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Darboux Transformations for Super-Symmetric KP Hierarchies Q.P. Liu1 and Manuel Manãs2 1Beijing Graduate School, China University of Mining and Technology 0 Beijing 100083, China 0 0 1The Abdus Salam International Centre for Theoretical Physics 2 n 34100 Trieste, Italy a J 2Departamento de F´ısica Teórica II, Universidad Complutense 2 1 28040 Madrid, Spain 2 v 6 Abstract 1 0 We construct Darboux transformations for the super-symmetric KP hierarchies of 9 0 Manin–Radul and Jacobian types. We also consider the binary Darboux transformation 9 for the hierarchies. The iterations of both type of Darboux transformations are briefly 9 discussed. / t n i - 1 Introduction v l o s The first super-symmetric KP (SKP) hierarchy was introduced by Manin and Radul [5] more : v than a decade ago. Another relevant super-symmetrization is due to Mulase and Rabin [8]. i X These systems have been subject of extensive studies from both mathematical and physical r viewpoints. Inparticularwestressthepossibleapplicationsintwodimensionalsuper-symmetric a quantum gravity [1]. There are now many results for SKP hierarchies such as the description of their solution space in the framework of the universal super-Grassmann manifold [12], the construction of their additional symmetries [6] and tau functions [7]. In our previous papers [3, 4], we constructed the Darboux transformations for the Manin– Radul’s super-symmetric KdV (SKdV) hierarchy and its reductions. We found that, as in ordinary case, Darboux transformations constitute a very efficient tool for the generation of solutions. The SKdV hierarchies are reductions of SKP hierarchies, so a natural task is to construct Darboux transformations for SKP hierarchies itself. It should be remarked that such generalization is not so straightforward as one may suppose. The reason is that the SKP hierarchies incorporates both even time and odd time flows while the SKdV hierarchy only has the even time flows. Very recently, Araytn et al have pointed out that the natural candidate for the Darboux transformation does not preserve the odd flows [2]. The purpose of the paper is to present proper Darboux transformations for the SKP hierarchies. For that aim we must recall, following Ueno and Yamada [12], that there two SKP hierarchies, saySKP , k = 0,1. Infact, by reversing thesigns oftheoddtimes wegofromSKP k 0 1 to SKP . We will show that the natural candidate for elementary Darboux transformation is 1 a map SKP → SKP , mod 2; thus, when composed with a reversion of odd times gives k k+1 new solution of the SKP . Observe also that when this elementary Darboux transformation is k composed twice we get a transformation SKP → SKP . We also consider the binary type of k k Darboux transformation for these hierarchies. The paper is organized as follows. In the next section, we recall all the relevant facts and formulae for the Manin–Radul and Jacobian SKP hierarchies. In §3 the reader may found our main results. We construct both the so called elementary and binary Darboux transformations. We show that while the binary one has a rather straightforward generalization, the elementary one needs to be modified. We discuss the iterations of both type of Darboux transformations briefly. Two appendices are included to present some details. 