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The Super W∞Symmetry of the Manin-Radul Super KP Hierarchy

arXiv:hep-th/9111054v1 25 Nov 1991UR–1232ER–13065–685CTP TAMU–54/91IC/91/383UFIFT–HEP–91–26November 1991The Super W∞Symmetry of the Manin-Radul Super KP HierarchyA. Das,1⋆E.

Sezgin,2† and S.J. Sin 3††1 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627,USA2 Center for Theoretical Physics, Texas A&M University, College Station,TX 77843–4242, USA3 Department of Physics, University of Florida, Gainesville, FL 32611, USAABSTRACTWe show that the Manin-Radul super KP hierarchy is invariant undersuper W∞transformations.

These transformations are characterized by timedependent flows which commute with the usual flows generated by the con-served quantities of the super KP hierarchy.⋆Supported in part by the U.S. Department of Energy, under grant DE-AC02-76ER-13065.† Supported in part by the U.S. National Science Foundation, under grant PHY-9106593.†† Supported in part by the U.S. Department of Energy, under grant DE-FG05-86ER-40272.

1. IntroductionThere has been a great deal of interest in the study of conformal symmetries in thepast several years.

In the context of high energy physics, the motivation for such studiescame from string theories and has already led to some remarkable results. The conformalsymmetries, in turn, have led in a natural way to the study of WN algebras [1].

While WNalgebras do not define a Lie algebra, W∞or W1+∞algebras [2] are proper Lie algebras andare expected to play a role in the understanding of string theories [3][4].In a completely parallel development, the integrable models have also attracted a lotof attention in recent years. For example, it is known now that one of the Hamiltonianstructures (Poisson brackets) of the KdV equation is nothing other than the Virasoro algebra[5].

Similarly, it has become clear that the WN algebras arise as Hamiltonian structuresof various integrable models in 1+1 dimensions [6].The integrable models also play animportant role in the study of various 2D gravity theories. For example, the gravitationalWard identity for pure 2D gravity in the light-cone gauge turns out to be none other thanthe KdV hierarchy equation [7] and similarly for other gravity theories.The KP (Kadomtsev-Petviashvili) equation [8] is a 2+1 dimensional integrable systemwhich leads to a large class of 1+1 dimensional integrable models upon appropriate reduction.In terms of the dynamical variable u(x, y, t), the KP equation reads∂∂x∂u∂t −14∂3u∂x3 −3u∂u∂x= 34∂2u∂y2(1.1)As is obvious, in the absence of y-dependence, Eq.

(1.1) reduces to the KdV equation which,we have argued, plays an important role in the study of strings. It is, therefore, naturalto expect that the KP equation may provide further understanding of the various stringtheories.

In fact, it has already been pointed out that one of the Hamiltonian structures ofthe KP equation is isomorphic to the W1+∞algebra [9] which is expected to play a significantrole in the understanding of strings. It has also been argued that the KP hierarchy admitssymmetries which satisfy the W1+∞algebra [10][3].One can, of course, supersymmetrize various integrable systems.

In fact, the KP equa-tion allows for more than one supersymmetric generalization [11][12]. And it is, of course,ultimately the superstring theory which is of physical interest.

Therefore, it is the structureof the supersymmetries and the algebra of the symmetries for the supersymmetric systemsthat will be of direct physical significance. It is with this goal that we have undertaken, inthis paper, the study of symmetries for the simplest of the super KP hierarchies, namely,the Manin-Radul hierarchy [11].

In Sec. II, we describe our notation and recapitulate, verybriefly, the essential features of the Manin-Radul hierarchy.

The symmetry conditions arediscussed in detail in Sec. III and explicit symmetry generators as well as the symmetrytransformations for the Manin-Radul hierarchy are given in Sec.

IV. We would like to em-phasize here that the usual conserved quantities associated with an integrable system, of2

course, generate symmetries of the system. But what we are interested in are additionalsymmetries–in general time dependent.We find an infinite set of bosonic and fermionicgenerators of symmetry for the super KP hierarchy.

In Sec. V, we show that the algebragenerated by these generators is precisely the super W∞.

Our conclusions are presented inSec. VI.2.

The Manin-Radul Super KP HierarchyThe supersymmetric extension of the KP hierarchy introduced by Manin and Radul [11]is a system of nonlinear equations for an infinite set of even and odd functions, dependingon a pair of odd and even space variables (ξ, x) and the odd-even times (τ1, t2, τ3, t4, ...) †.The manifold on which the solutions are defined, in this case, is a graded manifold. On thismanifold, we can, of course, define the usual supercovariant derivativeθ = ∂∂ξ + ξ ∂∂x(2.1)which satisfies[θ, θ] = 2 ∂∂x(2.2)where the graded commutator is defined by [A, B] = AB −(−1)abBA with a, b denoting thegradings of A and B and taking values 0 and 1 depending on whether the variable is bosonicor fermionic.

