New nonlocal charges in SUSY-B integrable models

11 September 2000

Ž .Physics Letters A 274 2000 30–36www.elsevier.nlrlocaterpla

New nonlocal charges in SUSY-B integrable models

Ashok Das a, Ziemowit Popowicz b,)

a Department of Physics and Astronomy, UniÕersity of Rochester, Rochester, NY 14627-0171, USAb Institute of Theoretical Physics, UniÕersity of Wroclaw, 50-205 Wroclaw, Poland

Received 4 May 2000; received in revised form 4 August 2000; accepted 9 August 2000Communicated by A.P. Fordy

Abstract

In this Letter, we study systematically the general properties of the B-extension of any integrable model. In addition todiscussing the general properties of Hamiltonians, Hamiltonian structures etc, we also clarify the origin of AexoticB charges

Žin such models. We show that, in such models, there exist at least two sets of non-local conserved charges and more if.N)1 supersymmetry is present and that the AexoticB charges are part of this non-local charge hierarchy. The construction

of these non-local charges from the Lax operator is explained. q 2000 Published by Elsevier Science B.V.

1. Introduction

w xIntegrable models 1–3 appear naturally in thestudy of strings in the matrix model approach. Thus,while the KdV hierarchy is obtained in the double

w xscaling limit of the one matrix model 4–7 , thesupersymmetric matrix models lead to a particularsupersymmetric version of the KdV hierarchy known

w xas the Ns1 supersymmetric KdV-B hierarchy 8 .In simple terms, if u denotes the dynamical variableof the KdV equation

u su q6uu ,t x x x x

where the subscripts represent differentiation withrespect to the corresponding variables, then, the Ns1 supersymmetric KdV-B hierarchy is given by

2F sF q3 D DF .Ž .Ž .t x x x

) Corresponding author.Ž .E-mail address: [email protected] Z. Popowicz .

Ž . Ž . Ž .Here, F x,u sc x qu u x represents the dy-namical variable which is a Ns1 fermionic super-field with u denoting the Grassmann coordinate and

E EDs qu .

Eu E x

There are, of course, other supersymmetrizations ofw xthe KdV hierarchy that are integrable 9–11 , but it

w xis this particular supersymmetrization 12 that mani-fests in the study of string theories. Therefore, in thisLetter, we undertake a systematic study of the prop-erties of such a supersymmetrization. In particular,we show, in Section 2, that this particular method ofsupersymmetrization can be applied to any integrablemodel, although the original study involved thebosonic KdV hierarchy. In Section 3, we bring outsome general properties of such models, such as theHamiltonians, Hamiltonian structures, recursion op-erators etc. In these models, there arise local con-served charges which have opposite Grassmann par-ity relative to the Hamiltonians of the system. The

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0375-9601 00 00523-5

( )A. Das, Z. PopowiczrPhysics Letters A 274 2000 30–36 31

w xorigin of such AexoticB charges 12 is explained inSection 4, where we identify that such local chargesbelong to the hierarchy of an infinite set of non-localcharges. In fact, we show that, in such models, thereexist, at least, two infinite sets of non-local chargesand may be more. Explicitly, in the Ns2 supersym-metric KdV-B hierarchy, we show that there existthree infinite sets of non-local charges and present amethod for constructing them. In Section 5, wepresent briefly an alternate description for such asupersymmetrization which allows the constructionof B-extensions of systems such as the NLS and theAKNS hierarchies. A brief conclusion is presented inSection 6. We used the symbolic computer language

w x w xREDUCE 13 and the special package 14 in allcalculations presented in this Letter.

2. Model

Let us consider a general integrable model of theform

w xf s A f , 1Ž .Ž . xt

where the subscripts refer to differentiation withrespect to the corresponding variables. Here, f is ageneral dynamical variable. It can be a purely bosonicfunction of x alone, in which case, the equation willrepresent the dynamics of a bosonic integrable sys-

Ž .tem Ns0 supersymmetry . Alternately, f mayŽ .represent a superfield bosonic or fermionic depend-

ing on x as well as N fermionic coordinates, u ,isi

1,2, PPP , N, in which case, the dynamical equationwill describe an integrable model with N-extendedsupersymmetry. Let us denote the covariant deriva-tives with respect to the N fermionic coordinates by