2 Super-symmetric KP Hierarchies In this section we remind the reader the basic aspects of the two main SKP hierarchies: the Manin–Radul and Jacobian hierarchies. The proper setting for the SKP hierarchies is the Sato’s formalism which we now recall [12, 8, 7]. We consider the algebra X of super pseudo-differential operators in the following form P = a (x,θ,t,τ)Dn, N ∈ N n n≤N X here the coefficients a are taken from a super-commutative algebra n Y = C[[x,t]]⊗Λ(θ,τ)⊗A where x is an even variable, θ an odd variable, t = {t }∞ , τ = {τ }∞ , and A is a finite or n n=1 n n=1 infinite dimensional Grassmann algebra over C. The super-differential operator D is given by ∂ ∂ D = θ + ∂x ∂θ which satisfies ∂ ∂ ∂ −1 D2 = ∂ := , D−1 = θ+ . ∂x ∂θ ∂x (cid:16) (cid:17) For any operator L = a Di ∈ X we denote the truncations to various orders by i i P L = a Di, L = a Di, L = a + i − i 0 0 i≥0 i<0 X X and its super-residue is res(L) = a . −1 We also follow the convention: • Parentheses is intended to indicate that an operator has acted on an argument. 2 • Juxtaposition of operators to indicate an operator product. The parity of a super quantity s is denoted by |s|; i.e., if the quantity is even, |s| = 0, otherwise |s| = 1. To work out the so called binary Darboux transformation, we need the adjoint operation ∗, which is defined as D∗ = −D, (Dn)∗ = (−1)n(n2−1)Dn, (fg)∗ = (−1)|f||g|g∗f∗ for any f,g ∈ X. The super-vector fields on Y generating the SKP flows are given by ∂ ∂ − τ for the Manin–Radul SKP, ∂ ∂τ m≥1 m∂t ∂ := , D := n n+m−1 n ∂tn n  ∂ P for the Jacobian SKP.  ∂τ n  The action of ∂ on an objectf is sometimes written as f . They generate a Lie super-algebra n tn with relations −2∂ for the Manin–Radul SKP, n+m−1 [∂ ,∂ ] = 0, [∂ ,D ] = 0, {D ,D } = (1) n m n m n m (0 for the Jacobian SKP, where [·,·] denotes the usual commutator and {·,·} the anti-commutator. Now we introduce Sato’s operator ∞ W = 1+ w (x,θ,t,τ)D−n, (2) n n=1 X where the parity of the field w is given by (1 − (−1)n)/2; i. e., w are even and w are n 2k 2k−1 odd, so that W is a homogeneous even operator. Sato’s equation for the SKP hierarchies of type k = 0,1 are given by k ∂ W = −(W∂nW−1) W, n − D2n−1, n ≥ 1 for the Manin–Radul SKP, DnW = −(−1)k(WAnW−1)−W, with An := ∂  D2n−2, n ≥ 1 for the Jacobian SKP. ∂θ Twoobservationsareinordernow: NoticefirstthatgivenaSatooperatorW(x,θ,t,τ)forSKP k the operator W(x,θ,t,−τ) is a Sato operator of SKP (mod 2); i. e. both are essentially k+1 the same hierarchy. Secondly, for the Jacobian hierarchy we have A = D2n−1 −θD2n. n An essential ingredient in our forthcoming considerations are the wave functions, which are solutions ϕ ∈ Y → A of the following equations ∂ ϕ = P ϕ, (−1)kD ϕ = Q ϕ, (3) n n n n where P and Q are defined in terms of the dressing operator W as follows n n P := (W∂nW−1) , Q := (WA W−1) , n + n n + 3 and accordingly we also have their adjoint counterparts ∂ ϕ∗ := −P∗ϕ∗, (−1)kD ϕ∗ := −Q∗ϕ∗. (4) n n n n A second equivalent formulation of the SKP hierarchies is the Zakharov–Shabat formalism [12], which can be regarded as the compatibility condition of the linear systems (3) ∂ P −∂ P +[P ,P ] = 0 n m m n m n ∂ Q −(−1)kD P +[Q ,P ] = 0 n m m n m n −2P for the Manin–Radul SKP, (−1)kD Q +(−1)kD Q −{Q ,Q } = n+m−1 m n n m n m (0 for the Jacobian SKP. FortheManin–RadulSKPhierachythereisathirdformulation: theLaxform. Itisobtained by dressing the operator D with W, getting thus the Lax operator ∞ L = WDW−1 = D + u D−n+1 n n=1 X and the corresponding Lax representation ∂ L = −[(L2n) ,L] = [(L2n) ,L], n − + (−1)kD L = −{(L2n−1) ,L} = {(L2n−1) ,L}−2L2n. n − + 3 Darboux Transformations Inthissection, wediscuss theDarbouxtransformationsfortheSKPhierarchy. Wefirstconsider so called elementary Darboux transformation. We will see that the naive candidate does not qualify as a proper one, but by examining the failure we show how to remedy it and obtain a meaningful Darboux transformation. The iteration of such transformation leads us to super ‘Wron´ski’ determinant representation for the solution of the susy KP hierarchies. This is the generalization of the one found by Ueno et al [12] in the framework of the universal super- Grassmann manifold. Next we discuss the binary Darboux transformation. We show that unlike the elementary one, the binary Darboux transformation does enjoy a straightforward generalization. As in the SKdV case, the iteration of such transformation provides us the solution of the SKP hierarchy in the form of super-Gramm determinants. 3.1 Elementary Darboux Transformations We first introduce odd gauge operator T := ϕDϕ−1 = D −ϕ−1(Dϕ) (5) where ϕ is an even invertible wave function of the linear system (3). Indeed, as shown in previous papers [3, 4], this very operator supplies us the Darboux transformation for the SKdV equation. We first observe 4 Lemma 1. The gauge operator T satisfies (∂ T)T−1 = −(TP T−1) , n n − (−1)k(D T)T−1 = (TQ T−1) . n n − Proof. First observe ∂ T = −∂ (ϕ−1(Dϕ)) = −D(ϕ−1∂ ϕ) = −D(ϕ−1P ϕ) n n n n and (TP T−1) =(ϕDϕ−1P ϕD−1ϕ−1) n − n − =ϕ(Dϕ−1P ϕD−1) ϕ−1 n − =ϕ(Dϕ−1P ϕ)D−1ϕ−1. n Second, D T = −D (ϕ−1(Dϕ)) = D(ϕ−1(D ϕ)) n n n and (TQ T−1) =(ϕDϕ−1Q ϕD−1ϕ−1) = ϕ(Dϕ−1Q ϕD−1) ϕ−1 n − n − n − =ϕ(Dϕ−1Q ϕ)D−1ϕ. n This gauge operator allows us to define, in the realm of the standard Darboux transformation, the transformed Sato’s operator Wˆ := TWD−1. (6) The susy KP flows on this transformed operator are Lemma 2. The operator Wˆ satisfies ∂ Wˆ = −(Wˆ ∂nWˆ −1) Wˆ , n − (−1)k+1D Wˆ = −(Wˆ A Wˆ −1) Wˆ . n n − Proof. For the even evolution we have ∂ Wˆ =−(TP T−1) TWD−1 −T(W∂nW−1) WD−1 n n − − =(TP T−1) TWD−1−TW∂nD−1 n + =(T(W∂nW−1) T−1) TWD−1 −TW∂nD−1 + + =(TW∂nW−1T−1) TWD−1 −TW∂nD−1 + =−(Wˆ ∂nWˆ −1) Wˆ , − while for the odd time evolution (−1)kD Wˆ =(TQ T−1) TWD−1+T(WA W−1) WD−1 n n − n − =−(TQ T−1) TWD−1+TWA D−1 n + n =−(T(WA W−1) T−1) TWD−1 +TWA D−1 n + + n =−(TWA W−1T−1) TWD−1 +TWA D−1 n + n =(Wˆ A Wˆ −1) Wˆ . n − 5 We see then that while the even evolutions are preserved by the suggested transformation the odd time evolutions are not. ˆ At the level of wave functions for any given solution ψ of (3) a new function ψ as ψˆ := Tψ, Pˆ := (TP T−1) , Qˆ = (TQ T−1) (7) n n + n n + then Proposition 1. The new function ψˆ satisfies ∂ ψˆ = Pˆ ψˆ, n n (−1)k+1D ψˆ = Qˆ ψˆ. n n Proof. We first compute ∂ ψˆ = ((∂ T)T−1 +TP T−1)ψˆ = (TP T−1) ψˆ n n n n + and then we calculate (−1)kD ψˆ = ((−1)k(D T)T−1 −TQ T−1)ψˆ = −(TQ T−1) ψˆ. n n n n + Hence, we observe that is true that the SKP odd flows will not be preserved because there k is a change τ → −τ involved within the transformation. In fact, this Darboux transformation n n generates a mapSKP → SKP fromsolutions of theSKP tosolutions of SKP , k mod 2. k k+1 k k+1 That is: Theorem 1. Given a Sato operator W of the SKP hierarchy then Wˆ is a Sato operator of k the SKP (k mod 2). k+1 