N=2 supersymmetric unconstrained matrix GNLS hierarchies are consistent

ITP-UH-26/06 JINR-E2-2006- 170 N=2 sup ersymmetric unconstrained matrix GNLS hierarc hies are consisten t F. Delduc a, 1 , O. Lec htenfeld b, 2 , and A.S. Sorin c, 3 ( a ) L ab or atoir e de Physique † , Gr oup e de Physique Th ´ eorique, ENS Lyon, 46 A l l ´ ee d’Italie, 69364 Ly on, F r anc e ( b ) Institut f¨ ur The or etische Physik, L eibniz Universit¨ at Hannover, App elstr aße 2, D-30 167 Hannover, Germany ( c ) Bo goliub ov L ab or atory of Th e or etic al Physics, Joint I nstitute for Nucle ar R ese ar ch , 141980 D ubna, Mosc ow R e gion, Russia Abstract W e dev elop a pseudo–differen tial approac h t o the N =2 sup ersymmetric unconstrained matrix ( k | n, m )–Generalized Nonlinear Sc hr¨ odinger hierarc hies and prov e consistency of the corre- sp onding Lax–pair represen tation (nlin.SI/0201026). F urthermore, w e establish their equiv a- lence to the in tegrable hierarc hies deriv ed in the sup er–algebraic approach of t he homogeneously- graded lo op sup eralg ebra sl (2 k + n | 2 k + m ) ⊗ C [ λ, λ − 1 ] (nlin.SI/0206037). W e in tro duce an uncon ven tional definition of N = 2 supersymmetric strictly pseudo–differen tial o p erators so as to close their algebra among themselv es. P ACS : 02.2 0.Sv; 02 .3 0.Jr; 11.30.Pb Keywor ds : Completely integrable syste ms; Sup ersymmetry; D iscrete symmetries 1) E-Mail: fr anc ois.delduc@ens-lyon. fr 2) E-Mail: le chtenf@itp.uni-hannover.de 3) E-Mail: sorin@the or.jinr.ru † ) UMR 56 7 2 du CNRS, asso ci´ ee ` a l’Ecole Normale Sup ´ erieure de Ly on. 1 In tro duct i o n and Summary The N =2 sup ersymmetric unconstrained matrix ( k | n, m )–Generalized Nonlinear Sc hr¨ odinger (( k | n, m )–GNLS) hierarc hies w ere pro p osed in [1] b y exhibiting the corresp o nding matrix ps e udo– differ ential Lax–pair represen tation L = I ∂ + F D D∂ − 1 F in terms of a k × ( m + n ) matrix F with N = 2 unc onstr aine d sup erfield en tries fo r the bosonic isosp ectral flo ws. Their sup er–alg ebraic form ulation and recursion relations w ere pro p osed in [2] on the basis o f t he homogeneously-graded lo op sup era lg ebra sl (2 k + n | 2 k + m ) ⊗ C [ λ, λ − 1 ]. These hierarc hies generalize and con ta in as limiting cases man y other in t eresting N =2 sup er- symmetric hierarc hies discussed in the literature: When matrix en tries are chiral and an tic hiral N =2 s up erfields, these hierarc hies re pro duce the N = 2 c hiral matrix ( k | n, m )-GNLS hierar- c hies [3, 4], and in turn the latter coincide with the N =2 GNLS hierarc hies of references [5, 6] in t he scalar case k =1. When matrix entries are unconstrained N =2 superfields and k =1 , these hierarc hies are equiv alen t to the N =2 sup ersymmetric multicomponent hierarc hies [7]. The b osonic limit of the N =2 unconstrained matrix ( k | 0 , m )–GNLS hierarc h y repro duces the b osonic matrix NLS equation elab orated in [8] via the g l (2 k + m ) / ( g l (2 k ) × g l ( m ))–coset con- struction. The N = 2 matrix (1 | 1 , 0)–G NLS hierarc h y is related to one of three differen t existing N =2 sup ersymmetric KdV hierarc hies – t he N =2 