THE SUPERCOMPLEXIFICATIONS AND ODD BIHAMILTONIANS STRUCTURES Z. Popowicz 1 Institute of Theoretical Physics, University of Wroc law pl.M.Borna 9, 50-204 Wroclaw, Poland 0 [email protected] 0 0 2 n a J Abstract. The general method of the complex supersymmetrization (su- 3 1 percomplexifications) of the soliton equations with the odd (bi) hamiltonian structure is established. New version of the supercomplexified Kadomtsev- ] I Petvishvili hierarchy is given. The second odd Hamiltonian operator of the S . SUSY KdV equation generates the odd N=2 SUSY Virasoro - like algebra. n i l n [ 1 v 8 2 0 1 Introduction 1 0 0 The Kadomtsev-Petviashvili (KP) hierarchy of integrable soliton nonlinear 0 evolution equations [1,2] is among the most important physically relevant inte- / n grable systems. Quite recently a new class of integrable systems motivated by i l the Toda field theory appeared both in the mathematical and in the physical n : literature . v i On the other side the applications of the supersymmetry (SUSY) to the X soliton theory provide us a possibility of the generalization of the integrable r a systems. The supersymmetric integrable equations [3-14] have drawn a lot of attention in recent years for a variety of reasons. In order to get a supersym- metric theory we have to add to a system of k bosonic equations kN fermions 1E-mail: [email protected] 1 and k(N-1) boson fields (k=1,2,..N=1,2,..) in such a way that the final the- ory becomes SUSY invariant. Interestingly enough, the supersymmetrizations, leads to new effects (not present in the bosonic soliton’s theory). In this paper we describe a new method of the N = 2 supersymmetriza- tion of the KP hierarchy, first time presented for (M)KdV equation in [13]. Our method contains the N = 1 supersymmetric KdV equation presented by Beckers [7] as a special case. Herewe wouldlike topresent newresults onthesupercomplexified method. We construct the supersymmetric Lax operator for our supercomplex KP hier- archy. This operator however does not generate the supersymmetric conserved currents defined in the whole superspace. We describe therefore special pro- cedure which allows us to obtain whole chain of conserved currents for our supercomplex hierarchy. The supercomplex Korteweg de Vries equation, considered as a special element of our KP hierarchy, constitue a bi-hamiltonian equation and is com- pletely integrable. The second Hamiltonian operator of the supercomplexified KdV equation generates some odd N = 2 SUSY algebra in contrast to the even algebra SUSY N = 2 Virasoro algebra considered in [8-10]. This supercomplexification is a general method. In order to obtain the N = 4 supercomplex version of soliton’s equation it is possible to use two nonequivalent methods. In both cases we obtain two different generalizations of N = 4 KdV equation. The model proposed in this paper, differs from the one considered in [9,12]. The idea of introducing an odd hamiltonian structure is not new. Leites noticed almost 20 years ago [15], that in the superspace one can consider both even and odd sympletic structures, with even and odd Poisson brackets respectively. The odd brackets (also known as antibrackets) have recently drawn some interest in the context of BRST formalism in the Lagrangian framework [16], in the supersymmetrical quantum mechanics [17], and in the classical mechanics [18,19]. 2 Supercomplexification We shall consider an N = 2 superspace with the space coordinates x and the Grassman coordinates θ ,θ ,θ θ = −θ θ ,θ2 = θ2 = 0. 