Volume 262, number 2,3 PHYSICS LETTERS B 20 June 1991

W-superalgebras from supersymmetric Lax operators

Jos6 M. Figueroa-O'Farrill 1,2 and Eduardo Ramos 3

Instituut voor Theoretische Fysica, Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Heverlee, Belgium

Received 11 March 1991

We construct an infinite series of classical W-superalgebras via a supersymmetric Miura transformation. These algebras appear as the "second hamiltonian structure" in the space of Lax operators for supersymmetric generalized KdV hierarchies.

I. Introduction

W-algebras [ 1,2 ] are playing an increasing role in two-dimensional conformal field theory and in string theory. In particular, unexpected relationships have been unveiled between W-algebras and noncritical strings [ 3 ] via their matrix model formulation. This seems to lie at the heart of the relationship between string theory and classical integrable systems [ 4 ].

Classical W-algebras, in fact, first appeared in the context o f integrable systems, where they arise natu- rally as the "second hamiltonian structure" o f the generalized KdV hierarchies [ 5 ]. Indeed, the nth or- der KdV hierarchy (for n > 2 ) is hamiltonian with respect to a classical version o f W,, generalizing the well-known fact that the KdV equation is hamilto- nian with respect to a classical version of the Vira- soro algebra.

The fact that the Poisson bracket defined by the second hamiltonian structure does indeed obey the Jacobi identity was originally notoriously cumber- some to prove: most proofs being more or less refined versions of a direct computat ion of the Jacobi iden- tity. In ref. [ 6 ] Kupershmidt and Wilson showed that the second hamiltonian structure is induced from a much simpler one - namely that o f" f ree fields" - via

E-mail addresses: fgbda 11 @blekul 11 .BITNET, fgbda 11 @cc 1. kuleuven.ac.be.

2 Address after October 1991 ; Physikalisches Institut der Univ- ersit~it Bonn, W-5300 Bonn, FRG.

3 E-mail addresses: fgbda06@blekulll.BITNET, fgbda06@ cc I .kuleuven.ac.be.

a generalization o f the celebrated Miura transforma- tion of the KdV theory. But their proof was still rather involved and it was not until Dickey found a short and elementary proof [7] of the Kupershmidt - Wilson theorem that the conceptual and computa- tional advantage (in this context) of the Miura trans- formation became clear. From the point o f view of conformal field theory the Miura transformation is particularly useful [2], since the free field realiza- tions implicit in this method lend themselves natu- rally to the quantization of these algebras and to the study of their representations.

It is certainly an interesting question to ask if this method survives supersymmetrization, yielding W- superalgebras [ 8,10 ]. Partial results have already been obtained by the Komaba group [ 11,12 ] who, in the context of super-Toda field theory, has explicitly constructed examples o f W-superalgebras from su- persymmetric Miura transformations. This seems to suggest an affirmative answer to this question.

Indeed in this letter we start such a supersymme- trization program: we show that the space of super- symmetric differential operators of the KdV type is endowed with a bi-hamiltonian structure analogous to the Gel ' fand-Dickey brackets o f the bosonic case and that the second bracket is induced, via a super- symmetric version of the Miura transformation, by that o f free superfields.

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2. Bi-hamiltonian structure of generalized KdV hierarchies

In this section we briefly review the bi-hamiltonian structure of the generalized KdV hierarchies. We do this to set up the notation and to motivate the objects we will use in the main body of this paper.

The KdV hierarchy is the isospectral problem as- sociated to the differential (Lax) operator L = 0 2 + u. The KdV flows

OL [A, ,L] (2.1) 0t~

are hamiltonian with respect to two Poisson brack- ets. The KdV equation itseff f t=u" +6uu' can be written as fi = {H1, u}l = {//2, u}2, with

HI= f [½(U')2-U3], H 2 = - I ½uZ , (2.2)

and Poisson brackets

{u(x), u(y)}, = O 8 ( x - y ) , (2.3)

{u(x) , u(Y)}2 = ( 0 3 + 2 u O + 2 O u ) g ( x - Y ) , (2.4)

where, here and in the sequel, the differential opera- tors appearing on the right-hand side of the brackets are taken at the point x. Moreover, these brackets are coordinated: any linear combination is again a Pois- son bracket. This gives the KdV hierarchy its bi-ham- iltonian structure.