α =1 KdV hierarc hy – by a reduction [7, 1, 9]. Self-consistency of the Lax–pair represe n tation for t he N =2 sup ersymmetric unconstrained matrix ( k | n, m )–GNLS hierarc hies w as actually prov en in [1] only for the first four flows , but conjectured for the general case. The equiv alence of their sup er–alg ebraic and pseudo– differen tial for mulations w as established in [2], but a gain for the first few flo ws only . The presen t letter completes these pro ofs. In Section 2 w e de v elop a pse udo–differen tial approac h to the N = 2 sup ersymmetric unc onstrained ma t r ix ( k | n, m )–GNLS hierarc hies in N =2 sup er- space and r igorously construct their Lax–pair represe n tation ∂ ∂ t p L = [( L p ) ⊕ , L ] with ∂ ∂ t p F =  ( L p ) ⊕ F  and ∂ ∂ t p F = −  F ( ← L p ) ⊕  , where w e introduce a n uncon ve n tional definition of N = 2 supersymmetric strictly pseudo– differen tial op erators so as to close their algebra among themse lv es 1 . F urthermore, w e pro duce the recursion relations for the corresp onding isosp ectral flo ws. As w e establish in Section 3, this Lax–pair represen tation agrees with the one deriv ed in the sup er–a lgebraic approach o f the homogeneously-graded lo op sup eralgebra sl (2 k + n | 2 k + m ) ⊗ C [ λ, λ − 1 ]. Th us, w e finally prov e the conjectured equiv alence of the tw o hierarc hies. Apart from the Lax–pair represen tation fo r the isospectral flows and the recursion rela- tions of these hierarc hies, we prese n tly do not know other c har a cteristic prop erties lik e their (bi)Hamiltonian structures , discrete symmetries etc., althoug h part of these are kno wn for some limiting cases. W e hop e to address these problems in future. 1 The precise notation will b e explained ther e. 1 2 Pseudo– d ifferen tial approac h Our starting p oin t is the Lax op erator for the N =2 sup ersymmetric unconstrained matrix ( k | n, m )–GNLS hierarc hies introduced in [1] 2 L = I ∂ + F D D∂ − 1 F . (1) Here, F ≡ F Aa ( Z ) and F ≡ F aA ( Z ) ( A, B = 1 , . . . , k ; a, b = 1 , . . . , n + m ) a re rectan- gular matrices whic h en tries are unconstrained N = 2 sup erfields, I is the unity matrix, I AB ≡ δ AB , and t he matrix pro duct is implied, for example ( F F ) AB ≡ P n + m a =1 F Aa F aB and ( F F ) ab ≡ P k A =1 F aA F Ab . The mat r ix en tries are Grassmann even sup erfields for a = 1 , . . . , n and Grassmann o dd sup erfields for a = n + 1 , . . . , n + m . Th us, fields do not comm ute, but rather satisfy F Aa F bB = ( − 1) d a d b F bB F Aa where d a and d b are the Grassmann pa rities of the matrix elemen ts F Aa and F bB , r espectiv ely , d a = 1 ( d a = 0) for o dd (ev en) entries. Sup erfields dep end on the co ordinates Z = ( z , θ, θ ) of N = 2 sup erspace, and the N = 2 sup ersymmetric fermionic co v ariant deriv atives D , D are D = ∂ ∂ θ − 1 2 θ ∂ ∂ z , D = ∂ ∂ θ − 1 2 θ ∂ ∂ z , D 2 = D 2 = 0 ,  D , D  = − ∂ ∂ z ≡ − ∂ . (2) The set of op erators { ˆ o i , i ∈ Z } := { D n D n ∂ m , n, n = 0 , 1 , m ∈ Z } (3) forms a ba sis in the asso ciative algebra o f the sup ermatrix v alued pseudo–differen tial o p erators on N = 2 superspace O = ∞ X i = −∞ f i ˆ o i := N max X m = −∞ 1 X n, n =0 f n,n,m D n D n ∂ m , d O = d ˆ o i + d f i (4) where f i is a sup ermatrix v alued N = 2 sup erfield and ˆ o i is a basis op erator with the G rassmann parities d f i and d ˆ o i , respective ly , and w e unde rstand that the op erator O p ossesses a definite Grassmann parity d O . W e shall sa y that O ab ov e is a differential op era t o r if the sum o v er m is restricted t o p ositive or zero v alues only , and that O is strictly pseudo–differen tial if the sum o v er m is restricted to negat ive v alues o f m . The set of the differen tial op erators a nd the set of the strictly pseudo–differen tial op erators b oth form a subalgebra of the whole space of pseudo– differen tial op erators. An y pseudo–differen tial op erator O is the sum of a differen tial op erator O ⊕ and a strictly pseudo–differen t ia l op erator O ⊖ . Hereafter, the notation ( L p ) ˆ o i denotes t he sup ermatrix co efficien t of the basis elemen t ˆ o i , i.e. ( L p ) ˆ o i ˆ o i b elongs to the expans ion (4) of L p ; ( O f ) has t he meaning of a sup ermatrix v alued pseudo–differen tial op erator O acting only o n a sup ermatrix v alued function f inside the brac k ets 3 . 2 Note, the sup ermatrices { F , F } are re - scaled b y √ 2 comparing to [1]. 3 In order to avoid mis under standing, let us remark the difference in the notations ( O f ) and { ( O f ) ⊕ , ( O f ) ⊖ } : the former represents a sup ermatrix v alued sup erfield, while the latter cor resp onds to a differential and strictly pseudo–differential par ts of the o pe r ator O f , resp ectively . 2 Remark. The definition whic h is used in this pap er of a strictly pseudo–differen tia l op erat or sligh tly differs from the one which was used in the articles [1, 2 ]. There, instead of the basis elemen ts D D ∂ n , one w as r a ther using as basis elemen ts the op erators [ D , D ] ∂ n . Differential and strictly pseudo–differen tial op erators w ere defined with resp ect t o this basis. Any pseudo– differen tial op erator O may b e separated into a differen tia l op erator O + and a strictly pseudo– differen tial op erator O − according to this basis. This other definition has the dr awbac k that strictly pseudo–differen tial op erators do not form a closed algebra, b ecause of the relation ([ D , D ] ∂ − 1 ) 2 = 1. The relatio n b et w een the t w o definitions is easily obta ined b y using the relation D D ∂ − 1 = − 1 2 + 1 2 [ D , D ] ∂ − 1 . Defining the residue of the op erat or O as the co efficien t of D D ∂ − 1 in the new basis ( t his differs by a factor of 2 from the definition in [1, 2], where it w as defined as the co efficien t of [ D , D ] ∂ − 1 ), one finds t he following r elation O ⊕ = O + + 1 2 r es O , (5) whic h a llo ws one to relate calculations in previous articles and in the presen t art icle. Definition 1. W e define the in v olutiv e automor phism ∗ of the sec ond order of the sup er- symmetry algebra { ∂ , D , D } ∗ = −{ ∂ , D , D } , ( D D ) ∗ = − D D , ( D D ) ∗ = − D D . (6) It can b e e xtended to all basis elemen ts in (3) using the rule ( ˆ o i ˆ o j ) ∗ = ( − 1) d ˆ o i d ˆ o j ˆ o ∗ j ˆ o ∗ i . When applied to a sup