A formal inverse of θ is defined to beθ−1 = ξ + ∂∂ξ ∂∂x−1(2.3)On this manifold, we can also define the even and odd time derivatives asθ2i =∂∂t2ii, j = 1, 2, . .

.θ2i−1 =∂∂τ2i−1+∞Xj=1τ2j−1∂∂t2i+2j−2(2.4)These time derivatives satisfy the algebraθ2i, θ2j= 0θ2i, θ2j−1= 0θ2i−1, θ2j−1= 2θ2i+2j−2(2.5)†t2 and t4 are to be identified with y and t, respectively. In the bosonic case, the KP equation (1.1)then arises as the lowest member of a hierarchy of equations.3

Furthermore, it is easy to see that both the even and the odd time derivatives have vanishingcommutator with θ, namely,[θ, θ2i] = 0 = [θ, θ2i−1](2.6)In other words, these derivatives are covariant with respect to supersymmetry transforma-tions.With these preliminaries, we can define the Lax operator for the Manin-Radul super KPhierarchy as a pseudo-differential operator on this graded manifold with the formL = θ +∞Xi=1Uiθ−i(2.7)where the Ui’s are functions of all the even and odd variables with the grading (i + 1). Thisis completely parallel to the bosonic KP hierarchy where the Lax operator is defined as apseudo-differential operator involving∂∂x.

The generalized Leibnitz rule for the supercovari-ant derivatives is given byθiU =∞Xj=0(−1)u(i−j)ij θjUθi−j(2.8)where the super-binomial coefficients are defined for i ≥0 byij=0for j < 0 or j > i or (i, j) = (0, 1) mod 2 i2 j2!for 0 ≤j ≤i and (i, j) ̸= (0, 1) mod 2(2.9)For i < 0, we defineij= (−1)[ j2]i + j −1j(2.10)The Manin-Radul super KP hierarchy can, then, be described in terms of the Lax equa-tions [11]θiL =hLi−, Lii = 1, 2, . .

. (2.11)where by A+, A−, we will understand the parts of the pseudo differential operator A con-taining nonnegative and only negative powers of θ respectively.

The structure of Eq. (2.11)is quite analogous to the Lax equation for the bosonic KP hierarchy.

Let us note here thatthe equations in Eq. (2.11) can also be written in an equivalent alternate form asθ2i = −L2i+, Lθ2i−1 = −L2i−1+, L+ 2L2i(2.12)4

Furthermore, we can introduce a dressing operator, as in the bosonic case, byL = KθK−1(2.13)withK = 1 +∞Xi=1Kiθ−i. (2.14)Consistency would then require that K1 + 12K0 = 0.

The Lax equation (2.11) can now bewritten asθiK =Li−K(2.15)With a little bit of algebra, one can easily show that the Lax equation (2.11) or equiv-alent Eq. (2.15) are consistent with the algebraic structure in Eq.(2.5).

For the even timederivatives, then, we obtain the zero curvature conditionθ2iL2j−−θ2jL2i−−hL2i−,L2j−i= 0.(2.16)3. The Symmetries of the Manin-Radul Super KP HierarchyThe symmetries of the super KP hierarchy can be discussed equivalently at various levels.For example, one can study the symmetries of the linear equation or the symmetries of theevolution of the dressing operator or the symmetries of the Lax equation.

From our point ofview, it is most useful to study the evolution of the dressing operator. We know that (seeEq.

(2.15))θiK =Li−K(3.1)Consider an infinitesimal deformation of the dressing operator K, such thatδK = ǫQ−K(3.2)where ǫ is an infinitesimal constant parameter of deformation. Note that we have restrictedthe operator Q to its negative part.

As we shall see, this restriction turns out to simplifyconsiderably the symmetry condition on Q which we derive below. One can think of thedeformation as being generated through a flow equationφK = Q−K(3.3)This deformation will be a symmetry of the super KP equation (namely, of Eq.

(3.1) if δKsatisfiesθiδK =δLi−K +Li−δK(3.4)5

One can show quite easily from the relations in Eqs. (2.13) and (3.2) thatδLi−= ǫQ−, Li−(3.5)The symmetry condition, Eq.

(3.4), can now be equivalently written asθiQ−= −Li, Q−−+hLi−, Q−i(3.6)This can be easily seen to take the simple formθiQ−= −hLi+ , Q−i−(3.7)In obtaining this simple form, it was essential that the symmetry operator Q is restricted toits negative part in (3.2). It is often convenient to express the symmetry generator asQ = KV K−1(3.8)The symmetry condition, Eq.

(3.7), can then be shown to be equivalent toθiV = −θi, V(3.9)The construction of symmetries of the super-KP hierarchy is, then, equivalent to determiningV ’s which satisfy Eq. (3.9).

