N U CLEARNuclear PhysicsB 368 (1992)361—376 P H Y S I C S BNorth-Holland _________________

ClassicalN = 2 W-superalgebrasandsupersymmetricGel’fand—Dickeybrackets

JoséM. Figueroa-O’Farrill * andEduardoRamos* *

Instituut L’oor TheoretischeFysica, UnicersiteitLeuien, Celestijnenlaan200 D, B-3001 Hererlee,Belgium

Received11 April 1991(Revised6 September1991)

Acceptedfor publication 6 September1991

We constructan infinite seriesof classicalN = 2 W-superalgebrasasa reductionof a recentlyconstructedsupersymmetricversionof the Gel’fand—Dickey bracketsfor the generalizedsuperKdV hierarchies.This is achievedvia a supersymmetricMiura transformationwhich yields, asaby-product,classicalfree-field realizations.

1. Introduction

W-algebras[1,2] play an increasingrole in two-dimensional conformal fieldtheory and in string theory. In the former context it has been shown that theyappearas the symmetry [3] algebrasof statisticalmechanicssystemsat criticality;and in the latter case,their minimal modelscan be usedfor the constructionofclassicalvacua.Furthermore,unexpectedrelationshipshavebeenrecentlyunveiledbetweenW-algebrasandnon-critical strings[4] via their matrix model formulation,where the “string equation”correspondsto generalizedKdV flows with specificboundaryconditions[51.This observationallows one to translatecomputationsofcorrelation functions in non-critical string theory to more manageablealgebraiccomputationsin terms of the associatedLax system, without having to appealdirectly to the matrix model formulation. This has a varietyof advantages,not theleastof which is that it allows for generalizations— onevery importantgeneraliza-tion being theoneto non-critical superstrings,wherea direct treatmentin termsof(super)matrixmodelsis lacking[6].

W-superalgebrasappearnaturally in the contextof supersymmetricTodafieldtheory [7] as the algebrasof conservedcurrents. In this paper we presentasystematictreatmentof N= 2 W-superalgebrasbasedon the formalism of super-symmetric KdV hierarchies,which seemsespeciallywell suitedfor theapplicationsto superstringtheory.

* E-mail: jmf@iks.kuleuven.ac.beandfgbdall(ãblekulll.BITNET

Presentaddress:PhysikalischesInstitut der Universität Bonn, Germany.** E-mail: fgbdaO6@blekulll.BITNETand fgbda06@cc1.kuleuven.ac.be

0550-3213/92/$05.OO© 1992 — Elsevier SciencePublishersB.V. All rights reserved

362 J.M. Figueroa-O‘Farrill, E. Ramos/ W-superalgebras

The formalism of supersymmetricLax operatorsis, however,not as developedas its bosonic counterpart,where the relationshipwith W-algebrasis well under-stood. Classical W-algebras,in fact, first appearedin the context of scalarLaxoperatorsfor the generalizedKdV hierarchies[8], where they arise naturally astheir “secondhamiltonianstructure”. Indeed,the nth-order KdV hierarchy(for

n > 2) is hamiltonianwith respectto a classicalversion of W,~,generalizingthewell-known fact [9] that the KdV equationis hamiltonianwith respectto a classical

version of the Virasoro algebra. Recently we [10] have defined a hamiltonianstructurein the spaceof supersymmetricLax operatorsyielding a supersymmetricversion of the Gel’fand—Dickey bracketsfor the generalizedKdV hierarchies.Inthis paperwe construct reductionsof this hamiltonianstructureyielding N = 2W-superalgebras.

In the non-supersymmetriccase,severalreductions of the secondGel’fand—Dickey bracketare known to exist. In fact, it follows from the work of Drinfel’d

and Sokolov, that the secondGel’fand—Dickeybracketis the naturalhamiltonianstructurefor an integrablesystemof evolutionequationsassociatedto the generallinear algebra.Reductionsof this bracket are related to someof its subalgebras;notably the A~,B~,and C~series.The former series is obtained by demandingthat a particular coefficient in the Lax operatorvanish:whereasthe latter seriesareobtainedby imposingthat the Lax operatorhavedefinite symmetrypropertiesundera naturalinvolution in the spaceof (pseudo)differentialoperators.All thesereductionsyield extensionsof the Virasoroalgebrawhich aregeneratedby a finitenumberof primary fields of conformal weights equal to the exponentsof therelevant Lie algebra.For the A~seriesthis is nothing but the W~algebrasofFateevandLykyanov.

In thispaperweconstructthe reductionof thesupersymmetricGel’fand—Dickeybracketswhich is analogousto the one yielding W~in the non-supersymmetriccase.Wefind that this only makessensefor supersymmetricLax operatorsof oddorder, since for evenorder the constraintis not secondclassand thus doesnotdefine a symplectic submanifold.Other reductions analogousto the ones con-nectedto the B,1 and C,~seriesin the non-supersymmetriccaseare also possible

and yield N= 1 W-superalgebras.This requires different techniquesin order toprove the Jacobi identity of the induced bracketsand thus has been left for aseparatepublication [11].