ermatrix f ⇒ f ∗ simply amounts to a c hange in the sign of its Gr assmann–o dd en tries. Its k-fold action on the sup ermatrix f will b e denoted f ∗ ( k ) ( k = 0 , 1 mod 2). Remark. Relations (6) repro duce the conv en tio nal op erato r-conjugation rules for the fermionic and b osonic cov aria n t deriv a t iv es, although the star–op eration ∗ , b eing applied to a sup ermatrix f ∗ , differs comparing to the conv en t io nal op eration of the super–tr ansp osition of a s up erma- trix f T , i.e. f ∗ 6 = f T . W e also r emark that ( F F ) ∗ ≡ F ∗ F ∗ = F F , F F ∗ = F ∗ F , while ( F F ) ∗ ≡ F ∗ F ∗ 6 = F F . Definition 2. W e in tro duce the adjoint op erator ← O b y defining its action on the sup ermatrix v alued superfield f with the Grassmann parit y d f f ← O := ∞ X i = −∞ ˆ o ∗ i ( f f i ) ∗ ( d ˆ o i ) . (7) Remark. This definition generalizes the definition of the adjoint op erator to the non- ab elian, noncommu tativ e case. F or t he ab elian, commu tativ e case, i.e. when f and f i are not (sup er)matrices, but comm utative functions, it repro duces the con v en tio nal definition of the adjoint op erator. Due to this reason w e call the op erator ← O noncomm uta t ively –adjoint op erator. Equation (7) defines a pro duct of a noncomm uta tiv ely–adjoint op erator and a supermatrix– v alued sup erfield. In order to consisten tly define a pro duct of different noncomm utativ ely– adjoin t op erators with themselv es, w e firstly need to prov e: 3 Prop osition 1. ( f ← − − − − O 1 ... O k ) = ( f ← O 1 ... ← O k ) ≡ ((( f ← O 1 ) ... ) ← O k ) . (8) Pro of. By induc tion, it is sufficien t to c hec k (8) for tw o op erators O 1 and O 2 , whic h has b een done with the help of the rules (6,7).  Remark. Prop osition 1 giv es the result of the a ction of a pro duct of noncomm utative ly- adjoin t op erators on a sup ermatrix–v alued sup erfield. Using the latter w e define a pro duct Q k i =1 ← O i of noncomm utativ ely–adjoin t op erators whic h generalizes the definition of the pro duct of the conjugated op erators in the comm utativ e case to the noncomm utat iv e case. Definition 3. f ← O 1 ... ← O k := f ← − − − − O 1 ... O k . (9) Remark. It is ob vious that the r.h.s. of eq. ( 9) can b e calculated using eq. (7), if one tak es in t o accoun t that due to the asso ciativit y of the algebra o f pseudo–differen tial op erato r s the pro duct O 1 ... O k = O , where O is a pseudo–differen tia l op erator in the canonical fo rm (4), therefore one can use (7). The consistenc y of eq. (9) with eq. (7 ) is provided by eq. (8). Lemma 1. D D ∂ − 1 f D D ∂ − 1 = ( D D ∂ − 1 f ) D D ∂ − 1 + D D ∂ − 1 ( f ← − − − − D D ∂ − 1 ) , (10) ( O ⊕ f D D ∂ − 1 ) ⊖ = ( O ⊕ f ) D D ∂ − 1 , (11) ( D D∂ − 1 f O ⊕ ) ⊖ = D D ∂ − 1 ( f ← − O ⊕ ) . (12) Pro of. Equalit y (10) results from the follo wing simple relation : ∂ − 1 f ∂ − 1 = ( ∂ − 1 f ) ∂ − 1 + ∂ − 1 ( f ← − ∂ − 1 ) (13) One acts with D D on b oth sides of (13), then tries to push D and D to the righ t in the first term, and to t he left in the second term. Equalit y (11) is obvious if one tak es in to a ccoun t t hat the pseudo–differen t ial op erator D D∂ − 1 b eing multiply either b y D or D from the right or left b ecomes a differen tial op erator . Equalit y (12) is an op erator-adjoin t coun terpart o f equalit y ( 1 1).  