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On the classical WN(l) algebras

arXiv:hep-th/9111046v1 22 Nov 1991LAVAL PHY-27/91On the classical WN(l) algebrasDidier A. Depireux and Pierre MathieuD´epartement de Physique,Universit´e LavalQu´ebec, Canada, G1K 7P4We analyze the WN (l) algebras according to their conjectured realization as the secondHamiltonian structure of the integrable hierarchy resulting from the interchange of x andt in the lth flow of the sl(N) KdV hierarchy. The W4(3) algebra is derived explicitly alongthese lines, thus providing further support for the conjecture.

This algebra is found tobe equivalent to that obtained by the method of Hamiltonian reduction. Furthermore, itstwisted version reproduces the algebra associated to a certain non-principal embedding ofsl(2) into sl(4), or equivalently, the u(2) quasi-superconformal algebra.

General aspectsof the WN (l) algebras are also presented.We point out in particular that the x ↔tinterchange approach of the WN (l) algebra appears straightforward only when N and l arecoprime.Nov. 1991

1. IntroductionGiven a hierarchy of two-dimensional evolution equations, one can interchange the roleof the independent variables x and t for any member of the hierarchy, thus producing anew integrable hierarchy of evolution equations.

Furthermore, such an interchange in twodifferent equations of a given hierarchy produces two new independent hierarchies. Theidea of interchanging the roles of x and t for integrable equations goes back to [1].

There itwas shown with simple examples (KdV, mKdV, sine-Gordon) that the resulting hierarchyis also integrable and bi-Hamiltonian. For the KdV and mKdV cases, Kupershmidt [2]has obtained the same conclusion independently by considering a more general transfor-mation (GL(2)) of the independent variables.

The Boussinesq equation in x–evolution hasbeen constructed in [3] and was shown to be bi-Hamiltonian and in fact equivalent to thefractional KdV hierarchy of [4].These statements have now been fully generalized in [5][6] where various extensions ofthe Drinfeld-Sokolov [7] approach to KdV-type equations have been worked out, includinghierarchies obtained by x ↔t interchange.The interest for these new hierarchies obtained by x ↔t interchange is motivated bythe potential conformal character of their second Hamiltonian structure. Since the Hamil-tonian character of an evolution equation depends crucially upon which of the variables ischosen for the evolution, one expects a priori that the interchange of x and t will substan-tially modify the Hamiltonian properties (i.e.

the form of the Poisson brackets) of a givenintegrable system. New conformal Poisson algebras would yield new extended conformalalgebras upon quantization, in the same way as the second Hamiltonian structure of theusual generalized KdV hierarchies are related to W-algebras [8].

Recall that a Hamilto-nian structure is said to be conformal if it contains the Poisson bracket characterizing thesecond Hamiltonian structure of the KdV equation, namely{u(x) , u(y)} = (∂3 + 4u∂+ 2ux) δ(x −y) ,(1.1)where the fields on the RHS are evaluated at x and ∂≡∂x. The Fourier transform ofthis bracket yields the Virasoro algebra realized in terms of Poisson brackets [9], hence thename conformal.

Actually, the classical W3(2) algebra of Polyakov [10], has been shown[3] to be equivalent to the second Hamiltonian structure of the Boussinesq (or sl(3) KdV)hierarchy with x and t interchanged at the level of the Boussinesq equation itself.1

It was conjectured in [3] that the second Hamiltonian structure of the hierarchy ob-tained by interchanging x and t in the lth flow of the sl(N) KdV hierarchy (with l < N)would produce a new conformal algebra, called WN (l) (with WN (1) ≡WN). See also [5][6]for similar results and conjectures.

We will call this new hierarchy the sl(N)l hierarchy.It is simple to show that the restriction l < N is necessary to produce a conformalWN (l) algebra. The sl(N) KdV hierarchy is characterized by the scalar Lax operatorL = ∂N + u2∂N−2 + .

. .

+ uN . (1.2)The evolution equations are∂tlL = [(Ll/N)+ , L] ,(1.3)where + denotes the differential part of a pseudo-differential operator.

Interchanging xand t in the lth flow amounts to interchanging tl and t1 = x. In the newly producedhierarchy tl plays the role of the space variable.

In the normalization where dim(x) = −1,tl has dimensions −l. To renormalize the dimensions of the new space variable to −1, onehas to divide all the dimensions by l. Thus the new algebra will contain bosonic fields offractional dimension (multiples of 1/l).

In order to be conformal, it must contain a fieldof spin 2 (after dimensional renormalization). Now in (1.3), the evolution of the highestspin field takes the form∂tluN = cu2(N+l−2) + .

. .,c = constant, u(i) = (∂iu) .

(1.4)As it will become clear below, the new set of independent fields required to describethe system obtained by interchange of t1 and tl can be chosen generically to includeu2, u2x, u2xx, . .

., u2(N+l−3). Since u2(i) originally had dimension 2 + i, its new dimensionwill be (2 + i)/l.

Thus the field u2(2l−2) will have new dimension 2, and it belongs to theabove sequence only if l < N.Before pursuing the discussion of the second Hamiltonian structure of the sl(N) hier-archy, one should settle the question of its integrability. This is most naturally discussedin terms of a zero curvature condition.

The usual sl(N) hierarchy can be described by thescattering problemΦtl = V (l)Φ(1.5)where Φ = (φ, φx, . .

. )T , V (l) = (L)l/N and Lφ = λφ, λ being the spectral parameter, i.e.V (l)x −V (1)tl+ [V (1), V (l)] = 0(1.6)2

with x = t1.The interchange t1 ↔tl amounts to interchange the roles of V (1) andV (l). But the point is that we stick to a zero curvature formulation, hence this operationmanifestly preserves the integrability property (this is proved rigorously in [5]).

In V (l)there is a constant piece, equal toΛ(l)N = λi0IN+j−1λIj0Ni + j = l(1.7)where Ik is the k × k unit matrix. Hence for l < N, we find i = 0 and j = l and the twomatrices entering in (1.6) are linear in λ.One strategy to obtain the second Hamiltonian structure of the sl(N)l KdV hierarchiesis the following [3].One first writes down the lth flow of the sl(N) KdV and mKdVhierarchies.