The generalized nth-order (n > 2) KdV hierarchy is defined as the isospectral problem of the differen- tial operator L = 0 n + un_ ~0"- 1 + ... + Uo. Again one can show that its flows are bi-hamiltonian, where the Poisson brackets are now given by expressions of the form

{l/i(X), u j (y ) } -~-Jijt~( x - Y ) , (2.5)

where for one of the structures the Ju are differential operators whose coefficients are nonlinear expres- sions in the u~. In particular, the reduction to the case u,_ ~ = 0 yields classical versions of the Wn algebras.

It turns out that in terms of L, rather than in terms of its coefficients, these brackets can be written in an elegant and compact form. But for this we need to develop some differential calculus with the L's.

We will define the Poisson brackets on functionals of the form

F [ L ] = I f ( u , ) , (2.6)

where f (u i ) is a differential polynomial in the ui (i.e., a polynomial in the ui and their derivatives). The precise meaning of integration depends on the con- text: it denotes integration over the real line if we take the ui to be rapidly decreasing functions; integration over one period if we take the u~ to be periodic func- tions; or, more abstractly, a linear map annihilating derivatives so that we can "integrate by parts" #l

It is familiar from classical mechanics that to every functionfone can associate a hamiltonian vector field ~f in such a way that ~fg= ~, g}, and where ~f= 12df with t2 a linear map from one-forms to vector fields. In local coordinates, ~2 coincides with the fundamen- tal Poisson brackets. This formalism is the most suit- able to define Poisson brackets in the infinite-dimen- sional space of the operators L.

Vector fields are parametrized by infinitesimal de- formations L - L+eA where A= ZatO t is a differen- tial operator of order at most n - 1. We denote the space of such operators by Rn. To such an operator AeR, we associate a vector field 0A as follows. If F = f f is a functional then

OAF= d F [ L + e A ] ,=o

ol = ~ a~O (2.7) l=o i=o 0u}" "

Integrating by parts we can write this as

f , , - i 8f (2.8) OAF= ,=~oat~-~ut'

where the Euler variational derivative is given by

8 ~o ( - ~ 0 (2.9) 5 u l - 0) ~u}i w

Since vector fields are parametrized by Rn, it is natural to think of one-forms as parametrized by the dual space to Rn. This turns out to be given by pseudo- differential operators (5VDOs) with the dual pairing

#l Notice that whereas differential polynomials can be multi- plied, this does not induce a multiplication on the functionals we are considering. Thus the Poisson brackets will not enjoy the usual derivation property. This, however, does not affect the formalism.

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given by the Adler trace to be defined below. We in- troduce a formal inverse 0-~ of 0 and define ~tDOs as formal Laurent series in O- 1 whose coefficients are differential polynomials in the u~. The multiplication of ~DOs is given by the following composition law (for any keZ):

okf=fok'~- i=, ~ (~) f(i)Ok-i'

(~ ) = k ( k - l ) . . . ( k - i + l ) (2.10)

i!

extending the usual Leibniz rule for positive k. Given a 7JDO P = Zpi0 * we define its residue as res P=p_ and its (Adler) trace as Tr P = fres P. One can show that the residue of a commutator is a total derivative so that its trace vanishes. This justifies the name. The Adler trace defines a symmetric bilinear form on the space of 7~)Os given by Tr(PQ), thus allowing us to identify the dual space to Rn as the space R* of ~ttDOs of the form v v , - ~ a--k-- A=~k=O- IXk. In fact, i fA=Zal0/ , then

Tr(AX) = aiXi, (2.11) i=0

which is clearly nondegenerate. We will then think of one-forms as elements of R*~ and their pairing with vector fields as given by

(OA, X) = T r ( A X ) . (2.12)

In particular, if we define dF, for F = f f by (OA, dF) = OAF, it follows from (2.8) that

n--I ~ f dF= Y~ 0 - k - ~ - (2.13)

k=o 8Uk "

Finally we can define the Poisson bracket of two functionals. The map Q taking one-forms to vector fields is induced from a linear map J: R*~Rn by ~(X) = Ojtx). The Poisson bracket is then defined by