1 2 2 1 1 2 1 2 The supersymmetric covariant derivatives are defined by ∂ ∂ ∂ ∂ = , D = +θ ∂, D = +θ ∂, (1) ∂x 1 ∂θ 1 2 ∂θ 2 1 2 2 with the properties D2 = D2 = ∂, D D +D D = 0. (2) 1 2 1 2 2 1 D−1 := D ∂−1, D−1 := D ∂−1. (3) 1 1 2 2 We define the integration over the N = 2 superspace to be dXH(x,θ ,θ ) = dxdθ dθ H(x,θ ,θ ), (4) Z 1 2 Z 1 2 1 2 where Berezin’s convention are assumed dθ θ := δ , dθ := 0. (5) Z i j i,j Z i We always assume that the components of the superfileds and their derivatives vanish rapidly enough. Let us now consider some classical evolution equation in the form u = F(u,u ,u ,...) = P ∗grad H(u,u ,u ..), (6) t x xx x xx where u is considered evolution function, P is the Poisson tensor, H is Hamil- tonian of some dynamical system and grad denotes the functional gradient. In order to get the N = 2 supersymmetric version of the equation (6) let us consider the following ansatz which in the next we will call as supercom- plexificiation u = (D D U)+iU , (7) 1 2 x where now U is some N = 2 superfield and i is imaginary quantity i2 = −1. Introducing this ansatz to equation (6) we obtain that superfield U evolves in the time as U = G(U,U ,U ,(D U),(D U),....) = D−1D−1Re(F) = ∂−1Im(F), (8) t x xx 1 2 2 1 where Re and Im denotes the real and imaginary part respectively. In order to guarantee validity of (8) we assume that F is chosen in such a way that (D Re(F)) = (D Im(F)), (9) 1 2 (D Re(F)) = −(D Im(F)), (10) 2 1 always holds. It is a condition of solvability of our construction. Examples. 3 Let us consider the famous KdV equation u = −u −6u u. (11) t xxx x The supercomplex version of this equation is U = −U −6(D D U)U , (12) t xxx 1 2 x and takes the following form in the components f = −f +6gf , (13) t xxx x g = ∂(−g +3g2−3f2), (14) t xx x ξ = −ξ +6ξ g +6ξ f , (15) 1t 1xxx 1x 2x x ξ = −ξ −6ξ f +6ξ g, (16) 2t 2xxx 1x x 2x where U = f +θ ξ +θ ξ +θ θ g. (17) 1 1 2 2 2 1 Notice that the bosonic part of the equations (13,14) does not contain any fermions fields. Hence in some sense this ”supersymmetrization is without supersymmetry”. Now assuming that U = Π+θ Φ, (18) 2 where Π and Φ are N = 1 superbosonic and superfermionic functions respec- tively, our supercomplex KdV equation (12) reduces to Π = −Π −6Π (D Φ), (19) t xxx x 1 Φ = −Φ +6Π (D Π )−6Φ (D Φ). (20) t xxx x 1 x x 1 Notice that when Π = 0 then our equations (20) reduces to the N = 1 super- symmetric KdV equation considered by Beckers in [7]. Let us consider the supercomplexified version of the Nonlinear Schr˝odinger equation as a second example f = −f −2gf2, (21) t xx g = g +2fg2, (22) t xx as the second example. Assuming that f = (D D F)+iF , (23) 1 2 x g = (D D G)+iG , (24) 1 2 x we obtain F = −F −2∂−1 2(D D G)(D D F)F +G (D D F)2 −G F2 ,(25) t xx 1 2 1 2 x x 1 2 x x (cid:16) (cid:17) G = G +2∂−1 2(D D G)(D D F)G +F (D D G)2 −F G2 . (26) t xx 1 2 1 2 x x 1 2 x x (cid:16) (cid:17) 4 3 N=2 Supercomplexified Kadomtsev-Petviashvili hierarchy ItwellknownthatKortewegdeVriesequationaswellastheNonlinearSchr˝odinger equation are the memebrs of the KP hierarchy. This hierarchy is defined by the following Lax operator Lax := ∂ +f∂−1g, (27) which generates the equations by the so called Lax pair representation d Lax = Lax,(Lax)n , (28) dt h +i where n is an arbitrary natural number and (+) denotes the projection onto purely superdifferential part of the pseudosuperdifferential element. In order to define the Lax operator which generate our supercomplexified solitonic equations let us first define the supercomplexified algebra of pseudo- superdifferential elements Υ as a set of the following elements +∞ ((D D F )+F ∂−1D D )∂n, (29) 1 2 n nx 1 2 n=X−∞ where F is an arbitrary N = 2 superfield. There are three different projection n in this algebra +∞ P (g) = ((D D F )+F ∂−1D D )∂n, (30) k 1 2 n nx 1 2 X n=k where g is an arbitrary element belonging to Υ while k can take three values k = 0,1,2 only. Now in ordedr to obtain the supercomplexified version of the Lax operator (27) it is enought to replace f and g by the following substitution f ⇒ (D D F)+F D D ∂−1, (31) 1 2 x 1 2 g ⇒ (D D G)+G D D ∂−1, (32) 1 2 x 1 2 in (27) and replace the projection (+) by the projection P defined