These equations are related by a Miura transformation which characterizesthe usual sl(N) hierarchy. The Miura map gives the free field representation of the classicalWN algebra, which is the second Hamiltonian structure of the sl(N) KdV hierarchy.

Let usdenote the corresponding Hamiltonian operator by P2. Similarly, denote by Θ the naturalHamiltonian operator of the sl(N) mKdV hierarchy1.

The canonical character of the Miuramap translates into the statement that [11][12]P2 = D ΘD†(1.8)where D is the Fr´echet derivative of the KdV fields with respect to the modified KdVfields. It is computed from the Miura map.

D† is its formal adjoint. Thus the lth flows ofthe sl(N) KdV and mKdV hierarchies read asutl = P2∇uH,u = (u2, u3, .

. .

, uN)T,ptl = Θ ∇pHm ,p = (p1, p2, . .

., pN−1)T ,(1.9)where ∇u = (δ/δu2, . .

., δ/δuN)T and similarly for ∇p. H is the appropriate Hamiltonianfor the lth flow and Hm is the expression of the same Hamiltonian in terms of the modifiedfields.

By interchanging t1 and tl, one gets˜utl = ˜P2∇˜u ˜H,˜ptl = ˜Θ ∇˜p ˜Hm ,(1.10)1 It is also called the mKdV first Hamiltonian structure for historical reasons (except in [7]).Also, it has lower dimensions than the second one. Its naturalness is due to the fact that thesecond Hamiltonian structure is both complicated and non-local.3

where ˜u = (˜u2, ˜u3, . .

., ˜ulN−l+1)T and ˜p = (˜p1, ˜p2, . .

., ˜plN−l)T are the new independentfields, whose number depends on l (a canonical set of independent fields will be displayedlater). We want to find ˜P2.

Notice that we know ˜H since any conserved density h for thesl(N) hierarchy satisfies ∂tlh = ∂x˜h. Thus ˜h is a conserved density of the new system.

In(1.10), ˜H is the conservation law of appropriate dimension. We also know the Miura maprelating ˜p to ˜u: it is simply a rewriting of the usual Miura map where the x-derivatives ofthe modified fields are eliminated by means of the lth sl(N) mKdV equation.

Hence wealso know ˜Hm. Now as it will be illustrated below, there is a natural way to write thefirst equation of the sl(N)l mKdV hierarchy (i.e.

a choice of new modified fields ˜p) whichmakes ˜Θ obtainable by inspection in a totally straightforward way. (It only contains ∂tand constants, which makes its Hamiltonian character manifest).

Having ˜Θ and ˜D, theFr´echet derivative of ˜u with respect to ˜p, one can reconstruct ˜P2 by˜P2 = ˜D ˜Θ ˜D†. (1.11)The Hamiltonian property of ˜P2 is thus inherited from that of ˜Θ.

˜P2 corresponds to theclassical WN (l) algebra.The advantage of this construction, apart from its conceptual simplicity, is that itgives directly the free field representation of the WN (l) algebra. A minor drawback is that˜Θ must be obtained by inspection.

Now the above procedure is totally straightforward inthe cases where N and l are coprime ((N, l) = 1). However, when such is not the case,the first sl(N)l equation appears under the form of a constrained system (see e.g.

(5.7)).We defer to another publication the detailed analysis of such cases, for which the simplestexample is W4(2).2Here we work out in details a new example of a WN (l) algebra, namely the W4(3) case.This gives the first explicit form of a WN (l) algebra for N > 3, derived from thepoint of view of integrable hierarchies. We will also check that the same algebra can beobtained directly by the method of Hamiltonian reduction and the corresponding flows canbe extracted by reduction from sl(4) self-dual Yang-Mills equations, as was the case forW3(2) .

After having worked out a new non-trivial example of WN (l) algebra, we will be inposition to present a set of general remarks concerning the structure of these algebras for(N, l) = 1, including their spin content. But before, we illustrate the method by a simpleexample.2 W4(2) is presented from the point of view of Hamiltonian reduction in [13].After fieldredefinition and twisting, it is equivalent to the particular W algebra derived in [14] by consideringthe embedding of sl(2) into sl(4) fixed by the decomposition 4 →2 + 2 of the fundamentalrepresentation.4

2. A simple example: Interchange of x and t for the usual KdV equationLet us write the KdV equation under the formut = uxxx + 6uux .

(2.1)Its two Hamiltonian structures areut = P2∇12Zu2dx(2.2a)= P1∇12Z(2u3 −ux2)dx ,(2.2b)withP2 = ∂3 + 4u∂+ 2uxandP1 = ∂,(2.3)The KdV equation in x-evolution is obtained as follows: one first introduces two newindependent fieldsv = uxandw = uxx(2.4)so that (2.1) can be rewritten asuvwx=vwut −6uv. (2.5)This is the KdV equation in x-evolution or equivalently the first flow in the sl(2)3 hierarchy.One proceeds similarly with the mKdV equationpt =pxxx −6p2px(2.6a)=Θ1∇12Z(p2x + p4)(2.6b)=Θ2∇12Zp2 ,(2.6c)withΘ1 = −∂andΘ2 = ∂3 −4∂p ∂−1p ∂.

(2.7)The Miura transformationu = px −p2(2.8)5

is a canonical map from Θ1 to P2. Indeed the Fr´echet derivative of u with respect to p is∂−2p so that D† = −∂−2p and [11]P2 = (∂−2p)(−∂)(−∂−2p) .

(2.9)Introducingq = px, r = pxx ,(2.10)one can rewrite the mKdV equations in the formpqrx=qrpt + 6p2q. (2.11)Now in order to find ˜P2, the second Hamiltonian structure for (2.5), we will need ˜Θ1,the first Hamiltonian structure for (2.11).

As already pointed out this must be found byinspection. We now show that with a simple field redefinition, this step is straightforward.The trick is to look for the field transformation which simplifies maximally the equationof the highest degree in (2.11).

Here this amounts to introducing a new variable s linearlyrelated to r such that sx = pt. Thus we choose s = r −2p3 and (2.11) becomespqsx=qs + 2p3pt.

(2.12)We want to write the RHS under the form ˜Θ1∇˜Hm. ˜Hm is the x ↔t interchanged versionofR(p4 + p2x)dx, that isR(p4 + .

. .

)dt.Hence we look for the density (p4 + . .

.) suchthat ∂x(p4 + .

