 . 36Ž .a a gg˙ ˙

After substituting the explicit expressions for theŽ . Ž . Ž .anti commutators from 30 , we observe that 36 issatisfied identically, i.e. it does not imply any further

a arestrictions on c ,c . It can be also explicitlya achecked, in a few lowest orders in c ,c , that the

Ž . w xhigher-order corrections to 32 found in 15 areŽ .vanishing on the shell of Eq. 33 .

Thus the full set of equations describing the dy-namics of the D3-brane supposedly consists of the

Ž . Ž .generalized chirality constraint 29 and Eqs. 33 .To prove its equivalence to the Ns1 superfield

w xdescription of D3-brane proposed in 15 , recall thatw xthe latter is the Ns1 supersymmetrization 17 of

the ds4 Born–Infeld action with one extra nonlin-early realized Ns1 supersymmetry. So, let us con-sider the bosonic part of the proposed set of equa-tions. Our superfields c ,c contain the followingbosonic components:

ab ba a b <V sV 'DD c ,us0

˙ ˙ ˙ab ba a b˙ ˙ ˙ <V sV 'DD c , 37Ž .us0

Ž .which, owing to 33 , obey the following simpleequations

ab g g a b˙E V yV V E V s0 ,aa a a gg˙ ˙ ˙

˙ ˙ab g g a b˙ ˙ ˙E V yV V E V s0 . 38Ž .aa a a gg˙ ˙ ˙

Ž .Like in the D2-brane case, in Eqs. 38 nothingreminds us of the Born–Infeld equations. Neverthe-less, it is possible to rewrite these equations in thestandard Born–Infeld form.

Ž .The first step is to rewrite Eq. 38 as

˙1 1 12 2 b 2 b 2 b 21y V V E V q V V E V q V E VŽ . ˙ba a a ba a a b4 4 2˙ ˙ ˙

s0 , 39Ž .˙ ˙1 1 12 2 b 2 b 2 b 21y V V E V q V V E V q V E VŽ . ˙ ˙ab a a a b a a b4 4 2˙ ˙ ˙

s0 . 40Ž .After some algebra, one can bring them into thefollowing equivalent form

˙b bE fV yE fV s0 ,Ž . ˙ž /ba a a b a˙ ˙

˙b bE gV qE gV s0 , 41Ž .Ž . ˙ž /ba a a b a˙ ˙

where

2 2V y2 V q2fs , gs . 42Ž .1 12 2 2 21y V V 1y V V4 4

After introducing the ‘‘genuine’’ field strengths

1 1˙ ˙b b b bF ' f V , F ' f V , 43Ž .a a a a˙ ˙' '2 2 2 2

Ž .first of Eq. 41 is recognized as the Bianchi identity

˙b bE F yE F s0 , 44Ž .˙ba a a b a˙ ˙

while the second one acquires the familiar form ofthe Born–Infeld equation

2 21qF yFbE Fba a˙ 2� 02 2 2 2( F yF y2 F qF q1Ž . Ž .

2 21yF qFbqE F˙ab a2� 02 2 2 2( F yF y2 F qF q1Ž . Ž .

s0 . 45Ž .Thus, in this new basis the action for our bosonicsystem is the Born–Infeld action:

24 2 2 2 2(Ss d x F yF y2 F qF q1 . 46Ž . Ž . Ž .HŽ .Now the equivalence of the system 33 to the

w xequations corresponding to the action of Ref. 15 ,like in the D2-brane case, can be established pro-

Ž .ceeding from the following two arguments: i It isNs1 supersymmetrization of the ds4 Born–Infeld


Ž .equations; ii It possesses the second hidden nonlin-early realized supersymmetry lifting Ns1, ds4 to

w xNs2, ds4. The action given in 15 provides theunique extension of the ds4 Born–Infeld actionwith both these requirements satisfied. Hence, bothrepresentations should be equivalent to each other.