It should b e noted that, although there is an arbitrariness in the definition of the action of ∂ − 1 on a function f , this a rbitrariness do es not show up in (10) b ecause it comp ensates b etw een b oth terms o n the righ t- hand side. Prop osition 2. ( L p ) ⊖ = p − 1 X k =0 ( L p − k − 1 F ) D D∂ − 1 ( F ← − L k ) , p ∈ N . (14) Pro of. The pro of is by induction with the use of relatio ns (10 – 12). Equation (14) is ob vi- ously correct at p = 1 (compare with eq. (1)). If it is correct for the p = n case, then w e hav e for the p = n + 1 case ( L n +1 ) ⊖ ≡ ( LL n ) ⊖ 4 =  I ∂ + F D D ∂ − 1 F  ( L n ) ⊕ + n − 1 X k =0 ( L p − k − 1 F ) D D∂ − 1 ( F ← − L k )  ! ⊖ =  F D D ∂ − 1 F ( L n ) ⊕ + n − 1 X k =0 ( ∂ L n − k − 1 F ) D D ∂ − 1 ( F ← − L k ) + F D D ∂ − 1 F n − 1 X k =0 ( L n − k − 1 F ) D D ∂ − 1 ( F ← − L k )  ⊖ = F D D ∂ − 1  F ← − − − ( L n ) ⊕ + n − 1 X k =0 F ( L n − k − 1 F ) ← − − − − D D ∂ − 1 ( F ← − L k )  + n − 1 X k =0  ( ∂ + F D D∂ − 1 F ) L n − k − 1 F  D D ∂ − 1 ( F ← − L k ) = F D D ∂ − 1 ( F ← − L n ) + n − 1 X k =0 ( L n − k F ) D D ∂ − 1 ( F ← − L k ) = n X k =0 ( L n − k F ) D D∂ − 1 ( F ← − L k ) . (15)  Prop osition 3. ( L p +1 ) ⊕ = ( L p ) ⊕ L − (( L p ) ⊕ F ) D D ∂ − 1 F + p − 1 X k =0 ( L p − k − 1 F ) D D ( F ← − L k ) (16) = L ( L p ) ⊕ − F D D ∂ − 1 ( F ← − − − ( L p ) ⊕ ) + p − 1 X k =0 ( L p − k − 1 F ) D D ( F ← − L k ) . (17) Pro of. This is an easy calculation using eqs. (11,12,14) and ob vious identities ( L p +1 ) ⊕ = ( L p ) ⊕ L − (( L p ) ⊕ L ⊖ ) ⊖ + (( L p ) ⊖ L ⊕ ) ⊕ = L ( L p ) ⊕ − ( L ⊖ ( L p ) ⊕ ) ⊖ + ( L ⊕ ( L p ) ⊖ ) ⊕ . (18)  Corollary . Subtracting eq. (17 ) from eq. (16) w e obtain [( L p ) ⊕ , L ] = (( L p ) ⊕ F ) D D∂ − 1 F − F D D ∂ − 1 ( F ( ← L p ) ⊕ ) . (19) If one intro duces ev olution deriv atives (flo ws) ∂ ∂ t p according to the form ula ∂ ∂ t p F = (( L p ) ⊕ F ) , ∂ ∂ t p F = − ( F (( ← L p ) ⊕ ) , (20) then eq. (19) tak es the form of the Lax pair represen tation ∂ ∂ t p L = [( L p ) ⊕ , L ] (21) 5 whic h w as prop osed in [1]. Actually , its self-consistency was prov en in [1] only for the first few flo ws p=0,1,2 a nd 3, t hen conjectured for the general case there, and the correspo nding in t egrable hierarc hies w ere called the N = 2 sup ersymmetric unconstrained matrix ( k | n, m )– Generalized Nonlinear Sc h¨ odinger hierarc hies. The algebra of t he flows in (21) can easily b e calculated [ ∂ ∂ t m , ∂ ∂ t n ] = 0 , (22) it is ab elian algebra of the isosp ectral flo ws. The Lax–pair represen tation (21 ) ma y b e seen as the in tegra bility condition for the fo llo wing linear syste m: Lψ 1 = λψ 1 , ( 2 3) ∂ ∂ t p ψ 1 = (( L p ) ⊕ ψ 1 ) (24) where λ is the sp ectral parameter and the eigenfunction ψ 1 is the Bak er-Akhiezer f unction of the hierarc hy . Pro jecting the Lax–pair represen tation (2 1) on D D∂ − 1 , D ∂ − 1 , D ∂ − 1 and ∂ − 1 parts, one can straig h tf orw a r dly extract the follo wing ev olution equations ( ∂ ∂ t p F F ) = ( r es ( L p )) ′ + [ r es ( L p ) , F F ] , (25) ( ∂ ∂ t p F ∗ D F ) = − ( L p ) ′ D ∂ − 1 + F F ( L p ) D ∂ − 1 + r es ( L p ) ( F ∗ D F ) , (26) ( ∂ ∂ t p F ∗ D F ) = ( L p ) ′ D ∂ − 1 + ( L p ) D ∂ − 1 F F − ( F ∗ D F ) r es ( L p ) − F F ( D r es ( L p )) + r es ( L p )( D F F ) , (27) ( ∂ ∂ t p F D D F ) = ( L p ) ′ ∂ − 1 + ( F ∗ D F ( L p ) D ∂ − 1 ) + ( L p ) D ∂ − 1 ( F ∗ D F ) + r es ( L p )( D F ∗ D F ) (28) whic h can b e used t o express r es ( L p ), ( L p ) D ∂ − 1 , ( L p ) D ∂ − 1 and ( L p ) ∂ − 1 , en tering these equations, in terms of the t ime deriv ativ e ∂ ∂ t p of differen t f unctionals of F and F . With this aim w e need to in tro duce a k × k matrix g by the consisten t set of equations. Definition 4. g ′ = − g F F , ( D g ) = −  ∂ − 1 g ( D F F ) g − 1  g , ( D g ) = −  ∂ − 1 g ( D F F ) g − 1  g . (29) With the help of g the resolution of eqs. ( 2 5 – 28) with respect to r es ( L p ), ( L p ) D ∂ − 1 , ( L p ) D ∂ − 1 and ( L p ) ∂ − 1 is rather simple r es ( L p ) = − ( g − 1 ∂ ∂ t p g ) ≡ ( ∂ − 1 ∂ ∂ t p F F g − 1 ) g , (30) ( L p ) D ∂ − 1 = − g − 1 ( ∂ − 1 ∂ ∂ t p g F ∗ D F ) ≡ − ( ∂ − 1 ∂ ∂ t p F ∗ D F ) + ( ∂ − 1 ∂ ∂ t p F F g − 1 )( ∂ − 1 g F ∗ D F ) − ( ∂ − 1 ∂ ∂ t p F F g − 1 ∂ − 1 g F ∗ D F ) , (31) ( L p ) D∂ − 1 = [( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ) g ] , (32) ( L p ) ∂ − 1 = ( ∂ − 1 ∂ ∂ t p F D D F ) − [( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 )( ∂ − 1 g F ∗ D F )] + ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ∂ − 1 g F ∗ D F ) (33) 6 where in eqs. (32 – 33) the fermionic deriv ativ e D entering the square brack ets a cts on the right inside these brac k ets. This can easily b e v erified by directly substituting these expressions in to the original equations (25 – 28) and using eqs. (29). Prop osition 4. ( L p +1 ) ⊕ = ( L p ) ⊕ L − ( ∂ ∂ t p F ) D D∂ − 1 F + ( ∂ − 1 ∂ ∂ t p F D D F ) − ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 )( ∂ − 1 g F ∗ D F ) + ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ∂ − 1 g F ∗ D F ) ( 3 4) = L ( L p ) ⊕ + F D D∂ − 1 ( ∂ ∂ t p F ) + ( ∂ − 1 ∂ ∂ t p F D D F ) − ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 )( ∂ − 1 g F ∗ D F ) + ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ∂ − 1 g F ∗ D F ) ( 3 5) where in these equations the fermionic deriv ativ es D a nd D entering the brac k ets act as op er- ators on the r igh t b oth inside and outside the brac k ets. Pro of. T aking ( L p +1 ) ⊕ in (16) and using eqs . (20) as well as the identit y p − 1 X k =0 ( L p − k − 1 F ) D D ( F ← L k ) = r es ( L p ) D D + ( L p ) D ∂ − 1 D + ( L p ) D ∂ − 1 D + ( L p ) ∂ − 1 (36) whic h f ollo ws from eq. (14), one can easily o btain the follo wing express ion: ( L p +1 ) ⊕ = (( L p ) ⊕ ) L − ( ∂ ∂ t p F ) D D∂ − 1 F + r es ( L p ) DD + ( L p ) D ∂ − 1 D + ( L p ) D ∂ − 1 D + ( L p ) ∂ − 1 . (37) Substituting r es ( L p ) (30), ( L p ) D ∂ − 1 (31), ( L p ) D ∂ − 1 (32) and ( L p ) ∂ − 1 (33) into eq. (37), w e arrive at the first equalit y (34). The second equalit y (35) can obv iously b e deriv ed fro m the first one if one substitutes ( L p ) ⊕ L b y L ( L p ) ⊕ + ∂ ∂ t p L there, a ccording to eq. (21).  