. .) = ∂t(.

. .

), where in the RHS one has a usual mKdV conserved density.Dimensionally one sees that it is p2. Since ∂tp2 = ∂x(2ppxx −p2x −3p4), and ˜Hm is onlydefined up to a multiplicative constant, we choose ˜Hm as˜Hm =Z(32p4 + 12p2x −ppxx)dt =Z(−12p4 + 12q2 −ps)dt.

(2.13)Of course, using (2.12), it is simple to check explicitly that ( ˜Hm)x = 0. An even moredirect approach is the following: we know that the KdV conservation laws can be obtainedasRResLk/2dx with L = ∂2 + u.

Writing ResLk/2 in terms of the new fields, one getsdirectly the new conserved densities. For example, ResL3/2 = 18(u2 + 13uxx) so that onecan take ˜H to be 32R(u2 + 13w)dt which gives directly the above ˜Hm, using the Miura6

transformation presented below in (2.16). We now search for a matrix differential operator˜Θ1 such thatqs + 2p3pt= ˜Θ1−s −2p3q−p.

(2.14)The elements of ˜Θ1 are easily found by inspection to be˜Θ1 =010−10000−∂t. (2.15)We claim that once the fields are chosen such that the highest degree modified equationhas the form φx = ψt, ˜Θ1 is always obtained as simply as above.

Now let us work out theMiura transformation:u =px −p2 = q −p2 ,v =ux = pxx −2ppx = s + 2p3 −2pq ,w =uxx = pxxx −2p2x −2ppxx = pt −2q2 −2ps −4p4 + 6p2q . (2.16)The Fr´echet derivative of (u, v, w)T with respect to (p, q, s)T is found to be˜D =−2p106p2 −2q−2p1∂t −2s −16p3 + 12pq−4q + 6p2−2p(2.17)and˜D† =−2p6p2 −2q−∂t −2s −16p3 + 12pq1−2p−4q + 6p201−2p.

(2.18)Now it is simple to obtain ˜P2, from the matrix product (1.11). The result is [1][2][6]˜P2 =02u∂t + 2v−2u−∂t2w + 12u2∂t −2v−2w −12u2−8u∂t −4w.

(2.19)The first Hamiltonian structure can be obtained by shifting u by a constant factor, i.e.with u →u + λ, ˜P2 →˜P2 + 2λ ˜P1 where˜P1 =010−1012u0−12u−4∂t. (2.20)7

The field redefinition w →w + 6u2 transforms ˜P1 into a manifestly Hamiltonian formsimilar to (2.15). Notice that to recover the first Hamiltonian structure, one should notshift the field of highest spin in the new set of independent fields, but merely the highestspin field of the original set.

On the other hand, given ˜Θ1, one can calculate ˜Θ2 as follows[1]: the master-symmetries for (2.12) can be obtained directly from those of the mKdVequation with x and t interchanged. Let T be the analogue, for (2.12), of the first timedependent symmetry for the mKdV equation.

Then ˜Θ2 is, up to a multiplicative factor,the Lie derivative of ˜Θ1 with respect to T.Thus the operator (2.19) characterizes the second Hamiltonian structure of the sl(2)3KdV hierarchy. This operator clearly does not define a W-algebra, since it is not conformal.Indeed, by rescaling the dimensions such that dim(∂t) = 1, so that one must divide thedimension of all the fields by 3, one gets dim(u, v, w) = 2/3, 1, 4/3.

Hence there is no spin-2field.As a final remark, we stress that all the equations of the sl(2)3 hierarchy can beobtained systematically from those of the sl(2) hierarchy by interchanging tl = t3 andt1 = x at the level of the KdV equation itself. For instance, the jth flow in the ordinaryKdV hierarchy takes the formutj = fj(u, ux, uxx, uxxx, .

. .

)(2.21)The x-derivatives are eliminated by means of (2.4) and (2.1) with the result that (2.2) istransformed intoutj = gj(u, v, w, ut3, vt3, wt3, ...)(2.22)and similar expressions for vtj, wtj.3. The classical W4(3) algebra by x ↔t interchange.3.1.

Generalities on the second Hamiltonian structure of scalar Lax equations.Introduce the pseudo-differential operatorF (l) =NXk=1∂−kf (l)N−k+1(3.1)where f (l)1is fixed by the conditionRes [F (l), L] = 0 ,(3.2)8

with L the scalar Lax operator (1.1) and Res Pi ai∂i = a−1. The second Hamiltonianstructure of the sl(N) KdV hierarchy takes the form [11][7]∂tlL = (LF (l))+L −L(F (l)L)+ ,(3.3)which translates into(ui)tl = (P2)ijf (l)j.

(3.4)P2 gives the Poisson brackets of the different fields, i.e. {ui(x), uj(y)} = (P2(x))ijδ(x −y) .

(3.5)The Miura tranformation, which furnished the free field realization of this Poisson algebra,can be obtained as follows: one first factorizes L asL = (∂+ φN−1)(∂+ φN−2) . .

. (∂+ φ1)(∂+ φ0) ,(3.6)where PN−10φi = 0.

The φi’s are then expressed in terms of a set of linearly independentfields pi’s byφk =N−1Xi=1ωkipi,ω = e2iπ/N . (3.7)The Poisson structure for the pi’s is [11][7]{pi(x), pj(y)} = −1N δN−i,jδx(x −y) .

(3.8)These brackets can be diagonalized, by introducing the fieldsrj =rN2 (pj + pN−j) ,j < N2√Npj,j = N2−irN2 (pj −pN−j) ,j > N2(3.9)so that{ri(x), rj(y)} = −δijδx(x −y) . (3.10)9