{F, G}= (g2(dF), d G ) = T r ( J ( d F ) d G ) . (2.14)

Demanding that this be antisymmetric and obey the Jacobi identity, constrains the allowed maps J. Gel'fand and Dickey constructed two such maps:

J , (X) = [X, L ] + , (2.15)

J:(X) = ( L X ) + L - L ( X L ) + , (2.16)

where for P any 7090, P+ denotes its differential part. The first and second Gel'fand-Dickey brackets are then

{F, G}l =Tr ( [dF, L ] + d G ) , (2.17)

{F, G}2 =Tr ( (L dE) + L d G - L ( d F L ) + d G ) . (2.18)

In order to obtain the fundamental Poisson brackets of the u~ it is clearly sufficient to consider linear func- tionals F [L ] = fZXiui.

The first bracket has a natural interpretation as the Kirillov-Kostant Poisson bracket on a coadjoint or- bit of the Volterra group. A natural interpretation of the second bracket was provided by Kupershmidt and Wilson. Let us factorize the Lax operator L as L=(O--1)n)(O--Un_l)...(O--U1). This allows us to write the u~ as the Miura transforms of the vj. Kupershmidt and Wilson showed that if the funda- mental Poisson brackets of the vj are given by those of "free fields"

{v,, v j }=-~ i jO~(x -y ) , (2.19)

the induced brackets on the ui coincide with (2.18 ). Notice that whereas the fundamental Poisson brack- ets of the ui obviously obey the Jacobi identity when defined in terms of the vj, the fact that they close among the u~ is far from obvious. On the other hand closure is obvious from (2.18 ), but the Jacobi iden- tity is not.

We end with an example. Factorizing the KdV op- erator L = (O+v)(O-v)=O2+u yields u= - v ' - v 2 which is the classical analog of the Coulomb gas re- alization of the Virasoro algebra. Indeed, computing {u(x), u(y)} from v reproduces (2.4).

3. Supersymmetric Miura transformation

In this section we exhibit a bi-hamiltonian struc- ture on the space of supersymmetric Lax operators and we prove that the second structure is induced, via a supersymmetric Miura transformation, from free superfields. Our proof follows very closely that of Dickey [ 7 ] for the bosonic case.

We will consider differential operators on a ( 1 I 1 ) superspace with coordinates (x, 0). These operators are polynomials in the supercovariant derivative D = 0o+ 00 whose coefficients are superfields. A super-

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symmetric Lax operator has the form L = D ' + U,_ t D ' - t + ... + U0 which is homogeneous under the usual 712 grading; that is, I f d =-n+i (mod2) . In analogy to the bosonic case discussed in the previous section, we will define Poisson brackets on function- als of the form

F [ L ] = ~ f ( U ) , (3.1) B

where f ( U ) is a homogeneous (under the 77 2 grad- ing) differential polynomial of the U and f8 is de- fined as follows: if Ui=ui+Ov~, and f ( U ) = a ( u , v) + Ob(u, v), then f B f ( U ) =fb(u , v).

Vector fields are parametrized by the space S, of differential operators of order at most n - 1. We do not demand that the A have the same parity as L since we can have either odd or even flows. IfA = ZA~D ~ is one such operator and if F[L ] = fB f i s a functional, then the vector field D~ defined by A is given by

I " - ' ~f (3.2) D A F = ( - 1 ) IAl+n ~=oAkSUk ' B

where the variational derivative is defined by

3Uk -- ,=oZ ( - -1 ) ~uk~'+m+L)/2D~ OU~i---"-T, (3.3)

with U~ d =D~Uk. As before, to define the one-forms we introduce

super-pseudo-differential operators (S 7tDOs) [ 13 ]. These are defined as formal Laurent series in D - where the composition law is now given by

£r l( Dkq~= i=oLk- id --l )i~l(k-i)qb[i]Dk-i' (3.4)

where the superbinomial coefficients are given by

for i<0 or (k,i)=-(O, 1) (rood 2),

(t½k]) = \ [ ½ ( k - i ) ]

for i>~Oand (k, i )~ (0, 1) (rood 2). (3.5)

Given a S 7530 P = Zp~D ~ we define its super-residue as sresP=p_~ and its (Adler) supertrace as S t rP

= fBsres P. Again it can be shown that the supertrace vanishes on graded commutators: Str[P, Q] =0, for [ P, Q] = P Q - ( - 1 ) wnQi Qp. This then defines a su- persymmetric bilinear form on ST~DOs: Str(PQ) = ( - 1 )wuQiStr(QP).