Corollary: recursion relations. Applying the noncomm utativ ely–adjoin t of op erator re- lation (34) t o t he su p ermatrix v alued sup erfield F from the right and s imilarly applying ( 35) to F from the left as w ell as using eqs. (20) it is not complicated to obtain recurrence relations relating flo ws with the ev o lution deriv ativ es ∂ ∂ t p +1 and ∂ ∂ t p ( ∂ ∂ t p +1 F ) = − ( ∂ ∂ t p F ) ′ + ( D D ∂ − 1 ∂ ∂ t p F F ) F − [ F ( ∂ − 1 ∂ ∂ t p F ← D ← D F ) − F ( ∂ − 1 ∂ ∂ t p F ∗ ← D F g − 1 )( ∂ − 1 g F ∗ ← D F ) + F ( ∂ − 1 ∂ ∂ t p F ∗ ← D F g − 1 ∂ − 1 g F ∗ ← D F )] , (38) ( ∂ ∂ t p +1 F ) = ( ∂ ∂ t p F ) ′ + ( F D D∂ − 1 ∂ ∂ t p F F ) + [( ∂ − 1 ∂ ∂ t p F D D F ) F − ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 )( ∂ − 1 g F ∗ D F ) F + ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ∂ − 1 g F ∗ D F ) F ] (39) where the fermionic deriv ative s D and D , en tering the square brac kets in eqs. (38) and (3 9), act inside t hese brack ets on the left and righ t, resp ectiv ely . 7 3 Sup er–algebraic approac h F ollow ing the sup er–algebraic approach 4 of ref. [1 0], in [2] a wide class o f in tegrable hierarc hies w as constructed whic h corresp onds to the homogeneous gradation of the lo op sup eralgebra sl ( 2 k + n | 2 k + m ) ⊗ C [ λ , λ − 1 ] with the splitting matr ix E and the grading op erato r d , E =       1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1       , d = λ ∂ ∂ λ . (40) The corresponding isosp ectral flo ws are [2] ( ∂ ∂ t p L z ) = [(Θ λ p E Θ − 1 ) + − ( G − 1 ∂ ∂ t p G ) , L z ] (41) where t he dressing matrix Θ is obta ined from dressing the Lax op erator L z L z := ∂ − λE + A = Θ − 1  ∂ − λE  Θ , Θ = 1 + ∞ X k =1 λ − k θ ( − k ) . (42) Hereafter, the subsc ript + denotes the pro jection on the p ositiv e ho mogeneous grading (40), A =       0 0 F 0 0 0 0 ( D F ) 0 0 − ( D D F ) ( D F ∗ ) 0 − ( D F ∗ ) − F − ( F ∗ D F ) 0 ( D F ) − F F 0 − ( D F ∗ D F ) ( F ∗ D F ) ( D D F ) − ( D F F ) − F F       (43) and G =       1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 − ( ∂ − 1 g F ∗ D F ) 0 0 g 0 − ( D ∂ − 1 g F ∗ D F ) ( ∂ − 1 g F ∗ D F ) 0 ( D g ) g       . (44) It is easily seen that t he matrix G (44) en tering into the Lax–pair represen tation (41) is nonlo cal. Moreo v er, the N = 2 sup erfield en tries of the connection A (43) are not indep enden t quan tit ies, i.e. they are sub jected to constrain ts. Wh y in this case do isosp ectral matrix flo ws (41) b e lo cal, as it is ob viously the case f or the flows (21)? Why are they sup ersymmetric, or in other w ords, wh y do t hese flo ws prese rv e the ab ov e–men tioned constrain ts? Finally , how can one see in g eneral that these flo ws coincide with the isospectral flow s (2 1). These questions w ere raised in [2], but clarified only partly there. Based on t he pseudo–differen tial approa ch 4 F or mor e recent developmen t of the supe r–algebr aic appro ach, see ref. [1 1] and references therein. 