The plan of this paper is the following. In sect. 2 we describe the basicformalism associatedto the spaceof supersymmetricLax operatorsandsummarizethe resultsof ref. [10]. In sect.3 we work out explicitly the simplestexampleof ourreductionsyielding the N= 2 superVirasoroalgebraitself. In sect. 4 we proceedto define the general reduction. There we prove that the reducedbracketsareindeed Poissonbrackets and that they define W-superalgebrascontaining theN = 2 superVirasoro algebraas a subalgebra.Finally sect.5 offers someconclud-ing remarks.

f.M. Figueroa-O’Farrill, E. Ramos/ W-superalgebras 363

2. Supersymmetric Lax operators

In this sectionwe set up the formalism and the notation for the rest of thepaper. It is our purposeto define Poissonbracketsin certain spacesof (super)differential operators.Thesespacesare infinite-dimensionalmanifolds and theirrigorousgeometrictreatmentis beyondour scope.Fortunatelythesespacescanbeendowedwith a “formal” geometrywhich is sufficient for the rigoroustreatmentof Poissonbracketsand their flows. As usual, this formal geometryconsistsin thealgebraizationof the necessarygeometricnotions.Thereforewe proceedto definealgebraicallythe ingredientsneededto definePoissonstructures:functions,vectorfields, andone-forms.

We will considerthe spaceof differential operatorson a (1 1) superspacewithcoordinates(x, 6). Theseoperatorsarepolynomialsin the supercovariantderiva-

tive D = 85 + 68 whosecoefficients are superfields.The supercovariantderivativeobeys D2 = 0. A supersymmetricLax operatorhas the form L = D’1 + U~

1D~1

+ ... + U1~and is homogeneousunder the usual ~2 grading; that is U~ n + i

(mod 2). We will define Poissonbracketson functionsof the form

F[L] =ff(U), (2.1)

where f(U) is a homogeneous(under the Z2 grading)differential polynomial ofthe U and fB is definedas follows: if Liç = u1 + Ov,, and f(U) = a(u, v) + 6b(u,v),then fB f(U) = f b(u, v), wherethe precisemeaningof integrationwill dependonthe context.It denotesintegrationover the realline if we takenthe u1 andv1 to berapidly decreasingfunctions; integration over one period if we take them to beperiodicfunctions;or, moreabstractly,a linearmapannihilatingderivativessothatwe can “integrate by parts”. It is worth remarking that whereasdifferentialpolynomials can be multiplied, this does not induce a multiplication on thefunctions we are considering.Thus the Poissonbracketswill not enjoy the usualderivationproperty.This, however,doesnot affect the formalism.

Vector fields are parametrizedby infinitesimal deformationsL i—’ L + EA whereA = EA1D’ is a homogeneousdifferential operator of order at most n — 1. Wedenote the spaceof such operatorsby S,~.We do not demandthat A havethesameparity as L sincewe canhaveeitheroddor evenflows. To such an operatorA E S,~we associateavector field DA as follows. If F = JB f is a function then

dDAF~—F[L +�A]

dE

n--i 0O

(_1)IAI+nf ~ E(_l)(IAI+niA~L~ (2.2)Bk010 OUk

364 f.M Figueroa-O’Farril4 E. Ramos/ W-superalgebras

with ~~A’1= D1Uk and the samefor ~ Integratingby partswe can write this as

n—i

D4F=(_1)~~f ZAk—, (2.3)

Bk0 tS(Jk

where the Euler variationalderivativeis given by

= ~ ~ (2.4)SUk 1=0

Sincevectorfields areparametrizedby S,, it is naturalto think of one-formsasparametrizedby the dual space to S,~.This turns out to be given by superpseudo-differentialoperators[121(S~I’DOs)with the dual pairing given by theAdler supertraceto be definedbelow. We introducea formal inverse D — to Dand define S~I’DOsas formal Laurent series in D’ whose coefficients aredifferential polynomials in the Liç. The multiplication of S~I’DOsis given by thefollowing compositionlaw (for any k ~ 7/):

= £ {k~~](—1) (k_l)~Ei]Dk_~ (2.5)

where the superbinomialcoefficientsaregiven by

0 fori<0or(k,i)~(0,1) (mod2),[ k ] = [k/2] (2.6)k—i ([(ki)/2]) fori~0and(k,i)~(0,1) (mod2).