We define one-forms as the space S*~ of ST'DOs of the form X= Z'--lD--k-~Xk,k=O whose pairing with a vector field DA, A = ZAkD k is given by

( DA, X) = ( - 1 )tAI + txl +,+ lstr (AX)

= ( - 1 ) tAl+" ( - 1)kAkXk, (3.6) k=0

B

which is again nondegenerate. The choice of signs has been made to avoid undesirable signs later on. Given a functional F = f a f w e define its gradient dFby (DA, dF) = DAF whence, comparing with (3.2), yields

n--I ~ f d F = ~ ( - -1)kD - k - ' k=o ~ G " (3.7)

Finally, to define a Poisson bracket we need a map J: S* ~ S, in such a way that the Poisson bracket of two functionals F and G is given by (cf. (2.14) )

{ F, G} =DstdF> G= (Dj<~F), dG)

=(- l ) tS l+ lF t+f r l+ '+ lS t r ( J (dF)dG) . (3.8)

Instead of giving the map J directly, we will follow a constructive approach starting from a supersymmet- ric version of the Miura transformation and, only at the end, write down the resulting map J.

Let us factorize L = ( D - q~,) ( D - q~,_ ~ )... (D -@~). This defines the U~ as differential polyno- mials in the q~j. We define the fundamental Poisson brackets of the q~j as follows. I f we let X= (x, 0) and Y= (y, o9), then

{qbi(X), @ j ( Y ) } = ( - 1 ) ' r q D O ( X - Y ) , (3.9)

where 6 ( X - Y ) = 8 ( x - y ) ( O - o g ) . Notice that the signs in the Poisson brackets are alternating. This turns out to be crucial: whereas any choice of signs would yield a Poisson bracket, it is only this choice which will give a closed algebra among the Uk. In fact, writing the Uk as the Miura transforms of the q~i al- lows us to write their Poisson brackets from (3.9). It is clear that these Poisson brackets will obey the Jacobi identity, but it is far from obvious that they will close in terms of the Uk. We will now show, how-

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ever, that this is indeed the case, yielding a super- symmetric version of the Kupershmidt -Wilson theorem.

Let F = f s f a n d G f a g be two functionals w i t h f a n d g differential polynomials in the Uk. Via the Miura transformation we can think of them as differential polynomials in the ~g. Their Poisson brackets can then be read o f f f rom (3.9):

310, B

cyclicity o f the supertrace in the first set of terms we can write

{F, G}= ( - 1 ) ( IFI+IGI)(n+I)

X ~ S t r [ ( - 1 ) (IFI+IGI+I)(i+I) i = 1

X (Vi_ l ...dF...Vi) + V i_ 1 " 'dG"'Vi

- ( - 1 )(IFI +IGI+ l) i

X (Vi...dF... Vi+ t ) + Vi...dG...Vi+ l ] • ( 3.16 )

We must first calculate 8f /6~ , . The variation of F can be computed in two ways:

8 F = i= t~ ~l~)i -~ii = k~= ~ Uk ~ Uk B B

= ( - 1 )lFI +n+ IStr(SL d F ) . (3.11 )

Defining Vi--- D - ~ , one computes

8 L = - ~ V,...Vi+l 8q~iVi_l...Vl • (3.12) i = l

Inserting this into (3.11 ) one finds after some reor- dering inside the integrals

8f - - ( - - 1 ) IFI ( n + i + 1 ) + i

Xsres(Vi_l...Vl dF~n...Vi+ l ) . (3.13)