E ED s qu , is1,2, PPP , N , 2Ž .i iEu E xi

which satisfy

ED , D sd sd E .i j i j i jq E x

˜ŽWe can now introduce a new superfield, f and f

are superfields depending on the original N fermionic.coordinates

˜GsG x ,u , PPP ,u sfqu f , 3Ž . Ž .1 Nq1 Nq1

which depends on one extra fermionic coordinateŽand has opposite Grassmann parity relative to f the

.original dynamical variable , and define the dynami-cal equation

D G s A D G , 4Ž . Ž . Ž .Ž .Nq1 Nq1t x

which would represent an integrable system withŽ .Nq1 -extended supersymmetry. This, therefore,describes the generalization of Beckers’ extension toŽ . Žextended supersymmetric models. Basically, theoriginal f equation remains unchanged under this

.extension since D G sf.Ž .Nq1 u s0Nq 1 Ž . w xThus, for example, with fsu x and A u s2 Žu q3u , we have the bosonic KdV equation Ns0x x

. Ž .supersymmetry while, with GsF x,u , where F

is a fermionic superfield, the equation

2DF s DF q3 DF ,Ž . Ž . Ž .Ž .t x x x

2or, F sF q3D DF , 5Ž . Ž .Ž .t x x x

represents the Ns1 supersymmetric KdV-B equa-w x Ž .tion 8 . Similarly, for F x,u a fermionic super-1

w x Ž Ž ..field and A F sy F q3F D F , the dynami-x x 1

cal equation represents the Ns1 supersymmetricw x Ž .KdV equation 9–11 , while, with G x,u ,u a1 2

bosonic superfield, the equation

D G sy D G q3 D G D D G ,Ž . Ž . Ž . Ž .Ž .2 2 x x 2 1 2t x

or, G syG y3D D G D D G , 6Ž . Ž . Ž .Ž .t x x x 2 2 1 2

would give rise to an Ns2 extended supersymmet-ric KdV equation of the B-type. Similarly, if f

represents an Ns2 superfield and the dynamicalequation gives the Ns2 supersymmetric KdV equa-

w xtion 15–18 , then, the G equation would correspondto the Ns3 supersymetric KdV-B equation and soon. While this procedure is quite general, in this

Ž .Letter, we would study the specific model in Eq. 6and bring out properties of this model which arenonetheless common to all such models. We alsonote here that, as described above, this extension,when applied twice to a given equation, would seemto lead to a nonlocal dynamical equation and, there-fore, is not useful. Similarly, if the right hand side of

Ž .Eq. 1 is not a total space derivative, this methodwill also appear to fail. We would come back to this

( )A. Das, Z. PopowiczrPhysics Letters A 274 2000 30–3632

point in Section 5, where we would describe analternate, but equivalent generalization, which can beapplied to any given equation and as many times,without introducing non-locality.

3. Hamiltonians and Hamiltonian structures

Ž .Let us next look at the general model of Eq. 1 .We note that if

ŽN . ŽN .w xH s dx du PPP du h f , ns1,2, PPP ,Hn 1 N n

7Ž .

represent the Hamiltonians of the original model,then,

ŽNq1. ŽN .H s dx du PPP du h D G ,Ž .Hn 1 Nq1 n Nq1

ns1,2, PPP , 8Ž .

would correspond to the Hamiltonians of the ex-w xtended B-model 12 . These are conserved local

quantities which would be invariant under the ex-tended supersymmetry and we note that, since theintegration, in the second case, is over an additionalfermionic variable relative to the definition of theoriginal charges, the Hamiltonians of the new systemwould have an opposite Grassmann parity comparedto those of the original system. It is also not hard tosee that the Hamiltonian densities can be written as

ŽN . ŽN . ŽNq1.˜ ˜w xh D G sh f qu h f ,f ,Ž .n Nq1 n Nq1 n

9Ž .

so that each of the two parts of the Hamiltonians,namely, the u independent term as well as theNq1

linear term in u , will be independently con-Nq1

served. However, the u independent term in theNq1Ždensity would give a conserved charge when inte-

.grated over appropriate coordinates which is invari-ant only under the lower, N-extended supersymme-try.