Given a S1I~DOP= Ep1D’ we define its super-residueas sres P=p1 and its(Adler) supertraceas Str P = lB sresP. One can show [12] that the super-residueof a gradedcommutator is a perfect derivative so that its supertracevanishes:Str [P, Q] = 0, for [P, Q] PQ — (— 1) QP. This then definesa supersym-metric bilinear form on SIIIDOs: Str (pQ)(j)~PIIQ~Str (QP). If P is anyS1IIDO we defineits differential part P~as the part of P which is polynomial inD (including free terms) and its “integral” part P_ as simply ~ — P~.It then

follows that Str P~Q~=0 for any two S~I’DOs.Let us then defineone-formsas the spaceS~’of “integral” S~I’DOsof the form

X= E~ii~D~’’ Xk, whosepairing with avector field D4, with A = ~AkDk, isgiven by

(DA, X) (1)L4H~XI+n+i Str(AX) = ( 1)IAI+flf ~(_1)kAkXk, (2.7)Bk = 0

f.M Figueroa-O’Farrill, E. Ramos/ W-superalgebras 365

which is non-degenerate.The choiceof signs hasbeenmadeto avoid undesirablesignslateron. Given a function F = lB f we defineits gradientdF by (DA, dF) =

DAF whence,comparingwith (2.3), yields

n—idF= ~ (l)kD_k_i (2.8)

k=0 vUk

It is familiar from classicalmechanicsthat to everyfunction f onecan associatea hamiltonian vector field in such a way that ~fg = {f, g}. The hamiltonianvector field is obtainedfrom df by a tensor12 mapping one-formsto vectorfields, so that the Poissonbracketof two functionsis given by {f, g} = (Xf, dg) =

(Q(df), dg),with (,) beingthe naturalpairing betweenvectorfields andone-forms.In local coordinates,12 coincideswith the fundamentalPoissonbrackets.In other

words, the map 12 carriesthe sameinformation as the Poissonbrackets.In analogy, we define Poissonbracketson the spaceof supersymmetricLax

operatorsby defininga map J : S~’ Sn in sucha way that the Poissonbracketoftwo functions F andG is givenby

{F,G}=DJ(dF)G=(DJ(dF),dG)=(—1)~~’~ Str(J(dF)dG).

(2.9)

The map 12 in this caseis dF —~ DJ(dF); although, becauseof the ratherformalnatureof our geometricalsetting, it is the mapJ that plays themore relevantrole.Demandingthat thePoissonbracketsdefinedby J obey the correct(anti)symmetrypropertiesandthe Jacobiidentity imposesstrongrestrictionson the allowedmapsJ. Maps obeyingtheseconditionsare often called “hamiltonian”. A hamiltonianmapJ wasconstructedin ref. [101via a supersymmetricMiura transformation.Webriefly review this construction.

Let us factorize L = (D — cIn)(D — ~n—i)~ . .(D — iI~). This defines the U, asdifferential polynomialsin the cPa.We definethe fundamentalPoissonbracketsofthe as follows. If we let X=(x, 6) and Y=(y, w), then

{~1(x),cP~(Y)}= (—1)’ S1~D~(X— Y), (2.10)

where ~(X — Y) = 8(x — y) (0 —

Let F = lB f and G = lB g be two functions with f and g differentialpolynomials in the Uk. Via the Miura transformationwe can think of them as

differential polynomials in the ‘I~.Their Poissonbracketscan then be read offfrom (2.10),

n ~5f ~g{F, G} = f ~ (-1)’ D~ ~. (2.11)

B11 ~ on,,

366 f.M. Figueroa-O’Farrill, E. Ramos/ W-superalgebras

We must first calculate of/0cI~.The variation of F can be computedin twoways:

0 n—iOF=f ~0~I~ =f ~ 8U~~= (—l)~”~ Str(OL dF). (2.12)

B1,j ~ Bk0 OUk

Defining V, D — ~,, onecomputes

OL= —~V11...V1+10V~...V1. (2.13)

Insertingthis into (2.12) onefinds after some reorderinginside the integrals

sres(V11...V1dF ~ (2.14)

Plugging this into (2.11) and after some standardmanipulations[13,10] withS~I’DOs,one finds

{F, G} = (—1) F1±IG1~Str[L(dF L) + dG — (L dF)±LdG], (2.15)

which is the supersymmetricanalogof the secondGel’fand—Dickey bracket [8].From this and(2.9), the map J : S~’~ Sn canbe readoff,

J(X) = (LX)~L—L(XL)±=L(XL)—(LX)L. (2.16)

The fundamentalPoissonbracketsof the Uk can,in turn, be readoff from the mapJ as follows.