by (30) 1 with k = 0 . As an example let us consider the Lax pair representation for the super- complexified N=2 KdV equation. The Lax operator for this case is Lax := ∂ +∂−1((D D F)+F D D ∂−1), (33) 1 2 x 1 2 5 and produces the supercomplexified KdV equation d Lax := Lax,P (Lax3) , (34) dt h 1 i where P (Lax3) = ∂3 +3((D D F)+F D D ∂−1)∂. (35) 1 1 2 x 1 2 It is interesting to notice that this Lax operator does not produce any conserved currents defined in the whole N = 2 superspace. The traditional supersymmetric residual definition as the coefficients standing in D D ∂−1 in 1 2 Laxn gives us that this coefficient after integration over whole superspace is zero. 4 Conserved currents for the supercomplexi- fied KdV equation It is easy to prove using the symbolic computer computations [20-21] that this equationdoesnotpossesses anysuperbosoniccurrents. Ontheothersideusing the same techinque it is easy to find superfermionic conserved currents. Let us explain the connections of these currents with the usual (classical) currents of the KdV equation. This connection is achieved in four steps. First step. Let us supercomplexify an arbitrary conserved current of the classical KdV equation. By H let us denote the real part of n-th conserved nr current after the supercomplexifications. Second step. We compute the usual integral of H . This can be denoted nr as H dx = K0 + K1dx. (36) Z nr n Z n Third step. Now we compute the supersymmetrical integral over first super- symmetrical variable from K1. It can be symbolicaly denoted as n (D K1)dx = H +(D S1dx). (37) Z 1 n 1n 1Z n Fourth step. Finally we compute the supersymmetrical integral over second supersymmetrical variable which is denoted as (D S1)dx = H . (38) Z 2 n 2n H and H are just the conserved superfermionic currents of the supercom- 1n 2n plexified KdV equation. 6 Let us presents several superfermionic conserved currents of the supercom- plexified KdV equation 1 H = dxdθ dθ U(D U ), (39) 12 2 Z 1 2 1 x 1 H = dxdθ dθ U(D U ), (40) 22 2 Z 1 2 2 x 1 H = dxdθ dθ (−(D U )U +4(D U)(D D U)U ), (41) 13 2 Z 1 2 2 xxx 2 1 2 x 1 H = dxdθ dθ (−(D U )U +4(D U)(D D U)U ). (42) 23 2 Z 1 2 1 xxx 1 1 2 x Interestingly these currents does not contain the classical currents of the KdV equation in the bosonic or fermionic parts. We have checked, using the symbolic computations [20-21] thatthis proce- dure could be aplied to the whole supercomplexified KP hierarchy as well. 5 Odd Virasoro - like algebra It is well known that Korteweg - de Vries equation constitute the so called bihamiltonian structure 1 1 u = ∂ ∗grad dx(u u −2u3) = Vir ∗grad dxu2 , (43) t (cid:16)2 Z x x (cid:17) (cid:16)2 Z (cid:17) where Vir := −∂3 −2u∂ −2∂u. (44) The operator ∂ and Vir are two Poissons tensors which constitue with two different hamiltonians the so called bihamiltonian structure. Tensor Vir is connected with the Virasoro algebra, realized as the Poison bracket algebra, through the Fourier decomposition of the field u[]. Indeed introducing +∞ 1 u := T exp(ikx)− , (45) k k=X−∞ 4 where T is some generator, we obtain that k u(x),u(y) = Vir ∗δ(x−y), (46) n o are equivalent with the following Virasoro algebra. T ,T = (n+m)T +δ (n3 −n). (47) n m n+m n+m,0 n o 7 We successed to find the supercomplexified version of the bihamiltonian structure of the supercomplexified KdV equation. Let us define four operators P := D ∂ −2∂−1(D D U)D −2(D D U)∂−1D 21 1 1 2 1 1 2 1 +2∂−1U D +2U ∂−1D , (48) x 2 x 2 P := D ∂ −2∂−1(D D U)D −2(D D U)∂−1D 22 2 1 2 2 1 2 2 −2∂−1U D −2U ∂−1D , (49) x 1 x 1 P := D−1, (50) 11 1 P := D−1. (51) 12 2 These operators generate the supercomplexified KdV equation (12) δH δH δH δH U := P 12 = P 22 = P 23 = P 13 (52) t 21 δU 22 δU 11 δU 12 δU where hamiltonians are defdined by (36-39). Wecanconstruct anO(2)invariantbihamiltonianstructureconsidering the linear combinationofP ±P ,P ±P withH ±H ,H ±H respectively. 