The Hamiltonian structures of the two systems areŽN .w xalso related in a simple manner. Suppose DD f

represents the Hamiltonian structure of the originalsystem so that we can write

ŽN .w xdH fnŽN .w xf sDD f . 10Ž .tdf

Then, it follows that, we can write

dH ŽNq1.nŽN .D G sDD D G ,Ž . Ž .Nq1 Nq1t

d D GŽ .Nq1

dH ŽNq1.nŽNq1.w xor, G sDD G , 11Ž .td G

where

ŽNq1. y1 y1w xDD G sD DD D G D . 12Ž . Ž .Nq1 Nq1 Nq1

We note here that the new Hamiltonian structurewould have an opposite Grassmann parity from theold one, simply because of the delta functions in-

Žvolving fermionic coordinates. Such Hamiltonianstructures are known as anti-brackets or Buttin

w x .brackets 19–21 . This is consistent with the changeof the Grassmann parity for the Hamiltonians that wehave already noted. The recursion operators for thetwo systems are similarly related and, without goinginto details, we simply note here that

ŽNq1. y1 ŽN .w xR G sD R D G D . 13Ž . Ž .Nq1 Nq1 Nq1

The Lax description for the two systems are alsosimply related. For example, we know that the Laxdescription for the KdV hierarchy is given in termsof the Lax operator of the form

LsE 2 qu ,

Ewhere E represents . It follows, then, that the LaxE x

operator

LsE 2 q DF , 14Ž . Ž .

would describe the Ns1 supersymmetric KdV-Bhierarchy through the same normal Lax representa-tion,

E L3r2s L, L .Ž .q

E t


Similarly, the Ns1 supersymmetric KdV equa-tion can be described either by a standard representa-

w xtion with the Lax operator 9–11

LsE 2 qD F ,1

w xor by a Lax operator 22,23

LsEqDy1F ,1

with the non-standard Lax representation

E L3s L, L .Ž .G1

E t

Correspondingly, the Ns2 supersymmetric B-ex-tension of this system can also have a standard aswell as a non-standard representation through theLax operators

Lstd sE 2 qD D G , Lnstd sEqDy1 D G .Ž . Ž .1 2 1 2

15Ž .

As opposed to these Lax operators it is alsopossible to define the Lax operator on the Ns2superspace. Indeed the method of supercomplexifica-

w xtion 24 provides a much more general procedurefor obtaining the B-extensions and applied to thismodel, it provides the Lax operator

Lsc sEqDy1 D G yDy1 G Dy1 ,Ž .1 2 1 x 2

which would describe the system with a non-stan-dard Lax representation where

L3r2 sE 3 q3D E D G y3D D G , 16Ž . Ž .G1 1 2 1 2 x

The conserved Hamiltonians of the system can beŽobtained from the super residues namely, the coeffi-

y1 .cient of D in the expansion of any of these three1

Lax operators. Thus, for example, the Hamiltoniansof the system can be written in terms of the non-standard Lax operator as

2 ny1nstdH s dx du du sRes LŽ .Hn 1 2

s dx du du h , 17Ž .H 1 2 n

and the first two nontrivial charges of the series havethe forms

H s dx du du G D G ,Ž .H3 1 2 1 x

H s dx du du G D GŽ .H5 1 2 1 x x x

q4G D G D D G .Ž . Ž .1 x 1 2

It is worth noting here that the Ns2 supersym-Ž Ž ..metric B-extension Eq. 6 has yet another Lax

w xrepresentation 25 . Namely, consider the Lax opera-tor

LsD qEy1D GyGD Ey1 . 18Ž .1 2 2

Then, it is straight forward to check that the non-standard Lax equation

E L6s L, L , 19Ž . Ž .G1

E t

Ž .gives the Ns2 equation of Eq. 6 . However, thisLax operator, surprisingly, does not yield any of theconserved charges of the system. This is indeed apuzzling feature which deserves further study.

4. Non-local charges

Let us now concentrate on the Ns2 model of theŽ Ž ..previous section Eq. 6 for concreteness, although

the features we are going to discuss are quite gen-eral. We note that although the Hamiltonians for thissystem are fermionic, as we have discussed, there arealso the following bosonic charges which can beexplicitly checked to be conserved, namely,

H̃ s dx du du G ,H1 1 2

˜ 2H s dx du du G ,H2 2

1 3H̃ s dx du du G yG D D G . 20Ž . Ž .Ž .H3 1 2 1 23

At first sight, the existence of such charges withopposite Grassmann parity would seem surprisingand, in fact, the existence of such a charge wasalready noted earlier in connection with the Ns1supersymmetric KdV-B system and was termed Aex-


w xoticB 12 . In what follows, we would try to clarifythe origin of such charges and show that, in suchsystems, there are, at least, two infinite sets of

Ž .conserved non-local charges and may be more andthat these AexoticB charges are part of an infinitehierarchy of non-local charges.