Supposewe write the fundamentalPoissonbracketsin the following form:

{t~.(x),U~(Y)}=f2,~0(X— Y), (2.17)

where the 12,~are differential operatorsat the point X. If F = lB ~~UA1, G =

lB LJUJBJ are homogeneouslinear functions, then their Poissonbracket can becomputedfrom (2.17)yielding

{F, G} = ~ ( — 1)(IFI +ixn+l)+1+JJADB (2.18)B

On the other hand we can computethis from (2.9). The gradientof F is givensimply by dF = Ek(— 1)kD~~~Ak,and similarly for G with Bk replacing Ak.Since J(dF) E Sn is linear in dF, it definesdifferential operatorsJ,~by J(dF) =

f.M. Figueroa-O’Farril4 E. Ramos/ W-superalgebras 367

L~(—1)lF(~h)+t1~(J,1(_1)JA1)Dt. Pluggingthis into (2.9) onereadilyobtains

(F, G) = (~1)n ~ f (_1FI+~t+~1+n1A1J~7B1, (2.19)B

where the adjoint K * of a homogeneousdifferential operatorK is definedby

l~(K4)B=(_1)tK~I~hll~A(K*B)for A and B any two homogeneoussuperfields.Comparing(2.19)with (2.18) we get

= (— 1)’©~J~=J~1. (2.20)

From (2.15) it follows immediately that the Q,~are at most quadratic in the U,.Thus, in terms of the U,, we obtain associative superalgebraswith quadraticrelations.Theseare the supersymmetricanaloguesof the W-algebrasassociatedtoGL(n) in the Drinfel’d—Sokolov scheme[141;and thus we tentatively call themSWGL(n).

Drinfel’d and Sokolov constructedmany other classicalW-algebrasas reduc-tions of WGL(n). In particular,W~was obtainedby imposing that the next-to-highestorder term in the nth-orderLax operatorvanish.Onecan easily computethat this is a second-classconstraint so that the space of such Lax operatorsinherits a well-defined Poissonbracket. In our casewe see a sharpdistinctionbetweenthe evenandodd-n cases.It is easyto seethat if n is even,the constraintUni = 0 is actually first class; in fact, if andonly if n is even

12n1n1 vanishes.This meansthat the inducedPoissonbracket in the constrainedsubmanifoldis notwell definedunlesswe impose an additional constraintthat has a non-degeneratePoissonbracket with U,, — On the other hand,if n is odd, then the constraintU,, — = 0 is secondclass and the induced Poisson bracket is well defined andyields, as we will see,classicalW-superalgebrasextendingthe N = 2 superconfor-mal algebra.

3. An explicit example: from SWGL(3) to N= 2 superVirasoro

In this sectionwe computeexplicitly the simplestof the seriesof algebrasweconstruct:the reductionof SWGL(3)by the constraint~‘2 = 0. It is easily seenthatfU

2(X), U2(Y)} = —0’(X—Y) whence it defines a second-classconstraint.It iscustomarywhen facedwith second-classconstraintsto write down immediatelytheDirac bracket.However, since our geometricsettingis rather formal, this proce-dure is not guaranteedto be consistent.We will, aided by analogywith thesymplecticsetting, derive the induced Poissonbracket from first principles. Forthiswe digressmomentarilyto review the finite-dimensionalsymplecticcase.

368 f.M Figueroa-O’Farrill, E. Ramos/ W-superalgebras

SupposeM is a finite-dimensionalsymplecticmanifold andM0 is a submanifoldgiven by somesecond-classconstraints.Becausethe constraintsare secondclass,the symplecticform 12 on M restrictsto a symplecticform on M0: to evaluateit ontwo vector fields on M0 we simply view the vector fields as vector fields on M

tangentto M0 and apply the symplectic form on M. This is possible since onM0 ~ M thereis a canonicalmapembeddingthe tangentbundleTM0 of M0 intothe one of M. On the otherhand, in order to write down the Poissonbracketof

two functionson M0 we need to first define their associatedhamiltonianvectorfields andthis requiresknowledgeof their gradients.The gradientof a function isa one-form and, unlike vectorfields, thereis no canonicalway to view one-formson M0 as one-forms on M. In fact, such a choice is equivalentto a choice ofcomplementto TM0 in TM on M0 in otherwords, to a choiceof normal bundle.Once a complementis chosen,T * M0 c T * M can be identified with thoseone-formsannihilatingthe normalvectors.In generalthereis no naturalchoicefor thenormalbundle,but in the symplecticcasethereis. Wedefine thenormal bundleasthe symplecticcomplementof TM0. This is to be comparedwith the riemanniansetting.If we hada metricon M thenwe could alwaysdefinethe normal bundleasthose tangentvectorson M perpendicularto M0. In the symplecticcase,this onlyworks if the symplecticform is non-degeneratewhenrestrictedto M0 sothat thereareno vectorsthat areboth tangentto M0 and symplecticallyperpendicularto it.Luckily when M0 is definedby second-classconstraintsthis is the case.Then todefinethe Poissonbracketsof two functions on M0, we first extendthe functionsto M, we computetheir gradientsin M andfrom them their hamiltonianvectorfields. We finally apply the symplecticform to their perpendicularprojections.Theresultingobject is a function on M0 which canbe shownto be independentof theextension.In symbols,if we write TM = TM0 ® TMI±on M0, and ~ = ~ + ~‘ asthe associateddecomposition of a vector, then the induced bracket of twofunctions f and g on M0 is given by

