Table Of Content

Quantization of N=1 and N=2 SUSY KdV models 1 2 6 Petr P. Kulish , Anton M. Zeitlin 0 St. Petersburg Department of Steklov Mathematical 0 2 Institute, Fontanka, 27, St. Petersburg, 191023, Russia 1 [email protected] 2 [email protected] n a 2 http://www.ipme.ru/zam.html J 3 February 1, 2008 1 v 9 Abstract 1 The quantization procedure for both N=1 and N=2 supersymmetric 0 1 Korteweg-de Vries (SUSY KdV) hierarchies is constructed. Namely, the 0 quantum counterparts of the monodromy matrices, built by means of 6 the integrated vertex operators, are shown to satisfy a specialization of 0 reflectionequation,leadingtothequantumintegrabletheory. Therelation / of such models to the study of integrable perturbed superconformal and h topological models is discussed. t - p e 1 Introduction h : v The Superconformal Field Theory (SCFT) is very important in many areas i X ofMathematicalPhysics,suchasSuperstringtheory,whereitappearsasabasic r partofthe StringWorldsheetPhysicsorthe theoryoftwo-dimensionalsolvable a latticemodels,e.g. tricriticalIsingmodelasthewellknownexample. Thesym- metriesofSCFTaredescribedbythegeneratorsoftheSuperconformalAlgebra (SCA). This algebra is a supersymmetric extension of the infinite dimensional Virasoroalgebra,whichinitsturnisacentralextensionofthealgebraofvector fields on a circle. TherearetwotypesofSCAwhichareespeciallyimportantinphysicalprob- lems, these are the so-called N=1 and N=2 SCA, named by the number of fermionic degrees of freedom in the string worldsheet superspace. The infinite dimensionality of this symmetry allows to look for integrable structures inside SCFT, that is infinite families of pairwise commuting integrals of motion (IM). Motivated by the “physical” reasons we impose also the condition of the supersymmetric (SUSY) invariance of these families, i.e. the SUSY generator(s) should be included in these families. In this paper we review some results concerning the construction of these 1 families for both N=1 and N=2 SCFT. The key idea is that on the classical level the integrable structures are provided by the so-called second Hamilto- nian structure of N=1 and N=2 SUSY KdV correspondingly. In order to build infinite family of pairwise commuting quantum IM, we need to quantize the corresponding integrable models. Here we review the quantization procedure for both N=1 and N=2 SUSY KdV accordingto the results from the papers [1], [2], [3]. In the first part (Sec. 2) we consider the classical theory of SUSY N=1 KdV system, based on the twisted affine superalgebra C(2)(2) sl(12)(2) osp(22)(2). We introduce ≃ | ≃ | the supersymmetric Miura transformation and monodromy matrix associated with the corresponding L-operator. Then the auxiliary L-matrices are constructed, which satisfy the quadratic Poisson bracket relation. In Sec. 4 we will show that quantum counterparts of these matrices coincide with a vertex- operator-represented quantum R-matrix. The quantum version of the Miura transformation, i.e. the free field representation of the superconformal algebra, is given in Sec. 3. In Sec. 4 the