8 dev elop ed in the previous Section w e are able to pro v e here that the sup er–algebraic isosp ectral matrix flo ws (41) are equiv alen t t o the pseudo–differen tial isosp ectral flo ws (21), therefore the former are lo cal a nd sup ersymmetric as w ell, b ecause it is the case for the latter. The Lax–pair represen tat io n (41) may b e seen as the integrabilit y conditio n for t he follo wing linear system: L z Ψ = 0 , (45) ( ∂ ∂ t p G Ψ) = ( G Θ λ p E ( G Θ) − 1 ) + G Ψ (46) where Ψ T = ( ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 ). In order to pro v e eq uiv alence of the L a x–pair represen tat io ns (41) and (2 1) it is enough to prov e equiv alence of the corresp onding linear systems (45 – 46) and (23 – 24). Prop osition 5. The linear systems (45 – 46) and (23 – 24) ar e equiv alen t. Pro of. The first equation (45) of the linear system (45 – 46) is equiv alen t to the first equation (23) of the linear system (23 – 24) and p ossesses the solution Ψ =       ψ 1 ( D ψ 1 ) ( D D∂ − 1 F ψ 1 ) ( Dψ 1 ) ( D Dψ 1 )       (47) whic h w as actually o bserv ed in [2 ], and it w as the starting p oin t for the super–alg ebraic con- struction dev elop ed there. In order to demonstrate that t he second equation (46) of the linear system (45 – 4 6) is equiv a len t to the second equation (24) o f the linear sys tem ( 23 – 24) as we ll, w e use the equality [10] ( G Θ λ p +1 E ( G Θ) − 1 ) + = λ ( G Θ λ p E ( G Θ) − 1 ) + + ( ∂ ∂ t p Gθ ( − 1) ) (48) and rewrite eq. (46) in the follo wing equiv alent form: ( ∂ ∂ t p +1 G Ψ) = λ ( ∂ ∂ t p G Ψ) + ( ∂ ∂ t p Gθ ( − 1) ) G Ψ . (49) Then, substituting G (4 4), Ψ (4 7), and θ ( − 1) , deriv ed fr o m t he dressing condition ( 4 2), in to eq. (49), the latter b ecomes ( ∂ ∂ t p +1 ψ 1 ) = λ ( ∂ ∂ t p ψ 1 ) + [(( ∂ − 1 ∂ ∂ t p F D D F ) − ( ∂ ∂ t p F ) D D∂ − 1 F − ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 )( ∂ − 1 g F ∗ D F ) + ( ∂ − 1 ∂ ∂ t p F ∗ D F g − 1 ∂ − 1 g F ∗ D F )) ψ 1 ] (50) where the fermionic deriv ativ es D and D , en tering t he square brac k ets, act inside these brac kets. Eq. (50) is satisfied if and o nly if eq. (24) is satisfied, a nd the latter is ob vious if one tak es in to accoun t eq. (23) and relation (34) of the Prop osition 4.  9 Let us discus s shortly the lo calit y of the isosp ectral flo ws in t he super–a lg ebraic Lax–pair represen tation (41). The connection A (43) en tering in to the Lax op erator L ( 42) is a lo cal functional of the sup ermatrix–v alued sup erfields F , F and their deriv atives . It is a lso kno wn [10] that the mat rix (Θ λ p E Θ − 1 ) + is a lo cal functional. Using (44), (30 – 31) o ne can calculate the second term of the Lax represen tation (41) ( G − 1 ∂ ∂ t p G ) =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( L p ) D ∂ − 1 0 0 − r es ( L p ) 0 ( D ( L p ) D ∂ − 1 ) − ( L p ) D ∂ − 1 0 − ( D r es ( L p )) − r es ( L p )       (51) whic h is a lo cal functional as we ll. Th us, all the ob jects inv olv ed in to the Lax–pair repres en- tation (41 ) are local, therefore t he same is true with respect to the correspo nding isosp ectral flo ws. Ac kno wledgmen ts. A.S. w ould lik e to t hank the Lab or atoire de Ph ysique de l’ENS Ly on and Institut f ¨ ur Theoretisc he Phy sik, Univ ersit¨ at Ha nnov er, fo r the hospitality during the course of this w ork. 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