3.2. Specialization to the sl(4) case.The sl(4) scalar Lax operator isL = ∂4 + u2∂2 + u3∂+ u4 .

(3.11)The components (i, j) of the corresponding operator P2 are then found to be(2, 2) :5∂3 + u2∂+ ∂u2(2, 3) :−5∂4 −2∂2u2 + 2∂u3 + u3∂(2, 4) :32∂5 + 32∂3u2 −32∂2u3 + 3∂u4 + u4∂(3, 3) :−6∂5 −2(∂3u2 + u2∂3) + (∂2u3 −u3∂2) + 2(u4∂+ ∂u4)−12(u2∂+ ∂u2)(3, 4) :2∂6 + 2∂4u2 + 32u2∂4 −2∂3u3 + 3∂2u4 −u4∂2 + 12u2∂2u2 −12u2∂u3(4, 4) :34∂7 + 34(u2∂5 + ∂5u2) + 34(u3∂4 −∂4u3) + (u4∂3 + ∂3u4)+34u2∂3u2 + u2u4∂+ ∂u2u4 −34u3∂u3 + 34(u3∂2u2 −u2∂2u3)(3.12)Now by rewriting L under the formL = (∂+ φ3)(∂+ φ2)(∂+ φ1)(∂+ φ0) ,(3.13)one can express the fields ui in terms of the φi’s and ultimately using (3.7) and (3.9), interms of the fields ri’s. The result turns out to beu2 = −12(r21 + r22 + r23) +√2r1x + r2x −√2r3x ,u3 =3√2r1xx + r2xx −1√2r3xx −32r1r1x −1√2r1xr2 + 12r1xr3 −1√2r1r2x−r2r2x −1√6r2xr3 −12r1r3x −1√2r2r3x −12r3r3x + 12r21r2 −12r2r23 ,u4 =1√2r1xxx + 12r2xxx −12r1r1xx −12√2r2r1xx −1√2r1r2xx−12r2r2xx −12r1r3xx −12√2r2r3xx + 34r1r1xr2+12√2r1r1xr3 + 14r1xr2r3 −12√2r1xr23 + 14r21r2x +12√2r1r2r2x−10

14r22r2x −12r1r2xr3 −12√2r2r2xr3 −14r2xr23 +12√2r21r3x+14r1r2r3x −12√2r1r3r3x −14r2r3r3x −12r1xr1x −1√2r1xr2x−14r2xr2x −r1xr3x −1√2r2xr3x −18r21r22 + 116r42 + 14r21r23 −18r22r23. (3.14)The first Hamiltonian structure for these modified fields is given by the Hamiltonian op-erator Θ of (3.10) (we omit the subscript 1)Θ =−∂000−∂000−∂(3.15)Again one can check explicitly the canonical character of the above Miura map by checkingdirectly the identity P2 = DΘD†.In the following we will be interested more particularly in the third flow of the sl(4)KdV hierarchy.

It can be computed fromLt =[(L3/4)+, L]where t = t3andL3/4 =∂3 + 34u2∂+ (34u3 −38u2x)+34(u4 −12u3x + 112u2xx −18u22)∂−1 + . .

. (3.16)One obtainsu2t =14u2xxx −32u3xx + 3u4x −34u2u2x,u3t =34u2xxxx −2u3xxx + 3u4xx −34u2u3x −34u2xu3,u4t =38u2xxxxx −34u3xxxx + u4xxx + 38u2u2xxx −34u2u3xx+38u2xxu3 + 34u2u4x −34u3u3x.

(3.17)These equations have the following Hamiltonian formulation:u2u3u4t= P2∇uZ(u4 −18u22)dx ,11

the Hamiltonian density being Res L3/4 up to total derivatives. We will also be interestedin the modified version of this equation.

Rewriting the above Hamiltonian in terms of themodified fields, one hasHm =Z[−14r1xr1x + 18r2xr2x −14r3xr3x−34r1r2xr3 −132(r41 + 6r21r22 −6r21r23 −r42 + 6r22r23 + r43)] ,(3.18)and the corresponding mKdV equations arer1r2r3t= −12r1xx −34r2xr3 −18r31 −38r1r22 + 38r1r23−14r2xx + 34(r1r3)x −38r21r2 + 18r32 −38r2r2312r3xx −34r1r2x + 38r21r3 −38r22r3 −18r33x.(3.19)3.3. The Hamiltonian structure of the sl(4)3 hierarchy.We now want to rewrite (3.17) and (3.19) as evolution equations with respect to x.Let us start with (3.17), and introduce the fieldsuix = vi ,uixx = wi ,i = 2, 3, 4 .

(3.20)One findsuix =vi ,vix = wi ,w2x =4u2t + 6w3 −12v4 + 3u2v2 ,w3x = −65v2t + 25u3t + 125 w4 −910v22 −910u2w2 + 310u2v3 + 310v2u3 ,w4x =3w2t −6v3t + 10u4t −6u2u2t −6u2w3 + 212 u2v4−92u22v2 + 274 v2w2 −9v2v3 + 152 u3v3 −334 w2u3 . (3.21)The x ↔t interchange version of Hm read offfrom ResL3/4 in (3.16) and reexpressed interms of the above fields is˜H =Z(u4 −18u22 −12v3 + 112w2)dt ,(3.22)and we are looking for the corresponding Hamiltonian operator ˜P2 which allows the rewrit-ing of the above system in the form (1.10).

For this we need first the modified version of(3.21) and its natural Hamiltonian structure. To write (3.19) in x-evolution we introduce12

the variables rix = si, rixx = ˜qi. However, using the hindsight gained from the study ofthe KdV and the Boussinesq cases, we choose new variables qi so as to keep the highestfield equations in the form qix = rit.

The explicit form of the qi’s can then be read offdirectly from (3.19), i.e. q1 = −12r1xx + 34r2xr3 + .

. ., and the sl(4)3 version of (3.19) isr1x = s1s1x = −2q1 −2[−34s2r3 −18r31 −38r1r22 + 38r1r23]q1x = r1t(3.23a)r2x = s2s2x = 4q2 + 4[34s1r3 + 34r1s3 −38r21r2 + 18r32 −38r2r23]q2x = r2t(3.23b)r3x = s3s3x = −2q3 −2[−34r1s2 + 38r21r3 −38r22r3 −18r33]q3x = r3t(3.23c)These flows can be written in a Hamiltonian form as follows.

The Hamiltonian can beeasily obtained from the above ˜H, where we express the KdV fields in terms of the modifiedfields, using the Miura map (3.14) and the equations (3.23) to eliminate the x-derivativeof the modified KdV fields. We get˜Hm =Z[r1q1 + r2q2 + r3q3 + 14s21 −18s22 + 14s23−132r41 −316r21r22 + 316r21r23 + 132r42 −316r22r23 −132r43] dt(3.24)We determine ˜Θ by inspection, writing (r, s, q)Tx = ˜Θ(∇r ˜Hm, ∇s ˜Hm, ∇q ˜Hm)T , with theresult:˜Θ =rsqr02−420s−24−2−6r36r36r1−6r10q00∂t∂t∂t.