quantum C (2)(2) superalgebra [14] is in- q troduced. Then it is shown that the corresponding quantum R-matrix can be represented by two vertex operators, satisfying Serre relations of lower Borel subalgebraofC (2)(2) andinthe classicallimititcoincideswithL-matrix. The q vertex-operator-representedquantumR-matrixL(q) satisfiestheso-calledRTT- relation,which gives a possibility to consider the model from a point of view of Quantum Inverse Scattering Method (QISM) [4], [5]. In the next four sections we switch to the study of the SUSY N=2 KdV quantization procedure. The N=2 KdV model is based on the sl(1)(21) affine | superalgebra. The classicalversionofthe associatedmonodromy matrix is representedby means offamiliar P-exponentialbut the exponentconsists notonly ofgeneratorsofsimplerootsandthesquaresoffermioniconeswithappropriate exponentialmultipliers,butalsogenerators,correspondingtomorecomplicated composite roots (see Sec. 5). We prove that (as it was in the N=1 case) all these terms, corresponding to the composite roots disappear from the first it- eration of the quantum generalization of the P-exponential; we also show that this “quantum” P-exponential coincide with the reduced universal R-matrix of sˆl (21) quantum affine superalgebra (see Sec. 6,7). We demonstrate that the q | supertraces of the quantum version of the monodromy matrix in different evaluation representations, the so-called transfer-matrices (by the obvious analogy with the lattice case) commute with the supersymmetry generators. Hence, these generators can be included in the family of the integrals of motion both on classical and the quantum level (see Sec. 8). InthelastsectionwediscusstherelationofN=1andN=2SUSYKdVmodels to the superconformal/topologicaltheories and their integrable perturbations. 2 Integrable SUSY N=1 KdV hierarchy The SUSY N=1 KdV system can be constructed by means of the Drinfeld- Sokolov reduction applied to the C(2)(2) twisted affine superalgebra [6]. The 2 corresponding -operator has the following form: L ˜F =Du,θ Du,θΦ(h1+h2) λ(e1++e2++e1− e2−), (1) L − − − where D = ∂ +θ∂ is a superderivative, the variable u lies on a cylinder u,θ θ u of circumference 2π, θ is a Grassmann variable, Φ(u,θ) = φ(u) i θξ(u) is a − √2 bosonic superfield; h , h , e , e are the Chevalley generators of C(2) with 1 2 1± 2± the following commutation relations: [h ,h ]=0, [h ,e ]= e , [h ,e ]= e , (2) 1 2 1 2± 2± 2 1± 1± ± ± ad2 e =0, ad2 e =0, e1± 2± e2± 1± [hα,eα±]=0 (α=1,2), [eβ±,eβ′∓]=δβ,β′hβ (β,β′ =1,2), where the supercommutator [,] is defined as follows: [a,b] ad b ab a ≡ ≡ − ( 1)p(a)p(b)ba, where parity p is equal to 1 for odd elements and is equal to − 0 for even ones. In the particular case of C(2) h1,2 are even and e±1,2 are odd. Theoperator(1)canbeconsideredasmoregeneralone,takenintheevaluation representation of C(2)(2): =D D Φh (e +e ), (3) F u,θ u,θ α δ α α L − − − where h , e , e are the Chevalley generators of C(2)(2) with such commu- α δ α α − tation relations: [h ,h ]=0, [h ,e ]= e , [h ,e ]= e , (4) α1 α0 α0 ±α1 ∓ ±α1 α1 ±α0 ∓ ±α0 [h ,e ]= e , [e ,e ]=δ h , (i,j =0,1), αi ±αi ± ±αi ±αi ∓αj i,j αi ad3 e =0, ad3 e =0 e±α0 ±α1 e±α1 ±α0 wherep(h )=0,p(e )=1andα α,α δ α. ThePoissonbrackets α0,1 ±α0,1 1 ≡ 0 ≡ − for the field Φ, obtained by means of the Drinfeld-Sokolov reduction are: Du,θΦ(u,θ),Du′,θ′Φ(u′,θ′) =Du,θ(δ(u u′)(θ θ′)) (5) { } − − and the following boundary conditions are imposed on the components of Φ: φ(u+2π)=φ(u)+2πip,ξ(u+2π)= ξ(u). The -operatoris writtenin the F ± L Miura form, making a gauge transformation one can obtain a new superfield (u,θ) D Φ(u,θ)∂ Φ(u,θ) D3 Φ(u,θ) = θU(u) iα(u)/√2, where U U ≡ u,θ u − u,θ − − and α generate the superconformal algebra under the Poissonbrackets: U(u),U(v) = δ (u v)+2U (u)δ(u v)+4U(u)δ (u v), (6) ′′′ ′ ′ { } − − − U(u),α(v) = 3α(u)δ (u v)+α(u)δ(u v), ′ ′ { } − − α(u),α(v) = 2δ (u v)+2U(u)δ(u v). ′′ { } − − Using one of the corresponding infinite family of IM, which are in involution under the Poisson brackets (these IM could be extracted from the monodromy 3 matrix of -operator,see below)[7]: B L 1 (cl) I = U(u)du, (7) 1 2π Z 1 I(cl) = U2(u)+α(u)α(u)/2 du, 3 2π ′ Z (cid:16) (cid:17) 1 I(cl) = U3(u) (U )2(u)/2 α(u)α (u)/4 α(u)α(u)U(u) du, 5 2π − ′ − ′ ′′ − ′ . Z.(cid:16) . (cid:17) one can obtain an evolution equation; for example, taking I we get the SUSY 2 N=1 KdV equation [8]: = +3( D ) and in components: U = t uuu u,θ u t U −U U U U 6UU 3αα , α = 4α 3(Uα) . As we have noted in the in- − uuu− u− 2 uu t − uuu− u troductionone canshowthatthe IMareinvariantunder supersymmetrytrans- 2π formation generated by duα(u). 0 Inorderto constructthe so-calledmonodromymatrix we introducethe - B R L operator, equivalent to the one: F L i =∂ φ(u)h +(e +e ξh )2 (8) LB u− ′ α1 α1 α0 − √2 α1 Theequivalencecanbeeasilyestablishedifoneconsidersthelinearproblemas- sociatedwiththe -operator: χ(u,θ)=0(we considerthis operatoracting F F L L in some representation of C(2)(2) and χ(u,θ) is the vector in this representation). Then,expressingχ(u,θ)incomponents: χ(u,θ)=χ (u)+θχ (u),wefind: 0 1 χ =0 and χ =(e +e i ξh )χ . LB 0 1 α1 α0 − √2 α1 0 The solutiontothe equation χ =0canbewritteninthe followingway: B 0 L χ0(u) = eφ(u)hα1P exp udu′ √i2ξ(u′)e−φ(u′)eα1 (9) Z0 (cid:16) i ξ(u)eφ(u′)e e2 e 2φ(u′) e2 e2φ(u′) [e ,e ] η, − √2 ′ α0 − α1 − − α0 − α1 α0 (cid:17) where η is a constant vector in the corresponding representation of C(2)(2). Therefore we can define the monodromy matrix in the following way: 2π i M = e2πiphα1P exp du √2ξ(u)e−φ(u)eα1 (10) Z0 (cid:16) i − √2ξ(u)eφ(u)eα0 −e2α1e−2φ(u)−e2α0e2φ(u)−[eα1,eα0] . (cid:17) Introducing then (as in [9]) the auxiliary L-operators: L = e−πiphα1M we find that in the evaluation representation(when λ, the spectral parameter appears) the following Poisson bracket relation is satisfied [10]: L(λ) L(µ) =[r(λµ 1),L(λ) L(µ)], (11) , − { ⊗ } ⊗ where r(λµ 1) is trigonometric C(2)(2) r-matrix [11]. From this relation one − obtains that the supertraces of monodromy matrices t(λ)=strM(λ) commute 4 under the Poisson bracket: t(λ),t(µ) = 0. Expanding log(t(λ)) in λ in the { } evaluationrepresentationcorrespondingtothe defining3-dimensionalrepresen- tation of C(2) π we find: 1/2 log(t (λ))= ∞ c I(cl) λ 4n+2, λ (12) 1/2 − n 2n−1 − →∞ n=1 X where c1 = 21, cn = (22nn−n3!)