(f, g)0=fl(~f-~f’, ~g~~g’)’ (3.1)

where and ~g are, respectively,the hamiltonianvectorfields associatedto theextensionsof f and g to functionson M. A naturalbasisfor the normal bundletoM0 is given by the hamiltonianvectorfields associatedto the constraintsdefiningM0. Computing the abovebracket in that basisyields the familiar formula for theDirac bracket.

Anotherequivalentway of defining the one-formson M0 is to say that they arethe one-forms on M which get mappedto TM0 under the map T*M —‘ TMinducedby the symplecticform. This interpretationis bettersuitedfor our formalgeometricsetting,sincewe do not quite havea symplecticform, but rathera mapJtaking (formal) one-formsto (formal) vectorfields.

With thesewordsbehindus, we now resumethe caseat hand.Let M denotethe

f.M. Figueroa-O’Farrill, E. Ramos/ W-superalgebras 369

spaceof supersymmetricLax operatorsof order 3 and M0 the submanifolddefined

by U~=0. If L =D3 + U

1D + U0 is a point in M0, and X=D~X0+D2X

1 +

D —

3X2 is a one-form on M, we will seehow its coefficientsare related so that

J(X) = (LX)~L— L(XL)~ is a vector tangent to M0 at L. We will see thatdemandingthat the coefficient of D

2 in J(X) vanishes,fixes X2 in terms of X0,

X1, andthe U,-. A slightly tediouscomputationyields

+ [U1’X1— 2U0X1 —Xi’ — X[” — U1X~]D

+ ( — 1) [x~” — X~5~+ (U

0X1)’ — (U0 X0)” + (U1 X0)

— U~X1~”+ UJX~+ U1(U~X11)’— UQX(~’]. (3.2)

Demanding that the coefficient of D2 vanishesyields X

2 = X(’ + X1’ — U~X0.Notice that the equation actually involves X2’ but that we could solve for X2without havingto integratesincethewhole equationwasa perfectderivative.Thisis no accident,andwe will seethat this will alwaysbe the case.Whetheror not thisis important is not clear since in the rest of the expressionfor J(X), X, onlyappearsthrough its derivatives, but this may just be an accident of low n.Substitutingfor X2 in (3.2) yieldsafter somemore algebra

J(X) = {[—D4 — U~D+ D2UI] ‘X() — [2D3 + 2LJ~— U

1’] .x}D

—1) x([ —UQD2 — D2UO] x

0 + [D4 + U

1D2 + DUO] .x

1}, (3.3)

which define the operatorsJ,1 and,hence,by (2.20), the operatorsQ~1appearingin the fundamentalPoissonbracketsof the U,:

= U0D2 + D2U

0,

12~~=-D4-U

1D2-DU,),

= -D4 + D2U~- U11D,

Q~=—2D3—2U

0+U1’. (3.4)

Notice that fl~)= ~ Q~= ~ and Q~’~= as expectedfrom the gradingof the U,.

370 J.M Figueroa-O‘Earn/I, E. Ramos/ W-superalgebras

If wenow defineT U0 — ~-U~’,~fl U1 we find the classicalversionof the N = 2superVirasoroalgebra;

{T(X), T(Y)} = [~D~ + ~TD2 + ~T’D + T”I 0(X— Y),

{T(x), ~l(Y)) = [_/D2 + ~/‘D — il”] 0(X— Y),

{l(X),/(Y)} = —[2D3+27/] 0(X—Y). (3.5)

The first equationidentifiesT asthe superenergy—momentumtensor,whereasthe

secondidentifies / asan N= 1 superconformalprimary of weight 1.

4. The general reduction: from SWGL~2k+ lHto N= 2 W~uperalgebras

In this sectionwe describethe generalreductionof SWGL(2k+ 1) obtainedbyimposingthe constraint U

2k = 0 on the spaceM of supersymmetricLax operatorsof the form L =D

21~+ .... A quick calculationshowsthat (U2k(X), U2k(Y)} =

—D O(X— Y). Since the operator —D is formally invertible, the constraint issecondclass and we expect that the constrainedsubmanifold M0 inherits a

well-definedPoissonbracket from M. We will first show that this in indeedthecaseandthat the resulting fundamentalPoissonbracketdefinesaW-superalgebraextendingthe N = 2 superVirasoroalgebra.

Let L =D2k~+ U

2k1D + . . . U0 be a point in the constrainedsubmani-fold M0. We want to defineone-formson M0. According to the discussionin sect.