(3.25)13

This operator can be further simplified by introducing the new fields ˜si = si−3δ2ir1r3,2˜r1 = r1, −4˜r2 = r2, 2˜r3 = r3, so that ˜Θ takes the form˜Θ =˜r˜sq˜r0I30˜s−I300q00I3∂t. (3.26)Since this is antisymmetric and field independent (so that the Jacobi identities are auto-matically satisfied), this operator is manifestly Hamiltonian.

Now, having ˜Θ and the Miuramap, which leads to ˜D, one can calculate ˜P2 by (1.11). The explicit form of ˜P2 obtainedthis way is not manifestly conformal.

However, after some field redefinitions given in thenext section, it can be transformed into a conformal algebra, to be presented in that samesection. Exactly the same algebra can be derived by the method of Hamiltonian reductionto which we now turn.4.

The classical W4(3) algebra by the method of Hamiltonian reduction.4.1. W4(3) by Hamiltonian reduction.In this section, we derive the W4(3) algebra by the method of Hamiltonian reduction.This method is by now standard and will not be reviewed here [15].

We start from a 1-dimconnection that depends on a coordinate t, taking values in the algebras of sl(4). In thematrix representation we constrain the (1, 4) element of the connection to be -1, i.e.A(t) =J11J12J13−1J21J22J23J24J31J32J33J34J41J42J43J44,J44 = −3Xi=1Jii .

(4.1)The form of the constraint is preserved by the gauge transformationsg−1(∂t + A(t))g = ∂t + Ag(t) ,(4.2)where g is a lower triangular matrix with 1’s on the diagonal. We cannot completely fixthis gauge invariance in a local way.

However, restricting ourselves to the subalgebra with14

g23 = 0, we can fix the gauge invariance with respect to this subalgebra; using (4.2), webring A(t) to the canonical formQ(t) =000−1A1U1Z0B1A2U20TB2A3−U1 −U2. (4.3)Using the expression of the fields appearing in Q in terms of the original currents of A(t)(which form an sl(N) current algebra), we obtain an algebra which we call the W4(3)algebra.

The superscript 3 in this context corresponds to the fact that the third upperdiagonal has been set to -1. The spin content of the algebra can be read offdirectly from(4.3) since the dimensions are constant along the diagonals and by moving from the topright corner to the bottom left corner, they increase by units of 1/3.

Since a constant hasdimension 0, we find dim(Z, Ui, Ai, Bi, T) = (2/3, 1, 4/3, 5/3, 2).After introducing T0 = T −A2Z −U 21 −U 22 −U1U2 + 23U1t + 13U2t, which makes T0into a classical energy-momentum tensor, we get the algebra{Z, U1} = Zδ,{Z, U2} = −Zδ,{Z, A2} = −δt + (U2 −U1)δ{Z, B1} = −A1δ , {Z, B2} = A3δ , {U1, U1} = −34δt,{U1, U2} = 14δt{U1, A1} = −A1δ , {U1, A2} = A2δ , {U1, B2} = B2δ , {U2, U2} = −34δt{U2, A2} = −A2δ , {U2, A3} = A3δ , {U2, B1} = −B1δ , {A1, A2} = B1δ{A1, A3} = 2Zδt + (Zt + 2(U1 + U2)Z)δ,{A2, A3} = B2δ{A1, B2} = δtt + (3U1 + U2)δt+(43U1t + 23U2t + T0 + 2ZA2 + 3U 21 + 2U1U2 + U 22 )δ{A3, B1} = −δtt + (U1 + 3U2)δt+(23U1t + 43U2t −T0 −2ZA2 −U 21 −2U1U2 −3U 22 )δ{B1, B2} = 2A2δt + (A2t + 2(U1 + U2)A2)δ{T0, Z} = 23Zδt −13Ztδ{T0, U1} = −16δtt + U1δt,{T0, U2} = 16δtt + U2δt{T0, Ai} = 43Aiδt + 13Aitδ{T0, Bi} = 53Biδt + 23Bitδ{T0, T0} = 59δttt + 2T0δt + T0tδ(4.4)15

All other brackets vanish. Here the two fields in the Poisson brackets are evaluated at tand t′ respectively; all the fields on the RHS are evaluated at t and δ = δ(t −t′).

We havechecked that the Jacobi identities are satisfied for this algebra.4.2. A twisted version of the W4(3) algebra and the relation to covariantly coupled algebrasand quasi-superconformal algebras.Note the {T0, Ui} relations in (4.4), which display the non-primary character of theUi fields.

(Recall that a field φ is primary, with dimension h if it satisfies {T0, φ} =hφδt + (h −1)φδ). This could be cured by taking TN = T0 −16U1t + 16U2t as the energy-momentum tensor.

Then all the fields are primary, but their spins, as read offtheir Poissonbrackets with TN, is no longer equivalent to the grading under which the soliton equationsthat will be presented in the next section are homogeneous: Z, U1,2, A2 now have spin 1,A1,3, B1,2 spin 3/2 and TN spin 2.Actually, the twisted form of the algebra can be written in a rather compact way usingthe following notation:2J11 = −2J22 = U1 −U2 , J12 = A2 , J21 = Z , U = U1 + U2 ,G−1 =A1 , G+1 = B2 , G−2 = B1 , G+2 = A3 . (4.5)One finds{U, U} = −δt , {U, G±a } = ±G±a ,{Jab, Jcd} =(δcbJad −δadJcb)δ −(δadδcb −12δabδcd)δt ,{Jab, G+c } =(δabG+a −12δabG+c )δ ,{Jab, G−c } =(−δacG−b + 12δabG−c )δ ,{G−a , G+b } =2Jabδt + (Jabt + UJab)δ+δab[δtt + 2Uδt + (Ut + TN + 2JacJcb + 32U 2)δ] .

(4.6)The brackets with TN are those of primary fields with spins given above, and for TN thecentral term is now δ′′′/2. All other brackets vanish.

One sees that the algebra in theabove form contains an sl(2) and a u(1) Kac-Moody algebras. The spin 3/2 fields have adefinite u(1) charge and they transform in the defining representation of sl(2).One thus recovers a particular example of the algebras constructed in [14][16][17].In [17], it was obtained from the standard u(N −2) superconformal algebra [18] by16

changing the statistics of the fermionic fields (the resulting algebras were called quasi-superconformal). The present case corresponds to N = 4.