!! for n > 1. So, one can obtain the IM from the supertrace of the monodromy matrix. Using the Poisson bracket relation for thesesupertraceswithdifferentvaluesofspectralparameter(seeabove)wefind that infinite family of IM is involutive, as it was mentioned earlier. 3 Free field representation of superconformal algebra In this section we begin to build quantum counterparts of the introduced classical objects. We will start from the quantum Miura transformation, the free field representation of the superconformal algebra [12]: 1 ǫβ2 β2T(u) = :φ′2(u): (1 β2/2)φ′′(u)+ :ξξ′(u):+ (13) − − − 2 16 i1/2β2 G(u) = φ′ξ(u) (1 β2/2)ξ′(u), √2 − − where a φ(u)=iQ+iPu+ −neinu, ξ(u)=i−1/2 ξne−inu, (14) n n n X X i β2 [Q,P]= β2, [a ,a ]= nδ , ξ ,ξ =β2δ . n m n+m,0 n m n+m,0 2 2 { } Recallthattherearetwotypesofboundaryconditionsonξ: ξ(u+2π)= ξ(u). ± The sign “+” corresponds to the R sector,the case when ξ is integer modded, the “–” sign corresponds to the NS sector and ξ is half-integer modded. The variable ǫ in (13) is equal to zero in the R case and equal to 1 in the NS case. OnecanexpandT(u)andG(u)bymodesinsuchaway: T(u)= L einu cˆ, G(u) = G einu, where cˆ= 5 2(β2 + 2 ) and L ,G gnene−rnate th−e 16 n −n − 2 β2 n mP superconformal algebra: P cˆ n [L ,L ] = (n m)L + (n3 n)δ , [L ,G ]=( m)G n m n+m n, m n m m+n − 8 − − 2 − cˆ [G ,G ] = 2L +δ (n2 1/4). (15) n m n+m n, m − 2 − In the classical limit c (the same is β2 0) the following substitution: →−∞ → T(u) cÛ(u), G(u) cˆ α(u), [,] 4π , reduce the above algebra to →−4 →−2√2i → icˆ{ } 5 the Poissonbracket algebra of SUSY N=1 KdV theory. LetnowF bethehighestweightmoduleovertheoscillatoralgebraofa ,ξ p n m with the highest weight vector (ground state) p determined by the eigenvalue | i of P and nilpotency condition of the action of the positive modes: P p = | i pp , a p = 0, ξ p = 0 where n,m > 0. In the case of the R sector the n m | i | i | i highest weight becomes doubly degenerate due to the presence of zero mode ξ . So, there are two ground states p,+ and p, : p,+ = ξ p, . Using 0 0 | i | −i | i | −i theabovefreefieldrepresentationofthesuperconformalalgebraonecanobtain thatforgenericcândp,F isisomorphictothesuper-Virasoromodulewiththe p highest weight vector p : L p = ∆ p , where ∆ = (p/β)2+(cˆ 1)/16 0 NS NS | i | i | i − in the NS sector and module with two highest weight vectors in the Ramond case: L p, =∆ p, , ∆ =(p/β)2+cˆ/16, p,+ =(β2/√2p)G p, . 0 R R 0 | ±i | ±i | i | −i The space F , now considered as super-Virasoro module, splits into the sum of p finite-dimensional subspaces, determined by the value of L : F = F(k), 0 p ⊕∞k=0 p L F(k) =(∆+k)F(k). Thequantumversionsoflocalintegralsofmotionshould 0 p p act invariantly on the subspaces F(k). Thus, the diagonalization of IM reduces p (inagivensubspaceF(k))tothefinitepurelyalgebraicproblem,whichhowever p rapidly become rather complicated for large k. It should be noted also that in the case of the Ramond sector supersymmetry generator G commute with 0 IM, so IM act in p,+ and p, independently, without mixing of these two | i | −i ground states (unlike the super-KdV case [13]). 