To begin with, let us recall that supersymmetricintegrable systems, in general, possess non-local

w xcharges 22,23,26–28 . However, in the study ofdispersionless limits of B-extended models, it was

w xalready noted 29 that there are two sets of con-served, non-local charges present. In fact, a little bitof analysis shows that the B-extensions of integrablemodels will always have at least two infinite sets ofnon-local conserved charges. For example, in theNs2 supersymmetric KdV-B hierarchy, let us notethat the charges

2 ny1y1 nstdH s dx du du D sRes LŽ .H ž /n 1 2 2

s dx du du Dy1 h , 21Ž .Ž .H 1 2 2 n

will be conserved simply because these correspondto the conserved charges of the original system andthe original equations are unmodified by this exten-sion. Such a hierarchy of charges will always be

Ž Ž .present. In the spirit of Eq. 9 , these charges wouldbe obtained from the u independent part of the2

.densities. They are manifestly non-local and, conse-quently, are not invariant under the Ns2 extendedsupersymmetry. Let us also note that, by definition,

2 ny1y1 nstddx du du E D D sRes L s0.Ž .H ž /1 2 1 2

There is also a second set of non-local charges whichone can construct. Namely, let us evaluate the squareroot of the Lax operator for the non-standard repre-sentation. Conventionally, the super residues of theodd powers of the square root of the Lax operatorgives rise to conserved non-local charges in super-symmetric integrable models. As we will now show,the system under study presents a novel feature and,consequently, leads to new charges. Let us note thatsince both D and D satisfy1 2

D2 sEsD2 ,1 2

it follows that the general form of the square rootcan be determined to have the form

1r2nstd y1L saD qbD q2a D GŽ . Ž .1 2 2

y a Ey1D D G qbG Dy1Ž .Ž .1 2 1

q a D G yb D GŽ . Ž .Ž 2 1

yb Dy1 G2 Ey1Ž . .2

21 y1q a E D D GŽ .ž 1 22

qb Ey1D D G2 Dy3 q PPP , 22Ž .Ž . /1 2 1

where the constant parameters a and b are con-strained to satisfy a 2 qb 2 s1. While non-localcharges have been constructed earlier from squareŽ . w xand quartic roots 22,23,27,28 , here we have thenovel feature that there is a one parameter family ofsquare roots of the Lax operator. We think thisfeature would exist in extended supersymmetricmodels with N)1. In fact, let us note that for as1and bs0, this square root coincides with what has

w xbeen calculated earlier 22,23,28 . But, this is, infact, the more general form of the square root with aricher structure.

From the structure of this general square root, letus note that we can construct conserved charges bytaking the AsResB of odd powers of the square rootand would, in general, give non-local charges. Infact, let us note that the first few of these charges

Ž .have the forms dzsdx du du1 2

1r2nstddz sRes L syb dz G ,Ž .H H3a3r2nstd 2dz sRes L sy dz G ,Ž .H H2

5r2 1nstd 3dz sRes L s dz 2b G yG D D GŽ . Ž .Ž .H H 1 23

qb Dy1 G D GŽ .Ž .Ž .2 x 1

a 31 y1y D D GŽ .ž 1 232

q2 Dy1 D GŽ .ŽŽ 1 2

= D D G . 23Ž . Ž .. . /1 2

Thus, we see that the AsResB of the odd powers ofthe square root leads to conserved charges which are

Ža combination of new non-local charges the first.few of which are local as well as old ones of the


Ž .form Eq. 20 . In fact, if we neglect the old non-localcharges in these expressions, we see that the oneparameter family of charges really leads to twodistinct sets of conserved charges. Thus, for exam-

Ž .ple, when as1 and, therefore, bs0 , the non-lo-cal charges coincide with what has been obtained

w xearlier 22,23,27,28 . However, when bs1, we havea new set of non-local conserved charges for thesystem. Thus, we conclude that, in this Ns2 super-symmetric model, we have, in fact, three sets ofconserved non-local charges. Furthermore, we nowrecognize that the three AexoticB charges belong tothis hierarchy of non-local charges and can only be

Žobtained if we take the general square root. In otherwords, the first few members of the non-local hierar-chy of charges is really local, even though higherorder ones are truly non-local. We have explicitlyverified with REDUCE that there are no more local