On the other hand, in [14][16],the general structure was inferred by considering the embeddings of sl(2) in sl(N) associ-ated with the decomposition N →2 + (N −2)1 of the defining representation.It is natural to ask whether there is a KdV-type hierarchy related directly to thequasi-superconformal algebras. By a direct relation we mean a hierarchy homogeneouswith respect to the grading fixed by the quasi-superconformal algebra (4.6).

In fact thereexists such a hierarchy: this is exactly the bosonic version of the u(N) super KdV hierarchyintroduced in the third reference of [18] and which we will discuss in more details elsewhere.We just mention that its first Hamiltonian structure is deduced from the second one bythe shift TN →TN + λ, so that it reads{TN, TN} = 2δ′,{G−i , G+i } = δ . (4.7)In particular, the u(1) quasi-super KdV hierarchy corresponds to the hierarchy constructedin [6][19] starting from a gradation intermediate between the principal and the homoge-neous ones.4.3.

The sl(4)3 flows by reduction of the self-dual Yang–Mills equations.Following the method of [20][21] (see also [22]), we can use a reduction of self-dualYang-Mills equations in 4-dim to obtain the fractional KdV equations corresponding toW4(3). To this end we start from a four dimensional space with signature (2,2) and metricds2 = 2dx dy + 2dz dt.

With ǫxyzt = −1, the self-duality equations[Dx, Dt] =0[Dx, Dy] =[Dz, Dt][Dy, Dz] =0become[∂t + Q, ∂x + H] = 0[∂t + Q, P] = [B, ∂x + H][P, B] = 0(4.8)where we have performed a reduction with respect to the two null Killing symmetries ∂yand ∂z.For the matrix Q we take (4.3), whereas for B, we takeB =000010000100Z010(4.9)17

The reason for this choice will be given section 4.4.Let us denote the matrix elements of H by hij.We get, by consistency of (4.8),h14t = h24t = h34t = 0 andh13 = h24,h12 = h34,h23 = h34 −h14Z,h11 = h24Z,h22 = −h14U1 −h24Z,h33 = −h14U2 −h24Z,h21 = −h14A1 −h24(2U1 + U2) −h34Z,h32 = −h14A2 −2h24(U1 + U2) −2h34Z,h43 = −h14A3 −h24U2 −h34Z,h31 = −h14B1 −h24A2 −h34(U1 + 2U2),h42 = −h14B2 −h24A2 −h34U1,h41 = −h14T + h24(Zt −B1) −h34A1. (4.10)The form of P is given implicitly at the end of section 4.4.

For the fields we find theevolutionsZx = −h14Zt + h24(A1 −A3) + h34(U1 −U2)U1x = −h14U1t + h24(−Zt −2(U1 + U2)Z −B2) + h34(A1 −A2 −2Z2)U2x = −h14U2t + h24(−Zt + 2(U1 + U2)Z + B1) + h34(A2 −A3 + 2Z2)A1x = −h14A1t+h24(−43U1t −23U2t + 2(A1 −A2)Z −3U 21 −2U1U2 −U 22 −T0)+h34(−Zt −2(U1 + U2)Z −B1)A2x = −h14A2t + h24(−2U1t −2U2t −2U 22 + 2U 21 )+h34(−2Zt + B1 −B2 −2(U2 −U1)Z)A3x = −h14A3t+h24(−23U1t −43U2t + 2(A2 −A3)Z + U 21 + 2U1U2 + 3U 22 + T0)+h34(−Zt + B2 + 2(U1 + U2)Z)B1x = −h14B1t + h24(−A2t + 2ZB1 + 2(U1 + U2)(A1 −A2)+ h34(−12U1t −32U2t + 2(A1 −A2)Z −U 21 −2U1U2 −3U 22 −T0)B2x = −h14B2t + h24(−A2t −2ZB2 + 2(U1 + U2)(A2 −A3)+ h34(−32U1t −12U2t + 2(A2 −A3)Z + 3U 21 + 2U1U2 + U 22 + T0)18

T0x = −h14T0t + h24(−23(B1 + B2) + 43(U1 + U2)Z)t+ h34(−13(A1 + A2 + A3) + 13Z2)t(4.11)Inspection of the equations shows that the spins of h14, h24 and h34 differ in ascendingorder by 1/3. If all three coefficients are zero, all the flows are trivial.

If we set h14 = 1,then since h24 and h34 have spins 1/3 and 2/3 resp.and they are constant, we mustset them equal to zero. For the same reason, the only other two solutions are (h14 = 0,h24 = 1, h34 = 0) and (h14 = 0, h24 = 0, h34 = 1).

Each different solution gives rise to aset of equations which are Hamiltonian. Explicitly, we haveh14 = 1 ,H1=R−T0dth24 = 1 ,H2/3 =R[−(B1 + B2) + 2Z(U1 + U2)]dth34 = 1 ,H1/3 =R[−(A1 + A2 + A3) + Z2]dt(4.12)Note that the conformal dimension of these Hamiltonians is well defined.

If we had modifiedthe energy-momentum tensor, as in the previous section, in order to get a conformal algebrawith all fields primary, we would have found H1/3 =R−(A1 + A2 + A3) + . .

., but sincethe fields Ai’s have different spins, the Hamiltonian is not dimensionally homogeneous.4.4. Relation with the results obtained by x ↔t interchange.As already stated, the W4(3) algebra obtained by x ↔t interchange and the oneobtained by Hamiltonian reduction are fully equivalent.

The fields in the two approachesare related by4Z =u2,8U1 = −v2 + 2u3,8U2 = −3v2 + 2u3,16A1 =16u4 −2w2 −4v3 −u22,16A2 = 16u4 −8v3 + 3u2216A3 =16u4 + 6w2 −12v3 −u224B1 = −3u2t + 4w3 −10v4 + 3u2v2 −u2u34B2 = −5u2t + 6w3 −14v4 + 5u2v2 −u2u3T0 = 110w4 + 740v2t −110u23 −61160v22 + 780u2w2 + 340u2v3+3340v2u3 + 116u23 −38u32. (4.13)With these field redefinitions and t →−t, the flow associated to H1/3 is easily checked tobe equivalent to the first flow of the sl(4)3 KdV hierarchy (3.21).