4 Quantum monodromy matrix and RTT-relation Inthispartofthe workwewillconsiderthequantumC (2)(2) R-matrixand q show that the vertex operator representation of the lower Borel subalgebra of C (2)(2) allows to representthis R-matrix in the P-exponentlike form which in q theclassicallimitcoincidewiththe auxiliaryL-operator. C (2)(2) isaquantum q superalgebra with the following commutation relations [14]: [h ,h ]=0, [h ,e ]= e , [h ,e ]= e , (16) α0 α1 α0 ±α1 ∓ ±α1 α1 ±α0 ∓ ±α0 [h ,e ]= e (i=0,1), [e ,e ]=δ [h ] (i,j =0,1), αi ±αi ± ±αi ±αi ∓αj i,j αi [e ,[e ,[e ,e ] ] ] =0, [[[e ,e ] ,e ] ,e ] =0, ±α1 ±α1 ±α1 ±α0 q q q ±α1 ±α0 q ±α0 q ±α0 q wdehfienreed[xin] t=heqqfxo−−lqlq−o−w1xi,ngp(whαay0,:1)[e=γ,e0γ,′]pq(e±eαγ0e,1γ)′ =(11a)npd(eγq)p-s(euγp′)eqr(cγo,mγ′)meγu′teaγt,oqr =is ≡ − − eiπβ22. The corresponding coproducts are: ∆(hαj)=hαj ⊗1+1⊗hαj, ∆(eαj)=eαj ⊗qhαj +1⊗eαj, (17) ∆(e−αj)=e−αj ⊗1+q−hαj ⊗e−αj. The associatedR-matrix canbe expressedinsuch a way[14]: R=KR R R , + 0 wqh1e)re K =d(qnh)αe⊗hα, eR+ =).He→nr≥e0RRn=δ+eαx,pR− =(A(←nγ≥)1(qRnδq−α1,)(Re0 =eexp)()(aqn−−d − n>0 nδ⊗ −nδ Q γ (−q−1) Q − − γ⊗ −γ P 6 coefficientsAanddaredefinedasfollows: A(γ)= ( 1)nifγ =nδ+α;( 1)n 1 − iqf-cγo=mmnδut−atαo}rs, do(fnC)h=evnaq(lnql−e−yqq−−gn1e)n.eTrahteorgse.neInrattohresfeo{nllδo−,weinnδg±αwearweidllefinneeedd−tvhiae tehxe- pressionsonlyforsimplestones: eδ =[eα0,eα1]q−1 ande−δ =[e−α1,e−α0]q. The elementse areexpressedasmultiplecommutatorsofe withcorresponding nδ α δ ± Chevalley generators,e ones have more complicated form [14]. nδ Let’s introduce the reduced R-matrix R¯ K 1R. Using all previous infor- − mationonecanwriteR¯asR¯(e¯ ,e¯ ),wher≡ee¯ =e 1ande¯ =1 e , αi −αi αi αi⊗ −αi ⊗ −αi because it is representedas power series of these elements. After this necessary backgroundwewillintroducevertexoperatorsandusingthefactthatthey satisfy the Serre relations of the lower Borel subalgebra of C (2)(2) we will prove q thatthe reducedR-matrix,representedby the vertex operatorshas the proper- ties of the P-exponent. So, the vertex operators are: 1 u1 1 u1 V = dθ du:e Φ :, V = dθ du:eΦ :, (18) 1 q 1 q − 0 q 1 q − − Z Zu2 − − Z Zu2 where 2π u u 0, Φ = φ(u) i θξ(u) is a superfield and normal ≥ 1 ≥ 2 ≥ − √2 ordering here means that : e±φ(u) := exp ± ∞n=1 a−nneinu exp ± i(Q + Pu) exp ∓ ∞n=1 anne−inu . One can sh(cid:16)ow vPia the standar(cid:17)d con(cid:16)tour tech- niqu(cid:17)e that(cid:16)thePse operators sa(cid:17)tisfy the same commutation relations as eα1, eα0 correspondingly. Then, following [15] one can show (using the fundamental property of the universal R-matrix: (I ∆)R = R13R12) that the reduced R-matrix has the ⊗ following property: R¯(e¯ ,e +e )=R¯(e¯ ,e )R¯(e¯ ,e ), (19) αi ′−αi ′−′αi αi ′−αi αi ′−′αi wcohmermeuet′−aαtiion=r1el⊗atiqo−nhsαbie⊗twee−eαni,thee′−′mαia=re:1⊗e−αi ⊗1, e¯αi = eαi ⊗1⊗1. The e′−αie¯αj = −e¯αje′−αi, e′−′αie¯αj =−e¯αje′−′αi, (20) e′−αie′−′αj = −qbije′−′αje′−αi, where b is the symmetric matrix with the following