Moreover, for W˜ (x,θ,t,τ) := Wˆ (x,θ,t,−τ) we have Corolary 1. Given a Sato operator W of the SKP hierarchy then W˜ is again a Sato operator k of the SKP hierarchy. k Let us remark that, on the level of Lax formulation of the Manin–Radul SKP hierarchy, 0 this point is treated in §III of [2]. Their observation is that the Darboux transformation does preserve only the bosonic flows. But, as we have seen this can be remedied by reversing the sign of the fermionic flows. However, there is even another solution to this problem: the idea is that the single step of the transformation preserves the even flows but changes the odd flows in the way τ → −τ . Therefore, we do successively two steps of such transformation which do n n preserve the odd flows as well. Given two distinct solutions of the system (3) θ (even) and θ (odd), we construct an 0 1 operator T = ∂ +αD+a (8) e 6 where the coefficients α and a are given in terms of θ and θ 0 1 ˆ sdetF α = DlnsdetF, a = − (9) sdetF with the super-matrices F and Fˆ defined as θ θ ∂θ ∂θ F := 0 1 , Fˆ := 0 1 . Dθ Dθ Dθ Dθ 0 1 0 1 (cid:18) (cid:19) (cid:18) (cid:19) Here sdet means the super-determinant or Berezian of a super-matrix. We define as well Wˆ := T W∂−1 (10) e so that wˆ := w +α, wˆ := w −αw +a, 1 1 2 2 1 wˆ := w +(−1)|wn+1|αw +aw +α(Dw )+w , (n ≥ 1). n+2 n+2 n+1 n n nx Then Theorem 2. The operator Wˆ defined by (10) satisfies the SKP hierarchy k ∂ Wˆ = −(Wˆ ∂nWˆ −1) Wˆ , D Wˆ = −(Wˆ A Wˆ −1) Wˆ , (n ≥ 1). n − n n − So in the SKP case, the proper elementary Darboux transformation is generated by the compound operator T . e This Darboux transformation can be iterated and the solutions can be formed in terms of super ‘Wron´ski’ determinants as in super KdV case. Proposition 2. Given 2n distinct solutions θ (i = 0,··· ,2n−1) of the linear system (3) with i parities |θ | = (−1)i, then the new Sato’s operator is given by i 2n−1 Wˆ = T [n]W∂−n, T [n] = ∂n + a Di (11) e e i i=0 X where the coefficients a of the operator T [n] are given by solving the linear equations i e T [n](θ ) = 0, i = 0,··· ,2n−1. (12) e i To obtain the explicit transformations between fields, one has to solve the linear algebraic system (12) first, then compare the coefficients of the different powers Di of the equation (11). The system (12) can be solved easily as we did in [3, 4]. To this end, we introduce new variables a(0) := (a ,a ,··· ,a ), a(1) := (a ,a ,··· ,a ), 0 2 2n−2 1 3 2n−1 θ(0) := (θ ,θ ,··· ,θ ), θ(1) := (θ ,θ ,··· ,θ ), 0 2 2n−2 1 3 2n−1 θ(i) ∂θ(i) b(i) := ∂nθ(i), W(i) :=  . , i = 0,1 . .   ∂n−1θ(i)     7 so that the linear algebraic system (12) can be reformulated as (a(0),a(1))W = −(b(0),b(1)) (13) where W(0) W(1) W = DW(0) DW(1) (cid:18) (cid:19) Then, we obtain det W(0) −W(1)(DW(1))−1(DW(0)) i i sdetW i a = − = − , i = 1,··· ,n−1, 2i+2 (cid:16) (cid:17) sdetW det W(0) −W(1)(DW(1))−1(DW(0)) (cid:16) (cid:17) det (DW(1)) −(DW(0)) (W(0))−1W(1) i i a = − , i = 1,··· ,n. 2i−1 (cid:16) (cid:17) det DW(1) −(DW(0))(W(0))−1W(1) (cid:16) (cid:17) (0) (1) wherethematrixW(j) istheW(j) withitsi-throwreplacedbyb(j) andW = Wi Wi , i DW(0) DW(1) (cid:18) (cid:19) the matrix (DW(1)) is the matrix DW(1) with its i-th row replaced by b(1) and the matrix i (DW(0)) is the matrix DW(0) with its i-th row replaced by b(0). i By considering the equation (11), we obtain the general formulae 2n j−k wˆ = a C , k = 1,··· ,2n, 2n−k j i,j−k−i,j