At this point, we recall19

that when regarding the W4 algebra as a second Hamiltonian structure, we obtain the firstHamiltonian structure by shifting u4 by a constant. Inspection of the field redefinitionsshows that such a shift corresponds to shifting each of the Ai fields by the same constant.So the first Hamiltonian structure for the sl(4)3 flows can be obtained by such a shift, andthen the usual procedure can be employed to generate an infinite hierarchy of flows andconserved quantities, recovering the integrability properties from another point of view.Also, in [4] it was noticed that the same shift relates Q to B, and H to P, in thefollowing sense: if the first and the second Hamiltonian structures are related by a shiftof, say, the fields q, {., .

}2,q+λ = {., . }2,q + λ{., .

}1,q, then Q(q + λ) = Q(q) + λB andH(q + λ) = H(q) −λP. Here we should remember that the field T which enters Q isrelated to the energy-momentum tensor T0 by T = T0 + A2Z + .

. .

Therefore, we can seethat a similar relation exists here between Q and B. We observed that the same relationholds true for H and P. This remark gives a heuristic justification for the form of the Bmatrix we took in (4.9).5.

On the general structure of WN (l) algebras.In this section we want to present some general characteristics of the WN (l) algebraswhich can be extracted from the x ↔t interchange and the Hamitonian reduction methods.5.1. Canonical basis of independent fields for the sl(N)l hierarchy and spin content of theWN (l) algebraIn this section, N and l will be taken to be coprime.

To perform the x ↔t interchangein the lth flow of the sl(N) KdV hierarchy, one has to introduce a certain number of newindependent fields which are the first few derivatives of the sl(N) KdV fields u2, u3, . .

. uN.This set must be chosen such that the x derivative of every field is either another field ofthe set or can be expressed in terms of the time derivative of other fields of the set usingthe lth sl(N) KdV equation.

A convenient basis for these new independent fields is givenby{ui, uix, uixx, . .

., ui(l−1)} , i = 2, . .

.N. (5.1)This will be called the canonical basis.

It consists of l(N −1) fields. From such a basisone can read offdirectly the spin content of the fields in the WN (l) algebra.

These arespins WN(l) = {i + kl, i = 2, . .

., N; k = 0, . .

., l −1}(5.2)20

(Recall that spin ui(k) = (i+k)/l.) This spin content satisfies the sum rule (as conjecturedin [13])Xspins s(2s −1) =NXi=2l−1Xk=0(2(i + k)l−1) = N 2 −1 = dim sl(N) .

(5.3)The basis (5.1) is of course not unique but any other basis has exactly the same spincontent. For instance, in the sl(3) case, one can write the second flow under the formu2t = −u2xx + u3xu3t = u3xx + u2u2x(5.4)The canonical basis corresponds to the choice of the new fieldsv2 = u2x,v3 = u3x(5.5)but one could also have consideredv2 = u2x,w2 = u2xx(5.6)However these two choices yield equivalent sl(3)2 hierarchies and in particular ˜P2 is thesame for both cases, up to simple field redefinitions.

This generalized to more complicatedcases.5.2. What happens when (l, N) ̸= 1?When l and N are coprime, it always appear to be possible to construct the sl(N)lhierarchy by direct x ↔t interchange.

However, this is not the case when (l, N) ̸= 1. Forinstance, consider the second flow of the sl(4) hierarchy:u2t = −2u2xx + 2u3x ,u3t = −2u2xxx + u3xx + 2u4x −u2u2x ,u4t = −12u2xxxx + u4xx −12u2u2xx −12u3u2x ,(5.7)It turns out here that for any choice of independent fields, the x-derivative of one of thefields is not determined by the above equation.

This problem persists at the level of themodified equations. The Hamiltonian for the flow (5.7) is simplyRu3dx.

When reexpressedin terms of modified fields, one easily gets the modified equationsr1t =r3xx −(r1r2)x ,r2t = −12(r21 −r23)x ,r3t = −r1xx + (r2r3)x . (5.8)21

Therefore if we choose the new independent fields to be si = rix, one sees that s2x is notdetermined by the above equations due to the absence of a term r2xx.It is simple to show that this situation is generic for l = 2 and N even. Indeed, up tonon-linear terms, the evolution equation for ui readsuit = uixx + 2ui+1 x −2NNiu2(i) + .

. .

(5.9)whereNidenotes a binomial coefficient. Let us fix the basis to be{u2, u2x, .

. ., u2(N−1), u3, u4, .

. ., uN}, in terms of which the argument is simpler.

We wantto show that the x-derivative of u2(N−1) is not determined by (5.7). From the N −3 firstequations in (5.7), one expresses ukx (k ̸= 2) in terms of the other fields of the above set:ukx = 1N u2(k−1)k−1Xi=1(−)i+1 NN −i 12i−1(5.10)In the final equation, the coefficient of u2(N) is1nN−1Xi=1Ni(−)N−i+12N−i+1−2N = (−)N −1N 2N−1(5.11)This vanishes when N is even, in which case u2(N) is not determined.

The argument worksfor other choices of the basis. We expect this to be generic to the cases where N and lare not coprime, but we haven’t found a direct check of this statement within the aboveapproach.5.3.

A comment on the Hamiltonian structure of the modified fields for the sl(N)l hierarchy.Let us introduce the following basis for the modified fields:{φi(k/l), i = 1, . .

., N −1; k = 1, . .

., l −1}(5.12)where φi(1/l) are the usual modified fields ri introduced previously, up to a possible scalingfactor, and φi(k/l) is linearly related to ri(k+1). Notice that the new dimension of φi(k/l) isjust k/l.

From the first few examples which have been worked out, it is tempting to guessthat for suitably chosen φi(k/l), the Hamiltonian structure of the modified fields will read{φi(k/l), φj((l−k)/l)} = −δijδk ̸= l/2{φi(1/2), φj(1/2)} = −δi,N−i−1δ{φi(1), φj(1)} = −δijδt(5.13)This structure is certainly ill defined for N and l even. For instance, for W4(2), this wouldgive {r2, r2} = −δ, which is impossible.22

5.4. Results from the Hamiltonian reductionFor the WN-algebras (i.e.

WN (1) in our notation), the matrix A(t) corresponding to(4.1) is constrained by setting the first upper diagonal to −1. It can always be brought byHamiltonian reduction to the form of a matrix with zero entries everywhere except for thefirst upper diagonal which is set to −1 and the lowest row which takes the form000−1uN.