elements: b = b = ij 00 11 b = 1. Now, denoting L¯(q)(u ,u ) the reduced R-matrix with e rep- r−es0e1nted by V , we find, using th2e a1bove property of R¯ with e −reαpilaced by appropriatei vertex operators: L¯(q)(u ,u ) = L¯(q)(u ,u )L¯(q)(′−uαi,u ) with 3 1 2 1 3 2 u u u . So, L¯(q) has the property of P-exponent. But because of sin- 1 2 3 ≥ ≥ gularities in the operator products of vertex operators it can not be written in the usual P-ordered form. Thus, we propose a new notion, the quantum P-exponent: L¯(q)(u ,u )=Pexp(q) u1du dθ(e :e Φ :+e :eΦ :). (21) 1 2 α1 − α0 Zu2 Z 7 Introducing new object: L(q) eiπPhα1L¯(q)(0,2π), which coincides with R- ≡ matrix with 1 h replaced by 2P/β2 and 1 e , 1 e replaced by V ⊗ α1 ⊗ −α1 ⊗ −α0 1 and V (with integrationfrom 0 to 2π) correspondingly,we find that it satisfies 0 the well-known RTT-relation: R(λµ−1) L(q)(λ) I I L(q)(µ) (22) ⊗ ⊗ (cid:16) (cid:17)(cid:16) (cid:17) =(I L(q)(µ) L(q)(λ) I R(λµ 1), − ⊗ ⊗ (cid:17)(cid:16) (cid:17) where the dependence on λ,µ means that we are considering L(q)-operators in the evaluation representation of C (2)(2). The quantum counterpart of the q monodromy matrix M(q) = e i=1,2πipihαiL¯(q) satisfies the reflection equation [21]: P R˜12(λµ−1)M1(q)(λ)F1−21M2(q)(µ)=M2(q)(µ)F1−21M1(q)(λ)R12(λµ−1), (23) whereF =K 1,theCartan’sfactorfromtheuniversalR-matrix,R˜ (λµ 1)= − 12 − F1−21R12(λµ−1)F12 andlabels1,2denotethepositionofmultipliersinthetensor product. Thus the supertraces of monodromy matrices, the transfer matrices t(q)(λ) str(M(q)(λ)) commute ≡ [t(q)(λ),t(q)(µ)]=0, (24) giving the quantum integrability. Now we will show that in the classical limit (q 1) the L(q)-operator will give the auxiliary L-matrix defined in the Sec. 2. →We will use the P-exponent property of L¯(q)(0,2π). Let’s decompose L¯(q)(0,2π) in the following way: L¯(q)(0,2π) = lim N L¯(q)(x ,x ), N m=1 m 1 m →∞ − wherewe dividedthe interval[0,2π]intoinfinitesimalintervals [x ,x ]with m m+1 Q x x =ǫ=2π/N. Let’s find the terms that can give contribution of the m+1 m first o−rder in ǫ in L¯(q)(x ,x ). In this analysis we will need the operator m 1 m − product expansion of vertex operators: iβ2 ∞ ξ(u)ξ(u)= + c (u)(iu iu)k, (25) ′ k ′ −(iu iu) − − ′ k=1 X :eaφ(u) ::ebφ(u′) :=(iu iu′)ab2β2(:e(a+b)φ(u) :+ ∞ dk(u)(iu iu′)k), − − k=1 X wherec (u)andd (u)areoperator-valuedfunctionsofu. Nowonecanseethat k k onlytwotypesoftermscangivethecontributionoftheorderǫinL¯(q)(x ,x ) m 1 m − when q 1. The first type consists of operatorsof the first order in V and the i → secondtypeisformedbytheoperators,quadraticinV ,whichgivecontribution i of the order ǫ1 β2 by virtue of operator product expansion. Let’s look on the ± terms of the second type in detail. At first we consider the terms appearing from the R -part of R-matrix, represented by vertex operators: 0 e xm xm δ du :e φ :ξ(u i0) du :eφ :ξ(u +i0)+ 2(q−q−1) Zxm−1 1 − 1− Zxm−1 2 2 8 xm xm q 1 du :eφ :ξ(u i0) du :e φ :ξ(u +i0) (26) − 2 2 1 − 1 Zxm−1 − Zxm−1 ! Neglecting the terms, which give rise to O(ǫ2) contribution, we obtain, using the operator products of vertex operators: e xm xm