Note that at the full superfield level the redefini-Ž .tion 43 should correspond to passing from the

Goldstone fermions c , c which have the simplea a

transformation properties in the nonlinear realizationof Ns1, ds4 supersymmetry but obey the nonlin-ear irreducibility constraints, to the ordinary Maxwell

Ž . Ž .superfield strength W , W defined by Eq. 26 , 27 .a a

w xThe nonlinear action in 15 was written just in termsŽ .of this latter object. The equivalent form 33 of the

equations of motion and Bianchi identity is advanta-geous in that it is manifestly covariant under the

Ž .second hidden supersymmetry, being constructedout of the covariant objects.

4. Conclusions

In this Letter we demonstrated that in many casesone can simplify the analysis of the equations ofmotion which follow from the coset approach bytaking no account of the automorphism group at all.We showed that the equations of motion for theNs1, Ds4 supermembrane, D2- and D3-branes ina flat background have a very simple form whenwritten in terms of Goldstone superfields of nonlin-ear realizations and the corresponding nonlinear co-variant derivatives. As a by-product, we got a newsimple form for the ds3 and ds4 Born–Infeldtheory equations of motion combined with the appro-priate Bianchi identities. The remarkable property ofthis representation is that it involves only a thirdorder nonlinearity in the gauge field strength.

Note that the idea to use the geometric and sym-metry principles to derive the dynamical equations isnot new, of course. For instance, the completelyintegrable ds2 equations admit the geometricalinterpretation as the vanishing of some curvatures. In

Ž w xthe superembedding approach see 18 and refer-.ences therein the equations of motion for super-

branes in a number of important cases amount to the

so-called ‘‘geometro-dynamical’’ constraint which,in the PBGS language, is just a kind of the inverseHiggs constraints. For instance, this applies to the

w xNs1, Ds10 5-brane 1,2 . In this case the condi-Ž .tion like 9 , besides eliminating the Goldstone

fermion superfield in terms of the appropriate analogŽof the ds3 superfield q ds6 hypermultiplet su-

.perfield , also yields the equation of motion for thelatter 7. However, as we saw in the above examples,

Žin other interesting cases the inverse Higgs or ge-.ometro-dynamical constraints do not imply any dy-

namics which, however, can still be implemented ina manifestly covariant way using the approach pro-posed here.

It still remains to fully understand why in thePBGS scheme the dynamical worldvolume super-field equations are not sensitive to the presence orabsence of the automorphism generators in the initialcoset construction. This is in contrast with the caseof purely bosonic p-branes. For the self-consistentdescription of them in terms of nonlinear realizationsone should necessarily make use of the cosets of thefull target Minkowski space Poincare group includ-´

Ž .ing the Lorentz automorphism part of the latterw x3,12 . A possible explanation of this apparent dis-agreement is that the Goldstone fermion superfieldsor Goldstone superfields associated with the central

Žcharges andror with the transverse components of.the full momenta already accommodate the Lorentz

and other automorphism groups Goldstone fields.These come out as component fields in the u -expan-sion of the Goldstone superfields. So the automor-phism groups Goldstone fields are implicitly presentin the superbrane superfield equations of motion.

The most interesting practical application of theapproach exemplified here is the possibility to con-struct, more or less straightforwardly, the equationsfor the Ns4 and Ns8 supersymmetric Born–In-

w xfeld theory. This work is in progress now 19 .

7 It is curious that this equation, in accord with the generalreasoning of the present work, proved to be finally written interms of the covariant quantities of nonlinear realization of thepure Ns1, Ds10 Poincare superalgebra, despite the fact that´

w xwe started in 1,2 from the coset of the extended supergroupinvolving Ds10 Lorentz group.


Acknowledgements

This work was supported in part by the FondoAffari Internazionali Convenzione ParticellareINFN-JINR, grants RFBR-CNRS 98-02-22034,RFBR 99-02-18417, INTAS-96-0538, INTAS-96-0308 and NATO Grant PST.CLG 974874.

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