Ž . .AexoticB bosonic conserved charges present.To complete the story of the AexoticB charges, let

us look at the simpler system of Ns1 supersym-metric KdV-B hierarchy. Here the Lax operator, as

Ž .we have seen in Eq. 14 , has the form

LsE 2 q DF .Ž .The fermionic Hamiltonians of the system are givenby

H s dx du Res LŽ2 ny1.r2 s dx du h ,Ž .H Hn n

ns1,2, PPP , 24Ž .and the first set of non-local charges are given by

y1 Ž2 ny1.r2H s dx du D Res LŽ .Ž .Hn

s dx du Dy1 h . 25Ž .Ž .H n

Since, in this case, we have only one fermioniccoordinate, the quartic root is without any arbitraryparameter and the residues of the odd powers of itgive rise to a linear combination of new non-local

Ž .conserved charges and the ones in Eq. 24 . Ignoringthese old charges, we can write the new set ofnon-local charges to be coming from

˜ Ž2 ny1.r4H s dx du Res L , 26Ž . Ž .Hn

These are bosonic charges and explicitly, the firstfew of them have the form

1H̃ s dx du F ,H1 2

1 2H̃ s dx du D F s0,Ž .H2 4

1H̃ s dx du F DF ,Ž .H3 4

3 3H̃ s dx du F D F ,Ž .H4 8

21 5H̃ s dx du F D F q2 DF . 27Ž . Ž . Ž .Ž .H5 8

˜Of these, only H was found earlier and termed3w xAexoticB 12 . We see that it belongs to a hierarchy

of non-local charges, the first four nontrivial ones ofŽwhich are, in fact, local. Although these charges

should, in general, be non-local, we suspect that, inthis particular case, namely, sKdV-B, this new set ofcharges is indeed local. This follows from an analy-sis of the structure of the charges in the dispersion-

w x .less limit 29 . However, this is not a general feature.

5. Alternate description

As we had noted earlier, the conventional B-ex-tension cannot be applied to equations where thetime evolution of the dynamical variable is not aspace derivative. Furthermore, even when the B-ex-tension exists, it gives non-local equations if appliedmore than once. In this section, we will describevery briefly, an alternate extension which does notsuffer from this problem. Let us consider an inte-grable system of the form

w xf sB f . 28Ž .t

If we now define a superfield

G x ,u , PPP ,u sfqu f , 29Ž . Ž .1 Nq1 Nq1

then, we note that the new superfield depends on oneextra fermionic coordinate and has the same Grass-mann parity as the original variable f. If we nowdefine a dynamical system described by

w xG sB G . 30Ž .t


then, this system would be integrable. We note thatthe u independent part of this equation wouldNq1

Ž .correspond to Eq. 27 , the original equation, so thatit provides an alternate description of the B-exten-

w x Ž w x.sion. In fact, when we can write B f s A f ,x

the two descriptions would be equivalent and willmap into each other under

G™ D G . 31Ž . Ž .Nq1

All the discussions of the earlier sections can becarried out in this framework as well. However, theadvantage of such a description may lie in the fact

Ž .that the B-extended equation, in Eq. 29 , is exactlylike the original equation independent of whether theright hand side is a total derivative or not. Conse-quently, we can apply B-extension to any givenequation and as many times without running into the

Žproblems of non-locality. In simple terms, the newvariables in the alternate representation are more

.local than the older ones. As a result, systems, suchas the NLS and the AKNS hierarchies, which were

Žthought not to have a local B-extension in the. w xstandard approach 30,31 , can actually have one in

this alternate description.

6. Conclusion

In this Letter, we have studied systematically thegeneral features of B-extension of any given inte-grable system. We have brought out general featuressuch as the Hamiltonians, Hamiltonian structures andrecursion operators. We have clarified the origin ofAexoticB charges in such models and have identifiedthem as belonging to an infinite set of non-localcharges. We have shown that, in such models, therenaturally exist, at least, two infinite sets of non-localcharges and, for N)1 supersymmetry, even more.We have explicitly shown that the Ns2 supersym-metric KdV-B hierarchy has three sets of non-localconserved quantities and have discussed their con-struction starting from the Lax operator.

Acknowledgements

A.D. acknowledges support in part by the U.S.Dept. of Energy Grant DE-FG 02-91ER40685 whileZ.P. is supported in part by the Polish KBN Grant 2P0 3B 136 16.

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New nonlocal charges in SUSY-B integrable models