. .u20(5.14)There are field redefinitions so that the corresponding W-algebra is conformal.

It is well-known that he energy-momentum tensor takes the formT0 = u2 = −12trJ2 + 1l ∂((n −1)J11 + (n −2)J22 + . .

. + Jn−1,n−1)(5.15)with l = 1.For the WN (2) algebras, we performed the Hamiltonian reductions up to N = 8 andfound the following results.

The constraint we impose is to set the second upper diagonalto −1. This constraint is preserved by gauge transformations generated by strictly lowertriangular matrices.However, for N even, it is not possible to fix this gauge freedomcompletely in a local way.

So when considering gauge transformations of the form (4.2),we set g21 = 0 if N is even. The hamiltonian reduction gives the following: the “Q” matrixcan always be reduced so that it is zero everywhere, except for the second upper diagonalwhich is −1, and the lower two rows.

These rows appear as follows:0. .

.000−1∗. .

.T1G(+)UZ∗. .

.∗T2G(−)−U(5.16)The field Z is identically zero if N is odd. We find that T0 = T1 +T2 −U 2 −ZG(−) + 12U ′ isan energy-momentum tensor, whose expression in terms of the original fields is (5.15) withl = 2 modulo terms involving derivatives of the fields above the diagonal.

With respectto this energy-momentum tensor, Z has spin 1/2, U is quasi-primary of spin 1, etc. T0is the energy-momentum that respects the original grading of the KdV fields, as given bythe x ↔t interchange.The presence of the Z field for N even is a reflection, in the context of Hamiltonianreduction, of the presence of constraints in the approach where x and t are interchanged.From this latter point of view, since the Z field has spin 1/2, that means that it originally23

had spin 1 before the interchange. Since there is no spin 1 field in the sl(4) KdV hierarchy,this indicates that such a field has to be introduced to take care of the constraints.For the more general WN (l) algebras, let us know restrict ourselves for convenienceto the cases where N and l are coprime.

For the WN (3) cases, we find by Hamiltonianreduction a “Q” matrix that has zeros everywhere but for the third upper diagonal setto -1 and the lowest three rows, in which we find one spin-2/3 field, two spin-2 fields, . .

.They are arranged as follows:0. .

.00000−10. .

.T1B1A1U1Z0∗. .

.∗T2B2A2U20∗. .

.∗∗T3B3A3−U1 −U2(5.17)for the W3n+1(3), whereas for the W3n+2(3), the spin 2/3 field Z is in position (N −1, N).The combination T0 = T1 + T2 + T3 −U 21 −U 22 −U1U2 −AiZ + 23U ′1 + 13U ′2 (with i = 2for W3n+1(3) and i=3 for W3n+2(3)) forms the energy-momentum tensor.The explicitexpression of this tensor in terms of the original currents is again (5.15) with l = 3 andterms involving derivatives of the fields above the diagonal.It is clear from the above examples and their natural extension that for N and lcoprime, Hamiltonian reduction leads to a WN (l) spectrum that corresponds identically tothe one dictated by the x ↔t interchange. In addition, the energy-momentum tensor thatrespects the natural grading of the KdV equations takes the formT0(WN(l)) = −12trJ2 −1l (N−1X1(N −i)J′i,i)+(derivatives of non −diagonal elements)(5.18)We conclude that {T0, T0} = (N3−N)12l2δttt + 2T0δt + T0tδ.6.

Conclusions.In this work we have constructed a new explicit example of a WN (l) algebra, W4(3),from the point of view of generalized KdV hierarchies ([3],[4],[5],[6]). We have thus pro-vided further support to the conjecture that the WN (l) algebras correspond to the secondHamiltonian structure of the sl(N)l KdV hierarchy.

The latter refers to the new hierarchyobtained from the standard sl(N) KdV hierarchy by interchanging the roles of the vari-ables x and t in the lth flow. Granted this conjecture, another original motivation was to24

advocate this approach as being a simple and systematic way of constructing the WN (l)algebras. From this point of view, the present analysis shows that it is not as simple asinitially expected.

At first we found from the outset that this approach works straight-forwardly only when N and l are coprime. We plan to return to the cases (N, l) ̸= 1elsewhere, but these are certainly more complicated since one has to deal with constrainedsystems.

Second, for the new example we have worked out, the expression we obtain forthis second Hamiltonian structure is quite complicated. This Poisson structure can besomewhat simplified by introducing a new set of independent fields, namely those fieldswhich appear naturally in a zero curvature formulation, or equivalently, the method ofHamiltonian reduction.

If we modify the energy-momentum tensor to make them primary,then the spin content of the algebra is modified. At this step, it appears that the algebraacquires a much nicer form once we recognize the existence of an underlying Kac-Moodyalgebra organizing the whole conformal algebra.

In fact we recover an example of a W al-gebra one obtains by considering a non-principal embedding of sl(2) (the sum-embedding)into sl(N) [14], or equivalently of a u(N −2) quasi-conformal superalgebra [17]. We pointout that the latter is also related to the bosonic form of the u(N −2) super KdV hierarchy.It is certainly interesting and satisfying to display explicitly the perfect equivalence of themethod of x ↔t interchange and that of non-principal sl(2) embeddings, which from theoutset looks conceptually rather remote.Finally let us emphasize some favorable aspects of the method of x ↔t interchange.That we derive the second Hamiltonian structure via the modified fields gives us in oneshot both the algebra and its free field representation.

Also the method emphasizes the factthat everything we want to know about WN (l) and the sl(N)l hierarchy can be extractedsystematically from WN and the usual sl(N) hierarchy. Hence, although there is no directrelation between the various WN (l) algebras for N fixed, the method unravels one suchhidden relationship: all these algebras have the same Miura transformation, which ishowever written differently for different values of l.AcknowledgementsWe would like to thank I.Bakas for a lot of help in the early stage of this project andsubsequent discussions, and P. van Driel for discussions and comments.

This work wassupported by NSERC (Canada), FCAR (Qu´ebec) and BSR (U.Laval).25

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