3, we define them as one-forms on M which get mappedunder J to tangentvectors to M0 at L, i.e. to differential operatorsof order at most 2k — 1. LetX= ED~X1 be a one-formon M. UnderJ it getsmappedto the tangentvectorX —, J(X) = (LX)~L— L(XL)~,which is, in general,a differential operator oforder at most2k. X will be a one-form on M0 if we demandthat thecoefficient of

J(X) multiplying D2k vanish.The coefficient of J(X) of order 2k is given by the

super-residueof J(X)D21’~~. Now, using that J(X) is also given by L(XL)_we find

J(X)D2~t = sres[L(XL)D_2k_i — (LX)LD2k_u]

= sres[L(XL) D2”~ — LX( LD_2k_t)+ 1’

wherewe haveusedthat sresAB = sresA_B~=sres AB~for any two S1I’DOs

f.M. Eiguenoa-O’Farnill, E. Ramos/ W-superalgebras 371

A and B. Using further that (LD21’t)~= 1, we find

0 = sresJ(X)D21’~

= sres[L(XL)_D2k_i —LX]

ix~

=sres[(—1) XL—LX

= —sres[L, X], (4.1)

wherefor the next-to-laststepwe havesimply noticedthat sresL(XLLD2k~ =

sresD2~’1(XLLD2~ andwe havethen computedit. Thereforeone-formsonM

0 are one-formsX on M which satisfy the additional conditionthat sres[L, Xl= 0. A closerlook at this relationshows that this determinesX2k in termsof theother coefficientsand of the U,. Indeed,

0=sres[L, X]

2k—i= sresLD

2k~X2k+ sres L, ~ D’

1X1

1=0

2k—i

whence

2k—i

X~k=(—1)~sresL, ~ D’~X1 . (4.2)1=1

Notice that although this is an equationfor X2’k, we can actually solve for X2ksince the r.h.s.,being the super-residueof a gradedcommutator,is also a perfectderivative.Thus in solving for X2k we simply drop the “constantof integration”.

We now write down the fundamentalPoissonbracketsinduced on M0 fromthose of M. Let us write J(X)= E11(_1)(Ix ‘~

11JXD’ which defines thedifferential operators~ Demandingthat the top termvanishesbecomesa linearrelation EIJ

2k1X1 = 0, from which it follows that

2k —

X2k = ~2k,2k E J2k.IXJ. (4.3)I I)

Pluggingthis backinto J(X) we find

J( X) = 2~ 1 ~— 1)~~ + i)(,+ i)jXD~ (4.4)i .1 1)

372 1M. Eiguenoa-O’Earnill, E. Ramos/ W-supenalgebras

with

~i) = ~ij — ~i,2k~2k,2k~2k,j, (4.5)

which implies that the new fundamentalPoissonbracket is none other than the

“naïve” Dirac bracket. In the usual geometricsetting the Jacobi identity for theDirac bracketis automaticsincethe Jacobiidentity is equivalentto the symplecticform being closedon M0 and this follows trivially from the fact that the inducedsymplecticform on the constrainedsubmanifoldis the pull-back of the one on M

via the embeddingM0 ‘-~ M. In our formal setting,however,we cannotappealtoomuch to the geometry unlessthe necessarygeometricfacts have been suitablyalgebraizedandthis, althoughinterestingandpossible,is beyondthe scopeof thispaper.

Therefore to prove that the induced bracket does indeed obey the Jacobiidentity we will proceedas in ref. [13] andmakesure that the proof in ref. [101ofthe analogousfact for the unconstrainedmanifold goes throughessentiallyunmod-ified. Since the proof invoked the Miura transformation,we needto understandthe constraint U2k = 0 in this context. A simple computation shows that U2k =

E1(—1)”~I~.Thus, in termsof the fundamentalfields 1,, the constrainedsubmani-fold is a “hyperplane”.Our first task is to definethe gradientof a function on thishyperplane.As before,we define it in such a waythat the associatedhamiltonianvector field is tangentto the hyperplane.For this we have to find out what thehamiltonianvectorfield of a functionlooks like. Let F = lB f be any function.Wewrite its gradientas a (2k + 1)-tupledF = (of/OCD,). Wealso write vectorfields as(2k + 1)-tuples ~ = (~‘),with pairing given by

(~dF)f~’~-~-~ (4.6)

Now if F = lB f andG = lB g are any two functions, their Poissonbracket is given

by ~ G, where ~F is the hamiltonianvector field associatedto F. Comparing(4.6) with (2.11) we read off the coefficients of the hamiltonian vector field~=(—1)’(0f/8c~1)’. We now demandthat ~F be tangent to the constrainedsubmanifold; in other words, that the hamiltonian vector field preservestheconstraint.This translatesinto

~ ( - 1)’~P~=0 ~ ~ ( - 1)’~=0I I

(4.7)

f.M. Eiguenoa-O’Eannill,E. Ramos/ W-supenalgebras 373

Observing that the proof in ref. [10] was essentially combinatorial starting fromthe expression(2.14)for the gradientof a function, we can appealto it providedthat (2.14) is consistentwith the constraint;that is, providedthat it obey (4.7). Thiswill provide a constraint on dF which will turn out to be precisely (4.1). Weproceedto provethis now.