Plugging this into (3.10), replacing the supercovar- iant derivative o f 8 g / 8 ~ by the graded commuta tor [V~, 8 g / 8 ~ ], and using that for any S 7590 P its super- residue can be written as

s r e s P = ( P _ V i ) + = ( - l ) l P l + l ( V i P _ ) + , (3.14)

with P_ = P - P+ we obtain

{ F , G } = ( - 1 ) (IFl+lal)(n+l) ~ ( - 1 ) (IFl+lal+l)i i = 1

× Str [Vi( (V~_ ~...dF...Vi+ ~ ) _ V~) + Vi_ t...dG...Vi+ L

- - ( V i ( V i _ I . . . d F . . . V i + 1 ) - ) + ViVi-l...dG...Vi+ 1 ] •

(3.15)

Now notice that we can drop the - subscripts since if we replace them with a + we can then drop the outer + ' s and the terms cancel pairwise. Using graded

Notice that all terms cancel pairwise except for the first term of the first sum and the last term of the sec- ond sum, yielding after some reordering,

{F, G}= ( - 1 )IFI+IGI +n

× S t r [ L ( d F L )+ d G - ( L d F ) + L d G ] , (3.17)

which is the supersymmetric analog of the second Ger fand-Dickey bracket (2.18). I f F and G are lin- ear functionals then from this bracket one can re- cover the fundamental Poisson brackets of the Ui. It is clear from the expression that they close quadrati- cally among themselves giving classical versions of W- superalgebras.

Using (3.8) we can read off the map J: S*--, Sn

J ( X ) = ( L X ) + L - L ( X L ) + . (3.18)

As in the bosonic case the first hamiltonian structure can be obtained by deforming the second structure. I f we change Uo ~ Uo+2, where 121---n ( m o d e 2 ) , then J becomes

J x ( X ) = J ( X ) - 2 [ ( X L ) + - ( - 1 ) ,+, lx l ( L X ) + ] .

(3.19)

The term proportional to 2 gives us the first hamil- tonian structure. Several remarks are in order. When n is even, the first hamiltonian structure coincides with the Kiri l lov-Kostant structure on the coadjoint orbit of L under the super-Volterra group and the two brackets are coordinated in the usual way. On the other hand, when n is odd, the bracket has an odd grading (since 2 is an anticommuting parameter) and the connection with the Kiri l lov-Kostant structure seems to be missing. In this case the Poisson brackets can still be considered coordinated if we allow linear

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combinat ions with an t icommut ing parameters . [ 11,12 ] since Cartan matrices o f simple superalge- bras are generally indefinite.

4. Conclusion and outlook

In this letter we have constructed a b i -hami l tonian structure in the space of supersymmetr ic Lax opera- tors. In the process we have generated an infini te se- ries of classical W-superalgebras generated by super- fields of "weights" {½, 1, 3, ..., ½n} and we have provided free field real izat ions via a supersymmetr ic Miura t ransformat ion. This series is the supersym- metr ic analog of the G L ( n ) series o f Dr infe l 'd and Sokolov. The first nontr ivial algebra in the series ( n = 2 ) can be checked to be superconformal: al- though the basic fields do not include the super-en- e r g y - m o m e n t u m tensor, this can be buil t out of dif- ferential polynomials in them.

In analogy with the bosonic case, we also expect that reduct ion schemes related to s imple superalge- bras will generalize the results ob ta ined by Drinfe l 'd and Sokolov. In part icular , some of these reduct ions [ 14 ] will yield W-superalgebras extending the super- Virasoro algebra, thus making them more interest ing from a physical point of view. For example, one can check that the reduced Lax opera tor L = D 3 + T = ( D + ~ ) D ( D - ~ ) yields the super-Virasoro algebra.

Final ly let us comment on the free-field realiza- t ions associated to the Miura t ransformations. As we ment ioned in section 3, the free fields have to be cho- sen with al ternating signs in their fundamenta l brackets. This means that in the quant isa t ion of these algebras we will be generically forced to consider in- definite Fock spaces, unless, as in the example com- mented upon above, reduct ion leaves only fields with the same sign. The emergence of the indefini te Fock spaces is a lready present in the Toda approach

Acknowledgement

We are grateful to Leonid Dickey for his corre- spondence and for sending us his unpubl ished notes [ 15 ] which have been inst rumental in setting up the right framework. We also thank Stany Schrans for a careful reading of the manuscript .

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