As a familiar example, let us note thatVn = θ2n(3.10)automatically satisfies the above relation. These are nothing other than the familiar flowsgenerated by the bosonic conserved quantities of the theory.However, we are interested in additional time dependent symmetries of the theory.

Thus,we make an ansatz for V of the following formV = αξ + x(βθ + γξ∂) +Xn≥1α1nθ + α2nξ∂t2n∂n−1+Xn≥1α3n + α4nξθτ2n−1∂n−1+Xn,k≥1α5nkθ + α6nkξ∂τ2n−1τ2k−1∂n+k−2(3.11)where α, β, γ, α1n, . .

. α6nk are constant coefficients to be determined.

Requiring the symmetrycondition to hold for even flows, we obtainα1n = −βnα2n = −γn(3.12)6

Similarly, requiring the symmetry condition to hold for the odd flows we determine all buttwo constants. Thus, a symmetry generator satisfying Eq.

(3.9) can be written asV = αξ + βxθ −βXn≥1nt2n∂n−1θ +Xn≥1(−α + (n −1)β)τ2n−1∂n−1−βXn≥1(2n −1)τ2n−1∂n−1ξθ + βXn,k≥1nτ2n−1τ2k−1∂n+k−2θ(3.13)where α, β are the two arbitrary constants andθ = ∂∂ξ −ξ ∂∂x(3.14)which satisfies[θ, θ] = 0[θ, θ] = −2∂(3.15)4. Algebraic Structure of the Symmetry TransformationsIn this section, we would like to construct a canonical basis of the symmetry operatorswhich would lead to a natural (graded) Lie-algebraic structure.

From experience, we realizethat in the coordinate basis, it is most convenient to isolate the symmetry operators intothe form x + . .

. , ξ + .

. .

, ∂+ . .

. ,∂∂ξ + .

. ..

It is trivially seen from the symmetry conditionin Eq. (3.9) that the operator ∂=∂∂x is a symmetry operator.

Furthermore, by choosingspecial values of the parameters α and β in Eq. (3.13) we obtainα = 1, β = 0 :V1 = ξ −Xn≥1τ2n−1∂n−1 ≡T(4.1)α = 0, β = 1 :V2 = Xθ(4.2)whereX = x−Xnt2n∂n−1 −12Xn≥1τ2n−1∂n−2(2n −1)θ −θ+Xn,k≥1nτ2n−1τ2k−1∂n+k−2(4.3)While T is already in the desired form, we need to reduce V2 to the canonical form.

Tothis end, we note that the commutator of two symmetry operators is, itself, a symmetryoperator. Therefore, since[∂, V2] = θ(4.4)we conclude that θ is a symmetry operator.

(In fact, it generates space supersymmetry. )Furthermore, from the structure of V2, which is a symmetry operator, it is then obvious that7

X must also be a symmetry operator. To find a symmetry operator with the coordinateform∂∂ξ + .

. ., we note thatS ≡θ + T∂= ∂∂ξ −Xτ2n−1∂n(4.5)is also a symmetry operator.From the structures of the symmetry operators ∂, X, S, T – two of which are bosonicand the other two fermionic – it is easy to verify that[∂, X] = 1 = [S, T](4.6)with all other graded commutators vanishing.

One can, therefore, make the following corre-spondence now, namely,X ↔z∂↔∂∂zT ↔κS ↔∂∂κ(4.7)where z is a bosonic variable while κ is a Grassmann variable. With this identification, wecan now show that the algebra of the symmetry operators is nothing other than the superW +∞algebra.

We do this in the next section.5. The Super W∞StructureThe super W∞algebra was constructed in Ref.

[13]. The bosonic subalgebra is W∞⊕W1+∞generated by V im and ˜V im, respectively, where i + 2 labels the (quasi) conformal spinof the generator, and −∞≤m ≤∞.For V im, i = 0, 1, .. while for ˜V im, i = −1, 0, ...The fermionic generators are Gi±m , where the (quasi) conformal spin of the generator is nowi + 32, i = 0, 1, ... A realisation of this algebra in terms of super differential operators wasgiven in Ref.

[14]. These operators are as follows †V im =i+1Xℓ=01i + 1aimℓhℓ+ (i + 1 −ℓ)κ ∂∂κizk−m ∂∂zℓ,˜V im =i+1Xℓ=012i + 1aimℓh−ℓ+ (i + ℓ+ 2)κ ∂∂κizk−m ∂∂zℓ,Gi±m−12 =iXℓ=0zℓ+1−m ∂∂zℓ2ℓi + ℓ+ 1 aim,ℓ+1 κ ∂∂z ± i + ℓ+ 12i + 1ai−1m−1,ℓ∂∂κ,(5.1)†In Ref.