Replacingthe supercovariantderivativeof Of/011 by the graded commutator

[V1-~8f/8~1],and usingthat for any S~I’DOP its super-residue can be written as

sresP=(P)+=(_1)t~(V,P)+, (4.8)

we find

~ (~)= ~( — 1) +1~V1(sresV11 ... V~dF~,...

—(—1)~’(sresV11...V1 dFV,,...V1±~)V1]

= ~(—1) FI+i)~[V((V1 V1 dFV,,...V1+~)V1)+

-(V1(V1~1...V1dFV~...V1+1))+V1I.

Notice that we can drop the — subscripts because, if we replace them by +, wecould drop theouter + subscripts and the terms cancel pairwise. Dropping the —

subscripts,we find

= E(—1 ~~°‘[V1(V1_1...v,dFV~...V1)+

= E(—1) F~t[(V1...V, dFV,,...V1+1)_V1

Since this is a zeroth-order differential operator we can project to its + part forfree andusingthe expressionsabovefor the super-residuewe obtain

~ (~)‘= E(—1) ~°‘[sres V1...V1~

_(_1)+1sresV~_i...VidFVn...V1]

=(_1)IF~sresLdF_sresdFL

= —sres[dF, L].

374 JM Figuenoa-O’Fannill, E. Ramos/ W-supenalgebnas

Therefore as long as the gradient of a function is defined so that it obeyssres[dF, L] = 0, the proof in ref. [10] goesthroughandthe reducedsupersymmet-

nc Gel’fand—Dickey bracketsobeythe Jacobiidentity.Finally we provethat the fundamentalPoissonbrackets(4.5) on the constrained

submanifold contain the N = 2 superVirasoro algebra as a subalgebra.This isdone by computation.After reducingthe supersymmetricLax operator,the firsttwo fields appearingare U2k_i and U2k_2 which have weights 1 and 3/2,respectively. It can be shown quite generally that, before reduction, the fundamen-tal Poissonbracketsof U2k, U2k_~,and U2k2 close amongthemselves.It then

follows that after reductionthe inducedPoissonbracketsof U2k_ and U2k2 willstill close into them, so that thesetwo fields generatea subalgebra.We will seethat this is (after somefield redefinition)the N = 2 superVirasorosubalgebra.

There are two (at least ) different ways we could compute the induced

fuundamental Poisson brackets of ~j2k_i and U2k_2. We could compute thefundamentalPoissonbrackets fl,~ for i, j = 2k, 2k — 1, 2k — 2 and then useformula (4.5) or, equivalently,computethe mappingJ(X) for X a one-formon theconstrainedsubmanifold,i.e. after imposing sres[L, Xl = 0; or we could computethe bracketsdirectly via the Miura transformationfrom the fundamentalbracketsof the basic fields D, after taking into accountthe reduction to the hyperplane~,(— 1)’~= 0. Both ways are, of course,equivalentand equallycomputationallyinvolved. We chooseto presentthe former computationsinceat least it has theadvantageof keepingthings manifestly in terms of the U,.; whereasthe Miura

calculationwould, at the end of the day, involve recombiningthe basic fields n,backinto the U~:a slightly tedioustask.

Thus let L =D21’~+ U

2k ~D21~’+ U

2k_2D21~2+ ... andlet X= ~

1DJ_iX~.be a one-form. Because we are interested in fundamentalPoisson brackets

involving only U2k_i and U2k_2 it suffices to extract the terms in J(X) whichdependon X2k_i and X2~_2.Part of this dependencewill comefrom imposingsres[L, XI = 0. Computingthis we find

sres[L, X]=O~X2k=kX~k_t+kX~_2—X2k_2U2ki+..., (4.9)

where the ... stand for terms involving XJ<2k_2. We now computeJ(X) andkeep only those terms involving X2k — and X2k —2~After a little algebraonefinds

~2k-i,2k-i = —k(k + 1)D3+ U~k_i—

.i,f, 1’,n4..j..fT~2k—2,2k—i —

2”-i, ~) 2k—i 2k—2’

~2k—i,2k—2 = — ~k(k + 1)D4+ D2U

2k_! — U2k_2D,

I — rr r~2 2,’i

~2k—2,2k—2 — ~-12k—2

f.M Figueroa-O’Fannil4 E. Ramos/ W-supenalgebnas 375

from wherewe can deducethe fundamentalPoissonbrackets ñ via eq. (2.20).