iβ2 δ du du − + (27) 2(q−q−1) Zxm−1 1Zxm−1 2(i(u1 u2 i0))β22+1 − − xm xm iβ2 q−1Zxm−1du2Zxm−1du1(i(u2 u−1 i0))β22+1! − − In the β2 0 limit we get: [eα1,eα0] xm du xm du 1 1 which,by→thewellknownformu2laiπ: 1xm=−1P11 ixπmδ−(x1)gi2veus:1−u2+xim0 −duu1[−eu2−,ei0 ]. x+Ri0 x−R (cid:0) − xm−1 α1 α(cid:1)0 AnothertermsarisefromtheR andR partsofR-matrixandareverysimilar + − R to each other: e2 xm xm α0 du :eφ :ξ(u i0) du :eφ :ξ(u +i0), (28) 1 1 2 2 2(2)(−q−1) Zxm−1 − Zxm−1 e2 xm xm α1 du1 :e−φ :ξ(u1 i0) du2 :e−φ :ξ(u2+i0). 2(2)(−q−1) Zxm−1 − Zxm−1 The integrals can be reduced to the ordered ones: e2 xm u1 α0 du :eφ :ξ(u ) du :eφ :ξ(u ), (29) 1 1 2 2 2 Zxm−1 Zxm−1 e2 xm u1 α1 du1 :e−φ :ξ(u1) du2 :e−φ :ξ(u2). 2 Zxm−1 Zxm−1 Following [13] we find that their contribution (of order ǫ) in the classical limit is: xm xm −e2α0 due2φ(u), −e2α1 due−2φ(u). (30) Zxm−1 Zxm−1 Gathering now all the terms of order ǫ we find: L¯(1)(x ,x )=1+ xm du( i ξ(u)e φ(u)e (31) m−1 m Zxm−1 √2 − α1 − i ξ(u)eφ(u)e e2 e 2φ(u) e2 e2φ(u) [e ,e ])+O(ǫ2) √2 α0 − α1 − − α0 − α1 α0 andcollectingallL¯(1)(xm 1,xm)wefindthatL¯(1)=e−iπphα1L. ThereforeL(1)=L. − 9 5 N = 2 SUSY KdV hierarchy in nonstandard form The matrix L-operator of the N=2 SUSY KdV has the following explicit form [16]: DΦ 1 0 1 =D λ DΦ DΦ 1 , F 1 2 L − −  λD(Φ Φ ) λ DΦ 1 2 2 − − where D = ∂ + θ∂ and u is a variable on a cylinder of circumference 2π θ u and Φ (u,θ) = φ (u) i θξ (u) are the superfields with the following Poisson i i − √2 i brackets: Φ (u ,θ ),Φ (u ,θ ) =0 (i=1,2) i 1 1 i 2 2 { } D Φ (u ,θ ),D Φ (u ,θ ) = D (δ(u u )(θ θ ))(32) { u1,θ1 1 1 1 u2,θ2 2 2 2 } − u1,θ1 1− 2 1− 2 ThisisoddL-operator,relatedtothesl(1)(21)affinesuperalgebra(theelements | in the associatedcolumn of the representationvector are graded from upper to the lower in such an order: even, odd, even). There is also one (canonical) related to the sl(1)(22) [6]. The quantization of this higher rank case can be | performedbymeansoftheprocedureoutlinedin[2]anddoesnotaddsomething new inthe quantizationprocedurewith respectto [13], [2]. Moreoverif we look forwardtostudyandpossiblysolvethecorrespondingintegrablequantummodel wewillfindthattheassociatedrepresentationtheoryofsl(1)(22)(bothclassical | and quantum) is very complicated. However the introduced lower rank form allows us to work with much more simple representation theory of sl(1)(21) | and, as we will see, it provides very interesting features of the quantization formalism. In order to build the scalar L-operator related with this matrix one, let’s consider the corresponding linear problem ( Ψ = 0) and express the second L an third element in the vector Ψ in terms of the first (upper) one. The linear equation for this element is the following one: ((D+DΦ )(D D(Φ Φ ))(D DΦ )+2λD)Ψ =0 (33) 1 1 2 2 1 − − − Therefore the scalar linear operator is: L=D3+( +2λ)D+ , (34) V U where the Miura map is given by: i = DΦ2DΦ1+∂Φ2 ∂Φ1 = θ(α+ α−)+V, V − − √2 − i = ∂Φ DΦ D∂Φ =θU + α+ 2 1 1 U − − √2 1 U = φ φ ξ ξ φ , α+ =ξ φ +ξ , − ′1 ′2− 2 1 2′ − ′1′ 1 ′2 1′ 1 α =ξ φ +ξ , V =φ φ + ξ ξ (35) − 2 ′1 2′ ′2− ′1 2 2 1 10