11 12 21 22 12 22 14 24 These structures define the same supercomplexified KdV equation (12). Notice that the operators P ,P or P ± P play the same role as the 21 22 21 22 Virasoro algebra in the usual KdV equation. There is a basic difference - our Hamiltonian operators generates the odd Poisson brackets in the odd super- space. In order to obtain the explicit realization of this algebra we connect the Hamiltonian operator P −P with the Poisson bracket 21 22 ′ ′ ′ ′ {U(x,θ ,θ ),U(y,θ ,θ } = P −P (θ −θ )(θ −θ )δ(x−y), (53) 1 2 1 2 21 22 1 1 2 2 (cid:16) (cid:17) where U(x,θ ,θ ) = u +θ ξ +θ ξ +θ θ u . (54) 1 2 0 1 1 2 2 2 1 1 Introducing the Fourier decomposition of u ,ξ ,ξ ,u 0 1 2 1 ∞ ξ := Gjeisx, j := 1,2, (55) j s s=X−∞ ∞ ∞ 1 u := i L eisx, u := T eisx − , (56) 0 s 1 s s=X−∞ s=X−∞ 4 in (64) we obtain {T ,T } = {L ,L } = {L ,T } = 0, (57) n m n m n m 8 n−m n2 −m2 {T ,Gi } = (n2 −1)δ +(−1)i2 T −2 L , (58) n m n+m,0 m n+m m n+m 1 m−n m2 −n2 {Gi,L } = (n− )δ +2 T +(−1)i2 L , (59) n m n n+m,0 nm n+m nm n+m m2 −n2 {Gi,Gi } = (−1)i2 G1 +G2 , (60) n m nm (cid:16) n+m n+m(cid:17) m2 −n2 {G1,G2 } = −2 G1 −G2 . (61) n m nm (cid:16) n+m n+m(cid:17) These formulae define the closed algebra with the graded Jacobi identity [22] (−1)[1+a][1+c]{a,{b,c}} = 0, (62) X cycl(a,b,c) where [a] denotes the parity of a. It is the desired odd Virasoro - like algebra. 6 Supercomplexified N=4 KdV equation It is possible to obtain in two different manner the supercomplexified N=4 version of the soliton equation. In the first manner we strictly speaking make dublesupercomplexifications oftheusualsolitonsequationswhileinthesecond methodwesupercomplexifythewellknowN=2supersymmetric equations. Let us describe these methods. In the first method we assume that the superfunction U which satisfy the equation (12) could be presented as U := (D D W)+iW . (63) 3 4 x where now W is some N = 4 bosonic superfunction. Introducing this ansatz to (12) we obtain that W satisfy the dubly supercomplexified KdV equation W := −W −6∂−1((D4W)W +(D D W)(D D W)), (64) t xxx xx 1 2 3 4 where D4 = D D D D . 1 2 3 4 For the second method we consider the N = 2 supersymmetric KdV equa- tion 1 Φ := ∂(−Φ +3Φ(D D Φ)+ (α−1)(D D Φ2)+αΦ3). (65) t xx 1 2 1 2 2 9 whereαcantakethreevalues−2,1,4ifwewouldliketoconsider theintegrable extensions. We assume that the superfield Φ satisfy the N = 2,α = 4 SUSY KdV equation (62), takes after the supercomplexification the following form Φ := (D D Υ)+iΥ , (66) 3 4 x where Υ is some N = 4 superboson field. Substituting this form in (65) we obtain Υ := −Υ +3(D D (Υ (D D Υ)))−4Υ3+ t xxx 1 2 x 3 4 x Υ (D4Υ)+(D D Υ)(D D Υ ) +12(D D Υ)2Υ . (67) x 3 4 1 2 x 3 4 x (cid:16) (cid:17) It is the desired generalization of the N = 4 SUSY KdV equation, which is different from the one considered in [9,12]. Let us remark that usuing the similar methods descrbided earlier it is pos- sible to obtain the supercomplexification of the bihamiltonians structures, the Lax pair representations and the conserved currents. As an example let me present the Lax operator for the equations (64) and (67). For equation (64) the Lax operator is L := ∂2 +(D4W)+(D D W )i +(D D W )i +W i i (68) 1 2 x 2 3 4 x 1 xx 1 2 where i = ∂−1D D , i = ∂−1D D and i2 = i2 = −1. 1 1 2 2 3 4 1 2 For equation (67) the Lax operator is L := − D D +(D D Υ)+Υ ∂−1D D 2 (69) 1 2 3 4 x 3 4 (cid:16) (cid:17) Acknowledgement The author wish to thank organizer for the scientific discussion and warm hospitability during the conference. This paper has been supported by the KBN grant 2 P03B 136 16 and by Bogoliubow-Infeld found. References [1] Dickey L A, 1991 Soliton Equations and Hamiltonian System (Singapore: World Scientific). [2] B laszak M, 1998 Multi - Hamiltonian Theory of Dynamical System (Berlin: Springer Verlag). [3] Kupershmidt B A, Phys.Lett A 109 (1985) 417. 10