Defining T U2k_2— ~U2’k_I and di = U2k_i, we find that they obey

{T(X), T(Y)} = [~k(k + 1)D5 + 4TD2 + ~T’D + T”] 0(X— Y),

{T(X), di(Y)} = [—JD2+ ~di’D— ~di”] 5(X— Y),

= —[k(k+1)D3+2T] 0(X—Y), (4.10)

which, onceagain, is the classicalN = 2 superVirasoro algebra.One can checkthat for the specialcasek = 1 we do, in fact, recovereq. (3.5).

5. Conclusions

In this paper we have obtained an infinite seriesof N= 2 extendedsuperalgebrasby reductionof a recentlyconstructedPoissonstructureon the spaceofsupersymmetricLax operatorsof odd order.This reductionis the supersymmetricanalogue of the Drinfel’d—Sokolov reduction of the second Gel’fand—Dickeybracket which is associatedto the A~seriesof classicalLie algebrasand whichyields the W~algebras of Fateev and Lykyanov.

For the Lax operatorof order 2k + 1, the spectrumof the resulting algebraconsistsof (N = 1) superfieldsU, for j = 0, 1, . . . ,2k — 1, of naïveweights k —

(j — 1)/2. In particular,U2k_l and U2k_2 havebeenshown to generatea N = 2

super Virasoro algebra. It is then a natural conjecture to expect that eachremainingfield U, for j odd gives riseto an N= 2 superconformalprimary field U,.obtainedby deforming U,. via the additionof differential polynomials in the U,. >~.

Similarly the remainingU,. with j evengive rise to their partners.Wehavecheckedthis explicitly for the simplestexample, but we have not proven it in general.Nevertheless,we haveno doubt of the validity of this statementandwe hope toreturn to this point in a future paper. It should be remarkedthat the bosonicspectrumof thesealgebrasdoesnot in generalagreewith the spectrumof thenonsupersymmetricW,, algebras.Thereforeone does not recover thesealgebras

upon truncation,like thesimple caseof SWGL(2)might suggest.It seemsplausible— like the readercancheck explicitly for SWGL(3) — that further reductionsmightyield conformal algebraswith the spectrumof W,,, but thesereductionsdo notseemnatural in this context, since they involve constrainingpart of the bosonicspectrumas well.

As mentionedin sect. 1, this seriesof algebrashasbeenalso obtainedin ref. [7]from the Sl(n + 1 n) Toda field theory, where it appearsas the algebra ofconservedquantities.It is a very interestingopenquestionto obtain this algebraashamiltonian reductionof the correspondingaffine algebra,thus recoveringthe

376 f.M. Figuenoa-O‘Fannill, E. Ramos/ W-superalgebnas

connectionbetweenLie superalgebrasandW-superalgebrasexistingin the bosoniccase.Work on this is in progess.

We are grateful to Leonid Dickey for his correspondenceand for sendingussomeof his unpublishedlecture notes as well as excerptsfrom his forthcomingbook. It is a pleasureto acknowledgeveryfruitful discussionswith JavierMas andhis careful readingof the manuscript.

References

[11A.B. Zamolodchikov,Theor. Mat. Phys.65 (1986)1205[2] V.A. FateevandS.L. Lykyanov, Int. J. Mod. Phys.A3 (1988)507[3] VS. Dotsenko,NucI. Phys.B235 [FS11] (1984)54[4] J. Goeree,NucI. Phys.B358 (1991)737[5] M. Douglas,Phys.Lett. B248 (1990) 176[6] L. Alvarez-Gauméand J.L. Manes,Supermatrixmodels,CERN-TH. 6067/91[7] S. Komata, K. Mohri and H. Nohara,Classicaland quantumextendedsuperconformalalgebra,

Tokyo preprint UT-Komaba90-21 (July 1990)[8] I.M. Gel’fand and L.A. Dickey, A family of hamiltonian structuresconnectedwith integrable

nonlineardifferential equations,preprint 136, 1PM AN SSSR,Moscow (1978)[9] F. Magri, J. Math. Phys.19 (1978) 1156

[10] J.M. Figueroa-O’Farrill andE. Ramos,Phys.Lett. B262 (1991)265[11] J.M. Figueroa-O’Farrill and E. Ramos, ClassicalW-superalgebrasfrom hamiltonian reduction,

preprint-KUL-TF-91/ 14 (March 1991)[12] Yu.I. Manin andA.O. Radul, Commun.Math.Phys.98 (1985)65[13] L.A. Dickey, Commun. Math.Phys.87 (1982) 127[14] V.G. Drinfel’d and V.V. Soko!ov, J. Soy. Math. 30 (1984) 1975