j=k i=0 X X 2n j wˆ = a C , k = 0,1,··· 2n+k j i,j+k−i,j j=0 i=0 X X with w = a = 1 and the a are given by (3.1)-(3.1) and 0 2n i k C = (−1)|wj|(k−i)(Diw ) i,j,k k −i j (cid:20) (cid:21) · where denotes the super-binomial coefficients. · (cid:20) (cid:21) 3.2 Binary Darboux Transformation In this section we consider the extension of the well known binary Darboux transformation for the KP hierarchy to its super-symmetrizations. For an eigenfunction ϕ and an adjoint eigenfunction ϕ∗ satisfying the linear system (3) and (4) respectively, we may introduce a potential operator Ω as follows DΩ(ϕ∗,ϕ) = ϕ∗ϕ,∂ Ω = res(D−1ϕ∗P ϕD−1), D Ω = res(D−1ϕ∗Q ϕD−1) (14) n n n n where we choose ϕ as an even quantity and ϕ∗ as an odd one, so that our potential operator Ω is even. 8 By lengthy calculation, see Appendix A, one can show that Ω is well defined, i.e., the equations (14) are consistent. In terms of Ω, we introduce the gauge operator T := 1−ϕΩ−1D−1ϕ∗ (15) where we assume the invertibility of Ω. The formal inverse of T is T−1 = 1+ϕD−1Ω−1ϕ∗. (16) In Appendix B we prove that Proposition 3. The gauge operator T solves (∂ T)T−1 = −(TP T−1) , (D T)T−1 = −(TQ T−1) . (17) n n − n n − ThispropertyqualifiesT forgeneratingaDarbouxtransformation[9]. GivenaSatooperator W we introduce its transformed Wˆ as follows Wˆ := TW. (18) Proposition 4. The operator Wˆ is again a Sato operator. Proof. We begun with ∂ Wˆ = = −(TP T−1) TW −T(W∂nW−1) W n n − − =(TP T−1) TW −TP W +T(W∂nW−1) W −TW∂n n + n + =(T(W∂ W−1) T−1) TW −TW∂n n + + =(TW∂nW−1T−1) TW −TW∂n + =−(Wˆ ∂nWˆ −1) Wˆ − and D Wˆ = −(TQ T−1) TW −T(WA W−1) W n n − n − = (TQ T−1) TW −TQ W +T(WA W−1) W −TWA n + n n + n = (T(WA W−1) W−1) TW −TWA n + + n = −(Wˆ A Wˆ −1) Wˆ n − Therefore, the transformed Sato’s operator indeed is the solution of Sato’s equation. The explicit transformation is wˆ = w +ϕΩ−1ϕ∗, 1 1 wˆ = w −ϕΩ−1 (Dϕ∗)+ϕ∗w , 2 2 1 j−1 (cid:16) (cid:17) wˆ = w +(−1)jϕΩ−1 ϕ∗(j) + (−1)k (Dϕ∗w )(j−k−1) −(ϕ∗w )(j−k−1) , 2j+1 2j+1 2k+1 2k+2 (cid:16) Xk=0 (cid:16) (cid:17) j wˆ = w +(−1)j+1ϕΩ−1 (Dϕ∗)(j) +(ϕ∗w )(j) + (−1)k (Dϕ∗w )(j−k) 2j+2 2j+2 1 2k (cid:16) Xk=1 (cid:16) +(ϕ∗w )(j−k) , 2k+1 (cid:17)(cid:17) 9 ∂if for j = 1,2,··· and where f(i) = . ∂xi For any given solution ψ of (3) we define ψˆ := Tψ, Pˆ := (TP T−1) , Qˆ = (TQ T−1) n n + n n + and as before Proposition 5. The function ψˆ satisfies ∂ ψˆ = Pˆ ψˆ, n n D ψˆ = Qˆ ψˆ. n n For the Manin–Radul SKP hierarchy the transformation on the Lax level can be easily obtained as Lˆ = TLT−1 and we found that Lˆ satisfies Lˆ = [(Lˆ2n) ,Lˆ], D Lˆ = {(Lˆ2n−1) ,Lˆ}−2Lˆ2n. tn + n + Indeed, we have Lˆ =[T T−1 +T(L2n) T−1,Lˆ] tn tn + =[−(TP T−1) +T(L2n) T−1,Lˆ] n − + =[(TP T−1) ,Lˆ] n + =[(TL2nT−1) ,Lˆ] = [(Lˆ2n) ,Lˆ] + + since (T(L2n) T−1) = 0 − + Likewise, D Lˆ ={(D T)T−1 +T(L2n−1) T−1,Lˆ}−2Lˆ2n n n + ={−(TQ T−1) +T(L2n−1) T−1,Lˆ}−2Lˆ2n n − + ={(T(L2n−1) T−1) ,Lˆ}−2Lˆ2n + + ={(Lˆ2n−1) ,Lˆ}−2Lˆ2n + because of (T(L2n−1) T−1) = 0. The binary Darboux transformation can be iterated. To do − + this, we need to consider the effect of this transformation for adjoint wave functions ψ∗. It is indeed not difficult to find they are transformed as follows ψˆ∗ = ψ∗ −ϕ∗(Ω(ϕ∗,ϕ))−1Ω(ψ∗,ϕ). Another important fact is the following Ω(ψˆ∗,ψˆ) = Ω(ψ∗,ψ)−Ω(ψ∗,ϕ)Ω(ϕ∗,ϕ)−1Ω(ϕ∗,ψ) then as in the pure bosonic case [10], by iterations of the binary Darboux transformation leads to the following result 10