[14], one parameter (denoted by λ ) family of super W∞algebras are given, which correspondto different choices of basis. Here we have made the choice λ = 0, which produces the super W∞algebraconstructed explicitly in Ref.

[13]).8

whereaimℓ=i + 1ℓ (m −i −1)i+1−ℓ(−i −1)i+1−ℓ(i + ℓ+ 2)i+1−ℓ,(5.2)where we have used the definition(a)n ≡Γ(a + n)Γ(a)= a(a + 1)(a + 2) · · ·(a + n −1),with (a)0 = 1. (5.3)The reason for complicated choices for coefficients is to ensure that the resulting generatorhas a definite transformation property under the SL(2, R) subalgebra of the Virasoro algebragenerated by V 0m.

This SL(2, R) covariance property allows one to assign (quasi) conformalspin to each genarator, thus facilitating the use of many conformal field theoretic techniquesin dealing with this algebra.It is clear now how to identify the super W∞structure of the symmetry transformationsof the previous section. Since any power of the basic symmetry operators ∂, X, S and T isalso a symmetry operator, using the correspondence (4.7), we have the following symmetryflows with manifest super W∞structure:∂∂εimK = (KV imK−1)−K,∂∂˜εimK = (K ˜V imK−1)−K,∂∂εi±m−12K = (KGi±m−12K−1)−K,(5.4)These are time dependent, non-isospectral flows which commute with the isospectral superKP flows (2.11).

We emphasize that in (5.4), the replacements z →X,∂∂z →θ2, κ →T,∂∂κ →S are to be made.For example, some of the low lying generators are givenexplicitly as followsV 0m = X1−mθ2 −12(m −1)X−mTS,V 1m = X2−mθ4 −12(m −2)(1 + TS)X1−mθ2 + 13(m −1)(m −2)X−mTS,˜V −1m= −X−mTS,˜V 0m = (−1 + 3TS)X1−mθ2 −(m −1)X−mTS,G0±m−12 = X1−m(Tθ2 ± S),G1±m−12 = X2−mθ2(Tθ2 ± S) −13(m −2)X1−m(2Tθ2 ± S). (5.5)It is straightforward to show that V 0m obey the Virasoro algebra.

Other interesting subalge-bras of the full super W∞are discussed in Ref. [13], and contraction down to the classical9

version super w∞which is equivalent to super symplectic diffeomorphism of a suitable su-permanifold are discussed in Refs. [13] and [15].

In particular, there exists the subalgebrasuper W −∞where m ≤i + 1 in V im, in which case no negative powers of z occur in (5.1).6. ConclusionsWe have constructed flows (5.4) which commute with the Manin-Radul super KP flows(2.11).

These, therefore, represent the symmetries of the Manin-Radul super KP hierarchy.We would like to emphasize here that the usual conserved quantities associated with anintegrable system also generate symmetries of the system, but what we have obtained hereare additional symmetries which are in general time dependent.The action of the basic symmetry operators ∂, and T, X, S defined in (4.1), (4.3), (4.5),respectively, on the vacuum solution (2.20) can not all be representated in terms of differentialoperators involving the spectral parameters λ and η alone. In the case of of the bosonic KPhierarchy this is possible.

It may as well be possible, and it would be useful, to constructa similar representation of the symmetry operators in terms of the spectral parameters inthe case of super KP hierarchy as well, by taking suitable combinations of the symmetryoperators given in this paper.We conclude by pointing out some further interesting problems which deserve study. InRef.

[16] it has been shown that the KP hierarchy has additional symmetries which obey aKac-Moody-Virasoro based on a subalgebra of V ir ⊕ˆsℓ(5, R). It is not clear whether thismeans that the symmetry ring of the Manin-Radul system admits a Kac-Moody extendedversion of W +∞which contains the symmetry algebra of Ref.

[16] as a subalgebra. To under-stand this, it will be useful first to realize such an extended algebra in terms of differentialoperators in a manner similar to Eq.

(5.1).The KP hierarchy has other supersymmetric extensions as well [12]. It would be inter-esting to explore the symmetry properties of these extensions in a systematic manner and tocompare with one another.

What would be very interesting in this context is to find a theoryof a relativistic membrane or even higher extended objects whose equations of motion wouldbelong to a KP hierarchy, in which case the large symmetry of the hierarchy, such as thosefound in this paper, could be utilized to generate infinitely many solutions, and infinitelymany relations between scattering matrix elements, thus rendering the theory, if not soluble,at least manageable.ACKNOWLEDMENTSWe would like to thank Professor Abdus Salam, the International Atomic Energy Agencyand UNESCO for hospitality at the International Center for Theoretical Physics where thiswork was done.We would also like to thank M. G¨urses for bringing Ref. [16] into ourattention.10

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